Stochastic Methods and Their Applications to Communications

Stochastic Methods and Their Applications to Communications Stochastic Differential Equations Approach Serguei Primak U...

Author: Serguei Primak | Valeri Kontorovitch | Vladimir Lyandres

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Stochastic Methods and Their Applications to Communications Stochastic Differential Equations Approach

Serguei Primak University of Western Ontario, Canada

Valeri Kontorovich Cinvestav-IPN, Mexico

Vladimir Lyandres Ben-Gurion University of the Negev, Israel

Copyright # 2004

John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester, West Sussex PO19 8SQ, England Telephone (+44) 1243 779777

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To our loved ones

Contents 1.

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Digital Communication Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Random Variables and Their Description. . . . . . . . . . . . . . . . . . . 2.1 Random Variables and Their Description . . . . . . . . . . . . . . . . . 2.1.1 Definitions and Method of Description . . . . . . . . . . . . . 2.1.1.1 Classification . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.2 Cumulative Distribution Function. . . . . . . . . . . 2.1.1.3 Probability Density Function . . . . . . . . . . . . . . 2.1.1.4 The Characteristic Function and the Log-Characteristic Function. . . . . . . . . . . . . . . 2.1.1.5 Statistical Averages . . . . . . . . . . . . . . . . . . . . 2.1.1.6 Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.7 Central Moments . . . . . . . . . . . . . . . . . . . . . . 2.1.1.8 Other Quantities. . . . . . . . . . . . . . . . . . . . . . . 2.1.1.9 Moment and Cumulant Generating Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1.10 Cumulants . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Orthogonal Expansions of Probability Densities: Edgeworth and Laguerre Series. . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The Edgeworth Series . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.2 The Laguerre Series . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.3 Gram–Charlier Series . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Transformation of Random Variables. . . . . . . . . . . . . . . . . . . . 2.3.1 Transformation of a Given PDF into an Arbitrary PDF . . 2.3.2 PDF of a Harmonic Signal with Random Phase . . . . . . . 2.4 Random Vectors and Their Description . . . . . . . . . . . . . . . . . . 2.4.1 CDF, PDF and the Characteristic Function . . . . . . . . . . . 2.4.2 Conditional PDF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Numerical Characteristics of a Random Vector. . . . . . . . 2.5 Gaussian Random Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Transformation of Random Vectors . . . . . . . . . . . . . . . . . . . . . 2.6.1 PDF of a Sum, Difference, Product and Ratio of Two Random Variables . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Probability Density of the Magnitude and the Phase of a Complex Random Vector with Jointly Gaussian Components . . . . . . . . . . . . . . . . . . . . 2.6.2.1 Zero Mean Uncorrelated Gaussian Components of Equal Variance. . . . . . . . . . . . . . . . . . . . . .

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2.6.2.2 Case of Uncorrelated Components with Equal Variances and Non-Zero Mean. . . . . . . . . . . . . . . . 2.6.3 PDF of the Maximum (Minimum) of two Random Variables . 2.6.4 PDF of the Maximum (Minimum) of n Independent Random Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Additional Properties of Cumulants . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Moment and Cumulant Brackets . . . . . . . . . . . . . . . . . . . . . 2.7.2 Properties of Cumulant Brackets. . . . . . . . . . . . . . . . . . . . . 2.7.3 More on the Statistical Meaning of Cumulants . . . . . . . . . . . 2.8 Cumulant Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8.1 Non-Linear Transformation of a Random Variable: Cumulant Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Appendix: Cumulant Brackets and Their Calculations. . . . . . . . . . . . . . 3.

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Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 General Remarks. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Probability Density Function (PDF). . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Characteristic Functions and Cumulative Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Moment Functions and Correlation Functions . . . . . . . . . . . . . . . . . 3.5 Stationary and Non-Stationary Processes . . . . . . . . . . . . . . . . . . . . 3.6 Covariance Functions and Their Properties. . . . . . . . . . . . . . . . . . . 3.7 Correlation Coefficient. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Cumulant Functions. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 Ergodicity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Power Spectral Density (PSD) . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Mutual PSD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.11.1 PSD of a Sum of Two Stationary and Stationary Related Random Processes. . . . . . . . . . . . . . . . . . . . . . . . 3.11.2 PSD of a Product of Two Stationary Uncorrelated Processes 3.12 Covariance Function of a Periodic Random Process . . . . . . . . . . . . 3.12.1 Harmonic Signal with a Constant Magnitude . . . . . . . . . . . 3.12.2 A Mixture of Harmonic Signals . . . . . . . . . . . . . . . . . . . . 3.12.3 Harmonic Signal with Random Magnitude and Phase . . . . . 3.13 Frequently Used Covariance Functions . . . . . . . . . . . . . . . . . . . . . 3.14 Normal (Gaussian) Random Processes . . . . . . . . . . . . . . . . . . . . . . 3.15 White Gaussian Noise (WGN) . . . . . . . . . . . . . . . . . . . . . . . . . . . Advanced Topics in Random Processes. . . . . . . 4.1 Continuity, Differentiability and Integrability 4.1.1 Convergence and Continuity. . . . . . . 4.1.2 Differentiability . . . . . . . . . . . . . . . 4.1.3 Integrability . . . . . . . . . . . . . . . . . . 4.2 Elements of System Theory . . . . . . . . . . . . 4.2.1 General Remarks . . . . . . . . . . . . . . 4.2.2 Continuous SISO Systems . . . . . . . . 4.2.3 Discrete Linear Systems . . . . . . . . . 4.2.4 MIMO Systems . . . . . . . . . . . . . . .

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4.3

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4.2.5 Description of Non-Linear Systems. . . . . . . . . . . . . . . . . . . . . . Zero Memory Non-Linear Transformation of Random Processes . . . . . . 4.3.1 Transformation of Moments and Cumulants . . . . . . . . . . . . . . . . 4.3.1.1 Direct Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1.2 The Rice Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Cumulant Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cumulant Analysis of Non-Linear Transformation of Random Processes . 4.4.1 Cumulants of the Marginal PDF . . . . . . . . . . . . . . . . . . . . . . . . 4.4.2 Cumulant Method of Analysis of Non-Gaussian Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Linear Transformation of Random Processes . . . . . . . . . . . . . . . . . . . . 4.5.1 General Expression for Moment and Cumulant Functions at the Output of a Linear System . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.1.1 Transformation of Moment and Cumulant Functions . . . 4.5.1.2 Linear Time-Invariant System Driven by a Stationary Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.2 Analysis of Linear MIMO Systems . . . . . . . . . . . . . . . . . . . . . . 4.5.3 Cumulant Method of Analysis of Linear Transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5.4 Normalization of the Output Process by a Linear System . . . . . . Outages of Random Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 General Considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Average Level Crossing Rate and the Average Duration of the Upward Excursions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Level Crossing Rate of a Gaussian Random Process . . . . . . . . . . 4.6.4 Level Crossing Rate of the Nakagami Process . . . . . . . . . . . . . . 4.6.5 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Narrow Band Random Processes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Definition of the Envelope and Phase of Narrow Band Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2 The Envelope and the Phase Characteristics . . . . . . . . . . . . . . . . 4.7.2.1 Blanc-Lapierre Transformation. . . . . . . . . . . . . . . . . . . 4.7.2.2 Kluyver Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.2.3 Relations Between Moments of pAn ðan Þ and pi ðIÞ . . . . . 4.7.2.4 The Gram–Charlier Series for pR ðxÞ and pi ðIÞ . . . . . . . 4.7.3 Gaussian Narrow Band Process . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3.1 First Order Statistics . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3.2 Correlation Function of the In-phase and Quadrature Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3.3 Second Order Statistics of the Envelope . . . . . . . . . . . . 4.7.3.4 Level Crossing Rate . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 Examples of Non-Gaussian Narrow Band Random Processes . . . . 4.7.4.1 K Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4.2 Gamma Distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4.3 Log-Normal Distribution . . . . . . . . . . . . . . . . . . . . . . . 4.7.4.4 A Narrow Band Process with Nakagami Distributed Envelope . . . . . . . . . . . . . . . . . . . . . . . . .

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4.8 Spherically Invariant Processes . . . . . 4.8.1 Definitions . . . . . . . . . . . . . . 4.8.2 Properties. . . . . . . . . . . . . . . 4.8.2.1 Joint PDF of a SIRV 4.8.2.2 Narrow Band SIRVs . 4.8.3 Examples . . . . . . . . . . . . . . . 5.

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Markov Processes and Their Description . . . . . . . . . . . . . . . . . . . 5.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.1 Markov Chains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.2 Markov Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1.3 A Discrete Markov Process . . . . . . . . . . . . . . . . . . . . . 5.1.4 Continuous Markov Processes . . . . . . . . . . . . . . . . . . . 5.1.5 Differential Form of the Kolmogorov–Chapman Equation 5.2 Some Important Markov Random Processes . . . . . . . . . . . . . . . 5.2.1 One-Dimensional Random Walk . . . . . . . . . . . . . . . . . . 5.2.1.1 Unrestricted Random Walk . . . . . . . . . . . . . . . 5.2.2 Markov Processes with Jumps . . . . . . . . . . . . . . . . . . . 5.2.2.1 The Poisson Process . . . . . . . . . . . . . . . . . . . . 5.2.2.2 A Birth Process . . . . . . . . . . . . . . . . . . . . . . . 5.2.2.3 A Death Process . . . . . . . . . . . . . . . . . . . . . . 5.2.2.4 A Death and Birth Process . . . . . . . . . . . . . . . 5.3 The Fokker–Planck Equation . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.2 Derivation of the Fokker–Planck Equation . . . . . . . . . . . 5.3.3 Boundary Conditions. . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.4 Discrete Model of a Continuous Homogeneous Markov Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3.5 On the Forward and Backward Kolmogorov Equations . . 5.3.6 Methods of Solution of the Fokker–Planck Equation . . . . 5.3.6.1 Method of Separation of Variables . . . . . . . . . . 5.3.6.2 The Laplace Transform Method . . . . . . . . . . . . 5.3.6.3 Transformation to the Schro¨dinger Equations. . . 5.4 Stochastic Differential Equations. . . . . . . . . . . . . . . . . . . . . . . 5.4.1 Stochastic Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.5 Temporal Symmetry of the Diffusion Markov Process . . . . . . . . 5.6 High Order Spectra of Markov Diffusion Processes. . . . . . . . . . 5.7 Vector Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7.1.1 A Gaussian Process with a Rational Spectrum . . 5.8 On Properties of Correlation Functions of One-Dimensional Markov Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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6.2.1 Processes with Random Structure and Their Classification . . . . . . 6.2.2 Statistical Description of Markov Processes with Random Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Generalized Fokker–Planck Equation for Random Processes with Random Structure and Distributed Transitions . . . . . . . . . . . . . . 6.2.4 Moment and Cumulant Equations of a Markov Process with Random Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Approximate Solution of the Generalized Fokker–Planck Equations . . . . 6.3.1 Gram–Charlier Series Expansion. . . . . . . . . . . . . . . . . . . . . . . . 6.3.1.1 Eigenfunction Expansion. . . . . . . . . . . . . . . . . . . . . . . 6.3.1.2 Small Intensity Approximation . . . . . . . . . . . . . . . . . . 6.3.1.3 Form of the Solution for Large Intensity. . . . . . . . . . . . 6.3.2 Solution by the Perturbation Method for the Case of Low Intensities of Switching . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.1 General Small Parameter Expansion of Eigenvalues and Eigenfunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.2.2 Perturbation of 0 ðxÞ . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 High Intensity Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3.1 Zero Average Current Condition . . . . . . . . . . . . . . . . . 6.3.3.2 Asymptotic Solution P1 ðxÞ . . . . . . . . . . . . . . . . . . . . . 6.3.3.3 Case of a Finite Intensity v . . . . . . . . . . . . . . . . . . . . 6.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7.

Synthesis of Stochastic Differential Equations. . . . . . . . . . . . . . . . . . . . . 7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Modeling of a Scalar Random Process Using a First Order SDE . . . . . 7.2.1 General Synthesis Procedure for the First Order SDE . . . . . . . . 7.2.2 Synthesis of an SDE with PDF Defined on a Part of the Real Axis. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.3 Synthesis of Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4 Non-Diffusion Markov Models of Non-Gaussian Exponentially Correlated Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2.4.1 Exponentially Correlated Markov Chain—DAR(1) and Its Continuous Equivalent . . . . . . . . . . . . . . . . . . . . . 7.2.4.2 A Mixed Process with Exponential Correlation . . . . . . 7.3 Modeling of a One-Dimensional Random Process on the Basis of a Vector SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.1 Preliminary Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.2 Synthesis Procedure of a ð; !Þ Process. . . . . . . . . . . . . . . . . . 7.3.3 Synthesis of a Narrow Band Process Using a Second Order SDE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3.1 Synthesis of a Narrow Band Random Process Using a Duffing Type SDE . . . . . . . . . . . . . . . . . . . . . . . . . . 7.3.3.2 An SDE of the Van Der Pol Type . . . . . . . . . . . . . . . 7.4 Synthesis of a One-Dimensional Process with a Gaussian Marginal PDF and Non-Exponential Correlation. . . . . . . . . . . . . . . . . . . . . . . .

. . . .

.

279

.

280

.

281

. . . . . .

288 295 296 296 297 302

.

304

. 304 . 305 . 310 . 310 . 311 . 314 . 317

. 321 . 321 . 322 . 322

. . 326 . . 329 ..

334

. . 335 . . 341 . . 347 . . 347 . . 347 ..

351

.. ..

352 356

..

361

xii

CONTENTS

7.5 Synthesis of Compound Processes. . . . . . . . . . . . . . . . . . . . . . . . 7.5.1 Compound Process . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.5.2 Synthesis of a Compound Process with a Symmetrical PDF. 7.6 Synthesis of Impulse Processes . . . . . . . . . . . . . . . . . . . . . . . . . . 7.6.1 Constant Magnitude Excitation . . . . . . . . . . . . . . . . . . . . . 7.6.2 Exponentially Distributed Excitation . . . . . . . . . . . . . . . . . 7.7 Synthesis of an SDE with Random Structure . . . . . . . . . . . . . . . . 8.

Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.1 Continuous Communication Channels . . . . . . . . . . . . . . . . . . . . 8.1.1 A Mathematical Model of a Mobile Satellite Communication Channel . . . . . . . . . . . . . . . . . . . . . . . . 8.1.2 Modeling of a Single-Path Propagation . . . . . . . . . . . . . . 8.1.2.1 A Process with a Given PDF of the Envelope and Given Correlation Interval. . . . . . . . . . . . . . 8.1.2.2 A Process with a Given Spectrum and Sub-Rayleigh PDF . . . . . . . . . . . . . . . . . . . . . . 8.2 An Error Flow Simulator for Digital Communication Channels . . 8.2.1 Error Flow in Digital Communication Systems. . . . . . . . . 8.2.2 A Model of Error Flow in a Digital Channel with Fading . 8.2.3 SDE Model of a Buoyant Antenna–Satellite Link . . . . . . . 8.2.3.1 Physical Model . . . . . . . . . . . . . . . . . . . . . . . . 8.2.3.2 Phenomenological Model . . . . . . . . . . . . . . . . . 8.2.3.3 Numerical Simulation . . . . . . . . . . . . . . . . . . . . 8.3 A Simulator of Radar Sea Clutter with a Non-Rayleigh Envelope . 8.3.1 Modeling and Simulation of the K-Distributed Clutter. . . . 8.3.2 Modeling and Simulation of the Weibull Clutter. . . . . . . . 8.4 Markov Chain Models in Communications. . . . . . . . . . . . . . . . . 8.4.1 Two-State Markov Chain—Gilbert Model . . . . . . . . . . . . 8.4.2 Wang–Moayeri Model . . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.3 Independence of the Channel State Model on the Actual Fading Distribution . . . . . . . . . . . . . . . . . . . . . . . 8.4.4 A Rayleigh Channel with Diversity . . . . . . . . . . . . . . . . . 8.4.5 Fading Channel Models . . . . . . . . . . . . . . . . . . . . . . . . . 8.4.6 Higher Order Models . . . . . . . . . . . . . . . . . . . . . . . . . . 8.5 Markov Chain for Different Conditions of the Channel . . . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

. . . . . . .

364 365 367 369 370 371 371

. . . . . . 377 . . . . . . 377 . . . . . . 377 . . . . . . 380 ......

380

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

. . . . . . . . . . . . . .

383 388 389 389 391 391 392 395 397 397 404 408 408 409

. . . . .

. . . . .

. . . . .

. . . . .

. . . . .

. 418 . 418 . 419 . 421 . 422

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

433

As an extra resource we have set up a companion website for our book containing supplementary material devoted to the numerical simulation of stochastic differential equations and description, modeling and simulation of impulse random processes. Additional reference information is also available on the website. Please go to the following URL and have a look: ftp://ftp.wiley.co.uk/pub/books/primak/

1 Introduction 1.1

PREFACE

A statistical approach to consideration of most problems related to information transmission became dominant during the past three decades. This can be easily explained if one takes into account that practically any real signal, propagation media, interference and even information itself all have an intrinsically random nature.1 This is why D. Middleton, one of the founders of modern communication theory, coined a term ‘‘statistical theory of communications’’ [2]. The recent spectacular achievements in information technology are based mainly on progress in three fundamental areas: communication and information theory, signal processing and computer and related technologies. All these allow the transmission of information at a rate close to that limited by the Shannon theorem [2]. In principle, the limitation on the speed of the transmission of information is defined by the noise and interference which are inherently present in all communication systems. An accurate description of such impairments is very important for a proper organization and noise immunity of communication systems. The choice of a relevant interference model is a crucial moment in design of communication systems and is an important step in their testing and performance evaluation. The requirements, formulated to improve the performance of systems are often conflicting and hard to formulate in a way convenient for optimization. On one side, such models must accurately reflect the main features of the interference under investigation. On the other hand, a maximally simple description is needed to be applicable for the massive numerical simulation required to test modern communication system designs. Historically, two simple processes, so-called White Gaussian Noise (WGN) and the Poisson Point Process (PPP) have been widely used to obtain first rough estimates of a system performance: the WGN model of the noise is a good approximation of an additive wide band noise, caused by a variety of natural phenomena, while the PPP is a good model for event modeling in a discrete communication channel. Unfortunately, in the majority of realistic situations these basic models of the noise and errors are not adequate. In realistic communication channels, a non-stationary non-Gaussian interference and noise are often present. In addition, these processes are often band-limited, thus showing significant time correlation, which cannot be represented by WGN and PPP. The following phenomena can be considered as examples when the simplest models fail to provide an accurate description: 1

Having said that, we would like to acknowledge an exponentially increasing body of literature which describes the same phenomena using a rather different approach, based on the chaotic description of signals [1].

