Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Algorithms for Communications Systems and their Applications
Algorithms for Communications Systems and their Applications
Nevio Benvenuto University of Padova, Italy
Giovanni Cherubini IBM Zurich Research Laboratory, Switzerland
c 2002 Copyright
John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (C44) 1243 779777
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To Adriana, and to Antonio, Claudia, and Mariuccia
Contents
Preface
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Acknowledgements
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1 Elements of signal theory 1.1 Signal space : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Properties of a linear space : : : : : : : : : : : : : : : : Inner product : : : : : : : : : : : : : : : : : : : : : : : 1.2 Discrete signal representation : : : : : : : : : : : : : : : : : : : : The principle of orthogonality : : : : : : : : : : : : : : Signal representation : : : : : : : : : : : : : : : : : : : Gram–Schmidt orthonormalization procedure : : : : : : 1.3 Continuous-time linear systems : : : : : : : : : : : : : : : : : : : 1.4 Discrete-time linear systems : : : : : : : : : : : : : : : : : : : : : Discrete Fourier transform (DFT) : : : : : : : : : : : : The DFT operator : : : : : : : : : : : : : : : : : : : : : Circular and linear convolution via DFT : : : : : : : : : Convolution by the overlap-save method : : : : : : : : : IIR and FIR filters : : : : : : : : : : : : : : : : : : : : 1.5 Signal bandwidth : : : : : : : : : : : : : : : : : : : : : : : : : : The sampling theorem : : : : : : : : : : : : : : : : : : Heaviside conditions for the absence of signal distortion 1.6 Passband signals : : : : : : : : : : : : : : : : : : : : : : : : : : : Complex representation : : : : : : : : : : : : : : : : : : Relation between x and x .bb/ : : : : : : : : : : : : : : : Baseband equivalent of a transformation : : : : : : : : : Envelope and instantaneous phase and frequency : : : : 1.7 Second-order analysis of random processes : : : : : : : : : : : : : 1.7.1 Correlation : : : : : : : : : : : : : : : : : : : : : : : : : : Properties of the autocorrelation function : : : : : : : : 1.7.2 Power spectral density : : : : : : : : : : : : : : : : : : : : Spectral lines in the PSD : : : : : : : : : : : : : : : : : Cross-power spectral density : : : : : : : : : : : : : : :
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Properties of the PSD : : : : : : : : : : : : : : : PSD of processes through linear transformations : PSD of processes through filtering : : : : : : : : 1.7.3 PSD of discrete-time random processes : : : : : : : Spectral lines in the PSD : : : : : : : : : : : : : PSD of processes through filtering : : : : : : : : Minimum-phase spectral factorization : : : : : : 1.7.4 PSD of passband processes : : : : : : : : : : : : : PSD of the quadrature components of a random process : : : : : : : : : Cyclostationary processes : : : : : : : : : : : : : The autocorrelation matrix : : : : : : : : : : : : : : : : : : Definition : : : : : : : : : : : : : : : : : : : : : Properties : : : : : : : : : : : : : : : : : : : : : Eigenvalues : : : : : : : : : : : : : : : : : : : : Other properties : : : : : : : : : : : : : : : : : : Eigenvalue analysis for Hermitian matrices : : : Examples of random processes : : : : : : : : : : : : : : : Matched filter : : : : : : : : : : : : : : : : : : : : : : : : Matched filter in the presence of white noise : : Ergodic random processes : : : : : : : : : : : : : : : : : : 1.11.1 Mean value estimators : : : : : : : : : : : : : : : : Rectangular window : : : : : : : : : : : : : : : Exponential filter : : : : : : : : : : : : : : : : : General window : : : : : : : : : : : : : : : : : : 1.11.2 Correlation estimators : : : : : : : : : : : : : : : : Unbiased estimate : : : : : : : : : : : : : : : : : Biased estimate : : : : : : : : : : : : : : : : : : 1.11.3 Power spectral density estimators : : : : : : : : : : Periodogram or instantaneous spectrum : : : : : Welch periodogram : : : : : : : : : : : : : : : : Blackman and Tukey correlogram : : : : : : : : Windowing and window closing : : : : : : : : : Parametric models of random processes : : : : : : : : : : : ARMA. p; q/ model : : : : : : : : : : : : : : : MA(q) model : : : : : : : : : : : : : : : : : : : AR(N ) model : : : : : : : : : : : : : : : : : : : Spectral factorization of an AR(N ) model : : : : Whitening filter : : : : : : : : : : : : : : : : : : Relation between ARMA, MA and AR models : 1.12.1 Autocorrelation of AR processes : : : : : : : : : : 1.12.2 Spectral estimation of an AR.N / process : : : : : : Some useful relations : : : : : : : : : : : : : : : AR model of sinusoidal processes : : : : : : : : Guide to the bibliography : : : : : : : : : : : : : : : : : :
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Bibliography : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : 1.A Multirate systems : : : : : : : : : : 1.A.1 Fundamentals : : : : : : : : 1.A.2 Decimation : : : : : : : : : 1.A.3 Interpolation : : : : : : : : : 1.A.4 Decimator filter : : : : : : : 1.A.5 Interpolator filter : : : : : : : 1.A.6 Rate conversion : : : : : : : 1.A.7 Time interpolation : : : : : : Linear interpolation : : : : Quadratic interpolation : : 1.A.8 The noble identities : : : : : 1.A.9 The polyphase representation Efficient implementations : 1.B Generation of Gaussian noise : : : :
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2 The Wiener filter and linear prediction 2.1 The Wiener filter : : : : : : : : : : : : : : : : : : : : : : : : Matrix formulation : : : : : : : : : : : : : : : : : Determination of the optimum filter coefficients : : The principle of orthogonality : : : : : : : : : : : Expression of the minimum mean-square error : : : Characterization of the cost function surface : : : : The Wiener filter in the z-domain : : : : : : : : : 2.2 Linear prediction : : : : : : : : : : : : : : : : : : : : : : : : Forward linear predictor : : : : : : : : : : : : : : Optimum predictor coefficients : : : : : : : : : : : Forward “prediction error filter” : : : : : : : : : : Relation between linear prediction and AR models First and second order solutions : : : : : : : : : : 2.2.1 The Levinson–Durbin algorithm : : : : : : : : : : : Lattice filters : : : : : : : : : : : : : : : : : : : : 2.2.2 The Delsarte–Genin algorithm : : : : : : : : : : : : 2.3 The least squares (LS) method : : : : : : : : : : : : : : : : Data windowing : : : : : : : : : : : : : : : : : : : Matrix formulation : : : : : : : : : : : : : : : : : Correlation matrix : : : : : : : : : : : : : : : : Determination of the optimum filter coefficients : : 2.3.1 The principle of orthogonality : : : : : : : : : : : : : Expressions of the minimum cost function : : : : : The normal equation using the T matrix : : : : : : Geometric interpretation: the projection operator : : 2.3.2 Solutions to the LS problem : : : : : : : : : : : : : Singular value decomposition of T : : : : : : : : : Minimum norm solution : : : : : : : : : : : : : :
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Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : 2.A The estimation problem : : : : : : : : : : : : : : : : : The estimation problem for random variables MMSE estimation : : : : : : : : : : : : : : : Extension to multiple observations : : : : : : MMSE linear estimation : : : : : : : : : : : MMSE linear estimation for random vectors : 3
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Adaptive transversal filters 3.1 Adaptive transversal filter: MSE criterion : : : : : : : : : : : : : : 3.1.1 Steepest descent or gradient algorithm : : : : : : : : : : : Stability of the steepest descent algorithm : : : : : : : : Conditions for convergence : : : : : : : : : : : : : : : : Choice of the adaptation gain for fastest convergence : : Transient behavior of the MSE : : : : : : : : : : : : : : 3.1.2 The least mean-square (LMS) algorithm : : : : : : : : : : Implementation : : : : : : : : : : : : : : : : : : : : : : Computational complexity : : : : : : : : : : : : : : : : Canonical model : : : : : : : : : : : : : : : : : : : : : Conditions for convergence : : : : : : : : : : : : : : : : 3.1.3 Convergence analysis of the LMS algorithm : : : : : : : : Convergence of the mean : : : : : : : : : : : : : : : : : Convergence in the mean-square sense (real scalar case) Convergence in the mean-square sense (general case) : : Basic results : : : : : : : : : : : : : : : : : : : : : : : : Observations : : : : : : : : : : : : : : : : : : : : : : : Final remarks : : : : : : : : : : : : : : : : : : : : : : : 3.1.4 Other versions of the LMS algorithm : : : : : : : : : : : : Leaky LMS : : : : : : : : : : : : : : : : : : : : : : : : Sign algorithm : : : : : : : : : : : : : : : : : : : : : : Sigmoidal algorithm : : : : : : : : : : : : : : : : : : : Normalized LMS : : : : : : : : : : : : : : : : : : : : : Variable adaptation gain : : : : : : : : : : : : : : : : : LMS for lattice filters : : : : : : : : : : : : : : : : : : : 3.1.5 Example of application: the predictor : : : : : : : : : : : : 3.2 The recursive least squares (RLS) algorithm : : : : : : : : : : : : Normal equation : : : : : : : : : : : : : : : : : : : : : Derivation of the RLS algorithm : : : : : : : : : : : : : Initialization of the RLS algorithm : : : : : : : : : : : : Recursive form of E min : : : : : : : : : : : : : : : : : : Convergence of the RLS algorithm : : : : : : : : : : : : Computational complexity of the RLS algorithm : : : : : Example of application: the predictor : : : : : : : : : : 3.3 Fast recursive algorithms : : : : : : : : : : : : : : : : : : : : : : 3.3.1 Comparison of the various algorithms : : : : : : : : : : :
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Block adaptive algorithms in the frequency domain : : : : : : : : : 3.4.1 Block LMS algorithm in the frequency domain: the basic scheme : : : : : : : : : : : : : : : : : : : : : : : : Computational complexity of the block LMS algorithm via FFT : : : : : : : : : : : : : : : 3.4.2 Block LMS algorithm in the frequency domain: the FLMS algorithm : : : : : : : : : : : : : : : : : : : : : : Computational complexity of the FLMS algorithm : : : : : Convergence in the mean of the coefficients for the FLMS algorithm : : : : : : : : : : : : 3.5 LMS algorithm in a transformed domain : : : : : : : : : : : : : : : 3.5.1 Basic scheme : : : : : : : : : : : : : : : : : : : : : : : : : On the speed of convergence : : : : : : : : : : : : : : : : 3.5.2 Normalized FLMS algorithm : : : : : : : : : : : : : : : : : 3.5.3 LMS algorithm in the frequency domain : : : : : : : : : : : 3.5.4 LMS algorithm in the DCT domain : : : : : : : : : : : : : : 3.5.5 General observations : : : : : : : : : : : : : : : : : : : : : 3.6 Examples of application : : : : : : : : : : : : : : : : : : : : : : : 3.6.1 System identification : : : : : : : : : : : : : : : : : : : : : Linear case : : : : : : : : : : : : : : : : : : : : : : : : : Finite alphabet case : : : : : : : : : : : : : : : : : : : : : 3.6.2 Adaptive cancellation of interfering signals : : : : : : : : : : General solution : : : : : : : : : : : : : : : : : : : : : : : 3.6.3 Cancellation of a sinusoidal interferer with known frequency 3.6.4 Disturbance cancellation for speech signals : : : : : : : : : : 3.6.5 Echo cancellation in subscriber loops : : : : : : : : : : : : : 3.6.6 Adaptive antenna arrays : : : : : : : : : : : : : : : : : : : : 3.6.7 Cancellation of a periodic interfering signal : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 3.A PN sequences : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Maximal-length sequences : : : : : : : : : : : : : : : : : CAZAC sequences : : : : : : : : : : : : : : : : : : : : : Gold sequences : : : : : : : : : : : : : : : : : : : : : : : 3.B Identification of a FIR system by PN sequences : : : : : : : : : : : 3.B.1 Correlation method : : : : : : : : : : : : : : : : : : : : : : Signal-to-estimation error ratio : : : : : : : : : : : : : : : 3.B.2 Methods in the frequency domain : : : : : : : : : : : : : : : System identification in the absence of noise : : : : : : : : System identification in the presence of noise : : : : : : : 3.B.3 The LS method : : : : : : : : : : : : : : : : : : : : : : : : Formulation using the data matrix : : : : : : : : : : : : : Computation of the signal-to-estimation error ratio : : : : 3.B.4 The LMMSE method : : : : : : : : : : : : : : : : : : : : : 3.B.5 Identification of a continuous-time system : : : : : : : : : : 3.4
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Transmission media 4.1 Electrical characterization of a transmission system : : : : : Simplified scheme of a transmission system : : : Characterization of an active device : : : : : : : Conditions for the absence of signal distortion : : Characterization of a 2-port network : : : : : : : Measurement of signal power : : : : : : : : : : 4.2 Noise generated by electrical devices and networks : : : : : Thermal noise : : : : : : : : : : : : : : : : : : : Shot noise : : : : : : : : : : : : : : : : : : : : : Noise in diodes and transistors : : : : : : : : : : Noise temperature of a two-terminal device : : : Noise temperature of a 2-port network : : : : : : Equivalent-noise models : : : : : : : : : : : : : Noise figure of a 2-port network : : : : : : : : : Cascade of 2-port networks : : : : : : : : : : : : 4.3 Signal-to-noise ratio (SNR) : : : : : : : : : : : : : : : : : SNR for a two-terminal device : : : : : : : : : : SNR for a 2-port network : : : : : : : : : : : : Relation between noise figure and SNR : : : : : 4.4 Transmission lines : : : : : : : : : : : : : : : : : : : : : : 4.4.1 Fundamentals of transmission line theory : : : : : : Ideal transmission line : : : : : : : : : : : : : : Non-ideal transmission line : : : : : : : : : : : : Frequency response : : : : : : : : : : : : : : : : Conditions for the absence of signal distortion : : Impulse response of a non-ideal transmission line Secondary constants of some transmission lines : 4.4.2 Cross-talk : : : : : : : : : : : : : : : : : : : : : : Near-end cross-talk : : : : : : : : : : : : : : : : Far-end cross-talk : : : : : : : : : : : : : : : : : 4.5 Optical fibers : : : : : : : : : : : : : : : : : : : : : : : : : Description of a fiber-optic transmission system : 4.6 Radio links : : : : : : : : : : : : : : : : : : : : : : : : : : 4.6.1 Frequency ranges for radio transmission : : : : : : Radiation masks : : : : : : : : : : : : : : : : : : 4.6.2 Narrowband radio channel model : : : : : : : : : : Equivalent circuit at the receiver : : : : : : : : : Multipath : : : : : : : : : : : : : : : : : : : : : 4.6.3 Doppler shift : : : : : : : : : : : : : : : : : : : : : 4.6.4 Propagation of wideband signals : : : : : : : : : : Channel parameters in the presence of multipath : Statistical description of fading channels : : : : : 4.6.5 Continuous-time channel model : : : : : : : : : : : Power delay profile : : : : : : : : : : : : : : : :
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Doppler spectrum : : : : : : : : : : : : : : : : : : : Doppler spectrum models : : : : : : : : : : : : : : : Shadowing : : : : : : : : : : : : : : : : : : : : : : Final remarks : : : : : : : : : : : : : : : : : : : : : 4.6.6 Discrete-time model for fading channels : : : : : : : : Generation of a process with a pre-assigned spectrum 4.7 Telephone channel : : : : : : : : : : : : : : : : : : : : : : : : 4.7.1 Characteristics : : : : : : : : : : : : : : : : : : : : : : Linear distortion : : : : : : : : : : : : : : : : : : : Noise sources : : : : : : : : : : : : : : : : : : : : : Non-linear distortion : : : : : : : : : : : : : : : : : Frequency offset : : : : : : : : : : : : : : : : : : : Phase jitter : : : : : : : : : : : : : : : : : : : : : : Echo : : : : : : : : : : : : : : : : : : : : : : : : : : 4.8 Transmission channel: general model : : : : : : : : : : : : : : Power amplifier (HPA) : : : : : : : : : : : : : : : : Transmission medium : : : : : : : : : : : : : : : : : Additive noise : : : : : : : : : : : : : : : : : : : : : Phase noise : : : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 5 Digital representation of waveforms 5.1 Analog and digital access : : : : : : : : : : : : : : : : : 5.1.1 Digital representation of speech : : : : : : : : : : Some waveforms : : : : : : : : : : : : : : : : Speech coding : : : : : : : : : : : : : : : : : : The interpolator filter as a holder : : : : : : : : Sizing of the binary channel parameters : : : : 5.1.2 Coding techniques and applications : : : : : : : : 5.2 Instantaneous quantization : : : : : : : : : : : : : : : : : 5.2.1 Parameters of a quantizer : : : : : : : : : : : : : 5.2.2 Uniform quantizers : : : : : : : : : : : : : : : : Quantization error : : : : : : : : : : : : : : : : Relation between 1, b and −sat : : : : : : : : Statistical description of the quantization noise Statistical power of the quantization error : : : Design of a uniform quantizer : : : : : : : : : Signal-to-quantization error ratio : : : : : : : : Implementations of uniform PCM encoders : : 5.3 Non-uniform quantizers : : : : : : : : : : : : : : : : : : Three examples of implementation : : : : : : : 5.3.1 Companding techniques : : : : : : : : : : : : : : Signal-to-quantization error ratio : : : : : : : : Digital compression : : : : : : : : : : : : : : : Signal-to-quantization noise ratio mask : : : : :
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Optimum quantizer in the MSE sense : : : : : : : : : : : : Max algorithm : : : : : : : : : : : : : : : : : : : : : : Lloyd algorithm : : : : : : : : : : : : : : : : : : : : : : Expression of 3q for a very fine quantization : : : : : : Performance of non-uniform quantizers : : : : : : : : : Adaptive quantization : : : : : : : : : : : : : : : : : : : : : : : : General scheme : : : : : : : : : : : : : : : : : : : : : : 5.4.1 Feedforward adaptive quantizer : : : : : : : : : : : : : : : Performance : : : : : : : : : : : : : : : : : : : : : : : : 5.4.2 Feedback adaptive quantizers : : : : : : : : : : : : : : : : Estimate of ¦s .k/ : : : : : : : : : : : : : : : : : : : : : Differential coding (DPCM) : : : : : : : : : : : : : : : : : : : : : 5.5.1 Configuration with feedback quantizer : : : : : : : : : : : 5.5.2 Alternative configuration : : : : : : : : : : : : : : : : : : 5.5.3 Expression of the optimum coefficients : : : : : : : : : : : Effects due to the presence of the quantizer : : : : : : : 5.5.4 Adaptive predictors : : : : : : : : : : : : : : : : : : : : : Adaptive feedforward predictors : : : : : : : : : : : : : Sequential adaptive feedback predictors : : : : : : : : : Performance : : : : : : : : : : : : : : : : : : : : : : : : 5.5.5 Alternative structures for the predictor : : : : : : : : : : : All-pole predictor : : : : : : : : : : : : : : : : : : : : : All-zero predictor : : : : : : : : : : : : : : : : : : : : : Pole-zero predictor : : : : : : : : : : : : : : : : : : : : Pitch predictor : : : : : : : : : : : : : : : : : : : : : : APC : : : : : : : : : : : : : : : : : : : : : : : : : : : : Delta modulation : : : : : : : : : : : : : : : : : : : : : : : : : : 5.6.1 Oversampling and quantization error : : : : : : : : : : : : 5.6.2 Linear delta modulation (LDM) : : : : : : : : : : : : : : : LDM implementation : : : : : : : : : : : : : : : : : : : Choice of system parameters : : : : : : : : : : : : : : : 5.6.3 Adaptive delta modulation (ADM) : : : : : : : : : : : : : Continuously variable slope delta modulation (CVSDM) ADM with second-order predictors : : : : : : : : : : : : 5.6.4 PCM encoder via LDM : : : : : : : : : : : : : : : : : : 5.6.5 Sigma delta modulation (6DM) : : : : : : : : : : : : : : : Coding by modeling : : : : : : : : : : : : : : : : : : : : : : : : : Vocoder or LPC : : : : : : : : : : : : : : : : : : : : : : RPE coding : : : : : : : : : : : : : : : : : : : : : : : : CELP coding : : : : : : : : : : : : : : : : : : : : : : : Multipulse coding : : : : : : : : : : : : : : : : : : : : : Vector quantization (VQ) : : : : : : : : : : : : : : : : : : : : : : 5.8.1 Characterization of VQ : : : : : : : : : : : : : : : : : : : Parameters determining VQ performance : : : : : : : : : Comparison between VQ and scalar quantization : : : : 5.3.2
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Optimum quantization : : : : : : : : : : : : : : : Generalized Lloyd algorithm : : : : : : : : : : 5.8.3 LBG algorithm : : : : : : : : : : : : : : : : : : Choice of the initial codebook : : : : : : : : : Description of the LBG algorithm with splitting Selection of the training sequence : : : : : : : 5.8.4 Variants of VQ : : : : : : : : : : : : : : : : : : Tree search VQ : : : : : : : : : : : : : : : : : Multistage VQ : : : : : : : : : : : : : : : : : Product code VQ : : : : : : : : : : : : : : : : 5.9 Other coding techniques : : : : : : : : : : : : : : : : : : Adaptive transform coding (ATC) : : : : : : : Sub-band coding (SBC) : : : : : : : : : : : : : 5.10 Source coding : : : : : : : : : : : : : : : : : : : : : : : 5.11 Speech and audio standards : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : 5.8.2
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6 Modulation theory 6.1 Theory of optimum detection : : : : : : : : : : : : : : : : : : Statistics of the random variables fwi g : : : : : : : : Sufficient statistics : : : : : : : : : : : : : : : : : : Decision criterion : : : : : : : : : : : : : : : : : : : Theorem of irrelevance : : : : : : : : : : : : : : : : Implementations of the maximum likelihood criterion Error probability : : : : : : : : : : : : : : : : : : : 6.1.1 Examples of binary signalling : : : : : : : : : : : : : : Antipodal signals (² D 1) : : : : : : : : : : : : : : Orthogonal signals (² D 0) : : : : : : : : : : : : : : Binary FSK : : : : : : : : : : : : : : : : : : : : : : 6.1.2 Limits on the probability of error : : : : : : : : : : : : Upper limit : : : : : : : : : : : : : : : : : : : : : : Lower limit : : : : : : : : : : : : : : : : : : : : : : 6.2 Simplified model of a transmission system and definition of binary channel : : : : : : : : : : : : : : : : : : : : : : : : Parameters of a transmission system : : : : : : : : : Relations among parameters : : : : : : : : : : : : : 6.3 Pulse amplitude modulation (PAM) : : : : : : : : : : : : : : : 6.4 Phase-shift keying (PSK) : : : : : : : : : : : : : : : : : : : : Binary PSK (BPSK) : : : : : : : : : : : : : : : : : Quadrature PSK (QPSK) : : : : : : : : : : : : : : : 6.5 Differential PSK (DPSK) : : : : : : : : : : : : : : : : : : : : 6.5.1 Error probability for an M-DPSK system : : : : : : : : 6.5.2 Differential encoding and coherent demodulation : : : : Binary case (M D 2, differentially encoded BPSK) : Multilevel case : : : : : : : : : : : : : : : : : : : :
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AM-PM or quadrature amplitude modulation (QAM) : : : : : : : : : : Comparison between PSK and QAM : : : : : : : : : : : : : 6.7 Modulation methods using orthogonal and biorthogonal signals : : : : 6.7.1 Modulation with orthogonal signals : : : : : : : : : : : : : : : Probability of error : : : : : : : : : : : : : : : : : : : : : : Limit of the probability of error for M increasing to infinity 6.7.2 Modulation with biorthogonal signals : : : : : : : : : : : : : : Probability of error : : : : : : : : : : : : : : : : : : : : : : 6.8 Binary sequences and coding : : : : : : : : : : : : : : : : : : : : : : Optimum receiver : : : : : : : : : : : : : : : : : : : : : : 6.9 Comparison between coherent modulation methods : : : : : : : : : : : Trade-offs for QAM systems : : : : : : : : : : : : : : : : : Comparison of modulation methods : : : : : : : : : : : : : 6.10 Limits imposed by information theory : : : : : : : : : : : : : : : : : Capacity of a system using amplitude modulation : : : : : : Coding strategies depending on the signal-to-noise ratio : : : Coding gain : : : : : : : : : : : : : : : : : : : : : : : : : : Cut-off rate : : : : : : : : : : : : : : : : : : : : : : : : : : 6.11 Optimum receivers for signals with random phase : : : : : : : : : : : ML criterion : : : : : : : : : : : : : : : : : : : : : : : : : Implementation of a non-coherent ML receiver : : : : : : : Error probability for a non-coherent binary FSK system : : : Performance comparison of binary systems : : : : : : : : : 6.12 Binary modulation systems in the presence of flat fading : : : : : : : : Diversity : : : : : : : : : : : : : : : : : : : : : : : : : : : 6.13 Transmission methods : : : : : : : : : : : : : : : : : : : : : : : : : : 6.13.1 Transmission methods between two users : : : : : : : : : : : : Three methods : : : : : : : : : : : : : : : : : : : : : : : : 6.13.2 Channel sharing: deterministic access methods : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6.A Gaussian distribution function and Marcum function : : : : : : : : : : 6.A.1 The Q function : : : : : : : : : : : : : : : : : : : : : : : : : 6.A.2 The Marcum function : : : : : : : : : : : : : : : : : : : : : : 6.B Gray coding : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6.C Baseband PPM and PDM : : : : : : : : : : : : : : : : : : : : : : : : Signal-to-noise ratio : : : : : : : : : : : : : : : : : : : : : 6.D Walsh codes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 6.6
7 Transmission over dispersive channels 7.1 Baseband digital transmission (PAM systems) : : Transmitter : : : : : : : : : : : : : : : Transmission channel : : : : : : : : : : Receiver : : : : : : : : : : : : : : : : : Power spectral density of a PAM signal
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Passband digital transmission (QAM systems) : : : : : : : : : Transmitter : : : : : : : : : : : : : : : : : : : : : : Power spectral density of a QAM signal : : : : : : : Three equivalent representations of the modulator : : Coherent receiver : : : : : : : : : : : : : : : : : : : 7.3 Baseband equivalent model of a QAM system : : : : : : : : : 7.3.1 Signal analysis : : : : : : : : : : : : : : : : : : : : : : Signal-to-noise ratio : : : : : : : : : : : : : : : : : 7.3.2 Characterization of system elements : : : : : : : : : : Transmitter : : : : : : : : : : : : : : : : : : : : : : Transmission channel : : : : : : : : : : : : : : : : : Receiver : : : : : : : : : : : : : : : : : : : : : : : : 7.3.3 Intersymbol interference : : : : : : : : : : : : : : : : : Discrete-time equivalent system : : : : : : : : : : : Nyquist pulses : : : : : : : : : : : : : : : : : : : : Eye diagram : : : : : : : : : : : : : : : : : : : : : : 7.3.4 Performance analysis : : : : : : : : : : : : : : : : : : Symbol error probability in the absence of ISI : : : : Matched filter receiver : : : : : : : : : : : : : : : : 7.4 Carrierless AM/PM (CAP) modulation : : : : : : : : : : : : : 7.5 Regenerative PCM repeaters : : : : : : : : : : : : : : : : : : : 7.5.1 PCM signals over a binary channel : : : : : : : : : : : Linear PCM coding of waveforms : : : : : : : : : : Overall system performance : : : : : : : : : : : : : 7.5.2 Regenerative repeaters : : : : : : : : : : : : : : : : : : Analog transmission : : : : : : : : : : : : : : : : : Digital transmission : : : : : : : : : : : : : : : : : : Comparison between analog and digital transmission Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 7.A Line codes for PAM systems : : : : : : : : : : : : : : : : : : 7.A.1 Line codes : : : : : : : : : : : : : : : : : : : : : : : : Non-return-to-zero (NRZ) format : : : : : : : : : : : Return-to-zero (RZ) format : : : : : : : : : : : : : : Biphase (B-) format : : : : : : : : : : : : : : : : : Delay modulation or Miller code : : : : : : : : : : : Block line codes : : : : : : : : : : : : : : : : : : : Alternate mark inversion (AMI) : : : : : : : : : : : 7.A.2 Partial response systems : : : : : : : : : : : : : : : : : The choice of the PR polynomial : : : : : : : : : : : Symbol detection and error probability : : : : : : : : Precoding : : : : : : : : : : : : : : : : : : : : : : : Error probability with precoding : : : : : : : : : : : Alternative interpretation of PR systems : : : : : : :
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7.B Computation of Pe for some cases of interest 7.B.1 Pe in the absence of ISI : : : : : : : 7.B.2 Pe in the presence of ISI : : : : : : Exhaustive method : : : : : : : : Gaussian approximation : : : : : Worst-case limit : : : : : : : : : : Saltzberg limit : : : : : : : : : : GQR method : : : : : : : : : : : 7.C Coherent PAM-DSB transmission : : : : : : General scheme : : : : : : : : : : Transmit signal PSD : : : : : : : Signal-to-noise ratio : : : : : : : 7.D Implementation of a QAM transmitter : : : 7.E Simulation of a QAM system : : : : : : : :
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8 Channel equalization and symbol detection 8.1 Zero-forcing equalizer (LE-ZF) : : : : : : : : : : : : : : : : : 8.2 Linear equalizer (LE) : : : : : : : : : : : : : : : : : : : : : : 8.2.1 Optimum receiver in the presence of noise and ISI : : : Alternative derivation of the IIR equalizer : : : : : : Signal-to-noise ratio : : : : : : : : : : : : : : : : 8.3 LE with a finite number of coefficients : : : : : : : : : : : : : Adaptive LE : : : : : : : : : : : : : : : : : : : : : 8.4 Fractionally spaced equalizer (FSE) : : : : : : : : : : : : : : : Adaptive FSE : : : : : : : : : : : : : : : : : : : : : 8.5 Decision feedback equalizer (DFE) : : : : : : : : : : : : : : : Adaptive DFE : : : : : : : : : : : : : : : : : : : : : Design of a DFE with a finite number of coefficients Design of a fractionally spaced DFE (FS-DFE) : : : Signal-to-noise ratio : : : : : : : : : : : : : : : : Remarks : : : : : : : : : : : : : : : : : : : : : : : : 8.6 Convergence behavior of adaptive equalizers : : : : : : : : : : Adaptive LE : : : : : : : : : : : : : : : : : : : : : Adaptive DFE : : : : : : : : : : : : : : : : : : : : : 8.7 LE-ZF with a finite number of coefficients : : : : : : : : : : : 8.8 DFE: alternative configurations : : : : : : : : : : : : : : : : : DFE-ZF : : : : : : : : : : : : : : : : : : : : : : : : DFE-ZF as a noise predictor : : : : : : : : : : : : : DFE as ISI and noise predictor : : : : : : : : : : : : 8.9 Benchmark performance for two equalizers : : : : : : : : : : : Performance comparison : : : : : : : : : : : : : : : Equalizer performance for two channel models : : : 8.10 Optimum methods for data detection : : : : : : : : : : : : : : 8.10.1 Maximum likelihood sequence detection : : : : : : : : Lower limit to error probability using the MLSD criterion : : : : : : : : : : :
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The Viterbi algorithm (VA) : : : : : : : : : : : : : : : : : : : Computational complexity of the VA : : : : : : : : : : : : : : 8.10.2 Maximum a posteriori probability detector : : : : : : : : : : : : Statistical description of a sequential machine : : : : : : : : : The forward-backward algorithm (FBA) : : : : : : : : : : : : Scaling : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Likelihood function in the absence of ISI : : : : : : : : : : : Simplified version of the MAP algorithm (Max-Log-MAP) : : Relation between Max-Log-MAP and Log-MAP : : : : : : : : 8.11 Optimum receivers for transmission over dispersive channels : : : : : : Ungerboeck’s formulation of the MLSD : : : : : : : : : : : : 8.12 Error probability achieved by MLSD : : : : : : : : : : : : : : : : : : : Computation of the minimum distance : : : : : : : : : : : : : 8.13 Reduced state sequence detection : : : : : : : : : : : : : : : : : : : : : Reduced state trellis diagram : : : : : : : : : : : : : : : : : : RSSE algorithm : : : : : : : : : : : : : : : : : : : : : : : : : Further simplification: DFSE : : : : : : : : : : : : : : : : : : 8.14 Passband equalizers : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8.14.1 Passband receiver structure : : : : : : : : : : : : : : : : : : : : Joint optimization of equalizer coefficients and carrier phase offset : : : : : : : : : : : : : : Adaptive method : : : : : : : : : : : : : : : : : : : : : : : : 8.14.2 Efficient implementations of voiceband modems : : : : : : : : : 8.15 LE for voiceband modems : : : : : : : : : : : : : : : : : : : : : : : : : Detection of the training sequence : : : : : : : : : : : : : : : Computations of the coefficients of a cyclic equalizer : : : : : Transition from training to data mode : : : : : : : : : : : : : Example of application: a simple modem : : : : : : : : : : : 8.16 LE and DFE in the frequency domain with data frames using cyclic prefix 8.17 Numerical results obtained by simulations : : : : : : : : : : : : : : : : QPSK transmission over a minimum phase channel : : : : : : QPSK transmission over a non-minimum phase channel : : : : 8-PSK transmission over a minimum phase channel : : : : : : 8-PSK transmission over a non-minimum phase channel : : : 8.18 Diversity combining techniques : : : : : : : : : : : : : : : : : : : : : : Antenna arrays : : : : : : : : : : : : : : : : : : : : : : : : : Combining techniques : : : : : : : : : : : : : : : : : : : : : Equalization and diversity : : : : : : : : : : : : : : : : : : : Diversity in transmission : : : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 8.A Calculus of variations and receiver optimization : : : : : : : : : : : : : 8.A.1 Calculus of variations : : : : : : : : : : : : : : : : : : : : : : : Linear functional : : : : : : : : : : : : : : : : : : : : : : : : Quadratic functional : : : : : : : : : : : : : : : : : : : : : :
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8.A.2 Receiver optimization : : : : : : : : : : : : : : 8.A.3 Joint optimization of transmitter and receiver : : 8.B DFE design: matrix formulations : : : : : : : : : : : : 8.B.1 Method based on correlation sequences : : : : : 8.B.2 Method based on the channel impulse response and i.i.d. symbols : : : : : : : : : : : : : : : : 8.B.3 Method based on the channel impulse response and any symbol statistic : : : : : : : : : : : : : 8.B.4 FS-DFE : : : : : : : : : : : : : : : : : : : : : 8.C Equalization based on the peak value of ISI : : : : : : 8.D Description of a finite state machine (FSM) : : : : : : :
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10 Spread spectrum systems 10.1 Spread spectrum techniques : : : : : : : : : : : : : : : : : : : : : : : : 10.1.1 Direct sequence systems : : : : : : : : : : : : : : : : : : : : : :
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9 Orthogonal frequency division multiplexing 9.1 OFDM systems : : : : : : : : : : : : : : : : : : : : : : : : 9.2 Orthogonality conditions : : : : : : : : : : : : : : : : : : : : Time domain : : : : : : : : : : : : : : : : : : : : Frequency domain : : : : : : : : : : : : : : : : : z-transform domain : : : : : : : : : : : : : : : : : 9.3 Efficient implementation of OFDM systems : : : : : : : : : OFDM implementation employing matched filters : Orthogonality conditions in terms of the polyphase components : : : : : OFDM implementation employing a prototype filter 9.4 Non-critically sampled filter banks : : : : : : : : : : : : : : 9.5 Examples of OFDM systems : : : : : : : : : : : : : : : : : Discrete multitone (DMT) : : : : : : : : : : : : : Filtered multitone (FMT) : : : : : : : : : : : : : : Discrete wavelet multitone (DWMT) : : : : : : : : 9.6 Equalization of OFDM systems : : : : : : : : : : : : : : : : Interpolator filter and virtual subchannels : : : : : Equalization of DMT systems : : : : : : : : : : : Equalization of FMT systems : : : : : : : : : : : : 9.7 Synchronization of OFDM systems : : : : : : : : : : : : : : 9.8 Passband OFDM systems : : : : : : : : : : : : : : : : : : : Passband DWMT systems : : : : : : : : : : : : : Passband DMT and FMT systems : : : : : : : : : Comparison between OFDM and QAM systems : : 9.9 DWMT modulation : : : : : : : : : : : : : : : : : : : : : : Transmit and receive filter banks : : : : : : : : : : Approximate interchannel interference suppression Perfect interchannel interference suppression : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
Contents
Classification of CDMA systems : : : : : : : : : Synchronization : : : : : : : : : : : : : : : : : : 10.1.2 Frequency hopping systems : : : : : : : : : : : : : Classification of FH systems : : : : : : : : : : : 10.2 Applications of spread spectrum systems : : : : : : : : : : 10.2.1 Anti-jam communications : : : : : : : : : : : : : : 10.2.2 Multiple-access systems : : : : : : : : : : : : : : : 10.2.3 Interference rejection : : : : : : : : : : : : : : : : 10.3 Chip matched filter and rake receiver : : : : : : : : : : : : Number of resolvable rays in a multipath channel Chip matched filter (CMF) : : : : : : : : : : : : 10.4 Interference : : : : : : : : : : : : : : : : : : : : : : : : : Detection strategies for multiple-access systems : 10.5 Equalizers for single-user detection : : : : : : : : : : : : : Chip equalizer (CE) : : : : : : : : : : : : : : : : Symbol equalizer (SE) : : : : : : : : : : : : : : 10.6 Block equalizer for multiuser detection : : : : : : : : : : : 10.7 Maximum likelihood multiuser detector : : : : : : : : : : : Correlation matrix approach : : : : : : : : : : : Whitening filter approach : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : :
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11 Channel codes 11.1 System model : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11.2 Block codes : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 11.2.1 Theory of binary codes with group structure : : : : : : : : Properties : : : : : : : : : : : : : : : : : : : : : : : : : Parity check matrix : : : : : : : : : : : : : : : : : : : : Code generator matrix : : : : : : : : : : : : : : : : : : Decoding of binary parity check codes : : : : : : : : : : Cosets : : : : : : : : : : : : : : : : : : : : : : : : : : : Two conceptually simple decoding methods : : : : : : : Syndrome decoding : : : : : : : : : : : : : : : : : : : : 11.2.2 Fundamentals of algebra : : : : : : : : : : : : : : : : : : Modulo q arithmetic : : : : : : : : : : : : : : : : : : : Polynomials with coefficients from a field : : : : : : : : The concept of modulo in the arithmetic of polynomials Devices to sum and multiply elements in a finite field : : Remarks on finite fields : : : : : : : : : : : : : : : : : : Roots of a polynomial : : : : : : : : : : : : : : : : : : Minimum function : : : : : : : : : : : : : : : : : : : : Methods to determine the minimum function : : : : : : Properties of the minimum function : : : : : : : : : : : 11.2.3 Cyclic codes : : : : : : : : : : : : : : : : : : : : : : : : : The algebra of cyclic codes : : : : : : : : : : : : : : : :
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Properties of cyclic codes : : : : : : : : : : : : : : : : Encoding method using a shift register of length r : : Encoding method using a shift register of length k : : Hard decoding of cyclic codes : : : : : : : : : : : : : Hamming codes : : : : : : : : : : : : : : : : : : : : : Burst error detection : : : : : : : : : : : : : : : : : : 11.2.4 Simplex cyclic codes : : : : : : : : : : : : : : : : : : : Relation to PN sequences : : : : : : : : : : : : : : : : 11.2.5 BCH codes : : : : : : : : : : : : : : : : : : : : : : : : An alternative method to specify the code polynomials Bose–Chaudhuri–Hocquenhem (BCH) codes : : : : : : Binary BCH codes : : : : : : : : : : : : : : : : : : : Reed–Solomon codes : : : : : : : : : : : : : : : : : : Decoding of BCH codes : : : : : : : : : : : : : : : : Efficient decoding of BCH codes : : : : : : : : : : : : 11.2.6 Performance of block codes : : : : : : : : : : : : : : : : 11.3 Convolutional codes : : : : : : : : : : : : : : : : : : : : : : : : 11.3.1 General description of convolutional codes : : : : : : : : Parity check matrix : : : : : : : : : : : : : : : : : : : Generator matrix : : : : : : : : : : : : : : : : : : : : Transfer function : : : : : : : : : : : : : : : : : : : : Catastrophic error propagation : : : : : : : : : : : : : 11.3.2 Decoding of convolutional codes : : : : : : : : : : : : : Interleaving : : : : : : : : : : : : : : : : : : : : : : : Two decoding models : : : : : : : : : : : : : : : : : : Viterbi algorithm : : : : : : : : : : : : : : : : : : : : Forward-backward algorithm : : : : : : : : : : : : : : Sequential decoding : : : : : : : : : : : : : : : : : : : 11.3.3 Performance of convolutional codes : : : : : : : : : : : 11.4 Concatenated codes : : : : : : : : : : : : : : : : : : : : : : : : Soft-output Viterbi algorithm (SOVA) : : : : : : : : : 11.5 Turbo codes : : : : : : : : : : : : : : : : : : : : : : : : : : : : Encoding : : : : : : : : : : : : : : : : : : : : : : : : The basic principle of iterative decoding : : : : : : : : The forward-backward algorithm revisited : : : : : : : Iterative decoding : : : : : : : : : : : : : : : : : : : : Performance evaluation : : : : : : : : : : : : : : : : : 11.6 Iterative detection and decoding : : : : : : : : : : : : : : : : : 11.7 Low-density parity check codes : : : : : : : : : : : : : : : : : : Encoding procedure : : : : : : : : : : : : : : : : : : : Decoding algorithm : : : : : : : : : : : : : : : : : : : Example of application : : : : : : : : : : : : : : : : : Performance and coding gain : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
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11.A Nonbinary parity check codes : : : : : : : : : : : : : Linear codes : : : : : : : : : : : : : : : : Parity check matrix : : : : : : : : : : : : : Code generator matrix : : : : : : : : : : : Decoding of nonbinary parity check codes : Coset : : : : : : : : : : : : : : : : : : : : Two conceptually simple decoding methods Syndrome decoding : : : : : : : : : : : : :
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12 Trellis coded modulation 12.1 Linear TCM for one- and two-dimensional signal sets : : : : 12.1.1 Fundamental elements : : : : : : : : : : : : : : : : : Basic TCM scheme : : : : : : : : : : : : : : : : : Example : : : : : : : : : : : : : : : : : : : : : : : 12.1.2 Set partitioning : : : : : : : : : : : : : : : : : : : : 12.1.3 Lattices : : : : : : : : : : : : : : : : : : : : : : : : 12.1.4 Assignment of symbols to the transitions in the trellis 12.1.5 General structure of the encoder/bit-mapper : : : : : Computation of dfree : : : : : : : : : : : : : : : : 12.2 Multidimensional TCM : : : : : : : : : : : : : : : : : : : : Encoding : : : : : : : : : : : : : : : : : : : : : : Decoding : : : : : : : : : : : : : : : : : : : : : : 12.3 Rotationally invariant TCM schemes : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
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13 Precoding and coding techniques for dispersive channels 13.1 Capacity of a dispersive channel : : : : : : : : : : : : : 13.2 Techniques to achieve capacity : : : : : : : : : : : : : : Bit loading for OFDM : : : : : : : : : : : : : Discrete-time model of a single carrier system : Achieving capacity with a single carrier system 13.3 Precoding and coding for dispersive channels : : : : : : : 13.3.1 Tomlinson–Harashima (TH) precoding : : : : : : 13.3.2 TH precoding and TCM : : : : : : : : : : : : : : 13.3.3 Flexible precoding : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : :
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14 Synchronization 14.1 The problem of synchronization for QAM systems : : : : : : 14.2 The phase-locked loop : : : : : : : : : : : : : : : : : : : : : 14.2.1 PLL baseband model : : : : : : : : : : : : : : : : : Linear approximation : : : : : : : : : : : : : : : : 14.2.2 Analysis of the PLL in the presence of additive noise Noise analysis using the linearity assumption : : : 14.2.3 Analysis of a second-order PLL : : : : : : : : : : : : 14.3 Costas loop : : : : : : : : : : : : : : : : : : : : : : : : : :
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14.3.1 PAM signals : : : : : : : : : : : : : : : : : : : : : : : : 14.3.2 QAM signals : : : : : : : : : : : : : : : : : : : : : : : 14.4 The optimum receiver : : : : : : : : : : : : : : : : : : : : : : : Timing recovery : : : : : : : : : : : : : : : : : : : : Carrier phase recovery : : : : : : : : : : : : : : : : : 14.5 Algorithms for timing and carrier phase recovery : : : : : : : : : 14.5.1 ML criterion : : : : : : : : : : : : : : : : : : : : : : : : Assumption of slow time varying channel : : : : : : : 14.5.2 Taxonomy of algorithms using the ML criterion : : : : : Feedback estimators : : : : : : : : : : : : : : : : : : Early-late estimators : : : : : : : : : : : : : : : : : : 14.5.3 Timing estimators : : : : : : : : : : : : : : : : : : : : : Non-data aided : : : : : : : : : : : : : : : : : : : : : Non-data aided via spectral estimation : : : : : : : : : Data-aided and data-directed : : : : : : : : : : : : : : Data- and phase-directed with feedback: differentiator scheme : : : : : : : : : : : : Data- and phase-directed with feedback: Mueller & Muller scheme : : : : : : : : : Non-data aided with feedback : : : : : : : : : : : : : 14.5.4 Phasor estimators : : : : : : : : : : : : : : : : : : : : : Data- and timing-directed : : : : : : : : : : : : : : : : Non-data aided for M-PSK signals : : : : : : : : : : : Data- and timing-directed with feedback : : : : : : : : 14.6 Algorithms for carrier frequency recovery : : : : : : : : : : : : 14.6.1 Frequency offset estimators : : : : : : : : : : : : : : : : Non-data aided : : : : : : : : : : : : : : : : : : : : : Non-data aided and timing-independent with feedback : Non-data aided and timing-directed with feedback : : : 14.6.2 Estimators operating at the modulation rate : : : : : : : : Data-aided and data-directed : : : : : : : : : : : : : : Non-data aided for M-PSK : : : : : : : : : : : : : : : 14.7 Second-order digital PLL : : : : : : : : : : : : : : : : : : : : : 14.8 Synchronization in spread spectrum systems : : : : : : : : : : : 14.8.1 The transmission system : : : : : : : : : : : : : : : : : Transmitter : : : : : : : : : : : : : : : : : : : : : : : Optimum receiver : : : : : : : : : : : : : : : : : : : : 14.8.2 Timing estimators with feedback : : : : : : : : : : : : : Non-data aided: non-coherent DLL : : : : : : : : : : : Non-data aided MCTL : : : : : : : : : : : : : : : : : Data- and phase-directed: coherent DLL : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
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15 Self-training equalization 15.1 Problem definition and fundamentals : : : : : : : : : : : : : : : : : : : Minimization of a special function : : : : : : : : : : : : : : :
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15.2 Three algorithms for PAM systems : : : : : : : : : : : : : The Sato algorithm : : : : : : : : : : : : : : : : Benveniste–Goursat algorithm : : : : : : : : : : Stop-and-go algorithm : : : : : : : : : : : : : : Remarks : : : : : : : : : : : : : : : : : : : : : : 15.3 The contour algorithm for PAM systems : : : : : : : : : : Simplified realization of the contour algorithm : : 15.4 Self-training equalization for partial response systems : : : The Sato algorithm for partial response systems : Contour algorithm for partial response systems : 15.5 Self-training equalization for QAM systems : : : : : : : : The Sato algorithm for QAM systems : : : : : : 15.5.1 Constant modulus algorithm : : : : : : : : : : : : : The contour algorithm for QAM systems : : : : Joint contour algorithm and carrier phase tracking 15.6 Examples of applications : : : : : : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : 15.A On the convergence of the contour algorithm : : : : : : : :
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16 Applications of interference cancellation 16.1 Echo and near–end cross-talk cancellation for PAM systems : Cross-talk cancellation and full duplex transmission Polyphase structure of the canceller : : : : : : : : Canceller at symbol rate : : : : : : : : : : : : : : Adaptive canceller : : : : : : : : : : : : : : : : : Canceller structure with distributed arithmetic : : : 16.2 Echo cancellation for QAM systems : : : : : : : : : : : : : 16.3 Echo cancellation for OFDM systems : : : : : : : : : : : : : 16.4 Multiuser detection for VDSL : : : : : : : : : : : : : : : : : 16.4.1 Upstream power back-off : : : : : : : : : : : : : : : 16.4.2 Comparison of PBO methods : : : : : : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : :
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1152 1156
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17 Wired and wireless network technologies 17.1 Wired network technologies : : : : : : : : : : : : : : : : : : : : : 17.1.1 Transmission over unshielded twisted pairs in the customer service area : : : : : : : : : : : : : : : : : : : : : : : : : Modem : : : : : : : : : : : : : : : : : : : : : : : : : : Digital subscriber line : : : : : : : : : : : : : : : : : : 17.1.2 High speed transmission over unshielded twisted pairs in local area networks : : : : : : : : : : : : : : : : : : : : 17.1.3 Hybrid fiber/coaxial cable networks : : : : : : : : : : : : : Ranging and power adjustment for uplink transmission : : : : : : : : : : : : : : : : :
xxvi
17.2 Wireless network technologies : : : : : : : 17.2.1 Wireless local area networks : : : Medium access control protocols 17.2.2 MMDS and LMDS : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : 17.A Standards for wireless systems : : : : : : 17.A.1 General observations : : : : : : : Wireless systems : : : : : : : : Modulation techniques : : : : : Parameters of the modulator : : Cells in a wireless system : : : 17.A.2 GSM standard : : : : : : : : : : : System characteristics : : : : : : Radio subsystem : : : : : : : : GSM-EDGE : : : : : : : : : : : 17.A.3 IS-136 standard : : : : : : : : : : 17.A.4 JDC standard : : : : : : : : : : : 17.A.5 IS-95 standard : : : : : : : : : : : 17.A.6 DECT standard : : : : : : : : : : 17.A.7 HIPERLAN standard : : : : : : :
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1161 1162 1164 1165 1167 1170 1170 1171 1171 1171 1171 1172 1172 1172 1175 1177 1177 1180 1180 1182 1185
18 Modulation techniques for wireless systems 18.1 Analog front-end architectures : : : : : : : : : : : : : : : : : Conventional superheterodyne receiver : : : : : : : Alternative architectures : : : : : : : : : : : : : : Direct conversion receiver : : : : : : : : : : : : : Single conversion to low-IF : : : : : : : : : : : : Double conversion and wideband IF : : : : : : : : 18.2 Three non-coherent receivers for phase modulation systems : 18.2.1 Baseband differential detector : : : : : : : : : : : : : 18.2.2 IF-band (1 Bit) differential detector (1BDD) : : : : : Performance of M-DPSK : : : : : : : : : : : : : : 18.2.3 FM discriminator with integrate and dump filter (LDI) 18.3 Variants of QPSK : : : : : : : : : : : : : : : : : : : : : : : 18.3.1 Basic schemes : : : : : : : : : : : : : : : : : : : : : QPSK : : : : : : : : : : : : : : : : : : : : : : : : Offset QPSK or staggered QPSK : : : : : : : : : : Differential QPSK (DQPSK) : : : : : : : : : : : : ³=4-DQPSK : : : : : : : : : : : : : : : : : : : : 18.3.2 Implementations : : : : : : : : : : : : : : : : : : : : QPSK, OQPSK, and DQPSK modulators : : : : : ³=4-DQPSK modulators : : : : : : : : : : : : : : 18.4 Frequency shift keying (FSK) : : : : : : : : : : : : : : : : : 18.4.1 Power spectrum of M-FSK : : : : : : : : : : : : : :
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Contents
xxvii
Power spectrum of non-coherent binary FSK : : Power spectrum of coherent M-FSK : : : : : : : 18.4.2 FSK receivers and corresponding performance : : : Coherent demodulator : : : : : : : : : : : : : : Non-coherent demodulator : : : : : : : : : : : : Limiter-discriminator FM demodulator : : : : : : 18.5 Minimum shift keying (MSK) : : : : : : : : : : : : : : : : 18.5.1 Power spectrum of continuous-phase FSK (CPFSK) 18.5.2 The MSK signal viewed from two perspectives : : Phase of an MSK signal : : : : : : : : : : : : : MSK as binary CPFSK : : : : : : : : : : : : : : MSK as OQPSK : : : : : : : : : : : : : : : : : Complex notation of an MSK signal : : : : : : : 18.5.3 Implementations of an MSK scheme : : : : : : : : 18.5.4 Performance of MSK demodulators : : : : : : : : : MSK with differential precoding : : : : : : : : : 18.5.5 Remarks on spectral containment : : : : : : : : : : 18.6 Gaussian MSK (GMSK) : : : : : : : : : : : : : : : : : : 18.6.1 GMSK via CPFSK : : : : : : : : : : : : : : : : : 18.6.2 Power spectrum of GMSK : : : : : : : : : : : : : 18.6.3 Implementation of a GMSK scheme : : : : : : : : Configuration I : : : : : : : : : : : : : : : : : : Configuration II : : : : : : : : : : : : : : : : : : Configuration III : : : : : : : : : : : : : : : : : 18.6.4 Linear approximation of a GMSK signal : : : : : : Performance of GMSK demodulators : : : : : : Performance of a GSM receiver in the presence of multipath : : : : : : : : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : 18.A Continuous phase modulation (CPM) : : : : : : : : : : : : Alternative definition of CPM : : : : : : : : : : Advantages of CPM : : : : : : : : : : : : : : : 19 Design of high speed transmission systems over unshielded twisted pair cables 19.1 Design of a quaternary partial response class-IV system sion at 125 Mbit/s : : : : : : : : : : : : : : : : : : : : Analog filter design : : : : : : : : : : : : : : Received signal and adaptive gain control : : Near-end cross-talk cancellation : : : : : : : Decorrelation filter : : : : : : : : : : : : : : Adaptive equalizer : : : : : : : : : : : : : : Compensation of the timing phase drift : : :
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Adaptive equalizer coefficient adaptation : : : : : : Convergence behavior of the various algorithms : : 19.1.1 VLSI implementation : : : : : : : : : : : : : : : : : Adaptive digital NEXT canceller : : : : : : : : : : Adaptive digital equalizer : : : : : : : : : : : : : : Timing control : : : : : : : : : : : : : : : : : : : Viterbi detector : : : : : : : : : : : : : : : : : : : 19.2 Design of a dual duplex transmission system at 100 Mbit/s : : Dual duplex transmission : : : : : : : : : : : : : : Physical layer control : : : : : : : : : : : : : : : : Coding and decoding : : : : : : : : : : : : : : : : 19.2.1 Signal processing functions : : : : : : : : : : : : : : The 100BASE-T2 transmitter : : : : : : : : : : : : The 100BASE-T2 receiver : : : : : : : : : : : : : Computational complexity of digital receive filters : Bibliography : : : : : : : : : : : : : : : : : : : : : : : : : : : : : Appendices : : : : : : : : : : : : : : : : : : : : : : : : : : : : : 19.A Interference suppression : : : : : : : : : : : : : : : : : : : : Index
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Preface
The motivation for this book is twofold. On the one hand, we provide a didactic tool to students of communications systems. On the other hand, we present a discussion of fundamental algorithms and structures for telecommunication technologies. The contents reflect our experience in teaching courses on Algorithms for Telecommunications at the University of Padova, Italy, as well as our professional experience acquired in industrial research laboratories. The text explains the procedures for solving problems posed by the design of systems for reliable communications over wired or wireless channels. In particular, we focus on fundamental developments in the field in order to provide the reader with the necessary insight to design essential elements of various communications systems. The book is divided into nineteen chapters. We briefly indicate four tracks corresponding to specific areas and course work offered. Track 1. Track 1 includes the basic elements for a first course on telecommunications, which we regard as an introduction to the remaining tracks. It covers Chapter 1, which recalls fundamental concepts on signals and random processes, with an emphasis on secondorder statistical descriptions. A discussion of the characteristics of transmission media follows in Chapter 4. In this track we focus on the description of noise in electronic devices and on the laws of propagation in transmission lines and radio channels. The representation of waveforms by sequences of binary symbols is treated in Chapter 5; for a first course it is suggested that emphasis be placed on PCM. Next, Chapter 6 examines the fundamental principles of a digital transmission system, where a sequence of information symbols is sent over a transmission channel. We refer to Shannon theorem to establish the maximum bit rate that can be transmitted reliably over a noisy channel. Signal dispersion caused by a transmission channel is then analyzed in Chapter 7. Examples of elementary and practical implementations of transmission systems are presented, together with a brief introduction to computer simulations. The first three sections of Chapter 11, where we introduce methods for increasing transmission reliability by exploiting the redundancy added to the information bits, conclude the first track. Track 2. Track 2, which is an extension of Track 1, focuses on modulation techniques. First, parametric models of random processes are analyzed in Chapter 1. The Wiener filter and the linear prediction theory, which constitute fundamental elements for receiver design, are dealt with in Chapter 2. Chapter 3 lists iterative methods to achieve the objectives stated
xxx
Preface
in Chapter 2, as well as various applications of the Wiener filter, for example channel identification and interference cancellation. These applications are further developed in the first two sections of Chapter 16. In the first part of Chapter 8, channel equalization is examined as a further application of the Wiener filter. In the second part of the chapter, more sophisticated methods of equalization and symbol detection, which rely on the Viterbi algorithm and on the forwardbackward algorithm, are analyzed. Initially single-carrier modulation systems are considered. In Chapter 9, we introduce multicarrier modulation techniques, which are preferable for transmission over very dispersive channels and/or applications that require flexibility in spectral allocation. In Chapter 10 spread spectrum systems are examined, with emphasis to applications for simultaneous channel access by several users that share a wideband channel. The inherent narrowband interference rejection capabilities of spread spectrum systems, as well as their implementations, are also discussed. This is followed by Chapter 18, which illustrates specific modulation techniques developed for mobile radio applications. Track 3. We observe the trend towards implementing transceiver functions using digital signal processors. Therefore the algorithmic aspects of a transmission system are becoming increasingly important. Hardware devices are assigned wherever possible only the functions of analog front-end, fixed filtering, and digital-to-analog and analog-to-digital conversion. This approach enhances the flexibility of transceivers, which can be utilized for more than one transmission standard, and considerably reduces development time. In line with the above considerations, Track 3 begins with a review of Chapters 2 and 3, which illustrate the fundamental principles of transmission system design, and of Chapter 8, which investigates individual building blocks for channel equalization and symbol detection. The assumption that the transmission channel characteristics are known a priori is removed in Chapter 15, where blind equalization techniques are discussed. Channel coding techniques to improve the reliability of transmission are investigated in depth in Chapters 11 and 12. A further method to mitigate channel dispersion is precoding. The operations of systems that employ joint precoding and channel coding are explained in Chapter 13. Because of electromagnetic coupling, the desired signal at the receiver is often disturbed by other transmissions taking place simultaneously. Cancellation techniques to suppress interference signals are treated in Chapter 16. Track 4. Track 4 addresses various challenges encountered in designing wired and wireless communications systems. The elements introduced in Chapters 2 and 3, as well as the algorithms introduced in Chapter 8, are essential for this track. The principles of multicarrier and spread spectrum modulation techniques, which are increasingly being adopted in communications systems, are investigated in depth in Chapters 9 and 10, respectively. The design of the receiver front-end, as well as various methods for timing and carrier recovery, are dealt with in Chapter 14. Applications of interference cancellation and multi-user detection are addressed in Chapter 16. An overview of wired and wireless access technologies appears in Chapter 17, and specific examples of system design are given in Chapters 18 and 19.
Acknowledgements
We gratefully acknowledge all who have made the realization of this book possible. In particular, the editing of the various chapters would never have been completed without the contributions of numerous students in our courses on Algorithms for Telecommunications. Although space limitations preclude mentioning them all by name, we nevertheless express our sincere gratitude. We also thank Christian Bolis and Chiara Paci for their support in developing the software for the book, Charlotte Bolliger and Lilli M. Pavka for their assistance in administering the project, and Urs Bitterli and Darja Kropaci for their help with the graphics editing. For text processing of the Italian version, the contribution of Barbara Sicoli was indispensable; our thanks also go to Jane Frankenfield Zanin for her help in translating the text into English. We are pleased to thank the following colleagues for their invaluable assistance throughout the revision of the book: Antonio Assalini, Paola Bisaglia, Alberto Bononi, Giancarlo Calvagno, Giulio Colavolpe, Roberto Corvaja, Elena Costa, Andrea Galtarossa, Antonio Mian, Carlo Monti, Ezio Obetti, Riccardo Rahely, Roberto Rinaldo, Antonio Salloum, Fortunato Santucci, Andrea Scaggiante, Giovanna Sostrato, Stefano Tomasin, and Luciano Tomba. We gratefully acknowledge our colleague and mentor Jack Wolf for letting us include his lecture notes in the chapter on channel codes. A special acknowledgment goes to our colleagues Werner Bux and Evangelos Eleftheriou of the IBM Zurich Research Laboratory, and Silvano Pupolin of the University of Padua, for their continuing support. Nevio Benvenuto Giovanni Cherubini
To make the reading of the adopted symbols easier, a table containing the Greek alphabet is included. The Greek alphabet Þ
A
alpha
¹
N
nu
þ
B
beta
¾
4
xi
0
gamma
o
O
omicron
Ž
1
delta
³
5
pi
ž, "
E
epsilon
², %
rho
Z
zeta
¦, &
P P
H
eta
−
T
tau
, #
2
theta
×
Y
upsilon
I
iota
, '
8
phi
K
kappa
X
chi
½
3
lambda
9
psi
¼
M
mu
omega
!
sigma
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 1
Elements of signal theory
In the present chapter we recall fundamental concepts of signal theory and random processes. A majority of readers will simply find this chapter a review of known principles, while others will find it a useful incentive for further in-depth study, for which we recommend the items in the bibliography. In any event, we will begin with the definition of signal space and its discrete representation, then move to the study of discrete-time linear systems (discrete Fourier transforms, IIR and FIR impulse responses) and signals (complex representation of passband signals and the baseband equivalent). We will conclude with the study of random processes, with emphasis on the statistical estimation of first- and second-order ergodic processes (periodogram, correlogram, ARMA, MA and especially AR models).
1.1
Signal space
Definition 1.1 A linear space is a set of elements called vectors, together with two operators defined over the elements of the set, the sum between vectors and the multiplication of a vector by a scalar. The Euclidean space is an example of linear space in which the sum of two vectors coincides with the vector obtained by adding the individual components, and the product of a vector by a scalar coincides with the vector obtained by multiplying each component for that scalar. In our case of particular interest is the set of complex vectors, i.e., those with complex-valued components, in an Euclidean space.
Properties of a linear space Let x, y, z and 0 be elements of a linear space, and Þ and þ be complex numbers (scalars). 1. Addition is commutative xCyDyCx
(1.1)
x C .y C z/ D .x C y/ C z
(1.2)
2. Addition is associative
3. There exists a unique vector 0, called null, such that 0CxDx
(1.3)
2
Chapter 1. Elements of signal theory
4. For each x, there is a unique vector x, called additive inverse, such that x C .x/ D 0
(1.4)
5. Multiplication by scalars is associative Þ.þx/ D .Þþ/x
(1.5)
In particular, we have 1x D x
0x D 0
(1.6)
6. Distributive laws Þ.x C y/ D Þx C Þy .Þ C þ/x D Þx C þx
(1.7) (1.8)
A geometrical interpretation of the two elementary operations in a two-dimensional Euclidean space is given in Figure 1.1. As previously mentioned, the Euclidean space is an example of a linear space. Two other examples of linear spaces are: the discrete-time signal space (an Euclidean space with infinite dimensions), whose elements are the signals fx.kTc /g
k integer
(1.9)
where Tc is the sampling period or interval,1 and the continuous-time signal space, whose elements are the signals x.t/
t 2
0
8x 6D 0.
5. (Schwarz inequality) jhx; yij jjxjjjjyjj. Equality holds if and only if x D Ky, with K a complex scalar.
1.2
Discrete signal representation
Let us consider the problem of associating a sequence (possibly finite) of numbers with a continuous-time signal.3 A basis of orthonormal signals (orthogonal signals with unit norm) fi .t/g, t 2 B, then the analytic signal of x is related to that of c by: x .a/ .t/ D a.t/ c.a/ .t/
(1.181)
Proof. We consider the general relation (1.157), valid for every real signal c.t/ D
1 .a/ 1 .a/Ł .t/ 2 c .t/ C 2 c
(1.182)
Substituting (1.182) in (1.180) yields x.t/ D a.t/ 12 c.a/ .t/ C a.t/ 12 c.a/Ł .t/
(1.183)
In the frequency domain the support of the first term in (1.183) is given by the interval [ f 0 B; C1/, while that of the second is equal to .1; f 0 C B]. Under the hypothesis that f 0 ½ B, the two terms in (1.183) have disjoint support in the frequency domain and (1.181) is immediately obtained. Corollary 1.1 From (1.181) we obtain x .h/ .t/ D a.t/c.h/ .t/
(1.184)
x .bb/ .t/ D a.t/c.bb/ .t/
(1.185)
and
1.6. Passband signals
41
In fact, from (1.169) we get x .h/ .t/ D Im[x .a/ .t/]
(1.186)
which substituted in (1.181) yields (1.184). Finally, (1.185) is obtained by substituting (1.152), x .bb/ .t/ D x .a/ .t/e j2³ f 0 t
(1.187)
in (1.181). An interesting application of (1.186) is in the design of a Hilbert filter h .h/ starting from a lowpass filter h. In fact, from (1.155) and (1.186) we get h .h/ .t/ D h.t/ sin.2³ f 0 t/
(1.188)
Example 1.6.3 Let a modulated double sideband (DSB) signal be expressed as x.t/ D a.t/ cos.2³ f 0 t C '0 /
(1.189)
where a is a BB signal with bandwidth B. Then, if f 0 > B, from the above theorem we have the following relations: x .a/ .t/ D a.t/e j .2³ f 0 tC'0 /
(1.190)
x .h/ .t/ D a.t/ sin.2³ f 0 t C '0 /
(1.191)
.bb/
(1.192)
x
.t/ D a.t/e
j'0
We list in Table 1.5 some properties of the Hilbert transformation (1.168) that are easily obtained by using the Fourier transform and the properties of Table 1.1. Table 1.5 Some properties of the Hilbert transform.
Property
(Real) signal
(Real) Hilbert transform
x.t/
x .h/ .t/
duality
x .h/ .t/
x.t/
inverse time
x.t/
x .h/ .t/
even signal
x.t/ D x.t/
x .h/ .t/ D x .h/ .t/, odd
odd signal
x.t/ D x.t/
x .h/ .t/ D x .h/ .t/, even
product (see Theorem 1.1) cosinusoidal signal
a.t/ c.t/ a.t/ c.h/ .t/ sin.2³ f 0 t C '0 / cos.2³ f 0 t C '0 / Z C1 Z C1 Ex D jx.t/j2 dt D jx .h/ .t/j2 dt D E x .h/
energy
1
orthogonality
hx; x .h/ i D
1
Z
C1
1
x.t/ x .h/ .t/ dt D 0
42
Chapter 1. Elements of signal theory
Baseband equivalent of a transformation Given a transformation involving also passband signals, it is often useful to determine an equivalent relation between baseband complex representations of input and output signals. Three transformations are given in Figure 1.30, together with their baseband equivalent. We will prove the relation illustrated in Figure 1.30b. Assuming h is the real-valued impulse response of an LPF and using (1.158), y.t/ D fh.− / Ł Re[x .bb/ .− /e j2³ f 0 − .cos.2³ f 0 − C '1 //]g.t/
Figure 1.30. Passband transformations and their baseband equivalent.
1.6. Passband signals
"
43
e j'1 eC j .2³ 2 f 0 − C'1 / C h.− / Ł x .bb/ .− / D Re h.− / Ł x .− / 2 2 ½ e j'1 D Re h Ł x .bb/ .t/ 2 .bb/
!
# .t/
(1.193)
where the last equality follows because the term with frequency components around 2 f 0 is filtered by the LPF. .bb/ We note, moreover, that the filter h .bb/ in Figure 1.30c has in-phase component h I .bb/ and quadrature component h Q that are related to H.a/ by (see (1.160) and (1.161)) .bb/ 1 H.bb/ . f / C H.bb/Ł . f /] I . f / D 2 [H
D 12 [H.a/ . f C f 0 / C H.a/Ł . f C f 0 /]
(1.194)
and H.bb/ Q .f/ D
1 [H.bb/ . f / H.bb/Ł . f /] 2j
1 [H.a/ . f C f 0 / H.a/Ł . f C f 0 /] D 2j
(1.195)
Consequently, if H.a/ has Hermitian symmetry around f 0 then .a/ H.bb/ I . f / D Ha . f C f 0 /
and
H.bb/ Q .f/ D 0
1 .bb/ is simplified. In other words h .bb/ .t/ D h .bb/ I .t/ is real and the realization of the filter 2 h .a/ In practice this condition is verified by ensuring that the filter h has symmetrical frequency specifications around f 0 .
Envelope and instantaneous phase and frequency We will conclude this section with a few definitions. Given a PB signal x.t/, with reference to the analytic signal we define: 1. Envelope Mx .t/ D jx .a/ .t/j
(1.196)
'x .t/ D arg x .a/ .t/
(1.197)
2. Instantaneous phase
44
Chapter 1. Elements of signal theory
3. Instantaneous frequency f x .t/ D
1 d 'x .t/ 2³ dt
(1.198)
In terms of the complex envelope signal x .bb/ , from (1.152) the equivalent relations follow: Mx .t/ D jx .bb/ .t/j
(1.199)
'x .t/ D arg x .bb/ .t/ C 2³ f 0 t
(1.200)
f x .t/ D
1 d [arg x .bb/ .t/] C f 0 2³ dt
(1.201)
Then, from the polar representation x .a/ .t/ D Mx .t/ e j'x .t/ and from (1.157), a PB signal x can be written as x.t/ D Re[x .a/ .t/] D Mx .t/ cos.'x .t//
(1.202)
Two simplified methods to get the envelope Mx .t/ from the PB signal x.t/ are given in Figure 6.58 on page 514. For example if x.t/ D A cos.2³ f 0 t C '0 / it follows that Mx .t/ D A
(1.203)
'x .t/ D 2³ f 0 t C '0
(1.204)
f x .t/ D f 0
(1.205)
With reference to the above relations, three other definitions follow. 1. Envelope deviation 1Mx .t/ D jx .a/ .t/j A D jx .bb/ .t/j A
(1.206)
1'x .t/ D 'x .t/ .2³ f 0 t C '0 / D arg x .bb/ .t/ '0
(1.207)
2. Phase deviation
3. Frequency deviation 1 f x .t/ D f x .t/ f 0 D
1 d 1'x .t/ 2³ dt
(1.208)
Then (1.202) becomes x.t/ D [A C 1Mx .t/] cos.2³ f 0 t C '0 C 1'x .t//
1.7
(1.209)
Second-order analysis of random processes
We recall the functions related to the statistical description of random processes, especially those functions concerning second-order analysis.
1.7. Second-order analysis of random processes
1.7.1
45
Correlation
Let x.t/ and y.t/, t 2 < K T 1 þ t T þ 0 < t < 2T þ T þ gu .t/ D > :0 elsewhere
(1.440)
(1.441)
76
Chapter 1. Elements of signal theory
g(t)
tg
0 t0 = 0
-tg t0 = t g
t
gM (t)
t
0 gM (t)
0
tg
t
tg
t
r g (t)
-tg
0
Figure 1.39. Various pulse shapes related to a matched filter.
1.11
Ergodic random processes
The functions that have been introduced in the previous sections for the analysis of random processes give a valid statistical description of an ensemble of realizations of a random process. We investigate now the possibility of moving from ensemble averaging to time averaging, that is we consider the problem of estimating a statistical descriptor of a random process from the observation of a single realization. Let x be a discrete-time WSS random process having mean mx . If in the limit it holds18 1 1 KX lim x.k/ D E[x.k/] D mx (1.442) K !1 K kD0
18 The limit is meant in the mean-square sense, that is the variance of the r.v.
for K ! 1.
1 P K 1 x.k/ m x vanishes K kD0
1.11. Ergodic random processes
77
then x is said to be ergodic in the mean. In other words, for a process for which the above limit is true, the time-average of samples tends to the statistical mean as the number of samples increases. We note that the existence of the limit (1.442) implies the condition 2þ þ2 3 1 þ 1 KX þ þ þ lim E 4þ (1.443) x.k/ mx þ 5 D 0 þ K kD0 þ K !1 or equivalently 1 lim K !1 K
K 1 X nD.K 1/
½ jnj 1 cx .n/ D 0 K
(1.444)
From (1.444) we see that for a random process to be ergodic in the mean, some conditions on the second-order statistics must be verified. Analogously to definition (1.442), we say that x is ergodic in correlation if in the limits it holds: lim
K !1
1 1 KX x.k/x Ł .k n/ D E[x.k/x Ł .k n/] D rx .n/ K kD0
(1.445)
Also for processes that are ergodic in correlation one could get a condition of ergodicity similar to that expressed by the limit (1.444). Let y.k/ D x.k/x Ł .k n/. Observing (1.445) and (1.442), we find that the ergodicity in correlation of the process x is equivalent to the ergodicity in the mean of the process y. Therefore it is easy to deduce that the condition (1.444) for y translates into a condition on the statistical moments of the fourth order for x. In practice, we will assume all stationary processes to be ergodic; ergodicity is, however, difficult to prove for non-Gaussian random processes. We will not consider particular processes that are not ergodic such as x.k/ D A, where A is a random variable, or x.k/ equal to the sum of sinusoidal signals (see (1.386)). The property of ergodicity assumes a fundamental importance if we observe that from a single realization it is possible to obtain an estimate of the autocorrelation function and from this, the power spectral density. Alternatively, one could prove that under the hypothesis19 C1 X
jnj rx .n/ < 1
(1.446)
nD1
the following limit holds:
2
1 lim E 4 K !1 K Tc
þ þ2 3 1 þ KX þ þ þ x.k/ e j2³ f kTc þ 5 D Px . f / þTc þ kD0 þ
(1.447)
Then, exploiting the ergodicity of a WSS random process, one obtains the relations among the process itself, its autocorrelation function and power spectral density shown
19 We note that for random processes with non-zero mean and/or sinusoidal components this property is not
verified. Therefore it is usually recommended that the deterministic components of the process be removed before the spectral estimation is performed.
78
Chapter 1. Elements of signal theory
Figure 1.40. Relation between ergodic processes and their statistical description.
in Figure 1.40. We note how the direct computation of the PSD, given by (1.447), makes use of a statistical ensemble of the Fourier transform of process x, while the indirect method via ACS makes use of a single realization. If we let XQK Tc . f / D Tc F[x.k/ w K .k/]
(1.448)
where w K is the rectangular window of length K (see (1.474)) and Td D K Tc , (1.447) becomes Px . f / D lim
Td !1
E[jXQTd . f /j2 ] Td
(1.449)
The relation (1.449) holds also for continuous-time ergodic random processes, where XQTd . f / denotes the Fourier transform of the windowed realization of the process, with a rectangular window of duration Td .
1.11.1
Mean value estimators
Given the random process fx.k/g, we wish to estimate the mean value of a related process fy.k/g: for example, to estimate the statistical power of x we set y.k/ D jx.k/j2 , while for the estimation of the correlation of x with lag n we set y.k/ D x.k/x Ł .k n/. Based on a realization of fy.k/g, from (1.442) an estimate of the mean value of y is given by the expression mO y D
1 1 KX y.k/ K kD0
(1.450)
1.11. Ergodic random processes
79
In fact, (1.450) attempts to determine the average component of the signal fy.k/g. As illustrated in Figure 1.41a, in general we can think of extracting the average component of fy.k/g using an LPF filter h having unit gain, i.e. H.0/ D 1, and suitable bandwidth B. Let K be the length of the impulse response with support from k D 0 to k D K 1. Note that for a unit step input signal the transient part of the output signal will last K 1 time instants. Therefore we assume mO y D z.k/ D h Ł y.k/
for k ½ K 1
(1.451)
We now compute mean and variance of the estimate. From (1.451), the mean value is given by E[mO y ] D m y H.0/ D m y
(1.452)
0.035
0.03
0.025
h(k)
0.02
0.015
0.01
0.005
0
0
5
10
15
(a)
20 k
25
30
35
40
(b) 1.2
1
0.8
|H(f)|
0.6
0.4
0.2
0
−0.2
0
0.05
0.1
0.15
0.2
0.25 fT
0.3
0.35
0.4
0.45
0.5
c
(c)
Figure 1.41. (a) Time average as output of a narrow band lowpass filter. (b) Typical impulse responses: exponential filter with parameter a D 125 and rectangular window with K D 33. (c) Corresponding frequency responses.
80
Chapter 1. Elements of signal theory
as H.0/ D 1. Using the expression in Table 1.6 of the correlation of a filter output signal given the input, the variance of the estimate is given by var[mO y ] D ¦ y2 D
C1 X
rh .n/c y .n/
(1.453)
nD1
Assuming C1 X
SD
jc y .n/j D ¦ y2
nD1
C1 X jc y .n/j
kTc0 t2 Tc
then, letting
¾ n1 D
n
<x k k D 0; šL ; š2L ; : : : L (1.592) yk D > : 0 otherwise where L, the interpolation factor, is an integer number. We will show that the input–output relation in terms of the z-transforms Y .z/ and X .z/ is given by Y .z/ D X .z L /
(1.593)
Equivalently, in terms of radian frequency normalized by the sampling frequency, !0 D 2³ f =Fc0 , then (1.593) can be expressed as 0
0
Y .e j! / D X .e j! L /
(1.594)
The graphical interpretation of (1.594) is illustrated in Figure 1.59: Y .e j! / is the compressed version by a factor L of X .e j! /; moreover, there are L 1 replicas of the compressed spectrum, called images. The creation of images implies that a lowpass signal does not remain lowpass after interpolation. It is also useful to give the expression of the output sequence in the frequency domain; we get Y. f / D X . f /
(1.595)
X . f / D X .e j2³ f Tc / Tc Y. f / D Y e j2³ f L
(1.596)
where
(1.597)
The (1.595) for the signal of Figure 1.59 is illustrated in Figure 1.60. We note that the only effect of the interpolation is that the signal X must be regarded as periodic with period Fc0 rather than Fc . xn Tc 1 Fc = Tc
yk L T’c =
Tc L
F’c =LF c
Figure 1.58. Interpolation or upsampling transformation by a factor L.
110
Chapter 1. Elements of signal theory
¼
¼
Figure 1.59. Interpolation by a factor L D 3: (a) in the time domain, (b) in the normalized radian frequency domain.
´ µ
̽
¼
´ µ
̽
¼
¾
½
Ì
Ì
½
¾
Ì
Ì
¿
Ì
¿
Ì
Figure 1.60. Effect of interpolation in the frequency domain.
Proof of (1.593). Y .z/ D
Observing (1.592) we get C1 X
yk z k D
kD1
1.A.4
C1 X nD1
yn L z n L D
C1 X
xn z n L D X .z L /
(1.598)
nD1
Decimator filter
In most applications, a downsampler is preceded by a lowpass digital filter, to form a decimator filter as illustrated in Figure 1.61. The filter ensures that the signal vn is bandlimited, to avoid aliasing in the downsampling process.
1.A. Multirate systems
111
¼
¼
Figure 1.61. Decimator filter.
Let h n D h.nTc /. Then we have yk D vk M
(1.599)
and vn D
C1 X
h i xni
(1.600)
i D1
The output can be expressed as C1 X
yk D
C1 X
h i x k Mi D
h k Mn xn
(1.601)
nD1
i D1
Using definition (1.577) we get gk;i D h i
8k; i
(1.602)
Note that the overall system is not time invariant, unless the delay applied to the input is constrained to be a multiple of M. From V .z/ D X .z/H .z/ it follows that Y .z/ D
X 1 M1 m m H .z 1=M W M /X .z 1=M W M / M mD0
(1.603)
or, equivalently, recalling that !0 D 2³ f M Tc , 0
Y .e j! / D
X 1 M1 H .e M mD0
j
!0 2³ m !0 2³ m M /X .e j M /
(1.604)
If ( H .e / D j!
1 0
³ M otherwise
j!j
(1.605)
we obtain 0
Y .e j! / D
0 1 j! X eM M
j!0 j ³
(1.606)
In this case h is a lowpass filter that avoids aliasing caused by sampling; if x is bandlimited, the specifications of h can be made less stringent. The decimator filter transformations are illustrated in Figure 1.62 for M D 4.
112
Chapter 1. Elements of signal theory
| X (f)|
| H (f)|
| V (f)|
| Y (f)|
0
Fc /2
Fc f
0
Fc /2
Fc f
0
Fc /2
Fc f
0 F’c /2 F’c
Fc /2
Fc
f
Figure 1.62. Frequency responses related to the transformations in a decimator filter for M D 4.
1.A.5
Interpolator filter
An interpolator filter is given by the cascade of an upsampler and a digital filter, as illustrated in Figure 1.63; the task of the digital filter is to suppress images created by upsampling [17]. Let h n D h.nTc0 /. Then we have the following input–output relations: yk D
C1 X
h k j w j
(1.607)
jD1
wk D
8
< 1 X e M 00 M Y .e j! / D > : 0
or Y. f / D
1 X . f / for M
³M j!00 j min ³; L elsewhere 1 L j f j min ; 2Tc 2M Tc
(1.620)
(1.621)
(1.622)
Example 1.A.1 (M > L: M D 5, L D 4) Transformations for M D 5 and L D 4 are illustrated in Figure 1.67. Observing the fact 0 ³ ³ that W .e j! / is zero for M !0 2 ³L M , the desired result is obtained by a response 0 j! H .e / that has the stopband cut-off frequency within this interval. Example 1.A.2 (M < L: M D 4, L D 5) The inverse transformation of the above example is obtained by a transformation with M D 4 and L D 5, as depicted in Figure 1.68.
116
Chapter 1. Elements of signal theory
X(e j ω)
0
W(e j ω’)
π /L Fc /2
0
H(e j ω’)
4.2 π 4Fc
ω = 2π f T f
2π 4Fc
ω’ = ω L f
0
π /M LFc 2M
2π LFc
ω’ f
0
π /M LFc
2π LFc
ω’ f
V(e j ω’)
Y(e j ω")
2M M=5
0
5.2 π LFc
π LFc
ω" = ω ’ M f
2M
Figure 1.67. Rate conversion by a rational factor L=M where M > L.
1.A.7
Time interpolation
Referring to the interpolator filter h of Figure 1.63, one finds that if L is large the filter implementation may require non-negligible complexity; in fact, the number of coefficients required for an FIR filter implementation can be very large. Consequently, in the case of a very large interpolation factor L, after a first interpolator filter with a moderate value of the interpolation factor, the samples fyk D y.kTc0 /g may be further time interpolated until the desired sampling accuracy is reached [17]. As shown in Figure 1.69, let fyk g be the sequence that we need to interpolate to produce the signal z.t/, t 2 0 P[A a] D (1.652) 0 a > : A 1
per a < A (3.134)
per A a A per a > A
where A is a positive parameter.
1
β =6 β =12 β =24 β =48
0.8 0.6 0.4
ϕ (a)
0.2 0 −0.2 −0.4 −0.6 −0.8 −1 −1
−0.8
−0.6
−0.4
−0.2
0 a
0.2
0.4
0.6
0.8
Figure 3.10. Sigmoidal function for various values of the parameter þ.
1
3.1. Adaptive transversal filter: MSE criterion
189
Normalized LMS In the LMS algorithm, if some x.k/ assume large values, the adaptation algorithm is affected by strong noise in the gradient. This problem can be overcome by choosing an adaptation gain ¼ of the type: ¼D
¼Q p C MO x .k/
(3.135)
where 0 < ¼Q < 2, and MO x .k/ D jjxjj2 D
N 1 X
jx.k i/j2
(3.136)
i D0
or, alternatively, MO x .k/ D N MO x .k/
(3.137)
where MO x .k/ is the estimate of the statistical power of x.k/. A simple estimate is obtained by the iterative equation (see (1.468)): MO x .k/ D a MO x .k 1/ C .1 a/jx.k/j2
(3.138)
where 0 < a < 1, with time constant given by −D
1 1 ' ln a 1a
(3.139)
for a ' 1. In (3.135), p is a positive parameter that is introduced to avoid the denominator becoming too small; typically p'
1 Mx 10
(3.140)
The normalized LMS algorithm has a speed of convergence that is potentially higher than the standard algorithm, for uncorrelated as well as correlated input signals [10]. To be able to apply the normalized algorithm, however, some knowledge of the input process is necessary, in order to assign the values of Mx and p so that the adaptation process does not become unstable.
Variable adaptation gain In the following variants of the LMS algorithms the coefficient ¼ varies with time. 1. Two values of ¼. a. Initially a large value of ¼ is chosen for fast convergence, for example, ¼ D 1=.N rx .0//. b. Subsequently ¼ is reduced to achieve a smaller J .1/.
190
Chapter 3. Adaptive transversal filters
For a choice of ¼ of the type ( ¼D
¼1 ¼2
per 0 k K 1 per k ½ K 1
(3.141)
the behavior of J will be illustrated in Figure 3.11. 2. Decreasing ¼. For a time-invariant system, the adaptation gain usually selected for application with the sign algorithm (3.130) is given by ¼.k/ D
¼1 ¼2 C k
k½0
(3.142)
3. ¼ proportional to e.k/. The following expression of ¼ is used: ¼.k C 1/ D Þ1 ¼.k/ C Þ2 je.k/j2
(3.143)
with ¼ limited to the range [¼min ; ¼max ]. Typical values are Þ1 ' 1 and Þ2 − 1. 4. Vector of values of ¼. Let µT D [¼0 ; : : : ; ¼ N 1 ]; two approaches are possible. a. Initially larger values ¼i are chosen in correspondence of those coefficients ci that have larger amplitude.
J(k)
µ1
µ2 = µ1 / 2 Jmin
0
K
1
k
Figure 3.11. Behavior of J.k/ obtained by using two values of ¼.
3.1. Adaptive transversal filter: MSE criterion
191
b. ¼i changes with time following the rule 8 1 if the i-th component of the gradient has al> > < ¼i .k/ Þ ways changed sign in the last m 0 iterations ¼i .k C 1/ D > > if the i-th component of the gradient has : ¼i .k/Þ never changed sign in the last m 1 iterations (3.144) with ¼ limited to the range [¼min ; ¼max ]. Typical values are m 0 ; m 1 2 f1; 3g and Þ D 2.
LMS for lattice filters We saw in Section 2.2.1 that filters with a lattice structure have some interesting properties. The application of the LMS algorithm for lattice filters, however, is not as simple as for transversal filters. For this reason such filters are now rarely used, although they were popular in the past when fast hardware implementations were rather costly. For the study of the LMS algorithm for lattice filters we refer the reader to [11, 12].
3.1.5
Example of application: the predictor
We consider a real AR(2) process of unit power, described by the equation x.k/ D a1 x.k 1/ a2 x.k 2/ C w.k/
(3.145)
with w additive white Gaussian noise (AWGN), and a2 D 0:995
(3.146)
From (1.547), the roots of A.z/ are given by %e š'0 , where p % D a2 D 0:997
(3.147)
a1 D 1:3
and a1 D 2:28 rad '0 D cos1 2%
(3.148)
Being rx .0/ D ¦x2 D 1, from the (1.552) we find that the statistical power of w is given by ¦w2 D
1 a2 [.1 C a2 /2 a21 ] D 0:0057 D 22:4 dB 1 C a2
(3.149)
We construct a predictor for x of order N D 2 with coefficients cT D [c1 ; c2 ], as illustrated in Figure 3.12, using the LMS algorithm and some of its variants [13]. From (2.83) we expect to find in steady state c ' a
(3.150)
192
Chapter 3. Adaptive transversal filters
Figure 3.12. Predictor of order N D 2.
that is c1 ' a1 , c2 ' a2 , and ¦e2 ' ¦w2 . In any case, the predictor output is given by y.k/ D cT .k/x.k 1/ D c1 .k/x.k 1/ C c2 .k/ x.k 2/
(3.151)
with prediction error e.k/ D x.k/ y.k/
(3.152)
For the predictor of Figure 3.12 we now consider various versions of the adaptive LMS algorithm and their relative performance. Example 3.1.1 (Standard LMS) The equation for updating the coefficient vector is c.k C 1/ D c.k/ C ¼e.k/x.k 1/
(3.153)
Convergence curves are plotted in Figure 3.13 for a single realization and for the mean (estimated over 500 realizations) of the coefficients and of the squared prediction error, for ¼ D 0:04. In Figure 3.14 a comparison is made between the curves of convergence of the mean for three values of ¼. We observe that, by decreasing ¼, the excess error decreases, thus giving a more accurate solution, but the convergence time increases. Example 3.1.2 (Leaky LMS) The equation for updating the coefficient vector is c.k C 1/ D .1 ¼Þ/c.k/ C ¼e.k/x.k 1/
(3.154)
Convergence curves are plotted in Figure 3.15 for a single realization and for the mean (estimated over 500 realizations) of the coefficients and of the squared prediction error,
3.1. Adaptive transversal filter: MSE criterion
193
Figure 3.13. Convergence curves for the predictor of order N D 2, obtained by the standard LMS algorithm. 0.2
−a
1
1.2
0
1
µ =0.01 µ =0.04
0.6
−0.4
µ =0.1
0.4
µ =0.04
−0.6
0.2
µ =00.1
−0.8
0
−0.2
µ =0.1
c2(k)
c1(k)
0.8
−0.2
−a −1 2 0
200
400
600
800
0
1000
200
k
400
600
800
1000
k
0
µ =0.1
J(k) (dB)
−5
µ =0.04
−10
µ =0.01
−15
−20
σ2
w
−25
0
100
200
300
400
500
600
700
800
900
1000
k
Figure 3.14. Comparison among curves of convergence of the mean obtained by the standard LMS algorithm for three values of ¼.
194
Chapter 3. Adaptive transversal filters
Figure 3.15. Convergence curves for the predictor of order N D 2, obtained by the leaky LMS.
for ¼ D 0:04 and Þ D 0:01. We note that the steady-state values are worse than in the previous case. Example 3.1.3 (Normalized LMS ) The equation for updating the coefficient vector is c.k C 1/ D c.k/ C ¼.k/e.k/x.k 1/
(3.155)
The adaptation gain ¼ is of the type ¼.k/ D
¼Q p C N ¦O x2 .k/
(3.156)
where ¦O x2 .k/ D a ¦O x2 .k 1/ C .1 a/jx.k/j2 ¦O x2 .1/ D 12 [jx.1/j2 C jx.2/j2 ]
k½0
(3.157) (3.158)
with a D 1 25 D 0:97
(3.159)
and pD
1 E[jjxjj2 ] D 0:2 10
(3.160)
3.1. Adaptive transversal filter: MSE criterion
195
Figure 3.16. Convergence curves for the predictor of order N D 2, obtained by the normalized LMS algorithm.
Convergence curves are plotted in Figure 3.16 for a single realization and for the mean (estimated over 500 realizations) of the coefficients and of the squared prediction error, for ¼Q D 0:08. We note that, with respect to the standard LMS algorithm, the convergence is considerably faster. A direct comparison of the convergence curves obtained in the previous examples is given in Figure 3.17. Example 3.1.4 (Sign LMS algorithm) We consider the three versions of the sign LMS algorithm: (1) c.k C 1/ D c.k/ C ¼ sgn.e.k//x.k 1/, (2) c.k C 1/ D c.k/ C ¼e.k/ sgn.x.k 1//, (3) c.k C 1/ D c.k/ C ¼ sgn.e.k// sgn.x.k 1//. A comparison of convergence curves is given in Figure 3.18 for the three versions of the sign LMS algorithm, for ¼ D 0:04. It turns out that version (2), where the estimation error in the adaptation equation is not quantized, yields the best performance in steady state. Version (3), however, yields fastest convergence. To decrease the prediction error in steady state for versions (1) and (3), the value of ¼ could be further lowered, at the expense of reducing the speed of convergence.
196
Chapter 3. Adaptive transversal filters
Figure 3.17. Comparison of convergence curves for the predictor of order N D 2, obtained by three versions of the LMS algorithm. 0.2
−a1 1.2
0
1
ver.3 ver.1
0.6
0.4
−0.4
ver.1
−0.6
ver.2
ver.3
0.2
−0.8
0
−0.2
ver.2
c2(k)
c1(k)
0.8
−0.2
−a −1 2 0
200
400
600
800
0
1000
200
400
k
600
800
1000
k
0
ver.3
J(k) (dB)
−5
ver.1 −10
ver.2 −15
−20
σ2
w
−25
0
100
200
300
400
500
600
700
800
900
1000
k
Figure 3.18. Comparison of convergence curves obtained by three versions of the sign LMS algorithm.
3.2. The recursive least squares (RLS) algorithm
197
Observation 3.2 As observed on page 97, for an AR process x, if the order of the predictor is greater than the required minimum, the correlation matrix result is ill-conditioned with a large eigenvalue spread. Thus the convergence of the LMS prediction algorithm can be extremely slow and can lead to a solution quite different from the Yule–Walker solution. In this case it is necessary to adopt a method that ensures the stability of the error prediction filter, such as the leaky LMS.
3.2
The recursive least squares (RLS) algorithm
We now consider a recursive algorithm to estimate the vector of coefficients c by an LS method, named recursive least squares (RLS) algorithm. The RLS algorithm is characterized by a speed of convergence that can be one order of magnitude faster than the LMS algorithm, obtained at the expense of a larger computational complexity. With reference to the system illustrated in Figure 3.19, we introduce the following quantities: 1. Input vector at instant i xT .i/ D [x.i/; x.i 1/; : : : ; x.i N C 1/]
(3.161)
2. Coefficient vector at instant k cT .k/ D [c0 .k/; c1 .k/; : : : ; c N 1 .k/]
(3.162)
3. Filter output signal at instant i, obtained for the vector of coefficients c.k/ y.i/ D cT .k/x.i/ D xT .i/c.k/ x(i)
x(i-1)
x(i-N+1)
x(i-2)
Tc
Tc
c0 (k)
c1 (k)
(3.163)
Tc
c2 (k)
cN-1 (k)
+ -
y(i) e(i)
+ d(i) Figure 3.19. Reference system for a RLS adaptive algorithm.
198
Chapter 3. Adaptive transversal filters
4. Desired output at instant i d.i/
(3.164)
At instant k, based on the observation of the sequences fx.i/g
i D 1; 2; : : : ; k
fd.i/g
(3.165)
the criterion for the optimization of the vector of coefficients c.k/ is the minimum sum of squared errors up to instant k. Defining E.k/ D
k X
½ki je.i/j2
(3.166)
i D1
we want to find min E.k/ c.k/
(3.167)
where the error signal is e.i/ D d.i/ xT .i/c.k/. Two observations arise: ž ½ is a forgetting factor, that enables proper filtering operations even with nonstationary signals or slowly time-varying systems. The memory of the algorithm is approximately 1=.1 ½/. ž This problem is the classical LS problem (2.128), applied to a sequence of prewindowed samples with the exponential weighting factor ½k .
Normal equation Using the gradient method, the optimum value of c.k/ satisfies the normal equation .k/c.k/ D ϑ .k/
(3.168)
where .k/ D
k X
½ki xŁ .i/xT .i/
(3.169)
½ki d.i/xŁ .i/
(3.170)
i D1
ϑ .k/ D
k X i D1
From (3.168), if 1 .k/ exists, the solution is given by c.k/ D 1 .k/ϑ .k/
(3.171)
3.2. The recursive least squares (RLS) algorithm
199
Derivation of the RLS algorithm To solve the normal equation by the inversion of .k/ may be too hard, especially if N is large. Therefore we seek a recursive algorithm for k D 1; 2; : : : . Both expressions of .k/ and ϑ .k/ can be written recursively. From .k/ D
k1 X
½ki xŁ .i/xT .i/ C xŁ .k/xT .k/
(3.172)
i D1
it follows that .k/ D ½.k 1/ C xŁ .k/xT .k/
(3.173)
ϑ .k/ D ½ϑ .k 1/ C d.k/xŁ .k/
(3.174)
and similarly
We now recall the following identity known as matrix inversion lemma [12]. Let A D B1 C CD1 C H
(3.175)
where A, B and D are positive definite matrices. Then A1 D B BC.D C C H BC/1 C H B
(3.176)
For B1 D ½.k 1/
A D .k/
C D xŁ .k/
DD1
(3.177)
the equation (3.176) becomes 1 .k/ D ½1 1 .k 1/
½1 1 .k 1/xŁ .k/xT .k/½1 1 .k 1/ 1 C xT .k/½1 1 .k 1/xŁ .k/
(3.178)
We introduce two quantities: P.k/ D 1 .k/
(3.179)
and kŁ .k/ D
½1 1 .k 1/xŁ .k/ 1 C ½1 xT .k/1 .k 1/xŁ .k/
(3.180)
also called the Kalman vector gain. From (3.178) we have the recursive relation P.k/ D ½1 P.k 1/ ½1 kŁ .k/xT .k/P.k 1/
(3.181)
200
Chapter 3. Adaptive transversal filters
We derive now a simpler expression for kŁ .k/. From (3.180) we obtain kŁ .k/[1 C ½1 xT .k/1 .k 1/xŁ .k/] D ½1 1 .k 1/xŁ .k/
(3.182)
from which we get kŁ .k/ D ½1 P.k 1/xŁ .k/ ½1 kŁ .k/xT .k/P.k 1/xŁ .k/ (3.183) D [½1 P.k 1/ ½1 kŁ .k/xT .k/P.k 1/]xŁ .k/ Using (3.181), it follows kŁ .k/ D P.k/xŁ .k/
(3.184)
Using the (3.174), the recursive equation to update the estimate of c is given by c.k/ D 1 .k/ϑ .k/ D P.k/ϑ .k/
(3.185)
D ½P.k/ϑ .k 1/ C P.k/x .k/d.k/ Ł
Substituting the recursive expression for P.k/ in the first term, we get c.k/ D ½[½1 P.k 1/ ½1 kŁ .k/xT .k/P.k 1/]ϑ .k 1/ C P.k/xŁ .k/d.k/ D P.k 1/ϑ .k 1/ kŁ .k/xT .k/P.k 1/ϑ .k 1/ C P.k/xŁ .k/d.k/
(3.186)
D c.k 1/ C kŁ .k/[d.k/ xT .k/c.k 1/] where in the last step (3.184) has been used. Defining the a priori estimation error, ž.k/ D d.k/ xT .k/c.k 1/
(3.187)
we note that xT .k/c.k 1/ is the filter output at instant k obtained by using the old coefficient estimate. In other words, from the a posteriori estimation error e.k/ D d.k/ xT .k/c.k/
(3.188)
we could say that ž.k/ is an approximated value of e.k/, that is computed before updating c. In any case the relation holds c.k/ D c.k 1/ C kŁ .k/ž.k/
(3.189)
In summary, the RLS algorithm consists of four equations: kŁ .k/ D
P.k 1/xŁ .k/ ½ C xT .k/P.k 1/xŁ .k/
(3.190)
ž.k/ D d.k/ xT .k/c.k 1/
(3.191)
c.k/ D c.k 1/ C ž.k/kŁ .k/
(3.192)
P.k/ D ½1 P.k 1/ ½1 kŁ .k/xT .k/P.k 1/
(3.193)
3.2. The recursive least squares (RLS) algorithm
201
In (3.190), k.k/ is the input vector filtered by P.k 1/ and normalized by the ½ C xT .k/ P.k 1/xŁ .k/. The term xT .k/P.k 1/xŁ .k/ may be interpreted as the energy of the filtered input.
Initialization of the RLS algorithm We need to assign a value to P.0/. We modify the definition of .k/ in .k/ D
k X
½ki xŁ .i/xT .i/ C Ž½k I
with Ž − 1
(3.194)
i D1
so that .0/ D ŽI
(3.195)
This is equivalent to having for k 0 an all zero input with the exception of x.N C 1/ D .½N C1 Ž/1=2 . Consequently P.0/ D Ž 1 I
Ž − rx .0/
(3.196)
Typically Ž 1 D
100 rx .0/
(3.197)
where rx .0/ is the statistical power of the input signal. In Table 3.1 we give a version of the RLS algorithm that exploits the fact that P.k/ (inverse of the Hermitian matrix .k/) is Hermitian, hence xT .k/P.k 1/ D [P.k 1/xŁ .k/] H D π T .k/ Table 3.1 RLS algorithm.
Initialization For k D 1; 2; : : :
c.0/ D 0 P.0/ D Ž 1 I π Ł .k/ D P.k 1/xŁ .k/ 1 r.k/ D T ½ C x .k/π Ł .k/ Ł k .k/ D r.k/π Ł .k/ ž.k/ D d.k/ xT .k/c.k 1/ c.k/ D c.k 1/ C ž.k/kŁ .k/ P.k/ D ½1 .P.k 1/ kŁ .k/π T .k//
(3.198)
202
Chapter 3. Adaptive transversal filters
Recursive form of E min We set Ed .k/ D
k X
½ki jd.i/j2 D ½Ed .k 1/ C jd.k/j2
(3.199)
i D1
From the general LS expression (2.150), Emin .k/ D Ed .k/ ϑ H .k/c.k/
(3.200)
observing (3.174) and (3.192) we get Emin .k/ D ½Ed .k 1/ C jd.k/j2 [½ϑ H .k 1/ C xT .k/d Ł .k/][c.k 1/ C ž.k/kŁ .k/] D ½Ed .k 1/ ½ϑ H .k 1/c.k 1/
(3.201)
C d.k/d Ł .k/ d Ł .k/xT .k/c.k 1/ ϑ H .k/kŁ .k/ž.k/ D ½Emin .k 1/ C d Ł .k/ž.k/ ϑ H .k/kŁ .k/ž.k/ Using the expression (3.179), and recalling that .k/ is Hermitian, from (3.184) we obtain ϑ H .k/kŁ .k/ D ϑ H .k/1 .k/xŁ .k/ D [1 .k/ϑ .k/] H xŁ .k/
(3.202)
Moreover from (3.184) and (3.171) it follows that ϑ H .k/kŁ .k/ D c H .k/xŁ .k/ D x H .k/cŁ .k/
(3.203)
Then (3.201) becomes Emin .k/ D ½Emin .k 1/ C d Ł .k/ž.k/ c H .k/xŁ .k/ž.k/ D ½Emin .k 1/ C ž.k/[d Ł .k/ .xT .k/c.k//Ł ]
(3.204)
Finally, the recursive relation is given by Emin .k/ D ½Emin .k 1/ C ž.k/eŁ .k/
(3.205)
We note that, as Emin .k/ is real, we get ž.k/eŁ .k/ D ž Ł .k/e.k/ that is ž.k/eŁ .k/ is a real scalar value.
(3.206)
3.2. The recursive least squares (RLS) algorithm
203
Convergence of the RLS algorithm We make some remarks on the convergence of the RLS algorithm. ž The RLS algorithm converges in the mean-square sense in about 2N iterations, independently of the eigenvalue spread of R. ž For k ! 1 there is no excess error and the misadjustment MSD is zero. This is true for ½ D 1. ž In any case, when ½ < 1 the “memory” of the algorithm is approximately 1=.1 ½/ and 1½ N (3.207) MSD D 1C½ ž From the above observation it follows that the RLS algorithm for ½ < 1 gives origin to noisy estimates. ž On the other hand the RLS algorithm for ½ < 1 can be used for tracking slowly time-varying systems.
Computational complexity of the RLS algorithm Exploiting the symmetry of P.k/, the computational complexity of the RLS algorithm, expressed as the number of complex multiplications per output sample, is given by CCRLS D 2N 2 C 4N
(3.208)
For a number of .K N C 1/ output samples, the direct method (3.171) requires instead CCDIR D N 2 C N C
N3 K N C1
(3.209)
We note that, if K × N , the direct method is more convenient. In any case the RLS solution has other advantages: 1. It can be numerically more stable than the direct method. 2. It provides an estimate of the coefficients at each step and not only at the end of the data sequence. 3. For ½ < 1 and 1=.1 ½/ much less than the time interval it takes for the input samples to change statistics, the algorithm is capable of “tracking” the changes.
Example of application: the predictor With reference to the AR(2) process considered in Section 3.1.5, convergence curves for the RLS algorithm are plotted in Figure 3.20 for a single realization and for the mean (estimated over 500 realizations) of the coefficients and of the squared estimation error, for ½ D 1. We note that a different scale is used for the abscissa as compared to the LMS method; in fact the RLS algorithm converges in a number of iterations of the order of N .
204
Chapter 3. Adaptive transversal filters
Figure 3.20. Convergence curves for the predictor of order N D 2, obtained by the RLS algorithm.
3.3
Fast recursive algorithms
As observed in the previous section, the RLS algorithm has the disadvantage of requiring .2N 2 C 4N / multiplications per iteration. Therefore we will list a few fast algorithms, whose computational complexity increases linearly with N , the number of dimensions of the coefficient vector c. 1. Algorithms for transversal filters. The fast Kalman algorithm has the same speed of convergence as the RLS, but with a computational complexity comparable to that of the LMS algorithm. Exploiting some properties of the correlation matrix .k/, Falconer and Ljung [14] have shown that the recursive equation (3.193) requires only 10.2N C 1/ multiplications. Cioffi and Kailath [15], with their fast transversal filter (FTF), have further reduced the number of multiplications to 7.2N C 1/. The implementation of these algorithms still remains relatively simple; their weak point resides in the sensitivity of the operations to round off errors in the various coefficients and signals. As a consequence the fast algorithms may become numerically unstable. 2. Algorithms for lattice filters. There are versions of the RLS algorithm for lattice structures that in the literature are called recursive least squares lattice (LSL) that have, in addition to a lower computational complexity than the standard RLS form, strong and weak points similar to those already discussed in the case of the LMS algorithm for lattice structures [12, 16].
3.4. Block adaptive algorithms in the frequency domain
205
Table 3.2 Comparison of three adaptive algorithms in terms of computational complexity.
cost function algorithm multiplications MSE LS
LMS RLS FTF
divisions
2N C 1 2N 2
C 7N C 5 7.2N C 1/
additions subtractions
0 N2
C 4N C 3 4
2N 2N 2
C 6N C 4 6.2N C 1/
3. Algorithms for filters based on systolic structures. A particular structure is the QR decomposition-based LSL. The name comes from the use of an orthogonal triangularization process, usually known as QR decomposition, that leads to a systolic-type structure with the following characteristics: ž high speed of convergence; ž numerical stability, owing to the QR decomposition and lattice structure; ž a very efficient and modular structure, which does not require the a priori knowledge of the filter order and is suitable for implementation in very largescale integration (VLSI) technology. For further study on the subject we refer the reader to [17, 18, 19, 20, 21, 22].
3.3.1
Comparison of the various algorithms
In practice the choice of an algorithm must be made bearing in mind some fundamental aspects: ž computational complexity; ž performance in terms of speed of convergence, error in steady state, and tracking capabilities under non-stationary conditions; ž robustness, that is good performance achieved in the presence of a large eigenvalue spread and finite-precision arithmetic [5, 23]. Regarding the computational complexity per output sample, a brief comparison among LMS, RLS and FTF is given in Table 3.2. Although the FTF method exhibits a lower computational complexity than the RLS method, its implementation is rather laborious, therefore it is rarely used.
3.4
Block adaptive algorithms in the frequency domain
In this section some algorithms are examined that transform the input signal, for example from the time to the frequency domain, before adaptive filtering. With respect to the LMS algorithm, this approach may exhibit: a) lower computational complexity, or b) improved
206
Chapter 3. Adaptive transversal filters
convergence properties of the adaptive process. We will first consider some adaptive algorithms in the frequency domain that offer some advantages from the standpoint of computational complexity [24, 25, 26, 27].
3.4.1
Block LMS algorithm in the frequency domain: the basic scheme
The basic scheme includes a filter that performs the equivalent operation of a circular convolution in the frequency domain. As illustrated in Figure 3.21, the method operates over blocks of N samples. The instant at which a block is processed is k D n N , where n is an integer number. Each input block is transformed using the DFT (see Section 1.4). The samples of the transformed sequence are denoted by fX i .n N /g, i D 0; 1; : : : ; N 1. We indicate with fDi .n N /g and fYi .n N /g, i D 0; 1; : : : ; N 1, respectively, the DFT of the desired output and of the adaptive filter output. Defining E i .n N / D Di .n N / Yi .n N /, the LMS adaptation algorithm is expressed as: Ci ..n C 1/N / D Ci .n N / C ¼E i .n N /X iŁ .n N /
i D 0; 1; : : : ; N 1
(3.210)
In the following, lower case letters will be used to indicate sequences in the time domain, while upper case letters will denote sequences in the frequency domain.
Computational complexity of the block LMS algorithm via FFT We consider the computational complexity of the scheme of Figure 3.21 for N -sample real input vectors. The algorithm requires three N -point FFTs and 2N complex multiplications to update fCi g and compute fYi g. As for real data the complexity of an N -point FFT in
Figure 3.21. Adaptive transversal filter in the frequency domain.
3.4. Block adaptive algorithms in the frequency domain
207
Table 3.3 Comparison between the computational complexity of the LMS algorithm via FFT and the standard LMS for various values of the filter length N.
N
CCLMS f =CCLMSt
16 64 1024
0.41 0.15 0.015
terms of complex multiplications is given by N N -point FFT + 2 2 N N N N D C log2 4 2 2 2
N -point FFT of N real samples D
(3.211)
then the algorithm requires a number of complex multiplications per output sample equal to N 1 C1 (3.212) log2 CCLMS f D 3 4 2 using the fact that fYi g and fCi g, i D 0; 1; : : : ; N 1 are Hermitian sequences. As each complex multiplication requires four real multiplications, the complexity in terms of real multiplications per output sample becomes N 3 log2 C1 (3.213) CCLMS f D 4 4 2 We note that the complexity in terms of real multiplications per output sample of the standard LMS algorithm is CCLMSt D 2N C 1 ' 2N
(3.214)
A comparison between the computational complexity of the LMS algorithm via FFT and the standard LMS algorithm is given in Table 3.3. We note that the advantage of the LMS algorithm via FFT is non negligible even for small values of N . However, as the product between DFTs of two time sequences is equivalent to a circular convolution, the direct application of the scheme of Figure 3.21 is appropriate only if the relation between y and x is a circular convolution rather than a linear convolution.
3.4.2
Block LMS algorithm in the frequency domain: the FLMS algorithm
We consider a block LMS adaptive algorithm in the time domain, for blocks of N input samples. Let us define:
208
Chapter 3. Adaptive transversal filters
1. input vector at instant k xT .k/ D [x.k/; x.k 1/; : : : ; x.k N C 1/]
(3.215)
2. coefficient vector at instant n N cT .n N / D [c0 .n N /; c1 .n N /; : : : ; c N 1 .n N /]
(3.216)
3. filter output signal at instant n N C i y.n N C i/ D cT .n N /x.n N C i/
(3.217)
4. error at instant n N C i e.n N C i/ D d.n N C i/ y.n N C i/
i D 0; 1; : : : N 1
(3.218)
The equation for updating the coefficients according to the block LMS algorithm is given by c..n C 1/N / D c.n N / C ¼
N 1 X
e.n N C i/xŁ .n N C i/
(3.219)
i D0
As in the case of the standard LMS algorithm, the updating term is the estimate of the gradient at instant n N , ∇.n N /. The above equations can be efficiently implemented in the frequency domain by the overlap-save technique (see (1.112)). Assuming L-point blocks, where for example L D 2N , we define10 C0 T .n N / D DFT[cT .n N /; 0; : : : ; 0] | {z }
(3.220)
N zeros
² ¦ X0 .n N / D diag DFT[x.n N N /; : : : ; x.n N 1/; x.n N /; : : : ; x.n N C N 1/] {z } | {z } | block n1
block n
(3.221) and Y0 .n N / D X0 .n N /C0 .n N / then the filter output at instants k 2 y.n N / 6 y.n N C 1/ 6 y.n N / D 6 :: 4 :
(3.222) D n N ; n N C 1; : : : ; n N C N 1, is given by 3 7 7 7 D last N elements of DFT1 [Y0 .n N /] 5
y.n N C N 1/
10 The superscript 0 denotes a vector of 2N elements.
(3.223)
3.4. Block adaptive algorithms in the frequency domain
209
We give now the equations to update the coefficients in the frequency domain. Let us consider the m-th component of the gradient, [∇.n N /]m D
N 1 X
e.n N C i/x Ł .n N C i m/
m D 0; 1; : : : ; N 1
(3.224)
i D0
This component is given by the correlation between the error sequence fe.k/g and input fx.k/g, which is also equal to the convolution between e.k/ and x Ł .k/. Let E0 T .n N / D DFT[0; : : : ; 0; d.n N / y.n N /; : : : ; d.n N C N 1/ y.n N C N 1/] | {z } | {z } N zeros
errors in block n
(3.225) then ∇.n N / D first N elements of DFT1 [X0 Ł .n N /E0 .n N /] In the frequency domain, the adaptation equation (3.219) becomes ½ ∇.n N / C0 ..n C 1/N / D C0 .n N / C ¼DFT 0
(3.226)
(3.227)
where 0 is the null vector with N elements. In summary, if 0 N ðN is the N ð N all zero matrix, I N ðN the N ð N identity matrix, and F the 2N ð 2N DFT matrix, then the following equations define the fast LMS (FLMS): d0 T .n N /D[0T ; d.n N /; : : : ; d.n N C N 1/] ½ 0 0 y0 .n N /D N ðN N ðN F1 [X0 .n N /C0 .n N /] 0 N ðN I N ðN E0 .n N /DF[d0 .n N / y0 .n N /] ½ I 0 C0 ..n C1/N /DC0 .n N / C ¼F N ðN N ðN F1 [X0 Ł .n N /E0 .n N /] 0 N ðN 0 N ðN
(3.228) (3.229) (3.230) (3.231)
The implementation of the FLMS algorithm is illustrated in Figure 3.22.
Computational complexity of the FLMS algorithm For N output samples we have to evaluate five 2N -point FFTs and 4N complex multiplications. For real input samples, referring to the scheme in Figure 3.22, the complexity in terms of real multiplications per output sample is given by CCFLMS D 10 log2 N C 8
(3.232)
A comparison between the computational complexity of the FLMS algorithm and the standard LMS is given in Table 3.4.
Figure 3.22. Implementation of the FLMS algorithm.
210 Chapter 3. Adaptive transversal filters
3.5. LMS algorithm in a transformed domain
211
Table 3.4 Computational complexity comparison between FLMS and LMS.
N
CCFLMS =CCLMS
16 32 64 1024
1.5 0.85 0.53 0.05
Convergence in the mean of the coefficients for the FLMS algorithm Observing (3.217) and (3.218), and taking the expectation of both members of the adaptation equation (3.219), we get E[c..n C 1/N /] D E[c.n N /] C ¼N .p R E[c.n N /]/ D .I ¼N R/E[c.n N /] C ¼N p
(3.233)
where, as usual, R D E[xŁ .k/xT .k/] and p D rdx D E[d.k/xŁ .k/]. Recalling the analysis of the convergence of the steepest-descent algorithm of Section 3.1.1, we have lim E[c..n C 1/N /] D R1 p
n!1
(3.234)
for 0 < ¼ < 2=.N ½max /, where ½max is the maximum eigenvalue of R. From these equations we can conclude: 1. The FLMS algorithm converges in the mean to the same solution of the LMS, however, ¼ must be smaller by a factor N in order to guarantee stability. 2. The time constant for the convergence of the i-th mode (for ¼ − 1) is −i D
1 ¼½i N
1 D ¼½i
blocks (3.235) samples
equal to that of the LMS algorithm. 3. For ¼ − 2=N ½max , it can be seen that the misadjustment is equal to that of the LMS algorithm: ¼ ¼ (3.236) MSD D tr[R] D N rx .0/ 2 2
3.5
LMS algorithm in a transformed domain
We consider now some adaptive algorithms in the frequency domain that offer some advantages in terms of speed of convergence [28].
212
Chapter 3. Adaptive transversal filters
3.5.1
Basic scheme
Referring to Figure 3.23, we define the following quantities. 1. Input vector at instant k xT .k/ D [x.k/; x.k 1/; : : : ; x.k N C 1/]
(3.237)
with correlation matrix Rx D E[xŁ .k/xT .k/]. 2. Transformed vector zT .k/ D [z 0 .k/; z 1 .k/; : : : ; z N 1 .k/]
(3.238)
z.k/ D Gx.k/
(3.239)
In general,
where G is a unitary matrix of rank N : G1 D G H
(3.240)
cT .k/ D [c0 .k/; c1 .k/; : : : ; c N 1 .k/]
(3.241)
3. Coefficient vector at instant k
Figure 3.23. General scheme for a LMS algorithm in a transformed domain.
3.5. LMS algorithm in a transformed domain
213
4. Filter output signal y.k/ D zT .k/c.k/ D cT .k/z.k/
(3.242)
e.k/ D d.k/ y.k/
(3.243)
5. Estimation error
6. Equation for updating the coefficients, LMS type: ci .k C 1/ D ci .k/ C ¼i e.k/z iŁ .k/
i D 0; 1; : : : ; N 1
(3.244)
where ¼i D
¼Q E[jz i .k/j2 ]
(3.245)
We note that each component of the adaptation gain vector has been normalized using the statistical power of the corresponding component of the transformed input vector. The various powers can be estimated, e.g., by considering a small window of input samples or recursively. Let N D diagfE[jz 0 .k/j2 ]; E[jz 1 .k/j2 ]; : : : ; E[jz N 1 .k/j2 ]g
(3.246)
Then (3.244) can be written in vector notation as 1 Ł c.k C 1/ D c.k/ C ¼e.k/ Q N z .k/
(3.247)
We find that, for a suitable choice of ¼, Q lim c.k C 1/ D copt D R1 z rdz
(3.248)
Rz D E[zŁ .k/zT .k/] D E[GŁ xŁ .k/xT .k/GT ] D GŁ Rx GT
(3.249)
k!1
where
and rdz D E[d.k/zŁ .k/] D GŁ E[d.k/xŁ .k/] D GŁ rdx
(3.250)
Then copt D .GŁ Rx G/1 GŁ rdx Ł 1 Ł D G1 R1 G rdx x G
D G H R1 x rdx D G H .R1 x rdx / where R1 x rdx is the optimum Wiener solution without transformation.
(3.251)
214
Chapter 3. Adaptive transversal filters
On the speed of convergence The speed of convergence depends on the eigenvalue spread of the matrix Rz . If Rz is diagonal, then the eigenvalue spread of 1 N Rz is equal to one. Consequently, a transformation with these characteristics exhibits the best convergence properties. In this case the adaptation algorithm reduces to N independent scalar adaptation algorithms in the transformed domain, and the N modes of convergence do not influence each other. Common choices for G are the following: 1. Karhunen-Lo`eve transform (KLT). The KLT depends on Rx , and consequently is difficult to evaluate in real time. 2. Lower triangular matrix transformation, used in lattice filters. 3. DFT and discrete cosine transform (DCT). They reduce the number of computations to evaluate z.k/ in (3.239) from O.N 2 / to O.N log2 N /. Moreover, recalling the definition (1.376) of the eigenvalue spread, these two transformations, for the normalization 1 N , whiten the signal x by operating on the different sub-bands; the resulting signal, with reduced spectral variations, is used for the adaptation process.
3.5.2
Normalized FLMS algorithm
The convergence of the FLMS algorithm can be improved by dividing each component of the vector [X0 Ł .n N /E0 .n N /] in (3.231) by the power of the respective component of X0 .n N /. Consequently the adaptation gain ¼ is adjusted to the various modes. This procedure, however, requires that the components of X0 .n N / are indeed uncorrelated.
3.5.3
LMS algorithm in the frequency domain
In this case GDF
(3.252)
N ð N DFT matrix. Then z i .k/ D
N 1 X
x.k m/e j2³
mi N
i D 0; 1; : : : ; N 1
(3.253)
i C x.k/ x.k N / z i .k/ D z i .k 1/ exp j2³ N
(3.254)
mD0
or, in a simpler recursive form,
The filters are of passband comb type, implemented by either 1) FFT with parallel input or 2) recursively with serial input to implement equations (3.254), as illustrated in Figure 3.24. In both cases the computational complexity to evaluate the output sample y.k/ is O.N log2 N /. Observation 3.3 ž A filter bank can be more effective in separating the various subchannels in frequency, even if more costly from the point of view of the computational complexity.
3.5. LMS algorithm in a transformed domain
215
Figure 3.24. Adaptive filter in the frequency domain.
ž There are versions of the algorithm where each output z i .k/ is decimated, with the aim of reducing the number of operations. ž If fx.k/g and fd.k/g are real-valued signals, the filter coefficients satisfy the Hermitian property: ci .k/ D cŁN 1i .k/
3.5.4
i D 0; 1; : : : ;
N 1 2
(3.255)
LMS algorithm in the DCT domain
The LMS algorithm in the DCT domain is obtained by filtering the input by the filter bank of Figure 3.24, where the i-th filter has impulse response and transfer function given by, respectively, gi .k/ D cos
³.2k C 1/i 2N
k D 0; 1; : : : ; N 1
(3.256)
and ³ .1 z 1 /.1 .1/i z N / cos i 2N ³ G i .z/ D Z[g i .k/] D z 1 C z 2 1 2 cos N
(3.257)
216
Chapter 3. Adaptive transversal filters
Correspondingly, we have p N 1 2X x.k m/ z 0 .k/ D N mD0 z i .k/ D
1 2 NX ³.2m C 1/i x.k m/ cos N mD0 2N
i D0
(3.258)
i D 1; 2; : : : ; N 1
(3.259)
Ignoring the gain factor cos..³=2N /i/, that can be included in the coefficient ci , even the filtering operation determined by G i .z/ can be implemented recursively [12]. We note that, if all the signals are real, the scheme can be implemented by using real arithmetic.
3.5.5
General observations
ž Orthogonalization algorithms are useful if the input has a large eigenvalue spread and fast adaptation is required. ž If the signals exhibit time-varying statistical parameters, usually these methods do not offer any advantage over the standard LMS algorithm. ž In general, they require larger computational complexity than the standard LMS.
3.6
Examples of application
We give now some examples of applications of the algorithms investigated in this chapter [1, 25, 29, 30].
3.6.1
System identification
We want to determine the relation between the input x and the output z of the system illustrated in Figure 3.25. We note that the observation d is affected by additive noise w, having zero mean and variance ¦w2 , assumed statistically independent of x.
Figure 3.25. System model in which we want to identify the relation between x and z.
3.6. Examples of application
217
Linear case Assuming the system between z.k/ and x.k/ can be modelled as a FIR filter, the experiment illustrated in Figure 3.26 can be adopted to estimate the filter impulse response. Using an input x, known to both systems, we determine the output of the transversal filter c with N coefficients y.k/ D
N 1 X
ci .k/x.k i/ D cT .k/x.k/
(3.260)
i D0
and the estimation error e.k/ D d.k/ y.k/
(3.261)
The LMS adaptation equation follows, c.k C 1/ D c.k/ C ¼e.k/xŁ .k/
(3.262)
We analyze the specific case of an unknown linear FIR system whose impulse response has Nh coefficients. Assuming N ½ Nh , we introduce the vector h with N components, hT D [h 0 ; h 1 ; : : : ; h Nh 1 ; 0; : : : ; 0]
(3.263)
In this case, d.k/ D h 0 x.k/ C h 1 x.k 1/ C Ð Ð Ð C h Nh 1 x.k .Nh 1// C w.k/ D hT x.k/ C w.k/
(3.264)
For N ½ Nh , and assuming the input x is white noise11 with statistical power rx .0/, we get R D E[xŁ .k/xT .k/] D rx .0/I
(3.265)
Figure 3.26. Adaptive scheme to estimate the impulse response of the unknown system.
11 Typically x is generated by repeating a PN sequence of length L > N (see Appendix 3.A). h
218
Chapter 3. Adaptive transversal filters
and p D E[d.k/xŁ .k/] D rx .0/h
(3.266)
Then the Wiener–Hopf solution to the system identification problem is given by copt D R1 p D h
(3.267)
Jmin D ¦w2
(3.268)
and
From (3.267) we see that the noise w does not affect the solution copt , consequently the expectation of (3.262) for k ! 1 (equal to copt ) is also not affected by w. Anyway, as seen in Section 3.1.3, the noise influences the convergence process and the solution obtained by the adaptive LMS algorithm. The larger the power of the noise, the smaller ¼ must be so that c.k/ approaches E[c.k/]. In any case J .1/ 6D 0. On the other hand, if N < Nh then copt in (3.267) coincides with the first N coefficients of h, and Jmin D ¦w2 C rx .0/ jj h.1/jj2
(3.269)
where h.1/ represents the residual error vector, h.1/ D [0; : : : ; 0; h N ; h N C1 ; : : : ; h Nh 1 ]T
(3.270)
As the input x is white, the convergence behavior of the LMS algorithm (3.262) is easily determined. Let be defined as in (3.79): D 1 C rx .0/.¼2 N rx .0/ 2¼/ Let c.k/ D c.k/ copt ; then we get J .k/ D E[je.k/j2 ] D Jmin C rx .0/E[jj c.k/jj2 ]
(3.271)
where E[jj c.k/jj2 ] D k E[jj c.0/jj2 ] C ¼2 N rx .0/ Jmin
1 k 1
k½0
The result (3.272) is obtained by (3.70) and the following assumptions: 1. c.k/ is statistically independent of x.k/; 2. emin .k/ is orthogonal to x.k/; 3. the approximation xT .k/ xŁ .k/ ' N rx .0/ holds.
(3.272)
3.6. Examples of application
219
Indeed, (3.272) is an extension of (3.78). At convergence, for ¼ rx .0/ − 1, it results in E[jj c.1/jj2 ] D ¼
N Jmin 2
(3.273)
and N J .1/ D Jmin 1 C ¼ rx .0/ 2
(3.274)
A faster convergence and a more accurate estimate, for fixed ¼, are obtained by choosing a smaller value of N ; this, however, may increase the residual estimation error (3.269). Example 3.6.1 Consider an unknown system whose impulse response, given in Table 1.4 on page 26 as h 1 , has energy equal to 1.06. The noise is additive, white, and Gaussian with statistical power ¦w2 D 0:01. Identification via standard LMS and RLS adaptive algorithms is obtained using as input a maximal-length PN sequence of length L D 31 and unit power, Mx D 1. For a filter with N D 5 coefficients, the convergence curves of the mean-square error (estimated over 500 realizations) are shown in Figure 3.27. For the LMS algorithm, ¼ D 0:1 is chosen, which leads to a misadjustment equal to MSD D 0:26. As discussed in Appendix 3.B, as index of the estimate quality we adopt the ratio: 3n D
¦w2 E[jj hjj2 ]
(3.275)
Figure 3.27. Convergence curves of the mean-square error for system identification using LMS and RLS.
220
Chapter 3. Adaptive transversal filters
where h D c h is the estimate error vector. At convergence, that is for k D 30 in our example, it results: ( 3:9 for LMS (3.276) 3n D 7:8 for RLS We note that, even if the input signal is white, the RLS algorithm usually yields a better estimate than the LMS. However, for systems with a large noise power and/or slow timevarying impulse responses, the two methods tend to give the same performance in terms of speed of convergence and error in steady state. As a result it is usually preferable to adopt the LMS algorithm, as it leads to easier implementation.
Finite alphabet case Assume a more general, non-linear relation between z.k/ and x.k/, given by z.k/ D g[x.k/; x.k 1/; x.k 2/] D g.x.k//
(3.277)
where x.i/ 2 A, finite alphabet with M elements. Then z.k/ assumes values in an alphabet with at most M 3 values, which can be identified by a table or random-access memory (RAM) method, as illustrated in Figure 3.28. The cost function to be minimized is expressed as 2 E[je.k/j2 ] D E[jd.k/ g.x.k//j O ]
(3.278)
and the gradient estimate is given by rgO je.k/j2 D 2e.k/
(3.279)
Figure 3.28. Adaptive scheme to estimate the input--output relation of a system.
3.6. Examples of application
221
Therefore the LMS adaptation equation becomes g.x.k// O D g.x.k// O C ¼e.k/
(3.280)
In other words, the input vector x.k/ identifies a particular RAM location whose content is updated by adding a term proportional to the error. In the absence of noise, if the RAM is initialized to zero, the content of a memory location can be immediately identified by looking at the output. In practice, however, it is necessary to access each memory location several times to average out the noise. We note that, if the sequence fx.k/g is i.i.d., x.k/ selects in the average each RAM location the same number of times. An alternative method consists of setting y.k/ D 0 during the entire time interval devoted to system identification, and to update the RAM with the values of fd.k/g, according to the equation g.x.k// O D g.x.k// O C d.k/
k D 0; 1; : : :
(3.281)
To complete the identification process, the value at each RAM location is scaled by the number of updates that have taken place for that location. This is equivalent to considering g.x/ O D E[g.x/ C w]
(3.282)
We note that this method is a block version of the LMS algorithm with block length equal to the input sequence, where the RAM is initialized to zero, so that e.k/ D d.k/, and ¼ is given by the relative frequency of each address. Observation 3.4 In this section and in Appendix 3.B, the observation d and the input x are determined on the same time domain with sampling period Tc . Often, however, the input is determined on the domain with sampling period Tc , and the system output signal is determined on Tc =F0 . Using the polyphase representation (see Section 1.A.9) of d, it is convenient to represent the estimate of h determined on Tc =F0 as F0 estimates determined on Tc .
3.6.2
Adaptive cancellation of interfering signals
With reference to Figure 3.29, we consider two sensors: 1. Primary input, consisting of the desired signal s corrupted by additive noise w0 , d.k/ D s.k/ C w0 .k/
with s ? w0
(3.283)
2. Reference input, consisting of the noise signal w1 , with s ? w1 . We assume that w0 and w1 are in general correlated. w1 is filtered by an adaptive filter with coefficients fci g, i D 0; 1; : : : ; N 1, so that the filter output, given by y.k/ D
N 1 X i D0
ci .k/w1 .k i/
(3.284)
222
Chapter 3. Adaptive transversal filters
Figure 3.29. General configuration of an interference canceller.
is the most accurate replica of w0 .k/. Defining the error e.k/ D d.k/ y.k/ D s.k/ C w0 .k/ y.k/
(3.285)
the cost function, assuming real-valued signals and recalling that s is orthogonal to the noise signals, is given by J D E[e2 .k/] D E[s 2 .k/] C E[.w0 .k/ y.k//2 ]
(3.286)
We have two cases. 1. w1 and w0 are correlated: min J D rs .0/ C min E[.w0 .k/ y.k//2 ] D rs .0/ c
c
(3.287)
for y.k/ D w0 .k/. In this case e.k/ D s.k/. 2. w1 and w0 are uncorrelated: min J D E[.s.k/ C w0 .k//2 ] C min E[y 2 .k/] c
c
(3.288)
D E[.s.k/ C w0 .k//2 ] for y.k/ D 0. In this case e.k/ D d.k/ and the noise w0 is not cancelled.
General solution With reference to Figure 3.30, for a general input x to the adaptive filter, the Wiener–Hopf solution in the z-transform domain is given by (see (2.50)) Copt .z/ D
Pdx .z/ Px .z/
(3.289)
3.6. Examples of application
223
Figure 3.30. Block diagram of an adaptive cancellation scheme.
Figure 3.31. Specific configuration of an interference canceller.
Adopting for d and x the model of Figure 3.31, in which w00 and w10 are additive noise signals uncorrelated with w and s, and using Table 1.3, (3.289) becomes Copt .z/ D
Pw .z/H Ł .1=z Ł / Pw10 .z/ C Pw .z/H .z/H Ł .1=z Ł /
(3.290)
1 H .z/
(3.291)
If w10 0, (3.290) becomes Copt .z/ D
224
Chapter 3. Adaptive transversal filters
3.6.3
Cancellation of a sinusoidal interferer with known frequency
Let d.k/ D s.k/ C A cos.2³ f 0 kTc C '0 /
(3.292)
where s is the desired signal, and the sinusoidal term is the interferer. As shown in Figure 3.32, we take as reference signals x1 .k/ D B cos.2³ f 0 kTc C '/
(3.293)
x2 .k/ D B sin.2³ f 0 kTc C '/
(3.294)
and
The adaptation equations of the LMS algorithm are c1 .k C 1/ D c1 .k/ C ¼e.k/x 1 .k/
(3.295)
c2 .k C 1/ D c2 .k/ C ¼e.k/x 2 .k/
(3.296)
At convergence, the two coefficients c1 and c2 change the amplitude and phase of the reference signal to cancel the interfering tone. The relation between d and output e corresponds to a notch filter as illustrated in Figure 3.33. It is easy to see that x2 is obtained from x1 via a Hilbert filter (see Figure 1.28). We note that in this case x2 can be obtained as a delayed version of x1 .
3.6.4
Disturbance cancellation for speech signals
With reference to Figure 3.34, the primary signal is a speech waveform affected by interference signals such as echoes and/or environmental disturbances. The reference signal
Figure 3.32. Configuration to cancel a sinusoidal interferer of known frequency.
3.6. Examples of application
225
Figure 3.33. Frequency response of a notch filter.
Figure 3.34. Disturbance cancellation for speech signals.
consists of a replica of the disturbances. At convergence, the adaptive filter output will attempt to subtract the interference signal, which is correlated to the reference signal, from the primary signal. The output signal is a replica of the speech waveform, obtained by removing to the best possible extent the disturbances from the input signal.
3.6.5
Echo cancellation in subscriber loops
With reference to the simplified scheme of Figure 3.35, the speech signal of user A is transmitted over a transmission line consisting of a pair of wires (local loop) [31] to the central office A, where the signals in the two directions of transmission, i.e. the signal transmitted by user A and the signal received from user B, are separated by a device called
226
Chapter 3. Adaptive transversal filters
Figure 3.35. Transmission between two users in the public network.
Figure 3.36. Configuration to remove the echo of signal A caused by the hybrid B.
hybrid. A similar situation takes place at the central office B, with the roles of the signals A and B reversed. Because of impedance mismatch, the hybrids give origin to echo signals that are added to the desired speech signals. For speech waveforms, the echo of signal A that is generated at the hybrid A can be ignored because it is not perceived by the human ear. The case for digital transmission is different, as will be discussed in Chapter 16. A method to remove echo signals is illustrated in Figure 3.36, where y is a replica of the echo. At convergence, e will consist of the speech signal B only.
3.6.6
Adaptive antenna arrays
In radio systems, to equalize the desired signal and remove interference, it is convenient to use several sensors, i.e. an antenna array, with the task of filtering signals in space, discriminating them through their angle of arrival. The signals of the array are then equalized to compensate for linear distortion introduced by the radio channel. A general scheme for wideband signals is illustrated in Figure 3.37. For narrowband signals, it is sufficient to substitute for each sensor the filter with a single complex-valued coefficient [32, 33] (see Section 8.18).
3.6. Examples of application
227
Figure 3.37. Antenna array to filter and equalize wideband radio signals.
3.6.7
Cancellation of a periodic interfering signal
For the cancellation of a periodic interfering signal, we can use the scheme of Figure 3.38, where: ž we note the absence of an external reference signal; the reference signal is generated by delaying the primary input; ž a delay 1 D DTc , where D is an integer, is needed to decorrelate the desired component of the primary signal from that of the reference signal, otherwise part of the desired signal would also be cancelled. On the other hand, to cancel a wideband interferer from a periodic signal it is sufficient to take the output of the adaptive filter (see Figure 3.39).
228
Chapter 3. Adaptive transversal filters
Figure 3.38. Scheme to remove a periodic interferer from a wideband desired signal.
Figure 3.39. Scheme to remove a wideband interferer from a periodic desired signal.
Figure 3.40. Scheme to remove a sinusoidal interferer from a wideband signal.
3. Bibliography
229
Note that in both schemes the adaptive filter acts as a predictor. Exploiting the general concept described above, an alternative scheme to that of Figure 3.32 is illustrated in Figure 3.40, where the knowledge of the frequency of the interfering signal is not required. In general, for D > 1 the scheme of Figure 3.40 requires many more than two coefficients, therefore it has a higher implementation complexity than that of the scheme of Figure 3.32. However, if the wideband signal can be modeled as white noise, then D D 1; hence, observing (1.555), for a sinusoidal interferer a secondorder predictor is sufficient.
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[12] S. Haykin, Adaptive filter theory. Englewood Cliffs, NJ: Prentice-Hall, 3rd ed., 1996. [13] J. R. Zeidler, “Performance analysis of LMS adaptive prediction filters”, IEEE Proceedings, vol. 78, pp. 1781–1806, Dec. 1990. [14] D. Falconer and L. Ljung, “Application of fast Kalman estimation to adaptive equalization”, IEEE Trans. on Communications, vol. 26, pp. 1439–1446, Oct. 1978. [15] J. M. Cioffi and T. Kailath, “Fast, recursive-least-squares transversal filter for adaptive filtering”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 32, pp. 304– 337, Apr. 1984. [16] M. L. Honig and D. G. Messerschmitt, Adaptive filters: structures, algorithms and applications. Boston, MA: Kluwer Academic Publishers, 1984. [17] J. M. Cioffi, “High speed systolic implementation of fast QR adaptive filters”, in Proc. ICASSP, pp. 1584–1588, 1988. [18] F. Ling, D. Manolakis, and J. G. Proakis, “Numerically robust least-squares latticeladder algorithm with direct updating of the reflection coefficients”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 34, pp. 837–845, Aug. 1986. [19] P. A. Regalia, “Numerical stability properties of a QR-based fast least squares algorithm”, IEEE Trans. on Signal Processing, vol. 41, pp. 2096–2109, June 1993. [20] S. T. Alexander and A. L. Ghirnikar, “A method for recursive least-squares filtering based upon an inverse QR decomposition”, IEEE Trans. on Signal Processing, vol. 41, pp. 20–30, Jan. 1993. [21] J. M. Cioffi, “The fast adaptive rotor’s RLS algorithm”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 38, pp. 631–653, Apr. 1990. [22] Z.-S. Liu, “QR methods of O.N / complexity in adaptive parameter estimation”, IEEE Trans. on Signal Processing, vol. 43, pp. 720–729, Mar. 1995. [23] J. A. Bucklew, T. G. Kurtz, and W. A. Sethares, “Weak convergence and local stability properties of fixed step size recursive algorithms”, IEEE Trans. on Information Theory, vol. 39, pp. 966–978, May 1993. [24] B. Widrow, “Fundamental relations between the LMS algorithm and the DFT”, IEEE Trans. on Circuits and Systems, vol. 34, pp. 814–820, July 1987. [25] C. F. N. Cowan and P. M. Grant, Adaptive filters. Englewood Cliffs, NJ: Prentice-Hall, 1985. [26] N. J. Bershad and P. L. Feintuch, “A normalized frequency domain LMS adaptive algorithm”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 34, pp. 452– 461, June 1986. [27] R. P. Bitmead and B. D. O. Anderson, “Adaptive frequency sampling filters”, IEEE Trans. on Circuits and Systems, vol. 28, pp. 524–543, June 1981.
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[28] D. F. Marshall, W. K. Jenkins, and J. J. Murphy, “The use of orthogonal transforms for improving performance of adaptive filters”, IEEE Trans. on Circuits and Systems, vol. 36, pp. 474–484, Apr. 1989. [29] B. Widrow and S. D. Stearns, Adaptive signal processing. Englewood Cliffs, NJ: Prentice-Hall, 1985. [30] O. Macchi, Adaptive processing: the LMS approach with applications in transmission. New York: John Wiley & Sons, 1995. [31] D. G. Messerschmitt, “Echo cancellation in speech and data transmission”, IEEE Journal on Selected Areas in Communications, vol. 2, pp. 283–297, Mar. 1984. [32] I. J. Gupta and A. A. Ksienski, “Adaptive antenna array for weak interfering signals”, IEEE Trans. on Antennas Propag., vol. 34, pp. 420–426, Mar. 1986. [33] L. L. Horowitz and K. D. Senne, “Performance advantage of complex LMS for controlling narrow-band adaptive array”, IEEE Trans. on Circuits and Systems, vol. 28, pp. 562–576, June 1981. [34] D. G. Messerschmitt and E. A. Lee, Digital communication. Boston, MA: Kluwer Academic Publishers, 2nd ed., 1994. [35] S. W. Golomb, Shift register sequences. San Francisco: Holden-Day, 1967. [36] P. Fan and M. Darnell, Sequence design for communications applications. Taunton: Research Studies Press, 1996. [37] D. C. Chu, “Polyphase codes with good periodic correlation properties”, IEEE Trans. on Information Theory, vol. 18, pp. 531–532, July 1972. [38] R. L. Frank and S. A. Zadoff, “Phase shift pulse codes with good periodic correlation properties”, IRE Trans. on Information Theory, vol. 8, pp. 381–382, Oct. 1962. [39] A. Milewsky, “Periodic sequences with optimal properties for channel estimation and fast start-up equalization”, IBM Journal of Research and Development, vol. 27, pp. 426–431, Sept. 1983. [40] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to spread spectrum communications. Englewood Cliffs, NJ: Prentice-Hall, 1995. [41] R. Gold, “Optimal binary sequences for spread spectrum multiplexing”, IEEE Trans. on Information Theory, vol. 13, pp. 619–621, Oct. 1967. [42] R. Gold, “Maximal recursive sequences with 3-valued recursive cross-correlation functions”, IEEE Trans. on Information Theory, vol. 14, pp. 154–155, Jan. 1968. [43] S. L. Marple Jr., “Efficient least squares FIR system identification”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 29, pp. 62–73, Feb. 1981.
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3.A. PN sequences
Appendix 3.A
233
PN sequences
In this Appendix we introduce three classes of deterministic periodic sequences having spectral characteristics similar to those of a white noise signal, hence the name pseudonoise (PN) sequences.
Maximal-length sequences Maximal-length sequences are binary PN sequences, also called r-sequences, that are generated recursively, e.g., using a shift-register (see page 877), and have period equal to L D 2r 1. Let f p.`/g, ` D 0; 1; : : : ; L 1, p.`/ 2 f0; 1g, be the values assumed by the sequence in a period. It can be shown that the maximal-length sequences enjoy the following properties [34, 35]. ž Every non-zero sequence of r bits appears exactly once in each period; therefore all binary sequences of r bits are generated, except the all zero sequence. ž The number of bits equal to “1” in a period is 2r 1 , and the number of bits equal to “0” is 2r 1 1. ž A subsequence is intended here as a set of consecutive bits of the r-sequence. The relative frequency of any non-zero subsequence of length i r is 2r i ' 2i 2r 1
(3.297)
and the relative frequency of a subsequence of length i < r with all bits equal to zero is 2r i 1 ' 2i 2r 1
(3.298)
In both formulae the approximation is valid for a sufficiently large r. ž The sum of two r-sequences, which are generated by the same shift-register, but with different initial conditions, is still an r-sequence. ž The linear span, that determines the predictability of a sequence, is equal to r [36]. In other words, the elements of a sequence can be determined by any 2r consecutive elements of the sequence itself, while the remaining elements can be produced by a recursive algorithm (see, e.g., the Berlekamp-Massey algorithm on page 891). A practical example is given in Figure 3.41 for a sequence with L D 15 (r D 4), which is generated by the recursive equation p.`/ D p.` 3/ ý p.` 4/
(3.299)
234
Chapter 3. Adaptive transversal filters
Figure 3.41. Generation of a PN sequence with period L D 15.
where ý denotes modulo 2 sum. Assuming initial conditions p.1/ D p.2/ D p.3/ D p.4/ D 1, applying (3.299) we obtain the sequence 0 0 1 0 0 1 1 0 1 0 1 1 1 |{z} 1 ::: 0 |{z} |{z}
(3.300)
p.L1/
p.0/ p.1/
Obviously, the all zero initial condition must be avoided. To generate sequences with a larger period L we refer to Table 3.5. The above properties make an r-sequence, even if deterministic and periodic, appear as a random i.i.d. sequence from the point of view of the relative frequency of subsequences of bits. It turns out that an r-sequence appears as random i.i.d. also from the point of view of the autocorrelation function. In fact, mapping “0” to “1” and “1” to “C1”, we get the following correlation properties. 1. Mean L1 1 X 1 p.`/ D L `D0 L
(3.301)
2. Correlation (periodic of period L) 8 L1 < 1 1 X Ł r p .n/ D p.`/ p .` n/mod L D : 1 L `D0 L
for .n/mod L D 0 (3.302) otherwise
3. Spectral density (periodic of period L) 8 1 > > L1 < Tc 1 X 1 j2³ m L T nTc L c D Tc Pp m r p .n/e D 1 > L Tc > nD0 : Tc 1 C L
for .m/mod L D 0 otherwise (3.303)
We note that, with the exception of the values assumed for .m/mod L D 0, the spectral density of maximal length sequences is constant.
3.A. PN sequences
235
Table 3.5 Recursive equations to generate PN sequences of length L D 2r 1, for different values of r.
Period L D 2r 1
r 1 2
p.`/ D p.` 1/ p.`/ D p.` 1/ ý p.` 2/
3
p.`/ D p.` 2/ ý p.` 3/
4
p.`/ D p.` 3/ ý p.` 4/
5
p.`/ D p.` 3/ ý p.` 5/
6
p.`/ D p.` 5/ ý p.` 6/
7
p.`/ D p.` 6/ ý p.` 7/
8
p.`/ D p.` 2/ ý p.` 3/ ý p.` 4/ ý p.` 8/
9
p.`/ D p.` 5/ ý p.` 9/
10
p.`/ D p.` 7/ ý p.` 10/
11
p.`/ D p.` 9/ ý p.` 11/
12
p.`/ D p.` 2/ ý p.` 10/ ý p.` 11/ ý p.` 12/
13
p.`/ D p.` 1/ ý p.` 11/ ý p.` 12/ ý p.` 13/
14
p.`/ D p.` 2/ ý p.` 12/ ý p.` 13/ ý p.` 14/
15
p.`/ D p.` 14/ ý p.` 15/
16
p.`/ D p.` 11/ ý p.` 13/ ý p.` 14/ ý p.` 16/
17
p.`/ D p.` 14/ ý p.` 17/
18
p.`/ D p.` 11/ ý p.` 18/
19
p.`/ D p.` 14/ ý p.` 17/ ý p.` 18/ ý p.` 19/
20
p.`/ D p.` 17/ ý p.` 20/
CAZAC sequences The constant amplitude zero autocorrelation (CAZAC) sequences are complex-valued PN sequences with constant amplitude (assuming values on the unit circle) and autocorrelation function r p .n/ equal to zero for .n/mod L 6D 0. Because of these characteristics they are also called polyphase sequences [37, 38, 39]. Let L and M be two integer numbers that are relatively prime. The CAZAC sequences are defined as, for L even for L odd
p.`/ D e j p.`/ D
M³ `2 L
M³ `.`C1/ L ej
` D 0; 1; : : : ; L 1
(3.304)
` D 0; 1; : : : ; L 1
(3.305)
It can be shown that, in both cases, these sequences have the following properties.
236
Chapter 3. Adaptive transversal filters
1. Mean L1 1 X p.`/ D 0 L `D0
(3.306)
2. Correlation ( r p .n/ D
1 0
for .n/mod L D 0 otherwise
(3.307)
3. Spectral density 1 D Tc Pp m L Tc
(3.308)
Gold sequences In a large number of applications, as for example in spread-spectrum systems with codedivision multiple access (see Chapter 10), sets of sequences having one or both of the following properties [40] are required. ž Each sequence of the set must be easily distinguishable from its own time shifted versions. ž Each sequence of the set must be easily distinguishable from any other sequence of the set and from its time-shifted versions. An important class of periodic binary sequences that satisfy these properties, or, in other words, that have good autocorrelation and cross-correlation characteristics, is the set of Gold sequences [41, 42]. Construction of pairs of preferred r-sequences. In general the cross-correlation sequence (CCS) between two r-sequences may assume three, four or maybe even a greater number of values. We show now the construction of a pair of r-sequences, called preferred r-sequences [36], whose CCS assumes only three values. Let a D fa.`/g be an r-sequence with period L D 2r 1. We define now another r-sequence of length L D 2r 1 obtained from the sequence a by decimation by a factor M, that is: (3.309)
b D fb.`/g D fa.M`/mod L g We make the following assumptions.
ž rmod 4 6D 0, that is r must be odd or equal to odd multiples of 2, i.e. rmod 4 D 2. ž The factor M satisfies one of the following properties: M D 2k C 1 or
M D 22k 2k C 1
k integer
(3.310)
3.A. PN sequences
237
ž For k determined as in the (3.310), defining g:c:d:.r; k/ as the greatest common divisor of r and k, let ( 1 r odd e D g:c:d:.r; k/ D (3.311) 2 rmod 4 D 2 Then the CCS between the two r-sequences a and b assumes only three values [35, 36]: rab .n/ D
L1 1 X a.`/bŁ .` n/mod L L `D0
8 r Ce > 1 C 2 2 > < 1 1 D > L> r Ce : 1 2 2
r Ce2
(value assumed 2r e1 C 2 2 times) (value assumed 2r 2r e 1 times) (value assumed 2r e1 2
r Ce2 2
(3.312)
times)
Example 3.A.1 (Construction of a pair of preferred r-sequences) Let the following r-sequence of period L D 25 1 D 31 be given: fa.`/g D .0000100101100111110001101110101/
(3.313)
As r D 5 and rmod 4 D 1, we take k D 1. Therefore e D g:c:d:.r; k/ D g:c:d:.5; 1/ D 1 and M D 2k C 1 D 21 C 1 D 3. The sequence fb.`/g obtained by decimation of the sequence fa.`/g is then given by fb.`/g D fa.3`/mod L g D .0001010110100001100100111110111/
(3.314)
The CCS between the two sequences, assuming “0” is mapped to “1”, is: frab .n/g D
1 .7; 7; 1; 1; 1; 9; 7; 9; 7; 7; 1; 1; 7; 7; 1; 7; 1; 31
(3.315)
1; 9; 1; 1; 1; 1; 9; 1; 7; 1; 9; 9; 7; 1/ We note that, if we had chosen k D 2, then e D g:c:d:.5; 2/ D 1 and M D 22 C 1 D 5, or else M D 22Ð2 22 C 1 D 13. Construction of a set of Gold sequences. A set of Gold sequences can be constructed from any pair fa.`/g and fb.`/g of preferred r-sequences of period L D 2r 1. We define the set of sequences: G.a; b/ D fa; b; a ý b; a ý Z b; a ý Z 2 b; : : : ; a ý Z L1 bg
(3.316)
where Z is the shift operator that cyclically shifts a sequence to the left by a position. The set (3.316) contains L C 2 D 2r C 1 sequences of length L D 2r 1 and is called the set of Gold sequences. It can be proved [41, 42] that, for the two sequences fa 0 .`/g and fb0 .`/g
238
Chapter 3. Adaptive transversal filters
belonging to the set G.a; b/, the CCS as well as the ACS, with the exception of zero lag, assume only three values: 8 r C1 r C1 > 1C2 2 r odd 1 < 1 1 2 2 ra 0 b0 .n/ D (3.317) r C2 L> : 1 1 2 r C2 2 1C2 2 rmod 4 D 2 Clearly, the ACS of a Gold sequence no longer has the characteristics of an r-sequence, as is seen in the next example. Example 3.A.2 (Gold sequence properties) Let r D 5, hence L D 25 1 D 31. From Example 3.A.1, the two sequences (3.313) and (3.314) are a pair of preferred r-sequences, from which it is possible to generate the whole set of Gold sequences. For example we calculate the ACS of fa.`/g and fb0 .`/g D fa.`/ ý b.` 2/g D a ý Z 2 b, and the CCS between fa.`/g and fb0 .`/g: fa.`/gD.1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1/ fb0 .`/gD.1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1/
(3.318)
(3.319)
fra .n/gD
1 .31; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 31 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1; 1/ (3.320)
frb0 .n/gD
1 .31; 1; 9; 7; 7; 9; 1; 7; 1; 9; 7; 7; 1 1; 7; 1; 1; 7; 31 1; 1; 7; 7; 9; 1; 7; 1; 9; 7; 7; 9; 1/ (3.321)
frab0 .n/gD
1 .1; 7; 7; 7; 1; 1; 1; 1; 1; 7; 9; 1; 1; 7; 1; 9; 7; 31 1 9; 7; 7; 9; 1; 7; 1; 9; 1; 1; 1; 9; 1/ (3.322)
3.B. Identification of a FIR system by PN sequences
Appendix 3.B
3.B.1
239
Identification of a FIR system by PN sequences
Correlation method
With reference to (3.264), which describes the relation between input and output of an unknown system with impulse response fh i g, i D 0; 1; : : : ; N 1, we take as an input signal white noise with statistical power rx .0/. To estimate the impulse response of a linear system, we observe that the cross-correlation between d and x is then proportional, with a factor rx .0/, to the impulse response fh i g. In fact, we have rdx .n/ D rzx .n/ D rx Ł h.n/ D rx .0/h n
(3.323)
In practice, instead of noise a PN sequence with period L, f p.i/g, i D 0; 1; : : : ; L 1, is used as input. We recall that the autocorrelation of a PN sequence is also periodic with period L and is given by (see Appendix 3.A): ( r p .n/
D1 '0
n D 0; L ; 2L ; : : : n D 1; : : : ; L 1; L C 1; : : :
(3.324)
Moreover, we recall that if the input to a time-invariant filter is periodic with period L, the output will also be periodic with period L. To estimate the impulse response fh i g, i D 0; 1; : : : ; N 1, we consider the scheme illustrated in Figure 3.42, where we choose L ½ N , and an input sequence x with length of at least .L C 1/N samples, obtained by repeating f p.i/g; in other words, x.k/ D p.k/mod L . We assume a delay m 2 f0; 1; : : : ; N 1g, a rectangular window g Rc .k/ D .1=L/w L .k/, and that the system is started at instant k D 0. For k ½ .N 1/ C .L 1/, the output v.k/ is given by
Figure 3.42. Correlation method to estimate the impulse response of an unknown system.
240
Chapter 3. Adaptive transversal filters
v.k/ D
L1 L1 X X 1 1 u.k `/ D d.k `/ p Ł .k ` m/mod L L L `D0 `D0
L1 X 1 D L `D0
"
N 1 X
h i p.k ` i/mod L pŁ .k ` m/mod L
i D0
#
Cw.k `/ p .k ` m/mod L Ł
D
N 1 X i D0
C
hi
(3.325)
L1 1 X p.k ` i/mod L pŁ .k ` m/mod L L `D0
L1 1 X w.k `/ p Ł .k ` m/mod L L `D0
As L1 1 X p.k ` i/mod L pŁ .k ` m/mod L D r p .m i/mod L L `D0
(3.326)
(3.325) becomes v.k/ D
N 1 X
L1 1 X w.k `/ p Ł .k ` m/mod L L `D0
h i r p .m i/mod L C
i D0
(3.327)
If L × 1, the second term on the right-hand side of (3.327) can be ignored, hence observing (3.324) we get v.k/ ' h m
(3.328)
Mean and variance of the estimate of h m given by (3.327) are obtained as follows. 1. Mean E[v.k/] D
N 1 X
h i r p .m i/mod L
(3.329)
i D0
assuming w has zero mean. 2. Variance "
L1 1 X var[v.k/] D var w.k `/ p Ł .k ` m/mod L L `D0
assuming w white and j p.`/j 1.
# '
¦w2 L
(3.330)
3.B. Identification of a FIR system by PN sequences
241
Figure 3.43. Correlation method via correlator to estimate the impulse response of an unknown system.
Using the scheme of Figure 3.42, varying m from 0 to N 1 it is possible to get an estimate of the samples of the impulse response of the unknown system fh i g at the output of the filter g Rc . However, this scheme has two disadvantages: 1. it requires a very long computation time (N L); 2. it requires synchronization between the two PN sequences, at transmitter and receiver. Both problems can be resolved by memorizing, after a transient equal to N 1 instants, L consecutive output samples fd.k/g in a buffer and computing the correlation off-line: rO dx .m/ D
1 L
.N 1/C.L1/ X
d.k/ pŁ .k m/mod L ' h m
m D 0; 1; : : : ; N 1
(3.331)
kD.N 1/
An alternative scheme is represented in Figure 3.43: with steps analogous to those of the preceding scheme, we get v.k/ D
L1 1 X d.k .L 1/ C `/ p Ł .` C .N 1//mod L ' h .k.N 1/.L1//mod L (3.332) L `D0
After a transient of .N 1/ C .L 1/ samples, from (3.332) we get hO i D v.i C .N 1/ C .L 1//
i D 0; 1; : : : ; N 1
(3.333)
In other words, the samples at the correlator output from instant k D .N 1/ C .L 1/ to 2.N 1/ C .L 1/, give an estimate of the samples of the impulse response of the unknown system fh i g.
Signal-to-estimation error ratio Let hT D [h 0 ; h 1 ; : : : ; h N 1 ] be the filter coefficients to be estimated and hO T D [hO 0 ; hO 1 ; : : : ; hO N 1 ] those estimated. Let h be the estimation error vector h D hO h
242
Chapter 3. Adaptive transversal filters
The quality of the estimate is measured by the signal-to-estimation error ratio jjhjj2 E[jj hjj2 ]
3e D
(3.334)
On one hand, we have to take into consideration the noise present in the observed system and measured by (see Figure 3.42): 3D
Mx jjhjj2 ¦w2
(3.335)
where Mx is the statistical power of the input signal. In our case Mx D 1. Finally, we refer to the normalized ratio 3n D
¦w2 3e D 3 Mx E[jj hjj2 ]
(3.336)
O We note that if we indicate with d.k/ the output of the identified system, O d.k/ D
N 1 X
hO i x.k i/
(3.337)
i D0
O the fact that hO 6D h causes d.k/ 6D z.k/, with an error given by O z.k/ d.k/ D
N 1 X
.h i hO i / x.k i/
(3.338)
i D0
having variance Mx E[jj hjj2 ] for a white noise input. As a consequence, (3.336) measures the ratio between the variance of the additive noise of the observed system and the variance of the error at the output of the identified system. From (3.338) we note that the difference O O d.k/ d.k/ D .z.k/ d.k// C w.k/
(3.339)
consists of two terms, one due to the estimation error and the other due to the noise of the system.
3.B.2
Methods in the frequency domain
System identification in the absence of noise In the absence of noise (w D 0), the output signal of the unknown system, represented in Figure 3.42, is z.k/ D
L1 X
x.k n/mod L h n
(3.340)
nD0
where a PN sequence of period L D N , equal to the length of the impulse response to be estimated, is assumed as input signal x. Let us consider the vector zT D [z.k/;
3.B. Identification of a FIR system by PN sequences
243
z.k C 1/; : : : ; z.k C .L 1//], and a circulant matrix M whose first row is [x.k/ mod L ; x.k 1/mod L ; : : : ; x.k .L 1//mod L ]. After an initial transient of L 1 samples, using the output samples fz.L 1/; : : : ; z.2.L 1//g we obtain a system of L linear equations in L unknowns, which in matrix notation can be written as z D Mh
(3.341)
assuming k D L 1 in the definition of z and M. The system of equations (3.341) admits a unique solution if and only if the matrix M is non-singular. Because the input sequence is periodic, the system (3.341) can be solved very efficiently, from a computational complexity point of view, in the frequency domain, rather than inverting the matrix M. Being M circulant, the product in (3.341) can be substituted by the circular convolution (see (1.105)) L
k D L 1; : : : ; 2.L 1/
z.k/ D x h.k/
(3.342)
Letting Zm D DFT[z.k/], X m D DFT[x.k/], and Hm D DFT[h k ], (3.342) can be rewritten in terms of the discrete Fourier transforms as Zm D Xm Hm
m D 0; : : : ; L 1
(3.343)
from which we get h k D DFT1
Zm Xm
½
k D 0; : : : ; L 1
(3.344)
or, setting s.k/ D DFT1 [1=Xm ], L
h k D s z.k/
(3.345)
System identification in the presence of noise Substituting in (3.345) the expression of the output signal obtained in the presence of noise, d.k/ D z.k/ C w.k/, for k D L 1; : : : ; 2.L 1/, the estimate of the coefficients of the unknown system is given by L
L
L
L
hO k D s d.k/ D s z.k/ C s w.k/ D h k C s w.k/
(3.346)
Assuming that w is zero-mean white noise with power ¦w2 , mean and variance of the estimate (3.346) are obtained as follows. 1. Mean O Dh E[h]
(3.347)
2. Variance " E[jj hjj2 ] D E
L1 X kD0
# jhO k h k j2 D L E[jhO k h k j2 ] D L ¦w2
L1 X i D0
js.i/j2
(3.348)
244
Chapter 3. Adaptive transversal filters
Using the Parseval theorem L1 X
L1 L1 1 1 X 1 X jS j j2 D L jD0 L jD0 jX j j2
js.i/j2 D
i D0
(3.349)
it is possible to particularize (3.336) for PN maximal-length and CAZAC sequences. In the first case, from (3.303), it turns out X0 D 1 jX1 j2 D jX2 j2 D Ð Ð Ð D jX L1 j2 D L C 1
(3.350)
hence, observing (3.348), (3.336) becomes 3n D
L C1 2L
(3.351)
For CAZAC sequences, from (3.308), we have that all terms jX j j2 are equal jX j j2 D L
j D 0; 1; : : : ; L 1
(3.352)
and the minimum of (3.348) is equal to ¦w2 , therefore 3n D 1. In other words, if L is large, CAZAC sequences yield 3 dB improvement with respect to the maximal-length sequences. Although this method is very simple, it has the disadvantage that, in the best case, it gives an estimate with variance equal to the noise variance of the original system.
3.B.3
The LS method
With reference to the system of Figure 3.42, letting xT .k/ D [x.k/; x.k 1/; : : : ; x.k .N 1//], the noisy output of the unknown system can be written as d.k/ D hT x.k/ C w.k/
k D .N 1/; : : : ; .N 1/ C .L 1/
(3.353)
From (3.353) we see that the observation of L samples of the received signal requires the transmission of L T S D L C N 1 symbols of the training sequence fx.0/; x.1/; : : : ; x..N 1/ C .L 1//g. The unknown system can be identified using the LS criterion [43, 44, 45]. For a certain estimate hO of the unknown system, the sum of squared errors at the output is given by ED
N 1CL1 X
2 O jd.k/ d.k/j
(3.354)
kDN 1
where, from (3.337), O d.k/ D hO T x.k/ As for the analysis of Section 2.3, we introduce the following quantities.
(3.355)
3.B. Identification of a FIR system by PN sequences
245
1. Energy of the desired signal Ed D
N 1CL1 X
jd.k/j2
(3.356)
kDN 1
2. Correlation matrix of the input signal i; n D 0; : : : ; N 1
D [8.i; n/]
(3.357)
where N 1CL1 X
8.i; n/ D
x Ł .k i/ x.k n/
(3.358)
kDN 1
3. Cross-correlation vector ϑ T D [#.0/; : : : ; #.N 1/]
(3.359)
where #.n/ D
N 1CL1 X
d.k/ x Ł .k n/
(3.360)
kDN 1
Then the cost function (3.354) becomes E D Ed hO H ϑ ϑ H hO C hO H hO
(3.361)
As the matrix is determined by a suitably chosen training sequence, we can assume that is positive definite and therefore the inverse exists. The solution to the LS problem yields hO ls D 1 ϑ
(3.362)
Emin D Ed ϑ H hO ls
(3.363)
with a corresponding error equal to
We observe that the matrix 1 in the (3.362) can be pre-computed and memorized, because it depends only on the training sequence. In some applications it is useful to estimate the variance of the noise signal w that, observing (3.339), for hO ' h can be assumed equal to .¦O w2 / D
1 Emin L
(3.364)
246
Chapter 3. Adaptive transversal filters
Formulation using the data matrix From the general analysis given on page 152, we recall the following definitions. 1. L ð N observation matrix 2
3 ::: x.0/ 6 7 :: :: I D4 5 : : x..N 1/ C .L 1// : : : x.L 1/ x.N 1/ :: :
(3.365)
2. Desired sample vector o T D [d.N 1/; : : : ; d..N 1/ C .L 1//]
(3.366)
where d.k/ is given by (3.353). Observing (2.139), (2.131), and (2.160), we have D IHI
ϑ D IHo
(3.367)
and hO ls D .I H I/1 I H o
(3.368)
which coincides with (3.362). We note the introduction of the new symbols I and o, in relation to an alternative LMMSE estimation method, which will be given in Section 3.B.4.
Computation of the signal-to-estimation error ratio We now evaluate the performance of the LS method for the estimation of h. From (3.360), (3.359) can be rewritten as ϑ D
N 1CL1 X
d.k/ xŁ .k/
(3.369)
w.k/ xŁ .k/
(3.370)
kDN 1
Substituting (3.353) in (3.369), and letting ξD
N 1CL1 X kDN 1
observing (3.357), we obtain the relation ϑ D h C ξ
(3.371)
Consequently, substituting (3.371) in (3.362), the estimation error vector can be expressed as h D 1 ξ
(3.372)
3.B. Identification of a FIR system by PN sequences
247
If w is zero-mean white noise with variance ¦w2 , ξ Ł is a zero-mean random vector with correlation matrix Rξ D E[ξ ∗ ξ T ] D ¦w2 Ł
(3.373)
Therefore, h has mean zero and correlation matrix R h D ¦w2 .Ł /1
(3.374)
E[jj hjj2 ] D ¦w2 tr[.Ł /1 ]
(3.375)
3n D .tr[1 ]/1
(3.376)
In particular,
and, from (3.336), we get
Using as training sequence a CAZAC sequence, the matrix is diagonal, D LI
(3.377)
where I is the N ð N identity matrix. The elements on the diagonal of 1 are equal to 1=L, and (3.376) yields 3n D
L N
(3.378)
The (3.378) gives a good indication of the relation between the number of observations L, the number of system coefficients N , and 3n . For example, doubling the length of the training sequence, 3n also doubles. Now, using as training sequence a maximal-length sequence of periodicity L, and indicating with 1 N ðN the matrix with all elements equal to 1, the correlation matrix can be written as D .L C 1/I 1 N ðN
(3.379)
From (3.379) the inverse is given by 1
1 N ðN 1 IC D L C1 L C1 N
(3.380)
which, substituted in (3.376), yields 3n D
.L C 1/.L C 1 N / N .L C 2 N /
(3.381)
In Figure 3.44 the behavior of 3n is represented as a function of N , for CAZAC sequences (solid line) and for maximal-length sequences (dotted-dashed line), with parameter L. We make the following observations.
248
Chapter 3. Adaptive transversal filters
Figure 3.44. 3n vs. N for CAZAC sequences (solid line) and maximal-length sequences (dotted-dashed line), for various values of L.
ž For a given N , choosing L × N , the two sequences yield approximately the same 3n . The worst case is obtained for L D N ; for example, for L D 15 the maximallength sequence yields a value of 3n that is about 3 dB lower than the upper bound (3.378). We note that the frequency method operates for L D N . ž For a given value of L, because of the presence of the noise w, the estimate of the coefficients becomes worse if the number of coefficients N is larger than the number of coefficients of the system Nh . On the other hand, if N is smaller than Nh , the estimation error may assume large values (see (3.270)). ž For sparse systems, where the number of coefficients may be large, but only a few of them are non-zero, the estimate is usually very noisy. Therefore, after obtaining the estimate, it is necessary to set to zero all coefficients whose amplitude is below a certain threshold. ž If the correlation method (3.331) is adopted, we get 1 hO D ϑ L where ϑ is given by (3.359). Observing (3.371), we get 1 1 h D I hC ξ L L
(3.382)
3.B. Identification of a FIR system by PN sequences
249
Consequently the estimate is affected by a BIAS term equal to ..1=L/ I/h, and has a covariance matrix equal to .1=L 2 / Rξ . In particular, using (3.373), it turns out 2 1 ¦w2 I h C tr[] E[jj hjj ] D L L2 2
(3.383)
and 3n D
1 2 1 1 1 I h tr[] C ¦2 2 L L w
(3.384)
Using a CAZAC sequence, from (3.377) the second term of the denominator in (3.384) vanishes, and 3n is given by (3.378). In fact, for a CAZAC sequence, as (3.324) is strictly true and 1 is diagonal, the LS method (3.362) coincides with the correlation method (3.331). Using instead a maximal-length sequence, from (3.379) we get 2 N 1 X 1 D 1 jj.1 I/ hjj2 D 1 h I jh i H.0/j2 L L2 L 2 i D0 D where H.0/ D
P N 1 i D0
(3.385)
1 .jjhjj2 C .N 2/ jH.0/j2 / L2
h i . Moreover, we have tr[] D N L
(3.386)
hence 3n D
L
1 jH.0/j2 NC 3 C .N 2/ L ¦w2
½
(3.387)
where 3 is defined in (3.335). We observe that using the correlation method, we obtain the same values 3n (3.381) as the LS method, if L is large enough to satisfy the condition 3 C .N 2/
3.B.4
jH.0/j2 1 the network is said to be active. If instead gd < 1 then the network is passive; in this case, we speak of available attenuation of the network : 1 (4.32) ad D gd In dB, .gd /dB D 10 log10 gd
(4.33)
.ad /dB D .gd /dB
(4.34)
and
Definition 4.6 Apart from a possible delay, for an ideal distortionless network with power gain (attenuation) gd .ad /, we will assume that the frequency response of the network is GCh . f / D Go
constant
f 2B
(4.35)
Consequently, the impulse response is given by gCh .t/ D Go Ž.t/
(4.36)
where, observing (4.31), Go D
1 p gd D p ad
(4.37)
We note that, in case the conditions leading to (4.31) are not verified, the relation between Go and gd is more complicated (see (4.25)).
262
Chapter 4. Transmission media
Measurement of signal power Typically gd and ad are expressed in dB; the power P is expressed in W, mW (103 W), or pW (1012 W), or in dBW, dBm, or dBrn: .P/dBW D 10 log10 .P/dBm D 10 log10 .P/dBrn D 10 log10
.P in W/ .P in mW/ .P in pW/
(4.38) (4.39) (4.40)
Some relations are .P/dBW D .P/dBrn 120 D .P/dBm 30
(4.41)
Example 4.1.1 For P D 0:5 W we have .P/dBW D 3 dBW, and .P/dBm D 27 dBm. With reference to (4.37), we note that .Go /d B D .gd /d B . In fact, as Go denotes a ratio of voltages, it follows that .Go /d B D 20 log10 Go D 10 log10 gd D .gd /d B
(4.42)
For telephone signals, a further power unit is given by dBrnc, which expresses the power in dBrn of a signal filtered according to the mask given in Figure 4.5 [2]. The filter reflects the perception of the human ear and is known as C-message weighting.
c 1982 Bell Telephone Figure 4.5. Frequency weighting known as C-message weighting. [ Laboratories. Reproduced with permission of Lucent Technologies, Inc./Bell Labs.]
4.2. Noise generated by electrical devices and networks
4.2
263
Noise generated by electrical devices and networks
Various noise and disturbance signals are added to the desired signal at different points of a transmission system. In addition to interference caused by electromagnetic coupling between various system elements and noise coming from the surrounding environment, there is also noise generated by the transmission devices themselves. Such noise is very important because it determines the limits of the system. We will analyze two types of noise generated by transmission devices: thermal noise and shot noise.
Thermal noise Thermal noise is a phenomenon associated with Brownian or random motion of electrons in a conductor. As each electron carries a unit charge, its motion between collisions with atoms produces a short impulse of current. Actually, if we represent the motion of an electron within a conductor in a two-dimensional plane, the typical behavior is represented in Figure 4.6a where the changes in the direction of the electron motion are determined by random collisions with atoms at the set of instants ftk g. Between two consecutive collisions the electron produces a current that is proportional to the projection of the velocity onto the axis of the conductor. For example, the behavior of instantaneous current for the path of Figure 4.6a is illustrated in Figure 4.6b. Although the average value (DC component) is zero, the large number of electrons and collisions gives origin to a measurable alternating component. If a current flows through the conductor, an orderly motion is superimposed on the disorderly motion of electrons; the sources of the two motions do not interact with each other. For a conductor of resistance R, at an absolute temperature of T Kelvin, the power spectral density of the open circuit voltage w at the conductor terminals is given by Pw . f / D 2kTR . f / where k D 1:3805 Ð 1023 J/K is the Boltzmann constant and hf 1 hf . f/ D e kT 1 kT
(4.43)
(4.44)
Figure 4.6. Representation of electron motion and current produced by the motion.
264
Chapter 4. Transmission media
where h D 6:6262 Ð 1034 Js is the Planck constant. We note that, for f − kT= h D 6 Ð 1012 Hz (at room temperature T D 290 K), we get . f / ' 1. Therefore the PSD of w is approximately white, i.e. Pw . f / D 2kTR
(4.45)
We adopt the electrical model of Figure 4.7, where a conductor is modelled as a noiseless device having in series a generator of noise voltage w.2 Because at each instant the noise voltage w.t/ is due to the superposition of several current pulses, a suitable model for the amplitude distribution of w.t/ is the Gaussian distribution with zero mean. Note that the variance is very large, because of the wide support of Pw . In the case of a linear two-terminal device with impedance Z D R C j X at absolute temperature T, the spectral density of the open circuit voltage w is still given by (4.43), where R D Re[Z ]. In other words, only the resistive component of the impedance gives origin to thermal noise. Let us consider the scheme of Figure 4.8, where a noisy impedance Z D R C j X is matched to the load for maximum transfer of power. Observing (4.11), the available noise power per
Figure 4.7. Electrical model of a noisy conductor.
(a) Electrical circuit
(b) Equivalent scheme
Figure 4.8. Electrical circuit to measure the available source noise power to the load.
2
An equivalent model assumes a noiseless conductor in parallel to a generator of noise current j .t/ with PSD P j . f / D 2kT R1 . f /.
4.2. Noise generated by electrical devices and networks
265
unit of frequency is given by pw;d . f / D
kT (W/Hz) 2
(4.46)
At room temperature T D 290 K, pw;d . f / D 2 Ð 1021 (W/Hz) and .pw;d . f //dBm D 177 (dBm/Hz)
(4.47)
If the circuit of Figure 4.8 has a bandwidth B, the power delivered to the load is equal to Pw D
kT 2B D kTB (W) 2
(4.48)
We note that a noisy impedance produces an open circuit voltage w with a root mean-square (rms) value equal to p p p ¦w D Pw 2B D pw;d 4R2B D kT 4R B (V) (4.49) We also note from (4.48) that the total available power of a thermal noise source is proportional to the product of the system bandwidth and the absolute temperature of the source. For T D 290 K, .Pw /dBm D 174 C 10 log10 B (dBm)
(4.50)
Shot noise Most devices are affected by shot noise, which is due to the discrete nature of electron flow: also in this case the noise represents the instantaneous random deviation of current or voltage from the average value. Shot noise, expressed as a current signal, can also be modelled as Gaussian noise with a constant PSD given by Pishot . f / D eI (A2 /Hz)
(4.51)
where e D 1:6 Ð 1019 C is the electron charge and I is the average current that flows through the device; in this case it is convenient to use the electrical model of Figure 4.4.
Noise in diodes and transistors Models are given in the literature to describe the different noise sources in electronic devices. Specifically, in [2] shot noise is evaluated for a junction diode and shot and thermal noise for a transistor. In any case, the total output noise power of a device is usually not described by p, but rather by an equivalent function called noise temperature.
Noise temperature of a two-terminal device Let pw;d be the available power per unit of frequency, due to the presence of noise in a device. The noise temperature is defined as Tw . f / D
pw;d . f / k=2
(4.52)
266
Chapter 4. Transmission media
In other words, Tw represents the absolute temperature that a thermal noise source should have in order to produce the same available noise power as the device. This concept can be extended and applied to the output of an amplifier or an antenna, expressing the noise power in terms of effective noise temperature. We note that if a device at absolute temperature T contains more than one noise source, then Tw > T.
Noise temperature of a 2-port network We will consider the circuit of Figure 4.9a, where both the source impedance and the load impedance are matched for maximum transfer of power. Assuming that the source, with noise temperature T S , generates a noise voltage wi .S/ , and gd is the available power gain of the 2-port network defined in (4.30), the noise voltage generated at the network output because of the presence of wi .S/ is equal to wo D wo.S/ , with available power at the load given by kT S (4.53) pwo.S/ . f / D gd . f / 2 If in addition to the source, the network also introduces noise, which if measured at the output is equal to wo.A/ , with available power pwo.A/ , we will have a total output noise
Figure 4.9. Noise source connected to a noisy 2-port network: three equivalent models.
4.2. Noise generated by electrical devices and networks
267
signal given by wo D wo.S/ C wo.A/ . Assuming the two noise signals wo.S/ and wo.A/ are uncorrelated, the available power at the load will be equal to the sum of the two powers, i.e. p wo . f / D g d . f /
kT S C pwo.A/ . f / 2
(4.54)
Definition 4.7 The effective noise temperature T A of the 2-port network is defined as TA. f / D
pwo.A/ . f / gd . f / k2
(4.55)
and denotes the temperature of a thermal noise source connected to a 2-port noiseless network that produces the same output noise power. Then (4.54) becomes:3 k pwo . f / D gd . f / [T S C T A ] 2
(4.56)
Definition 4.8 The effective input temperature of a system consisting of a source connected to a 2-port network is Twi D T S C T A
(4.57)
Definition 4.9 The effective output temperature of a system consisting of a source connected to a 2-port network is Two D gd . f /Twi
(4.58)
k Tw 2 o
(4.59)
Then p wo . f / D
Equivalent-noise models By the previous considerations, we introduce the equivalent circuits illustrated in Figure 4.9b and 4.9c. In particular the scheme of Figure 4.9b assumes the network to be noiseless and an equivalent noise source is considered at the input. The scheme of Figure 4.9c, on the other hand, considers all noise sources at the output. The effects on the load for the three schemes of Figure 4.9 are the same.
3
To simplify the notation we have omitted indicating the dependency on frequency of all noise temperatures. Note that the dependency on frequency of T A and T S is determined by gd . f /, pwo.A/ . f /, and pwi.S/ . f /.
268
Chapter 4. Transmission media
Noise figure of a 2-port network Usually the noise of a 2-port network is not directly characterized through T A , but through the noise figure F. Recognizing that pwo.A/ does not depend on T S , as the source and network noise signals are generated by uncorrelated phenomena, leads to the following experiment. We set the source at a noise temperature equal to the room temperature: T S 0 D T0 D 290 K. This is obtained by disconnecting the source and setting as an input to the 2-port network an impedance Z i equal to the source impedance; now the noise wi .S 0 / will only be thermal noise with noise temperature equal to T0 . The noise figure is given by the ratio between the available power at the load due to the total noise power pwo D pwo.A/ C pwo.S0 / and that due only to the source pwo.S0 / 4
F. f / D
p wo . f / pwo.S0 / . f /
D1C
pwo.A/ . f /
(4.61)
pwo.S0 / . f /
Being pwo.S0 / . f / D gd . f / kT2 0 , and substituting for pwo.A/ the expression (4.55), we obtain the important relation F. f / D 1 C
TA T0
(4.62)
We note that F is always greater than 1 and it equals 1 in the ideal case of a noiseless 2-port network. Moreover, F is a parameter of the network and does not depend on the noise temperature of the source to which it is connected. From (4.61) the noise power of the 2-port network can be expressed as pwo.A/ . f / D
k .F 1/T0 gd 2
(4.63)
From the above considerations we deduce that to describe the noise of an active 2-port network, we must assign the gain gd and the noise figure F (or equivalently the noise temperature T A ). We now see that for a passive network at temperature T0 , it is sufficient to assign only one of the two parameters. Let us consider a passive network at temperature T0 , as for example a transmission line, for which gd < 1. To determine the noise figure let us assume as source an impedance, which is matched to the network for maximum transfer of power, at temperature T0 . Applying Thevenin theorem to the network output, the system is equivalent to a two-terminal device with impedance Z 2 at temperature T0 .
4
Given an electrical circuit, a useful relation to determine F, equivalent to (4.61), that employs the PSDs of the output noise signals, is given by (see (4.9)) F. f / D
Pwo.A/ . f / Pwo . f / D1C Pw 0 . f / Pw 0 . f / o.S /
o.S /
(4.60)
4.2. Noise generated by electrical devices and networks
269
Assuming the load is matched for maximum transfer of power, i.e. Z 2 D Z ŁL , from (4.46) at the output we have pw0 . f / D .kT0 =2/. On the other hand, pwi.S0 / . f / D .kT0 =2/, and pw0.S0 / . f / D gd pwi.S0 / . Hence from the first of (4.61) we have F. f / D
1 D ad gd
(4.64)
where ad is the power attenuation of the network. Note that also in this case, given F, we can determine the effective noise temperature of the network, T A , according to (4.62). Summarizing, in a connection between a source and a 2-port network, the effective input temperature of the system can be expressed as Twi D T S C T A D T S C .F 1/T0
(4.65)
and the available noise power at the load is given by p w0 . f / D g d . f /
kTwi 2
(4.66)
Example 4.2.1 Let us consider the configuration of Figure 4.10a, where an antenna with noise temperature T S is connected to a pre-amplifier with available power gain g and noise figure F. An electrical model of the connection is given in Figure 4.10b, where the antenna is modelled as a resistance with noise temperature T S . If the impedances of the two devices are matched, Twi is given by (4.65).
(a)
(b)
Figure 4.10. Antenna-preamplifier configuration and electrical model.
270
Chapter 4. Transmission media
Cascade of 2-port networks As shown in Figure 4.11, we consider the cascade of two 2-port networks A1 and A2 , with available power gains g1 and g2 and noise figures F1 and F2 , respectively. Assuming the impedances are matched for maximum transfer of power between different networks, we wish to determine the parameters of a network equivalent to the cascade of the two networks. With regard to the power gain, the overall network has a gain g equal to the product of the gains of the individual networks: (4.67)
g D g1 g2
With regard to the noise characteristics, it is sufficient to determine the noise figure of the cascade of the two networks. For a source at room temperature T0 , from (4.66) for T S D T0 , the noise power at the output of the first network is given by pwo;1 . f / D
kT0 F1 g1 2
(4.68)
At the output of the second network we have pwo;2 . f / D pwo;1 . f /g2 C
k .F2 1/T0 g2 2
(4.69)
using (4.63) to express the noise power due to the second network only. Then the noise figure of the overall network is given by pwo;2 . f / FD D pwo;2.S0 / . f /
kT0 kT0 g1 g2 F1 C .F2 1/g2 2 2 kT0 g1 g2 2
(4.70)
Simplifying (4.70) we get F D F1 C
.F2 1/ g1
(4.71)
Extending this result to the cascade of N 2-port networks, Ai , i D 1; : : : ; N , characterized by gains gi and noise figures Fi , we obtain Friis formula of the total noise figure F D F1 C
.F2 1/ .F3 1/ .F N 1/ C C ÐÐÐC g1 g1 g2 g1 g2 : : : g N 1
Figure 4.11. Equivalent scheme of a cascade of two 2-port networks.
(4.72)
4.2. Noise generated by electrical devices and networks
271
We observe that F strongly depends on the gain and noise figure parameters of the first stages; in particular, the smaller F1 and the larger g1 , the more F will be reduced. Substituting (4.62), that relates the noise figure to the effective noise temperature, in (4.72), we have that the equivalent noise temperature of the cascade of N 2-port networks, characterized by noise temperatures T Ai , i D 1; : : : ; N , is given by T A D T A1 C
T A2 T A3 T AN C C ÐÐÐ C g1 g1 g2 g1 g2 : : : g N 1
(4.73)
Obviously the total gain of the cascade is given by g D g1 g2 : : : g N
(4.74)
Example 4.2.2 The idealized configuration of a transmission medium consisting of a very long cable where amplifiers are inserted at equally spaced points is illustrated in Figure 4.12. Each section of the cable, with power attenuation ac and noise figure Fc D ac (see (4.64)), cascaded with an amplifier, with gain g A and noise figure F A , is called a repeater section. To compensate for the attenuation of the cable we choose g A D ac . Then, each section has a gain gsr D
1 gA D 1 ac
(4.75)
and noise figure Fsr D Fc C
FA 1 D ac C ac .F A 1/ D g A F A gc
(4.76)
Therefore the N sections have overall unit gain and noise figure .Fsr 1/ .Fsr 1/ C ÐÐÐ C gsr gsr gsr : : : gsr D N .Fsr 1/ C 1 ' N Fsr
F D Fsr C
(4.77)
where Fsr is given by (4.76). We note that the output noise power of N repeater sections is N times the noise power introduced by an individual section.
Figure 4.12. Transmission channel composed of N repeater sections.
272
4.3
Chapter 4. Transmission media
Signal-to-noise ratio (SNR)
SNR for a two-terminal device Let us consider the circuit of Figure 4.3, where the source vb generates a desired signal s and a noise signal w: vb .t/ D s.t/ C w.t/
(4.78)
To measure the level of the desired signal with respect to the noise, one of the most widely used methods considers the signal-to-noise ratio (SNR), defined as the ratio of the statistical powers Z C1 Ps . f / d f Ms E[s 2 .t/] 1 3s D D Z C1 D (4.79) Mw E[w 2 .t/] Pw . f / d f 1
On the other hand, the effects of the two signals on a certain load Z c are measured by the average powers. Therefore we also introduce the following signal-to-noise ratio of average powers Z C1 ps . f / d f Ps D Z 1 3p D (4.80) C1 Pw pw . f / d f 1
where, from (4.9), ps . f / D Ps . f /
Rc jZ b C Z c j2
(4.81)
Rc jZ b C Z c j2
(4.82)
pw . f / D Pw . f /
Therefore 3s and 3 p are in general different. However, if the term Rc =jZ b C Z c j2 is a constant within the passband of s and w, then the two SNRs coincide. Note that if Z b D Z cŁ , that is the condition for maximum transfer of power is satisfied, then Rc =jZ b C Z c j2 D 1=.4Rb /. Hence it is sufficient that Rb is constant within the passband of s and w to have 3 D 3s D 3 p
(4.83)
Moreover, assuming pw is constant within the passband of w, with bandwidth B, we have k Pw D Tw 2B (4.84) 2 and, from (4.80), 3D
Ps E[s 2 .t/] D 2 kTw B E[w .t/]
(4.85)
where Ps is the available average power of the desired signal, and Tw is the noise temperature of w. Later we will often use this relation.
4.3. Signal-to-noise ratio (SNR)
273
SNR for a 2-port network Let us consider now the connection of a source to the linear 2-port network of Figure 4.2b, where vi has a desired component s and a noise component wi (see Figure 4.9b): vi .t/ D s.t/ C wi .t/
(4.86)
and wi has an effective noise temperature Twi D T S C T A . Therefore pwi . f / D kTwi =2. The open circuit voltage of the network output is given by vo .t/ D so .t/ C wo .t/
(4.87)
where so and wo depend on s and wi , respectively. Under matched load conditions (that is Z L D Z 2Ł ), and assuming (4.83) holds, at the network output we obtain 3out D
Ps E[so2 .t/] D o 2 P wo E[wo .t/]
(4.88)
We indicate with B the passband of the network frequency response, usually equal to or including the passband of s, and with B its bandwidth. From the expressions (4.4) and (4.30) Z C1 Z Pso D pso . f / d f D 2 ps . f /g. f / d f (4.89) B
1
and Z P wo D 2
B
pwi . f /g. f / d f
(4.90)
Assuming now that g. f / is constant within B, we have k Tw g2B (4.91) 2 i assuming that also the source is matched for maximum transfer of power. Finally we get Pso D Ps g and Pwo D
3out D
Ps E[so2 .t/] D 2 kTwi B E[wo .t/]
(4.92)
where Ps is the available power of the desired signal at the network input, and Twi D T S C .F 1/T0 is the effective noise temperature including both the source and the 2-port network. With reference to the above configuration, we observe that the power of wi could be very high if Twi is constant over a wide band, but wo has much smaller power since its passband coincides with that of the network frequency response. From (4.91) and (4.50), the effective input noise due to the connection source-network has an average power for T S D T0 .Twi D FT0 / equal to .Pwi /dBm D 114 C 10 log10 B jMHz C.F/d B .T S D T0 /
(4.93)
and the average power of the effective output noise is given by .Pwo /dBm D .Pwi /dBm C .g/d B .T S D T0 / In (4.93), BjMHz denotes the bandwidth in MHz.
(4.94)
274
Chapter 4. Transmission media
Example 4.3.1 A station, receiving signals from a satellite, has an antenna with gain gant of 40 dB and a noise temperature T S of 60 K (that is the antenna acts as a noisy resistor at a temperature of 60 K). The antenna feeds a preamplifier with a noise temperature T A1 of 125 K and a gain g1 of 20 dB. The preamplifier is followed by an amplifier with a noise figure F2 of 10 dB and a gain g2 of 80 dB. The transmitted signal bandwidth is 1 MHz. The satellite has an antenna with a power gain of gsat D 6 dB and the total attenuation a` due to the distance between transmitter and receiver is 190 dB. We want to find: 1. the average power of the thermal noise at the receiver output, 2. the minimum power of the signal transmitted by the satellite to obtain a SNR of 20 dB at the receiver output. The two receiver amplifiers can be modelled as one amplifier with gain: .g A /d B D .g1 /d B C .g2 /d B D 20 C 80 D 100 dB
(4.95)
and effective noise temperature: T A D T A1 C
T A2 .F2 1/T0 .1010=10 1/290 D T A1 C D 125 C D 151 K g1 g1 1020=10
(4.96)
1. From (4.91) the average power of the output noise is Pwo D k.T S C T A /g A B D 1:38 ð 1023 .60 C 151/ 10100=10 106 D 2:91 ð 105 W D 15:36 dBm
(4.97)
2. From 3out D .Pso =Pwo / ½ 20 dB we get .Pso =Pwo / ½ 100. As Pso D Ps gsat .1=a` / gant g A D Ps 1044=10 , it follows Ps ½ 73 W
(4.98)
Relation between noise figure and SNR For a source at room temperature T S D T0 , given that pwi.S0 / . f / D kT0 =2, it can be shown that FD
ps . f /=pwi.S0 / . f / pso . f /=pwo . f /
(4.99)
A more useful relation is obtained assuming that g. f / is a constant within the passband B of the network. Given the average power of the noise generated by the source at room temperature Pwi.S0 / D
kT0 2B 2
(4.100)
4.4. Transmission lines
275
Table 4.1 Parameters of three devices.
Device
F (dB)
maser TWT amplifier IC amplifier
0.16 2.7 7.0
T A (K)
g (dB)
Frequency
11 250 1163
20 ł 30 20 ł 30 50
6 GHz 3 GHz 70 MHz
and 3in D
Ps Pwi.S0 /
(4.101)
then, from (4.92), we have 3in (4.102) .T S D T0 / F In other words, F is a measure of the reduction of the SNR at the output due to the noise introduced by the network. In Table 4.1 the typical values of F, T A , and gain g are given for three devices. In the last column the frequency range usually considered for the operations of each device is also given. 3out D
4.4 4.4.1
Transmission lines Fundamentals of transmission line theory
In this section, the principles of signal propagation in transmission lines are briefly reviewed. A uniform transmission line consists of a two-conductor cable with a uniform cross-section, that supports the propagation of transverse electromagnetic (TEM) waves [3, 1]. Examples of transmission lines are twisted-pair cables and coaxial cables. We now develop the basic transmission line theory. With reference to Figure 4.13, which illustrates a uniform line, let
Figure 4.13. Uniform transmission line of length L.
276
Chapter 4. Transmission media
i
rdx
i+
ldx
v
ð i dx ðx
v+ gdx
cdx
ð v dx ðx
Figure 4.14. Line segment of infinitesimal length dx.
x denote the distance from the origin and L be the length of the line. The termination is found at distance x D 0 and the signal source at x D L. Let v D v.x; t/ and i D i.x; t/ be, respectively, the voltage and current at distance x at time t. To determine the law that establishes the voltage and current along the line, let us consider a uniform line segment of infinitesimal length that we assume to be time invariant, depicted in Figure 4.14. The parameters r; `; g; c are known as primary constants of the line. They define, respectively, resistance, inductance, conductance and capacitance of the line per unit length. Primary constants are in general slowly time-varying functions of the frequency; however, in this context, they will be considered time invariant. The model of Figure 4.14 is obtained using the first order Taylor series expansion of v.x; t/ and i.x; t/ as a function of distance x.
Ideal transmission line We initially assume an ideal lossless transmission line characterized by r D g D 0. Voltage and current variations in the segment dx are given by 8 @v @i > > < @ x dx D .` dx/ @t > > : @i dx D .cdx/ @v @x @t
(4.103)
Differentiating the first equation with respect to distance and the second with respect to time, we obtain 8 @ 2v @ 2i > > > < @ x 2 D ` @ x@t (4.104) > @ 2i > @ 2v > : D c 2 @t@ x @t Substituting @ 2 i=@t@ x in the first equation with the expression obtained from the second, we get the wave equation @ 2v 1 @ 2v @ 2v D `c D @x2 @t 2 ¹ 2 @t 2
(4.105)
4.4. Transmission lines
277
p where ¹ D 1= `c represents the velocity of propagation of the signal on a lossless transmission line. The general solution to the wave equation for a lossless transmission line is given by x x C '2 t C (4.106) v.x; t/ D '1 t ¹ ¹ where '1 and '2 are arbitrary functions. Noting that from (4.103) @i =@t D .1=`/@v=@ x, (4.106) yields `
1 x 1 0 x @i D '10 t C '2 t C @t ¹ ¹ ¹ ¹
(4.107)
where '10 and '20 are the derivatives of '1 and '2 , respectively. Integrating by parts (4.107) we get i.x; t/ D
x x i 1 h '1 t '2 t C C '.x/ `¹ ¹ ¹
(4.108)
where '.x/ is time independent and can therefore be ignored in the study of propagation. Defining the characteristic impedance of a lossless transmission line as r ` Z 0 D `¹ D (4.109) c the expression for the current is given by i.x; t/ D
1 h x x i '1 t '2 t C Z0 ¹ ¹
(4.110)
From the general solution to the wave equation we find that the voltage (or the current), considered as a function of distance along the line, consists of two waves that propagate in opposite directions: the wave that propagates from the source to the line termination is called the source or incident wave, that which propagates in the opposite direction is called reflected wave. We consider now the propagation of a sinusoidal wave with frequency f D !=2³ in an ideal transmission line. The voltage at distance x D 0 is given by v.0; t/ D V0 cos.!t/
(4.111)
The wave propagating in the positive direction of x is given by vC .x; t/ D jVC j cos[!.t x=¹/], that propagating in the negative direction is given by v .x; t/ D jV j cos[!.t C x=¹/ C p ]. The transmission line voltage is obtained as the sum of the two components and is given by i h h x i x (4.112) C jV j cos ! t C C p v.x; t/ D jVC j cos ! t ¹ ¹ The current has the expression i h h jVC j x i jV j x i.x; t/ D C p cos ! t cos ! t C Z0 ¹ Z0 ¹
(4.113)
278
Chapter 4. Transmission media
Let us consider a point on the x-axis individuated at each time instant t by the condition that the argument of the function F.t x=¹/ is a constant. This point is seen by an observer as moving at velocity ¹ in the positive direction of the x-axis. For sinuosoidal waves the velocity for which the phase is a constant is called phase velocity ¹. It is useful to write (4.112) and (4.113) in complex notation, where the phasors V and I represent amplitude and phase at distance x of the sinusoidal signals (4.112) and (4.113), respectively, V D VC e jþx C V e jþx 1 ID .VC e jþx V e jþx / Z0
(4.114) (4.115)
where þ D !=¹ denotes the phase constant. We define the wavelength as ½ D 2³=þ. We note that frequency f and wavelength ½ are related by ½D
¹ f
(4.116)
In particular, the propagation in free space is characterized by ¹ D c D 3 Ð 108 m/s. If VC is taken as the reference phasor with phase equal to zero, then V D jV je j p , where p is the phase rotation between the incident and the reflected waves at x D 0. Let us consider a transmission line having as termination an impedance Z L . By Kirchhoff laws, the voltage and current at the termination are given by 8 > < VL D VC C V (4.117) VL VC V > D : IL D ZL Z0 Z0 The reflection coefficient is defined as the ratio between the phasors representing, respectively, the reflected and incident waves, % D V =VC . The transmission coefficient is defined as the ratio between the phasors representing, respectively, the termination voltage and the incident wave − D VL =VC . From (4.117), it turns out þ þ þ V þ Z L Z0 %D D þþ þþ e j p (4.118) Z L C Z0 VC and −D
2Z L Z L C Z0
(4.119)
At the termination, defining the incident power as PC D jVC j2 =Z 0 and the reflected power as P D jV j2 =Z 0 , we obtain P =PC D j%j2 ; the ratio between the power delivered to the load and the incident power is hence given by 1 j%j2 . Let us consider some specific cases: ž if Z L D Z 0 , % D 0 and there is no reflection; ž if Z L D 1, the line is open-circuited, % D 1 and V D VC ; ž if Z L D 0, the line is short-circuited, % D 1 and V D VC .
4.4. Transmission lines
279
Non-ideal transmission line Typically, in a transmission line the primary constants r and g are different from zero. For sinusoidal waves in steady state, the changes in voltage and current in a line segment of infinitesimal length characterized by an impedance Z and an admittance Y per unit length can be expressed using complex notation as 8 dV > > < dx D Z I > > : d I D Y V dx
(4.120)
Differentiating and substituting in the first equation the expression of d I =dx obtained from the second, we get d2V D 2V dx 2
(4.121)
where D
p ZY
(4.122)
is a characteristic constant of the transmission line called propagation constant. Let Þ and þ be, respectively, the real and imaginary parts of : Þ is the attenuation constant measured in neper per unit of length, and þ is the phase constant measured in radians per unit of length. The solution of the differential equation for the voltage can be expressed in terms of exponential functions as V D VC e x C V e x D VC eÞx e jþx C V eÞx e jþx
(4.123)
The expression of the current is given by I D
Ð 1 VC e x V e x Z0
(4.124)
where r Z0 D
Z Y
(4.125)
is the characteristic impedance of the transmission line. The propagation constant and the characteristic impedance are also known as secondary constants of the transmission line.
Frequency response Let us consider the transmission line of Figure 4.15, with a sinusoidal voltage source vi and a load Z L . From (4.123) the voltage at the load can be expressed as VL D VC .1 C %/. Recalling that V = VC D %, we define the voltage Vo D VL j Z LD1 D VC .1 C %/ j%D1 D 2VC .
280
Chapter 4. Transmission media
i(t) 1
Z
i
v 1 (t)
v L(t)
ZL
v(t) i
x = -L
x=0
Figure 4.15. Transmission line with sinusoidal voltage generator vi and load ZL .
For the voltage V1 and current I1 we find 8 L L / > < V1 D Vi Z i I1 D VC .e C %e > I D VC .e L %e L / : 1 Z0
(4.126)
where Z i denotes the generator impedance. The input and output impedances of the 2-port network are, respectively, given by: Z1 D
1 C %e2 L V1 D Z0 I1 1 %e2 L
(4.127)
Z2 D
VC .1 C %/ j%D1 Vooc D V D Z0 C I L sc Z 0 .1 %/ j%D1
(4.128)
where I L sc D I L j Z L D0 and Vooc D VL j Z L D1 . We now want to determine the ratio between the voltage VL and the voltage V1 , defined as GCh D VL = V1 . Observing the above relations we find the following frequency responses: GL D G1o D Gi D
ZL 1 VL 1C% D D VC .1 C %/ D Vo 2VC 2 Z L C Z0
(4.129)
Vo 2e L D V1 1 C %e2 L
(4.130)
V1 Z1 Z 0 .1 C %e2 L / D D Vi Zi C Z1 Z 0 .1 C %e2 L / C Z i .1 %e2 L /
(4.131)
4.4. Transmission lines
281
Then, from (4.1), the channel frequency response is given by: GCh D G1o G L D
.1 C %/e L 1 C %e2 L
(4.132)
Let us consider some specific cases: ž Matched transmission line: % D 0 for Z i D Z L D Z 0 .Gi D 1=2/ GCh D e L
(4.133)
ž Short-circuited transmission line: % D 1 GCh D 0
(4.134)
ž Open-circuited transmission line: % D 1 GCh D
2e L 1 D 2 L cosh. L/ 1Ce
(4.135)
To determine the power gain of the network, we can use the general equation (4.25), or observe (4.23); in any case, we obtain g. f / D
1 j%j2 e2ÞL 1 j%j2 e4ÞL
(4.136)
where Þ D Re[ ]. We note that, for a matched transmission line, the available attenuation is given by ad . f / D
1 D e2ÞL je L j2
(4.137)
In (4.137), Þ expresses the attenuation in neper per unit of length. Alternatively, one can introduce an attenuation in dB per unit of length, .aQ d . f //d B , as 1
ad . f / D 10 10 .aQ d . f //d B L
(4.138)
The relation between Þ and .aQ d . f //d B is given by .aQ d . f //d B D 8:68Þ
(4.139)
From (4.139), the attentuation in dB introduced by the transmission line is equal to .ad . f //d B D .aQ d . f //d B L
(4.140)
In a transmission line with a non-matched resistive load that satisfies the condition Z L − Z 0 , from (4.118) we get 1 C % ' 2Z L =Z 0 , %2 e4ÞL ' 0, and %2 ' 1 4Z L =Z 0 . Therefore (4.136) yields þ þ þ ZL þ þ .ad . f //d B D .aQ d . f //d B L 10 log10 4 þþ (4.141) Z0 þ
282
Chapter 4. Transmission media
Conditions for the absence of signal distortion We recall that Heaviside conditions for the absence of signal distortion are satisfied if GCh . f / has a constant amplitude and a linear phase, at least within the passband of the source. For a matched transmission line, these conditions are satisfied if Þ is a constant and þ is a linear function of the frequency. p transmission line can p The secondary parameters of the be expressed as D Þ C jþ D .r C j!`/.g C j!c/, and Z 0 D .r C j!`/=.g C j!c/. For a matched transmission line, it can be shown that Heaviside conditions are equivalent to the condition r c D g`
(4.142)
In the special case g D 0, we obtain r
`c ÞD! 2
(
r2 1C 2 2 ! `
)1=2
1=2
1
(4.143)
and r
`c þD! 2
(
r2 1C 2 2 ! `
)1=2
1=2
C1
(4.144)
For frequencies at which r − !`, using the approximation 1=2 r2 1 r2 1C 2 2 '1C 2 ! 2 `2 ! `
(4.145)
we find r Þ' 2
r
c `
p and þ ' ! `c
(4.146)
Impulse response of a non-ideal transmission line For commonly used transmission lines, a more accurate model of the propagation constants, that takes into account the variation of r with the frequency due to the skin effect, shows that both the attenuation constant and the phase constant must include a term proportional to the square root of frequency. An expression of the propagation constant generally used to characterize the propagation of TEM waves over a metallic transmission line [1] is r r p ! ! C jK C j! `c . f/ D K (4.147) 2 2 where K is a constant that depends on the transmission line. The expression (4.147) is valid for both coaxial and twisted-pair cables insulated with plastic material. The attenuation constant of the transmission line is therefore given by p Þ. f / D K ³ f (neper/m) (4.148)
4.4. Transmission lines
283
and the attenuation introduced by the transmission line can be expressed as p .aQ d . f //d B D 8:68K ³ f (dB/m)
(4.149)
We note that, given the value of Þ. f / at a certain frequency f D f 0 , we can obtain the value of K . Therefore it is possible to determine the attenuation constant at every other frequency. From the expression (4.133) of the frequency response p of a matched transmission `c introduced by the term line, with given by (4.147), without considering the delay p j! `c, the impulse response has the following expression K L .K L/2 gCh .t/ D p e 4t 1.t/ 2 ³t3
(4.150)
The pulse signal gCh is shown in Figure 4.16 for various values of the product K L. We note a larger dispersion of gCh for increasing values of K L.
Secondary constants of some transmission lines In Table 4.2 we give the values of Z 0 and D Þ C jþ experimentally measured for some telephone transmission lines characterized by a certain diameter, which is usually indicated by a parameter called gauge. The behavior of Þ as a function of frequency ispgiven in Figure 4.17 for four telef law in the range of frequencies phone lines [2]; we may note that it follows the f < 10 kHz. For some transmission lines this law is followed also for f > 100 kHz, 0.12
KL=2
0.1
gCh(t)
0.08
0.06
KL=3
0.04
KL=4 KL=5
0.02
KL=6
0
0
2
4
6
8 t (s)
10
12
14
16
Figure 4.16. Impulse response of a matched transmission line for various values of KL.
284
Chapter 4. Transmission media
Table 4.2 Secondary constants of some telephone lines.
Gauge diameter (mm) 19 .0:9119/ 22 .0:6426/ 24 .0:5105/ 26 .0:4039/
Frequency (Hz) 1000 2000 3000 1000 2000 3000 1000 2000 3000 1000 2000 3000
Characteristic impedance Z 0 () 297 217 183 414 297 247 518 370 306 654 466 383
j278 j190 j150 j401 j279 j224 j507 j355 j286 j645 j453 j367
Propagation constant Þ C jþ (neper/km) (rad/km) 0:09 C 0:12 C 0:15 C 0:13 C 0:18 C 0:22 C 0:16 C 0:23 C 0:28 C 0:21 C 0:29 C 0:35 C
Attenuation aQ d D 8:68Þ (dB/km)
j0:09 j0:14 j0:18 j0:14 j0:19 j0:24 j0:17 j0:24 j0:30 j0:21 j0:30 j0:37
0:78 1:07 1:27 1:13 1:57 1:90 1:43 2:00 2:42 1:81 2:55 3:10
c 1982 Telephone Laboratories. Reproduced with permission of Lucent Technologies, Inc./Bell Labs.
Figure 4.17. Attenuation as a function of frequency for some telephone transmission lines: three are polyethylene-insulated cables (PIC) and one is a coaxial cable with a diameter c 1982 Bell Telephone Laboratories. Reproduced with permission of Lucent of 9.525 mm. [ Technologies, Inc./Bell Labs.]
4.4. Transmission lines
285
albeit with a different constant of proportionality. In any case in the local-loop,5 to force the primary constants to satisfy Heaviside conditions in the voice band, which goes from 300 to 3400 Hz, formerly some lump inductors were placed at equidistant points along the transmission line. This procedure, called inductive loading, causes Þ. f / to be flat in the voice band, but considerably increases the attenuation outside of the voice band. Moreover, the phase þ. f / may result very distorted in the passband. Typical behavior of Þ and þ in the frequency band 0 ł 4000 Hz, with and without loading, are given in Figure 4.18 for a transmission line with gauge 22 [2]. The digital subscriber line (DSL) technologies, introduced for data transmission in the local loop, require a bandwidth much greater than 4 kHz, up to about 20 MHz for the VDSL technology (see Chapter 17). For DSL applications it is therefore necessary to remove possible loading coils that are present in the local loops. The frequency response of a DSL transmission line can also be modified by the presence of one or more bridged-taps. A bridged-tap consists of a twisted pair cable of a certain length L BT , terminated by an open circuit and connected in parallel to a local loop. At the connection point, the incident signal separates into two components. The component propagating along the bridged-tap is reflected at the point of the open circuit: the component propagating on the transmission line must therefore be calculated taking also into consideration this reflected component. At the frequencies f BT D ¹=½ BT , where ½ BT satisfies the condition .2n C 1/½ BT =4 D L BT , n D 0; 1; : : : , at the connection point we get destructive interference between the reflected and incident component: this interference reveals itself as a notch in the frequency response of the transmission line. Given
Figure 4.18. Attenuation constant Þ and phase constant þ for a telephone transmission line c 1982 Bell Telephone Laboratories. Reproduced with permission with and without loading. [ of Lucent Technologies, Inc./Bell Labs.]
5
By local-loop we intend the transmission line that goes from the user telephone set to the central office.
286
Chapter 4. Transmission media
Table 4.3 Transmission characteristics defined by the EIA/TIA for unshielded twisted pair (UTP) cables.
Signal attenuation at 16 MHz
NEXT attenuation at 16 MHz
Characteristic impedance
13.15 dB/100 m 8.85 dB/100 m 8.20 dB/100 m
½23 dB ½38 dB ½44 dB
100 š 15% 100 š 15% 100 š 15%
UTP-3 UTP-4 UTP-5
the large number of transmission lines actually in use, to evaluate the performance of DSL systems we usually refer to a limited number of loop characteristics, which can be viewed as samples taken from the ensemble of frequency responses. On the other hand, the transmission characteristics of unshielded twisted-pair (UTP) cables commonly used for data transmission over local area networks are defined by the EIA/TIA and ISO/IEC standards. As illustrated in Table 4.3, the cables are divided into different categories according to the values of 1) the signal attenuation per unit of length, 2) the attenuation of the near-end cross-talk signal, or NEXT, that will be defined in the next section, and 3) the characteristic impedance. Cables of category three (UTP-3) are commonly called voice-grade, those of categories four and five (UTP-4 and UTP-5) are data-grade. We note that the signal attenuation and the intensity of NEXT are substantially larger for UTP-3 cables than for UTP-4 and UTP-5 cables.
4.4.2
Cross-talk
The interference signal that is commonly referred to as cross-talk is determined by magnetic coupling and unbalanced capacitance between two adjacent transmission lines. Let us consider the two transmission lines of Figure 4.19, where the terminals .1; 10 / belong to the disturbing transmission line and the terminals .2; 20 / belong to the disturbed transmission line. In the study of the interference signal produced by magnetic coupling, we consider
Figure 4.19. Transmission lines configuration for the study of cross-talk.
4.4. Transmission lines
287
i1 1 v1 Z0 1’ im
m
2 Z0
Z0 2’
Figure 4.20. Interference signal produced by magnetic coupling. 1
1 c 11Ȁ v1
c 12Ȁ
c 12 1Ȁ 2
c 12
Z0 v1 Z 0
c 1Ȁ2
c 11Ȁ
2
ic
Z0
c 12Ȁ
c 22Ȁ
2Ȁ
c 1Ȁ2Ȁ Z0
Z0
c 22Ȁ
c 1Ȁ2
Z0
c 1Ȁ2Ȁ
2Ȁ 1Ȁ (a)
(b)
Figure 4.21. Interference signal produced by unbalanced capacitance.
the circuit of Figure 4.20. We will assume that the length of the transmission line is much longer than the wavelength corresponding to the maximum transmitted frequency and that the impedance Z 0 is much higher than the inductor reactance. The induced electromagnetic force (EMF) is given by E D j2³ f m I1 , where I1 ' V1 =Z 0 . The EMF produces a current j2³ f m E D .1=.2Z I1 , that can be expressed as Im D j2³ f m2 V1 . Im D .1=.2Z 0 // 0 // .1=.2Z 0 //
To study the interference signal due to unbalanced capacitance, we consider the circuit of Figure 4.21a, that can be redrawn in an equivalent way as illustrated in Figure 4.21b. We assume that the impedance Z 0 is much smaller than the reactance of the capacitors that can be found on the bridge. Applying the principle of the equivalent generator we find 0 Ic D
1
1
1
C B V220 j Ic D0 1 c10 20 c10 2 C D B j2³ f V1 A @ 1 1 1 1 1 1 Z 220 C C C c10 20 c120 c10 2 c12 c12 C c10 2 c120 C c10 20 (4.151)
288
Chapter 4. Transmission media
Figure 4.22. Illustration of near-end cross-talk (NEXT) and far-end cross-talk (FEXT) signals.
from which we obtain Ic D
c12 c10 20 c120 c10 2 j2³ f V1 D j2³ 1cV1 c12 C c10 2 C c120 C c10 20
(4.152)
Recalling that the current Ic is equally divided between the impedances Z 0 on which the transmission line terminates, we find that the cross-talk current produced at the transmitter side termination is I p D Im C Ic =2, and the cross-talk current produced at the receiver side termination is It D Im C Ic =2. As illustrated in Figure 4.22, the interference signals are called near-end cross-talk or NEXT, or far-end cross-talk or FEXT, depending on whether the receiver side of the disturbed line is the same as the transmitter side of the disturbing line, or the opposite side, respectively. We now evaluate the total contribution of the near and far-end cross-talk signals for lines with distributed impedances.
Near-end cross-talk Let a p .x/ D
m.x/ 1c.x/ Z0 C 2Z 0 2
(4.153)
be the near-end cross-talk coupling function at distance x from the origin. In complex notation, the NEXT signal is expressed as Z
L
Vp D Z0 I p D
V1 e2 x j2³ f a p .x/ dx
(4.154)
0
To calculate the power spectral density of NEXT we need to know the autocorrelation function of the random process a p .x/. A model commonly used in practice assumes that a p .x/ is a white stationary random process, with autocorrelation ra p .z/ D E[a p .x C z/a Łp .x/] D r p .0/Ž.z/
(4.155)
4.4. Transmission lines
289
For NEXT the following relation holds p ³ 3=2 r p .0/ f 3=2 .1 e4K ³ f L / ' E[jV1 . f /j2 ]k p f 3=2 K (4.156) where K is defined by (4.148), and
E[jV p . f /j2 ] D E[jV1 . f /j2 ]
kp D
³ 3=2 r p .0/ K
(4.157)
Using (1.449), the level of NEXT coupling is given by6 jG p . f /j2 D
E[jV p . f /j2 ] ' k p f 3=2 E[jV1 . f /j2 ]
(4.158)
To perform computer simulations of data transmission systems over metallic lines in the presence of NEXT, it is required to characterize not only the amplitude, but also the phase of NEXT coupling. In addition to experimental models obtained through laboratory measurements, the following stochastic model is used:
a p .x/ D
L 1 1x X
ai w1x .x i1x/
(4.159)
if x 2 [0; 1x/ otherwise
(4.160)
i D0
with ( w1x .x/ D
1 0
where ai , i D 0; : : : ; L=1x 1, denote statistically independent Gaussian random variables with zero mean and variance E[ai2 ] D
r p .0/ 1x
(4.161)
A NEXT coupling function is thus given by
GNE X T . f / D
L 1 1x X
j2³ f ai w1x .x i1x/e2.K
p
p p 1 ³ f C j K ³ f C j2³ f `c/.i C 2 /1x
(4.162)
i D0
If we know the parameters of the transmission line K and k p , then from (4.157) and (4.161) the variance of ai to be used in the simulations is given by E[ai2 ] D
6
K kp ³ 3=2 1x
Observing (1.449), jG p . f /j2 is also equal to the ratio between the PSDs of v p and v1 .
(4.163)
290
Chapter 4. Transmission media
Far-end cross-talk Let at .x/ D
m.x/ 1c.x/ Z0 C 2Z 0 2
(4.164)
be the far-end cross-talk coupling function at distance x from the origin. In complex notation, the FEXT signal is given by Z L V1 e L j2³ f at .x/ dx (4.165) Vt D Z 0 It D 0
Analogously to the case of NEXT, we assume that at is a white stationary random process, with autocorrelation rat .z/ D E[a t .x C z/a tŁ .x/] D rt .0/Ž.z/
(4.166)
For the FEXT signal the following relation holds E[jVt . f /j2 ] D E[jV1 . f /j2 ]e2K
p
³f L
.2³ f /2 rt .0/L
(4.167)
where L is the length of the transmission line. The level of FEXT coupling is given by jGt . f /j2 D
p E[jVt . f /j2 ] 2 2K ³ f L D k f Le t E[jV1 . f /j2 ]
(4.168)
where kt D .2³ /2 rt .0/. We note that for high-speed data transmission systems over unshielded twisted-pair cables, NEXT usually represents the dominant source of interference. Example 4.4.1 For local-area network (LAN) applications, the maximum length of cables connecting stations is typically limited to 100 m. Deviations from the characteristic expressed by (4.147) may be caused by losses in the dielectric material of the cable, the presence of connectors, non-homogeneity of the transmission line, etc. For the IEEE Standard 100BASE-T2, which defines the physical layer for data transmission at 100 Mb/s over UTP-3 cables in Ethernet LANs (see Chapter 17), the following worst-case frequency response is considered: 1:2
GCh . f / D 10 20 e.0:00385
p
j f C0:00028 f /L
(4.169) p
where f is expressed in MHz and L in meters. In (4.169), the term e j2³ f `cL is ignored, as it indicates a constant propagation delay. A frequency independent attenuation of 1.2 dB has been included to take into account the attenuation caused by the possible presence of connectors. The amplitude of the frequency response obtained for a cable length L D 100 m is shown in Figure 4.23 [4]. We note that the signal attenuation at the frequency of 16 MHz is equal to 14.6 dB, a higher value than that indicated in Table 4.3 for UTP-3 cables.
4.5. Optical fibers
291
0 Amplitude characteristic for 100 m cable length
16 MHz NEXT coupling envelope curve –21 + 15 log10 ( f/16 ) , f in MHz
–10 –14.6 dB –21.0 dB
(dB)
–20
Amplitude
–30
–40 Four NEXT coupling functions –50
–60
0
5
10
15
20 f (MHz)
25
30
35
40
Figure 4.23. Amplitude of the frequency response for a voice-grade twisted-pair cable with c 1997 IEEE.] length equal to 100 m, and four realizations of NEXT coupling function. [
The level of NEXT coupling (4.158) is illustrated in Figure 4.23 as a dotted line; we note the increase as a function of frequency of 15 dB/decade, due to the factor f 3=2 . The level of NEXT coupling equal to 21 dB at the frequency of 16 MHz is larger than that given in Table 4.3 for UTP-3 cables. The amplitude characteristics of four realizations of the NEXT coupling function (4.162) are also shown in Figure 4.23.
4.5
Optical fibers
Transmission systems using light pulses that propagate over thin glass fibers were introduced in the 1970s and have since then undergone continuous development and experienced an increasing penetration, to the point that they now constitute a fundamental element of modern information highways. For in-depth study of optical fiber properties and of optical component characteristics we refer the reader to the vast literature existing on the subject [5, 6, 7]; in this section we limit ourselves to introducing some fundamental concepts. The term “optical communications” is used to indicate the transmission of information by the propagation of electromagnetic fields at frequencies typically of the order of 1014 ł 1015 Hz, that are found in the optical band and are much higher than the frequency of radio waves or microwaves; to identify a transmission band, the wavelength rather than the frequency is normally used. We recall that for electromagnetic wave propagation in free space, the relation (4.116) holds: a frequency of 3 Ð 1014 Hz corresponds therefore to a wavelength of 1 µm for transmission over optical fibers. The signal attenuation as a function of the wavelength exhibits the behavior shown in Figure 4.24 [8, 9]; we note that the useful interval for transmission is in the range from 800 to 1600 nm, that corresponds
292
Chapter 4. Transmission media
Figure 4.24. Attenuation curve as a function of wavelength for an optical fiber. [From Li c 1980 IEEE.] (1980), see also Miya et al. (1979),
to a bandwidth of 2 Ð 1014 Hz. Three regions are typically used for transmission: the first window goes from 800 to 900 nm, the second from 1250 to 1350 nm, and the third from 1500 to 1600 nm. We immediately realize the enormous capacity of fiber transmission systems: for example, a system that uses only 1% of the 2 Ð 1014 Hz bandwidth mentioned above, has an available bandwidth of 2 Ð 1012 Hz, equivalent to that needed for the transmission of ¾300:000 television signals, each with a bandwidth of 6 MHz. To efficiently use the band in the optical spectrum, multiplexing techniques using optical devices have been developed, such as wavelength-division multiplexing (WDM) and optical frequency-division multiplexing (O-FDM). Moreover, we note that, although the propagation of electromagnetic fields in the atmosphere at these frequencies is also considered for transmission (see Section 17.2.1), the majority of optical communication systems employ as transmission medium an optical fiber, which acts as a waveguide. A fundamental device in optical communications is represented by the laser, which, beginning in the 1970s, made coherent light sources available for the transmission of signals.
Description of a fiber-optic transmission system The main components of a fiber-optic transmission system are illustrated in Figure 4.25 [10]. Optical transmission lines with lengths of over a few hundred meters use fiber glass, because they present less attenuation with respect to fibers using plastic material. Dispersion in the transmission medium causes “spreading” of the transmitted pulses; this phenomenon in turn causes intersymbol interference and limits the available bandwidth of the transmission
4.5. Optical fibers
293
Figure 4.25. Elements of a typical fiber-optic transmission system.
medium. A measure of the pulse dispersion is given by 1− D .M C Mg / L 1½
(4.170)
where M is the dispersion coefficient of the material, Mg is the dispersion coefficient related to the geometry of the waveguide, L denotes the length of the fiber and 1½ denotes the spectral width of the light source. The total dispersion .M C Mg / has values near 120, 0, and 15 ps/(nmðkm) at wavelengths of 850, 1300, and 1550 nm, respectively. The bandwidth of the transmission medium is inversely proportional to the dispersion; we note that the dispersion is minimum in the second window, with values near zero around the wavelength of 1300 nm for conventional fibers. Special fibers are designed to compensate for the dispersion introduced by the material; because of the low attenuation and dispersion, these fibers are normally used in very long distance connections. Multimode fibers allow the propagation of more than one mode of the electromagnetic field. In this case the medium introduces signal distortion caused by the fact that propagation of energy for different modes has different speeds: for this reason multimodal fibers are used in applications where the transmission bandwidth and the length of the transmission line are not large. Monomode fibers limit the propagation to a single mode, thus eliminating the dispersion caused by multimode propagation. Because in this case the dispersion is due only to the material and the geometry of the waveguide, monomodal fibers are preferred for applications that require wide transmission bandwidth and very long transmission lines. In Table 4.4 typical values of the transmission bandwidth, normalized by the length of the optical fiber, are given for different types of fibers. The step-index (SI) fiber is characterized by a constant value of the refraction index, whereas the graded-index (GRIN) fiber has a refraction index decreasing with the distance from the fiber axis. As noticed previously, the monomodal fibers are characterized by larger bandwidths; to limit the number of modes
294
Chapter 4. Transmission media
Table 4.4 Characteristic parameters of various types of optical fibers.
Fiber
Wavelength (nm)
Source
Bandwidth (MHzÐkm)
850 850 1300 1300 1550
LED LD LD o LED LD LD
30 500 1000 >10000 >10000
multimode SI multimode GRIN multimode GRIN monomode monomode
to one, the diameter of the monomodal fiber is related to the wavelength and is normally about one order of magnitude smaller than that of multimodal fibers. Semiconductor laser diodes (LD) or light-emitting diodes (LED) are used as signal light sources in most applications; these sources are usually modulated by electronic devices. The conversion from a current signal to an electromagnetic field that propagates along the fiber can be described in terms of light signal power by the relation PT x D k0 C k1 i
(4.171)
where k0 and k1 are constants. The transmitted waveform can therefore be seen as a replica of the modulation signal, in this case the current signal. Laser diodes are characterized by a smaller spectral width 1½ as compared to that of LEDs, and therefore lead to a lower dispersion (see (4.170)). The more widely used photodetector devices are semiconductor photodiodes, which convert the optical signal into a current signal according to the relation i D ² P Rc
(4.172)
where i is the device output current, P Rc is the power of the incident optical signal and ² is the photodetector response. Typical values of ² are of the order of 0.5 mA/mW. Signal quality is measured by the signal-to-noise ratio expressed as 3D
gin
.gi ² P Rc /2 R L 2e R L B.I D C ² P Rc / C 4kTw B
(4.173)
where gi is the photodetector current gain, n is a parameter that indicates the photodetector excess noise, B is the receiver bandwidth, k is Boltzmann constant, e is the charge of the electron, Tw is the effective noise temperature in Kelvin, I D is the photodetector dark current, and R L is the resistance of the load that follows the photodetector. We note that in the denominator of (4.173) the first term is due to shot noise and the second term to thermal noise.
4.6
Radio links
The term radio is used to indicate the transmission of an electromagnetic field that propagates in free space. Some examples of radio transmission systems are: ž point-to-point terrestrial links [11]; ž mobile terrestrial communication systems [12, 13, 14, 15, 16];
4.6. Radio links
295
Figure 4.26. Radio link model.
ž earth-satellite links (with satellites employed as signal repeaters) [17]; ž deep-space communication systems (with space probes at a large distance from earth). A radio link model is illustrated in Figure 4.26, where we assume that the transmit antenna input impedance and the receive antenna output impedance are matched for maximum transfer of power.
4.6.1
Frequency ranges for radio transmission
Frequencies used for radio transmission are in the range from about 100 kHz to some tens of GHz. The choice of the carrier frequency depends on various factors, among which the dimensions of the transmit antenna play an important role. In fact, to achieve an efficient radiation of electromagnetic energy, one of the dimensions of the antenna must be at least equal to 1=10 of the carrier wavelength. This means that an AM radio station, with carrier frequency f 0 D 1 MHz and wavelength ½ D c= f 0 D 300 m, where c is the speed of light in free space, requires an antenna of at least 30 m. A radio wave usually propagates as a ground wave (or surface wave), via reflection and scattering in the atmosphere (or via tropospheric scattering), or as a direct wave. Recall that, if the atmosphere is non-homogeneous (in terms of temperature, pressure, humidity, : : : ), the electromagnetic propagation depends on the changes of the refraction index of the medium. In particular, this gives origin to the reflection of electromagnetic waves. We speak of diffusion or scattering phenomena if molecules that are present in the atmosphere absorb part of the power of the incident wave and then re-emit it in all directions. Obstacles such as mountains, buildings, etc., give also origin to signal reflection and/or diffusion. In any case, these are phenomena that permit transmission between two points that are not in line-of-sight (LOS). We will now consider the types of propagation associated with frequency bands. Very low frequency (VLF) for f 0 < 0:3 MHz. The earth and the ionosphere form a waveguide for the electromagnetic waves. At these frequencies the signals propagate around the earth.
296
Chapter 4. Transmission media
Medium frequency (MF) for 0:3 < f 0 < 3 MHz. The waves propagate as ground waves up to a distance of 160 km. High frequency (HF) for 3 < f 0 < 30 MHz. The waves are reflected by the ionosphere at an altitude that may vary between 50 and 400 km. Very high frequency (VHF) for 30 < f 0 < 300 MHz. For f 0 > 30 MHz, the signal propagates through the ionosphere with small attenuation. Therefore these frequencies are adopted for satellite communications. They are also employed for line-of-sight transmissions, using high towers where the antennas are positioned to cover a wide area. The limit to the coverage is set by the earth curvature. p If h is the height of the tower in meters, the range covered expressed in km is r D 1:3 h: for example, if h D 100 m, coverage is up to about r D 13 km. However, ionospheric and tropospheric scattering (at an altitude of 16 km or less) are present at frequencies in the range 30–60 MHz and 40–300 MHz, respectively, which cause the signal to propagate over long distances with large attenuations. Ultra high frequency (UHF) for 300 MHz < f 0 < 3 GHz. Super high frequency (SHF) for 3 < f 0 < 30 GHz. At frequencies of about 10 GHz, atmospheric conditions play an important role in signal propagation. We note the following absorption phenomena, which cause additional signal attenuation: 1. due to oxygen: for f 0 > 30 GHz, with peak attenuation at 60 GHz; 2. due to water vapor: for f 0 > 20 GHz, with peak attenuation at around 20 GHz; 3. due to rain: for f 0 > 10 GHz, assuming the diameter of the rain drops is of the order of the signal wavelength. We note that, if the antennas are not positioned high enough above the ground, the electromagnetic field propagates not only into the free space but also through ground waves. Extremely high frequency (EHF) for f 0 > 30 GHz.
Radiation masks A radio channel by itself does not set constraints on the frequency band that can be used for transmission. In any case, to prevent interference among radio transmissions, regulatory bodies specify power radiation masks: a typical example is given in Figure 4.27, where the plot represents the limit on the power spectrum of the transmitted signal with reference to the power of a non-modulated carrier. To comply with these limits, a filter is usually employed at the transmitter front-end.
4.6.2
Narrowband radio channel model
The propagation of electromagnetic waves should be studied using Maxwell equations with appropriate boundary conditions. Nevertheless, for our purposes a very simple model, which
4.6. Radio links
297
Figure 4.27. Radiation mask of the GSM system with a bandwidth of 200 kHz around the carrier.
consists in approximating an electromagnetic wave as a ray (in the optical sense), is often adequate. The deterministic model is used to evaluate the power of the received signal when there are no obstacles between the transmitter and receiver, that is in the presence of line of sight: in this case we can think of only one wave that propagates from the transmitter to the receiver. This situation is typical of transmissions between satellites and terrestrial radio stations in the microwave frequency range (3 < f 0 < 70 GHz). Let PT x be the power of the signal transmitted by an ideal isotropic antenna, which uniformly radiates in all directions in the free space. At a distance d from the antenna, the power density is 80 D
PT x (W/m2 ) 4³ d 2
(4.174)
where 4³ d 2 is the surface of a sphere of radius d that is uniformly illuminated by the antenna. We observe that the power density decreases with the square of the distance. On a logarithmic scale (dB) this is equivalent to a decrease of 20 dB-per-decade with the distance. In the case of a directional antenna, the power density is concentrated within a cone and is given by 8 D G T x 80 D
GT x PT x 4³ d 2
(4.175)
where GT x is the transmit antenna gain. Obviously, GT x D 1 for an isotropic antenna; usually, GT x × 1 for a directional antenna.
298
Chapter 4. Transmission media
At the receive antenna, the available power in conditions of matched impedance is given by P Rc D 8A Rc Rc
(4.176)
where P Rc is the received power, A Rc is the effective area of the receive antenna and Rc is the efficiency of the receive antenna. The factor Rc < 1 takes into account the fact that the antenna does not capture all the incident radiation, because a part is reflected or lost. To conclude, the power of the received signal is given by P Rc D PT x
A Rc GT x Rc 4³ d 2
(4.177)
The antenna gain can be expressed as [1] 4³ A (4.178) ½2 where A is the effective area of the antenna, ½ D c= f 0 is the wavelength of the transmitted signal, f 0 is the carrier frequency and is the efficiency factor. The (4.178) holds for the transmit as well as for the receive antenna. We note that, because of the factor A=½2 , working at higher frequencies presents the advantage of being able to use smaller antennas, for a given G. Usually 2 [0:5; 0:6] for parabolic antennas, while ' 0:8 for horn antennas. Observing (4.178), we get ½ 2 (4.179) P Rc D PT x GT x G Rc 4³ d GD
The (4.179) is known as the Friis transmission equation and is valid in conditions of maximum transfer of power. The term .½=4³ d/2 is called free space path loss. Later, we will use the following definition: 2 ½ P0 D PT x GT x G Rc (4.180) 4³ which represents the power of a signal received at the distance of 1 meter from the transmitter. In any case, (4.179) does not take into account attenuation due to rain or other environmental factors, nor the possibility that the antennas may not be correctly positioned. The available attenuation of the medium, expressed in dB, is PT x D 32:4 C 20 log10 djkm C 20 log10 f 0 jMHz .GT x /d B .G Rc /d B P Rc (4.181) 2 where 32:4 D 10 log10 .4³=c/ , d is expressed in km, f 0 in MHz, and .GT x /d B and .G Rc /d B in dB. It is worthwhile making the following observations on the attenuation ad expressed by (4.181): a) it increases with distance as log10 d, whereas for metallic transmission lines the dependency is linear (see (4.140)); b) it increases with frequency as log10 f 0 . For GT x D G Rc D 1, .ad /d B coincides with the free space path loss. .ad /d B D 10 log10
4.6. Radio links
299
Equivalent circuit at the receiver We redraw in Figure 4.28 the electrical equivalent circuit at the receiver, using a slightly different notation from that of Figure 4.10. The antenna produces the desired signal s, and w represents the total noise due to the antenna and the amplifier. The amplifier has a bandwidth B around the carrier frequency f 0 . The spectral density of the open circuit noise voltage is Pw . f / D 2kTw Ri , and the available noise power per unit of frequency is pw . f / D .k=2/Tw . The effective noise temperature at the input is Tw D T S C .F 1/T0 , where T S is the effective noise temperature of the antenna, and T A D .F 1/T0 is the noise temperature of the amplifier; T0 is the room temperature and F is the noise figure of the amplifier. From (4.92), for matched input and output circuits, the signal-to-noise ratio at the amplifier output is equal to 3D
available power of received desired signal P Rc D kTw B available power of effective input noise
(4.182)
We note that there are two noise sources, introduced by the antenna (w S ) and by the receiver (w A ). The noise temperature of the antenna depends on the direction in which the antenna is pointed: for example T S;Sun > T S;atmosphere
(4.183)
Multipath It is useful to study the propagation of a sinusoidal signal hypothesizing that the one-ray model is adequate, which implies using a directional antenna. Let sT x be a narrowband
Figure 4.28. Electrical equivalent circuit at the receiver.
300
Chapter 4. Transmission media
transmitted signal, that is sT x .t/ D Re[A T x e j2³ f 0 t ]
(4.184)
The received signal at a distance d from the transmitter is given by s Rc .t/ D Re[A Rc e j2³ f 0 .t−1 / ] D Re[A Rc e j' Rc e j2³ f 0 t ]
(4.185)
where −1 D d=c denotes the propagation delay, A Rc is the amplitude of the received signal, and ' Rc D 2³ f 0 −1 D 2³ f 0 d=c is the phase of the received signal. Using the definition (1.150) of h .a/ .t/, the radio channel associated with (4.185) has impulse response ½ A Rc .a/ h .− −1 / (4.186) gCh .− / D Re AT x that is the channel attenuates the signal and introduces a delay equal to −1 . Choosing f 0 as the carrier frequency, the baseband equivalent of gCh is given by7 .bb/ gCh .− / D
2A Rc j2³ f 0 −1 e Ž.− −1 / AT x
Limited to signals sT x of the type (4.184), (4.186) can be rewritten as ½ A Rc j' Rc .a/ e h .− / gCh .− / D Re AT x
(4.187)
(4.188)
Thus, (4.188) indicates that the received signal exhibits a phase shift of ' Rc D 2³ f 0 −1 with respect to the transmitted signal, because of the propagation delay. As the propagation delay is given by − D d=c, the delay per unit of distance is equal to 3.3 ns/m. As the power decreases with the square of the distance between transmitter and receiver, the amplitude of the received signal decreases linearly with the distance, hence A Rc / A T x =d; in particular, if A0 is the amplitude of the received signal at the distance of 1 meter from the transmitter, then A Rc D A0 =d, and the power of the received signal is given by P Rc D A2Rc =2. Reflection and scattering phenomena imply that the one-ray model is applicable only to propagation in free space, and is not adequate to characterize radio channels, such as for example the channel between a fixed radio station and a mobile receiver. We will now consider the propagation of a narrowband signal in the presence of reflections. If a ray undergoes a reflection caused by a surface, a part of its power is absorbed by the surface while the rest is re-transmitted in another direction. If the i-th ray has undergone K i reflections before arriving at the receiver and if ai j is a complex number denoting the reflection coefficient of the j-th reflection of the i-th ray, the total reflection factor is ai D
Ki Y
ai j
(4.189)
jD1
7
.bb/
The constraint that GCh . f / D 0 for f < f 0 was removed because the input already satisfies the condition .bb/ ST x . f / D 0 for f < f 0 .
4.6. Radio links
301
Therefore signal amplitudes, corresponding to rays that are not the direct or line of sight ray, undergo an attenuation due to reflections that is added to the attenuation due to distance. The total phase shift asociated with each ray is obtained by summing the phase shifts introduced by the various reflections and the phase shift due to the distance traveled. If Nc is the number of paths and di is the distance traveled by the i-th ray, extending the channel model (4.186) we get " # Nc ai .a/ A0 X gCh .− / D Re h .− −i / (4.190) A T x i D1 di where −i D di =c is the delay of the i-th ray. The complex envelope of the channel impulse response (4.190) around f 0 is equal to .bb/ .− / D gCh
Nc ai j2³ f 0 −i 2A0 X e Ž.− −i / A T x i D1 di
(4.191)
We note that the only difference between the passband model and its baseband equivalent is constituted by the additional phase term e j2³ f 0 −i for the i-th ray. Limited to narrowband signals, extending the channel model (4.188) to the case of many reflections, the received signal can still be written as s Rc .t/ D Re[A Rc e j' Rc e j2³ f 0 t ]
(4.192)
where now amplitude and phase are given by A Rc e j' Rc D A0
Nc X ai j'i e d i D1 i
(4.193)
with 'i D 2³ f 0 −i . Let Ai and i be amplitude and phase, respectively, of the term A0 .ai =di /e j'i ; from (4.193) the resulting signal is given by the sum of Ai e j i , i D 1; : : : ; Nc , as represented in Figure 4.29. As P0 D A20 =2, the received power is
ψ3 ARc
ψ2 φR
c
ψ
1
Figure 4.29. Representation of (4.193) in the complex plane.
302
Chapter 4. Transmission media
P Rc
þ þ2 Nc þX ai j'i þþ þ D P0 þ e þ þ i D1 di þ
(4.194)
and is independent of the total phase of the first ray. We will now give two examples of application of the previous results. Example 4.6.1 (Power attenuation as a function of distance in mobile radio channels) We consider two antennas, one transmitting and the other receiving, with height respectively h 1 and h 2 , that are placed at a distance d. Moreover, it is assumed that d × h 1 and d × h 2 (see Figure 4.30). We consider the case of two paths: one is the straight path (LOS), and the other is reflected by the earth surface with reflection coefficient a1 D 1, i.e. the earth acts as an ideal reflecting surface and does not absorb power. Observing (4.194), and considering that for the above assumptions the lengths of the two paths are both approximately equal to d, the received power is given by P0 (4.195) P Rc ' 2 j1 e j1' j2 d where 1' D 2³ f 0 1d=c D 2³ 1d=½ is the phase shift between the two paths, and 1d D 2h 1 h 2 =d is the difference between the lengths of the two paths. For small values of 1' we obtain: j1 e j1' j2 ' j1'j2 D 16³ 2
h 21 h 22 ½2 d 2
(4.196)
from which, by substituting (4.180) in (4.195), we get h 21 h 22 P0 2 j1'j D P G G (4.197) T x T x Rc d2 d4 We note that the received power decreases as the fourth-power of the distance d, that is 40 dB/decade instead of 20 dB/decade as in the case of free space. Therefore the law of power attenuation as a function of distance changes in the presence of multipath with respect to the case of propagation in free space. P Rc D
Example 4.6.2 (Fading caused by multipath) Consider again the previous example, but assume that transmitter and receiver are positioned in a room, so that the inequalities between the antenna heights and the distance d are no LOS h1
h2
d Figure 4.30. Two-ray propagation model.
4.6. Radio links
303
longer valid. It is assumed, moreover, that the rays that reach the receive antenna are due, respectively, to LOS, reflection from the floor, and reflection from the ceiling. As a result the received power is given by þ þ2 3 þX ai e j'i þþ þ (4.198) P Rc D P0 þ þ þ i D1 di þ where the reflection coefficients are a1 D 1 for the LOS path, and a2 D a3 D 0:7. With these assumptions, one finds that the power decreases with the distance in an erratic way, in the sense that by varying the position of the antennas the received power presents fluctuations of about 20ł30 dB. In fact, depending on the position, the phases of the various rays change and the sum in (4.193) also varies: in some positions all rays are aligned in phase and the received power is high, whereas in others the rays cancel each other and the received power is low. In the previous example this phenomenon is not observed because the distance d is much larger than the antenna heights, and the phase difference between the two rays remains always small.
4.6.3
Doppler shift
In the presence of relative motion between transmitter and receiver, the frequency of the received signal undergoes a shift with respect to the frequency of the transmitted signal, known as a Doppler shift. We now analyze in detail the Doppler shift. With reference to Figure 4.31, we consider a transmitter radio Tx and a receiver radio that moves with speed v p from a point P to a point Q. The variation in distance between the transmitter and the receiver is 1` D v p 1t cos , where v p is the speed of the receiver relative to the transmitter, 1t is the time required for the receiver to go from P to Q, and is the angle of incidence of the signal with respect to the direction of motion ( is assumed to be the same in P and in Q). The phase variation of the received signal because of the different path length in P and Q is 1' D
2³ v p 1t 2³ 1` D cos ½ ½
(4.199)
and hence the apparent change in frequency or Doppler shift is fs D
vp 1 1' D cos 2³ 1t ½
(4.200)
Tx
θ
∆l P
Rc Q
Figure 4.31. Illustration of the Doppler shift.
304
Chapter 4. Transmission media
This implies that if a narrowband signal given by (4.184) is transmitted, the received signal is s Rc .t/ D Re[A Rc e j2³. f 0 f s /t ]
(4.201)
The (4.200) relates the Doppler shift to the speed of the receiver and the angle ; in particular, for D 0 we get f s D 9:259 104 v p jkm=h f 0 jMHz (Hz) (4.202) þ where v p þkm=h is the speed of the mobile in km/h, and f 0 jMHz is the carrier frequency in MHz. For example, if v p D 100 km/h and f 0 D 900 MHz we have f s D 83 Hz. We note that if the receiver moves towards the transmitter the Doppler shift is positive, if it moves away from the transmitter the Doppler shift is negative. We now consider a narrowband signal transmitted in an indoor environment8 where the signal received by the antenna is given by the contribution of many rays, each with a different length. If the signal propagation were taking place through only one ray, the received signal would undergo only one Doppler shift. But according to (4.200) the frequency shift f s depends on the angle . Therefore, because of the different paths, the received signal is no longer monochromatic, and we speak of a Doppler spectrum to indicate the spectrum of the received signal around f 0 . This phenomenon manifests itself also if both the transmitter and the receiver are static, but a person or an object moves modifying the signal propagation. The Doppler spectrum is characterized by the Doppler spread, which measures the dispersion in the frequency domain that is experienced by a transmitted sinusoidal signal. It is intuitive that the more the characteristics of the radio channel vary with time, the larger the Doppler spread will be. An important consequence of this observation is that the convergence time of algorithms used in receivers, e.g., to perform adaptive equalization, must be much smaller than the inverse of the Doppler spread of the channel, thus enabling the adaptive algorithms to follow the channel variations. Example 4.6.3 (Doppler shift) Consider a transmitter that radiates a sinusoidal carrier at the frequency of f 0 D 1850 MHz. For a vehicle traveling at 96.55 km/h (26.82 m/s), we want to evaluate the frequency of the received carrier if the vehicle is moving: a) approaching the transmitter, b) going away from the transmitter, c) perpendicular to the direction of arrival of the transmitted signal. The wavelength is 3 ð 108 c D D 0:162 m f0 1850 ð 106 a) The Doppler shift is positive; the received frequency is 26:82 f Rc D f 0 C f s D 1850 ð 106 C D 1850:000166 MHz 0:162 ½D
8
(4.203)
(4.204)
The term indoor is usually referred to areas inside buildings, possibly separated by walls of various thickness, material, and height. The term outdoor, instead, is usually referred to areas outside of buildings: these environments can be of various types, for example, urban, suburban, rural, etc.
4.6. Radio links
305
b) The Doppler shift is negative; the received frequency is 26:82 f Rc D f 0 f s D 1850 ð 106 D 1849:999834 MHz 0:162 c) In this case cos. / D 0; therefore there is no Doppler shift.
4.6.4
(4.205)
Propagation of wideband signals
For a wideband signal with spectrum centered around the carrier frequency f 0 , the channel model (4.191) is still valid; we rewrite the channel impulse response as a function of both the time variable t and the delay − for a given t: .bb/ gCh .t; − / D
Nc X
gi .t/Ž.− −i .t//
(4.206)
i D1
where gi represents the complex-valued gain of the i-th ray that arrives with delay −i . For a given receiver location, (4.206) models the channel as a linear filter having time-varying impulse response, where the channel variability is due to the motion of transmitter and/or receiver, or to changes in the surrounding environment, or to both factors. If the channel is time-invariant, or at least it is time-invariant within a short time interval, in this time interval the impulse response is only a function of − . The transmitted signal undergoes three phenomena: a) fading of some gains gi due to multipath, which implies rapid changes of the received signal power over short distances (of the order of the carrier wavelength) and brief time intervals, b) time dispersion of the impulse response caused by diverse propagation delays of multipath rays, c) Doppler shift, which introduces a random frequency modulation that is in general different for different rays. In a digital transmission system the effect of multipath depends on the relative duration of the symbol period and the channel impulse response. If the duration of the channel impulse response is very small with respect to the duration of the symbol period, i.e. the transmitted signal is narrowband with respect to the channel, then the one-ray model is a suitable channel model; if the gain of the single ray varies in time we speak of a flat fading channel. Otherwise, an adequate model must include several rays: in this case if the gains vary in time we speak of a frequency selective fading channel. Neglecting the absolute delay −1 .t/, letting −Q2 D −2 −1 a simple two-ray radio channel model has impulse response .bb/ .t; − / D g1 .t/Ž.− / C g2 .t/Ž.− −Q2 .t// gCh
(4.207)
At a given instant t, the channel is equivalent to a filter with impulse response illustrated in Figure 4.32 and frequency response given by:9 .bb/ GCh .t; f / D g1 .t/ C g2 .t/e j2³ f −Q2 .t/
9
(4.209)
If we normalize the coefficients with respect to g1 , (4.209) becomes .bb/
GCh . f / D 1 C b e j2³ f − where b is a complex number. In the literature (4.208) is called Rummler model of the radio channel.
(4.208)
306
Chapter 4. Transmission media
Figure 4.32. Physical representation and model of a two-ray radio channel, where g1 and g2 are assumed to be positive.
It is evident that the channel has a selective frequency behavior, as the attenuation depends on frequency. For g1 and g2 real-valued, from (4.209) the following frequency response is obtained þ2 þ þ þ .bb/ (4.210) þGCh .t; f /þ D g12 .t/ C g22 .t/ C 2g1 .t/g2 .t/ cos.2³ f −Q2 .t// shown in Figure 4.32. In any case, the signal distortion depends on the signal bandwidth in comparison to 1=−Q2 . Going back to the general case, for wideband communications, rays with different delays are assumed to be independent, that is they do not interact with each other. In this case from (4.206) the received power is P Rc D PT x
Nc X
jgi j2
(4.211)
i D1
From (4.211) we note that the received power is given by the sum of the squared amplitude of all the rays. Conversely, in the transmission of narrowband signals the received power is the square of the vector amplitude resulting from the vector sum of all the received rays. Therefore, for a given transmitted power, the received power will be lower for a narrowband signal as compared to a wideband signal.
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Channel parameters in the presence of multipath To study the performance of mobile radio systems it is convenient to introduce a measure of the channel dispersion in the time domain known as multipath delay spread (MDS). The MDS is the measure of the time interval that elapses between the arrival of the first and the last ray; the simplest measure is the delay time that it takes for the amplitude of the ray to decrease by x dB below the maximum value; this time is also called excess delay spread (EDS). However, the EDS is not a very meaningful parameter, because channels that exhibit considerably different distributions of the gains gi may have the same value of EDS. A parameter that is normally used to define conveniently the MDS of the channel is the root-mean square (rms) delay spread, −r ms , which corresponds to the second-order central moment of the channel impulse response, that is −r ms D
q −2 −2
(4.212)
where Nc X
−n D
jgi j2 −in
i D1 Nc X
n D 1; 2
(4.213)
2
jgi j
i D1
The above formulae give the rms delay spread for an instantaneous channel impulse response. With reference to the time-varying characteristics of the channels, we use the (average) rms delay spread − r ms obtained by substituting in (4.213) in place of jgi j2 its expectation. In this case − r ms measures the mean time dispersion that a signal undergoes because of multipath. Typical values of (average) rms delay spread are of the order of µs in outdoor mobile radio channels, and of the order of some tenths of ns in indoor channels. We define as power delay profile, also called delay power spectrum or multipath intensity profile, the expectation of the squared amplitude of the channel impulse response, E[jgi j2 ], as a function of delay −i . In Table 4.5 power delay profiles are given for some typical channels.
Statistical description of fading channels The most widely used statistical description of the gains fgi g is given by g1 D C C gQ 1 gi D gQi
i D1 i D 2; : : : ; Nc
(4.214)
where C is a real-valued constant and gQi is a complex-valued random variable with zero mean and Gaussian distribution (see Example 1.9.3 on page 67). In other words, whereas the first ray contains a direct (deterministic) component in addition to a random component, all the other rays are assumed to have only a random component: therefore the distribution
308
Chapter 4. Transmission media
Table 4.5 Values of E[jgi j2 ] (in dB) and −i (in ns) for three typical channels.
Standard GSM
Indoor offices
Indoor business
−i
E[jgi j2 ]
−i
E[jgi j2 ]
−i
E[jgi j2 ]
0 200 500 1600 2300 5000
3:0 0 2:0 6:0 8:0 10:0
0 50 150 325 550 700
0:0 1:6 4:7 10:1 17:1 21:7
0 50 150 225 400 525 750
4:6 0 4:3 6:5 3:0 15:2 21:7
of jgi j will be a Rice distribution for jg1 j and a Rayleigh distribution for jgi j, i 6D 1. In p 2 particular, letting gNi D gi = E[jgi j ], we have p pjgN 1 j .a/ D 2.1 C K / a exp[K .1 C K /a 2 ]I0 [2a K .1 C K /]1.a/ (4.215) 2 pjgNi j .a/ D 2a ea 1.a/ where I0 is the modified Bessel function of the first type and order zero, Z ³ 1 e x cos Þ dÞ I0 .x/ D 2³ ³
(4.216)
The probability density (4.215) is given in Figure 4.33 for various values of K . In (4.214) the phase of gQi is uniformly distributed in [0; 2³ /. For a one-ray channel model, the parameter K D C 2 =E[jgQ 1 j2 ], known as the Rice factor, is equal to the ratio between the power of the direct component and the power of the reflected and/or scattered component. In general for a model with more rays we take K D C 2 =Md , where Md is the P c statistical power of all reflected and/or scattered components, that is Md D iND1 E[jgQ i j2 ]. Assuming that the power delay profile is normalized such that Nc X
E[jgi j2 ] D 1
(4.217)
i D1
p we obtain C D K =.K C 1/. Typical reference values for K are 3 and 10 dB. If C D 0, i.e. no direct component exists, it is K D 0, and the Rayleigh distribution is obtained for all the gains fgi g. For K ! 1, i.e. with no reflected and/or scattered components and, hence, C D 1, we find the model having only the deterministic component. To justify the Rice model for jg1 j we consider the transmission of a sinusoidal signal (4.184). In this case the expression of the received signal is given by (4.192), which we rewrite as follows: s Rc .t/ D [gQ 1;I .t/ C C] cos 2³ f 0 t gQ 1;Q .t/ sin 2³ f 0 t
(4.218)
4.6. Radio links
309
2 K=10 1.8
1.6
K=5 1.4
(a)
1.2
p
1
|g |
K=2 1
K=0 0.8
0.6
0.4
0.2
0
0
0.5
1
1.5 a
2
2.5
3
Figure 4.33. The Rice probability density function for various values of K. The Rayleigh density function is obtained for K D 0.
where C represents the contribution of the possible direct component of the propagation signal, and gQ 1;I and gQ 1;Q are due to the scattered component. As the gains gQ1;I and gQ 1;Q are given by the sum of a large number of random components, they can be approximated by independent Gaussian random processes with zero mean. The instantaneous envelope of the received signal is then given by q 2 .t/ [gQ 1;I .t/ C C]2 C gQ 1;Q (4.219) which, in the assumption just formulated, is a Rice random variable for each instant t.
4.6.5
Continuous-time channel model
The channel model previously studied is especially useful for system simulations, as will be discussed later. A general continuous-time model is now presented. Assuming that the signal propagation occurs through a large number of paths, which in turn are subject to a very large number of random phenomena, the (baseband equivalent) channel impulse response can be represented with good approximation as a time-varying complex-valued Gaussian random process g.t; − /. In particular g.t; − / represents the channel output at the instant t in response to an impulse applied at the instant .t − /. We now evaluate the autocorrelation function of the impulse response evaluated at two different instants and two different delays, rg .t; t 1t;−; − 1− / D E[g.t; − /g Ł .t 1t; − 1− /]
(4.220)
310
Chapter 4. Transmission media
According to the model known as the wide-sense stationary uncorrelated scattering (WSSUS), the values of g for rays that arrive with different delays are uncorrelated, and g is stationary in t. Therefore we have rg .t; t 1t;−; − 1− / D rg .1t;− /Ž.1− /
(4.221)
In other words, the autocorrelation is non-zero only for impulse responses that are considered for the same delay time. Moreover, as g is stationary in t, if the delay time is the same, the autocorrelation only depends on the difference of the times at which the two impulse responses are evaluated.
Power delay profile For 1t D 0 we define the function M.− / D E[jg.t; − /j2 ], that is called channel power delay profile and represents the statistical power of the gain g.t; − / for a given delay − . For a Rayleigh channel model, three typical curves are now given for M.− /, where − r ms is the parameter defined in (4.225): 1. Two rays, with equal power M.− / D 12 Ž.− / C 12 Ž.− 2− r ms /
(4.222)
2. Gaussian, unilateral r M.− / D
2 1 − 2 =.2− r2ms / e ³ − r ms
− ½0
(4.223)
3. Exponential, unilateral M.− / D
1 − r ms
e−=.− r ms /
− ½0
(4.224)
The measure of the set of values − for which M.− / is above a certain threshold is called (average) excess delay spread of the channel. As in the case of the discrete channel model previously studied, we define the (average) rms delay spread as Z 1 .− − /2 M.− / d− 1 Z 1 (4.225) − r2ms D M.− / d− 1
where −D
Z
1
− M.− / d−
(4.226)
1
The inverse of the (average) rms delay spread is called the coherence bandwidth of the channel.
4.6. Radio links
311
For digital transmission over such channels, we observe that if − r ms is of the order of 20% of the symbol period, or larger then signal distortion is non-negligible. Equivalently, if the coherence bandwidth of the channel is lower than 5 times the modulation rate of the transmission system, then we speak of a frequency selective fading channel, otherwise the channel is flat fading. However, in the presence of flat fading the received signal may vanish completely, whereas frequency selective fading produces several replicas of the transmitted signal at the receiver, so that a suitably designed receiver can recover the transmitted information. Example 4.6.4 (Power delay profile) We compute the average rms delay spread for the multipath delay profile of Figure 4.34, and determine the coherence bandwidth, defined as Bc D 5−¯r1ms . From (4.213) we have .1/.5/ C .0:1/.1/ C .0:1/.2/ C .0:01/.0/ D 4:38 µ s 0:01 C 0:1 C 0:1 C 1
(4.227)
.1/.5/2 C .0:1/.1/2 C .0:1/.2/2 C .0:01/.0/ D 21:07 .µ s/2 −N2 D 0:01 C 0:1 C 0:1 C 1
(4.228)
−N D and
Therefore we get −Nr ms D
p 21:07 .4:38/2 D 1:37 µ s
(4.229)
Consequently the coherence bandwidth of the channel is equal to Bc D 146 kHz.
Doppler spectrum We now analyze the WSSUS channel model with reference to time variations. First we introduce the correlation function of the channel frequency response taken at instants t and t 1t, and, respectively, at frequencies f and f 1 f , rG .t; t 1t; f; f 1 f / D E[G.t; f /G Ł .t 1t; f 1 f /]
M( τ ) (dB)
0 -10 -20 -30 0
1
2
5
τ ( µs)
Figure 4.34. Multipath delay profile.
(4.230)
312
Chapter 4. Transmission media
Substituting in (4.230) the relation G.t; f / D
Z
C1
g.t; − / e j2³ f − d−
(4.231)
1
we find that rG depends only on 1t and 1 f ; moreover it holds Z C1 rG .1t;1 f / D rg .1t;− / e j2³.1 f /− d−
(4.232)
1
that is rG .1t;1 f / is the Fourier transform of rg .1t;− /. The Fourier transform of rG is given by Z 1 PG .½; 1 f / D rG .1t;1 f /e j2³ ½.1t/ d.1t/ (4.233) 1
The time variation of the frequency response is measured by PG .½; 0/. Now we introduce the Doppler spectrum D.½/, which represents the power of the Doppler shift for different values of the frequency ½. We recall that the Doppler shift is caused by the motion of terminals or surrounding objects. We define D.½/ as the Fourier transform of the autocorrelation function of the impulse response, in correspondence of the same delay − , evaluated at two different instants, that is:10 Z C1 rg .1t;− / j2³ ½1t e D.½/ D d.1t/ (4.234) 1 rg .0;− / The term rg .0;− / in (4.234) represents a normalization factor such that Z C1 D.½/ d½ D 1
(4.235)
1
We note that (4.234) implies that rg .1t;− / is a separable function, rg .1t;− / D d.1t/ Ð rg .0;− / D d.1t/ M.− /
(4.236)
where d.1t/ D F 1 [D.½/], with d.0/ D 1 and M.− / is the power delay profile, so that Z C1 M.− / d− D 1
(4.237)
(4.238)
1
With the above assumptions the following equality holds: D.½/ D PG .½;0/
(4.239)
10 In very general terms, we could have a different Doppler spectrum for each path, or gain g.t; − /, of the channel.
4.6. Radio links
313
Therefore, D.½/ can also be obtained as the Fourier transform of rG .1t;0/, which in turn can be determined by transmitting a sinusoidal signal (hence 1 f D 0) and estimating the autocorrelation function of the amplitude of the received signal. The maximum frequency f d of the Doppler spectrum support is called the Doppler spread of the channel and gives a measure of the fading rate of the channel. Another measure of the support of D.½/ can be obtained through the rms Doppler spread or second order central moment of the Doppler spectrum. The inverse of the Doppler spread is called coherence time: it gives a measure of the time interval within which a channel can be assumed to be time invariant or static. Let T be the symbol period in a digital transmission system; we usually say that the channel is fast fading if f d T > 102 , and slow fading if f d T < 103 .
Doppler spectrum models A widely used model, known as the Jakes model or classical Doppler spectrum, to represent the Doppler spectrum is due to Clarke. If f d denotes the Doppler spread, then 8 1 < 1 p j f j fd ³ f D. f / D (4.240) d 1 . f = f d /2 : 0 otherwise For the channel model (4.206), the corresponding autocorrelation function of the channel impulse response is given by rg .1t;− / D
Nc X
J0 .2³ f d 1t/ M.−i / Ž.− −i /
(4.241)
i D1
where J0 is the Bessel function of the first type and order zero. The model of the Doppler spectrum described above agrees with the experimental results obtained for mobile radio channels. For indoor radio channels, thanks to the study conducted by a special commission (JTC), it was demonstrated that the Doppler spectrum can be modelled as 8 < 1 j f j fd (4.242) D. f / D 2 f d : 0 elsewhere with a corresponding autocorrelation function given by rg .1t;− / D
Nc X
sinc.2 f d 1t/ M.−i / Ž.− −i /
(4.243)
i D1
A further model assumes that the Doppler spectrum is described by a second or third-order Butterworth filter with the 3 dB cutoff frequency equal to f d .
Shadowing The simplest relation between average transmitted power and average received power is P Rc D
P0 dÞ
(4.244)
314
Chapter 4. Transmission media
where Þ is equal to 2 for propagation in free space and to 4 for the simple 2-ray model described before. For indoor and urban outdoor radio channels the relation depends on the environment, according to the number of buildings, their dimensions, and also the material used for their construction; in general, however, variations of the average received power are lower in outdoor environments than in indoor environments. Shadowing takes into account the fact that the average received power may present fluctuations around the value obtained by deterministic models. These fluctuations are modelled as a log-normal random variable, that is e¾ , where ¾ is a Gaussian random variable with zero mean and variance ¦¾2 . If P Rc is the average received power obtained by deterministic rules, in the presence of shadowing it becomes e¾ P Rc ; in practice shadowing provides a measure of the adequacy of the adopted deterministic model. A propagation model that completely ignores any information on land configuration, and therefore is based only on the distance between transmitter and receiver, has a shadowing with ¦.¾ /d B D 12 dB. The relation between ¦¾ and ¦.¾ /d B is ¦¾ D 0:23¦.¾ /d B . Improving the accuracy of the propagation model, for example, by using more details regarding the environmental configuration, the shadowing can be reduced; in case we had an enormous amount of topographic data and the means to elaborate them, we would have a model with ¦¾ D 0. Hence, shadowing should be considered in the performance evaluation of mobile radio systems, whereas for the correct design of a network it is good practice to make use of the largest possible quantity of topographic data.
Final remarks A signal that propagates in a radio channel for mobile communications undergoes a type of fading that depends on the signal as well as on the channel characteristics. In particular, whereas the delay spread due to multipath leads to dispersion in the time domain and therefore frequency selective fading, the Doppler spread causes dispersion in the domain of the variable ½ and therefore time selective fading. The first type of fading can be divided into flat fading and frequency selective fading. In the first case the channel has a constant gain; in other words, the inverse of the transmitted signal bandwidth is much larger than the delay spread of the channel and g.t; − / can be approximated by a delta function, with random amplitude and phase, centered at − D 0. In the second case instead the channel has a time-varying frequency response within the passband of the transmitted signal and consequently the signal undergoes frequency selective fading; these conditions occur when the inverse of the transmitted signal bandwidth is of the same order or smaller than the delay spread of the channel. The received signal consists of several attenuated and delayed versions of the transmitted signal. A channel can be fast fading or slow fading. In a fast fading channel, the impulse response of the channel changes within a symbol period, that is the coherence time of the channel is smaller than the symbol period; this condition leads to signal distortion, which increases with increasing Doppler spread. Usually there are no remedies to compensate for such distortion unless the symbol period is decreased; on the other hand, this choice leads to larger intersymbol interference. In a slow fading channel, the impulse response changes much more slowly with respect to the symbol period. In general, the channel can be assumed as time invariant for a time interval that is proportional to the inverse of the Doppler spread.
4.6. Radio links
4.6.6
315
Discrete-time model for fading channels
Our aim is to approximate a transmission channel defined in the continuous-time domain by a channel in the discrete-time domain characterized by sampling period TQ . We immediately notice that the various delays in (4.206) must be multiples of TQ and consequently we need to approximate the delays of the power delay profile (see, e.g., Table 4.5). Starting from a continuous-time model of the power delay profile (see, e.g., (4.224)), we need to obtain a sampled version of M.− /. The discrete-time model of the radio channel is represented, as illustrated in Figure 4.35, by a time-varying linear filter where the coefficient gi corresponds to the complex gain of the ray with delay i TQ , i D 0; 1; : : : ; Nc 1; in the case of flat fading we choose Nc D 1. If the channel is time invariant ( f d D 0), all coefficients fgi g, i D 0; : : : ; Nc 1, are constant, and are obtained as realizations of Nc random variables. In general, however, fgi g are random processes. To generate each process gi .kTQ /, the scheme of Figure 4.36 is used, where wN i .`T P / is complex-valued Gaussian white noise with zero mean and unit variance, h ds is a narrrowband filter that produces a signal gi0 with the desired Doppler spectrum, and h int is an interpolator filter (see Section 1.A.7). f d TP 1=5. The interpolator output signal Usually we choose f d TQ − 1, and 1=10 p is then multiplied by a constant ¦i D M.i TQ /, which imposes the desired power delay profile. x(kTQ) TQ
TQ
g (kTQ)
g (kTQ)
0
TQ
g (kTQ )
1
N-1 c
+ y(kTQ) Figure 4.35. Discrete time model of a radio channel.
Figure 4.36. Model to generate the i-th coefficient of a time-varying channel.
316
Chapter 4. Transmission media
If the channel model includes a deterministic component for the ray with delay −i D i TQ , a constant Ci must be added to the random component gNi . Furthermore, if the channel model includes a Doppler shift f si for the i-th branch, then we need to multiply the term Ci C gNi by the exponential function exp. j2³ f si kTQ /. Observing (4.211), to avoid modifying the average transmitted power, the coefficients fgi g, i D 0; 1; : : : ; Nc 1, are scaled, so that NX c 1
E[jgi .kTQ /j2 ] D 1
(4.245)
i D0
For example, the above condition is satisfied if each signal gi0 has unit statistical power11 and f¦i g satisfy the condition NX c 1
.¦i2 C Ci2 / D 1
(4.246)
i D0
Generation of a process with a pre-assigned spectrum The procedure can be generalized for a signal gi0 with a generic Doppler spectrum of the type (4.240) or (4.242) in two ways: 1) implement a filter h ds such that jHds . f /j2 D D. f /, 2) generate a set of N f (at least 10) complex sinusoids with frequencies fš f m g, m D 1; : : : ; N f , in the range from f d to f d . We analyze the two methods. 1) Using a filter. We give the description of h ds for two cases 1.a) Second-order Butterworth filter. Given !d D 2³ f d , where f d is the Doppler spread, the transfer function of the discrete-time filter is c0 .1 C z 1 /2 ! (4.247) Hds .z/ D 2 X n an z 1C nD1
where, defining !0 D tan.!d TP =2/ where TP is the sampling period, we have [18] 2.1 !02 / p 1 C !02 C 2 !0
(4.248)
a2 D
1 C !04 p .1 C !02 C 2 !0 /2
(4.249)
c0 D
1 4
a1 D
.1 C a1 C a2 /
(4.250)
11 Based on the Example 1.9.10 on page 72, it is M 0 D 1 if M wN i D 1; the equivalent interpolator filter, given by gi
the cascade of h ds and h int , has energy equal to the interpolation factor T P =TQ .
4.6. Radio links
317
The filter output gives gi0 .`T P / D a1 gi0 ..` 1/T P / a2 gi0 ..` 2/T P / C c0 .wQ i .`T P / C 2wQ i ..` 1/T P / C wQ i ..` 2/T P //
(4.251)
1.b) IIR filter with classical Doppler spectrum. Now h ds is implemented as the cascade of two filters. The first, Hds 1 .z/, is an FIR shaping filter with amplitude characteristic of the frequency response given by the square root of the function in (4.240). The second, Hds 2 .z/, is a Chebychev lowpass filter, with cutoff frequency f d . Table 4.6 reports values of the overall filter parameters for f d TP D 0:1 [19]. 2) Using sinusoidal signals. Let gi0 .`T P / D
Nf X
Ai;m [e j .2³ f m `TP C'i;m / e j8i;I C e j .2³ f m `TP C'i;m / e j8i;Q ]
(4.252)
mD1
The spacing between the different frequencies is 1 f m ; letting f 1 D 1 f 1 =2, for m > 1 we have f m D f m1 C 1 f m . Each 1 f m can be chosen as a constant, 1 fm D Z
fd
or, defining K d D
fd Nf
(4.253)
D1=3 . f /d f , as
0
1 fm D
Kd N f D1=3 . f m /
m D 1; : : : ; N f
Table 4.6 Parameters of an IIR filter which implements a classical Doppler spectrum. [From Anastasopoulos and Chugg (1997). c 1997 IEEE.]
Hds .z/ D B.z/=A.z/
f d TP D 0:1
fan g; n D 0; : : : ; 11 : 1:0000 e C 0 4:4153 e C 0 8:6283 e C 0 9:4592 e C 0 6:1051 e C 0 1:3542 e C 0 3:3622 e C 0 7:2390 e C 0 7:9361 e C 0 5:1221 e C 0 1:8401 e C 0 2:8706 e 1 fbn g; n D 0; : : : ; 21 : 1:3651 e 4 8:1905 e 4 2:0476 e 3 2:7302 e 3 2:0476 e 3 9:0939 e 4 6:7852 e 4 1:3550 e 3 1:8067 e 3 1:3550 e 3 5:3726 e 4 6:1818 e 5 7:1294 e 5 9:5058 e 5 7:1294 e 5 2:5505 e 5 1:3321 e 5 4:5186 e 5 6:0248 e 5 4:5186 e 5 1:8074 e 5 3:0124 e 6
(4.254)
318
Chapter 4. Transmission media
(bb)
Figure 4.37. Nine realizations of jgCh .t; − /j for a Rayleigh channel with an exponential power delay profile having − rms D 0:5 T.
Suppose f 0 D 0 and f m D f m1 C 1 f m , m D 1; : : : ; N f , the choice (4.254) corresponds to minimizing the error Nf Z X mD1
fm
. f m f /2 D. f / d f
(4.255)
f m1
The phases 'i;m , 8i;I and 8i;Q are uniformly distributed in [0; 2³ / and statistically independent. This choice for 8i;I and 8i;Q ensures that the real and imaginary parts of gi0 are statistically independent. p The amplitude is given by Ai;m D D. f m /1 f m . If D. f / is flat, by the central limit theorem we can claim that gi0 is a Gaussian process; if instead D. f / presents some frequencies with large amplitude, Ai;m must be generated as a Gaussian random variable with zero mean and variance D. f m /1 f m . In Figure 4.37 are represented nine realizations of the amplitude of the impulse response of a Rayleigh channel obtained by the simulation model of Figure 4.35, for an exponential power delay profile with − r ms D 0:5 T . The Doppler frequency f d was assumed to be zero. We point out that the parameter − r ms provides scarce information on the actual behavior .bb/ of gCh , which can scatter for a duration equal to 4-5 times − r ms .
4.7 4.7.1
Telephone channel Characteristics
Telephone channels, originally conceived for the transmission of voice, today are extensively used also for the transmission of data. Transmission of a signal over a telephone
4.7. Telephone channel
319
channel is achieved by utilizing several transmission media, such as symmetrical transmission lines, coaxial cables, optical fibers, radio, and satellite links. Therefore channel characteristics depend on the particular connection established. As a statistical analysis made in 1983 indicated [2], a telephone channel is characterized by the following disturbances and distortions.
Linear distortion The frequency response GCh . f / of a telephone channel can be approximated by a passband filter with band in the range of frequencies from 300 to 3400 Hz. The plots of the attenuation a. f / D 20 log10 jGCh . f /j
(4.256)
and of the group delay or envelope delay (see (1.149)) −. f / D
1 d arg GCh . f / 2³ d f
(4.257)
are illustrated in Figure 4.38 for two typical channels. The attenuation and envelope delay distortion are normalized by the values obtained for f D 1004 Hz and f D 1704 Hz, respectively.
Noise sources Impulse noise. It is caused by electromechanical switching devices and is measured by the number of times the noise level exceeds a certain threshold per unit of time. Quantization noise. It is introduced by the digital representation of voice signals and is the dominant noise in telephone channels (see Chapter 5). For a single quantizer, the signal-to-quantization noise ratio 3q has the behavior illustrated in Figure 4.39. Thermal noise. It is described in Section 4.2 and is present at a level of 20 ł 30 dB below the desired signal.
Non-linear distortion It is caused by amplifiers and by non-linear A-law and ¼-law converters (see Chapter 5).
Frequency offset It is caused by the use of carriers for frequency up and downconversion. The relation between the channel input x.t/ and output y.t/ is given by ( X . f f off / f >0 Y. f/ D (4.258) X . f C f off / f −3 ) c.k/ D 11
stop
Generally the number of comparisons depends on V and it is at most equal to 2b1 . 2. A second possible implementation, which encodes one bit at a time, is given in Figure 5.29. In this case b 1 comparisons are made: it is as if we were to explore a complete binary tree whose 2b1 leaves represent the output levels. For example, for b D 3, neglecting the sign bit, the code word length is 2, and c.k/ D .c1 ; c0 /. To determine the bits c0 and c1 we can operate as follows: if V < −2 ) c1 D 0
otherwise c1 D 1
if V < −1 C c1 21 1 ) c0 D 0
otherwise c0 D 1
Only two comparisons are made, but the decision thresholds now depend on the choice of the previous bits. 3. The last implementation, which encodes one code word of (b 1) bit at a time, is given in Figure 5.30. In this scheme V is compared simultaneously with the 2b1 quantizer thresholds: the outcome of this comparison is a word of 2b1 bit formed by a sequence of “0” followed by a sequence of “1”; through a logic network this word is mapped to a binary word of b 1 bits that yields the PCM encoding of s.k/. These encoders are called flash converters. We conclude this section explaining that the acronym PCM stands for pulse code modulation. We waited until the end of the section to avoid confusion about the term modulation: in fact, PCM is not a modulation, but rather a coding method.
5.3
Non-uniform quantizers
There are two observations that suggest the choice of a non-uniform quantizer. The first refers to stationary signals with a non-uniform probability density function: for such signals
5.3. Non-uniform quantizers
359
τ1 s(k)=V τ2 b-1
2 τ3
to b-1 decoding logic
(b-1)-bit code word
τ2b-1
Figure 5.30. Flash converter: encoding one word at a time.
uniform quantizers are suboptimum. The second refers to non-stationary signals, e.g., speech, for which the ratio between instantaneous power (estimated over windows of tenths of milliseconds) and average power (estimated over the whole signal) can exhibit variations of several dB; moreover, the variation of the average power over different links is also of the order of 40 dB. Under these conditions a quantizer with nonuniform characteristics, as that depicted for example in Figure 5.31, is more effective because the signal-to-quantization error ratio 3q is almost independent of the instantaneous power. As also illustrated in Figure 5.31, for a non-uniform quantizer the quantization error is large if the signal is large, whereas it is small if the signal is small: as a result the ratio 3q tends to remain constant for a wide dynamic range of the input signal.
Three examples of implementation 1. The characteristic of Figure 5.31 can be implemented directly, for example, with the techniques illustrated in Figures 5.29 and 5.30. 2. As shown in Figure 5.32, a compression function may precede a uniform quantizer: at the decoder it is therefore necessary to have an expansion of the quantized signal. 3. The most popular method, depicted in Figure 5.33, employs a uniform quantizer having a large number of levels, with a step size equal to the minimum step size of the desired non-uniform characteristic. Encoding of the non-uniformly quantized signal yq is obtained by a look-up table whose input is the uniformly quantized value xq . In Section 5.3.1 we will analyze in detail the last two methods.
360
Chapter 5. Digital representation of waveforms
Figure 5.31. Non-uniform quantizer characteristic with L D 8 levels.
5.3.1
Companding techniques
Figure 5.32b illustrates in detail the principle of Figure 5.32a. The signal is first compressed through a non-linear function F, that yields the signal y D F.s/
(5.50)
In Figure 5.32 we assume −sat D 1. If −sat 6D 1 we need to normalize s to −sat . The signal y is uniformly quantized and the code word given by the inverse bit mapper is transmitted. At the receiver the bit mapper gives yq , that must be expanded to yield a quantized version of s sq D F 1 [Q[y]]
(5.51)
This quantization technique takes the name of companding from the steps of compressing and expanding. We find that the ideal characteristics of F[Ð] should be logarithmic, F[s] D ln s We consider the two blocks shown in Figure 5.34.
(5.52)
5.3. Non-uniform quantizers
361
Figure 5.32. (a) Use of a compression function F to implement a non-uniform quantizer; (b) non-uniform quantizer characteristic implemented by companding and uniform quantization. Here −sat D 1 is assumed.
Encoding.
Let s.k/ D e y.k/ sgn[s.k/]
(5.53)
y.k/ D ln js.k/j
(5.54)
that is
362
Chapter 5. Digital representation of waveforms
Figure 5.33. Non-uniform quantizer implemented digitally using a uniform quantizer with small step size followed by a look-up table.
Figure 5.34. Non-uniform quantization by companding and uniform quantization: (a) PCM encoder, (b) decoder.
and assume the sign of the quantized signal is equal to that of s.k/. The quantization of y.k/ yields yq .k/ D Q[y.k/] D ln js.k/j C eq .k/
(5.55)
The value c.k/ is given by the inverse bit mapping of yq .k/ and the sign of s.k/. Decoder. Assuming c.k/ is correctly received, observing (5.55), the quantized version of s.k/ is given by sq .k/ D e yq .k/ sgn[s.k/] D js.k/j sgn[s.k/]eeq .k/
(5.56)
D s.k/eeq .k/ If eq − 1, then eeq .k/ ' 1 C eq .k/
(5.57)
5.3. Non-uniform quantizers
363
and sq .k/ D s.k/ C eq .k/s.k/
(5.58)
where eq .k/s.k/ represents the output error of the system. As eq .k/ is uncorrelated with the signal ln js.k/j, and hence with s.k/ (see (5.28)), we get 3q D
Ms 2 E[eq .k/s 2 .k/]
D
1 1 D 2 Meq E[eq .k/]
(5.59)
where from (5.41) we have that Meq depends only on the quantization step size 1. Consequently 3q does not depend on Ms . We note that a logarithmic compression function generates a signal y with unbounded amplitude, thus an approximation of the logarithmic law is usually adopted. Regulatory bodies have defined two compression functions: 1. A-law (A D 87:56). For −sat D 1, 8 Ajsj 1 > > 0 jsj < 1 C ln.A/ A (5.60) y D F[s] D > 1 1 C ln.Ajsj/ > : jsj 1 1 C ln.A/ A This law, illustrated in Figure 5.35 for two values of A, is adopted in Europe. The sign is considered separately: sgn[y] D sgn[s]
(5.61)
1
0.9
0.8
0.7
A=87.56
F(s)
0.6
0.5 A=1 0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5 s
0.6
Figure 5.35. A-law.
0.7
0.8
0.9
1
364
Chapter 5. Digital representation of waveforms
1
0.9 µ =255 0.8 µ =50 0.7
F(s)
0.6
µ =5
0.5
0.4 µ =0 0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5 s
0.6
0.7
0.8
0.9
1
Figure 5.36. ¼-law.
2. ¼-law (¼ D 255). For −sat D 1, ln.1 C ¼jsj/ (5.62) ln.1 C ¼/ This law, illustrated in Figure 5.36 for four values of ¼, is adopted in the United States and Canada. The compression increases for higher values of ¼; the standard value of ¼ is equal to 255. We note that, for µ s × 1, we have ln[µ s] F[s] D (5.63) ln.1 C ¼/ as in the ideal case (5.54). Similar behavior is exhibited by (5.60). y D F[s] D
Signal-to-quantization error ratio Assuming the quantization error uniform within each decision interval, which is well verified for a uniform input in the interval [−sat ; −sat ], we can see that for ¼-law, considering only the granular error, we have ( ) −sat −sat 2 p .3q /d B D 6:02b C 4:77 20 log10 [ln.1 C ¼/] 10 log10 1 C C 3 ¼¦s ¼¦s (5.64) Curves of 3q versus the statistical power of the input signal are plotted for ¼ D 255 in Figure 5.25. Note that in the saturation region they coincide with the curves obtained for a
5.3. Non-uniform quantizers
365
uniform quantizer with Laplacian input. We emphasize that also in this case 3q increases by 6 dB with the increase of b by one. We also note that, if b D 8, 3q ' 38 dB for a wide range of values of ¦s . An effect not shown in Figure 5.25 is that, by increasing ¼, the plot of 3q becomes “flatter”, but the maximum value decreases. Observation 5.2 In the standard non-linear PCM, a quantizer with 128 levels (7 bit/sample) is employed after the compression; including also the sign we have 8 bit/sample. For a sampling frequency of Fc D 8 kHz, this leads to a bit rate of the system equal to Rb D 64 kbit/s.
Digital compression An alternative method to the compression-quantization scheme is illustrated by an example in Figure 5.37. The relation between s.k/ and yq .k/ is obtained through a first multi-bit (5 in figure) quantization to generate xq ; then we have a mapping of the 5 bits of xq to the 3 bits of yq using the mapper (sign omitted) of Table 5.6. For decoding, for each code word yq we select only one code word xq , which represents the reconstructed value sq . Using the standard compression laws, we need to approximate the compression functions by piecewise linear functions, as shown in Figure 5.39. For encoding, a mapper with 12-bit input and 8-bit output is given in Table 5.7. For decoding, we select for each compressed
Figure 5.37. Distribution of quantization levels for a 3-bit ¼-law quantizer with ¼ D 40.
366
Chapter 5. Digital representation of waveforms
Table 5.6 Example of non-linear PCM from 4 to 2 bits (sign omitted).
Coding of xq
Coding of yq
Coding of sq
0000 0001 0010 0011 0100 0101 1000 1001 1010 1011 1100 1101 1110 1111
00 01
0000 0001
10
0100
11
1011
code word a corresponding linear code word, as given in the third column of Table 5.7. In the literature there are other non-linear PCM tables, that differ in the compression law or in the accuracy of the codes [4].
Signal-to-quantization noise ratio mask We conclude this section by giving in Figure 5.39 two masks that indicate the minimum tolerable values of 3q (dB) for an A-law quantizer (A D 87:6), −sat D 3:14 dBm, and b D 8 (sign included), as a function of ¦s (dBm) for input signals with Gaussian and sinusoidal distribution, respectively; these masks are useful to verify the quantizer performance.
5.3.2
Optimum quantizer in the MSE sense
Assuming we know the probability density function of the input signal s.k/, stationary with variance ¦s2 , we desire to determine the parameters of the non-uniform quantizer that optimizes 3q . The problem, illustrated in Figure 5.40, consists in choosing the decision thresholds ² ¦ − L ; : : : ; −1 ; −0 ; −1 ; : : : ; − L (5.65) − L D 1 − L D C1 2 1
2 1
2
2
and the quantization levels L L (5.66) i D ; : : : ; 1; 1; : : : ; 2 2 that minimize the statistical power of the error (minimum mean-square error criterion) fQi g
Meq D E[.sq .k/ s.k//2 ] D E[.Q[s.k/] s.k//2 ]
(5.67)
5.3. Non-uniform quantizers
367
Figure 5.38. Piecewise linear approximation of the A-law compression function (A D 87:6). The 12-bit encoded input signals are mapped into 8-bit signals. Table 5.7 Non linear PCM from 11 to 7 bits (sign omitted).
Linear code .xq /
Compressed code .yq /
Coding of sq
1WXYZ-----01WXYZ----001WXYZ---0001WXYZ--00001WXYZ-000001WXYZ0000001WXYZ 0000000WXYZ
111WXYZ 110WXYZ 101WXYZ 100WXYZ 011WXYZ 010WXYZ 001WXYZ 000WXYZ
1WXYZ011111 01WXYZ01111 001WXYZ0111 0001WXYZ011 00001WXYZ01 000001WXYZ0 0000001WXYZ 0000000WXYZ
Assuming ps .a/ even, because of the symmetry of the problem we can halve the number of variables to be determined ( by setting L −i D −i i D 1; : : : ; 1 (5.68) 2 −0 D 0 L (5.69) i D 1; : : : ; Qi D Qi 2 and L=2 Z −i X Meq D 2 .Qi a/2 ps .a/ da (5.70) i D1
−i1
368
Chapter 5. Digital representation of waveforms
(a) Gaussian test signal
(b) Sinusoidal test signal Figure 5.39. 3q versus ¦s2 for an A-law quantizer (A D 87:56) and b D 8.
Necessary but not sufficient conditions for minimizing (5.67) are @Meq @−i @Meq @Qi
D0
i D 1; : : : ;
L 1 2
(5.71)
D0
i D 1; : : : ;
L 2
(5.72)
5.3. Non-uniform quantizers
369
τ-3 τ-2 τ-1 τ0
8
τ-4 =-
Q-4
τ1
τ2
τ3
Q-3 Q-2 Q-1 Q1 Q2 Q3
τ4=+
8
ps (a)
a
Q4
Figure 5.40. Decision thresholds and output levels for a particular ps .a/ (b D 3).
From 1 @Meq D .Qi −i /2 ps .−i / .Qi C1 −i /2 ps .−i / 2 @−i
(5.73)
ps .−i /[Qi2 C −i2 2Qi −i Qi2C1 −i2 C 2Qi C1 −i ] D 0
(5.74)
(5.71) gives
that is −i D
Qi C Qi C1 2
(5.75)
.Qi a/ ps .a/ da D 0
(5.76)
Conversely, the equation 1 @Meq D2 2 @Qi
Z
−i
−i1
yields Z Qi D
−i
− Z i1 −i −i1
aps .a/ da (5.77) ps .a/ da
In other words, (5.75) establishes that the optimal threshold lies in the middle of the interval between two adjacent output values, and (5.77) sets Qi as the centroid of ps .Ð/ in the interval [−i 1 ; −i ]. These two rules are illustrated in Figure 5.41.
Max algorithm We present now the Max algorithm to determine the decision thresholds and the optimum quantization levels. 1. Fixed Q1 “at random”, we use (5.77) to get −1 from the integral equation Z −1 aps .a/ da −0 Q1 D Z −1 ps .a/ da −0
(5.78)
370
Chapter 5. Digital representation of waveforms
ps(a)
τi-1
τi
τi+1
Qi
a
Qi+1
Figure 5.41. Optimum decision thresholds and output levels for a given ps .a/.
2. From (5.75) we obtain Qi C1 D 2−i C Qi for i D 1. 3. We use (5.77) to obtain −i C1 by the equation Z −iC1 aps .a/ da −i Qi C1 D Z −iC1 ps .a/ da
(5.79)
−i
The procedure is iterated for i D 2; 3; : : : ; .L=2/ 2. For i D .L=2/ 1 we obtain Q L D 2− L 2
2 1
C Q L 1 2
(5.80)
Now, if − L=2 D C1 satisfies the last equation (5.77) Z C1 aps .a/ da QL D 2
−L 1 Z 2C1
−L 2 1
(5.81) ps .a/ da
then the parameters determined are optimum. Otherwise, if (5.81) is not satisfied we must change our choice of Q1 in step 1) and repeat the procedure.
Lloyd algorithm This algorithm uses (5.75) and (5.77), but in a different order. 1. We set a relative error ž > 0 and D0 D 1. 2. We choose an initial partition of the positive real axis: P1 D f−0 ; −1 ; : : : ; − L=2 D C1g such that −0 D 0 < −1 < Ð Ð Ð < − L=2 D C1.
(5.82)
5.3. Non-uniform quantizers
371
3. We set the iteration index j at 1. 4. We obtain the optimum alphabet A j D fQ1 ; : : : ; Q L=2 g for the partition P j using (5.77). 5. We evaluate the distortion associated with the choice of P j and A j , D j D E[eq2 ] D 2
L=2 Z X
−i
−i1
i D1
.Qi a/2 ps .a/ da
(5.83)
6. If D j1 D j 0 is non-increasing: hence the algorithm is converging, however, not necessarily to the absolute minimum, unless some assumptions are made about ps .Ð/.
Expression of q for a very fine quantization For both algorithms it is important to initialize the various parameters near the optimum values. The considerations that follow have this objective, in addition to determining the optimum value of 3q for a non-uniform quantizer, at least approximately for a number of bits sufficiently high. From (5.70), assuming that ps .a/ ' ps .−i 1 / for −i 1 a < −i
(5.85)
we have that Meq D 2
L=2 Z X i D1
'2
L=2 X
−i
−i1
.Qi a/2 ps .a/ da
ps .−i 1 /
i D1
Z
−i
−i1
(5.86) .Qi a/ da 2
If the Qi s are optimum, it must be @Meq @Qi
D0
i D 1; : : : ;
L 2
(5.87)
372
Chapter 5. Digital representation of waveforms
and Z Qi D
−i
− Zi1 −i
a da
−i1
−i C −i 1 2
D da
(5.88)
Correspondingly, introducing the length of the i-th decision interval .1−i / D −i −i 1
i D 1; : : : ;
L 2
(5.89)
where −0 D 0 and − L=2 D C1, it follows that
Meq D 2
L=2 X
ps .−i 1 /
i D1
.1−i /3 12
(5.90)
It is now a matter of finding the minimum of (5.90) with respect to .1−i /, with the constraint that the decision intervals cover the whole positive axis; this is obtained by imposing that
2
L=2 X
1=3
ps .−i 1 / .1−i / ' 2
Z
C1
1=3
ps .a/ da D K
(5.91)
0
i D1
Using the Lagrange multiplier method, the cost function is " min
½;f1−i g
Meq C ½ K 2
L=2 X
!# 1=3 ps .−i 1 / .1−i /
(5.92)
i D1
with Meq given by (5.90). By setting to zero the partial derivative of (5.92) with respect to .1−i /, we obtain ps .−i 1 /
.1−i /2 1=3 C ½. ps .−i 1 // D 0 4
i D 1; : : : ;
L 1 2
(5.93)
that yields p 1=3 .1−i / D 2 ½ ps .−i 1 /
(5.94)
Substituting (5.94) in (5.91) yields p K ½D 2L
(5.95)
5.3. Non-uniform quantizers
373
hence .1−i / D
K 1=3 ps .−i 1 / L
i D 1; : : : ;
L 1 2
(5.96)
and the minimum value of Meq is given by K3 12L 2
Meq;opt D
(5.97)
For a quantizer optimized for a certain probability density function, and for a high number of levels L D 2b (so that (5.85) holds), we have 3q D
Ms
D
Meq;opt
22b ff
(5.98)
where f f is a form factor related to the amplitude distribution of the normalized signal sQ .k/ D s.k/=¦s , KQ 3 ff D 12
KQ D
Z
C1 1
1=3
psQ .a/ da
(5.99)
p In the Gaussian case, s.k/ 2 N .0; ¦s2 /, sQ .k/ 2 N .0; 1/, and f f D 2=. 3³ /. Actually (5.96) indicates that the optimal thresholds are concentrated around the peak of the probability density; moreover, the optimum value of 3q , according to (5.98), follows the increment law of 6 dB per bit, as in the case of a quantizer granular error. Observation 5.3 Equation (5.90) can be used to evaluate approximately Meq for a general quantizer characteristic, even of the A-law and ¼-law types. In this case, from Figure 5.32, the quantization step size 1 of the uniform quantizer is related to the compression law according to the relation 1 D F.−i / F.−i 1 / ' .1−i / F 0 .−i 1 /
(5.100)
where F 0 is the derivative of F. Obviously (5.100) assumes that F 0 does not vary considerably in the interval .−i 1 ; −i ]. Substituting (5.100) in (5.90) we have Meq D 2
L=2 X
ps .−i 1 /
i D1
1 F 0 .−i 1 /
½2
.1−i / 12
(5.101)
For L sufficiently large, the intervals become small and we get Meq '
12 2 12
Z
−sat 0
ps .a/ da [F 0 .a/]2
(5.102)
374
Chapter 5. Digital representation of waveforms
where 1 is related to a uniform quantizer parameters according to (5.25). It is left to the reader to show that for a uniform signal s in [−sat ; −sat ], quantized according to the ¼-law, the ratio 3q D Ms =Meq has the expression given in (5.64).
Performance of non-uniform quantizers A quantizer takes the name uniform, Gaussian, Laplacian, or gamma, if it is optimized for input signals having the corresponding distribution. In Tables 5.8, 5.9, and 5.10 are given parameter values of three optimum quantizers obtained by the Max or Lloyd method, for Gaussian, Laplacian, and gamma input signal, respectively, and various numbers of levels [4]. Note that, even for a small number of levels, a more dispersive distribution, that is with longer tails, leads to less closely spaced thresholds and levels, and consequently to a decrease of 3q . Concerning the increment of .3q /d B according to the 6b law, we show in Figure 5.42 the deviation .13q /d B D 6:02 b maxf3q gd B
(5.103)
for both uniform and non-uniform quantizers [4]. The optimum value of 3q follows the 6b law only in the case of non-uniform quantizers for b ½ 4. In the case of uniform quantizers, with increasing b the maximum of 3q occurs for a smaller ratio ¦s =−sat (due to the saturation error): this makes 13q vary with b and in fact it increases. Finally, we consider what happens if a quantizer, optimized for a specific input distribution, has a different type of input. For example, a uniform quantizer, best Table 5.8 Optimum quantizers for a signal with Gaussian distribution (ms D 0, ¦s2 D 1). [From Jayant and Noll (1984).]
L 2
4
8
16
i −i Qi −i Qi −i Qi −i Qi 1 1 0.798 0.453 0.982 0.501 0.245 0.258 0.128 2 1 1.510 1.050 0.756 0.522 0.388 3 1.748 1.344 0.800 0.657 4 1 2.152 1.099 0.942 5 1.437 1.256 6 1.844 1.618 7 2.401 2.069 8 1 2.733 Meq 0.363 0.117 0.0345 0.00955 3q (dB) 4.40 9.30 14.62 20.20
5.3. Non-uniform quantizers
375
Table 5.9 Optimum quantizers for a signal with Laplacian distribution (ms D 0, ¦s2 D 1). [From Jayant and Noll (1984).]
L 2
4
8
16
i −i Qi −i Qi −i Qi −i Qi 1 1 0.707 1.127 0.420 0.533 0.233 0.264 0.124 2 1 1.834 1.253 0.833 0.567 0.405 3 2.380 1.673 0.920 0.729 4 1 3.087 1.345 1.111 5 1.878 1.578 6 2.597 2.178 7 3.725 3.017 8 1 4.432 Meq 0.500 0.1761 0.0545 0.0154 3q (dB) 3.01 7.54 12.64 18.12 Table 5.10 Optimum quantizers for a signal with gamma distribution (ms D 0, ¦s2 D 1). [From Jayant and Noll (1984).]
L 2
4
8
16
i −i Qi −i Qi −i Qi −i Qi 1 1 0.577 1.268 0.313 0.527 0.155 0.230 0.073 2 1 2.223 1.478 0.899 0.591 0.387 3 3.089 2.057 1.051 0.795 4 1 4.121 1.633 1.307 5 2.390 1.959 6 3.422 2.822 7 5.128 4.061 8 1 6.195 Meq 0.6680 0.2326 0.0712 0.0196 3q (dB) 1.77 6.33 11.47 17.07
for a uniform input, will have very low performance for an input signal with a very dispersive distribution; on the contrary, a non-uniform quantizer, optimized for a specific distribution, can have even higher performance for a less dispersive input signal.
376
Chapter 5. Digital representation of waveforms
16 Γ
14
12
∆ Λq (dB)
10
L Γ
8
L 6
G 4
G
2
U 0
1
2
3
4 b
5
6
7
Figure 5.42. Performance comparison of uniform (dashed line) and non-uniform (continuous line) quantizers, optimized for a specific probability density function of the input signal. Input type: uniform (U), Laplacian (L), Gaussian (G) and gamma (0) [4]. [From Jayant and Noll (1984).]
Figure 5.43. Comparison of the signal-to-quantization error ratio for uniform quantizer (dashed-dotted line), ¼-law (continuous line) and optimum non-uniform quantizer (dotted line), for Laplacian input. All quantizers have 32 levels (b D 5) and are optimized for ¦s D 1.
5.4. Adaptive quantization
377
The 0 quantizers have performance that is almost independent of the type of input. The performance also does not change for a wide range of the signal variance, as their characteristic is of logarithmic type (see Section 5.3.1). A comparison between uniform and non-uniform quantizers with Laplacian input is given in Figure 5.43. All quantizers have 32 levels (b D 5) and are determined using: a) Table 5.5 for the uniform Laplacian type quantizer; b) Table 5.9 for the non-uniform Laplacian type quantizer; c) the ¼ (¼ D 255) compression law of Figure 5.36 with −sat =¦s D 1. We note that the optimum non-uniform quantizer gives best performance, even if this happens in a short range of values ¦s ; for a decrease in the input statistical power, performance decreases according to the law 10 log Ms D 20 log ¦s (dB), as we can see from (5.98). Only a logarithmic quantizer is independent of the input signal level.
5.4
Adaptive quantization
An alternative method to quantize a non-stationary signal consists in using an adaptive quantizer. The corresponding coding scheme, which has parameters that are adapted (over short periods) to the level of the input signal, is called an adaptive PCM or APCM.
General scheme The overall scheme is given in Figure 5.44, where c.k/ Q 6D c.k/ if errors are introduced by the binary channel. For a uniform quantizer, the idea is that of varying with time the quantization step size 1.k/ so that the quantizer characteristic adapts to the statistical power of the input signal. If 1.k/ is the quantization step size at instant k, with reference to Figure 5.21 the quantizer characteristic is defined as
Figure 5.44. Adaptive quantization and mapping: general scheme.
378
Chapter 5. Digital representation of waveforms 8 1 > > i C 1.k/ < 2 output levels: Qi .k/ D 1 > > : i 1.k/ 2 thresholds:
−i .k/ D i 1.k/
L i D ; : : : ; 1 2 L i D 1; : : : ; 2
i D
(5.104)
L L C 1; : : : ; 1; 0; 1; : : : ; 1 2 2
If 1opt is the optimum value of 1 for a given amplitude distribution of the input signal assuming ¦s D 1 (see Table 5.5), and ¦s .k/ is the standard deviation of the signal at instant k, then we can use the following rule 1.k/ D 1opt ¦s .k/
(5.105)
For a non-uniform quantizer, we need to change the levels and thresholds according to the relations: Qi .k/ D Qi;opt ¦s .k/ −i .k/ D −i;opt ¦s .k/
(5.106)
where fQi;opt g and f−i;opt g are given in Tables 5.8, 5.9, and 5.10 for various input amplitude distributions. As illustrated in Figure 5.45, an alternative to the scheme of Figure 5.44 is the following: the quantizer is fixed and the input is scaled by an adaptive gain g, so that a signal fy.k/g is generated with a constant statistical power, for example, ¦ y2 D 1. Therefore we let g.k/ D
1 ¦s .k/
(5.107)
However, both methods require computing the statistical power ¦s2 of the input signal. The adaptive quantizers are classified as:
Figure 5.45. Adaptive gain, fixed quantization and mapping.
5.4. Adaptive quantization
379
ž feedforward, if ¦s is estimated by observing the signal fs.k/g itself; ž feedback, if ¦s is estimated by observing fsq .k/ D Q[s.k/]g or fc.k/g, i.e. the signals at the output of the quantizer.
5.4.1
Feedforward adaptive quantizer
The feedforward methods for the two adaptive schemes of Figure 5.44 and Figure 5.45 are shown, respectively, in Figure 5.46 and Figure 5.47. The main difficulty in the two methods is that we need to quantize also the value of ¦s .k/ so that it can be coded and transmitted over a binary channel. We emphasize that: 1. because of digital channel errors on both c.k/ and .¦s .k//q (or gq .k/) it may happen that sQq .k/ 6D sq .k/; 2. we need to determine the update frequency of ¦s .k/, that is what frequency is required to sample ¦s , and how many bits must be used to represent .¦s .k//q ; 3. the system bit rate is now the sum of the bit rate of c.k/ and .¦s .k//q (or gq .k/).
Figure 5.46. APCM scheme with feedforward adaptive quantizer: a) encoder, b) decoder.
(a)
Figure 5.47. APCM scheme with feedforward adaptive gain and fixed quantizer: a) encoder, b) decoder.
380
Chapter 5. Digital representation of waveforms
The data sequence that represents f.¦s .k//q g or fgq .k/g is called side information. Two methods to estimate ¦s2 .k/ are given in Section 1.11.1. For example, using a rectangular window of K samples, from (1.462) we have ¦s2 .k D/ D
1 K
k X
s 2 .n/
(5.108)
nDk.K 1/
where D expresses a certain lead of the estimate with respect to the last available sample: typically D D .K 1/=2 or D D K 1. If D D K 1, K samples need to be stored in a buffer and then the average power must be computed: obviously, this introduces a latency in the coding system that is not always tolerable. Moreover, windows usually do not overlap, hence ¦s is updated every K samples. For an exponential filter instead, from (1.468) we have ¦s2 .k D/ D a¦s2 .k 1 D/ C .1 a/s 2 .k/
(5.109)
Typically in this case we choose D D 0. To determine the update frequency of ¦s2 .k/, we .1a/ recall that the 3 dB bandwidth of ¦s2 .k/ in (5.109) is equal to B¦ D .2³ Tc / , for a > 0:9. Typically, however, we prefer to determine a from the equivalence (1.471) with the length of the rectangular window, that gives a D 1 K 11 : this means decimating, quantizing, and coding the values given by (5.109) every K instants. In Table 5.11 we give, for three values of a, the corresponding values of K 1 and B¦ for 1=Tc D 8 kHz.
Performance With the constraint that ¦s varies within a specific range, ¦min ¦s ¦max , in order to keep 3q relatively constant for a change of 40 dB in the input level, it must be ¦max ½ 100 ¦min
(5.110)
Actually ¦min controls the quantization error level for small input values (idle noise), whereas ¦max controls the saturation error level. For speech signals sampled at 8 kHz, Table 5.12 shows the performance of different fixed and adaptive 8-level (b D 3) quantizers. The estimate of the signal power is obtained by a rectangular window with D D K 1; the decimation and quantization of ¦s2 .k/ Table 5.11 Time constant and bandwidth of a discrete-time exponential filter with parameter a and sampling frequency 8 kHz.
a
Time constant K 1 D 1=.1 a/ (samples)
Filter bandwidth B¦ D .1 a/=.2³ Tc / (Hz)
1 25 D 0:9688 1 26 D 0:9844 1 27 D 0:9922
32 64 128
40 20 10
5.4. Adaptive quantization
381
Table 5.12 Performance comparison of fixed and adaptive quantizers for speech.
Speech s.k/ b=3
3q (dB) Non-adaptive
Adaptive K D 128 (16 ms)
Adaptive K D 1024 (128 ms)
¼ law (¼ D 100, −sat =¦s D 8) Gaussian (3q;opt D 14:6 dB) Laplacian (3q;opt D 12:6 dB) uniform Q
9.5 7.3 9.9
– 15 13.3
– 12.1 12.8
Gaussian (3q;opt D 14:3 dB) Laplacian (3q;opt D 11:4 dB)
6.7 7.4
14.7 13.4
11.3 11.5
non-uniform Q
are not considered. Although b D 3 is a small value to draw conclusions, we note that using an adaptive Gaussian quantizer with K D 128 we get 8 dB improvement over a non-adaptive quantizer. If K − 128 the side information becomes excessive, conversely there is a performance loss of 3 dB for K D 1024.
5.4.2
Feedback adaptive quantizers
As illustrated in Figure 5.48, the feedback method estimates ¦s from the knowledge of fsq .k/ D Q[s.k/]g or fc.k/g. We make the following observations: ž there is no need to transmit ¦s .k/; therefore feedback methods do not require the transmission of side information; ž a transmission error on c.k/ affects not only the identification of the quantized level, but also the scaling factor ¦s .k/. Concerning the estimate of ¦s , a possible method consists in applying (5.108) or (5.109), where fs.n/g is substituted by fsq .n/g. However, this signal is available only for n k 1:
Figure 5.48. APCM scheme with feedforward adaptive quantizer.
382
Chapter 5. Digital representation of waveforms
Pk1 2 this implies that the estimate (5.108) becomes ¦s2q .k/ D 1=K nD.k1/.K 1/ sq .n/, with a lag of one sample. Likewise, the recursive estimate (5.109) becomes ¦s2q .k/ D a¦s2q .k 1/ C .1 a/sq2 .k 1/. Because of the lag in estimating the level of the input signal and the computational complexity of the method itself, we present now an alternative method to estimate ¦s adaptively.
Estimate of σs (k) For an input with ¦s D 1 we compute the discrete amplitude distribution of the code words for a quantizer with 2b levels and jc.k/j 2 f1; 2; : : : ; L=2g. As illustrated in Figure 5.49 for b D 3, let Z −opt;1 8 > > P[jc.k/j D 1] D 2 ps .a/ da D pc1 > > > −0 D0 < :: :: (5.111) : : > Z C1 > > > > ps .a/ da D pc4 : P[jc.k/j D 4] D 2 −opt;3
If ¦s changes suddenly, the distribution of jc.k/j will be very different with respect to (5.111). For example, if ¦s < 1 it will be P[jc.k/j D 1] × pc1 , while P[jc.k/j D 4] − pc4 . 0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 0
2
σs =1
ps (a)
Q1,opt Q2,opt
Q3,opt
τ0,opt τ1,opt τ2,opt τ3,opt
Q4,opt
a
ps (a) 2
σs 1. In practice, what we do is vary ¦sq by small steps imposing bounds to the variations, that is: ¦min ¦sq .k/ ¦max
(5.113)
The problem consists now in choosing the parameters fp[i]g, i D 1; : : : ; L=2. Intuitively it should be .p[1]/ pc1 .p[2]/ pc2 : : : .p[L=2]/
pc L=2
D1
(5.114)
In fact, from (5.112) it follows that ln ¦sq .k/ D ln p[jc.k 1/j] C ln ¦sq .k 1/
(5.115)
E[ln ¦sq ].k/ D E[ln p[jc.k 1/j]] C E[ln ¦sq .k 1/]
(5.116)
from which
In steady state we expect that E[ln ¦sq .k/] D E[ln ¦sq .k 1/], therefore it must be E[ln p[jc.k 1/j]] D
L=2 X i D1
as in (5.114).
pci ln p[i] D 0
(5.117)
384
Chapter 5. Digital representation of waveforms
Based on numerous tests on speech signals, Jayant also gave the values of the parameters fp[i]g, i D 1; : : : ; L=2. Let 2i 1 q.i/ D 2 L 1
²
1 3 ; ;:::;1 L 1 L 1
¦ (5.118)
In Figure 5.51 the values of fp[i]g are given in correspondence of fq.i/g, i D 1; : : : ; L=2 [4]. For example, for L D 8 the values of fp[i]g, i D 1; : : : ; 4, are in correspondence of the values of fq.i/g D f1=7; 3=7; 5=7; 1g. Therefore p[1] is in the range from 0.8 to 0.9, and p[4] in the range from 1.8 to 2.9; we note that there is a large interval of possible values for p[i], especially if the index i is large. Summarizing, at instant k, ¦sq .k/ is known, and by (5.106) the decision thresholds f−i .k/g are also known. From the input sample s.k/, c.k/ is produced by the quantizer characteristic (see Figure 5.52). Then ¦sq .k C 1/ is computed by (5.112) and the thresholds are updated: the quantizer is now ready for the next sample s.k C 1/. At the receiver, the possible output values are also known from the knowledge of ¦sq .k/ (see (5.106)). At the reception of c.k/ the index i in (5.106) is determined and, consequently, sq .k/; in turn the receiver updates the value of ¦sq .k C 1/ by (5.112). Experimental measurements on speech signals indicate that this feedback adaptive scheme offers performance similar to that of a feedforward scheme. An advantage of the algorithm of Jayant is that it is sequential, thus it can adapt very quickly to changes in the mean signal level; on the other hand, it is strongly affected by the errors introduced by the binary channel. p 3
2
1
0 0
1
q
Figure 5.51. Interval of the multiplier parameters in the quantization of the speech signal as a function of the parameters fq.i/g [4]. [From Jayant and Noll (1984).]
5.5. Differential coding (DPCM)
385
sq=Q[s]
Q (k)
011
p[4]
4
010
Q (k)
p[3]
3
001
Q (k)
p[2]
2
000
Q (k)
p[1]
1
- τ3(k)
- τ2(k)
- τ1(k)
100
p[1] 101
- Q (k)
τ1(k)
τ2(k)
τ3(k)
s
1
- Q (k)
p[2]
2
110
- Q (k)
p[3]
3
111
- Q (k)
p[4]
4
Figure 5.52. Input--output characteristic of a 3-bit adaptive quantizer. For each output level the PCM code word and the corresponding value of p are given.
5.5
Differential coding (DPCM)
The basic idea consists in quantizing the prediction error signal rather than the signal itself.7 With reference to Figure 5.53, for a linear predictor with N coefficients, let sO .k/ be the prediction signal :8 sO .k/ D
N X
ci s.k i/
(5.119)
i D1
From (2.81) the prediction error is defined as f .k/ D s.k/ sO .k/
(5.120)
Considering the z-transform, let C.z/ D
N X
ci z i
(5.121)
i D1
7
In the following sections, as well as in some schemes of the previous section on adaptive quantization, when processing of the input samples fs.k/g is involved, it is desirable to perform the various operations in the digital domain on a linear PCM binary representation of the various samples, obtained by an ADC. Obviously, the finite number of bits of this preliminary quantization should not affect further processing. To avoid introducing a new signal, the preliminary conversion by an ADC is omitted in all our schemes. 8 The considerations presented in this section are valid for any predictor, even non-linear predictors.
386
Chapter 5. Digital representation of waveforms
s(k)
f(k)
+ -
c
^s(k)
Figure 5.53. Computation of the prediction error signal f.k/.
s(k) +
f(k)
f(k)
-
^s(k)
sq(k)
f(k) + +
sq(k) + c
^s(k)
c
+
(a)
(b)
Figure 5.54. (a) Prediction error filter; (b) Inverse prediction error filter.
then O S.z/ D C.z/S.z/
(5.122)
F.z/ D S.z/[1 C.z/]
(5.123)
and
Recalling (2.81), [1 C.z/] is the prediction error filter. In the case C.z/ D z 1 , that is for a predictor with a single coefficient equal to one, f .k/ coincides with the difference between two consecutive input samples. It is interesting to re-arrange the scheme of Figure 5.53 in the equivalent scheme of Figure 5.54, where sq .k/ is called the reconstruction signal and is given by sq .k/ D sO .k/ C f .k/
(5.124)
Figure 5.54a illustrates how the prediction error is obtained starting from input s.k/ and prediction sO .k/, Figure 5.54b shows how to obtain the reconstruction signal from f .k/ and sO .k/, according to (5.124). From (5.120) and (5.124), it is easy to prove that in the scheme of Figure 5.54 we have sq .k/ D s.k/ that is the reconstructed signal coincides with the input signal. We will now quantize the signal f f .k/g.
5.5.1
Configuration with feedback quantizer
With reference to the scheme of Figure 5.55, the following relations hold.
(5.125)
5.5. Differential coding (DPCM)
387
Figure 5.55. DPCM scheme with quantizer inserted in the feedback loop: (a) encoder, (b) decoder.
Encoder: f .k/ D s.k/ sO .k/
(5.126)
f q .k/ D Q[ f .k/]
(5.127)
sq .k/ D sO .k/ C f q .k/
(5.128)
sO .k C 1/ D
N X
ci sq .k C 1 i/
(5.129)
i D1
Decoder: sq .k/ D sO .k/ C f q .k/ sO .k C 1/ D
N X
ci sq .k C 1 i/
(5.130) (5.131)
i D1
In other words, the quantized prediction error is transmitted over the binary channel. Let eq; f .k/ D f q .k/ f .k/
(5.132)
be the quantization error and 3q; f the signal-to-quantization error ratio 3q; f D
E[ f 2 .k/] 2 .k/] E[eq; f
(5.133)
388
Chapter 5. Digital representation of waveforms
Recalling (5.98) we know that for an optimum quantizer, with the normalization by the standard deviation of f f .k/g, 3q; f is only a function of the number of bits and of the probability density function of f f .k/g. From (5.128) and (5.132), using (5.126), we have sq .k/ D sO .k/ C f .k/ C eq; f .k/ D s.k/ C eq; f .k/
(5.134)
To summarize, the reconstruction signal is different from the input signal, sq .k/ 6D s.k/, and the reconstruction error (or noise) depends on the quantization of f .k/, not of s.k/. Consequently, if M f < Ms then also Meq; f < Meq and the DPCM scheme presents an advantage over PCM. Observing (5.134) the signal-to-noise ratio is given by 3q D
Ms Ms M f D Meq; f M f Meq; f
(5.135)
Ms Mf
(5.136)
Given Gp D called prediction gain, it follows that 3q D G p 3q; f
(5.137)
where observing (5.133) 3q; f depends on the number of quantizer levels, which in turn determine the transmission bit rate, whereas G p depends on the predictor complexity and on the correlation sequence of the input fs.k/g. We observe that the input to the filter that yields sO .k/ in (5.129) is fsq .k/g and not fs.k/g; this will cause a deterioration of G p with respect to the ideal case fsq .k/ D s.k/g. This decrease will be more prominent the larger feq; f g will be with respect to fs.k/g. If we ignore the dependence of G p on feq; f .k/g, (5.137) shows that to obtain a given 3q we can use a quantizer with a few levels, provided the input fs.k/g is highly predictable. Therefore G p can be sufficiently high also for a predictor with a reduced complexity. For the quantizer, assuming the distribution of f f .k/g is known, 3q; f is maximized by selecting the thresholds and the output values according to the techniques given in Section 5.3. In particular the statistical power of f f .k/g, useful in scaling the quantizer characteristic, can be derived from (5.136), assuming known Ms and G p , Mf D
Ms Gp
(5.138)
Regarding the predictor, once the number of coefficients N is fixed, we need to determine the coefficients fci g, i D 1; : : : ; N , that minimize M f . For example in the case N D 1, recalling (2.91), the optimum value of c1 is given by ².1/, the correlation coefficient of the input signal at lag 1. Then we have Gp D
1 1 ² 2 .1/
ignoring the effect of the quantizer, that is for fsq .k/ D s.k/g.
(5.139)
5.5. Differential coding (DPCM)
389
Table 5.13 Prediction gain with N D 1 for three values of ².1/.
c1 D ².1/
G p D 1=.1 ² 2 .1// (dB)
0.85 0.90 0.95
5.6 7.2 10.1
Figure 5.56. (a) Reconstruction signal for a DPCM, with (b) a 6 level quantizer.
We give in Table 5.13 the values of G p for three values of ².1/. We note that, for an input having ².1/ D 0:85, a simple predictor with one coefficient yields a prediction gain equivalent to one bit of the quantizer: consequently, given the total 3q , the DPCM scheme allows us to use a transmission bit rate lower than that of PCM. Evidently, for an input with ².1/ D 0 there is no advantage in using the DPCM scheme. For a simple predictor with N D 1 and c1 D 1, hence sO .k/ D sq .k 1/, Figure 5.56a illustrates the behavior of the reconstruction signal after DPCM with the six-level quantizer shown in Figure 5.56b. We note that the minimum level of the quantizer still determines the statistical power of the granular noise in fsq .k/g; the maximum level of the quantizer is instead related to the slope overload distortion in the sense that if Q L=2 is not sufficiently large, as shown in Figure 5.56a, the output signal cannot follow the rapid changes of the input signal. In the specific case, being Q L=2 =Tc < max js.k/ s.k 1/j, fsq .k/g presents a slope different from that of fs.k/g in the instants of maximum variation.
5.5.2
Alternative configuration
If we use few quantization levels, the predictor of the scheme of Figure 5.55, having as input fsq .k/g instead of fs.k/g, can give poor performance because of the large quantization
390
Chapter 5. Digital representation of waveforms
Figure 5.57. DPCM scheme with quantizer inserted after the feedback loop: a) encoder, b) decoder.
noise present in fsq .k/g. An alternative consists in using the scheme of Figure 5.57, where the following relations hold. Encoder: f .k/ D s.k/ sO .k/
(5.140)
f q .k/ D Q[ f .k/]
(5.141)
sq .k/ D s.k/
(5.142)
sO .k C 1/ D
N X
ci s.k C 1 i/
(5.143)
i D1
Decoder: sq;o .k/ D f q .k/ C sOo .k/ sOo .k C 1/ D
N X
ci sq;o .k C 1 i/
(5.144) (5.145)
i D1
At the encoder, sO .k/ is obtained from the input signal without errors. However, the prediction signal reconstructed at the decoder is sOo .k/ 6D sO .k/. In fact, from (5.144) and
5.5. Differential coding (DPCM)
391
(5.145), even if by chance sOo .i/ D sO .i/, for i k 1, as f q .k 1/ 6D f .k 1/ then sq;o .k 1/ 6D s.k 1/, and consequently sOo .k/ 6D sO .k/. A difficulty of the scheme is that, depending on the function C.z/, the difference between the prediction signals, sOo .k/ sO .k/, may be non-negligible. As a result the output sq;o .k/ D sOo .k/ C f .k/ C eq; f .k/ D sOo .k/ C s.k/ sO .k/ C eq; f .k/
(5.146)
D s.k/ C [Oso .k/ sO .k/] C eq; f .k/ can assume values that are quite different from s.k/. Observation 5.4 Note that the same problem mentioned above may occur also in the scheme of Figure 5.55 because of errors introduced by the binary channel, though to a lesser extent as compared to the scheme of Figure 5.57, as the signal f f q .k/g at the encoder is affected by a smaller disturbance. For both configurations, however, the inverse prediction error filter must suppress the propagation of such errors in a short time interval. This is difficult to achieve if the transfer function 1=[1 C.z/] has poles near the unit circle, and consequently the impulse response is very long.
5.5.3
Expression of the optimum coefficients
For linear predictors, the prediction signal sO .k/ is given by sO .k/ D
N X
ci sq .k i/
(5.147)
i D1
where sq .k/ is the reconstruction signal, which in the case of a feedback quantizer system is given by sq .k/ D s.k/ C eq; f .k/. For the design of the predictor, we choose the coefficients fci g that minimize the statistical power of the prediction error, M f D E[.s.k/ sO .k//2 ]
(5.148)
We introduce the following vectors and matrices. Vector of prediction coefficients c D [c1 ; : : : ; c N ]T
(5.149)
Vector of correlation coefficients of fs.k/g ρ D [².1/; : : : ; ².N /]T where ².i/ is defined in (1.540).
(5.150)
392
Chapter 5. Digital representation of waveforms
Correlation matrix of sq , normalized by Ms 9 2 3 1 ².1/ : : : ².N 1/ 7 6 1 C 3q 6 7 6 7 1 6 ².1/ 1C : : : ².N 2/ 7 6 7 3q D6 7 6 7 : : : : :: :: :: :: 6 7 6 7 4 1 5 ².N 1/ ::: ².1/ 1 C 3q
(5.153)
Recalling the analysis of Section 2.2, the optimum prediction coefficients are given by the matrix equation (2.78) copt D ρ
(5.154)
The corresponding minimum value of M f is obtained from (2.79), T M f D Ms .1 copt ρ/
(5.155)
The difficulty of this formulation is that to determine the solution we need to know the value of 3q (see (5.153)). We may consider the solution with the quantizer omitted, hence 3q D 1, and depends only on the second order statistic of fs.k/g. In this case some efficient algorithms to determine c and M f in (5.154) and (5.155) are given in Sections 2.2.1 and 2.2.2.
Effects due to the presence of the quantizer Observing (5.155), the prediction gain is given by Gp D
Ms 1 D T ρ Mf 1 copt
(5.156)
In general it is very difficult to analyze the effects of feq; f .k/g on G p , except in the case N D 1, for which (5.154) becomes 1 D ².1/ (5.157) copt;1 1 C 3q Then copt;1 D
8
².1/ 1 C .1=3q /
(5.158)
Assuming for fs.k/g and feq; f .k/g the correlations are expressed by (5.27) and (5.28), we get Dividing by Ms we obtain
rsq .n/ D rs .n/ C Meq; f Žn rsq .n/ Ms
D ².n/ C 3q1 Žn
(5.151)
(5.152)
5.5. Differential coding (DPCM)
393
and Gp D
1 1 copt;1 ².1/
D
1 ² 2 .1/ 1 1 C 1=3q
(5.159)
The above relations show that if 3q is small, that is, if the system is very noisy, then copt;1 is small and G p tends to 1. Only for 3q D 1 it is copt;1 D ².1/. It may occasionally happen that a suboptimum value is assigned to c1 : we will try to evaluate the corresponding value of G p . For N D 1 and any c1 it is M f D E[.s.k/ c1 .s.k 1/ C eq; f .k 1///2 ] ' Ms .1 2c1 ².1/ C c12 / C c12 Meq; f
(5.160)
As from (5.135) it follows Meq; f D Ms =3q , where from (5.137) 3q D G p 3q; f , observing (5.160) we obtain Gp D
1 .c12 =3q; f / 1 2c1 ².1/ C c12
(5.161)
hence 3q; f depends only on the number of quantizer levels. Note that (5.161) allows the computation of the optimum value of c1 for a predictor with N D 1 in the presence of the quantizer: however, the expression is complicate and will not be given here. Rather we will derive G p for two values of c1 . 1. For c1 D ².1/ we have Gp D
² 2 .1/ 1 1 3q; f 1 ² 2 .1/
(5.162)
where the factor .1 ² 2 .1/=3q; f / is due only to the presence of the quantizer. 2. For c1 D 1 we have
1 1 1 Gp D 2.1 ².1// 3q; f
(5.163)
We note that the choice c1 D 1 leads to a simple implementation of the predictor: however, this choice results in G p > 1 only if ².1/ > 1=2. Various experiments with speech have demonstrated that for very long observations, of the order of one second, the prediction gain for a fixed predictor is between 5 and 7 dB, and saturates for N ½ 2; in fact, speech is a non-stationary signal and adaptive predictors should be used.
5.5.4
Adaptive predictors
In adaptive differential PCM (ADPCM), the predictor is time-varying. Therefore we have sO .k/ D
N X i D1
ci .k/sq .k i/
(5.164)
394
Chapter 5. Digital representation of waveforms
The vector c D [c1 ; : : : ; c N ]T is chosen to minimize M f over short intervals within which the signal fs.k/g is quasi-stationary. Speech signals have slowly-varying spectral characteristics and can be assumed as stationary over intervals of the order of 5–25 ms. Also for ADPCM two strategies emerge.
Adaptive feedforward predictors The general scheme is illustrated in Figure 5.58. We consider an observation window for the signal fs.k/g of K samples. Based on these samples the input autocorrelation function is estimated up to lag N using (1.478); then we solve the system of equations (5.154) to obtain the coefficients c and the statistical power of the prediction error. These quantities, after being appropriately quantized for finite precision representation, give the parameters of the predictor cq and of the quantizer .¦ f /q : the system is now ready to encode the samples of the observation window in sequence. The digital representation of f f q .k/g, together with the quantized parameters of the system, must be sent to the receiver to reconstruct the signal. For the next K samples of fs.k/g the procedure is repeated. In general, for speech we choose K Tc ' 10–20 ms, and N ' 10. Not considering the computation time, this system introduces a minimum delay from fs.k/g to the decoder output fQsq .k/g equal to K samples. The performance improvement obtained by using an adaptive scheme is illustrated in Figure 5.59. In particular, for speech signals sampled at 8 kHz, the power measured on windows of 128 samples is shown in Figure 5.59a: we note that the speech level exhibits a dynamic range of 30–40 dB and rapidly changes value. The prediction gain in the absence of the quantizer is shown for a fixed predictor with N D 3 and an adaptive predictor with N D 10 in Figure 5.59b and in Figure 5.59c, respectively. The fixed predictor is determined by considering the statistic of the whole signal, thus within certain windows the prediction gain is even less than 1. The adaptive predictor is estimated at every window by the feedforward method and yields G p > 1, even for unvoiced spurts that present small correlation. We note that, for some voiced spurts, G p can reach the value of 20–30 dB.
Sequential adaptive feedback predictors Also for adaptive feedback predictors we could observe fsq .i/g for a window of K samples and apply the same procedure as the feedforward method; however, the observation is now available only for instants i < k, and consequently this method is not suitable to track rapid changes of the input statistic. An alternative could be that of estimating at every instant k the correlation of fsq .i/g, i < k, and calculate c.k/; however, this method requires too many computations. Another simple alternative, of the sequential adaptive type, is illustrated in Figure 5.60, where the predictor is adapted by the LMS algorithm (see Section 3.1.2). Defining sq .k/ D [sq .k 1/; : : : ; sq .k N /]
(5.165)
coefficient adaptation is given by c.k C 1/ D ¼1 c.k/C µ f q .k/sq .k/
0 ; wQ > D 1 Q (6.152) 2 2 2N0 As from (6.134) E s D E h =2 and 0 D E s =N0 , we get p h p i Pe D 1 P[C] D 1 P[C j s1 ] D 2Q 0 1 12 Q 0 For 0 × 1, the following approximations are valid: p Pe ' 2Q 0
(6.153)
(6.154)
and Pbit ' Q
p 0
(6.155)
The QPSK transmitter is obtained by simplification of the general scheme (6.130), as illustrated in Figure 6.34. The binary bit maps are given in Table 6.2. The ML receiver for QPSK is illustrated in Figure 6.35. As the decision thresholds are set at .0; ³=2; ³; 3³=2/
6.4. Phase-shift keying (PSK)
473
φ2 b 1 b2 01
s2
s1
11
E s = Eh /2
φ1 s3
s4
00
10
Figure 6.33. Signal constellation of a QPSK system.
Figure 6.34. QPSK transmitter for an isolated pulse.
Table 6.2 Binary bit map for a QPSK system.
Binary bit map b1 (b2 ) 0 1
Þn;I (Þn;Q ) p 1=p2 1= 2
474
Chapter 6. Modulation theory
Figure 6.35. ML receiver for a QPSK system.
(see Figure 6.33), decisions can be made independently on r I and r Q , using a simple threshold detector with threshold set at zero. We observe that, for h Tx .t/ D K wT .t/, the transmitter filter is a simple holder. At the receiver the matched filter plus sampler becomes an integrator that is cleared before each integration over a symbol period of duration T . In other words, it consists of an integrateand-dump.
6.5
Differential PSK (DPSK)
We assume now that the receiver recovers the carrier signal, except for a phase offset of Figure 6.28, the reconstructed carrier is 'a . In particular, with reference to the scheme p cos.2³ f 0 t 'a /. In this case s n coincides with E s e j'a Þn , where Þn is given by (6.129). Consequently, it is as if the constellation at the receiver were rotated by 'a . To prevent this problem there are two strategies. By the coherent method, a receiver estimates 'a from the received signal, and considers the original constellation for detection, using the signal r e j 'Oa , where 'Oa is the estimate of 'a . By the differential non-coherent method, a receiver detects the data using the difference between the phases of signals at successive sampling instants. In other words ž for M-PSK, the phase of the transmitted signal at instant kT is given by (6.126), with k 2
²
.2M 1/³ ³ 3³ ; ;:::; M M M
¦ (6.156)
6.5. Differential PSK (DPSK)
475
ž for M-DPSK,12 the transmitted phase at instant kT is given by ¦ ² 2³ 2³ 0 0 ;:::; .M 1/ k 2 0; k D k1 C k M M
(6.157)
that is, the phase associated with the transmitted signal at instant kT is equal to that transmitted at the previous instant .k 1/T plus the increment k , which can assume one of M values. We note that the decision thresholds for k are now placed at .³=M/.2n 1/, n D 1; : : : ; M. For a phase offset equal to 'a introduced by the channel, the phase of the signal at the detection point becomes k
0 k
D
C 'a
(6.158)
In any case, k
k1
D k
(6.159)
and the ambiguity of 'a is removed. For phase-modulated signals, three differential noncoherent receivers that determine an estimate of (6.159) are discussed in Chapter 18.