Stochastic Methods and Their Applications to Communications. S. Primak, V. Kontorovich, V. Lyandres # 2004 John Wiley & Sons, Ltd ISBN: 0-470-84741-7

2

INTRODUCTION

fading in communication channels; interchannel/interuser interference; impulsive noise, including man-made noise. Of course this list can be greatly expanded [3]. Description of such phenomena using joint probability densities of order higher than two is difficult since a large amount of a priori information on the properties of the noise and interference is required. Such information is usually not available or difficult to obtain [4]. Furthermore, specification of higher order joint probability densities is usually not productive since it results in complicated expressions which cannot be effectively used at the current level of computer simulation. Substitution of non-Gaussian models by their equivalent Gaussian models may result in significant and unpredictable errors. All of this, coupled with increased complexity of the systems and reduced design time, requires simple and adequate models of the blocks, stages, complete systems and networks of the communication systems. Such a variety requires an approach which is flexible enough to cover the majority of these possibilities, and this is a major requirement for models of communication systems. This book describes techniques which, in the opinion of the authors, are well suited for the task of effective modeling of non-Gaussian phenomena in communication systems and related disciplines. It is based on the Markov approach to modeling of random processes encountered in applications. In particular, non-Gaussian processes are modeled as the solution of an appropriate Stochastic Differential Equation (SDE), i.e. a differential equation with a random excitation. This approach, in general phenomenological, is built on the idea of the system state, suggested by van Trees [5]. The essence of this method is to describe the process under investigation as a solution of some SDE synthesized based on some a priori information (such as marginal distribution and correlation functions). Such synthesis is an attempt to uncover hidden dynamics of the interference formation. Despite the fact that such an approach is very approximate,2 the SDE approach to modeling of random processes has significant advantages: universality: a single structure allows the modeling of a great variety of different processes by simple variation of the system parameters and excitation; effectiveness: a single structure allows the modeling and numerical simulation of a spectrum of the characteristics of the random process, such as marginal probability density, correlation function, etc; the suggested models suit well computer simulation. Unfortunately, the SDE approach to modeling is still not widely known as a tool in modeling of communication channels and related issues, despite its effective application in chemistry, physics and other areas [3]. Some work in this area first appeared more than thirty years ago in the Bonch–Bruevich Institute of Telecommunications (St Petersburg, Russia). The first attempt to summarize the results in the area of communication channel modeling based on the SDE approach resulted in a book [6], published in Russian and almost unavailable for international readers. Since then a number of new results have been obtained by a number of researchers, including the authors. A number of old approaches were also reconsidered. The majority of these results are scattered over a number of journal and conference proceedings, partially in Russian and are not always readily available. All of this 2

‘‘Pure’’ continuous time Markov processes cannot be physically implemented.

DIGITAL COMMUNICATION SYSTEMS

3

gave an impetus for a new book which offers a summary of the SDE approach to modeling and evaluation of communication and related systems, and provides solid background in the applied theory of random processes and systems with random excitation. Alongside classical issues, the book includes the cumulant method of Malakhov [7], often used for analysis of non-Gaussian processes, systems with random and some aspects of numerical simulation of SDE. The authors believe that this composition of the book will be helpful for both graduate students specializing in analysis of communication systems and researchers and practising engineers. Some chapters and sections can be omitted for the first reading; some of the topics formulate problems which have not been solved.3 The book is organized as follows. Chapters 2–4 present an introduction to the theory of random processes and form the basis of a first term course for graduate students at the Department of Electrical and Computer Engineering, The University of Western Ontario (UWO). Chapter 5 deals with the theory of Markov processes, stochastic differential equations and the Fokker–Planck equation. Synthesis of models in the form of SDE is described in Chapter 7, while Chapter 8 provides examples of the applications of SDE and other Markov models to practical communications problems. Four appendices detailing the numerical simulation of SDEs, impulse random processes, tables of distributions and orthogonal polynomials are published on the web. Some of the material presented in this book was originally developed in the State University of Telecommunications (St Petersburg, Russia) and is, to a great degree, augmented by the results of the research conducted in three groups at Ben-Gurion University of the Negev (Beer-Sheva, Israel), University of Western Ontario (London, Canada) and Cinvestav-IPN (Mexico City, Mexico). A large number of people have provided motivation, support and advice. The authors would like to thank Doctors R. Gut, A. Berdnikov, A. Brusentsov, M. Dotsenko, G. Kotlyar, J. LoVetri, J. Roy and D. Makrakis, who contributed greatly to the development of the techniques described in this book. We also would like to recognize the contribution of our graduate students, in particular Mr Mark Shahaf, Mrs My Pham, Mr Chris Snow, Mr Jeff Weaver and Ms Vanja Subotic who have provided contributions through a number of joint publications and valuable comments on the course notes which form the first three chapters of this book. This research has been supported in part by CRC Canada, NSERC Canada, the Ministry of Education of Israel, and the Ministry of Education of Mexico. The authors are also thankful to their families who were very considerate and patient. Without their love and support this book would have never been completed.

1.2

DIGITAL COMMUNICATION SYSTEMS

The following block diagram is used in this book as a generic model of a communication system.4 A detailed description of the blocks can be found in many textbooks, for example [8,9]. The main focus of this book is to model processes in a communication channel. Referring to Fig. 1.1, there are two ways in which the communication channel can be defined: the channel between points A and A0 is known as a continuous (analog) channel; the 3 4

At least, the authors are not aware of such solutions. This also includes such electronic systems as radar; the issue of immunity of electronic equipment to interference also can be treated as a communication problem.

4

INTRODUCTION

Figure 1.1

Generic communication system.

part of the system between the points B and B0 is called a discrete (digital) channel. In general, such separation is very subjective, especially in the light of recent advances in modulation and coding techniques, taking into account properties of the propagation channel already on the modulation stage. However, such separation is convenient, especially for a pedagogical purpose and computer simulation of separate blocks of the communications system [9,10]. In addition to propagation media, the continuous communication channel includes all linear blocks (such as filters and amplifiers) which cannot be identified with the propagation media. Such assignment is somewhat arbitrary, since by varying the parameters of the modulation, number of antennas and their directivity, it is possible to vary properties of the channel.5 A theoretical description of such a channel could be achieved by modeling the channel as a linear time and space varying filter [10–12]. This allows use of the concept of system functions, which is well suited for both narrow band and wide band channels. Recent advances in communication technology sparked greater interest in wide band channels [9–10,12], since they significantly increase the capacity of communication systems. In continuous channels one can find a great variety of noise and interference, which limit the information transmission rate subject to certain quality requirements (such as probability of error). Roughly speaking, the interference and noise can be classified as additive (such as thermal noise, antenna noise, man-made and natural impulsive noise, interchannel interference, etc.) and multiplicative (such as fading, intermodulations, etc.). Additive noise (interference) nðtÞ results in the received signal rðtÞ being an algebraic sum of the information signal sðtÞ and noise nðtÞ, rðtÞ ¼ sðtÞ þ nðtÞ

ð1:1Þ

while the multiplicative noise (such as fast and shadowing) changes the energy of the information signal. In the simplest case of frequency non-selective fading, the multiplicative 5

Sometimes modems are also included in the continuous channel. In this case the channel becomes non-linear, depending on the modulation–demodulation technique.

REFERENCES

5

noise is described simply as rðtÞ ¼ ðtÞsðtÞ

ð1:2Þ

where ðtÞ is the fading, sðtÞ is the transmitted signal and rðtÞ is the received signal. Multiplicative noise significantly transforms the spectrum of the received signal and thus severely impairs communication systems. The realistic situation is often much more complicated since different frequencies are transmitted with different gains [12], thus one has to distinguish between so-called flat (non-selective) and frequency selective fading. ‘‘Flatness’’ of fading is closely related to the product of the delay spread of the signal and its bandwidth [13–14]. Large-scale fading, or shadowing, is mainly defined by the absorption of radiowaves, defined by changes in the temperature, composition and other factors in the propagation media. Such changes are very slow compared to the period of the carrier frequency and the signal can be assumed constant for many cycles of the carrier. However, these changes are significant if a relatively long communication session is considered. On the contrary, fast fading is explained by the changing phase conditions on the propagation path, with corresponding destruction or reconstruction of coherency between possible multiple paths of propagation. As a result, the effective magnitude and the phase of the signals varies significantly, which is particularly important in cellular, personal and satellite communications [8,12]. Furthermore, these variations are comparable on the time scale with the duration of the bit, explaining the term ‘‘fast fading’’. A combination of fast and slow fading results in non-stationary conditions of the channel. Nevertheless, if the performance of a communication system is considered on the level of a bit, code block or sometimes a packet or frame, it could be assumed that the fading is at least locally stationary. However, if long term performance of the system is the concern, for example when evaluating the performance of the communication protocols by means of numerical simulation, the non-stationary nature must be taken into account. A digital communication channel (Section B B0 in Fig. 1.1) is characterized by a discrete set of the transmitted and received symbols and can usually be described by distribution of the errors, i.e. incorrectly received symbols. There are a great number of statistics which are useful in such description, for example average probability of error, average number of errors in a block of a given length, etc. It can be seen that the continuous channels can be statistically mapped into a discrete channel if the modulation and its characteristics are known. If simple modulation techniques are used, the performance of the modulation technique can be expressed analytically and corresponding statistics of the discrete channel can be derived. In this case, modeling of the continuous channel is unnecessary. However, modern communication algorithms often involve modulation and coding techniques whose performance cannot be described analytically. In this case, it is very important to be able to reproduce proper statistics of the underlying continuous channel. This book discusses techniques for modeling both continuous and discrete channels.

REFERENCES 1. E. Costamagna, L. Favalli, and P. Gamba, Multipath Channel Modeling with Chaotic Attractors, Proceedings of the IEEE, vol. 90, no. 5, May 2002, pp. 842–859.

6

INTRODUCTION

2. D. Middleton, An Introduction to Statistical Communication Theory, New York: McGraw-Hill, 1960. 3. C.W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences, Berlin: Springer, 1994, pp. 442. 4. D. Middleton, Non-Gaussian Noise Models in Signal Processing for Telecommunications: New Methods and Results For Class A and Class B Noise Models, IEEE Trans. Information Theory, vol. 45, no. 4, May 1999, pp. 1129–1149. 5. H. van Trees, Detection, Estimation, and modulation theory, New York: Wiley, 1968–1971. 6. D. Klovsky, V. Kontorovich, and S. Shirokov, Models of Continuous Communications Channels Based on Stochastic Differential Equations, Moscow: Radio i sviaz, 1984 (In Russian). 7. A.N. Malakhov, Kumuliantnyi Analiz Sluchainykh Negaussovykh Protsessov I Ikh Preobrazovanii, Moscow: Sov. Radio, 1978 (In Russian). 8. M. Jeruchim, Simulation of Communication Systems: Modeling, Methodology, and Techniques, New York: Kluwer Academic/Plenum Publishers, 2000. 9. J. Proakis, Digital Communications, Boston: McGraw-Hill, 2000. 10. S. Benedetto and E. Biglieri, Principles of Digital Transmission: with Wireless Applications, New York: Kluwer Academic/Plenum Press, 1999. 11. R. Steele and L. Hanzo, Mobile Radio Communications: Second And Third-generation Cellular And Watm Systems, New York: Wiley, 1999. 12. M. Simon and S. Alouini, Digital Communication Over Fading Channels: A Unified Approach To Performance Analysis, New York: Wiley, 2000. 13. T. Rappoport, Wireless communications: principles and practice. Upper Saddle River, NJ: London: Prentice Hall PTR, 2001. 14. P. Bello, Characterization of Randomly Time-Variant Linear Channels, IEEE Trans. Communications, vol. 11, no. 4, December, 1963, pp. 360–393.

2 Random Variables and Their Description This chapter briefly summarizes definitions and important results related to the description of random variables and random vectors. More detailed discussions can be found in [1,2]. A number of useful examples, important for applications, is also considered in this chapter.

2.1

RANDOM VARIABLES AND THEIR DESCRIPTION

2.1.1 2.1.1.1

Definitions and Method of Description Classification

A Random Variable (RV) 1 can be considered as an outcome of an experiment, physical or imaginary, such that this quantity has a different value from one run of the experiment to another, even if the experiment is repeated under the same conditions. The difference in the outcomes may come either as a result of unaccounted conditions (variables), or, as in the quantum theory, from the internal properties of the system under consideration. In order to describe a random variable one needs to specify a range of possible values which this variable can attain. In addition, some numerical measure of probability of these outcomes must be assigned. Based on the type of values the random variable under consideration can assume, it is possible to distinguish between three classes of random variable: (a) a discrete random variable; (b) a continuous random variable; and (c) a mixed random variable. For a discrete random variable there are a finite or infinite but countable number of values this random variable can attain. These possible values can be enumerated, i.e. a countable set, fn g, n ¼ 0, 1, . . ., covers all the possibilities. Continuous random variables assume values from a single or multiple non-intersecting intervals, ½ai ; bi , ai < bi < aiþ1 < . . . ; i ¼ 1; 2; . . . on the real axis. Both ends of the interval can be at infinity. Finally, a mixed random variable may assume values from both discrete and continuous sets. There is a number of characteristics which allow a complete description of a random variable . In particular, the Cumulative Distribution Function (CDF), P ðxÞ, Probability 1

In future, the Greek letters are used to denote a random variable. Latin letters a, b, c will be reserved for constants and x, y, z for variables.


8


Density Function (PDF) p ðxÞ and the characteristic function ð juÞ are the most important characteristics of random variables, allowing their complete description. 2.1.1.2

Cumulative Distribution Function

The cumulative distribution function P ðxÞ is defined as the probability of the event that the random variable does not exceed a certain threshold, x, i.e. P ðxÞ ¼ Probf < xg

ð2:1Þ

Any CDF P ðxÞ is a non-negative, non-decreasing, continuous on the left function which satisfies the following boundary conditions P ð1Þ ¼ 0;

P ð1Þ ¼ 1

ð2:2Þ

The converse is also valid: every function which satisfies the above listed conditions is a CDF of some random variable [1]. If CDF P ðxÞ is known, then the probability that the random variable falls inside a finite interval ½a; bÞ or intersection of non-overlapping intervals ½ai ; bi Þ is given by Probfa < bg ¼ P ðbÞ P ðaÞ X Probf 2 [i ½ai ; bi Þg ¼ ½P ðbi Þ P ðai Þ

ð2:3Þ

i

The advantage of the CDF method is the fact that the discrete, continuous and mixed variables are described using the same technique. However, such description is an integral representation and thus it is not easy to interpret. For a complete description of a discrete random variable , which assumes a set of values, xk , k ¼ 0; 1; . . ., one needs to know the distribution of probabilities Pk , i.e. Pk ¼ Probf ¼ xk g

ð2:4Þ

It is clear that Pk 0;

X

Pk ¼ 1

ð2:5Þ

k

Example: binomial distribution. This distribution arises when a given experiment with two possible outcomes, ‘‘SUCCESS’’ and ‘‘FAILURE’’, is repeated N times in a row. The probability of a ‘‘SUCCESS’’ outcome is 0 p 1, and the probability of a ‘‘FAILURE’’ is q ¼ 1 p. Probability PN ðkÞ of k successes in a set of N trials is then given by2 PN ðkÞ ¼ CNk pk qNk ¼

N! pk qNk ; k!ðN kÞ!

k ¼ 0; 1; . . . ; N

ð2:6Þ

Herethe following notation is used for a combination of k elements from a set of N elements: CNk ¼ N! N ½5. ¼ k k!ðN kÞ!

2


9

It can be seen that the probability PN ðkÞ is the coefficient in the expansion of ðq þ pxÞN into a power series with respect to x. This explains the term ‘‘binomial distribution’’. Situations described by the binomial law are often encountered in communication theory and many other applications. Examples include the number of errors (‘‘failures’’) in a group of N bits, the number of corrupted packets in a frame, number of rejected calls during a certain period of time, etc. Example: the Poisson random variable. In the case of the Poisson random variable , the random variable assumes integer numbers as values, with probabilities defined by Pk ¼ Probf ¼ kg ¼

k expðÞ k!

ð2:7Þ

This distribution can be obtained as a limiting case of the binomial distribution (2.6) when the number of experiments approaches infinity while the product ¼ N p remains constant [1,3]. This distribution also plays an important role in communication theory, reliability theory and networking.

2.1.1.3

Probability Density Function

A continuous random variable can be described by the probability density function p ðxÞ, defined such that d P ðxÞ; p ðxÞ ¼ dx

P ðxÞ ¼

ðx 1

p ðsÞ d s

ð2:8Þ

Any PDF satisfies the following conditions, which follow from the definition (2.8): 1. it is a non-negative function p ðxÞ 0 2. it is normalized to 1, i.e.