6.5.1
Error probability for an M-DPSK system
For E s =N0 × 1, using the definition of the Marcum function Q 1 .Ð; Ð/ (see Appendix 6.A) it can be shown that the error probability of an isolated symbol is approximated by the following bound [2, 3] s s ! Es Es ³ ³ ; 1 sin 1 C sin Pe 1 C Q 1 N0 M N0 M (6.160) s s ! ³ ³ Es Es Q1 1 C sin 1 sin ; N0 M N0 M Moreover, if M is large, the approximation (6.369) can be used and we get "s r r # Es ³ ³ 1 sin 1 C sin Pe ' 2Q N0 M M s ' 2Q
Es ³ sin N0 M
!
(6.161)
12 Note that we consider a differential non-coherent receiver with which is associated a differential symbol
encoder at the transmitter (see (6.157)) or ((6.169)). However, as we will see in the next section, a differential encoder and a coherent receiver can be used.
476
Chapter 6. Modulation theory
For Gray coding of the values of k in (6.156), the bit error probability is given by Pbit D
Pe log2 M
(6.162)
where Pe is given by (6.161). For M D 2, the exact formula of the error probability is [2, 3] Pbit D Pe D 12 e
E Ns
(6.163)
0
For M D 4, the exact formula is [2, 3] Pe D 2Q 1 .a; b/ I0 .ab/ e0:5.a
2 Cb2 /
(6.164)
where s aD
p Es .1 1=2/ N0
s bD
p Es .1 C 1=2/ N0
(6.165)
and where the function I0 is defined in (4.216). Using the previous results, a comparison in terms of Pbit between DPSK (6.161) and PSK (6.147) is given in Figure 6.36: we note that, for Pbit D 103 , DPSK presents a loss of only 1.2 dB in 0 for M D 2, that increases to 2.3 dB for M D 4, and to 3 dB for M > 4. As a DPSK receiver is simpler as compared to a coherent PSK receiver, in that it does not require recovery of the carrier phase, for M D 2 DPSK is usually preferred to PSK. −1
10
PSK DPSK −2
10
−3
Pbit
10
M=2
−4
10
M=4
M=8
M=16
M=32
−5
10
−6
10
5
10
15
20 Γ (dB)
25
30
Figure 6.36. Comparison between PSK and DPSK.
35
6.5. Differential PSK (DPSK)
477
Note that, if the previously received sample is used as a reference, DPSK gives lower performance with respect to PSK, especially for M ½ 4, because both the current sample and the reference sample are corrupted by noise. This drawback can be mitigated if the reference sample is constructed by using more than one previously received samples [4]. In this way we establish a gradual transition between differential phase demodulation and coherent demodulation. In particular, if the reference sample is constructed using the samples received in the two previous modulation intervals, DPSK and PSK yield similar performance [4].
6.5.2
Differential encoding and coherent demodulation
If 'a is a multiple of 2³=M, at the receiver the phase difference can be formed between the phases of two consecutive coherently detected symbols, instead of between the phases of two consecutive samples. In this case, symbols are differentially encoded before modulation.
Binary case (M = 2, differentially encoded BPSK) Let bk be the value of the information bit at instant kT , bk 2 f0; 1g. BPSK system without differential encoding. The phase k 2 f0; ³ g is associated with bk by the bit map of Table 6.3. Differential encoder. For any c1 2 f0; 1g, we encode the information bits as ck D ck1 ý bk
bk 2 f0; 1g
k½0
(6.166)
where ý denotes the modulo 2 sum; therefore ck D ck1 if bk D 0, and13 ck D cNk1 if bk D 1. For the bit map of Table 6.4 we have that bk D 1 causes a phase transition, and bk D 0 causes a phase repetition. Decoder. If fcOk g are the detected coded bits at the receiver, the information bits are recovered by bOk D cOk ý .cOk1 / D cOk ý cOk1 Table 6.3 Bit map for a BPSK system.
bk Transmitted phase k (rad) 0 1
0 ³
13 cN denotes the one’s complement of c: 1N D 0 and 0N D 1.
(6.167)
478
Chapter 6. Modulation theory
Table 6.4 Bit map for a differentially encoded BPSK system.
ck Transmitted phase 0 1
k
(rad)
0 ³
We note that a phase ambiguity 'a D ³ does not alter the recovered sequence fbOk g: in fact, in this case fcOk g becomes fcOk0 D cOk ý 1g and we have .cOk ý 1/ ý .cOk1 ý 1/ D cOk ý cOk1 D bOk
(6.168)
Multilevel case Let fdk g be a multilevel information sequence, with dk 2 f0; 1; : : : ; M 1g. In this case we have ck D ck1 ý dk
(6.169)
M
where ý denotes the modulo M sum. Because ck 2 f0; 1; : : : ; M 1g, the phase assoM
ciated with the bit map is k 2 f³=M; 3³ =M; : : : ; .2M 1/³ =Mg. This encoding and bit-mapping scheme are equivalent to (6.157). At the receiver the information sequence is recovered by (6.170) dOk D cOk ý .cOk1 / M
It is easy to see that an offset equal to j 2 f0; 1; : : : ; .M 1/g in the sequence fcOk g, corresponding to a phase offset equal to f0; 2³ =M; : : : ; .M 1/2³=Mg in f k g, does not cause errors in fdOk g. In fact, ½ cOk ý j ý cOk1 ý j D cOk ý .cOk1 / D dOk (6.171) M
M
M
M
Performance of a PSK system with differential encoding and coherent demodulation by the scheme of Figure 6.28, is worse as compared to a system with absolute phase encoding. However, for small Pe , up to values of the order of 0:1, we observe that an error in fcOk g causes two errors in fdOk g. Approximately, Pe increases by a factor 2,14 which causes a negligible loss in terms of 0. To combine Gray encoding of values of ck with the differential encoding (6.169), a two step procedure is adopted: 14 If we indicate with P e;Ch the channel error probability, then the error probability after decoding is given
by [2] Binary case Quaternary case
Pbit D 2Pbit;Ch [1 Pbit;Ch ] 2 3 4 Pe D 4Pe;Ch 8Pe;Ch C 8Pe;Ch 4Pe;Ch
(6.172) (6.173)
6.5. Differential PSK (DPSK)
479
Table 6.5 Gray coding for M D 8.
Three information bits 0 0 0 0 1 1 1 1
0 0 1 1 1 1 0 0
values of dk
0 1 1 0 0 1 1 0
0 1 2 3 4 5 6 7
1. represent the values of dk with a Gray encoder using a combinatorial table, as illustrated for example in Table 6.5 for M D 8; 2. determine the differentially encoded symbols according to (6.169).
Example 6.5.1 (Differential encoding 2B1Q) We consider a differential encoding scheme for a four-level system that makes the reception insensitive to a possible change of sign of the transmitted sequence. For M D 4 this implies insensitivity to a phase rotation equal to ³ in a 4-PSK signal or to a change of sign in a 4-PAM signal. For M D 4 we give the law between the binary representation of dk D .dk.1/ ; dk.0/ /, dk.i / 2 f0; 1g, and the binary representation of ck D .ck.1/ ; ck.0/ /, ck.i / 2 f0; 1g: .1/ ck.1/ D dk.1/ ý ck1
ck.0/ D dk.0/ ý ck.1/
(6.174)
The bit map is given in Table 6.6. The equations of the differential decoder are .1/ dOk.1/ D cOk.1/ ý cOk1
dOk.0/ D cOk.0/ ý cOk.1/ Table 6.6 Bit map for the differential encoder 2B1Q.
ck.1/ ck.0/ Transmitted symbol ak 0 0 1 1
0 1 0 1
3 1 1 3
(6.175)
480
6.6
Chapter 6. Modulation theory
AM-PM or quadrature amplitude modulation (QAM)
Quadrature amplitude modulation is another example of passband modulation. Consider choosing a bit mapper that associates to a sequence of log2 M bits a symbol from a constellation of cardinality M and elements given by the complex numbers n D 1; 2; : : : ; M
Þn
(6.176)
If we modulate a symbol of the constellation by a real baseband pulse h Tx with finite energy E h and support .0; t0 /, we obtain the isolated generic transmitted pulse given by sn .t/ D Þn;I h Tx .t/ cos.2³ f 0 t/ Þn;Q h Tx .t/ sin.2³ f 0 t/
t 2
1:59 dB N0
(6.238)
otherwise Pbit ! 1. Therefore 1:59 dB is the minimum value of E b =N0 that is M!1
necessary to reach an error probability that can be made arbitrarily small for M ! 1 (see Section 6.10).
6.7. Modulation methods using orthogonal and biorthogonal signals
493
Table 6.8 Values of Eb =N0 required to obtain Pbit D 106 for various values of M.
M
6.7.2
E b =N0 (dB)
23 24 25 26 210 215 220 :: :
9.4 8.3 7.5 7.0 5.4 4.5 3.9 :: :
1
1:59
Modulation with biorthogonal signals
The elements of a set of M biorthogonal signals are M=2 orthogonal signals and their antipodal signals: for example, 4-PSK is a biorthogonal signalling scheme. A further example of biorthogonal signalling with 2M signals is given by a signalling scheme using the M orthogonal signals in (6.212) and their antipodal signals. For biorthogonal signalling with M signals, the required bandwidth is proportional to M=2. We give the parameters of the system in the two cases of non-coherent and coherent demodulation. Passband signalling with non-coherent demodulation: Bmin D
M 1 2 T
¹D2 0D
log2 M M
2E s N0 M
(6.239) (6.240) (6.241)
and, as I D M=2, 0 I D 20
(6.242)
Baseband signalling or passband signalling with coherent demodulation: Bmin D
M 1 2 2T
(6.243)
494
Chapter 6. Modulation theory
¹D4 0D
log2 M M
(6.244)
4E s N0 M
(6.245)
and 0I D 0
(6.246)
Probability of error The receiver consists of M=2 correlators, or matched filters, which provide the decision variables n D 1; : : : ;
fUn g
M 2
(6.247)
The optimum receiver selects the output with the largest absolute value, jUi j; subsequently it selects si or si depending on the sign of Ui . To compute the probability of correct decision, we proceed as in the previous case. Assuming that sm is taken as one of the signals of the basis, then P[C j sm ] D P[Um > 0; jUm j > jU1 j; : : : ; jUm j > jUm1 j; jUm j > jUmC1 j; : : : ; jUm j > jU M=2 j] Z
C1
D 0
1 p e 2³
!2 r 2E 1 2 Þ N s 0
(6.248) [1 2Q.Þ/] M=21 dÞ
The symbol error probability is given by Pe D 1 P[C j sm ]
(6.249)
The bit error probability can be approximated as Pbit ' 12 Pe
(6.250)
Curves of Pbit as a function of 0 D 4E s =.N0 M/ and E b =N0 are plotted, respectively, in Figure 6.47 and in Figure 6.48, for various values of M. A bound to (6.249) for transmission with M biorthogonal signals is given by s s ! ! Es 2E s CQ (6.251) Pe .M 2/Q N0 N0 where the first term arises from the comparison with .M 2/ orthogonal signals, and the second arises from the comparison with an antipodal signal. Figure 6.49 shows a comparison between the error probability obtained by exact computation and the bound (6.251) for two values of M.
6.7. Modulation methods using orthogonal and biorthogonal signals
495
−1
10
−2
10
−3
Pbit
10
−4
10
M=128 M=32 M=16 M=8 M=4 M=2
−5
10
−6
10 −10
0
−5
Γ=4Es/(N0M) (dB)
15
10
5
Figure 6.47. Bit error probability as a function of 0 for transmission with M biorthogonal signals.
−1
10
M=128 M=32 M=16 M=8 M=4 M=2
−2
10
−3
Pbit
10
−4
10
−5
10
−6
10
−2
0
2
4
6 E / N (dB) b
8
10
12
14
0
Figure 6.48. Bit error probability as a function of Eb =N0 for transmission with M biorthogonal signals.
496
Chapter 6. Modulation theory
Figure 6.49. Comparison between the exact error probability and the limit (6.251) for transmission with M biorthogonal signals.
6.8
Binary sequences and coding
We consider a baseband signalling scheme where the transmitted signal is given by sn .t/ D
p
Ew
nX 0 1
cn; j wQ T .t j T /
n D 1; : : : ; M
0 < t < n 0 T D Ts
(6.252)
jD0 =2 is the normalized rectangular window of where cn; j ž f1; C1g, and wQ T .t/ D p1 rect tT T T duration T (see (1.456)) with unit energy. Then E w is the energy of the pulse sn evaluated 1 . on a generic subperiod T . Moreover, we have Bmin D 2T Interpreting the n 0 pulses
wQ T .t/; : : : ; wQ T .t .n 0 1/T /
(6.253)
as elements of an orthonormal basis, we derive the structure of the optimum receiver. Uncoded sequences. Every sequence of n 0 binary coefficients cn D [cn;0 ; : : : ; cn;n 0 1 ]T is allowed, hence M D I D n 0 , it follows
2n 0 .
cn; j ž f1; 1g
(6.254)
For a modulation interval Ts , we have L b D log2 M D n 0 . As RI D
log2 M D1 I
(6.255)
6.8. Binary sequences and coding
497
Es D n0 Ew Es D Ew I EI D Ew Eb D RI EI D
0I D
2E w EI 2E b D D N0 =2 N0 N0
0D
2E w Es D D 0I 1 N0 Ts N0 2T
(6.256) (6.257) (6.258) (6.259) (6.260)
Moreover, the minimum distance between two elements of the set of signals (6.252) is p equal to dmin D 4E w . The error probability is determined by the ratio (6.57) u D
2 2 dmin dmin 2E w 2E b D D D 2N0 N0 N0 .2¦ I /2
(6.261)
where in the last step equation (6.258) is used. Coded sequences. We consider a set of signals (6.252) corresponding to M D 2k0 binary sequences cn with n 0 components, assuming that only k0 components in (6.254), as for example those with index j D 0; 1; : : : ; k0 1, can assume values in f1; 1g arbitrarily: these components determine, through appropriate binary functions, also the remaining n 0 k0 components. Because the number of elements of the basis is always I D n 0 , we have RI D
k0 log2 M D I n0
Es D n0 Ew n0 Ew D Ew I EI n0 Eb D Ew D RI k0 EI D
0I D 0D
2E w EI k0 2E b D D N0 =2 N0 n 0 N0 Es D 0I 1 Ts N0 2T
(6.262) (6.263) (6.264) (6.265) (6.266) (6.267)
H the minimum number of positions in which two vectors c differ, Indicating with dmin n we find 2 H dmin D 4E w dmin
(6.268)
498
Chapter 6. Modulation theory
H D 1. An example of coding is given by the choice In the case of uncoded sequences dmin of the following vectors (code sequences or code words) for n 0 D 4 and k0 D 2, 2 3 2 3 2 3 2 3 1 1 C1 C1 6 1 7 6 1 7 6 C1 7 6 C1 7 7 7 7 7 c1 D 6 c2 D 6 c3 D 6 (6.269) c0 D 6 4 1 5 4 C1 5 4 1 5 4 C1 5 1 C1 1 C1 H D 2, and therefore d 2 For this signalling system, we have dmin min D 8E w . Using (6.265), the signal-to-noise ratio at the decision point is given by
c D
H E 4dmin d H R I 2E b w D min 2N0 N0
(6.270)
We note that for a given value of E b =N0 the coded system presents a larger , and H R > 1. consequently a lower bit error probability, if dmin I H for given values We will discuss in Chapter 11 the design of codes that yield a large dmin of the parameters n 0 and k0 . A drawback of these systems is represented by the reduction of the transmission bit rate Rb for a given modulation interval Ts ; alternative coding methods will be examined in Chapter 12.
Optimum receiver With reference to the implementation of Figure 6.8, as the elements of the orthonormal basis (6.253) are obtained by shifting the pulse wQ T .t/, the optimum receiver can be simplified as illustrated in Figure 6.50, where the projections of the received signal r.t/ onto the
Figure 6.50. ML receiver for the signal set (6.252).
6.9. Comparison between coherent modulation methods
499
Figure 6.51. ML receiver for the signal set (6.252) under the assumption of uncoded sequences.
components of the basis (6.253) are obtained sequentially. The vector components r D [r0 ; r1 ; : : : ; rn o 1 ]T are then used to compute the Euclidean distances with each of the possible code sequences. The scheme of Figure 6.50 yields the detected signal of the type (6.252), or equivalently the detected code sequence cO D [cO0 ; cO1 ; : : : ; cOn o 1 ]T , according to the ML criterion. This procedure is usually called soft-input decoding. For the uncoded system, the receiver can be simplified by computing the Euclidean distance component by component, as illustrated in Figure 6.51. In the binary case under examination, cOi D cQi D sgn.ri /
i D 0; : : : ; n 0 1
(6.271)
The resulting channel model (memoryless binary symmetric) is that of Figure 6.21. In some receivers for coded systems, a simplification of the scheme of Figure 6.50 is obtained by first detecting the single components cQi ž f1; 1g according to the scheme of Figure 6.51. Successively, the binary vector cQ D [cQ0 ; : : : ; cQn 0 1 ]T is formed. Then we choose among the possible code sequences cn , n D 1; : : : ; 2k0 , the one that differs in the smallest number of positions with respect to the sequence cQ . This scheme is usually called hard -input decoding and is clearly suboptimum as compared to the scheme with soft input.
6.9
Comparison between coherent modulation methods
Table 6.9 summarizes some important results derived in the previous sections. Passband PAM is considered as single sideband (SSB) modulation or double sideband (DSB) modulation (see Appendix 7.C). In the latter case Bmin is equal to 1=T , hence 0 D E s =N0 . We note that, for a given noise level, PAM, PAMCSSB and PAMCDSB methods require the same statistical power to achieve a certain Pe ; however the PAMCDSB technique has a Bmin that is double as compared to PAM or PAMCSSB methods. For a given value of the symbol error probability, we now derive 0 I as a function of R I for some multilevel modulations. The result will be compared with the Shannon limit given by 0 I D 22R I 1, that represents the minimum theoretical value of 0 I , in correspondence of a given R I , for which Pbit can be made arbitrarily small by using channel coding without constraints in complexity and latency (see Section 6.10). We note
500
Chapter 6. Modulation theory
Table 6.9 Comparison of various modulation methods in terms of performance, bandwidth, and spectral efficiency. Approximated symbol error probability Pe
Modulation
binary antipodal (BB) M-PAM M-PAM C SSB M-PAM C DSB M-QAM .M D L 2 /
Q
p 0
r 3 1 2Q 0 1 M M2 1 r 6 1 2Q 0 1 2 M M 1 ! r 3 1 0 1 p 4Q M 1 M
BPSK o 2-PSK QPSK o 4-PSK M-PSK .M > 2/
orthogonal (BB)
Q
.M 1/Q
biorthogonal (BB) .M 2/Q
Spectral efficiency ¹ D .1=Tb /=Bmin (bit/s/Hz)
1 2T
2
1 2T
2 log2 M
1 T
log2 M
r
M 0 2
1 T
log2 M
1 log2 M 2
1
1
2
1
1 T
!
! ! r M M 0 CQ 0 4 2
Encoder- Signalmodulator to-noise rate R I ratio 0 (bit/dim) 1
log2 M
p 20
r ³ 0 2 sin2 2Q M
r
Minimum bandwidth Bmin (Hz)
log2 M
1 log2 M 2
2E s N0 2E s N0 Es N0 Es N0
Es N0
M 2T
2
log2 M M
1 log2 M M
2E s N0 M
M 4T
4
log2 M M
2 log2 M M
4E s N0 M
that an equivalent approach often adopted in the literature is to give E b =N0 , related to 0 I through (6.107), as a function of ¹, related to R I through (6.104). A first comparison is made by assuming the same symbol error probability, Pe D 106 , p for all systems. As Q. z 0 / D 106 implies z 0 ' 22, considering only the argument of the Q function in Table 6.9, we have the following results. 1. M-PAM. From 3 0 D z0 1
M2
(6.272)
and 0I D 0
R I D log2 M
(6.273)
we obtain the following relation 0I D
z 0 2R I 1/ .2 3
(6.274)
6.9. Comparison between coherent modulation methods
501
2. M-QAM. From 3 0 D z0 M 1
(6.275)
and 0I D 0
RI D
1 2
log2 M
(6.276)
we obtain 0I D
z 0 2R I .2 1/ 3
(6.277)
We note that for QAM a certain R I is obtained with a number of symbols equal 2R I to MQAM p D 2 , whereas for PAM the same efficiency is reached for MPAM D R I 2 D MQAM . 3. M-PSK. It turns out 0I D
z 0 4R I 2 20
(6.278)
Equation (6.278) holds for M ½ 4, and is obtained by approximating sin.³=M/ with ³=M, and ³ 2 with 10. 4. Orthogonal modulation. Using the approximation r Pe ' .M 1/Q
M 0 2
! (6.279)
we note that the multiplicative constant in front of the Q function cannot be ignored: therefore a closed-form analytical expression for 0 I as a function of R I for a given Pe cannot be found. 5. Biorthogonal modulation. The symbol error probability is approximately the same as that of orthogonal modulation for half the number of signals. Both R I and ¹ are doubled. We note that, for a given value of R I , PAM and QAM require the same value of 0 I , whereas PSK requires a much larger value of 0 I . An exact comparison is now made for a given bit error probability. Using the Pbit curves previously obtained, the behavior of R I as a function of 0 I for Pbit D 106 is illustrated in Figure 6.52. We observe that the required 0 I is much larger than the minimum value obtained by the Shannon limit. As will be discussed in Section 6.10, the gap can be reduced by channel coding. We also note that, for large R I , PAM and QAM allow a lower 0 I with respect to PSK; moreover, orthogonal and biorthogonal modulation operate with R I < 1, and corresponding very small values of 0 I .
502
Chapter 6. Modulation theory
Shannon limit
Figure 6.52. 0I required for a given rate RI , for different modulation methods and bit error probability equal to Pbit D 106 . The parameter in the figure denotes the number of symbols M of the constellation.
Trade-offs for QAM systems There are various trade-offs that are possible among the parameters of a modulation method. We consider for example Figure 6.41 for M-QAM, where the parameter is ¹ D log2 M D .1=Tb /=Bmin . We assume that 1=Tb is fixed. For a given Pbit , we obtain 0 as a function of ¹, from which the required bandwidth is also obtained; given ¹ (and the bandwidth), the trade-off is between Pbit and 0; finally, fixed 0, we get ¹ as a function of Pbit . We note that to modify ¹ a modulator with a different constellation must be adopted.
Comparison of modulation methods PAM, QAM, and PSK are bandwidth efficient modulation methods as they cover the region for R I > 1, or equivalently ¹ > 2, or Bmin < 1=.2Tb /, as illustrated in Figure 6.52. The bandwidth is traded off with the power, that is 0, by increasing the number of levels: we note that, in this region, higher values of 0 are required to increase ¹. Orthogonal and biorthogonal modulation are not very efficient in bandwidth (R I < 1), but require much lower values of 0. As illustrated in Figure 6.52, biorthogonal modulation (see (6.249)) has the same performance as orthogonal modulation (see (6.234)), but requires half the bandwidth; in this region, by increasing the bandwidth it is possible to decrease 0. However, a slight decrease in 0 may determine a large increase of the bandwidth. The Pbit of orthogonal or biorthogonal modulation is almost independent of M
6.10. Limits imposed by information theory
503
and depends mainly on the energy E s of the signal and on the spectral density N0 =2 of the noise. In addition to the required power and bandwidth, the choice of a modulation scheme is based on the channel characteristics and on the cost of the implementation: until recently, for example, non-coherent receivers were preferred in radio mobile systems because of their simplicity, even though the performance is inferior to that of coherent receivers (see Chapter 18) [2].
6.10
Limits imposed by information theory
We consider the transmission of signals with a given power over an AWGN channel having noise power spectral density equal to N0 =2. We recall the definition (6.93) of the encoder-modulator rate, R I D L b =I , where I is the number of signal space dimensions. For example, the encoder-modulator for the 8-PAM system with bit map defined in Table 6.1 has rate R I D 3 (bit/dim), as L b D 3 and I D 1. From (6.95), we have the cardinality of alphabet A is equal to M D 2 R I D 8. Let us consider for example a monodimensional transmission system (PAM) with an alphabet of cardinality A for a given rate R I , such that L b < log2 M, that is M > 2 R I from (6.93); the redundancy of the alphabet can be used to encode sequences of information bits: in this case we speak of coded systems (see Example 6.7.5). Let us take a PAM system with R I D 3 and M D 16: redundancy may be introduced in the sequence of transmitted symbols. The mapping of sequences of information bits into sequences of coded output symbols may be described by a finite state sequential machine. Some specific examples will be illustrated in Chapter 12. We recall the definition (1.135) of the passband B associated with the frequency response R of a channel, with bandwidth given by (1.140), B D B d f . Channel capacity is defined as the maximum of the average mutual information between the input and output signals of the channel [6, 7]. For transmission over an ideal AWGN channel, channel capacity is given in bits per second by C[b=s] D B log2 .1 C 0/ (bit/s)
(6.280)
where 0 is obtained from (6.105) by choosing Bmin D B. Equation (6.280) is a limit derived by Shannon assuming the transmitted signal s.t/ is a Gaussian random process with zero mean and constant power spectral density in the passband B. Using (6.280) and (6.103), we define the maximum spectral efficiency as ¹max D
C[b=s] D log2 .1 C 0/ (bit/s/Hz) B
(6.281)
With reference to a message composed of a sequence of symbols, which belong to an I -dimensional space, the capacity can be expressed in bits per dimension as CD
1 2
log2 .1 C 0 I / (bit/dim)
(6.282)
504
Chapter 6. Modulation theory
obtained assuming a Gaussian distribution of the transmitted symbol sequence, where 0 I is given by (6.106). We give without proof the following fundamental theorem [8, 6]. Theorem 6.2 (Shannon’s theorem) For any rate R I < C, there exists channel coding that allows transmission of information with an arbitrarily small probability of error; such coding does not exist if R I > C. We note that Shannon’s theorem indicates the limits, in terms of encoder-modulator rate or, equivalently, in terms of transmission bit rate (see (6.280)), within which we can develop systems that allow reliable transmission of information, but it does not give any indication about the practical realization of channel coding. The capacity can be upper limited and approximated for small values of 0 I by a linear function, and also lower limited and approximated for large values of 0 I by a logarithmic function as follows: 0I − 1 : C
1 2
log2 .e/ 0 I
(6.283)
0I × 1 : C ½
1 2
log2 .0 I /
(6.284)
Extension of the capacity formula for an AWGN channel to multi-input multi-output (MIMO) systems can be found in [9, 10].
Capacity of a system using amplitude modulation Let us consider an M-PAM system with M ½ 2. The capacity of a real-valued AWGN channel having as input an M-PAM signal is given in bits per dimension by [11] Z C1 3 2 M X pr ja0 . j Þn / (6.285) C D max pn p r ja0 . j Þn / log2 6 M 7 p1 ;:::; p M 1 5 d 4X nD1 pi pr ja0 . j Þi / i D1
where pn indicates the probability of transmission of the symbol a0 D Þn . By the hypothesis of white Gaussian noise, we have ( ) . Þn /2 pr ja0 . j Þn / / exp (6.286) 2¦ I2 With the further hypothesis that only codes with equally likely symbols are of practical interest, the computation of the maximum of C with respect to the probability distribution of the input signal can be omitted. The channel capacity is therefore given by " !# ¾2 M Z C1 M X 2 1 1 X .Þn C ¾ Þi /2 ¾ 2 2¦ Q I C D log2 M e log2 exp d¾ p M nD1 1 2¦ I2 2³¦ I i D1 (6.287) Q The capacity C is illustrated in Figure 6.53, where the Shannon limit given by (6.282), as well as the signal-to-noise ratio given by (6.124) for which a symbol error probability
6.10. Limits imposed by information theory
505
Figure 6.53. Capacity of an ideal AWGN channel for Gaussian and M-PAM input signals. c 1998 IEEE.] [From Forney and Ungerboeck (1998).
equal to 106 is obtained for uncoded transmission, are also indicated [12]. We note that the curves saturate as information cannot be transmitted with a rate larger than R I D log2 M. Let us consider, for example, the uncoded transmission of 1 bit of information per modulation interval by a 2-PAM system, where we have a symbol error probability equal to 106 for 0 I D 13:5 dB. If the number of symbols in the alphabet A doubles, choosing 4-PAM modulation, we see that the coded transmission of 1 bit of information per modulation interval with rate R I D 1 is possible, and an arbitrarily small error probability can be obtained for 0 I D 5 dB. This indicates that a coded 4-PAM system may achieve a gain of about 8:5 dB in signal-to-noise ratio over an uncoded 2-PAM system, at an error probability of 106 . If the number of symbols is further increased, the additional achievable gain is negligible. Therefore we conclude that, by doubling the number of symbols with respect to an uncoded system, we obtain in practice the entire gain that would be expected from the expansion of the input alphabet. We see from Figure 6.53 that for small values of 0 I the choice of a binary alphabet is almost optimum: in fact for 0 I < 1 (0 dB) the capacity given by (6.282) is essentially equivalent to the capacity given by (6.287) with a binary alphabet of input symbols. For large values of 0 I , the capacity of multilevel systems asymptotically approximates a straight line that is parallel to the capacity of the AWGN channel. The asymptotic loss of ³ e=6 (1.53 dB) is due to the choice of a uniform distribution rather than Gaussian for the set of input symbols. To achieve the Shannon limit it is not sufficient to use coding techniques with equally likely input symbols, no matter how sophisticated they are: to bridge the gap
506
Chapter 6. Modulation theory
of 1.53 dB, shaping techniques are required [13] that produce a distribution of the input symbols similar to a Gaussian distribution. Coding techniques for small 0 I and large 0 I are therefore quite different: for low 0 I , the binary codes are almost optimum and the shaping of the constellation is not necessary; for high 0 I instead constellations with more than two elements must be used. To reach capacity, coding must be extended with shaping techniques; moreover, to reach the capacity in channels with limited bandwidth, techniques are required that combine coding, shaping and equalization, as we will see in Chapter 13.
Coding strategies depending on the signal-to-noise ratio The formula of the capacity (6.282) can be expressed as 0 I =.22C 1/ D 1. This relation suggests the definition of the normalized signal-to-noise ratio 0I D
0I 1
22R I
(6.288)
for a given R I given by (6.93). For a scheme that achieves the capacity, R I is equal to the capacity of the channel C and 0 I D 1 (0 dB); if R I < C, as it must be in practice, then 0 I > 1. Therefore the value of 0 I indicates how far from the Shannon limit a system operates, or, in other words, the gap that separates the system from capacity. We now consider two cases. High signal-to-noise ratios. We note from Figure 6.53 that for high values of 0 I it is possible to find coding methods that allow reliable transmission of several bits per dimension. For an uncoded M-PAM system, R I D log2 M
(6.289)
bits of information are mapped into each transmitted symbol. The average symbol error probability is given by (6.124), ! r 3 1 Pe D 2 1 Q 0I (6.290) M M2 1 We note that Pe is function only of M and 0 I . Moreover, using (6.289) and (6.288) we obtain 0I D
0I M2 1
For large M, Pe can therefore be expressed as q q 1 N Q Pe D 2 1 30 I ' 2Q 30 I M
(6.291)
(6.292)
We note that the relation between Pe and 0 I is almost independent of M, if M is large. This relation is used in the comparison illustrated in Figure 6.54 between uncoded systems and the Shannon limit given by 0 I D 1.
6.10. Limits imposed by information theory
507
Figure 6.54. Bit error probability as a function of Eb =N0 for an uncoded 2-PAM system, and symbol error probability as a function of 0 I for an uncoded M-PAM system. [From Forney c 1998 IEEE.] and Ungerboeck (1998).
Low signal-to-noise ratios. For low values of 0 I the capacity is less than 1 and can be approximated by binary transmission systems: consequently we refer to coding methods that employ more binary symbols to obtain the reliable transmission of 1 bit (see Section 6.8). For low values of 0 I it is customary to introduce the following ratio (see (6.107)): 22R I 1 Eb D 0I N0 2R I
(6.293)
We note the following particular cases: ž if R I − 1, then E b =N0 ³ .ln 2/ 0 I ; ž if R I D 1=2, then E b =N0 D 0 I ; ž if R I D 1, then E b =N0 D .3=2/ 0 I . For low 0 I , if the bandwidth can be extended without limit for a given power, for example, by using an orthogonal modulation with T ! 0 (see Example 6.7.3), then by increasing the bandwidth, or equivalently the number of dimensions M of input signals, both 0 I and R I tend to zero. For systems with limited power and unlimited bandwidth, usually E b =N0 is adopted as a figure of merit.
508
Chapter 6. Modulation theory
From (6.293) and the Shannon limit 0 I > 1, we obtain the Shannon limit in terms of E b =N0 for a given rate R I as Eb 22R I 1 > 2R I N0
(6.294)
This lower limit monotonically decreases with R I . In particular, we examine again the three cases: ž if R I tends to zero, the ultimate Shannon limit is given by Eb > ln 2 N0
.1:59 dB/
(6.295)
in other words, equation (6.295) affirms that even though an infinitely large bandwidth is used, reliable transmission can be achieved only if E b =N0 > 1:59 dB; ž if the bandwidth is limited, from (6.294) we find that the Shannon limit in terms of E b =N0 is higher; for example, if R I D 1=2 the limit becomes E b =N0 > 1 (0 dB); ž if R I D 1, as E b =N0 D .3=2/ 0 I , the symbol error probability or bit error probability for an uncoded 2-PAM system can be expressed in two equivalent ways: s ! q 2E b Pbit ³ Q 30 I D Q (6.296) N0
Coding gain Definition 6.2 The coding gain of a coded modulation scheme is equal to the reduction in the value of E b =N0 , or in the value of 0 or 0 I (see (11.9)), that is required to obtain a given probability of error relative to a reference uncoded system. If the modulation rate of the coded system remains unchanged, we typically refer to 0 or 0 I . Let us consider as reference systems a 2-PAM system and an M-PAM system with M × 1, for small and large values of 0 I , respectively. Figure 6.54 illustrates the bit error probability for an uncoded 2-PAM system as a function of both E b =N0 and 0 I . For Pbit D 106 , the reference uncoded 2-PAM system operates at about 12.5 dB from the ultimate Shannon limit. Thus a coding gain up to 12.5 dB is possible, in principle, at this probability of error, if the bandwidth can be sufficiently extended to allow the use of binary codes with R I − 1; if, instead, the bandwidth can be extended only by a factor 2 with respect to an uncoded system, then a binary code with rate R I D 1=2 can yield a coding gain up to about 10.8 dB. Figure 6.54 also shows the symbol error probability for an uncoded M-PAM system as a function of 0 I for large M. For Pe D 106 , a reference uncoded M-PAM system operates at about 9 dB from the Shannon limit: in other words, assuming a limited bandwidth system, the Shannon limit can be achieved by a code having a gain of about 9 dB.
6.11. Optimum receivers for signals with random phase
509
Cut-off rate It is useful to introduce the notion of cut-off rate R0 associated with a channel, for a given modulation and class of codes [2]. We sometimes refer to R0 as a practical upper bound of the transmission bit rate. Therefore for a given channel we can determine the minimum signal-to-noise ratio .E b =N0 /0 below which reliable transmission is not possible, assuming a certain class of coding and decoding techniques. Typically, for codes with rate Rc D 12 (see Chapter 11), .E b =N0 /0 is about 2 dB above the signal-to-noise ratio at which capacity is achieved.
6.11
Optimum receivers for signals with random phase
Let us consider transmission over an AWGN channel of one of the signals sn .t/ D Re[sn.bb/ .t/ e j2³ f 0 t ]
n D 1; 2; : : : ; M
(6.297)
where sn.bb/ is the complex envelope of sn , relative to the carrier frequency f 0 , with support .0; t0 /. If in (6.297) every signal sn.bb/ has a bandwidth smaller than f 0 , then the energy of sn is given by Z t0 Z t0 1 .bb/ 2 jsn .t/j dt En D sn2 .t/ dt D (6.298) 0 0 2 At the receiver, we observe the signal r.t/ D sn .t;'/ C w.t/
(6.299)
where sn .t;'/ D Re[sn.bb/ .t/e j' e j2³ f 0 t ] D Re[sn.bb/Ł .t/e j' e j2³ f 0 t ]
(6.300) n D 1; 2; : : : ; M
In other words, at the receiver we assume the carrier is known, except, however, for a phase ' that we assume to be a uniform r.v. in [³; ³ /. Receivers, which do not rely on the knowledge of the carrier phase, are called non-coherent receivers. We give three examples of signalling schemes that employ non-coherent receivers. Example 6.11.1 (Non-coherent binary FSK) The received signals are expressed as (see also (6.204)): s1 .t;'1 / D A cos.2³ f 1 t C '1 /
0UTh , a0 =1 Ua>0
(6.370)
530
Chapter 6. Modulation theory
and # " .aCb/2 1 .ab/2 2 2 e 1 Q 1 .a; b/ e 2
a>b½0
(6.371)
We observe that in (6.370) the upper bound is very tight, and the lower bound for a given value of b becomes looser as a increases. In (6.371) the lower bound is very tight. A recursive method for computing the Marcum function is given in [19].
6.B. Gray coding
Appendix 6.B
531
Gray coding
In this appendix we give the procedure to construct a list of 2n binary words of n bits, where adjacent words differ in only one bit. The case for n D 1 is immediate. We have two words with two possible values 0 1
(6.372)
The list for n D 2 is constructed by considering first the list of .1=2/22 D 2 words that are obtained by appending a 0 in front of the words of the list (6.372): 0 0 0 1
(6.373)
The remaining two words are obtained by inverting the order of the words in (6.372) and appending a 1 in front: 1 1 1 0
(6.374)
The final result is the following list of words: 0 0 1 1
0 1 1 0
(6.375)
Iterating the procedure for n D 3, the first 4 words are obtained by repeating the list (6.375) and appending a 0 in front of the words of the list. Inverting then the order of the list (6.375) and appending a 1 in front, the final result is the list of 8 words 0 0 0 0 1 1 1 1
0 0 1 1 1 1 0 0
0 1 1 0 0 1 1 0
(6.376)
It is easy to extend this procedure to any value of n. By induction it is just as easy to prove that two adjacent words in each list differ by one bit at most.
532
Chapter 6. Modulation theory
Appendix 6.C
Baseband PPM and PDM
In addition to the widely known PAM, two other baseband pulse modulation techniques are pulse position modulation (PPM) and pulse duration modulation (PDM). PPM consists of a set of pulses whose shift, with respect to a given time reference, depends on the value of the transmitted symbol. We consider the fundamental pulse shape of Figure 6.67 and an alphabet given by A D f0; 1; 2; 3; : : : ; M 1g
(6.377)
The transmitted isolated pulse is T sn .t/ D g0 t n M
n2A
(6.378)
For M D 4 the set of waveforms is represented in Figure 6.68. In PDM, instead, the input symbol determines the duration of the transmitted pulse, that is a multiple of a minimum time duration equal to T =M. For an alphabet A D f1; 2; : : : ; Mg
(6.379)
the transmitted isolated pulse is given by sn .t/ D g0
t n
n2A
(6.380)
where g0 is given in Figure 6.67. The set of PDM waveforms for M D 4 is illustrated in Figure 6.69.
Signal-to-noise ratio In both PPM and PDM the information lies in the position of the fronts of the transmitted pulses: therefore demodulation consists in finding the fronts of the pulses, which are disturbed by noise. If the channel bandwidth were infinite, one could receive perfectly rectangular pulses. In practice the received pulse, with amplitude equal to A, has a rise time t R different from g (t) 0
0 T/M
T t
Figure 6.67. Fundamental pulse shape of PPM.
6.C. Baseband PPM and PDM
533
n =1
t
T/4
n =2
t
2T/4
n =3
t
3T/4
n =4
T t
0
Figure 6.68. PPM waveforms for M D 4.
n =1
t
T/4
n=2
t
2T/4
n=3
t
3T/4
n =4
0
T
t
Figure 6.69. PDM waveforms for M D 4.
534
Chapter 6. Modulation theory
Figure 6.70. PPM and PDM demodulation in the presence of noise.
zero, as the channel has a finite bandwidth B, and noise disturbs the reception of the pulse, as illustrated in Figure 6.70. The detection of the front of a pulse is obtained by establishing the instant ti in which the received signal, pulse plus noise, crosses a given threshold. The error ".ti / is related to the noise w.ti /, amplitude A, and rise time t R of the received pulse: w.ti / ".ti / D tR A
(6.381)
Assuming the noise stationary with PSD N0 =2 over the channel passband with bandwidth B, the mean-square error is given by 2
E[" ] D
tR A
2
2
E[w ] D
tR A
2 N0 B
(6.382)
We consider the following approximated expression that links the rise time to the bandwidth of the pulse tR '
1 2B
(6.383)
Substitution of the above result in (6.382) yields E[" 2 ] '
N0 4A2 B
(6.384)
On the other hand, assuming the average duration of the pulses is −0 , the signal-to-noise ratio is given by (6.105) with Bmin D 1=.2T /, that is 0D
2E sCh N0
(6.385)
6.C. Baseband PPM and PDM
535
where E sCh D −0 A2
(6.386)
Finally, substitution of (6.385) in (6.384) yields E[" 2 ] D
−0 1 2B 0
(6.387)
For a more in-depth analysis we refer to [20], where a trade-off between the channel bandwidth and the signal-to-noise ratio at the decision point (see (6.387)) is illustrated.
536
Chapter 6. Modulation theory
Appendix 6.D
Walsh codes
We illustrate a procedure to obtain orthogonal binary sequences, with values f1; 1g, of length 2m . We consider 2m ð 2m Hadamard matrices Am , with binary elements from the set f0; 1g. For the first orders, we have A0D[0] ½ 0 0 A1D 0 1
1
(6.388) (6.389)
1
0
0
-1
-1 0
8Tc
1
0
8Tc
0
8Tc
0
8Tc
0
8Tc
1
0
0
-1
-1 0
8Tc
1
1
0
0
-1
-1 0
8Tc
1
1
0
0
-1
-1 0
8Tc
Figure 6.71. Eight orthogonal signals obtained from the Walsh code of length 8.
6.D. Walsh codes
537 2
0 60 A2D6 40 0 2 0 60 6 60 6 60 A3D6 60 6 60 6 40 0
0 1 0 1
0 0 1 1
3 0 17 7 15 0
0 1 0 1 0 1 0 1
0 0 1 1 0 0 1 1
0 1 1 0 0 1 1 0
0 0 0 0 1 1 1 1
(6.390)
0 1 0 1 1 0 1 0
0 0 1 1 1 1 0 0
0 1 1 0 1 0 0 1
3 7 7 7 7 7 7 7 7 7 7 5
(6.391)
In general the construction is recursive AmC1 D
Am Am Nm Am A
½ (6.392)
N m denotes the matrix that is obtained by taking the 1’s complement of the elements where A of Am . A Walsh code of length 2m is obtained by taking the rows (or columns) of the Hadamard matrix Am and by mapping 0 into 1. From the construction of Hadamard matrices, it is easily seen that two words of a Walsh code are orthogonal. Figure 6.71 shows the 8 signals obtained with the Walsh code of length 8: the signals are obtained by interpolating the Walsh code sequences by a filter having impulse response wTc .t/ D rect
t Tc =2 Tc
(6.393)
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 7
Transmission over dispersive channels
In this chapter we will reconsider amplitude modulation (PAM and QAM, see Chapter 6) for continuous transmission, taking into account the possibility that the transmission channel may distort the transmitted signal [1, 2]. We will also consider the effects of errors introduced by the digital transmission on a PCM encoded message (see Chapter 5) [3].
7.1
Baseband digital transmission (PAM systems)
Let us briefly examine the fundamental blocks of the baseband digital transmission system illustrated in Figure 7.1. The signals at various points of a ternary PAM system are shown in Figure 7.2. Source. We assume that a source, not explicitly indicated in Figure 7.1, generates a message fb` g composed of a sequence of binary symbols b` 2 f0; 1g, that are emitted every Tb seconds: fb` g D f: : : ; b1 ; b0 ; b1 ; b2 ; : : : g
(7.1)
Usually fb` g is a sequence of i.i.d. symbols. The system bit rate, which is associated with the message fb` g, is equal to (see Section 6.2) 1 Rb D (bit/s) (7.2) Tb
Transmitter Bit mapper. The bit mapper uses a one-to-one map to match a multilevel symbol to an input bit pattern. Let us consider, for example, symbols fak g from a quaternary alphabet, ak 2 A D f3; 1; 1; 3g. To select the values of ak we consider pairs of input bits and map them into quaternary symbols as indicated in Table 7.1. Note that bits are mapped into symbols without introducing redundancy, therefore we speak of uncoded transmission, or in other words the sequence of symbols fak g is not obtained by applying channel coding.1 This situation will be maintained throughout the chapter. 1
We distinguish three types of coding: 1) source or entropy coding; 2) channel coding; and 3) line coding. Their objectives are respectively: 1) “compress” the digital message by lowering the bit rate without losing the original signal information (see Chapter 5); 2) increase the “reliability” of the transmission by inserting redundancy in the transmitted message, so that errors can be detected and/or corrected at the receiver (see Chapters 11 and 12); and 3) “shape” the spectrum of the transmitted signal (see Appendix 7.A).
540
Chapter 7. Transmission over dispersive channels
Figure 7.1. Block diagram of a baseband digital transmission system.
Figure 7.2. Signals at various points of a ternary PAM transmission system with alphabet A D f1; 0; 1g.
For uncoded quaternary transmission the symbol period or modulation interval T is given by T D 2Tb . 1=T is the modulation rate or symbol rate of the system and is measured in Baud: it indicates the number of symbols per second that are transmitted. In general, if the values of ak belong to an alphabet A with M elements, then 1 1 1 D (Baud) (7.3) T log2 M Tb
7.1. Baseband digital transmission (PAM systems)
541
Table 7.1 Example of quaternary bit map.
b2k
b2kC1
ak
0 0 1 1
3 1 1 3
0 1 1 0
We note that in Section 6.3 we considered an alphabet whose elements were indices, that is ak 2 f1; 2; : : : ; Mg. Now the values of ak are associated with fÞn g, n D 1; : : : ; M, that is ak 2 A D f.M 1/; : : : ; 1; 1; : : : ; .M 1/g. Modulator. For a PAM system, see (6.109), the modulator associates the symbol ak with the amplitude of a given pulse h T x : ak ! ak h T x .t kT /
(7.4)
Therefore the modulated signal s.t/ that is input to the transmission channel is given by C1 X
s.t/ D
ak h T x .t kT /
(7.5)
kD1
Transmission channel The transmission channel is assumed to be linear and time invariant, with impulse response gCh . Therefore the desired signal at the output of the transmission channel still has a PAM structure. From the relation sCh .t/ D gCh Ł s.t/
(7.6)
qCh .t/ D h T x Ł gCh .t/
(7.7)
we define
then we have sCh .t/ D
C1 X
ak qCh .t kT /
(7.8)
kD1
The transmission channel introduces an effective noise w. Therefore the signal at the input of the receive filter is given by: r.t/ D sCh .t/ C w.t/
(7.9)
542
Chapter 7. Transmission over dispersive channels
Receiver The receiver consists of three functional blocks: 1. Amplifier-equalizer filter. This block is assumed linear and time invariant with impulse response g Rc . Then the desired signal is given by: s R .t/ D g Rc Ł sCh .t/
(7.10)
Let the overall impulse response of the system be q R .t/ D qCh Ł g Rc .t/ D h T x Ł gCh Ł g Rc .t/
(7.11)
then s R .t/ D
C1 X
ak q R .t kT /
(7.12)
kD1
In the presence of noise, r R .t/ D s R .t/ C w R .t/
(7.13)
where w R .t/ D w Ł g Rc .t/. 2. Sampler. If the duration of q R .t/ is confined to a modulation interval, then the various pulses do not overlap and, in the absence of noise, sampling at instants t0 CkT yields:2 rk D r R .t0 C kT / D ak q R .t0 / D ak h 0
(7.14)
where h 0 D q R .t0 / is the amplitude of the overall impulse response at the sampling instant t0 . The parameter t0 is called timing phase, and its choice is fundamental for system performance. 3. Threshold detector. From the sequence frk g we detect the transmitted sequence fak g. The simplest structure is the instantaneous non-linear threshold detector: aO k D Q[rk ]
(7.15)
where Q[rk ] is the quantizer characteristic with rk 2 < and aO k 2 A, alphabet of ak . An example of quantizer characteristic for A D f1; 0; 1g is given in Figure 7.3. From the sequence faO k g, using an inverse bit mapper, the detected binary information message fbO` g is obtained. 2
To simplify the notation, the sample index k, associated with the instant t0 C kT , here appears as a subscript.
7.1. Baseband digital transmission (PAM systems)
543
^a = Q[r ] k k 1
h0 2 0
−1
h0 2
rk
Figure 7.3. Characteristic of a threshold detector for ternary symbols with alphabet A D f1; 0; 1g, and amplitude h0 of the overall impulse response.
We recall that the receiver structure described above was optimized in Chapter 6 for an ideal AWGN channel.
Power spectral density of a PAM signal The PAM signals found in various points of the system, s R .t/, sCh .t/, and s.t/, have the following structure: s.t/ D
C1 X
ak q.t kT /
(7.16)
kD1
where q.t/ is the impulse response of a suitable filter. In other words, a PAM signal s.t/ may be regarded as a signal generated by an interpolator filter with impulse response q.t/, t 2 0 the spectral density of w, Pw . f /, is an even function around the frequency f 0 , then the real and imaginary parts of w.bb/ .t/ D w I .t/ C jw Q .t/ have a power spectral density that is given by ( 2Pw . f C f 0 / f ½ f0 1 (7.36) Pw I . f / D Pw Q . f / D Pw.bb/ . f / D 2 0 elsewhere Moreover, Pw I w Q . f / D 0, that is w I ? w Q , hence ¦w2 I D ¦w2 Q D 12 ¦w2 .bb/ D ¦w2
(7.37)
To simplify the analysis, for the study of a QAM system we will adopt the PAM model of Figure 7.1, assuming that all signals and filters are in general complex. We note that p the factor .1=2/e j .'1 '0 / appears in Figure 7.11. We will include the factor e j .'p1 '0 / = 2 in the impulse response of the transmission channel gCh , and the factor 1= 2 in the impulse response g Rc . Consequently the additive noise has a spectral density equal to .1=2/Pw.bb/ . f / D 2Pw . f C f 0 / for f ½ f 0 . Therefore the scheme of Figure 7.1 holds also for QAM in the presence of additive noise: the only difference is that in the case of a QAM system the noise is complex-valued with orthogonal in-phase and quadrature components, each having spectral density Pw . f C f 0 / for f ½ f 0 . Hence the scheme of Figure 7.12 is a reference scheme for both PAM and QAM, where 8 for PAM > < GCh . f / GC . f / D e j .'1 '0 / (7.38) > : GCh . f C f 0 /1. f C f 0 / for QAM p 2 We note that for QAM we have gC .t/ D
e j .'1 '0 / .bb/ gCh .t/ p 2 2
(7.39)
7.3. Baseband equivalent model of a QAM system
551
Figure 7.12. Baseband equivalent model of PAM and QAM transmission systems.
With reference to the scheme of Figure 7.9, the relation between the impulse responses of the receive filters is given by g Rc .t/ D
p1 g Rc .t/ 2
(7.40)
In the following, to simplify the notation, the filter g Rc will be indicated in many passband schemes simply as g Rc . We summarize the definitions of the various signals in QAM systems. 1. Sequence of input symbols, fak g, sequence of symbols with values from a complexvalued alphabet A. In PAM systems, the symbols of the sequence fak g assume real values. 2. Modulated signal,5 s.t/ D
C1 X
ak h T x .t kT /
(7.41)
kD1
3. Signal at the channel output, sC .t/ D
C1 X
ak qC .t kT /
qC .t/ D h T x Ł gC .t/
(7.42)
kD1
4. Circularly-symmetric, complex-valued, additive Gaussian noise, wC .t/ D w0I .t/ C jw0Q .t/ , with spectral density PwC . In the case of white noise it is:6 Pw0I . f / D Pw0Q . f / D
N0 (V2 /Hz) 2
(7.43)
and PwC . f / D N0 (V2 /Hz)
(7.44)
In the model of PAM systems, only the component w0I is considered. 5 6
We point out that for QAM s.t/ is in fact s .bb/ .t/. In fact (7.43) should include the condition f > f 0 . Because the bandwidth of g Rc is smaller than f 0 , this condition can be omitted.
552
Chapter 7. Transmission over dispersive channels
5. Received or observed signal, rC .t/ D sC .t/ C wC .t/
(7.45)
6. Signal at the output of the complex-valued amplifier-equalizer filter g Rc , r R .t/ D s R .t/ C w R .t/
(7.46)
where C1 X
s R .t/ D
ak q R .t kT /
(7.47)
kD1
with q R .t/ D qC Ł g Rc .t/ and w R .t/ D wC Ł g Rc .t/
(7.48)
In PAM systems, g Rc is a real-valued filter. 7. Signal at the decision point at instant t0 C kT , yk D r R .t0 C kT /
(7.49)
faO k g
(7.50)
8. Sequence of detected symbols,
Signal-to-noise ratio The performance of a system is expressed as a function of the signal-to-noise ratio 0 defined in (6.105), that we recall here. In general, with reference to the schemes of Figures 7.1 and 7.5, for a channel output signal, sCh , having minimum bandwidth Bmin , and assuming the noise w white with Pw . f / D N0 =2, we have 2 .t/] E[sCh MsCh E sCh D D .N0 =2/2Bmin N0 Bmin N0 .Bmin T / in the cases of PAM and QAM systems.
0D We express now E sCh
(7.51)
PAM systems. For an i.i.d. input symbol sequence, using (1.399) we have E sCh D Ma E qCh
(7.52)
and, for Bmin D 1=.2T /, we obtain Ma E qCh N0 =2 where Ma is the statistical power of the data and E qCh is the energy of the pulse h T x Ł gCh . Because for PAM, observing (7.38), we get qCh D qC , then (7.53) expressed as Ma E qC 0D N0 =2 0D
(7.53) qCh D can be (7.54)
7.3. Baseband equivalent model of a QAM system
553
QAM systems. From (1.295), using (7.38), we obtain .bb/ 2 .t/] D 12 E[jsCh .t/j2 ] D E[jsC .t/j2 ] E[sCh
(7.55)
Hence, as Bmin D 1=T , (7.51) becomes Ma E qC N0
(7.56)
Ma E qC =2 N0 =2
(7.57)
0D We note that (7.56), expressed as7 0D
represents the ratio between the energy per component of sC , given by T E[.Re[sC .t/]/2 ] and T E[.Im[sC .t/]/2 ], and the PSD of the noise components, equal to N0 =2. Then (7.56), observing also (7.38), coincides with (7.53) of PAM systems.
7.3.2
Characterization of system elements
We consider some characteristics of the signals in the scheme of Figure 7.12.
Transmitter The choice of the transmit pulse is quite important because it determines the bandwidth of the system (see (7.17) and (7.28)). Two choices are shown in Figure 7.13, where t T2 1. h T x .t/ D wT .t/ D rect T , with wide spectrum; 2. h T x .t/ with longer duration and smaller bandwidth.
Transmission channel The transmission channel is modelled as a time invariant linear system. Therefore it is represented by a filter having impulse response gCh . As described in Chapter 4, the majority of channels are characterized by frequency responses having a null at DC. Therefore the shape of the frequency response GCh . f / is as represented in Figure 7.14, where the passband goes from f 1 to f 2 . For transmission over cables, f 1 may be of the order of a few hundred Hertz, whereas for radio links, f 1 may be in the range of MHz or GHz. Consequently, PAM (possibly using a line code) as well as QAM transmission systems may be considered over cables; for transmission over radio, instead, a PAM signal needs to be translated in frequency (PAM-DSB or PAM-SSB), or a QAM system may be used, assuming as carrier frequency f 0 the center frequency of the passband ( f 1 ; f 2 ). In any case, the channel is bandlimited with a finite bandwidth f 2 f 1 . 7
The term Ma E qC =2 represents the energy of both Re[sC .t/] and Im[sC .t/], assuming that sC .t/ is circularly symmetric (see (1.407)).
554
Chapter 7. Transmission over dispersive channels
Figure 7.13. Two examples of transmit pulse hTx .
With reference to the general model of Figure 7.12, we adopt the polar notation for GC : GC . f / D jGC . f /je j arg GC . f /
(7.58)
Let B be the bandwidth of s.t/. According to (1.144), a channel presents ideal characteristics, known as Heaviside conditions for the absence of distortion, if the following two properties are satisfied: 1. the magnitude response is a constant for j f j < B, jGC . f /j D G0
for j f j < B
(7.59)
2. the phase response is proportional to f for j f j < B, arg GC . f / D 2³ f t0
for j f j < B
(7.60)
Under these conditions, s is reproduced at the output of the channel without distortion, that is: sC .t/ D G0 s.t t0 /
(7.61)
In practice, channels introduce both “amplitude distortion” and “phase distortion”. An example of frequency response of a radio channel is given in Figure 4.32: the overall effect is that the signal sC .t/ may be very different from s.t/. In short, for channels encountered in practice conditions (7.59) and (7.60) are too stringent; for PAM and QAM transmission systems we will refer instead to the Nyquist criterion (7.79).
7.3. Baseband equivalent model of a QAM system
555
Figure 7.14. Frequency response of the transmission channel.
Receiver We return to the receiver structure of Figure 7.12, consisting of a filter g Rc followed by a sampler with sampling rate 1=T , and a data detector. In general, if the frequency response of the receive filter G Rc . f / contains a factor C.e j2³ f T /, periodic of period 1=T , such that the following factorization holds: G Rc . f / D G M . f /C.e j2³ f T /
(7.62)
where G M . f / is a generic function, then the filter-sampler block before the data detector of Figure 7.12 can be represented as in Figure 7.15, where the sampler is followed by a discrete-time filter. It is easy to prove that in the two systems the relation between rC .t/ and yk is the same. Ideally, in the system of Figure 7.15 yk should be equal to ak . In practice, as illustrated in Figure 7.16, linear distortion and additive noise, the only disturbances considered here, may determine a significant deviation of yk from the desired symbol ak .
556
Chapter 7. Transmission over dispersive channels
Figure 7.15. Receiver structure with analog and discrete-time filters. 5
3
4 2
3
2 1
k,Q
0
y
y
k,Q
1
0
−1 −1
−2
−3 −2
−4
−3 −3
−2
0
−1
y k,I
2
1
3
−5 −5
−4
−3
−2
−1
0
y k,I
1
2
3
4
5
(b) 16-QAM.
(a) QPSK.
Figure 7.16. Values of yk D yk,I C jyk,Q , in the presence of noise and linear distortion.
The last element in the receiver is the data detector. One of the simplest data detectors is the threshold detector, that associates with each value of yk a possible value of ak in the constellation. Using the rule of deciding for the symbol closest to the sample yk , the decision regions for a QPSK system and a 16-QAM system are illustrated in Figure 7.17.
7.3.3
Intersymbol interference
Discrete-time equivalent system From (7.46) we define s R;k D s R .t0 C kT / D
C1 X
ai q R .t0 C .k i/T /
(7.63)
i D1
and w R;k D w R .t0 C kT /
(7.64)
Then, from (7.49), at the decision point the generic sample is expressed as yk D s R;k C w R:k
(7.65)
7.3. Baseband equivalent model of a QAM system
557
yk,Q 3
1 yk,Q
-3
-1
1
3 yk,I
-1 yk,I -3
(a) QPSK.
(b) 16-QAM.
Figure 7.17. Decision regions for a QPSK system and a 16-QAM system.
Introducing the version of q R , time-shifted by t0 , as h.t/ D q R .t0 C t/
(7.66)
h i D h.i T / D q R .t0 C i T /
(7.67)
and defining
it follows that s R;k D
C1 X
ai h ki D ak h 0 C ik
(7.68)
i D1
where ik D
C1 X
ai h ki D Ð Ð Ð C h 1 akC1 C h 1 ak1 C h 2 ak2 C Ð Ð Ð
(7.69)
i D1; i 6Dk
represents the intersymbol interference (ISI). The coefficients fh i gi 6D0 are called interferers. Moreover (7.65) becomes yk D ak h 0 C ik C w R;k
(7.70)
We observe that, even in the absence of noise, the detection of ak from yk by a threshold detector takes place in the presence of the term ik , which behaves as a disturbance with respect to the desired term ak h 0 . For the analysis, it is often convenient to approximate ik as noise with a Gaussian distribution: the more numerous and similar in amplitude are the interferers, the more valid
558
Chapter 7. Transmission over dispersive channels
Figure 7.18. Discrete-time equivalent scheme, with period T, of a QAM system.
is this approximation. In the case of i.i.d. symbols, the first two moments of ik are easily determined. C1 X Mean value of ik : m i D ma hi (7.71) i D1; i 6D0
¦i2 D ¦a2
Variance of ik :
C1 X
jh i j2
(7.72)
i D1; i 6D0
From (7.65), with fs R;k g given by (7.68), we derive the discrete-time equivalent scheme, with period T (see Figure 7.18), that relates the signal at the decision point to the data transmitted over a discrete-time channel with impulse response given by the sequence fh i g, called overall discrete-time equivalent impulse response of the system. Concerning the additive noise fw R;k g,8 being Pw R . f / D PwC . f /jG Rc . f /j2
(7.73)
the PSD of fw R;k g is given by Pw R;k . f / D
C1 X `D1
Pw R
f `
1 T
In any case, the variance of w R;k is equal to that of w R and is given by Z C1 ¦w2 R;k D ¦w2 R D PwC . f /jG Rc . f /j2 d f
(7.74)
(7.75)
1
In particular, the variance per dimension of the noise is given by PAM
¦ I2 D E[w 2R;k ] D ¦w2 R
(7.76)
QAM
¦ I2 D E[.Re[w R;k ]/2 ] D E[.I m[w R;k ]/2 ] D 12 ¦w2 R
(7.77)
In the case of PAM (QAM) transmission over a channel with white noise, where PwC . f / D N0 =2 (N0 ), (7.75) yields a variance per dimension equal to ¦ I2 D
8
See Observation 1.6 on page 62.
N0 E g Rc 2
(7.78)
7.3. Baseband equivalent model of a QAM system
559
where E g Rc is the energy of the receive filter. We observe that (7.78) holds for PAM as well as for QAM.
Nyquist pulses The problem we wish to address consists in finding the conditions on the various filters of the system, so that, in the absence of noise, yk is a replica of ak . The solution is the Nyquist criterion for the absence of distortion in digital transmission. From (7.68), to obtain yk D ak it must be: Nyquist criterion in the time domain (
h0 D 1 hi D 0
(7.79)
i 6D 0
and ISI vanishes. A pulse h.t/ that satisfies the conditions (7.79) is said to be a Nyquist pulse with modulation interval T . The conditions (7.79) have their equivalent in the frequency domain. They can be derived using the Fourier transform of the sequence fh i g (1.90), C1 X
h i e j2³ f i T D
i D1
C1 1 X ` H f T `D1 T
(7.80)
where H. f / is the Fourier transform of h.t/. From the conditions (7.79) the left-hand side of (7.80) is equal to 1, hence the condition for the absence of ISI is formulated in the frequency domain for the generic pulse h as: Nyquist criterion in the frequency domain C1 X
H
`D1
` f T
DT
(7.81)
From (7.81) we deduce an important fact: the Nyquist pulse with minimum bandwidth is given by: h.t/ D h 0 sinc
t T
F
! H. f / D Th 0 rect
f 1=T
(7.82)
Definition 7.1 The frequency 1=.2T /, which coincides with half of the modulation frequency, is called Nyquist frequency. A family of Nyquist pulses widely used in telecommunications is composed of the raised cosine pulses whose time and frequency plots, for three values of the parameter ², are illustrated in Figure 7.19a.
560
Chapter 7. Transmission over dispersive channels
Figure 7.19. Time and frequency plots of raised cosine and square root raised cosine pulses for three values of the roll-off factor ².
We define 8 > > 1 > > > > > >
2 ² > > > > > > > :0
1² 2
1² 1C² < jxj 2 2 jxj >
(7.83)
1C² 2
then H. f / D T rcos
f ;² 1=T
(7.84)
7.3. Baseband equivalent model of a QAM system
561
with inverse Fourier transform ½ t 1 t 1 t ³ sinc ² C C sinc ² h.t/ D sinc T 4 T 2 T 2 t t 1 D sinc cos ³² T T t 2 1 2² T
(7.85)
It is easily proven that, from (7.83), the area of H. f / in (7.84) is equal to one, that is Z C1 f (7.86) ;² df D 1 T rcos h.0/ D 1=T 1 and the energy is Z C1 f ² ² 2 2 Eh D ;² df D T 1 D H.0/h.0/ 1 T rcos 1=T 4 4 1
(7.87)
We note that, from (7.84), the bandwidth of the baseband equivalent system is equal to .1 C ²/=.2T /. Consequently, for a QAM system the required bandwidth is .1 C ²/=T . Later we will also refer to square root raised cosine pulses, with frequency response given by s f (7.88) ;² H. f / D T rcos 1=T and inverse Fourier transform ½ ½ t 1 t 1 t C ² cos ³ C sinc ² C h.t/ D .1 ²/ sinc .1 ²/ T T 4 T 4 ½ t 1 1 t C ² cos ³ sinc ² T 4 T 4 ½ ½ t t t C 4² cos ³.1 C ²/ sin ³ .1 ²/ T T T " D 2 # t t ³ 1 4² T T
(7.89)
In this case Z
C1
h.0/ D 1
s
4 f ;² df D 1 ² 1 T rcos 1=T ³
and the pulse energy is given by Z Eh D
C1 1
T 2 rcos
f ;² 1=T
(7.90)
df D T
(7.91)
562
Chapter 7. Transmission over dispersive channels
We note that H. f / in (7.88) is not the frequency response of a Nyquist pulse. Plots of h.t/ and H. f /, given respectively by (7.89) and (7.88), for various values of ² are shown in Figure 7.19b. The parameter ², called excess bandwidth parameter or roll-off factor, is in the range between 0 and 1. We note that ² determines how fast the pulse decays in time. Observation 7.1 From the Nyquist conditions we deduce that: 1. a data sequence can be transmitted with modulation rate 1=T without errors if H. f / satisfies the Nyquist criterion and there is no noise; 2. the channel, with frequency response GC , must have a bandwidth equal to at least 1=.2T /, otherwise intersymbol interference cannot be avoided.
Eye diagram From (7.68) we observe that if the samples fh i g, for i 6D 0, are not sufficiently small with respect to h 0 , the ISI may result a dominant disturbance with respect to noise and impair the performance of the system. On the other hand, from (7.66) and (7.67) the discrete-time impulse response fh i g depends on the choice of the timing phase t0 (see Chapter 14) and on the pulse shape q R . In the absence of noise, at the decision point the sample y0 , as a function of t0 , is given by y0 D y.t0 / D
C1 X
ai q R .t0 i T /
i D1
(7.92)
D a0 q R .t0 / C i0 .t0 / where i0 .t0 / D
C1 X
ai q R .t0 i T /
i D1; i 6D0
(7.93)
D Ð Ð Ð C a1 q R .t0 C T / C a1 q R .t0 T / C a2 q R .t0 2T / C Ð Ð Ð is the ISI. We now illustrate, through an example, a graphic method to represent the effect of the choice of t0 for a given pulse q R . We consider a PAM transmission system where y.t0 / is real: for a QAM system, both Re[y.t0 /] and Im[y.t0 /] need to be represented. We consider quaternary transmission with ak D Þn 2 A D f3; 1; 1; 3g
(7.94)
and pulse q R as shown in Figure 7.20. In the absence of ISI, i0 .t0 / D 0 and y0 D a0 q R .t0 /. In relation to each possible value Þn of a0 , the pattern of y0 as a function of t0 is shown in Figure 7.21: it is seen that the possible values of y0 , for Þn 2 A, are further apart, therefore they offer a greater
7.3. Baseband equivalent model of a QAM system
563
1.5
R
q (t)
1
0.5
0
−0.5 −T
0
T
2T
3T
4T
t
Figure 7.20. Pulse shape for the computation of the eye diagram.
3
αn=3 2
αn=1
αnqR(t0)
1
0
α =−1 n
−1
−2
αn=−3 −3
−T
0
T
2T
3T
4T
t0
Figure 7.21. Desired component Þn qR .t0 / as a function of t0 , Þn 2 f3; 1; 1; 3g.
564
Chapter 7. Transmission over dispersive channels
margin against noise in relation to the peak of q R , which in this example occurs at instant t0 D 1:5T . In fact, for a given t0 and for a given message : : : ; a1 ; a1 ; a2 ; : : : , it may result in i0 .t0 / 6D 0, and this value is added to the desired sample a0 q R .t0 /. The range of variations of y0 .t0 / around the desired sample Þn q R .t0 / is determined by the values imax .t0 ; Þn / D imin .t0 ; Þn / D
max
i0 .t0 /
(7.95)
min
i0 .t0 /
(7.96)
fak g; a0 DÞn fak g; a0 DÞn
The eye diagram is characterized by the 2M profiles ( imax .t0 ; Þn / Þn q R .t0 / C Þn 2 A imin .t0 ; Þn /
(7.97)
If the symbols fak g are statistically independent with balanced values, that is both Þn and Þn belong to A, defining Þmax D max Þn n
(7.98)
and iabs .t0 / D Þmax
C1 X
jq R .t0 i T /j
(7.99)
i D1; i 6D0
we have that imax .t0 / D iabs .t0 / imin .t0 / D iabs .t0 /
(7.100) (7.101)
We note that both functions do not depend on a0 D Þn . For the considered pulse, the eye diagram is given in Figure 7.22. We observe that as a result of the presence of ISI, the values of y0 may be very close to each other, and therefore reduce considerably the margin against noise. We also note that, in general, the timing phase that offers the largest margin against noise is not necessarily found in relation to the peak of q R . In this example, however, the choice t0 D 1:5T guarantees the largest margin against noise. In the general case, where there exists correlation between the symbols of the sequence fak g, it is easy to show that imax .t0 ; Þn / iabs .t0 / and imin .t0 ; Þn / ½ iabs .t0 /. Consequently the eye may be wider as compared to the case of i.i.d. symbols. For quaternary transmission, we show in Figure 7.23 the eye diagram obtained with a raised cosine pulse q R , for two values of the roll-off factor. In general, the M 1 “pupils” of the eye diagram have a shape as illustrated in Figure 7.24, where two parameters are identified: the height a and the width b. The height a is an indicator of the noise immunity of the system. The width b indicates the immunity with respect to deviations from the optimum timing phase. For example a raised cosine pulse with ² D 1 offers greater immunity against errors in the choice of t0 as compared to the case ² D 0:125. The price we pay is a larger bandwidth of the transmission channel.
7.3. Baseband equivalent model of a QAM system
565
αnqR(t0)+imax(t0;αn)
3
αnqR(t0)+imin(t0;αn)
2
y0(t0)
1
0
−1
−2
−3
−T
0
T
2T
3T
4T
t0
Figure 7.22. Eye diagram for quaternary transmission and pulse qR of Figure 7.20.
We now illustrate an alternative method to obtain the eye diagram. A long random sequence of symbols fak g is transmitted over the channel, and the portions of the curve y.t/ D s R .t/ relative to the various intervals [t1 ; t1 C T /; [t1 C T; t1 C 2T /; [t1 C 2T; t1 C 3T /; : : : ], are mapped on the same interval, for example, on [t1 ; t1 C T /. Typically, we select t1 so that the center of the eye falls in the center of the interval [t1 ; t1 C T /. Then the contours of the obtained eye diagram correspond to the different profiles (7.97). If the contours of the eye do not appear, it means that for all values of t0 the worst case ISI is larger than the desired component and the eye is shut. We note that, if the pulse q R .t/, t 2 < i D X c .k/2 j j jD0 > : Q i D −sat C .i C 12 /1
(7.125)
(7.126)
In (7.125) c.k/ is the word of b bits transmitted over the binary channel, with components c j .k/ 2 f0; 1g, j D 0; 1; : : : ; b 1.
Figure 7.30. Composite transmission scheme of an analog signal via a binary channel.
7.5. Regenerative PCM repeaters
573
We assume the binary channel symmetric and memoryless. Hence, if we express the generic detected bit at the output of the binary channel as cQ j .k/ 2 f0; 1g, the error probability is given by: P[cQ j .k/ 6D c j .k/] D Pbit
(7.127)
as illustrated in Figure 7.29. If we denote with c.k/ Q D [cQb1 .k/; : : : ; cQ0 .k/] the received word, the bit mapper of the DAC performs the inverse operation of (7.125): c.k/ Q ! sQq .kTc / D Q ıQ
(7.128)
8 b1 X > < ıQ D cQ j .k/2 j jD0 > : Q ıQ D −sat C .Qı C 12 /1
(7.129)
where
Given the one-to-one map between words and quantizer levels, using (7.124) it follows that P[Qsq .kTc / 6D sq .kTc /] D Pe;c ' bPbit
(7.130)
Overall system performance In Figure 7.30 the reconstructed analog signal sQ is different from the transmitted signal s for two reasons: 1. the presence of the quantization noise in the ADC; 2. the errors on the detection of the binary sequence at the output of the binary channel. The quantizer introduces an error eq such that sq .kTc / D s.kTc / C eq .kTc /
(7.131)
and the binary channel reconstructs sq with a certain error eCh , sQq .kTc / D sq .kTc / C eCh .kTc /
(7.132)
Therefore the overall relation is sQq .kTc / D s.kTc / C eq .kTc / C eCh .kTc /
(7.133)
where the two error terms are assumed uncorrelated, as they are to be ascribed to different phenomena. In particular, assuming the quantization noise uniform, from (5.41) it follows Meq D
−2 12 D sat2b 12 3Ð2
(7.134)
574
Chapter 7. Transmission over dispersive channels
The computation of MeCh is somewhat more difficult. First, from (7.126) and (7.132) we have b1 X .cQ j .k/ c j .k//2 j (7.135) eCh .kTc / D 1 jD0
Let the error on the j-th transmitted bit be " j .k/ D cQ j .k/ c j .k/
(7.136)
then (7.135) becomes eCh .kTc / D 1
b1 X
" j .k/2 j
(7.137)
jD0
We note that " j .k/ 2 f1; 0; 1g, with probabilities given by 8 P[" j D 1] D 12 Pbit > > < > > :
P[" j D 1] D
1 2
Pbit
(7.138)
P[" j D 0] D 1 Pbit
Then, observing (7.138), we get E[" j .k/] D 0
(7.139)
and E[" 2j .k/] D 1P[" j .k/ 6D 0] C 0P[" j .k/ D 0] D P[cQ j .k/ 6D c j .k/] D Pbit For a memoryless binary channel E[" j1 .k/" j2 .k/] D
(
E[" 2j1 ] D Pbit
for j1 D j2
0
for j1 6D j2
(7.140)
(7.141)
hence from (7.137) 2 E[eCh .kTc /] D 12 Pbit
b1 X
22 j D 12 Pbit
jD0
22b 1 3
(7.142)
We note that, recalling footnote 3 on page 338, the statistical power of the output signal of an interpolator filter in a DAC is equal to the statistical power of the input samples. Consequently, from (7.133) the output signal-to-noise ratio is given by 3PCM D D D
E[s 2 .t/] E[jQs .t/ s.t/j2 ] E[s 2 .kTc /] E[jQsq .kTc / s.kTc /j2 ] Meq
Ms C MeCh
(7.143)
7.5. Regenerative PCM repeaters
575
55
50
b=8
45
40 b=6
(dB)
35
Λ
PCM
30
25
b=4
20
15 b=2
10
5
0 −8 10
−7
10
−6
10
−5
10
−4
10 P bit
−3
10
−2
10
−1
10
0
10
Figure 7.31. Signal-to-noise ratio of a PCM system as a function of Pbit .
Using (7.134) and (7.142), and for a signal-to-quantization noise ratio 3q D Ms =.12 =12/ (see (5.33)), we get 3PCM D
3q 1 C 4Pbit .22b 1/
(7.144)
We note that usually Pbit is such that Pbit 22b − 1: thus it results 3PCM ' 3q , that is the output error is mainly due to the quantization error. In particular, for a signal s 2 U.−sat ; −sat ] whereby 3q D 22b , equation (7.144) is represented in Figure 7.31 for various values of b. For Pbit < 1=.4 Ð 22b / the output signal is corrupted mainly by the quantization noise, whereas for Pbit > 1=.4 Ð 22b / the output is affected mainly by errors introduced by the binary channel. For example for Pbit D 104 , going from b D 6 to b D 8 bits per sample yields an increment of 3PCM of only 2 dB. We observe that in the general case of non-uniform quantization there are no simple expressions similar to (7.142) and (7.144); however, the above observations remain valid.
7.5.2
Regenerative repeaters
The signal sent over a transmission line is attenuated and corrupted by noise. To cover long distances it is therefore necessary to place repeaters along the transmission line to restore the signal.
576
Chapter 7. Transmission over dispersive channels
Analog transmission The only solution possible in an analog transmission system is to place analog repeaters consisting of amplifiers with suitable filters to restore the level of the signal and eliminate the noise outside the passband of the desired signal. The cascade of amplifiers along a transmission line, however, deteriorates the signal-to-noise ratio. We consider the simplified scheme of Example 4.2.2 on page 271 with ž s.t/, transmitted signal with bandwidth B and available power Ps ; ž sCh .t/, desired signal at the output of transmission section i, with available power PsCh ; ž w.t/, effective noise at the input of repeater i; ž r.t/ D sCh .t/ C w.t/, overall signal at the amplifier input of repeater i; ž sQ .t/, signal at the output of a system with N repeaters. We note that, if ac is the attenuation of the generic section i, then PsCh D
1 Ps ac
(7.145)
In this example both the transmission channel and the various amplifiers do not introduce distortion; the only disturbance in sQ .t/ is due to additive noise introduced by the various devices. For a source at noise temperature T0 , if F A is the noise figure of a single amplifier, the signal-to-noise ratio at the amplifier output of a single section is given by (4.92): 3D
PsCh kT0 F A B
(7.146)
Analogously for N analog repeater sections, as the overall noise figure is equal to F D N Fsr (see (4.77)), the overall signal-to-noise ratio, expressed as 3a D
E[s 2 .t/] E[jQs .t/ s.t/j2 ]
(7.147)
3 Ps D kT0 FB N
(7.148)
is given by 3a D
Obviously in the derivation of (7.148) it is assumed that (4.83) holds, as a statistical power ratio is equated with an effective power ratio. Hence, in a system with analog repeaters, the noise builds up repeater after repeater and the overall signal-to-noise ratio worsens as the number of repeaters increases. Moreover, it must be remembered that in practical systems, possible distortion experienced by the desired signal through the various transmission channels and amplifiers also accumulates, contributing to an increase of the disturbance in sQ .t/.
7.5. Regenerative PCM repeaters
577
Digital transmission In a digital transmission system, as an alternative to the simple amplification of the received signal r.t/, we can resort to the regeneration of the signal. With reference to the scheme of Figure 7.32, given the signal r.t/, the digital message fbO` g is first reconstructed, and then re-transmitted by a modulator. Modeling each regenerative repeater by a memoryless binary symmetric channel (see Definition 6.1 on page 457) with error probability Pbit , and ignoring the probability that a bit undergoes more errors along the various repeaters, the bit error probability at the output of N regenerative repeaters is equal to10 Pbit;N ' 1 .1 Pbit / N ' N Pbit
(7.149)
assuming Pbit − 1, and errors of the different repeaters statistically independent. To obtain an expression of Pbit , it is necessary to specify the type of modulator. Let us consider an M-PAM system; then from (6.125) we get ! r 3 2.M 1/ Q Pbit D 0 (7.150) M log2 M M2 1 where from (6.108)11 PsCh (7.151) kT0 F A Bmin It is interesting to compare the bit error probability at the output of N repeaters in the two cases. r 1/ Q 3 0 (7.152) Analog repeaters: Pbit;N D 2.M M log2 M M2 1 N r 2.M 1/ 3 Regenerative repeaters: Pbit;N D M log M N Q 0 (7.153) M2 1 2 0D
Note that in (7.152) we used (7.148). Even if a regenerative repeater is much more complex than an analog repeater, for a given overall Pbit , regeneration allows a significant saving in the power of the transmitted signal.
Figure 7.32. Basic scheme of digital regeneration.
10 We note that a more accurate study shows that the errors have a Bernoulli distribution [4]. 11 To simplify the notation, we have indicated with the same symbol s Ch the desired signal at the amplifier
input for both analog transmission and digital transmission. Note, however, that in the first case sCh depends linearly on s, whereas in the second it represents the modulated signal that does not depend linearly on s.
578
Chapter 7. Transmission over dispersive channels
Comparison between analog and digital transmission We now compare the analog transmission of a signal s.t/ with the digital transmission, which includes PCM coding of s.t/ and modulation of the message. For PCM coding of s.t/ the bit rate of the message is given by Rb D b 2B
(7.154)
Consequently, for an M-PAM modulator, the modulation interval T is equal to log2 M=Rb , and the minimum bandwidth of the transmission channel is equal to Bmin D
b 1 D B 2T log2 M
(7.155)
We note that the digital transmission of an analog signal may require a considerable expansion of the required bandwidth, if M is small. Obviously, using a more efficient digital representation of waveforms, for example by CELP, and/or a modulator with higher spectral efficiency, for example, by resorting to multilevel transmission, Bmin may result very close to B or even smaller. Using (7.155) in (7.151), from (7.146) we have 0D
log2 M 3 b
(7.156)
The comparison between the two systems is based on the overall signal-to-noise ratio for the same transmitted power and transmission channel characteristics. To simplify the notation, initially we will consider a 2-PAM as modulator. Substituting the value of 0 given by (7.156) for M D 2 in (7.152) and (7.153), and recalling (7.144), valid for a uniform quantizer with 3q D 22b , that is assuming a uniform signal, see (5.44), we get 8 22b > > > q N analog repeaters > > > 3 > < 1 C 4.22b 1/Q bN (7.157) 3PCM D 2b 2 > > > N regenerative repeaters q > > > 3 > : 1 C 4.22b 1/N Q b Or else, using (7.148), we get
3PCM D
8 22b > > > q > > > 3a > < 1 C 4.22b 1/Q b
N analog repeaters
22b > > > q > > > 3a N > : 1 C 4.22b 1/N Q b
N regenerative repeaters
(7.158)
7.5. Regenerative PCM repeaters
579
45 b=7
40 b=6
35 b=5
b=4
25
Λ
PCM
(dB)
30
20
b=3
15 b=2
10
5
0
0
5
10
15
20
25 Λ
a
30
35
40
45
(dB)
Figure 7.33. 3PCM as a function of 3a for analog repeaters and 2-PAM. The parameter b denotes the number of bits for linear PCM representation.
In the case of analog repeaters, the plot of 3PCM as a function of 3a is given in Figure 7.33. We note that 3PCM is typically higher than 3a , as long as a sufficiently large number of bits and 3a larger than 17 dB are considered. However, the PCM system is penalized by the increment of the bandwidth of the transmission channel. Using regenerative repeaters, for example N D 20 in Figure 7.34, 3PCM is always much higher than 3a , assuming an adequate number of bits for PCM coding is used. We note the threshold effect of Pbit as a function of 0 in a digital transmission system: if the ratio 0 is higher than a certain threshold, then Pbit is very small. Consequently, the quantization error becomes predominant at the receiver. While the previous graphs relate 3PCM directly to 3a , in practice it is interesting to determine the minimum value of 3 (or 0) so that 3PCM and 3a reach a certain value, say, of the order of 20–40 dB, depending on the applications. We illustrate in Figure 7.35 these relations by varying the number N of repeaters and using a PCM encoder with b D 7. We show also a comparison for the same required bandwidth, which implies a modulator with M D 2b levels. In this case, with respect to 2-PAM, for the same Pbit the modulator requires an increment of about 6.b 1/ dB in terms of 0; therefore, from (7.156), the increment in terms of 3 is equal to 6.b 1/ 10 log10 .b1/. The curve of 3PCM as a function of 3 for 128-PAM, plotted in Figure 7.35, is shifted to the right by about 28 dB with respect to 2-PAM. Therefore also for the same bandwidth, digital transmission is more efficient than analog transmission if the number of repeaters is large.
580
Chapter 7. Transmission over dispersive channels
45 b=7
40 b=6
35 b=5
30
b=4
20
b=3
Λ PCM (dB)
25
15 b=2
10
5
0
0
5
10
15
20
Λ a (dB)
25
30
35
40
45
Figure 7.34. 3PCM as a function of 3a for 2-PAM transmission and N D 20 regenerative repeaters. The parameter b is the number of bits for linear PCM representation. 45
Λ
(N=10)
Λ
(N=100)
PCM
40 PCM
35
30
Λ PCM , Λ a (dB)
ΛPCM(N=1000)
Λa(N=10)
25
Λa(N=100)
20
Λ (N=1000)
15
a
10
5
0 10
15
20
25
30
35 40 Λ (dB)
45
50
55
60
65
Figure 7.35. 3a for analog transmission obtained by varying the number N of analog repeaters, and 3PCM for digital transmission with 2-PAM and b D 7, obtained by varying the number N of regenerative repeaters, as a function of 3 (signal-to-noise ratio of each repeater section). The dashed line represents 3PCM for 128-PAM and b D 7.
7. Bibliography
581
Figure 7.36. Minimum value of 3 as a function of the number N of regenerative repeaters required to guarantee an overall signal-to-noise ratio of 36 dB, for analog transmission and digital transmission with three different modulators. The number of bits for PCM coding is b D 7.
Finally, for a given objective, 3PCM D 3a D 36 dB
(7.159)
we illustrate in Figure 7.36 the minimum value of 3 as a function of the number of regenerative repeaters, for three different modulators.
Bibliography [1] L. W. Couch, Digital and analog communication systems. Upper Saddle River, NJ: Prentice-Hall, 1997. [2] J. G. Proakis and M. Salehi, Communication system engineering. Englewood Cliffs, NJ: Prentice-Hall, 1994. [3] M. S. Roden, Analog and digital communication systems. Upper Saddle River, NJ: Prentice-Hall, 1996. [4] A. Papoulis, Probability, random variables and stochastic processes. New York: McGraw-Hill, 3rd ed., 1991.
582
Chapter 7. Transmission over dispersive channels
[5] S. Benedetto and E. Biglieri, Principles of digital transmission with wireless applications. New York: Kluwer Academic Publishers, 1999. [6] P. Kabal and P. Pasupathy, “Partial-response signaling”, IEEE Trans. on Communications, vol. 23, pp. 921–934, Sept. 1975. [7] D. L. Duttweiler, J. E. Mazo, and D. G. Messerschmitt, “An upper bound on the error probability in decision-feedback equalization”, IEEE Trans. on Information Theory, vol. 20, pp. 490–497, July 1974. [8] G. Birkoff and S. MacLane, A survey of modern algebra. New York: Macmillan Publishing Company, 3rd ed., 1965. [9] D. G. Messerschmitt and E. A. Lee, Digital communication. Boston, MA: Kluwer Academic Publishers, 2nd ed., 1994. [10] B. R. Saltzberg, “Intersymbol interference error bounds with application to ideal bandlimited signaling”, IEEE Trans. on Information Theory, vol. 9, pp. 563–568, July 1968. [11] R. Gitlin, J. Hayes, and S. Weinstein, Data communication principles. New York: Plenum Press, 1992. [12] M. C. Jeruchim, P. Balaban, and K. S. Shanmugan, Simulation of communication systems. New York: Plenum Press, 1992.
7.A. Line codes for PAM systems
583
Line codes for PAM systems
Appendix 7.A
The functions of line codes are: 1. to shape the spectrum of the transmitted signals, and match it to the characteristics of the channel (see (7.17)); this task may be performed also by the transmit filter; 2. to facilitate synchronization at the receiver, especially in case the information message contains long sequences of ones or zeros; 3. to improve system performance in terms of Pe . This appendix is divided in two parts: in the first, several representations of binary symbols are listed; in the second, partial response systems are introduced. For in-depth study and analysis of spectral properties of line codes we refer to the bibliography, in particular [1, 5].
7.A.1
Line codes
With reference to Figure 7.37, the binary sequence fb` g, b` 2 f0; 1g, could be directly generated by a source, or be the output of a channel encoder. The sequence fak g is produced by a line encoder. The channel input is a PAM signal s.t/, obtained by modulating a rectangular pulse h T x .
Non-return-to-zero (NRZ) format The main feature of the NRZ family is that NRZ signals are antipodal signals: therefore NRZ line codes are characterized by the lowest error probability, for transmission over AWGN channels in the absence of ISI. Four formats are illustrated in Figure 7.38. 1. NRZ level (NRZ-L) or, simply, NRZ: “1” and “0” are represented by two different levels. 2. NRZ mark (NRZ-M): “1” is represented by a level transition, “0” by no level transition. 3. NRZ space (NRZ-S): “1” is represented by no level transition, “0” by a level transition. 4. Dicode NRZ: A change of polarity in the sequence fb` g, “1-0” or “0-1”, is represented by a level transition; every other case is represented by the zero level.
Figure 7.37. PAM transmitter with line encoder.
584
Chapter 7. Transmission over dispersive channels
NRZ−M
NRZ−L
2
2
1
1
0
1
1
0
0
0
1
1
0
1
1
0
0
−1
−1
−2
0
−2 0
10
8
6 t/T
4
2
0
1
1
1
0
0
1
1
0
1
10
8
6 t/T
4
2
Dicode NRZ
NRZ−S
2
2
1
1
0
1
1
0
0
0
1
1
0
1
1
0
0
−1
−1
−2
0
0
2
4
6 t/T
8
10
−2 0
1
0
1
2
1
0
4
0
6 t/T
0
1
1
8
0
1
10
Figure 7.38. NRZ line codes.
Return-to-zero (RZ) format 1. Unipolar RZ: “1” is represented by a pulse having duration equal to half a bit interval, “0” by a zero pulse; we observe that the signal does not have zero mean. This property is usually not desirable, as, for example, for transmission over coaxial cables. 2. Polar RZ: “1” and “0” are represented by opposite pulses with duration equal to half a bit interval. 3. Bipolar RZ or alternate mark inversion (AMI): Bits equal to “1” are represented by rectangular pulses having duration equal to half a bit interval, sequentially alternating in sign, bits equal to “0” by the zero level. 4. Dicode RZ: A change of polarity in the sequence fb` g, “1-0” or “0-1”, is represented by a level transition, using a pulse having duration equal to half a bit interval; every other case is represented by the zero level. RZ line codes are illustrated in Figure 7.39.
Biphase (B-φ) format 1. Biphase level (B--L) or Manchester NRZ: “1” is represented by a transition from high level to low level, “0” by a transition from low level to high level. Long sequences of ones or zeros in the sequence fb` g
7.A. Line codes for PAM systems
585
Polar RZ
Unipolar RZ
2
2 1.5
1
0
1
1
0
0
0
1
1
0
1
1
1
0
1
1
0
0
0
1
1
0
1
1
0
0.5 0
−1 −0.5 −1
−2 0
10
8
6 t/T
4
2
0
Dicode RZ
Bipolar RZ
2
2
1
1
0
1
1
0
0
0
1
1
0
1
1
0
0
−1
−1
−2
10
8
6 t/T
4
2
0
2
4
6 t/T
8
10
−2 0
1
0
1
2
1
0
4
0
6 t/T
0
1
1
8
0
1
10
Figure 7.39. RZ line codes.
do not create synchronization problems. It is easy to see, however, that this line code leads to a doubling of the transmission bandwidth. 2. Biphase mark (B--M) or Manchester 1: A transition occurs at the beginning of every bit interval; “1” is represented by a second transition within the bit interval, “0” is represented by a constant level. 3. Biphase space (B--S): A transition occurs at the beginning of every bit interval; “0” is represented by a second transition within the bit interval, “1” is represented by a constant level. Biphase line codes are illustrated in Figure 7.40.
Delay modulation or Miller code “1” is represented by a transition at midpoint of the bit interval, “0” is represented by a constant level; if “0” is followed by another “0”, a transition occurs at the end of the bit interval. This code shapes the spectrum similar to the Manchester code, but requires a lower bandwidth. The delay modulation line code is illustrated in Figure 7.40.
Block line codes The input sequence fb` g is divided into blocks of K bits. Each block of K bits is then mapped into a block of N symbols belonging to an alphabet of cardinality M, with
586
Chapter 7. Transmission over dispersive channels
Biphase−M
Biphase−L
2
2
1
1
0
1
1
0
0
0
1
1
0
1
1
0
0
−1
−1
−2
0
−2 0
10
8
6 t/T
4
2
0
1
1
0
0
0
1
1
0
1
10
8
6 t/T
4
2
Delay Modulation
Biphase−S
2
2
1
1
0
1
1
0
0
0
1
1
0
1
1
0
0
−1
−1
−2
1
0
2
4
6 t/T
8
−2 0
10
1
0
1
1
2
0
4
0
0
6 t/T
1
1
8
0
1
10
Figure 7.40. B- and delay modulation line codes.
the constraint 2K M N
(7.160)
The KBNT codes are an example of block line codes where the output symbol alphabet is ternary f1; 0; 1g.
Alternate mark inversion (AMI) We consider a differential binary encoder, that is ak D bk bk1
with
bk 2 f0; 1g
(7.161)
At the decoder the bits of the information sequence may be recovered by bOk D aO k C bOk1 Note that ak 2 f1; 0; 1g; in particular ( ak D
š1 0
if bk 6D bk1 if bk D bk1
(7.162)
(7.163)
From (7.161), the relation between the PSDs of the sequences fak g and fbk g is given by Pa . f / D Pb . f / j1 e j2³ f T j2 D Pb . f / 4 sin2 .³ f T /
7.A. Line codes for PAM systems
587
Therefore Pa . f / exhibits zeros at frequencies that are integer multiples of 1=T , in particular at f D 0. Moreover, from (7.161) we have ma D 0, independently of the distribution of fbk g. If the power of the transmitted signals is constrained, a disadvantage of the encoding method (7.161) is a reduced noise immunity with respect to antipodal transmission, that is for ak 2 f1; 1g, because a detector at the receiver must now decide among three levels. Moreover, long sequences of information bits fbOk g that are all equal to 1 or 0 generate sequences of symbols fak g that are all equal: this is not desirable for synchronization. In any case, the biggest problem is the error propagation at the decoder, which, observing (7.162), given that an error occurs in faO k g, generates a sequence of bits fbOk g that are in error until another error occurs in faO k g. This problem can be solved by precoding: from the sequence of bits fbk g we first generate the sequence of bits fck g, with ck 2 f0; 1g, by ck D bk ý ck1
(7.164)
where ý denotes the modulo 2 sum. Next, ak D ck ck1 with ak 2 f1; 0; 1g. Hence, it results in ( š1 ak D 0
if bk D 1 if bk D 0
(7.165)
(7.166)
In other words, a bit bk D 0 is mapped into the symbol ak D 0, and a bit bk D 1 is mapped alternately in ak D C1 or ak D 1. Consequently, from (7.166) decoding may be performed simply by taking the magnitude of the detected symbol: bOk D jaO k j
(7.167)
It is easy to prove that for a message fbk g with statistically independent symbols, and p D P[bk D 1], we have sin2 .³ f T / Pa e j2³ f T D 2 p.1 p/ (7.168) p2 C .1 2 p/ sin2 ³ f T Ð The plot of Pa e j2³ f T is shown in Figure 7.41 for different values of p. Note that the PSD presents a zero at f D 0. Also in this case ma D 0. We observe that the AMI line code is a particular case of the partial response system named dicode [6].
7.A.2
Partial response systems
From Section 7.1, we recall in Figure 7.42 the block diagram of a baseband transmission system, where the symbols fak g belong to the following alphabet12 of cardinality M: ak 2 A D f.M 1/; .M 3/; : : : ; .M 3/; .M 1/g
(7.169)
and w.t/ is an additive white Gaussian noise. 12 In the present analysis only M-PAM systems are considered; for M-QAM systems the results can be extended
to the signals on the I and Q branches.
588
Chapter 7. Transmission over dispersive channels
Figure 7.41. Power spectral density Pa .ej2³ fT / of an AMI encoded message.
ak T
h Tx
s(t)
g
sCh (t)
rCh (t)
g
rR (t)
yk
^a k
Rc
Ch
t 0 +kT w(t) Figure 7.42. Block diagram of a baseband transmission system.
We assume that the transmission channel is ideal: the overall system can then be represented as an interpolator filter having impulse response q.t/ D h T x Ł g Rc .t/
(7.170)
A noise signal w R .t/, obtained by filtering w.t/ by the receive filter, is added to the desired signal. Sampling the received signal at instants t0 CkT yields the sequence fyk g, as illustrated in Figure 7.43a. The discrete-time equivalent of the system is shown in Figure 7.43b, where fh i D q.t0 C i T /g, and w R;k D w R .t0 C kT /. We assume that fh i g is equal to zero for i < 0 and i ½ N . The partial response (PR) polynomial of the system is defined as l.D/ D
N 1 X
li D i
(7.171)
i D0
where the coefficients fli g are equal to the samples fh i g, and D is the unit delay operator.
7.A. Line codes for PAM systems
589
Figure 7.43. Equivalent schemes to the system of Figure 7.42.
ak T
(t)
l(D)
ak
yk
rR (t)
g
^a k
t 0 +kT w R (t) Figure 7.44. PR version of the system of Figure 7.42.
A PR system is illustrated in Figure 7.44, where l.D/ is defined in (7.171), and g is an analog filter satisfying the Nyquist criterion for the absence of ISI, m DT G f T mD1 C1 X
(7.172)
The symbols at the output of the filter l.D/ in Figure 7.44 are given by ak.t/ D
N 1 X
li aki
(7.173)
i D0
Note that the overall scheme of Figure 7.44 is equivalent to that of Figure 7.43a with q.t/ D
N 1 X
li g.t i T /
(7.174)
i D0
Also, observing (7.172), the equivalent discrete-time model is obtained for h i D li . In other words, from (7.174) the system of Figure 7.42 is decomposed into two parts: ž a filter with frequency response l.e j2³ f T /, periodic of period 1=T , that forces the system to have an overall discrete-time impulse response equal to fh i g; ž an analog filter g that does not modify the overall filter h.D/ and limits the system bandwidth. As it will be clear from the analysis, the decomposition of Figure 7.44, on one hand, allows simplification of the study of the properties of the filter h.D/, and, on the other, to design an efficient receiver. The scheme of Figure 7.44 suggests two possible ways to implement the system of Figure 7.42:
590
Chapter 7. Transmission over dispersive channels
ak T
l(D)
ak(t)
(PR)
h Tx
s(t)
g
g (PR)
Ch
rR (t)
Rc
yk
^a k
t 0 +kT
w(t) Figure 7.45. Implementation of a PR system using a digital filter.
1. Analog: the system is implemented in analog form; therefore the transmit filter h T x and the receive filter g Rc must satisfy the relation HT x . f / G Rc . f / D Q. f / D l.e j2³ f T / G. f /
(7.175)
2. Digital: the filter l.D/ is implemented as a component of the transmitter by a digital R/ R/ and receive filter g .P must satisfy the relation filter; then the transmit filter h .P Tx Rc R/ .P R/ H.P T x . f / G Rc . f / D G. f /
(7.176)
The implementation of a PR system using a digital filter is shown in Figure 7.45. Note from (7.172) that in both relations (7.175) and (7.176) g is a Nyquist filter.
The choice of the PR polynomial Several considerations lead to the selection of the polynomial l.D/. a) System bandwidth. With the aim of maximizing the transmission bit rate, many PR systems are designed for minimum bandwidth, i.e. from (7.175) it must be 1 (7.177) 2T Substitution of (7.177) into (7.172) yields the following conditions on the filter g: 8 1 < F 1 t T jfj G. f / D (7.178) ! g.t/ D sinc 2T : T 0 elsewhere l.e j2³ f T / G. f / D 0
jfj>
Correspondingly, observing (7.174) the filter q assumes the expression N 1 X t iT q.t/ D li sinc T i D0
(7.179)
b) Spectral zeros at f D 1=.2T /. From the theory of signals, it is known that if Q. f / and its first .n 1/ derivatives are continuous and the n-th derivative is discontinuous, then jq.t/j asymptotically decays as 1=jtjnC1 . The continuity of Q. f / and of its derivatives helps to reduce the portion of energy contained in the tails of q.t/. It is easily proven that in a minimum bandwidth system, the .n 1/-th derivative of Q. f / is continuous if and only if l.D/ has .1 C D/n as a factor. On the other hand, if l.D/ has a zero of multiplicity greater than one in D D 1, then the transition band of G. f / around f D 1=.2T / can be widened, thus simplifying the design of the analog filters.
7.A. Line codes for PAM systems
591
c) Spectral zeros at f D 0. A transmitted signal with attenuated spectral components at low frequencies is desirable in many cases, e.g. for the implementation of SSB modulators (see Example 1.7.4 on page 58), or for transmission over channels with frequency responses that exhibit a spectral null at the frequency f D 0. Note that a zero of l.D/ in D D 1 corresponds to a zero of l.e j2³ f T / at f D 0. d) Number of output levels. From (7.173), the symbols at the output of the filter l.D/ have an alphabet A.t/ of cardinality M .t/ . If we indicate with n l the number of coefficients of l.D/ different from zero, then the following inequality for M .t/ holds n l .M 1/ C 1 M .t/ M n l
(7.180)
In particular, if the coefficients fli g are all equal, then M .t/ D n l .M 1/ C 1. We note that, if l.D/ contains more than one factor .1 š D/, then n l increases and, observing (7.180), also the number of output levels increases. If the power of the transmitted signal is constrained, detection of the sequence fak.t/ g by a threshold detector will cause a loss in system performance. e) Some examples of minimum bandwidth systems. In the case of minimum bandwidth systems, it is possible to evaluate the expression of Q. f / and q.t/ once the polynomial l.D/ has been selected. As the coefficients fli g are generally symmetric or antisymmetric around i D .N 1/=2, it is convenient to consider the time-shifted pulse .N 1/T q.t/ Q Dq t 2 (7.181) Q f / D e j³ f .N 1/T Q. f / Q. In Table 7.2 the more common polynomials l.D/ are described, as well as the correQ f / and q.t/, sponding expressions of Q. Q and the cardinality M .t/ of the output alphabet A.t/ . In the next three examples, polynomials l.D/ that are often found in practical applications of PR systems are considered. Example 7.A.1 (Dicode filter) The dicode filter introduces a zero at frequency f D 0 and has the following expression l.D/ D 1 D
(7.182)
The frequency response, obtained by setting D D e j2³ f T , is given by l.e j2³ f T / D 2 j e j³ f T sin.³ f T /
(7.183)
Example 7.A.2 (Duobinary filter) The duobinary filter introduces a zero at frequency f D 1=.2T / and has the following expression l.D/ D 1 C D
(7.184)
592
Chapter 7. Transmission over dispersive channels
Table 7.2 Properties of several minimum bandwidth systems. l.D/
Q f / for j f j 1=.2T / Q.
q.t/ Q
M .t/
1C D
2T cos.³ f T /
4T 2 cos.³t=T / ³ T 2 4t 2
2M 1
1 D
j 2T sin.³ f T /
8T t cos.³t=T / ³ 4t 2 T 2
2M 1
1 D2
j 2T sin.2³ f T /
2T 2 sin.³t=T / ³ t2 T 2
2M 1
1 C 2D C D 2
4T cos2 .³ f T /
2T 3 sin.³t=T / ³t T 2 t 2
4M 3
1 C D D2 D3
j 4T cos.³ f T / sin.2³ f T /
1 D D2 C D3
4T sin.³ f T / sin.2³ f T /
16T 2 cos.³t=T /.4t 2 3T 2 / ³ .4t 2 9T 2 /.4t 2 T 2 /
1 2D 2 C D 4
4T sin2 .2³ f T /
2 C D D2
T C T cos.2³ f T / C j 3T sin.2³ f T /
2 D2 D4
T C T cos.4³ f T / C j 3T sin.4³ f T /
cos.³t=T / 64T 3 t ³ .4t 2 9T 2 /.4t 2 T 2 /
8T 3 sin.³t=T / ³t t 2 4T 2 3t T T2 sin.³t=T / 2 ³t t T2 2 2T 3t 2T sin.³t=T / 2 2 ³t t 4T
4M 3
4M 3
4M 3
4M 3
4M 3
The frequency response is given by l.e j2³ f T / D 2e j³ f T cos.³ f T / Observing (7.179) we have q.t/ D sinc
tT t C sinc T T
(7.185)
(7.186)
The plot of the impulse response of a duobinary filter is shown in Figure 7.46 with a continuous line. We notice that the tails of the two sinc functions cancel each other, in line with what was stated at point b) regarding the aymptotical decay of the pulse of a PR system with a zero in D D 1. Example 7.A.3 (Modified duobinary filter) The modified duobinary filter combines the characteristics of duobinary and dicode filters, and has the following expression l.D/ D .1 D/ .1 C D/ D 1 D 2
(7.187)
The frequency response becomes l.e j2³ f T / D 1 e j4³ f T D 2 j e j2³ f T sin.2³ f T /
(7.188)
7.A. Line codes for PAM systems
593
1.5
1
q(t)
0.5
0
−0.5
−1
−3
−2
−1
0
1 t/T
2
3
4
5
Figure 7.46. Plot of q.t/ for duobinary () and modified duobinary (- -) filters.
Using (7.179) it results in q.t/ D sinc
t t 2T sinc T T
(7.189)
The plot of the impulse response of a modified duobinary filter is shown in Figure 7.46 with a dashed line.
f) Transmitted signal spectrum. With reference to the PR system of Figure 7.45, the spectrum of the transmitted signal is given by (see (7.17)) þ þ2 þ1 þ .P R/ j2³ f T þ Ps . f / D þ l.e / HT x . f /þþ Pa . f / (7.190) T R/ For a minimum bandwidth system, with H.P T x . f / given by (7.178), (7.190) simplifies into 8 1 > < jl.e j2³ f T /j2 Pa . f / jfj 2T Ps . f / D (7.191) 1 > :0 jfj > 2T
In Figure 7.47 the PSD of a minimum bandwidth PR system is compared with that of a PAM system. The spectrum of the sequence of symbols fak g is assumed white. For the PR system, a modified duobinary filter is considered, so that the spectrum is obtained as the
594
Chapter 7. Transmission over dispersive channels
Figure 7.47. PSD of a modified duobinary PR system and of a PAM system. R/ 2 2 product of the functions jl.e j2³ f T /j2 D j2 sin.2³ f T /j2 and jH.P T x . f /j D T rect. f T /, plotted with continuous lines. For the PAM system, the transmit filter h T x is a square root raised cosine with roll-off factor ² D 0:5, and the spectrum is plotted with a dashed line.
Symbol detection and error probability We consider the discrete-time equivalent scheme of Figure 7.43b; the signal s R;k can be expressed as a function of symbols fak g and coefficients fli g of the filter l.D/ in the following form s R;k D ak.t/ D l0 ak C
N 1 X
li aki
(7.192)
i D1
The term l0 ak is the desired part of the signal s R;k , whereas the summation represents the ISI term that is often designated as “controlled ISI”, as it is deliberately introduced. The receiver detects the symbols fak g using the sequence of samples fyk D ak.t/ C w R;k g. We discuss four possible solutions.13 1. LE-ZF. A zero-forcing linear equalizer (LE-ZF) having D transform equal to 1=l.D/ is used. At the equalizer output, at instant k the symbol ak plus a noise term is 13 For a first reading it is suggested that only solution 3 is considered. The study of the other solutions should
be postponed until the equalization methods of Chapter 8 are examined.
7.A. Line codes for PAM systems
595
Figure 7.48. Four possible solutions to the detection problem in the presence of controlled ISI.
obtained; the detected symbols faO k g are obtained by an M-level threshold detector, as illustrated in Figure 7.48a. We note, however, that the amplification of noise by the filter 1=l.D/ is infinite at frequencies f such that l.e j2³ f T / D 0. 2. DFE. A second solution resorts to a decision-feedback equalizer (DFE), as shown in Figure 7.48b. An M-level threshold detector is also employed by the DFE, but there is no noise amplification as the ISI is removed by the feedback filter, having D transform equal to 1 l.D/=l0 . We observe that at the decision point the signal yNk has the expression 1 yQk D l0
ak.t/
C w R;k
N 1 X
! li aO ki
(7.193)
i D1
If we indicate with ek D ak aO k a detection error, then substituting (7.192) in (7.193), we obtain ! N 1 X 1 li eki w R;k C (7.194) yQk D ak C l0 i D1 The equation (7.194) shows that a wrong decision negatively influence successive decisions: this phenomenon is known as error propagation. 3. Threshold detector with M .t/ levels. This solution, shown in Figure 7.48c, exploits .t/ the M .t/ -ary nature of the symbols ak , and makes use of a threshold detector with .t/ M levels followed by a LE-ZF. This structure does not lead to noise amplification as solution 1, because the noise is eliminated by the threshold detector; however, there is still the problem of error propagation. 4. Viterbi algorithm. This solution, shown in Figure 7.48d, corresponds to maximumlikelihood sequence detection (MLSD) of fak g. It yields the best performance.
596
Chapter 7. Transmission over dispersive channels
Solution 2 using the DFE is often adopted in practice: in fact it avoids noise amplification and is simpler to implement than the Viterbi algorithm. However, the problem of error propagation remains. In this case, using (7.194) the error probability can be written as þ # "þþ N 1 þ X 1 þ þ P þw R;k C Pe D 1 li eki þ > l0 (7.195) þ þ M i D1 A lower bound Pe;L can be computed for Pe by assuming the error propagation is absent, or setting fek g D 0, 8k, in (7.195). If we denote by ¦w R the standard deviation of the noise w R;k , we obtain l0 1 (7.196) Q Pe;L D 2 1 M ¦w R Assuming w R;k white noise, an upper bound Pe;U is given in [7] in terms of Pe;L : Pe;U D
M N 1 Pe;L .M=.M 1// Pe;L .M N 1 1/ C 1
(7.197)
From (7.197) we observe that the effect of the error propagation is that of increasing the error probability by a factor M N 1 with respect to Pe;L . A solution to the problem of error propagation is represented by precoding, which will be investigated in depth in Chapter 13.
Precoding We make use here of the following two simplifications: 1. the coefficients fli g are integer numbers; 2. the symbols fak g belong to the alphabet A D f0; 1; : : : ; M 1g; this choice is made because arithmetic modulo M is employed. . p/
We define the sequence of precoded symbols faN k g as: ! N 1 X . p/ . p/ li aN ki mod M aN k l0 D ak
(7.198)
i D1
We note that (7.198) has only one solution if and only if l0 and M are relatively prime [8]. In case l0 D Ð Ð Ð D l j1 D 0 mod M, and l j and M are relatively prime, (7.198) becomes ! N 1 X . p/ . p/ aN k j l j D ak li aN ki mod M (7.199) i D jC1
For example, if l.D/ D 2C D D 2 and M D 2, (7.198) is not applicable as l0 mod M D 0. Therefore (7.199) is used.
7.A. Line codes for PAM systems
597
. p/
Applying the PR filter to faN k g we obtain the sequence .t/
ak D
N 1 X
. p/
li aN ki
(7.200)
i D0
From the comparison between (7.198) and (7.200), or in general (7.199), we have the fundamental relation ak.t/ mod M D ak
(7.201)
Equation (7.201) shows that, as in the absence of noise we have yk D ak.t/ , the symbol ak can be detected by considering the received signal yk modulo M; this operation is memoryless, therefore the detection of aO k is independent of the previous detections faO ki g, i D 1; : : : ; N 1. Therefore the problem of error propagation is solved. Moreover, the desired signal is not affected by ISI. If the instantaneous transformation . p/
ak
. p/
D 2aN k
.M 1/
(7.202)
. p/
is applied to the symbols faN k g, then we obtain a sequence of symbols that belong to the . p/ alphabet A. p/ in (7.169). The sequence fak g is then input to the filter l.D/. Precoding consists of the operation (7.198) followed by the transformation (7.202). However, we note that (7.201) is no longer valid. From (7.202), (7.200), and (7.198), we obtain the new decoding operation, given by ! ak.t/ ak D C K mod M (7.203) 2 where K D .M 1/
N 1 X
(7.204)
li
i D0
A PR system with precoding is illustrated in Figure 7.49. The receiver is constituted by a threshold detector with M .t/ levels that provides the symbols faO k.t/ g, followed by a block that realizes (7.203) and yields the detected data faO k g.
Error probability with precoding To evaluate the error probability of a system with precoding, the statistics of the symbols fak.t/ g must be known; it is easy to prove that if the symbols fak g are i.i.d., the symbols fak.t/ g are also i.i.d. ak
a (p) precoder
k
a (t) l(D)
k
yk
^a (t) k
Figure 7.49. PR system with precoding.
decoder
a^ k
598
Chapter 7. Transmission over dispersive channels
If we assume that the cardinality of the set A.t/ is maximum, i.e. M .t/ D M n l , then the output levels are equally spaced and the symbols ak.t/ result equally likely with probability P[ak.t/ D Þ] D
1 M nl
Þ 2 A.t/
(7.205)
In general, however, the symbols fak.t/ g are not equiprobable, because several output levels are redundant, as can be deduced from the following example. Example 7.A.4 (Dicode filter) We assume M D 2, therefore ak D f0; 1g; the precoding law (7.198) is simply an exclusive or and . p/
aN k
. p/
D ak ý aN k1
(7.206)
. p/
The symbols fak g are obtained from (7.202), . p/
ak . p/
they are antipodal as ak are given by
1
(7.207)
D f1; C1g. Finally, the symbols at the output of the filter l.D/ . p/
ak.t/ D ak . p/
. p/
D 2aN k
. p/ . p/ . p/ ak1 D 2 aN k aN k1
(7.208)
. p/
The values of aN k1 , ak , aN k and ak.t/ are given in Table 7.3. We observe that both output levels š2 correspond to the symbol ak D 1 and therefore are redundant; the three levels are not equally likely. The symbol probabilities are given by P[ak.t/ D š2] D P[ak.t/
D 0] D
1 4
(7.209)
1 2
Figure 7.50a shows the precoder that realizes equations (7.206) and (7.207). The decoder, realized as a map that associates the symbol aO k D 1 to š2, and the symbol aO k D 0 to 0, is illustrated in Figure 7.50b. Table 7.3 Precoding for the dicode filter.
aN k1
. p/
ak
aN k
. p/
ak.t/
0 0 1 1
0 1 0 1
0 1 1 0
0 C2 0 2
7.A. Line codes for PAM systems
599
0 1
← ←
(p)
ak
ak
(p) -1 a k +1
D (a) precoder
k
0 2
← ←
^a (t)
0 1
^a k
(b) decoder
Figure 7.50. Precoder and decoder for a dicode filter l.D/ with M D 2.
Alternative interpretation of PR systems Up to now we have considered a general transmission system, and looked for an efficient design method. We now assume that the system is given, i.e. that the transmit filter as well as the receive filter are assigned. The scheme of Figure 7.44 can be regarded as a tool for the optimization of a given system where l.D/ includes the characteristics of the transmit and receive filters: as a result, the symbols fak.t/ g no longer are the transmitted symbols, but are to be interpreted as the symbols that are ideally received. In the light of these considerations, the assumption of an ideal channel can also be removed. In this case the filter l.D/ will also include the ISI introduced by the channel. We observe that the precoding/decoding technique is an alternative equalization method to the DFE that presents the advantage of eliminating error propagation, which can considerably deteriorate system performance. In the following two examples [9], additive white Gaussian noise w R;k D wQ k is assumed, and various systems are studied for the same signal-to-noise ratio at the receiver. Example 7.A.5 (Ideal channel g) a) Antipodal signals. We transmit a sequence of symbols from a binary alphabet, ak 2 f1; 1g. The received signal is yk D ak C wQ A;k
(7.210)
where the variance of the noise is given by ¦w2Q A D ¦ I2 . At the receiver, using a threshold detector with threshold set to zero, we obtain Pbit D Q
1 ¦I
(7.211)
600
Chapter 7. Transmission over dispersive channels
. p/
b) Duobinary signal with precoding. The transmitted signal is now given by ak.t/ D ak C . p/ . p/ ak1 2 f2; 0; 2g, where ak 2 f1; 1g is given by (7.202) and (7.198). The received signal is given by .t/
yk D ak C wQ B;k
(7.212)
where the variance of the noise is ¦w2Q B D 2¦ I2 , as ¦ 2.t/ D 2. ak
At the receiver, using a threshold detector with thresholds set at š1, we have the following conditional error probabilities: 1 .t/ P[E j ak D 0] D 2Q ¦wQ B 1 P[E j ak.t/ D 2] D P[E j ak.t/ D 2] D Q ¦wQ B Consequently, at the detector output we have Pbit D P[aO k 6D ak ] D P[E j ak.t/ D 0] 12 C P[E j ak.t/ D š2] 12 1 D 2Q p 2 ¦I We observe a worsening of about 3 dB in terms of the signal-to-noise ratio with respect to case a). c) Duobinary signal. given by
The transmitted signal is ak.t/ D ak C ak1 . The received signal is yk D ak C ak1 C wQ C;k
(7.213)
where ¦w2Q C D 2¦ I2 . We consider using a receiver that applies MLSD to recover the data; from Example 8.12.1 on page 687 it results in p ! 8 1 (7.214) Pbit D K Q DKQ 2¦wQ C ¦I where K is a constant. We note that the PR system employing MLSD at the receiver achieves a performance similar to that of a system transmitting antipodal signals, as MLSD exploits the correlation between symbols of the sequence fak.t/ g. Example 7.A.6 (Equivalent channel g of the type 1 C D) In this example it is the channel itself that forms a duobinary signal.
7.A. Line codes for PAM systems
601
d) Antipodal signals. Transmitting ak 2 f1; 1g, the received signal is given by yk D ak C ak1 C wQ D;k
(7.215)
where ¦w2Q D D 2¦ I2 . An attempt at pre-equalizing the signal at the transmitter by inserting a filter l.D/ D 1=.1 C D/ D 1 D C D 2 C Ð Ð Ð would yield symbols ak.t/ with unlimited amplitude; therefore such a configuration cannot be used. Equalization at the receiver using the scheme of Figure 7.48a would require a filter of the type 1=.1 C D/, which would lead to unlimited noise enhancement. Therefore we resort to the scheme of Figure 7.48c, where the threshold detector has thresholds set at š1. To avoid error propagation, we precode the message and transmit the . p/ sequence fak g instead of fak g. At the receiver we have . p/
yk D ak
. p/
C ak1 C wQ D;k
We are therefore in the same conditions as in case b), and 1 Pbit D 2Q p 2 ¦I
(7.216)
(7.217)
e) MLSD receiver. To detect the sequence of information bits from the received signal (7.215), MLSD can be adopted. Pbit is in this case given by (7.214).
602
Chapter 7. Transmission over dispersive channels
Appendix 7.B
7.B.1
Computation of Pe for some cases of interest
Pe in the absence of ISI
In the absence of ISI, the signal at the decision point is the type (7.102) yk D h 0 ak C w R;k
ak 2 A
(7.218)
where w R;k is the sample of an additive noise signal. Assuming fw R;k g stationary with probability density function pw .¾ /, from (7.218) for ak D Þn 2 A we have p yk jak .² j Þn / D pw .² h 0 Þn /
(7.219)
Therefore the MAP criterion (6.26) becomes ² 2 Rm
aO k D Þm
if
Þm D arg max pn pw .² h 0 Þn / Þn
(7.220)
We consider now the application of the MAP criterion to an M-PAM system, where Þn D 2n 1 M
n D 1; : : : ; M
(7.221)
The decision regions fRn g, n D 1; : : : ; M, are formed by intervals, or, in general, by the union of intervals, whose boundary points are called decision thresholds f−i g, i D 1; : : : ; M 1. Example 7.B.1 (Determination of the optimum decision threholds) We consider a 4-PAM system with the following symbol probabilities: ¦ ² 3 3 1 1 ; ; ; f p1 ; p2 ; p3 ; p4 g D 20 20 2 5
(7.222)
The noise is assumed to have an exponential probability density function pw .¾ / D
þ j¾ jþ e 2
(7.223)
where þ is a constant; the variance of the noise is given by ¦w2 D 2=þ 2 . The curves pn pw .² h 0 Þn /
n D 1; : : : ; 4
(7.224)
are illustrated in Figure 7.51. We note that, for the choice in (7.222) of the symbols probabilities, the decision thresholds, also shown in Figure 7.51, are obtained from the intersections between curves in (7.224) relative to two adjacent symbols; therefore they are given by the solutions of the M 1 equations pi pw .−i h 0 Þi / D pi C1 pw .−i h 0 Þi C1 /
i D 1; : : : ; M 1
(7.225)
7.B. Computation of Pe for some cases of interest
603
pnpw(ρh oαn), n=1,2,3,4
τ
1
τ
τ
2
ρ
3
Figure 7.51. Optimum thresholds for a 4-PAM system with non-equally likely symbols.
We point out that, if the probability that the symbol ` is sent is very small, p` − 1, the measure of the corresponding decision interval could be equal to zero, and consequently this symbol would never be detected. In this case the decision thresholds will be fewer than M 1. Example 7.B.2 (Computation of Pe for a 4-PAM system) We indicate with Fw .x/ the probability distribution of w R;k : Z x pw .¾ / d¾ Fw .x/ D
(7.226)
1
For a M-PAM system with thresholds −1 ; −2 , and −3 , the probability of correct decision is given by (6.18): Z 4 X pn pw .² h 0 Þn / d² P[C] D Rn
nD1
Z
−1
D p1 1
Z C p3
pw .² h 0 Þ1 / d² C p2
−3 −2
Z
pw .² h 0 Þ3 / d² C p4
−2
pw .² h 0 Þ2 / d²
−1
Z
C1 −3
(7.227) pw .² h 0 Þ4 / d²
D p1 [Fw .−1 h 0 Þ1 /] C p2 [Fw .−2 h 0 Þ2 / Fw .−1 h 0 Þ2 /] C p3 [Fw .−3 h 0 Þ3 / Fw .−2 h 0 Þ3 /] C p4 [1 Fw .−3 h 0 Þ4 /]
604
Chapter 7. Transmission over dispersive channels
We note that, if Fw is a continuous function, optimum thresholds can be obtained by equating to zero the derivative of the expression in (7.227) with respect to −1 ; −2 , and −3 . In the case of equally likely symbols and equidistant thresholds, i.e. −i D h 0 .2i M/
i D 1; : : : ; M 1
(7.228)
equation (7.227) yields 1 P[C] D 1 2 1 Fw .h 0 / M
(7.229)
We note that (7.229) is in agreement with (6.122) obtained for Gaussian noise.
7.B.2
Pe in the presence of ISI
We consider M-PAM transmission in the presence of ISI. We assume the symbols in (7.221) are equally likely and the decision thresholds are of the type given by (7.228). With reference to (7.65), the received signal at the decision point assumes the following expression: yk D h 0 ak C ik C w R;k where ik represents the ISI and is given by X ik D h i aki
(7.230)
(7.231)
i 6D0
and w R;k is Gaussian noise with statistical power ¦ 2 and statistically independent of the i.i.d. symbols of the message fak g. We examine various methods to compute the symbol error probability in the presence of ISI.
Exhaustive method We refer to the case of 4-PAM transmission with Ni D 2 interferers due to one non-zero precursor and one non-zero postcursor. Therefore we have ik D ak1 h 1 C akC1 h 1
(7.232)
We define the vector of symbols that contribute to ISI as a0k D [ak1 ; akC1 ]
(7.233)
Then ik can be written as a function of a 0k as ik D i.a0k /
(7.234)
Therefore, ik is a random variable that assumes values in an alphabet with cardinality L D M Ni D 16.
7.B. Computation of Pe for some cases of interest
605
Starting from (7.230) the error probability can be computed by conditioning with respect to the values assumed by a0k D [Þ .1/ ; Þ .2/ ] D α 2 A2 . For equally likely symbols and thresholds given by (7.228) we have h 0 i.a0k / 1 X P[a0k D α] Pe D 2 1 Q M ¦ α2A2 (7.235) 1 1 X h 0 i.α/ D2 1 Q M L ¦ 2 α2A
This method gives the exact value of the error probability in the presence of interferers, but requires the computation of L terms. This method can be costly, especially if the number of interferers is large: it is therefore convenient to consider approximations of the error probability obtained by simpler computational methods.
Gaussian approximation If interferers have a similar amplitude and their number is large, we can use the central limit theorem and approximate ik as a Gaussian random variable. As the process w R;k is Gaussian, the process z k D ik C w R;k
(7.236)
¦z2 D ¦i2 C ¦ 2
(7.237)
h0 1 Q Pe D 2 1 M ¦z
(7.238)
is also Gaussian with variance
where ¦i2 is given by (7.72). Then
It is seen that this method, although very convenient, is rather pessimistic, especially for large values of 0. As a matter of fact, we observe that the amplitude of ik is limited by the value X imax D .M 1/ jh i j (7.239) i 6D0
whereas the Gaussian approximation implies that the values of ik are unlimited.
Worst-case bound This method substitutes ik with the constant imax defined in (7.239). In this case Pe is equal to h 0 imax 1 Q (7.240) Pe D 2 1 M ¦ This bound is typically too pessimistic, however, it yields a good approximation if ik is mainly due to one dominant interferer.
606
Chapter 7. Transmission over dispersive channels
Saltzberg bound With reference to (7.230), defining z k as the total disturbance given by (7.236), in general we have 1 P[z k > h 0 ] Pe D 2 1 (7.241) M Let Þmax D maxfÞn g D M 1 n
(7.242)
in the specific case, and I be any subset of the integers Z 0 , excluding zero, such that X
jh i j
h 0 ] [10], obtaining !2 1 B h 0 Þmax jh i j C C B C B i 2I C 0 1 P[z k > h 0 ] < exp B C B C B X 2A A @ 2 @¦ 2 C ¦ 2 jh i j a 0
X
(7.244)
i 2I C
The bound is particularly simple in the case of binary signaling, where fak g 2 f1; 1g, !2 1 C B h0 jh i j C B C B I i 2 B 1C Pe < exp B 0 C C B X 2A A @ 2 @¦ 2 C jh i j 0
X
(7.245)
i 2I C
P where I is such that i 2I jh i j < h 0 . In this case it is rather simple to choose the set I so that the limit is tighter. We begin with I D Z 0 . Then we remove from I one by one the indices i that correspond to the larger values of jh i j; we stop when the exponent of (7.245) has reached the minimum. Considering the limit of the function Q given by (6.364), we observe that for I D Z 0 and I C D ;, the bound in (7.244) practically coincides with the worst-case limit in (7.240). Taking instead I D ; and I C D Z 0 we obtain again the limit given by the Gaussian approximation for z k that yields (7.238). For the mathematical details we refer to [10]; for a comparison between the Saltzberg bound and other bounds we refer to [5, 11].
7.B. Computation of Pe for some cases of interest
607
GQR method The GQR method is based on a technique for the approximate computation of integrals called Gauss quadrature rule (GQR). It offers a good compromise between computational complexity and approximation accuracy. If we assume a very large number of interferers, to the limit infinite, ik can be modelled as a continuous random variable. Then Pe assumes the expression Z C1 1 h0 ¾ 1 pik .¾ / d¾ D 2 1 I (7.246) Q Pe D 2 1 M ¦ M 1 By the GQR method we obtain an approximation of the integral, given by I D
Nw X jD1
wj Q
h0 ¾ j ¦
(7.247)
In this expression the parameters f¾ j g and fw j g are called, respectively, abscissae and weights of the quadrature rule, and are obtained by a numerical algorithm based on the first 2Nw moments of ik . The quality of the approximation depends on the choice of Nw [5].
608
Chapter 7. Transmission over dispersive channels
Coherent PAM-DSB transmission
Appendix 7.C General scheme
For transmission over a passband channel, a PAM signal must be suitably shifted in frequency by a sinusoidal carrier at frequency f 0 . This task is achieved by DSB modulation (see Example 1.6.3 on page 41) of the signal s.t/ at the output of the baseband PAM modulator filter. In the case of a coherent receiver, the passband scheme is given in Figure 7.52. For the baseband equivalent model, we refer to Figure 7.53a. Now we consider the study of the PAM-DSB transmission system in the unified framework of Figure 7.12. Assuming the receive filter g Rc real-valued, we apply the operator Re [ ] to the channel filter impulse response and to the noise signal, and we split the factor 1=2 evenly among the channel filter and the receive filter responses; setting g Rc .t/ D g Rc .t/ p1 , we thus obtain the simplified scheme of Figure 7.53b, where the noise 2 signal contains only the in-phase component w0I .t/ with PSD Pw0I . f / D
N0 (V2 /Hz) 2
(7.248)
and "
.bb/
e j .'1 '0 / gCh .t/ p gC .t/ D Re 2 2
# (7.249)
or, in the frequency domain, GC . f / D
Ł . f C f /1. f C f / e j .'1 '0 / GCh . f C f 0 /1. f C f 0 / C e j .'1 '0 / GCh 0 0 p 4 2 (7.250)
For a non-coherent receiver we refer to the scheme developed in Example 6.11.6 on page 516.
Figure 7.52. PAM-DSB passband transmission system.
7.C. Coherent PAM-DSB transmission
609
Figure 7.53. PAM-DSB system.
Transmit signal PSD Considering the PSD of the message sequence, the average PSD of the modulated signal s.t/ is given by (7.28), 1 PN s . f / D [Pa . f f 0 / jHT x . f f 0 /j2 C Pa . f C f 0 / jHT x . f f 0 /j2 ] 4T 2
(7.251)
Consequently the transmitted signal bandwidth is equal to twice the bandwidth of h T x . The minimum bandwidth is given by Bmin D
1 T
(7.252)
Recalling the definition (6.103), the spectral efficiency of the transmission system is given by ¹ D log2 M (bit/s/Hz)
(7.253)
which is halved with respect to M-PAM (see Table 6.9).
Signal-to-noise ratio We assume the function .bb/ e j .'1 '0 / gCh .t/ p 2 2
(7.254)
is real-valued; then from Figure 7.53a, using (1.295), we have the following relation: E[jsC .t/j2 ] D
.bb/ E[jsCh .t/j2 ] D E[jsCh .t/j2 ] 2
(7.255)
610
Chapter 7. Transmission over dispersive channels
Setting qC .t/ D h T x Ł gC .t/
(7.256)
from (6.105) and (7.252) we have 0D
Ma E qC N0
(7.257)
M2 1 3
(7.258)
where, for an M-PAM system (6.110), Ma D
In the absence of ISI, for defined in (7.106), (7.107) still holds; moreover, using (7.257), for a matched filter receiver, (7.113) yields M F D
E qC 20 D N0 =2 Ma
(7.259)
Then the error probability is given by ! r 60 1 Q Pe D 2 1 M M2 1
(7.260)
We observe that the performance of an M-PAM-DSB system and that of an M-PAM system are the same, in terms of Pe as a function of the received power. However, because of DSB modulation, the required bandwidth is doubled with respect to both baseband PAM transmission and PAM-SSB modulation.14 This explains the limited usage of PAM-DSB for digital transmission.
14 The PAM-SSB scheme presents in practice considerable difficulties because the filter for modulation is non-
ideal: in fact, this causes distortion of the signal s.t/ at low frequencies that may be compensated for only by resorting to line coding (see Appendix 7.A).
7.D. Implementation of a QAM transmitter
Appendix 7.D
611
Implementation of a QAM transmitter
Three structures, which differ by the position of the digital-to-analog converter, may be considered for the implementation of a QAM transmitter. In Figure 7.54 the modulator employs for both in-phase and quadrature signals a DAC after the interpolator filter h T x , followed by an analog mixer that shifts the signal to passband. This scheme works if the sampling frequency 1=Tc is much greater than twice the bandwidth B of h T x . For applications where the symbol rate is very high, the DAC is placed right after the bit mapper and the various filters are analog (see Chapter 19). In the implementation illustrated in Figure 7.55, the DAC is placed instead at an intermediate stage with respect to the case of Figure 7.54. Samples are premodulated by a digital mixer to an intermediate frequency f 1 , interpolated by the DAC and subsequently remodulated by a second analog mixer that shifts the signal to the desired band. The intermediate frequency f 1 must be greater than the bandwidth B and smaller than 1=.2Tc / B, thus avoiding overlap among spectral components. We observe that this scheme requires only one DAC, but the sampling frequency must be at least double as compared to the previous scheme.
Figure 7.54. QAM with analog mixer.
Figure 7.55. QAM with digital and analog mixers.
612
Chapter 7. Transmission over dispersive channels
Figure 7.56. Polyphase implementation of the filter hTx for Tc D T=8.
For the first implementation, as the system is typically oversampled with a sampling interval Tc D T =4 or Tc D T =8, the frequency response of the DAC, G I . f /, may be considered as a constant in the passband of both the in-phase and quadrature signals. For the second implementation, unless f 1 − 1=Tc , the distortion introduced by the DAC should be considered and equalized by one of these methods (see page 338): ž including the compensation for G I . f / in the frequency response of the filter h T x , ž inserting a digital filter before the DAC, ž inserting an analog filter after the DAC. We recall that an efficient implementation of interpolator filters h T x is obtained by the polyphase representation, as shown in Figure 7.56 for Tc D T =8, where T h .`/ .m/ D h T x mT C ` ` D 0; 1; : : : ; 7 m D 1; : : : ; C1 (7.261) 8 To implement the scheme of Figure 7.56, once the impulse response is known, it may be convenient to precompute the possible values of the filter output and store them in a table or RAM. The symbols fak;I g are then used as pointers for the table itself. The same approach may be followed to generate the values of the signals cos.2³ f 1 nTc / and sin.2³ f 1 nTc / in Figure 7.55, using an additional table and the index n as a cyclic pointer.
7.E. Simulation of a QAM system
Appendix 7.E
613
Simulation of a QAM system
In Figure 7.12 we consider the baseband equivalent scheme of a QAM system. The aim is to simulate the various transformations in the discrete-time domain and to estimate the bit error probability. This simulation method, also called Monte Carlo, is simple and general because it does not require any special assumption on the processes involved; however, it is intensive from the computational point of view. For alternative methods, for example semi-analytical, to estimate the error probability, we refer to specific texts on the subject [12]. We describe the various transformations in the overall discrete-time system depicted in Figure 7.57, where the only difference with respect to the scheme of Figure 7.12 is that the
(a) Transmitter and channel block diagram.
(b) Receiver block diagram. Figure 7.57. Baseband equivalent model of a QAM system with discrete-time filters and sampling period TQ D T=Q0 . At the receiver, in addition to the general scheme, a multirate structure to obtain samples of the received signal at the timing phase t0 is also shown.
614
Chapter 7. Transmission over dispersive channels
filters are discrete-time with quantum TQ D T =Q 0 , Which is chosen to accurately represent the various signals. Binary sequence fb` g. The sequence fb` g is generated as a random sequence or as a PN sequence (see Appendix 3.A), and has length K . Bit mapper. The bit mapper maps patterns of information bits to symbols; the symbol constellation depends on the modulator (see Figure 7.6 for two constellations). Interpolator filter h T x from period T to TQ . The interpolator filter is efficiently implemented by using the polyphase representation (see Appendix 1.A). For a bandlimited pulse of the raised cosine or square root raised cosine type, the maximum value of TQ , submultiple of T , is T =2. In any case, the implementation of filters, for example, the filter representing the channel, and non-linear transformations, for example, the transformation due to a power amplifier operating near saturation (not considered in Figure 7.57), typically require a larger bandwidth, leading, for example, to the choice TQ D T =4 or T =8. In the following examples we choose TQ D T =4. For the design of h T x the window method can be used (Nh odd): ½ Nh 1 TQ w Nh .q/ q D 0; 1; : : : ; Nh 1 (7.262) h T x .q TQ / D h i d q 2 where typically w Nh is the discrete-time rectangular window or the Hamming window, and h i d is the ideal impulse response. Frequency responses of h T x are illustrated in Figure 7.58 for h i d square root raised cosine pulse with roll-off factor ² D 0:3, and w Nh rectangular window of length Nh , for various values of Nh (TQ D T =4). The corresponding impulse responses are shown in Figure 7.59. Transmission channel. For a radio channel the discrete-time model of Figure 4.35 can be used, where in the case of channel affected by fading, the coefficients of the FIR filter that model the channel impulse response are random variables with a given power delay profile. For a transmission line the discrete-time model of (4.150) can be adopted. We assume the statistical power of the signal at output of the transmission channel is given by MsCh D MsC . Additive white Gaussian noise. Let wN I .q TQ / and wN Q .q TQ / be two Gaussian statistically independent r.v.s, each with zero mean and variance 1=2, generated according to (1.655). To generate the complex-valued noise signal fwC .q TQ /g with spectrum N0 , it is sufficient to use the relation wC .q TQ / D ¦wC [wN I .q TQ / C j wN Q .q TQ /]
(7.263)
7.E. Simulation of a QAM system
615
N = 17 h N = 25 h N = 33
0
h
−10
| HT (f) | (dB)
−20
x
−30
−40
−50
−60
0.6
0.4
0.2
0
2
1.8
1.6
1.4
1.2
1 fT
0.8
Figure 7.58. Magnitude of the transmit filter frequency response, for a windowed square root raised cosine pulse with roll-off factor ² D 0:3, for three values of Nh (TQ D T=4).
Tx
Q
h (q T )
0.3
Nh=17
0.2 0.1 0
−0.1
20
15
10
5
0
−5
q / TQ
Tx
Q
h (q T )
0.3
Nh=25
0.2 0.1 0
−0.1
25
20
15
10
5
0
q / TQ
Tx
Q
h (q T )
0.3
Nh=33
0.2 0.1 0
−0.1
0
5
10
15 q / TQ
20
25
30
Figure 7.59. Transmit filter impulse response, fhTx .qTQ /g, q D 0; : : : ; Nh1 , for a windowed square root raised cosine pulse with roll-off factor ² D 0:3, for three values of Nh (TQ D T=4).
616
Chapter 7. Transmission over dispersive channels
where ¦w2 C D N0
1 TQ
(7.264)
Usually the signal-to-noise ratio 0 given by (6.105) is given. For a QAM system, from (7.51) and (7.55) we have MsC MsC D 2 (7.265) 0D N0 .1=T / ¦wC .TQ =T / The standard deviation of the noise to be inserted in (7.263) is given by r MsC Q 0 ¦wC D (7.266) 0 We note that ¦wC is a function of MsC , of the oversampling ratio Q 0 D T =TQ , and of the given ratio 0. In place of 0, the ratio E b =N0 D 0= log2 M may be assigned. Receive filter. As will be discussed in Chapter 8, there are several possible solutions for the receive filter. The most common choice is a matched filter g M , matched to h T x , of the square root raised cosine type. Alternatively, the receive filter may be a simple antialiasing FIR filter g A A , with passband at least equal to that of the desired signal. The filter attenuation in the stopband must be such that the statistical power of the noise evaluated in the passband is larger by a factor of 5–10 with respect to the power of the noise evaluated in the stopband, so that we can ignore the contribution of the noise in the stopband at the output of the filter g A A . If we adopt as bandwidth of g A A the Nyquist frequency 1=.2T /, the stopband of an ideal filter with unit gain goes from 1=.2T / to 1=.2TQ /: therefore the ripple Žs in the stopband must satisfy the constraint 1 N0 2T > 10 1 Žs N0 2T1 2T Q
(7.267)
from which we get the condition Žs
2 is straightforward. We consider the scheme of Figure 8.12, where the FF filter c is now a fractionally spaced filter, as illustrated in Figure 8.7. The overall receiver structure is shown in Figure 8.14, where the filter gAA may be more sophisticated than a simple anti-aliasing filter and partly perform the function of the MF. Otherwise the function of the MF may be performed by a discrete-time filter placed in front of the filter c. We recall that the MF, besides reducing the complexity of c (see task 2 on page 628), facilitates the optimum choice of t0 (see Chapter 14). As usual, the symbols are assumed i.i.d. and statistically independent of the noise signal. If fxn g is the input signal of the FF filter, we have C1 X h n2k ak C wQ n (8.101) xn D kD1
The signal fyk g at the DFE output is given by yk D
M 1 1 X i D0
ci x2ki C
M2 X jD1
b j akD j
(8.102)
8.5. Decision feedback equalizer (DFE)
643
-
Figure 8.14. FS-DFE structure.
Let p
M 1 1 X
D h Ł cp D
c` h p`
(8.103)
`D0
then the optimum choice of the coefficients fbi g is given by bi D
2.i CD/
D
M 1 1 X
c` h 2.i CD/`
i D 1; : : : ; M2
(8.104)
`D0
With this choice (8.102) becomes yk D
M 1 1 X
ci
x2ki
i D0
M2 X
! h 2. jCD/i ak. jCD/
(8.105)
jD1
Using the following relations (see also Example 1.9.10 on page 72) Ł E[akK x2ki ] D ¦a2 h Ł2K i Ł 2 E[x 2kq x2k p ] D ¦a
C1 X
(8.106) h 2nq h Ł2n p C rwQ . p q/
(8.107)
nD1
where, from (8.57), T rwQ .m/ D N0 rgAA m 2
(8.108)
the components of the vector p and the matrix R of the Wiener problem associated with (8.105) are given by [p] p D ¦a2 h Ł2D p [R] p;q D
¦a2
C1 X nD1
p D 0; 1; : : : ; M1 1 h 2nq h Ł2n p
M2 X
(8.109) !
h 2. jCD/q h Ł2. jCD/ p
C rwQ . p q/
jD1
p; q D 0; 1; : : : ; M1 1
(8.110)
644
Chapter 8. Channel equalization and symbol detection
The feedforward filter is obtained by solving the system of equations R copt D p
(8.111)
and the feedback filter is determined from (8.104). The minimum value of the cost function is given by Jmin D ¦a2
M 1 1 X `D0
D ¦a2 1
copt;` [p]Ł`
M 1 1 X
!
(8.112)
copt;` h 2D`
`D0
A problem encountered with this method is the inversion of the matrix R in (8.111), because it may be ill-conditioned. Similarly to the procedure outlined on page 187, a solution consists in adding a positive constant to the elements on the diagonal of R, so that R becomes invertible; obviously the value of this constant must be rather small, so that the performance of the optimum solution does not change significantly. Observation 8.7 Observations similar to the observations 8.1–8.3 hold for a FS-DFE, with appropriate changes. In this case the timing phase t0 after the filter gAA can be determined with accuracy T =2, for example by the correlation method (7.269). For an FSE, or FS-LE, the equations to determine copt are given by (8.109)–(8.111) with M2 D 0. Note that the matrix R is Hermitian but in general it is no longer Toeplitz. Observation 8.8 Two matrix formulations of the direct method to determine the coefficients of a DFE and an FS-DFE are given in Appendix 8.B. In particular, a formulation uses the correlation of the equalizer input signal fxn g, the correlation of the sequence fak g, and the cross-correlation of the two signals: using suitable estimates of the various correlations (see the correlation method and the covariance method considered in Section 2.3), this method avoids the need for the estimate of the overall channel impulse response; however, it requires a greater computational complexity with respect to the method described in this section.
Signal-to-noise ratio γ Using FF and FB filters with an infinite number of coefficients, it is possible to achieve the minimum value of Jmin . Salz derived the expression of Jmin for this case, given by [3] 1 0 Z 1 2T N0 Jmin D ¦a2 exp @T dfA (8.113) ln 2 8.e j2³ f T / 1 N C ¦ 0 a 2T
where 8 is defined in (8.27).
8.6. Convergence behavior of adaptive equalizers
645
Applying the Jensen’s inequality,
e
R
f .a/ da
Z
e f .a/ da
(8.114)
to (8.113), we can compare the performance of a linear equalizer given by (8.40) with that of a DFE given by (8.113): the result is that, assuming 8.e j2³ f T / 6D constant and the absence of detection errors in the DFE, for infinite-order filters the value Jmin of a DFE is always smaller than Jmin of a LE. If FF and FB filters with a finite number of coefficients are employed, in analogy with (8.44), also for a DFE we have DFE
2 Jmin
(8.115)
where Jmin is given by (8.98). An analogous relation holds for an FS-DFE, with Jmin given by (8.112).
Remarks 1. In the absence of errors of the data detector, the DFE has better asymptotic (M1 ; M2 ! 1) performance than the linear equalizer. However, for a given finite number of coefficients M1 C M2 , the performance is a function of the channel and of the choice of M1 . 2. The DFE is definitely superior to the linear equalizer for channels that exhibit large variations of the attenuation in the passband, as in such cases the linear equalizer tends to enhance the noise. 3. Detection errors tend to spread, because they produce incorrect cancellations. Error propagation leads to an increase of the error probability. However, simulations indicate that for typical channels and symbol error probability smaller than 5 Ð 102 , error propagation is not catastrophic. 4. For channels with impulse response fh i g, such that detection errors may spread catastrophically, instead of the DFE structure it is better to implement the linear FF equalizer at the receiver, and the FB filter at the transmitter as a precoder, using the precoding method discussed in Appendix 7.A and Chapter 13.
8.6
Convergence behavior of adaptive equalizers
We consider the digital transmission model of Figure 8.5, and observe the performance of adaptive LE and DFE by resorting to a specific channel realization. In particular, we analyze two cases in which the discrete-time overall impulse response of the system, fh n g, is either minimum phase, h min , as in Figure 1.15a, or non-minimum phase, h nom , as in Figure 1.15c. The additive channel noise wQ k is AWGN with statistical power ¦w2Q , such that
646
Chapter 8. Channel equalization and symbol detection
the signal-to-noise ratio 0 D ¦a2 Ð rh .0/=¦w2Q at the equalizer input is equal to 20 dB. The sequence of symbols ak 2 f1; 1g is a PN training sequence of length L D 63.
Adaptive LE With reference to the scheme of Figure 8.6, we consider a LE with N D 15 coefficients. In terms of mean-square error at convergence, the best results are obtained for a delay D D 0 in the case of h min and D D 8 in the case of h nom ; we observe that the overall impulse response h nom is not centered at the origin, and the delay D is the sum of the delays introduced by fh n g and fcn g. Figures 8.15c, d and 8.16c, d show curves of mean-square error convergence for standard LMS and RLS algorithms (see Section 3.1.2) for minimum and non-minimum phase channels, respectively; in the plots, Jmin represents the minimum value of J achieved with optimum coefficients computed by the direct method. We note that Jmin is 4 dB higher than the value given by ¦w2Q , because of the noise and residual ISI at the decision point. The impulse response fcopt;n g of the optimum LE and the overall system impulse response f n D h Ł copt;n g are depicted in Figures 8.15a, b and 8.16a, b for the two channels. The curves of convergence of J .k/ indicate that the RLS algorithm succeeds in achieving convergence by the end of the training sequence, whereas the LMS algorithm still presents a considerable offset from the optimum conditions, even though a large adaptation gain ¼ is chosen.
(a)
1
|ψn|
1
|
1.5
opt,n
|c
(b)
1.5
0.5
0.5
0
10
5
0
0
15
10
5
0
(c)
n
15
n
J(k) (dB)
0
LMS −5 −10
J
−15 −20
min
10
0
20
30
40
60
50
(d)
J(k) (dB)
−15
Jmin
−20
RLS −25 −30 −35
0
10
20
30
40
50
60
k
Figure 8.15. System impulse responses and curves of mean-square error convergence, estimated over 500 realizations, for a channel with minimum phase impulse response, and a LE employing the LMS with ¼ D 0:062 or the RLS.
8.6. Convergence behavior of adaptive equalizers
647
(b)
(a) 1.5
1.5
|ψn|
2
| copt,n |
2
1
1
0.5
0.5 0
0
15
10
5
0
15
10
5
0
(c)
n
n
J(k) (dB)
0
LMS −5 −10
Jmin
−15 −20
20
10
0
60
50
40
30
(d) Jmin
J(k) (dB)
−15 −20
RLS −25 −30 −35
20
10
0
60
50
40
30
k
Figure 8.16. System impulse responses and curves of mean-square error convergence, estimated over 500 realizations, for a channel with non-minimum phase impulse response, and a LE employing the LMS with ¼ D 0:343 or the RLS. (a)
(b) 1
|c
opt,n
|ψn|
|
1
0.5
0.5
0
10
5
0
n
0
(c)
10
5
0
n
J(k) (dB)
0
LMS
−5 −10 −15
Jmin
−20 10
0
J(k) (dB)
−15
20
30
40
60
50
(d)
Jmin
−20 −25
RLS
−30 −35 0
10
20
30
40
50
60
k
Figure 8.17. System impulse responses and curves of mean-square error convergence, estimated over 500 realizations, for a channel with minimum phase impulse response, and a DFE employing the LMS with ¼ D 0:063 or the RLS.
648
Chapter 8. Channel equalization and symbol detection
(b)
(a) 1.5
1.5
|ψn|
2
| copt,n |
2
1
1
0.5
0.5 0
10
5
0
n
0
(c)
10
5
0
n
J(k) (dB)
0
LMS
−5 −10
Jmin
−15 −20
30
(d)
40
60
50
Jmin
−15
J(k) (dB)
20
10
0
−20 −25
RLS
−30 −35 0
20
10
40
30
50
60
k
Figure 8.18. System impulse responses and curves of mean-square error convergence, estimated over 500 realizations, for a channel with non-minimum phase impulse response, and a DFE employing the LMS with ¼ D 0:143 or the RLS.
Adaptive DFE We consider now the performance of a DFE as illustrated in Figure 8.13, with parameters M1 D 10, M2 D 5, and D D 9, for both h min and h nom . Also in this case the chosen value of D gives the best results in terms of the value of J at convergence. Figures 8.17c, d and 8.18c, d show curves of mean-square error convergence for standard LMS and RLS algorithms, for minimum and non-minimum phase channels, respectively; The impulse response fcopt;n g of the optimum FF filter and the overall system impulse response f n D h Ł copt;n g are depicted in Figures 8.17a, b and 8.18a, b, for the two channels.
8.7
LE-ZF with a finite number of coefficients
Ignoring the noise, the signal at the output of a LE with N coefficients (see (8.91)) is given by
yk D
N 1 X i D0
ci x ki D
CN 1 N2X pDN1
p ak p
(8.116)
8.8. DFE: alternative configurations
649
where, from (8.90),
p
D
N 1 X
c` h p` D c0 h p C c1 h p1 C Ð Ð Ð C c N 1 h p.N 1/
`D0
(8.117)
p D N1 ; : : : ; 0; : : : ; N 2 C N 1 For a LE-ZF it must be p
D Ž pD
(8.118)
where D is a suitable delay. If the overall impulse response fh n g, n D N1 ; : : : ; N2 , is known, a method to determine the coefficients of the LE-ZF consists in considering the system (8.117) with Nt D N1 C N2 C N equations and N unknowns, that can be solved by the method of the pseudoinverse (see (2.185)); alternatively, the solution can be found in the frequency domain by taking the Nt -point DFT of the various signals (see (1.108)), and the result windowed in the time domain so that the filter coefficients are given by the N consecutive coefficients that maximize the energy of the filter impulse response. An approximate solution is obtained by forcing the condition (8.118) only for N values of p in (8.117) centered around D; then the matrix of the system (8.117) is square and, if the determinant is different from zero, it can be inverted. Note that all these methods require an accurate estimate of the overall impulse response, otherwise the equalizer coefficients may deviate considerably from the desired values. An alternative robust method, which does not require the knowledge of fh n g and can be extended to FSE-ZF systems, will be presented in Section 8.15. An adaptive ZF equalization method is discussed in Appendix 8.C.
8.8
DFE: alternative configurations
We determine the expressions of the FF and FB filters of a DFE in the case of IIR filter structure.
DFE-ZF We consider a receiver with a matched filter g M (see Figure 8.1) followed by the DFE illustrated in Figure 8.19, where, to simplify the notation, we assume t0 D 0 and D D 0. The z-transform of the QAM system impulse response is given by 8.z/, as defined in (8.25). With reference to Figure 8.19, the matched filter output x k is input to a linear equalizer zero forcing (LE-ZF) with transfer function 1=8.z/ to remove the ISI: therefore the LE-ZF output is given by x E;k D ak C w E;k
(8.119)
650
Chapter 8. Channel equalization and symbol detection
Figure 8.19. DFE zero-forcing.
From (8.29), using the property (8.26), we obtain that the spectrum Pw E .z/ of w E;k is given by Pw E .z/ D N 0 8.z/
D N0
1 8.z/8 Ł
1 zŁ
(8.120)
1 8.z/
As 8.z/ is the z-transform of a correlation sequence, it can be factorized as (see page 53) 1 Ł (8.121) 8.z/ D F.z/ F zŁ where F.z/ D
C1 X
fn z n
(8.122)
nD0
is a minimum-phase function, that is with poles and zeros inside the unit circle, associated with a causal sequence ffn g. Observation 8.9 A useful method to determine the filter F.z/ in (8.121), with a computational complexity that is proportional to the square of the number of filter coefficients, is obtained by considering a minimum-phase prediction error filter A.z/ D 1 C a 01;N z 1 C Ð Ð Ð C a0N ;N z N (see page 147), designed using the ACS frqC .nT /g defined by (8.23). The equation to determine the coefficients of A.z/ is given by (2.85), where fr x .n/g is now substituted by frqC .nT /g. The final result is F.z/ D f 0 =A.z/. On the other hand, F Ł .1=z Ł / is a function with zeros and poles outside the unit circle, Ł g. associated with an anticausal sequence f f n
8.8. DFE: alternative configurations
651
We choose as transfer function of the filter w in Figure 8.19 the function W .z/ D F.z/
1 f0
(8.123)
The ISI term in z k is determined by W .z/1, hence there are no precursors; the noise is white with statistical power.N0 =f20 /. Therefore the filter w is called whitening filter (WF). In any case, the filter composed of the cascade of LE-ZF and w is also a WF. If aO k D ak then, for B.z/ D 1 W .z/
(8.124)
the FB filter removes the ISI present in z k and leaves the white noise unchanged. As yk is not affected by ISI, this structure is called DFE-ZF, for which we obtain DFEZF D
2jf0 j2 N0
(8.125)
Summarizing, the relation between x k and z k is given by 1 1 F.z/ D 8.z/ f0
1 f0
FŁ
1 zŁ
(8.126)
With this filter the noise in z k is white. The relation between ak and the desired signal in z k is instead governed by 9.z/ D F.z/=f 0 and B.z/ D 1 9.z/. In other words, the overall discrete-time system is causal and minimum phase, that is the energy of the impulse response is mostly concentrated at the beginning of the pulse. The overall receiver structure is illustrated in Figure 8.20, where the block including the matched filter, sampler, and whitening filter, is called whitened matched filter (WMF). Note that the impulse response at the WF output has no precursors. In principle, the WF of Figure 8.20 is non-realizable, because it is anticausal. In practice we can implement it in two ways:
Figure 8.20. DFE-ZF as whitened matched filter followed by a canceller of ISI.
652
Chapter 8. Channel equalization and symbol detection
a) by introducing an appropriate delay in the impulse response of an FIR WF, and processing the output samples in the forward mode for k D 0; 1; : : : ; b) by processing the output samples of the IIR WF in the backward mode, for k D K 1; K 2; : : : ; 0, starting from the end of the block of samples. We observe that the choice F.z/ D f 0 =A.z/, where A.z/ is discussed in Observation 8.9 on page 650, leads to an FIR WF with transfer function 12 AŁ .1=z Ł /. f0
Observation 8.10 With reference to the scheme of Figure 8.20, using a LE-ZF instead of a DFE structure means that a filter with transfer function f0 =F.z/ is placed after the WF to produce the signal x E;k given by (8.119). For a data detector based on x E;k , the ratio is L EZ F D
2 rw E .0/
(8.127)
where rw E .0/ is determined as the coefficient of z 0 in N0 =8.z/. This expression is alternative to (8.3). Example 8.8.1 (WF for a channel with exponential impulse response) A method to determine the WF in the scheme of Figure 8.20 is illustrated by an example. Let p (8.128) qC .t/ D E qC 2þeþt 1.t/ be the overall system impulse response at the MF input; in (8.128) E qC is the energy of qC . The autocorrelation of qC , sampled at instant nT, is given by rqC .nT / D E qC a jnj
a D eþT < 1
(8.129)
Then 8.z/ D Z[r qC .nT /] D E qC
.1 a 2 / az 1 C .1 C a 2 / az
D E qC
.1 a 2 / .1 az 1 /.1 az/
(8.130)
We note that the frequency response of (8.129) is 8.e j2³ f T / D E qC and presents a minimum for f D 1=.2T /.
.1 a 2 / 1 C a 2 2a cos.2³ f T /
(8.131)
8.8. DFE: alternative configurations
653
With reference to the factorization (8.121), it is easy to identify the poles and zeros of 8.z/ inside the unit circle, hence F.z/ D D
q q
E qC .1 a 2 / E qC
.1 a 2 /
1 1 az 1 C1 X
(8.132) n n
a z
nD0
In particular, the coefficient of z 0 is given by f0 D
q
E qC .1 a 2 /
(8.133)
The WF of Figure 8.20 is expressed as 1 f0 F Ł
1 zŁ
D
1 .1 az/ E qC .1 a 2 / (8.134)
1 z.a C z 1 / D E qC .1 a 2 / In this case the WF, apart from a delay of one sample .D D 1/, can be implemented by a simple FIR with two coefficients, whose values are equal to a=.E qC .1 a 2 // and 1=.E qC .1 a 2 //. The FB filter is a first-order IIR filter with transfer function 1
C1 X 1 F.z/ D a n z n f0 nD1
D1 D
1 1 az 1
(8.135)
az 1 1 az 1
Example 8.8.2 (WF for a two-ray channel) In this case we directly specify the autocorrelation sequence at the matched filter output: 1 Ł (8.136) 8.z/ D Q CC .z/Q CC Ł z where Q CC .z/ D
p
E qC .q0 C q1 z 1 /
(8.137)
654
Chapter 8. Channel equalization and symbol detection
with q0 and q1 such that jq0 j2 C jq1 j2 D 1
(8.138)
jq0 j > jq1 j
(8.139)
In this way E qC is the energy of fqCC .nT /g and Q CC .z/ is minimum phase. Equation (8.137) represents the discrete-time model of a wireless system with a two-ray channel. The impulse response is given by p (8.140) qCC .nT / D E qC .q0 Žn C q1 Žn1 / The frequency response is given by QCC . f / D
p
E qC .q0 C q1 e j2³ f T /
(8.141)
We note that if q0 D q1 , the frequency response has a zero for f D 1=.2T /. From (8.136) and (8.137) we get 8.z/ D E qC .q0 C q1 z 1 /.q0Ł C q1Ł z/ hence, recalling assumption (8.139), p F.z/ D E qC .q0 C q1 z 1 / D Q CC .z/
(8.142)
(8.143)
and f0 D
p
E qC q 0
(8.144)
The WF is given by 1 f0 F Ł
1 zŁ
D
1 E qC jq0 j2 .1 bz/
(8.145)
where
q1 bD q0
Ł
We note that the WF has a pole for z D b 1 , which, recalling (8.139), lies outside the unit circle. In this case, in order to have a stable filter, it is convenient to associate the z-transform 1=.1 bz/ with an anticausal sequence, 0 1 X X 1 D .bz/ i D .bz/ n 1 bz i D1 nD0
(8.146)
8.8. DFE: alternative configurations
655
On the other hand, as jbj < 1, we can approximate the series by considering only the first .N 1/ terms, obtaining 1 f0 F Ł
1 zŁ
'
N X 1 .bz/ n E qC jq0 j2 nD0
(8.147) D
1 z N [b N C Ð Ð Ð C bz .N 1/ C z N ] E qC jq0 j2
Consequently the WF, apart from a delay D D N , can be implemented by an FIR filter with N C 1 coefficients. The FB filter in this case is a simple FIR filter with one coefficient 1
q1 1 F.z/ D z 1 f0 q0
(8.148)
DFE-ZF as a noise predictor Let A.z/ D Z[a k ]. From the identity Y .z/ D X E .z/W .z/ C .1 W .z//A.z/ D X E .z/ C .1 W .z//.A.z/ X E .z//
(8.149)
the scheme of Figure 8.19 is redrawn as in Figure 8.21, where the FB filter acts as a noise predictor. In fact, for aO k D ak , the FB filter input is colored noise. By removing the correlated noise from x E;k , we obtain yk that is composed of white noise, with minimum variance, plus the desired symbol ak .
DFE as ISI and noise predictor A variation of the scheme of Figure 8.21 consists in using as FF filter, a minimumMSE linear equalizer. We refer to the scheme of Figure 8.22, where the filter c is given
Figure 8.21. Predictive DFE: the FF filter is a linear equalizer zero forcing.
656
Chapter 8. Channel equalization and symbol detection
Figure 8.22. Predictive DFE with the FF filter as a minimum-MSE linear equalizer.
by (8.38). The z-transform of the overall impulse response at the FF filter output is given by: 8.z/C.z/ D
¦a2 8.z/ N0 C ¦a2 8.z/
(8.150)
As t0 D 0, the ISI in z k is given by: 8.z/C.z/ 1 D
N0 N0 C ¦a2 8.z/
(8.151)
Hence, the spectral density of the ISI has the following expression: PI S I .z/ D Pa .z/
D
N0 N0 C ¦a2 8.z/
N0 N0 C ¦a2 8Ł
1 zŁ
(8.152)
¦a2 .N0 /2 .N0 C ¦a2 8.z// 2
using (8.26) and the fact that the symbols are uncorrelated with Pa .z/ D ¦ a2 . The spectrum of the noise in z k is given by 1 Pnoise .z/ D N 0 8.z/C.z/C Ł Ł z 8.z/¦ a4 D N0 .N0 C ¦a2 8.z// 2
(8.153)
Therefore the spectrum of the disturbance vk in z k , composed of ISI and noise, is given by Pv .z/ D
N0 ¦a2 N0 C ¦a2 8.z/
(8.154)
We note that the FF filter could be an FSE and the result (8.154) would not change.
8.9. Benchmark performance for two equalizers
657
To minimize the power of the disturbance in yk , the FB filter, with input aO k z k D ak z k D vk (assuming aO k D ak ), needs to remove the predictable components of z k . For a predictor of infinite length, we set C1 X
bn z n
(8.155)
B.z/ D 1 A.z/
(8.156)
B.z/ D
nD1
An alternative form is
where A.z/ D
C1 X
an0 z n
a00 D 1
(8.157)
nD0
is the forward prediction error filter defined in (2.83). To determine B.z/ we use the spectral factorization in (1.526): Pv .z/ D
D
¦ y2 1 Ł A.z/A zŁ ¦ y2
(8.158)
½ 1 Ł [1 B.z/] 1 B zŁ
with Pv .z/ given by (8.154). In conclusion, it results that the prediction error signal yk is a white noise process with statistical power equal to ¦ y2 . An adaptive version of the basic scheme of Figure 8.22 suggests that the two filters, c and b, are separately adapted through the error signals fe F;k g and fe B;k g, respectively. This configuration, although sub-optimum with respect to the DFE, is used in conjunction with trellis-coded modulation (see Chapter 12) [4].
8.9
Benchmark performance for two equalizers
We compare limits on the performance of the two equalizers, LE-ZF and DFE-ZF, in terms of the signal-to-noise ratio at the decision point, .
Performance comparison From (8.120), the noise sequence fw E;k g can be modeled as the output of a filter having transfer function C F .z/ D
1 F.z/
(8.159)
658
Chapter 8. Channel equalization and symbol detection
and input given by white noise with PSD N0 . Because F.z/ is causal, also C F .z/ is causal: C F .z/ D
1 X
c F;n z n
(8.160)
nD0
where fc F;n g, n ½ 0, is the filter impulse response. Then we can express the statistical power of w E;k as: rw E .0/ D N0
1 X
jc F;n j2
(8.161)
1 1 D F.1/ f0
(8.162)
1 jf0 j2
(8.163)
nD0
where, from (8.159), c F;0 D C.1/ D Using the inequality 1 X
jc F;n j2 ½ jc F;0 j2 D
nD0
the comparison between (8.125) and (8.127) yields LEZF DFEZF
(8.164)
Equalizer performance for two channel models We now analyze the value of for the two simple systems introduced in Examples 8.8.1 and 8.8.2. LE-ZF Channel with exponential impulse response. From (8.130) the coefficient of z 0 in N0 =8.z/ is equal to rw E .0/ D
N0 1 C a 2 E qC 1 a 2
(8.165)
and, consequently, from (8.127), L EZ F D 2
E qC 1 a 2 N0 1 C a 2
(8.166)
Using the expression of MF (7.113), obtained for a MF receiver in the absence of ISI, we get L EZ F D MF
1 a2 1 C a2
(8.167)
Therefore the loss due to the ISI, given by the factor .1 a 2 /=.1 C a 2 /, can be very large if a is close to 1. In this case the frequency response in (8.131) assumes a minimum value close to zero.
8.10. Optimum methods for data detection
659
Two-ray channel. From (8.142) we have 1 1 D 8.z/ E qC .q0 C q1 z 1 /.q0Ł C q1Ł z/
(8.168)
By a partial fraction expansion, we find only the pole for z D q 1 =q0 lies inside the unit circle, hence rw E .0/ D
N0 N0 D 2 2 q E .jq 1 qC 0 j jq1 j / E qC q0 q0Ł q1Ł q0
(8.169)
Then L EZ F D MF .jq0 j2 jq1 j2 /
(8.170)
Also in this case we find that the LE is unable to equalize channels with a spectral zero. DFE-ZF Channel with exponential impulse response. Substituting the expression of f0 given by (8.133) in (8.125), we get D F EZ F D
2 E q .1 a 2 / N0 C
(8.171)
D MF .1 a / 2
We note that D F EZ F is better than L EZ F by the factor .1 C a 2 /. Two-ray channel. Substitution of (8.144) in (8.125) yields D F EZ F D
2 E q jq0 j2 D MF jq0 j2 N0 C
(8.172)
In this case the advantage with respect to LE-ZF is given by the factor jq0 j2 =.jq0 j2 jq1 j2 /, which may be substantial if jq1 j ' jq0 j. We recall that in case E qC × N0 , that is for low noise levels, the performance of LE and DFE are similar to the performance of LE-ZF and DFE-ZF, respectively. Anyway, for the two systems of Examples 8.8.1 and 8.8.2 the values of in terms of Jmin are given in [4], for both LE and DFE.
8.10
Optimum methods for data detection
Adopting an MSE criterion at the decision point, we have derived the configuration of Figure 8.1 for an LE, and that of Figure 8.12 for a DFE. In both cases, the decision on a transmitted symbol akD is based only on yk through a memoryless threshold detector.
660
Chapter 8. Channel equalization and symbol detection
Actually, the decision criterion that minimizes the probability that an error occurs in the detection of a symbol of the sequence fak g requires in general that the entire sequence of received samples is considered for symbol detection. We assume a sampled signal having the following structure: z k D u k C wk
k D 0; 1; : : : ; K 1
(8.173)
where: ž u k is the desired signal that carries the information, uk D
L2 X
n akn
(8.174)
nDL 1
where 0 is the sample of the overall system impulse response, obtained in correspondence of the optimum timing phase; fL 1 ; : : : ; 1 g are the precursors, which are typically negligible with respect to 0 . Recalling the expression of the pulse f p g given by (8.90), we have n D nCD . We assume the coefficients fn g are known; in practice, however, they are estimated by the methods discussed in Appendix 3.B; ž wk is a circularly symmetric white Gaussian noise, with equal statistical power in the two I and Q components given by ¦ I2 D ¦w2 =2; hence the samples fwk g are uncorrelated and therefore statistically independent. In this section, a general derivation of optimum detection methods is considered; possible applications span, e.g., decoding of convolutional codes (see Chapter 11), coherent demodulation of CPM signals (see Appendix 18.A), and obviously detection of sequences transmitted over channels with ISI. We introduce the following vectors with K components (K may be very large): 1. Sequence of transmitted symbols, or information message, modeled as a sequence of r.v.s from a finite alphabet a D [a L 1 ; a L 1 C1 ; : : : ; a L 1 CK 1 ]T
ai 2 A
(8.175)
2. Sequence of detected symbols, modeled as a sequence of r.v.s from a finite alphabet aO D [aO L 1 ; aO L 1 C1 ; : : : ; aO L 1 CK 1 ]T
aO i 2 A
(8.176)
Þi 2 A
(8.177)
3. Sequence of detected symbol values α D [Þ L 1 ; Þ L 1 C1 ; : : : ; Þ L 1 CK 1 ]T
4. Sequence of received samples, modeled as a sequence of complex r.v.s z D [z 0 ; z 1 ; : : : ; z K 1 ]
(8.178)
8.10. Optimum methods for data detection
661
5. Sequence of received sample values, or observed sequence ρ D [²0 ; ²1 ; : : : ; ² K 1 ]T
²i 2 C
ρ 2 CK
(8.179)
Let M be the cardinality of the alphabet A. By analogy with the analysis of Section 6.1, we divide the vector space of the received samples, C K , into M K non-overlapping regions Rα
α 2 AK
(8.180)
such that, if ρ belongs to Rα , then the sequence α is detected: if ρ 2 Rα H) aO D α
(8.181)
The probability of a correct decision is computed as follows: P[C] D P[Oa D a] X D P[Oa D α j a D α]P[a D α] α2A K
D
X
P[z 2 Rα j a D α]P[a D α]
(8.182)
α2A K
D
X Z
α2A K
Rα
pzja .ρ j α/dρ P[a D α]
As in (6.18), the following criteria may be adopted to minimize (8.182). Maximum a posteriori probability (MAP) criterion ρ 2 Rα
.and aO D α/ if α : max pzja .ρ j α/P[a D α] α
(8.183)
Using the identity pzja .ρ j α/ D pz .ρ/
P[a D α j z D ρ] P[a D α]
(8.184)
the decision criterion becomes: aO D arg max P[a D α j z D ρ] α
(8.185)
An efficient realization of the MAP criterion will be developed in Section 8.10.2. If all data sequences are equally likely, the MAP criterion coincides with the maximum likelihood sequence detection (MLSD) criterion aO D arg max pzja .ρ j α/ α
(8.186)
In other words, the sequence α is chosen, for which the probability to observe z D ρ is maximum.
662
8.10.1
Chapter 8. Channel equalization and symbol detection
Maximum likelihood sequence detection
We now discuss a computationally efficient method to find the solution indicated by (8.186). As the vector a of transmitted symbols is hypothesized to assume the value α, both a and u D [u 0 ; : : : ; u K 1 ]T are fixed. Recalling that the noise samples are statistically independent, from (8.173) we get:3 pzja .ρ j α/ D
KY 1
pzk ja .²k j α/
(8.187)
kD0
As wk is a complex-valued Gaussian r.v. with zero mean and variance ¦w2 , it follows pzja .ρ j α/ D
KY 1 kD0
1 ¦12 j²k u k j2 e w ³ ¦w2
(8.188)
Taking the logarithm, which is a monotonic increasing function, of both members we get ln pzja .ρ j α/ /
K 1 X
j²k u k j2
(8.189)
kD0
where non-essential constant terms have been neglected. Then the MLSD criterion is formulated as aO D arg min α
K 1 X
j²k u k j2
(8.190)
kD0
where u k , defined by (8.174), is a function of the transmitted symbols expressed by the general relation u k D fQ.akCL 1 ; : : : ; ak ; : : : ; akL 2 /
(8.191)
We note that (8.190) is a particular case of the minimum distance criterion (6.30), and it suggests a detecting the vector u that is closest to the observed vector ρ. However, we are interested in detecting the symbols fak g and not the components fu k g. A direct computation method requires that, given the sequence of observed samples, for each possible data sequence α of length K , the corresponding K output samples, elements of the vector u, should be determined, and the relative distance, or metric, should be computed as 0.K 1/ D
K 1 X
j²k u k j2
(8.192)
kD0
The detected sequence is the sequence that yields the smallest value of 0.K 1/; as in the case of i.i.d. symbols there are M K possible sequences, this method has a complexity O.M K /. 3
We note that (8.187) formally requires that the vector a is extended to include the symbols aL 2 ; : : : ; a L 1 1 .
8.10. Optimum methods for data detection
663
Lower limit to error probability using the MLSD criterion We interpret the vector u as a function of the sequence a, that is u D u.a/. In the signal space spanned by u, we compute the distances d 2 .u.α/; u.β// D jju.α/ u.β/jj2
(8.193)
for each possible pair of distinct α and β in A K . As the noise is white, we define 2 D min d 2 .u.α/; u.β// dmin α;β
Then the lower limit (6.86) can be used, and we get Nmin dmin Pe ½ Q MK 2¦ I
(8.194)
(8.195)
where M K is the number of vectors u.a/, and Nmin is the number of vectors u.a/ whose distance from another vector is dmin . In practice, the exhaustive method for the computation of the expressions in (8.192) and (8.194) is not used; the Viterbi algorithm, that will be discussed in the next section, is utilized instead.
The Viterbi algorithm (VA) The Viterbi algorithm efficiently implements the ML criterion. With reference to (8.191), it is convenient to describe fu k g as the output sequence of a finite state machine (FSM), as discussed in Appendix 8.D. In this case the input is akCL 1 , the state is sk D .akCL 1 ; akCL 1 1 ; : : : ; ak ; : : : ; akL 2 C1 /
(8.196)
and the output is given in general by (8.191). We denote by S the set of the states, that is the set of possible values of sk : sk 2 S D fσ 1 ; σ 2 ; : : : ; σ Ns g
(8.197)
With the assumption of i.i.d. symbols, the number of states is equal to Ns D M L 1 CL 2 . Observation 8.11 We denote by s 0k1 the vector that is obtained by removing from s k1 the oldest symbol, akL 2 . Then sk D .akCL 1 ; s 0k1 /
(8.198)
From (8.191) and (8.196) we may define u k as a function of s k and s k1 as u k D f .sk ; sk1 /
(8.199)
664
Chapter 8. Channel equalization and symbol detection
Defining the metric 0k D
k X
j²i u i j2
(8.200)
i D0
the following recursive equation holds: 0k D 0k1 C j²k u k j2
(8.201)
0k D 0k1 C j²k f .sk ; sk1 /j2
(8.202)
or, using (8.199),
Thus we have interpreted fu k g as the output sequence of a finite state machine, and we have expressed recursively the metric (8.192). Note that the metric is a function of the sequence of states s0 ; s1 ; : : : ; sk , associated with the sequence of output samples u 0 ; u 1 ; : : : ; u k . The following example illustrates how to describe the transitions between states of the finite state machine. Example 8.10.1 Let us consider a transmission system with symbols taken from a binary alphabet, that is M D 2, ak 2 f1; 1g, and overall impulse response characterized by L 1 D L 2 D 1. In this case we have sk D .akC1 ; ak /
(8.203)
and the set of states contains Ns D 22 D 4 elements: S D fσ 1 D .1; 1/; σ 2 D .1; 1/; σ 3 D .1; 1/; σ 4 D .1; 1/g
(8.204)
The possible transitions sk1 D σ i ! sk D σ j
(8.205)
are represented in Figure 8.23, where a dot indicates a possible value of the state at a certain instant, and a branch indicates a possible transition between two states at consecutive instants. According to (8.198), the variable that determines a transition is akCL 1 . Figure 8.23, extended for all instants k, is called a trellis diagram. We note that in this case there are exactly M transitions that leave each state sk1 ; likewise there are M transitions that arrive to each state sk . With each state σ j , j D 1; : : : ; Ns , at instant k we associate two quantities: 1. the path metric, or cost function, defined as: 0.sk D σ j / D
min
s0 ;s1 ;:::;sk Dσ j
0k
(8.206)
2. the survivor sequence, defined as the sequence of symbols that ends in that state and determines 0.sk D σ j /: L.sk D σ j / D .s0 ; s1 ; : : : ; sk D σ j / D .a L 1 ; : : : ; akCL 1 /
(8.207)
8.10. Optimum methods for data detection
s k-1 =(a k ,ak-1 )
665
s k=(ak+1,ak )
σ 1 =(-1,-1)
-1
σ 2 =(-1,1)
-1 1
(-1,1)
σ 3 =(1,-1)
-1 1
(1,-1)
σ 4=(1,1)
-1
(1,1)
(-1,-1)
1
1
a k+N
1
A
Figure 8.23. Portion of the trellis diagram showing the possible transitions from state sk1 to state sk , as a function of the symbol akCL1 2 A.
Note that the notion of survivor sequence can be equivalently applied to a sequence of symbols or to a sequence of states. These two quantities are determined recursively. In fact, it is easy to verify that if, at instant k, a survivor sequence of states includes sk D σ j then, at instant k 1, the same sequence includes sk1 D σ iopt , which is determined as follows: σ iopt D arg
min
sk1 Dσ i 2S !sk Dσ j
0.sk1 D σ i / C j²k f .σ j ; σ i /j2
(8.208)
The term j²k f .σ j ; σ i /j2 is called branch metric. Therefore, we obtain: 0.sk D σ j / D 0.sk1 D σ iopt / C j²k f .σ j ; σ iopt /j2
(8.209)
and the survivor sequence is augmented as follows: L.sk D σ j / D .L.sk1 D σ iopt /; σ j /
(8.210)
Starting from k D 0, with initial state s1 , which may be known or arbitrary, the procedure is repeated until k D K 1. The optimum sequence of states is given by the survivor sequence L.s K 1 D σ jopt / associated with s K 1 D σ jopt having minimum cost. If the state s1 is known and equal to σ i0 , it is convenient to assign to the states s1 the following costs: ( 0 for σ i D σ i0 0.s1 D σ i / D (8.211) 1 otherwise Analogously, if the final state s K is equal to σ f 0 , the optimum sequence of states coincides with the survivor sequence associated with the state s K 1 D σ j having minimum cost among those that admit a transition into s K D σ f 0 .
666
Chapter 8. Channel equalization and symbol detection
Example 8.10.2 Let us consider a system with the following characteristics: ak 2 f1; 1g, L 1 D 0, L 2 D 2, K D 4, and s1 D .1; 1/. The development of the survivor sequences on the trellis diagram from k D 0 to k D K 1 D 3 is represented in Figure 8.24a. The branch metric, j²k f .σ j ; σ i /j2 , associated with each transition is given in this example and is written above each branch. The survivor paths associated with each state are represented in bold; we note that some paths are abruptly interrupted and not extended at the following instant: for example the path ending at state σ 3 at instant k D 1. Figure 8.24b illustrates how the survivor sequences of Figure 8.24a are determined. Starting with s1 D .1; 1/ we have 0.s1 D σ 1 / D 0 0.s1 D σ 2 / D 1 0.s1 D σ 3 / D 1 0.s1 D σ 4 / D 1
Figure 8.24. Trellis diagram and determination of the survivor sequences.
(8.212)
8.10. Optimum methods for data detection
667
We apply (8.208) for k D 0; starting with s0 D σ 1 , we obtain 0.s0 D σ 1 / D minf0.s1 D σ 1 / C 1; 0.s1 D σ 2 / C 3g D minf1; 1g
(8.213)
D1 We observe that the result (8.213) is obtained for s1 D σ 1 . Then the survivor sequence associated with s0 D σ 1 is L.s0 D σ 1 / D .1/, expressed as a sequence of symbols rather than states. Considering now s0 D σ 2 , σ 3 , and σ 4 , in sequence, and applying (8.208), the first iteration is completed. Obviously, there is no interest in determining the survivor sequence for states with metric equal to 1, because the corresponding path will not be extended. Next, for k D 1, the final metrics and the survivor sequences are shown in the second diagram of Figure 8.24b, where we have 0.s1 D σ 1 / D minf1 C 1; 1g D 2
(8.214)
0.s1 D σ 2 / D minf1 C 2; 1g D 3
(8.215)
0.s1 D σ 3 / D minf1 C 1; 1g D 2
(8.216)
0.s1 D σ 4 / D minf1 C 0; 1g D 1
(8.217)
The same procedure is repeated for k D 2; 3, and the trellis diagram is completed. The minimum among the values assumed by 0.s3 / is min
i D1;:::;Ns
0.s3 D σ i / D 0.s3 D σ 2 / D 3
(8.218)
The associated optimum survivor sequence is a0 D 1
a1 D 1
a2 D 1
a3 D 1
(8.219)
In practice, if the parameter K is large, the length of the survivor sequences is limited to a value K d , called trellis depth or path memory depth, that typically is between 3 and 10 times the length of the channel impulse response: this means that at every instant k we decide on akK d CL 1 , a value that is then removed from the diagram. The decision is based on the survivor sequence associated with the minimum among the values of 0.sk /. In practice, k and 0.sk / may become very large; then the latter value is usually normalized by subtracting the same amount from all the metrics, for example the smallest of the metrics 0.sk D σ i /, i D 1; : : : ; Ns .
Computational complexity of the VA Memory. The memory to store the metrics 0.sk /, 0.sk1 / and the survivor sequences is proportional to the number of states Ns . Computational complexity. The number of additions and comparisons is proportional to the number of transitions, M Ns D M .L 1 CL 2 C1/ . In any case, the complexity is linear in K .
668
Chapter 8. Channel equalization and symbol detection
We note that the values of f .σ j ; σ i /, that determine u k in (8.199) can be memorized in a table, limited to the possible transitions sk1 D σ i ! sk D σ j , for σ i ; σ j 2 S. However, at every instant k and for every transition σ i ! σ j , the branch metric j²k f .σ j ; σ i /j2 needs to be evaluated: this computation, however, can be done outside the recursive algorithm.
8.10.2
Maximum a posteriori probability detector
We now discuss an efficient method to compute the a posteriori probability (APP) in (8.185) for an isolated symbol. Given a vector z D [z 0 ; z 1 ; : : : ; z K 1 ]T , the notation zm ` indicates the vector formed only by the components [z ` ; z `C1 ; : : : ; z m ]T . We introduce the likelihood function, defined as Lk .þ/ D P[akCL 1 D þ j z0K 1 D ρ 0K 1 ]
þ2A
(8.220)
Then we have aO kCL 1 D arg max Lk .þ/ þ2A
k D 0; 1; : : : ; K 1
(8.221)
As for the VA, it is convenient to define the state sk D .akCL 1 ; akCL 1 1 ; : : : ; akL 2 C1 /
(8.222)
so that the desired signal at instant k, u k , given by (8.174), turns out to be only a function of sk and sk1 (see (8.199)). The formulation that we give for the solution of the problem (8.221) is called a forwardbackward algorithm (FBA) and it follows the work by Rabiner [5]: it is seen that it coincides with the BCJR algorithm [6] for the decoding of convolutional codes. We observe that, unlike in (8.222), in the two formulations [5, 6] the definition of state also includes the symbol akL 2 and consequently u k is only a function of the state at the instant k.
Statistical description of a sequential machine Briefly, we give a statistical description of the sequential machine associated with the state sk . 1. Let Ns be the number of values that sk can assume. For a sequence of i.i.d. symbols fak g, we have Ns D M L 1 CL 2 ; as in (8.198), the values assumed by the state are denoted by σ j , j D 1; : : : ; Ns . 2. The sequence fsk g is obtained by a time invariant sequential machine with transition probabilities 5. j j i/ D P[sk D σ j j sk1 D σ i ]
(8.223)
Now, if there is a transition from sk1 D σ i to sk D σ j , determined by the symbol akCL 1 D þ, þ 2 A, then 5. j j i/ D P[akCL 1 D þ]
(8.224)
8.10. Optimum methods for data detection
669
which represents the a priori probability of the generic symbol. There are algorithms to iteratively estimate this probability from the output of another decoder or equalizer (see Section 11.5). Here, for the time being, we assume there is no a priori knowledge on the symbols. Consequently, for i.i.d. symbols, we have that every state sk D σ j can be reached by M states, and P[akCL 1 D þ] D
1 M
þ2A
(8.225)
If there is no transition from sk1 D σ i to sk D σ j , we set 5. j j i/ D 0
(8.226)
3. The channel transition probabilities are given by pzk .²k j j; i/ D P[z k D ²k j sk D σ j ; sk1 D σ i ]
(8.227)
assuming that there is a transition from sk1 D σ i to sk D σ j . For a channel with complex-valued additive Gaussian noise, (8.188) holds, and pzk .²k j j; i/ D
1 ¦12 e w ³ ¦w2
j²k u k j2
(8.228)
where u k D f .σ j ; σ i /. 4. We merge (8.223) and (8.227) by defining the variable C k . j j i/ D P[z k D ²k ; sk D σ j j sk1 D σ i ] D P[z k D ²k j sk D σ j ; sk1 D σ i ] P[sk D σ j j sk1 D σ i ]
(8.229)
D pzk .²k j j; i/ 5. j j i/ 5. Initial and final conditions are given by pN j D P[s1 D σ j ] D qN j D P[s K D σ j ] D C K . j j i/ D 5. j j i/
1 Ns 1 Ns
j D 1; : : : ; Ns
(8.230)
j D 1; : : : ; Ns
(8.231)
i; j D 1; : : : ; Ns
(8.232)
If the initial and/or final state are known, for example s1 D σ i0 we set
( pN j D
1 0
s K D σ f0
(8.233)
for σ j D σ i0 otherwise
(8.234)
670
Chapter 8. Channel equalization and symbol detection
and ( qN j D
1 0
for σ j D σ otherwise
f0
(8.235)
The forward-backward algorithm (FBA) We consider the following four metrics. a) Forward metric Fk . j/ D P[z0k D ρ 0k ; sk D σ j ]
(8.236)
Equation (8.236) gives the probability of observing the sequence ²0 ; ²1 ; : : : ; ²k up to instant k, and the state σ j at instant k. A recursive expression exists for Fk . j/. 1. Initialization F1 . j/ D pN j
j D 1; : : : ; Ns
(8.237)
2. Updating for k D 0; 1; : : : ; K 1, Fk . j/ D
Ns X
C k . j j `/ Fk1 .`/
j D 1; : : : ; Ns
(8.238)
`D1
Proof. Using the total probability theorem, and conditioning the event on the possible values of s k1 , we express the probability in (8.236) as Fk . j/ D
Ns X `D1
D
Ns X `D1
P[z0k1 D ρ 0k1 ; z k D ²k ; s k D σ j ; s k1 D σ ` ]
P[z0k1 D ρ 0k1 ; z k D ²k j s k D σ j ; s k1 D σ ` ] P[s k D σ j ; s k1 D σ ` ]
(8.239) Because the noise samples are i.i.d., once the values of s k and s k1 are assigned, the event [z0k1 D ρ 0k1 ] is independent of the event [z k D ²k ], and it results in Fk . j/ D
Ns X `D1
P[z0k1 D ρ 0k1 j s k D σ j ; s k1 D σ ` ] P[z k D ²k j s k D σ j ; s k1 D σ ` ] P[s k D σ j ; s k1 D σ ` ]
(8.240)
8.10. Optimum methods for data detection
671
Moreover, given s k1 , the event [z0k1 D ρ 0k1 ] is independent of s k , and we have Fk . j/ D
Ns X `D1
P[z0k1 D ρ 0k1 j s k1 D σ ` ]
(8.241)
P[z k D ²k j s k D σ j ; s k1 D σ ` ] P[s k D σ j ; s k1 D σ ` ] By applying Bayes’ rule, (8.241) becomes Fk . j/ D
Ns X `D1
D
Ns X `D1
P[z0k1 D ρ 0k1 ; s k1 D σ ` ]
1 P[z k D ²k ; s k D σ j ; s k1 D σ ` ] P[s k1 D σ ` ]
P[z0k1 D ρ 0k1 ; s k1 D σ ` ] P[z k D ²k ; s k D σ j j s k1 D σ ` ] (8.242)
Substitution of (8.229) in (8.242) yields (8.238). b) Backward metric K 1 K 1 Bk .i/ D P[zkC1 D ρ kC1 j sk D σ i ]
(8.243)
Equation (8.243) is the probability of observing the sequence ²kC1 ; : : : ; ² K 1 , from instant k C 1 onwards, given the state σ i at instant k. A recursive expression also exists for Bk .i/. 1. Initialization B K .i/ D qNi
i D 1; : : : ; Ns
(8.244)
2. Updating for k D K 1; K 2; : : : ; 0, Bk .i/ D
Ns X
BkC1 .m/ C kC1 .m j i/
i D 1; : : : ; Ns
(8.245)
mD1
Proof. Using the total probability theorem, and conditioning the event on the possible values of s kC1 , we express the probability in (8.243) as Bk .i/ D
Ns X
K 1 K 1 P[zkC1 D ρ kC1 ; s kC1 D σ m j s k D σ i ]
mD1
(8.246) D
Ns X mD1
K 1 K 1 P[zkC1 D ρ kC1 j s kC1 D σ m ; s k D σ i ] P[s kC1 D σ m j s k D σ i ]
672
Chapter 8. Channel equalization and symbol detection
K 1 K 1 Now, given the values of s kC1 and s k , the event [zkC2 D ρ kC2 ] is independent of [z kC1 D K 1 K 1 ²kC1 ]. In turn, assigned the value of s kC1 , the event [zkC2 D ρ kC2 ] is independent of s k . Then (8.246) becomes
Bk .i/ D
Ns X
K 1 K 1 P[zkC2 D ρ kC2 j s kC1 D σ m ; s k D σ i ]
mD1
P[z kC1 D ²kC1 j s kC1 D σ m ; s k D σ i ] P[s kC1 D σ m j s k D σ i ] (8.247) D
Ns X
K 1 K 1 P[zkC2 D ρ kC2 j s kC1 D σ m ]
mD1
P[z kC1 D ²kC1 j s kC1 D σ m ; s k D σ i ] P[s kC1 D σ m j s k D σ i ] Observing (8.243) and (8.229), (8.245) follows. c) State metric Vk .i/ D P[sk D σ i j z0K 1 D ρ 0K 1 ]
(8.248)
Equation (8.248) expresses the probability of being in the state σ i at instant k, given the whole observation ρ 0K 1 . It can be expressed as a function of the forward and backward metrics, Vk .i/ D
Fk .i/ Bk .i/ Ns X Fk .n/ Bk .n/
i D 1; : : : ; Ns
(8.249)
nD1
Proof. Using the fact that, given the value of s k , the r.v.s fz t g with t > k are statistically independent of fz t g with t k, from (8.248) it follows K 1 K 1 Vk .i/ D P[z0k D ρ 0k ; zkC1 D ρ kC1 ; sk D σ i ]
D
P[z0k
D
ρ 0k ; s k
D
K 1 σ i ] P[zkC1
D
1 P[z0K 1
K 1 ρ kC1
D ρ 0K 1 ]
j sk D σ i ]
1
(8.250)
P[z0K 1 D ρ 0K 1 ]
Observing the definitions of forward and backward metrics, (8.249) follows. We note that the normalization factor P[z0K 1 D ρ 0K 1 ] D
Ns X
Fk .n/ Bk .n/
(8.251)
nD1
makes Vk .i/ a probability, so that Ns X i D1
Vk .i/ D 1
(8.252)
8.10. Optimum methods for data detection
673
d) Likelihood function of the generic symbol. Applying the total probability theorem to (8.220) we obtain the relation Lk .þ/ D
Ns X
P[akCL 1 D þ; sk D σ i j z0K 1 D ρ 0K 1 ]
(8.253)
i D1
From the comparison of (8.253) with (8.248), indicating with [σ i ]m , m D 1; : : : ; L 1 C L 2 , the mth component of the state σ i (see (8.222)), we have Ns X
Lk .þ/ D
Vk .i/
þ2A
(8.254)
i D1 condition [σ i ]1 D þ
In other words, at instant k the likelihood function coincides with the sum of the metrics Vk .i/ associated with the states whose first component is equal to the symbol of value þ. Note that Lk .þ/ can also be obtained using the state metrics evaluated at different instants, that is Lk .þ/ D
Ns X
VkC.m1/ .i/
þ2A
(8.255)
i D1 [σ i ]m D þ
for m 2 f1; : : : ; L 1 C L 2 g
Scaling We see that, due to the exponential form of pzk .²k j j; i/ in Ck . j j i/, in a few iterations the forward and backward metrics may assume very small values; this leads to numerical problems in the computation of the metrics: therefore we need to substitute equations (8.238) and (8.245) with analogous expressions that are scaled by a suitable coefficient. We note that the state metric (8.249) does not change if we multiply Fk .i/ and Bk .i/, i D 1; : : : ; Ns , by the same coefficient Kk . The idea [5] is to choose Kk D
1 Ns X
(8.256)
Fk .n/
nD1
Indicating with FNk .i/ and BN k .i/ the normalized metrics, for Fk . j/ D
Ns X `D1
C k . j j `/ FNk1 .`/
(8.257)
674
Chapter 8. Channel equalization and symbol detection
equation (8.238) becomes Ns X
FNk . j/ D
C k . j j `/ FNk1 .`/
`D1 N Ns s XX
j D 1; : : : ; Ns k D 0; 1; : : : ; K 1
(8.258)
i D 1; : : : ; Ns k D K 1; K 2; : : : ; 0
(8.259)
C k .n j `/ FNk1 .`/
nD1 `D1
Correspondingly (8.245) becomes Ns X
BN k .i/ D
BN kC1 .m/ C kC1 .m mD1 Ns X Ns X
j i/
C k .n j `/ FNk1 .`/
nD1 `D1
Hence, Vk .i/ D
FNk .i/ BN k .i/ Ns X FNk .n/ BN k .n/
i D 1; : : : ; Ns k D 0; 1; : : : ; K 1
(8.260)
nD1
Likelihood function in the absence of ISI In the absence of ISI, u k D ak . Therefore the state is expressed as sk D .ak /, and is identified by the symbol value þ. The metric in (8.250), for i.i.d. symbols and channel transition probabilities (8.228) becomes Vk .þ/ D P[z k D ²k ; sk D ¦i ] D P[z k D ²k ; ak D þ] DKe
1 j² þj2 ¦w2 k
þ2A
(8.261)
where K is a constant. As Vk . þ/ is a probability, (8.261) can be written as
e Vk .þ/ D X
1 j² þj2 ¦w2 k
e
1 j² Þj2 ¦w2 k
þ2A
k D 0; 1; : : : ; K 1
(8.262)
þ2A
(8.263)
Þ2A
Then (8.254) simply becomes Lk .þ/ D Vk .þ/
8.10. Optimum methods for data detection
675
Simplified version of the MAP algorithm (Max-Log-MAP) We introduce the Log-MAP criterion, which employs the logarithm of the variables Fk .i/, Bk .i/, C k . j j i/, Vk .i/, and Lk .þ/. The logarithmic variables are indicated with the corresponding lower-case letters; in particular, from (8.249) we have vk .i/ D ln Vk .i/ and (8.254) becomes
(8.264) 1
0
B `k .þ/ D ln Lk .þ/ D ln B @
Ns X
C evk .i / C A
i D1 [σ i ]1 D þ
þ2A
(8.265)
The function `k .þ/ is called log-likelihood. The exponential emphasizes the difference between the metrics vk .i/: typically a term dominates within each sum; this suggests the approximation ln
Ns X
evk .i / '
i D1
max
i 2f1;:::;Ns g
vk .i/
(8.266)
Consequently (8.265) is approximated as `Qk .þ/ D
max
i 2 f1; : : : ; Ns g [σ i ]1 D þ
vk .i/
(8.267)
and the Log-MAP criterion is replaced by the Max-Log-MAP criterion aO kCL 1 D arg max `Qk .þ/
(8.268)
þ2A
Observation 8.12 In the particular case of absence of ISI, substitution of (8.263) in (8.265) yields `k .þ/ D
j²k þj2 ¦w2
(8.269)
and (8.268) becomes aO k D arg min j²k þj2
(8.270)
þ2A
which corresponds to the minimum distance decision criterion. Example 8.10.3 (Binary case) In the binary case, þ 2 f1; 1g, it is convenient to introduce the likelihood ratio given by (see (8.220)) Lk D
P[akCL 1 D 1 j z0K 1 D ρ 0K 1 ] P[akCL 1 D
1 j z0K 1
D
ρ 0K 1 ]
D
Lk .1/ Lk .1/
(8.271)
676
Chapter 8. Channel equalization and symbol detection
Then the decision rule (8.221) becomes ² aO kCL 1 D
1 1
if Lk 1 if Lk < 1
(8.272)
We recall that in the absence of ISI the expression of Lk .þ/ is given by (8.262). The analysis is simplified by the introduction of the log-likelihood ratio (LLR), `k D ln Lk D `k .1/ `k .1/
(8.273)
where `k .þ/ is given by (8.265). Then (8.272) becomes aO kCL 1 D sgn.`k /
(8.274)
`k D aO kCL 1 j`k j
(8.275)
Observing (8.274), we can write
In other words, the sign of the log-likelihood ratio yields a hard-decision on the symbol being detected, while the magnitude indicates the reliability of the decision. The function `k is used as a soft-decision parameter in some detection algorithms (see page 923). In the Max-Log-MAP formulation, the LLR is given by `Qk D `Qk .1/ `Qk .1/ D
max
i 2 f1; : : : ; Ns g [σ i ]1 D 1
vk .i/
max
i 2 f1; : : : ; Ns g [σ i ]1 D 1
vk .i/
(8.276)
and aO kCL 1 D sgn.`Qk /
(8.277)
Apart from non-essential constants, the Max-Log-MAP algorithm in the case of transmission of i.i.d. symbols over a channel with additive white Gaussian noise is formulated as follows. 1. Computation of channel transition metrics. For k D 0; 1; : : : ; K 1, ck . j j i/ D j²k u k j2
i; j D 1; : : : ; Ns
(8.278)
where u k D f .σ j ; σ i /, assuming there is a transition between sk1 D σ i and sk D σ j . For k D K , we let c K . j j i/ D 0
i; j D 1; : : : ; Ns
(8.279)
again, assuming there is a transition between σ i and σ j . 2. Backward procedure. For k D K 1; K 2; : : : ; 0, bQk .i/ D
max
[bQkC1 .m/ C ckC1 .m j i/]
m2f1;:::;N s g
i D 1; : : : ; Ns
(8.280)
8.10. Optimum methods for data detection
677
If the final state is known, then ( bQ K .i/ D
0 1
σ i D σ f0 otherwise
(8.281)
If the final state is unknown, we set bQ K .i/ D 0, i D 1; : : : ; Ns . 3. Forward procedure. For k D 0; 1; : : : ; K 1, fQk . j/ D
max
`2f1;:::;Ns g
[ fQk1 .`/ C ck . j j `/]
j D 1; : : : ; Ns
(8.282)
If the initial state is known, then ( fQ1 . j/ D
0 1
σ j D σ i0 otherwise
(8.283)
If the initial state is unknown, we set fQ1 . j/ D 0, j D 1; : : : ; Ns . 4. State metric. For k D 0; 1; : : : ; K 1, vQk .i/ D fQk .i/ C bQk .i/
i D 1; : : : ; Ns
(8.284)
5. Log-likelihood function of an isolated symbol. For k D 0; 1; : : : ; K 1, the loglikelihood function is given by (8.267), with vQk .i/ in substitution of vk .i/; the decision rule is given by (8.268). In practical implementations of the algorithm, steps 3, 4, and 5 can be carried out in sequence for each value of k: this saves memory locations. To avoid overflow, for each value of k a common value can be added to all variables fQk .i/ and bQk .i/, i D 1; : : : ; Ns . We observe that the two procedures, backward and forward, can be efficiently implemented by the Viterbi algorithm, using a trellis diagram run both in backward and forward directions. The simplified MAP algorithm requires about twice the complexity of the VA implementing the MLSD criterion. Memory requirements are considerably increased with respect to the VA, because the backward metrics must be stored before evaluating the state metrics. However, methods for an efficient use of the memory are proposed in [7].
Relation between Max-Log-MAP and Log-MAP We define the following function of two variables [7] maxŁ .x; y/ D ln.e x C e y /
(8.285)
it can be verified that the following relation holds: maxŁ .x; y/ D max.x; y/ C ln.1 C ejxyj /
(8.286)
We now extend the above definition to the case of three variables, maxŁ .x; y; z/ D ln.e x C e y C e z /
(8.287)
678
Chapter 8. Channel equalization and symbol detection
then we have maxŁ .x; y; z/ D max Ł .maxŁ .x; y/; z/
(8.288)
The extension to more variables is readily obtained by induction. So, if in the backward and forward procedures of page 676 we substitute the max function with the maxŁ function, we obtain the exact Log-MAP formulation that relates vk .i/ D ln Vk .i/ to bk .i/ D ln Bk .i/ and f k .i/ D ln Fk .i/, using the branch metric ck . j j i/.
8.11
Optimum receivers for transmission over dispersive channels
It is possible to identify two different receiver structures that supply the signal z k given by (8.173). 1. The first, illustrated in Figure 8.25a, is considered for the low implementation complexity; it refers to the receiver of Figure 7.12, where s f (8.289) ;² GRc . f / D rcos 1=T and wC .t/ is white Gaussian noise with spectral density N0 . Recalling that r R .t/ D s R .t/ C w R .t/, with f (8.290) ;² Pw R . f / D Pw . f / jGRc . f /j2 D N0 rcos 1=T
G R (f)= rcos C
ak
q
T
sC(t)
rC(t)
C
(1/Tf ,ρ) gRc
t 0+kT rR (t) zk T
wC (t) (AWGN)
(a)
gM (t)=q*C (t0 -t) ak T
q
sC(t) C
rC(t)
g
x(t) M
t 0+kT WF xk zk w T T
wC (t) (AWGN) (b)
Figure 8.25. Two receiver structures with i.i.d. noise samples at the decision point.
8.11. Optimum receivers for transmission over dispersive channels
679
it is important to verify that the noise sequence fwk D w R .t0 C kT /g has a constant spectral density equal to N0 , and the variance of the noise samples is ¦w2 D N0 =T . Although the filter defined by (8.289) does not necessarily yield a sufficient statistic (see Observation 8.13 on page 681), it considerably reduces the noise and this may be useful in estimating the channel impulse response. Another problem concerns the optimum timing phase, which may be difficult to determine for non-minimum phase channels. 2. An alternative, also known as the Forney receiver, is represented in Figure 8.20 and repeated in Figure 8.25b. To construct the WF, however, it is necessary to determine poles and zeros of the function 8.z/; this can be rather complicated in real time applications. A practical method is based on Observation 8.9 on page 650. From the knowledge of the autocorrelation sequence of the channel impulse response, the prediction error filter A.z/ is determined. The WF of Figure 8.25 is given by W .z/ D 12 AŁ .1=z Ł /; therefore f0
it is an FIR filter. The impulse response fn g is given by the inverse z-transform of 1 f0 F.z/ D 1=A.z/. For the realization of symbol detection algorithms, a windowed version of the impulse response is considered. A further method, which usually requires a filter w with a smaller number of coefficients than the previous method, is based on the observation that, for channels with low noise level, the DFE solution determined by the MSE method coincides with the DFE-ZF solution; in this case the FF filter plays the role of the filter w in Figure 8.25b. We consider two cases. a. At the output of the MF g M , let fh n g be the system impulse response with sampling period T , determined for example through the method described in the Observation 8.5 on page 641. Using (8.96), a DFE is designed with filter parameters .M1 ; M2 /, and consequently w D copt in Figure 8.25b. At the output of the filter w, the ideally minimum phase impulse response fn g corresponds to the translated, by D sampling intervals, and windowed, with L 1 D 0 and L 2 D M2 , version of D h Ł copt . b. If the impulse response of the system is unknown, we can resort to the FSDFE structure of Figure 8.14. Using now an adaptive method to determine the coefficients of the DFE, at convergence we get ( 1 nD0 n D nCD ' (8.291) bn n D 1; : : : ; M2 Actually, unless the length of fn g is shorter than that of the impulse response qC , it is convenient to use Ungerboeck’s formulation of the MLSD [8] that utilizes only samples fx k g at the MF output; now, however, the metric is no longer Euclidean. The derivation of the non-Euclidean metric is the subject of the next section. We note that, as it is not important to obtain the likelihood of an isolated symbol from the non-Euclidean metric, there are cases in which this method is not adequate. We refer in particular to the case in which decoding with soft input is performed separately from
680
Chapter 8. Channel equalization and symbol detection
symbol detection in the presence of ISI (see Section 11.3.2). However, joint decoding and detection are always possible using a suitable trellis (see Section 11.3.2).
Ungerboeck’s formulation of the MLSD We refer to the transmission of K symbols and to an observation interval TK D K T sufficiently large, so that the transient of filters at the beginning and at the end of the transmission has a negligible effect. The derivation of the likelihood is based on the received signal (7.45), rC .t/ D sC .t/ C wC .t/
(8.292)
where wC is white noise with PSD N0 , and sC .t/ D
K 1 X
ak qC .t kT /
(8.293)
kD0
For a suitable basis, we consider for (8.292) the following vector representation: rDsCw Assuming that the transmitted symbol sequence a D [a0 ; a1 ; : : : ; a K 1 ] is equal to α D [Þ0 ; Þ1 ; : : : ; Þ K 1 ], the probability density function of r is given by (6.16), 1 2 (8.294) prja .ρ j α/ D K exp jjρ sjj N0 Using (1.36), and observing rC .t/ D ².t/, we get Z 1 prja .ρ j α/ D K exp j².t/ sC .t/j2 dt N 0 TK Taking the logarithm in (8.295), the log-likelihood (to be maximized) is þ2 Z þþ K 1 þ X þ þ `.α/ D Þk qC .t kT /þ dt þ².t/ þ TK þ
(8.295)
(8.296)
kD0
Correspondingly the detected sequence is given by aO k D arg max `.α/
(8.297)
α
Expanding the squared term in (8.296), we obtain (Z "Z # K 1 X 2 Ł Ł `.α/ D j².t/j dt 2Re ².t/ Þk qC .t kT / dt TK
Z C
TK
1 K 1 K X X TK k1 D0 k2 D0
kD0
) Þk1 ÞkŁ2 qC .t k1 T / qCŁ .t k2 T / dt
(8.298)
8.11. Optimum receivers for transmission over dispersive channels
681
We now introduce the MF g M .t/ D qCŁ .t0 t/
(8.299)
and the overall impulse response at the MF output q.t/ D .qC Ł g M /.t/ D rqC .t t0 /
(8.300)
where rqC is the autocorrelation of qC , whose samples are given by (see (8.23)) h n D q.t0 C nT / D rqC .nT /
(8.301)
Let x.t/ be the MF output signal expressed as x.t/ D .rC Ł g M /.t/
(8.302)
x k D x.t0 C kT /
(8.303)
with samples given by
In (8.298) the first term can be ignored since it does not depend on α, while the other two terms are rewritten in the following form: ( " # ) K 1 K 1 K 1 X X X Ł Ł `.α/ D 2Re Þk x k C Þk1 Þk2 h k2 k1 (8.304) kD0
k1 D0 k2 D0
Observing (8.304), we obtain the following important result. Observation 8.13 The sequence of samples fx k g, taken by sampling the MF output signal with sampling period equal to the symbol period T , forms a sufficient statistic to detect the message fak g associated with the signal rC defined in (8.292). We express the double summation in (8.304) as the sum of three terms, the first for k1 D k2 , the second for k1 < k2 , and the third for k1 > k2 : AD
K 1 K 1 X X
Þk1 ÞkŁ2 h k2 k1
k1 D0 k2 D0
(8.305) D
K 1 X k1 D0
Þk1 ÞkŁ1 h 0 C
K 1 kX 1 1 X k1 D1 k2 D0
Þk1 ÞkŁ2 h k2 k1 C
K 1 kX 2 1 X
Þk1 ÞkŁ2 h k2 k1
k2 D1 k1 D0
Because the sequence fh n g is an autocorrelation, it enjoys the Hermitian property, i.e. h n D h nŁ ; consequently, the third term in (8.305) is the complex conjugate of the second, and " # k1 K 1 K 1 X X X Ł Ł AD Þk Þk h 0 C 2Re Þk Þk2 h kk2 (8.306) kD0
kD1 k2 D0
682
Chapter 8. Channel equalization and symbol detection
By the change of indices n D k k2 , assuming Þk D 0 for k < 0, we get ( " #) K 1 K 1 X k X X A D Re ÞkŁ Þk h 0 C 2 ÞkŁ Þkn h n kD0
( D Re
K 1 X
kD1 nD1
" ÞkŁ h 0 Þk C 2
kD0
k X
#)
(8.307)
h n Þkn
nD1
In particular, if jh n j ' 0
for jnj > Nh
(8.308)
(8.307) is simplified in the following expression ( " #) Nh K 1 X X A D Re ÞkŁ h 0 Þk C 2 h n Þkn kD0
(8.309)
nD1
Then the log-likelihood (8.304) becomes ( " #) Nh K 1 X X Ł `.α/ D Re Þk 2x k C h 0 Þk C 2 h n Þkn kD0
(8.310)
nD1
To maximize `.α/ or, equivalently, to minimize `.α/ with respect to α we apply the Viterbi algorithm (see page 663) with the state vector defined as sk D .ak ; ak1 ; : : : ; akNh C1 / and branch metric given by ( Re
akŁ
" 2x k C h 0 ak C 2
Nh X
(8.311) #)
h n akn
(8.312)
nD1
Extensions of Ungerboeck’s approach to time variant radio channels are proposed in [9].
8.12
Error probability achieved by MLSD
In the Viterbi algorithm, we have an error if a sequence of states that is different from the correct sequence is chosen as maximum likelihood sequence in the trellis diagram; the probability that one or more states of the detected ML sequence are in error is interesting. The error probability is dominated by the probability that a sequence at the minimum Euclidean distance from the correct sequence is chosen as ML sequence. We note, however, that increasing the sequence length K also increases the number of different paths in the trellis diagram associated with sequences that are at the minimum distance. Therefore, by increasing K , the probability that the chosen sequence is in error usually tends to 1. The probability that the whole sequence of states is not received correctly is rarely of interest; instead, we consider the probability that the detection of a generic symbol is in
8.12. Error probability achieved by MLSD
683
error. For the purpose of determining the symbol error probability, the concept of error event is introduced. Let fσ g D .σ i0 ; : : : ; σ i K 1 / be the realization of the state sequence associated with the information sequence, and let fσO g be the sequence chosen by the Viterbi algorithm. In a sufficiently long time interval, the paths in the trellis diagram associated with fσ g and fσO g diverge and converge several times: every distinct separation from the correct path is called an error event. Definition 8.1 An error event e is defined as a path in the trellis diagram that has only the initial and final states in common with the correct path; the length of an error event is equal to the number of nodes visited in the trellis before rejoining with the correct path. Error events of length one and two are illustrated in a trellis diagram with two states, where the correct path is represented by a continuous line, in Figure 8.26a and Figure 8.26b, respectively. Let E be the set of all error events beginning at instant i. Each element e of E is characterized by a correct path fσ g and a wrong path fσO g, which diverges from fσ g at instant i and converges at fσ g after a certain number of steps in the trellis diagram. We assume that the probability P[e] is independent of instant i: this hypothesis is verified with good approximation if the length of the trellis diagram is much greater than the length of the significant error events. An error event produces one or more errors in the detection of symbols of the input sequence. We have a detection error at instant k if the detection of the input at the k-th stage of the trellis diagram is not correct. We define the function [10] (
if e causes a detection error at the instant i C m ; with m 0 otherwise (8.313) The probability of a particular error event that starts at instant i and causes a detection error at instant k is given by cki .e/P[e]. Because the error events in E are disjointed, we have cm .e/ D
1 0
Pe D P[aO k 6D ak ] D
k X X
cki .e/ P[e]
(8.314)
i D1 e2E
k
k+1
(a)
k+2
k
k+1
k+2
k+3
(b)
Figure 8.26. Error events of length (a) one and (b) two in a trellis diagram with two states.
684
Chapter 8. Channel equalization and symbol detection
Assuming that the two equations can be exchanged, we obtain Pe D
X
P[e]
e2E
k X
cki .e/
(8.315)
cm .e/ D N .e/
(8.316)
i D1
With a change of variables it turns out k X i D1
cki .e/ D
1 X mD0
which indicates the total number of detection errors caused by the error event e. Therefore, X Pe D N .e/P[e] (8.317) e2E
where the dependence on the time index k vanishes. We therefore find that the detection error probability is equal to the average number of errors caused by all the possible error events initiating at a given instant i; this result is expected, because the detection error probability at a particular instant k must take into consideration all error events that initiate at previous instants and are not yet terminated. If fsg D .s0 ; : : : ; s K 1 / denotes the random variable sequence of states at the transmitter and fOs g D .Os 0 ; : : : ; sO K 1 / denotes the random variable sequence of states selected by the ML receiver, the probability of an error event e beginning at a given instant i depends on the joint probability of the correct and incorrect path, and it can be written as P[e] D P[fOs g D fσO g j fsg D fσ g]P[fsg D fσ g]
(8.318)
Because it is usually difficult to find the exact expression for P[fOs g D fσO g j fsg D fσ g], we resort to upper and lower limits. Upper limit. Because detection of the sequence of states fsg is obtained by observing the sequence fug; for the signal in (8.173) with zero mean additive white Gaussian noise having variance ¦ I2 per dimension, we have the upper limit d[u.fσ g/; u.fσO g/] P[fOs g D fσO g j fsg D fσ g] Q (8.319) 2¦ I where d[u.fσ g/; u.fσO g/] is the Euclidean distance between signals u.fσ g/ and u.fσO g/, given by (8.193). Substitution of the upper limit in (8.317) yields X d[u.fσ g/; u.fσO g/] Pe (8.320) N .e/ P[fsg D fσ g]Q 2¦ I e2E which can be rewritten as follows, by giving prominence to the more significant terms, X dmin C other terms (8.321) Pe N .e/ P[fsg D fσ g]Q 2¦ I e2Emin
8.12. Error probability achieved by MLSD
685
where Emin is the set of error events at minimum distance dmin defined in (8.194), and the remaining terms are characterized by arguments of the Q function larger than dmin =.2¦ I /. For higher values of the signal-to-noise ratio these terms are negligible and the following approximation holds dmin (8.322) Pe K1 Q 2¦ I where K1 D
X
N .e/ P[fsg D fσ g]
(8.323)
e2Emin
Lower limit. A lower limit to the error probability is obtained by considering the probability that any error event may occur rather than the probability of a particular error event. Since N .e/ ½ 1 for all the error events e, from (8.317) we have X Pe ½ P[e] (8.324) e2E
Let us consider a particular path in the trellis diagram determined by the sequence of states fσ g. We set dmin .fσ g/ D min d[u.fσ g/; u.fσQ g/]
(8.325)
fσQ g
i.e., for this path, dmin .fσ g/ is the Euclidean distance of the minimum distance error event. We have dmin .fσ g/ ½ dmin , where dmin is the minimum distance obtained considering all the possible state sequences. If fσ g is the correct state sequence, the probability of an error event is lower limited by dmin .fσ g/ (8.326) P[e j fsg D fσ g] ½ Q 2¦ I Consequently, Pe ½
X
P[fsg D fσ g]Q
fσ g
dmin .fσ g/ 2¦ I
(8.327)
If some terms are omitted in the equation, the lower limit is still valid, because the terms are non-negative. Therefore, taking into consideration only those state sequences fσ g for which dmin .fσ g/ D dmin , we obtain X dmin (8.328) Pe ½ P[fsg D fσ g]Q 2¦ I fσ g2A where A is the set of state sequences that admit an error event with minimum distance dmin , for an arbitrarily chosen initial instant of the given error event. Defining X K2 D P[fsg D fσ g] (8.329) fσ g2A
686
Chapter 8. Channel equalization and symbol detection
as the probability that a path fσ g admits an error event with minimum distance, it is dmin (8.330) Pe ½ K2 Q 2¦ I Combining upper and lower limits we obtain dmin dmin K2 Q Pe K1 Q 2¦ I 2¦ I For large values of the signal-to-noise ratio, therefore we have dmin Pe ' K Q 2¦ I
(8.331)
(8.332)
for some value of the constant K between K1 and K2 . We stress that the error probability, expressed by (8.332) and (8.195), is determined by the ratio between the minimum distance dmin and the standard deviation of the noise ¦ I . Here the expressions of the constants K1 and K2 are obtained by resorting to various approximations. An accurate method to calculate upper and lower limits of the error probability is proposed in [11].
Computation of the minimum distance The application of the Viterbi algorithm to maximum likelihood sequence detection in transmission systems with ISI requires that the overall impulse response is FIR, otherwise the number of states, and hence also the complexity of the detector, becomes infinite. From (8.173), the samples at the detector input, conditioned on the event that the sequence of symbols fak g is transmitted, are statistically independent Gaussian random variables with mean L2 X
n akn
(8.333)
nDL 1
and variance ¦ I2 per dimension. The metric that the Viterbi algorithm attributes to the sequence of states corresponding to the sequence of input symbols fak g is given by the squared Euclidean distance between the sequence of samples fz k g at the detector input and its mean value, which is known, given the sequence of symbols (see (8.189)), þ þ2 L2 1 þ þ X X þ þ n akn þ þz k þ þ nDL kD0
(8.334)
1
In the previous section it was demonstrated that the symbol error probability is given by (8.332). In particularly simple cases, the minimum distance can be determined by direct inspection of the trellis diagram; in practice, however, this situation is rarely verified in channels with ISI. To evaluate the minimum distance it is necessary to resort to simulations. To find the minimum distance error event with initial instant k D 0, we consider the
8.12. Error probability achieved by MLSD
687
desired signal u k under the condition that the sequence fak g is transmitted, and we compute the squared Euclidean distance between this signal and the signal obtained for another sequence faQ k g, þ þ2 L2 L2 1 þ X þ X X þ þ d [u.fak g/; u.faQ k g/] D n akn n aQ kn þ þ þ þ nDL kD0 nDL 2
1
(8.335)
1
where it is assumed that the two paths identifying the state sequences are identical for k < 0. It is possible to avoid computing the minimum distance for each sequence fak g if we exploit the linearity of the ISI. Defining žk D ak aQ k
(8.336)
we have þ þ2 L2 1 þ X þ X þ þ d .fžk g/ D d [u.fak g/; u.faQ k g/] D n žkn þ þ þ þ kD0 nDL 2
2
(8.337)
1
The minimum among the squared Euclidean distances relative to all error events that initiate at k D 0 is 2 dmin D
min
fžk g: žk D0; k PwC . f / ½ 3. Compute the magnitude of HT x . f /: 8 p T Pw . f / < T ½PwC . f / 2 C 2 jHT x . f /j2 D ¦a jGC . f /j2 : ¦a jGC . f /j 0
f 2 B½
(8.543)
f 2 = B½
jG Rc . f /j is determined using (8.537); the phase of HT x . f / and of G Rc . f / must be such that Q R . f / D HT x . f /GC . f /G Rc . f / is a positive real-valued signal. 4. Substituting (8.543) in (8.539), compute MT x . Verify whether MT x D V , otherwise steps 2–4 must be repeated for a different value of ½ until MT x D V . As observed in [58], the physical interpretation of (8.543) is that HT x . f / has non-zero components in relation to frequencies at which the channel does not attenuate too much with respect to the noise level. However, this ratio is defined by 1=½, where ½ is a Lagrange multiplier used in the optimization of problem (8.512) under the constraint (8.539).
8.B. DFE design: matrix formulations
Appendix 8.B 8.B.1
741
DFE design: matrix formulations
Method based on correlation sequences
We discuss a DFE design method that requires knowledge of the correlation sequences of the equalizer input signal and of the detected message, as well as their cross-correlation sequence [60]. With reference to Figure 8.13, we introduce the vectors whose elements are given by the coefficients of the feedforward and feedback filters, c D [c0 ; c1 ; : : : ; c M1 1 ]T b D [b1 ; b2 ; : : : ; b M2 ]T
(8.544)
bQ D [1; bT ]T and the vectors xk D [x k ; x k1 ; : : : ; x kM1 C1 ]T aO k D [aO kD ; : : : ; aO kDM2 ]T ak D [akD ; : : : ; akDM2 ]T
(8.545)
aO 0k D [aO kD1 ; : : : ; aO kDM2 ]T a0k D [akD1 ; : : : ; akDM2 ]T We note the presence of a delay D due to the feedforward filter. From (8.91), the sample at the decision point, yk , can be expressed using the following vector notation: yk D cT xk C bT aO 0k
(8.546)
Using (8.546), also the estimation error can be expressed in vector form as ek D yk akD D cT xk C bT aO 0k akD
(8.547)
Assuming that the detection of past symbols is correct, we have aO 0k D a0k . Then, introducing Q (8.547) can be written as the vector b, ek D cT xk C [1; bT ]ak D cT xk C bQ T ak
(8.548)
The filter coefficients c and bQ are determined by minimizing the cost function J D E[jyk akD j2 ]
(8.549)
742
Chapter 8. Channel equalization and symbol detection
Introducing the correlation matrices Ra D E[aŁk akT ] Rx D E[xŁk xkT ] Rax D E[aŁk xkT ]
(8.550)
Rxa D E[xŁk akT ] the cost function J can be written as J D E[ekŁ ek ] D E[.cT xk C bQ T ak /Ł .cT xk C bQ T ak /T ]
(8.551)
D c H Rx c C bQ H Ra bQ C c H Rxa bQ C bQ H Rax c On the other hand, minimization of the MSE implies orthogonality of the error ek with respect to the input xk . Consequently, using the expression (8.548) for ek it must be E[ekŁ xkT ] D c H Rx C bQ H Rax D 0
(8.552)
bQ H Rax D c H Rx
(8.553)
or
Assuming Rx is strictly positive definite and therefore invertible, we obtain H D bQ H Rax R1 copt x
(8.554)
This relation, inserted in (8.551), yields Q QH J D bQ H Rax c C bQ H Ra bQ bQ H Rax R1 x Rxa b C b Rax c
(8.555)
Q D bQ H .Ra Rax R1 x Rxa /b We recognize that Rajx D Ra Rax R1 x Rxa
(8.556)
is the correlation matrix of the estimation error vector (see Appendix 2.A). The problem thus reduces to finding the vector bQ that minimizes the quadratic form J D bQ H Rajx bQ
(8.557)
Q must be Defining e1 D [1; 01ðM2 ]T and remembering that bQ1 , the first component of b, equal to 1, the solution is obtained by the minimization of the quadratic function J D bQ H Rajx bQ
(8.558)
8.B. DFE design: matrix formulations
743
b
3
-1
b
1
b2 Figure 8.56. Constrained cost function in the case M2 D 2.
with the constraint v D e1 bQ1 C e1 D 0
(8.559)
Figure 8.56 illustrates the problem in the case M2 D 2. The vector bQ that minimizes J subject to the constraint v D 0 is obtained by the method of Lagrange multipliers, ( rbQ J C ½rbQ v D 0 (8.560) vD0 As rbQ J D 2Rajx bQ and rbQ v D e1 , (8.560) yields ( 2Rajx bQ D ½e1
(8.561)
vD0 whose solution is given by 1 bQ opt D .Rajx /1 e1 ½
(8.562)
where ½ is the factor that forces the first component of bQ opt equal to 1. We note that, apart from the coefficient 1=½, bQ opt coincides with the first column of the matrix .Rajx /1 . With this choice of bQ opt , (8.554) yields the optimum FF filter coefficients Q copt D R1 x Rxa bopt
(8.563)
Although this procedure is very general, valid for a general statistic of the information message, it is rather computationally expensive because it requires two matrix inversions, for Rajx and Rx .
744
Chapter 8. Channel equalization and symbol detection
Observation 8.15 For an LE, the formulation is as in (8.563), however, without the filter b, and with aO k D [aO kD ] and bQ D [1].
8.B.2
Method based on the channel impulse response and i.i.d. symbols
This method is obtained by applying a matrix formulation similar to that of Section 8.5. We introduce a definition that will simplify the notation. Definition 8.4 Let q D [q1 ; : : : ; q N ]T be a vector with N elements, and qi j D [qi ; qi C1 ; : : : ; q j ]T
(8.564)
denote the vector containing a subsequence of consecutive elements of q. Let A D [An;m ], n D 1; : : : ; N R , m D 1; : : : ; NC , be a N R ð NC matrix. If Až;m denotes the m-th column, then Až;i j D [Až;i ; Až;i C1 ; : : : ; Až; j ]
(8.565)
denotes the matrix containing a subsequence of consecutive columns of the matrix A. Figure 8.57 illustrates the vector representation of the scheme of Figure 8.5, extended to include a DFE. Let fh i g, i D N1 ; : : : ; N2 , be the overall impulse response at the equalizer input. Introducing the vectors xk D [x k ; x k1 ; : : : ; x kM1 C1 ]T ak D [akCN1 ; : : : ; ak ; : : : ; akN2 M1 C1 ]T
(8.566)
Q k D [wQ k ; wQ k1 ; : : : ; wQ kM1 C1 ]T w and using (8.544), it is possible to express the input vector xk of the FF filter as a linear combination of the transmitted symbols ak plus noise Qk xk D Hak C w
Figure 8.57. Overall system with DFE: vector formulation.
(8.567)
8.B. DFE design: matrix formulations
745
where 2 6 6 6 6 HD6 6 6 4
h N1 h N1 C1 h N1 C2 0 h N1 h N1 C1 0 0 h N1 :: : 0 0 0
: : : h N2 0 0 : : : h N2 1 h N2 0 : : : h N2 2 h N2 1 h N2 :: :
::: ::: ::: :: :
: : : h N1 h N1 C1
0 0 0 :: :
3 7 7 7 7 7 7 7 5
(8.568)
h N2
is an M1 ð .N1 C N2 C M1 / Toeplitz matrix . We note that in (8.566) the definition of ak refers to the system memory at the output of the filter c: It is useful to express the operation of the FB filter assuming as an input the vector ak , given by (8.566). Therefore we introduce the vector b0 D [01ð.N1 CDC1/ ; bT ; 01ðM3 ]T
(8.569)
where M3 D M1 C N2 D 1 M2 . The sample yk at the decision point can thus be expressed as Q k D .cT H C b0 T /ak C cT w Qk yk D cT xk C b0 T ak D cT Hak C b0 T ak C cT w
(8.570)
Equation (8.570) suggests that the cancellation of ISI from the sample yk , due to the symbols fakDi g, i D 1; : : : ; M2 , is obtained by setting N1 CDCM2 C1 b0 D [01ð.N1 CDC1/ ; .cT H/ N ; 01ðM3 ]T 1 CDC2
(8.571)
Fixed b0 , (8.570) becomes Q k/ yk D cT .H0 ak C w
(8.572)
where N1 CDC1 N1 CN2 CM1 ; 0 M1 ðM2 ; Hž;N ] H0 D [Hž;1 1 CDCM2 C2
2 6 6 D6 6 4
h N1 h N1 C1 0 h N1 :: : 0 0
: : : hD : : : h D1 :: :: : : ::: 0
0 ::: 0 ::: :: : 0 :::
0 0 ::: 0 0 0 ::: 0 :: :: :: : : : 0 h N2 M3 C1 : : : h N2
3 7 7 7 7 5
(8.573)
Q k /, whose We are now back to the case of one filter c, having as input vector .H0 ak C w coefficients are determined by minimizing the cost function J D E[jyk akD j2 ]
(8.574)
746
Chapter 8. Channel equalization and symbol detection
From the Wiener theory, we know that to find the solution we must compute the autocorrelation matrix R of the input signal and the cross-correlation vector p between the desired output signal and the input signal. From Q k /Ł .H0 ak C w Q k /T ] R D E[.H0 ak C w
(8.575)
assuming the symbols of the sequence fak g are i.i.d. with variance ¦a2 , we obtain R D ¦a2 H0 Ł H0 T C RwQ
(8.576)
where RwQ is the autocorrelation matrix of the noise, assumed known. Moreover, we have Q k /Ł ] D H0 Ł E[akD aŁk ] p D E[akD .H0 ak C w
(8.577)
given the statistical independence between transmitted symbols and noise. As the symbols are i.i.d., the vector E[akD aŁk ] has all zero elements except that in position N1 C D C 1, with value ¦a2 , and p corresponds to the .N1 C DC1/-th column of the matrix H0 Ł multiplied by the scalar ¦a2 , 0Ł
p D ¦a2 Hž;N1 CDC1 D ¦a2 [h D ; h D1 ; : : : ; h N1 ; 01ðM1 .N1 CDC1/ ] H
(8.578)
According to the Wiener theory, the optimum FF filter coefficients are given by copt D R1 p 0Ł
D .¦a2 H0 Ł H0 T C RwQ /1 ¦a2 Hž;N1 CDC1
(8.579)
and the minimum value of the cost function J is Jmin D ¦a2 p H copt
(8.580)
The feedback filter is obtained by substituting (8.579) in (8.571). Observation 8.16 For an LE, the formulation is as in (8.579) and (8.580), however, without the filter b, and with H0 equal to H.
8.B.3
Method based on the channel impulse response and any symbol statistic
For the signal (8.567), the correlation matrices (8.550) are given by Ra D E[aŁk akT ] Rx D E[xŁk xkT ] D HŁ Ra HT C RwQ Rax D E[aŁk xkT ] D Ra HT Rxa D E[xŁk akT ] D HŁ Ra
(8.581)
8.B. DFE design: matrix formulations
747
Using (8.581) and (8.569) in (8.555), from (3.176) we obtain Q J D bQ H [Ra Rax R1 x Rxa ]b
(8.582)
T 1 Ł 1 Q D bQ H [R1 a C H Rw Q H ] b
Setting 2 T 1 Ł 1 6 R D D [0.M2 C1/ð.N1 CD/ ; I M2 C1 ; 0.M2 C1/ðM3 ][R1 4 a C H Rw Q H ]
0.M2 C1/ð.N1 CD/ I M2 C1
3 7 5
0.M2 C1/ðM3 (8.583) (8.582) becomes J D [1; b H ]R D
1 b
½ (8.584)
This quadratic form is minimized by choosing as FB filter coefficients (see (8.562)) [1; bopt;1 ; : : : ; bopt;M2 ]T D
R1 D e1 e1T R1 D e1
(8.585)
With this choice for the FB filter, the minimum value of the MSE is given by Jmin D
1 e1T R1 D e1
(8.586)
and the optimum FF filter can be obtained by (8.563). Observation 8.17 We note that if the number of coefficients M2 of the FB filter is chosen equal to the length of the overall channel impulse response at the decision point, that is if M2 D N1 C N2 C 1, the performance is improved; moreover, it is possible to reduce the computational complexity by exploiting the special structure of the matrices [60].
8.B.4
FS-DFE
Although the FB filter necessarily operates with input samples having sampling period equal to T , the FF filter can be fractionally spaced. According to the Wiener theory, the optimum coefficients of c and b are obtained by one of the two methods presented in the previous sections; however, care must be taken in accounting for the fact that the FF filter operates with input samples having sampling period equal to T =F0 , where F0 is the oversampling factor.
748
Chapter 8. Channel equalization and symbol detection
Therefore we introduce the vector notation hi D [h i F0 C.F0 1/ ; : : : ; h i F0 C1 ; h i F0 ]T
i D N1 ; : : : ; N2
Q k D [wQ k F0 C.F0 1/ ; : : : ; wQ k F0 C1 ; wQ k F0 ]T w xk D [x k F0 C.F0 1/ ; : : : ; x k F0 ]T D
N2 X
(8.587) (8.588)
Qk hi aki C w
(8.589)
i DN1
Now the FF filter c has a number of coefficients equal to M1 F0 , cT D [c1T ; : : : ; cTM1 ]
(8.590)
ciT D [c0;i ; c1;i ; : : : ; c.F0 1/;i ]
(8.591)
where
The input signal to the filter c is still denoted by the vector xk , whose components are the vectors fxi g, i D k; : : : ; k M1 1, each with dimension F0 . Then we can write 3 2 xk 7 6 :: Qk xk D 4 (8.592) 5 D Hak C w : xkM1 1 Q k follows the same structure of xk , and where w 2 0 0 ::: hN1 hN1 C1 hN1 C2 : : : h N2 6 hN1 hN1 C1 : : : h N2 1 h N2 0 ::: 6 0 6 6 0 h N2 2 h N2 1 h N2 0 hN1 HD6 6 : :: : 6 : :: : 4 : 0
0
0
: : : hN1 hN1 C1
0 0 0 :: : h N2
3 7 7 7 7 7 7 7 5
(8.593)
Note that the above relation is formally identical to (8.567). Therefore, thanks to this property, we can obtain the optimum coefficients of the FF and FB filters by the methods introduced in the previous sections.
8.C. Equalization based on the peak value of ISI
Appendix 8.C
749
Equalization based on the peak value of ISI
The equalization algorithm discussed in this appendix is related to the eye diagram at the decision point [4]. For an equalizer with N coefficients, let n be the overall impulse response at the decision point: n
D
N 1 X
n D N1 ; : : : ; N2 C N 1
c j h n j
(8.594)
jD0
We form the lead version of
n
n D
(8.595)
nCD
such that 0 is the desired sample at the decision point. Typically, DD
N 1 2
(8.596)
To simplify the notation, we assume that 0 coincides with the peak 0 D max jn j
(8.597)
L 1 D N1 C D L 2 D N2 C N 1 D L D L2 C L1 C 1
(8.598) (8.599)
n
Set
the cost function J to be minimized considers the peak value of ISI, that is JD
1 0
L2 X
jn j
(8.600)
nD0 n 6D 0
(8.601)
nDL 1 ; n6D0
We note that J D 0 for ( n D
0 6D 0 0
and the system is equalized, hence c is a zero forcing filter. On the other hand, if J > 1, then the eye is completely shut. In any event, the effect of the noise is not considered. As illustrated in Figure 8.58, the equalizer coefficients are determined iteratively in relation to a repetitive input sequence, with repetition index i, composed of a symbol equal to one followed by L 1 zeros: fak ; k D 0; : : : ; L 1g D f1; 0; : : : ; 0g.
750
Chapter 8. Channel equalization and symbol detection
ak
i=0
0 1
i=1
(L−1)
i=2
0
0
k
{h n(i)}
0
1
0
1
0
1
n
{ηn(i)}
0
1
0
0
1
1
n
Figure 8.58. Illustration of the zero forcing iterative equalization method.
If we denote by fn.i / g the impulse response at the i-th iteration, the law of coefficient update by the gradient method is given by / c.ij C1/ D c.ij / ¼ sgn .ijD
c.iDC1/ D 1
j D 0; : : : ; N 1 N 1 X
j 6D D
c.ij C1/ h D j
(8.602) (8.603)
jD0; j6D D
The last equation is obtained from the constraint 0 D 1. It can be proved that if the eye is open at the equalizer input, that is 1 h0
N2 X
jh n j < 1
(8.604)
nDN1 ; n6D0
then (8.602) converges and (8.601) is verified. It is useful to observe that in the absence of noise it would be sufficient to send only one sequence fak g, k D 0; : : : ; L 1: in practice, the various estimates of fh n g are used for averaging with respect to noise. We note the simplicity of implementing this algorithm: indeed, it was one of the first algorithms to equalize transmission lines. Now its use is justified only in systems with very large modulation rates, for example, optical fiber systems. In any case, for the algorithm to converge, the eye needs to be already opened, a constraint that did not exist in the MSE criterion. Another distinct characteristic of this method derives from the fact that it uses test sequences fak g, different from those typically used for transmission.
8.D. Description of a finite state machine (FSM)
Appendix 8.D
751
Description of a finite state machine (FSM)
We consider a discrete-time system with input sequence fi k g and output sequence fok g, evaluated at instants kT: We say that the output sequence is generated by an FSM if there are a sequence of states fsk g and two functions f o and f s , such that ok D f o .i k ; sk1 /
(8.605)
sk D f s .i k ; sk1 /
(8.606)
as illustrated in Figure 8.59. The first equation describes the fact that the output sample depends on the current input and the state of the system. The second equation represents the memory part of the FSM and describes the state evolution. We note that if f s is a one-to-one function of i k , that is if every transition between two states is determined by a unique input value, then (8.605) can be written as ok D f .sk ; sk1 /
Figure 8.59. General block diagram of a finite state sequential machine.
(8.607)
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 9
Orthogonal frequency division multiplexing
For channels that exhibit high signal attenuation at frequencies within the passband, a valid alternative to CAP/QAM is represented by a modulation technique based on filter banks, known as orthogonal frequency division multiplexing (OFDM), or multicarrier modulation. As the term implies, multicarrier modulation is obtained in principle by modulating several carriers in parallel using blocks of symbols, therefore using a symbol period that is typically much longer than the symbol period of a single-carrier system transmitting at the same bit rate. The resulting narrowband signals around the frequencies of the carriers are then added and transmitted over the channel. The narrowband signals are usually referred to as subchannel signals. An advantage of OFDM with respect to single-carrier systems is represented by the lower complexity required for equalization, that under certain conditions can be performed by a filter with a single coefficient per subchannel. A long symbol period also yields a greater immunity of an OFDM system to impulse noise; however the symbol duration, and hence the number of subchannels, are limited for transmission over time-variant channels. As we will see in this chapter, another important aspect is represented by the efficient implementation of modulator and demodulator, obtained by sophisticated signal processing algorithms.1
9.1
OFDM systems
Orthogonal frequency division multiplexing is an efficient modulation technique by which blocks of M symbols are transmitted in parallel over M subchannels, using M modulation filters [1] with frequency responses Hi . f /, i D 0; : : : ; M 1. We consider the baseband equivalent OFDM system illustrated in Figure 9.1. The M input symbols at the k-th modulation interval are represented by the vector ak D [ak [0]; : : : ; ak [M 1]]T
1
(9.1)
In this text, we refer to OFDM as a general technique for multicarrier transmission. In particular, this applies to both wired and wireless systems. When multicarrier transmission is achieved without filtering of the individual subchannel signals, other authors use the term discrete multitone (DMT) modulation for wired transmission systems, whereas they reserve the term OFDM for wireless systems.
754
Chapter 9. Orthogonal frequency division multiplexing
Figure 9.1. Block diagram of an OFDM system.
where ak [i] 2 A[i], i D 0; : : : ; M 1. The alphabets A[i], i D 0; : : : ; M 1, correspond to two dimensional constellations, which are not necessarily identical. The symbol rate of each subchannel is equal to 1=T and corresponds to the rate of the input symbol vectors fak g: 1=T is also called modulation rate. The sampling period of each input sequence is changed from T to T =M by upsampling, that is by inserting M 1 zeros between consecutive symbols (see Appendix 1.A); each sequence thus obtained is then filtered by a bandlimited filter properly allocated in frequency. The filter on the i-th branch is characterized by the impulse response fh n [i]g, with transfer function Hi .z/ D
C1 X
h n [i] z n
(9.2)
nD1
and frequency response Hi . f / D Hi .z/j zDe j2³ f T =M , i D 0; : : : ; M 1. The transmit filters, also called interpolator filters, work in parallel at the rate M=T , called transmission rate. The transmitter output signal sn is given by the sum of the filter output signals, that is sn D
M 1 X C1 X
ak [i] h nk M [i]
(9.3)
i D0 kD1
The received signal rNn , suitably delayed of D0 samples, is filtered by M filters in parallel at the transmission rate. The receive filters havePimpulse responses fgn [i]g, i D 0; : : : ; M 1, and transfer functions given by G i .z/ D n gn [i]z n , i D 0; : : : ; M 1. By downsampling the M output signals of the receive filters at the modulation rate 1=T , we obtain the vector sequence fyk D [yk [0]; : : : ; yk [M 1]]T g, which is used to detect the vector sequence fak g; receive filters are also known as decimator filters. When the channel is ideal and noiseless, we get rNn D sn . The delay D0 is chosen to obtain proper synchronization of the receive filter bank. We consider as transmit filters FIR causal filters having length M, and support f0; 1; : : : ; M 1g, that is h n [i] may be non-zero for n D 0; : : : ; M 1. We assume matched receive filters; as we are dealing with causal filters, we consider the expression of
9.2. Orthogonality conditions
755
the matched filters fgn [i]g, for i D 0; : : : ; M 1, given by gn [i] D h Ł Mn [i]
(9.4)
8n
We observe that the support of fgn [i]g is f1; 2; : : : ; Mg, and, for an ideal channel, D0 D 0.
9.2
Orthogonality conditions
Assuming the channel is ideal and noiseless, if the transmit and receive filter banks are designed such that certain orthogonality conditions are satisfied, the subchannel output signals are delayed versions of the transmitted symbol sequences at the corresponding subchannel inputs. The orthogonality conditions, also called perfect reconstruction conditions, can be interpreted as a more general form of the Nyquist criterion.
Time domain With reference to the general scheme of Figure 9.1, at the output of the j-th receive subchannel, before downsampling, the impulse response relative to the i-th input is given by M X1 pD0
h p [i] gn p [ j] D
M X1
h p [i] h Ł MC pn [ j]
8n
(9.5)
pD0
We note that, for j D i, the peak of the sequence given by (9.5) is obtained for n D M. Observing (9.5), transmission in the absence of intersymbol interference over a subchannel, as well as absence of interchannel interference (ICI) between subchannels, is achieved if orthogonality conditions are satisfied, that in the time domain are expressed as M X1
h p [i] h ŁpCM. k/ [ j] D Ži j Žk
i; j D 0; : : : ; M 1
(9.6)
pD0
Hence, in the ideal channel case considered here, the vector sequence at the output of the decimator filter bank is a replica of the transmitted vector sequence with a delay of modulation intervals, that is fyk g D fak g. Sometimes the elements of a set of orthogonal impulse responses that satisfy (9.6) are called wavelets.
Frequency domain In the frequency domain, the conditions (9.6) are expressed as M 1 X `D0
Hi
M M Ł f ` Hj f ` D Ži j T T
i; j D 0; : : : ; M 1
(9.7)
z-transform domain In the ideal channel case, the relation between the inputs ak [i], i D 0; : : : ; M 1, and the j-th output yk [ j] is represented by the block diagram of Figure 9.2. We note that every
756
Chapter 9. Orthogonal frequency division multiplexing
a k [0]
M
T a k [1]
T
a k [M-1]
T
M
. .. M
T M T M
T M
H0 (z)
Gj (z)
T M
H1 (z)
Gj (z)
T M
HM
-1
(z)
Gj (z)
M
M
M
T M
T
T
. .. y k [j]
T
Figure 9.2. Block diagram that illustrates the relation between the M inputs and the j-th output.
branch in the diagram represents a time-invariant linear system. Setting a.z/ D
" X
ak [0]z
k
;:::;
k
X
#T ak [M 1]z
k
yk [M 1]z
k
(9.8)
k
and
y.z/ D
" X
yk [0]z
k
k
;:::;
X
#T (9.9)
k
in general the input–output relation is given by (see Figure 9.6) y.z/ D S.z/ a.z/
(9.10)
where the element [S.z/] p;q of the matrix S.z/ is the transfer function between the q-th input and the p-th output. We note that the orthogonality conditions, expressed in the time domain by (9.6), are satisfied if S.z/ D z I
(9.11)
where I is the M ð M identity matrix.
9.3
Efficient implementation of OFDM systems
For large values of M, a direct implementation of an OFDM system as shown in Figure 9.1 would require an exceedingly large computational complexity, as all filtering operations are performed at a high rate equal to M=T . A reduction in the number of operations per unit of time is obtained by resorting to the polyphase representation of the various filters [2].
9.3. Efficient implementation of OFDM systems
757
OFDM implementation employing matched filters Using the polyphase representation of both transmit and receive filter banks introduced in Appendix 1.A, we have (see (1.635)) Hi .z/ D
M 1 X `D0
z ` E i.`/ .z M /
i D 0; : : : ; M 1
(9.12)
where E i.`/ .z/ is the transfer function of `-th polyphase component of Hi .z/. We define the vector of the transfer functions of transmit filters as h.z/ D [H0 .z/; : : : ; HM1 .z/] T
(9.13)
Let e.z/ be the vector of delay elements given by e.z/ D [z .M1/ ; z .M2/ ; : : : ; 1]T and E.z/ the matrix of the transfer functions of the polyphase components filters, given by 2 .0/ .0/ .0/ E 0 .z/ E 1 .z/ ÐÐÐ E M1 .z/ 6 .1/ 6 E 1.1/ .z/ ÐÐÐ EM E 0.1/ .z/ 6 1 .z/ E.z/ D 6 6 :: :: :: 6 : : : 4 .M1/ E 0.M1/ .z/ E 1.M1/ .z/ ÐÐÐ EM 1 .z/
(9.14) of the transmit 3 7 7 7 7 7 7 5
(9.15)
From the identity z .M1/ e H
1 zŁ
D [1; z 1 ; : : : ; z .M1/ ]
(9.16)
h.z/ can be expressed as hT .z/ D z .M1/ e H
1 zŁ
E.z M /
(9.17)
Observing (9.17), the M interpolated input sequences are filtered by filters whose transfer functions are represented in terms of the polyphase components, expressed as columns of the matrix E.z M /. Because the transmitted signal fsn g is given by the sum of the filter outputs, the operations of the transmit filter bank are illustrated by the scheme of Figure 9.3, where the signal fsn g is obtained from a delay line that collects the outputs of the M vectors of filters with transfer functions given by the rows of E.z M /, and input given by the vector of interpolated input symbols. At the receiver, using a representation equivalent to (1.637), that is obtained by a permutation of the polyphase components, we get G i .z/ D
M 1 X `D0
z .M1`/ Ri.`/ .z M /
i D 0; : : : ; M 1
(9.18)
758
Chapter 9. Orthogonal frequency division multiplexing
Figure 9.3. Implementation of an OFDM system using the polyphase representation.
where Ri.`/ .z/ is the transfer function of the .M 1 `/-th polyphase component of G i .z/. Then the vector of the transfer functions of the receive filters g.z/ D [G 0 .z/; : : : ; G M1 .z/] T can be expressed as g.z/ D R.z M / e.z/
(9.19)
where R.z/ is the matrix of the transfer functions of the polyphase components of the receive filters, and is given by 2 3 R 0.1/ .z/ ÐÐÐ R 0.M1/ .z/ R0.0/ .z/ 6 7 6 R1.0/ .z/ R 1.1/ .z/ ÐÐÐ R 1.M1/ .z/ 7 6 7 R.z/ D 6 (9.20) 7 :: :: :: 6 7 : : : 4 5 .0/ RM 1 .z/
.1/ RM 1 .z/
ÐÐÐ
.M1/ RM 1 .z/
Observing (9.19), the M signals at the output of the receive filter bank before downsampling are obtained by filtering in parallel the received signal by filters whose transfer functions are represented in terms of the polyphase components, expressed as rows of the matrix R.z M /. Therefore the M output signals are equivalently obtained by filtering the vector of received signal samples [Nrn.M1/ ; rNnMC2 , : : : ; rNn ]T by the M vectors of filters with transfer functions given by the rows of the matrix R.z M /. In particular, recalling that the receive filters have impulse responses given by (9.4), we obtain 1 0i M1 (9.21) G i .z/ D z M HiŁ Ł z Substituting (9.21) in (9.19), and using (9.17), the expression of the vector g.z/ of the transfer functions of the receive filters becomes 1 1 MCM1 H g.z/ D z M h H D z e.z/ (9.22) E zŁ z ŁM
9.3. Efficient implementation of OFDM systems
759
a k [0]
yk [0]
T
T
yk [1]
a k [1] T
T
E(z)
P/S
. .
rn = s n T M
z
-D
rn 0
S/P
z
−(γ−1)
H
E 1 z*
( )
. . .
.
a [M -1]
yk M [ -1]
k
T
T
Figure 9.4. Equivalent OFDM system implementation.
a)
b)
Figure 9.5. Block diagrams of (a) parallel to serial converter, (b) serial to parallel converter.
Apart from a delay term z 1 , the operations of the receive filter bank are illustrated by the scheme of Figure 9.3. From Figure 9.3 using (9.16), (9.22), and applying the noble identities given in Figure 1.70, we obtain the system represented in Figure 9.4, where parallel to serial (P/S) and serial to parallel (S/P) converters are illustrated in Figures 9.5a and 9.5b, respectively.
Orthogonality conditions in terms of the polyphase components We observe that the input–output relation of the cascade of the P/S converter, the delay z D0 D z 1 , and the S/P converter is expressed by the matrix z 1 I, that means that any input is obtained at the output on the same branch with a delay equal to T : therefore the system of Figure 9.4 is equivalent to that of Figure 9.6, with matrix transfer functiongiven by 1 1 . 1/ H 1 H S.z/ D z z I E.z/ D z E E.z/ (9.23) E Ł z zŁ
760
Chapter 9. Orthogonal frequency division multiplexing
y k[0]
a k[0] T
T y k[1]
a k[1] T
T S(z)
a k[
–1]
y k[
T
–1]
T
Figure 9.6. Equivalent OFDM system with input–output relation expressed in terms of the matrix S.z/.
From (9.11), in terms of the polyphase components of the transmit filters in (9.15), the orthogonality conditions in the z-transform domain are therefore expressed as EH
1 zŁ
E.z/ D I
(9.24)
Equivalently, using the polyphase representation of the receive filters in (9.20), we find the conditions 1 H R.z/ R DI (9.25) zŁ
OFDM implementation employing a prototype filter The complexity of an OFDM system can be further reduced by resorting to uniform filter banks [1, 2]. In this case the frequency responses of the various filters are obtained by shifting the frequency response of a prototype filter around the carrier frequencies, given by fi D
i T
i D 0; : : : ; M 1
(9.26)
In other words, the spacing in frequency between the subcarriers is 1 f D 1=T . To derive an efficient implementation of uniform filter banks we consider the scheme represented in Figure 9.7a, where H .z/ and G.z/ are the transfer functions of the transmit and receive prototype filters, respectively. We consider as transmit prototype filter a causal FIR filter with impulse response fh n g, having length M and support f0; 1; : : : ; M 1g; the receive prototype filter is a causal FIR filter matched to the transmit prototype filter, with impulse response given by gn D h Ł Mn
8n
(9.27)
9.3. Efficient implementation of OFDM systems
761
Figure 9.7. Block diagram of an OFDM system with uniform filter banks: (a) general scheme, (b) equivalent scheme for fi D i=T, i D 0; : : : ; M 1.
With reference to Figure 9.7a, the i-th subchannel signal at the channel input is given by 1 X
in
sn [i] D e j2³ M
ak [i] h nk M
(9.28)
kD1
As in
e j2³ M D e j2³
i .nk M/
M
(9.29)
we obtain sn [i] D
1 X
ak [i] h nk M e j2³
i .nk M/
M
D
kD1
1 X
ak [i] h nk M [i]
(9.30)
kD1
where in
h n [i] D h n e j2³ M
(9.31)
2³
Recalling the definition WM D e j M , the z-transform of fh n [i]g is expressed as i Hi .z/ D H .zW M /
i D 0; : : : ; M 1
Observing (9.30) and (9.32), we obtain the equivalent scheme of Figure 9.7b.
(9.32)
762
Chapter 9. Orthogonal frequency division multiplexing
The scheme of Figure 9.7b may be considered as a particular case of the general OFDM scheme represented in Figure 9.1; in particular, the transfer functions of the filters can be expressed using the polyphase representations, which are, however, simplified with respect to the general case expressed by (9.15) and (9.20), as we show in the following. Observing (9.28) we express the overall signal sn as M 1 X
sn D
C1 X
in
e j2³ M
h nk M ak [i]
(9.33)
kD1
i D0
With the change of variables n D mM C `, for m D 1; : : : ; C1, and ` D 0; 1; : : : ; M 1, we get sm MC` D
M 1 X
C1 X
i
e j2³ M .m MC`/
h .mk/MC` ak [i]
(9.34)
kD1
i D0
Observing that e j2³i m D 1, setting sm.`/ D sm MC` and h .`/ m D h m MC` , and interchanging the order of equations, we express the `-th polyphase component of fsn g as sm.`/ D
C1 X
h .`/ mk
kD1
M 1 X i D0
i ` WM ak [i]
(9.35)
The sequences fh .`/ m g, ` D 0; : : : ; M1, denote the polyphase components of the prototype filter impulse response, with transfer functions given by H .`/ .z/ D
X 1
m h .`/ m z
` D 0; : : : ; M 1
(9.36)
mD0
Recalling the definition (1.94) of the DFT operator as M ð M matrix, the IDFT of the vector ak is expressed as T F1 M ak D Ak D [Ak [0]; : : : ; A k [M 1]]
(9.37)
We find that the inner summation in (9.35) yields 1 X 1 M i ` WM ak [i] D Ak [`] M i D0
` D 0; 1; : : : ; M 1
(9.38)
and sm.`/ D
1 X kD1
M h .`/ mk Ak [`] D
1 X
M h .`/ p Am p [`]
(9.39)
pD1
Including the factor M in the definition of the prototype filter impulse response, or in M gain factors that establish the statistical power levels to be assigned to each subchannel signal, an efficient implementation of a uniform transmit filter bank is given by an IDFT, a polyphase network with M branches, and a P/S converter, as illustrated in Figure 9.8.
9.3. Efficient implementation of OFDM systems
Ak [0]
a k [0] T
Ak [1]
a k [1] T
763
(0)
(0)
H (z)
sk
T (1)
(1)
H (z)
sk
T
IDFT
P/S
Ak [M-1]
a k [M -1] T
(0)
rk T
(1)
rk rn
z -D 0
rn
T
rn
sn T M
GC(z) w
n
(M-1)
H
(M -1)
(z)
sk
T
z (γ−1) H (0)* 1 z*
y [0]
z (γ−1) H (1)* 1 z*
y k [1]
S/P
T
(M-1)
T
T
^a [1] k
T
DFT
rk
^a [0] k
k
(M-1)* 1 z (γ−1) H z*
T . .
. .
.
.
yk [ M -1] T
^a [M -1] k T
Figure 9.8. Block diagram of an OFDM system with efficient implementation.
Observing (9.35), we find that the matrix of the transfer functions of the polyphase components of the transmit filters is given by E.z/ D diagfH .0/ .z/; : : : ; H .M1/ .z/gF 1 M
(9.40)
Therefore the vector of the transfer functions of the transmit filters is expressed as 1 T .M1/ H h .z/ D z diagfH .0/ .z M /; : : : ; H .M1/ .z M /gF1 e (9.41) M zŁ We note that we would arrive at the same result by applying the notion of prototype filter with the condition (9.26) to (9.15). The vector of the transfer functions of a receive filter bank, which employs a prototype filter with impulse response given by (9.27), is immediately obtained by applying (9.22), with the matrix of the transfer functions of the polyphase components given by (9.40). Therefore we get ¦ ² 1 1 Ł .M1/ Ł ; : : : ; H e.z/ (9.42) g.z/ D z MCM1 FM diag H .0/ zŁM zŁM
764
Chapter 9. Orthogonal frequency division multiplexing
Hence an efficient implementation of a uniform receive filter bank, also illustrated in Figure 9.8, is given by a S/P converter, a polyphase network with M branches, and a DFT; we note that the filter of the i-th branch at the receiver is matched to the filter of the corresponding branch at the transmitter. With reference to Figure 9.7a, it is interesting to derive (9.42) by observing the relation between the received sequence rn D rNnD0 and the output of the i-th subchannel yk [i], given by C1 X
yk [i] D
i
T
gk Mn e j2³ T n M rn
(9.43)
nD1
With the change of variables n D mM C `, for m D 1; : : : ; C1, and ` D 0; : : : ; M 1, and recalling the expression of the prototype filter impulse response given by (9.27), we obtain yk [i] D
M 1 X C1 X
i
`D0 mD1
h Ł. kCm/MC` e j2³ M .m MC`/ rm MC`
i
(9.44)
Observing that e j2³ M m M D 1, setting rm.`/ D rm MC` , and h .`/ D h Łm MC` , and interm changing the order of equations, we get Ł
yk [i] D
M 1 X
2³
e j M i `
`D0
1 X
.`/ h .`/ Cmk r m Ł
mD1
(9.45)
2³
i ` , we finally find the expression Using the relation e j M i ` D WM
yk [i] D
M 1 X `D0
i` WM
1 X mD1
.`/ h .`/ Cmk r m Ł
(9.46)
Provided the orthogonality conditions are satisfied, from the output samples yk [i], i D 0; : : : ; M 1, threshold detectors may be employed to yield the detected symbols aO k [i], i D 0; : : : ; M 1, with a certain delay D. As illustrated in Figure 9.8, all filtering operations are carried out at the low rate 1=T . Also note that in practice the FFT and the inverse FFT are used in place of the DFT and the IDFT, respectively, thus further reducing the computational complexity.
9.4
Non-critically sampled filter banks
For the above discussion on the efficient realization of OFDM systems, we referred to Figure 9.8, where the number of subchannels, M, coincides with the interpolation factor of the modulation filters in the transmit filter bank. These systems are called filter bank systems with critical sampling or, in short, critically sampled filter banks. We now examine the general case of M modulators where each use an interpolation filter by a factor K > M: this system is called non-critically sampled. In principle the schemes
9.4. Non-critically sampled filter banks
765
Figure 9.9. Block diagram of (a) transmitter and (b) receiver in a transmission system employing non-critically sampled filter banks, with K > M and fi D .iK/=.MT/ D i=.MTc /.
of transmit and receive non-critically sampled filter banks are illustrated in Figure 9.9. As in critically sampled systems, also in non-critically sampled systems it is advantageous to choose each transmit filter as the frequency-shifted version of a prototype filter with impulse response fh n g, defined over a discrete-time domain with sampling period Tc D T =K. At the receiver, each filter is the frequency-shifted version of a prototype filter with impulse response fgn g, also defined over a discrete-time domain with sampling period Tc D T =K. As depicted in Figure 9.10, each subchannel filter has a bandwidth equal to K=.MT /, larger than 1=T . Maintaining a spacing between subcarriers of 1 f D K=.MT /, it is easier to avoid spectral overlapping between subchannels and consequently to avoid ICI. It is also possible to choose fh n g, e.g., as the impulse response of a square root raised cosine filter, such that, at least for an ideal channel, the orthogonality conditions are satisfied and ISI is also avoided. We note that this advantage is obtained at the expense of a larger
766
Chapter 9. Orthogonal frequency division multiplexing
Figure 9.10. Filter frequency responses in a non-critically sampled system.
bandwidth required for the transmission channel, that changes from M=T for critically sampled systems to K=T for non-critically sampled systems. Therefore the system requires an excess bandwidth given by .K M/=M. Also for non-critically sampled filter banks it is possible to obtain an efficient implementation using the discrete Fourier transform [3, 4]. The transmitted signal is expressed as a function of the input symbol sequences as sn D
M 1 X
iK
T
e j2³ MT n K
1 X
ak [i] h nk K
(9.47)
kD1
i D0
or, equivalently, sn D
1 X
h nk K
kD1
1 M X i D0
in ak [i] WM
(9.48)
With the change of indices n D mM C `
m2Z
` D 0; 1; : : : ; M 1
(9.49)
(9.48) becomes sm MC` D
1 X
h m Mk KC`
M 1 X
kD1
i D0
i ` WM ak [i]
(9.50)
Using the definition of the IDFT (9.38), apart from a factor M that can be included in the impulse response of the filter, and introducing the following polyphase representation of the transmitted signal sm.`/ D sm MC`
(9.51)
we obtain sm.`/ D
1 X
h m Mk KC` Ak [`]
(9.52)
kD1
By analogy with (1.561), (9.52) is obtained by interpolation of the sequence fAk [`]g by a factor K, followed by decimation by a factor M. From (1.569) and (1.570), we introduce the change of indices ¹ ¼ mM k (9.53) pD K
9.4. Non-critically sampled filter banks
767
and 1m D
¼ ¹ mM .mM/mod K mM D K K K
(9.54)
Using (1.576) it results in sm.`/ D
C1 X
h . pC1m /KC` Aj mM k K
pD1
D
C1 X
p
[`]
h pKC`C.m M/mod K Aj mM k K
pD1
p
[`]
Letting h .`/ p;m D h p KC`C.m M/mod K
p; m 2 Z
` D 0; 1; : : : ; M 1
(9.55)
[`]
(9.56)
we obtain sm.`/ D
1 X pD0
j k h .`/ p;m A mM K
p
The efficient implementation of the transmit filter bank is illustrated in Figure 9.11. We note that the system is now periodically time-varying, i.e. the impulse response of the filter components cyclically changes. The M elements of an IDFT output vector are input to M delay lines. Also note that within a modulation interval of duration T , the samples stored in some of the delay lines are used to produce more than one sample of the transmitted signal. Therefore the P/S element used for the realization of critically sampled filter banks needs to be replaced by a commutator. At instant nT =K, the commutator is linked to the ` D n mod M -th filtering element. The transmit signal sn is then computed by
Figure 9.11. Efficient implementation of the transmitter of a system employing non-critically sampled filter banks; the filter components are periodically time-varying.
768
Chapter 9. Orthogonal frequency division multiplexing
convolving the signal samples stored in the `-th delay line with the n mod K -th polyphase component of the T =K-spaced-coefficients prototype filter. In other terms, each element of the IDFT output frame is filtered by a periodically time-varying filter with period equal to [l :c:m:.M; K/]T =K, where l :c:m:.M; K/ denotes the least common multiple of M and K. Likewise, the non-critically sampled filter bank at the receiver can also be efficiently implemented using the DFT. In particular, we consider the case of downsampling of the subchannel output signals by a factor K=2, which yields samples at each subchannel output at an (over)sampling rate equal to 2=T . With reference to Figure 9.9b, we observe that the output sequence of the i-th subchannel is given by 1 X
yn 0 [i] D
in
gn 0 K n e j2³ M rn
(9.57)
2
nD1
where gn D h Ł Mn . With the change of indices n D mM C `
m2Z
` D 0; 1; : : : ; M 1
and letting rm.`/ D rm MC` , from (9.57) we get yn 0 [i] D
M 1 X
1 X
`D0
mD1
gn 0 K m M` rm.`/ 2
!
i` WM
(9.58)
(9.59)
We note that in (9.59) the term within parenthesis may be viewed as an interpolation by a factor M followed by a decimation by a factor K=2. Letting ¼ 0 ¹ nK m (9.60) qD 2M and ¼ 0 ¹ nK .n 0 K=2/mod M n0K D 2M 2M M
1n 0 D
(9.61)
the terms within parenthesis in (9.59) can be written as 1 X qD1
gq MC.n 0 K /mod M ` rj.`/n0 K=2 k 2
M
q
(9.62)
Introducing the M periodically time-varying filters, .`/ gq;n 0 D gq MC.n 0 K / 2 mod M `
q; n 0 2 Z
` D 0; 1; : : : ; M 1
(9.63)
and defining the DFT input samples u n.`/ 0 D
1 X qD1
.`/ j.`/ k gq;n 0 r n0 K 2M
q
(9.64)
9.5. Examples of OFDM systems
769
Figure 9.12. Efficient implementation of the receiver of a system employing non-critically sampled filter banks; the filter components are periodically time-varying (see (9.63)).
(9.59) becomes yn 0 [i] D
M 1 X `D0
i` u n.`/ 0 WM
(9.65)
The efficient implementation of the receive filter bank is illustrated in Figure 9.12, where we assume for the received signal the same sampling rate of K=T as for the transmitted signal, and a downsampling factor K=2, so that the samples at each subchannel output are obtained at a sampling rate equal to 2=T . Note that the delay element z D0 at the receiver input has been omitted, as the optimum timing phase for each subchannel can be recovered by using per-subchannel fractionally spaced equalization, as discussed in Section 8.4 for single-carrier modulation. Also note that within a modulation interval of duration T , more than one sample is stored in some of the delay lines to produce the DFT input vectors. Therefore the S/P element used for the realization of critically sampled filter banks needs to be replaced by a commutator. After the M elements of a DFT input vector are produced, the commutator is circularly rotated K=2 steps clockwise from its current position, allowing a set of K=2 consecutive received samples to be input into the delay lines. The content of each delay line is then convolved with one of the M polyphase components of the T =K-spaced-coefficients receive prototype filter. A similar structure is obtained if in general a downsampling factor K0 K is considered.
9.5
Examples of OFDM systems
We consider three simple examples of critically sampled filter bank modulation systems. For practical applications, equalization techniques and possibly non-critically sampled filter bank realizations are required, as will be discussed in the following sections.
770
Chapter 9. Orthogonal frequency division multiplexing
Discrete multitone (DMT) The transmit and receive filter banks use a prototype filter with impulse response given by [5, 6, 7, 8] ( 1 if 0 n M 1 hn D (9.66) 0 otherwise The impulse responses of the polyphase components of the prototype filter are given by o n ` D 0; : : : ; M 1 (9.67) h n.`/ D fŽn g and we can easily verify that the orthogonality conditions (9.6) are satisfied. As shown in Figure 9.13, because the frequency responses of the polyphase components are constant, we obtain directly the transmit signal by applying a P/S conversion at the output of the IDFT. Assuming an ideal channel, at the receiver a S/P converter forms blocks of M samples, with boundaries between blocks placed so that each block at the output of the IDFT at the transmitter is presented unchanged at the input of the DFT. At the DFT output, the input blocks of M symbols are reproduced without distortion with a delay equal to T . We note, however, that the orthogonality conditions are satisfied only if the channel is ideal. From the frequency response of the prototype filter, M1 T T sin.³ f T / (9.68) H e j2³ f M D e j2³ f 2 M sin.³ f T =M/ using (9.32) the frequency responses of the individual subchannel filters Hi .z/ are obtained. Figure 9.14 shows the amplitude of the frequency responses of adjacent subchannel filters, M obtained for f 2 0; 0:06 T and M D 64. We note that the choice of a rectangular window of length M as impulse response of the baseband prototype filter leads to a significant overlapping of spectral components of transmitted signals in adjacent subchannels. a k [0]
rn(0)
Ak [0]
y [0] k
T
T
a k [1]
rn(1)
Ak [1]
y k [1]
T
. .
T
P/S
IDFT
sn T M
rn
GC(z) w
z
-D
rn 0
S/P
DFT
. .
n
.
.
a k [M -1] T
Ak [M-1]
rn( M -1)
yk [ M -1] T
Figure 9.13. Block diagram of an OFDM system with impulse response of the prototype filter given by a rectangular window of length M.
9.5. Examples of OFDM systems
771
Figure 9.14. Amplitude of the frequency responses of adjacent subchannel filters in a DMT Ð c system for f 2 0; 0:06 M T and M D 64. [From [4], 2002 IEEE]
Filtered multitone (FMT) The transmit and receive filter banks by [3, 4] 8 > < T H e j2³ f M D > :
use a prototype filter with frequency response given þ þ þ 1 C e j2³ f T þ þ þ þ 1 C ²e j2³ f T þ 0
1 2T otherwise
if j f j
(9.69)
where the parameter 0 ² 1 controls the spectral roll-off of the filter. The frequency response exhibits spectral nulls at the band edges and, when used as the prototype filter characteristic, leads to transmission free of ICI but with ISI within a subchannel. For ² ! 1, the frequency characteristic of each subchannel exhibits steep roll-off towards the band edge frequencies. On the other hand, for ² D 0 the partial-response class I characteristic is obtained. In general, it is required that at the output of each subchannel the ICI is negligible with respect to the noise. The amplitude of the frequency responses ofÐsubchannel filters obtained with a minimum phase prototype FIR filter for f 2 0; 0:06 M T , and design parameters M D 64, D 10, and ² D 0:1 are illustrated in Figure 9.15.
Discrete wavelet multitone (DWMT) As illustrated by Ð the subchannel frequency responses in Figure 9.16, obtained for and M D 64, in general, DWMT [9] has a higher spectral containment f 2 0; 0:06 M T
772
Chapter 9. Orthogonal frequency division multiplexing
Figure 9.15. Amplitude ofthe frequency responses of adjacent subchannel filters in an FMT M system for f 2 0; 0:06 T , and design parameters M D 64, D 10, and ² D 0:1. [From [4], c 2002 IEEE]
Figure 9.16. Amplitude of frequency responses of adjacent subchannel filters in a DWMT c 2002 IEEE] and M D 64. [From [4], system for f 2 0; 0:06 M T
9.6. Equalization of OFDM systems
773
of individual subchannel signals as compared to DMT. The orthogonality conditions are satisfied; each subchannel, however, requires a bandwidth larger than 1=T . In DWMT modulation, all signal processing operations involve real signals. Therefore, for the same number of dimensions per modulation interval of the transmitted signal, the minimum bandwidth of a subchannel for DWMT is half the minimum bandwidth of a subchannel for DMT or FMT modulation. The implementation of filter banks for DWMT is examined in Section 9.9.
9.6
Equalization of OFDM systems
Interpolator filter and virtual subchannels In order to simplify the analysis, in this section we consider the case of a modulated signal with a sampling frequency of M=T . The extension to the general case is obtained by considering a sampling frequency of K=T , with K ½ M. The signal fsn g with sampling rate M=T that we obtain at the output of the transmit filter bank must be converted into an analog signal before being sent over the transmission channel with impulse response gCh .t/, t 2 1. In this case the orthogonality conditions for the DMT system described in Section 9.2 are no longer satisfied: indeed, the transfer matrix S.z/, defined by (9.10) and evaluated for a non-ideal channel, has in general elements different from a delay factor along the main diagonal, meaning the presence of ISI for transmission over the individual subchannels, and non-zero elements off the main diagonal, meaning the presence of ICI. A simple equalization method is based on the concept of circular convolution introduced in Section 1.4, that allows a expression of a convolution in the time domain as a product of finite length vectors in the frequency domain (see (1.107)). Using the method indicated as Relation 2 on page 23, we extend the block of samples Ak by repeating Nc 1 elements: in this way we obtain the DMT system illustrated in Figure 9.19. For the same channel bandwidth and hence for a given transmission rate M=T , the M < T1 . After the IDFT (modulation) must be carried out at the rate T10 D .MCN c 1/T modulation, each block of samples is cyclically extended by copying the Nc 1 samples Ak [M Nc C 1]; : : : ; Ak [M 1] in front of the block, as shown in Figure 9.19. After
Figure 9.19. Block diagram of a DMT system with cyclic prefix and frequency-domain equalizer.
776
Chapter 9. Orthogonal frequency division multiplexing
the P/S conversion, where the Nc 1 samples of the cyclic extension are the first to be sent, the Nc 1 C M samples are transmitted over the channel. At the receiver, blocks of samples of length Nc 1 C M are taken; the boundaries between blocks are set so that the last M samples depend only on the elements of only one cyclically extended block of samples. The first Nc 1 samples of a block are discarded. We now recall the result (1.116). The vector rk of the last M samples of the block received at the k-th modulation interval is expressed as rk D k gC C wk
(9.72)
where gC D [gC;0 ; : : : ; gC;Nc 1 ; 0; : : : ; 0]T is the M-component vector of the channel impulse response extended with M Nc zeros, wk is a vector of additive white Gaussian noise samples, and k is an M ð M circulant matrix, given by 2 3 Ak [0] Ak [M 1] ÐÐÐ Ak [1] 6 Ak [1] Ak [0] ÐÐÐ Ak [2] 7 6 7 6 7 (9.73) k D 6 :: :: :: 7 4 5 : : : Ak [M 1]
Ak [M 2]
ÐÐÐ
Ak [0]
Equation (9.72) is obtained by observing that only the elements of the first Nc columns of the matrix k contribute to the convolution that determines the vector rk , as the last M Nc elements of gC are equal to zero. The elements of the last M Nc columns of the matrix k are chosen so that the matrix is circulant, even though they might have been chosen arbitrarily. Moreover, we observe that the matrix k , being circulant, satisfies the relation 2 3 ak [0] 0 ÐÐÐ 0 6 0 7 0 ak [1] Ð Ð Ð 6 7 6 7 D diagfak g FM k F1 D (9.74) :: :: :: 6 7 M 4 5 : : : 0
0
ÐÐÐ
ak [M 1]
Defining the DFT of the vector gC as GC D [GC;0 ; GC;1 ; : : : ; GC;M1 ]T D FM gC
(9.75)
and using (9.74), we find that the demodulator output is given by xk D FM rk D diagfak g GC C Wk
(9.76)
where Wk D FM wk is given by the DFT of the vector wk . Recalling the properties of wk , Wk is a vector of independent Gaussian r.v.s. Equalizing the channel using the zero-forcing criterion, the signal xk (9.76) is multiplied by the diagonal matrix K, whose elements on the diagonal are given by2 Ki D [K]i;i D
2
1 GC;i
i D 0; 1; : : : ; M 1
(9.77)
To be precise, the operation indicated by (9.77), rather than equalizing the signal, that is received in the absence of ISI, normalizes the amplitude and adjusts the phase of the desired signal.
9.6. Equalization of OFDM systems
777
Therefore the input to the data detector is given by yk D Kxk D ak C KWk
(9.78)
We assume that the sequence of input symbol vectors fak g is a sequence of i.i.d. random vectors. Equation (9.78) shows that the sequence fak g can be detected by assuming transmission over M independent and orthogonal subchannels in the presence of additive white Gaussian noise. A drawback of this simple equalization scheme is the reduction in the modulation rate by a factor .M C Nc 1/=M. Therefore it is essential that the length of the channel impulse response is much smaller than the number of subchannels, so that the reduction of the modulation rate due to the cyclic extension can be considered negligible. To reduce the length of the channel impulse response one approach is to equalize the channel before demodulation [8, 10, 11]. With reference to Figure 9.18b, a linear equalizer with input rn is used; it is usually chosen as the FF filter of a DFE that is determined by imposing a prefixed length of the feedback filter, smaller than the length of the cyclic prefix.
Equalization of FMT systems We analyze three schemes. Per-subchannel fractionally spaced equalization. We consider an FMT system with noncritically sampled transmit and receive filter banks, so that transmission within individual subchannels with non-zero excess bandwidth is achieved, and subchannel output signals obtained at a sampling rate equal to 2=T , as discussed in Section 9.4. We recall that the frequency responses of FMT subchannels are characterized by steep roll-off towards the band-edge frequencies, where they exhibit near spectral nulls. This suggests that persubchannel decision-feedback equalization be performed to recover the transmitted symbols. The block diagram of an FMT receiver employing per-subchannel fractionally spaced equalization is illustrated in Figure 9.20. Over the i-th subchannel, the DFE is designed for an overall impulse response given by h overall ;n [i] D
C1 X n 1 D1
gn Mn 1 [i]
NX c 1
h n 1 n 2 [i] gC;n 2
n2Z
(9.79)
n 2 D0
In the given scheme, the M DFEs depend on the transmission channel. If the transmission channel is time variant, each DFE must be able to track the channel variations, or it must be recomputed periodically. Error propagation inherent to decision-feedback equalization can be avoided by resorting to precoding techniques, as discussed in Chapter 13. The application of precoding techniques in conjunction with trellis-coded modulation (TCM) for FMT transmission is addressed in [4]. Per-subchannel T -spaced equalization. We consider now an FMT system with critically sampled filter banks, and subchannel output signals obtained at the sampling rate of 1=T . The high level of spectral containment of the transmit filters suggests that, if the number
778
Chapter 9. Orthogonal frequency division multiplexing
Figure 9.20. Per-subchannel equalization for an FMT system with non-critically sampled filter banks.
of subchannels is sufficiently high, and the group delay in the passband of the transmission channel is approximately constant, the frequency response of every subchannel becomes approximately a constant. In this case, the effect of the transmission channel is that of multiplying every subchannel signal by a complex value. Therefore, as for DMT systems with cyclic prefix, equalization of the transmission channel can be performed by choosing a suitable constant for every subchannel. We note, however, that, whereas for a DMT system with cyclic prefix the model of the transmission channel as a multiplicative constant for each subchannel is exact if the length of the cyclic prefix is larger than the length of the channel impulse response, for an FMT system such a model is valid only as an approximation. The degree of the approximation depends on the dispersion of the transmission channel and on the number M of subchannels. Assuming a constant frequency response for transmission over each subchannel, the equalization scheme is given in Figure 9.21a, where K i is defined in (9.77), and the DFE is designed to equalize only the cascade of the transmit and receive filters. Using (9.27) and (9.31) we find that the convolution of transmit and receive filters is independent of the subchannel index: in fact, we obtain 1 X
h n 1 [i] gn Mn 1 [i] D
n 1 D0
1 X
h n 1 gn Mn 1 D h eq;n
(9.80)
n 1 D0
In this case, all DFEs are equal. Simplified per-subchannel T -spaced equalization. A further simplification is obtained by using the implementation of Figure 9.21b. The idea is that, in the presence of a transmission channel with flat frequency response for each subchannel, a reconstruction of the signal is achieved by designing the `-th polyphase component of the receive prototype filter, g .`/ , to equalize the corresponding polyphase component h .`/ of the transmit prototype filter. In
9.7. Synchronization of OFDM systems
779
Figure 9.21. (a) Equalization scheme for FMT in the case of approximately constant frequency response for transmission over each subchannel; (b) simplified scheme.
general, a DFE scheme can be used, where the `-th polyphase components of the receive .`/ .`/ of the filters, g .`/ F F and g F B , equalize the corresponding i-th polyphase component h overall subchannel impulse response.
9.7
Synchronization of OFDM systems
Various algorithms may be applied to achieve synchronization of OFDM systems for transmission over dispersive channels, depending on the system and on the type of equalization adopted. For DMT systems two synchronization processes are identified: synchronization of the clock of the A/D converter at the receiver front-end, or clock synchronization, and synchronization of the vector rk at the output of the S/P element, or frame synchronization.
780
Chapter 9. Orthogonal frequency division multiplexing
Clock synchronization guarantees alignment of the timing phase at the receiver with that at the transmitter; frame synchronization, on the other hand, extracts from the sequence of received samples the blocks of M C Nc 1 samples that form the received frames, and determines the boundaries of the sequence of vectors rk that are presented at the input of the DFT. In principle, for a channel input sequence given by s0 D 1, and sn D 0, n 6D 0, the channel impulse response of length Nc must appear in the first Nc positions of the receive vector (see Figure 9.19). For the initial convergence of both synchronization processes, training sequences without cyclic prefix are usually employed [12]. For FMT systems with non-critically sampled filter banks and fractionally-spaced equalization, the synchronization is limited to clock recovery (see Section 8.4).
9.8
Passband OFDM systems
For a passband OFDM transmission system, the signal fsn g is in general complex valued and is shifted to passband adopting for example the scheme illustrated in Figure 9.18.
Passband DWMT systems Suppose a passband OFDM transmission system adopts DWMT modulation. Then, as described in detail in Section 9.9, the filters have real-valued impulse responses, and the baseband transmitted signal is generated by real-valued input symbols. Therefore the signal spectrum has Hermitian symmetry around DC; hence it is sufficient to consider the spectrum in the band [0; M=.2T /], that corresponds to the analytic signal. In principle, the passband signal is obtained by the scheme illustrated in Figure 9.22a, where the discrete-time signal generated by the DWMT modulator is converted into a continuous time signal by a D/A converter. The continuous-time signal is then filtered by a phase splitter, gT.a/x , which yields the analytic signal. This signal is shifted to high frequency by a modulator with carrier frequency f 0 , and the real part of the resulting signal forms the input signal of the passband transmission channel. In practice, the scheme illustrated in Figure 9.22b is adopted, where gT.a/x;I D Re[gT.a/x ] and gT.a/x;Q D Im[gT.a/x ], that is equivalent to a single side band (SSB) modulator (see Example 1.7.4 on page 58). If the phase splitter filter exhibits a suitable
Figure 9.22. Block diagram of an SSB modulator for a passband DWMT signal.
9.8. Passband OFDM systems
781
roll-off around DC we obtain a vestigial side band (VSB) modulator.3 However, because of the difficult recovery of the phase and frequency of the carrier, digital transmission systems using SSB and VSB modulators are characterized by lower performance as compared to systems that consider transmission of the double-sided signal spectrum. To overcome this difficulty and preserve the spectral efficiency of the transmission scheme, a pilot tone may be used to provide the required information for the carrier recovery. The transmission of pilot tones, however, does not represent in many cases a practical solution, as it reduces the power efficiency of the system and introduces one or more spectral lines in the signal spectrum.
Passband DMT and FMT systems For DMT and FMT systems it is not required that the baseband output signal at the output of the modulator be real-valued. Therefore we remove the constraint that complex-valued input symbols satisfy the Hermitian symmetry conditions (9.70), and we obtain a complexvalued baseband signal. Consequently, the passband signal is given in principle by the scheme illustrated in Figure 9.18a, where gT x is the real-valued impulse response of a lowpass filter with cut-off frequency equal to M=.2T /. This scheme is equivalent to a modulator for complex-valued signals, sometimes called double sideband modulator with amplitude and phase modulation (DSB-AM/PM); in this case carrier recovery does not represent a difficult problem. Multiple access DMT and FMT systems. Other difficulties arise, however, in the case of transmission in multiple-access networks. Then two or more users transmit signals simultaneously over subsets of the available subchannels. We recall that in DMT systems the channel impulse response needs to be shortened to reduce the length of the cyclic extension. Consequently in a multiple-access system the impulse response of each user’s channel must be shortened. We observe that, even if a cyclic extension of sufficient length is used, the orthogonality conditions are satisfied only if the subchannel signals are synchronous. Because of the spectral overlapping between signals on adjacent subchannels in a DMT system, a signal that is presented at the receiver input with an incorrect timing phase violates the orthogonality conditions, and disturbs many other subchannels: this situation cannot be avoided, for example, when a station sends a signal over a given subchannel without knowledge of the propagation delay. To solve the problems raised by the transmission of DMT signals in a multiple-access network, we resort to FMT systems, which present large attenuation of the signal spectrum outside the allocated subchannels. In this manner, ICI is avoided even if the various subchannel signals are received from stations without knowledge of the propagation delay.
Comparison between OFDM and QAM systems It can be shown that OFDM, or multicarrier, systems and QAM, or single carrier, systems achieve the same theoretical performance for transmission over ideal AWGN channels [13].
3
SSB and VSB modulations are used, for example, for the analog transmission of video signals and can also be considered for digital communication systems.
782
Chapter 9. Orthogonal frequency division multiplexing
In practice, however, OFDM systems offer some considerable advantages with respect to CAP/QAM systems. ž OFDM systems achieve higher spectral efficiency if the channel frequency response exhibits large attenuations at frequencies within the passband. In fact, the band used for transmission can be varied by increments equal to the modulation rate 1=T Hz, and optimized for each channel. Moreover, if the noise exhibits strong components in certain regions of the spectrum, the total band can be subdivided in two or more sub-bands. ž OFDM systems guarantee a higher robustness with respect to impulse noise. If the average arrival rate of the pulses is lower than the modulation rate, the margin against the impulse noise is of the order of 10 log10 .M/ dB. ž For typical values of M, OFDM systems achieve the same performance as QAM systems with a complexity that can be considerably lower. ž In multiple-access systems, the finer granularity of OFDM systems allows a greater flexibility in the spectrum allocation. On the other hand, OFDM systems present also a few drawbacks with respect to QAM systems. ž In OFDM systems the transmitted signals exhibit a higher peak-to-average power ratio, that contributes to an increase in the susceptibility of these systems to nonlinear distortion. ž Because of the block processing of samples, a higher latency is introduced by OFDM systems in the transmission of information.
9.9
DWMT modulation
In DMT systems the reduction of the modulation rate, equal to the factor .MC Nc 1/=M, and the consequent need to reduce this penalty by shortening the impulse response of the transmission channel are due to the non-negligible spectral overlap of signals on adjacent subchannels. As a cyclic prefix of length Nc 1 is used for equalization, the orthogonality between signals on different subchannels is verified only for channels with length of the impulse response smaller than or equal to Nc . Ideally, if the frequency response of the prototype filter is chosen such that signals of different subchannels do not overlap, the orthogonality is maintained independently of the impulse response of the transmission channel. Therefore it is interesting to consider a filter bank modulation scheme where the FIR filters are such that a large attenuation of the filter frequency responses outside the assigned subchannel bands is achieved and that the conditions (9.6) are satisfied. These objectives are achieved by OFDM systems that are usually known as discrete wavelet multitone modulation (DWMT). In a DWMT system, the elements of the input vector ak are realvalued symbols and the impulse response of the transmit and receive filter banks are also real-valued.
9.9. DWMT modulation
783
Transmit and receive filter banks To investigate the principles of a DWMT system, we initially consider a uniform filter bank with 2M filters. Let P0 .z/ be the transfer function of the prototype filter and Pi .z/ the transfer functions of the subchannel filters, Pi .z/ D P0 .zW 2i M /
i D 0; : : : ; 2M 1
(9.81)
also let P .i / .z/, i D 0; 1; : : : ; 2M1, be the transfer functions of the polyphase components of P0 .z/. The amplitude characteristics of the filters are illustrated in Figure 9.23. The basic idea of DWMT consists in combining pairs of these 2M filters so that they are used with M real input signals. We assume that the impulse response of the prototype filter f pn [0]g is real-valued. Thereþ 2³ f T Ðþ fore the function þ P0 e j 2M þ is symmetric with respect to f D 0. Ideally, the prototype filter is a lowpass filter with bandwidth equal to 1=.2T / Hz. From (9.31) we find that the impulse response of the i-th filter is given by pn [i] D pn [0] W2in M . We consider now a shifted version of 1=.2T / Hz of the original set of 2M frequency responses, obtained by 1=2 the change of variable z ! zW 2M . We define iC 1 Ð 0 i 2M 1 (9.82) Q i .z/ D P0 zW 2M2 The amplitude characteristics of the shifted frequency responses are illustrated in Figure 9.24. As the coefficients of P0 .z/ are real, the property Q 2M1i .z/ D Q iŁ .z Ł /
Figure 9.23. Amplitude of the frequency responses of the filters of an OFDM system with 2M subchannels.
Figure 9.24. Amplitude of the frequency responses of the filters shifted in frequency.
784
Chapter 9. Orthogonal frequency division multiplexing
holds, and consequently we get 2³ f T þ þ 2³ f T þ þ þ Q 2M1i e j 2M þ D þ Q i e j 2M þ
(9.83)
We set 1 iC Ui .z/ D þ i P0 z W 2M2 D þi Q i .z/
Vi .z/ D
þ iŁ
1 iC P0 z W 2M 2
Ð
D þiŁ Q 2M1i .z/
0i M1
(9.84)
0i M1
(9.85)
D þiŁ Q iŁ .z Ł /
(9.86)
and we define the transfer functions of a new filter bank with M transmit filters, real input symbol sequences, and modulation rate equal to 2=T , as Hi .z/ D Þ i Ui .z/ C Þ iŁ Vi .z/
0i M1
(9.87)
In the previous equations Þi and þi are constants with absolute value equal to one. The amplitude of the frequency response of the filter Hi .z/ is illustrated in Figure 9.25. We note that Hi .z/ has a frequency response with positive frequency content due to U i .z/, and negative frequency content due to Vi .z/. We assume that the original prototype filter P0 .z/ is an FIR filter with length M and transfer function given by P0 .z/ D
M X1
pn .0/ z n
(9.88)
nD0
The M filters defined by (9.87) are also FIR filters of length M and transfer functions defined as Hi .z/ D
M X1
h n [i] z n
0i M1
(9.89)
nD0
Because the coefficients of P0 .z/ are real-valued, the coefficients of U i .z/ are obtained as the complex conjugate of the coefficients of Vi .z/. Consequently in (9.87) the coefficients h n [i], i D 0; : : : ; M 1, are real-valued.
Figure 9.25. Amplitude of the frequency response of the filter Hi .z/.
9.9. DWMT modulation
785
We assume, moreover, that the prototype filter P0 .z/ is a linear-phase filter and that the relation p M1n [0] D pn [0] holds. Therefore we get 1 (9.90) P0Ł Ł D z . M1/ P0 .z/ z The frequency response of the filter can be expressed as 2³ f T 2³ f T M1 T P0 e j 2M D e j2³ f 2 2M PR e j 2M
(9.91)
2³ f T where PR e j 2M is a real-valued function. We choose the values of the constants þi so that Ui .z/ and Vi .z/ have the same linear phase as P0 .z/; observing that
1 fT iC 2 j2³ 2M D þi W2M Ui e
M1 2
e
j2³ f
M1 T 2 2M
PR e
f T i C1=2 j2³ 2M 2M
(9.92)
we let 1 M1 iC 2 2
þi D W2M
(9.93)
Therefore we get
2³ f T Ui e j 2M 2³ f T Vi e j 2M
De
D
j2³ f
M1 T 2 2M
M1 T e j2³ f 2 2M
f T i C1=2 j2³ 2M 2M
f T i C1=2 j2³ 2M C 2M
PR e
PR e
(9.94)
and the functions Ui .z/ and Vi .z/ indeed exhibit the same linear phase as P0 .z/. Because Ui .z/ and Vi .z/ have a linear phase, analogously to (9.90), the following relations hold: 1 Ł Ui D z . M1/ Ui .z/ zŁ (9.95) 1 Ł . M1/ Vi Vi .z/ Dz zŁ Moreover, we assume that the receive filters are matched, that is gn [i] D h Ł Mn [i] D h Mn [i], 0 i M 1. Hence the transfer functions of the receive filters are given by 1 0 j M1 (9.96) G j .z/ D z M H jŁ Ł D z M H j .z 1 / z From (9.87) we get G j .z/ D z 1 [ÞiŁ U j .z/ C Þ i V j .z/]
0 j M1
(9.97)
786
Chapter 9. Orthogonal frequency division multiplexing
Approximate interchannel interference suppression From (9.11), we recall that to obtain a system without ICI at the output of the j-th subchannel it is necessary that the polyphase components with index 0 of the filters G j .z/ Hi .z/, j 6D i, are zero. These components will be denoted by [G j .z/ Hi .z/] # M . In practice, as the prototype filter P0 .z/ is a lowpass filter, an approximate suppression of ICI can be obtained in many cases under the condition that only the components [G j .z/ H jš1 .z/] # M are cancelled, as shown in Figure 9.26 for M D 4, j D 1 and i D 2. Taking, for example, the case i D j C 1, from (1.580) we get [G j .z/H jC1 .z/] # M D
D
1 1 1 X 1 M ` ` H jC1 z M WM G j z M WM M `D0 1 1 1 X 1 1 1 M ` ` ` z M WM Þ Łj U j .z M WM / C Þ j V j .z M WM / M `D0
1 1 ` ` C Þ ŁjC1 V jC1 z M WM ð Þ jC1 U jC1 z M WM
H3 H2 H1 H0
1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
1 2
l =0
1111 0000 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111 0000 1111
1 2
l =0
0
f
T 2M
(a) l =3
1 4
0
l =1
l =1
l =0
l =2
2 4
l =3
l =3
l =2
3 4
l =0
1
fT 2
l =1
(b)
Figure 9.26. (a) Amplitude of the filter frequency responses for M D 4; (b) spectral components of ICI evaluated from the i-th input to the j-th output, for j D 1 and i D 2, after downsampling (see (9.98)).
9.9. DWMT modulation
'
787
1 1 1 X 1 1 1 M ` ` ` z M WM U jC1 z M WM Þ Łj Þ jC1 U j z M WM M `D0 1 1 1 1 ` ` ` V jC1 z M WM (9.98) C z M WM Þ j Þ ŁjC1 V j z M WM
where we have used the observation that in the case of ideal filters the functions U j1 and V j2 do not overlap in frequency. Therefore we assume that their product is negligible. From the definition (9.85) of the function V j .z/ and from (9.82) we get 1 1 `C jC1 ` D þ Łj Q jC1 z M WM V j z M WM V jC1
1 zM
`C jC1
WM
D
þ ŁjC1
Qj
1 zM
`
(9.99)
WM
Substituting the previous equations in (9.98), observing (9.85) and (9.86), and using the ` , we obtain periodicity of WM 1 1 1 X 1 M ` z M WM [Þ Łj Þ jC1 þ j þ jC1 [G j .z/H jC1 .z/] # M D M `D0 . jC1/
C Þ j Þ ŁjC1 þ Łj þ ŁjC1 WM
1 1 ` ` Q jC1 z M WM ]Q j z M WM
(9.100) The condition to suppress the ICI in the j-th subchannel due to the signal transmitted on the . j C 1/-th subchannel is therefore given by . jC1/
Þ Łj Þ jC1 þ j þ jC1 C Þ j Þ ŁjC1 þ Łj þ ŁjC1 WM
D0
(9.101)
Substitution of (9.93) in (9.101) yields the condition Þ Łj Þ jC1 D Þ j Þ ŁjC1
(9.102)
Analogously, for the suppression of the ICI in the j-th subchannel due to the signal transmitted on the . j 1/-th channel, the condition Þ Łj Þ j1 D Þ j Þ Łj1 is found. We note that, setting Þ j D e j' j , we find that the condition for the approximate suppression of the ICI can be expressed as ' j D ' j1 š
³ 2
(9.103)
Equation (9.103) sets a constraint on the sequence of the phases of the constants Þ j , 0 j M 1; to define the whole sequence it is necessary to determine the phase of Þ0 . From (9.87) and (9.97), we observe that G j .z/ H j .z/ D z 1 [U 2j .z/ C V j2 .z/ C .Þ 2j C Þ Ł2 j / U j .z/ V j .z/]
(9.104)
788
Chapter 9. Orthogonal frequency division multiplexing
where the products U j .z/ V j .z/ are negligible except for j D 0 and j D M 1. In these þ 2³ f T þ þ þ cases, to avoid that the function þ H0 e j 2M þ is distorted at frequencies near zero and þ 2³ f T þ þ þ that the function þHM1 e j 2M þ is distorted at frequencies near M=T , it must be Þ 2j C Þ Ł2 j D0
j D 0; M 1
(9.105)
Therefore we choose 4 a04 D aM 1 D 1
(9.106)
1 For example, a sequence f' j gM jD0 that satisfies both the (9.103) and the further condition (9.106) is given by
' j D .1/ j
³ 4
j D 0; 1; : : : ; M 1
(9.107)
Moreover, from (9.104), the condition for the absence of ISI is expressed as [z 1 .U 2j .z/ C V j2 .z//] # M D constant
j D 0; 1; : : : ; M 1
(9.108)
Summarizing, for the design of the system we may start from a prototype filter P0 .z/ that approximates a square root raised cosine filter with Nyquist frequency 1=.2T /. This leads to verifying the condition (9.108). The M subchannel filters are obtained using (9.84), (9.85), and (9.87), where þi is defined in (9.93) and the phase of Þi is given in (9.107). An efficient implementation of the transmit filter bank is illustrated in Figure 9.27, where P .i / .z/ are the 2M polyphase components of the prototype filter P0 .z/, and d i D Þi þi , i D 0; 1; : : : ; M 1. An efficient implementation of the receive filter bank is illustrated in Figure 9.28.
Perfect interchannel interference suppression We derive now the conditions on the FIR filters of a DWMT system for the absence of intersymbol as well as interchannel interference in the case of an ideal transmission channel, i.e. the orthogonality conditions. We consider the system of Figure 9.27. The relation between the vector of real-valued input symbols akT D [ak [0]; : : : ; ak [M 1]] and the vector of real-valued samples AkT D [Ak [0]; : : : ; Ak [2M 1]] can be expressed in terms of the matrix ( ) 1 2M1 2 2 I 1 T Ł Ł (9.109) F2M diagfd0 ; d1 ; : : : ; d1 ; d0 g T D diag 1; W2M ; : : : ; W2M JM where di D Þi þi D
i ³ e j .1/ 4
1 M1 iC 2 2
W2M
i D 0; : : : ; M 1
(9.110)
9.9. DWMT modulation
ak [0]
M
T 2 ak [1]
789
P (0)(-z
T 2M
sn 2M
) -1/2
z -1 W2M
d0 M
P (1) (-z
T 2M
T 2
2M
) -1/2
z -1 W2M
d1
ak [M -1]
M
IDFT
T 2M
T 2
T 2M
dM -1
* dM -1
-1/2
z -1 W2M
d 1* P
( 2M -1)
(-z
2M
)
d 0*
Figure 9.27. OFDM system with approximate suppression of the ICI: transmit filter bank.
while JM denotes the M ð M matrix that has elements equal to one on the antidiagonal and all other elements equal to zero: 3 2 0 ÐÐÐ 0 0 1 6 0 ÐÐÐ 0 1 0 7 7 6 (9.111) JM D 6 : :: :: :: 7 4 :: : : : 5 1
0
0
ÐÐÐ
0
We assume that the parameters and M that determine the length M of the prototype filter are even numbers. The element tin of matrix T is then given by n
n
2 W ni d C W 2 W n.2M1i / d Ł tin D W2M i 2M i 2M 2M ½ 1 M 1 ³ i ³ iC n C .1/ D 2 cos M 2 2 4
D 2.cOin cos i sOin sin i /
0 i M 1 0 n 2M 1
(9.112)
790
Chapter 9. Orthogonal frequency division multiplexing
Figure 9.28. OFDM system with approximate suppression of the ICI: receive filter bank.
where
1 1 ³ iC nC M 2 2 1 1 ³ iC nC sOin D sin M 2 2 ³ 1 i D ³ i C C .1/i 2 2 4
cOin D cos
(9.113)
Therefore using the matrix TT we obtain an equivalent block diagram to that of Figure 9.27, as illustrated in Figure 9.29. We give the following definitions: 1. C is the matrix of the M-point discrete cosine transform (DCT), whose element in the position i; n is given by "r # ³ 1 1 2 cos iC nC [C]i;n D M M 2 2 (9.114) i D 0; : : : ; M 1 n D 0; : : : ; M 1
9.9. DWMT modulation
791
Figure 9.29. DWMT system implemented by the matrix T.
2. S is the matrix of the M-point discrete sine transform (DST), whose element in the position i; n is given by "r # ³ 1 1 2 iC nC sin (9.115) [S]i;n D M M 2 2 3. c and s are diagonal matrices given by 1 [c ]ii D cos ³ i C 2 2
(9.116)
and 1 [s ]ii D sin ³ i C 2 2
(9.117)
respectively; 4.
2 6 6 M D 6 4
1 0 :: :
0 1 :: :
0 0 :: :
ÐÐÐ ÐÐÐ
0
0
0
ÐÐÐ
0 0 :: :
.1/M1
3 7 7 7 5
(9.118)
5. T D [A0 ; A1 ]
(9.119)
792
Chapter 9. Orthogonal frequency division multiplexing
6. A0 and A1 are M ð M matrices given by p p A0 D M c .C M S/ A1 D M c .C C M S/ A0 D
p
M c .C C M S/
A1 D
p M c .C M S/
even 2 odd 2 (9.120)
We consider the polyphase representation of the prototype filter P0 .z/ by the 2M polyphase components P .`/ .z/, ` D 0; 1; : : : ; 2M 1, each having =2 coefficients, and define p0 .z/ D diagfP .0/ .z/; : : : ; P .M1/ .z/g p1 .z/ D diagfP .M/ .z/; : : : ; P .2M1/ .z/g
(9.121)
Using (9.119) and (9.120), the vector of the transfer functions of the transmit filters hT .z/ can be expressed as #" T # ½ " A0 p0 .z 2M / 0M 1 1 T .2M1/ H M H e ; z e h .z/ D z Ł Ł z z 0M p1 .z 2M / A1T (9.122) Comparing (9.122) with (9.17), we find that the matrix of the transfer functions of the polyphase components of the transmit filters is given by " T # A0 1 2 2 E.z/ D [z p0 .z /; p1 .z /] (9.123) A1T We recall from (9.24) that we get a system without ISI and ICI if E H .1=z Ł /E.z/ D I. We consider the product 3 2 1 Ł " T # z p A0 6 0 zŁ2 7 1 1 2 2 6 [A ] E.z/ E H D [z p .z /; p .z /] ; A 7 0 1 0 1 5 4 zŁ 1 A1T Ł p1 Ł 2 z (9.124) 1 1 D p0 .z 2 /A0T A0 pŁ0 Ł 2 C z 1 p0 .z 2 /A0T A1 pŁ1 Ł 2 z z 1 1 ð zp 1 .z 2 /A1T A0 pŁ0 Ł 2 C p1 .z 2 /A1T A1 pŁ1 Ł 2 z z Using definition (9.120) of matrices A0 and A1 we obtain 1 1 1 H 2 Ł 2 Ł E.z/ E D 2M p0 .z / p0 Ł 2 C p1 .z / p1 Ł 2 zŁ z z (9.125) 1 1 2 Ł 2 Ł 2M.1/ 2 p0 .z / JM p0 Ł 2 p1 .z /JM p1 Ł 2 z z
9. Bibliography
793
We recall that each polyphase component P .`/ .z/ has length =2. Moreover, from (9.90), the relation pn [0] D p M1n [0] implies the following constraints on the polyphase components: ½Ł 1 2 1 .`/ P .2M1`/ Ł ` D 0; : : : ; 2M 1 (9.126) P .z/ D z z From property (9.126), we find that the diagonal matrices p0 .z/ and p 1 .z/ satisfy the relation 1 2 1 1 Ł 2 .1/ JM p0 Ł JM (9.127) p1 .z/ D z z Then, using (9.127), we find that the second term in the (9.125) vanishes. Therefore we obtain 1 1 1 2 Ł 2 Ł D 2M p C p (9.128) .z / p .z / p E.z/ E H 0 1 0 1 zŁ zŁ2 zŁ2 Recalling that for two matrices whose product is the identity matrix the commutative property holds, we get E H .1=z Ł / E.z/ D E.z/ E H .1=z Ł / D I if and only if 1 1 1 Ł Ł I (9.129) p0 .z/ p 0 Ł C p1 .z/ p 1 Ł D z z 2M Using (9.129), we find the conditions on the polyphase components of the prototype filter for perfect suppression of ISI and ICI, given by ½Ł ½Ł 1 1 1 .`/ .`/ .MC`/ P 0`M1 P .z/ C P P .MC`/ .z/ D zŁ zŁ 2M (9.130) The conditions (9.130) can be used for the design of filters for DWMT systems. An efficient filter bank implementation is obtained by the DCT [1].
Bibliography [1] P. P. Vaidyanathan, Multirate systems and filter banks. Englewood Cliffs, NJ: PrenticeHall, 1993. [2] M. G. Bellanger, G. Bonnerot, and M. Coudreuse, “Digital filtering by polyphase network: application to sample-rate alteration and filter banks”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. ASSP-24, pp. 109–114, Apr. 1976. ¨ ¸ er, and J. M. Cioffi, “Filter bank modulation tech[3] G. Cherubini, E. Eleftheriou, S. Olc niques for very high-speed digital subscriber lines”, IEEE Communications Magazine, vol. 38, pp. 98–104, May 2000.
794
Chapter 9. Orthogonal frequency division multiplexing
R [4] G. Cherubini, E. Eleftheriou, and S. Olcer, “Filtered multitone modulation for veryhigh-speed digital subscriber lines”, IEEE Journal on Selected Areas in Communications, June 2002. [5] S. B. Weinstein and P. M. Ebert, “Data transmission by frequency-division multiplexing using the discrete Fourier transform”, IEEE Trans. on Communications, vol. 19, pp. 628–634, Oct. 1971. [6] J. A. C. Bingham, “Multicarrier modulation for data transmission: an idea whose time has come”, IEEE Communications Magazine, vol. 28, pp. 5–14, May 1990. [7] H. Sari, G. Karam, and I. Jeanclaude, “Transmission techniques for digital terrestrial TV broadcasting”, IEEE Communications Magazine, vol. 33, pp. 100–109, Feb. 1995. [8] J. S. Chow, J. C. Tu, and J. M. Cioffi, “A discrete multitone transceiver system for HDSL applications”, IEEE Journal on Selected Areas in Communications, vol. 9, pp. 895–908, Aug. 1991. [9] S. D. Sandberg and M. A. Tzannes, “Overlapped discrete multitone modulation for high speed copper wire communications”, IEEE Journal on Selected Areas in Communications, vol. 13, pp. 1571–1585, Dec. 1995. [10] P. J. W. Melsa, R. C. Younce, and C. E. Rohrs, “Impulse response shortening for discrete multitone transceivers”, IEEE Trans. on Communications, vol. 44, pp. 1662– 1672, Dec. 1996. [11] R. Baldemair and P. Frenger, “A time-domain equalizer minimizing intersymbol and intercarrier interference in DMT systems”, in Proc. GLOBECOM ’01, San Antonio, TX, Nov. 2001. [12] T. Pollet and M. Peeters, “Synchronization with DMT modulation”, IEEE Communications Magazine, vol. 37, pp. 80–86, Apr. 1999. [13] J. M. Cioffi, G. P. Dudevoir, M. V. Eyuboglu, and G. D. Forney, Jr., “MMSE decisionfeedback equalizers and coding. Part I and Part II”, IEEE Trans. on Communications, vol. 43, pp. 2582–2604, Oct. 1995.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 10
Spread spectrum systems
The term spread spectrum systems [1, 2, 3, 4, 5, 6, 7] was coined to indicate communication systems in which the bandwidth of the signal obtained by a standard modulation method (see Chapter 6) is spread by a certain factor before transmission over the channel, and then despread, by the same factor, at the receiver. The operations of spreading and despreading are the inverse of each other, i.e. for an ideal and noiseless channel the received signal after despreading is equivalent to the transmitted signal before spreading. For transmission over an ideal AWGN channel these operations do not offer therefore any improvement in performance with respect to a system that does not use spread spectrum. However, the practical applications of spread spectrum systems are numerous, for example, in multipleaccess systems, narrowband interference rejection, and transmission over channels with fading (see Section 10.2).
10.1
Spread spectrum techniques
We consider the two most common spread spectrum techniques: direct sequence (DS) and frequency hopping (FH).
10.1.1
Direct sequence systems
The baseband equivalent model of a DS system is illustrated in Figure 10.1. We consider the possibility that U users in a multiple-access system simultaneously transmit, using the same frequency band, by code division multiple access (CDMA) (see Section 6.13.2). The sequence of information bits fb`.u/ g of user u undergoes the following transformations. 1) Bit-mapper. From the sequence of information bits, a sequence of i.i.d. symbols fak.u/ g with statistical power Ma is produced. The symbols assume values in an M-ary constellation, using one of the maps described in Chapter 6; typically, in mobile radio systems, BPSK or QPSK modulation is used. Let T be the symbol period. 2) Spreading. We indicate by the integer N S F the spreading factor, and by Tchi p the chip period. These two parameters are related to the symbol period T by the relation Tchi p D
T NSF
(10.1)
796
Chapter 10. Spread spectrum systems
Figure 10.1. Baseband equivalent model of a DS system: (a) transmitter, (b) multiuser channel.
We recall from Appendix 6.D the definition of the Walsh–Hadamard sequences of length .u/ N S F . Here we refer to these sequences as channelization code fcCh;m g, m D 0; 1; : : : ;
.u/ .u/ NSF 1, u 2 f1; : : : ; U g. Moreover, cCh;m 2 f1; 1g, and jcCh;m j D 1. The Walsh– Hadamard sequences are orthogonal, that is ( SF 1 1 if u 1 D u 2 1 NX .u 1 / .u 2 /Ł c c D (10.2) NSF mD0 Ch;m Ch;m 0 if u 1 6D u 2 .u/ We assume that the sequences of the channelization code are periodic, that is cCh;m D .u/ cCh;m . mod N SF
.u/ g, also called signature sequence or spreading We now introduce the user code fcm sequence, that we initially assume to be equal to the channelization code, .u/ .u/ cm D cCh;m
(10.3)
.u/ Consequently, fcm g is also a periodic sequence of period NSF . The operation of spreading consists of associating with each symbol ak.u/ a sequence of NSF symbols of period Tchi p , that is obtained as follows. First, each symbol ak.u/ is repeated NSF times with period Tchi p : as illustrated in Figure 10.2, this operation is equivalent to upsampling fak.u/ g, so that .NSF 1/ zeros are inserted between two consecutive symbols,
10.1. Spread spectrum techniques
a (u) k
797
NSF holder
T
am(u)
(u) dm
Tchip (u)
cm
(a)
a (u) k
(u) dm
g (u) sp
T
Tchip (b)
Figure 10.2. Spreading operation: (a) correlator, (b) interpolator filter.
and using a holder of NSF values. The obtained sequence is then multiplied by the user code. Formally we have .u/ aN m D ak.u/
m D k NSF ; : : : ; k NSF C NSF 1 (10.4)
.u/ .u/ cm dm.u/ D aN m
If we introduce the filter .u/ gsp .i Tchi p / D ci.u/
i D 0; : : : ; NSF 1
(10.5)
the correlation of Figure 10.2a can be substituted by an interpolation with the interpolator .u/ filter gsp , as illustrated in Figure 10.2b. .u/ j D 1, from (10.4) we get Recalling that jcm (10.6)
Md D Ma
3) Pulse-shaping. Let h Tx be the modulation pulse, typically a square root raised cosine function or rectangular window. The baseband equivalent of the transmitted signal of user u is expressed as s .u/ .t/ D A.u/
C1 X
dm.u/ h Tx .t m Tchi p /
(10.7)
mD1
where A.u/ accounts for the transmit signal power. In fact, if E h is the energy of h Tx and fdm.u/ g is assumed i.i.d., the average statistical power of s .u/ .t/ is given by (see (1.399)) MN s .u/ D .A.u/ /2 Md
Eh Tchi p
(10.8)
798
Chapter 10. Spread spectrum systems
Using (10.4), an alternative expression for (10.7) is given by s .u/ .t/ D A.u/
C1 X
ak.u/
kD1
NX SF 1 `D0
.u/ c`Ck NSF h Tx .t .` C k NSF / Tchi p /
(10.9)
.u/ In the scheme of Figure 10.1a we note that, if condition (10.3) holds, then gsp is invariant .u/ with respect to the symbol period, and the two filters gsp and h Tx can be combined into one filter (see also (10.9))
h .u/ T .t/ D
NX SF 1 `D0
c`.u/ h Tx .t ` Tchi p /
(10.10)
and s .u/ .t/ D A.u/
C1 X
ak.u/ h .u/ T .t kT /
(10.11)
kD1
As shown in Figure 10.3, the equivalent scheme to the cascade of spreader and pulse-shaping filter is still a QAM modulator. The peculiarity is that the filter h .u/ T has a bandwidth much larger than the Nyquist frequency 1=.2T /. Therefore a DS system can be interpreted as a QAM system either with input symbols fdm.u/ g and transmit pulse h Tx , or with input symbols fak.u/ g and pulse h .u/ T ; later both interpretations will be used. 4) Transmission channel. Modeling the transmission channel as a filter having impulse response gC.u/ , the output signal is given by sC.u/ .t/ D .s .u/ Ł gC.u/ /.t/
(10.12)
The possibility that many users transmit simultaneously over the same frequency band leads to a total signal sC .t/ D
U X
sC.u/ .t/
(10.13)
uD1
We assume we are interested in reconstructing the message fak.1/ g of user u D 1, identified as desired user. If Ms .u/ is the statistical power of sC.u/ , the following signal-to-interference C ratios (SIRs) define the relative powers of the user signals: Ms .1/ 0i.u/ D C u D 2; 3; : : : ; U (10.14) Ms .u/ C
a (u) k T
h (u)
A
(u)
s (u)(t)
T
Figure 10.3. Equivalent scheme of spreader and pulse-shaping filter in a DS system.
10.1. Spread spectrum techniques
799
5) Noise. In Figure 10.1b the term wC includes both the noise of the receiver and possible additional interference, such as the interference due to signals coming from other cells in a wireless system; wC is modeled as white noise with PSD equal to N0 . Two signal-to-noise ratios are of interest. To measure the performance of the system in terms of Pbit , it is convenient to refer to the signal-to-noise ratio defined in Chapter 6 for passband transmission, Ms .1/
0s D
C
(10.15)
N0 =T
We recall that for an uncoded sequence of symbols fak.1/ g, the following relation holds: 0s Eb D N0 log2 M
(10.16)
However, there are cases, as for example in the evaluation of the performance of the channel impulse response estimation algorithm, when it is useful to measure the power of the noise over the whole transmission bandwidth. Hence we define the ratio 0c D
Ms .1/ C
N0 =Tchip
D
0s NSF
(10.17)
6) Receiver. The receiver structure varies according to the channel model and number of users. Deferring until Section 10.3 the analysis of more complicated system configurations, here we limit ourselves to considering the case of an ideal AWGN channel with gC.u/ .t/ D Ž.t/ and synchronous users. The latter assumption implies that the transmitters of the various users are synchronized and transmit at the same instant. For an ideal AWGN channel this means that at the receiver the optimum timing phase of signals of different users is the same. With these assumptions, we verify that the optimum receiver is simply given by the .1/ matched filter to h T .t/. According to the analog or discrete-time implementation of the matched filters, we get the schemes of Figure 10.4 or Figure 10.5, respectively; note that in Figure 10.5 the receiver front-end comprises an anti-aliasing filter followed by a sampler with sampling period Tc D Tchip =2. Let t0 be the optimum timing phase at the matched filter output (see Section 14.7). For an ideal AWGN channel it results in rC .t/ D
U X
s .u/ .t/ C wC .t/
uD1
(10.18) D
U X uD1
A.u/
C1 X i D1
ai.u/
NX SF 1 `D0
c`.u/ h Tx .t .` C i NSF / Tchip / C wC .t/
In the presence only of the desired user, that is for U D 1, it is clear that in the absence of ISI the structure with the matched filter to h .1/ T .t/ is optimum. We verify that the presence of other users is cancelled at the receiver, given that the various user codes are orthogonal.
800
Chapter 10. Spread spectrum systems
(1) M
g (t) =h (1)*(−t) T
r (t) C
g
(1) a^ k
yk
(1) M
T (a)
g (t) =h * (−t) Tx
M
r (t) C
t 0 +m Tchip kNSF +NSF −1 (1)* xm c m m=kNSF
xm
g
M
Σ
Tchip
(1) a^ k
yk T
(1)*
cm (b)
Figure 10.4. Optimum receiver with analog filters for a DS-CDMA system with ideal AWGN channel and synchronous users. Two equivalent structures: (a) overall matched filter, (b) matched filter to hTx and despreading correlator.
rC (t)
(1) M
g (t) =h (1)*(−t)
t 0’ +nTc g
r
AA (t)
AA
Tc =
T
(1) g M
Tchip
2NSF
(1) a^ k
yk
T
T a)
2 g (t) =h * (−t) Tx
M
g Tchip
M
2
M
2
xm Tchip
g (1)
N SF
ds
yk b)
2 g Tchip 2
kNSF +NSF −1 (1)* xm c m m=kNSF
xm
Σ
Tchip
yk c)
c m(1)* despreading
Figure 10.5. Optimum receiver with discrete-time filters for a DS-CDMA system with ideal AWGN channel and synchronous users. Three equivalent structures: (a) overall matched filter, (b) matched filter of hTx and despreading filter, (c) matched filter to hTx and despreading correlator.
10.1. Spread spectrum techniques
801
We assume that the overall analog impulse response of the system is a Nyquist pulse, hence .h Tx Ł gC.u/ Ł g A A Ł g M /.t/jtDt0 C j Tchip D .h T x Ł g M /.t/jtDt0 C j Tchi p D E h Ž j
(10.19)
We note that, if t0 is the instant at which the peak of the overall pulse at the output of g M is observed, then t00 in Figure 10.5 is given by t00 D t0 tg M , where tg M is the duration of g M . Moreover, from (10.19), we get that the noise at the output of g M , sampled with sampling rate 1=Tchip , wQ m D .wC Ł g M /.t/jtDt0 Cm Tchip
(10.20)
is an i.i.d. sequence with variance N0 E h . Hence, from (10.9) and (10.19), the signal at the output of g M , sampled with sampling rate 1=Tchip , has the following expression: xm D E h
U X
C1 X
A.u/
ai.u/
i D1
uD1
NX SF 1 `D0
.u/ c`Ci Qm NSF Žm`i NSF C w
(10.21)
With the change of indices m D j C k NSF , j D 0; 1; : : : ; NSF 1, k integer, we get x jCk NSF D E h
U X
A.u/ ak.u/ c.u/ Q jCk NSF jCk NSF C w
(10.22)
uD1
7) Despreading. We now correlate the sequence of samples fxm g, suitably synchronized, with the code sequence of the desired user, and we form the signal yk D
NX SF 1
.1/Ł
x jCk NSF c jCk NSF
(10.23)
jD0 .u/ As usual, introducing the filter gds given by .u/ gds .i Tchip / D c.1/Ł NSF 1i
i D 0; 1; : : : ; NSF 1
(10.24)
the correlation (10.23) is implemented through the filter (10.24), followed by a downsampler, as illustrated in Figure 10.5b. Substitution of (10.22) in (10.23) yields yk D NSF E h
U X
A.u/ ak.u/ rc.u/ c.1/ .0/ C wk
(10.25)
uD1
where in general rc.u 1 / c.u 2 / .n D / D
1 NSF jn D j
NSF 1jn X Dj
.u /
.u /Ł
1 2 c jCk NSF Cn D c jCk NSF
jD0
n D D .NSF 1/; : : : ; 1; 0; 1; : : : ; N SF 1 is the cross-correlation at lag n D between the user codes u 1 and u 2 .
(10.26)
802
Chapter 10. Spread spectrum systems
In the considered case, from (10.3) and (10.2) we get rc.u 1 / c.u 2 / .0/ D Žu 1 u 2 . Therefore (10.25) simply becomes yk D NSF E h A.1/ ak.1/ C wk
(10.27)
where N S F E h is the energy of the pulse associated with ak.1/ (see (10.10)). In (10.27) the noise is given by wk D
NX SF 1
wQ jCk NSF c.1/Ł jCk NSF
(10.28)
jD0
therefore assuming fwQ m g i.i.d., the variance of wk is given by ¦w2 D NSF ¦w2Q D NSF N0 E h
(10.29)
8) Data detector. Using a threshold detector, from (10.27) the signal-to-noise ratio at the decision point is given by (see (7.106)) D
dmin 2¦ I
2 D
NSF E h .A.1/ /2 .NSF E h A.1/ /2 D NSF N0 E h =2 N0 =2
(10.30)
On the other hand, from (10.8) and (10.15) we get 0s D
.A.1/ /2 Ma E h =Tchip NSF E h .A.1/ /2 Ma Es D D N0 =.NSF Tchip / N0 N0
(10.31)
where E s is the average energy per symbol of the transmitted signal. In other words, the relation between and 0s is optimum, as given by (7.114). Therefore, with regard to user u D 1, at the decision point the system is equivalent to an M-QAM system. However, as observed before, the transmit pulse h .1/ T has a bandwidth much larger than 1=.2T /. 9) Multi-user receiver. The derivation of the optimum receiver carried out for user 1 can be repeated for each user. Therefore we obtain the multiuser receiver of Figure 10.6, composed of a matched filter to the transmit pulse and a despreader bank, where each branch employs a distinct user code. We observe that for the ideal AWGN channel case, spreading the bandwidth of the transmit signal by a factor U allows the simultaneous transmission of U messages using the same frequency band.
Classification of CDMA systems Synchronous systems. This is the case just examined, in which the user codes are orthogonal and the user signals are time-aligned. In a wireless cellular radio system, this situation occurs in the forward or downlink transmission from the base station to the mobile stations.
10.1. Spread spectrum techniques
803
(1) a^ k
g (1) ds
r (t) C
(2) a^ k
g (2)
g (t) =h * (−t)
ds
Tx
M
T
T
g
M
Tchip (U) g ds
(U) a^ k
T
Figure 10.6. Multiuser receiver for a CDMA synchronous system with an ideal AWGN channel.
From the point of view of each mobile station, all U users share the same channel. Therefore, although the channel impulse response depends on the site of the mobile, we have gC.u/ .t/ D gC .t/
u D 1; : : : ; U
(10.32)
and the residual interference is due to signals originating from adjacent cells in addition to the multipath interference introduced by the channel. In general, interference due to the other users within the same cell is called multi-user interference (MUI) or co-channel interference (CCI). Asynchronous systems. In this case the various user signals are not time-aligned. In a wireless system, this situation typically occurs in the reverse or uplink transmission from the mobile stations to the base station. Because the Walsh–Hadamard codes do not exhibit good cross-correlation properties for lags different from zero, PN scrambling sequences are used (see Appendix 3.A). The user code is then given by .u/ .u/ D cCh;m cscr;m cm
(10.33)
where fcscr;m g may be the same for all users in a cell. It is necessary now to make an important observation. In some systems the period of fcscr;m g is equal to the length of fcCh;m g, that is NSF , whereas in other systems it is much larger than NSF .1 In the latter case, spreading and despreading operations remain unchanged, .u/ even if they are symbol time varying, as fcm g changes from symbol to symbol; note that consequently the receiver is also symbol time varying.
1
This observation must not be confused with the distinction between the use of short (of period ' 215 ) or long (of period ' 242 ) PN scrambling sequences, which are employed to identify the base stations or the users and to synchronize the system [8].
804
Chapter 10. Spread spectrum systems
Asynchronous systems are characterized by codes with low cross-correlation for nonzero lags; however, there is always a residual non-zero correlation among the various user signals. Especially in the presence of multipath channels, the residual correlation is the major cause of interference in the system, which now originates from signals within the cell: for this reason the MUI is usually characterized as intracell MUI.
Synchronization The despreading operation requires that the receiver is capable of reproducing a user code sequence synchronous with that used for spreading. Therefore the receiver must first per.u/ form acquisition, that is the code sequence fcm g produced by the local generator must be synchronized with the code sequence of the desired user, so that the error in the time alignment between the two sequences is less than one chip interval. As described in Section 14.7, acquisition of the desired user code sequence is generally obtained by a sequential searching algorithm that, at each step, delays the local code generator by a fraction of a chip, typically half a chip, and determines the correlation between the .u/ g; the search terminates when the correlation level exceeds a certain signals fxm g and fcm threshold value, indicating that the desired time alignment is attained. Following the acquisition process, a tracking algorithm is used to achieve, in the steady state, a time alignment .u/ between the signals fxm g and fcm g that has the desired accuracy; the more commonly used tracking algorithms are the delay-locked loop and the tau-dither loop. The synchronization method also suggests the use of PN sequences as user code sequences. In practice, the chip frequency is limited to values of the order of hundreds of Mchip/s because of the difficulty in obtaining an accuracy of the order of a fraction of a nanosecond in the synchronization of the code generator. In turn, this determines the limit in the bandwidth of a DS signal.
10.1.2
Frequency hopping systems
The FH spread spectrum technique is typically used for the spreading of M-FSK signals. We consider an M-FSK signal (see Example 6.7.1 on page 486) with carrier frequency complex form, i.e. we consider the analytic signal, as A e j2³. f 0 C1 f .t//t , f 0 expressed in P where 1 f .t/ D C1 kD1 ak wT .t kT /, with fak g sequence of i.i.d. symbols taken from the alphabet A D f.M 1/; : : : ; 1; C1; : : : ; M 1g, at the symbol rate 1=T . An FH/M-FSK signal is obtained by multiplying the M-FSK signal by a signal cFH .t/ given by c F H .t/ D
C1 X
e j .2³ f 0;i tC'0;i / wThop .t i Thop /
(10.34)
i D1
where f f 0;i g is a pseudorandom sequence that determines shifts in frequency of the FH/MFSK signal, f'0;i g is a sequence of random phases associated with the sequence of frequency shifts, and wThop is a rectangular window of duration equal to a hop interval Thop . In an FH/M-FSK system, the transmitted signal is then given by s.t/ D Re[cFH .t/ e j2³. f 0 C1 f .t//t ]
(10.35)
10.1. Spread spectrum techniques
805
Figure 10.7. Block diagram of an FH/M-FSK system.
In practice, the signal cFH .t/ is not generated at the transmitter; the transmitted signal s.t/ is obtained by applying the sequence of pseudorandom frequency shifts f f 0;i g directly to the frequency synthesizer that generates the carrier at frequency f 0 . With reference to the implementation illustrated in Figure 10.7, segments of L consecutive chips from a PN sequence, not necessarily disjoint, are applied to a frequency synthesizer that makes the carrier frequency hop over a set of 2 L frequencies. As the band over which the synthesizer must operate is large, it is difficult to maintain the carrier phase coherent between two consecutive hops [9]; if the synthesizer is not equipped with any device to maintain a coherent phase, it is necessary to include a random phase '0;i as in the expression (10.34). In a time interval that is long with respect to Thop , the bandwidth of the signal s.t/, BSS , can be in practice of the order of several GHz. However, in a short time interval during which no frequency hopping occurs, the bandwidth of an FH/M-FSK signal is the same as the bandwidth of the M-FSK signal that carries the information, usually much lower than BSS . Despreading, in this case also called dehopping, is ideally carried out by multiplying the received signal r.t/ by a signal cOFH .t/ equal to that used for spreading, apart from the sequence of random phases associated with the frequency shifts. For non-coherent demodulation, the sequence of random phases can be modelled as a sequence of i.i.d. random variables with uniform probability density in [0; 2³ /. The operation of despreading yields the signal x.t/, given by the sum of the M-FSK signal, the noise and possibly interference. The signal x.t/ is then filtered by a lowpass filter and presented to the input of the receive section comprising a non-coherent demodulator for M-FSK signals. As in the case of DS systems, the receiver must perform acquisition and tracking of the FH signal, so that the waveform generated by the synthesizer for dehopping reproduces as accurately as possible the signal cFH .t/.
806
Chapter 10. Spread spectrum systems
Classification of FH systems FH systems are traditionally classified according to the relation between Thop and T . Fast frequency-hopped (FFH) systems are characterized by one or more frequency hops per symbol interval, that is T D N Thop , N integer, and slow frequency-hopped (SFH) systems are characterized by the transmission of several symbols per hop interval, that is Thop D N T . Moreover, a chip frequency Fchip is defined also for FH systems, and is given by the largest value among Fhop D 1=Thop and F D 1=T . Therefore the chip frequency Fchip corresponds to the highest among the clock frequencies used by the system. The frequency spacing between the tones of an FH/M-FSK signal is related to the chip frequency and is therefore determined differently for FFH and SFH systems. SFH systems. For SFH systems, Fchip D F, and the spacing between FH/M-FSK tones is equal to the spacing between the M-FSK tones themselves. In a system that uses a non-coherent receiver for M-FSK signals, orthogonality of tones corresponding to M-FSK symbols is obtained if the frequency spacing is an integer multiple of 1=T . Assuming the minimum spacing is equal to F, the bandwidth BSS of an FH/M-FSK signal is partitioned into N f D BSS =F D BSS =Fchip sub-bands with equally spaced center frequencies; in the most commonly used FH scheme the N f tones are grouped into Nb D N f =M adjacent bands without overlap in frequency, each one having a bandwidth equal to M F D M Fchip , as illustrated in Figure 10.8. Assuming M-FSK modulation symmetric around the carrier frequency, the center frequencies of the Nb D 2 L bands represent the set of carrier frequencies generated by the synthesizer, each associated with an L-uple of binary symbols. According to this scheme, each of the N f tones of the FH/M-FSK signal corresponds to a unique combination of carrier frequency and M-FSK symbol. BSS MF
1
2
3
MF
4
5
6
7
MF
8
MF
4i−3 4i−2 4i−1 4i F
1
2
MF
Nf −3 Nf −2 Nf −1 Nf =4Nb Nb
i
frequency
Figure 10.8. Frequency distribution for an FH/4-FSK system with bands non-overlapping in frequency; the dashed lines indicate the carrier frequencies.