ð1 1

p ðxÞd x ¼ 1

3. probability Probfx1 < x2 g is given by ð x2 p ðxÞd x ¼ P ðx2 Þ P ðx1 Þ Probfx1 < x2 g ¼

ð2:9Þ

ð2:10Þ

ð2:11Þ

x1

Formally, the PDF for a discrete random variable can be defined as a sum of delta functions weighted by the probability of each discrete even, i.e. p ðxÞ ¼

X k

Pk ðx xk Þ

ð2:12Þ

10


A great variety of probablity densities is used in applications. Discussion of particular distributions is postponed until later. However, it is important to mention the so-called Gaussian (normal) distribution with PDF " # 2 1 ðx mÞ pðxÞ ¼ Nðx; n; DÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi exp ð2:13Þ 2D 2D and the PDF of the so-called uniform distribution 1 for a x b pðxÞ ¼ ba 0 otherwise

ð2:14Þ

A much wider class of PDFs belong to the so-called Pearson family [4], which is described through a differential equation for its PDF d pðxÞ d xa dx ¼ ln pðxÞ ¼ dx b0 þ b1 x þ b2 x 2 pðxÞ

ð2:15Þ

It follows from eqs. (2.13) and (2.14) that the CDFs of Gaussian and uniform distributions are given by ðx xm ð2:16Þ Nðx; n; DÞd x ¼ pffiffiffiffi PN ðxÞ ¼ D 1 8 0 xa > <xa axb ð2:17Þ Pu ðxÞ ¼ > :b a 1 xb respectively. Here 1 ðxÞ ¼ pffiffiffiffiffiffi 2

2 t 1 x d t ¼ 1 erf exp 2 2 2 1

ðx

ð2:18Þ

is the probability integral which in turn can be expressed in terms of the error function [5].

2.1.1.4

The Characteristic Function and the Log-Characteristic Function

Instead of considering the CDF P ðxÞ or the PDF p ðxÞ one can consider an equivalent description by means of the Characteristic Function (CF). The characteristic function ð j uÞ is defined as the Fourier transform of the corresponding PDF3 ð ð1 1 1 jux ð j uÞ ¼ p ðxÞe d x; p ð j uÞ ¼ ð j uÞej u x d u ð2:19Þ 2 1 1 Thus, one can restore the PDF p ðxÞ from the corresponding characteristic function ð j uÞ. 3

Note the plus sign in the direct transform and the minus sign in the inverse transform.


11

There are a number of properties of the CF which follow from the definition (2.19). In particular ð1 ð1 jux jp ðxÞjje jd x ¼ p ðxÞd x ¼ ð0Þ ¼ 1 ð2:20Þ j ð j uÞj 1

1

and ðj uÞ ¼

ð1 1

p ðxÞej u x d x ¼ ð j uÞ

ð2:21Þ

Additional properties of the characteristic function follow from the properties of the Fourier transform pair and can be found in [1,6]. Tables of the Fourier transforms of PDFs are also available [7]. In addition to the characteristic function, one can define the so-called logcharacteristic function (or cumulant generating function) as ð j uÞ ¼ ln ð j uÞ

ð2:22Þ

This function is later used to define the cumulants of a random variable. Example: Gaussian distribution. The characteristic function of a Gaussian distribution can be obtained by using a standard integral (3.323, page 333 in [8])

ð1

q2 expðp x q xÞd x ¼ exp 4 p2 1 2 2

pffiffiffi p

ð2:23Þ

with q ¼ j u, p2 ¼ D=2. This results in

D u2 N ð j uÞ ¼ exp j u m 2

ð2:24Þ

The Log-Characteristic Function (LCF) n ð j uÞ of the Gaussian distribution is then N ð j uÞ ¼ j u m

2.1.1.5

D u2 2

ð2:25Þ

Statistical Averages

The CDF, PDF and CF each provide a complete description of any random variable. However, in many practical problems it is possible to limit consideration to a less complete but simpler description of random variables. This can be achieved, for example, through use of statistical averages. Let ¼ gðÞ be a deterministic function of a random variable , described by the PDF p ðxÞ. The ensemble average of the function gðÞ is defined as EfgðÞg ¼ hgðÞi ¼

ð1 1

gðxÞp ðxÞd x

ð2:26Þ

12


where the subscript indicates over which variable the averaging is performed. In future, the subscript will be dropped if it does not create uncertainty. By specifying a particular form of the function gðÞ it is possible to obtain various numerical characteristics of the random variable. Because averaging is a linear operation it is possible to interchange the order of averaging and summation, i.e. ( E

K X

) ck gk ðÞ

k¼1

2.1.1.6

¼

K X

ck Efgk ðÞg

ð2:27Þ

k¼1

Moments

A moment mn ¼ mn of order n of a random variable is obtained if the function gðÞ in eq. (2.26) is chosen as gðÞ ¼ n , thus giving mn ¼

ð1 1

xn p ðxÞ d x

ð2:28Þ

The first order moment4 m1 ¼ m ¼ m is called the average value of the random variable . It follows from its definition that the dimension of the average coincides with that of the random variable; the average of a deterministic variable coincides with the value of the variable itself; the average of a random variable whose PDF is a symmetric function around x ¼ a is equal to m ¼ a.

2.1.1.7

Central Moments

A central moment n can be obtained from eq. (2.26) by setting gðÞ ¼ ð m1 Þn , i.e. n

n ¼ hð m1 Þ i ¼

ð1 1

ðx m1 Þn p ðxÞd x

ð2:29Þ

The first central moment is always zero: 1 ¼ 0. The second central moment 2 ¼ D ¼ 2 is called the variance, and represents the degree of variation of the random variable around its mean. The quantity is known as the standard deviation. The following properties of the variance can be easily obtained from definition (2.29) the variance D has dimension of the square of the random variable ; D 0; D ¼ 0 if and only if is deterministic; 4

In the following the subindex indicating a random variable is dropped if it does not cause confusion.

13


the variance D of ¼ c is equal to c2 D , where c is a deterministic constant; the variance of ¼ þ c is equal to D ¼ D ; the following inequality is valid (Tschebychev inequality) [3]. Probfj m1 j "g

D "2

ð2:30Þ

Here " is an arbitrary positive number. The last property states that the probability of large deviations from the average value are very rare. It is possible to obtain relations between the moments and the central moments of the same random variable using the binomial expansion. Indeed n ¼

ð1 1

n

ðx mÞ pðxÞd x ¼

ð1 X n 1 k¼0

½Cnk ð1Þk xnk mk pðxÞd x ¼

n X

Cnk ð1Þk mnk mk

k¼0

ð2:31Þ Conversely, mn ¼

n X

Cnk nk mk

ð2:32Þ

k¼0

It is interesting to mention that the moment (central moment) of order n depends on all n central moments (moments) of lower or equal order. In particular m2 ¼ D þ m21

ð2:33Þ

Example: Gaussian PDF. In the case of the Gaussian distribution the mean value and the variance are " # 1 ðx mÞ2 x pffiffiffiffiffiffiffiffiffiffiffi exp m1 ¼ dx ¼ m 2D 2D 1 " # ð1 2 1 ðx mÞ d x ¼ D ¼ 2 ðx mÞ2 pffiffiffiffiffiffiffiffiffiffiffi exp 2 ¼ 2D 2D 1 ð1

ð2:34Þ ð2:35Þ

respectively. Thus, the parameters m and D of the Gaussian distribution are actually the mean and variance of the distribution.

2.1.1.8

Other Quantities

There is a great number of other numerical characteristics which are useful in describing a random variable. A few of them are listed below. Much more detailed discussions and applications can be found in [1,2].

14


If the PDF pðxÞ has a (local) maximum at x ¼ xm , the value xm is called a mode of the distribution. A distribution with a single mode is called a unimodal distribution, while a distribution with more than one mode is called multimodal. The median x1=2 of the distribution is a value such that Pðx1=2 Þ ¼

ð x1=2 1

pðxÞd x ¼

ð1

pðxÞd x ¼

x1=2

1 2

ð2:36Þ

This can be generalized to define ðn 1Þ values fxk g, k ¼ 0; . . . ; n 2, called the quantiles, which divide the real axis into n equally probable regions ð1 1

pðxÞd x ¼

ð x1

pðxÞd x ¼ ¼

ð1

pðxÞd x ¼

xn2

x0

1 n

ð2:37Þ

Absolute moments can be defined by setting gðÞ ¼ jjn and gðÞ ¼ j mjn in eq. (2.26). In addition, the requirement that the power n is an integer can also be dropped to obtain moments of a fractional order. Finally, the entropy (differential entropy) of the distribution can be defined as HðÞ ¼

K X

pk log pk ; HðÞ ¼

ð1

k¼1

1

p ðxÞ log p ðxÞd x

ð2:38Þ

The entropy is an important concept which is discussed in detail in textbooks on information theory. It also has significant application as a measure of difference between two PDFs.

2.1.1.9

Moment and Cumulant Generating Functions

The characteristic function ð j uÞ was formally defined in Section 2.1.1.4 as the Fourier transform of the corresponding PDF p ðxÞ. It also can be defined in the framework of the statistical averaging of eq. (2.26). Indeed, let gðÞ be an exponential function with a parameter s: gðÞ ¼ expðs Þ. In this case the average value MðsÞ of gðÞ is defined as M ðsÞ ¼ hexpðs Þi ¼

ð1 1

expðs xÞp ðxÞd x

ð2:39Þ

and is called the moment generating function. Expanding the exponential term expðs xÞ into a power series, one obtains the relation between the moment generating function and the moments of the distribution of random variable : * M ðsÞ ¼

1 k X k¼0

k!

+ sk

¼

1 X mk k¼0

k!

sk

ð2:40Þ


15

if all moments exist and are finite. In turn, the coefficients of the Taylor expansion (2.40), i.e. the moments mk , can be found as dk ð2:41Þ mk ¼ k M ðsÞ ds s¼0 The transform variable s can be a complex one, i.e. s ¼ þ j u. In a particular case of s ¼ j u, the moment generating function coincides with the characteristic function, i.e. ð j uÞ ¼ M ðsÞjs¼j u and expansion (2.40) can be rewritten as * + 1 k 1 X X k mk s ¼ j k uk ð2:42Þ ð j uÞ ¼ k! k! k¼0 k¼0 with the moments defined according to mk ¼ ðjÞk

dk ð j uÞju¼0 d uk

ð2:43Þ

Thus, under certain conditions, the characteristic function ð j uÞ can be restored from its moments. It is interesting to investigate what additional conditions must be imposed on the moments such that the restored characteristic function uniquely defines the distribution. This problem is known as the moments problem. It is possible to show that if all moments mk are finite and the series (2.42) absolutely converges for some u > 0, then the series (2.42) defines a unique distribution [2]. It should be noted that this is not true for an arbitrary distribution. For example, the log-normal distribution, considered below, is not uniquely defined by its moments [2]. Usually, the non-uniqueness arises when the moments mk increase rapidly with the index k, thus not allowing absolute convergence of the series (2.42). Such problems are often found for distributions with heavy tails. In many cases the higher order moments do not even exist [2]. A slight modification allows the characteristic function to be expressed in terms of central moments by rewriting the series (2.42) as " # 1 X k ð j uÞk

ð j uÞ ¼ e j u m1 1 þ ð2:44Þ k! k¼1 Here we use the property that the moments of random variable ¼ m1 are the central moments of the random variable , and the property that the characteristic function of p ðx m1 Þ is ð j uÞ expð j u m1 Þ.

2.1.1.10

Cumulants

Instead of considering the Taylor expansion for the characteristic function, one can construct a similar expansion of the log-characteristic function ð j uÞ, i.e. it is possible to formally write 1 X k ð j uÞ ¼ jk uk k! k¼1

ð2:45Þ

16


Coefficients of this expansion are called cumulants of the distribution p ðxÞ of the random variable , with k being the cumulants of the k-th order. Since both coefficients mk and k describe the same function, there is a close relation between the moments and the cumulants. Indeed, provided that both expansions (2.42) and (2.45) are possible, one can write " #

1 1 1 1 X 1 Y Y X X m r j r ur 1 k k k r r r r r r u ¼ exp u ¼ u ¼ j j exp j ð j uÞ ¼ s! k! r! r! r! r¼1 r¼1 r¼1 l¼0 k¼0 ð2:46Þ Collecting terms with the same power of u on both sides of this expansion, one obtains the expression of the moment of order k in terms of cumulants of order up to k m r X X p1 1 p2 2 pm r! ð2:47Þ mk ¼ 1 ! 2 ! . . . m ! p1 ! p2 ! pm ! m¼1 where the inner summation is taken over all non-negative values of indices i , such that 1 p1 þ 2 p2 þ þ m pm ¼ r

ð2:48Þ

In a similar way, the expression of the cumulant k can be written in terms of the moments of order up to k [2] k ¼ r!

r X X mp 1 mp 2 m¼1

1

2

p1 !

p2 !

mpm m ð1Þ 1 ð 1Þ! 1 ! 2 ! . . . m ! pm !

ð2:49Þ

The inner summation is extended over all the indices and such that 1 þ 2 þ þ m ¼

ð2:50Þ

There are a number of tables which contain explicit expressions for cumulants and moments up to order 12 [2] while [9] provides a convenient means of deriving relations between the cumulants and the moments.

2.2

ORTHOGONAL EXPANSIONS OF PROBABILITY DENSITIES: EDGEWORTH AND LAGUERRE SERIES

In many practical cases one has to deal with a probability density p1 ðxÞ which looks similar to a Gaussian one defined by eq. (2.13). Two characteristic features of such distributions can be summarized as follows: 1. Unimodality, i.e. the PDF has a single maximum, and 2. The PDF has tails extending to infinity on both sides of the maximum, which decay fast when the magnitude of the argument approaches infinity.

17

ORTHOGONAL EXPANSIONS OF PROBABILITY DENSITIES

In this case it is often possible to approximate such PDFs using series of Hermitian or Laguerre polynomials. 2.2.1

The Edgeworth Series

In this case a PDF pðxÞ under consideration is approximated by the following series 1 X 1 bn x m pðxÞ ¼ p0 ðxÞ Hn n! n n¼0

ð2:51Þ

Here " # 1 ðx mÞ2 p0 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp 2 2 2 2

ð2:52Þ

is the Gaussian PDF with the mean value m and variance 2 , and Hn ðzÞ stands for the Hermitian polynomials [5] 2 n 2

z d z ; Hn ðzÞ ¼ ð1Þ exp exp 2 d zn 2 n

n ¼ 0; 1; . . .

ð2:53Þ

Since the Hermitian polynomials are orthogonal with weight p0 ðxÞ, i.e. ð1 1

Hn ðzÞHm ðzÞp0 ðzÞd z ¼ n!m n ¼

n! 0

m¼n m 6¼ n

ð2:54Þ

the coefficients bn in the expression (2.51), called quasi-moments, can be calculated as bn ¼

x m m n pðxÞHn d x ¼ Hn 1

ð1 n

ð2:55Þ

This expansion is based on the well known theorem of functional analysis [10] stating that if pðxÞ is an arbitrary function such that ð1 1

jpðxÞj2 d x < 1

ð2:56Þ

then 2 ð 1 N X 1 bn x m lim Hn dx ¼ 0 pðxÞ p0 ðxÞ N!1 1 n! n n¼0

ð2:57Þ

In practice the function pðxÞ is known only with a certain degree of accuracy. Thus, the sum (2.51) can be truncated after a finite number, N, of terms. The number N depends on the

18


choice of m and 2 . In most cases the best choice of m and 2 for a given N is to choose them to coincide with the first two cumulants derived directly from pðxÞ. Then it is easy to show that b0 ¼ 1;

b1 ¼ b2 ¼ 0

ð2:58Þ

Indeed, using eq. (2.53) we find that H0 ðzÞ ¼ 1;

H1 ðzÞ ¼ z;

H2 ðzÞ ¼ z2 1;

H3 ðzÞ ¼ z3 3 z;

H4 ðzÞ ¼ z4 6 z2 þ 3 ð2:59Þ

Thus, according to eq. (2.55) D x mE b 0 ¼ 0 H 0 ¼ h1i ¼ 1 D x mE Dx mE ¼ ¼0 b 1 ¼ 1 H 1 * + 2 2 D x mE ðx mÞ 2 2 2 b 2 ¼ H2 ¼ 1 ¼ 1 ¼0 2 2

ð2:60Þ

The Edgeworth series is obtained from eq. (2.51) by taking a finite number N of terms and choosing m and to coincide with those obtained from p1 ðxÞ, i.e. "

N x m X bn Hn pðxÞ p0 ðxÞ 1 þ n!n n¼3

# ð2:61Þ

The first term in the expansion (2.61) corresponds to a Gaussian distribution. Thus, for a Gaussian distribution all quasi-moments bn are equal to zero. Coefficients b3 and b4 in the series (2.61) describe the departure of pðxÞ from a Gaussian PDF and are known as the skewness and the curtosis [2] b3 3 3 ¼ 3 ¼ 3=2 3 2 b4 4 4

2 ¼ 4 ¼ 4 3 ¼ 2 2

1 ¼

ð2:62Þ

Here 2 ; 3 ; 4 are the cumulants of the distribution pðxÞ as defined by eq. (2.49), and 3 ; 4 are the central moments of third and fourth order. The skewness coefficient describes the deviation of the PDF pðxÞ from a symmetric shape. For any symmetrical distribution this coefficient is equal to zero. Fig 2.1 shows two cases of PDF. The first one (a) has a shallow tail on the right of the mean value, thus in the expression for ðx mÞ3 the positive deviations overpower the negative deviations and give 3 > 0, 1 > 0. A case 2 < 0 leads to a situation where the left tail of the distribution is heavier than the right one.