10.2. Applications of spread spectrum systems
807
BSS MF
MF
MF
MF
MF
1
2
3
MF
4
5
6
7
8
MF
4i−3 4i−2 4i−1 4i F
Nf −3 Nf −2 Nf −1 Nf =4Nb
frequency
Figure 10.9. Frequency distribution for an FH/4-FSK system with bands overlapping in frequency.
In a different scheme, that yields a better protection against an intentional jammer using a sophisticated disturbance strategy, adjacent bands exhibit an overlap in frequency equal to .M 1/Fchip Hz, as illustrated in Figure 10.9. Assuming that the center frequency of each band corresponds to a possible carrier frequency, as all N f tones except .M 1/ are available as center frequencies, the number of carrier frequencies increases from N f =M to N f .M 1/, which for N f × M represents an increase by a factor M of the randomness in the choice of the carrier frequency. FFH systems. For FFH systems, where Fchip D Fhop , the spacing between tones of an FH/M-FSK signal is equal to the hop frequency. Therefore the bandwidth of the spread spectrum signal is partitioned into a total of N f D BSS =Fhop D BSS =Fchip sub-bands with equally spaced center frequencies, each corresponding to a unique L-uple of binary symbols. Because there are Fhop =F hops per symbol, the metric used to decide upon the symbol with a non-coherent receiver is suitably obtained by summing Fhop =F components of the received signal.
10.2
Applications of spread spectrum systems
The most common applications of spread spectrum systems, that will be discussed in the next sections, may be classified as follows. 1. Multiple access. In alternative to FDMA and TDMA systems, introduced in Section 6.13.2, spread spectrum systems allow the simultaneous transmission of messages by several users over the channel, as discussed in Section 10.1.1.
808
Chapter 10. Spread spectrum systems
2. Narrowband interference rejection. We consider the DS case. Because interference is introduced in the channel after signal spreading, at the receiver the despreading operation compresses the bandwidth of the desired signal to the original value, and at the same time it expands by the same factor the bandwidth of the interference, thus reducing the level of the interference power spectral density. After demodulation the ratio between the desired signal power and the interference power is therefore larger than that obtained without spreading the signal spectrum. 3. Robustness against fading. Widening the signal bandwidth allows exploitation of the multipath diversity of a radio channel affected by fading. Applying a DS spread spectrum technique, intuitively, has the effect of modifying a channel model that is adequate for transmission of narrowband signals in the presence of flat fading or multipath fading with a few rays, to a channel model with many rays. Using a receiver that combines the desired signal from the different propagation rays, the power of the desired signal at the decision point increases. In an FH system, on the other hand, we obtain diversity in the time domain, as the channel changes from one hop interval to the next. The probability that the signal is affected by strong fading during two consecutive hop intervals is usually low. To recover the transmitted message in a hop interval during which strong fading is experienced, error correction codes with very long interleaver and ARQ schemes are used (see Chapter 11).
10.2.1
Anti-jam communications
Narrowband interference We consider the baseband equivalent signals of an M-QAM passband communication system with symbol rate F D 1=T , transmitted signal power equal to Ms , and PSD with minimum bandwidth, i.e. Ps . f / D E s rect. f =F/, where E s F D Ms . We now consider the application of a DS spread spectrum modulation system. Due to spreading, the bandwidth of the transmitted signal is expanded from F to BSS D NSF F. Therefore, for the same transmitted signal power, the PSD of the transmitted signal becomes Ps 0 . f / D .E s =NSF / rect. f =BSS /, where E s =NSF D Ms =BSS . We note that spreading has decreased the amplitude of the PSD by the factor NSF , as illustrated in Figure 10.10. In the band of the spread spectrum signal, in addition to additive white Gaussian noise with PSD N0 , we assume the channel introduces an additive interference signal or jammer with power Mj , uniformly distributed on a bandwidth Bj , with Bj < 1=T . With regard to the operation of despreading, we consider the signals after the multiplication by the user code sequence. The interference signal spectrum is expanded and has a PSD equal to Pj0 . f / D Nj rect. f =BSS /, with Nj D Mj =BSS . The noise, that originally has a uniformly distributed power over all the frequencies, still has PSD equal to N0 , i.e. spreading has not changed the PSD of the noise. At the output of the despreading the desired signal exhibits the original PSD equal to E s rect. f =F/. Modeling the despreader filter as an ideal lowpass filter with bandwidth 1=.2T /, for the signal-to-noise ratio at the decision point the following relation holds:
Es 1 Ms =F Ma D D 2 N0 C Nj .N0 C Mj =BSS /
(10.36)
10.2. Applications of spread spectrum systems
809
Figure 10.10. Power spectral density of an M-QAM signal with minimum bandwidth and of a spread spectrum M-QAM signal with spreading factor NSF D 4.
In practice, performance is usually limited by interference and the presence of white noise can be ignored. Therefore, assuming Nj × N0 , (10.36) becomes Es 1 Ms =F Ms BSS Ma ' (10.37) D D 2 Nj Mj =BSS Mj F where Ms =Mj is the ratio between the power of the desired signal and the power of the jammer, and BSS =F is the spreading ratio N S F also defined as the processing gain of the system. The above considerations are now defined more precisely in the following case. Sinusoidal interference. We assume that the baseband equivalent received signal is expressed as rC .t/ D s.t/ C j.t/ C wC .t/
(10.38)
where s.t/ is a DS signal given by (10.9) with amplitude A.u/ D 1, wC .t/ is AWGN with spectral density N0 , and the interferer is given by j.t/ D Aj e j'
(10.39)
p In (10.39) Aj D Mj is the amplitude of the jammer and ' a random phase with uniform distribution in [0; 2³ /. We also assume a minimum bandwidth p p transmit pulse, h Tx .t/ D E h =Tchip sinc.t=Tchip /, hence g M .t/ D h Tx .t/, and G M .0/ D E h Tchip . For the coherent receiver of Figure 10.4, at the detection point the sample at instant kT is given by yk D NSF E h ak C wk C Aj e j' G M .0/
NX SF 1
cŁjCk NSF
(10.40)
jD0
Modeling the sequence fckŁNSF ; ckŁNSF C1 ; : : : ; ckŁNSF CNSF 1 g as a sequence of i.i.d. random variables, the variance of the summation in (10.40) is equal to NSF , and the ratio
810
Chapter 10. Spread spectrum systems
is given by D
.NSF E h /2 .NSF N0 E h C Mj E h Tchip NSF /=2
(10.41)
Using (10.8) and the relation E s D Ms T , we obtain
1 1 Ma D 2 N0 =E s C Mj =.NSF Ms /
(10.42)
We note that in the denominator of (10.42) the ratio Mj =Ms is divided by N S F . Recognizing that Mj =Ms is the ratio between the power of the jammer and the power of the desired signal before the despreading operation, and that Mj =.NSF Ms / is the same ratio after the despreading, we find that, by analogy with the previous case of narrowband interference, also in the case of a sinusoidal jammer the use of the DS technique reduces the effect of the jammer by a factor equal to the processing gain.
10.2.2
Multiple-access systems
Spread spectrum multiple-access communication systems represent an alternative to TDMA or FDMA systems and are normally referred to as CDMA systems (see Section 6.13.2 and Section 10.1.1). With CDMA, a particular spreading sequence is assigned to each user to access the channel; unlike FDMA, where users transmit simultaneously over nonoverlapping frequency bands, or TDMA, where users transmit over the same band but in disjoint time intervals, users in a CDMA system transmit simultaneously over the same frequency band. Because in CDMA systems correlation receivers are usually employed, it is important that the spreading sequences are characterized by low cross-correlation values. We have already observed that CDMA systems may be classified as synchronous or asynchronous. In the first case the symbol transition instants of all users are aligned; this allows the use of orthogonal sequences as spreading sequences and consequently the elimination of interference caused by one user signal to another; in the second case the interference caused by multiple access limits the channel capacity, but the system design is simplified. CDMA has received particular interest for applications in wireless communications systems, for example, cellular radio systems, personal communications services (PCS), and wireless local-area networks; this interest is mainly due to performance that spread spectrum systems achieve for the transmission over channels characterized by multipath fading. Other properties make CDMA interesting for application to cellular radio systems, for example the possibility of applying the concept of frequency reuse (see Chapter 17). In cellular radio systems based on FDMA or TDMA, to avoid excessive levels of interference from one cell onto neighboring cells, the frequencies used in one cell are not used in neighboring cells. In other words, the system is designed so that there is a certain spatial separation between cells that use the same frequencies. For CDMA, this spatial separation
10.3. Chip matched filter and rake receiver
811
is not necessary, making it possible, in principle, to reuse all frequencies. Moreover, as CDMA systems tend to be limited by interference, an increase in system capacity is obtained by detecting the speech signal activity. This gain is made possible by the fact that in every telephone conversation each user speaks only for about half the time, while in the silence intervals he does not contribute to instantaneous interference. If several users can be served by the system, on average only half of them are active at a given instant, and the effective capacity can be doubled.
10.2.3
Interference rejection
Besides the above described properties, that are relative to the application in multipleaccess systems, the robustness of spread spectrum systems in the presence of narrowband interferers is key in other applications, for example, in systems where interference is unintentionally generated by other users that transmit over the same channel. We have CCI when a certain number of services are simultaneously offered to users transmitting over the same frequency band. Although in these cases some form of spatial separation among signals interfering with each other is usually provided, for example, by using directional antennas, it is often desirable to use spread spectrum systems for their inherent interference suppression capability. In particular, we consider a scenario in which a frequency band is only partially occupied by a set of narrowband conventional signals: to increase the spectral efficiency of the system, a set of spread spectrum signals can simultaneously be transmitted over the same band, thus allowing two sets of users to access the transmission channel. Clearly, this scheme can be implemented only if the mutual interference, which a signal set imposes on the other, remains within tolerable limits.
10.3
Chip matched filter and rake receiver
Before introducing a structure that is often employed in receivers for DS spread spectrum signals, we make the following considerations on the radio channel model introduced in Section 4.6.
Number of resolvable rays in a multipath channel We want to represent a multipath radio channel with a number of rays having gains modeled as complex valued, Gaussian uncorrelated random processes. From (4.206), apart from a complex constant, the channel impulse response with infinite bandwidth is given by gC .− / D
Nc;1 X1
gi Ž.− i TQ /
(10.43)
i D0
where for simplicity we have assumed the absence of Doppler spread. Therefore the nonzero gains fgi g are uncorrelated random variables and the delays −i D i TQ are multiples of a sufficiently small period TQ .
812
Chapter 10. Spread spectrum systems
Hence, from (10.43), the channel output signal sC is related to the input signal s by sC .t/ D
Nc;1 X1
gi s.t i TQ /
(10.44)
i D0
Now the number of resolvable or uncorrelated rays in (10.44) is generally less than Nc;1 and is related to the bandwidth of s by the following rule: if s has a bandwidth B, the uncorrelated rays are spaced by a delay of the order of 1=B. Consequently, for a channel with a delay spread −r ms and bandwidth B / 1=Tchip , the number of resolvable rays is given by Nc;r es /
−r ms Tchip
(10.45)
Using the notion of channel coherence bandwidth, Bccb / 1=−r ms , (10.45) may be rewritten as Nc;r es /
B Bccb
(10.46)
We now give an example that illustrates the above considerations. Let fgC .nTQ /g be a realization of the channel impulse response with uncorrelated coefficients having a given power delay profile; the “infinite bandwidth” of the channel will be equal to B D 1=.2TQ /. We now filter fgC .nTQ /g with two filters having, respectively, bandwidth B D 0:1=.2TQ / and B D 0:01=.2TQ /, and we compare the three pulse shapes given by the input sequence and the two output sequences. We note that the output obtained in correspondence of the filter with the narrower bandwidth has fewer resolvable rays. In fact, in the limit for B ! 0 the output is modeled as a single random variable. Another way to derive (10.45) is to observe that, for t within an interval of duration 1=B, s does not vary much. Therefore, letting Ncor D
Nc;1 Nc;r es
(10.47)
equation (10.44) can be written as sC .t/ D
Nc;r es 1 X
gr es; j s.t j Ncor TQ /
(10.48)
jD0
where gr es; j '
NX cor 1
gi C j Ncor
(10.49)
i D0
are the gains of the resolvable rays. The conclusion is that, assuming the symbol period T is given and DS spread spectrum modulation is adopted, the larger, the NSF , the greater the resolution of the radio channel, that is, the channel can be modeled with a larger number of uncorrelated rays, with delays of the order of Tchip .
10.3. Chip matched filter and rake receiver
813
Chip matched filter (CMF) We consider the transmission of a DS signal (10.9) for U D 1 on a dispersive channel as described by (10.48). The receiver that maximizes the ratio between the amplitude of the pulse associated with the desired signal sampled with sampling rate 1=Tchip and the standard deviation of the noise is obtained by the filter matched to the received pulse. We define qC .t/ D .h Tx Ł gC Ł gAA /.t/
(10.50)
and let g M .t/ D qCŁ .t0 t/ be the corresponding matched filter. In practice, at the output of the filter gAA an estimate of qC with sampling period Tc D Tchip =2 is evaluated,2 which yields the corresponding discrete-time matched filter with sampling period of the input signal equal to Tc and sampling period of the output signal equal to Tchip (see Figure 10.11). If qC is sparse, that is, it has a large support but only a few non-zero coefficients, for the realization of g M we retain only the coefficients of qC with larger amplitude; it is better to set to zero the remaining coefficients because their estimate is usually very noisy (see Appendix 3.A). Figure 10.12a illustrates in detail the receiver of Figure 10.11 for a filter g M with at most NMF coefficients spaced of Tc D Tchip =2. If we now implement the despreader on every branch of the filter g M , we obtain the structure of Figure 10.12b. We observe that typically only 3 or 4 branches are active, that is they have a coefficient g M;i different from zero. Ideally, for an overall channel with Nr es resolvable paths, we assume Nr es X
qC .t/ D
qC;i Ž.t −i /
(10.51)
Ł qC; j Ž.t0 t − j /
(10.52)
i D1
hence g M .t/ D
Nr es X jD1
Defining t M; j D t0 − j
j D 1; : : : ; Nr es
(10.53)
the receiver scheme, analogous to that of Figure 10.12b, is illustrated in Figure 10.13.
rC (t)
t 0’ +nTc g
rAA (t)
g (t) =q * (t t) C 0 M r
AA,n
AA
Tc =
Tchip
g
M
despreader 2
xm Tchip
g (1) ds
yk
(1) a^ k
T
T
2
Figure 10.11. Chip matched filter receiver for a dispersive channel.
2
To determine the optimum sampling phase t0 , usually r A A is oversampled with a period TQ such that Tc =TQ D 2 or 4 for Tc D Tchip =2; among the 2 or 4 estimates of gC obtained with sampling period Tc , the one with the largest energy is selected (see Observation 8.5 on page 641).
814
Chapter 10. Spread spectrum systems
r
AA,2m
Tc =
Tc
Tchip
2
Tc
Tc
g
g
g
g
M,NSF 2
M,1
M,0
M,NMF1
2
xm
g (1) ds
Tchip
yk
T
^a (1) k
(a) r
AA,2NSF k
Tc
Tc
T Tc = chip 2
2
2
Tchip
Tchip
g (1)
despreader
g (1)
ds
ds
T
T g
g
M,NMF 1
M,0
yk
T
^a (1) k
(b)
Figure 10.12. Two receiver structures: (a) chip matched filter with despreader, (b) rake.
To simplify the analysis, we assume that the spreading sequence is a PN sequence with N S F sufficiently large, such that the following approximations hold: 1) the autocorrelation of the spreading sequence is a Kronecker delta: and 2) the delays f−i g are multiples of Tchip . From (10.51), in the absence of noise the signal r A A is given by
r A A .t/ D
Nr es X nD1
qC;n
C1 X i D1
ai.1/
NX SF 1 `D0
.1/ c`Ci NSF Ž.t −n .` C i NSF / Tchip /
(10.54)
10.3. Chip matched filter and rake receiver
815
despreader
Figure 10.13. Rake receiver for a channel with Nres resolvable paths.
and the output of the sampler on branch j is given by3 x j;m D
Nr es X nD1
qC;n
C1 X i D1
ai.1/
NX SF 1 `D0
.1/ c`Ci NSF Ž
− j −n mC T .`Ci NSF / chip
(10.55)
Correspondingly the despreader output, assuming rc.1/ .n D / D Žn D and the absence of noise, yields the signal NSF ak.1/ qC; j . The contributions from the various branches are then combined according to the MRC technique (see Section 6.13) to yield the sample ! Nr es X 2 yk D NSF jqC;n j ak.1/ (10.56) nD1
P Nr es
where E qC D nD1 jqC;n j2 is the energy per chip of the overall channel impulse response. The name rake originates from the structure of the receiver that is similar to a rake with Nr es fingers. In practice, near the rake receiver a correlator estimates the delays, with precision Tchip =2, and the gains of the various channel rays. The rake is initialized with the 3
Instead of using the Dirac delta in (10.51), a similar analysis assumes that 1) gAA .t/ D h ŁTx .t/, and 2) rh Tx .t/ is a Nyquist pulse. The result is the same as (10.55).
816
Chapter 10. Spread spectrum systems
coefficients of rays with larger gain. The delays and the coefficients are updated whenever a change in the channel impulse response is observed. However, after the initialization has taken place, on each finger of the rake the estimates of the amplitude and of the delay of the corresponding ray may be refined by using the correlator of the despreader, as indicated by the dotted line in Figure 10.13. We note that if the channel is static, the structure of Figure 10.12a with Tc D Tchip =2 yields a sufficient statistic.
10.4
Interference
For a dispersive channel and in the case of U users, we evaluate the expression of the signal yk at the decision point using the matched filter receiver of Figure 10.11. Similarly to (10.50), we define qC.u/ .t/ D .h Tx Ł gC.u/ Ł gAA /.t/
u D 1; : : : ; U
(10.57)
and let .v/Ł g .v/ M .t/ D qC .t0 t/
v D 1; : : : ; U
(10.58)
be the corresponding matched filter. Moreover, we introduce the correlation between qC.u/ and qC.v/ , expressed by rq .u/ q .v/ .− / D .qC.u/ .t/ Ł qC.v/Ł .t//.− / C
(10.59)
C
Assuming without loss of generality that the desired user signal has the index u D 1, we .1/Ł refer to the receiver of Figure 10.11, where we have g M .t/ D g .1/ M .t/ D qC .t0 t/, and xm D
U X uD1
A.u/
C1 X
ai.u/
NX SF 1 `D0
i D1
.u/ c`Ci Qm NSF rq .u/ q .1/ ..m ` i NSF / Tchip / C w C
(10.60)
C
where wQ m is given by (10.20). At the despreader output we obtain yk D
NX SF 1
x jCk NSF c.1/Ł jCk NSF C wk
jD0
D
U X
A.u/
C1 X
ai.u/
i D1
uD1
NX SF 1 NX SF 1 `D0
.u/ c`Ci NSF
jD0
rq .u/ q .1/ .. j ` C .k i/ NSF / Tchip /c.1/Ł jCk NSF C wk C
C
where wk is defined in (10.28).
(10.61)
10.4. Interference
817
Introducing the change of index n D ` j and recalling the definition of cross-correlation between two code sequences (10.26), the double equation in ` and j in (10.61) can be written as 1 X
rq .u/ q .1/ ..n C .k i/ NSF /Tchip /.NSF jnj/rŁc.1/ c.u/ .n/
nD.NSF 1/
C
C
C
(10.62)
NX SF 1
rq .u/ q .1/ ..n C .k i/ NSF /Tchip /.NSF jnj/rc.u/ c.1/ .n/
nD0
C
C
where, to simplify the notation, we have assumed that the user code sequences are periodic of period NSF . The desired term in (10.61) is obtained for u D 1; as rŁc.1/ .n/ D rc.1/ .n/, it has the following expression: A.1/
C1 X i D1
ai.1/
NX SF 1
.NSF jnj/ rc.1/ .n/ rq .1/ ..n C .k i/ NSF / Tchip /
(10.63)
C
nD.NSF 1/
Consequently, if the code sequences are orthogonal, that is rc.1/ .n/ D Žn
(10.64)
and in the absence of ISI, that is rq .1/ .i NSF Tchip / D Ži E q .1/ C
(10.65)
C
where E q .1/ is the energy per chip of the overall pulse at the output of the filter g A A , then C the desired term (10.63) becomes A.1/ NSF E qC ak.1/
(10.66)
which coincides with the case of an ideal AWGN channel (see (10.27)). Note that using the same assumptions we find the rake receiver behaves as an MRC (see (10.56)). If (10.64) is not verified, as happens in practice, and if rq .1/ .nTchip / 6D Žn E qC
(10.67)
C
the terms for n 6D 0 in (10.63) give rise to intersymbol interference, in this context also called inter-path interference (IPI). Usually the smaller the NSF , the larger the IPI. We note, however, that if the overall pulse at the output of the CMF is a Nyquist pulse, that is rq .1/ .nTchip / D Žn E qC
(10.68)
C
then there is no IPI, even if (10.64) is not verified. With reference to (10.62) we observe that, in the multiuser case, if rc.u/ c.1/ .n/ 6D 0 then yk is affected by MUI, whose value increases as the cross-correlation between the pulses qC.u/ and qC.1/ increases.
818
Chapter 10. Spread spectrum systems
x m(1)
g (1) M
Tchip
detector
^a (1) k
rC (t)
a) x m(U)
g (U) M
Tchip
detector
^a (U) k
g (i)(t) =q (i)* (t (i) t) M
C
g (1) M
0
x m(1) Tchip
multiuser
^a (1) k
rC (t)
b) g (U) M
x m(U)
detector
Tchip
^a (U) k
Figure 10.14. (a) Single-user receiver, and (b) multiuser receiver.
Detection strategies for multiple-access systems For detection of the user messages in CDMA systems, we make a distinction between two classes of receivers: single-user and multiuser. In the first class the receivers focus on detecting the data from a single user, and the other user signals are considered as uncancellable interference. In the second class the receivers seek to simultaneously detect all U messages. The performance of the multiuser receivers is substantially better than that of the single-user receivers, achieved at the expense of a higher computational complexity. Using as front-end a filter bank, where the filters are matched to the channel impulse responses of the U users, the structures of single-user and multiuser receivers are exemplified in Figure 10.14.
10.5
Equalizers for single-user detection
We consider two equalizers for single-user detection.
Chip equalizer (CE) To mitigate the interference in the signal sampled at the chip rate, after the CMF (see (10.68)) a ZF or an MSE equalizer can be used [10, 11, 12, 13]. As illustrated in Figure 10.15, let gCE be the equalizer filter with output fdQm g. For an MSE criterion the cost
10.5. Equalizers for single-user detection
wC (t) (1) dm
A
(U)
dm
Tchip
t ’ +nTc 0
(1)
h Tx *g (1) C
Tchip
819
g
g
AA
Tc =
A
Tchip
CE
2
~ dm Tchip
g (1) ds
yk
(1) a^ k−D
T
T
2
(U)
h Tx *g (U) C
Figure 10.15. Receiver as a fractionally-spaced chip equalizer.
function is given by J D E[jdQm dm j2 ]
(10.69)
where fdm g is assumed i.i.d. We distinguish the two following cases: 1) All code sequences are known. This is the case that may occur for downlink transmission in wireless networks. Then gC.u/ .t/ D gC .t/, u D 1; : : : ; U , and we assume dm D
U X
dm.u/
(10.70)
uD1
that is, for the equalizer design, all user signals are considered as desired signals. 2) Only the code sequence of the desired user signal is known. assume dm D dm.1/
In this case we need to
(10.71)
The other user signals are considered as white noise, with overall PSD Ni , that is added to wC . From the knowledge of qC.1/ and the overall noise PSD, the minimum of the cost function defined in (10.69) is obtained by following the same steps developed in Chapter 8. Obviously, if the level of interference is high, the solution corresponding to (10.71) yields a simple CMF, with low performance whenever the residual interference (MUI and IPI) at the decision point is high. A better structure for single-user detection is obtained by the following approach.
Symbol equalizer (SE) Recalling that we adopt the transmitter model of Figure 10.3, and that we are inter.1/ ested in the message fak.1/ g, the optimum receiver with linear filter gSE is illustrated in Figure 10.16.
820
Chapter 10. Spread spectrum systems
wC (t) a (1) k
A
t ’ +nTc 0
(1)
T
h (1) *g (1) T C
g
rAA,n
AA
Tc =
a (U) k
T
A
Tchip
g (1) SE
2NSF
yk
(1) a^ k−D
T
T
2
(U)
h
(U) *g (U) T C
Figure 10.16. Receiver as a fractionally-spaced symbol equalizer.
The cost function is now given by [14, 15, 16] J D E[jyk akD j2 ]
(10.72)
.1/ , that includes also the function of despreading, depends on the code sequence Note that gSE of the desired user. Therefore the length of the code sequence is usually not larger than NSF , otherwise we would find a different solution for every symbol period, even if gC.1/ is time invariant. Moreover, in this formulation the other user signals are seen as interference, and one of the tasks of gSE is to mitigate the MUI. In an adaptive approach, for example, using the LMS algorithm, the solution is simple to determine and does not require any particular a priori knowledge, except the training sequence in fak.1/ g for initial convergence. On the other hand, using a direct approach we need to identify the autocorrelation of rAA;n and the cross-correlation between rAA;n .1/ and akD . As usual these correlations are estimated directly or, assuming the messages
fak.u/ g, u D 1; : : : ; U , are i.i.d. and independent of each other, we can determine them .u/ using the knowledge of the various pulses fh .u/ T g and fgC g, that is the channel impulse responses and code sequences of all users; for the special case of downlink transmission, the knowledge of the code sequences is sufficient, as the channel is common to all user signals.
10.6
Block equalizer for multiuser detection
Multiuser detection techniques are essential for achieving near-optimum performance in communication systems where signals conveying the desired information are received in the presence of ambient noise plus multiple-user interference. The leitmotiv of developments in multiuser detection is represented by the reduction in complexity of practical receivers with respect to that of optimal receivers, which is known to increase exponentially with the number of active users and with the delay spread of the channel, while achieving nearoptimum performance. A further element that is being recognized as essential to reap the full benefits of interference suppression is the joint application of multiuser detection with other techniques such as spatial-temporal processing and iterative decoding.
10.6. Block equalizer for multiuser detection
821
Here we first consider the simplest among multiuser receivers. It comprises a bank of U filters gT.u/ , u D 1; : : : ; U , matched to the impulse responses4 qT.u/ .t/ D
NX SF 1 `D0
c`.u/ qC.u/ .t ` Tchip /
u D 1; : : : ; U
(10.73)
where the functions fqC.u/ .t/g are defined in (10.57). Decisions taken by threshold detectors on the U output signals, sampled at the symbol rate, yield the detected user symbol sequences. It is useful to introduce this receiver, that we denote as MF, as, substituting the threshold detectors with more sophisticated detection devices, it represents the first stage of several multiuser receivers, as illustrated in general in Figure 10.17. We introduce the following vector notation. The vector of symbols transmitted by U users in a symbol period T is expressed as ak D [ak.1/ ; : : : ; ak.U / ]T
(10.74)
and the vector that carries the information on the codes and the channel impulse responses of the U users is expressed as qT .t/ D [qT.1/ .t/; : : : ; qT.U / .t/]T
(10.75)
Joint detectors constitute an important class of multiuser receivers. They effectively mitigate both ISI and MUI, exploiting the knowledge of the vector qT . In particular we consider now block linear receivers: as the name suggests, a block linear receiver is a joint detector that recovers the information contained in a window of K symbol periods. Let a D [a0T ; : : : ; aTK 1 ]T .1/
.U /
.1/
(10.76)
.U /
D [a0 ; : : : ; a0 ; : : : ; a K 1 ; : : : ; a K 1 ]T (1) t 0 +m Tchip
g (1) M
rC (t)
Tchip
y (1) (1) g ds
k
T
multiuser
^a (1) k
(U)
t 0 +m Tchip (U)
g (U) M
y (U)
Tchip
g ds
detector
k
T
^a (U) k
Figure 10.17. Receiver as MF and multiuser detector.
4
.u/
We assume that the information on the power of the user signals is included in the impulse responses gC , u D 1; : : : ; U , so that A.u/ D 1, u D 1; : : : ; U .
822
Chapter 10. Spread spectrum systems
be the information transmitted by U users and let y be the corresponding vector of K U elements at the MF output. We define the following correlations: rq.u;v/ .k/ D .qT.u/ .t/ Ł qT.v/Ł .t//.− /j− DkT
(10.77)
Assuming rq.u;v/ .k/ D 0
for jkj > ¹
(10.78)
with ¹ < K , and following the approach in [17, 18, 19, 20], we introduce the K U ð K U matrix, 3 2 rq .1;1/ .0/ : : : rq .1;U / .0/ : : : rq .1;1/ .¹/ : : : rq .1;U / .¹/ : : : 6 r .2;1/ .0/ : : : r .2;U / .0/ : : : r .2;1/ .¹/ : : : r .2;U / .¹/ : : : 7 q q q 7 6 q 6 :: :: :: :: :: 7 :: :: :: 6 : : : : : 7 : : : 7 6 6 r .U;1/ .0/ : : : r .U;U / .0/ : : : r .U;1/ .¹/ : : : r .U;U / .¹/ : : : 7 7 6 q q q q 6 :: :: :: :: :: :: :: :: 7 6 : : 7 : : : : : : 7 6 (10.79) TD6 7 6 rq .1;1/ .¹/ : : : rq .1;U / .¹/ : : : rq .1;1/ .0/ : : : rq .1;U / .0/ : : : 7 7 6 6 rq .2;1/ .¹/ : : : rq .2;U / .¹/ : : : rq .2;1/ .0/ : : : rq .2;U / .0/ : : : 7 7 6 :: :: :: :: :: 7 :: :: :: 6 7 6 : : : : : : : : 7 6 6 rq .U;1/ .¹/ : : : rq .U;U / .¹/ : : : rq .U;1/ .0/ : : : rq .U;U / .0/ : : : 7 5 4 :: :: :: :: :: :: :: :: : : : : : : : : Let w be the vector of noise samples at the MF output. It is important to verify that its covariance matrix is N0 T. Then the matrix T is Hermitian and, assuming that it is definite positive, the Cholesky decomposition (2.174) can be applied T D LH L
(10.80)
where L H is a lower triangular matrix with positive real elements on the main diagonal. Using (10.76) and (10.79), we find that the vector y satisfies the linear relation y D Ta Cw
(10.81)
Once the expression (10.81) is obtained, the vector a can be detected by well-known techniques [20]. Applying the zero-forcing criterion, at the decision point we get the vector z D T1 y D a C T1 w
(10.82)
Equation (10.82) shows that the zero-forcing criterion completely eliminates both ISI and MUI, but it may enhance the noise. Applying instead the MSE criterion to the signal rC .t/ suitably sampled, leads to the solution (see (2.229)) z D .T C N0 I/1 y
(10.83)
10.7. Maximum likelihood multiuser detector
823
Both approaches require the inversion of a K U ð K U Hermitian matrix and therefore a large computational complexity. A scheme that is computationally efficient while maintaining comparable performance is described in [21]. A MMSE method with further reduced complexity operates on single output samples, that is for K D 1. However, the performance is lower because it does not exploit the correlation among the different observations. For the case K D 1, an alternative that yields performance near the optimum ML receiver is represented by a DFE structure (see Section 16.4).
10.7
Maximum likelihood multiuser detector
Correlation matrix approach Using the notation introduced in the previous section, the multiuser signal rC .t/ is expressed as rC .t/ D
U X
sC.u/ .t/ C wC .t/
(10.84)
aiT qT .t i T / C wC .t/
(10.85)
uD1
D
K 1 X i D0
The log-likelihood associated with (10.84) is [22] þ2 Z þþ U þ X þ þ .u/ sC .t/þ dt `C D þrC .t/ þ þ uD1
(10.86)
Defining the matrices Z Qk1 k2 D
qŁT .t k1 T / qTT .t k2 T / dt
(10.87)
after several steps, (10.86) can be written as `C D
K 1 X
MC .k/
(10.88)
kD0
where the branch metric is given by ( " MC .k/ D Re akH Q0 ak C
¹ X
#) 2Qm akm 2yk
(10.89)
mD1
having assumed that Qm D 0
jmj > ¹
(10.90)
We note that the first two terms within the brackets in (10.89) can be computed off-line. The sequence fOak g that maximizes (10.88) can be obtained using the Viterbi algorithm; the complexity of this scheme is, however, exceedingly large, because it requires O.42U ¹ / branch metric computations per detected symbol, assuming QPSK modulation.
824
Chapter 10. Spread spectrum systems
Whitening filter approach We now derive an alternative formulation of the ML multiuser detector; for this reason it is convenient to express the MF output using the D transform [22]. Defining ¹ X
Q.D/ D
Qk D k
(10.91)
kD¹
the MF output can be written as y.D/ D Q.D/ a.D/ C w.D/
(10.92)
where w.D/ is the noisy term with matrix spectral density N0 Q.D/. Assuming that it does not have poles on the unit circle, Q.D/ can be factorized in the form Q.D/ D F H .D 1 / F.D/
(10.93)
where F.D/ is minimum phase; in particular, F.D/ has the form F.D/ D
¹ X
Fk D k
(10.94)
kD0
where F0 is a lower triangular matrix. Now let .D/ D [F H .D 1 /]1 , an anticausal filter by construction. Applying .D/ to y.D/ in (10.92), we obtain z.D/ D .D/ y.D/ D F.D/ a.D/ C w0 .D/
(10.95)
where the noisy term w0 .D/ is a white Gaussian process. Consequently, in the time domain (10.95) becomes zk D
¹ X
Fm akn C w0k
(10.96)
mD0
With reference to [23], the expression (10.96) is an extension to the multidimensional case of Forney’s MLSD approach. In fact, the log-likelihood can be expressed as the sum of branch metrics defined as 2 ¹ X M E .k/ D zk Fm akm mD0 (10.97) þ þ2 U þ U þ X X þ .u/ .i / þ D .F0.u;i / ak.i / C Ð Ð Ð C F¹.u;i / ak¹ /þ þz k þ þ uD1 i D1 We note that, as F0 is a lower triangular matrix, the metric has a causal dependence also with regard to the ordering of the users. For further study on multiuser detection techniques we refer the reader to [24, 25, 26].
10. Bibliography
825
Bibliography [1] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread spectrum communications handbook. New York: McGraw-Hill, 1994. [2] R. C. Dixon, Spread spectrum systems. New York: John Wiley & Sons, 3rd ed., 1994. [3] L. B. Milstein and M. K. Simon, “Spread spectrum communications”, in The Mobile Communications Handbook (J. D. Gibson, ed.), ch. 11, pp. 152–165, New York: CRC/IEEE Press, 1996. [4] J. G. Proakis, Digital communications. New York: McGraw-Hill, 3rd ed., 1995. [5] R. Price and P. E. Green, “A communication technique for multipath channels”, IRE Proceedings, vol. 46, pp. 555–570, Mar. 1958. [6] A. J. Viterbi, CDMA: Principles of spread-spectrum communication. Reading, MA: Addison-Wesley, 1995. [7] R. L. Peterson, R. E. Ziemer, and D. E. Borth, Introduction to spread spectrum communications. Englewood Cliffs, NJ: Prentice-Hall, 1995. [8] “Wideband CDMA”, IEEE Communications Magazine, vol. 36, pp. 46–95, Sept. 1998. [9] G. Cherubini and L. B. Milstein, “Performance analysis of both hybrid and frequency– hopped phase–coherent spread–spectrum system. Part I and Part II”, IEEE Trans. on Communications, vol. 37, pp. 600–622, June 1989. [10] A. Klein, “Data detection algorithms specially designed for the downlink of CDMA mobile radio systems”, in Proc. 1997 IEEE Vehicular Technology Conference, Phoenix, USA, pp. 203–207, May 4–7 1997. [11] K. Li and H. Liu, “A new blind receiver for downlink DS-CDMA communications”, IEEE Communications Letters, vol. 3, pp. 193–195, July 1999. [12] S. Werner and J. Lilleberg, “Downlink channel decorrelation in CDMA systems with long codes”, in Proc. 1999 IEEE Vehicular Technology Conference, Houston, USA, pp. 1614–1617, May 16–20 1999. [13] K. Hooli, M. Latva-aho, and M. Juntti, “Multiple access interference suppression with linear chip equalizers in WCDMA downlink receivers”, in Proc. 1999 IEEE Global Telecommunications Conference, Rio de Janeiro, Brazil, pp. 467–471, Dec. 5–9 1999. [14] U. Madhow and M. L. Honig, “MMSE interference suppression for direct-sequence spread-spectrum CDMA”, IEEE Trans. on Communications, vol. 42, pp. 3178–3188, Dec. 1994.
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Chapter 10. Spread spectrum systems
[15] S. L. Miller, “An adaptive direct-sequence code-division multiple-access receiver for multiuser interference rejection”, IEEE Trans. on Communications, vol. 43, pp. 1746– 1755, Feb./Mar./Apr. 1995. [16] P. B. Rapajic and B. S. Vucetic, “Adaptive receiver structures for asynchronous CDMA systems”, IEEE Journal on Selected Areas in Communications, vol. 12, pp. 685–697, May 1994. [17] A. Klein and P. W. Baier, “Linear unbiased data estimation in mobile radio systems applying CDMA”, IEEE Journal on Selected Areas in Communications, vol. 11, pp. 1058–1066, Sept. 1993. [18] J. Blanz, A. Klein, M. Naßhan, and A. Steil, “Performance of a cellular hybrid C/TDMA mobile radio system applying joint detection and coherent receiver antenna diversity”, IEEE Journal on Selected Areas in Communications, vol. 12, pp. 568–579, May 1994. [19] G. K. Kaleh, “Channel equalization for block transmission systems”, IEEE Journal on Selected Areas in Communications, vol. 13, pp. 110–120, Jan. 1995. [20] A. Klein, G. K. Kaleh, and P. W. Baier, “Zero forcing and minimum mean-squareerror equalization for multiuser detection in code-division multiple-access channels”, IEEE Trans. on Vehicular Technology, vol. 45, pp. 276–287, May 1996. [21] N. Benvenuto and G. Sostrato, “Joint detection with low computational complexity for hybrid TD-CDMA systems”, IEEE Journal on Selected Areas in Communications, vol. 19, pp. 245–253, Jan. 2001. [22] G. E. Bottomley and S. Chennakeshu, “Unification of MLSE receivers and extension to time-varying channels”, IEEE Trans. on Communications, vol. 46, pp. 464–472, Apr. 1998. [23] A. Duel-Hallen, “A family of multiuser decision feedback detectors for asynchronous code-division multiple access channels”, IEEE Trans. on Communications, vol. 43, pp. 421–434, Feb./Mar./Apr. 1995. [24] S. Verd`u, Multiuser detection. Cambridge: Cambridge University Press, 1998. [25] “Multiuser detection techniques with application to wired and wireless communications systems I”, IEEE Journal on Selected Areas in Communications, vol. 19, Aug. 2001. [26] “Multiuser detection techniques with application to wired and wireless communications systems II”, IEEE Journal on Selected Areas in Communications, vol. 20, Feb. 2002.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 11
Channel codes
Forward error correction (FEC) is a widely used technique to achieve reliable data transmission. The redundancy introduced by an encoder for the transmission of data in coded form allows the decoder at the receiver to detect and partially correct errors. An alternative transmission technique, known as automatic repeat query or request (ARQ), consists in detecting the errors (usually by a check-sum transmitted with the data, see page 875) and requesting the retransmission of a data packet whenever it is received with errors. The FEC technique presents two advantages with respect to the ARQ technique. 1. In systems that make use of the ARQ technique the data packets do not necessarily have to be retransmitted until they are received without errors; however, for large values of the error probability, the aggregate traffic of the link is higher. 2. In systems that make use of the FEC technique the receiver does not have to request the retransmission of data packets, thus making possible the use of a simplex link (see Section 6.13); this feature represents a strong point in many applications like TDMA and video satellite links, where a central transmitter broadcasts to receiveonly terminals, which are unable to make a possible retransmission request. The FEC technique is also particularly useful in various satellite communication applications, in which the long round-trip delay of the link would cause serious traffic problems whenever the ARQ technique would be used. We distinguish two broad classes of FEC techniques, each with numerous subclasses, employing block codes or convolutional codes. All error correction techniques add redundancy, in the form of additional bits, to the information bits that must be transmitted. Redundancy makes the correction of errors possible and for the classes of codes considered in this chapter represents the coding overhead. The effectiveness of a coding technique is expressed in terms of the coding gain, G code , given by the difference between the signal-to-noise ratios, in dB, that are required to achieve a certain bit error probability for transmission without and with coding (see Definition 6.2 on page 508). The overhead is expressed in terms of the code rate, Rc , given by the ratio between the number of information bits and the number of code bits that are transmitted. The transmission bit rate is inversely proportional to Rc , and is larger than that necessary for uncoded data. If one of the modulation techniques of Chapter 6 is employed, the modulation rate is also larger. In Chapter 12 methods to
828
Chapter 11. Channel codes
transmit coded sequences of symbols without an increase in the modulation rate will be discussed. For further study on the topic of error correcting codes we refer to [1, 2, 3].
11.1
System model
With reference to the model of a transmission system with coding, illustrated in Figure 6.20, we introduce some fundamental parameters. A block code is composed of a set of vectors of given length called code words; the length of a code word is defined as the number of vector elements, indicated by n 0 . The elements of a code word are chosen from an alphabet of q elements: if the alphabet consists of two elements, for example 0 and 1, the code is a binary code and we refer to the elements of each code word as bits; if, on the other hand, the elements of a code word are chosen from an alphabet having q elements .q > 2/, the code is nonbinary. It is interesting to note that if q is a power of two, that is q D 2b , where b is a positive integer, each q-ary element has an equivalent binary representation of b bits and therefore a nonbinary code word of length N can be mapped to a binary code word of length n 0 D bN . There are 2n 0 possible code words in a binary code of length n 0 . From these 2n 0 possible code words, we choose 2k0 words (k0 < n 0 ) to form a code. Thus a block of k0 information bits is mapped to a code word of length n 0 chosen from a set of 2k0 code words; the resulting block code is indicated as .n 0 ; k0 / code and the ratio Rc D k0 =n 0 is the code rate.1 Observation 11.1 The code rate Rc is related to the encoder-modulator rate R I (6.93) by the following relation RI D
log M k0 log2 M D Rc 2 n0 I I
(11.1)
where M is the number of symbols of the I -dimensional constellation adopted by the bit-mapper. Because the number of bits per unit of time produced by the encoder is larger than that produced by the source, two transmission strategies are possible. Transmission for a given bit rate of the information message. With reference to Figure 6.20, from the relation k0 Tb D n 0 Tcod
(11.2)
1 1 1 D Tcod Rc Tb
(11.3)
we obtain
1
In this chapter a block code will sometimes be indicated also by the notation .n; k/.
11.1. System model
829
note that the bit rate at the modulator input is increased in the presence of the encoder. For a given modulator with M symbols, that is using the same bit mapper, this implies an increase of the modulation rate given by 1 1 1 1 D log2 M D 0 T Tcod T Rc
(11.4)
and therefore an increase of the bandwidth of the transmission channel by a factor 1=Rc . Moreover, for the same transmitted power, from (6.105) in the presence of the encoder the signal-to-noise ratio becomes (11.5) 0 0 D 0 Rc i.e. it decreases by a factor Rc with respect to the case of transmission of an uncoded message. Therefore, for a given information message bit rate 1=Tb , the system operates with a lower 0 0 : consequently the receiver is prone to introduce more bit errors at the decoder input. In spite of this, for a suitable choice of the code, in many cases the decoder produces a detected message fbO` g affected by fewer errors with respect to the case of transmission of an uncoded message. We note that the energy per information bit of the encoded message fcm g is equal to that of fb` g. In fact, for a given bit mapper, L 0b D
k0 log2 M D Rc L b n0
(11.6)
Assuming the same transmitted power, from (6.97) and (11.4) we get 0 D Rc E s E sCh Ch
(11.7)
Therefore (6.99) yields for the encoded message fcm g an energy per bit of information equal to 0 E sCh
L 0b
D
E sCh D Eb Lb
(11.8)
Since 0 0 6D 0, a comparison between the performance of the two systems, with and without coding, is made for the same E b =N0 . In this case the coding gain, in dB, is given by 10 .log10 0 log10 0 0 C log10 Rc /
(11.9)
Transmission for a given modulation rate. For given transmitted power and given transmission channel bandwidth, 0 remains unchanged in the presence of the encoder. Therefore, there are three possibilities. 1. The bit rate of the information message decreases by a factor Rc and becomes 1 1 D Rc 0 Tb Tb
(11.10)
830
Chapter 11. Channel codes
2. The source emits information bits in packets and each packet is followed by additional bits generated by the encoder, forming a code word; the resulting bits are transmitted at the rate 1 1 D (11.11) Tcod Tb 3. A block of m information bits is mapped to a transmitted symbol using a constellation with cardinality M > 2m . In this case transmission occurs without decreasing the bit rate of the information message. In the first two cases, for the same number of bits of the information message we have an increase in the duration of the transmission by a factor 1=Rc . For a given bit error probability in the sequence fbO` g, we expect that in the presence of coding a smaller 0 is required to achieve a certain error probability as compared to the case of transmission of an uncoded message; this reduction corresponds to the coding gain (see Definition 6.2 on page 508).
11.2
Block codes
We give the following general definition.2 Definition 11.1 The Hamming distance between two vectors v1 and v2 , d H .v1 ; v2 /, is given by the number of elements in which the two vectors differ.
11.2.1
Theory of binary codes with group structure
Properties A binary block code of length n is a subset containing Mc of the 2n possible binary sequences of length n, also called code words. The only requirement on the code words is that they are all of the same length. Definition 11.2 The minimum Hamming distance of a block code, to which we will refer in this chapter H and coincides with the smallest number simply as the minimum distance, is denoted by dmin of positions in which any two code words differ. H D 2 is given by (11.22). An example of a block code with n D 4; Mc D 4 and dmin For the binary symmetric channel model (6.90), assuming that the binary code word c of length n is transmitted, we observe at the receiver3
zDcýe
2
(11.12)
The material presented in Sections 11.2 and 11.3 is largely based on lectures given at the University of California, San Diego, by Professor Jack K. Wolf [4], whom the authors gratefully acknowledge. 3 In Figure 6.20, z is indicated as cQ .
11.2. Block codes
831
where ý denotes the modulo 2 sum of respective vector components; for example .0111/ ý .0010/ D .0101/. In (11.12), e is the binary error vector whose generic component is equal to 1 if the channel has introduced an error in the corresponding bit of c, and 0 otherwise. We note that z can assume all the 2n possible combinations of n bits. With reference to Figure 6.20, the function of the decoder consists in associating with each possible value z a code word. A commonly adopted criterion is to associate z with the code word cO that is closest according to the Hamming distance. From this code word the k0 information bits, which form the sequence fbOl g, are recovered by inverse mapping. Interpreting the code words as points in an n-dimensional space where the distance between points is given by the Hamming distance, we obtain the following properties. H can correct all patterns of 1. A binary block code with minimum distance dmin
tD
jdH 1k min 2
(11.13)
or fewer errors, where bxc denotes the integer value of x. H can detect all patterns of .d H 1/ 2. A binary block code with minimum distance dmin min or fewer errors.
3. In a binary erasure channel, the transmitted binary symbols are detected using a ternary alphabet f0; 1; erasureg; a symbol is detected as erasure if the reliability of a binary decision is low. In the absence of errors, a binary block code with minimum H can fill in .d H 1/ erasures. distance dmin min H we find that, for fixed n and odd d H , M 4. Seeking a relation among n, Mc , and dmin c min 4 is upper bounded by 9 8 = k j < n n n Mc MU B D 2n (11.15) 1C C C Ð Ð Ð C jdH 1k min ; : 1 2 2 H , it is always possible to find a code with M Ł words where 5. For fixed n and dmin c
¦³ ¾ ² n n n 1C C C ÐÐÐ C McŁ D 2n H 1 1 2 dmin
(11.16)
where dxe denotes the smallest integer greater than or equal to x. We will now consider a procedure for finding such a code.
4
We recall that the number of binary sequences of length n with m ‘ones’ is equal to n n! D m!.n m/! m where n! D n.n 1/ Ð Ð Ð 1.
(11.14)
832
Chapter 11. Channel codes
Step 1: choose any code word of length n and exclude from future choices that word H 1/ or fewer positions. The total number and all words that differ from it in .dmin of words excluded from future choices is n n n H (11.17) Nc .n; dmin 1/ D 1 C C C ÐÐÐ C H 1 1 2 dmin Step i: choose a word not previously excluded and exclude from future choices all words previously excluded plus the chosen word and those that differ from it in H 1/ or fewer positions. .dmin Continue this procedure until there are no more words available to choose from. At H 1/ additional words are excluded; each step, if still not excluded, at most Nc .n; dmin H 1/ therefore after step i, when i code words have been chosen, at most i Nc .n; dmin H n words have been excluded. Then, if 2 =Nc .n; dmin 1/ is an integer, we can choose at least that number of code words; if it is not an integer, we can choose at least a number of code words equal to the next largest integer. Definition 11.3 A binary code with group structure is a binary block code for which the following conditions are verified: 1. the all zero word is a code word (zero code word); 2. the modulo 2 sum of any two code words is also a code word. Definition 11.4 The weight of any binary vector x, denoted as w.x/, is the number of ones in the vector. H is given by Property 1 of a group code. The minimum distance of the code dmin H D min w.c/ dmin
(11.18)
where c can be any non-zero code word. Proof. The sum of any two distinct code words is a non-zero word. The weight of the resulting word is equal to the number of positions in which the two original words differ. H positions, there is a word of Because two words at the minimum distance differ in dmin H . If there were a non-zero word of weight less than d H , it would be different weight dmin min H positions. from the zero word in less than dmin Property 2 of a group code. If all code words in a group code are written as rows of an Mc ð n matrix, then every column is either zero or consists of half zeros and half ones. Proof. An all zero column is possible if all code words have a zero in that column. Suppose in column i there are m 1s and .Mc m/ 0s. Choose one of the words with a 1 in that column and add it to all words that have a 1 in that column, including the word itself: this
11.2. Block codes
833
operation produces m words with a 0 in that column, hence .Mc m/ ½ m. Now we add that word to each word that has a 0 in that column: this produces .Mc m/ words with a 1 in that column, hence .Mc m/ m. Therefore Mc m D m or m D Mc =2. Corollary 11.1 From Property 2 it turns out that the number of code words Mc must be even for a binary group code. Corollary 11.2 Excluding codes of no interest from the transmission point of view, for which all code words have a 0 in a given position, from Property 2 the average weight of a code word is equal to n=2.
Parity check matrix Let H be a binary r ð n matrix, which is called parity check matrix, of the form H D [A B]
(11.19)
where B is an r ð r matrix with det[B] 6D 0, i.e. the columns of B are linearly independent. A binary parity check code is a code consisting of all binary vectors c that are solutions of the equation Hc D 0 (11.20) The matrix product in (11.20) is computed using the modulo 2 arithmetic. Example 11.2.1 Let the matrix H be given by " HD
1 0 1 1 0 1 0 1
# (11.21)
There are four code words in the binary parity check code corresponding to the matrix H; they are 2 3 2 3 2 3 2 3 0 1 0 1 607 607 617 617 7 7 7 7 c1 D 6 c2 D 6 c3 D 6 (11.22) c0 D 6 405 415 415 405 0 0 1 1 Property 1 of a parity check code. A parity check code is a group code. Proof. The all zero word is always a code word, as H0 D 0
(11.23)
834
Chapter 11. Channel codes
Suppose that c1 and c2 are code words; then Hc1 D 0 and Hc2 D 0. It follows that H.c1 ý c2 / D Hc1 ý Hc2 D 0 ý 0 D 0
(11.24)
Therefore c1 ý c2 is also a code word. Property 2 of a parity check code. The code words corresponding to the parity check matrix H D [A B] are identical to the code words corresponding to the parity check matrix Q D [B1 A; I] D [A Q I], where I is the r ð r identity matrix. H nr ½ c1 Proof. Let c D be a code word corresponding to the matrix H D [A B], where cnnr C1 are the first .n r/ components of the vector and cnnr C1 are the last r components cnr 1 of the vector. Then ý Bcnnr C1 D 0 (11.25) Hc D Acnr 1 Multiplying by B1 we get B1 Acnr ý Icnnr C1 D 0 1
(11.26)
Q D 0. or Hc Q D [A Q I] are not less From Property 2 we see that parity check matrices of the form H general than parity check matrices of the form H D [A B], where det[B] 6D 0. In general, we can consider any r ð n matrix as a parity check matrix, provided that some set of r columns has a non-zero determinant. If we are not concerned with the order by which the elements of a code word are transmitted, then such a code would be equivalent to a code formed by a parity check matrix of the form H D [A I]
(11.27)
The form of the matrix (11.27) is called canonical or systematic form. We assume that the last r columns of H have a non-zero determinant and therefore that the parity check matrix can be expressed in canonical form. Property 3 of a parity check code. There are exactly 2nr D 2k code words in a parity check code. Proof. Referring to the proof of Property 2, we find that cnnr C1 D Acnr 1
(11.28)
For each of the 2nr D 2k possible binary vectors cnr it is possible to compute the 1 corresponding vector cnnr C1 . Each of these code words is unique as all of them differ in the first .n r/ D k positions. Assume that there are more than 2k code words; then at least two will agree in the first .n r/ D k positions. But from (11.28) we find that these two code words also agree in the last r positions and therefore they are identical.
11.2. Block codes
835
The code words have the following structure c D [m 0 : : : m k1 ; p0 : : : pr 1 ]T
(11.29)
where the first k D .n r/ bits are called information bits and the last r bits are called parity check bits. As mentioned in Section 11.1, a parity check code that has code words of length n that are obtained by encoding k information bits is an .n; k/ code. Property 4 of a parity check code. A code word of weight w exists if and only if the modulo 2 sum of w columns of H equals 0. Proof. c is a code word if and only if Hc D 0. Let hi be the i-th column of H and let c j be the j-th component of c. Therefore, if c is a code word, then n X
hjcj D 0
(11.30)
jD1
If c is a code word of weight w, then there are exactly w non-zero components of c, for example c j1 ; c j2 ; : : : ; c jw . Consequently h j1 ýh j2 ýÐ Ð Ðýh jw D 0, thus a code word of weight w implies that the sum of w columns of H equals 0. Conversely, if h j1 ýh j2 ýÐ Ð Ðýh jw D 0 then Hc D 0, where c is a binary vector with elements equal to 1 in positions j1 ; j2 ; : : : ; jw . From Property 1 of a group code and also from Properties 1 and 4 of a parity check code we obtain the following property. H if Property 5 of a parity check code. A parity check code has minimum distance dmin H some modulo 2 sum of dmin columns of H is equal to 0, but no modulo 2 sum of fewer H columns of H is equal to 0. than dmin Property 5 may be considered as the fundamental property of parity check codes, as it forms the basis for the design of almost all such codes. An important exception is constituted by low-density parity check codes, which will be discussed in Section 11.7. A limit on the H derives number of parity check bits required for a given block length n and given dmin directly from this property.
Property 6 of a parity check code. A binary parity check code exists of block length n H , having no more than r Ł parity check bits, where and minimum distance dmin 1 0 H dmin 2 j X n 1 k A C1 (11.31) r Ł D log2 @ i i D0 Proof. The proof derives from the following exhaustive construction procedure of the parity check matrix of the code. Step 1: choose as the first column of H any non-zero vector with r Ł components. Step 2: choose as the second column of H any non-zero vector different from the first.
836
Chapter 11. Channel codes
Step 3: choose as the i-th column of H any vector distinct from all vectors obtained by H 2/ or fewer previously chosen columns. modulo 2 sum of .dmin H 1/ or fewer Clearly such a procedure will result in a matrix H where no set of .dmin columns of H sum to 0. However, we must show that we can indeed continue this process for n columns. After applying this procedure for .n 1/ columns, there will be at most
Nc .n
H 1; dmin
n1 n1 n1 2/ D 1 C C C ÐÐÐ C H 2 2 dmin 1
(11.32) Ł
distinct vectors that are forbidden for the choice of the last column, but there are 2r vectors Ł H 2/. Thus n to choose from; observing (11.31) and (11.32) we get 2r > Nc .n 1; dmin H columns can always be chosen where no set of .dmin 1/ or fewer columns sums to zero. H . From Property 5, the code therefore has minimum distance at least dmin
Code generator matrix Using (11.28), we can write " cD
cnr 1 cnnr C1
#
" D
I A
# D GT cnr cnr 1 1
(11.33)
where G D [I AT ] is a k ð n binary matrix, and I is the k ð k identity matrix. Taking the transpose of (11.33), we obtain T (11.34) cT D .cnr 1 / G thus the code words, considered now as row vectors, are given as all linear combinations of the rows of G, which is called the generator matrix of the code. A parity check code can be specified by giving its parity check matrix H or its generator matrix G. Example 11.2.2 Consider the parity check code (7,4) with the parity check matrix 2
3 1 1 0 1 1 0 0 H D 4 1 1 1 0 0 1 0 5 D [A I] 1 0 1 1 0 0 1
(11.35)
Expressing a general code word according to (11.29), to every 4 information bits 3 parity check bits are added, related to the information bits by the equations (see (11.28)) p0 D m 0 ý m 1 ý m 3 p1 D m 0 ý m 1 ý m 2 p2 D m 0 ý m 2 ý m 3
(11.36)
11.2. Block codes
837
The generator matrix of this code is given by 2
1 60 GD6 40 0
0 1 0 0
0 0 1 0
0 0 0 1
1 1 0 1
1 1 1 0
3 1 07 7 D [I AT ] 15 1
(11.37)
There are 16 code words consisting of all linear combinations of the rows of G. By inspection, we find that the minimum weight of a non-zero code word is 3; hence, from (11.18) H D 3 and therefore is a single error correcting code. the code has dmin
Decoding of binary parity check codes Conceptually the simplest method for decoding a block code is to compare the received block of n bits with each code word and choose that code word that differs from the received word in the minimum number of positions; in case several code words satisfy this condition, choose amongst them at random. Although the simplest conceptually, the described method is out of the question practically because we usually employ codes with very many code words. It is, however, instructive to consider the application of this method, suitably modified, to decode group codes.
Cosets The 2n possible binary sequences of length n are partitioned into 2r sets, called cosets, by a group code with 2k D 2nr code words; this partitioning is done as follows: Step 1: choose the first set as the set of code words c1 ; c2 ; : : : ; c2k . Step 2: choose any vector, say, η2 , that is not a code word; then choose the second set as c 1 ý η 2 ; c 2 ý η 2 ; : : : ; c 2k ý η 2 . Step i: choose any vector, say, ηi , not included in any previous set; choose the i-th set, i.e. coset, as c1 ý ηi ; c2 ý ηi ; : : : ; c2k ý ηi . The partitioning continues until all 2n vectors are used. Note that each coset contains 2k vectors; if we show that no vector can appear in more than one coset, we will have demonstrated that there are 2r D 2nk cosets. Property 1 of cosets.
Every binary vector of length n appears in one and only one coset.
Proof. Every vector appears in at least one coset as the partitioning stops only when all vectors are used. Suppose that a vector appeared twice in one coset; then for some value of the index i we have c j1 ý ηi D c j2 ý ηi , or c j1 D c j2 , that is a contradiction as all code words are unique. Suppose that a vector appears in two cosets; then c j1 ý ηi1 D c j2 ý ηi2 , where we assume i 2 > i 1 . Then ηi2 D c j1 ý c j2 ý ηi1 D c j3 ý ηi1 , that is a contradiction as ηi2 would have appeared in a previous coset, against the hypothesis.
838
Chapter 11. Channel codes
Example 11.2.3 Consider partitioning the 24 binary vectors of length 4 into cosets using the group code with code words 0000, 0011, 1100, 1111, as follows:
η2 η3 η4
0 D 0 D 0 D 1
0 0 1 0
0 0 1 1
0 1 1 0
0 0 0 1
0 0 1 0
1 1 0 0
1 0 0 1
1 1 1 0
1 1 0 1
0 0 1 1
0 1 1 0
1 1 1 0
1 1 0 1
1 1 0 0
1 0 0 1
(11.38)
The vectors η1 D 0; η2 ; η3 ; : : : ; η2r , are called coset leaders; the partitioning (11.38) is called coset table or decoding table. Property 2 of cosets. Suppose that instead of choosing ηi as the coset leader of the i-th coset, we choose another element of that coset; the new coset formed by using this new coset leader contains exactly the same vectors as the old coset. Proof. Assume that the new coset leader is ηi ý c j1 , and that z is an element of the new coset; then z D ηi ý c j1 ý c j2 D ηi ý c j3 , so z is an element of the old coset. As the new and the old cosets both contain 2k vectors and all vectors in a coset are unique, every element of the new coset belongs to the old coset and vice versa. Example 11.2.4 Suppose that in the previous example we had chosen the third coset leader as 0100; then the table (11.38) would be 00 η2 D 0 0 η3 D 0 1 η4 D 1 0
0 0 0 1
0 1 0 0
0 0 0 1
0 0 1 0
1 1 1 0
1 0 1 1
1 1 1 0
1 1 0 1
0 0 0 1
0 1 0 0
1 1 1 0
1 1 0 1
1 1 1 0
1 0 1 1
(11.39)
Two conceptually simple decoding methods Assume that each coset leader is chosen as the minimum weight vector in its coset; in case several vectors in a coset have the same minimum weight, choose any one of them as the coset leader. Then a second method of decoding, using the decoding table, is as follows: Step 1: locate the received vector in the coset table. Step 2: choose the code word that appears as the first vector in the column containing the received vector. Proposition 11.1 Decoding using the decoding table decodes to the closest code word to the received word; in case several code words are at the same smallest distance from the received word, it decodes to one of these closest words.
11.2. Block codes
839
Proof. Assume that the received word is the j-th vector in the i-th coset. The received word, given by z D c j ý ηi , is corrected to the code word c j and the distance between the received word and the j-th code word is w.ηi /. Suppose that another code word, say, ck , is closer to the received vector: then w.ck ý c j ý ηi / < w.ηi /
(11.40)
w.c` ý ηi / < w.ηi /
(11.41)
or but this cannot be as w.ηi / is assumed to be the minimum weight vector in its coset and c` ý ηi is in that coset. We note that the coset leaders determine the only error patterns that can be corrected by the code. Coset leaders, moreover, have many other interesting properties: for example, H , all binary n-tuple of weight less than or equal to if a code has minimum distance dmin j H k dmin 1 are coset leaders. 2 Definition 11.5 A code for which coset leaders are all vectors of weight t or less, and no others, is called a perfect t-error correcting code. A code for which coset leaders are all vectors of weight t or less, and some vectors of weight t C 1 but not all, and no others, is called quasi-perfect t-error correcting code. The perfect binary codes are: 1. codes given by the repetition of n bits, with n odd: these codes contain only two code words, 000 : : : 0 (all zeros) and 111 : : : 1 (all ones), and correct t D .n 1/=2 H D n/; errors .dmin H D 3/ and have n D 2r 1, 2. Hamming codes: these codes correct t D 1 errors .dmin k D n r, r > 1; the columns of the matrix H are given by all non-zero vectors of length r; H D 7/, n D 23, k D 12, r D 11. 3. Golay code: t D 3 .dmin
The following modification of the decoding method dealt with in this section will be useful later on: Step 10 : locate the received vector in the coset table and identify the coset leader of the coset containing that vector. Step 20 : add the coset leader to the received vector to find the decoded code word.
Syndrome decoding A third method of decoding is based on the concept of syndrome. Among the methods described in this section, syndrome decoding is the only method of practical value for a code with a large number of code words.
840
Chapter 11. Channel codes
Definition 11.6 For any parity check matrix H, we define the syndrome s.z/ of a binary vector z of length n as s.z/ D Hz (11.42) We note that the syndrome is a vector of length r, whereas z is a vector of length n. Therefore many vectors will have the same syndrome. All code words have an all zero syndrome and these are the only vectors with this property. This property of the code words is a special case of the following: Property 3 of cosets. All vectors in the same coset have the same syndrome; vectors in different cosets have distinct syndromes. Proof. Assume that z1 and z2 are in the same coset, say, the i-th: then z1 D ηi ý c j1 and z2 D ηi ý c j2 . Moreover s.z1 / D Hz1 D H.ηi ý c j1 / D Hηi ý Hc j1 D Hηi ý 0 D s.ηi /. Similarly s.z2 / D s.ηi /, so s.z1 / D s.z2 / D s.ηi /: this proves the first part of the property. Now assume that z1 and z2 are in different cosets, say, the i 1 -th and i 2 -th: then z1 D ηi1 ý c j1 and z2 D ηi2 ý c j2 , so s.z1 / D s.ηi1 / and s.z2 / D s.ηi2 /. If s.z1 / D s.z2 / then s.ηi1 / D s.ηi2 /, which implies Hηi1 D Hηi2 . Consequently H.ηi1 ý ηi2 / D 0, or ηi1 ý ηi2 is a code word, say c j3 . Then ηi2 D ηi1 ý c j3 , which implies that ηi1 and ηi2 are in the same coset, that is a contradiction. Thus the assumption that s.z1 / D s.z2 / is incorrect. From Property 3 we see that there is a one-to-one relation between cosets and syndromes; this leads to the third method of decoding, which proceeds as follows: Step 100 : compute the syndrome of the received vector; this syndrome identifies the coset in which the received vector is. Identify then the leader of that coset. Step 200 : add the coset leader to the received vector to find the decoded code word. Example 11.2.5 Consider the parity check matrix 2
3 1 1 0 1 0 0 HD41 0 1 0 1 05 1 1 1 0 0 1
(11.43)
The coset leaders and their respective syndromes obtained using (11.42) are reported in Table 11.1. Suppose that the vector z D 000111 is received. To decode we 2 first 3 compute the syndrome 1 607 2 3 6 7 1 607 7 Hz D 4 1 5, then by Table 11.1 we identify the coset leader as 6 607, and obtain the decoded 6 7 1 405 0 code word . 1 0 0 1 1 1 /.
11.2. Block codes
841
Table 11.1 Coset leaders and respective syndromes for Example 11.2.5.
Coset leader
Syndrome
000000 000001 000010 000100 001000 010000 100000 100001
000 001 010 100 011 101 111 110
The advantage of syndrome decoding over the other decoding methods previously described is that there is no need to memorize the entire decoding table at the receiver. The first part of Step 100 , namely computing the syndrome, is trivial. The second part of Step 100 , namely identifying the coset leader corresponding to that syndrome, is the difficult part of the procedure; in general it requires a RAM with 2r memory locations, addressed by the syndrome of r bits and containing the coset leaders of n bits. Overall the memory bits are n2r . There is also an algebraic method to identify the coset leader. In fact, this problem is equivalent to finding the minimum set of columns of the parity check matrix which sum to the syndrome. In other words, we must find the vector z of minimum weight such that Hz D s. For a single error correcting Hamming code, all coset leaders are of weight 1 or 0, so a non-zero syndrome corresponds to a single column of H and the correspondence between syndrome and coset leader is simple. For a code with coset leaders of weight 0, 1, or 2, the syndrome is either 0, a single column of H, or the sum of two columns, etc. For a particular class of codes that will be considered later, the structure of the construction of H will allow identification of the coset leader starting from the syndrome by using algebraic procedures. In general, each class of codes leads to a different technique to perform this task. Property 7 of parity check codes. There are exactly 2r correctable error vectors for a parity check code with r parity check bits. Proof. Correctable error vectors are given by the coset leaders and there are 2nk D 2r of them, all of which are distinct. On the other hand, there are 2r distinct syndromes and each corresponds to a correctable error vector. For a binary symmetric channel (see Definition 6.1) we should correct all error vectors of weight i, i D 0; 1; 2; : : : , until we exhaust the capability of the code. Specifically, we should try to use a perfect code or a quasi-perfect code. For a quasi-perfect t-error correcting code, the coset leaders consist of all error vectors of weight i D 0; 1; 2; : : : ; t, and some vectors of weight t C 1. Nonbinary parity check codes are discussed in Appendix 11.A.
842
11.2.2
Chapter 11. Channel codes
Fundamentals of algebra
The calculation of parity check bits from information bits involves solving linear equations. This procedure is particularly easy for binary codes since we use modulo-2 arithmetic. An obvious question is whether or not the concepts of the previous section generalize to codes with symbols taken from alphabets with a larger cardinality, say, alphabets with q symbols. We will see that the answer can be yes or no according to the value of q; furthermore, even if the answer is yes we might not be able to use modulo q arithmetic. Consider the equation for the unknown x ax D b
(11.44)
where a and b are known coefficients, and all values are from the finite alphabet f0; 1; 2, : : : ; q 1g. First, we need to introduce the concept of multiplication, which is normally given in the form of a multiplication table, as the one given in Table 11.2 for the three elements f0; 1; 2g. Table 11.2 allows us to solve (11.44) for any values of a and b, except a D 0. For example, the solution to equation 2x D 1 is x D 2, as from the multiplication table we find 2 Ð 2 D 1. Let us now consider the case of an alphabet with four elements. A multiplication table for the four elements f0; 1; 2; 3g, resulting from the modulo 4 arithmetic, is given in Table 11.3. Note that the equation 2x D 2 has two solutions, x D 1 and x D 3, and equation 2x D 1 has no solution. It is possible to construct a multiplication table that allows the equation (11.44) to be solved uniquely for x, provided that a 6D 0, as shown in Table 11.4. Table 11.2 Multiplication table for 3 elements (modulo 3 arithmetic).
Ð
0
1
2
0 1 2
0 0 0
0 1 2
0 2 1
Table 11.3 Multiplication table for an alphabet with four elements (modulo 4 arithmetic).
Ð
0
1
2
3
0 1 2 3
0 0 0 0
0 1 2 3
0 2 0 2
0 3 2 1
11.2. Block codes
843
Table 11.4 Multiplication table for an alphabet with four elements.
Ð
0
1
2
3
0 1 2 3
0 0 0 0
0 1 2 3
0 2 3 1
0 3 1 2
Note that Table 11.4 is not obtained using modulo 4 arithmetic. For example, 2x D 3 has the solution x D 2, and 2x D 1 has the solution x D 3.
Modulo q arithmetic Consider the elements f0; 1; 2; : : : ; q 1g, where q is a positive integer larger than or equal to 2. We define two operations for combining pairs of elements from this set. The first, denoted by ý, is called modulo q addition and is defined as ( aCb if 0 a C b < q c Daýb D (11.45) aCbq if a C b ½ q Here a C b is the ordinary addition operation for integers that may produce an integer not in the set. In this case q is subtracted from a C b and a C b q is always an element in the set f0; 1; 2; : : : ; q 1g. The second operation, denoted by , is called modulo q multiplication and is defined as 8 > if 0 ab < q < ab j ab k (11.46) d Dab D > q if ab ½ q : ab q j k Note that ab ab q q, is the remainder or residue of the division of ab by q, and is always an integer in the set f0; 1; 2; : : : ; q 1g. Often we will omit the notation and write a b simply as ab. We recall that special names are given to sets which possess certain properties with respect to operations. Consider the general set G that contains the elements fÞ; þ; ; Ž; : : : g, and two operations for combining elements from the set. We denote the first operation 4 (addition), and the second operation ♦ (multiplication). Often we will omit the notation ♦ and write a♦b simply as ab. The properties we are interested in are: 1. Existence of additive identity. For every Þ 2 G, there exists an element ; 2 G, called additive identity, such that Þ4; D ;4Þ D Þ. 2. Existence of additive inverse. For every Þ 2 G, there exists an element þ 2 G, called additive inverse of Þ, and indicated by Þ, such that Þ4þ D þ4Þ D ;.
844
Chapter 11. Channel codes
3. Additive closure. For every Þ; þ 2 G, not necessarily distinct, Þ4þ 2 G. 4. Additive associative law. For every Þ; þ; 2 G, Þ4.þ4 / D .Þ4þ/4 . 5. Additive commutative law. For every Þ; þ 2 G, Þ4þ D þ4Þ. 6. Multiplicative closure. For every Þ; þ 2 G, not necessarily distinct, Þ♦þ 2 G. 7. Multiplicative associative law. For every Þ; þ; 2 G, Þ♦.þ♦ / D .Þ♦þ/♦ . 8. Distributive law. For every Þ; þ; 2 G, Þ♦.þ4 / D .Þ♦þ/4.Þ♦ / and .Þ4þ/♦ D .Þ♦ /4.þ♦ /. 9. Multiplicative commutative law. For every Þ; þ 2 G, Þ♦þ D þ♦Þ. 10. Existence of multiplicative identity. For every Þ 2 G, there exists an element I 2 G, called multiplicative identity, such that Þ♦I D I ♦Þ D Þ. 11. Existence of multiplicative inverse. For every Þ 2 G, except the element ;, there exists an element Ž 2 G, called multiplicative inverse of Þ, and indicated with Þ 1 , such that Þ♦Ž D Ž♦Þ D I . Any set G for which Properties 14 hold is called a group with respect to 4. If G has a finite number of elements, then G is called finite group and the number of elements of G is called the order of G. Any set G for which Properties 15 hold is called an Abelian group with respect to 4. Any set G for which Properties 18 hold is called a ring with respect to the operations 4 and ♦. Any set G for which Properties 19 hold is called a commutative ring with respect to the operations 4 and ♦. Any set G for which Properties 110 hold is called a commutative ring with identity. Any set G for which Properties 111 hold is called a field. It can be seen that the set f0; 1; 2; : : : ; q 1g is a commutative ring with identity with respect to the operations of addition ý defined in (11.45) and multiplication defined in (11.46). We will show by the next three properties that this set satisfies also Property 11 if and only if q is a prime: in other words, we will show that the set f0; 1; 2; : : : ; q 1g is a field with respect to the modulo q addition and modulo q multiplication if and only if q is a prime. Finite fields are called Galois fields; a field of q elements is usually denoted as G F.q/. Property 11a of modulo q arithmetic. If q is not a prime, each factor of q (less than q and greater than 1) does not have a multiplicative inverse. Proof. Let q D ab, where 1 < a; b < q; then, observing (11.46), a b D 0. Assume that a has a multiplicative inverse a 1 ; then a 1 .a b/ D a 1 0 D 0. Now, from a 1 .a b/ D 0 it is 1 b D 0; this implies b D 0, which is a contradiction as b > 1. Similarly we show that b does not have a multiplicative inverse.
11.2. Block codes
845
Property 11b of modulo q arithmetic. If q is a prime and a b D 0, then a D 0, or b D 0, or a D b D 0. Proof. Assume a b D 0 and a; b > 0; then ab D K q, where K < min.a; b/. If 1 < a q 1 and a has no factors in common with q, then it must divide K ; but this is impossible as K < min.a; b/. The only other possibility is that a D 1, but then a b 6D 0 as ab < q. Property 11c of modulo q arithmetic. If q is a prime, all non-zero elements of the set f0; 1; 2; : : : ; q 1g have multiplicative inverse. Proof. Assume the converse, that is the element j, with 1 j q 1, does not have a multiplicative inverse; then there must be two distinct elements a; b 2 f0; 1; 2; : : : ; q 1g such that a j D b j. This is a consequence of the fact that the product i j can only assume values in the set f0; 2; 3; : : : ; q 1g, as by assumption i j 6D 1; then .a j/ ý .q .b j// D 0
(11.47)
On the other hand, q .b j/ D .q b/ j, and .a ý .q b// j D 0
(11.48)
But j 6D 0 and consequently, by Property 11b, we have a ý .q b/ D 0. This implies a D b, which is a contradiction. Definition 11.7 An ideal I is a subset of elements of a ring R such that: 1. I is a subgroup of the additive group R, that is the elements of I form a group with respect to the addition defined in R; 2. for any element a of I and any element r of R, ar and ra are in I .
Polynomials with coefficients from a field We consider the set of polynomials in one variable; as this set is interesting in two distinct applications, to avoid confusion we will use a different notation for the two cases. The first application permits to extend our knowledge of finite fields. We have seen in Section 11.2.2 how to construct a field with a prime number of elements. Polynomials allow us to construct fields in which the number of elements is given by a power of a prime; for this purpose we will use polynomials in the variable y. The second application introduces an alternative method to describe code words. We will consider cyclic codes, a subclass of parity check codes, and in this context will use polynomials in the single variable x. Consider any two polynomials with coefficients from the set f0; 1; 2; : : : ; p 1g, where p is a prime: g.y/ D g0 C g1 y C g2 y 2 C Ð Ð Ð C gm y m f .y/ D f 0 C f 1 y C f 2 y 2 C Ð Ð Ð C f n y n
(11.49)
846
Chapter 11. Channel codes
We assume that gm 6D 0 and f n 6D 0. We define m as degree of the polynomial g.y/, and we write m D deg.g.y//; in particular, if g.y/ D a, a 2 f0; 1; 2; : : : ; p 1g, we say that deg.g.y// D 0. Similarly, it is n D deg. f .y//. If gm D 1, we say that g.y/ is a monic polynomial. Assume m n: then the addition among polynomials is defined as f .y/ C g.y/ D . f 0 ý g0 / C . f 1 ý g1 /y C . f 2 ý g2 /y 2 C Ð Ð Ð C . f m ý gm / y m C Ð Ð Ð C f n y n (11.50) Example 11.2.6 Let p D 5, f .y/ D 1C3y C2y 4 , and g.y/ D 4C3y C3y 2 ; then f .y/Cg.y/ D y C3y 2 C2y 4 . Note that deg. f .y/ C g.y// max.deg. f .y//; deg.g.y/// Multiplication among polynomials is defined as usual f .y/ g.y/ D d0 C d1 y C Ð Ð Ð C dmCn y mCn
(11.51)
where the arithmetic to perform operations with the various coefficients is modulo p, di D . f 0 gi / ý . f 1 gi 1 / ý Ð Ð Ð ý . f i 1 g1 / ý . f i g0 /
(11.52)
Example 11.2.7 Let p D 2, f .y/ D 1 C y C y 3 , and g.y/ D 1 C y 2 C y 3 ; then f .y/ g.y/ D 1 C y C y 2 C y3 C y4 C y5 C y6. Note that deg. f .y/ g.y// D deg. f .y// C deg.g.y// Definition 11.8 If f .y/ g.y/ D d.y/, we say that f .y/ divides d.y/, and g.y/ divides d.y/. We say that p.y/ is an irreducible polynomial if and only if, assuming another polynomial a.y/ divides p.y/, then a.y/ D a 2 f0; 1; : : : ; p 1g or a.y/ D k p.y/, with k 2 f0; 1; : : : ; p 1g. The concept of an irreducible polynomial plays the same role in the theory of polynomials as does the concept of a prime number in the number theory.
The concept of modulo in the arithmetic of polynomials We define a modulo arithmetic for polynomials, analogously to the modulo q arithmetic for integers. We choose a polynomial q.y/ D q0 C q1 y C Ð Ð Ð C qm y m with coefficients that are elements of the field f0; 1; 2; : : : ; p 1g. We consider the set P of all polynomials of degree less than m with coefficients from the field f0; 1; 2; : : : ; p 1g; this set consists of pm polynomials.
11.2. Block codes
847
Example 11.2.8 Let p D 2 and q.y/ D 1 C y C y 3 ; then the set P consists of 23 polynomials, f0; 1; y; y C 1; y 2 ; y 2 C 1; y 2 C y; y 2 C y C 1g. Example 11.2.9 Let p D 3 and q.y/ D 2y 2 ; then the set P consists of 32 polynomials, f0; 1; 2; y; y C 1; y+2; 2y; 2y C 1; 2y C 2g. We now define two operations among polynomials of the set P, namely modulo q.y/ addition, denoted by 4, and modulo q.y/ multiplication, denoted by ♦. Modulo q.y/ addition is defined for every pair of polynomials a.y/ and b.y/ from the set P as a.y/4b.y/ D a.y/ C b.y/
(11.53)
where a.y/ C b.y/ is defined in (11.50). The definition of modulo q.y/ multiplication requires the knowledge of the Euclidean division algorithm. Euclidean division algorithm. For every pair of polynomials Þ.y/ and þ.y/ with coefficients from some field, and deg.þ.y// ½ deg.Þ.y// > 0, there exists a unique pair of polynomials q.y/ and r.y/ such that þ.y/ D q.y/ Þ.y/ C r.y/
(11.54)
where 0 deg.r.y// < deg.Þ.y//; polynomials q.y/ and r.y/ are called, respectively, quotient polynomial and remainder or residue polynomial. In a notation analogous to that used for integers we can write j þ.y/ k (11.55) q.y/ D Þ.y/ and r.y/ D þ.y/
j þ.y/ k Þ.y/
Þ.y/
(11.56)
Example 11.2.10 Let p D 2, þ.y/ D y 4 C 1, and Þ.y/ D y 3 C y C 1; then y 4 C 1 D y.y 3 C y C 1/ C y 2 C y C 1, so q.y/ D y and r.y/ D y 2 C y C 1. We define modulo q.y/ multiplication, denoted by ♦, for polynomials a.y/ and b.y/ in the set P as 8 > if deg.a.y/ b.y// < deg.q.y// < a.y/ b.y/ k j a.y/♦b.y/ D a.y/ b.y/ > q.y/ otherwise : a.y/ b.y/ q.y/ (11.57)
848
Chapter 11. Channel codes
It is easier to think of (11.57) as a typical multiplication operation for polynomials whose coefficients are given according to (11.52). If in this multiplication there are terms of degree greater than or equal to deg.q.y//, then we use the relation q.y/ D 0 to lower the degree. Example 11.2.11 Let p D 2 and q.y/ D 1 C y C y 3 ; then .y 2 C 1/♦.y C 1/ D y 3 C y 2 C y C 1 D .1 y/ C y 2 C y C 1 D .1 C y/ C y 2 C y C 1 D y 2 . It can be shown that the set of polynomials with coefficients from some field and degree less than deg.q.y// is a commutative ring with identity with respect to the operations modulo q.y/ addition and modulo q.y/ multiplication. We now find under what conditions this set of polynomials and operations forms a field. Property 11a of modular polynomial arithmetic. If q.y/ is not irreducible, then the factors of q.y/, of degree greater than zero and less than deg.q.y//, do not have multiplicative inverses. Proof. Let q.y/ D a.y/ b.y/, where 0 < deg.a.y//; deg.b.y// < deg.q.y//; then a.y/♦b.y/ D 0. Assume a.y/ has a multiplicative inverse, a 1 .y/; then, from a 1 .y/ ♦ .a.y/♦b.y// D a 1 .y/♦0 D 0 it is .a 1 .y/♦a.y//♦b.y/ D 0, then 1♦b.y/ D 0, or b.y/ D 0. The last equation is a contradiction as by assumption deg.b.y// > 0. Similarly, we show that b.y/ does not have a multiplicative inverse. We give without proof the following properties. Property 11b of modular polynomial arithmetic. If q.y/ is irreducible and a.y/♦b.y/ D 0, then a.y/ D 0, or b.y/ D 0, or a.y/ D b.y/ D 0. Property 11c of modular polynomial arithmetic. If q.y/ is irreducible, all non-zero elements of the set of polynomials P of degree less than deg.q.y// have multiplicative inverses. We now have that the set of polynomials with coefficients from some field and degree less than deg.q.y// forms a field, with respect to the operations of modulo q.y/ addition and modulo q.y/ multiplication, if and only if q.y/ is irreducible. Furthermore it can be shown that there exists at least one irreducible polynomial of degree m, for every m ½ 1, with coefficients from a generic field f0; 1; 2; : : : ; p 1g. We now have a method of generating a field with pm elements. Example 11.2.12 Let p D 2 and q.y/ D y 2 C y C 1; we have that q.y/ is irreducible. Consider the set P with elements f0; 1; y; y C 1g. The addition and multiplication tables for these elements modulo y 2 C y C 1 are given in Table 11.5 and Table 11.6, respectively.
11.2. Block codes
849
Table 11.5 Modulo y2 C y C 1 addition table for p D 2.
4
0
1
y
yC1
0 1 y yC1
0 1 y yC1
1 0 yC1 y
y yC1 0 1
yC1 y 1 0
Table 11.6 Modulo y2 C y C 1 multiplication table for p D 2.
♦
0
1
y
yC1
0 1 y yC1
0 0 0 0
0 1 y yC1
0 y yC1 1
0 yC1 1 y
Devices to sum and multiply elements in a finite field For the G F. p m / obtained by an irreducible polynomial of degree m, m X
q.y/ D
qi y i
qi 2 G F. p/
(11.58)
ai y i
ai 2 G F. p/
(11.59)
bi y i
bi 2 G F. p/
(11.60)
i D0
let a.y/ and b.y/ be two elements of P: m1 X
a.y/ D
i D0
and m1 X
b.y/ D
i D0
The device to perform the addition (11.53), s.y/ D
m1 X
si y i D .a.y/ C b.y// mod q.y/
(11.61)
i D0
is illustrated in Figure 11.1. The implementation of a device to perform the multiplication is slightly more complicated, as illustrated in Figure 11.2, where Tc is the period of the clock applied to the shift-register (SR) with m elements, and all operations are modulo p. Let us define d.y/ D
m1 X i D0
di y i D .a.y/ b.y// mod q.y/
(11.62)
850
Chapter 11. Channel codes
am−1
b0
a0
bm−1
mod p
mod p s m−1
s0
Figure 11.1. Device for the sum of two elements .a0 ; : : : ; am1 / and .b0 ; : : : ; bm1 / of GF.pm /.
Figure 11.2. Device for the multiplication of two elements .a0 ; : : : ; am1 / and .b0 ; : : : ; bm1 / of GF.pm /. Tc is the clock period, and ACC denotes an accumulator. All additions and multiplications are modulo p.
The device is based on the following decomposition a.y/ b.y/ mod q.y/ D
m1 X
ai y i b.y/ mod q.y/
i D0
D a0 b.y/ C a1 .y b.y// mod q.y/ :: : C am1 .y m1 b.y// mod q.y/
(11.63)
11.2. Block codes
851
where additions and multiplications are modulo p. Now, using the identity 0 mod q.y/, note that the following relation holds:
Pm
i D0
qi y i D
y b.y/ D b0 y C b1 y 2 C Ð Ð Ð C bm2 y m1 C bm1 y m D .bm1 qm1 q0 / C .b0 bm1 qm1 q1 / y C Ð Ð Ð C .bm2 bm1 qm1 qm1 / y m1 : (11.64) The term .y i b.y// mod q.y/ is thus obtained by initializing the SR of Figure 11.2 to the sequence .b0 ; : : : ; bm1 /, and by applying i clock pulses; the desired result is then contained in the shift register. Observing (11.63), we find that it is necessary to multiply each element of the SR by ai and accumulate the result; after multiplications by all coefficients fai g have been performed, the final result is given by the content of the accumulators. Note that in the binary case, for p D 2, the operations of addition and multiplication are carried out by XOR and AND functions, respectively.
Remarks on finite fields 1. We have seen how to obtain finite fields with p ( p a prime) elements, given by f0; 1; : : : ; p 1g, or p m elements, using the Property 11c. These fields are also known as Galois fields and are usually denoted by G F. p/ or G F. p m /. It can be shown that there are no other fields with a finite number of elements. Moreover, all fields with the same number of elements are identical, that is all finite fields are generated by the procedures discussed in the previous sections. 2. The field, from which the coefficients of the irreducible polynomial are chosen, is called the ground field ; the field generated using the arithmetic of polynomials is called the extension field. 3. Every row of the addition table contains each field element once and only once; the same is true for the columns. 4. Every row of the multiplication table, except the row corresponding to the element 0, contains each field element once and only once; the same is true for the columns. 5. If we multiply any non-zero element by itself we get a non-zero element of the field (perhaps itself). As there are only .q 1/ non-zero elements, we must eventually find a situation for j > i such that an element Þ multiplied by itself j times will equal Þ multiplied by itself i times, that is j times i times Þ Þ Þ ÐÐÐ Þ D Þ Þ ÐÐÐ Þ D þ
(11.65)
j i times i times j times Þ Þ ÐÐÐ Þ Þ Þ ÐÐÐ Þ D Þ Þ ÐÐÐ Þ
(11.66)
We observe that
852
Chapter 11. Channel codes
Substituting (11.65) in (11.66), and observing that þ has a multiplicative inverse, we can multiply from the right by this inverse to obtain Þ ji
j i times D Þ Þ ÐÐÐ Þ D 1
(11.67)
Definition 11.9 For every non-zero field element, Þ, the order of Þ is the smallest integer ` such that Þ ` D 1. Example 11.2.13 Consider the field with elements f0; 1; 2; 3; 4g, and modulo 5 arithmetic. Then element 1 2 3 4
order 1 4 4 2
(11.68)
Example 11.2.14 Consider the field G F.22 / with 4 elements, f0; 1; y; y C 1g, and addition and multiplication modulo y 2 C y C 1. Then element 1 y yC1
order 1 3 3
(11.69)
6. An element from the field G F.q/ is said to be primitive if it has order q 1. For fields generated by arithmetic modulo a polynomial q.y/, if the field element y is primitive we say that q.y/ is a primitive irreducible polynomial. A property of finite fields that we give without proof is that every finite field has at least one primitive element; we note that once a primitive element has been identified, every other non-zero field element can be obtained by multiplying the primitive element by itself an appropriate number of times. A list of primitive polynomials for the ground field G F.2/ is given in Table 11.7. Example 11.2.15 For the field G F.4/ generated by the polynomial arithmetic modulo q.y/ D y 2 C y C 1, for the ground field G F.2/, y is a primitive element (see (11.69)); thus y 2 C y C 1 is a primitive polynomial. 7. The order of every non-zero element of G F.q/ must divide .q 1/.
11.2. Block codes
853
Table 11.7 List of primitive polynomials q.y/ of degree m for the ground field GF.2/.
m 2 3 4 5 6 7 8 9 10 11 12 13
m 1 C y C y2 1 C y C y3 1 C y C y4 1 C y2 C y5 1 C y C y6 1 C y3 C y7 1 C y2 C y3 C y4 C y8 1 C y4 C y9 1 C y 3 C y 10 1 C y 2 C y 11 1 C y C y 4 C y 6 C y 12 1 C y C y 3 C y 4 C y 13
14 15 16 17 18 19 20 21 22 23 24
1 C y C y 6 C y 10 C y 14 1 C y C y 15 1 C y C y 3 C y 12 C y 16 1 C y 3 C y 17 1 C y 7 C y 18 1 C y C y 2 C y 5 C y 19 1 C y 3 C y 20 1 C y 2 C y 21 1 C y C y 22 1 C y 5 C y 23 1 C y C y 2 C y 7 C y 24
Proof. Every non-zero element þ can be written as the power of a primitive element Þ p ; this implies that there is some i < .q 1/ such that i times þ D Þ p Þ p Ð Ð Ð Þ p D Þ ip
(11.70) q1
j
Note that from the definition of a primitive element we get Þ p D 1, but Þ p 6D 1 for j < .q 1/; furthermore there exists an integer ` such that þ ` D Þ ip` D 1. Consequently .`/.i/ is a multiple of .q 1/ and it is exactly the smallest multiple of i that is a multiple of .q 1/, thus .i/.`/ D l:c:m:.i; q 1/, i.e. the least common multiple of i and .q 1/. We recall that ab (11.71) l:c:m:.a; b/ D g:c:d:.a; b/ where g:c:d:.a; b/ is the greatest common divisor of a and b. Thus .i/.`/ D and
.i/.q 1/ g:c:d:.i; q 1/
q 1 D` g:c:d:.i; q 1/
(11.72)
(11.73)
Example 11.2.16 Let Þ p be a primitive element of G F.16/; from (11.73) the orders of the non-zero field elements are:
854
Chapter 11. Channel codes
field element
g:c:d:.i; q 1/
Þp
1
order of field element q 1 g.c.d..i; q 1/ 15
Þ 2p Þ 3p Þ 4p Þ 5p Þ 6p Þ 7p Þ 8p Þ 9p Þ 10 p 11 Þp Þ 12 p 13 Þp Þ 14 p 15 Þp
1
15
3
5
1
15
5
3
3
5
1
15
1
15
3
5
5
3
1
15
3
5
1
15
1
15
15
1
þ D Þi
(11.74)
8. A ground field can itself be generated as an extension field. For example G F.16/ can be generated by taking an irreducible polynomial of degree 4 with coefficients from G F.2/, which we would call G F.24 /, or by taking an irreducible polynomial of degree 2 with coefficients from G F.4/, which we would call G F.42 /. In either case we would have the same field, except for the names of the elements. Example 11.2.17 Consider the field G F.23 / generated by the primitive polynomial q.y/ D 1 C y C y 3 , with ground field G F.2/. As q.y/ is a primitive polynomial, each element of G F.23 /, except the zero element, can be expressed as a power of y. Recalling the polynomial representation P, we may attach to each polynomial a vector representation, with m components on G F. p/ given by the coefficients of the powers of the variable y. The three representations are reported in Table 11.8.
Roots of a polynomial Consider a polynomial of degree m with coefficients that are elements of some field. We will use the variable x, as the polynomials are now considered for a purpose that is not that of generating a finite field. In fact, the field of the coefficients may itself have a polynomial representation.
11.2. Block codes
855
Table 11.8 Three equivalent representations of the elements of GF.23 /.
Exponential
Polynomial
0 1 y y2 y3 y4 y5 y6
0 1 y y2 1Cy y C y2 1 C y C y2 1 C y2
Binary (y 0 y 1 y 2 ) 0 1 0 0 1 0 1 1
0 0 1 0 1 1 1 0
0 0 0 1 0 1 1 1
Consider, for example, a polynomial in x with coefficients from G F.4/. We immediately see that it is not worth using the notation f0; 1; y; yC1g to identify the 4 elements of G F.4/, as the notation f0; 1; Þ; þg would be much simpler. For example, a polynomial of degree three with coefficients from G F.4/ is given by f .x/ D Þx 3 C þx 2 C 1. Given any polynomial f .x/, we say that is a root of the equation f .x/ D 0 or, more simply, that it is a root of f .x/, if and only if f . / D 0. The definition is more complicated than it appears, as we must know the meaning of the two members of the equation f . / D 0. For example, we recall that the fundamental theorem of algebra states that every polynomial of degree m has exactly m roots, not necessarily distinct. If we take the polynomial f .x/ D x 2 C x C 1 with coefficients from f0; 1g, what are its roots? As f .0/ D f .1/ D 1, we have that neither 0 nor 1 are roots. Before proceeding, we recall a similar situation that we encounter in ordinary algebra. The polynomial x 2 C 3, with coefficients in the field of real numbers, has two roots in the field of complex numbers; however, no roots exist in the field of real numbers; therefore the polynomial does not have factors whose coefficients are real numbers. Thus we would say that the polynomial is irreducible, yet even the irreducible polynomial has complex-valued roots and can be factorized. This situation is due to the fact that, if we have a polynomial f .x/ with coefficients from some field, the roots of the polynomial are either from that field or from an extension field of that field. For example, take the polynomial f .x/ D x 2 C x C 1 with coefficients from G F.2/, and consider the extension field G F.4/ with elements f0; 1; Þ; þg that obey the addition and the multiplication rules given in Table 11.9 and Table 11.10, respectively. Then f .Þ/ D f .x/jxDÞ D Þ 2 4Þ41 D .þ4Þ/41 D 141 D 0, thus Þ is a root. Similarly we find f .þ/ D f .x/jxDþ D þ 2 4þ41 D .Þ4þ/41 D 141 D 0, thus the two roots of f .x/ are Þ and þ. We can factor f .x/ into two factors, each of which is a polynomial in x with coefficients from G F.4/. For this purpose we consider .x4 Þ/♦.x4 þ/ D .x4Þ/♦.x4þ/; leaving out the notations 4 and ♦ for C and ð we get .x4Þ/♦.x4þ/ D x 2 C .Þ C þ/ x C Þþ D x 2 C x C 1
(11.75)
856
Chapter 11. Channel codes
Table 11.9 Addition table for the elements of GF.4/.
4
0
1
Þ
þ
0 1 Þ þ
0 1 Þ þ
1 0 þ Þ
Þ þ 0 1
þ Þ 1 0
Table 11.10 Multiplication table for the elements of GF.4/.
♦
0
1
Þ
þ
0 1 Þ þ
0 0 0 0
0 1 Þ þ
0 Þ þ 1
0 þ 1 Þ
Thus if we use the operations defined in G F.4/, .x CÞ/ and .x Cþ/ are factors of x 2 Cx C1; it remains that x 2 C x C 1 is irreducible as it has no factors with coefficients from G F.2/. Property 1 of the roots of a polynomial. .x C . //, is a factor of f .x/.
If is a root of f .x/ D 0, then .x /, that is
Proof. Using the Euclidean division algorithm, we divide f .x/ by .x / to get f .x/ D Q.x/ .x / C r.x/
(11.76)
where deg.r.x// < deg.x / D 1. Therefore
But f . / D 0, so
f .x/ D Q.x/ .x / C r0
(11.77)
f . / D 0 D Q. / . / C r0 D r0
(11.78)
f .x/ D Q.x/ .x /
(11.79)
therefore
Property 2 of the roots of a polynomial. If f .x/ is an arbitrary polynomial with coefficients from G F. p/, p a prime, and þ is a root of f .x/, then þ p is also a root of f .x/. Proof. We consider the polynomial f .x/ D f 0 C f 1 xC f 2 x 2 CÐ Ð ÐC f m x m , where f i 2 G F. p/, and form the power . f .x// p . It results p
p
p
. f .x// p D . f 0 C f 1 x C f 2 x 2 C Ð Ð Ð C f m x m / p D f 0 C f 1 x p C Ð Ð Ð C f m x mp
(11.80)
11.2. Block codes
857
as the cross-terms contain a factor p, which is the same as 0 in G F. p/. On the other hand, p for f i 6D 0, f i D f i , as from Property 7 on page 853 the order of any non-zero element divides p 1; the equation is true also if f i is the zero element. Therefore . f .x// p D f .x p /
(11.81)
If þ is a root of f .x/ D 0, then f .þ/ D 0, and f p .þ/ D 0. But f p .þ/ D f .þ p /, so that f .þ p / D 0; therefore þ p is also a root of f .x/. A more general form of the property just introduced, that we will give without proof, is expressed by the following property. Property 2a of the roots of a polynomial. If f .x/ is an arbitrary polynomial having coefficients from G F.q/, with q a prime or a power of a prime, and þ is a root of f .x/ D 0, then þ q is also a root of f .x/ D 0, Example 11.2.18 Consider the polynomial x 2 C x C 1 with coefficients from G F.2/. We already have seen that Þ, element of G F.4/, is a root of x 2 C x C 1 D 0. Therefore Þ 2 is also a root; but Þ 2 D þ, so þ is a second root. The polynomial has degree two, thus it has two roots and they are Þ and þ, as previously seen. Note also that þ 2 is also a root, but þ 2 D Þ.
Minimum function Definition 11.10 Let þ be an element of an extension field of G F.q/; the minimum function of þ, m þ .x/, is the monic polynomial of least degree with coefficients from G F.q/ such that m þ .x/jxDþ D 0. We now list some properties of the minimum function. 1. The minimum function is unique. Proof. Assume there were two minimum functions, of the same degree and monic, m þ .x/ and m 0þ .x/. Form the new polynomial .m þ .x/ m 0þ .x// whose degree is less than the degree of m þ .x/ and m 0þ .x/; but .m þ .x/ m 0þ .x//jxDþ D 0, so we have a new polynomial, whose degree is less than that of the minimum function, that admits þ as root. Multiplying by a constant we can thus find a monic polynomial with this property, but this cannot be since the minimum function is the monic polynomial of least degree for which þ is a root. 2. The minimum function is irreducible. Proof. Assume the converse were true, that is m þ .x/ D a.x/ b.x/; then m þ .x/jxDþ D a.þ/ b.þ/ D 0. Then either a.þ/ D 0 or b.þ/ D 0, so that þ is a root of a polynomial of degree less than the degree of m þ .x/. By making this polynomial monic we arrive at a contradiction. 3. Let f .x/ be any polynomial with coefficients from G F.q/, and let f .x/jxDþ D 0; then f .x/ is divisible by m þ .x/.
858
Chapter 11. Channel codes
Proof. Use the Euclidean division algorithm to yield f .x/ D Q.x/ m þ .x/ C r.x/
(11.82)
where deg.r.x// < deg.m þ .x//. Then we have that f .þ/ D Q.þ/ m þ .þ/ C r.þ/
(11.83)
but as f .þ/ D 0 and m þ .þ/ D 0, then r.þ/ D 0. As deg.r.x// < deg.m þ .x//, the only possibility is r.x/ D 0; thus f .x/ D Q.x/ m þ .x/. 4. Let f .x/ be any irreducible monic polynomial with coefficients from G F.q/ for which f .þ/ D 0, where þ is an element of some extension field of G F.q/; then f .x/ D m þ .x/. Proof. From Property 3 f .x/ must be divisible by m þ .x/, but f .x/ is irreducible, so it is only trivially divisible by m þ .x/, that is f .x/ D K m þ .x/: but f .x/ and m þ .x/ are both monic polynomials, therefore K D 1. We now introduce some interesting propositions. 1. Let þ be an element of G F.q m /, with q prime; then the polynomial F.x/, defined as F.x/ D
m1 Y
i
2
.x þ q / D .x þ/ .x þ q / .x þ q / Ð Ð Ð .x þ q
m1
/
(11.84)
i D0
has all its coefficients from G F.q/. Proof. Observing Property 7 on page 853, we have that the order of þ divides q m 1, m therefore þ q D þ. Thus we can express F.x/ as F.x/ D
m Y
i
.x þ q /
(11.85)
i D1
therefore F.x q / D
m Y i D1
i
.x q þ q / D
m Y
.x þ q
i1
/q D
i D1
j
.x þ q /q D .F.x//q
(11.86)
jD0
Consider now the expression F.x/ D F.x q / D
m1 Y
Pm
i D0
m X
f i x i ; then
fi x i
q
(11.87)
i D0
and .F.x//q D
m X i D0
!q fi x i
D
m X i D0
q
fi x i
q
(11.88)
11.2. Block codes
859
q
Equating like coefficients in (11.87) and (11.88) we get f i D f i ; hence f i is a root of the equation x q x D 0. But on the basis of Property 7 on page 853 the q elements from G F.q/ all satisfy the equation x q x D 0, and this equation only has q roots; therefore the coefficients f i are elements from G F.q/. 2. If g.x/ is an irreducible polynomial of degree m with coefficients from G F.q/, and 2 g.þ/ D 0, where þ is an element of some extension field of G F.q/, then þ; þ q ; þ q , : : : , m1 þq are all the roots of g.x/. Proof. At least one root of g.x/ is in G F.q m /; this follows by observing that, if we form G F.q m / using the arithmetic modulo g.y/, then y will be a root of g.x/ D 0. From Q qi Proposition 1, if þ is an element from G F.q m / then F.x/ D im1 D0 .x þ / has all coefficients from G F.q/; thus F.x/ has degree m, and F.þ/ D 0. As g.x/ is irreducible, we know that g.x/ D K m þ .x/; but as F.þ/ D 0, and F.x/ and g.x/ have the same degree, 2 m1 then F.x/ D K 1 m þ .x/, and therefore g.x/ D K 2 F.x/. As þ; þ q ; þ q ; : : : ; þ q , are all roots of F.x/, then they must also be all the roots of g.x/. 3. Let g.x/ be a polynomial with coefficients from G F.q/ which is also irreducible in this field. Moreover, let g.þ/ D 0, where þ is an element of some extension field of G F.q/; then the degree of g.x/ equals the smallest integer k such that k
þq D þ 2
(11.89) k1
Proof. We have that deg.g.x// ½ k as þ; þ q ; þ q ; : : : ; þ q , are all roots of g.x/ and by assumption are distinct. Assume that deg.g.x// > k; from Proposition 2, we know that þ must be at least a double root of g.x/ D 0, and therefore g 0 .x/ D .d=dx/g.x/ D 0 must also have þ as a root. As g.x/ is irreducible we have that g.x/ D K m þ .x/, but m þ .x/ must divide g 0 .x/; we get a contradiction because deg.g 0 .x// < deg.g.x//.
Methods to determine the minimum function 1. Direct calculation. Example 11.2.19 Consider the field G F.23 / obtained by taking the polynomial arithmetic modulo the irreducible polynomial y 3 C y C 1 with coefficients from G F.2/; the field elements are f0; 1; y; y C 1; y 2 ; y 2 C 1; y 2 C y; y 2 C y C 1g. Assume we want to find the minimum function of þ D .y C 1/. If .y C 1/ is a root, also .y C 1/2 D y 2 C 1 and .y C 1/4 D y 2 C y C 1 are roots. Note that .y C 1/8 D .y C 1/ D þ, thus the minimum function is m yC1 .x/ D .x þ/ .x þ 2 / .x þ 4 / D .x C .y C 1//.x C .y 2 C 1//.x C .y 2 C y C 1// D x3 C x2 C 1 2. Solution of the system of the coefficient equations.
(11.90)
860
Chapter 11. Channel codes
Example 11.2.20 Consider the field G F.23 / of the previous example; as .y C 1/, .y C 1/2 D y 2 C 1, .y C 1/4 D y 2 C y C 1, .y C 1/8 D y C 1, the minimum function has degree three; as the minimum function is monic and irreducible, we have m yC1 .x/ D m 3 x 3 C m 2 x 2 C m 1 x C m 0 D x 3 C m 2 x 2 C m 1 x C 1
(11.91)
As m yC1 .y C 1/ D 0, then .y C 1/3 C m 2 .y C 1/2 C m 1 .y C 1/ C 1 D 0
(11.92)
y 2 .1 C m 2 / C ym 1 C .m 2 C m 1 C 1/ D 0
(11.93)
that can be written as
As all coefficients of the powers of y must be zero, we get a system of equations in the unknown m 1 and m 2 , whose solution is given by m 1 D 0 and m 2 D 1. Substitution of this solution in (11.91) yields m yC1 .x/ D x 3 C x 2 C 1
(11.94)
3. Using the minimum function of the multiplicative inverse. Definition 11.11 The reciprocal polynomial of any polynomial m Þ .x/ D m 0 C m 1 x C m 2 x 2 C Ð Ð Ð C m K x K is defined by m Þ .x/ D m 0 x K C m 1 x K 1 C Ð Ð Ð C m K 1 x C m K . We use the following proposition that we give without proof. The minimum function of the multiplicative inverse of a given element is equal to the reciprocal of the minimum function of the given element. In formulae: let Þþ D 1, then m þ .x/ D m Þ .x/. Example 11.2.21 Consider the field G F.26 / obtained by taking the polynomial arithmetic modulo the irreducible polynomial y 6 C y C 1 with coefficients from G F.2/; the polynomial y 6 C y C 1 is primitive, thus from Property 7 on page 853 any non-zero field element can be written as a power of the primitive element y. From Proposition 2, we have that the minimum function of y is also the minimum function of y 2 ; y 4 ; y 8 ; y 16 ; y 32 , the minimum function of y 3 is also the minimum function of y 6 ; y 12 ; y 24 ; y 48 ; y 33 , and so forth. We list in Table 11.11 the powers of y that have the same minimum function. Given the minimum function of y 11 , m y 11 D x 6 C x 5 C x 3 C x 2 C 1, we want to find the minimum function of y 13 . From Table 11.11 we note that y 13 has the same minimum function as y 52 ; furthermore we note that y 52 is the multiplicative inverse of y 11 , as .y 11 /.y 52 / D y 63 D 1. Therefore the minimum function of y 13 is the reciprocal polynomial of m y 11 , given by m y 13 D x 6 C x 4 C x 3 C x C 1.
11.2. Block codes
861
Table 11.11 Powers of a primitive element in GF.26 / with the same minimum function.
1 3 5 7 9 11 13 15 21 23 27 31
2 6 10 14 18 22 26 30 42 46 54 62
4 12 20 28 36 44 52 60 29 45 61
8 24 40 56
16 48 17 49
32 33 34 35
25 41 57
50 19 51
37 38 39
58
53
43
59
55
47
Properties of the minimum function 1. Let þ be an element of order n in an extension field of G F.q/, and let m þ .x/ be the minimum function of þ with coefficients from G F.q/; then x n 1 D m þ .x/ b.x/, but x i 1 6D m þ .x/ b.x/ for i < n. Proof. We show that þ is a root of x n 1, as þ n 1 D 0, but from Property 3 of the minimum function (see page 858) we know that m þ .x/ divides any polynomial f .x/ such that f .þ/ D 0; this proves the first part. Assume that x i 1 D m þ .x/ b.x/ for some i < n: then x i 1jxDþ D m þ .x/ b.x/jxDþ D 0
(11.95)
so þ i 1 D 0 for i < n. But from Definition 11.9 of the order of þ (see page 852), n is the smallest integer such that þ n D 1, hence we get a contradiction. 2. Let þ1 ; þ2 ; : : : ; þ L be elements of some extension field of G F.q/, and let `1 ; `2 ; : : : ; ` L be the orders of these elements, respectively. Moreover, let m þ1 .x/, m þ2 .x/; : : : , m þ L .x/ be the minimum functions of these elements with coefficients from G F.q/, and let g.x/ be the smallest monic polynomial with coefficients from G F.q/ that has þ1 ; þ2 ; : : : ; þ L as roots: then a) g.x/ D l:c:m:.m þ1 .x/; m þ2 .x/; : : : ; m þ L .x//; b) if the minimum functions are all distinct, that is they do not have factor polynomials in common, then g.x/ D m þ1 .x/ m þ2 .x/ : : : m þ L .x/; c) if n D l:c:m:.`1 ; `2 ; : : : ; ` L /, then x n 1 D h.x/ g.x/, and x i 1 6D h.x/ g.x/ for i < n.
862
Chapter 11. Channel codes
Proof. a) Noting that g.x/ must be divisible by each of the minimum functions, it must be the smallest degree monic polynomial divisible by m þ1 .x/; m þ2 .x/; : : : , m þ L .x/, but this is just the definition of the least common multiple. b) If all the minimum functions are distinct, as each is irreducible, the least common multiple is given by the product of the polynomials. c) As n is a multiple of the order of each element, þ nj 1 D 0, for j D 1; 2; : : : ; L; then x n 1 must be divisible by m þ j .x/, for j D 1; 2; : : : ; L, and therefore it must be divisible by the least common multiple of these polynomials. Assume now that g.x/ divides x i 1 for i < n; then þ ij 1 D 0 for each j D 1; 2; : : : ; L, and thus i is a multiple of `1 ; `2 ; : : : ; ` L . But n is the smallest integer multiple of `1 ; `2 ; : : : ; ` L , hence we get a contradiction. We note that if the extension field is G F.q k / and L D q k 1 D n, then g.x/ D x n 1 and h.x/ D 1.
11.2.3
Cyclic codes
In Section 11.2.1 we dealt with the theory of binary group codes. We now discuss a special class of linear codes. These codes, called cyclic codes, are based upon polynomial algebra and lead to particularly efficient implementations for encoding and decoding.
The algebra of cyclic codes We consider polynomials with coefficients from some field G F.q/; in particular we consider the polynomial x n 1, and assume it can be factorized as x n 1 D g.x/ h.x/
(11.96)
Many such factorizations are possible for a given polynomial x n 1; we will consider any one of them. We denote the degrees of g.x/ and h.x/ as r and k, respectively; thus n D k C r. The choice of the symbols n, k and r is intentional, as they assume the same meaning as in the previous sections. The polynomial arithmetic modulo q.x/ D x n 1 is particularly important in the discussion of cyclic codes. Proposition 11.2 Consider the set of all polynomials of the form c.x/ D a.x/ g.x/ modulo q.x/, as a.x/ ranges over all polynomials of all degrees with coefficients from G F.q/. This set must be finite as there are at most q n remainder polynomials that can be obtained by dividing a polynomial by x n 1. Now we show that there are exactly q k distinct polynomials. Proof. There are at least q k distinct polynomials a.x/ of degree less than or equal to k 1, and each such polynomial leads to a distinct polynomial a.x/ g.x/. In fact, as the degree
11.2. Block codes
863
of a.x/ g.x/ is less than r C k D n, no reduction modulo x n 1 is necessary for these polynomials. Now let a.x/ be a polynomial of degree greater than or equal to k. To reduce the polynomial a.x/ g.x/ modulo x n 1, we divide by x n 1 and keep the remainder; thus a.x/ g.x/ D Q.x/ .x n 1/ C r.x/
(11.97)
where 0 deg.r.x// < n. By using (11.96), we can express r.x/ as r.x/ D .a.x/ h.x/ Q.x//g.x/ D a 0 .x/ g.x/
(11.98)
As r.x/ is of degree less than n, a 0 .x/ is of degree less than k, but we have already considered all polynomials of this form; therefore r.x/ is one of the q k polynomials determined in the first part of the proof. Example 11.2.22 Let g.x/ D x C 1, G F.q/ D G F.2/, and n D 4; then all polynomials a.x/ g.x/ modulo x 4 1 D x 4 C 1 are given by a.x/ 0 1 x x C1 x2 2 x C1 x2 C x x2 C x C 1
a.x/ g.x/ mod .x 4 1/ code word 0 x C1 x2 C x x2 C 1 x3 C x2 3 x C x2 C x C 1 x3 C x x3 C 1
0000 1100 0110 1010 0011 1111 0101 1001
(11.99)
We associate with any polynomial of degree less than n and coefficients from G F.q/ a vector of length n with components equal to the coefficients of the polynomial, that is f .x/ D f 0 C f 1 x C f 2 x 2 C Ð Ð Ð C f n1 x n1
! f D . f 0 ; f 1 ; f 2 ; : : : ; f n1 /
(11.100)
Note that in the definition f n1 does not need to be non-zero. We can now define cyclic codes. The code words will be the vectors associated with a set of polynomials; alternatively, we speak of the polynomials themselves as being code words or code polynomials (see (11.99)). Definition 11.12 Choose a field G F.q/, a positive integer n and a polynomial g.x/ with coefficients from G F.q/ such that x n 1 D g.x/ h.x/; furthermore, let deg.g.x// D r D n k. Words of a cyclic code are the vectors of length n that are associated with all multiples of g.x/ reduced modulo x n 1. In formulae: c.x/ D a.x/ g.x/ mod .x n 1/, for a.x/ polynomial with coefficients from G F.q/. The polynomial g.x/ is called a generator polynomial.
864
Chapter 11. Channel codes
Properties of cyclic codes 1. In a cyclic code there are q k code words, as shown in the previous section. 2. A cyclic code is a linear code. Proof. The all zero word is a code word as 0 g.x/ D 0; any multiple of a code word is a code word, as if a1 .x/ g.x/ is a code word so is Þa1 .x/ g.x/. Let a1 .x/ g.x/ and a2 .x/ g.x/ be two code words; then Þ1 a1 .x/ g.x/ C Þ2 a2 .x/ g.x/ D .Þ1 a1 .x/ C Þ2 a2 .x//g.x/ D a3 .x/ g.x/
(11.101)
is a code word. 3. Every cyclic permutation of a code word is a code word. Proof. It is enough to show that if c.x/ D c0 Cc1 x CÐ Ð ÐCcn2 x n2 Ccn1 x n1 corresponds to a code word, then also cn1 C c0 x C Ð Ð Ð C cn3 x n2 C cn2 x n1 corresponds to a code word. But if c.x/ D a.x/ g.x/ D c0 C c1 x C Ð Ð Ð C cn2 x n2 C cn1 x n1 mod.x n 1/, then xc.x/ D xa.x/ g.x/ D cn1 C c0 x C Ð Ð Ð C cn3 x n2 C cn2 x n1 mod.x n 1/. Example 11.2.23 Let G F.q/ D G F.2/, g.x/ D x C 1, and n D 4. From the previous example we obtain the code words, which can be grouped by the number of cyclic shifts. code polynomials
code words
0
0000
1Cx x C x2 x2 C x3 1 C x3
1100 0110 0011 1001
1 C x C x2 C x3
1111
1 C x2 x C x3
1010 0101
o 9 > > = > > ; o
cyclic shifts 1
4
(11.102)
1
¦ 2
4. c.x/ is a code polynomial if and only if c.x/ h.x/ D 0 mod.x n 1/. Proof. If c.x/ is a code polynomial, then c.x/ D a.x/g .x/ mod.x n 1/, but h.x/ c.x/ D h.x/ a.x/ g.x/ D a.x/ .g.x/h.x// D a.x/.x n 1/ D 0 mod.x n 1/. Assume now h.x/ c.x/ D 0 mod.x n 1/; then h.x/ c.x/ D Q.x/.x n 1/ D Q.x/ h.x/ g.x/, or c.x/ D Q.x/ g.x/, therefore c.x/ is a code polynomial. 5. Let x n 1 D g.x/ h.x/, where g.x/ D g0 C g1 x C Ð Ð Ð C gr x r and h.x/ D h 0 C h 1 x C Ð Ð Ð C h k x k ; then the code corresponding to all multiples of g.x/ modulo x n 1 has the
11.2. Block codes
865
generator matrix 3 g0 g1 g2 : : : gr 0 0 : : : 0 6 0 g0 g1 : : : gr 1 gr 0 : : : 0 7 7 GD6 5 4 0 0 0 ::: : : : gr 2
(11.103)
and parity check matrix 2
3 0 0 : : : 0 h k h k1 : : : h 1 h 0 6 0 0 : : : h k h k1 h k2 : : : h 0 0 7 7 HD6 4 5 ::: 0 0 h k h k1 : : : : : :
(11.104)
Proof. We show that G is the generator matrix. The first row of G corresponds to the polynomial g.x/, the second to xg.x/ and the last row to x k1 g.x/, but the code words are all words of the form .a0 C a1 x C Ð Ð Ð C ak1 x k1 /g.x/ D a0 g.x/ C a1 .xg.x// C Ð Ð Ð C ak1 .x k1 g.x// (11.105) But (11.105) expresses all code words as linear combinations of the rows of G, therefore G is the generator matrix of the code. To show that H is the parity check matrix, we consider the product c.x/ h.x/. If we write c.x/ D c0 C c1 x C Ð Ð Ð C cn1 x n1
(11.106)
and h.x/ D h 0 C h 1 x C Ð Ð Ð C h k1 x k1 C h k x k C Ð Ð Ð C h n1 x n1 below this point where h kC1 D h kC2 D Ð Ð Ð D h n1 D 0, we get d.x/ D c.x/ h.x/ D d0 C d1 x C Ð Ð Ð C d2n2 x 2n2
(11.107)
(11.108)
where
di D
8 i X > > > c j hi j >
> > > :
if n i 2n 2
jD0 n1 X
c j hi j jDi .n1/ modulo x n 1, and
(11.109)
O denote the result as d.x/ D dO0 C dO1 x C Ð Ð Ð We consider reducing d.x/ C dOn1 x n1 ; then dOi D di C dnCi , i D 0; 1; 2; : : : ; n 1. If c.x/ h.x/ D 0 mod.x n 1/, then dOi D 0, i D 0; 1; 2; : : : ; n 1, therefore we get i X jD0
c j hi j C
n1 X jDi C1
c j h nCi j D 0
i D 0; 1; 2; : : : ; n 1
(11.110)
866
Chapter 11. Channel codes
For i D n 1, (11.110) becomes
n1 X
c j h n1 j D 0
(11.111)
jD0
or [h n1 h n2 : : : h 1 h 0 ] [c0 c1 : : : cn1 ]T D 0. For i D n 2, (11.110) becomes
n2 X
c j h n2 j C cn1 h n1 D 0
(11.112)
jD0
or [h n2 h n3 : : : h 0 h n1 ] [c0 c1 : : : cn1 ]T D 0. After r steps, for i D n r, (11.110) becomes
nr X
c j h nr j C
jD0
n1 X
c j h 2nr j D 0
(11.113)
jDnr C1
or [h nr h nr 1 : : : h nr C2 h nr C1 ] [c0 c1 : : : cn1 ]T D 0. The r equations can be written in matrix form as 2 6 6 6 4
h n1 h n2 :: :
h n2 h n3
::: :::
h1 h0
h0 h n1 :: :
32 76 76 76 54
c0 c1 :: :
3
2
7 6 7 6 7D6 5 4
0 0 :: :
3 7 7 7 5
(11.114)
0 h nr h nr 1 : : : h nr C2 h nr C1 cn1 therefore all code words are solutions of the equation Hc D 0, where H is given by (11.104). It still remains to be shown that all solutions of the equation Hc D 0 are code words. As h n1 D h n2 D Ð Ð Ð D h nr C1 D 0, and h 0 6D 0, from (11.104) H has rank r, and can be written as H D [A B], where B is an r ð r matrix with non-zero determinant; therefore 2
ck
3
2
c0 c1 :: :
3
7 6 ckC1 7 6 7 7 6 6 (11.115) 7 6 :: 7 D B1 A 6 5 4 : 5 4 cn1 ck1 so there are q k D q nr solutions of the equation Hc D 0. As there are q k code words, all solutions of the equation Hc D 0 are the code words in the cyclic code. Example 11.2.24 Let q D 2 and n D 7. As x 7 1 D x 7 C 1 D .x 3 C x C 1/.x 3 C x 2 C 1/.x C 1/, we can choose g.x/ D x 3 C x C 1 and h.x/ D .x 3 C x 2 C 1/.x C 1/ D x 4 C x 2 C x C 1; thus the
11.2. Block codes
867
matrices G and H of this code are given 2 1 6 60 GD6 60 4 0 2
0 HD40 1
by 1 0 1 0 0 0
3
7 1 1 0 1 0 07 7 0 1 1 0 1 07 5 0 0 1 1 0 1 3 0 1 0 1 1 1 1 0 1 1 1 05 0 1 1 1 0 0
(11.116)
(11.117)
Note that the columns of H are all possible non-zero vectors of length 3, so the code is a Hamming single error correcting (7,4) code. 6. In a code word, any string of r consecutive symbols, even taken cyclically, can identify the check positions. Proof. From (11.115) it follows that the last r positions can be check positions. Now, if we cyclically permute every code word of m positions, the resultant words are themselves code words; thus the r check positions can be cyclically permuted anywhere in the code words. 7. As the r check positions can be the first r positions, a simple encoding method in canonical form is given by the following steps. Step 1: represent the k information bits by the coefficients of the polynomial m.x/ D m 0 C m 1 x C Ð Ð Ð C m k1 x k1 . Step 2: multiply m.x/ by x r to obtain x r m.x/. Step 3: divide x r m.x/ by g.x/ to obtain the remainder r.x/ D r0 C r1 x C Ð Ð Ð C rr 1 x r 1 . Step 4: form the code word c.x/ D .x r m.x/ r.x//; note that the coefficients of .r.x// are the parity check bits. Proof. To show that .x r m.x/ r.x// is a code word, we must prove that it is a multiple of g.x/: from Step 3 we obtain x r m.x/ D Q.x/ g.x/ C r.x/
(11.118)
.x r m.x/ r.x// D Q.x/ g.x/
(11.119)
so that
Example 11.2.25 Let g.x/ D 1 C x C x 3 , for q D 2 and n D 7. We report in Table 11.12 the message words .m 0 ; : : : ; m 3 / and the corresponding code words .c0 ; : : : ; c6 / obtained by the generator polynomial according to Definition 11.12 on page 863 for a.x/ D m.x/; the same code in canonical form, obtained by (11.119), is reported in Table 11.13.
868
Chapter 11. Channel codes
Table 11.12 (7,4) binary cyclic code, generated by g.x/ D 1 C x C x3 .
Message .m 0 m 1 m 2 m 3 /
Code polynomial c.x/ D m.x/ g.x/ mod x 7 1
Code .c0 c1 c2 c3 c4 c5 c6 /
0000 1000 0100 1100 0010 1010 0110 1110 0001 1001 0101 1101 0011 1011 0111 1111
0g.x/ D 0 1g.x/ D 1 C x C x 3 xg.x/ D x C x 2 C x 4 .1 C x/g.x/ D 1 C x 2 C x 3 C x 4 x 2 g.x/ D x 2 C x 3 C x 5 .1 C x 2 /g.x/ D 1 C x C x 2 C x 5 .x C x 2 /g.x/ D x C x 3 C x 4 C x 5 .1 C x C x 2 /g.x/ D 1 C x 4 C x 5 x 3 g.x/ D x 3 C x 4 C x 6 .1 C x 3 /g.x/ D 1 C x C x 4 C x 6 .x C x 3 /g.x/ D x C x 2 C x 3 C x 6 .1 C x C x 3 /g.x/ D 1 C x 2 C x 6 .x 2 C x 3 /g.x/ D x 2 C x 4 C x 5 C x 6 2 .1 C x C x 3 /g.x/ D 1 C x C x 2 C x 3 C x 4 C x 5 C x 6 .x C x 2 C x 3 /g.x/ D x C x 5 C x 6 .1 C x C x 2 C x 3 /g.x/ D 1 C x 3 C x 5 C x 6
0000000 1101000 0110100 1011100 0011010 1110010 0101110 1000110 0001101 1100101 0111001 1010001 0010111 1111111 0100011 1001011
Table 11.13 (7,4) binary cyclic code in canonical form, generated by g.x/ D 1 C x C x3 .
Message .m 0 m 1 m 2 m 3 /
Code polynomial r.x/ D x r m.x/ mod g.x/ c.x/ D x r m.x/ r.x/
Code .c0 c1 c2 c3 c4 c5 c6 /
0000 1000 0100 1100 0010 1010 0110 1110 0001 1001 0101 1101 0011 1011 0111 1111
0 1 C x C x3 x C x2 C x4 1 C x2 C x3 C x4 1 C x C x2 C x5 x2 C x3 C x5 1 C x4 C x5 x C x3 C x4 C x5 1 C x2 C x6 x C x2 C x3 C x6 1 C x C x4 C x6 x3 C x4 C x6 x C x5 C x6 1 C x3 C x5 C x6 x2 C x4 C x5 C x6 1 C x C x2 C x3 C x4 C x5 C x6
0000000 1101000 0110100 1011100 1110010 0011010 1000110 0101110 1010001 0111001 1100101 0001101 0100011 1001011 0010111 1111111
11.2. Block codes
869
Encoding method using a shift register of length r We show that the steps of the encoding procedure can be accomplished by a linear shift register with r stages. We begin by showing how to divide m 0 x r by the generator polynomial g.x/ and obtain the remainder. As g.x/ D gr x r C gr 1 x r 1 C Ð Ð Ð C g1 x C g0
(11.120)
then x r D gr1 .gr 1 x r 1 C gr 2 x r 2 C Ð Ð Ð C g1 x C g0 / mod g.x/
(11.121)
m 0 x r D m 0 gr1 .gr 1 x r 1 C gr 2 x r 2 C Ð Ð Ð C g1 x C g0 / mod g.x/
(11.122)
and
is the remainder after dividing m 0 x r by g.x/. We now consider the scheme illustrated in Figure 11.3, where multiplications and additions are in G F.q/, and Tc denotes the clock period with which the message symbols fm i g, i D k 1; : : : ; 1; 0, are input to the shift register. In the binary case, the storage elements are flip flops, the addition is the modulo 2 addition, and multiplication by gi is performed by a switch that is open or closed depending upon whether gi D 0 or 1, respectively. Note that if m 0 is input, the storage elements of the shift register will contain the coefficients of the remainder upon dividing m 0 x r by g.x/. Let us suppose we want to compute the remainder upon dividing m 1 x r C1 by g.x/. We could first compute the remainder of the division of m 1 x r by g.x/, by presenting m 1 at the input, then multiplying the remainder by x, and again reduce the result modulo g.x/. But once the remainder of the first division is stored in the shift register, multiplication by x and division by g.x/ are obtained simply by clocking the register once with no input. In fact, if the shift register contains the polynomial b.x/ D b0 C b1 x C Ð Ð Ð C br 1 x r 1
g0
g
g r−1
g r−1
1
Tc
(11.123)
Tc
Tc IN OUT
mi
Figure 11.3. Scheme of an encoder for cyclic codes using a shift register with r elements.
870
Chapter 11. Channel codes
and we multiply by x and divide by g.x/, we obtain x b.x/ D b0 x C b1 x 2 C Ð Ð Ð C br 1 x r D b0 x C b1 x 2 C Ð Ð Ð C br 2 x r 1 C br 1 .gr1 .gr 1 x r 1 C Ð Ð Ð C g1 x C g0 // mod g.x/
(11.124)
D br 1 gr1 g0 C .b0 br 1 gr1 g1 / x C Ð Ð Ð C .br 2 br 1 gr1 gr 1 / x r 1 mod g.x/ that is just the result obtained by clocking the register once. Finally, we note that superimposition holds in computing remainders; in other words, if m 0 x r D r1 .x/ mod g.x/ and m 1 x r C1 D r2 .x/ mod g.x/, then m 0 x r C m 1 x r C1 D r1 .x/Cr2 .x/ mod g.x/. Therefore, to compute the remainder upon dividing m 0 x r Cm 1 x r C1 by g.x/ using the scheme of Figure 11.3, we would first input m 1 and then next input m 0 to the shift register. Hence, to compute the remainder upon dividing x r m.x/ D m 0 x r C m 1 x r C1 C Ð Ð Ð C m k1 x n1 by g.x/ we input the symbols m k1 ; m k2 ; : : : ; m 1 ; m 0 to the device of Figure 11.3; after the last symbol, m 0 , enters, the coefficients of the desired remainder will be contained in the storage elements. From (11.119) we note that the parity check bits are the inverse elements (with respect to addition) of the values contained in the register. In general for an input z.x/, polynomial with n coefficients, after n clock pulses the device of Figure 11.3 yields x r z.x/ mod g.x/.
Encoding method using a shift register of length k It is also possible to accomplish the encoding procedure for cyclic codes by using a shift register with k stages. Again we consider the first r positions of the code word as the parity check bits, p0 ; p1 ; : : : ; pr 1 ; utilizing the first row of the parity check matrix we obtain h k pr 1 C h k1 m 0 C Ð Ð Ð C h 1 m k2 C h 0 m k1 D 0
(11.125)
or pr 1 D h 1 k .h k1 m 0 C h k2 m 1 C Ð Ð Ð C h 1 m k2 C h 0 m k1 /
(11.126)
Similarly, using the second row we obtain pr 2 D h 1 k .h k1 pr 1 C h k2 m 0 C Ð Ð Ð C h 1 m k3 C h 0 m k2 /
(11.127)
and so forth. Let us consider the scheme of Figure 11.4 and assume that the register initially contains the symbols m 0 ; m 1 ; : : : ; m k1 . After one clock pulse m k1 will appear at the output, all information symbols will have moved by one place to the right and the parity check symbol pr 1 will appear in the first left-most storage element; after the second clock pulse, m k2 will appear at the output, all symbols contained in the storage elements will move one place to the right and the parity check symbol pr 2 will appear in the left-most storage element. It is easy to verify that, if we apply n clock pulses to the device, the output will be given by the k message symbols followed by the r parity check bits.
11.2. Block codes
hk−1
871
h k−1
h k−2
h k−3
Tc
Tc
Tc
h0
h1 Tc
OUT
IN Figure 11.4. Scheme of an encoder for cyclic codes using a shift register with k elements.
Hard decoding of cyclic codes We discover (see page 839) that all vectors in the same coset of the decoding table have the same syndrome and that vectors in different cosets have different syndromes. Proposition 11.3 All polynomials corresponding to vectors in the same coset have the same remainder if they are divided by g.x/; polynomials corresponding to vectors in different cosets have different remainders if they are divided by g.x/. Proof. Let a j .x/ g.x/, j D 0; 1; 2; : : : ; q k 1, be the code words, and i .x/, i D 0; 1; 2; : : : ; q r 1, be the coset leaders. Assume z 1 .x/ and z 2 .x/ are two arbitrary polynomials of degree n 1: if they are in the same coset, say, the i-th, then z 1 .x/ D i .x/ C a j1 .x/ g.x/
(11.128)
z 2 .x/ D i .x/ C a j2 .x/ g.x/
(11.129)
and
As upon dividing a j1 g.x/ and a j2 g.x/ by g.x/ we get 0 as a remainder, the division of z 1 .x/ and z 2 .x/ by g.x/ gives the same remainder, namely the polynomial ri .x/, where i .x/ D Q.x/ g.x/ C ri .x/
deg.ri .x// < deg.g.x// D r
(11.130)
Now assume z 1 .x/ and z 2 .x/ are in different cosets, say, the i 1 -th and i 2 -th cosets, but have the same remainder, say, r0 .x/, if they are divided by g.x/; then the coset leaders i1 .x/ and i2 .x/ of these cosets must give the same remainder r0 .x/ if they are divided by g.x/, i.e. i1 .x/ D Q 1 .x/ g.x/ C r0 .x/
(11.131)
872
Chapter 11. Channel codes
g0 IN
g Tc
g r−1
g r−1
1
Tc
Tc
z 0 z 1 ... z n−2 z n−1 Figure 11.5. Device to compute the division of the polynomial z.x/ D z0 Cz1 xCÐ Ð ÐCzn1 xn1 by g.x/. After n clock pulses the r storage elements contain the remainder r0 ; r1 ; : : : ; rr1 .
and i2 .x/ D Q 2 .x/ g.x/ C r0 .x/
(11.132)
therefore we get i2 .x/ D i1 .x/ C .Q 2 .x/ Q 1 .x//g.x/ D i1 .x/ C Q 3 .x/ g.x/
(11.133)
This implies that i1 .x/ and i2 .x/ are in the same coset, which is a contradiction. This result leads to the following decoding method for cyclic codes. Step 1: compute the remainder upon dividing the received polynomial z.x/ of degree n 1 by g.x/, for example, by the device of Figure 11.5 (see (11.124)), by presenting at the input the sequence of received symbols, and applying n clock pulses. The remainder identifies the coset leader of the coset where the received polynomial is located. Step 2: subtract the coset leader from the received polynomial to obtain the decoded code word.
Hamming codes Hamming codes are binary cyclic single error correcting codes. We consider cyclic codes over G F.2/, where g.x/ is an irreducible polynomial of degree r such that g.x/ divides r x 2 1 1, but not x ` 1 for ` < 2r 1. To show that g.x/ is a primitive irreducible polynomial, we choose n D 2r 1, thus n x 1 D g.x/ h.x/ and the corresponding cyclic code has parameters n D 2r 1, r, and k D 2r 1 r. Proposition 11.4 H D 3 and therefore is a single error correcting code. This code has minimum distance dmin H ½ 3 by showing that all single error polynomials have Proof. We first prove that dmin distinct, non-zero remainders if they are divided by g.x/.
11.2. Block codes
873
Assume that x i D 0 mod g.x/, for some 0 i n 1; then x i D Q.x/ g.x/, which is impossible since g.x/ is not divisible by x. Now assume that x i and x j give the same remainder upon division by g.x/, and that 0 i < j n 1; then x j x i D x i .x ji 1/ D Q.x/ g.x/
(11.134)
but g.x/ does not divide x i , so it must divide .x ji 1/. But 0 < j i n 1 and by H ½ 3. assumption g.x/ does not divide this polynomial. Hence dmin By the limit (11.15) we know that for a code with fixed n and k the following inequality holds: 2 3 n n n (11.135) C C Ð Ð Ð C j d H 1 k 5 2n 2k 41 C min 1 2 2 As n D 2r 1 and k D n r, we have 2
3 r 2r 1 r 1 2 1 2 41 C C C Ð Ð Ð C j d H 1 k 5 2r min 1 2 2 but 1C
r 2 1 D 2r 1
(11.136)
(11.137)
H 3. and therefore dmin
We have seen in the previous section how to implement an encoder for a cyclic code. We consider now the decoder device of Figure 11.6, whose operations are described as follows. 1. Initially all storage elements of the register contain zeros and the switch SW is in position 0. The received n-bit word z D .z 0 ; : : : ; z n1 / is sequentially clocked into the lower register, with n storage elements, and into the feedback register, with r storage elements, whose content is denoted by r0 ; r1 ; : : : ; rr 1 . 2. After n clock pulses, the behavior of the decoder depends on the value of v: if v D 0, the switch SW remains in the position 0 and both registers are clocked once. This procedure is repeated until v D 1, which occurs for r0 D r1 D Ð Ð Ð D rr 2 D 0; then SW moves to position 1 and the content of the last stage of the feedback shift register is added modulo 2 to the content of the last stage of the lower register; both registers are then clocked until the n bits of the entire word are obtained at the output of the decoder. Overall, 2n clock pulses are needed.
874
Chapter 11. Channel codes
g
g2
1
0 (v=0) SW 1 (v=1)
g r−1
x0
x1
x r−2
x r−1
Tc
Tc
Tc
Tc
NOR v z IN
x0
x1
x n−1
Tc
Tc
Tc
^c OUT
Figure 11.6. Scheme of a decoder for binary cyclic single error correcting codes (Hamming codes). All operations are in GF.2/.
We now illustrate the procedure of the scheme of Figure 11.6. First of all we note that for the first n clocks the device coincides with that of Figure 11.3, hence the content of the shift register is given by (11.138) r.x/ D x r z.x/ mod g.x/ We consider two cases. 1. The received word is correct, z.x/ D c.x/. After the first n clock pulses, from (11.138) we have (11.139) r.x/ D x r c.x/ D x r a.x/ g.x/ D 0 mod g.x/ and thus v D 1 and rr 1 D 0
(11.140)
In the successive n clock pulses we have cOi D z i C 0
i D 0; : : : ; n 1
(11.141)
therefore cO D c. 2. The received word is affected by one error, z.x/ D c.x/ C x i . In other words we assume that there is a single error in the i-th bit, 0 i n 1. After the first n clock pulses, it is r.x/ D x r x i mod g.x/
(11.142)
11.2. Block codes
875
If i D n 1, we have r.x/ D x r x n1 D x n x r 1 D .x n 1/ x r 1 C x r 1
(11.143)
D h.x/ g.x/ x r 1 C x r 1 mod g.x/ D x r 1 and consequently rr 1 D 1 and rr 2 D Ð Ð Ð D r0 D 0 .v D 1/
(11.144)
This leads to switching SW , therefore during the last n clock pulses we have cOn1 D z n1 C 1
i D n 2; : : : ; 0
cOi D z i C 0
Therefore the bit in the last stage of the buffer is corrected. If i D n j, then we have r.x/ D x r j
(11.145)
(11.146)
thus only at the (n C j 1)-th clock pulse the condition (11.144) that forces to switch SW from 0 to 1 occurs; therefore, at the next clock pulse the received bit in error will be corrected.
Burst error detection We assume that a burst error occurs in the received word and that this burst affects ` n k consecutive bits, that is the error pattern is bit j
e D .0; 0; 0; : : : ; 0; 1 ; : : : ; : : : ;
bit . jC`1/
1
; 0; : : : ; 0/
(11.147)
where within the two ‘1’s the values can be either ‘0’ or ‘1’. Then we can write the vector e in polynomial form, e.x/ D x j B.x/
(11.148)
where B.x/ is a polynomial of degree ` 1 n k 1. Thus e.x/ is divisible by the generator polynomial g.x/ if B.x/ is divisible by g.x/, as x is not a factor of g.x/; but B.x/ has a degree at most equal to .n k 1/, lower than the degree of g.x/, equal to n k; therefore e.x/ cannot be a code word. We have then that all burst errors of length ` less than or equal to r D n k are detectable by .n; k/ cyclic codes. This result leads to the introduction of the cyclic redundancy check (CRC) codes.
11.2.4
Simplex cyclic codes
We consider a class of cyclic codes over G F.q/ such that the Hamming distance between every pair of distinct code words is a constant; this is equivalent to stating that the weight
876
Chapter 11. Channel codes
Table 11.14 Parameters of some simplex binary codes.
n
k
r
dmin
7 15 31 63 127
3 4 5 6 7
4 11 26 57 120
4 8 16 32 64
of all non-zero code words is equal to the same constant. We show that in the binary case, for these codes the non-zero code words are related to the PN sequences of Appendix 3.A. Let n D q k 1, and x n 1 D g.x/ h.x/, where we choose h.x/ as a primitive polynomial of degree k; then the resultant code has minimum distance H dmin D .q 1/ q k1
(11.149)
The parameters of some binary codes in this class are listed in Table 11.14. To show that these codes have minimum distance given by (11.149), first we prove the following: Property. All non-zero code words have the same weight. Proof. We begin by showing that x i g.x/ 6D x j g.x/ mod.x n 1/
0i < j n1
(11.150)
Assume the converse is true, that is x i g.x/ D x j g.x/ mod.x n 1/; then x i .x ji 1/ g.x/ D Q.x/ g.x/ h.x/
(11.151)
x i .x ji 1/ D Q.x/ h.x/
(11.152)
or
But this is impossible since h.x/ is a primitive polynomial of degree k and cannot divide .x ji 1/, as . j i/ < n D .q k 1/. Relation (11.150) implies that all cyclic shifts of the code polynomial g.x/ are unique, but there are n D .q k 1/ cyclic shifts. Furthermore we know that there are only q k code words and one is the all-zero word; therefore all cyclic shifts of g.x/ are all the non-zero code words and they all have the same weight. Recall Property 2 of a group code (see page 832), that is if all code words of a linear code are written as rows of a matrix, every column is either formed by all zeros, or it consists of each field element repeated an equal number of times. If we apply this result to
11.2. Block codes
877
a simplex code, we find that no column can be all zero as the code is cyclic, so the sum of the weights of all code words is given by sum of weights D n.q 1/
qk D .q k 1/ .q 1/ q k1 q
(11.153)
But there are .q k 1/ non-zero code words, all of the same weight; the weight of each word is then given by weight of non-zero code words D .q 1/ q k1
(11.154)
Therefore the minimum weight of the non-zero code words is given by H dmin D .q 1/ q k1
(11.155)
Example 11.2.26 H D 8. Choose h.x/ as a primitive Let q D 2, n D 15, and k D 4; hence r D 11, and dmin irreducible polynomial of degree 4 over G F.2/, h.x/ D x 4 C x C 1. The generator polynomial g.x/ is obtained by dividing x 15 1 by h.x/ D x 4 C x C 1 in G F.2/, obtaining g.x/ D x 11 C x 8 C x 7 C x 5 C x 3 C x 2 C x C 1
(11.156)
Given an extension field G F.2k / and n D 2k 1, from Property 2 on page 861, x n 1 is given by the l.c.m. of the minimum functions of the elements of the extension field. As h.x/ is a primitive polynomial, g.x/ is therefore given by the l.c.m. of the minimum functions of the elements 1; Þ 3 ; Þ 5 ; Þ 7 , from G F.24 /. By a table similar to Table 11.11, obtained for G F.26 /, and using one of the three methods to determine the minimum function (see page 859), it turns out that the generator polynomial for this code is given by g.x/ D .x C 1/.x 4 C x 3 C x 2 C x C 1/.x 2 C x C 1/.x 4 C x 3 C 1/
(11.157)
Relation to PN sequences We consider a periodic binary sequence of period L, given by : : : ; p.1/; p.0/, p.1/, : : : , with p.`/ 2 f0; 1g. We define the normalized autocorrelation function of this sequence as # " L1 X 1 r p .m/ D . p.`/ ý p.` m// (11.158) L 2 L `D0 Note that with respect to (3.302), now p.`/ 2 f0; 1g rather than p.`/ 2 f1; 1g. Theorem 11.1 If the periodic binary sequence f p.`/g is formed by repeating any non-zero code word of a simplex binary code of length L D n D 2k 1, then 8 m D 0; šL ; š2L ; : : : : D k k 2 1 2 1
m D 0; šL ; š2L ; : : : (11.161) otherwise
If we recall the implementation of Figure 11.4, we find that the generation of such sequences is easy. We just need to determine the shift register associated with h.x/, load it with anything except all zeros, and let it run. For example, choosing h.x/ D x 4 C x C 1, we get the PN sequence of Figure 3.41, as illustrated in Figure 11.7, where L D n D 24 1 D 15.
11.2.5
BCH codes
An alternative method to specify the code polynomials Definition 11.13 Suppose we arbitrarily choose L elements from G F.q m / that we denote as Þ1 , Þ2 ; : : : ; Þ L (we will discuss later how to select these elements), and we consider polynomials, of degree n 1 or less, with coefficients from G F.q/. A polynomial is a code polynomial if each of the elements Þ1 ; Þ2 ; : : : ; Þ L is a root of the polynomial. The code then consists of the set of all the code polynomials. Using this method we see that c.x/ D c0 C cx C Ð Ð Ð C cn1 x n1 is a code polynomial if and only if c.Þ1 / D c.Þ2 / D Ð Ð Ð D c.Þ L / D 0; thus 2 3 3 2 2 3 c .Þ1 /0 .Þ1 /1 .Þ1 /2 : : : .Þ1 /n1 6 0 7 0 c 6 .Þ2 /0 .Þ2 /1 .Þ2 /2 : : : .Þ2 /n1 7 6 1 7 6 0 7 7 6 c2 7 6 7 6 (11.162) 76 6 :: 7D6 : 7 :: 5 6 :: 7 4 :: 5 4 : : 4 : 5 0 .Þ L /0 .Þ L /1 .Þ L /2 : : : .Þ L /n1 cn1 All vectors c D [c0 ; c1 ; : : : ; cn1 ]T with elements from G F.q/ that are solutions of this set of equations, where operations are performed according to the rules of G F.q m /, are
11.2. Block codes
879
p(l)
p(l−1)
p(l−2)
p(l−4)
p(l−3)
l
p(l−1)
p(l−2)
p(l−3)
p(l−4)
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
0 1 0 0 1 1 0 1 0 1 1 1 1 0 0 0
0 0 1 0 0 1 1 0 1 0 1 1 1 1 0 0
0 0 0 1 0 0 1 1 0 1 0 1 1 1 1 0
1 0 0 0 1 0 0 1 1 0 1 0 1 1 1 1
Figure 11.7. Generation of a PN sequence as a repetition of a code word of a simplex code with L D n D 15.
code words. The form of (11.162) resembles equation (11.20), where H is the generalized parity check matrix. One obvious difference is that in (11.20) H and c have elements from the same field, whereas this does not occur for the vector equation (11.162). However, this difference is not crucial as each element from G F.q m / can be written as a vector of length m with elements from G F.q/. Thus each element .Þi / j in the matrix is replaced by a column vector with m components. The resultant matrix, with Lm rows and n columns, consists of elements from G F.q/ and is therefore just a generalized parity check matrix for the considered code. From the above discussion it appears that, if L roots are specified, the resultant linear code has r D Lm parity check symbols, as the parity check matrix has r D Lm rows. However, not all rows of the matrix are necessarily independent; therefore the actual number of parity check symbols may be less than Lm. We now show that if n is properly chosen, the resultant codes are cyclic codes. Let m j .x/ be the minimum function of Þ j , j D 1; 2; : : : ; L, where Þ j 2 G F.q m / and m j .x/ has coefficients from G F.q/. For Property 3 on page 858, every code polynomial c.x/ must be divisible by m 1 .x/; m 2 .x/; : : : ; m L .x/, and is thus divisible by the least common multiple of such minimum functions, l:c:m:.m 1 .x/; m 2 .x/; : : : ; m L .x//. If we define g.x/ D l:c:m:.m 1 .x/; m 2 .x/; : : : ; m L .x//
(11.163)
880
Chapter 11. Channel codes
then all multiples of g.x/ are code words. In particular from Definition 11.12 the code is cyclic if (11.164) x n 1 D g.x/ h.x/ Let `i be the order of Þi , i D 1; 2; : : : ; L, and furthermore let n D l:c:m:.`1 ; `2 ; : : : ; ` L /
(11.165)
From the properties of the minimum function (see Property 2, page 861), we know that g.x/ divides x n 1; thus the code is cyclic if n is chosen as indicated by (11.165). We note that r D deg.g.x// m L
(11.166)
as deg.m i .x// m. We see that r is equal to m L if all minimum functions are distinct and are of degree m; conversely, r < m L if any minimum function has degree less than m or if two or more minimum functions are identical. Example 11.2.27 Choose q D 2 and let Þ be a primitive element of G F.24 /; furthermore let the code polynomials have as roots the elements Þ, Þ 2 , Þ 3 , Þ 4 . To derive the minimum functions of the chosen elements we look up for example the Appendix C of [3], where such functions are listed. Minimum functions and orders of elements chosen for this example are given in Table 11.15. Then g.x/ D .x 4 C x C 1/ .x 4 C x 3 C x 2 C x C 1/ (11.167) n D l:c:m:.15; 15; 5; 15/ D 15 H D 5. The resultant code is therefore a (15,7) code; later we will show that dmin
Bose–Chaudhuri–Hocquenhem (BCH) codes The BCH codes are error correcting codes with symbols from G F.q/ and roots of code polynomials from G F.q m /. Table 11.15 Minimum functions and orders of elements Þ, Þ 2 , Þ 3 , Þ 4 , in GF.24 /.
Roots
Minimum function
Order
Þ Þ2 Þ3 Þ4
x4 C x C 1 x4 C x C 1 4 x C x3 C x2 C x C 1 x4 C x C 1
15 15 5 15
11.2. Block codes
881
The basic mathematical fact required to prove the error correcting capability of BCH codes is that if Þ1 ; Þ2 ; : : : ; Þr are elements from any field, the determinant of the Vandermonde matrix, given by þ þ þ 1 1 ::: 1 þ þ þ þ Þ1 Þ2 : : : Þr þ þ 2 þ þ Þ22 : : : Þr2 þþ (11.168) det þ Þ1 þ :: :: þ þ : þ : þ þ þ Þr 1 Þr 1 : : : Þ r 1 þ 1
r
2
is non-zero if and only if Þi 6D Þ j , for all indices i 6D j. In particular, we prove the following result. Lemma.
The determinant (11.168) is given by þ þ þ 1 1 ::: 1 þ þ þ þ Þ1 Þ2 : : : Þr þ þ 2 þ r .r C1/ þ Þ22 : : : Þr2 þþ D .1/ 2 D D det þ Þ1 þ :: þ :: þ : : þþ þ þ Þr 1 Þr 1 : : : Þ r 1 þ r 1 2
Proof. Consider the polynomial P.x/ defined þ þ 1 þ þ x þ 2 þ P.x/ D det þ x þ :: þ : þ þ x r 1
r Y
.Þi Þ j /
(11.169)
i; j D 1 i< j
as 1 Þ2 Þ22
::: ::: :::
1 Þr Þr2 :: :
Þ2r 1 : : : Þrr 1
þ þ þ þ þ þ þ þ þ þ þ
(11.170)
so that D D P.Þ1 /. Now, P.x/ is a polynomial of degree at most r 1 whose zeros are x D Þ2 ; x D Þ3 ; : : : ; x D Þr , because if x D Þi , i D 2; 3; : : : ; r, the determinant D is equal to zero as two columns of the matrix are identical. Thus P.x/ D k1 .x Þ2 /.x Þ3 / : : : .x Þr /
(11.171)
and D D P.Þ1 / D k1 .Þ1 Þ2 /.Þ1 Þ3 / : : : .Þ1 Þr /
(11.172)
It remains to calculate k1 . The constant k1 is the coefficient of x r 1 ; therefore from (11.170) we get þ þ þ 1 1 : : : 1 þþ þ þ Þ2 Þ3 : : : Þr þþ þ .1/r k1 D det þ :: :: þ D k2 .Þ2 Þ3 /.Þ2 Þ4 / : : : .Þ2 Þr / (11.173) þ : : þþ þ r 2 r 2 r þÞ Þ : : : Þ 1 þ 2
3
using a result similar to (11.172).
r
882
Chapter 11. Channel codes
Proceeding we find .1/r 1 k2 D k3 .Þ3 Þ4 /.Þ3 Þ5 / Ð Ð Ð .Þ3 Þr / .1/r 2 k3 D k4 .Þ4 Þ5 /.Þ4 Þ6 / Ð Ð Ð .Þ4 Þr / :: : 2 .1/ kr 1 D .1/.Þr 1 Þr /
(11.174)
and therefore D D .1/r C.r 1/CÐÐÐC2C1
r Y i; j D 1 i< j
.Þi Þ j / D .1/
r .r C1/ 2
r Y
.Þi Þ j /
(11.175)
i; j D 1 i< j
We now prove the important Bose–Chaudhuri–Hocquenhem theorem.
Theorem 11.2 Consider a code with symbols from G F.q/, whose code polynomials have as zeros the elements Þ m 0 ; Þ m 0 C1 ; : : : ; Þ m 0 Cd2 , where Þ is any element from G F.q m / and m 0 is any integer. Then the resultant .n; k/ cyclic code has the following properties: H ½ d if the elements Þ m 0 ; Þ m 0 C1 ; : : : ; Þ m 0 Cd2 , are a) it has minimum distance dmin distinct; m l m; b) n k .d 1/m; if q D 2 and m 0 D 1, then n k d1 2
c) n is equal to the order of Þ, unless d D 2, in which case n is equal to the order of Þ m 0 ; d) g.x/ is equal to the least common multiple of the minimum functions of Þ m 0 , Þ m 0 C1 , : : : , Þ m 0 Cd2 . Proof. The proof of part d) has already been given (see (11.163)); the proof of part b) then follows by noting that each minimum function is at most of degree m, and there are at most .d 1/ distinct minimum functions. If q D 2 and m 0 D 1, the minimum function of function of Þ i Þ raised to an even power, for example Þ 2i , is the same aslthe minimum m d1 m distinct minimum (see Property 2 on page 859), therefore there are at most 2 functions. To prove part c) note that, if d D 2, we have only the root Þ m 0 , so that n is equal to the order of Þ m 0 . If there is more than one root, then n must be the least common multiple of the order of the roots. If Þ m 0 and Þ m 0 C1 are both roots, then .Þ m 0 /n D 1 and .Þ m 0 C1 /n D 1, so that Þ n D 1; thus n is a multiple of the order of Þ. On the other hand, if ` is the order of Þ, .Þ m 0 Ci /` D .Þ ` /m 0 Ci D 1m 0 Ci D 1; therefore ` is a multiple of the order of every root. Then n is the least common multiple of numbers all of which divide `, and therefore n `; thus n D `.
11.2. Block codes
883
Finally we prove part a). We note that the code words must satisfy the condition 3 2 3 2 2 3 c0 .Þ m 0 /2 ::: .Þ m 0 /n1 1 Þm0 0 7 6 c 1 m 0 C1 /2 m 0 C1 /n1 7 6 6 1 Þ m 0 C1 7 607 .Þ : : : .Þ 7 6 c2 7 6 7 6 (11.176) 76 6 :: 7D6 : 7 :: 5 6 :: 7 4 :: 5 4 : : 4 : 5 0 1 Þ m 0 Cd2 .Þ m 0 Cd2 /2 : : : .Þ m 0 Cd2 /n1 cn1 We now show that no linear combination of .d 1/ or fewer columns is equal to 0. We do this by showing that the determinant of any set of .d 1/ columns is non-zero. Choose columns j1 ; j2 ; : : : ; jd1 ; then þ þ þ .Þ m 0 / j1 .Þ m 0 / j2 ::: .Þ m 0 / jd1 þþ þ þ .Þ m 0 C1 / j1 .Þ m 0 C1 / j2 : : : .Þ m 0 C1 / jd1 þþ þ (11.177) det þ þ :: :: þ þ : : þ þ þ .Þ m 0 Cd2 / j1 .Þ m 0 Cd2 / j2 : : : .Þ m 0 Cd2 / jd1 þ þ þ þ þ 1 1 ::: 1 þ þ j2 jd1 þ þ Þ j1 Þ : : : Þ þ þ D Þ m 0 . j1 C j2 CÐÐÐC jd1 / det þ þ (11.178) :: :: þ þ : : þ þ þ .Þ j1 /d2 .Þ j2 /d2 : : : .Þ jd1 /d2 þ D Þ m 0 . j1 C j2 CÐÐÐC jd1 / .1/
.d1/d 2
d1 Y
.Þ ji Þ jk / 6D 0
(11.179)
i; k D 1 i r 1, Ž2 D 0, L 2 D L 1 D 1. .2/ ½ ½ ½ ½ .x/ 1 C Þ 12 x 1 Þ 7 x D 0 x Þ3 þ .2/ .x/ ½ ½ 1 C Þ 12 x C Þ 10 x 1 C Þ3 x D D Þ3 x Þ3 x Step 3 (r D 3): 13 D Þ 14 C Þ 3 D 1, and 2L 2 D 2 D r 1, Ž3 D 1, L 3 D 3 1 D 2. .3/ ½ ½ ½ ½ ½ .x/ 1 C Þ3 x C Þ3 x 2 1 x 1 C Þ3 x D D 1 0 Þ3 x 1 C Þ3 x þ .3/ .x/ Step 4 (r D 4): 14 D Þ 13 C Þ 3 Þ 14 C Þ 3 D 1, and 2L 3 D 4 > r 1, Ž4 D 0, L 4 D L 3 D 2. .4/ ½ ½ ½ ½ .x/ 1 x 1 C Þ3 x C Þ3 x 2 D 0 x 1 C Þ3 x þ .4/ .x/ ½ ½ 1 C Þ 14 x 1 C Þ3 x C Þ3 x 2 C x C Þ3 x 2 D D x C Þ3 x 2 x C Þ3 x 2
898
Chapter 11. Channel codes
Step 5 (r D 5): 15 D 1 C Þ 14 Þ 13 D Þ 11 , and 2L 4 D 4 D r 1, Ž5 D 1, L 5 D 5 2 D 3. .5/ ½ ½ ½ ½ ½ .x/ 1 C Þ 14 x 1 C Þ 14 x C Þ 11 x 2 C Þ 14 x 3 1 Þ 11 x D D Þ4 x C Þ3 x 2 0 Þ4 C Þ3 x þ .5/ .x/ Step 6 (r D 6): 16 D Þ 11 C Þ 14 C Þ 11 Þ 13 C Þ 14 Þ 14 D 0, Ž6 D 0, L 6 D L 5 D 3. .6/ ½ ½ ½ ½ .x/ 1 0 1 C Þ 14 x C Þ 11 x 2 C Þ 14 x 3 D 0 x Þ4 C Þ3 x þ .6/ .x/ ½ 1 C Þ 14 x C Þ 11 x 2 C Þ 14 x 3 D Þ4 x C Þ3 x 2 The error indicator polynomial is ½.x/ D ½.6/ .x/. By using the exhaustive method to find the three roots, we obtain ½.x/ D 1 C Þ 14 x C Þ 11 x 2 C Þ 14 x 3 D .1 Þ 7 x/.1 Þ 5 x/.1 Þ 2 x/
(11.238)
Consequently the three errors are at positions ` D 2; 5; and 7. To determine the values of the errors we use the Forney algorithm. The derivative of ½.x/ is given by ½0 .x/ D Þ 14 C Þ 14 x 2
(11.239)
The error evaluator polynomial is given by !.x/ D .Þ 12 x C x 2 C Þ 14 x 3 C Þ 13 x 4 C x 5 C Þ 11 x 6 / .1 C Þ 14 x C Þ 11 x 2 C Þ 14 x 3 / mod x 6 D
Þ 12 x
C Þ 12 x 2
(11.240)
C Þ8 x 3
Thus the values of the errors are "2 D "5 D "7 D
!.Þ 2 / Þ 2 ½0 .Þ 2 / !.Þ 5 / Þ 5 ½0 .Þ 5 /
D Þ 11 D Þ5
(11.241)
!.Þ 7 / DÞ Þ 7 ½0 .Þ 7 /
An alternative approach for the encoding and decoding of Reed–Solomon codes utilizes the concept of Fourier transform on a Galois field [2, 5]. Let Þ be a primitive element of the field G F.q/. The Fourier transform on the field G F.q/ (GFFT) of a vector c D .c0 ; c1 ; : : : ; cn1 / of n bits is defined as .C0 ; C1 ; : : : ; C n1 /, where Cj D
n1 X i D0
ci Þ i j
j D 0; : : : ; n 1
(11.242)
11.2. Block codes
899
Let us consider a code word c of n bits in the “time domain” from a Reed–Solomon cyclic code that corrects up to t errors; then c corresponds to a code polynomial that has as roots 2t D d 1 consecutive powers of Þ. If we take the GFFT of this word, we find that in the “frequency domain” the transform has 2t consecutive components equal to zero. Indeed from (11.176), specialized to Reed–Solomon codes, and from (11.242), we can show that the two conditions are equivalent, that is a polynomial has 2t consecutive powers of Þ as roots if and only if the transform has 2t consecutive components equal to zero. The approach that resorts to the GFFT is therefore the mirror of the approach that uses the generator polynomial. This observation leads to the development of efficient methods for encoding and decoding.
11.2.6
Performance of block codes
In this section we consider the probability of error in the decoding of block codes, in the case of decoding with hard or soft input (see Section 6.8). For an in-depth study of the subject we refer the reader, for example, to [6]. With reference to Figure 6.20, let Pbit be the bit error probability for the detection of the bits of the binary sequence fcQm g, or bit error probability of the channel, Pw the error .dec/ probability for a code word, and Pbit the error probability for a bit of the binary sequence H 1/=2 and hard input fbOl g obtained after decoding. For a .n; k/ block code with t D .dmin decoding the following inequality holds: n X n i Pbit Pw .1 Pbit /ni (11.243) i i DtC1 which, under the condition n Pbit − 1, can be approximated as n P tC1 .1 Pbit /nt1 Pw ' t C 1 bit
(11.244)
The inequality (11.243) follows from the channel model (11.12) assuming errors that are i.i.d., and from the consideration that the code may not be perfect (see page 839), and therefore it could correct also some received words with more than t errors. If a word error occurs, the most probable event is that the decoder decides for a code H D 2t C 1 from the transmitted code word, thus making d H bit word with distance dmin min errors in the sequence fcOm g. As c is formed of n bits, we have that at the decoder output the bit error probability is .dec/ Pbit '
2t C 1 Pw n
(11.245)
Example 11.2.34 H D 5 (see page 839), decoding with hard input yields For a (5,1) repetition code with dmin 5 5 3 5 Pw D Pbit .1 Pbit /2 C P 4 .1 Pbit / C Pbit (11.246) 3 4 bit
900
Chapter 11. Channel codes
Example 11.2.35 H D 3 (see page 839), (11.243) yields For an .n; k/ Hamming code with dmin Pw
n X n i D2
i
i Pbit .1 Pbit /ni
(11.247)
D 1 [.1 Pbit / C n Pbit .1 Pbit / n
n1
]
For example, for a (15,11) code, if Pbit D 103 then Pw ' 104 , and from (11.245) we .dec/ get Pbit ' 2 105 . The decoders that have been considered so far are classified as hard input decoders, as the demodulator output is quantized to the values of the coded symbols before decoding. In general, other decoding algorithms with soft input may be considered, that directly process the demodulated signal, and consequently the decoder input is real valued (see Section 11.3.2). In the case of antipodal binary signals and soft input decoding we obtain (see also Section 6.8 on page 496) 0s 1 H 2E R d c min b A (11.248) Pw ' .2k 1/ Q @ N0
11.3
Convolutional codes
Convolutional codes are a subclass of the class of tree codes, so called because their code words are conveniently represented as sequences of nodes in a tree. Tree codes are of great interest because decoding algorithms have been found that are easy to implement, and can be applied to the entire class of tree codes, in contrast to decoding algorithms for block codes, each designed for a specific class of codes, as for example BCH codes. Several approaches have been used in the literature for describing convolutional codes; here we will illustrate these approaches by first considering a specific example. Example 11.3.1 Consider a rate 1=2 binary convolutional code, obtained by the encoder illustrated in Figure 11.9a. For each bit bk that enters the encoder, two output bits, ck.1/ and ck.2/ , are transmitted. The first output ck.1/ is obtained if the switch at the output is in the upper position, and the second output ck.2/ is obtained if the switch is in the lower position; the two previous input bits, bk1 and bk2 , are stored in the memory of the encoder. As the information bit is not presented directly to one of the outputs, we say that the code is nonsystematic. The two coded bits are generated as linear combinations of the bits of the message; denoting the input sequence as : : : ; b0 ; b1 ; b2 ; b3 ; : : : , and the output sequence as : : : ; c0.1/ ; c0.2/ ; c1.1/ ; c1.2/ ; c2.1/ ; c2.2/ ; c3.1/ ; c3.2/ ; : : : , then the following
11.3. Convolutional codes
901
ck(1) bk
D
D
(2)
ck (a)
10 01
11
10 1
0
11
00
00
(b) (c)
00 10
(d) (a)
01 00
(b) (c)
11 01
(d) (a)
10 11
(b) (c)
00
(d)
(c)
(d) 00
10 11
(b)
(d) 11
(d) (a)
(a)
(c) 10
11 01
(d)
(d) 01
(b) (c)
(c)
(b) 11
01 00 (b)
(c) 00
(a)
(a)
(a) 01
10
(d)
(b)
Figure 11.9. (a) Encoder and (b) tree diagram for the convolutional code of Example 11.3.1.
902
Chapter 11. Channel codes
relations hold: ck.1/ D bk ý bk1 ý bk2 ck.2/ D bk ý bk2
(11.249)
A convolutional code may be described in terms of a tree, trellis, or state diagram; for the code defined by (11.249) these descriptions are illustrated in Figures 11.9b, 11.10a, and 11.10b, respectively. With reference to the tree diagram of Figure 11.9b, we begin at the left (root) node and proceed to the right by choosing an upper path if the input bit is equal to 1 and a lower path if the input bit is 0. We output the two bits represented by the label on the branch that takes us to the next node, and then repeat this process at the next node. The nodes or
(a)
(b)
Figure 11.10. (a) Trellis diagram and (b) state diagram for the convolutional code of Example 11.3.1.
11.3. Convolutional codes
903
states of the encoder are labeled with the letters a, b, c, and d, which indicate the relation with the four possible values assumed by the two bits stored in the encoder, according to the table: bk1
bk2
label
0 1 0 1
0 0 1 1
d c b a
(11.250)
If for example we input the sequence b0 ; b1 ; b2 ; b3 ; Ð Ð Ð D 1 1 0 1 : : : , we would then output the sequence c0.1/ ; c0.2/ ; c1.1/ ; c1.2/ ; c2.1/ ; c2.2/ ; c3.1/ ; c3.2/ ; Ð Ð Ð D 1 1 0 1 0 1 0 0 : : : . As, at any depth in the tree, nodes with the same label will have the same tree growing from them, we can superimpose these nodes on a single node. This results in the trellis diagram represented in Figure 11.10a, where solid and dashed lines correspond to transitions determined by input bits equal to 1 and 0, respectively. The state diagram for the encoder is illustrated in Figure 11.10b. The four states (a, b, c, d) correspond to the four possible combinations of bits stored in the encoder. If the encoder is in a certain state, a transition to one of two possible states occurs, depending on the value of the input bit. Possible transitions between states are represented as arcs, on which an arrow indicates the direction of the transition; with each arc is associated a label that indicates the value assumed by the input bit, and also the value of the resulting output bits. The description of the encoder by the state diagram is convenient for analyzing the properties of the code, as we will see later. It is also convenient to represent code sequences in terms of the D transform, as b.D/ D b0 C b1 D C b2 D 2 C b3 D 3 C Ð Ð Ð c.1/ .D/ D c0.1/ C c1.1/ D C c2.1/ D 2 C c3.1/ D 3 C Ð Ð Ð D g .1;1/ .D/ b.D/
(11.251)
c.2/ .D/ D c0.2/ C c1.2/ D C c2.2/ D 2 C c3.2/ D 3 C Ð Ð Ð D g .2;1/ .D/ b.D/ where g .1;1/ .D/ D 1 C D C D 2 , and g .2;1/ .D/ D 1 C D 2 .
11.3.1
General description of convolutional codes
In general we consider convolutional codes with symbols from G F.q/; assuming the encoder produces n 0 output code symbols for every k0 input message symbols, the code rate is equal to k0 =n 0 . It is convenient to think of the message sequence as being the interlaced version of k0 different message sequences, and to think of the code sequence as the interlaced version of n 0 different code sequences. In other words, given the information sequence fb` g we form the k0 subsequences bk.i / D bkk0 Ci 1
i D 1; : : : ; k0
(11.252)
904
Chapter 11. Channel codes
that have D transform defined as b0.1/ b1.1/ b2.1/ Ð Ð Ð () b.1/ .D/ D b0.1/ C b1.1/ D C b2.1/ D 2 C Ð Ð Ð b0.2/ b1.2/ b2.2/ Ð Ð Ð () b.2/ .D/ D b0.2/ C b1.2/ D C b2.2/ D 2 C Ð Ð Ð :: :: : :
(11.253)
b0.k0 / b1.k0 / b2.k0 / Ð Ð Ð () b.k0 / .D/ D b0.k0 / C b1.k0 / D C b2.k0 / D 2 C Ð Ð Ð Let c.1/ .D/; c.2/ .D/; : : : ; c.n 0 / .D/ be the D transforms of the n 0 output sequences; then c. j/ .D/ D
k0 X
g . j;i / .D/ b.i / .D/
j D 1; 2; : : : ; n 0
(11.254)
i D1
An .n 0 ; k0 / convolutional code is then specified by giving the coefficients of all the polynomials g . j;i / .D/, i D 1; 2; : : : ; k0 , j D 1; 2; : : : ; n 0 . If for all j D 1; 2; : : : ; k0 , we have ( 1 j Di (11.255) g . j;i / .D/ D 0 j 6D i then the code is systematic and k0 of the n 0 output sequences are just the message sequences. An encoder for a convolutional code needs storage elements. Let ¹ be the constraint length of the code,5 ¹ D max.deg g . j;i / .D// j;i
(11.256)
Therefore the encoder of a convolutional code must store ¹ previous blocks of k0 message symbols to form a block of n 0 output symbols. The general structure of an encoder for a code with k0 D 1 and n 0 D 2 is illustrated in Figure 11.11; for such an encoder, ¹k0 storage elements are necessary. If the code is systematic, then the encoder can be implemented with ¹.n 0 k0 / storage elements, as illustrated in Figure 11.12 for k0 D 2 and n 0 D 3. If we interpret the sequence fck g as the output of a sequential finite-state machine (see Appendix 8.D), at instant k the trellis of a nonsystematic code is defined by the three signals: 1. Input
[bk.1/ ; bk.2/ ; : : : ; bk.k0 / ]
(11.257)
2. State
.k / .k0 / .1/ [bk.1/ ; : : : ; bk 0 ; : : : ; bk.¹1/ ; : : : ; bk.¹1/ ]
(11.258)
[ck.1/ ; ck.2/ ; : : : ; ck.n 0 / ]
(11.259)
3. Output . j/
where ck ; j D 1; : : : ; n 0 , is given by (11.254). Then there are q k0 ¹ states in the trellis. There are q k0 branches departing from each state and q k0 branches merging into a state. The output vector consists of n 0 q–ary symbols. 5
Many authors define the constraint length as ¹ C 1, where ¹ is given by (11.256).
11.3. Convolutional codes
905
c (1)(D)
(1,1)
b
g1
(1)
(D)
D
3
D g (2,1)
g (2,1)
0
1
(1,1)
gν
g
g2
D g (2,1)
(1,1)
(1,1)
(1,1)
g0
D g (2,1)
g (2,1)
2
ν
3
c (2)(D) Figure 11.11. Block diagram of an encoder for a convolutional code with k0 D 1, n0 D 2, and constraint length ¹. (1) b (D)
c (1) (D)
(2)
b (D)
c (2) (D)
g
(3,1) ν
(3,2)
g (3,1)
gν
ν−1
D
(3,1)
(3,2)
g ν−1
g
D
D
0
g
(3,2) 0
c (3)(D)
Figure 11.12. Block diagram of an encoder for a systematic convolutional code with k0 D 2, n0 D 3, and constraint length ¹.
Parity check matrix A semi-infinite parity check matrix can be defined in general for convolutional codes; however, we note that it is only in the case of systematic codes that we can easily express the elements of this matrix in terms of the coefficients of the generator polynomials g . j;i / .D/.
906
Chapter 11. Channel codes
We write the coefficients of the generator polynomials in the form .n 0 ;1/
g0.1;1/ g0.2;1/ : : : g0
.n 0 ;2/
g0.1;2/ g0.2;2/ : : : g0 :: :
.1;k0 / .2;k0 / .n ;k / g0 : : : g0 0 0
g0
.n 0 ;1/
:::
g¹.1;1/ g¹.2;1/ : : : g¹
g1.1;2/ : : : g1
.n 0 ;2/
:::
.1;k0 /
g¹.1;2/ g¹.2;2/ : : : g¹ :: :
: : : g¹
g1.1;1/ : : : g1
g1
.n 0 ;k0 /
: : : g1
.n 0 ;1/ .n 0 ;2/
(11.260)
.1;k0 / .2;k0 / .n ;k / g¹ : : : g¹ 0 0
If the code is systematic, the parity matrix of the generator polynomials can be written as I P0 0 P1 : : : 0 P¹ where I and 0 are k0 ð k0 matrices and Pi , i The semi-infinite parity check matrix is then 2 0 P0T I 6 PT 0 PT 6 0 6 : : :1 ::: 6 H1 D 6 T 6 P¹T 0 P¹1 6 4 0 0 PT ¹ ::: :::
(11.261)
D 0; : : : ; ¹, are k0 ð .n 0 k0 / matrices. 0
0
0 :::
I
0 :::
0 ::: :::
T 0 P¹2 T 0 P¹1 :::
3
7 7 7 7 7 0 ::: 7 7 0 ::: 5 :::
(11.262)
Thus for any code word c of infinite length, H1 c D 0. Often, rather than considering the semi-infinite matrix H1 , we consider the finite matrix H defined as 2 3 P0T I 0 0 ::: 0 0 6 7 6 P1T 0 P0T I : : : 0 0 7 (11.263) HD6 7 ::: ::: ::: 4 ::: 5 T 0 : : : P0T I P¹T 0 P¹1 The bottom row of matrices of the matrix H is called the basic parity check matrix. From it we can see that the parity symbols in a block are given by the linear combination of information bits in that block, corresponding to non-zero terms in P0T , in the immediately preceding block, corresponding to non-zero terms in P1T , and so on until the ¹-th preceding block, corresponding to non-zero terms in P¹T .
Generator matrix From (11.260), we introduce the matrices 2 .1;1/ .n ;1/ 3 gi gi.2;1/ : : : gi 0 6 7 :: gi D 4 ::: 5 : .1;k0 / .2;k0 / .n 0 ;k0 / gi : : : gi gi
i D 0; : : : ; ¹
Hence the generator matrix is of the form 2 3 g0 g1 : : : g¹ 0 ::: G1 D 4 0 g0 : : : g¹1 g¹ : : : 5 ::: ::: ::: ::: ::: :::
(11.264)
(11.265)
11.3. Convolutional codes
907
Some examples of convolutional codes with the corresponding encoders and generator matrices are illustrated in Figure 11.13.
Transfer function H , that determines the performance An important parameter of a convolutional code is dfree of the code (see Section 11.3.3).
Definition 11.14 Let e.D/ D [e.n 0 / .D/; : : : ; e.1/ .D/], be any error sequence between two code words c1 .D/ D [c1.n 0 / .D/; : : : ; c1.1/ .D/] and c2 .D/ D [c2.n 0 / .D/; : : : ; c2.1/ .D/], that is c1 .D/ D c2 .D/ C e.D/, and ek D [ek.n 0 / ; : : : ; ek.1/ ] denotes the k-th element of the sequence. We define the free Hamming distance of the code as H dfree D min e.D/
1 X
w.ek /
(11.266)
kD0
H corresponds where w is introduced in Definition 11.4 on page 832. As the code is linear, dfree to the minimum number of symbols different from zero in a non-zero code word.
Next we consider a method to compute the weights of all code words in a convolutional code; to illustrate the method we examine the simple binary encoder of Figure 11.9a. We begin by reproducing the trellis diagram of the code in Figure 11.14, where each path is now labeled with the weight of the output bits corresponding to that path. We consider all paths that diverge from state (d) and return to state (d) for the first time after a number of steps j. By inspection, we find one such path of weight 5 returns to state (d) after 3 steps; moreover, we find two distinct paths of weight 6, one that returns to state (d) after 4 steps and another after 5 steps. Hence we find that this code has H D 5. dfree We now look for a method that enables us to find the weights of all code words as well as the lengths of the paths that give origin to the code words with these weights. Consider the state diagram for this code, redrawn in Figure 11.15 with branches labeled as D 2 , D, or D 0 D 1, where the exponent corresponds to the weight of the output bits corresponding to that branch. Next we split node (0,0) to obtain the state diagram of Figure 11.16, and we compute a generating function for the weights. The generating function is the transfer function of a signal flow graph with unit input. From Figure 11.16, we obtain this transfer function by solving the system of equations þ D D 2 Þ C 1 D Dþ C DŽ Ž D Dþ C DŽ D D2
(11.267)
908
Chapter 11. Channel codes
c (1) (D) g0 =(1,1) (1)
b (D)
D
g1 =(1,0)
D
g =(1,1) 2
c (2) (D)
k 0 =1, n0 =2, ν =2 (a)
b
b
(1)
(2)
(D)
D
(D)
c
(1)
c
(2)
c
(3)
(D) g0 = 1 1 1 010
(D)
g1 =
(D)
101 110
D k 0 =2, n0 =3, ν =1 (b)
b
(1)
(D)
b (2) (D)
c
(1)
(D)
c
(2)
(D)
D g0 =
1001 0101 0011
g1 =
0001 0000 0001
D c (3) (D)
b (3) (D)
D
c (4) (D)
k0 =3, n 0 =4, ν =1 (c)
Figure 11.13. Examples of encoders for three convolutional codes.
11.3. Convolutional codes
909
Figure 11.14. Trellis diagram of the code of Example 11.3.1; the labels represent the Hamming weight of the output bits.
Figure 11.15. State diagram of the code of Example 11.3.1; the labels represent the Hamming weight of the generated bits.
Figure 11.16. State diagram of the code of Example 11.3.1; node (0,0) is split to compute the transfer function of the code.
910
Chapter 11. Channel codes
Figure 11.17. State diagram of the code of Example 11.3.1; node (0,0) is split to compute the augmented transfer function.
Then we get t .D/ D
D5 D D D 5 C 2D 6 C 4D 7 C Ð Ð Ð C 2i D i C5 C Ð Ð Ð Þ 1 2D
(11.268)
From inspection of t .D/, we find there is one code word of weight 5, two of weight 6, four of weight 7, : : : . Equation (11.268) holds for code words of infinite length. If we want to find code words that return to state (d) after j steps we refer to the state diagram of Figure 11.17. The term L introduced in the label on each branch allows to keep track of the length of the sequence, as the power of L is augmented by 1 every time a transition occurs. Furthermore, we introduce the term I in the label on a branch if the corresponding transition is due to an information bit equal to 1; this allows computation for each path on the trellis diagram of the corresponding number of information bits equal to 1. The augmented transfer function is given by t .D; L ; I / D
D5 L 3 I 1 D L.1 C L/I
D D 5 L 3 I C D 6 L 4 .1 C L/I 2 C D 7 L 5 .1 C L/2 I 3 C Ð Ð Ð
(11.269)
C D 5Ci L 3Ci .1 C L/i I 1Ci C Ð Ð Ð Thus we see that the code word of weight 5 is of length 3 and is originated by a sequence of information bits that contains one bit equal to 1, there are two code words of weight 6, one of length 4 and the other of length 5, both of which are originated by a sequence of information bits that contain two bits equal to 1, : : : .
Catastrophic error propagation For certain codes a finite number of channel errors may lead to an infinite number of errors in the sequence of decoded bits. For example, consider the code with encoder and state diagram illustrated in Figure 11.18a and b, respectively. Note that in the state diagram the self-loop at
11.3. Convolutional codes
911
Figure 11.18. (a) Encoder and (b) state diagram for a catastrophic convolutional code.
state .1; 1/ does not increase the weight of the code word, so that a code word corresponding to a path passing through the states .0; 0/; .1; 0/; .1; 1/; .1; 1/; : : : ; .1; 1/; .0; 1/; .0; 0/ is of weight 6, independently of the number of times it passes through the self loop at state (1,1). In other words, long sequences of coded bits equal to zero may be obtained by remaining in the state .0; 0/ with a sequence of information bits equal to zero, or by remaining in the state .1; 1/ with a sequence of information bits equal to one. Therefore a limited number of channel errors, in this case 6, can cause a large number of errors in the sequence of decoded bits. Definition 11.15 A convolutional code is catastrophic if there exists a closed loop in the state diagram that has all branches with zero weight.
912
Chapter 11. Channel codes
~g (1,1) (D) g
(1,1)
~g (1,1) (D)
(D) ~g( n0 ,1)(D)
b (1)(D)=0 ( n0 ,1)
(D)
g
~g( n0 ,1)(D)
(a)
g b (1)(D)=
(1,1)
(D)
~g (1,1) (D)
1 gc (D) ( n0 ,1)
g
(D)
~g( n0 ,1)(D)
(b) Figure 11.19. Two distinct infinite sequences of information bits that produce the same output sequence with a finite number of errors.
For codes with rate 1=n 0 , it has been shown that a code is catastrophic if and only if all generator polynomials have a common polynomial factor. In the above example, the common factor is 1 C D. This can be proved using the following argument: suppose that g .1;1/ .D/; g .2;1/ .D/; : : : ; g .k0 ;1/ .D/ all have the common factor gc .D/, so that g .i;1/ .D/ D gc .D/ gQ .i;1/ .D/
(11.270)
Suppose the all zero sequence is sent, b.1/ .D/ D 0, and that the finite error sequence gQ .i;1/ .D/, equal to that defined in (11.270), occurs in the i-th subsequence output, for i D 1; 2; : : : ; n 0 , as illustrated in Figure 11.19a. The same output sequence is obtained if the sequence of information bits with infinite length b.1/ .D/ D 1=gc .D/ is sent, and no channel errors occur, as illustrated in Figure 11.19b. Thus a finite number of errors yields a decoded sequence of information bits that differ from the transmitted sequence in an infinite number of positions.
11.3.2
Decoding of convolutional codes
Various algorithms have been developed for the decoding of convolutional codes. One of the first decoding methods was algebraic decoding, which is similar to the methods developed for the decoding of block codes. However, this method has the disadvantages that it is applicable only to a limited number of codes having particular characteristics, and exhibits
11.3. Convolutional codes
913
performance that is lower as compared to decoding methods based on the observation of the whole received sequence. The latter methods, also called probabilistic decoding methods, include the Viterbi algorithm (VA), the sequential decoding algorithm by Fano [6], and the forward-backward algorithm by Bahl–Cocke–Jelinek–Raviv (BCJR). Before illustrating the various decoding methods, we consider an important function.
Interleaving The majority of block codes as well as convolutional codes are designed by assuming that the errors introduced by the noisy channel are statistically independent. This assumption is not always true in practice. To make the channel errors, at least approximately, statistically independent it is customary to resort to an interleaver, which performs a permutation of the bits of a sequence. For example, a block interleaver orders the coded bits in a matrix with M1 rows and M2 columns. The coded bits are usually written in the matrix by row and then read by column before being forwarded to the bit mapper. At the receiver, a deinterleaver stores the detected bits in a matrix of the same M1 ð M2 dimensions, where the writing is done by column and the reading by row. As a result, possible error bursts of length M1 B are broken up into bursts of shorter length B.
Two decoding models We consider a binary convolutional code with k0 D 1, n 0 D 2, and constraint length ¹. In general, from (11.254) we write the code sequence as a function of the message sequence as ck.1/ D g .1;1/ .bk ; : : : ; bk¹ / ck.2/ D g .2;1/ .bk ; : : : ; bk¹ /
(11.271)
At the receiver, two models may be adopted. Model with hard input. With reference to the transmission system of Figure 6.20, we consider the sequence at the output of the binary channel. In this case the demodulator has already detected the transmitted symbols, for example, by a threshold detector, and the inverse bit mapper provides the binary sequence fz m D cQm g to the decoder, from which we obtain the interlaced binary sequences z k.1/ D z 2k D ck.1/ ý ek.1/ z k.2/ D z 2kC1 D ck.2/ ý ek.2/
(11.272)
where the errors ek.i / 2 f0; 1g, for a memoryless binary symmetric channel, are i.i.d. (see (6.91)). From the description on page 904, introducing the state of the encoder at instant k as the vector with ¹ elements sk D [bk ; : : : ; bk.¹1/ ]
(11.273)
914
Chapter 11. Channel codes
the desired sequence in (11.272), that coincides with the encoder output, can be written as (see (11.271) and (11.273)) ck.1/ D f .1/ .sk ; sk1 / ck.2/ D f .2/ .sk ; sk1 /
(11.274)
Model with soft input. Again with reference to Figure 6.20, at the decision point of the receiver the signal can be written as (see (8.173)) z k D u k C wk
(11.275)
where we assume wk is white Gaussian noise with variance ¦w2 D 2¦ I2 , and u k is given by (8.174) uk D
L2 X
n akn
(11.276)
nDL 1
where fak g is the sequence of symbols at the output of the bit mapper. Note that in (11.276) the symbols fak g are in general not independent, as the input of the bit mapper is a code sequence according to the law (11.274). The relation between u k and the bits fb` g depends on the intersymbol interference in (11.276), the type of bit mapper and the encoder (11.271). We consider the case of absence of ISI, that is u k D ak
(11.277)
.1/ .2/ and a 16-PAM system where, without interleaving, four consecutive code bits c2k , c2k , .1/ .2/ c2k1 , c2k1 are mapped into a symbol of the constellation. For an encoder with constraint length ¹ we have .1/ .2/ .1/ .2/ ; c2k ; c2k1 ; c2k1 ]g] u k D fQ.ak / D fQ[BMAP f[c2k
D fQ[BMAP fg .1;1/ .b2k ; : : : ; b2k¹ /; g .2;1/ .b2k ; : : : ; b2k¹ /
(11.278)
ð g .1;1/ .b2k1 ; : : : ; b2k1¹ /; g .2;1/ .b2k1 ; : : : ; b2k1¹ /g] In other words, let sk D [b2k ; : : : ; b2k¹C1 ]
(11.279)
u k D f .sk ; sk1 /
(11.280)
we can write
We observe that in this example each state of the trellis admits four possible transitions. As we will see in Chapter 12, better performance is obtained by jointly optimizing the encoder and the bit mapper.
11.3. Convolutional codes
915
Viterbi algorithm The Viterbi algorithm, described in Section 8.10.1, is a probabilistic decoding method that implements the maximum likelihood criterion, which minimizes the probability of detecting a sequence that is different from the transmitted sequence. VA with hard input. The trellis diagram is obtained by using the definition (11.273), and the branch metric is the Hamming distance between zk D [z k.1/ ; z k.2/ ]T and ck D [ck.1/ ; ck.2/ ]T , (see Definition 11.1), d H .zk ; ck / D number of positions where zk differs from ck
(11.281)
where ck is generated according to the rule (11.274). VA with soft input. The trellis diagram is now obtained by using the definition (11.279), and the branch metric is the Euclidean distance between z k and u k , jz k u k j2
(11.282)
where u k , in the case of the previous example of absence of ISI and 16-PAM transmission, is given by (11.280). As an alternative to the VA we can use the FBA of Section 8.10.2.
Forward-backward algorithm The previous approach, which considers joint detection in the presence of ISI and convolutional decoding, requires a computational complexity that in many applications may turn out to be exceedingly large. In fact, the state (11.279), that takes into account both encoding and the presence of ISI, usually is difficult to define and is composed of several bits of the sequence fb` g. An approximate solution is obtained by considering the detection and the decoding problems separately, however, assuming that the detector passes the soft information on the detected bits to the decoder. Soft output detection by FBA. By using a trellis diagram that takes into account the ISI introduced by the channel, the code bits fcn g are detected assuming that they are i.i.d., and the reliability of the detection is computed (soft detection). For this purpose we use the FBA of page 670, that determines for each state a metric Vk .i/, i D 1; : : : ; Ns . Now, with reference to the example of the channel given by (11.277) and 16-PAM transmission, the state is identified by sk D .ak / D [c4k ; c4k1 ; c4k2 ; c4k3 ], where6 fcn g, cn 2 f1; 1g, is assumed to be a sequence of i.i.d. symbols. By considering the binary state representation, and by suitably adding the values Vk .i/, we get the MAP metric, or
6
It is sometimes convenient to view the encoder output cn and/or the encoder input bn as symbols from the alphabet f1; C1g, rather than f0; 1g. It will be clear from the context to which alphabet we refer.
916
Chapter 11. Channel codes
likelihood, associated with the bits fcm g, L.in/ 4kt .Þ/ D
Ns X
Vk .i/
Þ 2 f1; 1g
t D 0; 1; 2; 3
(11.283)
i D1 σ i with t-th binary component equal to Þ
or equivalently the Log-MAP metric, or log-likelihood, `n.in/ .Þ/ D ln Ln.in/ .Þ/
Þ 2 f1; 1g
(11.284)
By the above formulation, the soft decision associated with the bit cn is given by `n.in/ D `n.in/ .1/ `n.in/ .1/
(11.285)
also called log-likelihood ratio (LLR). Observation 11.3 For binary transmission in the absence of ISI, from (8.269) on page 675, we have, apart from a non-essential additive constant, `n.in/ .Þ/ D
.z n Þ/2 2¦ I2
Þ 2 f1; 1g
(11.286)
where ¦ I2 is the variance of real-valued noise samples. Then we get `n.in/ D
2 zn ¦ I2
(11.287)
In other words, apart from a constant factor, the LLR associated with the bit cn coincides with the demodulator output z n . Rather than (11.284) and (11.283), we can use the Max-Log-MAP criterion (8.267) that yields an approximate log-likelihood, `Q.in/ 4kt .Þ/ D
max
i 2 f1; : : : ; Ns g σ i with t-th binary component equal to Þ
vk .i/
(11.288)
An alternative to the FBA is obtained by modifying the VA to yield a soft output (SOVA), as discussed in the next section. Convolutional decoding with soft input (SI). The decoder for the convolutional code typically uses the VA with branch metric (associated with a cost function to be minimized) given by 2
ck jj jj.in/ k
(11.289)
11.3. Convolutional codes
917
where ck is given by (11.274) for a code with n 0 D 2, and .in/ D [`.in;1/ ; `.in;2/ ] are the k k k . j/ 2
LLR associated, respectively, with ck.1/ and ck.2/ . As jck j D 1, (11.289) can be rewritten as: 2
.`.in;1/ ck.1/ / C .`.in;2/ ck.2/ / k k
2
2
2
D .`.in;1/ / C .`.in;2/ / C 2 2ck.1/ `.in;1/ 2ck.2/ `.in;2/ k k k k
(11.290)
Leaving out the terms that do not depend on ck , and extending the formulation to a convolutional code with rate k0 =n 0 , the branch metric (associated with a cost function to be maximized) is expressed as (see also [7]) 2
n0 X
. j/ .in; j/
ck ` k
(11.291)
jD1
where the factor 2 can be omitted. Observation 11.4 As we have previously stated, best system performance is obtained by jointly designing the encoder and the bit mapper. However in some systems, typically radio, an interleaver is used between the encoder and the bit mapper. In this case joint detection and decoding are impossible to implement in practice. Detection with soft output followed by decoding with soft input remains a valid approach, obviously after re-ordering the LLR as determined by the deinterleaver. In applications that require a soft output (see Section 11.6), the decoder, that is called in this case soft-input soft-output (SISO), can use one of the versions of the FBA or the SOVA.7
Sequential decoding Sequential decoding of convolutional codes represented the first practical algorithm for ML decoding. It has been employed, for example, for the decoding of signals transmitted by deep-space probes, such as the Pioneer, 1968 [10]. There exist several variants of sequential decoding algorithms, that are characterized by the search of the optimum path in a tree diagram (see Figure 11.9b), instead of along a trellis diagram, as considered, e.g., by the VA. Sequential decoding is an attractive technique for the decoding of convolutional codes and trellis codes if the number of states of the encoder is large [11]. In fact, as the implementation complexity of ML decoders such as the Viterbi decoder grows exponentially with the constraint length of the code, ¹, the complexity of sequential decoding algorithms is essentially independent of ¹. On the other hand, sequential decoding presents the drawback that the number of computations Nop required for the decoding process to advance by one branch in the decoder tree is a random variable with a Pareto distribution, i.e. P[Nop > N ] D AN ² 7
An extension of SISO decoders for the decoding of block codes is found in [8, 9].
(11.292)
918
Chapter 11. Channel codes
where A and ² are constants that depend on the channel characteristics and on the specific code and the specific version of sequential decoding used. Real-time applications of sequential decoders require buffering of the received samples. As practical sequential decoders can perform only a finite number of operations in a given time interval, resynchronization of the decoder must take place if the maximum number of operations that is allowed for decoding without incurring buffer saturation is exceeded. To determine whether it is practical for a receiver to adopt sequential decoding, we recall the definition of cut-off rate of a transmission channel, and the associated minimum signalto-noise ratio .E b =N0 /0 (see page 509). Sequential decoders exhibit very good performance, with a reduced complexity as compared to the VA, if the constraint length of the code is sufficiently large and the signal-to-noise ratio is larger than .E b =N0 /0 . If the latter condition is not verified, the average number of operations required to produce one symbol at the decoder output is very large. The Fano Algorithm In this section we consider sequential decoding of trellis codes, a class of codes that will be studied in detail in Chapter 12. However, the algorithm can be readily extended to convolutional codes. At instant k, the k0 information bits bk D [bk.1/ ; : : : ; bk.k0 / ] are input to a rate k0 = .k0 C 1/ convolutional encoder with constraint length ¹ that outputs the coded bits ck D [ck.0/ ; : : : ; ck.k0 / ]. The k0 C 1 coded bits select from a constellation with M D 2k0 C1 elements a symbol ak , which is transmitted over an additive white Gaussian noise channel. Note that the encoder tree diagram has 2k0 branches that correspond to the values of bk stemming from each node. Each branch is labeled by the symbol ak selected by the vector ck . The received signal is given by (see (11.275)) z k D a k C wk
(11.293)
The received signal sequence is input to a sequential decoder. Using the notation of Section 8.10.1, in the absence of ISI and assuming a D α, the ML metric to be maximized can be written as [6] 2 3 0.α/ D
K 1 6 X kD0
7 6log X Pzk jak .²k jÞk / 7 B 2 4 5 Pzk jak .²k jÞi /Pak .Þi /
(11.294)
Þi 2A
where B is a suitable constant that determines a trade–off between computational complexity and performance and is related to the denominator in (11.294). Choosing B D k0 and Pak .Þi / D M1 D 2.k0 C1/ , Þi 2 A, we obtain 2 6
0.α/ D
K 1 6 X 6 kD0
3
j²k Þk j2 2¦ I2
e 6log 2 6 2 6 X j²k Þ2i j 2¦ 4 I e Þi 2A
7 7 7 C 17 7 7 5
(11.295)
11.3. Convolutional codes
919
Various algorithms have been proposed for sequential decoding [12, 13, 14]. We will restrict our attention here to the Fano algorithm [6, 11]. The Fano algorithm examines only one path of the decoder tree at any time using the metric in (11.294). The considered path extends to a certain node in the tree and corresponds to a segment of the entire code sequence α, up to symbol Þk . Three types of moves between consecutive nodes on the decoder tree are allowed: forward, lateral, and backward. On a forward move, the decoder goes one branch to the right in the decoder tree from the previously hypothesized node. This corresponds to the insertion of a new symbol ÞkC1 in (11.294). On a lateral move, the decoder goes from a path on the tree to another path differing only in the last branch. This corresponds to the selection of a different symbol Þk in (11.294). The ordering among the nodes is arbitrary, and a lateral move takes place to the next node in order after the current one. A backward move is a move one branch to the left on the tree. This corresponds to the removal of the symbol Þk from (11.294). To determine which move needs to be made after reaching a certain node, it is necessary to compute the metric 0k of the current node being hypothesized, and consider the value of the metric 0k1 of the node one branch to the left of the current node, as well as the current value of a threshold T h, which can assume values that are multiples of a given constant 1. The transition diagram describing the Fano algorithm is illustrated in Figure 11.20. Typically, 1 assumes values that are of the order of the minimum distance between symbols. As already mentioned, real-time applications of sequential decoding require buffering of the input samples with a buffer of size S. Furthermore, the depth of backward search is also finite and is usually chosen to be at least five times the constraint length of the code. To avoid erasures of output symbols in case of buffer saturation, in [15] a buffer looking algorithm (BLA) is proposed. The buffer is divided into L sections, each with size S j ; j D 1; : : : ; L. A conventional sequential decoder (primary decoder) and L 1 secondary decoders are used. The secondary decoders employ fast algorithms, such as the M-algorithm [16], or variations of the Fano algorithm that are obtained by changing the value of the bias B in the metric (11.294). Example 11.3.2 (Sequential decoding of a 512-state 16-PAM trellis code) We illustrate sequential decoding with reference to a 512-state 16-PAM trellis code specified for SHDSL transmission (see Chapter 17). The encoder for this trellis code comprises a rate 1=2 nonsystematic non-recursive convolutional encoder with constraint length ¹ D 9 and a bit mapper as specified in Figure 11.21. The symbol error probabilities versus signal-to-noise ratio 0 obtained by sequential decoding with infinite buffer size and depth of backward search of 64 and 128 symbols, and by a 512–state VA decoding with length of the path memory of 64 and 128 symbols are shown in Figure 11.22. Also shown for comparison are the error probabilities obtained for uncoded 8-PAM and 16-PAM transmission.
11.3.3
Performance of convolutional codes
H , and bit error probability of the For binary convolutional codes with free distance dfree channel equal to Pbit , decoding with hard input yields .dec/ Pbit ' A 2dfree Pbit H
(11.296)
920
Chapter 11. Channel codes
Figure 11.20. Transition diagram of the Fano algorithm.
and decoding with soft input, for a system with antipodal signals, yields 0s .dec/ Pbit
' AQ@
H 2E Rc dfree b
N0
1 A
(11.297)
where A is a constant [17]. In particular, we consider BPSK transmission over an ideal AWGN channel. Assuming an encoder with rate Rc D 1=2 and constraint length ¹ D 6, the coding gain for a soft Viterbi decoder is about 3.5 dB for Pbit D 103 ; it becomes about 4.6 dB for Pbit D 105 . Note that a soft decoder allows gain of about 2.4 dB with respect to a hard decoder, for Pbit < 103 .
11.4. Concatenated codes
921
Figure 11.21. (a) Block diagram of the encoder and bit mapper for a trellis code for 16-PAM transmission, (b) structure of the rate-1/2 convolutional encoder, and (c) bit mapping for the 16-PAM format.
11.4
Concatenated codes
Concatenated coding is usually introduced to achieve an improved error correction capability [18]. Interleaving is also commonly used in concatenated coding schemes, as illustrated in Figure 11.23, so that the decoding processes of the two codes (inner and outer) can be considered approximately independent. The first decoding stage is generally utilized to produce soft decisions on the information bits, that are passed to the second decoding stage. While the forward-backward algorithm directly provides a soft output (see (11.285)), the Viterbi algorithm must be slightly modified.
Soft-output Viterbi algorithm (SOVA) We have seen that the FBA in the original MAP version directly yields a soft output, at the expense of a large computational complexity (see page 916). The Max-Log-MAP criterion has a reduced complexity, but there remains the problem of having to perform the two procedures, forward and backward. We now illustrate how to modify the VA to obtain a soft output equivalent to that produced by the Max-Log-MAP.
922
Chapter 11. Channel codes
0
10
−1
16−PAM, uncoded
10
−2
10
−3
10 P
e
8−PAM, uncoded −4
10
−5
10
−6
10
−7
10
16
SD, depth=64 SD, depth=128 VA, path mem.=64 VA, path mem.=128 18
20
22 Γ (dB)
24
26
28
Figure 11.22. Symbol error probabilities for the 512-state 16-PAM trellis code with sequential decoding (depth of search limited to 64 or 128) and 512-state Viterbi decoding (length of path memory limited to 64 or 128). Symbol error probabilities for uncoded 8-PAM and 16-PAM transmission are also shown.
Figure 11.23. Transmission scheme with concatenated codes and interleaver.
In this section we consider different methods to generate the soft output. The difference metric algorithm (DMA). Figure 11.24 shows a section of a trellis diagram with four states, where we assume sk D .bk ; bk1 /. We consider two states at instant k 1 that differ in the least significant bit bk2 of the binary representation, that is s.0/ k1 D .00/
11.4. Concatenated codes
923
Figure 11.24. Soft-output Viterbi algorithm.
and s.1/ k1 D .01/. A transition from each of these two states to state sk D .00/ at instant k is allowed. According to the VA, we choose as survivor sequence the sequence that minimizes the metric ² ¦ .i/ sk1 !sk .i / (11.298) min 0k1 .sk1 / C k i 2f0;1g
/ / where 0k1 .s.ik1 / is the path metric associated with the survivor sequence up to state s.ik1 .i/
s
!sk
at instant k 1, and k k1 denotes the branch metric associated with the transition from / state s.ik1 to state sk . Let Ak D 0k1 .00/ C k00!00 and Bk D 0k1 .01/ C k01!00 , then we choose the upper or the lower transition according to whether 1k D Ak Bk is smaller or larger than zero, respectively. Note that j1k j is a reliability measure of the selection of a certain sequence as survivor sequence. 0 In other words, if j1k j is small, there is a non-negligible probability that the bit bk2
/ associated with the transition from state s.ik1 to sk on the survivor sequence is in error. The difference j1k j D ½k yields the value of the soft decision for bk2 , in case the final sequence chosen by the Viterbi algorithm includes the state sk ; conversely, this information is disregarded. Thus the DMA can be formulated as follows. For each state sk of the trellis diagram at instant k, the metric 0k .sk / and the most 0 g are memorized, recent .K d C 1/ bits of the survivor sequence b0 .sk / D fbk0 ; : : : ; bkK d where K d denotes the path memory depth of the VA. Furthermore, the reliability measures λ.sk / D f½k ; : : : ; ½kK d g associated with the bits b0 .sk / are also memorized. Interpreting bk and bOk as binary symbols in the alphabet f1; 1g (see note 6 on page 915), the soft output associated with bk is given by
`Qk D bOk ½kC2 where fbOk g is the sequence of information bits associated with the ML sequence.
(11.299)
924
Chapter 11. Channel codes
The soft-output VA (SOVA). As for the DMA, the SOVA determines the difference between the metrics of the survivor sequences on the paths that converge to each state of the trellis, and updates at every instant k the reliability information λ.sk / for each state of the trellis. To perform this update, the sequences on the paths that converge to a certain state are compared to identify the positions at which the information bits of the two sequences differ. With reference to Figure 11.24, we denote the two paths that converge to the state (00) at instant k as path 0 and path 1. Without loss of generality we assume that the sequence associated with path 0 is the survivor sequence, and thus the sequence with the smaller .0/ .0/ .1/ .1/ .1/ cost; furthermore we define λ.s.0/ k / D f½k ; : : : ; ½kK d g and λ.sk / D f½k ; : : : ; ½kK d g as the two reliability vectors associated with the information bits of two sequences. If one information bit along path 0 differs from the corresponding information bit along path 1, then its reliability is updated according to the rule for i D k K d ; : : : ; k 1 ½i D min.j1k j; ½i.0/ /
.1/ if bi.0/ 2 6D bi 2
(11.300)
With reference to Figure 11.24, the two sequences on path 0 and on path 1 diverge from state sk D .00/ at instant k 4. The two sequences differ in the associated information bits at the instants k and k 1; therefore, only ½k1 will be updated. Modified SOVA. In the modified version of the SOVA, the reliability of an information bit along the survivor path is also updated if the information bit is the same, according to the rule for i D k K d ; : : : ; k 1 ( min .j1k j; ½i.0/ / ½i D min .j1k j C ½i.1/ ; ½i.0/ /
.1/ if bi.0/ 2 6D bi 2
(11.301)
.1/ if bi.0/ 2 D bi 2
Note that (11.300) is still used to update the reliability if the information bits differ; this version of the SOVA gives a better estimate of ½i . As proved in [19], if the VA is used as decoder, the modified SOVA is equivalent to Max-Log-MAP decoding. An example of how the modified SOVA works is illustrated in Figure 11.25.
11.5
Turbo codes
Turbo codes, proposed in 1993 by Berrou and Glavieux [20, 21], constitute an evolution of concatenated codes, in the form of parallel concatenated convolutional codes (PCCC), and allow reliable transmission of information at rates near the Shannon limit [20, 21, 22]. As will be discussed in this section, the term turbo, even though it is used to qualify these codes, is rather tied to the decoder, whose principle is reminiscent of that of turbo engines.
Encoding For the description of turbo codes, we refer to the first code of this class that appeared in the scientific literature [20, 21]. A sequence of information bits is encoded by a simple
11.5. Turbo codes
925
Figure 11.25. Modified soft-output Viterbi algorithm.
Figure 11.26. Encoder of a turbo code with code rate Rc D 13 .
recursive systematic convolutional (RSC) binary encoder with code rate 1=2, to produce a first sequence of parity bits, as illustrated in Figure 11.26. The same sequence of information bits is permuted by a long interleaver and then encoded by a second recursive systematic convolutional encoder with code rate 1=2 to produce a second sequence of parity bits. Then the sequence of information bits and the two sequences of parity bits are transmitted. Note
926
Chapter 11. Channel codes
bk
ck(1)
●
ck(2) ●
●
D
●
D
●
D
●
interleaver
ck(3) ●
D
●
D
●
D
●
Figure 11.27. Turbo encoder adopted by the UMTS standard.
that the resulting code has rate Rc D 1=3. Higher code rates Rc are obtained by transmitting only some of the parity bits (puncturing). For example, for the turbo code in [20, 21], a code rate equal to 1/2 is obtained by transmitting only the bits of the parity sequence 1 with odd indices, and the bits of the parity sequence 2 with even indices. A specific example of turbo encoder is reported in Figure 11.27. The exceptional performance of turbo codes is due to one particular characteristic. We can think of a turbo code as being a block code for which an input word has a length equal to the interleaver length, and a code word is generated by initializing to zero the memory elements of the convolutional encoders before the arrival of each input word. This block code has a group structure. As for the usual block codes, the asymptotic performance, for large values of the signal-to-noise ratio, is determined by the code words of minimum weight and by their number. For low values of the signalto-noise ratio, also the code words of non-minimum weight and their multiplicity need to be taken into account. Before the introduction of turbo codes, the focus on designing codes was mainly on asymptotic performance, and thus on maximizing the minimum distance. With turbo codes, the approach is different. Because of the large ensemble of code words, the performance curve, in terms of bit error probability as a function of the signal-to-noise ratio, rapidly decreases for low values of the signal-to-noise ratio. For Pbit lower than 105 , where performance is determined by the minimum distance between code words, the bit error probability curve usually exhibits a reduction in the value of slope. The two encoders that compose the scheme of Figure 11.26 are called component encoders and they are usually identical. As mentioned above, Berrou and Glavieux proposed two recursive systematic convolutional encoders as component encoders. Later it was shown that it is not necessary to use systematic encoders [23, 17]. Recursive convolutional codes are characterized by the property that the code bits at a given instant do not depend only on the information bits at the present instant and the
11.5. Turbo codes
927
previous ¹ instants, where ¹ is the constraint length of the code, but on all the previous information bits, as the encoder exhibits a structure with feedback. Starting from a non-recursive nonsystematic convolutional encoder for a code with rate 1=n 0 , it is possible to obtain in a very simple way a recursive systematic encoder for a code with the same rate and the same code words, and hence with the same free distance H . Obviously, for a given input word, the output code words will be different in the dfree two cases. Consider for example a non-recursive nonsystematic convolutional encoder for a code with code rate 1=2. The code bits can be expressed in terms of the information bits as (see (11.254)) c .1/ .D/ D g .1;1/ .D/ b.D/ 0 c .2/ .D/ D g .2;1/ .D/ b.D/ 0
(11.302)
The corresponding recursive systematic encoder is obtained by dividing the polynomials in (11.302) by g .1;1/ .D/, and implementing the functions c.1/ .D/ D b.D/ g .2;1/ .D/ b.D/ c.2/ .D/ D .1;1/ g .D/
(11.303) (11.304)
Let us define d.D/ D
b.D/ c.2/ .D/ D .1;1/ .2;1/ g .D/ g .D/
(11.305)
then the code bits can be expressed as a function of the information bits and the bits of the sequence fdk g as ck.1/ D bk ¹ X gi.2;1/ dki ck.2/ D
(11.306) (11.307)
i D0
where, using the fact that g0.1;1/ .D/ D 1, from (11.305) we get d k D bk C
¹ X
gi.1;1/ dki
(11.308)
i D1
We recall that the operations in the above equations are in GF(2). Another recursive systematic encoder that generates a code with the same free distance is obtained by exchanging the role of the polynomials g .1;1/ .D/ and g .2;1/ .D/ in the above equations. One recursive systematic encoder corresponding to the non-recursive nonsystematic encoder of Figure 11.9(a) is illustrated in Figure 11.28. The 16-state component encoder for a code with code rate 1=2 used in the turbo code of Berrou and Glavieux [20, 21], is shown in Figure 11.29. The encoder in Figure 11.27, with an 8-state component encoder for a code with code rate 1/2, is adopted in the standard for third generation universal mobile telecommunications systems (UMTS) [24]. Turbo codes
928
Chapter 11. Channel codes
ck(1)
bk
dk
D
dk−1
D
dk−2
ck(2) Figure 11.28. Recursive systematic encoder that generates a code with the same free distance as the non-recursive nonsystematic encoder of Figure 11.9(a).
c (1) k
bk
D
D
D
D
c k(2) Figure 11.29. A 16-state component encoder for the turbo code of Berrou and Glavieux.
are also used in digital video broadcasting (DVB) [25] standards and in space telemetry applications as defined by the Consultative Committee for Space Data Systems (CCSDS) [26]. In [27] are listed generator polynomials of recursive systematic convolutional encoders for codes with rates 1/2, 1/3, 1/4, 2/3, 3/4, 4/5, and 2/4, that can be used for the construction of turbo codes. Another fundamental component in the structure of turbo codes is represented by a nonuniform interleaver. We recall that a uniform8 interleaver, as that described in Section 11.3.2, operates by writing input bits in a matrix by rows and reading them by columns. In practice, a non-uniform interleaver determines the permutation of the sequence of input bits so that adjacent bits in the input sequence are separated, after the permutation, by a number of bits that varies with the position of the bits in the input sequence. The interleaver determines directly the minimum distance of the code and therefore performance for high values of the signal-to-noise ratio. Nevertheless, the choice of the interleaver is not critical for low values of the signal-to-noise ratio. Beginning with the interleaver originally proposed 8
The adjective “uniform”, referred to an interleaver, is used with a different meaning in [23].
11.5. Turbo codes
929
in [20, 21], various interleavers have since been proposed (see [28] and references contained therein). One of the interleavers that yields better performance is the so-called spread interleaver [29]. Consider a block of M1 input bits. The integer numbers that indicate the position of these bits after the permutation are randomly generated with the following constraint: each integer randomly generated is compared with the S1 integers previously generated; if the distance from them is shorter than a prefixed threshold S2 , the generated integer is discarded and another one is generated until the condition is satisfied. The two parameters S1 and S2 must be larger than the memory of the two-component encoders. If the two-component encoders are equal, it is convenient to choose S1 D S2 . The computation time needed to generate the interleaver increases with S1 and S2 , and there is no guarantee that the procedure terminates successfully. Empirically, it has been verified that, p choosing both S1 and S2 equal to the closest integer to M1 =2, it is possible to generate the interleaver in a finite number of steps. Many variations of the basic idea of turbo codes have been proposed. For example, codes generated by serial concatenation of two convolutional encoders, connected by means of a non-uniform interleaver [30]. Parallel and serial concatenation schemes were then extended to the case of multilevel constellations to obtain coded modulation schemes with high spectral efficiency (see [31] and references contained therein).
The basic principle of iterative decoding The presence of the interleaver in the scheme of Figure 11.26 makes an encoder for a turbo code have a very large memory even if very simple component encoders are used. Therefore the optimum MLSD decoder would require a Viterbi decoder with an exceedingly large number of states and it would not be realizable in practice. For this reason we resort to a suboptimum iterative decoding scheme with a much lower complexity than the optimum scheme. However, as it was verified empirically, this scheme exhibits near optimum performance [23]. The decoder for the turbo encoder of Figure 11.26 is illustrated in Figure 11.30. The received sequences corresponding to the sequence of information bits and the first sequence of parity bits are decoded using a soft input soft output decoder, corresponding to the first convolutional encoder. Thus this decoder provides a soft decision for each information bit; these soft decisions are then used by a second decoder corresponding to the second convolutional encoder, together with the received sequence corresponding to the second sequence of parity bits. Soft decisions thus obtained are taken back to the input of the first decoder for a new iteration, where the additional information obtained by the second decoder is used to produce more reliable soft decisions. The procedure continues iteratively for about 10–20 cycles until final decisions are made on the information bits. The two component decoders of the scheme in Figure 11.30 are soft input soft output decoders that produce estimates on the reliability of the decisions. Therefore they implement the SOVA or the FBA (or one of its simplified realizations). The basic principle of iterative decoding is the following: each component decoder uses the “hints” of the other to produce more reliable decisions. In the next sections we will see in detail how this is achieved, and in particular how the reliability of the decisions is determined.
930
Chapter 11. Channel codes
Figure 11.30. Principle of the decoder for a turbo code with rate 1=3.
The algorithms for iterative decoding introduced with the turbo codes were also immediately applied in wider contexts. In fact, this iterative procedure may be used every time the transmission system includes multiple processing elements with memory interconnected by an interleaver. Iterative decoding procedures may be used, for example, for detection in the presence of intersymbol interference, also called turbo equalization or turbo detection [32] (see Section 11.6), for non-coherent decoding [33, 34], and for joint detection and decoding in the case of transmission over channels with fading [35]. Before discussing in detail iterative decoding, it is useful to revisit the FBA.
The forward-backward algorithm revisited The formulation of the FBA presented here is useful for the decoding of recursive systematic convolutional codes [36]. We consider a binary recursive systematic convolutional encoder for a code with rate k0 =n 0 , and constraint length ¹. Let the encoder input be given by a sequence of K vectors, each composed of k0 binary components. As described on page 903, each information vector to be encoded is denoted by (see (11.252)) bk D [bk.1/ ; bk.2/ ; : : : ; bk.k0 / ]
bk.i / 2 f1; 1g
k D 0; 1; : : : K 1
(11.309)
where k0 can be seen either as the number of encoder inputs or as the length of an information vector. As the convolutional encoder is systematic, at instant k the state of the
11.5. Turbo codes
931
convolutional encoder is given by the vector (see extension of (11.308)) .k0 C1/
s k D [ dk
.k C1/
.k0 C2/
0 ; : : : ; dk¹C1 ; dk
.k C2/
.n 0 /
0 ; : : : ; dk¹C1 ; : : : ; dk
.n /
0 ] ; : : : ; dk¹C1
(11.310)
which has a number of components N2 D ¹ Ð .n 0 k0 /, equal to the number of the encoder memory elements. The set of states S, that is the possible values assumed by sk , is given by sk 2 S D fσ 1 ; σ 2 ; : : : ; σ Ns g
(11.311)
where Ns D 2 N2 is the number of encoder states. It is important to observe that the convolutional encoder can be seen as a sequential finite-state machine with i.i.d. input bk , and state transition function sk D f s .bk ; sk1 /. Hence, for a given information vector bk , the transition from state sk1 D σ i to state sk D σ j is unique, in correspondence of which a code vector is generated, that is expressed as .k /
.k0 C1/
ck D [ck.1/ ; ck.2/ ; : : : ; ck 0 ; ck
.n 0 /
; : : : ; ck
. p/
] D [ c.s/ k ; ck ]
(11.312)
where the superscript .s/ denotes systematic bits, and . p/ denotes parity check bits. Then . p/ c.s/ k D bk , and from (11.307) we can express ck as a function of sk and sk1 as . p/
ck
D f . p/ .sk ; sk1 /
(11.313)
The values assumed by the code vectors are indicated by β D [ þ .1/ ; þ .2/ ; : : : ; þ .k0 / ; þ .k0 C1/ ; : : : ; þ .n 0 / ] D [ β .s/ ; β . p/ ]
(11.314)
For simplicity, we assume that the code binary symbols so determined are transmitted by a binary modulation scheme over an AWGN channel. In this case, at the decision point of the receiver, we get the signal (see (11.275)), z k D ck C wk
(11.315)
where ck 2 f1; 1g denotes a code bit, and fwk g is a sequence of real-valued i.i.d. Gaussian noise samples with variance ¦ I2 . It is useful to organize the samples fz k g into subsequences that follow the structure of the code vectors (11.312). Then we introduce the vectors .k /
.k0 C1/
zk D [ z k.1/ ; z k.2/ ; : : : ; z k 0 ; z k
.n 0 /
; : : : ; zk
. p/
] D [ z.s/ k ; zk ]
(11.316)
As usual we denote as ρ k an observation of zk , . p/
ρ k D [ ²k.1/ ; ²k.2/ ; : : : ; ²k.k0 / ; ²k.k0 C1/ ; : : : ; ²k.n 0 / ] D [ ρ .s/ k ; ρk ]
(11.317)
We recall from Section 8.10 that the FBA yields the detection of the single information vector bk , k D 0; 1; : : : ; K 1, expressed as .k / bO k D [ bOk.1/ ; bOk.2/ ; : : : ; bOk 0 ]
(11.318)
through the computation of the a posteriori probability. We also recall that in general for a sequence a D [a0 ; : : : ; ak ; : : : ; a K 1 ], with the notation alm we indicate the subsequence formed by the components [al ; alC1 ; : : : ; am ].
932
Chapter 11. Channel codes
Defining the likelihood of the generic information vector (see (8.220)), Lk .β .s/ / D P[bk D β .s/ j z0K 1 D ρ 0K 1 ]
(11.319)
detection by the MAP criterion is expressed as bO k D arg max Lk .β .s/ / β .s/
k D 0; 1; : : : ; K 1
(11.320)
We note that the likelihood associated with the individual bits of the information vector bk are obtained by suitably adding (11.319), as Lk;i .Þ/ D P[bk.i / D Þ j z0K 1 D ρ 0K 1 ] X Lk .β .s/ / Þ 2 f1; 1g D
(11.321)
β .s/ 2f1;1gk0 þ .i/ DÞ
In a manner similar to the analysis of page 668, we introduce the following quantities: 1. The state transition probability 5. j j i/ D P[sk D σ j j sk1 D σ i ], that assumes non-zero values only if there is a transition from the state sk1 D σ i to the state sk D σ j for a certain input β .s/ , and we write .s/ 5. j j i/ D P[bk D β .s/ ] D L.a/ k .β /
(11.322)
.s/ L.a/ k .β / is called the a priori information on the information vector bk D β .s/ , and is one of the soft inputs.
2. For an AWGN channel the channel transition probability pzk .ρ k j j; i/ can be separated into two contributions, one due to the systematic bits and the other to the parity check bits, pzk .ρ k j j; i/ D P[zk D ρ k j sk D σ j ; sk1 D σ i ] .s/ D P[z.s/ k D ρ k j sk D σ j ; sk1 D σ i ] . p/
P[zk
. p/
D ρk
j sk D σ j ; sk1 D σ i ] . p/
. p/
.s/ .s/ .s/ D P[z.s/ k D ρ k j ck D β ] P[zk D ρ k 00 1 1 k0
B D @@ q 00
1 2³ ¦ I2
A
e
1n 0 k0
B@ 1 A @ q 2³ ¦ I2
1 .s/ jjρ β .s/ jj2 C 2¦ I2 k
A 1
1 . p/ 2 jjρ k β . p/ jj2 C e 2¦ I
A
. p/
j ck
D β . p/ ] (11.323)
11.5. Turbo codes
933
3. We merge (11.322) and (11.323) into one variable (see (8.229)), C k . j j i/ D P[zk D ρ k ; sk D σ j j sk1 D σ i ] D pzk .ρ k j j; i/ 5. j j i/ 0 1n 0 1 A C .s/ . j j i/ C . p/ . j j i/ D @q k k 2 2³ ¦ I
(11.324)
where C k.s/ . j
j i/ D e
. p/
C k . j j i/ D
1 jjρ .s/ β .s/ jj2 2¦ I2 k
.s/ L.a/ k .β /
1 . p/ 2 jjρ k β . p/ jj2 e 2¦ I
(11.325) (11.326)
The two previous quantities are related, respectively, to the systematic bits and the parity check bits of a code vector. Observe that the exponential term in (11.325) .s/ represents the reliability of a certain a priori information L.a/ k .β / associated .s/ with β . 4. The computation of the forward and backward metrics is carried out as in the general case. - Forward metric, for k D 0; 1; : : : ; K 1: Fk . j/ D
Ns X
C k . j j `/ Fk1 .`/
j D 1; : : : ; Ns
(11.327)
`D1
- Backward metric, for k D K 1; K 2; : : : ; 0: Bk .i/ D
Ns X
BkC1 .m/ C kC1 .m j i/
i D 1; : : : ; Ns
(11.328)
mD1
Suitable initializations are obtained, respectively, through (8.237) and (8.244). 5. By using the total probability theorem, the likelihood (11.319) can be written as Lk .β .s/ / D A
Ns X
P[sk1 D σ i ; sk D σ j ; z0K 1 D ρ 0K 1 ] (11.329)
i D1 σ j D f s .β .s/ ; σ i / where f s is the state transition function, and the multiplicative constant A D 1=P[z0K 1 D ρ 0K 1 ] is irrelevant for vector detection as can be seen from (11.320). We note that the summation in (11.329) is over all transitions from the general state sk1 D σ i to the state sk D σ j D f s .β .s/ ; σ i / generated by the information vector bk D β .s/ . On the other hand, the probability in (11.329) can be
934
Chapter 11. Channel codes
written as P[sk1 D σ i ; sk D σ j ; z0K 1 D ρ 0K 1 ] K 1 K 1 D P[zkC1 D ρ kC1 j sk1 D σ i ; sk D σ j ; z0k D ρ 0k ]
P[sk D σ j ; zk D ρ k j sk1 D σ i ; z0k1 D ρ 0k1 ] P[sk1 D σ i ; z0k1 D ρ 0k1 ]
(11.330)
K 1 K 1 D P[zkC1 D ρ kC1 j sk D σ j ]
P[sk D σ j ; zk D ρ k j sk1 D σ i ] P[sk1 D σ i ; z0k1 D ρ 0k1 ] D Bk . j/ Ck . j j i/ Fk1 .i/ Substituting for C k . j j i/ the expression in (11.324), the likelihood becomes .a/
.int/
Lk .β .s/ / D B Lk .β .s/ / Lk
.ext/
.β .s/ / Lk
.β .s/ /
(11.331)
where B D A=.2³ ¦ I2 / is an irrelevant constant, .β .s/ / D e L.int/ k
1 .s/ jjρ β .s/ jj2 2¦ I2 k
(11.332)
and L.ext/ .β .s/ / D k
Ns X
. p/
Bk . j/ Ck . j j i/ Fk1 .i/
(11.333)
i D1 σ j D f s .β .s/ ; σ i / Observing each term in (11.331), we make the following considerations. .s/ .s/ i. L.a/ k .β / represents the a priori information on the information vector bk D β .
.β .s/ / depends on the received samples associated with the information vecii. L.int/ k tor and on the channel characteristics. .β .s/ / represents the extrinsic information extracted from the received samiii. L.ext/ k ples associated with the parity check bits. This is the incremental information on the information vector obtained by the decoding process. 6. Typically it is easier to work with the logarithm of the various likelihoods. We associate with each bit of the code vector ck a log-likelihood ratio (LLR) that depends on the channel (see (11.285)), that is .in; p/
0/ .in/ D [ `.in;1/ ; : : : ; `.in;n ] D [ .in;s/ ; k k k k k
]
(11.334)
11.5. Turbo codes
935
For binary modulation, from (11.315), we get (see (11.287)) 2 ρk ¦ I2
.in/ D k
(11.335)
where ρ k is the observation at the instant k. We define now two quantities that are related, respectively, to the systematic bits and the parity check bits of the code vector, as k0 1X `.in;m/ þ .m/ 2 mD1 k
(11.336)
n0 1 X `.in;m/ þ .m/ 2 mDk C1 k
(11.337)
.s/ `.s/ k .β / D
and . p/
`k . j; i/ D
0
where by (11.313) and (11.314) we have β . p/ D [ þ .k0 C1/ ; : : : ; þ .n 0 / ] D f . p/ .σ j ; σ i /
(11.338)
Expressing (11.325) and (11.326) as a function of the likelihoods (11.336) and (11.337), apart from factors that do not depend on fþ .m/ g; m D 1; : : : ; n 0 , we get .s/
C k.s/ . j j i/ D e`k 0
.a/
.β .s/ / `k .β .s/ /
e
(11.339)
and 0 . p/
Ck
. p/
. j j i/ D e`k
. j;i /
(11.340)
To compute the forward and backward metrics, we use, respectively, (11.327) and 0 . p/ 0 0 (11.328), where the variable C k . j j i/ is replaced by Ck . j j i/ D Ck.s/ . j j i/ Ck . j j . p/
0 . p/
i/. Similarly in (11.333) C k . j j i/ is replaced by Ck . j j i/. Taking the logarithm of (11.333) we obtain the extrinsic component `.ext/ .β .s/ /. k Finally, from (11.331), by ignoring non-essential terms, the log-likelihood associated with the information vector bk D β .s/ is given by .int/ .s/ .s/ .β / C `.ext/ .β .s/ / `k .β .s/ / D `.a/ k .β / C `k k
(11.341)
.s/ where `.int/ .β .s/ / D `.s/ k k .β / is usually called the intrinsic component.
Expression (11.341) suggests an alternative method to (11.333) to obtain .ext/ `k .β .s/ /, which uses the direct computation of `k .β .s/ / by (11.329) and (11.330), 0 where C k . j j i/ is replaced by Ck . j j i/, whose factors are given in (11.339) and
936
Chapter 11. Channel codes
.s/ (11.340). From the known a priori information `.a/ k .β / and the intrinsic information (11.336), from (11.341) we get .int/ .s/ .s/ `.ext/ .β .s/ / D `k .β .s/ / `.a/ .β / k k .β / `k
(11.342)
Going back to the expression (11.341), detection of the vector bk is performed according to the rule bO k D arg max `k .β .s/ /
(11.343)
β .s/
.β .s/ / (`k .β .s/ /) by the logarithm of (11.333) (or (11.329) and Note that to compute `.ext/ k (11.330)), we can use the Max-Log-MAP method discussed in Section 8.10.2. Example 11.5.1 (Systematic convolutional code with rate 1=2) For a convolutional code with rate Rc D 1=2 the information vector bk D [bk ] is composed . p/ of only one bit (k0 D 1), like the systematic part and the parity check part of ck D [ck.s/ ; ck ]. In this case it is sufficient to determine the log-likelihoods `k .1/ and `k .1/, or better the LLR Lk .1/ D `k .1/ `k .1/ (11.344) `k D ln Lk .1/ Detection of the information bit is performed according to the rule bOk D sgn.`k /
(11.345)
The a priori information at the decoder input is given by .a/
`k D ln
P[bk D 1] P[bk D 1]
(11.346)
from which we derive the a priori probabilities P[bk D þ
.s/
]De
.s/ `.a/ k .β /
eþ
D
1
.s/ `.a/ k
1Ce
þ .s/ `.a/ k
.a/
e 2 `k
D
1Ce
1
e2þ
`.a/ k
.s/ `.a/ k
þ .s/ D f1; 1g (11.347)
By using LLRs, (11.336) yields `.int/ D `.int/ .1/ `.int/ .1/ D `.in;1/ D `.in;s/ k k k k k
(11.348)
In turn (11.339) and (11.340) for k0 D 1 and n 0 D 2 simplify into 1
C k.s/ . j j i/ D e 2 0
0 . p/
Ck
.in;s/
þ .s/ .`k 1
. j j i/ D e 2
.a/
C`k /
.in; p/
þ . p/ `k
(11.349) (11.350)
11.5. Turbo codes
937
The extrinsic component is obtained starting from (11.333) and using the above variables `.ext/ k
D ln
L.ext/ .1/ k
L.ext/ .1/ k
D `.ext/ .1/ `.ext/ .1/ k k
(11.351)
From (11.341), apart from irrelevant terms, the LLR associated with the information bit bk can be written as .int/ `k D `.a/ C `.ext/ k C `k k
(11.352)
where the meaning of each of the three contributions is as follows. - A priori information `.a/ k . It is an a priori reliability measure on the bit bk . This value can be extracted either from the known statistic of the information sequence or, in the case of iterative decoding of turbo codes, from the previous analysis. D `.in;s/ . As it is evident from the case of binary modu- Channel information `.int/ k k 2 .s/ D ² , if the noise variance is low, the contribution of `.int/ lation, where `.in;s/ 2 k k k ¦I
usually dominates with respect to the other two terms; in this case bit detection is simply obtained by the sign of ²k.s/ . - Extrinsic information `.ext/ . It is a reliability measure that is determined by the k redundancy in the transmitted sequence. This contribution improves the reliability of transmission over a noisy channel using the parity check bits. The decomposition (11.352) forms the basis for the iterative decoding of turbo codes. Observation 11.5 In the case of multilevel modulation and/or for transmission over channels with ISI, the previous formulation of the decoding scheme remains unchanged, provided the expression (11.285) for f`.in;m/ g; m D 1; : : : ; n 0 , is used in place of (11.335). k Example 11.5.2 (Nonsystematic code and LLR associated with the code bits) Consider the case of a nonsystematic code. If the code is also non-recursive, for example as illustrated on page 915 for k0 D 1, we need to use in place of (11.310) the state definition (11.273). Now all bits are parity check bits and (11.312) and (11.316) become, respectively, . p/
ck D ck zk D
. p/ zk
.n 0 /
D [ ck.1/ ; : : : ; ck D
]
(11.353)
[ z k.1/ ; : : : ; z k.n 0 / ]
(11.354)
However, the information vector is still given by bk D [ bk.1/ ; : : : ; bk.k0 / ] with values α D [ Þ .1/ ; : : : ; Þ .k0 / ]; Þ .i / 2 f1; 1g. The likelihood (11.319) is given by Lk .α/ D P[bk D α j z0K 1 D ρ 0K 1 ]
(11.355)
938
Chapter 11. Channel codes
The various terms with superscript .s/ of the previous analysis vanish by setting k0 D 0. Therefore (11.336) and (11.337) become .s/ .β .s/ / D `.s/ `.int/ k k .β / D 0
(11.356)
and n0 1X `.in;m/ þ .m/ 2 mD1 k
. p/
`k . j; i/ D
(11.357)
where β D β . p/ D [ þ .1/ ; : : : ; þ .n 0 / ] D f .σ j ; σ i / is the code vector associated with the transition from state σ i to state σ j . Note that, apart from irrelevant factors, (11.357) coincides with (11.291). For k0 D 1, it is convenient to use LLRs; in particular (11.352) yields a LLR associated with the information bit bk that is given by .ext/ `k D `.a/ k C `k
(11.358)
can be obtained directly using (11.351), (11.340), and (11.333). where `.ext/ k .q/ In some applications it is useful to associate a LLR with the encoded bit ck ; q D 1; : : : ; n 0 , rather than to the information bit bk . We define .q/
`Nk;q D ln
P[ck
.q/
P[ck
D 1 j z0K 1 D ρ 0K 1 ]
D 1 j z0K 1 D ρ 0K 1 ]
(11.359)
Let `N.a/ k;q be the a priori information on the code bits. The analysis is similar to the previous .q/
case but now, with respect to the encoder output, ck is regarded as an information bit, while the remaining bits ck.m/ ; m D 1; : : : ; n 0 ; m 6D q, are regarded as parity check bits. Equations (11.336), (11.337), (11.349), and (11.350), are modified as follows: 1 .in;q/ .q/ .q/ þ `N.s/ k;q .β / D `k 2 n0 1 X . p/ `.in;m/ þ .m/ `Nk;q . j; i/ D 2 mD1 k
(11.360) (11.361)
m 6D q
and 0
1
0 . p/
. p/ `Nk;q . j;i /
.s/ C k;q . j j i/ D e 2
C k;q . j j i/ D e
.in;q/
þ .q/ .`k
C`N.a/ k;q /
(11.362) (11.363)
Associated with (11.363) we obtain `N.ext/ k;q by using (11.351) and (11.333). The overall result is given by .in;q/ `Nk;q D `N.a/ C `N.ext/ k;q C `k k;q
q D 1; : : : ; n 0
(11.364)
11.5. Turbo codes
939
Example 11.5.3 (Systematic code and LLR associated with the code bits) With reference to the previous example, if the code is systematic, whereas (11.352) holds .q/
for the systematic bit ck.1/ , for the parity check bits ck the following relations hold [37]. For k0 D 1 let Þ be the value of the information bit bk , bk D Þ, with Þ 2 f1; 1g, associated with the code vector ck D β D [Þ; þ .2/ ; : : : ; þ .n 0 / ]
(11.365)
where we assume þ .1/ D Þ. For q D 2; : : : ; n 0 , we get 1 .in;q/ .q/ .q/ `N.s/ þ k;q .þ / D `k 2
(11.366)
n0 1 X 1 . p/ `.in;m/ þ .m/ C `.a/ Þ `Nk;q . j; i/ D 2 mD1 k 2 k
(11.367)
m 6D q
.a/
where `k
is the a priori information of bk . Furthermore 0
1
0 . p/
N. p/
.s/ C k;q . j j i/ D e 2
.in;q/
þ .q/ `k
(11.368)
C k;q . j j i/ D e`k;q . j;i /
(11.369)
From (11.369) we get `N.ext/ k;q using (11.351) and (11.333). The overall result is given by .in;q/ C `N.ext/ `Nk;q D `k k;q
(11.370)
Iterative decoding In this section we consider the iterative decoding of turbo codes with k0 D 1. In this case, as seen in Example 11.5.1, using the LLRs simplifies the procedure. In general, for k0 > 1 we should refer to the formulation (11.341). We now give a step-by-step description of the decoding procedure of a turbo code with rate 1=3, of the type shown in Figure 11.27, where each of the two component decoders DEC1 and DEC2 implements the FBA for recursive systematic convolutional codes with rate 1=2. The decoder scheme is shown in Figure 11.30 where the subscript in LLR corresponds to the component decoder. In correspondence to the information bit bk , the turbo code generates the vector ck D [ck.1/ ; ck.2/ ; ck.3/ ]
(11.371)
where ck.1/ D bk . We now introduce the following notation for the observation vector .in/ k that relates to the considered decoder: .in; p/
D [`.in;s/ ; `k;1 .in/ k k
.in; p/
; `k;2
]
(11.372)
940
Chapter 11. Channel codes
.in; p/
.in; p/
where `.in;s/ corresponds to the systematic part, and `k;1 and `k;2 correspond to the k parity check parts generated by the first and second convolutional encoder, respectively. If some parity check bits are punctured to increase the rate of the code, at the receiver are set to zero. the corresponding LLRs `.in;m/ k 1. First iteration 1.1 Decoder DEC1 If the statistic of the information bits is unknown, then the bits of the information sequence are considered i.i.d. and the a priori information is zero, `.a/ k;1 D ln
P[bk D 1] D0 P[bk D 1] .in; p/
For k D 0; 1; 2; : : : ; K 1, observed `.in;s/ and `k;1 k 0 .s/
(11.373) , we compute according
0 . p/
to (11.349) and (11.350) the variables C k and C k , and from these the corresponding forward metric Fk . j/ (11.327). After the entire sequence has been received, we compute the backward metric Bk .i/ (11.328) and, using (11.333), .ext/ we find L.ext/ k;1 .1/ and Lk;1 .1/. The decoder soft output is the extrinsic information obtained by the LLR `.ext/ k;1 D ln
L.ext/ k;1 .1/ L.ext/ k;1 .1/
(11.374)
1.2 Interleaver Because of the presence of the interleaver, the parity check bit cn.3/ is obtained in correspondence to a transition of the convolutional encoder state determined by the information bit bn , where n depends on the interleaver pattern. In decoding, the extrinsic information `.ext/ k;1 , extracted from DEC1 , and the systematic obserare scrambled by the turbo code interleaver and associated with vation `.in;s/ k .in; p/ the corresponding observation `n;2 to form the input of the second component decoder.
1.3 Decoder DEC2 The extrinsic information generated by DEC1 is set as the a priori information .a/ `n;2 to the component decoder DEC2 , .a/ D ln `n;2
P[bn D 1] .ext/ D `n;1 P[bn D 1]
(11.375)
.ext/ of The basic idea consists in supplying DEC2 only with the extrinsic part `n;1 `n;1 , in order to minimize the correlation between the a priori information and the observations used by DEC2 . Ideally, the a priori information should be an independent estimate. .ext/ . As done for DEC1 , we extract the extrinsic information `n;2
11.5. Turbo codes
941
1.4 Deinterleaver The deinterleaver realizes the inverse function of the interleaver, .ext/ so that the extrinsic information extracted from DEC2 , `n;2 , is synchronized .in; p/
with the systematic part `.in;s/ and the parity check part `k;1 k
of the observation
of DEC1 . By a feedback loop the a posteriori information `.ext/ k;2 is placed at the input of DEC1 as a priori information `.a/ k;1 , giving origin to an iterative structure.
2. Successive iterations Starting from the second iteration each component decoder has at its input an a priori information. The information on the bits become more reliable as the a priori information stabilizes in sign and increases in amplitude. 3. Last iteration When the decoder achieves convergence, the iterative process can stop and form the overall LLR (11.352), .ext/ `k;overall D `.in;s/ C `.ext/ k k;1 C `k;2
k D 0; 1; : : : ; K 1
(11.376)
and detection of the information bits bk is obtained by bOk D sgn.`k;overall /
(11.377)
To make decoding more reliable, the final state of each component decoder is set to zero, thus enabling an initialization of the backward metric as in (8.244). As illustrated in Figure 11.31, at the instant following the input of the last information bit, that is for k D K , the commutator is switched to the lower position, and therefore we have dk D 0; after ¹ clock intervals the zero state is reached. The bits ck.1/ and ck.2/ , for k D K ; K C 1; : : : ; K C ¹ 1, are appended at the end of the code sequence to be transmitted.
Performance evaluation Performance of the turbo code with the encoder of Figure 11.27 is evaluated in terms of error probability and convergence of the iterative decoder implemented by the FBA. For the memoryless AWGN channel, error probability curves versus E b =N0 are plotted in Figure 11.32 for a sequence of information bits of length K D 640, and various numbers of ck(1) ck(2) bk ●
●
dk
D
dk−1
●
D
dk−2 ●
D
●
Figure 11.31. Termination of trellis.
●
942
Chapter 11. Channel codes
0
10
−1
10
−2
10
−3
bit
P(dec)
10
−4
10
−5
10
−6
10
−7
10 −0.25
1 2 3 4 6 8 0
0.25
0.5
0.75
Eb/N0 (dB)
1
1.25
1.5
1.75
Figure 11.32. Performance of the turbo code defined by the UMTS standard, with length of the information sequence K D 640, and various numbers of iterations of the iterative decoding process.
0
10
Eb/N0=0dB Eb/N0=0.5dB Eb/N0=1dB Eb/N0=1.5dB
−1
10
−2
bit
P(dec)
10
−3
10
−4
10
−5
10
1
2
3
4
5
6
7
8
9
10
11
12
Number of iterations
Figure 11.33. Curves of convergence of the decoder for the turbo code defined by the UMTS standard, for K D 320 and various values of Eb =N0 .
11.6. Iterative detection and decoding
943
0
10
K=40 K=320 K=640
−1
10
−2
10
−3
P(dec) bit
10
−4
10
−5
10
−6
10
−7
10 −0.75
−0.5
−0.25
0
0.25
0.5
Eb/N0 (dB)
0.75
1
1.25
1.5
1.75
Figure 11.34. Performance of the turbo code defined by the UMTS standard achieved after 12 iterations, for K D 40, 320 and 640.
iterations of the iterative decoding process. Note that performance improves as the number of iterations increases; however, the gain between consecutive iterations becomes smaller as the number of iterations increases. .dec/ is given as a function of the number of iterIn Figure 11.33, the error probability Pbit ations, for fixed values of E b =N0 , and K D 320. From the behavior of the error probability we deduce possible criteria for stopping the iterative decoding process at convergence [36]. A timely stop of the iterative decoding process leads to a reduction of the decoding delay and of the overall computational complexity of the system. Note, however, that convergence is not always guaranteed. The performance of the code depends on the length K of the information sequence. Figure 11.34 illustrates how the bit error probability decreases by increasing K , for a constant E b =N0 . A higher value of K corresponds to an interleaver on longer sequences and thus the assumption of independence among the inputs of each component decoder is better satisfied. Moreover, the burst errors introduced by the channel are distributed over all the original sequence, increasing the correction capability of the decoder. As the length of the interleaver grows, also the latency of the system increases.
11.6
Iterative detection and decoding
We consider the transmitter of Figure 11.35 composed of a convolutional encoder, interleaver, bit mapper and modulator for 16-PAM. Interpreting the channel as a finite-state machine, the overall structure may be interpreted as a serial concatenated convolutional
944
Chapter 11. Channel codes
bl
convolutional code
cm
interleaver
cn
S/P
ck
ak
BMAP
modulator
Figure 11.35. Encoder structure, bit mapper, and modulator; for 16-PAM: ck D [c4k ; c4k1 ; c4k2 ; c4k3 ]. (a)
`n,det
interleaver
bit likelihood (a)
`m,dec=0
(a,SYM)
`k,det (γ) rk
SISO detector
(ext)
`n,det
SISO decoder
`m,dec (in)
deinterleaver
`m,dec (ext)
SI decoder ^
bl
Figure 11.36. Iterative detection and decoding.
code (SCCC). The procedure of SISO detection and SI decoding of page 916 can be made iterative by applying the principles of the previous section, by including a SISO decoding stage. With reference to Figure 11.36, a step by step description follows. 0. Initialization Suppose we have no information on the a priori probability of the code bits, therefore we associate with cn a zero LLR, .a/ `n;det D0
(11.378)
1. Detector First we associate a log-likelihood with the two possible values of cn D , 2 f1; 1g, according to the rule .a/ .a/ .1/ D `n;det `n;det
.a/ `n;det .1/ D 0
(11.379)
Then we express the symbol ak as a function of the bits fcn g according to the bit mapper, for example, for 16-PAM, ak D BMAP fck D [c4k ; c4k1 ; c4k2 ; c4k3 ]g
(11.380)
11.6. Iterative detection and decoding
945
Assuming the sequence fcn g is a sequence of i.i.d. binary symbols, we associate with each value of the symbol ak the a priori information expressed by the log-likelihood M/ `.a;SY . / D k;det
3 X
`.a/ 4kt;det .t /
2A
(11.381)
tD0
where D BMAP f[0 ; : : : ; 3 ]gt 2 f1; 1g. For multilevel transmission over a channel with ISI, the FBA of Section 8.10.2 provides a log-likelihood for each value of ak . The new feature is that now in (8.224) we take into account the a priori information on the various values of akCL 1 ; then (8.229) becomes C k . j j i/ D q
1
e
1 j²k u k j2 `.a;SY M/ . / 2¦ I2 e kCL 1 ;det
(11.382)
2³ ¦ I2
where D f .σ j ; σ i / 2 A is the symbol associated with the transition from the state σ i to the state σ j on the trellis determined by the ISI. If Vk .i/; i D 1; : : : ; Ns , denotes the metric corresponding to the various states of the trellis, we associate with each value of the code bits fcn g the following likelihood: L4.kCL 1 /t;det .Þ/ D
Ns X
Vk .i/
Þ 2 f1; 1g
t D 0; : : : ; 3 (11.383)
i D1 σ i such that c4.kCL /t D Þ 1
Taking the logarithm of (11.383), we obtain the LLR `n;det D `n;det .1/ `n;det .1/
(11.384)
To determine the extrinsic information associated with fcn g, we subtract the a priori information from (11.384), .ext/ .a/ `n;det D `n;det `n;det
(11.385)
Note that in this application, the detector considers the bits fcn g as information bits and the log-likelihood associated with cn at the detector output is due to the channel information9 in addition to the a priori information. .a/ in (11.385) is weighted by a coefficient, which is In [38], the quantity `n;det initially chosen small, when the a priori information is not reliable, and is increased after each iteration.
2. Deinterleaver .ext/ are re-ordered according to the deinterleaver to provide the seThe metrics `n;det quence `.ext/ m;det .
9
For the iterative decoding of turbo codes, this information is defined as intrinsic.
946
Chapter 11. Channel codes
3. Decoder (SISO) As input LLR, we use .ext/ `.in/ m;dec D `m;det
(11.386)
`.a/ m;dec D 0
(11.387)
and we set
in the lack of an a priori information on the code bits fcm g. Indeed, we note that in .in/ the various formulae the roles of `.a/ m;dec and `m;dec can be interchanged. Depending on whether the code is systematic or not, we use the SISO decoding procedure reported in Example 11.5.1 and Example 11.5.2, respectively. In both cases we associate with the encoded bits cm the quantity .in/ `.ext/ m;dec D `m;dec `m;dec
(11.388)
that is passed to the SISO detector as a priori information, after reordering by the interleaver. 4. Last iteration After a suitable number of iterations, the various metrics stabilize and from the LLRs f`.in/ m;dec g associated with fcm g, the SI decoding of bits fbl g is performed, using the procedure of Example 11.5.1.
11.7
Low-density parity check codes
Low-density parity check (LDPC) codes were introduced by Gallager [6] as a family of linear block codes with parity check matrices containing mostly zeros and only a small number of ones. The “sparsity” of the parity check matrices defining LDPC codes is the key for the efficient decoding of these codes by a message-passing procedure also known as the “sum-product algorithm”. LDPC codes and their efficient decoding were “reinvented” by MacKay and Neal [39, 40] in the mid-1990s, shortly after Berrou and Glavieux introduced the turbo-codes discussed in Section 11.5. Subsequently, LDPC codes have generated interest from a theoretical as well as from a practical viewpoint and many new developments have taken place. It is today well acknowledged that LDPC codes are as good as turbo codes, as they are based on a similar design philosophy. Also the decoding techniques used for both methods can be viewed as different realizations of the same fundamental decoding process. However, the soft input soft output forward backward algorithm of Section 11.5, or suboptimal versions of it, used for turbo decoding is rather complex, whereas the sum-product algorithm used for LDPC decoding lends itself to parallel implementation and is computationally simpler. LDPC codes, on the other hand, may lead to more stringent requirements in terms of storage. Recall that a linear .n 0 ; k0 / block code, where n 0 and k0 denote the transmitted block length and the source block length, respectively, can be described in terms of a parity
11.7. Low-density parity check codes
947
check matrix H, such that the equation Hc D 0 is satisfied for all code words c (see (11.20)). Each row of the r0 ð n 0 parity check matrix, where r0 D n 0 k0 is the number of parity check bits, defines a parity check equation that is satisfied by each code word c. For example, the (7,4) Hamming code is defined by the following parity check equations 2 3 c1 6 c2 7 2 36 7 7 c5 D c1 ý c2 ý c3 1 1 1 0 1 0 0 6 (check 1) 6 c3 7 4 1 1 0 1 0 1 0 5 6 c4 7 D 0 ! c6 D c1 ý c2 ý c4 (check 2) (11.389) 6 7 7 1 0 1 1 0 0 1 6 c D c ý c ý c (check 3) c 7 1 3 4 6 57 4 c6 5 c7 LDPC codes differ in major ways with respect to the above simple example; they usually have long block lengths n 0 in order to achieve near Shannon-limit performance, their parity check matrices are defined in nonsystematic form and exhibit a number of ones that is much less than r0 Ð n 0 . A parity check matrix for a .J; K /-regular LDPC code has exactly J ones in each of its columns and K ones in each of its rows. A parity check matrix can generally be represented by a bipartite graph, also called a Tanner graph, with two types of nodes: the bit nodes and the parity check nodes (or check nodes) [41]. A bit node n, representing the code bit cn , is connected to the check node m only if the element .m; n/ of the parity check matrix is equal to 1. No bit (check) node is connected to a bit (check) node. For example, the (7,4) Hamming code can be represented by the graph shown in Figure 11.37. We note in this specific case that, because the parity check matrix is given in systematic form, bit nodes c5 , c6 , and c7 in the associated graph are connected to single distinct check nodes. The parity check matrix of a .J; K /-regular LDPC code leads to a graph where every bit node is connected to precisely J check nodes and every check node is connected to precisely K bit nodes. We emphasize that the performance of an LDPC code depends on the random realization of the parity check matrix H. Hence these codes form a constrained random code ensemble. Graphical representations of LDPC codes are useful for deriving and implementing the iterative decoding procedure introduced in [6]. Gallager decoder is a message-passing
Figure 11.37. Tanner graph corresponding to the parity check matrix of the (7,4) Hamming code.
948
Chapter 11. Channel codes
decoder, in a sense to be made clear below, based on the so-called sum-product algorithm, which is a general decoding algorithm for codes defined on graphs.10
Encoding procedure Encoding is performed by multiplying the vector of k0 information bits b by the generator matrix G of the LDPC code: cT D bT G
(11.390)
where the operations are in GF(2). Recall that generator and parity check matrices satisfy the relation HGT D 0
(11.391)
Q D [A; Q I], where I is the From (11.27), the parity check matrix in systematic form is H Q is a binary matrix. Recall also that any other r0 ð n 0 matrix r0 ð r0 identity matrix, and A Q is a valid parity check matrix. H whose rows span the same space as H Given the block length n 0 of the transmitted sequence and the block length k0 of the information sequence, we select a column weight J , greater than or equal to 3. To define the code, we generate a rectangular r0 ð n 0 matrix H D [A B] at random with exactly J ones per column and, assuming a proper choice of n 0 and k0 , exactly K ones per row. The r0 ð k0 matrix A and the square r0 ð r0 matrix B are very sparse. If the rows of H are independent, which is usually true with high probability if J is odd [40], by Gaussian Q elimination and reordering of columns we determine an equivalent parity check matrix H in systematic form. From (11.26), we obtain the generator matrix in systematic form as ½ ½ I I GT D Q D (11.392) B1 A A where I is the k0 ð k0 identity matrix. Assuming initially antipodal linear signaling over an ideal AWGN channel, for the vector of transmitted symbols a D [a1 ; : : : ; an 0 ]T , ak 2 f1; 1g, corresponding to the code word c D [c1 ; : : : ; cn 0 ]T , the received vector is given by (11.393)
zDaCw where w denotes a vector of Gaussian noise samples with variance ¦ I2 .
Decoding algorithm Adopting the MAP criterion (8.221), the optimal decoder returns the components of the vector bO D [bO1 ; : : : ; bOk0 ] that maximize the a posteriori probabilities bOk D arg max P[bk D þ j z D ρ; G] þ2f0;1g
k D 1; : : : ; k0
(11.394)
10 A wide variety of other algorithms (e.g., the Viterbi algorithm, the forward backward algorithm, the iterative
turbo decoding algorithm, the fast Fourier transform, : : : ) can also be derived as specific instances of the sum-product algorithm [42].
11.7. Low-density parity check codes
949
Note that (11.394) is equivalent to the MAP criterion expressed by (11.321). However, an attempt to evaluate (11.394) by the direct computation of the joint probability distribution of the components of b given the observation would be impractical. Assuming the probability of b uniform, and w statistically independent of b, we resort to the knowledge of the parity check matrix to simplify the decoding problem. We will find the most likely binary vector x such that (see (11.20)) (11.395)
s D Hx D 0
given the received noisy vector z and a valid parity check matrix H. We call checks the elements si ; i D 1; : : : ; r0 , of the vector s, which are represented by the check nodes in the corresponding Tanner graph. Then the aim is to compute the marginal a posteriori probabilities Ln .þ/ D P[x n D þ j z D ρ; s D 0; G]
þ 2 f0; 1g
n D 1; : : : ; n 0
(11.396)
The detected code bits will then be given by cOn D arg max Ln .þ/ þ2f0;1g
n D 1; : : : ; n 0
(11.397)
We define as Hi;n the element with indices .i; n/ of the parity check matrix H. Let L.i/ D fn : Hi;n D 1g; i D 1; : : : ; r0 , be the set of the bit nodes that participate in the check with index i. Also, let L.i/nnQ be the set L.i/ from which the element with index nQ has been removed. Similarly, let M.n/ D fi : Hi;n D 1g; n D 1; : : : ; n 0 , be the set of the check nodes in which the bit with index n participates. The algorithm consists of two alternating steps, illustrated in Figure 11.38, in which þ þ quantities qi;n and ri;n , associated with each non-zero element of the matrix H, are iteratively þ
updated. The quantity qi;n denotes the probability that xn D þ; þ 2 f0; 1g, given the information obtained via checks other than check i: þ
qi;n D P[xn D þ j fsi 0 D 0; i 0 2 M.n/nig; z D ρ]
(11.398)
Moreover, we define the a posteriori probabilities qnþ D P[xn D þ j s D 0; z D ρ]
Figure 11.38. Message-passing decoding.
(11.399)
950
Chapter 11. Channel codes
þ
Given xn D þ; þ 2 f0; 1g, the quantity ri;n denotes the probability of check i being þ
satisfied and the other bits having a known distribution (given by the probabilities fqi;n 0 : n 0 2 L.i/nn; þ 2 f0; 1gg): þ
ri;n D P[si D 0; fxn 0 ; n 0 2 L.i/nng j xn D þ; z D ρ]
(11.400)
þ
In the first step, the quantities ri;n associated with check node i are updated and passed as messages to the bit nodes checked by check node i. This operation is performed for all þ check nodes. In the second step, quantities qi;n associated with bit node n are updated and passed as messages to the check nodes that involve bit node n. This operation is performed for all bit nodes. From (11.395), we note the property of (11.400) that 0 D 1 P[si D 1; fxn 0 ; n 0 2 L.i/nng j xn D 0; z D ρ] ri;n
D 1 P[si D 0; fxn 0 ; n 0 2 L.i/nng j xn D 1; z D ρ] D
(11.401)
1 1 ri;n
The algorithm is described as follows. Initialization. Let pn0 D P[xn D 0 j z D ρ] denote the probability that xn D 0 given the observation, and pn1 D P[xn D 1 j z D ρ] D 1 pn0 . For the AWGN channel with binary antipodal input symbols considered in this section, we have (see (8.262)) pn0 D þ
1 2 1 C e2²n =¦ I
pn1 D
1
(11.402)
2 1 C e2²n =¦ I
þ
Let qi;n D pn ; n 2 L.i/; i D 1; : : : ; r0 ; þ 2 f0; 1g. First step. We run through the checks, and for the i-th check we compute for each n 2 L.i/ 0 that, given x D 0, s D 0 and the other bits fx 0 : n 0 6D ng have the probability ri;n n i n 0 1 g. a distribution fqi;n 0 ; qi;n 0 From (11.400) we obtain X 0 D P[si D 0 j xn D 0; fxn 0 D Þn 0 : n 0 2 L.i/nng] ri;n Þn 0 2 f0; 1g : n 0 2 L.i/nn
Y n 0 2L.i /nn
Þ
0
qi;nn 0 (11.403)
1 D 1 r0 . Moreover, ri;n i;n
The conditional probabilities in the above expression are either one or zero, depending on whether si D 0 or si D 1 is obtained for the hypothesized values of 0 ; r 1 g can be found efficiently by the FBA, as xn and fxn 0 g. The probabilities fri;n i;n illustrated by the following example.
11.7. Low-density parity check codes
951
Example 11.7.1 Assume K D 4 and L.i/ D fn 1 ; n 2 ; n 3 ; n 4 g. The observation si can be expressed in terms of the input variables xk , k 2 L.i/, as si D xn 1 C xn 2 C xn 3 C xn 4 D
K X
xnl
(11.404)
lD1
where the addition is in GF(2). Let us define the state as s nk D
k X
xnl D sn k1 C xn k
(11.405)
lD1
with s 0 D 0, and observe that s n K D si . Following the formulation of the FBA in Section 8.10.2 we define the quantities: 1. Forward metric: Fn k . j/ D P[s n k D j]
j 2 f0; 1g
(11.406)
Fn k . j/ D P[xn k D j]Fn k1 . j/ C P[xn k D j ]Fn k1 . j/
k D 1; : : : K (11.407)
From (11.405) we obtain the recursive equation
where j denotes the one’s complement of j, with the initial condition Fn 0 .0/ D 1. 2. Backward metric: Bn k . j/ D P[si D 0 j sn k D j] D
1 X
P[si D 0 j sn kC1 D m; sn k D j]P[sn kC1 D m j sn k D j]
mD0
D
1 X
P[si D 0 j sn kC1 D m]P[x n kC1 D m ý j]
j 2 f0; 1g
mD0
(11.408) using (11.405) and the fact that si is independent of sn k given sn kC1 . From (11.408), we obtain the recursive equation Bn k . j/ D P[xn kC1 D j]Bn kC1 .0/ C P[xn kC1 D j ]Bn kC1 .1/
k D 1; : : : K (11.409)
with the initial condition Bn K C1 .0/ D 1, which is obtained from the observation si D 0. þ
Therefore the probabilities ri;n k , þ 2 f0; 1g are given by (see (8.244)) 0 ri;n D Fn k .0/Bn k .0/ C Fn k .1/Bn k .1/ k 1 D 1 r0 . and ri;n i;n k k
k D 1; : : : ; K
(11.410)
952
Chapter 11. Channel codes
0 and r 1 we update the values of the probabilities q 0 Second step. After computing ri;n i;n i;n 1 . From (11.398) we find and qi;n þ
qi;n D
P[xn D þ; fsi 0 D 0; i 0 2 M.n/nig; z D ρ] P[fsi 0 D 0; i 0 2 M.n/nig; z D ρ]
(11.411)
Lumping in Þi;n the contribution of the terms that do not depend on þ and using the i.i.d. assumption on the code bits, we obtain þ
qi;n D Þi;n P[z n D ²n ; fsi 0 D 0; i 0 2 M.n/nig j xn D þ] Y þ D Þi;n pnþ ri 0 ;n
(11.412)
i 0 2M.n/ni
0 C q 1 D 1. Taking into account the information where Þi;n is chosen such that qi;n i;n from all check nodes, from (11.399) we can also compute the “pseudo a posteriori probabilities” qn0 and qn1 at this iteration, given by Y 0 ri;n (11.413) qn0 D Þn pn0 i 2M.n/
qn1
D
Y
Þn pn1
1 ri;n
(11.414)
i 2M.n/
where Þn is chosen such that qn0 C qn1 D 1. At this point, the algorithm repeats from the first step. At the end of the second step, one iteration of the decoding algorithm is completed. At each iteration, it is possible to detect a code word cO by the log-MAP criterion (8.277), i.e. detect qn1 cOk D sgn ln 0 n D 1; : : : ; n 0 (11.415) qn Decoding is stopped if HOc D 0, or if some other stopping criterion is met, e.g., maximum number of iterations is achieved. Messages passed between the nodes need not be probabilities but can be likelihood or log-likelihood ratios. In fact, various simplifications of the decoding algorithm have been explored and can be adopted for practical implementations [43, 44]. We note that the sum–product algorithm for the decoding of LDPC codes has been derived under the assumption that the check nodes si , i D 1; : : : ; r0 , are statistically independent given the bit nodes xn , n D 1; : : : ; n 0 , and vice versa, i.e. the variables of the vectors s and x form a Markov field [42]. Although this assumption is not strictly true, it turns out that the algorithm yields very good performance with low computational complexity. However, we note that parity check matrices leading to Tanner graphs that exhibit cycles of length four, such as the one depicted in Figure 11.39, should be avoided. In fact, this structure would introduce non-negligible statistical dependence between nodes. In graph theory, the length of the shortest cycle in a graph is referred to as girth. A general method for constructing Tanner graphs with large girth is described in [45].
11.7. Low-density parity check codes
953
Figure 11.39. Tanner graph with a cycle of length four.
Example of application We study in this section the application of binary LDPC codes to two-dimensional QAM transmission over an AWGN channel [46]. The block diagrams of the encoding and decoding processes are shown in Figure 11.40. For bit mapping, log2 M code bits are mapped into one QAM symbol taken from an M-point constellation using Gray mapping. At the receiver, the received samples, which represent noisy QAM symbols, are input to a soft detector that provides soft information on individual code bits in the form of a posteriori probabilities. These probabilities are employed to carry out the message-passing LDPC decoding procedure described in the previous section. Assuming that the employed QAM constellation is square, with log2 M equal to an even number, and that the in-phase and quadrature noise components are independent, it is computationally advantageous to perform soft detection independently for the real and imaginary parts of the received complex samples. We will therefore consider only square QAM constellations. Bit mapping for the real or the imaginary part of transmitted QAM ] symbols is performed by mapping a group of 12 log2 M code bits [c0 ; c1 ; : : : ; c 1 2 .log2 M/1 p that are part of a code word into one of the M real symbols within the set p p p A D f. M 1/; . M 3/; : : : ; 1; C1; : : : ; C. M 1/g
(11.416)
Denoting by z n the real or the imaginary part of a noisy received signal, we have z n D an C wn
Figure 11.40. Multilevel LDPC encoding and decoding.
(11.417)
954
Chapter 11. Channel codes
where an 2 A, and wn is an AWGN sample with variance ¦ I2 . The a posteriori probability that bit c` is zero or one is computed as (see (8.262)) X
e
.²n Þ/2 2¦ I2
Þ2A
P[c` D þ j z n D ²n ] D
c` Dþ .² Þ/2 n 2 2¦ I e
X
1 ` D 0; 1; : : : ; .log2 M/ 1 2
þ 2 f0; 1g
Þ2A
(11.418) where the summation in the numerator is taken over all symbols an 2 A for which c` D þ, þ 2 f0; 1g.
Performance and coding gain Recall from (6.197) the expression of the error probability for uncoded M-QAM transmission, ! r 3 0 (11.419) Pe ' 4Q M 1 where 0 is the signal-to-noise ratio given by (6.105). In general, the relation between M and the rate of the encoder-modulator is given by (11.1), RI D
k0 log2 M n0 2
(11.420)
Recall also, from (6.191), that the signal-to-noise ratio per dimension is given by 0 I D 0 D 2R I
Eb N0
(11.421)
Using (6.288) we introduce the normalized signal-to-noise ratio 0I D
2R I E b 0I D 2R 1 2 I 1 N0
22R I
Then for an uncoded M-QAM system we express (11.419) as q Pe ' 4Q 30 I
(11.422)
(11.423)
As illustrated in Figure 6.54, the curve of Pe versus 0 I indicates that the “gap to capacity” for uncoded QAM with M × 1 is equal to 0 gap;d B ' 9:8 dB at a symbol error probability of 107 . We therefore determine the value of the normalized signal-to-noise c ratio 0 I needed for the coded system to achieve a symbol error probability of 107 , and compute the coding gain at that symbol error probability as c
G code D 9:8 10 log10 .0 I / dB
(11.424)
11.7. Low-density parity check codes
955
Table 11.22 LDPC codes considered for the simulation and coding gains achieved at a symbol error probability of 107 for different QAM constellations. The spectral efficiencies ¹ are also indicated. n0
code rate k0 =n 0
495
0:8747
Code 2 1777 1998
0:8894
Code 3 4095 4376
0:9358
k0 Code 1 433
16-QAM
64-QAM
4096-QAM
4:9 dB 4:6 dB 3:5 dB (3:49 bit/s/Hz) (5:24 bit/s/Hz) (10:46 bit/s/Hz) 6:1 dB 5:9 dB 4:8 dB (3:55 bit/s/Hz) (5:33 bit/s/Hz) (10:62 bit/s/Hz) 6:2 dB 6:1 dB 5:6 dB (3:74 bit/s/Hz) (5:61 bit/s/Hz) (11:22 bit/s/Hz)
From Figure 6.54, as for large signal-to-noise ratios the Shannon limit cannot be approached to within less than 1.53 dB without shaping, we note that an upper limit to the coding gain measured in this manner is about 8:27 dB. Simulation results for three high-rate .n 0 ; k0 / binary LDPC codes are specified in Table 11.22 in terms of the coding gains obtained at a symbol error probability of 107 for transmission over an AWGN channel for 16, 64, and 4096-QAM modulation formats. Transmitted QAM symbols are obtained from coded bits via Gray mapping. To measure error probabilities, one code word is decoded using the message-passing (sum-product) algorithm for given maximum number of iterations. Figure 11.41 shows the effect on performance of the maximum number of
Figure 11.41. Performance of LDPC decoding with Code 2 and 16-QAM for various values of the maximum number of iterations.
956
Chapter 11. Channel codes
iterations allowed in the decoding process for code 2 specified in Table 11.22 and 16-QAM transmission. The codes given in Table 11.22 are due to MacKay and have been obtained by a random construction method. The results of Table 11.22 indicate that LDPC codes offer net coding gains that are similar to those that have been reported for turbo codes. Furthermore, LDPC codes achieve asymptotically an excellent performance without exhibiting “error floors” and admit a wide range of trade-offs between performance and decoding complexity.
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[28] O. Y. Takeshita and D. J. Costello Jr., “New deterministic interleaver designs for turbo codes”, IEEE Trans. on Information Theory, vol. 46, pp. 1988–2006, Sept. 2000. [29] D. Divsalar and F. Pollara, “Turbo codes for PCS applications”, in Proc. 1995 IEEE Int. Conference on Communications, Seattle, U.S.A., pp. 54–59, June 1995. [30] S. Benedetto, D. Divsalar, G. Montorsi, and F. Pollara, “Serial concatenation of interleaved codes: performance analysis, design, and iterative decoding”, IEEE Trans. on Information Theory, vol. 45, pp. 909–926, May 1998. [31] S. Benedetto and G. Montorsi, “Versatile bandwidth-efficient parallel and serial turbotrellis-coded modulation”, in Proc. 2000 Intern. Symp. on Turbo Codes & Relat. Topics, Brest, France, pp. 201–208, Sept. 2000. [32] C. Douillard, M. Jezequel, C. Berrou, A. Picart, P. Didier, and A. Glavieux, “Iterative correction of intersymbol interference turbo-equalization”, Europ. Trans. on Telecommun. (ETT), vol. 6, pp. 507–511, September/October 1995. [33] G. Colavolpe, G. Ferrari, and R. Raheli, “Noncoherent iterative (turbo) decoding”, IEEE Trans. on Communications, vol. 48, pp. 1488–1498, Sept. 2000. [34] G. Bauch, H. Khorram, and J. Hagenauer, “Iterative equalization and decoding in mobile communications system”, in Proc. European Personal Mobile Communications Conference, Bristol, UK, pp. 307–312, 1997. [35] P. Hoeher and J. Lodge, “Turbo-DPSK”: iterative differential PSK demodulation”, IEEE Trans. on Communications, vol. 47, pp. 837–843, June 1999. [36] J. Hagenauer, E. Offer, and L. Papke, “Iterative decoding of binary block and convolutional codes”, IEEE Trans. on Information Theory, vol. 42, pp. 429–445, Mar. 1996. [37] G. Bauch, H. Khorram, and J. Hagenauer, “Iterative equalization and decoding in mobile communications systems”, in Proc. EPMCC, pp. 307–312, Oct. 1997. [38] A. Picart, P. Didier, and G. Glavieux, “Turbo-detection: a new approach to combat channel frequency selectivity”, in Proc. IEEE Int. Conference on Communications, pp. 1498–1502, 1997. [39] D. MacKay and R. Neal, “Near Shannon limit performance of low density parity check codes”, Electron. Lett., vol. 32, pp. 1645–1646, Aug. 1996. [40] D. MacKay, “Good error-correcting codes based on very sparse matrices”, IEEE Trans. on Information Theory, vol. 45, pp. 399–431, Mar. 1999. [41] R. M. Tanner, “A recursive approach to low complexity codes”, IEEE Trans. on Information Theory, vol. 27, pp. 533–547, Sept. 1981. [42] B. J. Frey, Graphical models for machine learning and digital communications. Cambridge, MA: MIT Press, 1998.
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960
Appendix 11.A
Chapter 11. Channel codes
Nonbinary parity check codes
Assume that code words are sequences of symbols from the finite field G F.q/ (see Section 11.2.2), all of length n. As there are q n possible sequences, the introduction of redundancy in the transmitted sequences is possible if the number of code words Mc is less than q n . We denote by c a transmitted sequence of n symbols taken from G F.q/. We also assume that the symbols of the received sequence z are from the same alphabet. We define the error sequence e by the equation (see (11.12) for the binary case) zDcCe
(11.425)
where C denotes a component by component addition of the vectors in accordance with the rules of addition in the field G F.q/. Definition 11.16 The number of non-zero components of a vector x is defined as the weight of the vector, denoted by w.x/. Then w.e/ is equal to the number of errors occurred in transmitting the code word. Definition 11.17 H , is equal to the minimum Hamming distance The minimum distance of a code, denoted dmin between all pairs of code words; i.e. it is the same as for binary codes. We will give without proof the following propositions, similar to those for binary codes on page 830. H can correct all error sequences 1. A nonbinary block code with minimum distance dmin k j H dmin 1 or less. of weight 2 H can detect all error sequences 2. A nonbinary block code with minimum distance dmin H of weight .dmin 1/ or less. H , As in the binary case, we ask for a relation among the parameters of a code: n, Mc , dmin H , and q. It can be proved that for a block code with length n and minimum distance dmin Mc must satisfy the inequality 6 8 j H k 97 6 < dmin 1 =7 6 n 7 n n n 2 5 H 1 k .q 1/ M c 4q 1C .q 1/ C .q 1/2 C Ð Ð Ð C j dmin ; : 1 2 2
(11.426)
11.A. Nonbinary parity check codes
961
H given, it is always possible to find a code with M Ł words, Furthermore, for n and dmin c where ¦³ ¾ ² n n n H 1 Ł n 2 dmin 1C .q 1/ C .q 1/ C Ð Ð Ð C Mc D q H 1 .q 1/ 1 2 dmin (11.427)
Linear codes Definition 11.18 A linear code is a block code with symbols from G F.q/ for which: a) the all zero word is a code word; b) any multiple of a code word is a code word; c) any linear combination of any two code words is a code word. Example 11.A.1 A binary group code is a linear code with symbols from G F.2/. Example 11.A.2 Consider a block code of length 5 having symbols from G F.3/ with code words 0 1 0 2 1 2 0 1 2
0 0 1 0 1 1 2 2 2
0 0 1 0 1 1 2 2 2
0 2 2 1 1 0 1 0 2
0 1 2 2 0 1 1 2 0
(11.428)
It is easily verified that this code is a linear code. We give the following two properties of a linear code. H , is given as 1. The minimum distance of the code, dmin H D min w.Nc/ dmin
(11.429)
where cN can be any non-zero code word. Proof. By definition of the Hamming distance between two code words, we get d H .c1 ; c2 / D w.c1 C .c2 //
(11.430)
962
Chapter 11. Channel codes
By Property b), .c2 / is a code word if c2 is a code word; by Property c), c1 C .c2 / H positions, there must also be a code word. As two code words differ in at least dmin H ; if there were a code word of weight less than d H , is a code word of weight dmin min H positions. this word would be different from the zero word in fewer than dmin 2. If all code words in a linear code are written as rows of an Mc ð n matrix, every column is composed of all zeros, or contains all elements of the field, each repeated Mc =q times.
Parity check matrix Let H be an r ð n matrix with coefficients from G F.q/, expressed as H D [A B]
(11.431)
where the r ð r matrix B is such that det[B] 6D 0. A generalized nonbinary parity check code is a code composed of all vectors c of length n, with elements from G F.q/, that are the solutions of the equation Hc D 0
(11.432)
The matrix H is called the generalized parity check matrix. Propriety 1 of nonbinary generalized parity check codes. A nonbinary generalized parity check code is a linear code. Proof. a) The all zero word is a code word, as H0 D 0. b) Any multiple of a code word is a code word, because if c is a code word, then Hc D 0. But H.Þc/ D ÞHc D 0, and therefore Þc is a code word; here Þ is any element from G F.q/. c) Any linear combination of any two code words is a code word, because if c1 and c2 are two code words, then H.Þc1 C þc2 / D ÞHc1 C þHc2 D Þ0 C þ0 D 0, and therefore Þc1 C þc2 is a code word. Property 2 of nonbinary generalized parity check codes. The code words corresponding to the matrix H D [A B] are identical to the code words corresponding to the parity check Q D [B1 A; I]. matrix H Proof. Same as for the binary case. The matrices in the form [A I] are said to be in canonical or systematic form. Property 3 of nonbinary generalized parity check codes. A code consists of exactly q nr D q k code words. Proof. Same as for the binary case (see Property 3 on page 834). The first k D n r symbols are called information symbols, and the last r symbols are called generalized parity check symbols.
11.A. Nonbinary parity check codes
963
Property 4 of nonbinary generalized parity check codes. A code word of weight w exists if and only if some linear combination of w columns of the matrix H is equal to 0. Proof. c is a code word if and only if Hc D 0. Let c j be the j-th component of c and let hi be the i-th column of H; then if c is a code word we have n X
hj cj D 0
(11.433)
jD1
If c is a code word of weight w, there are exactly w non-zero components of c, say c j1 ; c j2 ; : : : ; c jw ; then c j1 h j1 C c j2 h j2 C Ð Ð Ð C c jw h jw D 0
(11.434)
thus, a linear combination of w columns of H is equal to 0. Conversely, if (11.434) is true, then Hc D 0, where c is a vector of weight w with non-zero components c j1 ; c j2 ; : : : ; c jw . Combining Property 1 of a linear code and Properties 1 and 4 of a nonbinary generalized parity check code, we obtain the following property. Property 5 of nonbinary generalized parity check codes. A code has minimum distance H if some linear combination of d H columns of H is equal to 0, but no linear combidmin min H number of columns of H is equal to 0. nation of fewer than dmin Property 5 is fundamental for the design of nonbinary codes. Example 11.A.3 Consider the field G F.4/, and let Þ be a primitive element of this field; moreover consider the generalized parity check matrix ½ 1 1 1 1 0 HD (11.435) 1 Þ Þ2 0 1 We find that no linear combination of two columns is equal to 0. However, there are many linear combinations of three columns that are equal to 0, for example, h1 C h4 C h5 D 0, H D 3. Þh2 C Þh4 C Þ 2 h5 D 0, ....; hence the minimum distance of this code is dmin
Code generator matrix We assume that the parity check matrix is in canonical form; then cnnr C1 D Acnr 1
(11.436)
and cD
cnr 1
cnnr C1
½
D
I A
½
D GT cnr cnr 1 1
(11.437)
The matrix G is called the generator matrix of the code and is expressed as G D [I; AT ]
(11.438)
964
Chapter 11. Channel codes
so that T cT D .cnr 1 / G
(11.439)
Thus the code words, considered as row vectors, are given as all linear combinations of the rows of the matrix G. A nonbinary generalized parity check code can be specified by giving its generalized parity check matrix or its generator matrix. Example 11.A.4 Consider the field G F.4/ and let Þ be a primitive element of this field; moreover, consider the generalized parity check matrix (11.435). The generator matrix of this code is given by 2 3 1 0 0 1 1 GD40 1 0 1 Þ 5 (11.440) 0 0 1 1 Þ2 There are 64 code words corresponding to all linear combinations of the rows of the matrix G.
Decoding of nonbinary parity check codes Methods for the decoding of nonbinary generalized parity check codes are similar to those for the binary case. Conceptually the simplest method consists in comparing the received block of n symbols with each code word and choosing that code word that differs from the received word in the fewest positions. An equivalent method for a linear code consists in partitioning the q n possible sequences into q r sets. The partitioning is done as follows. Step 1: choose the first set as the set of q nr D q k code words, c1 ; c2 ; : : : ; cq k . Step 2: choose any vector, say η2 , that is not a code word; then choose the second set as c1 C η 2 ; c2 C η 2 ; : : : ; c q k C η 2 . Step i: choose any vector, say ηi , not included in any previous set; choose the i-th set as c1 C η i ; c2 C η i ; : : : ; c q k C η i . The partitioning continues until all q n vectors are used; each set is called a coset, and the vectors ηi are called coset leaders. The all zero vector is the coset leader for the first set.
Coset We give the following properties of the cosets omitting the proofs. 1. Every one of the q n vectors occurs in one and only one coset. 2. Suppose that, instead of choosing ηi as coset leader of the i-th coset, we choose another element of that coset as the coset leader; then the coset formed by using the new coset leader contains exactly the same vectors as the old coset. 3. There are q r cosets.
11.A. Nonbinary parity check codes
965
Two conceptually simple decoding methods We now form a coset table by choosing as coset leader for each coset the vector of minimum weight in that coset. The table consists of an array of vectors, with the i-th row in the array being the i-th coset; the coset leaders make up the first column, and the j-th column consists of the vectors c j ; c j C η2 ; c j C η3 ; : : : ; c j C ηq r . A method for decoding consists of the following steps. Step 1: locate the received vector in the coset table. Step 2: choose the code word that appears as the first vector in the column containing the received vector. This decoding method decodes to the closest code word to the received word and the coset leaders are the correctable error patterns. A modified version of the described decoding method is: Step 10 : locate the received vector in the coset table and then identify the coset leader of the coset containing this vector. Step 20 : subtract the coset leader from the received vector to find the decoded code word.
Syndrome decoding Another method of decoding is the syndrome decoding. For any generalized parity check matrix H and all vectors z of length n, we define the syndrome of z, s.z/, as s.z/ D Hz
(11.441)
We can show that all vectors in the same coset have the same syndrome and vectors in different cosets have different syndromes. This leads to the following decoding method: Step 100 : compute the syndrome of the received vector, as this syndrome identifies the coset in which the received vector is in, and so identifies the leader of that coset. Step 200 : subtract the coset leader from the received vector to find the decoded code word. The difficulty with this decoding method is in the second part of step 100 , that is identifying the coset leader that corresponds to the computed syndrome; this step is equivalent to finding a linear combination of the columns of H which is equal to that syndrome, using the smallest number of columns. The algebraic structure of the generalized parity check matrix for certain classes of codes allows for algebraic means of finding the coset leader from the syndrome.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 12
Trellis coded modulation
During the 1980s an evolution in the methods to transmit data over channels with limited bandwidth took place, giving origin to techniques for joint coding and modulation that are generally known by the name of trellis coded modulation (TCM). The main characteristic of TCM lies in the fact that it yields coding gains with respect to conventional modulation techniques without requiring that the channel bandwidth be increased. The first article on TCM appeared in 1976 by Ungerboeck; later, a more detailed publication by the same author on the principles of TCM [1] spurred considerable interest in this topic [2, 3, 4, 5, 6, 7, 8], leading to a full development of the theory of TCM. TCM techniques use multilevel modulation with a set of signals from a one, two, or multidimensional space. The choice of the signals that generate a code sequence is determined by a finite-state encoder. In TCM, the set of modulation signals is expanded with respect to the set used by an uncoded, i.e. without redundancy, system; in this manner, it is possible to introduce redundancy in the transmitted signal without widening the bandwidth. At the receiver, the signals in the presence of additive noise and channel distortion are decoded by a maximum likelihood sequence decoder. By simple TCM techniques using a four-state encoder, it is possible to obtain a coding gain of 3 dB with respect to conventional uncoded modulation; with more sophisticated TCM techniques, coding gains of 6 dB or more can be achieved (see Chapter 6). Errors in the decoding of the received signal sequence are less likely to occur if the waveforms, which represent the code sequences, are easily distinguishable from each other; in mathematical terms, the signal sequences, represented in the Euclidean multidimensional space, need to be separated by large distances. The novelty of TCM is in postulating the expansion of the set of symbols1 in order to provide the redundancy necessary for the encoding process. The construction of modulation code sequences that are characterized by a free distance, i.e. the minimum Euclidean distance between code sequences, that is much larger than the minimum distance between uncoded modulation symbols, with the same information bit rate, and the same bandwidth and power of the modulated signal, is obtained by the joint design of encoder and bit mapper. The term trellis derives from the similarity between state transition diagrams of a TCM encoder and trellis diagrams 1
In the first part of this chapter we mainly use the notion of symbols of an alphabet with cardinality M, although the analysis could be conducted by referring to vectors in the signal space as modulation signals. We will use the term “signals” instead of “symbols” only in the multidimensional case.
968
Chapter 12. Trellis coded modulation
of binary convolutional codes; the difference lies in the fact that, in TCM schemes, the branches of the trellis are labeled with modulation symbols rather than binary symbols. Thanks to the use of sophisticated TCM schemes, it was possible to achieve reliable data transmission over telephone channels at rates much higher than 9.6 kbit/s, which for years was considered the practical limit. In the mid-1980s, the rate of 14.4 kbit/s was reached. Transmission at a maximum bit rate of 28.8 kbit/s was later specified in the standard CCITT V.34, and extensions were proposed to achieve the rates of 31.2 kbit/s and 33.6 kbit/s.
12.1 12.1.1
Linear TCM for one- and two-dimensional signal sets Fundamental elements
Consider the transmission system illustrated in Figure 6.20, that consists of the modulator, transmission channel, demodulator and data detector. Errors occasionally occur in the symbol detection, and aO k 6D ak . Usually the simplest data detector is a threshold detector that takes an instantaneous hard decision on the value aO k of the transmitted symbol, based on the observation of the sample z k at the demodulator output. Detection is of the nearest-neighbor type, i.e. the detector decides for the symbol of the constellation that is at the minimum Euclidean distance from the received sample z k . The objective of traditional channel coding techniques consists in detecting and/or correcting the errors present in the detected sequence of bits fcQm g. In the approach followed in Chapter 11, a binary encoder was used to map k0 information binary symbols fb` g in n 0 code binary symbols fcm g. As mentioned in Section 11.1, we note that, if we want to maintain the effective rate 1=Tb of the information message and at the same time the modulation rate 1=T of the system, we need to increase the cardinality of the modulation alphabet. If we do not consider the joint design of encoder and bit mapper, however, a reduction of the bit error probability cannot be efficiently achieved, as we see from the following example. Example 12.1.1 Consider an uncoded 4-PSK system and an 8-PSK system that uses a binary error correcting code with rate 2/3; both systems transmit two information bits per modulation interval, which corresponds to a spectral efficiency of 2 bit/s/Hz. If the 4-PSK system works with an error probability of 105 , for a given signal-to-noise ratio 0, the 8-PSK system works with an error probability larger than 102 , due to the smaller Euclidean distance between signals of the 8-PSK system. We must use an error correcting code with minimum Hamming H ½ 7 to reduce the error probability to the same value of the uncoded 4-PSK distance dmin system. A binary convolutional code with rate 2/3 and constraint length 6 has the required H D 7. Decoding requires a decoder with 64 states that implements the Viterbi value of dfree algorithm. However, even after increasing the complexity of the 8-PSK system, we have obtained an error probability only equal to that of the uncoded 4-PSK system. Two problems determine the unsatisfactory result obtained with the traditional approach. The first is originated by the use of independent hard decisions taken by the detector before
12.1. Linear TCM for one- and two-dimensional signal sets
969
Figure 12.1. Block diagram of a transmission system with trellis coded modulation.
decoding; hard input decoding leads to an irreversible loss of information; the remedy is the use of soft decoding (see page 912), whereby the decoder directly operates on the samples at the demodulator output. The second derives from the independent design of encoder and bit mapper. We now consider the transmission system of Figure 12.1, where the transmitted symbol sequence fak g is produced by a finite-state machine having the information bit sequence fb` g as input, possibly with a number of information bits per modulation interval larger than one. We denote by 1=T the modulation rate and by A the alphabet of ak . For an AWGN channel, at the decision point the received samples in the absence of ISI are given by (see (8.173)) z k D a k C wk
(12.1)
where fwk g is a sequence of white Gaussian noise samples. Maximum likelihood sequence detection represents the optimum strategy for decoding a sequence transmitted over a dispersive noisy channel. The decision rule consists in determining the sequence faO k g closest to the received sequence in terms of Euclidean distance (see (8.190)) in the set S of all possible code symbol sequences. MLSD is efficiently implemented by the Viterbi algorithm, provided that the generation of the code symbol sequences follows the rules of a finite-state machine. In relation to (8.194), we define as free distance, dfree , the minimum Euclidean distance between two code symbol sequences fÞk g and fþk g, that belong to the set S, given by X 2 D min jÞk þk j2 fÞk g; fþk g 2 S (12.2) dfree fÞk g6Dfþk g
k
The most probable error event is determined by two code symbol sequences of the set S at the minimum distance. The assignment of symbol sequences using a code that is optimized for Hamming distance does not guarantee an acceptable structure in terms of Euclidean distance, as in general the relation between the Hamming distance and the Euclidean distance is not monotonic. Encoder and modulator must then be jointly designed for the purpose of assigning to symbol sequences waveforms that are separated in the Euclidean signal space by a distance
970
Chapter 12. Trellis coded modulation
equal to at least dfree , where dfree is greater than the minimum distance between the symbols of an uncoded system. At the receiver, the demodulator-decoder does not make errors if the received signal in the Euclidean signal space is at a distance smaller than dfree =2 from the transmitted sequence.
Basic TCM scheme The objective of TCM is to obtain an error probability lower than that achievable with uncoded modulation, for the same bit rate of the system, channel bandwidth, transmitted signal power and noise power spectral density. The generation of code symbol sequences by a sequential finite-state machine (FSM) sets some constraints on the symbols of a sequence, thus introducing interdependence among them (see Appendix 8.D). The transmitted symbol at instant kT depends not only on the information bits generated by the source at the same instant, as in the case of memoryless modulation, but also on the previous symbols. We define as bk the vector of log2 M information bits at instant kT . We recall that for M-ary uncoded transmission there exists a one-to-one correspondence between bk and the symbol ak 2 A. We also introduce the state sk at the instant kT . According to the model of Appendix 8.D, the generation of a sequence of encoded symbols is obtained by the two functions ak D f .bk ; sk1 / sk D g.bk ; sk1 /
(12.3)
For an input vector bk and a state sk1 , the first equation describes the choice of the transmitted symbol ak from a certain constellation, the second the choice of the next state sk . Interdependence between the symbols fak g is introduced without a reduction of the bit rate by increasing the cardinality of the alphabet. For example, for a length K of the sequence of input vectors, if we change A of cardinality M with A0 ¦ A of cardinality M 0 > M, and we select M K sequences as a subset of .A0 / K , a better separation of the code sequences in the Euclidean space may be obtained. Hence, we can obtain a minimum distance dfree between any two sequences larger than the minimum distance between signals in A K . Note that this operation may cause an increase in the average symbol energy from E s;u for uncoded transmission to E s;c for coded transmission, and hence a loss in efficiency given by E s;c =E s;u . Furthermore, we define as Nfree the number of sequences that a code sequence has, on average, at the distance dfree in the Euclidean multidimensional space.
Example Suppose we want to transmit two bits of information per symbol. Instead of using QPSK modulation, we can use the scheme illustrated in Figure 12.2. The scheme has two parts. The first is a finite-state sequential machine with 8 states, where the state sk is defined by the content of the memory cells sk D [sk.2/ ; sk.1/ ; sk.0/ ]. The
12.1. Linear TCM for one- and two-dimensional signal sets
971
Figure 12.2. Eight-state trellis encoder and bit mapper for the transmission of 2 bits per modulation interval by 8-PSK.
two bits bk D [bk.2/ ; bk.1/ ] are input to the FSM, which undergoes a transition from state sk1 to one of four next possible states, sk , according to the function g. The second part is the bit mapper, which maps the two information bits and one bit that depends on the state, i.e. .0/ the three bits [bk.2/ ; bk.1/ ; sk1 ], in one of the symbols of an eight-ary constellation according to the function f , for example, an 8-PSK constellation using the map of Figure 12.5. Note that the transmission of two information bits per modulation interval is achieved. Therefore the constellation of the system is expanded by a factor 2 with respect to uncoded QPSK transmission. Recall from the discussion in Section 6.10 that most of the achievable coding gain for transmission over an ideal AWGN channel of two bits per modulation interval can be obtained by doubling the cardinality of the constellation from four to eight symbols. We will see that trellis coded modulation using the simple scheme of Figure 12.2 allows to achieve a coding gain of 3.6 dB. For the graphical representation of the functions f and g, it is convenient to use a trellis diagram; the nodes of the trellis represent the FSM states and the branches represent the possible transitions between states. For a given state sk1 , a branch is associated with each possible vector bk by the function g, that reaches a next state sk . Each branch is labeled with the corresponding value of the transmitted symbol ak . For the encoder of Figure 12.2 and the map of Figure 12.5, the corresponding trellis is shown in Figure 12.3, where the trellis is terminated by forcing the state of the FSM to zero at the instant k D 4. For a general representation of the trellis, see Figure 12.13. Each path of the trellis corresponds to only one message sequence fb` g and is associated with only one sequence of code symbols fak g. The optimum decoder searches the trellis for the most probable path, given the received sequence fz k g is observed at the output of the demodulator. This search is usually realized by the Viterbi algorithm (see Section 8.10). Because of the presence of noise, the chosen path may not coincide with the correct one, but diverge from it at the instant k D i and rejoin it at the instant k D i C L; in this case we say that an error event of length L has occurred, as illustrated in the example in Figure 12.4 for an error event of length two (see Definition 8.1 on page 683). Note that in a trellis diagram more branches may connect the same pair of nodes. In this case we speak of parallel transitions, and by the term free distance of the code
972
Chapter 12. Trellis coded modulation
k= 0
1 0 4 2 6
s0 = 0 1 2
2
3
4
5
40 6 2 5
3
1 7 3
4
1 5
5
3
6
7
7 Figure 12.3. Trellis diagram for the encoder of Figure 12.2 and the map of Figure 12.5. Each branch is labeled with the corresponding value of ak .
ak =
4
1
6
6
4
4
7
7
0
4
sk = 0 1 2 3 4 5 6 7 ^a = k
Figure 12.4. Section of the trellis for the decoder of an eight-state trellis code. The two continuous lines indicate two possible paths relative to two 8-PSK signal sequences, fak g and faˆk g.
we denote the minimum among the distances between symbols on parallel transitions and the distances between code sequences associated with pairs of paths in the trellis that originate from a common node and merge into a common node after L transitions, L > 1. By utilizing the sequence of samples fz k g, the decoding of a TCM signal is done in two phases. In the first phase, called subset decoding, within each subset of symbols assigned
12.1. Linear TCM for one- and two-dimensional signal sets
973
to the parallel transitions in the trellis diagram, the receiver determines the symbol closest to the received sample; these symbols are then memorized together with their squared distances from the received sample. In the second phase we apply the Viterbi algorithm to find the code sequence faO k g along the trellis such that the sum of the squared distances between the code sequence and the sequence fz k g is minimum. Recalling that the signal is obtained at the output of the demodulator in the presence of additive white Gaussian noise with variance ¦ I2 per dimension, the probability of an error event for large values of the signal-to-noise ratio is approximated by (see (8.195)) dfree (12.4) Pe ' Nfree Q 2¦ I where dfree is defined in (12.2). From the Definition 6.2 on page 508 and the relation (12.4) between Euclidean distance and error probability, we give the definition of asymptotic coding gain, G code ;2 as the ratio between the minimum distance, dfree , between code sequences and the minimum Euclidean distance for uncoded sequences, equal to the minimum distance between symbols of the Q 0 , normalized by the ratio between the average energy constellation of an uncoded system, 1 of the coded sequence, E s;c , and the average energy of the uncoded sequence, E s;u . The coding gain is then expressed in dB as G code D 10 log10
12.1.2
2 =1 Q2 dfree 0
E s;c =E s;u
(12.5)
Set partitioning
The design of trellis codes is based on a method called mapping by set partitioning. This method requires that the bit mapper assign symbol values to the input binary vectors so that the minimum Euclidean distance between possible code sequences fak g is maximum. For a given encoder the search of the optimum assignment is made by taking into consideration subsets of the symbol set A. These subsets are obtained by successive partitioning of the set A, and are characterized by the property that the minimum Euclidean distance between symbols in a subset corresponding to a certain level of partitioning is larger than or equal to the minimum distance obtained at the previous level. Consider the symbol alphabet A D A0 with 2n elements, that corresponds to level zero of partitioning. At the first level of partitioning, that is characterized by the index q D 1, the set A0 is subdivided into two disjoint subsets A1 .0/ and A1 .1/ with 2n1 elements each. Let 11 .0/ and 11 .1/ be the minimum Euclidean distances between elements of the subsets A1 .0/ and A1 .1/, respectively; define 11 as the minimum between the two Euclidean distances 11 .0/ and 11 .1/; we choose a partition for which 11 is maximum. At the level of partitioning characterized by the index q > 1, each of the 2q1 subsets Aq1 .`/, ` D 0; 1; : : : ; 2q1 1, is subdivided into two subsets, thus originating 2q subsets. During 2
To emphasize the dependence of the asymptotic coding gain on the choice of the symbol constellations of the coded and uncoded systems, sometimes the information on the considered modulation schemes is included as a subscript in the symbol used to denote the coding gain, e.g. G 8PSK/4PSK for the introductory example.
974
Chapter 12. Trellis coded modulation
the procedure, it is required that the minimum Euclidean distance at the q-th level of partitioning, 1q D
min
`2f0;1;:::;2q 1g
1q .`/
with
1q .`/ D
min
Þi ; Þm 2 Aq .`/ Þi 6D Þm
jÞi Þm j
(12.6)
is maximum. At the n-th level of partitioning the subsets An .`/ consist of only one element each; to subsets with only one element we assign the minimum distance 1n D 1; at the end of the procedure we obtain a tree diagram of binary partitioning for the symbol set. At the q-th level of partitioning, to the two subsets obtained by a subset at the .q 1/-th level we assign the binary symbols y .q1/ D 0 and y .q1/ D 1, respectively; in this manner, an n-tuple of binary symbols yi D .yi.n1/ ; : : : ; yi.1/ ; yi.0/ / is associated with each element Þi found at an end node of the tree diagram.3 Therefore the Euclidean distance between two elements of A, Þi and Þm , indicated by the binary vectors yi and ym that are equal in the first q components, satisfies the relation jÞi Þm j ½ 1q
for
. p/
yi
. p/
D ym
p D 0; : : : ; q 1
i 6D m
(12.7)
In fact, because of the equality of the components in the positions from .0/ up to .q 1/, we have that the two elements are in the same subset Aq .`/ at the q-th level of partitioning. Therefore their Euclidean distance is at least equal to 1q . Example 12.1.2 The partitioning of the set A0 of symbols with statistical power E[jak j2 ] D 1 for an 8-PSK system is illustrated in Figure 12.5. The minimum Euclidean distance between elements of the set A0 is given by 10 D 2 sin.³=8/ D 0:765. At the first level of partitioning the two subsets B0 D f.y .2/ ; y .1/ ; 0/; y .i / D 0; 1g and B1 D f.y .2/ ; y .1/ ; 1/;py .i / D 0; 1g are found, with four elements each and minimum Euclidean distance 11 D 2. At the second level of partitioning four subsets C0 D f.y .2/ ; 0; 0/; y .2/ D 0; 1g, C2 D f.y .2/ ; 1; 0/; y .2/ D 0; 1g, C1 D f.y .2/ ; 0; 1/; y .2/ D 0; 1g, and C3 D f.y .2/ ; 1; 1/; y .2/ D 0; 1g are found with two elements each and minimum Euclidean distance 12 D 2. Finally, at the last level eight subsets D0 ; : : : ; D7 are found, with one element each and minimum Euclidean distance 13 D 1. Example 12.1.3 The partitioning of the set A0 of symbols with statistical power E[jak j2 ] D 1 for a 16-QAM system is illustrated in Figure p 12.6. The minimum Euclidean distance between the elements of A0 is given by 10 D 2= 10 D 0:632. Note that at each successive partitioning level the minimum Euclidean distance among the elements of a subset increases by a factor equal to p 2. Therefore at the third level of partitioning the minimum Euclidean distance between p the elements of each of the subsets Di , i D 0; 1; : : : ; 7, is given by 13 D 810 .
3
For TCM encoders, the n-tuples of binary code symbols will be indicated by y D .y .n1/ ; : : : ; y .0/ / rather than by the notation c employed in the previous chapter.
12.1. Linear TCM for one- and two-dimensional signal sets
975
Figure 12.5. Partitioning of the symbol set for an 8-PSK system. [From Ungerboeck (1982). c 1982 IEEE.]
Figure 12.6. Partitioning of the symbol set for a 16-QAM system. [From Ungerboeck (1982). c 1982 IEEE.]
12.1.3
Lattices
Several constellations and the relative partitioning can be effectively described by lattices; furthermore, as we will see in the following sections, the formulation based on lattices is particularly convenient in the discussion on multidimensional trellis codes.
976
Chapter 12. Trellis coded modulation
In general, let Z D D Z D , where Z denotes the set of integers;4 a lattice 3 in < D is defined by the relation 3 D f.i 1 ; : : : ; i D / G j .i 1 ; : : : ; i D / 2 Z D g
(12.8)
where G is a non-singular D ð D matrix, called lattice generator matrix, by means of which we obtain a correspondence Z D ! 3. The vectors given by the rows of G form a basis for the lattice 3; the vectors of the basis define a parallelepiped whose volume V0 D j det.G/j represents the characteristic volume of the lattice. The volume V0 is equivalent to the volume of a Voronoi cell associated with an element or point of lattice 3 and defined as the set of points in < D whose distance from a given point of 3 is smaller than the distance from any other point of 3. The set of Voronoi cells associated with the points of 3 is equivalent to the space < D . A lattice is characterized by two parameters: 1. dmin , defined as the minimum distance between points of the lattice; 2. the kissing number, defined as the number of lattice points at minimum distance from a given point. We obtain a subgroup 3q .0/ if points of the lattice 3 are chosen as basis vectors in a matrix Gq , such that they give rise to a characteristic volume Vq D j det.Gq /j > V0 . Example 12.1.4 (Z p lattice) In general, as already mentioned, the notation Z p is used to define a lattice with an infinite number of points in the p-dimensional Euclidean space with coordinates given by integers. The generator matrix G for the lattice Z p is the p ð p identity matrix; the minimum distance is dmin D 1 and the kissing number is equal to 2 p. The Z2 type constellations (see Figure 12.7a) for QAM systems are finite subsets of Z2 , with center at the origin and minimum Euclidean distance equal to 10 . Example 12.1.5 (Dn lattice) Dn is the set of all n-dimensional points whose coordinates are integers that sum to an even number; it may be regarded as a version of the Zn lattice from which the points whose coordinates p are integers that sum to an odd number were removed. The minimum distance is dmin D 2 and the kissing number is 2n.n 1/. The lattice D2 is represented in Figure 12.7b. D4 , called the Schl¨afli lattice, constitutes the densest lattice in 2, we approximate ak as e j'k , where 'k is a uniform r.v. on .³ ; ³ ]; then (14.112) becomes L;" .z; e/ D
KY 1 Z C³ kD0
³
²
2 dvk exp Re[e jvk x k .e/ e j z ] N0 2³
¦ (14.113)
If we use the definition of the Bessel function (4.216), (14.113) is independent of the phase and we obtain KY 1 jx k .e/j L" .e/ D (14.114) I0 N0 =2 kD0 On the other hand, if we take the expectation of (14.95) only with respect to the phase we obtain KY 1 jx k .e/ ÞkŁ j (14.115) L";a .e; α/ D I0 N0 =2 kD0 We observe that, for M-PSK, L";a .e; α/ D L" .e/, as jÞk j is a constant, while this does not occur for M-QAM. To obtain estimates from the two likelihood functions just obtained, if the signal-to-noise ratio 0 is sufficiently high, we utilize the fact that I0 .Ð/ can be approximated as I0 . / ' 1 C
2 2
for j j − 1
(14.116)
Taking the logarithm of the likelihood and eliminating non-relevant terms, we obtain the following NDA estimator and DA estimator. NDA:
"O D arg max lnfL" .e/g e
' arg max e
K 1 X
jx k .e/j2
(14.117)
kD0
"O D arg max lnfL";a .e; α/g
DA:
e
' arg max e
K 1 X
jx k .e/j2 jÞk j2
(14.118)
kD0
On the other hand, if 0 − 1, (14.112) can be approximated using a power series expansion of the exponential function. Taking the logarithm of (14.112), using the hypothesis of i.i.d. symbols, with E[an ] D 0, and eliminating non-relevant terms, we obtain the following log-likelihood: " # K 1 K 1 X X `;" .z; e/ D E[ja n j2 ] jx k .e/j2 C Re E[an2 ] .x kŁ .e//2 e j2z (14.119) kD0
kD0
14.5. Algorithms for timing and carrier phase recovery
1057
Averaging with respect to we obtain the following phase independent log-likelihood: `" .e/ D
K 1 X
jx k .e/j2
(14.120)
kD0
which yields the same NDA estimator as (14.117). For a modulation technique characterized by E[an2 ] 6D 0, (14.119) may be used to obtain an NDA joint estimate of phase and timing. In fact, for a phase estimate given by ( ) K 1 X 1 2 Ł 2 .x k .e// O D arg E[an ] (14.121) 2 kD0 the second term of (14.119) is maximized. Substitution of (14.121) in (14.119) yields a new estimate "O given by þ þ K 1 K 1 þ þ X X þ þ 2 2 2 2 (14.122) jx k .e/j C þ E[an ] x k .e/þ "O D arg max E[jan j ] e þ þ kD0 kD0 The block diagram of the joint estimator is shown in Figure 14.24, where P values of the time shift ", ".m/ , m D 1; : : : ; P, equally spaced in [1=2; 1=2] are considered; usually the resolution obtained with P D 8 or 10 is sufficient. For each time shift ".m/ , the log-likelihood (14.122) is computed and the value of ".m/ associated with the largest value of the log-likelihood is selected as the timing estimate. Furthermore, we observe that in the generic branch m, filtering by the matched filter g M .i Tc C ".m/ T / and sampling at the instants kT can be implemented by the cascade of an interpolator filter h I .i Tc ; ¼.m/ / (where ¼.m/ depends on ".m/ ) and a filter g M .i Tc /, followed by a decimator that provides samples at the instants mk Tc , as illustrated in Figure 14.19 and described in Section 14.4.
Non-data aided via spectral estimation Let us consider the log-likelihood (14.120) limited to a symmetric observation time interval [L T; L T ]; thus we obtain `" .e/ D
L X
jx.kT C eT /j2
(14.123)
kDL
Now, as x is a QAM signal, the process jx.kT C eT /j2 is approximately cyclostationary in e of period 1 (see Section 7.2). We introduce the following Fourier series representation jx.kT C eT /j2 D
C1 X
.k/
ci e j2³i e
(14.124)
i D1
where the coefficients fci.k/ g are random variables given by Z 1 jx.kT C eT /j2 e j2³i e de ci.k/ D 0
(14.125)
Figure 14.24. NDA joint timing and phase (for E[a2n ] 6D 0) estimator. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
1058 Chapter 14. Synchronization
14.5. Algorithms for timing and carrier phase recovery
1059
Now (14.123) is equal to the average of the cyclostationary process jx.kT C eT /j2 in the interval [L; L]; defining L X
ci D
ci.k/
(14.126)
kDL
it results [4] that only c0 and c1 have non-zero mean, and (14.123) can be written as X `" .e/ D c0 C 2Re[c1 e j2³ e ] C 2Re[ci e j2³i e ] (14.127) ji j½2
|
{z
}
disturbance with zero mean for each value of e
As c0 and jc1 j are independent of e, the maximum of `" .e/ yields "O D
1 arg c1 2³
(14.128)
However, the coefficient c1 is obtained by integration, which in general is hard to implement in the digital domain; on the other hand, if the bandwidth of jx.lT /j2 satisfies the relation Bjxj2 D
1 1 .1 C ²/ < T 2Tc
(14.129)
where ² is the roll-off factor of the matched filter, then c1 can be computed by DFT. Let F0 D T =Tc , then we obtain c1 D
L X kDL
"
0 1 1 FX jx.[k F0 C l] Tc /j2 e j .2³=F0 /l F0 lD0
# (14.130)
A simple implementation of the estimator is possible for F0 D 4; in fact, in this case no multiplications are needed; as e j .2³=4/l D . j/l , (14.130) simplifies into " # L 3 X 1 X c1 D jx.[4k C l] Tc /j2 . j/l (14.131) 4 kDL lD0 Figure 14.25 illustrates the implementation of the estimator for F0 D 4.
Data-aided and data-directed If in (14.95) we substitute the parameters Þk and z with their estimates we obtain the phase independent DA (DD) log-likelihood ( " #) K 1 X 2 O L" .e/ D exp Re aO kŁ x k .e/ e j (14.132) N0 kD0
1060
Chapter 14. Synchronization
L
-
| x [ (4k-1)T
c]|
2
Σ
+
| z [(4k-3)T c ] |
Σ ( .) k=-L
Im(c 1 )
2
Tc
Tc
Tc
^ε
-
| x(nTc )| 2
1 arg (c1 ) 2π
(4kTc )
(4kTc )
| x(4kTc ) | 2
| x [(4k-2)T
c ]| 2
+
Σ
L
Σ ( .) k=-L
-
Re(c1 )
Figure 14.25. NDA timing estimator via spectral estimation for the case F0 D 4. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
Figure 14.26. Phase independent DA (DD) timing estimator. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
from which we immediately derive the estimate "O D arg max L" .e/ e
(14.133)
The block diagram of the estimator is shown in Figure 14.26; note that this algorithm can only be used in the case phase recovery is carried out before timing recovery. For a joint phase and timing estimator, from (14.95) we get ( L;" .z; e/ D exp
" #) K 1 X 2 Re aO kŁ x k .e/ e j z N0 kD0
(14.134)
14.5. Algorithms for timing and carrier phase recovery
1061
Defining r.e/ D
K 1 X
aO kŁ x k .e/
(14.135)
kD0
the estimation algorithm becomes O "O / D arg max Re[r.e/ e j z ] .; z;e
D arg max jr.e/j Re[e j .zarg.r.e/// ]
(14.136)
z;e
The two-variable search of the maximum reduces to a single-variable search; as a matter of fact, once the value of e that maximizes jr.e/j is obtained, which is independent of z, the second term Re[e j .zarg.r.e/// ]
(14.137)
is maximized by z D arg.r.e//. Therefore the joint estimation algorithm is given by þ þ 1 þ KX þ þ þ Ł aO k x k .e/þ "O D arg max jr.e/j D arg max þ e e þ þ kD0 (14.138) O D arg r.O"/ D arg
K 1 X
aO kŁ x k .O" /
kD0
Figure 14.27 illustrates the implementation of this second estimator; note that this scheme is a particular case of (7.269). For both estimators, estimation of the synchronization parameters is carried out every K samples, according to the assumption of slow parameter variations made at the beginning of the section.
x k (ε (1))
g (iTc +ε (1) ) M
Σ (.) k
kT
r (ε 1 )
r AA (t) nTc xk
g (iTc +ε (P) ) M kT
(ε (P) )
Σ (.) k
arg max r ( ε )
a^ k*
^ε
arg r ( ^ε )
^ θ
r (ε P )
Figure 14.27. DA (DD) joint phase and timing estimator. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
1062
Chapter 14. Synchronization
Observation 14.2 In the case the channel is not known, to implement the matched filter g M we need to estimate the overall impulse response qC ; then the estimation of qC , for example, by one of the methods presented in Appendix 3.A, and of timing can be performed jointly. Let F0 D T =Tc and Q 0 D T =TQ be integers, with Q 0 ½ F0 . From the signal fr A A .q TQ /g, obtained by oversampling r A A .t/ or by interpolation of fr A A .nTc /g, and the knowledge of the training sequence fak g, k D 0; : : : ; L T S 1, the estimate of qC with sampling period TQ , or equivalently the estimate of its Q 0 =F0 polyphase components with sampling period Tc (see Observation 8.5), is made. Limiting the estimate to the more significant consecutive samples around the peak, the determination of the timing phase with precision TQ coincides with the selection of the polyphase component with the largest energy among the Q 0 =F0 polyphase components. This determines the optimum filter g M with sampling period Tc . Typically for radio systems F0 D 2, and Q 0 D 4 or 8.
Data- and phase-directed with feedback: differentiator scheme Differentiating the log-likelihood (14.95) with respect to e, neglecting non-relevant terms, and evaluating the result at .O ; e; aO /, we obtain " # K 1 X @ Ł @ j O ln fL" .e/g / Re x.kT C eT / e aO k (14.139) @e @e kD0 With reference to the scheme of Figure 14.22, if we suppose that the sum in (14.139) is approximated by the filtering operation by the loop filter F.z/, the error signal e.kT / results ½ Ł @ j O (14.140) x.kT C eT /jeDO"k e e.kT / D Re aO k @e The partial derivative of x.kT C eT / with respect to e can be carried out in the digital domain by a differentiator filter with an ideal frequency response given by Hd . f / D j2³ f
jfj
1 2Tc
(14.141)
In practice, if T =Tc ½ 2 it is simpler to implement a differentiator filter by a finite difference filter having a symmetric impulse response given by h d .i Tc / D
1 .Ži C1 Ži 1 / 2Tc
(14.142)
Figure 14.28 illustrates the block diagram of the estimator, where the compact notation x.t/ P is used in place of .dx.t/=dt/; moreover, based on the analysis of Section 14.5.2, if u.kT / is the loop filter output, the estimate of " is given by "O kC1 D "O k C ¼" u.kT /
(14.143)
where ¼" is a suitable constant. Applying (14.88) to the value of "O kC1 , we obtain the values of ¼kC1 and mkC1 .
Figure 14.28. DD & D -FB timing estimator. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
14.5. Algorithms for timing and carrier phase recovery 1063
1064
Chapter 14. Synchronization
Data- and phase-directed with feedback: Mueller & Muller scheme The present algorithm gets its name from Mueller and Muller, who first proposed it in 1976 [6]. Consider the estimation error e" D "O " and the pulse q R .t/ D qC Ł g M .t/; the basic idea consists in generating an error signal whose mean value assumes one of the following two expressions: n o Type A: E[e.kT /] DRe 12 [q R .e" T C T / q R .e" T T /] (14.144) Type B:
E[e.kT /] DRefq R .e" T C T /g
(14.145)
Observe that, under the assumptions of Section 14.4, q R .t/ is a Nyquist pulse; moreover, we assume that in the absence of channel distortion, q R .t/ is an even function. Note that the signal (14.144) is an odd function of the estimation error e" for e" 2 .1; 1/, whereas the signal (14.145) is an odd function of e" only around e" D 0. Under lock conditions, i.e. for e" ! 0, the two versions of the algorithm exhibit a similar behavior. However, the type A algorithm gives better results than the type B algorithm in transient conditions because the mean value of the error signal for the type B algorithm is not symmetric. Moreover, the type A algorithm results effective also in the presence of signal distortion. The error signal for the type A algorithm is chosen equal to Ł x k .O" / aO kŁ x k1 .O" /] e j ] e.kT / D Re[[aO k1 O
(14.146)
where is a suitable constant whose value is discussed below. Assuming that aO k1 D ak1 , aO k D ak , and O D , from (14.71) and (14.79) for o D D 0 (14.146) can be written as (" C1 X Ł e.kT / D Re ak1 ai q R .kT C e" T i T / i D1
akŁ
C1 X
# ai q R ..k 1/T C e" T i T /
(14.147)
i D1
) Ł C ak1 wQ k akŁ wQ k1
where wQ k is the decimated noise signal at the matched filter output. We define qm .e" / D q R .mT C e" T /
(14.148)
then with a suitable change of variables (14.147) becomes (" # ) C1 C1 X X Ł Ł Ł Ł e.kT / D Re ak1 akm qm .e" / ak ak1m qm .e" / C ak wQ k ak wQ k1 mD1
mD1
(14.149)
14.5. Algorithms for timing and carrier phase recovery
1065
Taking the mean value of e.kT / we obtain E[e.kT /] D Ref.E[jak j2 ] jma j2 /[q1 .e" / q1 .e" /]g
(14.150)
D Ref.E[jak j2 ] jma j2 /[q R .e" T C T / q R .e" T T /]g For D
1 2.E[jak j2 ] jma j2 /
(14.151)
we obtain (14.144). Similarly, in the case of the type B algorithm the error signal assumes the expression e.kT / D
1
[aO k1 ma ]Ł x k .O" / e j
O
.E[jak j2 ] jma j2 /
(14.152)
Figure 14.29 illustrates the block diagram of the direct section of the type A estimator. The constant is included in the loop filter and is not explicitly shown.
Non-data aided with feedback We consider the log-likelihood (14.120), obtained for a NDA estimator, and differentiate it with respect to e to get 1 @ KX @`" .e/ D jx.kT C eT /j2 @e @e kD0
(14.153) D
K 1 X
2Re[x.kT C eT / xP Ł .kT C eT /]
kD0
a^k
xk ( ^ε)
( . )*
^ e -j θ
a^k* T
Σ
+ -
Re[ .]
T Figure 14.29. Mueller & Muller type A timing estimator.
e(kT)
1066
Chapter 14. Synchronization
If we assume that the sum is carried out by the loop filter, the error signal is given by e.kT / D Re[x.kT C "O k T / xP Ł .kT C "O k T /]
14.5.4
(14.154)
Phasor estimators
Data- and timing-directed We discuss an algorithm that directly yields the phasor exp. j O / in place of the phase O . Assuming that an estimate of aO and "O is available, the likelihood (14.95) becomes ( " #) K 1 X 2 Ł jz Re aO k x k .O" / e (14.155) L .z/ D exp N0 kD0 and is maximized by e j D e j arg O
P K 1 kD0
aO kŁ xk .O"/
(14.156)
Figure 14.30 illustrates the implementation of the estimator (14.156).
Non-data aided for M-PSK signals In an M-PSK system, to remove the data dependence from the decimator output signal in the scheme of Figure 14.19, we raise the samples x k .O" / to the M-th power. Assuming absence of ISI, we get x kM .O" / D [ak e j C wQ k ] M D akM e j M C w M;k
(14.157)
where wQ k represents the decimator output noise, and w M;k denotes the overall disturbance. As akM D .e j2³l=M / M D 1, (14.157) becomes x kM .O" / D e j M C w M;k
(14.158)
From (14.95), we substitute .x k .O" // M for ÞkŁ x k .O" / obtaining the likelihood ( " #) K 1 X 2 M j zM L .z/ D exp Re .x k .O" // e N0 kD0
(14.159)
which is maximized by the phasor " exp. j O M/ D exp j arg
K 1 X
# .x k .O" // M
kD0
a^ k* xk ( ^ε)
K-1
Σ a^ k* xk ( ^ε) k=0
^
e jθ
Figure 14.30. DD & D" estimator of the phasor ej .
(14.160)
14.5. Algorithms for timing and carrier phase recovery
xk ( ^ε)
j^ e θM
K-1
arg Σ ( xk ( ^ε) ) M
( .) M
1067
k=0
Figure 14.31. NDA estimator of the phasor e j for M-PSK.
We note that raising x k .O" / to the M-th power causes a phase ambiguity equal to a multiple of .2³ /=M; in fact, if O is a solution to (14.160), also .O C 2³l=M/ for l D 0; : : : ; M 1, are solutions. This ambiguity can be removed, for example, by differential encoding (see Section 6.5.2). The estimator block diagram is illustrated in Figure 14.31.
Data- and timing-directed with feedback Consider the likelihood (14.155) obtained for the DD & D" estimator of the phasor e j ; taking the logarithm, differentiating it with respect to z, and ignoring non-relevant terms, we obtain the error signal e.kT / D Im[aO kŁ x k .O" / e j k ] O
(14.161)
Observe that, in the absence of noise, x k .O" / D ak e j , and (14.161) becomes e.kT / D jaO k j2 sin. Ok / pk
ek -
a)
uk
F(z)
^p k+1
(14.162)
z -1
^p k
NCO
loop filter
~ xk ( ε^ )=ak e jθ+w k
a^ k
b)
( .) *
( .) *
a^ k* 1/ . pk
p^ k
PHLL Figure 14.32. (a) PHLL; (b) DD & D"-FB phasor estimator. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
1068
Chapter 14. Synchronization
Hence, we can use a digital version of the PLL to implement the estimator. However, the error signal (14.161) introduces a phase ambiguity; in fact it assumes the same value if we substitute (Ok ³ ) for Ok . An alternative to the digital PLL is given by the phasor-locked loop (PHLL), that provides an estimate of the phasor e j , rather than the estimate of , thus eliminating the ambiguity. The block diagram of the PHLL is illustrated in Figure 14.32a; it is a feedback structure O with the phasor pk D e jk as input and the estimate pO k D e j k as output. The error signal ek is obtained by subtracting the estimate pO k from pk ; then ek is input to the loop filter F.z/ that yields the signal u k , which is used to update the phasor estimate according to the recursive relation pO kC1 D pO k C . f Ł u/.k/
(14.163)
Figure 14.32b illustrates the block diagram of a DD & D" phasor estimator that implements the PHLL. Observe that the input phasor pk is obtained by multiplying x k .O" / by aO kŁ to remove the dependence on the data; the dashed block normalizes the estimate pO k in the QAM case.
14.6
Algorithms for carrier frequency recovery
As mentioned in Section 14.4, phase and timing estimation algorithms work correctly only if the frequency offset is small. Therefore the frequency offset must be compensated before the estimate of the other two synchronization parameters takes place. Hence the algorithms that we will present are mainly NDA and non-clock-aided (NCA); timingdirected algorithms are possible only in the case the frequency offset has a magnitude much smaller than 1=T . In Figure 14.33 we redraw part of the digital receiver scheme of Figure 14.19; observe that the position of the matched filter is interchanged with that of the interpolator filter. In O c / to remove the frequency this scheme, the samples r A A .nTc / are multiplied by exp. j nT offset. In [4] it is shown that, whenever satisfies the condition þ þ þ T þ þ þ (14.164) þ 2³ þ 0:15
Figure 14.33. Receiver of Figure 14.19 with interpolator and matched filter interchanged. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
14.6. Algorithms for carrier frequency recovery
1069
the following approximation holds: x.kT C eT; o/ ' ejokT x k .e/
(14.165)
Then the likelihood (14.80) can be written as ( ) 1 2 KX Ł jokT j z Re[Þk x k .e/ e e ] L;";;a .z; e; o; α/ D exp N0 kD0
(14.166)
Therefore in the schemes of Figure 14.19 and Figure 14.33 the frequency translator may be moved after the decimator, together with the phase rotator.
14.6.1
Frequency offset estimators
Non-data aided Suppose the receiver operates with a low signal-to-noise ratio 0; similarly to (14.123) the log-likelihood for the joint estimate of ."; / in the observation interval [L T; L T ] is given by `"; .e; o/ D
L X
jx.kT C eT; o/j2
(14.167)
kDL
By expanding `"; .e; o/ in Fourier series and using the notation introduced in the previous section we obtain X 2Re[ci e j2³i e ] (14.168) `"; .e; o/ D c0 C 2Re[c1 e j2³ e ] C ji j½2
|
{z
disturbance
}
Now the mean value of c0 , E[c0 ], depends on o, but is independent of e and furthermore O hence is maximized for o D ; O D arg max c0 o
(14.169)
As we did for the derivation of (14.131), starting with (14.169) and assuming the ratio F0 D T =Tc is an integer, we obtain the following joint estimate of .; "/ [4]: O D arg max o
LX F0 1
jx.nTc ; o/j2
nDL F0
(14.170) "O D arg
LX F0 1 nDL F0
jx.nTc ; o/j2 e j2³ n=F0
1070
Chapter 14. Synchronization
e -j
Ω (1)nTc
g (iTc ) M
( .) 2
Σn
arg max
r AA (t) nTc e -j
Ω (M) nTc
g (iTc ) M
( .) 2
^) x(nTc,Ω to timing and/or phase estimator
Σn
Figure 14.34. NDA frequency offset estimator. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
The implementation of the estimator is illustrated in Figure 14.34; observe that the signal x.nTc ; o/ can be rewritten as X x.nTc ; o/ D r A A .i Tc / ejoiTc g M .nTc i Tc / i
D e jonTc
X
r A A .i Tc / ejo.i n/Tc g M .nTc i Tc /
i
D e jonTc
X
(14.171)
. pb/
r A A .i Tc / g M .nTc i Tc ; o/
i
where the expression of the filter . pb/
g M .i Tc ; o/ D g M .i Tc / e joiTc
(14.172)
depends on the offset o. Defining xo .nTc / D
X
. pb/
r A A .i Tc / g M .nTc i Tc ; o/
(14.173)
i
we note that jx.nTc ; o/j D jxo .nTc /j, and hence in the m-th branch of Figure 14.34 the cascade of the frequency translator and the filter can be substituted with a simple filter with . pb/ impulse response g M .i Tc ; .m/ /.
14.6. Algorithms for carrier frequency recovery
1071
Non-data aided and timing-independent with feedback Differentiating the log-likelihood defined by (14.170), equal to c0 ./, with respect to we obtain the error signal ½þ þ @ Ł x .nTc ; o/ þþ (14.174) e.nTc / D 2Re x.nTc ; o/ @o On oD Observe that, as c0 ./ is independent of e, then also e.nTc / is independent of e. From the first of (14.171), the partial derivative of x.nTc ; o/ with respect to o is given by C1 X @ x.nTc ; o/ D .jiTc / r A A .i Tc / ejoiTc g M .nTc i Tc / @o i D1
(14.175)
We define the frequency matched filter as g F M .i Tc / D .jiTc / Ð g M .i Tc /
(14.176)
Observe now that, if the signal r D .nTc / D r A A .nTc / ejonTc is input to the filter g F M .i Tc /, then from (14.175) the output is given by x F M .nTc / D g F M Ł r D .nTc / D
@ x.nTc ; o/ C jnTc x.nTc ; o/ @o
(14.177)
from which we obtain @ x.nTc ; o/ D x F M .nTc / jnTc x.nTc ; o/ @o
(14.178)
Therefore the expression of the error signal (14.174) becomes O n / x FŁ M .nTc ; O n /] e.nTc / D 2Re[x.nTc ;
(14.179)
The block diagram of the resultant estimator is shown in Figure 14.35. The loop filter output u.kTc / is sent to the NCO that yields the frequency offset estimate according to the recursive equation O nC1 Tc D O n Tc C ¼ u.nTc /
(14.180)
where ¼ is the NCO gain.
Non-data aided and timing-directed with feedback Consider the log-likelihood (14.167); to get the D" estimator, we substitute e with the estimate "O obtaining the log-likelihood ` .o/ D
K 1 X
jx.kT C "O T; o/j2
(14.181)
kD0
Proceeding as in the previous section, we get the block diagram illustrated in Figure 14.36.
1072
Chapter 14. Synchronization
matched filter
rAA (nTc ) e-j
^ ) x(nTc ,Ω n
gM (iTc )
^ nTc Ω n
frequency matched filter
g
FM
NCO
(iTc )
^ ) xFM (nTc ,Ω n
loop filter
u(nTc )
F(z)
( .) *
2Re{ .}
e(nTc )
Figure 14.35. NDA-ND"-FB frequency offset estimator. [From Meyr, Moeneclaey, and Fechtel (1998). Reproduced by permission of Wiley.]
m k Ts
µk matched filter
rAA(nTc )
g (iTc ) M
^ ) x(nTc ,Ω n
frequency matched filter
g (iTc )
h I (iTc ,µ k )
µk
^
e-j Ωn nTc
decimator
interpolator
^ ) xFM (nTc ,Ω n
FM
m k Tc decimator
interpolator
h I (iTc ,µ k )
( .) *
loop filter NCO Tc T
u(kT)
F(z)
e(kT)
2Re{ .}
Figure 14.36. NDA-D"-FB frequency offset estimator.
14.6.2
Estimators operating at the modulation rate
As mentioned at the beginning of Section 14.6, whenever the condition (14.164) is satisfied, the frequency translator can be moved after the decimator and consequently frequency offset estimation may take place after timing estimation, thus obtaining D" algorithms. The likelihood (14.166) for an observation interval [.K 1/=2; .K 1/=2], with K odd, evaluated at e D "O becomes ( ) .KX 1/=2 2 Ł jokT j z Re[Þk x k .O" / e e ] (14.182) L;;a .z; o; α/ D exp N0 kD.K 1/=2
14.6. Algorithms for carrier frequency recovery
1073
Data-aided and data-directed We assume a known training sequence a D α 0 is transmitted during the acquisition phase, and denote as Þ0;k the k-th symbol of the sequence. The log-likelihood yields the joint estimate of .; / as O / O D arg max .; z;o
.KX 1/=2
Ł Re[Þ0;k x k .O" / ejokT e j z ]
(14.183)
kD.K 1/=2
Note that the joint estimate is computed by finding the maximum of a function of one variable; in fact, defining r.o/ D
.KX 1/=2
Ł Þ0;k x k .O" / ejokT
(14.184)
kD.K 1/=2
(14.183) can be rewritten as O / O D arg max jr.o/j Re[e j .zargfr.o/g/ ] .;
(14.185)
z;o
The maximum (14.183) is obtained by first finding the value of o that maximizes jr.o/j, O D arg max jr.o/j
(14.186)
o
and then finding the value of z for which the term within brackets in (14.185) becomes real valued, O O D argfr./g
(14.187)
We now want to solve (14.186) in close form; a necessary condition to get a maximum O is that the derivative of jr.o/j2 D r.o/ rŁ .o/ with respect to o is equal to zero for o D . Defining ½ 2 K 1 k.k C 1/ (14.188) bk D 4 we obtain
( O D arg T
.KX 1/=2
bk
kD.K 1/=2
Ł Þ0;kC1 Ł Þ0;k
) [x kC1 .O" / x kŁ .O" /]
(14.189)
The DD estimator is obtained by substituting α 0 with the estimate aO .