19

Figure 2.1 Different shapes of PDF similar to a Gaussian PDF (a) 1 > 0, (b) 1 < 0, (c) 2 > 0 and (d) 2 < 0. Dashed line corresponds to a Gaussian distribution.

The curtosis 2 describes how flat the PDF is around its peak value. For a Gaussian PDF this coefficient is equal to zero. Positive values of 2 show that the PDF curve has a narrower (sharper) top than a Gaussian, while 2 < 0 results in a flatter top. Figures 2.1(c) and 2.1(d) show following curves: 2 ¼ 0 (Gaussian), 2 < 0 and 2 > 0. It is common practice to approximate the distributions with small deviation from Gaussian using only 1 and 2 . In this case h

1 x m 2 x mi pðxÞ p0 ðxÞ 1 þ H3 þ H4 ð2:63Þ 3! 4! To obtain such an approximation one needs to estimate the first four moments (or cumulants) of the random process ðtÞ. It is important to mention that for such an approximation (2.63), the non-negativity property of the PDF is often violated. It can be shown that the approximating curve assumes negative values for large values of the argument jxj [9]. This is a consequence of the fact that eq. (2.63) is only an approximation and not an exact PDF. The characteristic function for the PDF given by the series (2.51) is equal to

X 1 1 1 bn 2 ðj uÞn ð2:64Þ

1 ðuÞ ¼ exp j m u þ ð uÞ n 2 n! n¼0 Expanding the exponent into the Taylor series and comparing with the series (2.61), consisting of the moments of the PDF pðxÞ, one can conclude that the moments are linearly expressed in terms of quasi-moments (and vice versa). This is the reason why the term

20


‘‘quasi-moment’’ is used. Quasi-moments can be used to describe a random process in exactly the same way as regular moments, correlation functions and cumulants. It is important to mention that despite the fact that eq. (2.63) provides an approximation which has negative values for large $ $, the importance of the quasi-moment technique lies in the ability to provide an easy algorithm that can be used to obtain a similar approximation at the output of non-linear and linear systems.

2.2.2

The Laguerre Series

If the PDF pðxÞ has zero values for x < 0, then the Edgeworth series converges slowly to pðxÞ. In this case, a more convenient approximation, called the Laguerre series, can be used pðxÞ ¼

1 X

Cu exp½xx Lð Þ n ðxÞ

ð2:65Þ

n¼0 ð Þ

Here Ln ðzÞ are the generalized Laguerre polynomials [5] Lð Þ n ðzÞ

z dn ½expðzÞznþ ; ¼ expðzÞ n!d z

> 1

ð2:66Þ

The first four of these polynomials are ð Þ

L0 ¼ 1 ð Þ

L1 ¼ 1 þ z ð Þ

2 L2 ¼ ð þ 1Þð þ 2Þ 2 zð þ 2Þ þ z2

ð2:67Þ

ð Þ

6 L3 ¼ ð þ 1Þð þ 2Þð þ 3Þ 3 zð þ 2Þð þ 3Þ þ 3 z2 ð þ 3Þ z3 The Laguerre polynomials are orthogonal in the interval ½0; 1Þ with the weight p0 ðxÞ ¼ z expðzÞ, i.e. ð1 1 z expðzÞL n ðzÞL m ðzÞd z ¼ ðn þ þ 1Þm n ð2:68Þ n! 0

Here ðzÞ is the Gamma function [5]. Taking the orthogonality (2.68) into account, the coefficients of the expansion (2.65) can be found as ð1 n! Lð Þ pðxÞd x ð2:69Þ Cn ¼ ðn þ þ 1Þ 0 n If, instead of the original process ðtÞ, one considers a normalized process ¼ = (with the coefficient yet to be defined), then its PDF p ðyÞ can be expressed in terms of the PDF p ðxÞ as 1 x p ðyÞ ¼ p

ð2:70Þ


21

In a similar manner to eq. (2.65) one can write an expansion of the new PDF in terms of the Laguerre polynomials pðyÞ ¼

1 X

bn exp½yy Lð Þ n ðyÞ

ð2:71Þ

n¼0

Here, the coefficients of the expansion can be calculated as n! bn ¼ ðn þ þ 1Þ

ð1 0

Lð Þ n ðyÞp ðyÞdy

ð2:72Þ

After substituting eq. (2.67) in eq. (2.72) and taking into account normalization conditions for PDF and the definition of moments, one can obtain m b1 ¼ ð þ 2Þ

1 2m m2 b2 ¼ ð þ 1Þð þ 2Þ ð þ 2Þ þ 2 ð þ 3Þ 1þ

ð2:73Þ

Since at this moment and are arbitrary constants, it is possible to choose them in such a manner that both b1 and b2 vanish (i.e. the expansion will have less terms) b1 ¼ b2 ¼ 0. For this purpose the coefficients and have to be chosen as m2 m2 ¼ 1¼ 2 1 m2 m2 2 2 m2 m ¼ ¼ m m

ð2:74Þ

With this choice of and the first four coefficients in eq. (2.71) become 1 ; b0 ¼ ð þ 1Þ

b1 ¼ b2 ¼ 0;

1 m2 m3 b3 ¼ ð þ 3Þ 3 ð þ 1Þ 2

ð2:75Þ

Higher order coefficients have more complicated expressions. The Laguerre series is often used when the first term provides a good approximation, i.e. when the function pðxÞ is close to

1 x x p0 ðxÞ ¼ exp ð þ 1Þ

ð2:76Þ

where and are defined through the mean and the variance of pðxÞ as in eq. (2.74). Thus, the first term in the Laguerre series is the Gamma distribution. Similar orthogonal expansions can be obtained for other weighting functions as long as they generate a complete system of basis functions. Not all probability densities have this property. A detailed discussion of this subject can be found in [2].

22

2.2.3


Gram–Charlier Series

There is another important method of representation of a PDF p ðxÞ of some random variable through a given PDF p ðxÞ of some standardized random variable . Without loss of generality one can assume that both random variables have zero mean and unit variance. This can always be achieved by considering a normalized and centred random variable 0 ¼

m1

ð2:77Þ

The PDFs of and 0 are related by x m1 1 p ðxÞ ¼ p0

ð2:78Þ

Thus, if the series for p0 is known then the series for p can be easily obtained by changing the variable. In this section the following approximation is considered p ðxÞ ¼

1 X k¼0

k

dk p ðxÞ d xk

ð2:79Þ

Such expansion is especially convenient when the derivative of the PDF p ðxÞ can be expressed in the following form dk p ðxÞ ¼ Qk ðxÞp ðxÞ d xk

ð2:80Þ

where Qk ðxÞ are well studied functions, for example polynomials5 of order k. Taking the (inverse) Fourier transform of both sides, one obtains the series relating the characteristic functions ð j uÞ and ð j uÞ:

ð j uÞ ¼

1þ

1 X

! ð j uÞk ð1Þk k ð j uÞ

ð2:81Þ

k¼1

Using representation of the characteristic functions through the moments of the corresponding distributions, one obtains the following relation between the unknown coefficients k of the expansion (2.79) and the moments of both distributions 1 X ð j uÞn n¼0

n!

mn

1 u2 X ð j uÞn m n ¼1 þ 2 n! n¼3

¼

1þ

1 X k¼1

5

k

k

ð j uÞ ð1Þ k

!

1 u2 X ð j uÞl m l 1 þ 2 l! l¼3

The existence of such densities is a consequence of the Rodriguez formula [5].

! ð2:82Þ

TRANSFORMATION OF RANDOM VARIABLES

23

Finally, equating coefficients of equal powers of u, it is possible to obtain the following expressions for the coefficients k : 1 ¼ 2 ¼ 0 1 3 ¼ ðm 3 m 3 Þ 3! 1 4 ¼ ðm 4 m4 Þ 4! 1 5 ¼ ½m 5 m 5 10ðm 3 m3 Þ 5! 1 6 ¼ ½m 6 m 6 15ðm 4 m 4 Þ 20 m 3 ðm 3 m 3 Þ 6!

ð2:83Þ

It can be seen that the Gram–Charlier series coincides with the Edgeworth series if p ¼ expðx2 =2Þ pffiffiffiffiffiffi . 2 Some properties of classical orthogonal polynomials are listed in Appendix B on the web; more detailed discussion can be found in [5,11].

2.3


Let a random variable ðtÞ be described by the PDF p ðxÞ which is assumed to be known. Applying a non-linear memoryless transformation ¼ gðÞ to this random variable, one obtains a new random variable , whose PDF p ðyÞ and moments mn are to be found. Moments and central moments can be easily found using the definition of a statistical average mn ¼ h i ¼ hg ðÞi n

ð1

n

1

gn ðxÞp ðxÞd x

n

n

n ¼ hð m1 Þ i ¼ hðgðÞ m1 Þ i ¼

ð2:84Þ ð1 1

½gðxÞ m1 n p ðxÞd x

ð2:85Þ

Thus, the calculation of moments of the random variable can be converted to the calculation of corresponding averages based on the known PDF p ðxÞ. No direct calculation of the PDF p ðyÞ is required. The next step is to calculate the PDF p ðyÞ itself. First, it is assumed that y ¼ gðxÞ is a monotonic function, and thus y ¼ gðxÞ is a one-to-one mapping. Furthermore, it should be noted that since ¼ gðÞ is a deterministic transformation, then if a value of the random variable falls inside a small interval ½x; x þ d x, then the random variable falls insides the region ½ y; y þ d y, where the boundaries of the interval are defined by the non-linear transformation and its derivative: y ¼ gðxÞ;

dy ¼

d gðxÞd x dx

ð2:86Þ

24


This means that probabilities of these two events are equal and thus d p ðyÞd y ¼ p ðxÞd x ¼ p ½hðyÞ hðyÞ dy

ð2:87Þ

Here x ¼ hðyÞ is a function inverse to y ¼ gðxÞ. For a monotonic y ¼ gðxÞ the inverse function is unique, i.e. has only one branch. If the function f ðxÞ is increasing then a positive increment of the argument d x > 0 results in a positive increment of the function dy > 0. However, if y ¼ gðxÞ is a decreasing function then the increment of the function is negative: dy < 0. However, the area under the curve between y and y þ d y, reflecting the probability, is a positive quantity, thus the absolute value must be used in eq. (2.87).

Figure 2.2 Transformation of a random variable: (a) one-to-one transformation; (b) two-to-one transformation.

If the inverse function has more than one branch the situation is more involved. In this case interval ½y; y þ d y is an image of as many non-intersecting intervals as there are branches of the inverse function. For example, if there are two branches x1 ¼ h1 ðyÞ and x2 ¼ g2 ðyÞ as in Fig. 2.2(b), then 2 ½y; y þ d y is satisfied when 2 ½x1 ; x1 þ d x1 or 2 ½x2 ; x2 þ d x2 . The equality of probabilities thus becomes d d p ðyÞjd yj ¼ p ðx1 Þjd x1 j þ p ðx2 Þjd x2 j ¼ p ðh1 ðyÞÞ h1 ðyÞ þ p ðh2 ðyÞÞ h2 ðyÞ ð2:88Þ dy dy The generalization to a case of M branches x ¼ hm ðyÞ, m ¼ 1; . . . ; M of the inverse function is straightforward d p ðhm ðyÞÞ hm ðyÞ p ðyÞ ¼ dy m¼1 M X

ð2:89Þ

It is also worth mentioning that while the probability of the transformed random variable p ðyÞ can be uniquely defined if the transformation and the input PDF are known, it is not


25

possible to uniquely restore the input distribution by observing the output random variable if the transformation is not one-to-one. The following examples demonstrate applications of the rule of transformation to a number of problems often encountered in applications. 2.3.1

Transformation of a Given PDF into an Arbitrary PDF

Let be a continuous random variable, uniformly distributed on an interval ½a; b. It is often desired to find a transformation y ¼ gðxÞ which transforms into another random variable whose PDF p ðyÞ is given. In this case, according to eq. (2.89) p ðyÞd y ¼

dx ba

ð2:90Þ

or x ¼ c þ ðb aÞ

ðy 1

p ðyÞd y ¼ c þ ðb aÞP ðyÞ

ð2:91Þ

where c is a constant of integration. Since P ð1Þ and P ð1Þ ¼ 1, this constant must be chosen as c ¼ a. In this case a $ $ b and, consequently xa 1 x a and gðxÞ ¼ P ð2:92Þ P ðyÞ ¼ ba ba Here P1 is a function, inverse to the CDF of the random variable . On the other hand, if one chooses y ¼ gðxÞ ¼ a þ ðb aÞP ðxÞ

ð2:93Þ

then eq. (2.89) results in the transformation of an arbitrary distribution p ðxÞ to a uniform distribution. Thus, in principle, any PDF can be transformed into another PDF. This fact is often used in numerical simulation. 2.3.2

PDF of a Harmonic Signal with Random Phase

The next important example is an ensemble of the harmonic functions with constant amplitude A, angular frequency !0 and random phase ’ sðt; ’Þ ¼ gð’Þ ¼ A0 sinð!0 t þ ’Þ

ð2:94Þ

Assuming that the PDF p’ ð’Þ is known, the distribution ps ðxÞ of the signal sðtÞ can be found for any given fixed moment of time t. However, the calculation is complicated by the fact that the inverse transformation ’ ¼ hðxÞ has infinitely many branches ’n ¼ a sinðx=A0 Þ !0 t;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi d ’n ¼ A20 x2 dx

ð2:95Þ

26


and thus the expression for the PDF ps ðxÞ is given by 8 1 X > 1 < pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p’ ð’n Þ ps ðxÞ ¼ A20 x2 n¼1 > : 0

jxj A0

ð2:96Þ

jxj > A0

since the resulting signal is confined to the interval ½A0 ; A0 . In many communication applications it is reasonable to assume that the phase ’ is uniformly distributed in the interval ½; . p’ ð’Þ ¼

1 ; j’j 2

ð2:97Þ

In this case the inverse transformation is limited to only two branches, each having the distribution (2.97). Thus, the distribution ps ðxÞ of the signal sðtÞ is ps ðxÞ ¼

2.4 2.4.1

1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ffi ; x A0 1 A0

jxj A0

ð2:98Þ

RANDOM VECTORS AND THEIR DESCRIPTION CDF, PDF and the Characteristic Function

A random vector ¼ ½1 ; 2 ; . . . ; n T is a set of random variables i , considered jointly. The number of components n is called the dimension of the random vector. Each component can be continuous, discrete or mixed. In order to describe the random vector one has to describe the range of values each component assumes and probabilities of each value. However, it is not sufficient to provide separate characteristics for each component—all components should be described jointly. Similarly to the case of a scalar random variable, considered above, it is possible to generalize a notion of the cumulative distribution function, probability density function, characteristic function and log-characteristic function for the vector case. The cumulative distribution function Pn ðxÞ and the joint PDF, suitable for description of any type of random vector, are defined as Pn ðxÞ ¼ Pn ðx1 ; x2 ; . . . ; xn Þ ¼ Probf1 < x1 ; 2 < x2 ; . . . ; n < xn g @n Pn ðx1 ; x2 ; . . . ; xn Þ pn ðxÞ ¼ pn ðx1 ; x2 ; . . . ; xn Þ ¼ @ x1 @ x2 @ xn

ð2:99Þ ð2:100Þ

The CDF defined by eq. (2.100) poses the following properties 1. The CDF Pn ðxÞ is equal to zero if at least one of its arguments is negative infinity and it approaches one if all the arguments are equal to positive infinity: Pn ðx1 ; . . . ; 1; . . . ; xn Þ ¼ 0 Pn ð1; . . . ; 1; . . . ; 1Þ ¼ 1

ð2:101Þ ð2:102Þ

27

RANDOM VECTORS AND THEIR DESCRIPTION

2. The CDF is a non-decreasing function in each of its arguments. 3. If the k-th component of the random vector n is excluded from consideration, the CDF Pnn1 of the remaining vector can be obtained as follows nn1 ¼ ½1 ; . . . ; k1 ; kþ1 ; . . . ; n T Pn ðx1 ; . . . ; 1; . . . ; xn Þ ¼ Pnn1 ðx1 ; . . . ; xk1 ; xkþ1 ; . . . ; xn Þ

ð2:103Þ

4. The CDF can be expressed as an n-fold repeated integral of the corresponding CDF ð x1 ð xn Pn ðxÞ ¼ pn ðxÞd x1 d x2 . . . d xn ð2:104Þ 1

1

The PDF pn ðxÞ of a random vector n must satisfy the following conditions, which follow from the properties of the CDF and relation (2.100): 1. The PDF is a non-negative function pn ðxÞ 0

ð2:105Þ

2. The PDF is normalized to unity, i.e. ð1 ð1 p ðxÞd x1 d x2 . . . d xn ¼ 1 1

ð2:106Þ

1

3. The PDF is a symmetric function with respect to its arguments. 4. If integrated over one of the arguments over the real axis, the PDF of the random vector of order n produces a PDF of a random vector of dimension ðn 1Þ obtained by excluding the component over which integration has been performed: ð1 pnn1 ðx1 ; . . . ; xk1 ; xkþ1 ; . . . ; xn Þ ¼ pn ðxÞd xk ð2:107Þ 1

If 1n ¼ n þ j n is a vector of complex random variables, this vector can be described as a vector with 2 n real components, allowing complex vectors to be handled within the framework developed above. The characteristic function n ð j uÞ of the random vector can be defined as the ndimensional Fourier transform of the corresponding PDF pn ðxÞ

n ð j uÞ ¼ n ð j u1 ; j u2 ; . . . ; j un Þ ð1 ð1 ¼ pn ðx1 ; . . . ; xn Þ exp½jðu1 x1 þ þ un xn Þd x1 d xn 1

1

ð2:108Þ

The characteristic function is a symmetric, continuous function of its arguments, j n ð j uÞj

n ð j0Þ ¼ 1. m n ð j u1 ; j u2 ; . . . ; j um Þ ¼ nn ð j u1 ; j u2 ; . . . ; j um ; 0; . . . ; 0Þ;

m m by setting the arguments to zero at coordinates not included in the sub-vector. If components f1 ; 2 ; . . . ; n g are independent then pn ðxÞ ¼

n Y

pk ðxk Þ;

and n ð j uÞ ¼

k¼1

n Y

k ð j uk Þ

ð2:110Þ

k¼1

The converse statement is also true: if the characteristic function is expressed as a product of characteristic functions of the components then the components of the random vector are independent.