Non-data aided for M-PSK Ł Ł /.x By raising .Þ0;kC1 =Þ0;k " / x kŁ .O" // to the M-th power we obtain the NDA version of kC1 .O the DA (DD) algorithm for M-PSK signals as ( .K 1/=2 ) X O D arg bk [x kC1 .O" / x kŁ .O" /] M T (14.190) kD.K 1/=2
1074
14.7
Chapter 14. Synchronization
Second-order digital PLL
In feedback systems, the recovery of the phase and the frequency can be jointly performed by a second-order digital PLL (DPLL), given by Ok OkC1 D Ok C ¼ e .kT / C O kC1 D O k C ¼;1 e .kT / C ¼;2 e .kT /
(14.191) (14.192)
where e and e are estimates of the phase error and of the frequency error, respectively, and ¼ , ¼;1 and ¼;2 are suitable constants. Typically (see Example 15.6.4 on page 1110) p ¼;1 ' ¼;2 ' ¼ . Observe that (14.191) and (14.192) form a digital version of the second-order analog PLL illustrated in Figure 14.7.
14.8
Synchronization in spread spectrum systems
In this section we discuss early-late FB schemes, named delay-locked loops (DLLs) [7], for the timing recovery in spread spectrum systems (see Chapter 10).
14.8.1
The transmission system
Transmitter In Figure 14.37 (see Figures 10.2 and 10.3) the (baseband equivalent) scheme of a spread spectrum transmitter is illustrated. The symbols fak g, generated at the modulation rate 1=T , are input to a holder, which outputs the symbols aN m at the chip rate 1=Tchi p . The chip period Tchi p is given by Tchi p D
T NS F
(14.193)
where N S F denotes the spreading factor. Then the symbols fak g are multiplied by the spreading code fcm g and input to a filter with impulse response h T x .t/ that includes the DAC. The (baseband equivalent) transmitted signal s.t/ is expressed as s.t/ D
C1 X
aN m cm h T x .t mTchi p /
mD1
Figure 14.37. Baseband transmitter for spread spectrum systems.
(14.194)
14.8. Synchronization in spread spectrum systems
1075
In terms of the symbols fak g, we obtain the following alternative representation that will be used next: C1 X
s.t/ D
ak
CN S F 1 k N S FX
kD1
cm h T x .t mTchi p /
(14.195)
mDk N S F
Optimum receiver With the same assumptions of Section 14.1 and for the transmission of K symbols, the received signal rC .t/ is expressed as rC .t/ D
K 1 X kD0
ak
CN S F 1 k N S FX
cm qC .t mTchi p "Tchi p / e j .tC / C wC' .t/
(14.196)
mDk N S F
The likelihood Lss D L;";;a .z; e; o; α/ can be computed as in (14.77); after a few steps we obtain ( " k N S FX CN S F 1 1 2 KX Ł Re ÞkŁ e j z cm Lss D exp N0 kD0 mDk N S F (14.197) #) Z ².t/ ejot g M .mTchi p C eTchi p t/ dt TK
Defining the two signals Z
².− / ejo− g M .t − / d−
x.t; o/ D TK
y.kT; e; o/ D
CN S F 1 k N S FX
(14.198) Ł cm
x.mTchi p C eTchi p ; o/
mDk N S F
the likelihood becomes ( L;";;a .z; e; o; α/ D exp
) 1 2 KX Ł jz Re[Þk y.kT; e; o/ e ] N0 kD0
(14.199)
To obtain the samples y.kT; e; o/ we can proceed as follows: 1. obtain the samples y.lTchi p ; e; o/ D
l X
Ł cm x.mTchi p C eTchi p ; o/
(14.200)
mDlN S F C1
2. decimate y.lTchi p ; e; o/ at l D .k C 1/N S F 1, i.e. evaluate y.lTchi p ; e; o/ for l D N S F 1; 2N S F 1; : : : ; K Ð N S F 1.
1076
Chapter 14. Synchronization
^ y(kT, ^ε , Ω )
m
^ x(mTchip + ^ε Tchip , Ω )
Σ i=m−N
a^ k
NSF
SF +1
^
^ y(mTchip , ε^ , Ω )
cm*
e−jθ (kT)
phase estimator
Figure 14.38. Digital receiver for spread spectrum systems.
By (14.199) it is possible to derive the optimum digital receiver. In particular, up to the decimator that outputs samples at instants fmm Tc g the receiver is identical to that of Figure 14.19;8 then in Figure 14.38 only part of the receiver is shown.
14.8.2
Timing estimators with feedback
Assume there is no frequency offset, i.e. D 0; the likelihood is obtained by letting o D 0 in (14.199) to yield ( ) 1 2 KX Ł jz Re[Þk yk .e/ e ] (14.201) L;";a .z; e; α/ D exp N0 kD0 where we use the compact notation yk .e/ D y.kT; e; 0/
(14.202)
The early-late FB estimators are obtained by approximating the derivative of (14.201) with respect to e with a finite difference, and evaluating it for e D "O k .
Non-data aided: non-coherent DLL In the NDA case, the log-likelihood is obtained from (14.120); using yk .e/ instead of x k .e/ yields `" .e/ D
K 1 X
jyk .e/j2
(14.203)
kD0
From (14.110) the derivative of `" .e/ is approximated as 1 1 KX @`" .e/ ' [jyk .e C Ž/j2 jyk .e Ž/j2 ] @e 2Ž kD0
(14.204)
By including the constant 1=.2Ž/ in the loop filter, and also assuming that the sum is performed by the loop filter we obtain the error signal e.kT / D jyk .O"k C Ž/j2 jyk .O"k Ž/j2 8
(14.205)
Note that now Tc ' Tchi p =2, and the sampling instants at the decimator are such that mTchi p C "O Tchi p D mm Tc C ¼m Tc . The estimate "O is updated at every symbol period T D Tchi p Ð N S F .
14.8. Synchronization in spread spectrum systems
1077
The block diagram of the estimator is shown in Figure 14.39; note that the lag and the lead equal to ŽTchi p are implemented by interpolator filters operating at the sampling Q where period Tc (see (14.91)) with parameter ¼ equal, respectively, to ŽQ and CŽ, ŽQ D Ž.Tchi p =Tc /. The estimator is called a non-coherent digital DLL [8, 9], as the dependence of the error signal on the pair .; a/ is eliminated without computing the estimates.
Non-data aided MCTL We compute now the derivative of (14.203) in exact form, " # K 1 X @`" .e/ d Ł D 2Re y .e/ yk .e/ @e de k kD0
(14.206)
Approximating the derivative of ykŁ .e/ as in (14.110) we obtain 1 @`" .e/ 1 KX ' Re[yk .e/.yk .e C Ž/ yk .e Ž//Ł ] @e Ž kD0
(14.207)
Assuming the loop filter performs the multiplication by 1=Ž and the sum, the error signal is given by e.kT / D Re[yk .O"k /.yk .O"k C Ž/ y.O"k Ž//Ł ]
(14.208)
The block diagram of the estimator is shown in Figure 14.40 and is called modified code tracking loop (MCTL) [10]; also in this case the estimator is non-coherent.
Data- and phase-directed: coherent DLL In case the estimates O and aO are given, the log-likelihood is expressed as `" .e/ D
K 1 X
Re[aO kŁ yk .e/ e j ] O
(14.209)
kD0
Approximating the derivative as in (14.110) and including both the multiplicative constant and the summation in the loop filter, the error signal is given by e.kT / D Re[aO kŁ e j [yk .O"k C Ž/ yk .O"k Ž/]] O
(14.210)
Figure 14.41 illustrates the block diagram of the estimator, which is called coherent DLL [9, 10, 11], as the error signal is obtained by the estimates O and aO . In the three schemes of Figure 14.39, 14.40, and 14.41 the direct section of the DLL gives estimates of mm and ¼m at every symbol period T , whereas the feedback loop may operate at the chip period Tchi p . Observe that by removing the decimation blocks the DLL is able to provide timing estimates at every chip period.
Chapter 14. Synchronization
Figure 14.39. Non-coherent DLL.
1078
Figure 14.40. Direct section of the non-coherent MCTL.
14.8. Synchronization in spread spectrum systems 1079
Figure 14.41. Direct section of the coherent DLL.
1080 Chapter 14. Synchronization
14. Bibliography
1081
Bibliography [1] H. Meyr and G. Ascheid, Synchronization in digital communications, vol. 1. New York: John Wiley & Sons, 1990. [2] L. E. Franks, “Carrier and bit synchronization in data communication—A tutorial review”, IEEE Trans. on Communications, vol. 28, pp. 1107–1120, Aug. 1980. [3] J. G. Proakis, Digital communications. New York: McGraw-Hill, 3rd ed., 1995. [4] H. Meyr, M. Moeneclaey, and S. A. Fechtel, Digital communication receivers. New York: John Wiley & Sons, 1998. [5] U. Mengali and A. N. D’Andrea, Synchronization techniques for digital receivers. New York: Plenum Press, 1997. [6] K. H. Mueller and M. S. Muller, “Timing recovery in digital synchronous data receivers”, IEEE Trans. on Communications, vol. 24, pp. 516–531, May 1976. [7] M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread spectrum communications handbook. New York: McGraw-Hill, 1994. [8] R. De Gaudenzi, M. Luise, and R. Viola, “A digital chip timing recovery loop for bandlimited direct-sequence spread-spectrum signals”, IEEE Trans. on Communications, vol. 41, pp. 1760–1769, Nov. 1993. [9] R. De Gaudenzi, “Direct-sequence spread-spectrum chip tracking loop in the presence of unresolvable multipath components”, IEEE Trans. on Vehicular Technology, vol. 48, pp. 1573–1583, Sept. 1999. [10] R. A. Yost and R. W. Boyd, “A modified PN code tracking loop: its performance analysis and comparative evaluation”, IEEE Trans. on Communications, vol. 30, pp. 1027– 1036, May 1982. [11] R. De Gaudenzi and M. Luise, “Decision-directed coherent delay-lock tracking loop for DS-spread spectrum signals”, IEEE Trans. on Communications, vol. 39, pp. 758– 765, May 1991.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 15
Self-training equalization
By the expression self-training equalization we refer to channel equalization techniques by which we obtain the initial convergence of the parameters of an adaptive equalizer without resorting to the transmission of a training sequence. Although these techniques generally achieve sub-optimum performance at convergence, they are applied in many cases of practical interest, where the transmission of known training sequences is not viable, as for example in broadcast systems, or may result in an undesirable increase of system complexity [1, 2, 3, 4, 5, 6]. In these cases it is necessary to consider self-training of the adaptive equalizer, where the terms for adjusting the coefficient parameters are obtained by processing the received signal. Usually the receiver performs self-training of the equalizer by referring to the knowledge of the input signal characteristics, for example, the probability distribution of the input symbols. The subject of self-training equalization in communications systems has received considerable attention since the publication of the article by Sato [7] in 1975; the proposed algorithms have been alternatively called self-recovering, self-adaptive, or blind.
15.1
Problem definition and fundamentals
With reference to the discrete-time equivalent scheme of Figure 7.18 shown in Figure 15.1, we consider a real-valued discrete-time system given by the cascade of a linear channel fh i g, in general with non-minimum phase transfer function1 H , and an equalizer C $ fci g; in general, the impulse responses fh i g and fci g are assumed with P unlimited duration. The overall system is given by 9 D H Ð C $ f i g, where i D 1 `D1 c` h i ` . The channel input symbol sequence fak g is modeled as a sequence of i.i.d. random variables with symmetric probability density function pa .Þ/. The channel output sequence is fx k g. In this section, additive noise introduced by the channel is ignored. We note that a non-minimum phase rational transfer function (see Definition 1.8 on page 26) can be expressed as H .z/ D H0
1
P1 .z/ P2 .z/ P3 .z/
In this chapter, to simplify notation, often the argument of the z-transform is not explicitly indicated.
(15.1)
1084
Chapter 15. Self-training equalization
Figure 15.1. Discrete-time equivalent channel and equalizer filter.
where H0 denotes the gain, P3 .z/ and P1 .z/ are monic polynomials with zeros inside the unit circle, and P2 .z/ is a monic polynomial with zeros outside the unit circle. We introduce the inverse functions with respect to the polynomials P1 .z/ and P2 .z/ given by P11 .z/ D 1=P1 .z/ and P21 .z/ D 1=P2 .z/, respectively. By expanding P11 .z/ and P21 .z/ in Laurent series we obtain, apart from lag factors, P11 .z/ D
C1 X
c1;n z n
nD0
(15.2) P21 .z/
D
0 X
c2;n z
n
nD1
both converging in a ring that includes the unit circle. Therefore we have ! ! C1 0 X X 1 1 n n H .z/ D P3 .z/ c1;n z c2;n z H0 nD1 nD0
(15.3)
In general from (15.3) we note that, as the system is non-minimum phase, the inverse system cannot be described by a finite number of parameters; in other words, the reconstruction of a transmitted symbol at the equalizer output at a certain instant requires the knowledge of the entire received sequence. In practice, to obtain an implementable system, the series in (15.3) are truncated to N terms, and an approximation of H 1 with a lag equal to .N1 1/ modulation intervals is given by ! ! N 1 0 X X 1 .N1 1/ 1 n n z P3 .z/ c1;n z c2;n z (15.4) H .z/ ' H0 nD0 nD.N 1/ Therefore the inverse system can only be defined apart from a lag factor. The problem of self-training equalization is formulated as follows: from the knowledge of the probability distribution of the channel input symbols fak g and from the observation of the channel output sequence fx k g, we want to find an equalizer C such that the overall system impulse response is ISI free. Observe that, if the channel H is minimum phase,2 both H and H 1 are causal and stable; in this case the problem of channel identification, and hence the determination of
2
We recall that all AR models (1.518) are minimum phase.
15.1. Problem definition and fundamentals
1085
the sequence fak g, can be solved by whitening (see page 143) the observation fx k g using known procedures that are based on the second-order statistical description of signals. If, as it happens in the general case, the channel H is non-minimum phase, by the second-order statistical description (1.263) it is possible to identify only the amplitude characteristic of the channel transfer function, but not its phase characteristic. In particular, if the probability density function of the input symbols is Gaussian, the output fx k g is also Gaussian and the process is completely described by a second-order analysis; therefore the above observations are valid and in general the problem of selftraining equalization cannot be solved for Gaussian processes. Note that we have referred to a system model in which the sampling frequency is equal to the symbol rate; a solution to the problem using a second-order description with reference to an oversampled model is obtained in [8]. Furthermore, we observe that, as the probability density function of the input symbols is symmetric, the sequence fak g has the same statistical description as the sequence fak g; consequently it is not possible to distinguish the desired equalizer H 1 from the equalizer H 1 . Hence, the inverse system can only be determined apart from the sign and a lag factor. Therefore the solution to the problem of self-training equalization is given by C D šH 1 , which yields the overall system 9 D šI , where I denotes the identity, with the exception of a possible lag. In this chapter we will refer to the following theorem, demonstrated in [9]. Theorem 15.1 Assuming that the probability density function of the input symbols is non-Gaussian, then 9 D šI if the output sample yk D
C1 X
cn x kn
(15.5)
nD1
has a probability density function p yk .b/ equal to the probability density function of the symbols fak g. Therefore, using this theorem to obtain the solution, it is necessary to determine an algorithm for the adaptation of the coefficients of the equalizer C such that the probability distribution of yk converges to the distribution of ak . We introduce the cost function J D E[8.yk /]
(15.6)
where yk is given by (15.5) and 8 is an even, real-valued function that must be chosen so that the optimum solution, determined by Copt .z/ D arg min J C.z/
(15.7)
is found at the points šH 1 , apart from a lag factor. In most applications, minimization is done by equalizers of finite length N having coefficients at instant k given by ck D [c0;k ; c1;k ; : : : ; c N 1;k ]T (15.8)
1086
Chapter 15. Self-training equalization
Let xk D [x k ; x k1 ; : : : ; x k.N 1/ ]T
(15.9)
then (15.5) becomes yk D
N 1 X
cn;k x kn D ckT xk
(15.10)
nD0
Then also (15.7) simplifies into copt D arg min J c
(15.11)
If 2 is the derivative of 8, the gradient of J with respect to c is given by (see (2.18)) rc J D E[xk 2.yk /]
(15.12)
For the minimization of J we use a stochastic gradient algorithm for which (see (3.40)) ckC1 D ck ¼ 2.yk / xk
(15.13)
where ¼ is the adaptation gain. Note that the convergence of C to H 1 or to H 1 depends on the initial choice of the coefficients. Before tackling the problem of choosing the function 8, we consider the following problem.
Minimization of a special function We consider an overall system ψ with impulse response having unlimited duration. We want to determine the values of the sequence f i g that minimize the following cost function: C1 X
VD
j
ij
(15.14)
i D1
subject to the constraint C1 X
2 i
D1
(15.15)
i D1
The function V characterizes the peak amplitude of the system output signal fyk g and the constraint can be interpreted as a requirement that the solution f i g belongs to the sphere with center the origin and radius r D 1 in the parameter space f i g. Letting ψ D [: : : ; 0 ; 1 ; : : : ]T , it results in rψ V D [: : : ; sgn. 0 /; sgn. 1 /; : : : ]T ; then, if max is the maximum value assumed by i , the cost function (15.14) presents stationary points for i D š max . Now, taking into account the constraint (15.15), it is easily seen that the minimum of (15.14) is reached only when one element of the sequence f i g is different from zero (with a value equal to max ) and the others are all zeros. In other words, the only points of minimum are given by 9 D šI , and the other stationary points correspond to saddle points. Figure 15.2 illustrates the cost function V along the unit circle for a system with two parameters.
15.1. Problem definition and fundamentals
1087
V (ψ0, ψ1) 1.5
1
0.5
+I
+I
0 1
−I
0.5
1
0.5 0
−0.5
ψ
1
−I
−∇ V
0
−0.5
−1
ψ
0
−1
Figure 15.2. Illustration of the cost function V for the system 9 $ f 0 ; 1 g and of the gradient rV projected onto the straight line tangent to the unit circle.
The minimization can also be obtained recursively, updating the parameters f i g as indicated by the direction of steepest descent3 of the cost function V. The projection of the gradient V onto the plane tangent to the unit sphere at the point ψ is given by rψ V [.rψ V/T ψ]ψ then the recursive equation yields 0 i;kC1
i;kC1
D
i;k
" ¼ sgn.
i;k /
i;k
(15.16) C1 X `D1
0 i;kC1
Dv u C1 u X t . P
`D1
# j
`;k j
(15.17)
(15.18)
0 2 `;kC1 /
Note that, if the term i;k ` j `;k j is omitted in (15.17), with good approximation the direction of the steepest descent is still followed, provided that the adaptation gain ¼ is sufficiently small. From (15.17), for each parameter i the updating consists of a correction toward zero of a fixed value and a correction in the opposite direction of a value proportional to the 3
The direction is defined by the gradient vector.
1088
Chapter 15. Self-training equalization
−∆
0
1
2
Reduce by − +∆
ψi,0
3
4
6
5
+∆
’ Rescale by ε ψi,1
’ = ψi,1 (1+ε)ψi,1 ψi,1 ’
0
1
2
3
4
Reduce by − +∆
ψi,1
0
1
2
3
6
5
4
5
6 Rescale by ε ψ ’i,2
0
1
2
3
4
5
6
4
5
6
ψi,k
0
1
2
3
Figure 15.3. Update of the parameters f i g, i D 0; : : : ; 6.
parameter amplitude. Assuming that the initial point is not a saddle point, by repeated iterations of the algorithm, one of the parameters i approaches the value one, while all others converge to zero, as shown in Figure 15.3. We now want to obtain the same adaptation rule for the parameters f i g using the output signal yk . We define ψ k as the vector of the parameters of ψ at instant k, ψ k D [: : : ;
0;k ;
1;k ; : : : ]
T
(15.19)
and ak D [: : : ; ak ; ak1 ; : : : ]T
(15.20)
15.1. Problem definition and fundamentals
1089
Therefore yk D ψ kT ak
(15.21)
Assume that at the beginning of the adaptation process the overall system 9 satisfies the condition jjψjj2 D 1, but it deviates significantly from the system identity; then the equalizer output signal yk will occasionally assume positive or negative values which are much P larger in magnitude than Þmax D max ak . The peak value of yk is given by šÞmax i j i;k j, obtained with symbols faki g equal to šÞmax , and indicates that the distortion is too large and must be reduced. In this case a correction of a fixed value towards zero is obtained using the error signal ek D yk Þmax sgn.yk /
(15.22)
and applying the stochastic gradient algorithm ψ kC1 D ψ k ¼ ek ak
(15.23)
If the coefficients are scaled so that the condition jjψ k jj2 D 1 is satisfied at every k, we obtain a coefficient updating algorithm that approximates algorithms (15.17) and (15.18). Obviously algorithm (15.23) cannot be directly applied, as the parameters of the overall system 9 are not available. However, observe that if the linear transformation H is non-singular, then formally C D H 1 9. Therefore the overall minima of V at the points 9 D šI are mapped into overall minima at the points C D šH 1 of a cost function J that is the image of V under the transformation given by H 1 , as illustrated in Figure 15.4. Furthermore, it is seen that the direction of steepest descent of V has not been modified by this transformation. Thus the updating terms for the equalizer coefficients are still given by (15.22) and (15.23), if symbols ak are replaced by the channel output samples x k . Then, a general algorithm that converges to the desired solution C D H 1 can be formulated as follows: ž observe the equalizer output signal fyk g and determine its peak value; Global minima of V ( ψ )
Global minima of J( C )
H −1 ψ
C
Figure 15.4. Illustration of the transformation C D H1 9.
1090
Chapter 15. Self-training equalization
ž whenever a peak value occurs, update the coefficients according to the algorithm ek D yk Þmax sgn.yk / ckC1 D ck ¼ ek xk
(15.24)
ž scale the coefficients so that the statistical power of the equalizer output samples is equal to the statistical power of the channel input symbols. We observe that it is not practical to implement an algorithm that requires computing the peak value of the equalizer output signal and updating the coefficients only when a peak value is observed. In the next sections we describe algorithms that allow the updating of the coefficients at every modulation interval, thus avoiding the need of computing the peak value of the signal at the equalizer output and of scaling the coefficients.
15.2
Three algorithms for PAM systems
The Sato algorithm The Sato cost function is defined as JDE
h
1 2
i yk2 S jyk j
(15.25)
where S D
E[ak2 ] E[jak j]
(15.26)
The gradient of J is given by rc J D E[xk .yk S sgn.yk //]
(15.27)
ž S;k D yk S sgn.yk /
(15.28)
We introduce the signal
which assumes the meaning of pseudo error that during self-training replaces the error used in the LMS decision directed algorithm. We recall that for the LMS algorithm the error signal is4 ek D yk aO k
(15.29)
where aO k is the detection of the symbol ak , obtained by a threshold detector from the sample yk . Figure 15.5 shows the pseudo error žS ;k as a function of the value of the equalizer output sample yk .
4
Note that in this chapter, the error signal is defined with opposite sign with respect to the previous chapters.
15.2. Three algorithms for PAM systems
1091
e S,k
gS
* gs
yk
Figure 15.5. Characteristic of the pseudo error žS,k as a function of the equalizer output.
Therefore the Sato algorithm for the coefficient updating of an adaptive equalizer assumes the expression ckC1 D ck ¼ ž S;k xk
(15.30)
It was proved by Benveniste, Goursat, and Ruget [9] that, if the probability density function of the output symbols fak g is sub-Gaussian,5 then the Sato cost function (15.25) admits as unique points of minimum the systems C D šH 1 , apart from a possible lag; however, note that the uniqueness of the points of minimum of the Sato cost function is obtained by assuming a continuous probability distribution of input symbols. In the case of a discrete probability distribution with an alphabet A D fš1; š3; : : : ; š.M 1/g, the convergence properties of the Sato algorithm are not always satisfactory. Another undesirable characteristic of the Sato algorithm is that the pseudo error ž S;k is not equal to zero for C D šH 1 , unless we consider binary transmission. In fact only the gradient of the cost function given by (15.27) is equal to zero for C D šH 1 ; moreover, we find that the variance of the pseudo error may assume non-negligible values in the neighborhood of the desired solution.
Benveniste–Goursat algorithm To mitigate the above mentioned inconvenience, we observe that the error ek used in the LMS algorithm in the absence of noise becomes zero for C D šH 1 . It is possible to combine the two error signals obtaining the pseudo error proposed by Benveniste and Goursat [10], given by žG;k D 1 ek C 2 jek j ž S;k 5
(15.31)
A probability density function pak .Þ/ is sub-Gaussian if it is uniform or if pak .Þ/ D K expfg.Þ/g, where dg g.Þ/ is an even function such that both g.Þ/ and Þ1 dÞ are strictly increasing in the domain [0; C1/.
1092
Chapter 15. Self-training equalization
where 1 and 2 are constants. If the distortion level is high, jek j assumes high values and the second term of (15.31) allows convergence of the algorithm during the self-training. Near convergence, for C ' šH 1 , the second term has the same order of magnitude as the first and the pseudo error assumes small values in the neighborhood of C D šH 1 . Note that an algorithm that uses the pseudo error (15.31) allows a smooth transition of the equalizer from the self-training mode to the decision-directed mode. In the case of a sudden change in channel characteristics, the equalizer is found working again in selftraining mode. Thus the transitions between the two modes occur without control on the level of distortion of the signal fyk g at the equalizer output.
Stop-and-go algorithm The stop-and-go algorithm proposed by Picchi and Prati [11] can be seen as a variant of the Sato algorithm that achieves the same objectives of the Benveniste–Goursat algorithm with better convergence properties. The pseudo error for the stop-and-go algorithm is formulated as ( ek if sgn.ek / D sgn.ž S;k / (15.32) ž P;k D 0 otherwise where ž S;k is the Sato pseudo error given by (15.28), and ek is the error used in the decision-directed algorithm given by (15.29). The basic idea is that the algorithm converges if updating of the equalizer coefficients is turned off with sufficiently high probability every time the sign of error (15.29) differs from the sign of error ei d;k D yk ak , that is sgn.ek / 6D sgn.ei d;k /. As ei d;k is not available in a self-training equalizer, with the stopand-go algorithm coefficient updating is turned off whenever the sign of error ek is different from the sign of Sato error ž S;k . Obviously, in this way we also get a non-zero probability that coefficient updating is inactive when the condition sgn.ek / D sgn.ei d;k / occurs, but this does not usually bias the convergence of the algorithm.
Remarks At this point we can make the following observations. ž Self-training algorithms based on the minimization of a cost function that includes the term E[jyk j p ], p ½ 2, can be explained referring to the algorithm (15.24), because the effect of raising to the p-th power the amplitude of the equalizer output sample is that of emphasizing the contribution of samples with large amplitude. ž Extension of the Sato cost function (15.25) to QAM systems, which we discuss in Section 15.5, is given by i h (15.33) J D E 12 jyk j2 S jyk j where S D E[jak j2 ]=E[jak j]. In general this term guarantees that, at convergence, the statistical power of the equalizer output samples is equal to the statistical power of the input symbols.
15.3. The contour algorithm for PAM systems
1093
ž In the algorithm (15.24), the equalizer coefficients are updated only when we observe a peak value of the equalizer output signal. As the peak value decreases with the progress of the equalization process, updating of the equalizer coefficients ideally depends on a threshold that varies depending on the level of distortion in the overall system impulse response.
15.3
The contour algorithm for PAM systems
The algorithm (15.24) suggests that the equalizer coefficients are updated when the equalizer output sample reaches a threshold value, which in turn depends on the level of distortion in the overall system impulse response. In practice, we define a threshold at instant k as Tk D Þmax C C A;k , where the term C A;k ½ 0 represents a suitable measure of distortion. Each time the absolute value of the equalizer output sample reaches or exceeds the threshold Tk , the coefficients are updated so that the peak value of the equalizer output is “driven” towards the constellation boundary šÞmax ; at convergence of the equalizer coefficients, C A;k vanishes. Figure 15.6 illustrates the evolution of the “contour” Tk for a two-dimensional constellation. Im[ yk ] T0 Tk α max
α max
Re[ yk ]
Figure 15.6. Evolution in time of the ‘‘contour’’ Tk for a two-dimensional constellation.
1094
Chapter 15. Self-training equalization
The updating of coefficients described above can be obtained by a stochastic gradient algorithm that is based on a cost function E[8.yk /] defined on the parameter space f i g. We assume that the overall system 9 initially corresponds to a point on a sphere of arbitrary radius r. With updating terms that on the average exhibit the same sign as the terms found in the general algorithm (15.24), the point on the sphere of radius r moves in such a way to reduce distortion. Moreover, if the radial component of the gradient, i.e. the component that is orthogonal to the surface of the sphere, is positive for r > 1, negative for r < 1 and vanishes on the sphere of radius r D 1, it is not necessary to scale the coefficients and the convergence will take place to the point of global minimum 9 D šI . Clearly, the derivative function of 8.yk / with respect to yk that defines the pseudo error can be determined in various ways. A suitable definition is given by ( yk [Þmax C C A;k ] sgn.yk / if jyk j ½ Þmax (15.34) 2.yk / D C A;k sgn.yk / otherwise For a self-training equalizer, the updating of coefficients as indicated by (15.8) is obtained by the algorithm ckC1 D ck ¼ 2.yk / xk
(15.35)
In this algorithm, to avoid the computation of the threshold Tk at each iteration, the amplitude of the equalizer output signal is compared with Þmax rather than with Tk ; note that the computation of 2.yk / depends on the event that yk falls inside or outside the constellation boundary. In the next section, for a two-dimensional constellation, we define in general the constellation boundary as the contour line that connects the outer points of the constellation; for this reason we refer to this algorithm as the contour algorithm [12]. To derive the algorithms (15.34) and (15.35) from the algorithm (15.24) several approximations are introduced; consequently the convergence properties cannot be directly derived from those of the algorithm (15.24). In Appendix 15.A we show how C A should be defined to obtain the desired behavior of the algorithms, (15.34) and (15.35) in the case of systems with input symbols having a uniform continuous distribution. An advantage of the functional introduced in this section with respect to the Sato cost function is that the variance of the pseudo error vanishes at the points of minimum 9 D šI ; this means that it is possible to obtain the convergence of the MSE to a steady state value that is close to the achievable minimum value. Furthermore, the radial component of the gradient of E[8.yk /] vanishes at every point on the unit sphere, whereas the radial component of the gradient in the Sato cost function vanishes on the unit sphere only at the points 9 D šI . As the direction of steepest descent does not intersect the unit sphere, the contour algorithm avoids overshooting of the convergence trajectories observed using the Sato algorithm; in other words, the stochastic gradient yields a coefficient updating that is made in the correct direction more often than in the case of the Sato algorithm. Therefore, substantially better convergence properties are expected for the contour algorithm even in systems with a discrete probability distribution of input symbols. The complexity of the algorithms (15.34) and (15.35) can be deemed prohibitive for practical implementations, especially for self-training equalization in high-speed communication systems, as the parameter C A;k must be estimated at each iteration. In the next
15.3. The contour algorithm for PAM systems
1095
section, we discuss a simplified algorithm that allows implementation with low complexity; we will see later how the simplified formulation of the contour algorithm can be extended to self-training equalization of partial response and QAM systems.
Simplified realization of the contour algorithm We assume that the input symbols fak g form a sequence of i.i.d. random variables with a uniform discrete probability density function. From the scheme of Figure 15.1, in the presence of noise, the channel output signal is given by 1 X
xk D
h i aki C wk
(15.36)
i D1
where fwk g denotes additive white Gaussian noise. The equalizer output is given by yk D ckT xk . To obtain an algorithm that does not require the knowledge of the parameter C A , the definition (15.34) suggests introducing the pseudo error ( yk Þmax sgn.yk / if jyk j ½ Þmax (15.37) žC A;k D Žk sgn.yk / otherwise where Žk is a non-negative parameter that is updated at every iteration as follows: 8 M 1 > > 1 if jyk j ½ Þmax < Žk M ŽkC1 D (15.38) > > : Žk C 1 1 otherwise M and 1 is a positive constant. The initial value Ž0 is not a critical system parameter and can be, for example, chosen equal to zero; the coefficient updating algorithm is thus given by ckC1 D ck ¼ žC A;k xk
(15.39)
In comparison to (15.34), now Žk does not provide a measure of distortion as C A . The definition (15.37) is justified by the fact that the term yk [Þmax C C A ] sgn.yk / in (15.34) can be approximated as yk Þmax sgn.yk /, because if the event jyk j ½ Þmax occurs the pseudo error yk Þmax sgn.yk / can be used for coefficient updating. Therefore, Žk should increase in the presence of distortion only in the case the event jyk j < Þmax occurs more frequently than expected. This behavior of the parameter Žk is obtained by applying (15.38). Moreover, (15.38) guarantees that Žk assumes values that approach zero at the convergence of the equalization process; in fact, in this case the probabilities of the events fjyk j < Þmax g and fjyk j ½ Þmax g assume approximately the values .M 1/=M and 1=M, respectively, that correspond to the probabilities of such events for a noisy PAM signal correctly equalized. Figure 15.7 shows the pseudo error žC A;k as a function of the value of the equalizer output sample yk . The contour algorithm has been described for the case of uniformly distributed input symbols; however, this assumption is not necessary. In general, if fyk g represents an equalized signal, the terms .M 1/=M and 1=M in (15.38) are, respectively, substituted by p0 D P[jak j < Þmax ] C
1 2
P[jak j D Þmax ] ' P[jyk j < Þmax ]
(15.40)
1096
Chapter 15. Self-training equalization
εCA,k
+δk −α max
+α max
−δk
yk
Figure 15.7. Characteristic of the pseudo error žCA,k as a function of the equalizer output.
and p1 D
1 2
P[jak j D Þmax ] ' P[jyk j ½ Þmax ]
(15.41)
We note that the considered receiver makes use of signal samples at the symbol rate; the algorithm can also be applied for the initial convergence of a fractionally spaced equalizer (see Section 8.4) in case a sampling rate higher than the symbol rate is adopted.
15.4
Self-training equalization for partial response systems
Self-training adaptive equalization has been mainly studied for full response systems; however, self-training equalization methods for partial response systems have been proposed for linear and non-linear equalizers [7, 13, 14]. In general, a self-training equalizer is more difficult to implement for partial response systems (see Appendix 7.A), especially if the input symbol alphabet has more than two elements. Moreover, as self-training is a slow process, the accuracy achieved in the recovery of the timing of the received signal before equalization plays an important role. In fact, if timing recovery is not accomplished, because of the difference between the sampling rate and the modulation rate we have that the sampling phase varies with respect to the timing phase of the remote transmitter clock; in this case we speak of phase drift of the received signal; self-training algorithms fail if the phase drift of the received signal is not sufficiently small. In this section, we discuss the extension to partial response systems of the algorithms for PAM systems presented in the previous section.
The Sato algorithm for partial response systems We consider a multilevel partial response class IV (PR-IV) system, also called a modified duobinary system. Using the D transform, the desired transfer function of the overall system
15.4. Self-training equalization for partial response systems
1097
is given by .D/ D .1 D 2 /. The objective of an adaptive equalizer for a PR-IV system consists in obtaining an equalized signal of the form yk D .ak ak2 / C w y;k D u k C w y;k
(15.42)
where w y;k is a disturbance due to noise and residual distortion. We consider the case of quaternary modulation. Then the input symbols ak are from the set f3; 1; C1, C3g, and the output signal u k D ak ak1 , for an ideal channel in the absence of noise, can assume one of the seven values f6; 4; 2; 0; C2; C4; C6g.6 As illustrated in Figure 15.8, to obtain a pseudo error to be employed in the equalizer coefficient updating algorithm, the equalizer output signal fyk g is transformed into a fullresponse signal v S;k by the linear transformation v S;k D yk C þ S v S;k2
(15.43)
where þ S is a constant that satisfies the condition 0 < þ S < 1. Then the signal v S;k is quantized by a quantizer with two levels corresponding to š S , where S is given by (15.26). The obtained signal is again transformed into a partial response signal that is subtracted from the equalizer output to generate the pseudo error ž S;k D yk S [sgn.v S;k / sgn.v S;k2 /]
(15.44)
Figure 15.8. Block diagram of a self-training equalizer for a PR-IV system using the Sato algorithm.
6
In general, for an ideal PR-IV channel in the absence of noise, if the alphabet of the input symbols is A D fš1; š3; : : : ; š.M 1/g, the output symbols assume one of the .2M 1/ values f0; š2; : : : ; š2.M 1/g.
1098
Chapter 15. Self-training equalization
Then the Sato algorithm for partial response systems is expressed as ckC1 D ck ¼ ž S;k xk
(15.45)
Contour algorithm for partial response systems In this case, first, the equalizer output is transformed into a full response signal and a pseudo error is computed. Then the error to compute the terms for coefficient updating is formed. The method differs in two ways from the Sato algorithm described above. First, the channel equalization, carried out to obtain the full response signals, is obtained by combining linear and non-linear feedback, whereas in the case of the Sato algorithm it is carried out by a linear filter. Second, the knowledge of the statistical properties of the input symbols is used to determine the pseudo error, as suggested by the contour algorithm. As mentioned above, to apply the contour algorithm the equalizer output signal is transformed into a full response signal using a combination of linear feedback and decision feedback, that is, we form the signal vk D yk C þC A vk2 C .1 þC A / aO k2
(15.46)
The equation (15.46) and the choice of the parameter þC A are justified in the following way. If þC A D 0 is selected, we obtain an equalization system with decision feedback, that presents the possibility of significant error propagation. The effect of the choice þC A D 1 is easily seen using the D transform. From (15.42) and assuming correct decisions, (15.46) can be expressed as v.D/ D a.D/ C
w y .D/ 1 þC A D 2
(15.47)
Therefore with þC A D 1 we get the linear inversion of the PR-IV channel with infinite noise enhancement at frequencies f D 0 and š1=.2T / Hz. The value of þC A is chosen in the interval 0 < þC A < 1, to obtain the best trade-off between linear feedback and decision feedback. We now apply the contour algorithm (15.37)–(15.39) using the signal vk rather than the equalizer output signal yk . The pseudo error is defined as ( vk Þmax sgn.vk / if jvk j ½ Þmax (15.48) žCv A;k D otherwise Žkv sgn.vk / where Žkv is a non-negative parameter that is updated at each iteration as 8 M 1 > > 1 if jvk j ½ Þmax < Žkv M v (15.49) ŽkC1 D > 1 > v : Žk C 1 otherwise M The stochastic gradient must be derived taking into consideration the channel equalization performed with linear feedback and decision feedback. We define the error on the M-ary symbol ak after channel equalization as ekv D vk ak
(15.50)
15.4. Self-training equalization for partial response systems
1099
Assuming correct decisions, by (15.46) it is possible to express the equalizer output as v yk D .ak ak2 / C ekv þC A ek2
ekv
(15.51) v þC A ek2
The equation (15.51) shows that an estimate of the term must be included as error signal in the expression of the stochastic gradient. After initial convergence of the equalizer coefficients, the estimate eOkv D vk aO k is reliable. Therefore decision-directed coefficient updating can be performed according to the algorithm v / xk ckC1 D ck ¼dd .eOkv þC A eOk2
(15.52)
The contour algorithm for coefficient updating during self-training is obtained by substituting the decision-directed error signal ekv with the pseudo error žCv A;k (15.48), v / xk ckC1 D ck ¼.žCv A;k þC A žk2
(15.53)
During self-training, satisfactory convergence behavior is usually obtained for þC A ' 1=2. Figure 15.9 shows the block diagram of an equalizer for a PR-IV system with a quaternary alphabet (QPR-IV), with the generation of the error signals to be used in decisiondirected and self-training mode.
Figure 15.9. Block diagram of a self-training equalizer with the contour algorithm for a QPR-IV system.
1100
Chapter 15. Self-training equalization
In the described scheme, the samples of the received signal can be initially filtered by a filter with transfer function 1=.1 a D 2 /, 0 < a < 1, to reduce the correlation among samples. The obtained signal is then input to the equalizer delay line.
15.5
Self-training equalization for QAM systems
We now describe various self-training algorithms for passband transmission systems that employ a two-dimensional constellation.
The Sato algorithm for QAM systems Consider a QAM transmission system with constellation A and a sequence fak g of i.i.d. symbols from A such that ak;I D Re[ak ] and ak;Q D Im[ak ] are independent and have the same probability distribution. We assume a receiver in which sampling of the received signal occurs at the symbol rate and tracking of the carrier phase is carried out at the equalizer output, as shown in Figure 15.10 (see scheme of Figure 8.37 with a baseband equalizer filter where fx k g is already demodulated and k D 'Ok ). If yQk D cT xk is the equalizer filter output, the sample at the decision point is then given by yk D yQk e j 'Ok
(15.54)
We let yk;I D Re[yk ] and
yk;Q D Im[yk ]
(15.55)
and introduce the Sato cost function for QAM systems, J D E[8.yk;I / C 8.yk;Q /]
(15.56)
where 8.v/ D
1 2
v 2 S jvj
(15.57)
Figure 15.10. Block diagram of a self-training equalizer with the Sato algorithm for a QAM system.
15.5. Self-training equalization for QAM systems
1101
and S D
2 ] E[ak;I
E[jak;I j]
D
2 ] E[ak;Q
E[jak;Q j]
(15.58)
The gradient of (15.56) with respect to c yields (see also (8.380)) rc J D rRe[c] J C j rIm[c] J D E[e j 'Ok xŁk .2.yk;I / C j 2.yk;Q //]
(15.59)
where 2.v/ D
d 8.v/ D v S sgn.v/ dv
(15.60)
The partial derivative with respect to the carrier phase estimate is given by (see also (8.385)) @ J D E[Im.yk .2.yk;I / C j 2.yk;Q //Ł /] @ 'O
(15.61)
Defining the Sato pseudo error for QAM systems as ž S;k D yk S sgn.yk /
(15.62)
Ł / D Im. S yk sgn.ykŁ // Im.yk ž S;k
(15.63)
and observing that
the equalizer coefficient updating and carrier phase estimate are given by ckC1 D ck ¼ ž S;k e j 'Ok xŁk
(15.64)
'OkC1 D 'Ok C ¼' Im[ S yk sgn.ykŁ /]
(15.65)
where ¼' is the adaptation gain of the carrier phase tracking loop. The equations (15.64) and (15.65) are analogous to (8.382) and (8.387) for the decision-directed case. The same observations made for self-training equalization of PAM systems using the Sato algorithm hold for QAM systems. In particular, assuming that the algorithm converges to a point of global minimum of the cost function, we recall that the variance of the pseudo error assumes high values in the neighborhood of the point of convergence. Therefore in the steady state it is necessary to adopt a decision directed algorithm. To obtain smooth transitions between the self-training mode and the decision-directed mode without the need of a further control of the distortion level, in QAM systems we can use extensions of the Benveniste–Goursat and stop-and-go algorithms considered for self-training of PAM systems.
15.5.1
Constant modulus algorithm
In the constant modulus algorithm (CMA) for QAM systems proposed by Godard [15], self-training equalization is based on the cost function J D E[.j yQk j p R p /2 ] D E[.jyk j p R p /2 ]
(15.66)
1102
Chapter 15. Self-training equalization
Figure 15.11. Block diagram of a self-training equalizer using the CMA.
where p is a parameter that usually assumes the value p D 1 or p D 2. We note that, as J depends on the absolute value of the equalizer output raised to the p-th power, the CMA does not require the knowledge of the carrier phase estimate. Figure 15.11 shows the block diagram of a receiver using the CMA. The gradient of (15.66) is given by rc J D 2 p E[.j yQk j p R p /j yQk j p2 yQk xŁk ]
(15.67)
The constant R p is chosen so that the gradient is equal to zero for a perfectly equalized system; therefore we have Rp D
E[jak j2 p ] E[jak j p ]
(15.68)
For example, for a 64-QAM constellation, we obtain R1 D 6:9 and R2 D 58. The 64-QAM constellation and the circle of radius R1 D 6:9 are illustrated in Figure 15.12. By using (15.67), we obtain the equalizer coefficient updating law ckC1 D ck ¼.j yQk j p R p /j yQk j p2 yQk xŁk
(15.69)
For p D 1, (15.69) becomes ckC1 D ck ¼.j yQk j R1 /
yQk Ł x j yQk j k
(15.70)
We note that the Sato algorithm, introduced in Section 15.2, can then be viewed as a particular case of the CMA.
The contour algorithm for QAM systems Let us consider a receiver in which the received signal is sampled at the symbol rate and the carrier phase recovery is ideally carried out before the equalizer. The scheme of Figure 15.1
15.5. Self-training equalization for QAM systems
1103
Im[ ak ]
R1 =6.9 R1
1
3
Re[ ak ]
Figure 15.12. The 64-QAM constellation and the circle of radius R1 D 6:9.
is still valid, and the complex-valued baseband equivalent channel output is given by xk D
C1 X
h i aki C wk
(15.71)
i D1
The equalizer output is expressed as yk D cT xk . To generalize the notion of pseudo error of the contour algorithm introduced in Section 15.3 for PAM systems, we define a contour line C that connects the outer points of the constellation. For simplicity, we assume a square constellation with L ð L points, as illustrated in Figure 15.13 for the case L D 8. Let S be the region of the complex plane enclosed by the contour line C and let C 2 =S = S, that is by definition. We denote by ykC the closest point to yk on C every time that yk 2 every time the point yk is found outside the region enclosed by C. The pseudo error (15.37) is now extended as follows: 8 C =S se yk 2 > < yk yk ) žC A;k D Žk sgn.yk;I / (15.72) if jyk;I j ½ jyk;Q j se yk 2 S > : j Žk sgn.yk;Q / if jyk;I j < jyk;Q j Also in this case Žk is a non-negative parameter, updated at each iteration as ( if yk 2 =S Žk pS 1 ŽkC1 D Žk C .1 pS / 1 if yk 2 S
(15.73)
where, by analogy with (15.40), the probability pS ' P[yk 2 S] is computed assuming that yk is an equalized signal in the presence of additive noise.
1104
Chapter 15. Self-training equalization
Im[ yk ]
~y C k
~y k
Re[ yk ]
S C
Figure 15.13. Illustration of the contour line and surface S for a 64-QAM constellation.
Let Þmax be the maximum absolute value of the real and imaginary parts of the square L ð L symbol constellation. If jyk;I j ½ Þmax or jyk;Q j ½ Þmax , but not both, the projection of the sample yk on the contour line C yields a non-zero pseudo error along one dimension and a zero error in the other dimension. If both jyk;I j and jyk;Q j are larger than Þmax , ykC is chosen as the corner point of the constellation closest to yk ; in this case we obtain a non-zero pseudo error in both dimensions. Thus the equalizer coefficients are updated according to the algorithm ckC1 D ck ¼ žC A;k xŁk
(15.74)
Clearly, the contour algorithm can also be applied to systems that use non-square constellations. In any case, the robust algorithm for carrier phase tracking that is described in the next section requires that the shape of the constellation is non-circular.
Joint contour algorithm and carrier phase tracking We now apply the idea of generating an error signal with respect to the contour line of a constellation to the problem of carrier phase recovery and frequency offset compensation [12]. With reference to the scheme of Figure 15.10, we denote as 'Ok the carrier phase estimate used for the received signal demodulation at instant k. If carrier recovery follows equalization, the complex equalizer output signal is given by yk D yQk e j 'Ok
(15.75)
As for equalizer coefficient updating, reliable information for updating the carrier phase estimate 'Ok is only available if yk falls outside of the region S. As illustrated in Figure 15.14,
15.5. Self-training equalization for QAM systems
1105
Im[ yk ]
(−αmax ,+αmax )
(+αmax ,+αmax )
yC ∆ϕk
k
yk
Re[yk ] region S region D C (+αmax ,−αmax )
(−αmax ,−αmax )
Figure 15.14. Illustration of the rotation of the symbol constellation in the presence of a phase error, and definition of 1'k .
the phase estimation error can then be computed as (see also (15.65)) 1'k ' Im.yk ykC Ł / D Im[yk .ykŁ ykC Ł /]
(15.76)
If yk falls within S the phase error is set to zero, that is 1'k D 0. From Figure 15.14 we note that the phase error 1'k is invariant with respect to a rotation of yk equal to an integer multiple of ³=2. Then to determine 1'k we can first rotate yk , yk0 D yk e j`³=2
(15.77)
where ` is chosen such that Re[yn0 ] > jIm[yn0 ]j (shaded region in Figure 15.14). Furthermore, we observe that the information on the phase error obtained by samples of the sequence fyk g that fall in the corner regions, where jyk;I j > Þmax and jyk;Q j > Þmax , is not important. Thus we calculate a phase error only if yk is outside of S, but not in the corner regions, that is if yk0 2 D, with D D fyk0 : Re[yk0 ] > Þmax ; jIm[yk0 ]j < Þmax g. Then (15.76) becomes ( Im[yk0 ][Re[yk0 ] Þmax ] if yk0 2 D (15.78) 1'k D 0 otherwise In the presence of a frequency offset equal to =.2³ /, the probability distribution of the samples fyk g rotates at a rate of =.2³ / revolutions per second. For large values of =.2³ /, the phase error 1'k does not provide sufficient information for the carrier phase tracking system to achieve a lock condition; therefore the update of 'Ok must be made by a secondorder PLL, where in the update of the second-order term a factor that is related to the value of the frequency offset must be included (see Section 14.7).
1106
Chapter 15. Self-training equalization
The needed information is obtained observing the statistical behavior of the term Im[yk0 ], conditioned by the event yk0 2 D. At instants in which the sampling distribution of yk is aligned with S, the distribution of Im[yk0 ] is uniform in the range [Þmax ; Þmax ]. Between these instants, the distribution of Im[yk0 ] exhibits a time varying behavior with a downward or upward trend depending on the sign of the frequency offset, with a minimum variance when the corners of the rotating probability distribution of yk , which we recall rotates at a rate of =.2³ / revolutions per second, cross the coordinate axes. Defining ( v if jvj < Þmax (15.79) Q.v/ D 0 otherwise from the observation of Figure 15.14 a simple method to extract information on =.2³ / consists in evaluating 1 Im[yk0 ] D QfIm[yk0 ] Im[ym0 ]g
yk0 2 D
(15.80)
where m < k denotes the last time index for which yk0 2 D. In the mean, 1 Im[yk0 ] exhibits the sign of the frequency offset. The equations for the updating of the parameters of a second-order phase-locked loop for the carrier phase recovery then become ( 'OkC1 D 'Ok C ¼' 1'k C 1'Oc;k if yk0 2 D 1'Oc;kC1 D 1'Oc;k C ¼ f 1 1'k C ¼ f 2 1 Im[yk0 ] (15.81) ( 'OkC1 D 'Ok otherwise 1'Oc;kC1 D 1'Oc;k where ¼' , ¼ f 1 and ¼ f 2 are suitable adaptation gains; typically, ¼' is in the range 104 103 , ¼ f 1 D .1=4/¼2' , and ¼ f 2 ' ¼ f 1 . The rotation of yk given by (15.77) to obtain yk0 also has the advantage of simplifying the error computation for self-training equalizer coefficient adaptation with the contour algorithm. With no significant effect on performance, we can introduce a simplification similar to that adopted to update the carrier phase, and let the pseudo error equal zero if yk is found in the corner regions, that is žC A;k D 0 if Im[yk0 ] > Þmax . By using (15.72) and (15.73) to compute the pseudo error, the coefficient updating equation (15.74) becomes (see (8.382)) ckC1 D ck ¼ žC A;k e j 'Ok xŁk
15.6
(15.82)
Examples of applications
In this section, we give examples of applications that illustrate the convergence behavior and steady state performance of self-training equalizers, with particular regard to the contour algorithm. We initially consider self-training equalization for PAM transmission systems over unshielded twisted-pair UTP cables with frequency response given by (4.169).
15.6. Examples of applications
1107
Example 15.6.1 As a first example, we consider a 16-PAM system (M D 16) with a uniform probability distribution of the input symbols and symbol rate equal to 25 MBaud; the transmit and receive filters are designed to yield an overall raised cosine channel characteristic for a cable length of 50 m. In the simulations, the cable length is chosen equal to 100 m, and the received signal is disturbed by additive white Gaussian noise. The signal-to-noise ratio at the receiver input is equal to 0 D 36 dB. Self-training equalization is achieved by a fractionally spaced equalizer having N D 32 coefficients, and input signal sampled with sampling period equal to T =2. Figure 15.15 shows the convergence of the contour algorithms (15.37) and
Figure 15.15. Illustration of the convergence of the contour algorithm for a 16-PAM system: (a) behavior of the parameter Žn , (b) MSE convergence, (c) relative frequency of equalizer output samples at the beginning and the end of the convergence process.
1108
Chapter 15. Self-training equalization
(15.38) for Ž0 D 0 and c0 chosen equal to the zero vector. The results are obtained for a cable with attenuation (4.148) equal to Þ. f / j f D1 D 3:85 ð 106 [m1 Hz1=2 ], parameters of the self-training equalizer given by ¼ D 105 and 1 D 2:5 ð 104 , and ideal timing recovery. Example 15.6.2 We consider self-training equalization for a baseband quaternary partial response class IV system (M D 4) for transmission at 125 Mbit/s over UTP cables; a VLSI transceiver implementation for this system will be described in Chapter 19. We compare the performance of the contour algorithm, described in Section 15.3, with the Sato algorithm for partial response systems. Various realizations of the MSE convergence of a self-training equalizer with N D 16 coefficients are shown in Figures 15.16 and 15.17 for the Sato algorithm and the contour algorithm, respectively. The curves are parameterized by t D 1T =T , where T D 16 ns, and 1T denotes the difference between the sampling phase of the channel output signal and the optimum sampling phase that yields the minimum MSE; we note that the contour algorithm has a faster convergence with respect to the Sato algorithm and yields significantly lower values of MSE in the steady state. The Sato algorithm can be applied only if timing recovery is achieved prior to equalization; note that the convergence characteristics of the contour algorithm makes self-training equalization possible even in the presence of considerable distortion and phase drift of the received signal.
Figure 15.16. MSE convergence with the Sato algorithm for a QPR-IV system [13].
15.6. Examples of applications
1109
Figure 15.17. MSE convergence with the contour algorithm for a QPR-IV system [13].
0.4 Real part Imaginary part
0.2
0
–0.2
–0.4 0
1
2
3 t/T
4
5
6
Figure 15.18. Overall baseband equivalent channel impulse response for simulations of a c 1998 Springer-Verlag London, Ltd.] 256-QAM system. [From [12],
1110
Chapter 15. Self-training equalization
Example 15.6.3 We now examine self-training equalization for a 256-QAM transmission system having a square constellation with L D 16 (M D 256), and symbol rate equal to 6 MBaud. Along each dimension, symbols š3, š1 have probability 2=20, and symbols š15, š13, š11, š9, š7, and š5 have probability 1=20. The overall baseband equivalent channel impulse response is illustrated in Figure 15.18. The received signal is disturbed by additive white Gaussian noise. The signal-to-noise ratio at the receiver input is equal to 0 D 39 dB. Signal equalization is obtained by a fractionally spaced equalizer having N D 32 coefficients, and input signal sampled with sampling period equal to T =2. Figure 15.19 shows the convergence of the contour algorithm and the behavior of the parameter Žk for pS D 361=400, various initial values of Ž0 , and c0 given by a vector with all elements equal to zero except for one element. Results are obtained for ¼ D 104 , 1 D 104 , and ideal timing and carrier phase recovery. Example 15.6.4 With reference to the previous example, we examine the behavior of the carrier phase recovery algorithm, assuming ideal timing recovery. Figure 15.20 illustrates the behavior of the MSE and of the second-order term 1'Oc;k for an initial frequency offset of C2:5 kHz, ¼' D 4 ð 104 , ¼ f 1 D 8 ð 108 , and ¼ f 2 D 2 ð 108 .
Figure 15.19. Convergence behavior of MSE and parameter Žk using the contour algorithm c 1998 for a 256-QAM system with non-uniform distribution of input symbols. [From [12], Springer-Verlag London, Ltd.]
15. Bibliography
1111
Figure 15.20. Illustration of the convergence behavior of MSE and second-order term 1 'ˆ c,k using the contour algorithm in the presence of an initial frequency offset equal to 500 ppm c 1998 for a 256-QAM system with non-uniform distribution of input symbols. [From [12], Springer-Verlag London, Ltd.]
Bibliography [1] H. Ichikawa, J. Sango, and T. Murase, “256 QAM multicarrier 400 Mb/s microwave radio system field tests”, in Proc. 1987 IEEE Int. Conference on Communications, pp. 1803–1808, 1987. [2] F. J. Ross and D. P. Taylor, “An enhancement to blind equalization algorithms”, IEEE Trans. on Communications, vol. 39, pp. 636–639, May 1991. [3] J. G. Proakis and C. L. Nikias, “Blind equalization”, in Proc. SPIE Adaptive Signal Processing, vol. 1565, pp. 76–87, July 22–24 1991. [4] S. Bellini, “Blind equalization and deconvolution”, in Proc. SPIE Adaptive Signal Processing, vol. 1565, pp. 88–101, July 22–24 1991. [5] N. Benvenuto and T. W. Goeddel, “Classification of voiceband data signals using the constellation magnitude”, IEEE Trans. on Communications, vol. 43, pp. 2759–2770, Nov. 1995.
1112
Chapter 15. Self-training equalization
[6] R. Liu and L. Tong, eds, “Special issue on blind systems identification and estimation”, IEEE Proceedings, vol. 86, Oct. 1998. [7] Y. Sato, “A method of self-recovering equalization for multilevel amplitudemodulation systems”, IEEE Trans. on Communications, vol. 23, pp. 679–682, June 1975. [8] L. Tong, G. Xu, B. Hassibi, and T. Kailath, “Blind channel identification based on second-order statistics: a frequency-domain approach”, IEEE Trans. on Information Theory, vol. 41, pp. 329–334, Jan. 1995. [9] A. Benveniste, M. Goursat, and G. Ruget, “Robust identification of a nonminimum phase system: blind adjustment of a linear equalizer in data communications”, IEEE Trans. on Automatic Control, vol. 25, pp. 385–399, June 1980. [10] A. Benveniste and M. Goursat, “Blind equalizers”, IEEE Trans. on Communications, vol. 32, pp. 871–883, Aug. 1984. [11] G. Picchi and G. Prati, “Blind equalization and carrier recovery using a ‘Stop-and-Go’ decision directed algorithm”, IEEE Trans. on Communications, vol. 35, pp. 877–887, Sept. 1987. ¨ ¸ er, and G. Ungerboeck, “The contour algorithm for self-training [12] G. Cherubini, S. Olc equalization”, in Broadband Wireless Communications, 9th Tyrrhenian Int. Workshop on Digital Communications (M. Luise and S. Pupolin, eds), Lerici, Italy, pp. 58–69, Sept. 7–10 1997. Berlin: Springer-Verlag, 1998. ¨ ¸ er, and G. Ungerboeck, “Self-training adaptive equalization for [13] G. Cherubini, S. Olc multilevel partial-response transmission systems”, in Proc. 1995 IEEE Int. Symposium on Information Theory, Whistler, Canada, p. 401, Sept. 17–22 1995. [14] G. Cherubini, “Nonlinear self-training adaptive equalization for partial-response systems”, IEEE Trans. on Communications, vol. 42, pp. 367–376, February/March/April 1994. [15] D. N. Godard, “Self recovering equalization and carrier tracking in two-dimensional data communication systems”, IEEE Trans. on Communications, vol. 28, pp. 1867– 1875, Nov. 1980.
15.A. On the convergence of the contour algorithm
Appendix 15.A
1113
On the convergence of the contour algorithm
PC1 Given y D y0 D i D1 ci x i , we show that the only minima of the cost function J D E[8.y/] correspond to the equalizer settings C D šH 1 , for which we obtain 9 D šI , except for a possible delay. If the systems H and C have finite energy, then also 9 has finite energy and P E[8.y/] may be regarded as a functional J .9/ D E[8.y/], where y is expressed as y D iC1 D1 i ai ; thus we must prove that the only minima of V.9/ are found at points 9 D šI . We consider input symbols with a probability density function pak .Þ/ uniform in the interval [Þmax ; Þmax ]. N where r ½ 0, and 9 N $ f N i g, ForPthe analysis, we express the system 9 as 9 D r 9, with i N i2 D 1, denotes the normalized overall system. We consider the cost function J P N as a functional V.9/ D E[8.r yN /], where yN D yN0 D iC1 D1 i ai denotes the output of N the system 9, and 8 has derivative 2 given by (15.34). Let ( x Þmax sgn.x/ if jxj ½ Þmax Q 2.x/ D (15.83) 0 otherwise and p yNk .x/ denote the probability density function of yN . We express the parameter C A as C A D 9N C r Z Q b 2.b/ p yNk .b/ db 9N D Z jbj p yNk .b/ db r D
²
1r 0
(15.84)
if r 1 otherwise
N on the unit sphere S. To claim that the only minima of V.9/ N Examine the function V.9/ are found at points 9 D šI , we apply Theorem 3.5 of [9]. Consider a pair of Pindices .i; j/, i 6D j, and a fixed system with coefficients f N ` g`6Di; j , such that R 2 D 1 `6Di; j N `2 > 0. N ' 2 S be the system with coefficients f N ` g`6Di; j , N i D R cos ', and For ' 2 [0; 2³ /, let 9 N j D R sin '; moreover, let .@=@'/V.9 N ' / be the derivative of V.9 N ' / with respect to ' at N D9 N ' . As pak .Þ/ is sub-Gaussian, it can be shown that point 9 @ N '/ D 0 V.9 @'
for ' D k
@ N '/ > 0 V.9 @'
for 0 < '
0. Therefore there exists a radius r0 such that the radial component of the gradient N 2 S, r0 is given by the is negative for r < r0 and positive for r > r0 . For a fixed point 9 solution of the equation Z Z Q b/ p yNk .b/ db .9N C r / jbj p yNk .b/ db D 0 (15.89) b 2.r N 2 S. Therefore Substituting the expressions of 9N and r in (15.89), we obtain r0 D 1, 89 the only minima of V are at šI . Furthermore, as the radial component of the gradient vanishes on S, the steepest descent lines of V do not cross the unit sphere. Using the same argument given in [9], we conclude that the points šI are the only “stable attractors” of the steepest descent lines of the function V, and that the unique stable attractors of the steepest descent lines of J are šH 1 . Note that the parameter 9N is related to the distortion of the distribution of the input sequence filtered by a normalized system; this parameter varies along the trajectories of the stochastic gradient algorithm and vanishes as a point of minimum is reached. Moreover, note that the parameter r indicates the deviation of the overall system gain from the desired unit value; if the gain is too small, the gradient is augmented with an additional driving term.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 16
Applications of interference cancellation
The algorithms and structures discussed in this chapter can be applied to both wired and wireless systems, even though transmission systems over twisted-pair cables will be considered to describe examples of applications. Full-duplex data transmission over a single twisted-pair cable permits the simultaneous flow of information in two directions using the same frequency band. Examples of applications of this technique are found in digital communications systems that operate over the telephone network. In a digital subscriber loop, at each end of the full-duplex link, a circuit called hybrid separates the two directions of transmission. To avoid signal reflections at the near and far-end hybrid, a precise knowledge of the line impedance would be required. As the line impedance depends on line parameters that, in general, are not exactly known, an attenuated and distorted replica of the transmit signal leaks to the receiver input as an echo signal. Data-driven adaptive echo cancellation mitigates the effects of impedance mismatch. A similar problem is caused by cross-talk in transmission systems over voice-grade unshielded twisted-pair cables for local-area network applications, where multipair cables are used to physically separate the two directions of transmission. Cross-talk is a statistical phenomenon due to randomly varying differential capacitive and inductive coupling between adjacent two-wire transmission lines (see Section 4.4.2). At the rates of several megabit-persecond that are usually considered for local-area network applications, near–end cross-talk (NEXT) represents the dominant disturbance; hence adaptive NEXT cancellation must be performed to ensure reliable communications. A different problem shows up in frequency–division duplexing transmission, where different frequency bands are used in the two directions of transmission. In such a case far–end cross-talk (FEXT) interference is dominant. This situation occurs, for instance, in very– high–speed digital subscriber line (VDSL) systems, where multiple users are connected to a central station via unshielded twisted-pairs located in the same cable binder. In voiceband data modems the model for the echo channel is considerably different from the echo channel model adopted for baseband transmission. The transmitted signal is a passband QAM signal, and the far-end echo may exhibit significant carrier-phase jitter and carrier-frequency shift, which are caused by signal processing at intermediate points in the telephone network. Therefore a digital adaptive echo canceller for voiceband modems needs to embody algorithms that account for the presence of such additional impairments.
1116
Chapter 16. Applications of interference cancellation
In the first three sections of this chapter,1 we describe the echo channel models and adaptive echo canceller structures for various digital communications systems, which are classified according to the employed modulation techniques. We also address the trade-offs between complexity, speed of adaptation, and accuracy of cancellation in adaptive echo cancellers. In the last section, we address the problem of FEXT interference cancellation for upstream transmission in a VDSL system. The system is modelled as a multi–input multi–output (MIMO) system, to which multi–user detection techniques (see Section 10.4) can be applied.
16.1
Echo and near–end cross-talk cancellation for PAM systems
The model of a full-duplex baseband PAM data transmission system employing adaptive echo cancellation is shown in Figure 16.1. To describe system operations, we consider one end of the full-duplex link. The configuration of an echo canceller for a PAM transmission system (see Section 6.13) is shown in Figure 16.2. The transmitted data consist of a sequence fak g of i.i.d. real-valued symbols from the M-ary alphabet A D fš1, š3; : : : ; š.M 1/g. The sequence fak g is converted into an analog signal by a digital-to-analog (D/A) converter (see Chapter 7). The conversion to a staircase signal by a zero-order hold D/A converter is described by the frequency response HD=A . f / D T sin.³ f T /=.³ f T /, where T denotes the modulation interval. The D/A converter output is filtered by the analog transmit filter and is input to the channel through the hybrid. The signal x.t/ at the output of the low-pass analog receive filter has three components, namely, the signal from the far-end transmitter r.t/, the echo u.t/, and additive Gaussian noise w R .t/. The signal x.t/ is given by x.t/ D r.t/ C u.t/ C w R .t/ D
C1 X kD1
akR h.t kT / C
C1 X
ak h E .t kT / C w R .t/
kD1
(16.1) where fakR g is the sequence of symbols from the remote transmitter, and h.t/ and h E .t/ D fh D=A Ł g E g.t/ are the impulse responses of the overall channel and of the echo channel,
Figure 16.1. Model of a full-duplex PAM transmission system.
1
The material presented in Sections 16.1–16.3 is reproduced with permission from G. Cherubini, “Echo Cancellation,” in The Mobile Communications Handbook (J. D. Gibson, ed.), Ch. 7, pp. 7.1–7.15, Boca Raton, FL: c 1999 CRC Press, Boca Raton, FL. CRC Press, 1999, 2nd ed.
16.1. Echo and near–end cross-talk cancellation for PAM systems
1117
Figure 16.2. Configuration of an echo canceller for a PAM transmission system.
respectively. In the expression of h E .t/, the function h D=A .t/ is the inverse Fourier transform of HD=A . f /: The signal obtained after echo cancellation is processed by a detector that outputs the sequence of detected symbols faO kR g.
Cross-talk cancellation and full duplex transmission In the case of full-duplex PAM data transmission over multi-pair cables for local-area network applications, where NEXT represents the main disturbance, the configuration of a digital NEXT canceller is also obtained as shown in Figure 16.2, with the echo channel replaced by the cross-talk channel. For these applications, however, instead of mono-duplex transmission, where one pair is used to transmit only in one direction and the other pair to transmit only in the reverse direction, dual-duplex transmission may be adopted. Bidirectional transmission at rate Rb over two pairs is then accomplished by full-duplex transmission of data streams at Rb /2 over each of the two pairs. The lower modulation rate and/or spectral efficiency required per pair for achieving an aggregate rate equal to Rb represents an advantage of dual-duplex over mono-duplex transmission. Dual-duplex transmission requires two transmitters and two receivers at each end of a link, as well as separation of the simultaneously transmitted and received signals on each pair, as illustrated in Figure 16.3. In dual-duplex transceivers it is therefore necessary to suppress echoes originated by reflections at the hybrids and at impedance discontinuities in the cable, as well as self NEXT, by adaptive digital echo and NEXT cancellation. Although a dualduplex scheme might appear to require higher implementation complexity than a monoduplex scheme, it turns out that the two schemes are equivalent in terms of the number of multiply-and-add operations per second that are needed to perform the various filtering operations. One of the transceivers in a full-duplex link will usually employ an externally provided reference clock for its transmit and receive operations. The other transceiver will extract timing from the receive signal, and use this timing for its transmitter operations. This is known as loop timing, also illustrated in Figure 16.3. If signals were transmitted in opposite directions with independent clocks, signals received from the remote transmitter would generally shift in phase relative to the also received echo signals. To cope with this effect, some
1118
Chapter 16. Applications of interference cancellation
Tx
Tx
rate R b /2
rate R b
rate R b H
H Rc
Rc
echo
external
loop timing
self−NEXT
timing Tx
Tx H
H rate R b
Rc
rate Rb
rate R b /2
echo
Rc
Figure 16.3. Model of a dual-duplex transmission system.
form of interpolation (see Chapter 14) would be required that can significantly increase the transceiver complexity.
Polyphase structure of the canceller In general, we consider baseband signalling techniques such that the signal at the output of the overall channel has non-negligible excess bandwidth, i.e., non-negligible spectral components at frequencies larger than half of the modulation rate, j f j ½ 1=.2T /. Therefore, to avoid aliasing, the signal x.t/ is sampled at twice the modulation rate or at a higher sampling rate. Assuming a sampling rate equal to F0 =T; F0 > 1; the i-th sample during the k-th modulation interval is given by T D x k F0 Ci D rk F0 Ci C u k F0 Ci C wk F0 Ci x .k F0 C i/ F0 D
C1 X nD1
R h n F0 Ci akn C
C1 X
i D 0; : : : ; F0 1
h E;n F0 Ci akn C wk F0 Ci
(16.2)
nD1
where fh k F0 Ci ; i D 0; : : : ; F0 1g and fh E;k F0 Ci ; i D 0; : : : ; F0 1g are the discretetime impulse responses of the overall channel and the echo channel, respectively, and fwk F0 Ci ; i D 0; : : : ; F0 1g is a sequence of Gaussian noise samples with zero mean and variance ¦w2 . Equation (16.2) suggests that the sequence of samples fx k F0 Ci ; i D 0; : : : ; F0 1g be regarded as a set of F0 interlaced sequences, each with a sampling rate equal to the modulation rate. Similarly, the sequence of echo samples fu k F0 Ci ; i D 0; : : : ; F0 1g can be regarded as a set of F0 interlaced sequences that are output by F0 independent echo channels with discrete-time impulse responses fh E;k F0 Ci g; i D 0; : : : ; F0 1, and an identical sequence fak g of input symbols [1]. Hence, echo cancellation can be performed by F0 interlaced echo cancellers, as shown in Figure 16.4. As the performance of each
16.1. Echo and near–end cross-talk cancellation for PAM systems
1119
{ak }
E.C. (0)
E.C. ( F0 −1)
E.C. (1)
{u^ k F } 0
{u^ k F +1 } 0
{u^ k F + F } 0 0 t=l T F0
{u^ k F +i , i=0,...,F0 −1} 0 {z k F +i , i=0,...,F0 −1} 0
− +
{x k F +i , i=0,...,F0 −1} 0
sampler
x(t)
Figure 16.4. A set of F0 interlaced echo cancellers.
canceller is independent of the other F0 1 units, in the remaining part of this section we will consider the operations of a single echo canceller.
Canceller at symbol rate The echo canceller generates an estimate uO k of the echo signal. If we consider a transversal filter implementation, uO k is obtained as the inner product of the vector of filter coefficients at time t D kT , ck D [c0;k ; : : : ; c N 1;k ]T , and the vector of signals stored in the echo canceller delay line at the same instant, ak D [ak ; : : : ; akN C1 ]T , expressed by uO k D ckT ak D
N 1 X
cn;k akn
(16.3)
nD0
The estimate of the echo is subtracted from the received signal. The result is defined as the cancellation error signal z k D x k uO k D x k ckT ak
(16.4)
The echo attenuation that must be provided by the echo canceller to achieve proper system operation depends on the application. For example, for the integrated services digital network (ISDN) U-Interface transceiver, the echo attenuation must be larger than 55 dB [2]. It is then required that the echo signals outside of the time span of the echo canceller delay line be negligible, i.e., h E;n ³ 0 for n < 0 and n > N 1: As a measure of system performance, we consider the mean-square error Jk at the output of the echo canceller at time t D kT , defined by Jk D E[z k2 ]
(16.5)
1120
Chapter 16. Applications of interference cancellation
For a particular coefficient vector ck , substitution of (16.4) into (16.5) yields (see (2.17)) Jk D E[x k2 ] 2ckT p C ckT Rck
(16.6)
where p D E[x k ak ] and R D E[ak akT ]. With the assumption of i.i.d. transmitted symbols, the correlation matrix R is diagonal. The elements on the diagonal are equal to the variance of the transmitted symbols, ¦a2 D .M 2 1/=3. From (2.40) the minimum mean-square error is given by T Jmin D E[x k2 ] copt Rcopt
(16.7)
where the optimum coefficient vector is copt D R1 p. We note that proper system operation is achieved only if the transmitted symbols are uncorrelated with the symbols from the remote transmitter. If this condition is satisfied, the optimum filter coefficients are given by the values of the discrete-time echo channel impulse response, i.e., cn;opt D h E;n , n D 0; : : : ; N 1.
Adaptive canceller By the LMS algorithm, the coefficients of the echo canceller converge in the mean to copt . The LMS algorithm (see Section 3.1.2) for an N -tap adaptive linear transversal filter is formulated as follows: ckC1 D ck C ¼ z k ak
(16.8)
where ¼ is the adaptation gain. The block diagram of an adaptive transversal filter echo canceller is shown in Figure 16.5. If we define the vector ck D ck copt , the mean-square error can be expressed by (2.40) Jk D Jmin C ckT R ck
(16.9)
where the term ckT R ck represents an excess mean-square error due to the misadjustment of the filter settings. Under the assumption that the vectors ck and ak are statistically independent, the dynamics of the mean-square error are given by (see (3.272)) Jk D ¦02 [1 ¼¦a2 .2 ¼N ¦a2 /]k C
2Jmin 2 ¼N ¦a2
(16.10)
where ¦02 is determined by the initial conditions. The mean-square error converges to a finite steady-state value J1 if the stability condition 0 < ¼ < 2=.N ¦a2 / is satisfied. The optimum adaptation gain that yields fastest convergence at the beginning of the adaptation process is ¼opt D 1=.N ¦a2 /. The corresponding time constant and asymptotic mean-square error are −opt D N and J1 D 2Jmin , respectively. We note that a fixed adaptation gain equal to ¼opt could not be adopted in practice, as after echo cancellation the signal from the remote transmitter would be embedded in a residual echo having approximately the same power. If the time constant of the convergence
16.1. Echo and near–end cross-talk cancellation for PAM systems
1121
ak
ak
Σ c0,k
µ zk
a k−1 T
T
Σ
Σ c1,k
a k−N+2
cN−2,k
T
a k−N+1
Σ c N−1,k
+ u^k xk
− +
Figure 16.5. Block diagram of an adaptive transversal filter echo canceller.
mode is not a critical system parameter, an adaptation gain smaller than ¼opt will be adopted to achieve an asymptotic mean-square error close to Jmin . On the other hand, if fast convergence is required, a variable adaptation gain will be chosen. Several techniques have been proposed to increase the speed of convergence of the LMS algorithm. In particular, for echo cancellation in data transmission, the speed of adaptation is reduced by the presence of the signal from the remote transmitter in the cancellation error. To mitigate this problem, the data signal can be adaptively removed from the cancellation error by a decision-directed algorithm [3]. Modified versions of the LMS algorithm have been also proposed to reduce system complexity. For example, the sign algorithm suggests that only the sign of the error signal be used to compute an approximation of the gradient [4]. An alternative means to reduce the implementation complexity of an adaptive echo canceller consists in the choice of a filter structure with a lower computational complexity than the transversal filter.
Canceller structure with distributed arithmetic At high rates, very large-scale integration (VLSI) technology is needed for the implementation of transceivers for full-duplex data transmission. High-speed echo cancellers and near-end cross-talk cancellers that do not require multiplications represent an attractive solution because of their low complexity. As an example of an architecture suitable for VLSI
1122
Chapter 16. Applications of interference cancellation
implementation, we consider echo cancellation by a distributed-arithmetic filter, where multiplications are replaced by table look-up and shift-and-add operations [5]. By segmenting the echo canceller into filter sections of shorter lengths, various trade-offs concerning the number of operations per modulation interval and the number of memory locations needed to store the look-up tables are possible. Adaptivity is achieved by updating the values stored in the look-up tables by the LMS algorithm. To describe the principles of operations of a distributed-arithmetic echo canceller, we assume that the number of elements in the alphabet of the input symbols is a power of two, M D 2W . Therefore, each symbol is represented by the vector [ak.0/ ; : : : ; ak.W 1/ ], where ak.i / 2 f0; 1g, i D 0; : : : ; W 1, are independent binary random variables, i.e., ak D
W 1 X wD0
.2ak.w/ 1/ 2w D
W 1 X wD0
bk.w/ 2w
(16.11)
where bk.w/ D .2ak.w/ 1/ 2 f1; C1g. By substituting (16.11) into (16.3) and segmenting the delay line of the echo canceller into L sections with K D N =L delay elements each, we obtain " # L1 W K 1 1 X X X .w/ w 2 bk`K m c`K Cm;k uO k D (16.12) `D0 wD0
mD0
Note that the summation within parenthesis in (16.12) may assume at most 2 K distinct real .w/ values, one for each binary sequence fak`K m g, m D 0; : : : ; K 1. If we precompute K these 2 values and store them in a look-up table addressed by the binary sequence, we can substitute the real time summation by a simple reading from the table. Equation (16.12) suggests that the filter output can be computed using a set of L2 K values that are stored in L tables with 2 K memory locations each. The binary vectors .w/ .w/ a.w/ k;` D [ak`K ; : : : ; ak`K K C1 ], w D 0; : : : ; W 1, ` D 0; : : : ; L 1, determine the addresses of the memory locations where the values that are needed to compute the filter output are stored. The filter output is obtained by W L table look-up and shift-and-add operations. N .w/ We observe that a.w/ k;` and its binary complement a k;` select two values that differ only in their sign. This symmetry is exploited to halve the number of values to be stored. To determine the output of a distributed-arithmetic filter with reduced memory size, we reformulate (16.12) as " # K 1 L1 W 1 X X X .w/ .w/ .w/ 2w bk`K c`K ;k C bk`K bk`K m c`K Cm;k (16.13) uO k D `D0 wD0
mD1
.w/ Then the binary symbol bk`K determines whether a selected value is to be added or subtracted. Each table has now 2 K 1 memory locations, and the filter output is given by
uO k D
L1 W 1 X X `D0 wD0
.w/ .w/ 2w bk`K dk .i k;` ; `/
(16.14)
16.1. Echo and near–end cross-talk cancellation for PAM systems
1123
where dk .n; `/, n D 0; : : : ; 2 K 1 1, are the look-up values stored in the `-th table, ` D 0; : : : ; L 1, whose values are .w/ dk .n; `/ D c`K ;k C bk`K
K 1 X
.w/ bk`K m c`K Cm;k
.w/ n D i k;`
mD1 .w/ and i k;` ; w D 0; : : : ; W 1; ` D 0; : : : ; L 1, are the look-up indices computed as follows: 8 K 1 X > > .w/ .w/ m1 > ak`K if ak`K D1 > m 2 < .w/ mD1 (16.15) i k;` D K 1 X .w/ > > .w/ m1 > aN k`K m 2 if ak`K D 0 > : mD1
We note that, as long as (16.12) and (16.13) hold for some coefficient vector [c0;k ; : : : ; c N 1;k ], a distributed-arithmetic filter emulates the operation of a linear transversal filter. For arbitrary values dk .n; `/, however, a non-linear filtering operation results. The expression of the LMS algorithm to update the values of a distributed-arithmetic echo canceller is derived as in (3.280). To simplify the notation we set uO k .`/ D
W 1 X wD0
.w/ .w/ 2w bk`K dk .i k;` ; `/
(16.16)
hence (16.14) can be written as uO k D
L1 X
uO k .`/
(16.17)
`D0
We also define the vector of the values in the `-th look-up table as dk .`/ D [dk .0; `/; : : : ; dk .2 K 1 1; `/]T
(16.18)
indexed by the variable (16.15). The values dk .`/ are updated according to the LMS criterion, i.e., dkC1 .`/ D dk .`/
1 2
¼ rdk .`/ z k2
(16.19)
where rdk .`/ z k2 D 2z k rdk .`/ z k D 2z k rdk .`/ uO k D 2z k rdk .`/ uO k .`/
(16.20)
The last expression has been obtained using (16.17) and the fact that only uO k .`/ depends on dk .`/. Defining yk .`/ D [yk .0; `/; : : : ; yk .2 K 1 1; `/]T D rdk .`/ uO k .`/ (16.19) becomes dkC1 .`/ D dk .`/ C ¼ z k yk .`/
(16.21)
1124
Chapter 16. Applications of interference cancellation
For a given value of k and `, we assign the following values to the W addresses (16.15): .w/ I .w/ D i k;`
w D 0; 1; : : : ; W 1
From (16.16) we get yk .n; `/ D
W 1 X wD0
.w/ 2w bk`K ŽnI .w/
(16.22)
In conclusion, in (16.21) for every instant k and for each value of the index w D .w/ 0; 1; : : : ; W 1, the product 2w bk`K ¼z k is added to the memory location indexed by I .w/ . The complexity of the implementation can be reduced by updating, at every iteration k, only the values corresponding to the addresses given by the most significant bits of the symbols in the filter delay line. In this case (16.22) simplifies into ( yk .n; `/ D
.W 1/ 2W 1 bk`K
0
n D I .W 1/ n 6D I .W 1/
(16.23)
The block diagram of an adaptive distributed-arithmetic echo canceller with input symbols from a quaternary alphabet is shown in Figure 16.6. The analysis of the mean-square error convergence behavior and steady-state performance can be extended to adaptive distributed-arithmetic echo cancellers [6]. The dynamics of the mean-square error are in this case given by Jk D
¦02
¼¦a2 1 K 1 .2 ¼L¦a2 / 2
½k C
2Jmin 2 ¼L¦a2
(16.24)
The stability condition for the echo canceller is 0 < ¼ < 2=.L¦a2 /. For a given adaptation gain, echo canceller stability depends on the number of tables and on the variance of the transmitted symbols. Therefore, the time span of the echo canceller can be increased without affecting system stability, provided that the number L of tables is kept constant. In that case, however, mean-square error convergence will be slower. From (16.24), we find that the optimum adaptation gain that permits the fastest mean-square error convergence at the beginning of the adaptation process is ¼opt D 1=.L¦a2 /. The time constant of the convergence mode is −opt D L2 K 1 . The smallest achievable time constant is therefore proportional to the total number of values. As mentioned above, the implementation of a distributed-arithmetic echo canceller can be simplified by updating at each iteration only the values that are addressed by the most significant bits of the symbols stored in the delay line. The complexity required for adaptation can thus be reduced at the price of a slower rate of convergence.
16.2
Echo cancellation for QAM systems
Although most of the concepts presented in the preceding section can be readily extended to echo cancellation for communications systems employing QAM, the case of
16.2. Echo cancellation for QAM systems
ak(0)
ak(1)
(0) a k−(L−1)K
address computation (1) i k,0
table
0
L−1 +
(1) d k (i k,0 ,0)
(0) d k (i k,0 ,0)
+1 −1
bk(1)
1 0
+1 −1
← ←
+1 −1
bk(0)
1 0
← ←
+1 −1
2
zk
+
← ←
1 0
,0)
(0) i k,L−1
µ zk
← ←
1 0
+
address computation (1) i k,L−1
table
+
(1)
a k−(L−1)K
(0) ik,0
µ zk (1) d k+1 (i k,0
1125
(1) d k(i k,L−1 ,L−1)
(0) d k(i k,L−1 ,L−1)
(1)
b k−(L−1)K
(0) b k−(L−1)K
2
u^k
xk
+
Figure 16.6. Block diagram of an adaptive distributed-arithmetic echo canceller.
full-duplex transmission over a voiceband data channel requires a specific discussion. We consider the system model shown in Figure 16.7. The transmitter generates a sequence fak g of i.i.d. complex-valued symbols from a two-dimensional constellation A, that are modulated by the carrier e j2³ f 0 kT , where T and f 0 denote the modulation interval and the carrier frequency, respectively. The discrete-time signal at the output of the transmit passband phase splitter filter may be regarded as an analytic signal, which is generated at the rate of Q 0 =T samples/s, Q 0 > 1. The real part of the analytic signal is converted into an analog signal by a D/A converter and input to the channel. We note that by transmitting the real part of a complex-valued signal, positive and negativefrequency components become folded. The attenuation in the image band of the transmit filter thus determines the achievable echo suppression. In fact, the receiver cannot extract aliasing image-band components from desired passband frequency components, and the echo canceller is able to suppress only echo arising from transmitted passband components.
1126
Chapter 16. Applications of interference cancellation
Figure 16.7. Configuration of an echo canceller for a QAM transmission system.
The output of the echo channel is represented as the sum of two contributions. The near-end echo u N E .t/ arises from the impedance mismatch between the hybrid and the transmission line, as in the case of baseband transmission. The far-end echo u F E .t/ represents the contribution due to echoes that are generated at intermediate points in the telephone network. These echoes are characterized by additional impairments, such as jitter and frequency shift, which are accounted for by introducing a carrier-phase rotation equal to '.t/ in the model of the far-end echo. At the receiver, samples of the signal at the channel output are obtained synchronously with the transmitter timing, at the sampling rate of Q 0 =T samples/s. The discrete-time received signal is converted to a complex-valued baseband signal fx k F0 Ci g, i D 0; : : : ; F0 1, at the rate of F0 =T samples/s, 1 < F0 < Q 0 , through filtering by the receive phase splitter filter, decimation, and demodulation. From delayed transmit symbols, estimates of the near and far-end echo signals after demodulation, fuO kNFE0 Ci g, i D 0; : : : ; F0 1, and fuO kFFE0 Ci g, i D 0; : : : ; F0 1, respectively, are generated using F0 interlaced near and far-end echo cancellers. The cancellation error is given by z ` D x` .uO `N E C uO `F E /
(16.25)
A different model is obtained if echo cancellation is accomplished before demodulation. In this case, two equivalent configurations for the echo canceller may be considered. In one configuration, the modulated symbols are input to the transversal filter, which approximates the passband echo response. Alternatively, the modulator can be placed after the transversal filter, which is then called a baseband transversal filter [7].
16.2. Echo cancellation for QAM systems
1127
In the considered implementation, the estimates of the echo signals after demodulation are given by uO kNFE0 Ci D
NX N E 1
cnNFE0 Ci;k akn
i D 0; : : : ; F0 1
(16.26)
i D 0; : : : ; F0 1
(16.27)
nD0
uO kFFE0 Ci D
NX F E 1
! cnFFE0 Ci;k aknD E
e j 'Ok F0 C1
nD0 N E ; : : : ; cN E FE FE where [c0;k F0 N N E 1;k ] and [c0;k ; : : : ; c F0 N F E 1;k ] are the coefficients of the F0 interlaced near and far-end echo cancellers, respectively, f'Ok F0 Ci g, i D 0; : : : ; F0 1, is the sequence of far-end echo phase estimates, and D F E denotes the bulk delay accounting for the round-trip delay from the transmitter to the point of echo generation. To prevent overlap of the time span of the near-end echo canceller with the time span of the far-end echo canceller, the condition D F E > N N E must be satisfied. We also note that, because of the different nature of near and far-end echo generation, the time span of the far-end echo canceller needs to be larger than the time span of the near-end canceller, i.e., N F E > N N E . Adaptation of the filter coefficients in the near and far-end echo cancellers by the LMS algorithm leads to
cnNFE0 Ci;kC1 D cnNFE0 Ci;k C ¼ z k F0 Ci .akn /Ł n D 0; : : : ; N N E 1
i D 0; : : : ; F0 1
(16.28)
and cnFFE0 Ci;kC1 D cnFFE0 Ci;k C ¼ z k F0 Ci .aknD F E /Ł e j 'Ok F0 Ci n D 0; : : : ; N F E 1
i D 0; : : : ; F0 1
(16.29)
respectively. The far-end echo phase estimate is computed by a second-order phase-lock loop algorithm (see Section 14.7), where the following gradient approach is adopted: ( 'O`C1 D 'O` 12 ¼' r'O jz ` j2 C 1'` .mod 2³ / (16.30) 1'`C1 D 1'` 12 ¼ f r'O jz ` j2 where ` D k F0 C i, i D 0; : : : ; F0 1, ¼' and ¼ f are parameters of the loop, and r'O jz ` j2 D
@jz ` j2 D 2 Imfz ` .uO `F E /Ł g @ 'O`
(16.31)
We note that the algorithm (16.30) requires F0 iterations per modulation interval, i.e., we cannot resort to interlacing to reduce the complexity of the computation of the far-end echo phase estimate.
1128
Chapter 16. Applications of interference cancellation
16.3
Echo cancellation for OFDM systems
We discuss echo cancellation for OFDM with reference to a DMT system (see Chapter 9), as shown in Figure 16.8. Let fh i g, i D 0; : : : ; Nc 1, be the channel impulse response, sampled with period T =M, with length Nc − M and fh E;i g, i D 0; : : : ; N 1, be the discrete-time echo impulse response whose length is N < M, where M denotes the number of subchannels of the DMT system. To simplify the notation, the length of the cyclic prefix will be set to L D Nc 1. Recall that in a DMT transmitter the block of M samples at the IDFT output in the k-th modulation interval, [Ak [0]; : : : ; Ak [M1]], is cyclically extended by copying the last L samples at the beginning of the block. After a P/S conversion, wherein the L samples of the cyclic extension are the first to be output, the L C M samples of the block are transmitted into the channel. At the receiver the sequence is split into blocks of length L C M, [x k.MCL/ ; : : : ; x .kC1/.MCL/1 ]. These blocks are separated in such a way that the last M samples depend only on a single cyclically extended block, then the first L samples are discarded.
Case N L C1. We initially assume that the length of the echo channel impulse response is N L C 1: Furthermore, we assume that the boundaries of the received blocks are placed such that the last M samples of the k-th received block are expressed by the vector (see (9.72)) xk D kR h C k h E C wk
(16.32)
Figure 16.8. Configuration of an echo canceller for a DMT transmission system.
16.3. Echo cancellation for OFDM systems
1129
where h D [h 0 ; : : : ; h L ; 0; : : : ; 0]T is the vector of the overall channel impulse response extended with M L 1 zeros, h E D [h E;0 ; : : : ; h E;N 1 ; 0; : : : ; 0]T is the vector of the overall echo impulse response extended with M N zeros, wk is a vector of AWGN samples, kR is the circulant matrix with elements given by the signals from the remote transmitter 3 2 AkR [0] AkR [M 1] : : : AkR [1] 6 AkR [1] AkR [0] : : : AkR [2] 7 7 6 (16.33) kR D 6 :: :: :: 7 4 : : : 5 AkR [M 1] AkR [M 2] : : : AkR [0]
and k is the circulant matrix with elements generated by the local transmitter 3 2 Ak [0] Ak [M 1] : : : Ak [1] 6 Ak [0] : : : Ak [2] 7 Ak [1] 7 6 k D 6 :: :: :: 7 4 : : : 5 Ak [M 1] Ak [M 2] : : : Ak [0]
(16.34)
In the frequency domain, the echo is expressed as Uk D diag.ak / H E
(16.35)
where H E denotes the DFT of the vector h E . In this case, the echo canceller provides an echo estimate that is given by O k D diag.ak / Ck U
(16.36)
where Ck denotes the DFT of the vector ck of the N coefficients of the echo canceller filter extended with M N zeros. In the time domain, (16.36) corresponds to the estimate (16.37)
uO k D k ck
Case N > L C 1. In practice, however, we need to consider the case N > L C 1. The expression of the cancellation error is then given by xk D kR h C k;k1 h E C wk
(16.38)
where k;k1 is a circulant matrix given by k;k1 D 2 Ak [0] 6 A k [1] 6 6 :: 4 :
3 : : : Ak1 [L C 1] : : : Ak1 [L C 2] 7 7 7 5 Ak [0] Ak [M 1] Ak [M 2] : : : Ak [M L 1] Ak [M L 2] : : : (16.39) Ak [M 1] : : : Ak [M L] Ak [0] : : : Ak [M L C 1]
Ak1 [M 1] Ak [M L]
1130
Chapter 16. Applications of interference cancellation
From (16.38) the expression of the cancellation error in the time domain is then given by zk D xk k;k1 ck
(16.40)
We now introduce the Toeplitz triangular matrix k;k1 D k;k1 k
(16.41)
Substitution of (16.41) into (16.40) yields zk D xk k;k1 ck k ck
(16.42)
In the frequency domain, (16.42) can be expressed as Zk D FM .xk k;k1 ck / diag.ak / Ck
(16.43)
Equation (16.43) suggests a computationally efficient, two-part echo cancellation technique. First, in the time domain, a short convolution is performed and the result subtracted from the received signals to compensate for the insufficient length of the cyclic extension. Second, in the frequency domain, cancellation of the residual echo is performed over a set of M independent echo subchannels. Observing that (16.43) is equivalent to Q k;k1 Ck Z k D Xk
(16.44)
Q k;k1 D FM k;k1 F1 ; the echo canceller adaptation by the LMS algorithm in where M the frequency domain takes the form H Q k;k1 Zk CkC1 D Ck C ¼
(16.45)
where ¼ is the adaptation gain. We note that, alternatively, echo canceller adaptation may also be performed by the simplified algorithm [8] CkC1 D Ck C ¼ diag.aŁk / Zk
(16.46)
which entails a substantially lower computational complexity than the LMS algorithm, at the price of a slower rate of convergence. In DMT systems it is essential that the length of the channel impulse response be much less than the number of subchannels, so that the reduction in rate due to the cyclic extension may be considered negligible. Therefore, time-domain equalization is adopted in practice to shorten the length of the channel impulse response. From (16.43), however, we observe that transceiver complexity depends on the relative lengths of the echo and of the channel impulse responses. To reduce the length of the cyclic extension as well as the computational complexity of the echo canceller, various methods have been proposed to shorten both the channel and the echo impulse responses jointly [9].
16.4. Multiuser detection for VDSL
16.4
1131
Multiuser detection for VDSL
In this section, we address the problem of multiuser detection for upstream VDSL transmission (see Chapter 17), where FEXT signals at the input of a VDSL receiver are viewed as interferers that share the same channel as the remote user signal [10]. We assume knowledge of the FEXT responses at the central office and consider a decision-feedback equalizer (DFE) structure with cross-coupled linear feedforward (FF) equalizers and feedback (FB) filters for cross-talk suppression. DFE structures with crosscoupled filters have also been considered for interference suppression in wireless CDMA communications [11] and fast Ethernet transmission (see Appendix 19.A). Here we determine the optimum DFE coefficients in a minimum mean-square error (MMSE) sense assuming that each user adopts OFDM modulation for upstream transmission. A system with reduced complexity may be considered for practical applications, in which for each user and each subchannel only the most significant interferers are suppressed. To obtain a receiver structure for multiuser detection that exhibits moderate complexity, we assume that each user adopts FMT modulation with M subchannels for upstream transmission (see Chapter 9). Hence the subchannel signals exhibit non-zero excess bandwidth as well as negligible spectral overlap. Assuming upstream transmission by U users, the system illustrated in Figure 16.9 is considered. In general, the sequences of subchannel signal samples at the multicarrier demodulator output are obtained at a sampling rate equal to a rational multiple F0 of the modulation rate 1=T . To simplify the analysis, here an integer F0 ½ 2 is assumed. We introduce the following definitions: 1. fak.u/ [i]g, sequence of i.i.d. complex-valued symbols from a QAM constellation A.u/ [i] transmitted by user u over subchannel i, with variance ¦a2.u/ [i ] ; 2. faO k.u/ [i]g, sequence of detected symbols of user u at the output of the decision element of subchannel i; 3. h n.u/ [i], overall impulse response of subchannel i of user u; 4. h .u;v/ FEXT;n [i], overall FEXT response of subchannel i, from user v to user u; 5. G .u/ [i], gain that determines the power of the signal of user u transmitted over subchannel i; 6. fwQ n.u/ [i]g, sequence of additive Gaussian noise samples with correlation function rw.u/ [i ] .m/.
1132
Chapter 16. Applications of interference cancellation
c 2001 Figure 16.9. Block diagram of transmission channel and DFE structure. [From [10], IEEE.]
At the output of subchannel i of the user-u demodulator, the complex baseband signal is given by xn.u/ [i] D G .u/ [i]
1 X
.u/ .u/ h nk F0 [i] ak [i] C
kD1
C
U X vD1
v6Du
G .v/ [i]
1 X
.v/ h .u;v/ Q n.u/ [i] FEXT;nk F0 [i]ak [i] C w
(16.47)
kD1
For user u, symbol detection at the output of subchannel i is achieved by a DFE structure such that the input to the decision element is obtained by combining the output signals of U linear filters and U feedback filters from all users, as illustrated in Figure 16.9. In practice, to reduce system complexity, for each user only a subset of all other user signals (interferers) is considered as an input to the DFE structure [10]. The selection of the subset signals is based on the power and the number of the interferers. This strategy, however, results in a loss of performance, as some strong interferers may not be considered.
16.4. Multiuser detection for VDSL
1133
This effect is similar to the near–far problem in CDMA systems. To alleviate this problem, it is necessary to introduce power control of the transmitted signals; a suitable method will be described in the next section. To determine the DFE filter coefficients, we assume M1 and M2 coefficients for each FF and FB filter, respectively. We define the following vectors: .u/ .u/ T 1. x.u/ k F0 [i] D [x k F0 [i]; : : : ; x k F0 Mi C1 [i]] , signal samples stored at instant k F0 in the delay lines of the FF filters with input given by the demodulator output of user u at subchannel i; T 2. c.u;v/ [i] D [c0.u;v/ [i]; : : : ; c.u;v/ M1 1 [i]] , coefficients of the FF filter from the demodulator output of user v to the decision element input of user u at subchannel i; T 3. b.u;v/ [i] D [b1.u;v/ [i]; : : : ; b.u;v/ M2 [i]] , coefficients of the FB filter from the decision element output of user v to the decision element input of user u at subchannel i; 2 3T
4 O .u/ 4. aO .u/ k [i] D a
¾
D .u/[i] k1 F0
³ [i]; : : : ; a ³ [i]5 ¾ O .u/ D .u/[i] kM2 F
, symbol decisions stored at
0
instant k in the delay lines of the FB filters with input given by the decision element output of user u at subchannel i, where D .u/ [i] is a suitable integer delay related to the DFE; we assume no decision errors, that is aO k.u/ [i] D ak.u/ [i]. The input to the decision element of user u at subchannel i at instant k is then expressed by (see Section 8.5) .u;u/ yk.u/ [i] D c.u;u/ [i] x.u/ [i] a.u/ k F0 [i] C b k [i] C T
C
U X vD1
T
.u;v/ fc.u;v/ [i] x.v/ [i] a.v/ k F0 [i] C b k [i]g T
T
(16.48)
v6Du
and the error signal is given by .u/
.u/
ek [i] D yk [i] a
.u/ l
k
D .u/ [i] F0
m
(16.49)
Without loss of generality, we extend the technique developed in Section 8.5 for the single-user fractionally-spaced DFE to determine the optimum coefficients of the DFE structure for user u D 1. We introduce the following vectors and matrices: .u/ .u/ .u/ T 1. h.u/ m [i] D G [i][h m F CM 1CD .u/ [i ] [i]; : : : ; h m F CD .u/ [i ] [i]] , vector of M1 samples 0 1 0 of the impulse response of subchannel i of user u. .u;v/ .u;v/ .v/ T 2. h.u;v/ FEXT;m [i] D G [i][h FEXT;m F0 CM1 1CD .u;v/ [i ] [i]; : : : ; h FEXT;m F0 CD .u;v/ [i ] [i]] , vector of M1 samples of the FEXT impulse response of subchannel i from user v to user u; assuming the differences between the propagation delays of the signal of user u and of the cross-talk signals originated by the other users are negligible, we have D .u;v/ [i] D D .u/ [i].
1134
Chapter 16. Applications of interference cancellation
3. .1/ R.1;1/ [i] D E[x.1/ [i]] k [i] xk T
Ł
M2 X
.1/Ł [i] h.1/ ¦a2.1/ [i ] hm m [i] T
mD1
C
V X vD2
!
Ł .1;v/T ¦a2.v/ [i ] h.1;v/ FEXT;m [i] hFEXT;m [i]
(16.50)
.l/ R.l;l/ [i] D E[x.l/ k [i] xk [i]] T
Ł
M2 X
.l;1/ .l;1/ .l/ 2 .¦a2.l/ [i ] h.l/ m [i] hm [i] C ¦a .1/ [i ] hFEXT;m [i] hFEXT;m [i]/ Ł
T
Ł
T
mD1
l D 2; : : : U
(16.51)
.1/ R.l;1/ [i] D E[x.l/ [i]] k [i] xk 0 T
Ł
M2 B X Ł .l/Ł .l;1/T .1/T 2 B¦ 2.1/ h.l;1/ @ a [i ] FEXT;m [i] hm [i] C ¦a .l/ [i ] hFEXT;m [i] hFEXT;m [i]
mD1
1 C
U X pD2
C .l; p/Ł .1; p/T ¦a2. p/ [i ] hFEXT;m [i] hFEXT;m [i]C A
p6Dl
l D 2; : : : U .l/Ł
. j/T
R.l; j/ [i] D E[xk [i] xk
M2 X
(16.52)
[i]] . j;1/T
.l; j/Ł
. j/T
2 .¦a2.1/ [i ] h.l;1/ FEXT;m [i] hFEXT;m [i] C ¦a . j/ [i ] hFEXT;m [i] hm [i]/ Ł
mD1
1 < j < l U (16.53) where all above matrices are M1 ð M1 square matrices. 4. 2
R.1;1/ [i] R.1;2/ [i] 6 R.2;1/ [i] R.2;2/ [i] 6 R.1/ [i] D 6 :: :: 4 : : R.U;1/ [i] R.U;2/ [i]
3 : : : R.1;U / [i] : : : R.2;U / [i] 7 7 7 :: :: 5 : : .U;U / ::: R [i]
(16.54)
16.4. Multiuser detection for VDSL
1135
where R.1/ [i] in general is a positive semi-definite Hermitian matrix, for which we assume here the inverse exists. 5. .2;1/ .U;1/ p.1/ [i] D ¦a2.1/ [i ] [h0.1/ [i]; hFEXT;0 [i]; : : : ; hFEXT;0 [i]]T
(16.55)
Defining the vectors c.1/ [i] D [c.1;1/ [i]; c.1;2/ [i]; : : : ; c.1;U / [i]]T , and b.1/ [i] D T T .1;1/ [i]; : : : ; b.1;U / [i]]T , the optimum coefficients are given by [b T
T
T
.1/ [i] D [R.1/ [i]]1 p.1/ [i] copt
(16.56)
and 2
.2;1/ .U;1/ h.1/ 1 [i]; hFEXT;1 [i]; : : : ; hFEXT;1 [i] : : : T
T
T
3
6 7 6 7 6 7 6 7 6 7 6 7 6 7 T T T .U;1/ 6 h.1/ [i]; h.2;1/ 7 6 7 M1 FEXT;M1 [i]; : : : ; hFEXT;M1 [i] 6 7 6 7 .1;2/T .2/T .U;2/T 6 7 hFEXT;1 [i]; h1 [i]; : : : ; h1 [i] 6 7 6 7 : 6 7 6 7 : 6 7 6 7 .1/ .1/ : bopt [i] D 6 7 copt [i] T T T .2/ .U;2/ 6 h.1;2/ 7 [i]; h [i]; : : : ; h [i] 6 FEXT;M1 7 FEXT;M1 FEXT;M1 6 7 : 6 7 6 7 : 6 7 6 7 : 6 7 6 7 T T T .1;U / .2;U / .U / 6 7 h [i]; h [i]; : : : ; h [i] 6 7 FEXT;1 FEXT;1 1 6 7 : 6 7 6 7 : 6 7 6 7 : 4 5 .1;U /T .2;U /T .U /T hFEXT;M1 [i]; hFEXT;M1 [i]; : : : ; hFEXT;M1 [i]
(16.57)
The MMSE value at the decision point of user 1 on subchannel i is thus given by .1/ .1/ Jmin [i] D ¦a2.1/ [i ] p.1/ [i]copt [i] H
(16.58)
The performance of an OFDM system is usually measured in terms of achievable bit rate for given channel and cross-talk characteristics (see Chapter 13). The number of bits per modulation interval than can be loaded with a bit-error probability of 107 on subchannel i is given by (see (13.15)) ! ¦a2.1/ [i ] 10.G code 0 gap;d B /=10 b.1/ [i] D log2 1 C .1/ (16.59) Jmin [i]
1136
Chapter 16. Applications of interference cancellation
where G code is the coding gain assumed to be the same for all users and all subchannels. The achievable bit rate for user 1 is therefore given by Rb.1/ D
16.4.1
1 X 1 M b.1/ [i] bit=s T i D0
(16.60)
Upstream power back-off
Upstream power back-off (PBO) methods are devised to allow remote users in a VDSL system to achieve a fair distribution of the available capacity in the presence of FEXT [12]. The upstream VDSL transmission rates, which are achievable with PBO methods, usually depend on parameters, for example, a reference length or the integral of the algorithm of the received signal power spectral density, that are obtained as the result of various trade-offs between services to be offered and allowed maximum line length. However, the application of such PBO methods results in a suboptimum allocation of the signal power for upstream transmission. It is desirable to devise a PBO algorithm with the following characteristics: ž for each individual user, the transmit signal PSD is determined by taking into account the distribution of known target rates and estimated line lengths of users in the network, and ž the total power for upstream signals is kept to a minimum to reduce interference with other services in the same cable binder. In the preceding section, we found the expression (16.60) of the achievable upstream rate for a user in a VDSL system with U users, assuming perfect knowledge of FEXT impulse responses and multiuser detection. However, PBO may be applied by assuming only the knowledge of the statistical behavior of FEXT coupling functions with no attempt to cancel interference. In this case, the achievable bit rate of user u is given by 2 3 Rb.u/
6 7 6 7 6 7 .u/ .u/ 2 P . f /jH . f /j 6 7 D log2 61 C 10.G code 0gap;d B /=10 7 d f U 6 7 X B .u;v/ 6 7 P .v/ . f /jHFEXT . f /j2 C N0 4 5 Z
(16.61)
vD1
v6Du
where P .u/ . f / denote the PSD of the signal transmitted by user u, H.u/ . f / is the frequency .u;v/ response of the channel for user u, HFEXT . f / is the FEXT frequency response from user v to user u, and N0 is the PSD of additive white Gaussian noise. From Section 4.4.2, the expression of the average FEXT power coupling function is given by .u;v/ . f /j2 D kt f 2 min.L u ; L v /jH.v/ . f /j2 jHFEXT
(16.62)
where L u and L v denote the lengths of the lines of user u and v, respectively, and kt is a constant.
16.4. Multiuser detection for VDSL
1137
Assuming that the various functions are constant within each subchannel band for OFDM modulation, we approximate (16.61) as 2 3 .u/
Rb D
M 1 X i D0
6 7 6 7 6 7 1 P .u/ . f i /jH.u/ . f i /j2 6 .G code 0gap;d B /=10 7 10 log2 61 C 7 (16.63) U 6 7 T X .u;v/ 6 7 P .v/ . f i /jHFEXT . f i /j2 C N0 4 5 vD1
v6Du
where f i denotes the center frequency of subchannel i. .u/ Let P . f i / denote the PSD of the signal transmitted by user u on subchannel i with .u/ gain G [i] D 1. Then the PBO problem can be formulated as follows: find the minimum of the function U Z U M 1 X X X 1 .u/ 2 .u/ .G [i]/ P . f i / P .u/ . f / d f ' (16.64) T uD1 B uD1 i D0 subject to the constraints: 1. 0 .G .u/ [i]/2 P
.u/
. f i / Pmax
u D 1; : : : ; U
i D 0; : : : ; M 1
(16.65)
and 2. .u/ Rb.u/ ½ Rb;target
u D 1; : : : ; U
(16.66)
.u/ where Pmax is a constant maximum PSD value and Rb;target is the target rate for user u.
In (16.66), Rb.u/ is given by (16.60) or (16.63), depending on the receiver implementation. Finding the optimum upstream transmit power distribution for each user is therefore equivalent to solving a non-linear programming problem in the U M parameters G .u/ [i], u D 1; : : : ; U , i D 0; : : : ; M 1. The optimum values of these parameters that minimize (16.64) can be found by simulated annealing [13, 14].
16.4.2
Comparison of PBO methods
ETSI has defined two modes of operation, named A and B, for PBO [12]. For a scenario using upstream VDSL transmission of two adjacent links with unequal lengths, mode A states that the signal-to-noise ratio degradation to either link shall not exceed 3 dB relative to the equal-length FEXT case. Mode B requires that the signal-to-noise ratio on the longer line shall not be degraded relative to the equal-length FEXT case; furthermore, degradation to the signal-to-noise ratio on the shorter line shall be bounded such that the shorter line can support at least the upstream rate supported on the longer line. Several methods compliant
1138
Chapter 16. Applications of interference cancellation
with either mode A or B have been proposed. PBO methods are also classified into methods that allow shaping of the PSD of the transmitted upstream VDSL signal, e.g., the equalized FEXT method, and methods that lead to an essentially flat PSD of the transmitted signal over each individual upstream band, e.g., the average log method. Both the equalized FEXT and the average log method, which are described below, comply with mode B. The equalized FEXT method requires that the PSD of user u be computed as [12] ½ L ref jHref . f /j2 .u/ (16.67) Pmax ; Pmax P . f / D min L v jH.u/ . f /j2 where L ref and Href denote a reference length and a reference channel frequency response, respectively. The average log method requires that, for an upstream channel in the frequency band . f 1 ; f 2 /, user u adopt a constant PSD given by [15] .u/
P .u/ . f / D P. f 1 ; f 2 /
f 2 . f1; f2/
(16.68)
where P..u/ f 1 ; f 2 / is a constant PSD level chosen such that it satisfies the condition Z
f2 f1
.u/ 2 log2 [P..u/ f 1 ; f 2 / jH . f / j] d f D K . f 1 ; f 2 /
(16.69)
where K . f 1 ; f 2 / is a constant. In this section, the achievable rates of VDSL upstream transmission using the optimum algorithm (16.64) and the average log method are compared for various distances and services. The numerical results presented in this section are derived assuming a 26-gauge telephone twisted-pair cable (see Table 4.2). The noise models for the alien-cross-talk disturbers at the line termination and at the network termination are taken as specified in [16] for the fiber-to-the-exchange case. Additive white Gaussian noise with a power spectral density of –140 dBm/Hz is assumed. We consider upstream VDSL transmission of U D 40 users over the frequency band given by the union of B1 = (2.9 MHz, 5.1 MHz) and B2 = (7.05 MHz, 12.0 MHz), similar to those specified in [12]. The maximum PSD value is Pmax D 60 dBm/Hz. FEXT power coupling functions are determined according to (16.62), where kt D 6:65ð1021 . Upstream transmission is assumed to be based on FMT modulation with bandwidth of the individual subchannels equal to 276 kHz and excess bandwidth of 12.5%; for an efficient implementation, a frequency band of (0, 17.664 MHz) is assumed, with M D 64 subchannels, of which only 26 are used. For the computation of the achievable rates, for an error probability of 107 a signal-to-noise ratio gap to capacity equal to 0 0 gap;dB D 0 gap;dB C 6 D 15:8 dB, which includes a 6 dB margin against additional noise sources that may be found in the DSL environment [17], and G code D 5:5 dB are assumed. For each of the methods and for given target rates we consider two scenarios: the users are i) all the same distance L from the central office, and ii) uniformly distributed at ten different nodes, having distances j L max =10, j D 1; : : : ; 10, from the central office. To assess the performance of each method, the maximum line length L max is found, such that all users can reliably achieve a given target rate Rb;target D 13 MBit/s. The achievable rates are also computed for the case that all users are at the same distance from the central office and no PBO is applied.
16.4. Multiuser detection for VDSL
1139
Figure 16.10. Achievable rates of individual users versus cable length using the optimum c 2001 IEEE.] upstream PBO algorithm for a target rate of 13 Mbit/s. [From [10],
For the optimum algorithm, the achievable rates are computed using (16.63). Furthermore, different subchannel gains may be chosen for the two bands, but transmission gains within each band are equal. Figure 16.10 shows the achievable rates for each group of four users with the optimum algorithm for the given target rate. The maximum line length L max for scenario ii) turns out to be 950 m. For application to scenario ii), the optimum algorithm requires the computation of 20 parameters. Note that for all users at the same distance from the central office, i.e., scenario i), the optimum algorithm requires the computation of two gains equal for all users. For scenario i), the achievable rate is equal to the target rate up to a certain characteristic length L max , which corresponds to the length for which the target rate is achieved without applying any PBO. Also note that L max for scenario ii) is larger than the characteristics length found for scenario i). Figure 16.11 illustrates the achievable rates with the average log algorithm (16.69). Joint optimization of the two parameters K B1 and K B2 for maximum reach under scenario ii) yields K B1 D 0:02 mW, K B2 D 0:05 mW, and L max D 780 m. By comparison with Figure 16.10, we note that for the VDSL transmission spectrum plan considered, optimum upstream PBO leads to an increase in the maximum reach of up to 20%. This increase depends on the distribution of target rates and line lengths of the users in the network. At this point, further observations can be made on the application of PBO. ž Equal upstream services have been assumed for all users. The optimum algorithm described is even better suited for mixed-service scenarios.
1140
Chapter 16. Applications of interference cancellation
Figure 16.11. Achievable rates of individual users versus cable length using the average log c 2001 IEEE.] upstream PBO method for a target rate of 13 Mbit/s. [From [10],
ž The application of PBO requires the transmit PSDs of the individual user signals to be recomputed at the central office whenever one or more users join the network or drop out of the network. To illustrate system performance achievable with multiuser detection, we consider U D 20 users, uniformly distributed at ten different nodes having distances j L max =10, j D 1; : : : ; 10, from the central office, where identification of FEXT impulse responses is performed. For the computation of the achievable rates of individual users, the FEXT impulse responses are generated by a statistical model, and L max D 500 m is assumed. Furthermore, to assess the relative merits of multiuser detection and coding, the achievable rates are computed for the two cases of uncoded and coded transmission. For coded transmission, a powerful coding technique yielding 8.5 dB coding gain for an error probability of 107 is assumed. Figure 16.12 illustrates the achievable rates for perfect suppression of all interferers, which corresponds to the single-user bound. For comparison, the achievable rates for the case that all users are the same distance away from the central office and neither multiuser detection nor coding are applied are also given. Figure 16.13 illustrates the achievable rates for perfect suppression of the ten worst interferers and no application of PBO. We observe that, without PBO, the partial application of multiuser detection does not lead to a significant increase of achievable rates for all users, even assuming large coding gains. Finally, Figure 16.14 depicts the achievable rates obtained for perfect suppression of the ten worst interferers and application of the optimum PBO algorithm with target rate of Rb;target D 75 Mbit/s for all users. The target rate is achieved by all users with the joint application of multiuser detection, coding, and power back-off.
16.4. Multiuser detection for VDSL
1141
Figure 16.12. Achievable rates of individual users versus cable length with all interferers c 2001 IEEE.] suppressed. [From [10],
Figure 16.13. Achievable rates of individual users versus cable length with ten interferers c 2001 IEEE.] suppressed and no PBO applied. [From [10],
1142
Chapter 16. Applications of interference cancellation
Figure 16.14. Achievable rates of individual users versus cable length with ten interferers c 2001 suppressed and optimum PBO applied for a target rate of 75 Mbit/s. [From [10], IEEE.]
To summarize the results of this section, a substantial increase in performance with respect to methods that do not require the identification of FEXT responses is achieved by resorting to reduced-complexity multiuser detection in conjunction with power back-off. This approach yields a performance close to the single-user bound shown in Figure 16.12.
Bibliography [1] D. G. Messerschmitt and E. A. Lee, Digital communication. Boston, MA: Kluwer Academic Publishers, 2nd ed., 1994. [2] D. G. Messerschmitt, “Design issues for the ISDN U-Interface transceiver”, IEEE Journal on Selected Areas in Communications, vol. 4, pp. 1281–1293, Nov. 1986. [3] D. D. Falconer, “Adaptive reference echo-cancellation”, IEEE Trans. on Communications, vol. 30, pp. 2083–2094, Sept. 1982. [4] D. L. Duttweiler, “Adaptive filter performance with nonlinearities in the correlation multiplier”, IEEE Trans. on Acoustics, Speech and Signal Processing, vol. 30, pp. 578– 586, Aug. 1982.
16. Bibliography
1143
[5] M. J. Smith, C. F. N. Cowan, and P. F. Adams, “Nonlinear echo cancelers based on transposed distributed arithmetic”, IEEE Trans. on Circuits and Systems, vol. 35, pp. 6–18, Jan. 1988. [6] G. Cherubini, “Analysis of the convergence behavior of adaptive distributed-arithmetic echo cancelers”, IEEE Trans. on Communications, vol. 41, pp. 1703–1714, Nov. 1993. [7] S. B. Weinstein, “A passband data-driven echo-canceler for full-duplex transmission on two-wire circuits”, IEEE Trans. on Communications, vol. 25, pp. 654–666, July 1977. [8] M. Ho, J. M. Cioffi, and J. A. C. Bingham, “Discrete multitone echo cancellation”, IEEE Trans. on Communications, vol. 44, pp. 817–825, July 1996. [9] P. J. W. Melsa, R. C. Younce, and C. E. Rohrs, “Impulse response shortening for discrete multitone transceivers”, IEEE Trans. on Communications, vol. 44, pp. 1662– 1672, Dec. 1996. [10] G. Cherubini, “Optimum upstream power back-off and multiuser detection for VDSL”, in Proc. GLOBECOM ’01, San Antonio, TX, Nov. 2001. [11] A. Duel-Hallen, “Decorrelating decision-feedback multiuser detector for synchronous code-division multiple-access channel”, IEEE Trans. on Communications, vol. 41, pp. 285–290, Feb. 1993. [12] “Access transmission systems on metallic access cables; Very high speed Digital Subscriber Line (VDSL); Part 2: Transceiver specification”, ETSI Technical Specification 101 270-2 V1.1.1, May 2000. [13] S. Kirkpatrik, C. D. Gelatt Jr., and M. P. Vecchi, “Optimization by simulated annealing approach”, Science, vol. 220, pp. 671–680, May 1983. [14] D. Vanderbilt and S. Louie, “A Monte Carlo simulated annealing approach to optimization over continuous variables”, J. Comp. Phys., vol. 56, pp. 259–271, 1984. [15] “Constant average log: robust new power back-off method”, Contribution D.815 (WP1/15), ITU-T SG 15, Question 4/15, Apr. 1999. [16] “ETSI VDSL specifications (Part 1) functional requirements”, Contribution D.535 (WP1/15), ITU-T SG 15, Question 4/15. June 21 V- July 2, 1999. [17] T. Starr, J. M. Cioffi, and P. J. Silverman, Digital subscriber line technology. Upper Saddle River, NJ: Prentice-Hall, 1999.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 17
Wired and wireless network technologies
Wired and wireless network technologies allow users to obtain services that are offered by various providers, for example, telephony, Internet access at a rate of several Megabit per second, on demand television programs, and telecommuting. Links make use of channels that allow full duplex transmission and exhibit sufficient capacity in the two directions of transmission. Wired network technologies make use of the existing cable infrastructure, consisting of 1) UTP cables originally laid in the customer service area (local loops) for access to the public switched telephone network (PSTN) over a telephone channel with band limited to about 4 kHz; 2) UTP cables installed in buildings or small geographic areas for the links between stations of a local-area network (LAN); and 3) coaxial cables currently used to distribute analog TV signals via cable TV networks. Wireless network technologies have gained widespread popularity among users; for example, technologies that allow the mobility of users are digital-enhanced cordless telecommunications (DECT) (see Chapter 18), personal access communications systems (PACS), and universal mobile telecommunications systems (UMTS). Wireless LANs have been developed to extend or replace wired LANs in environments where portable network stations are needed, allowing users to freely move around. The multichannel multipoint distribution service (MMDS) and the local multipoint distribution service (LMDS) are proposed as “wireless cable” networks that can offer high transmission capacity in geographical areas where the terrain does not present obstacles to the propagation of microwave radio signals. Hybrid structures are also considered, where the downstream transmission from a central office to the user and the upstream transmission in the opposite direction are performed using different transmission media, for example, services provided by direct broadcast satellite (DBS) in combination with upstream transmission over the PSTN.
17.1 17.1.1
Wired network technologies Transmission over unshielded twisted pairs in the customer service area
Modem The word “modem” is the contraction of mod ulator-demodulator. The function of a modem is to convert a data signal into an analog passband signal that can be transmitted over the
1146
Chapter 17. Wired and wireless network technologies
Local office Voiceband modem
end–to–end PSTN
Analog: 0 – 4 kHz e.g., V.34: { 2.4 – 28.8 (33.6) kbit/s
Voiceband modem
Figure 17.1. Illustration of a link between voiceband modems over the PSTN.
PSTN. A modem is defined full duplex if it can transmit and receive simultaneously over the same telephone channel, or half duplex in the other case (see Section 6.13.1 on page 522). A link between voiceband modems over the PSTN is illustrated in Figure 17.1. One of the technologies developed for data transmission over the PSTN is described by the ITU-T standard V.34. The V.34 modem uses a QAM scheme with 4-dimensional TCM (see Chapter 12) and flexible precoding for transmission over channels with ISI (see Chapter 13). We recall that flexible precoding permits achieving coding and shaping gains with minimum transmit power penalty for arbitrary constellations, provided that the transfer function of the overall discrete time system does not have zeros on the unit circle. Among the innovations introduced by the V.34 modem, we recall the initial probing of the channel, during which the passband is determined; based on the probing results, the symbol rate, which is in the range from 2400 to 3429 Baud, and the carrier frequency are defined. The information bit rate for the standard version V.34bis is in the range from 2.4 to 33.6 kbit/s; the selected value depends on the symbol rate and on the channel signal-to-noise ratio. The assumption that the link between two modems in the PSTN is completely “analog” does not lead to an accurate model for the majority of telephone channels available today; in fact the same network is essentially digital and transports signals that represent speech or data with a bit rate of 64 kbit/s. If modems used by service providers are linked to the PSTN by a digital channel that allows data transmission at a bit rate of 64 kbit/s, then only one “analog” local loop is found between the user and the rest of the network, as illustrated in Figure 17.2. Modems designed for the channel model just described are usually called PCM modems and can transmit data at a bit rate of 40–56 kbit/s over channels with a frequency band that goes from 0 Hz to about 4 kHz (see Section 17.1.1). This technology is described by the ITU-T standard X.90. In Table 17.1 the characteristics of some modems are briefly described. For full duplex modems frequency division duplexing (FDD) or echo cancellation (EC) is used. The column labelled “ITU-T std” identifies the international standards defined by the International Telecommunications Union—Telecommunications Standardization Sector. The acronym TC stands for trellis coding (see Chapter 12); for a description of the various modulation techniques we refer to Chapter 6.
Digital subscriber line In customer service areas UTP cables also represent a low cost alternative to optical fibers for links that allow data transmission at a considerably higher bit rates than those achievable by modems for transmission over the telephone channel, over distances that can reach 6 km; in fact, although optical fibers have substantially better transmission characteristics
17.1. Wired network technologies
1147
H
***
H v 7 km
8–bit x 8 ks/s=64 kbit/s
ADC
Analog subscriber loop
DSP DAC
DSP
PCM codecs
t 56 kbitńs
Full–duplex B–channel
(service provider)
Analog Client Modem
Goal: 56 kbit/s
Digital Server Modem
Ańm law
linear, high res.
(local telco switch)
(customer)
*** B–channels are not necessarily trans– parent: e.g., robbed–bit signalling (RBS) and digital attenuation (PAD) are possible
Figure 17.2. Illustration of a link between the PCM modem and the server modem. Table 17.1 Commercial modems.
Bit-rate Symbol Duplex (bit/s) rate (method)
ITU-T std
Modulation
300 300 full (FDD) 1200 1200 half 1200 600 full (FDD) 2400 1200 half 2400 600 full (FDD) 2400 1200 full (EC) 4800 1600 half 4800 2400 full (EC) 9600 2400 half 9600 2400 full (EC) 14400 2400 full (EC) 28800 3429 full (EC) 56000 100 km/h <wireline Outdoor vehicular GSM, IS-95, UMTS, IMT-2000
Small and inexpensive 4 hours) ¾50 km/h ¾wireline Indoor/outdoor pedestrian DECT, PACS, PHS, Bluetooth
Fixed part RJ 11
”Portable” part (cordless terminal adapter) NI Network interface
mobile
WLL applications
Figure 17.16. Illustration of the utilization of DECT in the wireless local loop.
technologies are summarized in Table 17.4. Appendix 17.A describes some of the widely adopted wireless technologies listed in Table 17.4. Low tier technologies are designed to achieve a quality of service (QoS) similar to that offered by wired networks. DECT in Europe and PACS in the United States were developed as technologies for the wireless local loop (WLL) to provide communication services also to mobile users [14, 15]. DECT was originally developed for application in wireless private branch exchanges (wireless PBXs) with low user mobility, and later extended as a technology for the WLL, as illustrated in Figure 17.16. PACS is illustrated in Figure 17.17.
17.2.1
Wireless local area networks
For the transmission at bit rates of the order of 1–10 Mbit/s, wireless local area networks (WLANs) normally use the industrial, scientific, and medical (ISM) frequency bands defined by the United States Federal Communications Commission (FCC), that is 902–928 MHz,
17.2. Wireless network technologies
1163
Radio port RJ 11
Wireless access fixed unit Switch
Radio port controller unit
mobile
NI Network interface
WLL applications
Figure 17.17. Illustration of the utilization of PACS in the wireless local loop.
Mobile nodes
Backbone LAN
Base stations
Router Wired LAN
In–building LAN
Figure 17.18. Illustration of the configuration of a wireless LAN inside a building.
2400–2483.5 MHz, and 5725–5850 MHz; these networks are also called RF WLANs to distinguish them from the IR WLANs that use the infrared (IR) frequency band. Specifications for PHY and MAC layers of WLANs are developed by various standardization organizations, among which we cite the IEEE 802.11 Working Group and the European Telecommunications Standard Institute (ETSI). The region within which mobile stations have the possibility of exchanging information with the network is divided into cells (see Figure 17.24). To reduce interference, neighboring cells use different frequencies. Within each cell, an access station or base station allocates frequencies and guarantees to the mobile stations the possibility of accessing fixed networks over metallic cables or optical fibers (see Figure 17.18). In the United States WLANs are allowed to operate in the IMS frequency bands without needing a license from the FCC, which, however, sets restrictions on the power of the radio signal that must be less than 1 W and specifies that spread-spectrum technology (see Chapter 10) must be used whenever the signal power is larger than 50 mW. Most RF WLANs employ direct sequence or frequency hopping spread-spectrum systems;
1164
Chapter 17. Wired and wireless network technologies
WLANs that use narrowband modulation systems usually operate in the band around 5.8 GHz, with a transmitted signal power lower than 50 mW, in compliance with FCC regulations. The coverage radius of RF WLANs is typically of the order of a few hundred meters. IR WLANs usually employ PAM-DSB modulation (see Appendix 7.C) with values of the carrier frequency of the order of the frequency of the infrared radiation, as also typically adopted in a fiber optic link; this system allows a simple and low cost implementation of the PHY layer [16]. However, it is necessary to consider that non-negligible interference signals can be generated by infrared radiation sources, like the sun or lighting systems. Two methods are usually considered in designing IR WLANs. In the first method the signal is transmitted along a well-defined direction; in this case an IR system can also be used outdoor with a coverage radius of a few kilometers. In the second method, the signal is irradiated in all directions and the coverage radius is limited to about 10 m.
Medium access control protocols Unlike cabled LANs, WLANs operate over channels with multipath fading, and channel characteristics typically vary over short distances. Channel monitoring to determine whether other stations are transmitting requires a larger time interval than that required by a similar operation in cabled LANs; this translates into an efficiency loss of the CSMA protocols, whenever they are used without modifications. The MAC layer specified by the IEEE 802.11 Working Group is based on the CSMA protocol with collision avoidance (CSMA/CA), in which four successive stages are foreseen for the transmission of a data packet, as illustrated in Figure 17.19 [17].
Base station
Mobile station RTS
CTS
DATA ACK
Figure 17.19. Illustration of the CSMA/CA protocol.
17.2. Wireless network technologies
1165
The CSMA/CA principle is simple. All mobile stations that have packets to transmit compete for channel access by sending ready to transmit (RTS) messages by a CSMA protocol. If the base stations recognize a RTS message sent by a mobile station, it sends a clear to transmit (CTS) message to the same mobile station and this one transmits the packet; if reception of this packet occurs correctly, then the base station sends an acknowledgement (ACK) message to the mobile station. With CSMA/CA, the only possibility of collision will occur during the RTS stage of the protocol; however, we note that also the efficiency of this protocol is reduced with respect to that of the simple CSMA/CD because of the presence of the RTS and CTS stages. As mobility is allowed, the number of mobile stations that are found inside a cell can change at any instant. Therefore it is necessary that each station informs the others of its presence as it moves around. A protocol used to solve this problem is the so-called hand-off, or hand-over, protocol, which can be described as follows: ž a switching station, or all base stations with a coordinated operation, registers the information relative to the signal levels of all mobile stations inside each cell; ž if a mobile station M is serviced by base station B1, but the signal level of station M becomes larger if received by another base station B2, the switching station proceeds to a hard hand-off operation whose final result is that mobile station M is considered part of the cell covered by base station B2.
17.2.2
MMDS and LMDS
The multichannel multipoint distribution service is presently used in the United States to distribute TV signals for entertainment and education programming. It can offer a maximum number of 33 channels for analog TV in the band 2.150–2.686 GHz, and the cell radius is typically in the range from 40 to 60 km, according to the environment characteristics and antenna positioning. The local multipoint distribution service is also considered for wireless access. The proposed spectrum allocation for this system comprises the bands 27.5–28.35 GHz and 29.1–29.25 GHz, and the cell radius is typically limited to 8 km. The configuration of MMDS and LMDS networks is illustrated in Figure 17.20. Assuming that most MMDS channels are dedicated to the distribution of TV signals, only a limited number of channels are available for downstream data transmission. An increase of system capacity can be obtained by methods of frequency reuse, for example “cellularization” or “sectorization”. “Cellularization” consists of subdividing the area into cells and designing the system so that there is a certain spatial separation between cells using the same frequencies. On the other hand, “sectorization” consists of using multiple directional antennas at the base station, where each antenna is used to distribute information to a certain group of users; frequency reuse means sending different signals to different user groups using the same RF channels. This method requires that the various user groups are located in geographical areas characterized by a sufficient azimuth separation and that antennas have a good rejection characteristic of the secondary lobes. An example of “sectorization” is illustrated in Figure 17.21, where six antennas are used; each antenna offers a 60-degree coverage, and to each antenna is assigned the same number of downstream channels. Moreover, two separate channel sets, labelled A and B,
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head end fiber
central office/
Video programming, high-speed data
Information highway, telephony
Figure 17.20. Illustration of the configuration of MMDS and LMDS networks.
A1 B3
B1
A3
A2 B2
Figure 17.21. Example of sectorization using six antennas.
are considered; the two sets include the entire frequency band for downstream transmission and they are reused three times, thus tripling the system capacity with respect to the case without “sectorization”. The required rejection level of secondary lobes can be reduced by the combination of “sectorization” with orthogonal polarization of adjacent sectors; we note, however, that orthogonal polarization can be effectively introduced only in the case of signal propagation in the absence of multipath, as in point-to-point links over short distances, using at each end of the link antennas placed at a large height with respect
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to the ground. For residential area installations where roof top antennas are used, usually multipath components due to reflected signals from the ground and surrounding buildings are present (see Section 4.6); the reflected signals may have components with orthogonal polarity with respect to the desired signal and this phenomenon introduces interference. In the design of transmission systems for LMDS it is necessary to consider the fact that, even if the transmitter and receiver are in line of sight, the effect of traffic and foliage movement determines a transmission channel with very hostile fading characteristics; for example, it is common to encounter in these environments a Doppler spread of 2 Hz or less, with a fade of over 40 dB. To obtain a reliable transmission for LMDS and to overcome effects due to the multipath, in addition to sectorized directional antenna systems various techniques may be considered, such as: ž signal coding and adaptive equalization, ž frequency diversity, ž combining of received signals by multiple antenna systems.
Bibliography [1] S. V. Ahamed, P. L. Gruber, and J.-J. Werner, “Digital subscriber line (HDSL and ADSL) capacity of the outside loop plant”, IEEE Journal on Selected Areas in Communications, vol. 13, pp. 1540–1549, Dec. 1995. [2] W. Y. Chen and D. L. Waring, “Applicability of ADSL to support video dial tone in the copper loop”, IEEE Communications Magazine, vol. 32, pp. 102–109, May 1994. [3] G. T. Hawley, “Systems considerations for the use of xDSL technology for data access”, IEEE Communications Magazine, vol. 35, pp. 56–60, Mar. 1997. [4] W. Y. Chen and D. L. Waring, “Applicability of ADSL to support video dial tone in the copper loop”, IEEE Communications Magazine, vol. 32, pp. 102–109, May 1994. [5] T. Starr, J. M. Cioffi, and P. J. Silverman, Understanding digital subscriber line technology. Englewood Cliffs, NJ: Prentice-Hall, 2000. ¨ [6] F. Sjoberg, M. Isaksson, R. Nilsson, P. Odling, S. K. Wilson, and P. O. Borjesson, “Zipper: a duplex method for WDSL based on DMT”, IEEE Trans. on Communications, vol. 47, pp. 1245–1252, Aug. 1999. ¨ ¸ er, and J. M. Cioffi, “Filter bank modulation tech[7] G. Cherubini, E. Eleftheriou, S. Olc niques for very high-speed digital subscriber lines”, IEEE Communications Magazine, vol. 38, pp. 98–104, May 2000.
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R [8] G. Cherubini, E. Eleftheriou, and S. Olcer, “Filtered multitone modulation for veryhigh-speed digital subscriber lines”, IEEE Journal on Selected Areas in Communications, June 2002. R [9] G. Cherubini, E. Eleftheriou, and S. Olcer, “Filtered multitone modulation for VDSL”, in Proc. IEEE GLOBECOM ’99, Rio de Janeiro, Brazil, pp. 1139–1144, Dec. 1999. [10] C. A. Eldering, N. Himayat, and F. M. Gardner, “CATV return path characterization for reliable communications”, IEEE Communications Magazine, vol. 33, pp. 62–69, Aug. 1995. [11] C. Bisdikian, K. Maruyama, D. I. Seidman, and D. N. Serpanos, “Cable access beyond the hype: on residential broadband data services over HFC networks”, IEEE Communications Magazine, vol. 34, pp. 128–135, Nov. 1996. [12] D. Fellows and D. Jones, “DOCSIS(tm) ”, IEEE Communications Magazine, vol. 39, pp. 202–209, Mar. 2001. [13] “Digital multi-programme systems for television sound and data services for cable distribution”, ITU-T Recommendation J.83, ITU-T Study Group 9, Oct. 24 1995. [14] W. Honcharenko, J. P. Kruys, D. Y. Lee, and N. J. Shah, “Broadband wireless access”, IEEE Communications Magazine, vol. 35, pp. 20–27, Jan. 1997. [15] C. C. Yu, D. Morton, C. Stumpf, R. G. White, J. E. Wilkes, and M. Ulema, “Lowtier wireless local loop radio systems. Part 1: Introduction”, IEEE Communications Magazine, vol. 35, pp. 84–92, Mar. 1997. [16] F. Gfeller and W. Hirt, “A robust wireless infrared system with channel reciprocity”, IEEE Communications Magazine, vol. 36, pp. 100–106, Dec. 1998. [17] S. Singh, “Wireless LANs”, in The Mobile Communications Handbook (J. D. Gibson, ed.), ch. 34, pp. 540–552, New York: CRC/IEEE Press, 1996. [18] R. O. LaMaire, A. Krishna, P. Bhagwat, and J. Panian, “Wireless LANs and mobile networking: standards and future directions”, IEEE Communications Magazine, vol. 34, pp. 86–94, Aug. 1996. [19] M. Rahnema, “Overview of the GSM system and protocol architecture”, IEEE Communications Magazine, vol. 31, pp. 92–100, Apr. 1993. [20] T. S. Rappaport, Wireless communications: principles and practice. Englewood Cliffs, NJ: Prentice-Hall, 1996. [21] D. J. Goodman, Wireless personal communication systems. Reading, MA: AddisonWesley, 1997. [22] R. Pirhonen, T. Rautava, and J. Penttinen, “TDMA convergence for packet data services”, IEEE Personal Communications Magazine, vol. 6, pp. 68–73, June 1999.
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[23] A. Furuskar, S. Mazur, F. Muller, and H. Olofsson, “EDGE: enhanced data rates for GSM and TDMA/136 evolution”, IEEE Personal Communications Magazine, vol. 6, pp. 56–66, 1999. [24] N. R. Sollenberger, N. Seshadri, and R. Cox, “The evolution of IS-136 TDMA for third-generation wireless services”, IEEE Personal Communications Magazine, vol. 6, pp. 8–18, 1999. [25] M. Austin, A. Buckley, C. Coursey, P. Hartman, R. Kobylinski, M. Majmundar, and J. P. Seymour, “Service and system enhancements for TDMA digital cellular systems”, IEEE Personal Communications Magazine, vol. 6, pp. 20–33, 1999. [26] K. Pahlavan and A. H. Levesque, Wireless information networks. New York: John Wiley & Sons, 1995. [27] H. Holma and A. Toskala, eds, WCDMA for UMTS radio access for third generation mobile communications. Chichester: John Wiley & Sons, 2001. [28] B. A. Miller and C. Bisdikian, Bluetooth revealed. Upper Saddle River, NJ: PrenticeHall, 2001. [29] J. C. Haartsen and S. Mattison, “Bluetooth — a new low–power radio interface providing short–range connectivity”, in Proc. of the IEEE, vol. 88, pp. 1651–1661, Oct. 2000. [30] T. Wilkinson, T. G. C. Phipps, and S. K. Barton, “A report on HIPERLAN standardization”, Int. Journal of Wireless Information Networks, vol. 2, pp. 99–120, Apr. 1995. [31] R. van Nee, G. Awater, M. Morikura, and H. Takanashi, “New high-rate wireless LAN standards”, IEEE Communications Magazine, vol. 37, pp. 82–88, Dec. 1999.
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Appendix 17.A 17.A.1
Standards for wireless systems
General observations
Before presenting some of the consolidated standards for wireless systems, we recall a few concepts that are of great interest in this field.
Wireless systems Although the trend, through third generation universal mobile telecommunications service (UMTS) systems, is to “merge” the different type of services into one large structure, a distinction must be made between two classes of wireless systems: voice-oriented systems and data-oriented systems. Figure 17.22 illustrates this aspect. From this scheme we note that both classes of systems can be further subdivided into two categories. Systems for voice transmission can be cordless, that is with low transmitted power and local area services [18], or cellular, that is with high transmitted power and wide area services. A similar distinction can be made for data transmission services: the wireless local area networks (WLANs) are low power systems with short range coverage, whereas the mobile-data networks are high power systems with long range coverage (see Sections 17.2 and 17.2.1).
Figure 17.22. Types of services offered by wireless systems.
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We will discuss here cellular systems, specifically the standards GSM, IS-136, JDC, and IS-95; we will also address the standard DECT for cordless systems and HIPERLAN for WLANs.
Modulation techniques The current second generation wireless systems are digital and use different types of modulation according to the required performance. For a discussion of modulation techniques for radio transmission systems, see Chapter 18. Using a QPSK (or a BPSK) system with a square root raised cosine transmit pulse yields a good trade-off between level of ISI and required bandwidth; in this case, however, the modulated signals do not have a constant envelope, which may represent a problem in the presence of a high power amplifier. An alternative is given by GMSK, whose basic scheme is given in Figure 18.45; in this case the required bandwidth increases for higher values of Bt T , where Bt is the bandwidth of the Gaussian filter; the choice of the parameter Bt T is a compromise between pulse duration and system bandwidth. Figure 17.23 illustrates estimated power spectra of baseband QPSK and GMSK signals for two values of Bt T . The frequency is normalized by bit rate 1=Tb of the system; in this way it is easy to realize that the bandwidth efficiency of QPSK is larger than that of GMSK.
Parameters of the modulator There are some essential parameters that must be taken into account for the design and definition of a wireless system.
Figure 17.23. Comparison between the PSD of QPSK and GMSK signals.
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1. Bandwidth efficiency or spectral efficiency: it is the ratio between the bit rate 1=Tb and the utilized bandwidth. 2. Power efficiency: for the same 1=Tb and receiver complexity, the lower the transmitted power for a certain bit error probability, the greater the efficiency. 3. Out-of-band radiation: it is the power outside of the main lobe of the power spectrum.1 4. Robustness to multipath: it expresses the insensitivity of the system to multipath channels. 5. Constant envelope: the use of power-efficient class C amplifiers, because of their strong non-linearity, requires input signals with a constant envelope.
Cells in a wireless system A fundamental concept when we speak of wireless systems is that of a cell. The coverage area of a certain system is not “served” by a single transmitter but by numerous transmitters, called base stations (BSs) or base station transceivers (BTSs): each one of them can “cover” only a small part of this area, called a cell. To each cell a set of carrier frequencies completely different from those of the neighboring cells is assigned, so that co-channel interference is as low as possible. The method of frequency reuse, illustrated in Figure 17.24, is related to the concept of a cell. To two cells separated by an adequate distance, the same set of carrier frequencies can be assigned with minimum co-channel interference; then there is a periodic repetition in assigning the frequencies, as seen in Figure 17.24. Cluster is a set of cells that subdivides the available bandwidth of the system. Obviously, each station will interfere with the other stations, especially with those in the same cell and in adjacent cells, as there is always a certain percentage of irradiated power outside of the nominal bandwidth of the assigned channel; hence the strategy in the choice of carrier frequencies assigned to the various cells is of fundamental importance. Related to the concept of cell is also the concept of hand-off or hand-over. When a user moves from one cell to another, the communication cannot be interrupted; hence, a mechanism must be present that, at every instant, keeps track of which BTS sends the “strongest” signal and possibly is capable of changing the BTS to which the mobile terminal is “linked”. Other specific aspects will be dealt with in the description of each standard.
17.A.2
GSM standard
System characteristics The global system for mobile communications (GSM) was started in the early 1990s with the aim of providing one standard within Europe. The services that this system offers are [19]: 1
For example, MSK has low bandwidth efficiency and high out-of-band radiation; the Gaussian filter of a GMSK system attempts to mitigate both these problems.
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Figure 17.24. Illustration of cell and frequency reuse concepts; cells with the same letter are assigned the same carrier frequencies.
ž Telephony service: digital telephone service with guarantee of service to users that moves at a speed of up to 250 km/h; ž Data service: can realize the transfer of data packets with bit rates in the range from 300 to 9600 bit/s. ž ISDN service: some services, as the identification of a user that sends a call and the possibility of sending short messages (SMS), are realized by taking advantage of the integrated services digital network (ISDN), whose description can be found in [20]. A characteristic of the GSM system is the use of the subscriber identity module (SIM) card together with a four-digit number (ID); inserting the card in any mobile terminal, it identifies the subscriber who wants to use the service. Important is also the protection of privacy offered to the subscribers of the system. Figure 17.25 represents the structure of a GSM system, that can be subdivided into three subsystems. The first subsystem, composed of the set of BTSs and mobile terminals or mobile stations (MSs), is called a radio subsystem; it allows communication between the MSs and the mobile switching center (MSC), that coordinates the calls and
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Figure 17.25. GSM system structure.
also other system control operations. To a MSC are linked many base station controllers (BSCs); each BSC is linked up to several hundreds BTSs, each of which identifies a cell and directly realizes the link with the mobile terminal. The hand-off procedure between two BTSs is assisted by the mobile terminal in the sense that it is a task of the MS to establish at any instant which BTS is sending the “strongest” signal. In the case of the hand-over between two BTS linked to the same BSC, the entire procedure is handled by the BSC itself and not by the MSC; in this way the MSC can save many operations. The second subsystem is the network switching subsystem (NSS), that in addition to the MSC includes: ž the home location register (HLR): this is a database that contains information regarding subscribers who reside in the same geographical area as the MSC; ž the visitor location register (VLR): this is a database that contains information regarding subscribers that are temporarily under the control of the MSC, but do not reside in the same geographical area;
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ž the authentication center (AUC): this controls codes and other information for correct communications management; ž the operation maintenance centers (OMC): they take care of the proper functioning of the various blocks of the structure. Finally, the MSC is directly linked to the public networks: PSTN for telephone services, ISDN for particular services as SMS, and data network for the transmission of data packets.
Radio subsystem We now give some details with regard to the radio subsystem. The total bandwidth allocated for the system is 50 MHz; frequencies that go from 890 to 915 MHz are reserved for MS-BTS communications, whereas the bandwidth 935–960 MHz is for communications in the opposite direction.2 In this way a full-duplex communication by frequency division duplexing (FDD) is realized. Within the total bandwidth there are 248 carriers allocated, that identify as many frequency channels called ARFCN; of these 124 are for uplink communications and 124 for downlink communications. The separation between two adjacent carriers is 200 kHz; the bandwidth subdivision is illustrated in Figure 17.26. Full-duplex communication is achieved by assigning two carriers to the user, one for transmission and one for reception, such that they are about 45 MHz apart. Each carrier is used for the transmission of an overall bit rate Rb of 270.833 kbit/s, corresponding to a bit period Tb D 3:692 µs. The system employs GMSK modulation with parameter Bt T equal to 0.3; the aim is to have a power-efficient system. However, the bandwidth efficiency is not very high; in fact we have 270:833 ' 1:354 bit/s/Hz (17.1) 200 which is smaller than that of other systems. Besides this FDM structure, there is also a TDMA structure; each transmission is divided into eight time intervals, or time slots, that identify the TDM frame. Figure 17.27 shows the structure of a frame as well as that of a single time slot. ¹D
Figure 17.26. Bandwidth allocation of the GSM system.
2
There also exists a version of the same system that operates at around the frequency of 1.8 GHz (in the USA the frequency is 1.9 GHz).
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Figure 17.27. TDM frame structure and slot structure of the GSM system.
As a slot is composed of 156.25 bits (not an integer number because of the guard time equal to 8.25 bits), its duration is about 576.92 µs; therefore the frame duration is about 4.615 ms. In Figure 17.27 it is important to note the training sequence of 26 bits, used to analyze the channel by the MS or BS. The flag bits signal if the 114 information bits are for voice transmission or for control of the system. Finally, the tail bits indicate the beginning and the end of the frame bits. Although transmissions in the two directions occur over different carriers, to each communication is dedicated a pair of time slots spaced 4 slots apart (one for the transmit station and one for the receive station); for example, the first and fifth or the second and sixth, etc. Considering the sequence of 26 consecutive frames, that have a duration of about 120 ms, the 13th and 26th frames are used for control; then in 120 ms a subscriber can transmit (or receive) 114 Ð 24 D 2736 bits, which corresponds to a bit rate of 22.8 kbit/s. Indeed, the net bit rate of the message can be 2.4, 4.8, 9.6, or 13 kbit/s. Redundancy bits are introduced by the channel encoder for protection against errors, so that we get a bit rate of 22.8 kbit/s in any case. The original speech encoder chosen for the system was a RELP (see Chapter 5), improved by a long-term predictor (LTP), with a bit rate of 13 kbit/s. The use of a voice activity detector (VAD) allows an improvement in system capacity by reducing the bit rate to a minimum value during silence intervals in the speech signal. For channel coding, a convolutional encoder with code rate 1/2 is used. In [20] the most widely used speech and channel encoders are described, together with the data interleavers. In Figure 17.28 a scheme is given that summarizes the protection mechanism against errors used by the GSM system. The speech encoder generates, in 20 ms, 260 bits; as the
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Figure 17.28. Channel coding for the GSM system.
bit rate per subscriber is 22.8 kbit/s, in 20 ms 456 bits must then be generated, introducing a few redundancy bits.3 To achieve reliable communications in the presence of multipath channels with delay spread up to 16 µs, at the receiver equalization by a DFE and/or detection by the Viterbi algorithm are implemented. In [20, 21] the calling procedure followed by the GSM system and other specific details are fully described.
GSM-EDGE Recently [22, 23], to support services with higher bit rates, the enhanced data for GSM evolution (EDGE) system was introduced, which employs 8-PSK, with a pulse represented in Figure 18.57, in place of GMSK. The bit rate is almost tripled to 69.2 kbit/s per subscriber; the channel encoder is also modified.
17.A.3
IS-136 standard
The standard IS-136 is an extension of the digital cellular IS-54 system; in turn IS-54 (called United States digital cellular (USDC)), was developed in the early 1990s as an extension of the analog standard AMPS, existing in the United States since the beginning of the 1980s. In many respects IS-136 is similar to GSM, in others is completely different. 3
The described slot structure and coding scheme refer to the transmission of user information, namely speech. Other types of communications, as for the control and management of the system, use different coding schemes and different time slot structures (see [21]).
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The structure of the system (see Figure 17.25) and types of services provided are similar to those of GSM; even IS-136 uses a combination of TDMA and FDMA, although with different specifications. A first substantial difference between the two systems is given by the modulation adopted: IS-136 uses a ³=4-DQPSK with a roll-off factor of 0.35. A considerable improvement in the bandwidth efficiency is thus obtained. The bandwidth efficiency is given by 48:6 ' 1:62 bit/s/Hz (17.2) ¹D 30 where the numerator is given by the bit rate per carrier and the denominator by the spacing between carriers. The price for a better bandwidth efficiency is a decrease of the power efficiency with respect to the GSM. We examine some parameters used by the system. IS-136 has also a bandwidth of 50 MHz; the band 824–849 MHz (or 1850–1865 MHz) is used for the MS-BS transmissions, whereas the band 869–894 MHz (or 1930–1945 MHz) is reserved for communications in the reverse direction; again, the FDD technique is used to achieve full-duplex communication. Moreover, as in GSM, a combination of FDMA and TDMA is used to realize multiple access; the overall bandwidth is divided into sub-bands of 30 kHz, each carrying an overall bit rate of 48.6 kbit/s, that corresponds to a bit period of Tb ' 20:576 µs. The structure of the TDM frame is illustrated in Figure 17.29, together with that of a single slot, which can be of two types, depending on whether it is used for communication from the mobile station to the base or vice versa. The frame is divided into 6 time slots, and each slot is composed of 324 bits, for a duration of about 6.667 ms, which means that the frame duration is of about 40 ms. As for GSM, for full-duplex communication two carriers, separated by about 45 MHz, are assigned: one for the MS-BS communication and the other for the reverse. This time, however, 2 time slots are assigned to the user, such that at most 3 users4 can use the same frame. We see from Figure 17.29 that, of the overall 324 bits per slot, only 260 bits are effective information, and the remaining serve as control and signaling for the system. Then in 40 ms there are only 260 Ð 2 D 520 information bits (transmitted or received), which corresponds to a bit rate of 13 kbit/s. The voice encoder is a vector-sum excitation linear predictive (VSELP) at 7.95 kbit/s, that generates only 159 bits in 20 ms (half a frame period). However, we have seen that in 20 ms there are 520=2 D 260 transmitted bits, hence the difference, that is 101 bits, is constituted by redundancy bits added by the channel encoder. Figure 17.30 illustrates channel coding for voice messages; for control messages the organization is different. The mechanism is similar to that of GSM; the 159 bits, generated by the voice encoder in 20 ms, are divided into two classes: the first comprises the 77 most significant bits (divided into a group of 65 bits and one of 12 bits), the second, the remaining 82 bits. After the procedure illustrated in the figure, the bits become 260, for a bit rate of 13 kbit/s. The type of demodulator/equalizer is not specified. For channels with a rms delay spread greater than 4:12 µs, a RLS adaptive DFE was proposed [20]. For information on other characteristics of the IS-136 system and on its evolution, as well as on convergence with GSM-EDGE, we recommend references [24, 25]. 4
In the GSM, 8 users share the same frame; in fact, to each user is assigned only one slot.
17.A. Standards for wireless systems
Figure 17.29. TDM frame structure and slot structure for the IS-136 system.
Figure 17.30. Channel coding for voice messages of the IS-136 system.
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17.A.4
Chapter 17. Wired and wireless network technologies
JDC standard
We discuss the Japanese digital cellular (JDC), which is very similar to the IS-136 standard. The JDC standard differs from the previous two standards mainly due to the fact that the total bandwidth of 80 MHz allocated for this system is much larger than that of GSM and IS-136. The frequency interval 940–956 MHz is reserved for base-mobile station communications, whereas the band 810–826 MHz is for communications in the opposite direction. To these two sub-bands two other sub-bands are added: the interval 1477–1501 MHz is for BS-MS communication, and the interval 1429–1453 MHz for MS-BS communications. The first pair of sub-bands is called low band (LB), the second high band (HB). The FDD technique is used to achieve full-duplex communication; if the LB is used, the transmit carrier and the receive carrier are separated by 130 MHz; if we use the HB, the separation is only 48 MHz. Channels have a bandwidth of 25 MHz, both in the LB and in the HB; each channel is used for transmission of an overall bit rate Rb D 42 kbit/s, that corresponds to a bit period of about 23:81 µs. Similar to the other two systems, besides the described FDMA structure, a TDMA structure is also used; a frame consists of 3 time slots and each one is composed of 280 bits (224 of information, 56 of control and signaling), which corresponds to a frame period of 3 Ð 280 Ð 23:81 D 20 ms. As one slot per frame is utilized, 280 bits (of which only 224 are of information) the user can transmit (or receive) information in 20 ms, which corresponds to a bit rate of 11.2 kbit/s. The speech encoder is a VSELP operating at 6.7 kbit/s and, after channel coding, the bit rate increases to 11.2 kbit/s. Channel coding is performed by a convolutional encoder with code rate 9/17. ³=4-DQPSK with a roll-off factor of 0.5 is adopted. As for the other two standards, the hand-off is assisted by the mobile terminal to reduce the MSC load.
17.A.5
IS-95 standard
As IS-136, the IS-95 system is a digital cellular system for North American geographical areas; it was also developed to be compatible with the AMPS standard, and is based on the CDMA spread spectrum technique, rather than on TDMA as the previously described systems [20, 26]. FDMA is still used: the total bandwidth allocated to the system is identical to that of IS-136 and has the same organization,5 but the channel bandwidth is 1.25 MHz. Also FDD is still used with a separation of 45 MHz between transmit and receive carriers. An important characteristic of this system is that the bit rate of the speech encoder can vary to lower power consumption, and is reduced to a minimum during silence intervals. The voice encoder is a Qualcomm codebook excited linear prediction (QCELP) with variable bit rate: 1.2, 2.4, 4.8, or 9.6 kbit/s.6 Interesting is the fact that the MS-BS communication greatly differs from the reverse: in particular, different sequences are employed to perform spreading. When the BS is
5
Within the total bandwidth, each of the two systems should use only a portion of the spectrum so that they do not interfere with each other. 6 In fact, the effective rate is from 1 to 8 kbit/s and the higher rates are due to the fact that additional redundancy bits have already been included.
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transmitting, data to all mobile terminals within the same cell (identified by a carrier) are sent using a different spreading sequence for each user. Regarding the MS, in transmission the communication occurs asynchronously, with the requirement that the power level of the received signals must be as uniform as possible. Whatever the bit rate used by the voice encoder, the channel encoder brings the bit rate to 19.2 kbit/s. The spreading procedure used for the BS-MS transmission is illustrated in Figure 17.31. Note that, after channel coding, interleaving is performed. Then the bit rate is increased to 1.2288 Mbit/s using one of 64 Walsh codes; to each user in a cell is assigned a different code, in order to maintain separation among signals of different users in the same cell and eliminate interference, at least in the absence of multipath. Moreover, to reduce the level of interference between two users that are in different cells, and to which the same Walsh code is assigned, scrambling is used; without going into details (see Chapter 10), the function of scrambling is to achieve quasi-orthogonality between the signals of the two users, also for different lags, so that the interference is kept at a low level [20]. Before the two modulation filters, a short pilot PN sequence is also inserted, that is used at the receiver for various purposes, e.g., channel identification. The system for the MS-BS communication has a different structure: channel coding is performed by a convolutional encoder with code rate 1/3, that yields a bit rate of 28.8 kbit/s. Spreading is still done by selecting one of 64 Walsh codes, but in a different way as compared to the BS-MS transmission [20]. QPSK is adopted; in particular, the modulation used by the mobile station is OQPSK. For detection, a RAKE receiver is used. We conclude this section by mentioning the third generation (3G) mobile radio system denoted universal mobile telecommunications system (UMTS), in the process of being standardized. 3G systems are intended for the transmission at rates from 16 kbit/s up to
Figure 17.31. Spreading procedure used by the IS-95 system for transmission from the base station to the mobile station.
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2 Mbit/s. Various approaches have been identified for the definition of the link layer; the most important is the wideband CDMA (WCDMA) system [27].
17.A.6
DECT standard
The digital European cordless telecommunications (DECT) system is a cordless digital system used in Europe. Unlike cellular systems that use cells with radius of the order of a few kilometers, the DECT system is mainly employed indoor and the cell radius is at most of a few tens of meters (typically 100 m). Services provided by the system are: ž speech transmission, ž data transmission, ž support to the ISDN network. The main characteristics of DECT are summarized in Table 17.5. For comparison, the main characteristics of PACS are summarized in Table 17.6. While originally the hand-off problem was not considered, as each MS corresponded to one BS only, now even for DECT we speak of hand-off assisted by the mobile terminal, such that the system configuration is similar7 to that of a cellular system (see Figure 17.25). An interesting characteristic is the use of the dynamic channel selection (DCS) algorithm, that allows the portable to know at every moment which channel (frequency) is the best (with the lowest level of interference) for communication and select it. We briefly illustrate the calling procedure followed by the system: 1. When a mobile terminal wishes to make a call, it first measures the received signals from the various BS8 and selects the one which yields the best signal level. 2. By the DCS algorithm, the mobile terminal selects the best free channels of the selected BS. Table 17.5 Table summarizing the main characteristics of DECT.
Frequency range RF channel spacing Modulation Transmission bit rate Voice encoding method Access method Frame duration Subscriber TX peak power Radius of service Frequency planning
7 8
1880ł1900 MHz 1728 kHz GMSK 1152 kbit/s 32 kbit/s ADPCM FDMA/TDMA/TDD 10 ms (24 time slots) 250 mW 100ł150 m Dynamic channel allocation
For a discussion of the differences between a cellular system and a cordless system, see Section 17.A.1. For DECT, the acronyms portable handset (PH) in place of MS and radio fixed part (RFP) in place of BS, are often used.
17.A. Standards for wireless systems
1183
Table 17.6 Table summarizing the main characteristics of PACS.
Frequency range
1850ł1910 MHz uplink 1930ł1990 MHz downlink RF channel spacing 300 kHz Modulation ³=4 QPSK Transmission bit rate 384 kbit/s Voice encoding method 32 kbit/s ADPCM Access method TDD/TDMA Frame duration 2.5 ms (8 time slots) Subscriber TX peak power 200 mW Radius of service 300ł500 m Frequency planning Quasi-static automatic frequency assignment f1
f2
f3
f4
f5
f6
f7
f8
f9
f10
1.728 MHz 1880 MHz
fi =1881.792+(i-1)x1.728 MHz
1900 MHz
Figure 17.32. FDMA structure of the DECT system.
3. MS sends a message, called access request, over the least interfered channel. 4. BS sends (or not) an answer: access granted . 5. If the MS receives this message, in turn it transmits the access confirm message and the communication starts. 6. If the MS does not receive the access granted signal on the selected channel, it abandons this channel and selects the second least interfered channel, repeating the procedure; after failing on 5 channels, the MS selects another BS and repeats all operations. The total band allocated to the system goes from 1880 to 1900 MHz and is subdivided into ten sub-bands, each with a width of 1.728 MHz; this is the FDMA structure of DECT represented in Figure 17.32. Each channel has an overall bit rate of 1.152 Mbit/s, that corresponds to a bit period of about 868 ns. Similar to the other systems, TDMA is used. The TDM frame, given in Figure 17.33, is composed of 24 slots: the first 12 are used for the communication BS-MS, the other
1184
Chapter 17. Wired and wireless network technologies
Figure 17.33. TDM frame structure and slot structure for the DECT system.
12 for the reverse communication; thus we realize a full-duplex communication by time division duplexing (TDD). In this DECT differs from all above considered wireless systems which use FDD; DECT allocates half of the frame for transmission and the other half for reception. In Figure 17.33 the slot structure is also shown; it is composed of 480 bits of which the first 32 are fixed and correspond to a synchronization word, the successive 388 are information bits, and the remaining 60 constitute the guard time. The frame has a duration of 480 Ð 24=1152 D 10 ms. The field of 388 bits reserved for information bits is subdivided into two subfields A and B: the first (64 bits) contains information for signaling and control of the system, the second (324 bits) contains user data. If the signal is speech, 4 of these 324 bits are parity bits, which translates into a net bit rate of 320 bits in a frame interval of 10 ms, and therefore in a bit rate of 32 kbit/s; an ADPCM voice encoder at 32 kbit/s is used and no channel coding is provided. For transmission, GMSK with parameter Bt T of 0.5 is adopted. At the MS receiver a pair of antennas are often used to realize switched antenna diversity. With this mechanism, fading and interference are mitigated. We summarize in Table 17.7 the characteristics of the various standards so far discussed. A standard that has in part overcome DECT is Bluetooth [28, 29], that operates around the frequency of 2:45 GHz, in the unlicensed and open industrial-scientific-medical (ISM) band. Bluetooth uses a FH/TDD scheme with Gaussian-shaped FSK (h 0:25). The modulation rate is 1 MBaud with a slot of 625 µs; a different hop channel is used for each slot. This gives a nominal hop rate of 1600 hops/s. There are 79 hop carriers spaced by 1 MHz.
17.A. Standards for wireless systems
1185
Table 17.7 Summary of characteristics of the standards GSM, IS-136, JDC, IS-95, and DECT.
System
GSM
IS-136
JDC
IS-95
DECT
Multiple access Band (MHz) Forward (BS!MS) Reverse (MS!BS) Duplexing Spacing between carriers Modulation Bit rate per carrier (kbit/s) Speech encoder bit rate (kbit/s) Convolutional encoderŁ code rate Frame duration (ms)
TDMA/ FDMA
TDMA/ FDMA
TDMA/ FDMA
FDMA/ CDMA
TDMA/ FDMA
935ł960 1805ł1880 890ł915 1710ł1785 FDD 200 kHz
869ł894 1930ł1945 824ł849 1850ł1865 FDD 30 kHz
940ł960 1477ł1501 810ł830 1429ł1453 FDD 25 kHZ
935ł960
1880ł1900
824ł849
1880ł1900
FDD 1250 kHz
TDD 1728 kHz
GMSK 0.3 270.833
³=4-DQPSK 48.6
³=4-DQPSK 42
QPSK 1228.8
GMSK 0.5 1152
RPE-LTP 13
VSELP 7.95
VSELP 6.7
QCELP 1.2ł9.6
ADPCM 32
1/2
1/2
none
4.615
40
20
1/2 (F) 1/3 (R) 20
10
Ł All these standards use a CRC code, possibly together with a convolutional code.
The maximum bit rate is of 723:2 kbit/s in one direction and 57:6 kbit/s in the reverse direction.
17.A.7
HIPERLAN standard
In 1991 ETSI started the standardization process for the realization of an European high speed and low power data transmission system for WLAN applications. As cordless systems, also HIPERLAN supports mobile users, up to a maximum speed of about 35 km/h. HIPERLAN has a double application: the first is that of replacing (see Figure 17.34) the last network link to the various users, in order to reduce costs due to the reconfiguration of the position of the terminal, the second is that of constituting, in indoor environments, a network among mobile (and fixed) terminals. There are two versions of the HIPERLAN standard. HIPERLAN type 1. The allocated band for the system is from 5.15 to 5.30 GHz and from 17.1 to 17.3 GHz. The first bandwidth of 150 MHz is still the most used and is further divided into 5 sub-bands; the first carrier is at 5.176468 GHz, with a channel bandwidth of 23.5294 MHz. Figure 17.35 shows the band allocation.
1186
Chapter 17. Wired and wireless network technologies
Figure 17.34. Configuration of a HIPERLAN network.
5150 MHz
5300 MHz 1
2
3
4
5
23.5294 MHz 5176.47 MHz 150 MHz Figure 17.35. Band allocation of the HIPERLAN type 1 system.
The maximum overall (raw) bit rate is of 23.5294 Mbit/s with a corresponding bit interval of about 42.5 ns. The system is also able to work at lower bit rates (the so-called low rate of 1.4706 Mbit/s), in order to reduce power consumption. The transmission takes place by data packets: m blocks of 496 bits, with m that can vary from 1 to 47, are transmitted. Before the transmission of these blocks, there is the transmission of a training sequence of 450 bits, which is needed for channel identification and synchronization. Figure 17.36 shows the block diagram of the transmitter.
17.A. Standards for wireless systems
1187
Figure 17.36. Block diagram of the transmitter for the HIPERLAN type 1 system.
The source generates a sequence of m blocks, each one of 416 bits; at the output of the channel encoder the block length is of 496 bits. After interleaving, differential precoding is performed and successively a training sequence is inserted; the generated symbols are then input to the modulator. For HIPERLAN type 1, GMSK with parameter Bt T equal to 0.3 is adopted.9 At the receiver, adaptive equalization is performed before decoding [30]. HIPERLAN type 2. A more recent version of HIPERLAN allows transmission at a raw bit rate of 54 Mbit/s, for a net bit rate of up to 45 Mbit/s. The modulation is DMT-OFDM, with 48 subchannels. Other WLAN standards are described in [31], among which we mention the IEEE 802.11 standard.
9
For low rate communication, the modulation is simple FSK.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 18
Modulation techniques for wireless systems
In this chapter, after an overview of front-end architectures for mobile radio receivers, we discuss modulation and demodulation schemes that are well suited for application to mobile radio systems because of their simplicity and robustness against disturbances introduced by the transmission channel. Appendix 17.A describes some of the standards where these schemes are adopted.
18.1
Analog front-end architectures
Conventional superheterodyne receiver Many receivers for radio frequency (RF) communications use the conventional superheterodyne scheme illustrated in Figure 18.1 [1]. A first mixer employs a local oscillator (LO), whose frequency f 1 is variable, to shift the signal to around a fixed intermediate frequency (IF) f IF . The output signal of the first mixer is filtered by a passband filter centered around the frequency f IF to eliminate out of band components. The cascade of RF filter, linear amplifier (LNA), and image rejection (IR) filter performs the tasks of amplifying the desired signal as well as eliminating the noise outside of the desired band and rejecting the spectral image components introduced by the LO. The IF filter output signal is shifted into baseband through a second mixer, which makes use of an oscillator with a fixed frequency f 2 , whose output is filtered with a lowpass filter to eliminate high frequency components. The signal is then sent to the analog-to-digital converter. One drawback of the superheterodyne receiver is the high selectivity in frequency (high Q) that the various elements must exhibit, which makes an integrated implementation at high frequency a difficult problem. Problems related to a full integration can be divided into two categories. 1. Integration of the acquisition structure and signal processing requires the elimination of both IR and IF filters. 2. For channel selection, a synthesizer can be integrated using an on-chip VCO with low Q, which, however, yields poor performance in terms of phase noise.
1190
Chapter 18. Modulation techniques for wireless systems
Figure 18.1. Conventional superheterodyne receiver.
Alternative architectures In this section three receiver architectures are considered that attempt to integrate the largest number of receiver elements. To reach this objective, the A/D conversion of the scheme of Figure 18.1 must be shifted from baseband to IF or RF. In this case implementation of the channel selection by an analog filter bank is not efficient; indeed, it is more convenient to use a wideband analog filter followed by a digital filter bank. Reference is also made to multi-standard software-defined radio (SDR) receivers, or to the possibility of receiving signals that have different bandwidths and carriers, that are defined according to different standard systems. We can identify two approaches to the A/D conversion. 1. Full bandwidth digital conversion: by using a very high sampling frequency, the whole bandwidth of the SDR system is available in the digital domain, i.e. all desired channels are considered. Considering that this bandwidth can easily achieve 100 MHz, and taking into account the characteristics of interference signals, the dynamic of the ADC should exceed 100 dB. This solution, even though it is the most elegant, cannot easily be implemented as it presents a high complexity and high power consumption. 2. Partial bandwidth digital conversion: this approach uses a sampling frequency that is determined by the radio channel that presents the most extended bandwidth within the different systems we want to implement. The second approach will be considered, as it leads to an SDR system architecture that can be implemented with moderate complexity.
Direct conversion receiver This architecture eliminates many off-chip components. In this approach, illustrated in Figure 18.2, all desired channels are shifted to baseband through a mixer that uses an oscillator with a varying frequency. Unwanted spectral components are removed by an on-chip baseband filter. Clearly this structure allows a higher integration level with respect to the superheterodyne scheme; in fact this architecture offers two important advantages: 1. the problem of image components is bypassed and the IR filter is not needed; 2. the IF filter and following operations were substituted with a lowpass filter and a baseband amplifier, which are suitable for monolitic integration. However, there are numerous problems with this structure; in fact, the oscillator is at the same frequency as the RF carrier and there is a possibility of leakage of the oscillator signal
18.1. Analog front-end architectures
1191
Figure 18.2. Direct conversion receiver.
towards the mixer input as well as towards the antenna, with consequent radiation. The interference signal generated by the oscillator could be reflected on surrounding objects and be “re-received”: therefore this spurious signal would produce a time variant DC offset [1] at the mixer output. To understand the origin and consequences of this offset, we can make the following two observations. 1. The isolation between the oscillator input and the inputs of the mixer and linear amplifier is not infinite; in fact an LO leakage is determined by both capacitive coupling and device substrate. The spurious signal appears at the linear amplifier and mixer inputs and is then mixed with the signal generated by the oscillator, creating a DC component; this phenomenon is called self-mixing. A similar effect is present when there is a strong signal coming from the linear amplifier or from the mixer that couples with the LO input: this signal would then be multiplied by itself. 2. In Figure 18.2 to amplify the input signal, that is of the order of microvolts, to a level such that it can be digitized by a low cost and low power ADC, the total gain, from the antenna to the LPF output, is about 100 dB. Of this gain, 30 dB are usually provided by the linear amplifier/mixer combination. With these data we can make a first computation of the offset due to self-mixing. We assume that the oscillator generates a signal with a peak-to-peak value of 0.63 V and undergoes an attenuation of 60 dB when it couples with the LNA input. If the gain of the linear amplifier/mixer combination is 30 dB, then the offset produced at the mixer output is of the order of 10 mV; if directly amplified with a gain of 70 dB, the voltage offset would saturate the circuits that follow. The problem is even worse if self-mixing is time variant. This event, as previously mentioned, occurs if the oscillator signal leaks to the antenna, thus being irradiated and reflected back to the receiver from moving objects. Finally, the direct conversion receiver needs a tuner for the channel frequency selection that works at high frequency and with low phase noise; this is hardly obtainable with an integrated VCO with low Q.
Single conversion to low-IF The low-IF architecture reduces the problem of DC offset present in the direct conversion receiver. This system has the same structure as Figure 18.2 and, similarly to direct
1192
Chapter 18. Modulation techniques for wireless systems
Figure 18.3. Double conversion with wideband IF.
conversion, it utilizes a single mixer stage; the principal difference is that the frequency shift is not made to DC but to a small intermediate frequency. The main advantage of the low-IF system is that the desired channel has no DC components; therefore, the usual problems that emerge from DC offset present in the direct conversion are avoided.
Double conversion and wideband IF This architecture, illustrated in Figure 18.3, takes all desired channels and shifts them from RF to IF using the first mixer by a fixed frequency oscillator; a simple lowpass filter is used to eliminate high frequency image components. All channels are then shifted down to baseband by a second mixer, this time with a variable frequency oscillator. The baseband filter that follows has a variable gain and eliminates spectral components not belonging to the desired signal band. Channel tuning is obtained by the second low frequency LO; in this case the first RF oscillator can be implemented by a quartz oscillator with a fixed frequency, while the second can be implemented by on-chip techniques that provide low-frequency oscillators with low phase noise and low Q. It is important to emphasize the absence of oscillators that operate at the same frequency of the RF carrier; this eliminates the potential problem of “re-radiation” of the oscillator. We can conclude by affirming that the double conversion architecture is the most adequate for the analog front-end.
18.2
Three non-coherent receivers for phase modulation systems
We introduce three non-coherent receivers to demodulate phase-modulated signals.
18.2.1
Baseband differential detector
In M-PSK, the transmitted signal (isolated pulse) is given by (6.128). For a continuous M-DPSK transmission with symbol period T , the transmitted signal is given by s.t/ D
C1 X
Re[e j
k
h T x .t kT /e j2³ f 0 t ]
(18.1)
kD1
where k is the phase associated with the transmitted symbol at instant kT given by the recursive equation (6.157).
18.2. Three non-coherent receivers for phase modulation systems
1193
Figure 18.4. Non-coherent baseband differential receiver. Thresholds are set at .2n 1/³=M, n D 1; 2; : : : ; M.
At the receiver the signal r is a version of s, filtered by the transmission channel and corrupted by additive noise. We denote as g A the cascade of passband filters used to amplify the desired signal and partially remove noise. As shown in Figure 18.4, let x be the passband received signal, centered around the frequency f 0 , equal to the carrier of the transmitted signal x.t/ D Re[x .bb/ .t/e j2³ f 0 t ]
(18.2)
where x .bb/ is the complex envelope of x with respect to f 0 . Using the polar notation x .bb/ .t/ D Mx .t/e j1'x .t/ , (18.2) can be written as x.t/ D Mx .t/ cos.2³ f 0 t C 1'x .t//
(18.3)
where 1'x .t/ is the instantaneous phase deviation of x with respect to the carrier phase (see Chapter 1, equation (1.207)). In the ideal case of absence of distortion and noise, sampling at suitable instants yields 1'x .t0 C kT / D k . Then for the recovery of the phase k we can use the receiver scheme of Figure 18.4, which, based on signal x, determines the baseband component y as y.t/ D y I .t/ C j y Q .t/ D 12 x .bb/ Ł g Rc .t/
(18.4)
The phase variation of the sampled signal yk D y.t0 C kT / between two consecutive symbol instants is obtained by means of the signal Ł z k D yk yk1
(18.5)
Always in the ideal case and assuming that g Rc does not distort the phase of x .bb/ , z k turns out to be proportional to e jk . The simplest data detector is the threshold detector based on
1194
Chapter 18. Modulation techniques for wireless systems
Figure 18.5. Baseband equivalent scheme of Figure 18.4.
the value of vk D arg z k D 1'x .t0 C kT / 1'x .t0 C .k 1/T /
(18.6)
Note that a possible phase offset 1'0 and a frequency offset 1 f 0 , introduced by the receive mixer, yields a signal y given by y.t/ D [ 12 x .bb/ .t/e j .1'0 C2³ 1 f 0 t/ ] Ł g Rc .t/
(18.7)
Assuming that g Rc does not distort the phase of x .bb/ , the signal vk becomes vk D 1'x .t0 C kT / 1'x .t0 C .k 1/T / 2³ 1 f 0 T
(18.8)
which shows that the phase offset 1'0 does not influence vk , while a frequency offset must be compensated by the data detector, summing the constant phase 2³ 1 f 0 T . The baseband equivalent scheme of the baseband differential receiver is given in Figure 18.5. The choice of h T x , g .bb/ A , and g Rc is governed by the same considerations as in the case of a QAM system; for an ideal channel, the convolution of these elements must be a Nyquist pulse.
18.2.2
IF-band (1 Bit) differential detector (1BDD)
The scheme of Figure 18.4 is introduced only to illustrate the basic principle, as its implementation complexity is similar to that of a coherent scheme. Indeed, this scheme has a reduced complexity because specifications on the carrier recovery can be less stringent; moreover, it does not need phase recovery. An alternative scheme that does not use carrier recovery is illustrated in Figure 18.6. In this case the signal is first delayed by a symbol period T and then multiplied by itself (I branch) and with its ³=2 phase-shifted version (Q branch) by a Hilbert filter. On the two branches the signals are given by I : x.t/x.t T / D Mx .t/ cos[2³ f 0 t C 1'x .t/]Mx .t T / cos[2³ f 0 .t T / C 1'x .t T /] Q : x .h/ .t/x.t T / D Mx .t/ sin[2³ f 0 t C 1'x .t/]Mx .t T / cos[2³ f 0 .t T / C 1'x .t T /]
(18.9)
18.2. Three non-coherent receivers for phase modulation systems
1195
Figure 18.6. Non-coherent 1 bit differential detector.
The filter g Rc removes the components around 2 f 0 ; the sampled filter outputs are then given by I : z k;I D Mx .t0 C kT /Mx .t0 C .k 1/T / 1 2
cos[2³ f 0 T C 1'x .t0 C kT / 1'x .t0 C .k 1/T /]
Q : z k;Q D Mx .t0 C kT /Mx .t0 C .k 1/T / 1 2
(18.10)
sin[2³ f 0 T C 1'x .t0 C kT / 1'x .t0 C .k 1/T /]
If f 0 T D n, n an integer, or by removing this phase offset by phase shifting x or z k , it results z k;Q vk D tan1 D 1'x .t0 C kT / 1'x .t0 C .k 1/T / (18.11) z k;I as in (18.6). The baseband equivalent scheme is shown in Figure 18.7, where, assuming that g Rc does not distort the desired signal, z k D 12 x .bb/ .t0 C kT /x .bb/Ł .t0 C .k 1/T /
(18.12)
Typically, in this modulation system, the transmit filter h T x is a rectangular pulse or a Nyquist pulse; instead, g A is a narrow band filter to eliminate out of band noise.
Figure 18.7. Baseband equivalent scheme of Figure 18.6.
1196
Chapter 18. Modulation techniques for wireless systems
For a simple DBPSK, with k 2 f0; ³ g, we only consider the I branch, and vk D z k;I is compared with a threshold set to 0.
Performance of M-DPSK With reference to the transmitted signal (18.1), we consider the isolated pulse k s.t/ D Re[e j
k
h T x .t kT /e j2³ f 0 t ]
(18.13)
where k
D
k1
²
2³.M 1/ 2³ ;:::; k 2 0; M M
C k
¦ (18.14)
In general, referring to the scheme of Figure 18.7, the filtered signal is given by .bb/
x .bb/ .t/ D u .bb/ .t/ C w R .t/
(18.15)
where u .bb/ is the desired signal at the demodulator input (isolated pulse k) u .bb/ .t/ D e j
k
.bb/ 1 .bb/ h T x Ł 12 gCh Ł 2 g A .t kT /
(18.16)
.bb/ is the complex envelope of the impulse response of the transmission channel, as .1=2/gCh .bb/ and w R .t/ is zero mean additive complex Gaussian noise with variance ¦ 2 D 2N0 Brn , where Z C1 Z C1 2 jG A . f /j d f D jg A .t/j2 dt Brn D 1
Z
C1
D 1
1
Z C1 þ 1 þþ .bb/ þþ2 1 þ .bb/ þþ2 þG A . f /þ d f D þg A .t/þ dt 4 1 4
(18.17)
is the equivalent bilateral noise bandwidth. We assume the following configuration: the transmit filter impulse response is given by r 2E s t T =2 h T x .t/ D rect (18.18) T T the channel introduces only a phase offset .bb/ .t/ D 2e j'a Ž.t/ gCh
and the receive filter is matched to the transmit filter, r t T =2 2 .bb/ g A .t/ D rect T T
(18.19)
(18.20)
At the sampling intervals t0 C kT D T C kT , let wk D w.bb/ R .t0 C kT /
(18.21)
18.2. Three non-coherent receivers for phase modulation systems
1197
and r AD
2E s 1 T 2
r
p 2 T D Es T
(18.22)
then x .bb/ .t0 C kT / D Ae j .
k C'a /
C wk
(18.23)
with E[jwk j2 ] D ¦ 2 D 2N0 Brn D N0
(18.24)
since Brn D 1=2. Moreover, it results in z k D 12 x .bb/ .T C kT /x .bb/Ł .T C .k 1/T / Ł D 12 [Ae j . k C'a / C wk ] [Ae j . k1 C'a / C wk1 ] ½ Ł wk wk1 1 Ł D A |Ae{zj}k C e j . k C'a / wk1 C e j . k1 C'a / wk C 2 A } {z desired term |
(18.25)
disturbance
The desired p term is similar to that obtained in the coherent case, M-phases on a circle of radius A D E s . The variance of wk is equal to N0 , and ¦ I2 D N0 =2. However, even Ł =A, if w and w ignoring the term wk wk1 k k1 are statistically independent it results in E[je j .
k C'a /
Ł wk1 C e j .
k1 C'a /
wk j2 ] D 2¦ 2 D 2N0
(18.26)
There is an asymptotic penalty, that is for E s =N0 ! 1, of 3 dB with respect to the coherent receiver case. Indeed, for a 4-DPSK, a more accurate analysis demonstrates that the penalty is only 2.3 dB for higher values of E s =N0 (see Section 6.5.1).
18.2.3
FM discriminator with integrate and dump filter (LDI)
The scheme of Figure 18.8 makes use of a limiter discriminator (LD), as for frequency modulated (FM) signals, followed by an integrator filter over a symbol period, or integrate and dump (I&D) filter. Ideally, the discriminator output provides the instantaneous frequency deviation of x, i.e. 1 f .t/ D
d P x .t/ 1'x .t/ D 1' dt
(18.27)
Then, integrating (18.27) over a symbol period, we have Z
t0 CkT t0 C.k1/T
1 f .− / d− D 1'x .t0 C kT / 1'x .t0 C .k 1/T / C 2n³
(18.28)
1198
Chapter 18. Modulation techniques for wireless systems
Figure 18.8. FM discriminator and integrate & dump filter.
Figure 18.9. Implementation of a limiter-discriminator.
r (bb)(t) 1 (bb) x (bb)(t) d g 2 A dt
I m[.]
∆ f(t)
t 0 +kT I&D
modulo 2π
vk
* 1 2π |.|2 Figure 18.10. Baseband equivalent scheme of a FM discriminator followed by an integrate & dump filter.
that coincides with (18.11) taking mod 2³ . An implementation of the limiter-discriminator is given in Figure 18.9, while the baseband equivalent scheme is given in Figure 18.10, which employs the general relation Ł ð Im xP .bb/ .t/x .bb/Ł .t/ 1 f .t/ D (18.29) 2³ jx .bb/ .t/j2 In conclusion, all three schemes (baseband differential detector, differential detector and FM discriminator) yield the same output.
18.3 18.3.1
Variants of QPSK Basic schemes
QPSK From (18.1), we write the expression of the QPSK modulated signal C1 i h X s.t/ D Re e jk h T x .t kT /e j2³ f 0 t e j'0 kD1
(18.30)
18.3. Variants of QPSK
1199
where k is the phase associated with the transmitted symbol at instant kT , and '0 is the carrier phase. We denote as ak the generic symbol at instant kT ak D e jk
(18.31)
As k 2 fš³=4; š3³=4g, we get 1 (18.32) ak 2 p fš1; š jg 2 For a modulation interval T D 2Tb , the map between bits and symbols is reproduced in Figure 18.11, with balanced symbols, b` 2 f1; 1g. In particular, the following bit map is adopted, Re [ak ] D b2k1 2 f1; 1g odd bits
(18.33)
Im [ak ] D b2k 2 f1; 1g
(18.34)
even bits
Let I .t/ D
C1 X
b2k1 h T x .t kT /
(18.35)
b2k h T x .t kT /
(18.36)
kD1
Q.t/ D
C1 X kD1
then s is given by s.t/ D I .t/ cos.2³ f 0 t C '0 / Q.t/ sin.2³ f 0 t C '0 / " # C1 X j .2³ f 0 tC'0 / D Re ak h T x .t kT /e
(18.37)
kD1
Q -1 1
1 1 ak
I b2k -1 -1
b 2k-1 1 -1
(b 2k-1 b2k ) Figure 18.11. QPSK constellation with corresponding bit map and possible transitions of phase k at instants kT.
1200
Chapter 18. Modulation techniques for wireless systems
ϕ +θ 0
0
k
−π
b−1=1 b =1 0
b1=−1 b =−1
−π/2
b3=1 b =−1
2
0
4
b5=1 b =1 6
s(t)
A
0
−A
0
3T
2T t
T
4T
Figure 18.12. Realization of a QPSK signal with f0 D 2=T and '0 D ³=4.
From Figure 18.11 we note that at successive instants, the phase fk g can undergo variations even equal to ³ ; this implies large discontinuities in s with a consequent very high peak-power/average-power ratio of s if h T x is narrow band. In radio systems this can create saturation problems of the transmit amplifier (see Section 4.8). Figure 18.12 shows the possible behavior of s for a wideband modulation pulse h T x .t/ D A rect
t T =2 T
(18.38)
for which s.t/ D A cos.2³ f 0 t C '0 C k /
kT t < .k C 1/T
(18.39)
In the figure we note phase jumps in s that are equal to ³ .
Offset QPSK or staggered QPSK The in-phase signal is anticipated of Tb D T =2 with respect to the quadrature signal 1 I .t/ D T b2k1 h T x t k 2 kD1 C1 X
Q.t/ D
C1 X kD1
b2k h T x .t kT /
(18.40)
(18.41)
18.3. Variants of QPSK
1201
Q -1 1
1 1 al
I b2k -1 -1
b 2k-1 1 -1
(b 2k-1 b2k ) Figure 18.13. OQPSK constellation with corresponding bit map and possible transitions of phase ` at instants `Tb .
and as usual we have s.t/ D I .t/ cos.2³ f 0 t C '0 / Q.t/ sin.2³ f 0 t C '0 /
(18.42)
In other words, the signal phase is varied alternatively, over the two branches, every Tb seconds; therefore the phase variation is now at most ³=2, as illustrated in Figure 18.13. For h T x as given by (18.38), we now have s.t/ D A cos.2³ f 0 t C '0 C ` /
for `Tb t < .` C 1/Tb
(18.43)
For the same binary sequence of Figure 18.12, we show in Figure 18.14 the behavior of s given by (18.43). The reduced phase variation in s implies a greater compatibility with nonlinear amplifiers. On the other hand, an OQPSK system requires a coherent demodulator.
Differential QPSK (DQPSK) Recall that DQPSK was introduced in (6.157). The transmitted symbol at instant kT is given by ak D e j
k
(18.44)
where k
D
k1 C k
² ¦ 3³ ³ k 2 0; ; ³; 2 2
(18.45)
The information is mapped in the phase variation between two successive symbols. At the receiver a non-coherent differential demodulator (see Section 18.2) is commonly employed. If we denote as e jk the information at instant kT , for the transmitted symbol ak the recursive relation ak D ak1 e jk holds, from which we again obtain e jk by the equation Ł . e jk D ak ak1
1202
Chapter 18. Modulation techniques for wireless systems
ϕ0+θk
0
π/2
0 b−1=1
b0=1
−π b1=−1
−π/2 b2=−1
−π/2
b3=1
−π/2
b4=−1
0
b5=1
b6=1
s(t)
A
0
−A
−Tb
0
2Tb
Tb
3Tb t
4Tb
5Tb
6Tb
7Tb
Figure 18.14. Realization of an OQPSK signal with f0 D 2=T and '0 D ³=4.
Figure 18.15. Possible values of the phase k for a ³=4-DQPSK.
π/4-DQPSK The transmitted symbol is ak D e j
k
, with k
D
k1
C k
(18.46)
where now k 2 f³=4; 3³=4; 5³=4; 7³=4g. Possible values of phase k are given in Figure 18.15. Note that for k even, k assumes values in f0; ³=2; ³; 3³=2g, and for k odd k 2 f³=4; 3³ =4; 5³=4; 7³=4g. Phase variations between two consecutive instants are š³=4 and š3³=4, with a good peak/average power ratio of the modulated signal.
18.3. Variants of QPSK
18.3.2
1203
Implementations
QPSK, OQPSK, and DQPSK modulators Extending the scheme of Figure 6.34, we obtain the general modulator scheme for QPSK, OQPSK, and DQPSK illustrated in Figure 18.16. To implement each of these modulators it is sufficient to change the encoder and possibly add a delay of Tb on the Q branch for OQPSK [2]. Starting from the scheme of Figure 6.35, a coherent demodulator for QPSK and OQPSK is shown in Figure 18.17. In this scheme the two functions of carrier recovery (CR) and symbol timing recovery (STR) have been added, that determine the sampling instants t0 C k2Tb (+ possible Tb ) for data detection. For OQPSK, on the I branch the delay Tb matches the delay introduced on the Q branch of the modulator. Finally, with reference to the demodulator for DQPSK, the 1BDD non-coherent scheme of Figure 18.6 is usually employed.
π/4-DQPSK modulators Due to its wide use, we examine in detail modulation and demodulation schemes for ³=4DQPSK. The modulator is similar to that for QPSK of Figure 18.16, differing only in the bit map. Now, for ak D ak;I C jak;Q
(18.47)
from (18.46) a recursive equation for ak;I and ak;Q is given by ak;I D ak1;I cos k ak1;Q sin k
(18.48)
ak;Q D ak1;Q cos k C ak1;I sin k
(18.49)
Figure 18.16. Modulator for QPSK, OQPSK, and DQPSK.
1204
Chapter 18. Modulation techniques for wireless systems
Figure 18.17. Coherent demodulator for QPSK and OQPSK (CR = carrier recovery, STR = symbol timing recovery).
As k 2 f³=4; 3³=4; 5³=4; 7³=4g, then ¦ ² 1 C j 1 C j 1 j 1 j jk e 2 p ; p ; p ; p 2 2 2 2 and
8 > < ak;I ; ak;Q 2 f0; š1g ¦ ² 1 > : ak;I ; ak;Q 2 š p 2
(18.50)
k even k odd
(18.51)
To demodulate a ³=4-DQPSK signal we can use one of the three differential schemes described in Section 18.2 or the coherent scheme illustrated in Figure 18.17. In any case, for an ideal AWGN channel performance of a ³=4-DQPSK is the same as for QPSK using a coherent receiver (see (6.153)) with differential encoding or a differential receiver (see (6.164)). For a 1BDD, for an ideal channel and using as transmit filter h T x and as receive filter g .bb/ a square root raised cosine pulse with roll-off ² D 0:3 (² D 1), Figure 18.18 A (Figure 18.19) illustrates the eye diagram at the decision point. We note that at the instant of maximum eye aperture the signal assumes the four possible values of e jk given by (18.50). Figures 18.20 and 18.21 illustrate eye diagrams for a LDI demodulator that are obtained using the same filters as in the previous case. As indicated by (18.28), at the instant of maximum eye aperture, the signal assumes the four possible values of k given by (18.46).
18.3. Variants of QPSK
1205
Figure 18.18. Eye diagram of the 1BDD for ³=4-DQPSK, for an ideal channel and square root raised cosine receive filter with ² D 0:3.
Figure 18.19. Eye diagram of the 1BDD for ³=4-DQPSK, for an ideal channel and square root raised cosine receive filter with ² D 1.
1206
Chapter 18. Modulation techniques for wireless systems
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1 −1
−0.8
−0.6
−0.4
−0.2
0 t/T
0.2
0.4
0.6
0.8
1
Figure 18.20. Eye diagram of the LDI demodulator for ³=4-DQPSK, for an ideal channel and square root raised cosine receive filter with ² D 0:3.
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1 −1
−0.8
−0.6
−0.4
−0.2
0 t/T
0.2
0.4
0.6
0.8
1
Figure 18.21. Eye diagram of the LDI demodulator for ³=4-DQPSK, for an ideal channel and square root raised cosine receive filter with ² D 1.
18.4. Frequency shift keying (FSK)
18.4
1207
Frequency shift keying (FSK)
The main advantage of FSK modulators consists in generating a signal having a constant envelope, therefore the distortion introduced by a HPA is usually negligible. However, they present two drawbacks: ž wider spectrum of the modulated signal as compared to amplitude and/or phase modulated systems, ž complexity of the optimum receiver in the presence of non-ideal channels. As already expressed in (6.72), a binary FSK modulator maps the information bits in frequency deviations (š f d ) around a carrier with frequency f 0 ; the possible transmitted waveforms are then given by s1 .t/ D A cos.2³. f 0 f d /t C '1 / s2 .t/ D A cos.2³. f 0 C f d /t C '2 /
kT < t < .k C 1/T
(18.52)
p where, if we denote by E s the average energy of an isolated pulse, A D 2E s =T . Figure 18.22 illustrates the generation of the above signals by two oscillators, with frequency f 1 D f 0 f d and f 2 D f 0 C f d , selected at instants kT by the variable ak 2 f1; 2g related to the information bits. A particular realization of s1 and s2 is shown in Figure 18.23a. The resultant signal is given by s.t/ D
C1 X
sak wT .t kT /
ak 2 f1; 2g
(18.53)
kD1
A realization of s is shown in Figure 18.23b.
18.4.1
Power spectrum of M-FSK
Following the derivation in [3], we consider the two cases of non-coherent and coherent FSK. Acos(2 π f2 t+ϕ 2 ) ~ ak ~
s(t)
Acos(2 π f1 t+ϕ 1 )
Figure 18.22. Generation of a binary (non-coherent) FSK signal by two oscillators.
1208
Chapter 18. Modulation techniques for wireless systems
Figure 18.23. Binary FSK waveforms and transmitted signal for a particular sequence fak g.
Power spectrum of non-coherent binary FSK In case the two oscillators maintain their original phases '1 and '2 , we can express the modulated signal as " # C1 X 1 C ak 1 ak j .2³ f 1 tC'1 / j .2³ f 2 tC'2 / wT .t kT / (18.54) Ae Ae s.t/ D Re C 2 2 kD1 where ak 2 f1; 1g represents the information symbol. The complex envelope of s, with respect to the carrier f 0 , is given by s .bb/ .t/ D
C1 X A j .2³ f d t'1 / [e C e j .2³ f d tC'2 / ] wT .t kT / 2 kD1
(18.55) C1 X A C [e j .2³ f d t'1 / e j .2³ f d tC'2 / ] ak wT .t kT / 2 kD1
We note that C1 X
wT .t kT / D 1
(18.56)
kD1
while the second term of (18.55) is a PAM signal; using the results of Example 7.1.1 on page 544 it is easy to derive the continuous and the discrete parts of the power spectrum of s .bb/ .
18.4. Frequency shift keying (FSK)
1209
If WT . f / D T sinc. f T /e j³ f T , we get A2 .c/ [jWT . f C f d /j2 C jWT . f f d /j2 PN s .bb/ . f / D 4T 2 .h/Re[e j .'2 '1 / WT . f C f d /WŁT . f f d /]] A2 [Ž. f C f d / C Ž. f f d /] PN s.d/ .bb/ . f / D 4 In (18.57), h D 2 f d T is the modulation index, and ² 1 h an integer .h/ D 0 otherwise
(18.57)
(18.58)
It can be seen that the power spectrum (18.57) contains different spurious components [3] and this is partially due to the fact that '1 6D '2 . Particular case. Applying the previous expressions to the binary case with '1 D '2 D 0, we have1 s1.bb/ .t/ D Ae j2³ f d t wT .t/ s2.bb/ .t/ D Ae j2³ f d t wT .t/
F
!
S1.bb/ . f / D AWT . f C f d / S2.bb/ . f / D AWT . f f d /
(18.59)
From (18.57), the continuous part becomes A2 [jWT . f C f d /j2 C jWT . f f d /j2 2 .h/ Re[WT . f C f d /WŁT . f f d /]] PN s.c/ .bb/ . f / D 4T (18.60) The discrete part is instead given by A2 [jWT . f C f d /j2 C jWT . f f d /j2 PN s.d/ .bb/ . f / D 4T 2 C1 X C 2Re[WT . f C f d /WŁT . f f d /]] Ž. f `=T /
(18.61)
`D1
PN s.c/ .bb/ . f / is illustrated in Figure 18.24 for three values of the modulation index; the amplitude in dB of the spectral lines of 12 PN .d/ .bb/ . f / is listed in Table 18.1. The behavior of
1 A2 T
A
s
Power spectrum of coherent M-FSK Signals in the various symbol intervals are assumed to be statistically independent; furthermore, they always have the same phase '1 D '2 D '; therefore, the generation of s can be rather different from the method of Figure 18.22. Let s .bb/ be the complex envelope of s, then the average power spectral density of s is given by (7.28). In general, let M be the number of possible transmitted waveforms, fsi .t/g, i D 1; : : : ; M, having complex 1
In any case, unless specific assumptions are made on f 1 and f 2 , the phase of the transmitted signal can undergo discontinuities at the instants kT .
1210
Chapter 18. Modulation techniques for wireless systems
Figure 18.24. Continuous part of the power spectral density of a non-coherent binary FSK for three values of the modulation index. Table 18.1 Amplitude in dB of several spectral lines of non-coherent binary FSK signals for three values of the modulation index.
h D 2 fd T fT
1.0
1.5
2.0
0 1 2 3 4 5 6 7
— 7:4 15:4 19:2 21:8 23:8 25:4 26:8
13:4 6:8 20:1 24:6 27:5 29:6 31:3 32:7
— 6:0 — — — — — —
envelope fsi.bb/ .t/g. As will be derived next, the continuous and discrete parts of the power spectrum are given by 8 þ þ2 9 M M þX þ = <X 1 1 þ þ .c/ .bb/ .bb/ jSi . f /j2 Si . f /þ PN s .bb/ . f / D (18.62) þ þ ; M T : i D1 M þ i D1
18.4. Frequency shift keying (FSK)
PN s.d/ .bb/ . f /
D
1 MT
1211
þ2 2 þþX C1 M þ X ` þ þ .bb/ S . f /þ Ž f þ þ i D1 i þ `D1 T
(18.63)
We derive now the expressions (18.62) and (18.63). s .bb/ is a cyclostationary process with autocorrelation rs .bb/ .t; t − / D E[s .bb/ .t/s .bb/Ł .t − /] 1 X
D
1 X
E[sa.bb/ .t k1 T /sa.bb/Ł .t − k2 T /] k k 1
k1 D1 k2 D1 1 X
D
1 X
E[sa.bb/ .t k1 T /sa.bb/Ł .t − k2 T /] k k 1
k1 D1 k D1 2 k1 6Dk2 1 X
C
2
(18.64)
2
E[sa.bb/ .t k1 T /sa.bb/Ł .t − k1 T /] k k 1
k1 D1
1
Assuming ak1 statistically independent of ak2 for k2 6D k1 , we have that 1 X
rs .bb/ .t; t − / D
"
1 X
k1 D1 k D1 2 k1 6Dk2
"
1 X
C
k1 D1
M M 1 X 1 X si.bb/ .t k T / s .bb/Ł .t − k2 T / 1 M i D1 1 M i D1 i2 1
#
2
M 1 X s .bb/ .t k1 T /si.bb/Ł .t − k1 T / M i D1 i
#
(18.65) The average autocorrelation is given by 1 rN s .bb/ .− / D T D
Z
T
0
C1 X mD1
rs .bb/ .t; t − / dt M M X 1 X r .bb/ .bb/ .− C mT / T M 2 i D1 i D1 si1 si2 1
(18.66)
2
M M M X 1 X 1 X r r .bb/ .bb/ .− / .bb/ .bb/ .− / C T M i D1 si si T M 2 i D1 i D1 si1 si2 1
2
.bb/ where r.bb/ and s2.bb/ , with Fourier transform s1 s2 is the cross-correlation between s1 S1.bb/ . f /S2.bb/Ł . f /. The second term of (18.66) is the term for m D 0 of the first summation. Of the three terms in (18.66) we identify the first as a periodic signal of period T ; it is this term that yields a discrete component in the power spectrum of s .bb/ . Taking the
1212
Chapter 18. Modulation techniques for wireless systems
Fourier transform of rN s .bb/ .− / and considering the property C1 X
x.− C mT / D repT x.− /
mD1
C1 ` 1 X ` Ž f ! X T `D1 T T F
(18.67)
where X . f / D F[x.− /], the result is given in (18.62) and (18.63).
18.4.2
FSK receivers and corresponding performance
We consider both coherent and non-coherent receiver types.
Coherent demodulator A coherent demodulator is used when the transmitter provides all M signals with a well defined phase (see Example 6.7.1 on page 486); at the receiver there must be a circuit for the recovery of the phase of the various carriers. In practice this coherent receiver is rarely used because of its implementation complexity. In any case, for orthogonal signals we can adopt the scheme of Figure 6.8, repeated in Figure 18.25 for the binary case. For this case ² D 0, and the bit error probability has already been derived in (6.71), s ! Es (18.68) Pbit D Q N0 From (6.76), we note that the signals are orthogonal if .2 f d /min D
1 2T
Figure 18.25. Coherent demodulator for orthogonal binary FSK.
(18.69)
18.4. Frequency shift keying (FSK)
1213
Non-coherent demodulator The transmitted waveforms can now have a different phase, and in any case unknown to the receiver (see Example 6.11.1 on page 509). For orthogonal signals, from the general scheme of Figure 6.59, we give in Figure 18.26 a possible non-coherent demodulator. In this case, for ² D 0, the bit error probability has already been derived in (6.341), Pbit D
Es 1 2N 0 e 2
(18.70)
From (6.306), we note that, in the non-coherent case, to have ² D 0 it must be .2 f d /min D
1 T
(18.71)
Therefore a non-coherent FSK system needs a double frequency deviation and has slightly lower performance (see Figure 6.62) as compared to coherent FSK, however, it does not need to acquire the carrier phase.
Limiter-discriminator FM demodulator A simple non-coherent receiver, with good performance, for f 1 T × 1, f 2 T × 1 and sufficiently high E s =N0 , is depicted in Figure 18.27. After a passband filter to partially eliminate noise, a classical FM demodulator is used to extract the instantaneous frequency deviation of r.t/, equal to š f d . Sampling at instants t0 C kT and using a threshold detector, we get the detected symbol aO k 2 f1; 1g. In general, for 2 f d T D 1, the performance of this scheme is very similar to that of a non-coherent orthogonal FSK.
Figure 18.26. Non-coherent demodulator for orthogonal binary FSK.
1214
Chapter 18. Modulation techniques for wireless systems
Figure 18.27. Limiter-discriminator FSK demodulator.
18.5
Minimum shift keying (MSK)
We recall the following characteristics of an ideal FSK signal. ž In order to avoid high frequency components in the power spectrum of s, the phase of an FSK signal should be continuous and not as represented in Figure 18.23. For signals as given by (18.52), with '1 D '2 , this implies that it must be f n T D In , n D 1; 2, In an integer. ž To minimize the error probability it must be sn .t/ ? sm .t/, m 6D n. ž To minimize the required bandwidth, the separation between the various frequencies must be minimum. For signals as given by (18.52), choosing jI1 I2 j D 1 we have f 2 D f 1 C 1=T . In this case s1 and s2 are as shown in Figure 18.28. Moreover, from (18.71) it is easy to prove that the signals are orthogonal. The result is that the frequency deviation is such that 2 f d D j f 2 f 1 j D 1=T , and the carrier frequency is f 0 D . f 1 C f 2 /=2 D f 1 C 1=.2T /. 1 0.8 0.6
s1(t), s2 (t)
0.4 0.2 0
−0.2 −0.4 −0.6 −0.8 −1 0
T/2 t
T
Figure 18.28. Waveforms as given by (18.52) with A D 1, f1 D 2=T, f2 D 3=T, and '1 D '2 D 0.
18.5. Minimum shift keying (MSK)
1215
In a digital implementation of an FSK system, the following observation is very useful. Independent of f 1 and f 2 , a method to obtain a phase continuous FSK signal given by s.t/ D A cos.'.t//
(18.72)
where '.t/ D 2³ f 0 t C ak 2³ f d t C k
kT t < .k C 1/T
ak 2 f1; 1g
(18.73)
is to employ a single oscillator, whose phase satisfies the constraint '..k C 1/T ./ / D '..k C 1/T .C/ /; thus it is sufficient that at the beginning of each symbol interval kC1 is set equal to kC1 D ..ak akC1 /2³ f d .k C 1/T C k / mod 2³
(18.74)
An alternative method is given by the scheme of Figure 18.29, in which the sequence fak g, with binary elements in f1; 1g, is filtered by g to produce a PAM signal x f .t/ D
C1 X
ak g.t kT /
ak 2 f1; 1g
(18.75)
The signal x f is input to a VCO, whose output is given by Z t s.t/ D A cos 2³ f 0 t C 2³ h x f .− / d−
(18.76)
kD1
1
In (18.76), f 0 is the carrier, h D 2 f d T is the modulation index and Z t x f .− / d− '.t/ D 2³ f 0 t C 2³ h 1
represents the phase of the modulated signal.
Figure 18.29. CPFSK modulator.
(18.77)
1216
Chapter 18. Modulation techniques for wireless systems
Choosing for g a rectangular pulse g.t/ D
1 wT .t/ 2T
(18.78)
we obtain a modulation scheme called continuous phase FSK (CPFSK). In turn, CPFSK is a particular case of the continuous phase modulation described in Appendix 18.A. We note that the area of g.t/ is equal to 0.5; in this case the information is in the instantaneous frequency of s, in fact f s .t/ D
1 d'.t/ hak D f 0 C hx f .t/ D f 0 C 2³ dt 2T
kT < t < .k C 1/T
(18.79)
The minimum shift keying (MSK) modulation is equivalent to CPFSK with modulation index h D 0:5, that is for f d D 1=.4T /. Summarizing: an MSK scheme is a binary FSK scheme (Tb D T ) in which, besides having f d D 1=.4T /, the modulated signal has continuous phase. Figure 18.30 illustrates a comparison between an FSK signal with f d D 1=.4T / as given by (18.52), and an MSK signal for a binary sequence fak g equal to f1; 1; 1; 1g. Note that in the MSK case, it is like having four waveforms s1 ; s1 ; s2 ; s2 , each pair related to ak D 1 and ak D 1, respectively. Signal selection within the pair is done in a way to guarantee the phase continuity of s.
Figure 18.30. Comparison between FSK and MSK signals.
18.5. Minimum shift keying (MSK)
18.5.1
1217
Power spectrum of continuous-phase FSK (CPFSK)
From the expressions given in [3], the behavior of the continuous part of the power spectral density is represented in Figure 18.31. Note that for higher values of h a peak will tend to emerge around f d D h=.2T /, showing the instantaneous frequency of waveforms. Comparing the spectra of Figure 18.31 with those of Figure 18.24 it is seen that phase continuity implies a lower bandwidth, at least if we use the definition of bandwidth based on a given percentage of signal power.
18.5.2
The MSK signal viewed from two perspectives
First, we consider the following aspect.
Phase of an MSK signal For an MSK signal, the pulse
Z
t
q.t/ D
g.− / d−
(18.80)
1
is illustrated in Figure 18.32. From (18.76) the phase deviation of the modulated signal is given by 1'.t/ D '.t/ 2³ f 0 t D ³
C1 X
ak q.t kTb /
(18.81)
kD1
Figure 18.31. Continuous part of the power spectral density of a CPFSK signal for five values of the modulation index.
1218
Chapter 18. Modulation techniques for wireless systems
q(t) 0.5
Tb
2Tb
3Tb
t
Figure 18.32. Pulse q.t/.
Let k1
then it follows ³ 1'.t/ D ak 2
D
t k C Tb
k1 ³ X ai 2 i D1
k1
kTb t < .k C 1/Tb
(18.82)
(18.83)
For t D .k C 1/Tb./ we get ³ C k1 D k 2 D 0, the distinct values assumed by k mod 2³ are only four: 8 ³ < ; ³ for k even 2 2 k D : 0; ³ for k odd 1'..k C 1/Tb./ / D ak
Assuming
1
(18.84)
(18.85)
From (18.84) we note how the information ak is also contained in the variation, in two successive instants, of the phase deviation 1'. The possible trajectories of 1' are shown in Figure 18.33. From (18.81), the complex envelope of the modulated signal is given by s .bb/ .t/ D Ae j1'.t/
(18.86)
where 1'.t/ is related to the message by (18.83). Correspondingly the modulated signal (18.76) becomes s.t/ D I .t/ cos.2³ f 0 t/ Q.t/ sin.2³ f 0 t/
(18.87)
where I .t/ D Re[s .bb/ .t/]
Q.t/ D Im[s .bb/ .t/]
(18.88)
The signal s .bb/ .t/ D I .t/ C j Q.t/
(18.89)
is represented in Figure 18.34. We note that the phase 1'.t/ continuously varies in time and assumes the values f0; ³=2; ³; 3³=2g at instants kTb .
18.5. Minimum shift keying (MSK)
2π
1219
∆ϕ (t)
3π/2 π π/2 0 6Tb 0 Tb 2Tb 3Tb 4Tb 5Tb a 0=1 a1 =-1 a 2=1 a 3=1 a 4=1 a5 =-1
t
Figure 18.33. Possible trajectories of the phase deviation 1'.t/.
Figure 18.34. MSK: values assumed by I.t/ C jQ.t/.
MSK as binary CPFSK From (18.73) for f d D 1=.4T / and T D Tb we obtain ak t C k per kTb t < .k C 1/Tb s.t/ D A cos 2³ f 0 C 4Tb
(18.90)
and in general s.t/ D A
ak t C k wTb .t kTb / cos 2³ f 0 C 4Tb kD1 C1 X
(18.91)
1220
Chapter 18. Modulation techniques for wireless systems
from which we can observe that an MSK signal is a binary FSK signal with frequencies f 1 D f 0 1=.4T / and f 2 D f 0 C 1=.4T /, where f 0 is the carrier frequency. In (18.91) the symbols of the sequence fak g, ak 2 f1; 1g, are information symbols, and k D .ak1 ak /2³
1 kTb C k1 mod 2³ 4Tb
³ D .ak1 ak / k C k1 mod 2³ 2 In particular we note that 8² k1 > > > < k1 š 2³ k D ² > k1 > > : k1 š ³
ak D ak1 ak 6D ak1 ak D ak1 ak 6D ak1
(18.92)
for k even (18.93) for k odd
hence k 2 f0; ³ g. From the comparison of (18.90) with (18.83) it is easy to derive the relation between k and k , k D
k1
kak
³ 2
(18.94)
MSK as OQPSK As ak 2 f1; 1g, it is easy to verify that ³t ³t D cos cos ak sin k D 0 2Tb 2Tb
³t ³t D ak sin sin ak 2Tb 2Tb
(18.95)
Moreover, from (18.91) we obtain ³t wTb .t kTb / cos ak C k 2Tb k X ³t wTb .t kTb / cos k Ð cos C0 DA 2Tb k
I .t/ D A
X
³t wTb .t kTb / sin ak C k Q.t/ D A 2Tb k X ³t wTb .t kTb / ak cos k Ð sin C0 DA 2Tb k X
(18.96)
As the signal s is continuous phase by construction, from (18.87) I and Q must be continuous functions; therefore ³t 1. cos k can change only for k odd, that is at instants .k 2Tb C Tb / in which cos 2T b vanishes;
18.5. Minimum shift keying (MSK)
1221
2. ak cos k can change only for k even, that is at instants .k 2Tb / in which sin vanishes.
³t 2Tb
Therefore, we note that the information symbols associated with I and Q components can change only every 2Tb and there is a lag Tb between the two branches. Defining the variable ( for k odd cos k ck D (18.97) ak cos k for k even from (18.93) the following relations hold: ck D ak cos k1 D ak ck1 ¦ ² cos k1 for ak D ak1 ck D cos k1 for ak 6D ak1 D ak ak1 cos k1 D ak ck1
for k even
(18.98)
for k odd
(18.99)
Indeed, the transformation that maps the bits of fak g into fck g corresponds to a differential encoder given by2 ck D ck1 ak
ak 2 f1; 1g
(18.100)
with ck 2 f1; 1g. Then the decoding rule is ak D ck ck1
(18.101)
From the previous observations, (18.96) becomes
C1 X
³t c2`1 w2Tb .t 2`Tb C Tb / cos I .t/ D A 2T b `D1
C1 X
³t c2` w2Tb .t 2`Tb / sin Q.t/ D A 2T b `D1
(18.102)
Another representation of (18.102) is obtained by recognizing the periodic behavior of the waveforms sin.³ t=2Tb / and cos.³ t=2Tb / D sin.³.t C Tb /=2Tb /, illustrated in Figure 18.35. If we window these waveforms to the intervals .0; 2Tb / and .Tb ; Tb /, respectively, in the other intervals it is sufficient to alternate the sign of the encoded symbol, and I .t/ D A
C1 X
.1/` c2`1 w2Tb .t 2`Tb C Tb / sin
`D1 C1 X
³.t `2Tb C Tb / 2Tb
³.t 2`Tb / Q.t/ D A .1/` c2` w2Tb .t 2`Tb / sin 2Tb `D1 2
It corresponds to the exclusive OR (6.166) if ak 2 f0; 1g.
(18.103)
1222
Chapter 18. Modulation techniques for wireless systems
1
cos(π t/2Tb )
0.5
0
−0.5
−1 −T
b
0
Tb
2T b t
3T
0
Tb
2T b t
3T b
b
4T
b
5T b
1
sin(π t/2Tb )
0.5
0
−0.5
−1
−T b
4T b
5T
b
Figure 18.35. Behavior of two sinusoidal waveforms.
Then it follows that an MSK scheme is an OQPSK scheme with modulation interval T D 2Tb and pulse t Tb ³t (18.104) h T x .t/ D A rect sin 2Tb 2Tb given in Figure 18.36, with Fourier transform HT x . f / D A
4Tb cos.2³ f Tb / j2³ f Tb e ³.1 .4 f Tb /2 /
(18.105)
We note that the transmitted symbols associated with the OQPSK interpretation are encoded with a suitable sign. The example in Table 18.2 shows how the sequence fak g is mapped into fck g and to the data on the I and Q branches. The modulated signals are shown in Figure 18.37; note the phase continuity of s.
Complex notation of an MSK signal A compact notation of (18.103) is given by s .bb/ .t/ D
C1 X kD1
where h T x is defined in (18.104).
j kC1 ck h T x .t kTb /
(18.106)
18.5. Minimum shift keying (MSK)
1223
Figure 18.36. Shape of the fundamental MSK pulse for A D 1.
Table 18.2 Example of a coded sequence associated with MSK transmission.
ak ck I data Q data
1 1 1
1 1
1 1 1
1 1
1
1 1 1
1
1 1 1
1 1 1
1 1 1
Furthermore, as for ak 2 f1; 1g it is e j .³=2/ak D jak , then from (18.100) we get j
kC1
k ³ X j ai 2 i D1
ck D exp
! D ej
k
(18.107)
using (18.82). Hence an alternative expression for (18.106) is given by s .bb/ .t/ D
C1 X
ej
k
h T x .t kTb /
(18.108)
kD1
We note the recursive structure of e j ej
k
k
,
D ej
k1
jak
(18.109)
Chapter 18. Modulation techniques for wireless systems
1
1
0.5
0.5 −Q(t)
I(t)
1224
0 −0.5 −1
0 −0.5
0
2
4
−1
6
0
2
t/T −Q(t)sin(2πf0 t)
0
I(t)cos(2πf t)
6
4
6
1
0.5 0
−0.5 −1
4 t/T
1
0.5 0
−0.5
0
2
4
−1
6
0
2
t/T
t/T
1
s(t)
0.5 0 −0.5 −1
0
2
4
6
t/T
Figure 18.37. MSK signal for the data sequence of Table 18.2. Using the modulated signals I and Q, s is formed by (18.87).
18.5.3
Implementations of an MSK scheme
Interpreting an MSK scheme as a CPFSK scheme, the modulator is as shown in Figure 18.29, with g.t/ D
1 wT .t/ 2Tb b
(18.110)
As a whole the modulator is represented in Figure 18.38. Interpreting instead an MSK scheme as an OQPSK scheme, we have the implementation of Figure 18.16 in which T D 2Tb , h T x .t/ is given by (18.104), and data are encoded and changed in sign according to (18.103). Considering its importance and widespread usage, this scheme is illustrated in Figure 18.39.
18.5.4
Performance of MSK demodulators
Among the different demodulation schemes listed in Figure 18.40, we show in Figure 18.41 a differential 1BDD non-coherent demodulator. This scheme is based on the fact that the phase deviation of an MSK signal can vary of š³=2 between two suitably instants spaced
18.5. Minimum shift keying (MSK)
1225
A cos[∆ϕ(t)]
Tb
1 w 2Tb Tb
xf (t)
π
t -
cos(2π f0 t)
∆ϕ(t)
~
8
ak
π/2 A sin[∆ϕ(t)]
-sin(2π f0 t)
Figure 18.38. MSK as CPFSK.
Figure 18.39. MSK as OQPSK.
serial demodulator
differential demodulator
Figure 18.40. MSK demodulator classification.
+ +
s(t)
1226
Chapter 18. Modulation techniques for wireless systems
Figure 18.41. Differential (1BDD) non-coherent MSK demodulator.
t 0+k2Tb
data detector
g Rc
1 −1
cos2 π f 0 t r(t)
CR
STR
Tb
delay
P/S +
a^ k
decoder
π /2 −sin2 πf 0 t 1
g Rc
−1
t 0 +k2Tb +Tb Figure 18.42. Coherent (OQPSK type) MSK demodulator.
of Tb (see (18.85)). In any case, the performance for an AWGN channel is that of a DBPSK scheme, but with half the phase variation; hence from (6.163), with E s that becomes E s =2, we get Pbit D
Es 1 2N 0 e 2
(18.111)
Note that this is also the performance of a non-coherent orthogonal binary FSK scheme (see (18.70)). In Figure 18.42 we illustrate a coherent (OQPSK type) demodulator that at alternate instants on the I branch and on the Q branch is of the BPSK type. In this case, from
18.5. Minimum shift keying (MSK)
1227
Figure 18.43. Comparison among various error probabilities. Table 18.3 Increment of 0 (in dB) for an MSK scheme with respect to a coherent BPSK demodulator for Pbit D 103 .
0
coherent BPSK
coherent MSK
non-coherent MSK
6.79
+1.1
+4.2
(6.151), the error probability for decisions on the symbols fck g is given by s ! 2E s Pbit;Ch D Q N0
(18.112)
As it is as if the bits fck g were differentially encoded, to obtain the bit error probability for decisions on the symbols fak g, we use (6.173): 2 3 4 C 8Pbit;Ch 4Pbit;Ch Pbit D 4Pbit;Ch 8Pbit;Ch
(18.113)
In Figure 18.43 we show error probability curves for various receiver types. From the graph we note that to obtain an error probability of 103 , going from a coherent system to a non-coherent one, it is necessary to increase the value of 0 D E s =N0 as indicated in Table 18.3.
MSK with differential precoding In the previously considered coherent scheme the transmitted symbols fak g are obtained from NRZ mapping of bits fbk g. If we now use a differential (pre)coding, where aQ k 2 f0; 1g,
1228
Chapter 18. Modulation techniques for wireless systems
aQ k D bk ý bk1 and ak D 1 2aQ k D 1 2.bk ý bk1 /, then from (18.100) with cQk 2 f0; 1g and ck D 1 2cQk , we get cQk D cQk1 ý aQ k D aQ 0 ý aQ 1 ý Ð Ð Ð ý aQ k
(18.114)
D b1 ý bk In other words, the symbol ck is directly related to the information bit bk ; the performance loss due to (18.113) is thus avoided and we obtain Pbit D Pbit;Ch
(18.115)
We emphasize that this differential (pre)coding scheme should be avoided if differential non-coherent receivers are employed, because one error in faO k g generates a long error sequence in fbOk D b aQ k ý bOk1 g.
18.5.5
Remarks on spectral containment
From the analogy with OQPSK, the power spectrum of the complex envelope s .bb/ of an MSK signal is directly obtained from (18.105), 8Tb .1 C cos 4³ f Tb / 1 N .c/ Ps .bb/ . f / D 2 A ³ 2 .1 16 f 2 Tb2 /2
(18.116)
Figure 18.44. Normalized power spectral density of the complex envelope of signals obtained by four modulation schemes.
18.6. Gaussian MSK (GMSK)
1229
Modulating both BPSK and QPSK signals with h T x given by a retangular pulse with duration equal to the symbol period, a comparison between the various power spectra is illustrated in Figure 18.44. We note that for limited bandwidth channels, it is convenient to choose h T x of the raised cosine or square root raised cosine type. However, in radio applications the choice of a rectangular pulse may be appropriate, as it generates a signal with a lower peak/average power ratio and therefore is more suitable to be amplified with a power amplifier that operates near saturation. Two observations on Figure 18.44 follow. ž For the same Tb the main lobe of QPSK extends up to 1=T D 0:5=Tb , whereas that of MSK extends up to 1=T D 0:75=Tb ; thus the lobe of MSK is 50% wider than that of QPSK, consequently requiring a larger bandwidth. ž At high frequencies the spectrum of MSK decays as 1= f 4 , whereas the spectrum of QPSK decays as 1= f 2 .
18.6
Gaussian MSK (GMSK)
18.6.1
GMSK via CPFSK
GMSK is a variation of MSK in which, to reduce the bandwidth of the modulated signal s, the PAM signal x f is filtered by a Gaussian filter. Consider the scheme illustrated in Figure 18.45, in which we have the following filters: ž interpolator filter g I .t/ D
1 wT .t/ 2T
(18.117)
ž shaping filter K 2 2 gG .t/ D p eK t =2 2³
with
2³ Bt K Dp ln.2/
.Bt is the 3 dB bandwidth) (18.118)
interpolator filter ak
gI
Gaussian filter
xI (t)
gG
xf (t)
FM VCO
Tb =T
ak
g
xf (t)
Figure 18.45. GMSK modulator.
s(t)
1230
Chapter 18. Modulation techniques for wireless systems
ž overall filter g.t/ D g I Ł gG .t/
(18.119)
ak 2 f1; 1g
(18.120)
Considering the signals we have: ž transmitted binary symbols
ž interpolated signal x I .t/ D
C1 X
C1 X
ak g I .t kT / D
kD1
kD1
ak
1 wT .t kT / 2T
(18.121)
ž PAM signal x f .t/ D
C1 X
ak g.t kT /
(18.122)
kD1
ž modulated signal Z s.t/ D A cos 2³ f 0 t C 2³ h
t
x f .− / d−
D A cos.2³ f 0 t C 1'.t//
(18.123)
1
where h is the modulation index, nominally equal to 0.5, and A D
p
2E s =T .
From the above expressions it is clearRthat the GMSK signal is a frequency modulated t signal with phase deviation 1'.t/ D ³ 1 x f .− / d− . An important parameter is the 3 dB bandwidth, Bt , of the Gaussian filter. However, a reduction in Bt , useful in making prefiltering more selective, corresponds to a broadening of the PAM pulse with a consequent increase in the intersymbol interference, as can be noted in the plots of Figure 18.46. Thus a trade-off between the two requirements is necessary. The product Bt T was chosen equal to 0.3 in the GSM and HIPERLAN standards, and equal to 0.5 in the DECT standard (see Appendix 17.A). The case Bt T D 1, i.e. without the Gaussian filter, corresponds to MSK. Analyzing g in the frequency domain we have g.t/ D g I Ł gG .t/
F
! G. f / D G I . f / Ð G G . f /
(18.124)
As GI. f / D
1 2
sinc. f T /e j³ f T
G G . f / D e2³
2 . f =K /2
(18.125)
18.6. Gaussian MSK (GMSK)
1231
0.6
0.5
T g(t)
0.4
0.3 B T=∞ t
Bt T = 0.5
0.2 Bt T = 0.3 Bt T = 0.1
0.1
0 −2
−1.5
−1
−0.5
0
0.5
1
1.5
2
2.5
3
t/T Figure 18.46. Overall pulse g.t/ D gI Ł gG .t/, with amplitude normalized to 1=T, for various values of the product Bt T.
it follows that G. f / D
1 2
sinc. f T /e2³
2 . f =K /2
e j³ f T
(18.126)
In Figure 18.47, the behavior of the phase deviation of a GMSK signal with Bt T D 0:3 is compared with the phase deviation of an MSK signal; note that in both cases the phase is continuous, but for GMSK we get a smoother curve, without discontinuities in the slope. Possible trajectories of the phase deviation for Bt T D 0:3 and Bt T D 1 are illustrated in Figure 18.48. Values of e j1'.t/ at the decision instants T C kT are illustrated in Figure 18.49 for a GMSK signal with Bt T D 0:3.
18.6.2
Power spectrum of GMSK
GMSK is a particular case of phase modulation which performs a non-linear transformation on the message, thus it is very difficult to evaluate analytically the power spectrum of a GMSK signal. Hence we resort to a discrete time spectral estimate, for example, the Welch method (see Section 1.11). The estimate is made with reference to the baseband equivalent of a GMSK signal, that is using the complex envelope of the modulated signal given by s .bb/ .t/ D Ae j1'.t/ . The result of the spectral estimate is illustrated in Figure 18.50. Note that the central lobe has an extension up to 0:75=T ; therefore the sampling frequency, FQ , to simulate the baseband equivalent scheme of Figure 18.45 may be chosen equal to FQ D 4=T or FQ D 8=T .
1232
Chapter 18. Modulation techniques for wireless systems
B T = 0.3 t
4
∆ ϕ (t)
2
0
−2
−4
2
4
6
8
10
12
14
16
10
12
14
16
Bt T = ∞
4
∆ ϕ (t)
2
0
−2
−4
2
4
6
8 t/T
Figure 18.47. Phase deviation 1' of a GMSK signal for Bt T D 0:3, compared with the phase deviation of an MSK signal.
1
0.8
0.6
0.4
∆ϕ (t) / π
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
0.2
0.4
0.6
0.8
1 t/T
1.2
1.4
1.6
1.8
2
Figure 18.48. Trajectories of the phase deviation of a GMSK signal for Bt T D 0:3 (solid line) and Bt T D 1 (dotted line).
18.6. Gaussian MSK (GMSK)
1233
1
0.8
0.6
0.4
Q
0.2
0
−0.2
−0.4
−0.6
−0.8
−1 −1
−0.8
−0.6
−0.4
−0.2
0 I
0.2
0.4
0.6
0.8
1
Figure 18.49. GMSK constellation for Bt T D 0:3.
10 B T=0.3 t
0
BtT=0.5 B T=1.0
−10
t
BtT=∞
PSD (dB)
−20
−30
−40
−50
−60
−70
−80
0
0.5
1
1.5
2 fT
2.5
3
3.5
4
Figure 18.50. Estimate of the power spectral density of a GMSK signal for various values of Bt T.
1234
18.6.3
Chapter 18. Modulation techniques for wireless systems
Implementation of a GMSK scheme
For the scheme of Figure 18.45 three possible configurations are given, depending on the position of the DAC.
Configuration I The first configuration, illustrated in Figure 18.51, is in the analog domain; in particular, an analog low pass Gaussian filter is employed. As shown in [1], it is possible to implement a good approximation of the Gaussian filter by resorting to simple devices, such as lattice LC filters. However, the weak point of this scheme is represented by the VCO, because in an open loop the voltage/frequency VCO characteristic is non-linear and as a consequence the modulation index can vary even by a factor 10 in the considered frequency range.
Configuration II The second configuration is represented in Figure 18.52. The digital filter g.nTQ / that approximates the analog filter g.t/ is designed by the window method [4, pag. 444]. For an oversampling factor of Q 0 D 8, letting TQ D T =Q 0 , we consider four filters that are obtained by windowing the pulse g.t/ to intervals .0; T /, .T =2; 3T =2/, .T; 2T /, and .3T =2; 5T =2/, respectively; the coefficients of the last filter are listed in Table 18.4, using the fact that g.nTQ / has even symmetry with respect to the peak at 4TQ D T =2. A comparison among the frequency responses of the four discrete-time filters and the continuous-time filter is illustrated in Figure 18.53. For a good approximation to the analog filter, the possible choices are limited to the two FIR filters with 23 and 31 coefficients, with support .T; 2T / and .3T =2; 5T =2/, respectively. From now on we will refer to the filter with 31 coefficients. We note from Figure 18.46 that, for Bt T ½ 0:3, most of the pulse energy is contained within the interval .T; 2T /; therefore the effect of interference does not extend over more than three symbol periods. With reference to Figure 18.52, the filter g is an interpolator filter
ak
xI (t)
DAC
gG
x f (t)
s(t)
VCO
T
Figure 18.51. GMSK modulator: configuration I.
ak T
g
xn TQ
DAC
s(t) VCO
Figure 18.52. GMSK modulator: configuration II.
18.6. Gaussian MSK (GMSK)
1235
Table 18.4 Digital filter coefficients obtained by windowing g.t/; TQ D T=8.
g.nTQ /
value
g.4TQ / g.5TQ / g.6TQ / g.7TQ / g.8TQ / g.9TQ / g.10TQ / g.11TQ / g.12TQ / g.13TQ / g.14TQ / g.15TQ / g.16TQ / g.17TQ / g.18TQ / g.19TQ /
0.37119 0.36177 0.33478 0.29381 0.24411 0.19158 0.14168 0.09850 0.06423 0.03921 0.02236 0.01189 0.00589 0.00271 0.00116 0.00046
Figure 18.53. Frequency responses of g.t/ and g.nTQ /, for TQ D T=8, and various lengths of the FIR filters.
1236
Chapter 18. Modulation techniques for wireless systems
from T to T =Q 0 , that can be efficiently implemented by using the polyphase representation of the impulse response (see Appendix 1.A). Then, recognizing that x k Q 0 C` D f .`/ .akC2 ; akC1 ; ak ; ak1 /, the input–output relation can be memorized in a RAM or lookup table, and let the input vector [akC2 ; akC1 ; ak ; ak1 ] address one of the possible 24 D 16 output values; this must be repeated for every phase ` 2 f0; 1; : : : ; Q 0 1g. Therefore, there are Q 0 RAMs, each with 16 memory locations.
Configuration III The weak point of the previous scheme is again represented by the analog VCO; thus it is convenient to partially implement in the digital domain also the frequency modulation stage. Real-valued scheme. The real-valued scheme is illustrated in Figure 18.54. The samples fu n g are given by u n D u.nTQ / D A cos.2³ f 1 nTQ C 1'.nTQ //
(18.127)
where f 1 is an intermediate frequency smaller than the sampling rate, f1 D
N1 1 N2 TQ
(18.128)
where N1 and N2 are relatively prime numbers, with N1 < N2 . Thus f 1 TQ D N1 =N2 , and N1 n C 1'n D A cos['n ] (18.129) u n D A cos 2³ N2 where 1'n D 1'.nTQ / D ³
Z
nTQ
n X
x.− / d− ' ³
1
TQ x.i TQ /
(18.130)
i D1
More simply, let xn D x f .nTQ / and X n D X n1 Cxn ; then it follows that 1'n D ³ TQ X n . Therefore in (18.129) 'n , with carrier frequency f 1 , becomes . f1 /
'n
D 2³
N1 N1 . f1 / n C ³ TQ X n D 'n1 C 2³ C ³ TQ x n N2 N2
.f /
(18.131)
that is the value 'n 1 is obtained by suitably scaling the accumulated values of xn . To .f / obtain u n , we map the value of 'n 1 into the memory address of a RAM which contains values of the cosine function (see Figure 18.55).
ak T
g
xn
digital
un
TQ
VCO
TQ
DAC
u(t)
RF stage
Figure 18.54. GMSK modulator: configuration III.
s(t)
18.6. Gaussian MSK (GMSK)
1237
Figure 18.55. Digital implementation of the VCO.
Obviously the size of the RAM depends on the accuracy with which u n and 'n are quantized.3 We note that u.nTQ / is a real-valued passband signal, with spectrum centered around the frequency f 1 ; the choice of f 1 is constrained by the bandwidth of the signal u.nTQ /, equal to about 1:5=T , and also by the sampling period, chosen in this example equal to T =8; then it must be
31 C f1 > 0 4T
31 4 C f1 < 4T T
(18.132)
or 3=.4T / < f 1 < 13=.4T /. A possible choice is f 1 D 1=.4TQ / assuming N1 D 1 and N2 D 4. With this choice we have a image spacing/signal bandwidth ratio equal to 4/3. Moreover, cos.'n / D cos.2.³=4/n C 1'n / becomes cos..³=2/n C 1'n /, which in turn is equal to š cos.1'n / for n even and š sin.1'n / for n odd. Therefore the scheme of Figure 18.54 can be further simplified. Complex-valued scheme. Instead of digitally shifting the signal to an intermediate frequency, it is possible to process it at baseband, thus simplifying the implementation of the DAC equalizer filter. Consider the scheme of Figure 18.56, where at the output of the exponential block we .bb/ .nTQ /, where sI.bb/ have the sampled signal s .bb/ .nTQ / D Ae j1'.nTQ / D sI.bb/ .nTQ /C jsQ
Figure 18.56. GMSK modulator with a complex-valued digital VCO.
3
To avoid quantization effects, the number of bits used to represent the accumulated values is usually much larger than the number of bits used to represent 'n . In practice 'n coincides with the most significant bits of the accumulated values.
1238
Chapter 18. Modulation techniques for wireless systems
.bb/ and sQ are the in phase and quadrature components. Then we obtain
sI.bb/ .nTQ / D A cos.1'n / .bb/
sQ .nTQ / D A sin.1'n / D A cos
³ 2
1'n
(18.133)
Once the two components have been interpolated by the two DACs, the signal s.t/ can be reconstructed as .bb/ s.t/ D sI.bb/ .t/ cos.2³ f 0 t/ sQ .t/ sin.2³ f 0 t/
(18.134)
With respect to the real-valued scheme, we still have a RAM which stores the values of the cosine function, but two DACs are now required.
18.6.4
Linear approximation of a GMSK signal
According to Laurent [5], if Bt T ½ 0:3, a GMSK signal can be approximated by a QAM signal given by s .bb/ .t/ D
C1 X
³
e j 2 6iD1 ai h T x .t kTb / k
kD1
(18.135) D
C1 X
j
kC1
ck h T x .t kTb /
ck D ck1 ak
kD1
where h T x .t/ is a suitable real-valued pulse that depends on the parameter Bt T and has a support equal to .L C 1/Tb , if L Tb is the support of g.t/. For example, we show in Figure 18.57 the plot of h T x for a GMSK signal with Bt T D 0:3. The linearization of s .bb/ , that leads to interpreting GMSK as a QAM extension of MSK with a different transmit pulse, is very useful for the design of the optimum receiver, which is the same as for QAM systems. Figure 18.58 illustrates the linear approximation of the GMSK model. As for MSK, also for GMSK it is useful to differentially (pre)code the data fak g if a coherent demodulator is employed.
Performance of GMSK demodulators We consider the performance of a GMSK system for an ideal AWGN channel, and compare it with the that of ³=4-DQPSK. Coherent demodulator. Assuming a coherent receiver, as illustrated in Figure 18.42, performance of the optimum receiver according to the MAP criterion, evaluated on the basis of the minimum distance of the received signals, is approximated by the following relation [6] p Pbit;Ch D Q. c0/ (18.136)
18.6. Gaussian MSK (GMSK)
1239
1
0.9
0.8
0.7
hTx(t)
0.6
0.5
0.4
0.3
0.2
0.1
0 −1
0
1
2 t/T
3
4
5
b
Figure 18.57. Pulse hTx for a GMSK signal with Bt T D 0:3.
j k+1 ak
c k = c k-1 ak
ck
c k j k+1
hT x
s
(bb)
(t)
Figure 18.58. Linear approximation of a GMSK signal. hTx is a suitable pulse which depends on the value of Bt T. Table 18.5 Values of coefficient c as a function of the modulation system.
Modulation system
c
MSK GMSK, Bt T D 0:5 GMSK, Bt T D 0:3 ³=4-DQPSK, 8²
2.0 1.93 1.78 1.0
where the coefficient c assumes the values given in Table 18.5 for four modulation systems. The plots of Pbit;Ch for the various cases are illustrated in Figure 18.59. As usual, if the data fak g are differentially (pre)coded, Pbit D Pbit;Ch holds, otherwise the relation (18.113) holds. From now on we assume that a differential precoder is employed in the presence of coherent demodulation.
1240
Chapter 18. Modulation techniques for wireless systems
0
10
MSK BtT=+∞ GMSK BtT=0.5 GMSK BtT=0.3
−1
10
π/4−DQPSK −2
Pbit,ch
10
−3
10
−4
10
−5
10
−6
10
0
2
4
6
8 Γ
10
12
14
(dB)
Figure 18.59. Pbit,Ch as a function of 0 for the four modulation systems of Table 18.5.
For the case Bt T D 0:3, and for an ideal AWGN channel, in Figure 18.60 we also give the performance obtained for a receive filter g Rc of the Gaussian type [7, 8], whose impulse response is given in (18.118), where the 3 dB bandwidth is now denoted by Br . Clearly the optimum value of Br T depends on the modulator type and in particular on Bt T . System performance is evaluated using a 4-state VA or a threshold detector (TD). The VA uses an estimated overall system impulse response obtained by the linear approximation of GMSK. The Gaussian receive filter is characterized by Br T D 0:3, chosen for best performance. We observe that the VA gains a fraction of dB as compared to the TD; furthermore, the performance is slightly better than the approximation (18.136). Non-coherent demodulator (1BDD). For a non-coherent receiver, as illustrated in Figure 18.41, without including the receive filter g A , we illustrate in Figure 18.61 the eye diagram at the decision point of a GMSK system, for three values of Bt T . We note that, for decreasing values of Bt T , the system exhibits an increasing level of ISI, and the eye tends to shut; including the receive filter g A with finite bandwidth this phenomenon is further emphasized. Simulation results obtained by considering a receive Gaussian filter g A , and a Gaussian baseband equivalent system, whose bandwidth is optimized for each different modulation, are shown in Figure 18.62. The sensitivity of the 1BDD to the parameter Bt T of GMSK is higher as compared to a coherent demodulator. With decreasing Bt T , we observe a considerable worsening of performance; in fact, to achieve a Pbit D 103 the GMSK with Bt T D 0:5 requires a signal-to-noise ratio 0 that is 2.5 dB higher with respect to MSK, whereas GMSK with
18.6. Gaussian MSK (GMSK)
1241
−1
10
TD VA
−2
Pbit
10
10
−3
−4
10
0
1
2
3
4
Γ (dB)
5
6
7
8
9
Figure 18.60. Pbit as a function of 0 for a coherently demodulated GMSK (Bt T D 0:3), for an ideal channel and Gaussian receive filter with Br T D 0:3. Two detectors are compared: 1) four-state Viterbi algorithm and 2) threshold detector.
Figure 18.61. Eye diagram at the decision point of the 1BDD for a GMSK system, for an ideal channel and without the filter gA : (a) Bt T D C1, (b) Bt T D 0:5, (c) Bt T D 0:3.
1242
Chapter 18. Modulation techniques for wireless systems
Figure 18.62. Pbit as a function of 0 obtained with the 1BDD for GMSK, for an ideal channel and Gaussian receive filter having a normalized bandwidth Br T.
Bt T D 0:3 requires an increment of 7.8 dB. In Figure 18.62 we also show for comparison purposes the performance of ³=4-DQPSK with a receive Gaussian filter. The performance of another widely used demodulator, LDI (see Section 18.2.3), is quite similar to that of 1BDD, showing a substantial equivalence between the two non-coherent demodulation techniques applied to GMSK and ³=4-DQPSK. Comparison. Always for an ideal AWGN channel, a comparison among the various modulators and demodulators is given in Table 18.6. As a first observation, we note that a coherent receiver with an optimized Gaussian receive filter provides the same performance, Table 18.6 Required values of 0, in dB, to achieve a Pbit D 103 for various modulation and demodulation schemes.
Modulation
Demodulation coherent coherent differential (MAP) (g A gauss. + TD) (g A gauss. + 1BDD)
³=4-DQPSK o QPSK (² D 0:3) MSK GMSK (Bt T D 0:5) GMSK (Bt T D 0:3)
9.8 6.8 6.9 7.3
9.8 6.8 6.9 7.3
(² D 0:3) (Br T D 0:25) (Br T D 0:3) (Br T D 0:3)
12.5 10.3 12.8 18.1
(Br T (Br T (Br T (Br T
D 0:5) D 0:5) D 0:625) D 0:625)
18.6. Gaussian MSK (GMSK)
1243
evaluated by the approximate relation (18.136), of the MAP criterion. Furthermore, we note that for GMSK with Bt T 0:5, because of strong ISI, a differential receiver undergoes a substantial penalty in terms of 0 to achieve a given Pbit with respect to a coherent receiver; this effect is mitigated by canceling ISI by suitable equalizers [9]. Another method is to detect the signal in the presence of ISI, in part due to the channel and in part to the differential receiver, by the Viterbi algorithm. Substantial improvements with respect to the simple threshold detector are obtained, as shown in [10]. In the previous comparison between amplitude modulation (³=4-DQPSK) and phase modulation (GMSK) schemes we did not take into account the non-linearity introduced by the power amplifier (see Section 4.8), which leads to: 1) signal distortion and 2) spectral spreading that creates interference in adjacent channels. Usually, the latter effect is dominant and is controlled using a HPA with a back-off that can be even of several dB. In some cases signal predistortion before the HPA allows a decrease of the OBO. Overall, the best system is the one that achieves, for the same Pbit , the smaller value of .0/d B C .OBO/d B
(18.137)
In other words (18.137), for the same Pbit , additive channel noise and transmit HPA, selects the system for which the transmitted signal has the lowest power. Obviously in (18.137) 0 depends on the OBO; at high frequencies, where the HPA usually introduces large levels of distortion, the OBO for a linear modulation scheme may be so large that a phase modulation scheme may be the best solution in terms of (18.137).
Performance of a GSM receiver in the presence of multipath We conclude this section giving the performance in terms of Pbit cdf (see Appendix 7.E) of a GMSK scheme with Bt T D 0:3 for transmission over channels in the presence of frequency selective Rayleigh fading (see Chapter 4). The receive filter is Gaussian with Br T D 0:3, implemented as a FIR filter with 24 T =8-spaced coefficients; the detector is a 32-state VA or a DFE with 13 coefficients of the T -spaced FF filter and 7 coefficients of the FB filter. For the DFE, a substantial performance improvement is observed by placing before the FF filter a matched filter (MF) that, to save computational complexity, operates at T ; the improvement is mainly due to a better acquisition of the timing phase. Table 18.7 Power delay profiles for the analyzed channels.
Coefficient Relative delay 0 1 2 3 4 5
0 T 2T 3T 4T 5T
Power delay profile EQ
HT
UA
RA
1/6 1/6 1/6 1/6 1/6 1/6
0.02 0.75 0.08 0.02 0.01 0.12
0.00566 0.01725 0.80256 0.15264 0.01668 0.00521
0.00086 0.99762 0.00109 0.00026 0.00011 0.00006
1244
Chapter 18. Modulation techniques for wireless systems
100 90
VA(32) MF+DFE(13,7)
80
bit
P cdf
70 60 50 VA(32)
40
MF+DFE(13,7)
VA(32): Γ=15 dB MF+DFE(13,7): Γ=15 dB VA(32): Γ=10 dB MF+DFE(13,7): Γ=10 dB
30 20 10
−3
10
−2
P
10
−1
bit
Figure 18.63. Comparison between Viterbi algorithm and DFE preceded by a MF for the multipath channel EQ6, in terms of BER cdf.
The channel model is also obtained by considering a T -spaced impulse response; in particular, the performance is evaluated for the four models given in Table 18.7: equal gain (EQ), hilly terrain (HT), urban area (UA), rural area (RA). The difference in performance between the two receivers is higher for the EQ channel; this is the case shown in Figure 18.63 for two values of the signal-to-noise ratio 0 at the receiver.
Bibliography [1] B. Razavi, RF microelectronics. Englewood Cliffs, NJ: Prentice-Hall, 1997. [2] K. Feher, Wireless digital communications. Upper Saddle River, NJ: Prentice-Hall, 1995. [3] S. Benedetto and E. Biglieri, Principles of digital transmission with wireless applications. New York: Kluwer Academic Publishers, 1999. [4] A. V. Oppenheim and R. W. Schafer, Discrete-time signal processing. Englewood Cliffs, NJ: Prentice-Hall, 1989. [5] P. Laurent, “Exact and approximate construction of digital phase modulations by superposition of amplitude modulated pulses (AMP)”, IEEE Trans. on Communications, vol. 34, pp. 150–160, Feb. 1986.
18. Bibliography
1245
[6] S. Ohmori, H. Wakana, and S. Kawase, Digital communications technologies. Boston, MA: Artech House, 1998. [7] K. Murota and K. H. Hirade, “GMSK modulation for digital mobile radio telephony”, IEEE Trans. on Communications, vol. 29, p. 1045, July 1991. [8] M. K. Simon and C. C. Wang, “Differential detection of Gaussian MSK in a mobile radio environment”, IEEE Trans. on Vehicular Technology, vol. 33, pp. 311–312, Nov. 1984. [9] N. Benvenuto, P. Bisaglia, A. Salloum, and L. Tomba, “Worst case equalizer for noncoherent HIPERLAN receivers”, IEEE Trans. on Communications, vol. 48, pp. 28–36, Jan. 2000. [10] N. Benvenuto, P. Bisaglia, and A. E. Jones, “Complex noncoherent receivers for GMSK signals”, IEEE Journal on Selected Areas in Communications, vol. 17, pp. 1876–1885, Nov. 1999.
1246
Chapter 18. Modulation techniques for wireless systems
Appendix 18.A
Continuous phase modulation (CPM)
A signal with constant envelope can be defined by its passband version as r 2E s s.t/ D cos.2³ f 0 t C 1'.t; a// T
(18.138)
where E s is the energy per symbol, T the symbol period, f 0 the carrier frequency and a denotes the symbol message fak g at the modulator input. For a continuous phase modulation scheme, the phase deviation 1'.t; a/ can be expressed as Z t 1'.t; a/ D 2³ h x f .− / d− (18.139) 1
with C1 X
x f .t/ D
ak g.t kT /
(18.140)
kD1
where g.t/ is called instantaneous frequency pulse. In general, the pulse g.t/ satisfies the following properties: for t < 0 and t > L T
limited duration
g.t/ D 0
symmetry
g.t/ D g.L T t/ Z
LT
normalization
g.− / d− D
0
(18.141) (18.142)
1 2
(18.143)
Alternative definition of CPM From (18.140) and (18.139) we can redefine the phase deviation as follows 1'.t; a/ D 2³ h
Z
t
k X
k X
ai g.− i T / d− D 2³ h
1 i D1
i D1
Z
ti T
ai
g.− / d−
1
(18.144)
kT t < .k C 1/T or 1'.t; a/ D 2³ h
k X
q.t i T /
kT t < .k C 1/T
(18.145)
i D1
with Z
t
q.t/ D 1
g.− / d−
(18.146)
18.A. Continuous phase modulation (CPM)
1247
PSK
1/2
T
t
2T
CPFSK
1/2
T
t
2T
BFSK
1/2
T
t
2T
Figure 18.64. Three examples of phase response pulses for CPM.
The pulse q.t/ is called phase response pulse and represents the most important part of the CPM signal because it indicates to what extent each information symbol contributes to the overall phase deviation. In Figure 18.64 the phase response pulse q.t/ is plotted for PSK, CPFSK, and BFSK. In general the maximum value of the slope of the pulse q.t/ is related to the width of the main lobe of the PSD of the modulated signal s.t/, and the number of continuous derivatives of q.t/ influences the shape of the secondary lobes. In general, information symbols fak g belong to an M-ary alphabet A, that for M even is given by fš1; š3; : : : ; š.M 1/g. The constant h is called modulation index and determines, together with the dimension of the alphabet, the maximum phase variation in a symbol period, equal to .M 1/h³ . By changing q.t/ (or g.t/), h, and M, we can generate several continuous phase modulation schemes. The modulation index h is always given by the ratio of two integers, h D `= p, because this implies that the phase deviation, evaluated modulo 2³ , assumes values in a finite alphabet. In fact, we can write 1'.t; a/ D 2³ h
k X
ai q.t i T / C
kT t < .k C 1/T
kL
(18.147)
i DkLC1
with "
kL
X ` kL ai D ³ p i D1
# (18.148) mod 2³
kL is called phase state; it represents the overall contribution given by the symbols : : : ; a1 ; a0 ; a1 ; : : : ; akL to the phase duration in the interval [kT; .k C 1/T /, and can only assume 2 p distinct values. The first term in (18.147) is called corrective state and, because it depends on L symbols akLC1 ; : : : ; ak , at a certain instant t it can only assume M L1 distinct values. The phase deviation is therefore characterized by a total number of values equal to 2 pM L1 . CPM schemes with L D 1, i.e. CPFSK, are called full response schemes and have a reduced complexity with a number of states equal to 2 p; schemes with L 6D 1 are instead
1248
Chapter 18. Modulation techniques for wireless systems
called partial response schemes. Because of the memory in the modulation process a partial response scheme allows a trade-off between the error probability at the receiver and the shaping of power spectra of the modulated signal. However, this advantage is obtained at the expense of a greater complexity of the receiver. For this reason the modulation index is usually a simple rational number as 1; 1=2; 1=4; 1=8.
Advantages of CPM The popularity of the continuous phase modulation technique derives first of all from the constant envelope property of the CPM signal; in fact, a signal with a constant envelope allows using very efficient power amplifiers. In the case of linear modulation techniques, as QAM or OFDM, it is necessary to compensate for the non-linearity of the amplifier by predistortion or to decrease the average power in order to work in linear conditions. Before the introduction of TCM, it was believed that source and/or channel coding would allow an improvement in performance only at the expense of a loss in transmission efficiency, and hence would require a larger bandwidth; CPM permits both good performance and highly efficient transmission. However, one of the drawbacks of the CPM is the implementation complexity of the optimum receiver, especially in the presence of dispersive channels.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Chapter 19
Design of high speed transmission systems over unshielded twisted pair cables
In this chapter we describe the design of two high speed data transmission systems over unshielded twisted pair cables [1, 2].
19.1
Design of a quaternary partial response class-IV system for data transmission at 125 Mbit/s
Figure 19.1 shows the block diagram of a transceiver for a quaternary partial response class-IV (QPR-IV), or quaternary modified duobinary (see Appendix 7.A), system for data transmission at 125 Mbit/s over unshielded twisted-pair cables [1]. In the transmitter, information bits are first scrambled and then input to a 2B1Q differential encoder that yields output symbols belonging to the quaternary alphabet A D f3; 1; C1; C3g (see Example 6.5.1 on page 479); differential encoding makes the transmission of information insensitive to the polarity of the received signals. Signal shaping into PR-IV form is accomplished by the cascade of the following elements: the D/A converter, the analog transmit filter (ATF), the cable, the analog receive filter (ARF) with automatic gain control (AGC), the A/D converter, and, in the digital domain, the fixed decorrelation filter (DCF) and the adaptive equalizer. After equalization, the sequence of transmitted quaternary symbols is detected by a Viterbi algorithm. As the Viterbi algorithm introduces a delay in the detection of the transmitted sequence, as an alternative it is possible to use a threshold detector that individually detects transmitted symbols with a negligible delay. If the mean-square error (MSE) is below a fixed value, threshold detection, or symbol-by-symbol detection (SSD), is selected as the corresponding symbol decisions are then sufficiently reliable. If, instead, the MSE exceeds the fixed value, the Viterbi algorithm is employed. Finally, differential 1Q2B decoding and descrambling are performed.
Analog filter design Figure 19.2 depicts the overall analog channel considered for the design of ATF and ARF. The impulse response is denoted by h.t; L ; u c /, where L is the cable length and u c is the control signal used for AGC. To determine ATF and ARF with low implementation
1250
Chapter 19. Design of high speed transmission systems
c 1995 IEEE.] Figure 19.1. Block diagram of a QPR-IV transceiver. [From [1],
Figure 19.2. Overall analog channel considered for the joint optimization of the analog c 1995 IEEE.] transmit (ATF) and receive (ARF) filters. [From [1],
complexity, the transfer functions of these filters are first expressed in terms of poles and zeros; the pole-zero configurations are then jointly optimized by simulated annealing as described in [3]. The cost function that is used for the optimization reflects two criteria: a) the mean-square error between the impulse response h.t; L ; u c / for a cable length equal to 50 m, and the ideal PR-IV response must be below a certain value and b) spectral components of the transmitted signal above 30 MHz should be well suppressed to achieve compliance with regulations on radiation limits. Investigations with various orders of the respective transfer functions have shown that a good approximation of an ideal PR-IV response is obtained with 5 poles and 3 zeros for the ATF, and 3 poles for the ARF.
Received signal and adaptive gain control The received signal at the input of the ADC is expressed as x.t/ D
C1 X kD1
ak h[t kT; L ; u c .L/] C
C1 X kD1
akN h N [t kT; L ; u c .L/] C w R .t/
(19.1)
19.1. Design of a quaternary partial response class-IV system
1251
where fak g and fakN g are the sequences of quaternary symbols generated by the local transmitter and remote transmitter, respectively, h N [t; L ; u c .L/] is the NEXT channel response and w R .t/ is additive Gaussian noise. The signal x.t/ is sampled by the ADC that operates synchronously with the DAC at the modulation rate of 1=T D 62:5 MBaud. The adjustment of the AGC circuit is computed digitally, using the sampled signal x k D x.kT /, such that the ADC output signal achieves a constant average statistical power M R , i.e., þ Z þ þ 1 T 2 þ 2 D E[x .t/] dt þþ D MR (19.2) E[x k ]þ u c Du c .L/ T 0 u c Du c .L/ This ensures that the received signal is converted with optimal precision independently of the cable length; moreover, a controlled level of the signal at the adaptive digital equalizer input is required for achieving optimal convergence properties.
Near-end cross-talk cancellation As illustrated in Chapter 16, near-end cross-talk (NEXT) cancellation is achieved by storing the transmit symbols akN in a delay line and computing an estimated NEXT signal uO kN , which is then subtracted from the received signal x k , that is xQk D x k uO kN D x k
NX N 1
N N ci;k aki
(19.3)
i D0 N g, i D 0; : : : ; N 1, are the coefficients of the adaptive NEXT canceller. Using where fci;k N the minimization of E[xQk2 ] as a criterion for updating the NEXT canceller coefficients (see Section 16.1), leads to the LMS algorithm N N N D ci;k C ¼ N xQk aki ci;kC1
0 i NN 1
(19.4)
where ¼ N is the adaptation gain. As discussed in Chapter 16, high-speed full duplex transmission over two separate wire pairs with NEXT cancellation and full-duplex transmission over a single pair with echo cancellation pose similar challenges. In the latter case, a hybrid is included to separate the two directions of transmission; a QPR-IV transceiver with NEXT cancellation can then be used also for full-duplex transmission over a single pair, as in this case the NEXT canceller acts as an echo canceller.
Decorrelation filter After NEXT cancellation, the signal is filtered by a decorrelation filter, which is used to improve the convergence properties of the adaptive digital equalizer by reducing the correlation between the samples of the sequence fxQk g. The filtering operation performed by the DCF represents an approximate inversion of the PR-IV frequency response. The DCF has frequency response 1=.1þ e j4³ f T /, with 0 < þ < 1, and provides at its output the signal z k D xQk C þ z k2
(19.5)
1252
Chapter 19. Design of high speed transmission systems
Adaptive equalizer The samples fz k g are stored in an elastic buffer, from which they are transferred into the equalizer delay line. Before describing this operation in more detail, we make some observations about the adaptive equalizer. As mentioned above, the received signal x.t/ is sampled in synchronism with the timing of the local transmitter. Due to the frequency offset between local and remote transmitter clocks, the phase of the remote transmitter clock will drift in time relative to the sampling phase. As the received signal is bandlimited to one half of the modulation rate, signal samples taken at the symbol rate are not affected by aliasing; hence, a fractionally-spaced equalizer is not required for a QPR-IV system. Furthermore, as the signal value x.t/ can be reconstructed from the T -spaced samples fz k g, an equalizer of sufficient length acts also as an interpolator. The adaptive equalizer output signal is given by yk D
NX E 1
E ci;k z ki
(19.6)
i D0 E g, i D 0; : : : ; N 1, denote the filter coefficients. where fci;k E
Compensation of the timing phase drift The effect of timing phase drift can be compensated by continuously adjusting the equalizer coefficients. As a result of these adjustments, for a positive frequency offset of CŽ=T Hz, i.e. for a frequency of the local transmitter clock larger than the frequency of the remote transmitter clock, the value of the i-th coefficient at time k is approximately equal to the value assumed by the .i C 1/-th coefficient 1=Ž modulation intervals earlier. In other words, the coefficients move to the left by one position relative to the equalizer delay line after 1=Ž modulation intervals; conversely, for a negative frequency offset of Ž=T Hz, the coefficients move to the right by one position relative to the equalizer delay line after 1=Ž modulation intervals. Hence, the center of gravity of the filter coefficients drifts. Proper operation of a finite-length equalizer requires that the coefficients be recentered; this is accomplished as follows: ž normally, for each equalizer output yk one new signal z k is transferred from the buffer into the equalizer delay line; ž periodically, after a given number of modulation intervals, the sums of the magnitudes of N 0 first and last coefficients are compared; if the first coefficients are too large, the coefficients are shifted by one position towards the end of the delay line (right shift) and two new signals z k and z kC1 are transferred from the buffer into the delay line; if the last coefficients are too large, the coefficients are shifted by one position towards the beginning of the delay line (left shift) and no new signal is retrieved from the buffer. To prevent buffer overflow or underflow, the rate of the remaining receiver operations is controlled so that the elastic buffer is kept half full on average, thus performing indirect timing recovery. The control algorithm used to adjust the VCO providing the timing signal for the remaining receiver operations is described in Chapter 14.
19.1. Design of a quaternary partial response class-IV system
1253
Adaptive equalizer coefficient adaptation Overall system complexity is reduced if self-training adaptive equalization is employed, as in this case a start-up procedure with the transmission of a known training sequence is not needed. We describe here the operation of the adaptive equalizer and omit the presentation of the self-training algorithm, discussed in Section 15.4. The MSE at the equalizer output is continuously monitored using seven-level tentative decisions. If the MSE is too large, self-training adaptive equalization is performed during a fixed time period TST . At the end of the self-training period, if the MSE is sufficiently small, equalizer operation is continued with the decision directed LMS algorithm E E D ci;k ¼ E z ki eOk ci;kC1
0 i NE 1
(19.7)
where eOk D yk .aO k aO k2 / is the error obtained using tentative decisions aO k on the transmitted quaternary symbols, and ¼ E is the adaptation gain.
Convergence behavior of the various algorithms We resort to computer simulations to study the convergence behavior of the adaptive digital NEXT canceller and of the adaptive equalizer. The length of the NEXT canceller is chosen equal to N N D 48 to ensure that, in the worst case signal attenuation, the power of the residual NEXT is with high probability more than 30 dB below the power of the signal from the remote transmitter. In Figure 19.3, the residual NEXT statistical power at the canceller output is plotted versus the number of iterations for a worst-case cable length of L D 100 m and an adaptation gain ¼ N D 218 . In the same figure the statistical power in dB of the portion of the NEXT signal that cannot be cancelled due to finite NEXT-canceller length
c 1995 IEEE.] Figure 19.3. Convergence of the adaptive NEXT canceller. [From [1],
1254
Chapter 19. Design of high speed transmission systems
Figure 19.4. Convergence of the adaptive equalizer for (a) best-case sampling phase and (b) c 1995 IEEE.] worst-case sampling phase. [From [1],
19.1. Design of a quaternary partial response class-IV system
1255
Figure 19.5. Convergence of the adaptive equalizer for worst-case timing phase drift. [From c 1995 IEEE.] [1],
is also indicated. For the simulations, the NEXT canceller was assumed to be realized in the distributed-arithmetic form (see Section 16.1). The convergence of the MSE at the output of the equalizer in the absence of timing phase drift is shown in Figure 19.4a and b for best and worst-case sampling phase, respectively, and a value 0T x D 2E T x =N0 of 43 dB, where E T x is the average energy per modulation interval of the transmitted signal. The NEXT canceller is assumed to have converged to the optimum setting. An equalizer length of N E D 24 is chosen, which guarantees that the mean-square interpolation error with respect to an ideal QPR-IV signal is less than 25 dB for the worst-case sampling phase. The self-training period is TST ³ 400 µs, corresponding to approximately 25000T . The adaptation gains for self-training and decision directed adjustment have the same value ¼ E D 29 , that is chosen for best performance in the presence of a worst-case timing phase drift Ž D 104 . Figure 19.5 shows the meansquare error convergence curves obtained for Ž D 104 .
19.1.1
VLSI implementation
Adaptive digital NEXT canceller As shown in Chapter 16, for the VLSI implementation of the adaptive digital NEXT canceller, a distributed-arithmetic filter presents significant advantages over a transversal filter in terms of implementation complexity. In a NEXT canceller distributed-arithmetic filter, the partial products that appear in the expression of a transversal filter output are not individually computed; evaluation of partial products is replaced by table look-up and shift-and-add operations of binary words. To compute the estimate of the NEXT signal to be subtracted
1256
Chapter 19. Design of high speed transmission systems
from the received signal, look-up values are selected by the bits in the NEXT canceller delay line and added by a carry-save adder. By segmenting the delay line of the NEXT canceller into sections of shorter lengths, a trade-off concerning the number of operations per modulation interval and the number of memory locations that are needed to store the look-up values is possible. The convergence of the look-up values to the optimum setting is achieved by an LMS algorithm. If the delay line of the NEXT canceller is segmented into L sections with K D N N =L delay elements each, the NEXT canceller output signal is given by uO kN D
NX N 1
N N aki ci;k D
i D0
L1 K 1 X X `D0 mD0
N N ak`K m c`K Cm;k
(19.8)
In a distributed-arithmetic implementation, the quaternary symbol akN is represented by .0/ .1/ the binary vector [ak ; ak ], that is akN D
1 X wD0
.2ak.w/ 1/2w D
1 X wD0
bk.w/ 2w
(19.9)
where bk.w/ D .2ak.w/ 1/ 2 f1; C1g. Introducing (19.9) into (19.8) we obtain (see (16.12)) " # L1 X K 1 1 X X .w/ N uO kN D 2w bk`K m c`K (19.10) Cm;k `D0 wD0
mD0
Equation (19.10) suggests that the filter output can be computed using a set of L2 K look-up values that are stored in L look-up tables with 2 K memory locations each. Extracting the .w/ out of the square bracket in (19.10), to determine the output of a distributedterm bk`K arithmetic filter with reduced memory size L2 K 1 we rewrite (19.10) as (see (16.13)) uO kN D
L1 X 1 X `D0 wD0
.w/ .w/ 2w bk`K dkN .i k;` ; `/
(19.11)
.w/ where fdkN .n; `/g, n D 0; : : : ; 2 K 1 1, ` D 0; : : : ; L 1, are the look-up values, and i k;` denotes the selected look-up address that is computed as follows: 8 K 1 X > > .w/ .w/ > a 2m1 if ak`K D 1 > > < mD1 k`K m .w/ i k;` (19.12) D > K 1 X > > .w/ .w/ m1 > aN k`K if ak`K D0 > : m 2 mD1
where aN n is the one’s complement of an . The expression of the LMS algorithm to update the look-up values of a distributed-arithmetic NEXT canceller takes the form N dkC1 .n; `/ D dkN .n; `/ C ¼ N xQk
1 X wD0
n D 0; : : : ; 2 K 1 1
.w/ 2w bk`K Žni .w/ k;`
` D 0; : : : ; L 1
(19.13)
19.1. Design of a quaternary partial response class-IV system
1257
where Žn is the Kronecker delta. We note that at each iteration only those look-up values that are selected to generate the filter output are updated. The implementation of the NEXT canceller is further simplified by updating at each iteration only the look-up values that are addressed by the most significant bits of the symbols, i.e. those with index w D 1, stored in the delay line (see (16.23)). The block diagram of an adaptive distributed-arithmetic NEXT canceller is shown in Figure 19.6. In the QPR-IV transceiver, for the implementation of a NEXT canceller with a time span of 48T , L D 16 segments with K D 3 delay elements each are employed. The look-up values are stored in 16 tables with four 16-bit registers each.
(0)
a k−(L−1)K
address computation (1) i k,0
(0)
(1) i k,L−1
N (1) (i k,0 ,0) d k+1
0
L−1 +
(1) ,0) d kN (i k,0
(0) ,0) d kN (i k,0
+1 −1
bk(1)
1 0
← ←
1 0
table
← ←
1 0
+1 −1
bk(0)
1 0
k
+
(1) ,L−1) d kN (i k,L−1
(0) ,L−1) d kN (i k,L−1
(1)
+1 −1
b k−(L−1)K
+1 −1
b k−(L−1)K
2
~x
(0)
i k,L−1
µ x~k
table
+
address computation
ik,0
µ~ xk
+
(1)
a k−(L−1)K
← ←
ak(1)
← ←
ak(0)
(0)
2
u^kN
xk
+ c 1995 IEEE.] Figure 19.6. Adaptive distributed-arithmetic NEXT canceller. [From [1],
1258
Chapter 19. Design of high speed transmission systems
Adaptive digital equalizer As discussed in the previous section, the effect of the drift in time of the phase of the remote transmitter clock relative to the sampling phase is compensated by continuously updating and occasionally recentering the coefficients of the digital equalizer. The need for fast equalizer adaptation excludes a distributed-arithmetic approach for the implementation of the digital equalizer. An efficient solution for occasional recentering of the N E equalizer coefficients is obtained by the following structure, which is based on an approach where N E multiply-accumulate (MAC) units are employed. According to (19.6), assuming that the coefficients are not time varying, at a given time instant k each MAC unit is presented with a different tap coefficient and carries out the multiplication of this tap coefficient with the signal sample z k . The result is a partial product that is added to a value stored in the MAC unit, which represents the sum of partial products up until time instant k. The MAC unit that has accumulated N E partial P E 1 E ci z ki , and products provides the equalizer output signal at time instant k, yk D iND0 its memory is cleared to allow for the accumulation of the next N E partial products. At this time instant, the MAC unit that has accumulated the result of .N E 1/ partial products P E 1 E ci z k.i 1/ . At time instant .k C 1/ this unit computes the term has stored the term iND1 E c0 z kC1 and provides the next equalizer output signal ykC1 D c0E z kC1 C
NX E 1
ciE z k.i 1/
(19.14)
i D1
This MAC unit is then reset and its output will be considered again N E time instants later. Figure 19.7 depicts the implementation of the digital equalizer. The N E coefficients fciE g, i D 0; : : : ; N E 1, normally circulate in the delay line shown at the top of the figure. Except when recentering of the equalizer coefficients is needed, N E coefficients in the delay line are presented each to a different MAC unit, and the signal sample z k is input to all MAC units. At the next time instant, the coefficients are cyclically shifted by one position and the new signal sample z kC1 is input to all the units. The multiplexer shown at the bottom of the figure selects in turn the MAC unit that provides the equalizer output signal. To explain the operations for recentering of the equalizer coefficients, we consider as an example a simple equalizer with N E D 4 coefficients; in Figure 19.8, where for simplicity the coefficients are denoted by fc0 ; c1 ; c2 ; c3 g, the coefficients and signal samples at the input of the 4 MAC units are given as a function of the time instant. At time instant k, the output of the MAC unit 0 is selected, at time instant k C 1 the output of the MAC unit 1 is selected, and so on. For a negative frequency offset between the local and remote transmitter clocks, a recentering operation corresponding to a left shift of the equalizer coefficients occasionally occurs as illustrated in the upper part of Figure 19.8. We note that as a result of this operation, a new coefficient c4 , initially set equal to zero, is introduced. We also note that signal samples with proper delay need to be input to the MAC units. A similar operation occurs for a right shift of the equalizer coefficients, as illustrated in the lower part of the figure; in this case a new coefficient c1 , initially set equal to zero, is introduced. In the equalizer implementation shown in Figure 19.7, the control operations to select the filter coefficients and the signal samples are implemented by the multiplexer
19.1. Design of a quaternary partial response class-IV system
10 11 01 00
c NE
c1E
c0E
E −1
+
+
1259
cE
cE
A
C
NE −2
NE −3
+ +
B
MUXC Nu coefficient updating terms
z k−1 zk
1 0 MUXS0
10 MUXS1
1 0
10 MUXS N
MUXS2
E −2
10 MUXS NE −1
z k(0) (1)
zk
MAC 0
MAC 1
MAC 2
MAC NE −2
MAC NE −1
MUX
y
k
Figure 19.7. Digital adaptive equalizer: coefficient circulation and updating, and computation c 1995 IEEE.] of output signal. [From [1],
MUXC at the input of the delay line and by the multiplexers MUXS(0), : : : , MUXS(N E 1) at the input of the MAC units, respectively. A left or right shift of the equalizer coefficients is completed in N E cycles. To perform a left shift, in the first cycle the multiplexer MUXC is controlled so that a new coefficient c NE E D 0 is inserted into the delay line. During the following .N E 1/ cycles, the input of the delay line is connected to point B. After inserting the coefficient c1E at the N E -th cycle, the input of the delay line is connected to point C and normal equalizer operations are restored. For a right shift, the multiplexer MUXC is controlled so that during the first N E 1 cycles the input of the delay line is E D 0 is inserted into the delay line at the connected to point A. A new coefficient c1 N E -th cycle and normal equalizer operations are thereafter restored. At the beginning of the equalizer operations, the equalizer coefficients are initialized by inserting the sequence f0; : : : ; 0; C1; 0; 1; 0; : : : ; 0g into the delay line. The adaptation of the equalizer coefficients in decision-directed mode is performed according to the LMS algorithm (19.7). However, to reduce implementation complexity, equalizer coefficients are not updated at every cycle; during normal equalizer operations, each coefficient is updated every N E =NU cycles by adding correction terms at NU equally spaced fixed positions in the delay line, as shown in Figure 19.9. The architecture adopted for the computation of the correction terms is similar to the architecture for the computation
1260
Chapter 19. Design of high speed transmission systems
MAC 0
MAC 1
MAC 2
MAC 3
c3
z k−3 z k−2
c0 c3
z k−3
c1
z k−3
z k−2
z k−2
z k−3 z k−2
c0 c3
z k−1 zk z k+1
c2 c1
z k−1 zk
c0 c3
z k−1 zk
c2 c1
z k+2 z k+3
z k−1 zk z k+1 z k+2 z k+3
c0 c3
c2 c1
z k+1 z k+2
c2 c1
c0 c4 c3
z k+4 z k+4
z k+3 z k+4
c0 c3
z k+1 z k+2 z k+3 z k+4
k k+1 k+2 k+3 k+4
z k+5 z
c2 c1
z k+5 z
k+5 k+6
c2 c1 c4 c3
z k+6 z k+7 z k+8 z k+9 z k+10 z k+11
c0 c4 c3
z k+7 z k+7 z k+8 z k+9 z k+10 z k+11
c2 c1
z k+12 z k+13
c2 c1
shift left
c2 c1 c4 c3 c2 c1
c3
k+6
z k+7 z k+8 z k+9 z k+10 z k+11
c2 c1 c0 c4 c3 c2 c1
z k+4 z k+5 z
k+5
z k+6 z k+7 z k+8
c4 c3
z k+9 z k+10
c2 c1
z k+11 z k+12
c2 c1 c0 c3 c2 c1 c0 c4 c3
z k−3 z k−2
c0 c3
z k−3
c1
z k−2
c0 c3
z k−1 zk z k+1
c2 c1
c2 c1
z k+2 z k+3
z k−1 zk z k+1 z k+2 z k+3
c0 c3
c0 c2
z k+4
c2 c1
shift right
z k+5 z
c0 c3
c1 c0 c−1 c2
z k+6 z k+7 z k+8 z k+9 z k+10 z k+11
c1 c0 z k+12 c−1 z k+13
c0 c3 c2 c1
c2 c1 c0 c3
z k+4 z k+5
c2 c1
z k+7 c1 z k+8 z k+9 c0 c−1 z k+10 z k+11 c
c0 c2
c1 z k+12 z k+13 c0 c−1 z k+14
c2
c0 c2
2
c1 c0 c−1
k+6
c2 c1 c4 c3
k+6
c2 c1
z k+12 z k+13 z k+14
z k−3 z k−2
c2 c1
z k−3 z k−2
z k−1 zk
c0 c3
z k−1 zk
z k+1 z k+2
c2 c1
z k+3 z k+4
c0 c3
z k+1 z k+2 z k+3
z k+5 z k+6 z k+8 z k+9 z k+10 z k+11
c2 c1
z k+12 c1 z k+13 z k+14 c0 c−1 z k+15
c0 c2
k−3 k−2 k−1
k+7 k+8 k+9 k+10 k+11 k+12 k+13 k+14 k+15
k−3 k−2 k−1 k k+1 k+2 k+3
z k+4
k+4
z k+5 z k+6 z k+7 z k+9 z k+10
k+5 k+6 k+7
c1 z k+11 c0 c−1 z k+12 c 2 z k+13 z k+14 c1 z k+15 c0 c−1 z k+16
k+8 k+9 k+10 k+11 k+12 k+13 k+14 k+15
Figure 19.8. Coefficients and signals at the input of the multiply-accumulate (MAC) units c 1995 IEEE.] during coefficient shifting. [From [1],
of the equalizer output signal. The gradient components ¼ E z ki eOk , i D 0; : : : ; N E 1, are accumulated in N E MAC units, as illustrated in the figure. The delay line stores the signal samples z k.0/ and z k.1/ . The inputs to each MAC unit are given by the error signal and a signal from the delay line. The multiplexers at the input of each register in the delay line allow selecting the appropriate inputs to the MAC units in connection with the recentering of the equalizer coefficients. At each cycle, the output of NU MAC units are selected and
19.1. Design of a quaternary partial response class-IV system
1261
z (0) k
z (1)
10 00 01
10 00 01
10 00 01
10 00 01
10 00 01
k
^e k
MAC
MAC
MAC
0
1
2
MAC NE −2
MAC NE −1
MUX
N U coefficient updating terms
Figure 19.9. Adaptive digital equalizer: computation of coefficient adaptation. [From [1], c 1995 IEEE.]
input to the NU adders in the delay line where the equalizer coefficients are circulating, as illustrated in Figure 19.7.
Timing control An elastic buffer is provided at the boundary between the transceiver sections that operate at the transmit and receive timings. The signal samples at the output of the DCF are obtained at a rate that is given by the transmit timing and stored at the same rate into the elastic buffer. Signal samples from the elastic buffer are read at the same rate that is given by the receive timing. The VCO that generates the receive timing signal is controlled in order to prevent buffer underflow or overflow. Let WPk and RPk denote the values of the two pointers that specify the write and read addresses, respectively, for the elastic buffer at the k-th cycle of the receiver clock. We consider a buffer with eight memory locations, so that WPk ; RPk 2 f0; 1; 2; : : : ; 7g. The write pointer is incremented by one unit at every cycle of the transmitter clock, while the read pointer is also incremented by one unit at every cycle of the receiver clock. The difference pointer, DPk D WPk RPk
.mod 8/
(19.15)
1262
Chapter 19. Design of high speed transmission systems
is used to generate a binary control signal 1k of the VCO must be increased or decreased: 8 C1 > > < 1k D 1 > > : 1k1
2 fš1g that indicates whether the frequency if DPk D 4; 5 if DPk D 2; 3
(19.16)
otherwise
The signal 1k is input to a digital loop filter which provides the control signal to adjust the VCO. If the loop filter comprises both a proportional and an integral term, with corresponding gains of ¼− and ¼1− , respectively, the resulting second-order phase-locked loop is described by (see Section 14.7) −kC1 D −k C ¼− 1k C 1−k 1−kC1 D 1−k C ¼1− 1k
(19.17)
where −k denotes the difference between the phases of the transmit and receive timing signals. With a proper setting of the gains ¼− and ¼1− , the algorithm (19.17) allows for correct initial frequency acquisition of the VCO and guarantees that the write and read pointers do not overrun each other during steady-state operations. For every time instant k, the two consecutive signal samples stored in the memory locations with the address values RPk and .RPk 1/ are read and transferred to the equalizer. These signal samples are denoted by z k and z k1 in Figure 19.10. When a recentering of the equalizer coefficients has to take place, for one cycle of the receiver clock the read pointer is either not incremented (left shift), or incremented by two units (right shift). These operations are illustrated in the figure, where the elastic buffer is represented as a circular memory. We note that by the combined effect of the timing control scheme and the recentering of the adaptive equalizer coefficients the frequency of the receive timing signal equals on average the modulation rate at the remote transceiver.
Viterbi detector For the efficient implementation of near MLSD of QPR-IV signals, we consider the reducedstate Viterbi detector of Example 8.12.1 on page 687. In other words, the signal samples at the output the of .1 D 2 / partial response channel are viewed as being generated by two interlaced .1 D 0 / dicode channels, where D 0 D D 2 corresponds to a delay of two modulation intervals, 2T . The received signal samples are hence deinterlaced into even and odd time-indexed sequences. The Viterbi algorithm using a 2-state trellis is performed independently for each sequence. This reduced-state Viterbi algorithm retains at any time instant k only the two states with the smallest and second smallest metrics and their survivor sequences, and propagates the difference between these metrics instead of two metrics. Because the minimum distance error events in the partialresponse trellis lead to quasi-catastrophic error propagation, a sufficiently long path memory depth is needed. A path memory depth of 64T has been found to be appropriate for this application.
19.2. Design of a dual duplex transmission system at 100 Mbit/s
1263
Normal operation
WPk
WPk+1 WP k+2 RPk+2 z k−1 RPk
z k+2
RPk+1
zk
z k+1
zk
z k+1
Shift left
WPk+1
WPk
WP k+2
z k−1
RPk
z k−1
RPk+2
RPk+1 zk
z k+1
zk
zk
Shift right
WPk+1
WPk
RPk+2 z k+3
WP k+2
RPk+1 z k−1
z k+2
z k+2
RPk zk
z k+1
Figure 19.10. Elastic buffer: control of the read pointer.
19.2
Design of a dual duplex transmission system at 100 Mbit/s
We now describe the 100BASE-T2 system for fast Ethernet mentioned in Section 17.1.2 [4, 2].
Dual duplex transmission The characteristics of the transmission channel used to develop the 100BASE-T2 standard are described in Section 4.4. Figure 4.23 indicates that near-end cross-talk (NEXT) represents the main disturbance for transmission at high data rates over UTP-3 cables. As illustrated in Figure 19.11, showing the principle of dual duplex transmission (see also Section 16.1), self NEXT is defined as NEXT from the transmitter output to the receiver input of the same transceiver, and can be cancelled by adaptive filters as discussed in Chapter 16. Alien NEXT is defined instead as NEXT from the transmitter output to the receiver input of another transceiver; this is generated in the case of simultaneous transmission over multiple links within one multi-pair cable, typically with 4 or 25 pairs. Suppression of alien NEXT
1264
Chapter 19. Design of high speed transmission systems
c 1997 IEEE.] Figure 19.11. Dual duplex transmission over two wire pairs. [From [2],
from other transmissions in multi-pair cables and far-end cross-talk (FEXT), although normally not very significant, requires specific structures (see, for example, Section 16.4). To achieve best performance for data transmission over UTP-3 cables, signal bandwidth must be confined to frequencies not exceeding 30 MHz. As shown in Chapter 17, this restriction is further mandated by the requirement to meet FCC and CENELEC class B limits on emitted radiation from communication systems. These limits are defined for frequencies above 30 MHz. Twisted pairs used in UTP-3 cables have fewer twists per unit of length and generally exhibit a lower degree of homogeneity than pairs in UTP-5 cables; therefore transmission over UTP-3 cables produces a higher level of radiation than over UTP-5 cables. Thus it is very difficult to comply with the class B limits if signals containing spectral components above 30 MHz are transmitted over UTP-3 cables. As illustrated in Figure 19.11, for 100BASE-T2 a dual duplex baseband transmission concept was adopted. Bidirectional 100 Mbit/s transmission over two pairs is accomplished by full duplex transmission of 50 Mbit/s streams over each of two wire pairs. The lower modulation rate and/or spectral modulation efficiency required per pair for achieving the 100 Mbit/s aggregate rate represents an obvious advantage over mono duplex transmission, where one pair would be used to transmit only in one direction and the other to transmit only in the reverse direction. Dual duplex transmission requires two transmitters and two receivers at each end of a link, as well as separation of the simultaneously transmitted and received signals on each wire pair. Sufficient separation cannot be accomplished by analog hybrid circuits only. In 100BASE-T2 transceivers it is necessary to suppress residual echoes returning from the hybrids and impedance discontinuities in the cable as well as self NEXT by adaptive digital echo and NEXT cancellation. Furthermore, by sending transmit signals with nearly 100% excess bandwidth, received 100BASE-T2 signals exhibit spectral redundancy that can be exploited to mitigate the effect of alien NEXT by adaptive digital equalization. It will be shown later in this chapter that, for digital NEXT cancellation and equalization as well as echo cancellation in the case of dual-duplex transmission, dual-duplex and mono-duplex schemes require a comparable number of multiply-add operations per second.
19.2. Design of a dual duplex transmission system at 100 Mbit/s
1265
The dual transmitters and receivers of a 100BASE-T2 transceiver will henceforth be referred to simply as transmitter and receiver. Signal transmission in 100BASE-T2 systems takes place in an uninterrupted fashion over both wire pairs in order to maintain timing synchronization and the settings of adaptive filters at all times. Quinary pulse-amplitude baseband modulation at the rate of 25 MBaud is employed for transmission over each wire pair. The transmitted quinary symbols are randomized by side-stream scrambling. The redundancy of the quinary symbol sets is needed to encode 4-bit data nibbles, to send between data packets an idle sequence that also conveys information about the status of the local receiver, and to insert special delimiters marking the beginning and end of data packets.
Physical layer control The diagram in Figure 19.12 shows in a simplified form the operational states defined for the 100BASE-T2 physical layer. Upon power-up or following a request to re-establish a link, an auto-negotiation process is executed during which two stations connected to a link segment advertise their transmission capabilities by a simple pulse-transmission technique. While auto-negotiation is in progress, the 100BASE-T2 transmitters remain silent. If the physical layers of both stations are capable of 100BASE-T2 operation, the auto-negotiation process further determines a master/slave relation between the two 100BASE-T2 transceivers: the master transceiver will employ an externally provided reference clock for its transmit and receive operations. The slave transceiver will extract timing from the received signal, and use this timing for its transmitter operations. This operation is usually referred to as loop timing. If signals were transmitted in opposite directions with independent clocks, signals
Auto-Negotiation Process TIMEOUT
100BASE-T2 available on both sides of the link
TRAINING STATE: blind receiver training, followed by decision-directed training; send idle loc_rcvr_status=NOT_OK
loc_rcvr_status=OK rem_rcvr_status=OK
loc_rcvr_status=OK rem_rcvr_status=NOT_OK
IDLE STATE: decision-directed receiver operation; send idle rem_rcvr_status=NOT_OK loc_rcvr_status=NOT_OK
rem_rcvr_status=OK
NORMAL STATE: decision-directed receiver operation; send idle or data
c 1997 IEEE.] Figure 19.12. State diagram of 100BASE-T2 physical layer control. [From [2],
1266
Chapter 19. Design of high speed transmission systems
received from the remote transmitter would generally shift in phase relative to the alsoreceived echo and self-NEXT signals, as discussed in the previous section. To cope with this effect some form of interpolation would be required, which can significantly increase the transceiver complexity. After auto-negotiation is completed, both 100BASE-T2 transceivers enter the TRAINING state. In this state a transceiver expects to receive an idle sequence and also sends an idle sequence, which indicates that its local receiver is not yet trained (loc_rcvr_status =NOT_OK). When proper local receiver operation has been achieved by blind training and then by further decision-directed training, a transition to the IDLE state occurs. In the IDLE state a transceiver sends an idle sequence expressing normal operation at its receiver (loc_rcvr_status = OK) and waits until the received idle sequence indicates correct operation of the remote receiver (rem_rcvr_status = OK). At this time a transceiver enters the NORMAL state, during which data nibbles or idle sequences are sent and received as demanded by the higher protocol layers. The remaining transitions shown in the state diagram of Figure 19.12 mainly define recovery functions. The medium independent interface (MII) between the 100BASE-T2 physical layer and higher protocol layers is the same as for the other 10/100 Mbit/s IEEE 802.3 physical layers. If the control line TX_EN is inactive, the transceiver sends an idle sequence. If TX_EN is asserted, 4-bit data nibbles TXD(3:0) are transferred from the MII to the transmitter at the transmit clock rate of 25 MHz. Similarly, reception of data results in transferring 4-bit data nibbles RXD(3:0) from the receiver to the MII at the receive clock of 25 MHz. Control line RX_DV is asserted to indicate valid data reception. Other control lines, such as CRS (carrier sense) and COL (collision), are required for CSMA/CD specific functions.
Coding and decoding The encoding and decoding rules for 100BASE-T2 are now described. During the k-th modulation interval, symbols akA and akB from the quinary set f2; 1; 0; C1; C2g are sent over pair A and pair B, respectively. The encoding functions are designed to meet the following objectives: ž the symbols 2; 1; 0; C1; C2 occur with probabilities 1/8, 1/4, 1/4, 1/4, 1/8, respectively; ž idle sequences and data sequences exhibit identical power spectral densities; ž reception of an idle sequence can rapidly be distinguished from reception of data; ž scrambler state, pair A and pair B assignment, and temporal alignment and polarities of signals received on these pairs can easily be recovered from a received idle sequence. At the core of idle sequence generation and side-stream scrambling is a binary maximumlength shift-register (MLSR) sequence f pk g (see Appendix 3.A) of period 233 1. One new bit of this sequence is produced at every modulation interval. The transmitters in the master and slave transceivers generate the sequence f pk g using feedback polynomials g M .x/ D 1 C x 13 C x 33 and g S .x/ D 1 C x 20 C x 33 , respectively. The encoding operations
19.2. Design of a dual duplex transmission system at 100 Mbit/s
1267
are otherwise identical for the master and slave transceivers. From delayed elements f pk g four derived bits are obtained at each modulation interval as follows: x k D pk3 ý pk8 yk D pk4 ý pk6
(19.18)
ak D pk1 ý pk5 bk D pk2 ý pk12
where ý denotes modulo 2 addition. The sequences fx k g, fyk g, fak g, and fbk g represent shifted versions of f pk g, that differ from f pk g and from each other only by large delays. When observed in a constrained time window, the five sequences appear as mutually uncorrelated sequences. Figures 19.13 and 19.14 illustrate the encoding process for the idle mode and data mode, respectively. Encoding is based in both cases on the generation of pairs of two-bit vectors .Xk ; Yk /, .Sak ; Sbk /, and .Tak ; Tbk /, and Gray-code mapping of .Tak ; Tbk / into symbol pairs .Dka ; Dkb /, where Dk 2 f2; 1; 0; C1g, D a; b. The generation of these quantities is determined by the sequences fx k g, fyk g and f pk g, the even/odd state of the time index k (equal to 2n or 2n C 1), and the local receiver status. Finally, pairs of transmit symbols .akA ; akB / are obtained by scrambling the signs of .Dka ; Dkb / with the sequences fak g and fbk g. In Figure 19.15, the symbol pairs transmitted in the idle and data modes are depicted as two-dimensional signal points. Master/Slave 1
S
S: 0 1
1
S
← ←
bk MLSR sequence ak generator
+1 −1
* x 2n+1 = x 2n+1 ( loc_rcvr_status =OK)
x 2n
X2n
* x 2n+1
X 2n+1
0 1
[1,0] [0.1]
0 1
[0.1] [1,0]
a
Xk
1
Sk = Tka M
a Dk
pk =0: a A k
{−2,0,+2}
akA
∋
xk
pk
n:odd
∋
n:even
{−1,+1}
pk =1: 0
Y2n+1
{−1,+1} {−2,0}
+1
M: [10] [11]
←
∋
M( Yk )
M
1
∋
M( X k )
Skb = Tkb
Yk
← ← ←
Y2n
0 [1,0] [1,1] 1 [0.1] [0,0] n:even even odd
−2
[01] [00]
b Dk
pk =0: a B k
{−1,+1}
akB
∋
y 2n
∋
yk
{−2,0,+2}
0 (Gray code mapping) −1
Figure 19.13. Signal encoding during idle mode.
pk =1:
1268
Chapter 19. Design of high speed transmission systems
Master/Slave
pk
ak
1
S
S: 0 1
1
S
← ←
bk
MLSR sequence generator
−1 +1
[TDXk (3),TDXk (2)]
Skb
Tkb
M
a Dk
akA
{−2,−1,0,+1,+2}
b Dk
akB
∋
Tka
∋
Ska
xk
{−2,−1,0,+1,+2}
(same as for idle mode)
yk
M
[TDXk (1),TDXk (0)]
Figure 19.14. Signal encoding during data mode. idle mode
2 1
a kA −2
−1
0
data mode
1−D symbol probabilities
a kB
1
a kB
1/8
2
1/4
1
a kA
1/4 −2
2
−1
0
−1
1/4
−1
−2
1/8
−2
2−D symbol probabilities:
1/8
1/16
1
2−D symbol probabilities:
2
1/16
1/32
1/64
Figure 19.15. Two-dimensional symbols sent during idle and data transmission.
We note that, in idle mode, if pk D 1 then symbols akA 2 Ax D f1; C1g and akB 2 A y D f2; 0; C2g are transmitted; if pk D 0 then akA 2 A y and akB 2 Ax are transmitted. This property enables a receiver to recover a local replica of f pk g from the two received quinary symbol sequences. The associations of the two sequences with pair A and pair B can be checked, and a possible temporal shift between these sequences can be corrected. Idle sequences have the
19.2. Design of a dual duplex transmission system at 100 Mbit/s
1269
further property that in every two-symbol interval with even and odd time indices k, two symbols š1, one symbol 0, and one symbol š2 occur. The signs depend on the receiver status of the transmitting transceiver and on the elements of the sequences fx k g, fyk g, fak g and fbk g. Once a receiver has recovered f pk g, these sequences are known, and correct signal polarities, the even/odd state of the time index k, and remote receiver status can be determined. In data mode, the two-bit vectors Sak and Sbk are employed to side-stream scramble the data nibble bits. Compared to the idle mode, the signs of the transmitted symbols are scrambled with opposite polarity. In the event that detection of the delimiter marking transitions between idle mode and data mode fails due to noise, a receiver can nevertheless rapidly distinguish an idle sequence from a data sequence by inspecting the signs of the two received š2 symbols. As mentioned above, during the transmission of idle sequences one symbol š2, i.e. with absolute value equal to 2, occurs in every two-symbol interval. The previous description does not yet explain the generation of delimiters. A start-ofstream delimiter (SSD) indicates a transition from idle-sequence transmission to sending packet data. Similarly, an end-of-stream delimiter (ESD) marks a transition from sending packet data to idle sequence transmission. These delimiters consist of two consecutive A B D š2; akC1 D 0/. The signs of symbols symbol pairs .akA D š2; akB D š2/ and .akC1 š2 in a SSD and in an ESD are selected opposite to the signs normally used in the idle mode and data mode, respectively. The choice of these delimiters allows detection of mode transitions with increased robustness against noise.
19.2.1
Signal processing functions
The principal signal processing functions performed in a 100BASE-T2 transmitter and receiver are illustrated in Figure 19.16. The digital-to-analog and analog-to-digital converters operate synchronously, although possibly at different multiples of 25 MBaud symbol rate. Timing recovery from the received signals, as required in slave transceivers, is not shown. This function can be achieved, for example, by exploiting the strongly cyclostationary nature of the received signals (see Chapter 14).
The 100BASE-T2 transmitter At the transmitter, pairs of quinary symbols .akAT ; akBT / are generated at the modulation rate of 1=T D 25 MBaud. The power spectral density of transmit signals s A .t/ and s B .t/ has to conform to the spectral template given in Figure 19.17. An excess bandwidth of about 100%, beyond the Nyquist frequency of 12.5 MHz is specified to allow for alien NEXT suppression in the receiver by adaptive fractionally spaced equalization, as explained below. The transmit signals are produced by digital pulse-shaping and interpolation filters (DFTs), conversion from digital to analog signals at a multiple rate of 25 Msamples/s, and analog transmit filters (ATFs). The PSD obtained with a particular implementation in which the digital-to-analog conversion occurs at a sampling rate of 100 Msamples/s (oversampling factor F0 D 4) is included in Figure 19.17.
1270
Chapter 19. Design of high speed transmission systems
Figure 19.16. Principal signal processing functions performed in a 100BASE-T2 transceiver. c 1997 IEEE.] [From [2]
The 100BASE-T2 receiver The receive signals r A .t/ and r B .t/ are bandlimited by analog receive filters (ARFs) to approximately 25 MHz, adjusted in amplitude by variable gain amplifiers (VGAs) and converted from analog-to-digital at a multiple rate of 25 Msamples/s. For the following discussion a sampling rate of 50 Msamples/s will be assumed (oversampling factor F0 D 2, or equivalently sampling at the rate 2=T ). The remaining receiver operations are performed digitally. Before detection of the pairs of quinary symbols .akA R ; akB R / transmitted by the remote transceiver, an adaptive decisionfeedback equalizer (DFE) structure together with adaptive echo and self NEXT cancellers is employed, as shown in Figure 19.16. The forward equalizer sections of the DFE operate on T =2-spaced input signals. The estimated echo and self NEXT signals are subtracted from the T -spaced equalizer output signals. When signal attenuation and disturbances increase with frequency, as is the case for 100BASE-T2 transmission, a DFE receiver provides noticeably higher noise immunity compared to that achieved by a receiver with linear forward equalization only. In Figure 19.16 additional feedback filters for DC restoration are not shown. In a complete receiver implementation, these filters are needed to compensate for a spectral null at DC introduced by linear transform coupling. This spectral notch may be broadened in a well-defined manner by the analog receive filters and compensated for by non-adaptive IIR filters.
19.2. Design of a dual duplex transmission system at 100 Mbit/s
1271
0
Power spectral density [dB]
-10 -20 -30 -40 -50 -60 -70
0
10
20 25 30
40
50 f [MHz]
60
70
80
90
100
Figure 19.17. Spectral template specified by the 100BASE-T2 standard for the power spectral density of transmit signals and achieved power spectral density for a particular transmitter implementation comprising a 5-tap digital transmit filter, 100 MHz D/A conversion, and a 3Ž c 1997 IEEE.] order Butterworth analog transmit filter. [From [2],
The use of forward equalizers with T =2-spaced coefficients serves two purposes. First, as illustrated in Section 8.4, equalization becomes essentially independent of the sampling phase. Second, when the received signals exhibit excess bandwidth, the superposition of spectral input-signal components at frequencies f and f 1=T , for 0 < f < 1=.2T /, in the T -sampled equalizer output signals, can mitigate the effects of synchronous interference and asynchronous disturbances, as shown in Appendix 19.A. Interference suppression achieved in this manner can be interpreted as a frequency diversity technique [5]. Inclusion of the optional cross-coupling feedforward and backward filters shown in Figure 19.16 significantly enhances the capability of suppressing alien NEXT. This corresponds to adding space diversity at the expense of higher implementation complexity. Mathematical explanations for the ability to suppress synchronous and asynchronous interference with the cross-coupled forward equalizer structure are given in the Appendix 19.A. This structure permits the complete suppression of the alien NEXT interferences stemming from another 100BASE-T2 transceiver operating in the same multi-pair cable at identical clock rate. Alternatively, the interference from a single asynchronous source, e.g. alien NEXT from 10BASE-T2 transmission over an adjacent pair, can also be eliminated. The 100BASE-T2 standard does not provide the transmission of specific training sequences. Hence, for initial receiver-filter adjustments, blind adaptation algorithms must be employed. When the mean-square errors at the symbol-decision points reach sufficiently low values, filter adaptation is continued in decision directed mode based on quinary symbol
1272
Chapter 19. Design of high speed transmission systems
decisions. The filter coefficients can henceforth be continuously updated by the LMS algorithm to track slow variations of channel and interference characteristics. The 100BASE-T2 Task Force adopted a symbol-error probability target value of 1010 that must not be exceeded under the worst-case channel attenuation and NEXT coupling conditions when two 100BASE-T2 links operate in a four-pair UTP-3 cable, that are illustrated in Figure 4.23. During the development of the standard, the performance of candidate 100BASE-T2 systems has been extensively investigated by computer simulation. For the scheme ultimately adopted, it was shown that by adopting time spans of 32T for the echo and self NEXT cancellers, 12T for the forward filters, and 10T for the feedback filters, the MSEs at the symbol-decision points remain consistently below a value corresponding to a symbol-error probability of 1012 .
Computational complexity of digital receive filters The digital receive filters account for most of the transceiver implementation cost. It is worthwhile comparing the filter complexities for a dual duplex and a mono duplex scheme. Intuitively, the dual-duplex scheme may appear to be more complex, because it requires two transceivers. We define the complexity of a finite-impulse response FIR as Filter complexity D time span ð input sampling rate ð output sampling rate D number of coefficients ð output sampling rate D number of multiply-and-adds per second
(19.19)
Note that the time span of an FIR filter is given in seconds by the product of the number of filter coefficients times the sampling period of the input signal. Transmission in a four-pair cable environment with suppression of alien NEXT from a similar transceiver is considered. Only the echo and self NEXT cancellers and forward equalizers will be compared. Updating of filter coefficients will be ignored. For dual duplex transmission, the modulation rate is 25 MBaud and signals are transmitted with about 100% excess bandwidth. Echo and self NEXT cancellation requires four FIR filters with time spans TC and input/output rates of 25 Msamples/s. For equalization and alien NEXT suppression, four forward FIR filters with time spans TE , an input rate of 50 Msamples/s and an output rate of 25 Msamples/s are needed. The modulation rate for mono duplex transmission is 1=T D 50 MBaud and signals are transmitted with no significant excess bandwidth. Hence, both schemes transmit within a comparable bandwidth ( 25 MHz). For an obvious receiver structure that does not allow alien NEXT suppression, one self NEXT canceller with time span TC and input/output rates of 50 Msamples/s, and one equalizer with time span TE and input/output rates of 50 Msamples/s will be needed. However, for a fair comparison, a mono duplex receiver must have the capability to suppress alien NEXT from another mono duplex transmission. This can be achieved by receiving signals not only from the receive pair but also in the reverse direction of the transmit pair, and combining this signal via a second equalizer with the output of the first equalizer. The additionally required equalizer exhibits the same complexity as the first equalizer. The filter complexities for the two schemes are summarized in Table 19.1. As the required time spans of the echo and self NEXT cancellers and the forward equalizers are similar for
19. Bibliography
1273
Table 19.1 Complexities of filtering for two transmission schemes.
dual duplex Echo and self NEXT cancellers 4 ð TC ð 25 ð 1012 Forward equalizers 4 ð TE ð 50 ð 25 ð 1012
mono duplex 1 ð TC ð 50 ð 50 ð 1012 2 ð TE ð 50 ð 50 ð 1012
the two schemes, it can be concluded that the two schemes have the same implementation complexity. The arguments can be extended to the feedback filters. Finally, we note that with the filter time spans considered in the preceding section (TC D 32T , TE D 12T and TFb D 10T ), in a 100BASE-T2 receiver on the order of 1010 multiply-and-add operations/s need to be executed.
Bibliography ¨ ¸ er, and G. Ungerboeck, “A quaternary partial response class-IV [1] G. Cherubini, S. Olc transceiver for 125 Mbit/s data transmission over unshielded twisted-pair cables: principles of operation and VLSI realization”, IEEE Journal on Selected Areas in Communications, vol. 13, pp. 1656–1669, Dec. 1995. ¨ ¸ er, G. Ungerboeck, J. Creigh, and S. K. Rao, “100BASE-T2: [2] G. Cherubini, S. Olc a new standard for 100 Mb/s ethernet transmission over voice-grade cables”, IEEE Communications Magazine, vol. 35, pp. 115–122, Nov. 1997. ¨ ¸ er, and G. Ungerboeck, “Adaptive analog equalization and receiver [3] G. Cherubini, S. Olc front-end control for multilevel partial-response transmission over metallic cables”, IEEE Trans. on Communications, vol. 44, pp. 675–685, June 1996. [4] “Supplement to carrier sense multiple access with collision detection (CSMA/CD) access method and physical layer specifications: physical layer specification for 100 Mb/s operation on two pairs of Category 3 or better balanced twisted pair cable (100BASET2, Clause 32)”, Standard IEEE 802.3y, IEEE, Mar. 1997. [5] B. R. Petersen and D. D. Falconer, “Minimum mean-square equalization in cyclostationary and stationary interference–Analysis and subscriber line calculations”, IEEE Journal on Selected Areas in Communications, vol. 9, pp. 931–940, Aug. 1991.
1274
Chapter 19. Design of high speed transmission systems
Interference suppression
Appendix 19.A
Figure 19.18 illustrates the interference situations considered here. Equalization by linear forward filters only is assumed. Reception of 100BASE-T2 signals is disturbed either by alien NEXT from another synchronous 100BASE-T2 transmitter or by cross-talk from a single asynchronous source. Only one of these disturbances may be present. The symbol sequences fakA R g and fakB R g denote the sequences transmitted by the remote 100BASEA0
B0
H Ac . f / C cA A . f / C
H Ac
H Bc . f / C Bc A . f / C
H Bc
T2 transceiver, whereas fak T g and fak T g denote the sequences transmitted by an adjacent synchronous 100BASE-T2 transmitter. The spectrum S. f / of the asynchronous source may be aperiodic or exhibit a period different from 1=T . The functions “H . f /” represent the spectral responses of the signal or cross-talk paths from the respective sources to the inputs of the forward equalizer filters with transfer functions C A A . f / and C B A . f /. Because of 2=T sampling rate, these functions exhibit 2=T -periodicity. All signals and filter coefficients are real-valued. It is therefore sufficient to consider only frequencies f and f 1=T , for 0 < f < 1=.2T /. We will concentrate on the signals arriving at decision point DPA; the analysis for signals at DPB is similar. Intersymbol-interference free reception of the symbol sequence fakA R g and the suppression of signal components stemming from fakB R g at DPA require
1 f T 1 f T
B’ (a) {a k T }
decimation 1 T DPA ~ {a A R } = k
*
CBA (f)
DPB
CAA
CB A
1 f T 1 f T
A’
HB’A (f)
nxt HA’A (f)
D1 (19.20)
D0
(b) S(f)
{a k T } nxt
2 T CAA (f)
xt
HSA (f) c
HA (f)
nxt
HB’B (f)
nxt
HA’B (f)
A
{a k R }
H xt (f) SB
c
HB (f)
B
{a k R }
Figure 19.18. Cross-talk disturbance by: (a) alien NEXT from another synchronous 100BASET2 transmitter, (b) an asynchronous single source, for example, a 10BASE-T transmitter. c 1997 IEEE.] [From [2],
19.A. Interference suppression
1275
To suppress alien NEXT from a 100BASE-T2 transmitter, two additional conditions must be met: ½ X ` ` ` ` nxt nxt H A0 A f CAA f C H A0 B f CB A f D0 T T T T `D0;1 (19.21) ½ X ` ` ` ` nxt nxt HB 0 A f CAA f C HB 0 B f CB A f D0 T T T T `D0;1 Alternatively, the additional conditions for the suppression of cross-talk caused by a single asynchronous source become HSxtA . f / C A A . f / C HSxtB . f / C B A . f / D 0 1 1 1 1 CAA f C HSxtB f CB A f D0 HSxtA f T T T T
(19.22)
Therefore in each case the interference is completely suppressed if for every frequency in the interval 0 < f < 1=.2T / the transfer function values C A A ( f ), C A A ( f .1=T /), C B A ( f ) and C B A . f .1=T // satisfy four linear equations. It will be highly unlikely that the cross-talk responses are such that the coefficient matrix of these equations becomes singular. Hence a solution will exist with high probability. In the absence of filter-length constraints, the T =2-spaced coefficients of these filters can be adjusted to achieve these transfer functions. For a practical implementation a trade-off between filter lengths and achieved interference suppression has to be made.
Algorithms for Communications Systems and Their Applications. Nevio Benvenuto and Giovanni Cherubini Copyright 2002 John Wiley & Sons, Ltd. ISBN: 0-470-84389-6
Index
Access methods, 523 Active device, 257 Adaptation gain, 168, 170, 177, 183–185, 189, 212, 645, 703, 1086, 1100, 1104, 1120, 1124, 1250 Adaptive differential pulse code modulation (ADPCM), 341, 393, 398, 456, 1183 Advanced Mobile Phone Service (AMPS), 1180 Algorithm Bahl–Cocke–Jelinek–Raviv (BCJR), 668 Benveniste–Goursat, 1091 Contour (CA), 1093, 1095, 1098, 1102, 1113 Delsarte–Genin, 110, 147 Fano, 917–918 forward backward (FBA), 670, 915, 930, 939, 941, 948 Godard, 1101 Jacobi, 158 Linde–Buzo–Gray (LBG), 424 least-mean-square (LMS), 173–179, 186–191, 205–216, 628, 633, 638, 648–649, 701, 722, 819, 1090, 1120, 1121, 1128, 1251, 1253, 1258, 1272 Lempel–Ziv, 434 Levinson–Durbin, 145 Lloyd, 422 Max, 369 Mueller–Muller, 1064 recursive least-squares (RLS), 197–203, 217, 645, 722, 1180
Sato, 1090, 1096, 1100, 1106 stochastic gradient (SGA), 1086, 1113 stop-and-go, 1092 Viterbi (VA), 663, 677, 682, 686, 691, 823, 915, 921–923, 968, 993, 1007, 1177, 1238, 1262 American National Standards Institute (ANSI), 1152 Analog-to-digital converter (ADC), 331, 338, 341, 385, 572–573, 1189–1191, 1249 Antenna array, 226 directional, 296, 299, 811, 1165 gain, 296 isotropic, 296 Autocorrelation average, 1212 matrix, 63–66, 132, 135, 149, 166, 420, 635, 719, 744 sequence, 50, 53, 86, 144, 401, 622, 655, 678 Automatic gain control (AGC), 1249 Automatic repeat query (ARQ), 807, 827 Autoregressive model (AR), 91–94, 96–101, 101, 115, 143, 197, 414 Autoregressive moving average model (ARMA), 90, 94 Bandwidth definition of, 29 excess, 764, 777, 1131, 1152, 1263, 1269
1278
Bandwidth (continued) minimum, 458, 552, 559, 578, 590, 609, 771, 808, 1003 Bit error probability, 340, 456, 475, 485, 494, 499, 578, 613, 899, 919, 941, 954, 1212, 1224 Bit loading, 1002, 1152 Blind equalization, 1083 Block codes Bose–Chaudhuri–Hocquenghem (BCH), 878–898 generator matrix, 836 generator polynomial, 864 Hamming, 872 low-density parity check (LDPC), 946–955 non binary, 960–965 parity-check matrix, 833 Reed–Solomon, 885 simplex, 875–877 syndrome decoding, 839 Calculus of variations, 731, 1000 Carrier sense multiple access (CSMA), 1164 Cancellation cross-talk, 1116, 1251 echo, 1116–1130, 1145, 1269 Cell radius, 1165, 1182 Cellular systems, 1171, 1182 Channel additive white Gaussian noise (AWGN), 326, 439, 458, 503–506, 543, 713, 999, binary symmetric (BSC), 456, 571 Channel model, 251, 296, 309, 313, 316, 322, 658, 811, 1116 Channel capacity, 503, 999 Cholesky decomposition, 155 Circular convolution, 21–23, 205, 242, 707 Code division multiple access (CDMA), 523, 795, 802, 810, 818, 1180 Code rate, 827, 828, 903, 1177, 1180, 1182, 1185
Index
Codes block, 827, 899, 946, 990 Bose–Chaudhuri–Hocquenghem (BCH), 878–898 channel, 827–965 concatenated, 921, 924 convolutional, 900–920 cyclic, 862–898 forward error correction (FEC), 827 linear, 830, 968 low-density parity check (LDPC), 946–955 Reed–Solomon, 885 turbo, 924–942 Walsh–Hadamard, 536 Coding adaptive predictive (APC), 341, 401 adaptive transform (ATC), 341, 433 code excited linear predictive (CELP), 341, 416, 433, 578 linear predictive (LPC), 341, 414 residual excited linear predictive (RELP), 341, 415, 1226 subband (SBC), 341, 456 vector sum excited liner predictive (VSELP), 457, 1177 Coding by modelling, 413 Coherence time, 311, 314 Combining equal gain (EGC), 720 maximal ratio (MRC), 720, 815 optimum (OC), 721 selective, 719 switched, 720 Convolution, 13 Convolutional codes, 900–920 catastrophic error propagation, 910 decoding algorithms, 912–918 general description, 903 transfer function, 907 Correlation coefficient partial (PARCOR), 146
Index
Correlogram, 86 Coset, 837–841, 871, 964–965, 977 Costas loop, 1040–1043 Criterion least squares (LS), 148, 197, 429 mean-square error (MSE), 166, 366 Cross-talk far-end (FEXT), 290, 1116 near-end (NEXT), 288–289, 1116, 1251, 1269 Cut-off rate, 509 Cyclostationary process, 56, 1057 Decimation, 106, 236, 380, 766, 1126 Decoding iterative, 929, 939 sequential, 958 Delay spread rms, 307, 1177 Delta modulation (DM), 343, 404 adaptive, 343 continuously variable slope (CVSDM), 411 linear, 407 Demodulation coherent, 499 non coherent, 487, 806, 1213 Despreading, 795, 801 Detection decision feedback sequence (DFSE), 695 maximum likelihood (MLSD), 662, 969, 1262 multiuser, 820, 823, 1131 reduced state sequence (RSSE), 691–696, 716 single-user, 818 threshold, 462, 474, 542, 555 Diagram state, 902, 909, 910 tree, 694, 901, 917, 974, 977 trellis, 664, 667, 677, 682, 903, 921, 968, 971, 981, 986, 990, 1014 Differential PCM (DPCM), 343, 385, 407
1279
Digital European Cordless Telephone (DECT), 523, 1145, 1162, 1182–1185 Digital signal processor (DSP), 709 Digital subscriber line (DSL), 285, 1131–1142, 1147 asymmetric (ADSL), 1148–1150 high bit rate (HDSL), 523, 1147–1148 single line high speed (SHDSL), 919, 1148 very high speed (VDSL), 285, 523, 1115, 1131–1142 Digital-to-analog converter (DAC), 113, 331, 339, 408, 413 572, 1234, 1251 Distance Euclidean, 7, 420, 499, 682, 915, 967 free Euclidean, 919, 967, 986 free Hamming, 907 Hamming, 830, 875, 960 Distortion envelope delay, 319 Diversity frequency, 521, 1167, 1271 polarization, 522 space, 522, 1271 time, 522 Division multiplexing frequency (FDM), 753, 1175 time (TDM), 524, 1175 wavelength (WDM), 292 Doppler frequency, 318 shift, 303 spectrum, 311, 313, 315 spread, 304, 313, 1167 Downlink, 802, 819, 1156, 1175, 1183 Duplex transmission dual, 1154, 1263 full, 522, 1117, 1145–1148, 1251, 1264 half, 522, 1146, 1154 mono, 1154, 1264, 1272
1280
Duplexing digital, 1149 frequency division (FDD), 523, 1146, 1175 time division (TDD), 522, 1184 Dynamic channel allocation (DCA), 1182 Dynamic channel selection (DCS), 1182 Echo cancellation, 225, 1116–1130 Envelope detector, 512 Equalization adaptive, 304, 645, 1018, 1083, 1252 fractionally spaced, 630, 642, 1269 self-training, 1083 Equalizer adaptive, 628, 1083, 1252 decision feedback (DFE), 595, 617, 635–645, 649–656, 695, 710, 741, 777, 823, 1010, 1131, 1270 decision feedback-zero forcing (DFE-ZF), 649, 657, 679, 695 fractionally spaced (FSE), 617, 630–634, 642, 699, 703, 1107, 1270 linear (LE), 594, 619–620, 627–628 linear zero forcing (LE-ZF), 619 passband, 697 Estimation error, 129, 151, 159, 165, 174, 200, 213, 241, 622, 741, 1064, 1105 Estimate biased, 83 unbiased, 82 Estimator early-late, 1055 feedback, 1053 phasor, 1066 timing, 1055 Ethernet, 290, 1152–1155, 1263 European Telecommunications Standards Institute (ETSI), 415, 435, 1137, 1148, 1163, 1185 Excess MSE, 854
Index
Extension field, 890, Eye diagram, 562 Fading channel flat, 305, 311, 718 frequency selective, 305, 311, 718 Rayleigh, 308, 518 Rice, 308 FCC, 1162, 1264 Fiber-to-the-curb architecture (FTTC), 1148 Fiber distributed data interface (FDDI), 1155 Filter allpass, 28 bank, 214, 433, 753–755, 757–773, 783, 818, 1152, 1190 Butterworth, 313 decimator, 110, 119–126, 628, 754–755 distributed arithmetic, 1121 finite impulse response (FIR), 25, 116, 129, 165, 399, 754, 1234, 1272 frequency response, 18, 543, 591, 760, 1028 highpass, 28 impulse response, 18, 542, 558, 754, 1005 infinite impulse response (IIR), 25, 90, 136, 317, 622 integrate and dump, 1197 interpolator, 112–126, 315, 754, 773, 1049, 1229 lattice, 146, 191, 204 loop, 1030, 1036, 1054, 1262 lowpass, 28 matched, 73, 462, 494, 567, 621, 698, 755, 799, 813, 1003, 1045, 1243 narrowband, 28 notch, 224 prediction error, 142, 386, 650, 679 transfer function, 16, 91 transversal, 165, 627, 1119–1120, 1255
Index
whitened matched (WMF), 651, 1003, 1007 whitening (WF), 94, 651, 1005 Wiener, 129–140, 165, 622 Filter bank critically sampled, 764–767, 769 non critically sampled, 764–769, 777 Finite field, 844 Finite state machine (FSM), 663, 751, 970 Flexible precoding, 1018–1025, 1146 Free space path loss, 298 Frequency deviation, 44, 453, 1197, 1207, 1214 Frequency division duplexing (FDD), 523, 1146, 1175 Frequency reuse, 810, 1165, 1172 Front-end architectures, 1189 Function Bessel, 308, 511, 1056 Marcum, 529 saw-tooth, 1048 Galois field, 844, 851 Gauss quadrature rule, 607 Gaussian random process, 68, 309, 503, 518 Gaussian random variable, 67, 314, 439 Generator matrix, 836, 865, 906 polynomial, 863, 869, 875, 905, 928 Geometric mean, 53, 1001 Global system for mobile communication (GSM), 523, 1172–1177 Gradient vector, 132, 150, 167, 1086 Gram–Schmidt orthonormalization procedure, 8 Granular error, 350, 354, 364 noise, 389, 408, 411 Graph, 907, 947, 952 Gray coding, 462, 531
1281
Group, 830, 961, 977 Guard time, 1176, 1184 Hard input decoding, 913 Heaviside conditions, 32, 259 Householder transformation, 158 Hybrid fiber/coax (HFC) networks, 1156–1160 IEEE 802 Working Group, 1152–1156, 1163–1164, 1266 Indoor environment, 1185 Inner product, 3 Integrated Services Digital Network (ISDN), 1119, 1147–1148, 1173 Intensity profile of multipath, 307 Interference cancellation, 1115 co-channel (CCI), 803 intersymbol (ISI), 456, 557, 604, 620, 655, 674, 749 multiuser (MUI), 803, 817 Interleaving, 913 International Telecommunications Union (ITU), 435, 1146, 1148 Interpolation, 31, 109, 116–118, 339, 764, 1255, 1269 Jakes model, 313 Japanese Digital Cellular (JDC), 1171, 1180, 1235 Lagrange interpolation, 118 Lattice Gosset, 977, 991 Schlaefli, 976, 991 Law A, 363 ¼, 364 Leakage, 86, 1190 Likelihood function, 511, 674, 1051 Likelihood ratio, 442, 675 Limiter-discriminator, 1197 Line coding, 583–601 Line codes biphase, 584
1282
Line codes (continued) block, 585 dicode, 583, 591 duobinary, 591 modified duobinary, 592 NRZ, 583 RZ, 584 Line-of-sight (LOS), 295, 302 LMS algorithm for lattice filters, 191 in a transformed domain, 211 leaky, 187, 634 normalized, 189 sign, 187 Local area network (LAN), 290, 523, 1152, 1155 Local multipoint distribution service (LMDS), 1165 Log-likelihood function, 512, 677 Log-likelihood ratio, 676, 916, 934 Matrix circulant, 243, 776 diagonal, 20, 22, 155 generator, 836, 865, 906 Hadamard, 536 Hermitian, 65 inverse, 145 parity check, 833–836, 840, 865, 870, 905, 906, 946–950, 962–965 Toeplitz, 63 triangular, 214 unitary, 66, 168, 182, 212 Vandermonde, 890 Maximum a posteriori (MAP) criterion, 160, 441, 661, 675, 677, 915, 921, 932, 948 Max-Log-MAP, 675, 677, 916, 921 Log-MAP, 677, 916, 952 Mean convergence of the, 178 Medium access control (MAC), 1156, 1163–1164 Message-passing decoding, 946, 953 Minimum function, 857–861
Index
Modulation amplitude (AM), 295, 461, 480, 539 amplitude and phase (AM-PM), 480 binary, 437, 487, 520 binary phase shift keying (BPSK), 450, 470, 477, 795, 919 biorthogonal, 493 carrierless AM/PM (CAP), 568–570 continuous phase (CPM), 1246–1248 continuous phase FSK (CPFSK), 1217, 1219 differential PSK (DPSK), 474–475 DMT, 770, 775, 781 double sideband (DSB), 41, 58, 608 DWMT, 782 FMT, 771, 777, 781 frequency (FM), 1197, 1213 frequency-shift keying (FSK), 452, 516, 519, 1207 Gaussian minimum-shift keying (GMSK), 1229–1243 index, 453, 1209 minimum-shift keying (MSK), 454, 1214–1228 multicarrier (MC), 753 offset QPSK (OQPSK), 1203, 1220 on-off keying (OOK), 510 orthogonal, 486 phase-shift keying (PSK), 465, 474, 995 pulse amplitude (PAM), 69, 72, 461–464, 539–543, 583, 1040 pulse duration (PDM), 464, 532, 534 pulse position (PPM), 464, 532 quadrature amplitude (QAM), 480–485, 502, 544–548, 611 quadrature PSK (QPSK), 472–473 single sideband (SSB), 58, 499 trellis coded (TCM), 967–998 vestigial sideband (VSB), 781 Moving average model (MA), 91, 94, 398
Index
Multichannel multipoint distribution service (MMDS), 1165 Multipath, 299, 302, 307, 521, 717, 803, 811, 1167, 1243 Multiple access code division (CDMA), 524, 795, 802, 810, 818, 1180 frequency division (FDMA), 523, 1178 time division (TDMA), 524, 1158, 1175 Noble identities, 118, 759 Noise figure, 268, 270, 274 impulse, 782, 1158 shot, 265, 294, 326 temperature, 265, 273 thermal, 263–264 Norm, 3 Numerically controlled oscillator (NCO), 1054 Nyquist criterion, 559, 562, 589 frequency, 462, 559, 616 pulse, 559, 801 OFDM, 753–794 passband, 780 synchronization of, 779 systems, 769, 773, 780, 1002, 1128 Optical fibers, 291–294, 1148 Orthogonality principle, 134, 151 Oversampling, 404, 627, 630, 699, 1062 Parseval theorem, 15, 84 Partial response systems, 587, 1096, 1249 Per survivor processing (PSP), 695 Periodogram, 84 Welch, 85 Personal communication services (PCS), 810 Phase detector (PD), 1030 Phase deviation, 44, 322, 1193
1283
Phase jitter, 321 Phase noise, 321, 697, 1189 Phase-locked loop (PLL), 326, 1029–1039, 1074, 1105, 1262 Pilot signal, 1027 Power amplifier (HPA), 322, 1171 Power back-off, 1136–1141 Power delay profile, 310, 812, 1243 Power spectral density (PSD), 46–62 Precoding, 596, 777, 599, 1008–1025 Prediction error, 141, 176, 192, 385, 399, 657 linear, 129, 140–147, 414, 1180 optimum, 143, 392 Predictor backward, 140 forward, 140 linear, 140, 385, 391 optimum, 141, 399 Probability a posteriori, 160, 441, 661, 668, 931, 954 a priori, 440, 669 conditional, 159, 442, 510 transition, 932 Probability density function Gaussian, 567 Rayleigh, 308 Rice, 308, 518 Probability of error, 454, 483, 489, 494, 899 Processing gain, 809 Projection operator, 153 Pseudo-inverse matrix, 155–157 Pseudo-noise (PN) sequences, 233–238 CAZAC, 235 Gold, 236 maximal length, 233 Public switched telephone network (PSTN), 331, 1018, 1145 Pulse code modulation (PCM), 345, 357, 377, 385, 571 QPR-IV transceiver, 1249–1262 Quantization adaptive, 377
1284
Quantization (continued) error, 345, 347, 350, 352, 387, 404, 575 scalar, 417, 420, 424 uniform, 355 vector, 417–432 Quantizer adaptive, 377–379, 381 non uniform, 358–377 uniform, 346, 353 vector, 418, 421 Quotient group, 978 Radio link, 294–318 Raised cosine pulse, 559–561 Rake receiver, 811, 815, 1181 Rate bit, 331, 338, 340, 379, 406, 457, 504, 539, 578, 827–830, 1002 sampling, 407, 555, 616, 626, 769 symbol, 458, 540, 754 Receiver direct conversion, 1190 superheterodyne, 1189 optimization, 731 Recovery carrier frequency, 1068 carrier phase, 1050, 1104 timing, 1046, 1053, 1060, 1074, 1252, 1269 Reflection coefficient, 146, 278 Regenerative repeaters, 575–581 Sampling theorem, 8, 30, 337, 627, 1046 Scattering, 295, 300, 310 Scrambling, 803, 1153, 1265 Self-training equalization 1083–1114 Set partitioning, 694, 973, 1014 Shadowing, 313 Shannon limit, 499, 504, 508, 924, 947 Signal analytic, 33, 38, 780, 804 bandwidth, 28–29 baseband, 31, 36, 459 baseband equivalent, 33–34, 549
Index
complex envelope, 34 envelope, 43, 322 passband, 33, 697, 780 space, 1–4, 439, 990, 1008 Signal constellation, 12, 458 Signal-to-interference ratio (SIR), 798 Signal-to-noise ratio (SNR), 272, 449, 463, 469, 552, 626 Signal-to-quantization error ratio, 352, 354, 364, 387 Simplex transmission, 523 Simplex cyclic codes, 875–878 Singular-value decomposition (SVD), 155 Slope overload distortion, 389 Soft output Viterbi algorithm (SOVA), 921–924 Source coding, 433 Spectral efficiency, 458, 460, 463, 468, 503, 782, 991 Spectral factorization, 53 Spread spectrum systems, 795–826 applications of, 807 chip equalizer, 818 direct sequence, 795 frequency hopping, 804 symbol equalizer, 819 synchronization, 1074 Soft input decoding, 900, 914 Spreading techniques, 795 factor, 795, 1074 Standard ADSL, 1148 DECT, 1182 Ethernet, 1152, 1156, 1263–1273 FDDI, 1155 HDSL, 1147 HIPERLAN, 1185 IS-136, 1177 JDC, 1180 modem, 1146–1147 SHDSL, 1148 speech and audio, 434 Token Ring, 1155
Index
VDSL, 1148 video, 435 Sufficient statistic, 440, 681, 816, 1007 Symbol error probability, 463, 469, 483, 508, 565, 604, 682 Synchronization, 1027–1082 Syndrome, 839–841, 871, 889, 895, 965 System bandwidth, 265, 590, 1171 baseband equivalent, 549, 775 causal, 18 continuous time, 13 discrete time, 17, 556, 1003 identification, 239–254 T1 carrier, 524 Tanner graph, 947 girth, 952 Telecommunications Industry Association (TIA), 286 Telephone channel, 318–322, 697, 703 Token Ring, 1155 Tomlinson–Harashima precoding, 1009 Transform D, 903 discrete cosine (DCT), 214–215, 433, 790 discrete Fourier (DFT), 19, 206, 214, 433 fast Fourier (FFT), 20 Fourier, 14–17
1285
Hilbert, 37–38 z, 18 Transmission lines, 274–291 Trellis coded modulation (TCM), 967–997 Union bound, 455 Universal mobile telecomunication service (UMTS), 1170, 1181 Uplink, 803, 1156, 1158, 1175 User code, 796, 803 Vector quantization (VQ), 417–432 Vocoder, 341, 414 Voltage controlled oscillator (VCO), 1030, 1042, 1189, 1252 Wiener–Khintchine theorem, 46 Window Hann, 82 raised cosine or Hamming, 82 rectangular, 78, 82, 239 triangular or Bartlett, 82 Window closing, 86 Wireless local area networks (WLANs), 1162–1163, 1170, 1185 Yule–Walker equations, 97 Zero-forcing equalizer LE-ZF, 619, 648–649 DFE-ZF, 649, 651, 655