2.4.2

Conditional PDF

Joint consideration of two or more random variables allows the introduction of a new concept not considered in the case of a scalar random variable—the concept of the conditional PDF (CDF or CF). Let be a random variable (or vector) which is observed simultaneously with a certain condition B. Then the probability that does not exceed x, given that the condition B is satisfied, is called the conditional CDF P ðx j BÞ of the random variable given the condition B: PjB ðx j BÞ ¼ Probf < x j Bg ¼

Probf < x; Bg ProbðBÞ

ð2:111Þ

The conditional PDF pjB ðxjBÞ is then defined as the derivative of the conditional CDF with respect to x. Its Fourier transform represents the conditional characteristic function jB ð j u j BÞ ð1 @ P ðx j BÞ; j B ð j u j BÞ ¼ p j B ðx j BÞ ¼ pjB ðx j BÞ expðj u xÞd x ð2:112Þ @x 1 Conditional CDF, PDF and CF satisfy all the properties obeyed by regular unconditional CDF, PDF and CF, as can be shown from the definitions (2.111) and (2.112). Upon generalization to the case of random vectors the definition (2.112) becomes pj ðx1 ; . . . ; xk jxkþ1 ; . . . ; xn Þ ¼

p; ðx1 ; . . . ; xk ; xkþ1 ; . . . ; xn Þ p ðxkþ1 ; . . . ; xn Þ

ð2:113Þ

or p; ðx1 ; . . . ; xk ; xkþ1 ; . . . ; xn Þ ¼ pj ðx1 ; . . . ; xk ; xkþ1 ; . . . ; xn Þp ðxkþ1 ; . . . ; xn Þ

ð2:114Þ

known as the formula of multiplication of probabilities. Applying it in a chain manner, one obtains pðx1 ; . . . ; xk ; xkþ1 ; . . . ; xn Þ ¼ pðx1 jx2 ; . . . ; xn Þpðx2 ; . . . ; xn Þ ¼ pðx1 jx2 ; . . . ; xn Þpðx2 jx3 ; . . . ; xn Þpðx3 ; . . . ; xn Þ ¼ pðx1 jx2 ; . . . ; xn Þpðx2 jx3 ; . . . ; xn Þ pðxn1 jxn Þpðxn Þ

ð2:115Þ

29


Two rules can be provided in order to eliminate variables from the expression for a conditional PDF. 1. To eliminate elements on the ‘‘left of the bar’’, i.e. to eliminate unconditional variables, one just has to integrate the conditional density over these variables ð1 pðx1 ; . . . ; xj1 ; xj ; xjþ1 ; . . . ; xk jxkþ1 ; . . . ; xn Þd xj 1

¼ pðx1 ; . . . ; xj1 ; xjþ1 ; . . . ; xk jxkþ1 ; . . . ; xn Þ

ð2:116Þ

2. To eliminate the variable on the ‘‘right of the bar’’, i.e. to eliminate condition variables, one has to multiply the conditional density by the conditional PDF of the conditions to be eliminated conditioned on all other conditions, and integrate over the variables to be eliminated ð1 pðx1 ; . . . ; xk jxkþ1 ; . . . ; xj1 ; xj ; xjþ1 ; . . . ; xn Þpðxj jxkþ1 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þd xj 1

¼ pðx1 ; . . . ; xk jxkþ1 ; . . . ; xj1 ; xjþ1 ; . . . ; xn Þ

ð2:117Þ

In particular, the following formula (playing a prominent role in the theory of Markov processes) can be obtained ð1 pðx1 jx2 ; x3 Þpðx2 jx3 Þd x2 ð2:118Þ pðx1 jx3 Þ ¼ 1

All the considered definitions and rules remain valid for the case of a discrete random variable, with integrals being reduced to sums. Random variables, 1 ; 2 ; . . . ; n are called mutually independent if events f1 < x1 g; f2 < x2 g; . . . ; fn < xn g are independent for any values of xi ; i ¼ 1; . . . ; n. In this case their joint PDF (CDF, CF) is a product of PDF (CDF, CF) of individual components, i.e. p ðx1 ; . . . ; xn Þ ¼

n Y

pi ðxi Þ

ð2:119Þ

i ðxi Þ

ð2:120Þ

i¼1

and ðx1 ; . . . ; xn Þ ¼

n Y i¼1

respectively. It is also interesting to note that it is possible to find a random vector whose components are pair-wise independent, i.e. pi j ðxi ; xj Þ ¼ pi ðxi Þpj ðxj Þ, but when considered jointly they are not independent p ðx1 ; x2 ; . . . ; xn Þ 6¼ p1 ðx1 Þp2 ðx2 Þ . . . pn ðxn Þ. Two complex variables 1 ¼ 1 þ j 1 ; . . . ; n ¼ n þ j n are mutually independent if pð1 ; 2 ; . . . ; n ; 1 ; 2 ; . . . ; n Þ ¼

n Y

pði ; i Þ

ð2:121Þ

i¼1

Note that independence of the real and imaginary parts of each variable is not required.

30


2.4.3

Numerical Characteristics of a Random Vector

The numerical characteristics considered in Sections 2.1.1.5–2.1.1.8 can be extended to the case of random vectors. In this section it is assumed that a random vector n ¼ ½1 ; 2 ; . . . ; n T is described by the joint PDF pn ðxÞ. In this case the n-dimensional moment m 1 ; 2 ;...; n of order ¼ 1 þ 2 þ þ n is defined as m 1 ; 2 ; ...; n ¼

ð1 1

ð1 1

x 1 1 x 2 2 . . . x n n pn ðxÞd x1 d x2 . . . d xn

ð2:122Þ

Similarly, the central moments can be defined as 1 ; 2 ; ...; n ¼

ð1 1

ð1 1

ðx1 m1 1 Þ 1 ðx2 m1 2 Þ 2 ðxn m1 n Þ n pn ðxÞd x1 d x2 . . . d xn ð2:123Þ

where m1 k ¼ m0; 0; ...;1;...;0 ¼ Efk g is the average of the k-th component k of the random vector n. Moments can be obtained as coefficients of the power series expansion of the characteristic function since (

"

n ð j u1 ; j u2 ; . . . ; j un Þ ¼ E exp j

n X

#) uk xk

k¼1 1 X

¼

1 ... n

m 1 ; 2 ;...; n ð j u1 Þ 1 ð j u2 Þ 2 ð j un Þ n ! ! n ! ¼0 1 2

ð2:124Þ

where, of course, m 1 ; 2 ;...; n ¼ ðjÞ

1 þ 2 þ þ n

@ 1 þ 2 þ þ n ðu ; u ; . . . ; u Þ n 1 2 n @ u 1 1 @ u 1 1 @ u 1 1 u1 ¼u2 ¼ ¼un ¼0 ð2:125Þ

In a similar manner, n-dimensional cumulants can be defined as coefficients of the power series of the log-characteristic function 1 þ 2 þ þ n

1 ; 2 ;...; n ¼ ðjÞ

@ 1 þ 2 þ þ n 1 1 1 ln n ðu1 ; u2 ; . . . ; un Þ @ u1 @ u1 @ u1 u1 ¼u2 ¼ ¼un ¼0 ð2:126Þ

A relation between moments and cumulants of a random vector can be established in a way similar to relations obtained for moments and cumulants of a random variable. Finally, conditional densities can be used to define conditional moments and cumulants.

31


As in the case of a random variable, lower order moments and cumulants play a prominent role in the description of a random vector. In particular, averages of components m1 k ¼ Efk g ¼ m0;...;1;...;0 ; k ¼ 1;

l ¼ 0;

k 6¼ l

ð2:127Þ

their variance 2k and m2 k 2k ¼ Dk ¼ Efðk m1 k Þ2 g ¼ 0;...;2;...;0 m2 k ¼ Efk2 g ¼ m0;...;2;...;0

ð2:128Þ

describe the main features of the individual components k . Statistical dependence between any two components is described by mixed moments, especially by the correlation Rk l and covariance Ck l , defined as non-central and central mixed moments of the second order k ;l ¼ Rk l ¼ Efk l g m1;1 k ;l 1;1 ¼ Ck l ¼ Efðk m1 k Þðl m1 l Þg

ð2:129Þ

with all sub-indices in the definition (2.122) and (2.123) equal to zero except for the k-th and l-th ones. There is a simple relation between correlation and covariance (resulting from the relation between the central moments and moments) Rk l ¼ Ck l þ m1 k m1 l

ð2:130Þ

Since the expectation of a non-negative random variable is non-negative, it is possible to write that, for any real , Ef½ðk m1 k Þ þ ðl m1 l Þ2 g ¼ 2 2k þ 2 Ck l þ 2l 0

ð2:131Þ

This can be satisfied for an arbitrary if and only if the discriminant of the quadratic form in eq. (2.131) is negative, i.e. C2k l 2k 2l 0

ð2:132Þ

This allows one to introduce a so-called correlation coefficient r between two components of the random vector r¼

Ck l ; k l

jrj 1

ð2:133Þ

Two random variables are called uncorrelated if r ¼ 0. This does not imply independence of these variables. It follows from eq. (2.131) that the maximum of the correlation coefficient is achieved when two random variables are linearly dependent, i.e. k ¼ a l þ b.

32


The notion of correlation can be extrapolated to a case of complex random variables 1 and 2 as C1 2 ¼ Efð1 m1 Þð2 m2 Þ g

ð2:134Þ

where the asterisk indicates complex conjugation.

2.5

GAUSSIAN RANDOM VECTORS

Since Gaussian random variables play a prominent role in statistical signal processing and communication theory, their most important properties are summarized in this section. A continuous random variable is called a Gaussian random variable if its PDF is given by " # 1 ðx mÞ2 p ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffi exp 2 2 2 2

ð2:135Þ

The corresponding characteristic function is then

2 u2 ð j uÞ ¼ exp j u m 2

ð2:136Þ

It is possible to show by direct integration that for a Gaussian random variable and an arbitrary deterministic function y ¼ gðxÞ [9,12] 2n

@n d n EfgðÞg ¼ 2 E gðÞ @ Dn d 2 n

d gðÞ Ef gðÞg ¼ m EfgðÞg þ D E d

ð2:137Þ ð2:138Þ

Here D ¼ 2 . Setting gðxÞ ¼ xn in eq. (2.137) allows easy computation of the moments of a Gaussian random variable. Two random variables 1 and 2 are called jointly Gaussian if their joint PDF has the following form " # 1 D2 ðx1 m1 Þ2 211 ðx1 m1 Þðx2 m2 Þþ D1 ðx2 m2 Þ2 ffi exp p2N ðx1 ;x2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2ðD1 D2 211 Þ 2 D1 D2 211 " # 1 22 ðx1 m1 Þ2 2r 1 2 ðx1 m1 Þðx2 m2 Þ þ 21 ðx2 m2 Þ2 pffiffiffiffiffiffiffiffiffiffiffi exp ¼ 221 22 ð1 r 2 Þ 21 2 1 r 2 ð2:139Þ

33

GAUSSIAN RANDOM VECTORS

The corresponding characteristic function is then

1 2 2 2 N ð j u1 ; j u2 Þ ¼ exp jðm1 u1 þ m2 u2 Þ ðD1 u1 þ 2 1 1 u1 u2 þ D2 u2 Þ 2

1 2 2 2 2 ¼ exp jðm1 u1 þ m2 u2 Þ ð1 u1 þ 2 r 1 2 u1 u2 þ 2 u2 Þ 2

ð2:140Þ

This distribution has five parameters: the averages of each component m1 , m2 , the variances of each component 21 , 22 , and the correlation coefficient r between the components m1 ¼ Ef1 g; 1 1

m2 ¼ Ef2 g;

21 ¼ Ef12 g; 1 1 ¼ Efð1 m1 Þð2 m2 Þg; r ¼ 1 2

22 ¼ Ef22 g

ð2:141Þ ð2:142Þ

Taking the derivative of both sides of eq. (2.140) with respect to r, one obtains a useful relationship @n ð j u1 ; j u2 Þ ¼ ð1Þn ð1 2 u1 u2 Þn ð j u1 ; j u2 Þ n @r

ð2:143Þ

Further properties of mutually Gaussian random variables are illustrated by examples of two Gaussian random variables. More detailed discussion can be found in [1,2,13]. 1. If 1 and 2 are uncorrelated, i.e. r ¼ 0, then according to eqs. (2.139) and (2.140) p2 N ðx1 ; x2 Þ ¼ p1 ðx1 Þp2 ðx2 Þ 2 N ð j u1 ; j u2 Þ ¼ 1 ð j u1 Þ2 ð j u2 Þ

ð2:144Þ

Thus, two uncorrelated Gaussian random variables are also independent. 2. Two correlated jointly Gaussian random variables can be transformed into a pair of independent zero mean Gaussian random variables by a linear transformation 1 ¼ ð1 m1 Þ cos þ ð2 m2 Þ sin 2 ¼ ð1 m1 Þ sin þ ð2 m2 Þ cos

ð2:145Þ

Here, the angle can be defined from the condition Ef1 ; 2 g ¼ 0 to be tan 2 ¼

2 r1 2 21 22

¼ =4

if

1 6¼ 2

if

1 ¼ 2

ð2:146Þ

3. Any linear transformation of jointly Gaussian random variables results in jointly Gaussian random variables. 4. If two random variables are jointly Gaussian, each variable by itself is also Gaussian. The inverse statement is not necessarily valid.

34


5. If two random variables 1 and 2 are jointly Gaussian, then both conditional densities p1 j2 ðx1 jx2 Þ and p2 j1 ðx2 jx1 Þ are Gaussian. For example 8

2 9 > > 2 > > > x2 m2 r ðx1 m1 Þ > < = p1 ;2 ðx1 ; x2 Þ 1 1 ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp p2 j1 ðx2 jx1 Þ ¼ > > p1 ðx1 Þ 2 22 ð1 r 2 Þ 2 2 ð1 r 2 Þ > > > > ; : ð2:147Þ It can be seen by inspection of eq. (2.147) that this PDF is a Gaussian PDF with the mean m ¼ m2 þ rð2 =1 Þðx1 m1 Þ

ð2:148Þ

D ¼ 22 ð1 r 2 Þ

ð2:149Þ

and the variance

Interestingly, the mean of the conditional PDF depends on the condition, while the variance does not, except for the correlation coefficient. 6. The joint PDF (2.139) can be expanded into the Miller series [13] " # 1 1 ðx1 m1 Þ2 ðx2 m2 Þ2 X x1 m 1 x2 m 2 r n p2 ðx1 ; x2 Þ ¼ exp Hn Hn 2 1 2 1 2 n! 2 21 2 22 n¼0 ð2:150Þ where Hn ðxÞ are Hermitian polynomials [5]. The advantage of this expansion is the fact that variables x1 , x2 and r are separated in each term. Most of the considered properties can be generalized to a case of a jointly Gaussian vector of dimension n. If mk and 2k ¼ Dk stand for the mean value and the variance of the k-th component and pffiffiffiffiffiffiffiffiffiffiffi Rk l ¼ Efðk mk Þðl ml Þg ¼ rk l Dk Dl ¼ rk l k l ; Rk k ¼ Dk ; Rk l ¼ Rl k ð2:151Þ The marginal joint PDF and the corresponding characteristic function are then given by " # 1 ðx mÞT R1 ðx mÞ pN ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð2:152Þ 2 ð2Þn jRj

uT Ru T ð2:153Þ N ð j uÞ ¼ exp j m u 2 Here 2

3 m1 6 m2 7 7 m¼6 4 . . . 5; mn

2

R11 6 R21 R¼6 4 ... Rn 1

R12 R22 ... Rn 2

... ... ... ...

3 2 R1 n r11 6 r21 R2 n 7 7 ¼ diagfi g6 4 ... ... 5 rn 1 Rn n

r12 r22 ... rn 2

... ... ... ...

3 r1 n r2 n 7 7diagfi g ... 5 rn n ð2:154Þ

35

TRANSFORMATION OF RANDOM VECTORS

are the vector of averages and the covariation matrix, respectively. Vectors x and u are vector-columns of dimension n. R1 represents the inverse matrix and T is used to denote transposition. Since rk l ¼ rl k , the covariation matrix R is a symmetric matrix and contains nðn þ 1Þ=2 independent elements. Altogether, nðn þ 1Þ=2 þ n parameters are needed to uniquely define Gaussian distribution. The conditional probability density can be easily obtained as pðx1 ; x2 ; . . . ; xk jxkþ1 ; . . . ; xn Þ ¼ pðX 1 jX2 Þ " # ðx mX1 jX2 ÞT R1 ðx m Þ 1 X jX X1 jX2 1 2 ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2 k ð2Þ jRX1 jX2 j

ð2:155Þ

where

R R ¼ 11 R21

R12 ¼ R11 R12 R1 R 22 R21 ; R22 X1 jX2

and mX1 jX2 ¼ m1 þ R12 R1 22 ðX 2 m2 Þ ð2:156Þ

and the mean vector and the correlation matrix are split according to conditional and nonconditional variables.

2.6


In this section the question of transformation of random vectors is treated. To simplify derivations, the case of a vector with two components is considered first and the results are generalized to the case of an arbitrary number of components. Consider two random variables 1 and 2 , described by the joint PDF pn ðx1 ; x2 Þ which are being transformed by two non-linear functions 1 ¼ g1 ð1 ; 2 Þ 2 ¼ g2 ð1 ; 2 Þ

ð2:157Þ

or, in the vector form g ¼ gðÞ, where g ¼ ½1 ; 2 T and g ¼ ½g1 ; g2 T . It is required to find the joint PDF p ðy1 ; y2 Þ of the new vector g. It is easier to approach this problem by finding the joint CDF Pg ðy1 ; y2 Þ. Let S be an area of the plane x1 x2 such that for any point ðx1 ; x2 Þ 2 S the following conditions are satisfied g1 ðx1 ; x2 Þ < y1 ; and g2 ðx1 ; x2 Þ < y2

ð2:158Þ

Events f1 < y1 ; 2 < y2 g and ff1 ; 2 g 2 Sg are then equivalent, since one implies another. Thus, their probabilities are equal as well, i.e. ð Pg ðy1 ; y2 Þ ¼

pn ðx1 ; x2 Þd x1 d x2 S

ð2:159Þ

36


In general, the area S can be a union of disjointed sets, thus the calculation of the integral may require integration over a few disjointed sets. If d Sy represents an area in the x1 x2 plane such that y1 g1 ðx1 ; x2 Þ < y1 þ d y1 ; and y2 g2 ðx1 ; x2 Þ < y2 þ d y2

ð2:160Þ

then ð Pg ðy1 þ d y1 ; y2 þ d y2 Þ Pg ðy1 ; y2 Þ ¼ pg ðy1 ; y2 Þd y1 d y2 ¼

pn ðx1 ; x2 Þd x1 d x2 ð2:161Þ d Sy

If the inverse vector function x ¼ hðyÞ has k branches, x ¼ hðiÞ ðyÞ, i ¼ 1; . . . ; k. Let x1 x2 and y1 y2 be two sets of Cartesian coordinates, then the area of the x1 x2 plane corresponding to a small rectangular d Sy ¼ d y1 d y2 is given by @ ðiÞ k k X X @ y1 h1 ðy1 ; y2 Þ d Sx ¼ jJk jd Sy ¼ ðiÞ i¼1 i¼1 @ h2 ðy1 ; y2 Þ @ y1

@ ðiÞ h1 ðy1 ; y2 Þ @ y2 d Sy @ ðiÞ h2 ðy1 ; y2 Þ @ y2

ð2:162Þ

Dðx1 ; x2 Þ is a Jacobian of the transformation of each branch of the function Dðy1 ; y2 Þ x ¼ hðyÞ. Thus, it follows from eq. (2.161) that the transformation of the PDF is where J ¼

@ ðiÞ h k X @ y1 1 ðiÞ ðiÞ pg ðy1 ; y2 Þ ¼ pn ðh1 ðy1 ; y2 Þ; h2 ðy1 ; y2 ÞÞ @ ðiÞ i¼1 h @ y1 2

@ ðiÞ h @ y2 1 @ ðiÞ h @ y2 2

ð2:163Þ

In general, if a random n ¼ ½1 ; 2 ; . . . ; n T vector of length n is transformed to another vector g ¼ ½1 ; 2 ; . . . ; n T by a non-linear transformation y ¼ gðxÞ having k branches of the inverse transformation x ¼ hðiÞ ð yÞ, then the probability density function of the transformed random variable g is given by p ð yÞ ¼

k X i¼1

DðxðiÞ Þ p ðh ðyÞÞ Dð yÞ ðiÞ

ð2:164Þ

where @ ðiÞ h ðyÞ @ y1 1 DðxðiÞ Þ ... ¼ Dð yÞ @ hðiÞ ðyÞ @ y1 n

@ ðiÞ ... h1 ðyÞ @ yn ... ... @ ðiÞ ... hn ðyÞ @ yn

is the Jacobian of the i-th branch of the inverse transformation x ¼ hðiÞ ð yÞ.

ð2:165Þ

37


As in the case of a scalar random variable, it is possible to calculate moments of the transformed random variable without directly calculating the PDF p ðyÞ. Indeed, the definition of moments of an arbitrary order implies that n m 11;; 2 ;...; 2 ; ...; n

¼

Ef1 1 2 2

. . . n n g

¼

ð1 1

ð1 1

h 1 1 ðyÞh 1 1 ð yÞ h 1 1 ð yÞd y

ð2:166Þ

If the transformation maps a vector of dimension n into a vector of dimension m < n, the vector g can be augmented by n m dummy variables, often by a sub-vector ½l1 ; . . . ; lnm of the vector n, to allow use of eq. (2.166). The resulting PDF can then be integrated over dummy variables to obtain the PDF of the original vector. Evaluation of moments and cumulants of non-linear transformation using moment and cumulant brackets are considered in Section 2.8. The application of the technique described in this section is illustrated by a number of examples, which have their own importance in applications.

2.6.1

PDF of a Sum, Difference, Product and Ratio of Two Random Variables

Let the joint PDFs pn ðx1 ; x2 Þ of two random variables 1 and 2 be known. The PDF of the sum (difference) of these two variables ¼ 1 2 is to be determined. Introducing a dummy variable 2 ¼ 2 , one can formulate this problem in a standard form. The inverse transformation is then 1 1 1 ¼ 2 ¼1 ð2:167Þ J ¼ 0 1 2 ¼ 2 Thus, according to eq. (2.164), the joint PDF p 2 ðy1 ; y2 Þ is equal to p 2 ðy1 ; y2 Þ ¼ pn ðy y2 ; y2 Þ

ð2:168Þ

Integration over the dummy variable 2 results in the PDF of the sum (difference) of two random variables ð1 pn ðy y2 ; y2 Þd y2 ; and ¼ 1 2 ð2:169Þ p ðyÞ ¼ 1

Equation (2.169) can be further simplified if 1 and 2 are independent. In this case pn ðx1 ; x2 Þ ¼ p1 ðx1 Þp2 ðx2 Þ and ð1 p1 ðy y2 Þp2 ðy2 Þd y2 ð2:170Þ p ðyÞ ¼ 1

In general, if there are n independent random variables 1 ; . . . ; n , the characteristic function of the weighted sum, ¼ a1 1 þ a2 2 þ þ an n ¼

n X k¼1

ak k

ð2:171Þ

38


of these variables can be easily found to be * !+ n n n Y X Y ak k hexpðj u ak k Þi ¼ k ð j u ak Þ ð j uÞ ¼ hexpðj u Þi ¼ exp j u ¼ k¼1

k¼1

k¼1

ð2:172Þ Similarly, the expression for the log-characteristic function becomes ð j uÞ ¼ ln ð j uÞ ¼

n X

ln k ð j u ak Þ ¼

k¼1

n X

k ð j u ak Þ

ð2:173Þ

k¼1

The resulting distribution is called the composition of the distributions of each variable. It can be seen from eq. (2.172) that ðjuÞ ¼

1 X l

l¼1

l!

l

ðjuÞ ¼

1 Pn n l k X X k¼1 ak l ðjuak Þ ¼ ðjuÞl ; l ¼ alk l k l! l! l¼1 k¼1

n X 1 k X

l

l

k¼1 l¼1

ð2:174Þ

Thus, the cumulant of an arbitrary order of the sum can be expressed as a weighted sum of the cumulants of the same order. In particular, eq. (2.174), written for the average m and the variance D ¼ 2 , produces the following rule m ¼

n X

ak mk ;

k¼1

D ¼

n X

a2k Dk

ð2:175Þ

k¼1

If all variables have the same distribution and all the weights are equal, ai ¼ a, then eq. (2.172) is reduced to ð j uÞ ¼ n ð j u aÞ; ð j uÞ ¼ n ð j u aÞ; k ¼ n ak k

ð2:176Þ

Since for an arbitrary Gaussian random variable all the cumulants of order higher than two are equal to zero, eq. (2.174) shows that all the cumulants of the resulting random variable are also equal to zero, i.e. the random variable is also a Gaussian random variable. Finally, eq. (2.174) allows the investigation of the limiting distribution of the average of identically distributed random variables ¼

1 þ 2 þ þ N N

ð2:177Þ

Setting n ¼ N and a ¼ 1=N in eq. (2.176), one obtains k

N k k ¼ k ¼ k1 N N

ð2:178Þ

Thus, it can be seen, that the transformation (2.177) preserves the mean value, since 1 ¼ 1 . However, all other cumulants are reduced by a factor N k1. For sufficiently large N

39


all cumulants of order higher than two can be neglected compared to the two first cumulants, i.e. pffiffiffiffi 2 2 ð= N Þ2 u2 ð2:179Þ ð j uÞ j u 1 u ¼ jum þ OðN 2 Þ 2N 2 Thus, the distribution of the average of a large number of identically distributed random variables differs little from a Gaussian distribution, with the mean p value ffiffiffiffi equal to the mean value of each component and the variance reduced by a factor of N . If random variables and are the product and ratio of two random variables 1 and 2 whose joint distribution is known, a similar approach allows one to obtain the PDF p ðyÞ of the result. Indeed, setting a dummy variable 2 ¼ 2 ; 2 ¼ 2 , one obtains ¼ 1 2 ; 1 ¼ =2 2 ¼ 2 ; 2 ¼ 2 y1 1 Dðx1 ; x2 Þ 2 1 ¼ y2 y2 ¼ ; J ¼ Dðy1 ; y2 Þ jy2 j 0 1 1 1 ¼ 2 ¼ ; 2 2 ¼ 2 2 ¼ 2 ; Dðx1 ; x2 Þ y2 ¼ J ¼ Dðy ; y Þ 0 1

2

y ¼ jy2 j; 1

p ðyÞ ¼

p ðyÞ ¼

ð1

ð1 1

1

p

y d y2 ; y2 y2 jy2 j

p ðy y2 ; y2 Þjy2 jd y2

ð2:180Þ

ð2:181Þ

Equations (2.180) and (2.181) are simplified if 1 and 2 are independent. In this case ð1 ð1 y d y2 ; and p ðyÞ ¼ p 1 p1 ðy y2 Þp2 ðy2 Þjy2 jd y2 ð2:182Þ p ðyÞ ¼ p2 ðy2 Þ y2 jy2 j 1 1

2.6.2

Probability Density of the Magnitude and the Phase of a Complex Random Vector with Jointly Gaussian Components

Let ¼ R þ j I be a complex random variable with jointly Gaussian real and imaginary parts. This distribution is described by five parameters: the mean values of each component mR and mI , their variances 2R and 2I and the correlation coefficient r. The magnitude A and the phase of can be obtained using the following non-linear transformation qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi I A ¼ R2 þ I2 ; ¼ atan ; A 0; j j ð2:183Þ R Such a problem arises in many cases when a modulated harmonic signal with random phase is studied. The first step is to transform R and I into new variables by a linear transformation (2.145) 1 ¼ R cos þ I sin 2 ¼ R sin þ I cos

ð2:184Þ

40


Here, the angle is defined as tan 2 ¼

2 r 1 2 21 22

¼ =4

if 1 6¼ 2

ð2:185Þ

if 1 ¼ 2

In this case the Gaussian components 1 and 2 are independent, with the joint distribution " # ðy1 m1 1 Þ2 ðy2 m1 2 Þ2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp p1 ;2 ðy1 ; y2 Þ ¼ 2 D1 2 D2 2 D1 D2

ð2:186Þ

where the new means and variances are defined as m1 1 ¼ mR cos þ mI sin

ð2:187Þ

m1 2 ¼ mI cos mR sin pffiffiffiffiffiffiffiffiffiffiffiffi D1 ¼ DR cos2 þ DI sin2 þ r DR DI sin 2 pffiffiffiffiffiffiffiffiffiffiffiffi D2 ¼ DI cos2 þ DR sin2 r DR DI sin 2

ð2:188Þ

The next step is to introduce polar coordinates qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A ¼ 12 þ 22 ¼ R2 þ I2 ;

¼ ¼ atan

2 1

ð2:189Þ

with the inverse transformation given by 1 ¼ A cos ; 2 ¼ A sin ;

cos J ¼ sin

A sin A cos

¼ A

ð2:190Þ

Using eq. (2.164) together with eqs. (2.186), (2.189) and (2.190), one readily obtains the joint PDF pA; ðA; Þ of the magnitude and the phase "

pA; ðA; Þ ¼

A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2 D1 D2

ðA cos

m1 1 Þ2 ðA sin m1 2 Þ2 2 D1 2 D2

# ð2:191Þ

The PDF pA ðAÞ of the magnitude and the PDF p ð Þ ¼ p ð Þ can be obtained by the integration of the joint density pA; ðA; Þ over the phase or magnitude A variables, respectively. Depending on relations between the parameters of the distribution pA; ðA; Þ a number of different distributions can arise from eq. (2.191). Detailed investigation of the particular cases can be found in [12]. Here, only two particularly important cases are considered.

41


2.6.2.1

Zero Mean Uncorrelated Gaussian Components of Equal Variance

In this case mR ¼ mI ¼ 0, r ¼ 0 and DR ¼ DI ¼ D ¼ 2 . Using eqs. (2.184) and (2.185) one obtains m1 1 ¼ m1 2 ¼ 0, D1 ¼ D2 ¼ D, ¼ =4. This reduces eq. (2.191) to pA;

A A2 A A2 1 ðA; Þ ¼ exp 2 ¼ exp 2 2 2 2 2 2

ð2:192Þ

Integration over the phase produces the Rayleigh distribution of the magnitude and the uniform distribution of phase

A A2 pA ðAÞ ¼ 2 exp 2 ; 2

A 0;

p ð Þ ¼

1 ; 2

j j

ð2:193Þ

In addition, it can be seen from eqs. (2.192) and (2.193) that pA; ðA; Þ ¼ pA ðAÞp ð Þ and thus the phase and the magnitude are independent.

2.6.2.2

Case of Uncorrelated Components with Equal Variances and Non-Zero Mean pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi In this case D1 ¼ D2 ¼ D, ¼ =4, m ¼ m2R þ m2I ¼ m21 1 þ m21 2 > 0 and the joint PDF becomes, (2.191), m 1 1 cos A þ m 2Am A m 4 ðA; Þ ¼ exp 2 2 2 2 2

pA;

2

2

3 m1 2 þ sin m 5

A A2 þ m2 2 A m cosð 0 Þ ¼ exp 2 2 2 2

ð2:194Þ

Here tan 0 ¼

m1 2 m1 1

ð2:195Þ

Further integration over the phase variable produces the Rice distribution for the magnitude

A A2 þ m2 2 A m cosð 0 Þ pA; ðA; Þd ¼ exp d

pA ðAÞ ¼ 2 2 2 2

ð

A A2 þ m 2 A m cosð 0 Þ A A2 þ m 2 Am exp exp exp d

¼ I 0 2 2 2 2 2 2 2 2 2 2 ð

ð

ð2:196Þ

42


Similarly, integration of pA; ðA; Þ over A produces a PDF of the phase with the following form

1 m2 m cosð 0 Þ m2 sin2 ð 0 Þ m cosð 0 Þ pffiffiffi pffiffiffiffiffiffiffiffiffiffiffi exp p’ ð Þ ¼ exp 2 þ 1 þ erf 2 22 2 2 2 22 ð2:197Þ Here erfðxÞ is the error function [5]: 2 erfðxÞ ¼ pffiffiffi

ðx

expðt2 Þd t:

ð2:198Þ

0

It can be seen that if m ¼ 0, eq. (2.196) produces the Rayleigh distribution since I0 ð0Þ ¼ 1, and eq. (2.197) describes the uniform distribution of the phase. Characteristically, the case of a large m= produces approximately Gaussian distributions for both the magnitude and the phase, as can be observed by inspection of Fig. 2.3.

Figure 2.3

2.6.3

Dependence of the magnitude and the phase distribution on the ratio.

PDF of the Maximum (Minimum) of two Random Variables

Once again, let p1 ;2 ðx1 ; x2 Þ be the joint PDF of two random variables 1 and 2 . In some applications, such as combining techniques, it is required to find the distribution of the following random variables ¼ maxf1 ; 2 g ¼ minf1 ; 2 g

ð2:199Þ

43


In this case, the method of non-linear transformation suggested above does not produce the desired result easily. However, this problem can be solved by finding the CDF of the result first. Indeed, based on the definition of CDF, one can write P ðyÞ ¼ Probf ¼ maxf1 ; 2 g < yg ¼ Probf1 < y; 2 < yg ¼

ðy 1

ðy 1

p1 ;2 ðx1 ; x2 Þd x1 d x2 ð2:200Þ

Taking the derivative with respect to y of both sides of eq. (2.200), the following expression for the PDF of can be obtained d d P ðyÞ ¼ p ðyÞ ¼ dy dy

ðy 1

ðy 1

p1 ;2 ðx1 ; x2 Þd x1 d x2

ð2:201Þ

If 1 and 2 are independent, this expression can be further simplified to produce p ðyÞ ¼ p1 ðyÞP2 ðyÞ þ p2 ðyÞP1 ðyÞ

ð2:202Þ

where Pi ðyÞ is the CDF of i , i ¼ 1; 2. In the same manner one can obtain the PDF of the random variable ¼ minf1 ; 2 g. In this case P ðyÞ ¼ Probf ¼ minf1 ; 2 g < yg ¼ Probf1 < y; x1 x2 g þ Probf2 < y; x2 x1 g ðy ð1 ðy ð1 ¼ d x1 p1 ;2 ðx1 ; x2 Þd x2 þ dx p1 ;2 ðx1 ; x2 Þd x1 ð2:203Þ 1

1

x1

x2

Taking the derivative of CDF, (2.203), the desired PDF is obtained in the following form d p ðyÞ ¼ dy

ðy

ð1

1

d x1 x1

p1 ;2 ðx1 ; x2 Þd x2 þ

ðy 1

ð1 d x2

p1 ; ðx1 ; x2 Þd x1

ð2:204Þ

x2

Finally, for independent 1 and 2 , eq. (2.204) reduces to p ðyÞ ¼ p1 ðyÞ½1 P2 ðyÞ þ ½1 P1 ðyÞp2 ðyÞ

ð2:205Þ

Example: selection combining. Two antennas, separated by a relatively large distance, are used to receive a signal from the same source. In each antenna the signal-to-noise ratio (SNR) is described by the so-called Gamma distribution i mi 1 mm mi i i exp pi ð i Þ ¼ m ;

i ðmi Þ

i

i ¼ 1; 2

ð2:206Þ

where mi and i are the parameters of the distribution (severity of fading and the average SNR, respectively). The antenna with the highest SNR is chosen to connect to the receiver. It is possible to assume that the levels of signals in both antennas are independent, thus

44


eq. (2.202) can be used to obtain the distribution of the SNR at the input of the receiver. It produces 1 2 mm m 1 1 m2 2 mm m2 2 m1 1 1 m11 2 m21 exp exp pð Þ ¼ m1 P m2 ; þ m2 P m1 ;

1 ðm1 Þ

1

2

2 ðm2 Þ

2

1 ð2:207Þ where 1 Pðm; xÞ ¼ ðmÞ

ðx tm1 expðtÞd t

ð2:208Þ

0

is one of the forms of incomplete Gamma function [5]. For the case of independent Rayleigh fading with equal average SNR in both antennas, eq. (2.207) can be simplified to produce

2

pð Þ ¼ exp 1 exp

2.6.4

ð2:209Þ

PDF of the Maximum (Minimum) of n Independent Random Variables

The equations obtained in the previous section can be generalized to the case of n independent random variables 1 ; 2 ; . . . ; n , each described by their PDF p1 ðx1 Þ; . . . ; pn ðxn Þ. New variables are defined as ¼ maxf1 ; 2 ; . . . ; n g ¼ maxf1 ; 2 ; . . . ; n g

ð2:210Þ

Following the same steps as in Section 2.6.3 one obtains n n n Y X d d Y pk ðyÞ P ðyÞ ¼ p ðyÞ ¼ Pk ðyÞ ¼ Pk ðyÞ dy d y k¼1 P ðyÞ k¼1 k¼1 k

ð2:211Þ

Similarly the PDF p ðyÞ can be expressed as n n n Y X d d Y pk ðyÞ P ðyÞ ¼ ½1 Pk ðyÞ ¼ ½1 Pk ðyÞ p ðyÞ ¼ dy d y k¼1 1 Pk ðyÞ k¼1 k¼1

2.7

ð2:212Þ

ADDITIONAL PROPERTIES OF CUMULANTS

It was shown earlier (see eq. (2.42)) that the characteristic function ð j uÞ of a process with finite moments mk can be expressed in terms of an infinite series of j u

ð j uÞ ¼ 1 þ

1 X mk k¼1

k!

ð j uÞk

ð2:213Þ


45

Here moments mk are defined by eq. (2.28). Instead of a characteristic function, one can consider the log-characteristic function ln ð j uÞ and its expansion in the Taylor series ln ð j uÞ ¼

1 X k k¼1

k!

ð j uÞk

ð2:214Þ

Coefficients of this expansion are called the cumulants. Equations (2.213) and (2.214) allow relations between moments and cumulants to be obtained. Furthermore, it is possible to relate central moments and cumulants by noting that "

ð j uÞ ¼ ej u m1 1 þ

1 X k k¼1

k!

# ð j uÞk

ð2:215Þ

Moments and cumulants of random vectors can be introduced in a similar manner by considering corresponding characteristic functions. n ðuÞ ¼

1 X 1 X

...

n1 ¼0 n2 ¼0

1 X mn1 ;n2 ;...;nN ð j u1 Þn1 ð j u2 Þn2 . . . ð j uN ÞnN n !n ! n ! N n ¼0 1 2

ð2:216Þ

N

and ln n ðuÞ ¼

1 X 1 X n1 ¼0 n2

1 X n1 ;n2 ;...;nN ð j u1 Þ n 1 ð j u2 Þ n 2 . . . ð j uN Þ n N ... n !n ! nN ! ¼0 n ¼0 1 2

ð2:217Þ

N

Using the expansion of lnð1 þ zÞ into the series of powers of z, it is possible to find relations between cumulants of any order, and moments, or central moments, of any order. It is worth noting that the elements of the vector n can repeat themselves. It is reasonable to ask why such a variety of descriptions is needed. One of the possible interpretations is the following: if a Gaussian random process is analyzed, then all the descriptions are equally convenient since the first two moments (or cumulants) completely describe a Gaussian random variable; however, description of a non-Gaussian random variable can be often more conveniently accomplished by using cumulant description. The advantages of cumulant description become obvious when orthogonal expansions of unknown densities, such as Edgeworth and Gram–Charlier series, are used. On the other n=2 hand, it can be shown, that, in contrast to moments, the relative value of cumulants n =2 is decreasing with the order, thus providing more reasonable approximations for an unknown density. Yet another important property of the cumulants is the fact that they are better suited to describe statistical dependence between elements of random vectors. This statement can be illustrated by considering the following example. Let n ¼ ½1 ; 2 T be a vector with two components, described by the joint probability density p ðx; yÞ, such that m1 x ¼ m1 y ¼ 0. Three possible relations between the components are considered here: (a) the components 1 and 2 are completely independent; (b) the components 1 and 2 are linearly dependent, i.e. 2 ¼ a 1 þ b; (c) the components 1 and 2 are related through a non-linear transformation

46


2 ¼ a 12 . It is also assumed that 1 ðtÞ is a symmetric random process, thus implying that all its odd moments are equal to zero mð2 nþ1Þ ¼ h12 nþ1 i ¼ 0

ð2:218Þ

In the case (a) of independent random variables, the characteristic function of the second is a product of the characteristic function of each component, and thus ln 2 ð j u1 ; j u2 Þ ¼ ln 1 1 ð j u1 Þ þ ln 1 2 ð j u2 Þ ¼

1 X n n¼0

n!

1

ð j u1 Þ n þ

1 X n n¼0

n!

2

ð j u2 Þ ð2:219Þ

Equation (2.219) indicates that all the cross cumulants are equal to zero, since there are no terms containing ð j u1 Þn1 ð j u2 Þn2 , with both n1 and n2 greater than zero. In the case (b) the first mixed cumulant 11 attains its maximum possible value, since 11 ¼ m11 ¼ h1 2 i ¼ a 21

ð2:220Þ

thus indicating maximum possible correlation. However, the same cumulant is equal to zero in the case (c): 11 ¼ h1 a 12 i ¼ ah13 i ¼ 0

ð2:221Þ

i.e. the first mixed cumulant does not completely describe dependence between two variables. Indeed, such dependence can be traced through the higher order cumulants, for example 21 : 21 ¼ h12 a 12 i ¼ ah14 i > 0

ð2:222Þ

The three cases above emphasize the following meaning of cumulants: the first mixed cumulant (or correlation function) measures the degree of linear dependence between two variables. Higher order cumulants are needed to describe non-linear dependence of two random variables.6

2.7.1

Moment and Cumulant Brackets

So far the angle brackets have been used as an alternative notation to the averaging operator over all random variables under consideration, i.e. hgðxÞi ¼ EfgðÞg ¼

ð1 1

gðxÞp ðxÞdx; hgðx;yÞi ¼ Efgð;Þg ð1 ð1 gðx;yÞp ðx;yÞdxdy; etc: ð2:223Þ ¼ 1 1

6

See [14,18] for more detailed discussions.

47


The following property of the moment brackets can be obtained from eq. (2.223) * + N N X X n fn ðxÞ ¼ n h fn ðxÞi ð2:224Þ n¼1

n¼1

where n are deterministic constants. Up to now such notation has been used mainly to shorten equations. However, it can be extended to allow a convenient tool for manipulation of cumulant and moment equations, considered below. For this purpose the following convention is used for mixed cumulants [9] ½n

½n

1 2 ½nm m n11;;n22;...; ;...;nm ¼ h1 ; . . . ; 1 ; 2 ; . . . ; 2 ; . . . ; m ; . . . ; m i ¼ h1 ; 2 ; . . . ; m i

ð2:225Þ

Here, the random variable 1 appears inside the brackets n1 times, 2 appears n2 times, etc. In future, brackets which contain at least one comma are called cumulant brackets. Otherwise the term moment brackets is used. For example ½

2 ¼ 2 1 ¼ h1 ; 1 i;

1 ;2 12 ¼ 1;2 ¼ h1 ; 2 ; 2 i; etc:

ð2:226Þ

The importance of this notation arises from the fact that it is possible to formalize the relationship between the moment and cumulant brackets in an easy way. In addition, it will be shown later in Section 2.8 that it is relatively easy to obtain systems of equations for cumulant functions of random processes, based on the dynamic equations describing the system under consideration. Moreover, the notation for random variables 1 , 2 , etc. and their transformations f ð1 Þ, gð2 Þ, etc. can be simply substituted by their ordinary number equivalents, as shown below. For example, the following relations between cumulant and moment brackets can be easily formalized through the use of Sratonovich symmetrization brackets [9]. 1 ;2 1 ;2 ¼ h1 ; 2 i ¼ h1 2 i h1 ih2 i h1; 2i ¼ h1 2i h1ih2i ¼ m1;1 m1 1 m1 2 1;1 1 ;2 ;3 1;1;1 ¼ h1; 2; 3i 3fh1ih2 3igs þ 2h1ih2ih3i 1 ;2 ;3 2 ;3 1 ;3 1 ;2 m1 1 m1;1 m1; 2 m1;1 m1 3 m1;1 þ 2 m1 1 m1 2 m1 3 ¼ m1;1;1

ð2:227Þ Symmetrization brackets together with an integer in front of the brackets represent the sum of all possible permutations of the arguments inside the brackets. For example 3fh1ih2; 3igs ¼ h1ih2; 3i þ h2ih1; 3i þ h3ih1; 2i 3fh1 2ih3 4igs ¼ h1 2ih3 4i þ h1 3ih2 4i þ h1 4ih2 3i

ð2:228Þ

In a similar manner, moment brackets can be expressed in terms of corresponding cumulants. For example mn ¼ N þ N 1 N1 þ þ

N! ð2 N2 þ 21 N2 Þ 2!ðN 2Þ!

N! ð3 N3 þ 1 2 N3 þ 31 N3 Þ þ þ N1 3!ðN 3Þ!

ð2:229Þ

48


If one sets 1 ¼ 0 in eq. (2.229), the relation between central moments and cumulants can be derived. Similar relations can be obtained for the mixed cumulants of two variables. For example, one can show that m11 ¼ 11 þ 10 01 m21 ¼ 21 þ 3 20 01 þ 2 11 10 þ 3 210 01 m31 ¼ 31 þ 3 20 11 þ 3 21 10 þ 3 20 10 01 þ 3 11 210 þ 310 01

ð2:230Þ

m22 ¼ 22 þ 20 02 þ 3 10 12 þ 2 21 01 þ 210 02 þ 20 201 þ 2 211 þ 4 10 11 01 þ 210 201 Equations (2.227), (2.229) and (2.230) are useful in the sense that they allow the transformation of a problem of analysis of moments of non-linear memoryless transformation random variables into a problem of evaluating its cumulants. The latter is preferable since the cumulant series converges faster. It is important to note that the cumulant brackets, in contrast to moment brackets, do not represent averaging of the quantity inside the brackets. Indeed, it can be seen that cumulants are obtained by non-linear transformation of the moments, thus they cannot be expressed as a linear combination of averaged quantities. Since, in any cumulant brackets, there are at least two arguments separated by a comma, there is no confusion between these two. However, of course, both types of brackets are closely related, since it is possible to express cumulants through moments and vice versa. It can also be shown that cumulant brackets are linear functions of their arguments, i.e. functions separated by two sequential commas. 2.7.2

Properties of Cumulant Brackets

In order to be useful, some formal rules of manipulation with cumulant brackets should be obtained. As earlier, Greek letters are used as a notation for random variables, while Latin letters are used for deterministic variables and constants. The following six properties can be easily obtained from the definition of cumulants. 1: h; ; . . . ; !i is a symmetric function of its arguments; 2: ha ; b ; . . . ; g !i ¼ ab gh; ; . . . ; !i; 3: h; ; . . . ; 1 þ 2 ; . . . ; !i ¼ h; ; . . . ; 1 ; . . . ; !i þ h; ; . . . ; 2 ; . . . ; !i; 4: h; ; . . . ; ; . . . ; !i ¼ 0; if is independent of f; ; . . .g; 5: h; ; . . . ; a; . . . ; !i ¼ 0;

ð2:231Þ

6: h þ a; þ b; . . . ; þ c; . . . ; ! þ gi ¼ h; ; . . . ; ; . . . ; !i The first three properties of cumulant brackets coincide with those of moment brackets, however, the last three properties are unique for cumulant brackets. The fourth property shows that if two groups of random processes are independent, any mutual cumulants are equal to zero (it is not true for corresponding moments). This is yet another indication that information about statistical dependence is encoded more evidently in cumulants rather than in moments. The sixth property indicates that the cumulants are invariant with respect to any deterministic shift in the random variables.

CUMULANT EQUATIONS

2.7.3

49

More on the Statistical Meaning of Cumulants

It has been shown earlier (see eqs. (2.220)–(2.222) and related discussions) that the first 1 ;2 mixed cumulant 1;1 is a measure of linear correlation between two random processes 1 and 2 . Such dependence is also often called first order dependence [14–18]. 1 ;2 ;3 ¼ h1 ; 2 ; 3 i. This cumulant can be The next step is to consider cumulant 1;1;1 expressed through the moment of the same order and mixed cumulants and moments of the lower order as h1 ;2 ;3 i ¼ h1 2 3 i m11 h2 ;3 i m12 h1 ;3 i m13 h1 ;2 i m11 m12 m13

ð2:232Þ

Two interesting particular cases of eq. (2.232) can be further considered: all processes are pairwise uncorrelated, i.e. h1 ; 2 i ¼ h1 ; 3 i ¼ h2 ; 3 i ¼ 0, i.e. there is no linear dependence between all these elements. If, in addition, these processes are independent, then 1 ;2 ;3 1 ;2 ;3 ¼ 0. Thus, cumulant 1;1;1 is not equal to zero if there is dependence between the 1;1;1 components of a more complicated nature than just linear dependence. This (non-linear) dependence can be obtained, according to eq. (2.232), by subtracting linear dependence and deterministic average components. This dependence is present or is absent, no matter if there is linear dependence or not. Thus, in some sense, cumulants are independent coordinates which describe a set of dependent random variables. In this sense, a set of cumulants is preferable to an equivalent set of moments. In general, so-called cumulant coefficients N

p11 ;;p22;...; ;...;pN ¼

N p11;;p22;...; ;...;pN p11 p22 . . . pNN

ð2:233Þ

of order p ¼ p1 þ p2 þ þ pN , describes a non-linear statistical dependence of ðN 1Þ between a set of N variables. It is interesting that there is only a limited freedom in choosing cumulants of a random variable or vector. Indeed, it is shown in [9] that cumulant coefficients n of a random variable

n ¼

n n=2 2

¼

n n

ð2:234Þ

must satisfy certain (non-linear) inequalities. For example, skewness 3 and curtosis 4 must satisfy the condition [9]

4 32 þ 2 0

ð2:235Þ

While there is no restriction placed on 3 , i.e. 3 2 ð1; 1Þ; 4 must exceed 2. Restrictions on higher order cumulants are still an area of active research.

2.8

CUMULANT EQUATIONS

It was shown earlier that the characteristic function ð j uÞ can be defined if an infinite set of cumulants k ; k ¼ 1; 2; . . ., is given. In other words, the characteristic function, and thus the

50


PDF, is a function of an infinite cumulant vector j ¼ f1 ; 2 ; . . .g: pðxÞ ¼ pðx; 1 ; 2 ; . . .Þ ¼ pðx; jÞ

ð2:236Þ

It was also mentioned that the cumulants can be chosen independently from a certain region of values and this choice can be made independently for each cumulant as long as they satisfy certain inequalities [9]. This fact allows differentiation of the PDF pðx; 1 ; 2 ; . . .Þ with respect to cumulants, i.e. expressions of the type @@s pðx; jÞ are justified since it only requires an infinitely small variation of cumulant js around its nominal value. Taking a derivative of both sides of the representation of the PDF through the characteristic function with respect to s one obtains ð @ 1 1 j u x @ pðx; jÞ ¼ e ð j u; jÞd u ð2:237Þ @ s 2 1 @ s Furthermore, using representation (2.19) it is possible to calculate the partial derivative inside the integral to be @ ð j uÞs ð j u; jÞ ð j u; jÞ ¼ @ s s!

ð2:238Þ

and, thus eq. (2.237) becomes @s 1 pðx; jÞ ¼ 2 @ s

ð1 1

ð j vÞs ej v x ð j v; jÞd v

ð2:239Þ

On the other hand, taking derivatives of both sides of eq. (2.19) s times with respect to x one obtains ð @s 1 1 pðx; jÞ ¼ ðj vÞs ej v x ð j v; jÞd v ð2:240Þ s 2 1 @x Comparing this with expression (2.239) one can obtain a partial differential equation of the form @ ð1Þs @ s pðx; jÞ ¼ pðx; jÞ @ s s! @ xs

ð2:241Þ

This relation between derivatives of the PDF with respect to the argument and the parameters is very useful in the derivation of cumulant equations [9]. The next step is to derive a set of equations which describe evolution of averages of a deterministic function f ðxÞ. In this case the average, defined as h f ðxÞi ¼

ð1 1

f ðxÞpðx; jÞd x

ð2:242Þ

can also be considered as a function of the cumulants of the underlying PDF h f ðxÞi hf ðxÞiðjÞ

ð2:243Þ

51

CUMULANT EQUATIONS

Taking a derivative of both sides of eq. (2.242) with respect to js it is easy to obtain ð1 ð @ @ ð1Þs 1 @s h f ðxÞi ¼ f ðxÞ pðx; jÞd x ¼ f ðxÞ s pðx; jÞd x ð2:244Þ @ s @ s s! @x 1 1 Here the relationship (2.242) has been used to substitute the derivative with respect to cumulants by the derivative with respect to the argument x. Further simplification of eq. (2.244) can be achieved by integrating the right hand side by parts s times and taking into account that for most distributions lim pðx; jÞ ¼ 0;

x!1

@s pðx; jÞ ¼ 0 x!1 @ xs

ð2:245Þ

lim

As a result, a simple relation is obtained between the derivative of the average with respect to the s-th cumulant and the average of the s-th derivative of the function f ðxÞ ð @ 1 1 ds 1 ds h f ðxÞi ¼ pðx; jÞ s f ðxÞd x ¼ f ðxÞ ð2:246Þ @ s s! 1 s! d xs dx Taking the derivative of both sides of eq. (2.246) with respect to js once more, one obtains @2 @ @ 1 @ h f ðxÞi ¼ h f ðxÞi ¼ 2 @ s @ s s! @ s @s

ds f ðxÞ d xs

¼

1 ðs!Þ2

d2 s f ðxÞ d x2 s

ð2:247Þ

Repeating this procedure n times, an expression for the n-th derivative can be obtained, ns @n 1 d h f ðxÞi ¼ f ðxÞ ð2:248Þ ðs!Þn d xn s @ ns By the same token, mixed partial derivatives of the average of a function can be obtained [9] to be * + @ ðnþmÞ 1 dðnkþmpÞ h f ðxÞi ¼ f ðxÞ ð2:249Þ ðk!Þn ðp!Þm d xnkþmp @ nk @ m p Equation (2.249) are called the cumulant equations [9]. Their usefulness becomes obvious from the examples below. Example: Relation between the fourth moment m4 and cumulants. It follows from the definition of averages that m4 ¼ hx4 i. Setting f ðxÞ ¼ x4 in eq. (2.249) one can easily obtain 4 @2 1 d 4 4! 4 hx i ¼ x ¼ ¼4 4 ð1!Þð3!Þ d x 3! @ 1 @ 3 4 3 1 @ 1 1 d 4 4! ¼6 ðhx4 iÞ ¼ x ¼ 2 2 4 2! @ 1 @ 2 2 ð1!Þ ð2!Þ d x 2 2! 4 1 @3 1 1 d 4 4! ðhx4 iÞ ¼ x ¼ ¼1 4 4 4 4! @ 1 4! ð1!Þ d x 4!

ð2:250Þ

52


These quantities can easily be identified with the coefficients of the expansion of the fourth moment in terms of corresponding cumulants. In a similar way, expansion of an arbitrary moment can be obtained. Of course, the applications of eq. (2.249) are much wider than just calculation of moments [9].

2.8.1

Non-linear Transformation of a Random Variable: Cumulant Method

Equations (2.249) are particularly important when it is desired to the obtain the relationship between the moments and cumulants of a transformed random variable ¼ f ðÞ given that the cumulants of the input process are known. In order to obtain this dependence, eq. (2.249) must be first modified. For example, if the task is to find the variance 2 of the output process; 2 ¼ h; i ¼ hf 2 ðÞi hf ðÞi2

ð2:251Þ

it can be seen that this equation already contains an expression for the average h f ðÞi of the output variable which itself may depend on all cumulants of . In general, the cumulant of an arbitrary order jNn ¼ ð; j1 Þ also depends on all cumulants. In order to take this fact into account, the expression for the n-th cumulant can be written as hð; h f ðÞiÞi ¼

ð1 1

ðx; h f ðxÞiÞpðx; jÞd x

ð2:252Þ

Taking a derivative with respect to s , one obtains an equation similar to eq. (2.249). @ hð; h f ðÞiÞi ¼ @ s

ð1

@ ðx; h f ðxÞiÞpðx; jÞd x þ 1 @ s

ð1 1

ðx; h f ðxÞiÞ

@ pðx; jÞd x @ s ð2:253Þ

Taking advantage of the fact that ðx; h f ðxÞiÞ depends on the cumulants only through h f ðxÞi, and using the relationship (2.241), the equation for the derivative of the output cumulant can be written as @ ð; h f ðÞiÞ ¼ @ js

@ @ 1 ds ð; h f ðÞiÞ h f ðÞi þ ð; h f ðÞiÞ @h f i @ s s! d s

ð2:254Þ

In turn, this can be further advanced by using eq. (2.248) to obtain

@ 1 ds @ ð; h f ðÞiÞ f ðÞ ð; h f ðÞiÞ ¼ ½ð; hðÞiÞ þ @ s s! d s @h f i

ð2:255Þ

The latter equation is a generalization of eq. (2.246) and can be used to find cumulants n of the output process in terms of the cumulants ks of the input process. Using the relation between the cumulant and the moments it is possible to obtain an expression for the

CUMULANT EQUATIONS

53

differentials of the cumulants as d 1 ¼ d m 1 d 2 ¼ d m 2 2 d m 1

ð2:256Þ

...: This, coupled with eq. (2.246), produces a set of differential equations for cumulants of the output process @ 1

1 ¼ @ s s!

ds f ðÞ d s

1 ds 2 ¼ ½ f ðÞ 2hf ðÞif ðÞ @ s s! d s

@ 2

ð2:257Þ

...: Taking into account that f ðÞ ¼ , eq. (2.257) can be rewritten as 1 ds ¼ @ s s! d s D s E d 2 ½ 2hi s d @ 2 ¼ s! @ s @ 1

ð2:258Þ

...: Some important relationships, often used in the following text, can be obtained using eqs. (2.258) and the rules of the simplification of cumulant brackets, described in the appendix. In particular, dependence between lower order cumulants is provided by @ D @ d m1 1 d2 ¼ 2 ; ¼ ; @ 2 d 2 @ m1 d D2 2 @ D d d d d ; þ ; 2 þ ¼ d d d d @ D

@ m1 ¼ @ m1

d ; d

ð2:259Þ ð2:260Þ

Finally, the derived equations can be further simplified if transformation of a Gaussian random variable is considered. In this case the input variable is completely described by its first two cumulants m1 ¼ 1 and D ¼ 2 . Thus, the output cumulants and any average h’ðÞi will depend only on the two quantities. As a result, eqs. (2.259) and (2.260) can be rewritten as @n h f ðÞi ¼ h’ðnÞ ðÞi @ mn1 @n 1 ð2nÞ h f ðÞi ¼ h’ ðÞi 2n @ Dn

ð2:261Þ

54


If D ¼ 0 then m1 and the following initial condition can be obtained for eq. (2.261) @n 1 ð2nÞ h’ðÞi ¼ h’ ðm1 Þi ð2:262Þ 2n @ Dn D¼0 Finally, using moments mn as h f ðÞi one obtains @ s mn 1 d2 s ðnÞ ¼ s f ðÞ 2 d 2 s @ Ds @ s mn 1 d2 s ðnÞ ¼ f ðm1 Þ @ Ds D¼0 2s d m21 s

ð2:263Þ

Solving differential equations (2.263) with the initial conditions (2.262) it is possible to obtain the expression for the moments of the transformed variable in terms of the mean and variance of the input Gaussian process. It can be seen that such moments are functions of D with coefficients depending only on the average m1 . Thus, an alternative way of calculating the output moments can be suggested. In this case the moments mn can be represented as a series of powers of D with undetermined coefficients mn ¼ A0 þ A1 D þ A2 D2 þ þ As Ds þ Taking derivatives of eq. (2.264) one obtains @ n mn ¼ n!An @ Dn D¼0 Comparing eqs. (2.263) and (2.265) the following expansion is obtained s 1 X 1 d2 s ðnÞ D mn ¼ ½f ðm1 Þ 2 s s! d m1 2 s¼0

ð2:264Þ

ð2:265Þ

ð2:266Þ

The latter equation can be considered as an alternative for non-linear analysis of non-linear transformations of Gaussian (and only Gaussian) random variables.

APPENDIX: A.1

CUMULANT BRACKETS AND THEIR CALCULATIONS [9]

Evaluation of Averages

A.1.1

Averages of Delta Functions and Their Derivatives ð1 hð xÞi ¼ p ðsÞðs xÞd s ¼ p ðxÞ 1 ð1 ð1 p; ðs1 ; s2 Þðs1 xÞðs2 yÞd s1 d s2 ¼ p; ðx; yÞ hð xÞð yÞi; ¼ 1 1 s ð1 s d ds s d ð xÞ ¼ ðy xÞp ðyÞd y ¼ ð1Þ p ðxÞ s d s d xs 1 d y

ðA:1Þ ðA:2Þ ðA:3Þ

CUMULANT BRACKETS AND THEIR CALCULATIONS

55

If and are jointly Gaussian and both have zero mean hi ¼ hi ¼ 0 then the following equations are valid

A.1.2

d2 sþ1 ð xÞ ¼ 0 d 2 sþ1 2s s d 1 ð1Þs ð2 s 1Þ!! 1 s d pffiffiffiffiffiffiffiffiffiffiffi ¼ pffiffiffiffiffiffiffiffiffiffiffi ð xÞ ¼ 2 Ds d 2 s d Ds 2 D 2D 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hðÞðÞi; ¼ 2 ½D D B2

ðA:4Þ ðA:5Þ

Averages of Arbitrary Functions

For an arbitrary function f ðÞ of one variable and Fð; Þ of two variables the following equalities are valid hf ðÞi ¼ h f ðhiÞ þ

1 X ðx hiÞs

s!

s¼2 s

f

3

ðsÞ

ðhiÞi ¼ f ðhiÞ þ

1 X s s¼2 5

4

s!

f ðsÞ ðhiÞ ðA:6Þ

1 df 1d f 1d f 1d f 2 þ þ ð þ 3 Þ þ ð5 þ 10 2 3 Þ þ f ðhiÞ þ 2 3 4 2 2! d 2 3! d 3 4! d 4 5! d 5 In particular p ðxÞ ¼

1 X m¼0

ð1Þm

s ðmÞ ðx mÞ m!

ðA:7Þ

1 d2 d2 d2 hFð; Þi ¼ Fðhi; hiÞ þ Fð; Þ11 þ 2 Fð; Þ02 Fð; Þ20 þ 2 2! d 2 dd d 3 3 3 1 d d d þ Fð; Þ Fð; Þ þ 3 þ 3 Fð; Þ12 30 21 3! d 3 d 2 d d d 2 d3 1 d4 þ 3 Fð; Þ03 þ Fð; Þ½40 þ 3 220 4! d 4 d d4 Fð; Þ½31 þ 3 11 20 d 3 d d4 þ 6 2 2 Fð; Þ½22 þ 2 211 þ 20 02 d d þ4

! d4 d4 4 Fð; Þ½13 þ 3 11 02 þ 4 Fð; Þ½04 þ 3 202 d d 3 d

ðA:8Þ

In the above equation, derivatives are calculated at ¼ hi and ¼ hi. Both equations are obtained by averaging corresponding Taylor series.

56


A.2

Rules of Manipulations with Cumulant Brackets

A.2.1

Arbitrary Distribution at the Same Moment in Time h;2 i ¼ 3 þ 21 2

ðA:9Þ

h;;2 i ¼ 4 þ 21 3 þ 222

ðA:10Þ

h;;;2 i ¼ 5 þ 21 4 þ 62 3

ðA:11Þ

h;;;;2 i ¼ 6 þ 21 5 þ 82 4 þ 622 s X ½s 2 h ; i ¼ sþ2 þ Cs þ1 sþ1

ðA:12Þ ðA:13Þ

¼0

h2 ;i ¼ h;;i þ 2hih;i ¼ 21 þ 210 11

ðA:14Þ

h2 ;;i ¼ h;;;i þ 2hih;;i þ 2h;i2 ¼ 22 þ 210 12 þ 2211

ðA:15Þ

h2 ;;;i ¼ h;;;;i þ 2hih;;;;iþ 6h;ih;;i ¼ 23 þ 210 13 þ 611 12 s s X X Cs h;½ ih;½s i ¼ 2;s þ Cs 1 ;1;s h2 ;½s i ¼ h;;½s i þ ¼0

ðA:16Þ ðA:17Þ

¼0

h; i ¼ 4 þ 31 3 þ 322 þ 321 2 h;;3 i ¼ 5 þ 31 4 þ 92 3 þ 321 3 þ 61 22 h;;;3 i ¼ 6 þ 31 5 þ 122 4 þ 321 4 þ 321 4 þ 923 þ 181 2 3 þ 623 h3 ;i ¼ 3;1 þ 31;0 2;1 þ 32;0 1;1 þ 321;0 1;1

ðA:18Þ

h2 ;;i ¼ 3;2 þ 31;0 2;2 þ 32;0 1;2 þ 61;1 2;1 þ 321;0 1;2 þ 61;0 21;1

ðA:22Þ

3

h½k i ¼ 1;k þ

k X

Cks s ks

ðA:19Þ ðA:20Þ ðA:21Þ

ðA:23Þ

s¼1

# " s X 1 ds l h;f ð’Þi ¼ f ð’Þ 1;1;s þ Cs 1;0;l 0;1;sl s! d’s s¼1 l¼0 " 1 s X X 1 dn h’;f ð Þi ¼ f ð Þ þ Cnk f31;0;0;k 0;1;1;sl gs 1;1;1;n n n! d n¼0 k¼0 # n nk XX l Cnk Cnk 1;0;0;k 0;1;0;l 0;0;1;nkl þ 1 X

ðA:24Þ

ðA:25Þ

k¼0 l¼0

If 1 ¼ hi ¼ 0; the following series can be obtained ( * + 1 n ðnkÞ k X 1 X d d h;f ðÞ;gðÞi ¼ Cnk f ðÞ; k gðÞ n! dnk d n¼0 k¼0 * + ) n1 ðnkÞ k X d d Cnk f ðÞ gðÞ 1;n þ nk k d d k¼1

ðA:26Þ

57

CUMULANT BRACKETS AND THEIR CALCULATIONS

kþ1 1 X 1 X 1 @ h; ; f ð; Þi ¼ f ð; Þ k @ l k!l! @ k ¼0 l¼0 k þl>0

2

4;;; 1;1;k;l þ

k l X X i¼0 i þ j 6¼ 0; k þ 1

3 ;; 5 Cki Clj ;; 1;i;j 1;ki;lj

ðA:27Þ

j¼0

Here the following shortened notations are used ;; ;;; n ¼ n ; n;m ¼ ; n;m ; m;n;k ¼ m;n;k ; n;m;k;l ¼ n;m;k;l

A.2.2

Two Stationary Related Processes

The following notations are used ¼ ðtÞ; ¼ ðt þ Þ; ¼ ðtÞ; ¼ ðt þ Þ.

d h Fð Þi ¼ h FðÞihi þ FðÞ þ FðÞ h; i d 1 d d2 2 FðÞ þ 2 FðÞ h; ; i þ 2 d d 2 1 d d3 3 2 FðÞ þ 3 FðÞ h; ; ; i þ þ 6 d d d FðÞ h; i þ hFðÞih; i h Fð Þi ¼ h FðÞihi þ d 2 1 d d 2 FðÞ h; ; i þ FðÞ h; ; i þ 2 d d 3 1 d 1 d2 3 FðÞ h; ; ; i þ þ FðÞ h; ; ; i þ 6 2 d 2 d

ðA:29Þ

h 2 Fð Þi

d FðÞ h; i ¼ h Fð Þihi þ 2 Fð Þ þ d 2 1 d 2 d 2 FðÞ þ 4 FðÞ þ þ FðÞ h; ; i 2 d d 2 3 1 d d2 2 d 6 FðÞ þ 6 2 FðÞ þ þ FðÞ h; ; ; i 6 d d d 3

ðA:28Þ

2

2

d h ; Fð Þi ¼ h; FðÞihi þ hFðxÞ þ ; FðÞ h; i d

1 d d2 FðÞ þ ; 2 FðÞ h; ; i þ 2 2 d d 2

1 d d3 FðÞ þ ; 3 FðÞ h; ; ; i þ 3 6 d 2 d

ðA:30Þ

ðA:31Þ

58


REFERENCES 1. W. Feller, An Introduction to Probability Theory and Its Applications, 3rd Edition, New York: Wiley, 1968. 2. M. Kendall and A. Stuart, The Advanced Theory of Statistics, 4th Edition, New York: Macmillan, 1977–1983. 3. A. Papoulis, Probability, Random Variables, and Stochastic Processes, Boston: McGraw-Hill, 2001. 4. J. Ord, Families of Frequency Distributions, London: Griffin, 1972. 5. A. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1972. 6. Iu. Linnik, Decomposition of Probability Distributions, New York: Dover, 1965. 7. F. Oberhettinger, Tables of Fourier Transforms and Fourier Transforms of Distributions, New York: Springer-Verlag, 1990. 8. I.S. Gradsteyn and I.M. Ryzhik, Table of Integrals, Series, and Products, New York: Academic Press, 1980. 9. A.N. Malakhov, Cumulant Analysis of Non-Gaussian Random Processes and Their Transformations, Moscow: Sovetskoe Radio, 1978 (in Russian). 10. W. Rudin, Functional Analysis, New York: McGraw-Hill, 1991. 11. G. Szego, Orthogonal Polynomials, Providence: American Mathematical Society, 1975. 12. D. Middleton, Topics in Communication Theory, New York: McGraw-Hill, 1965. 13. K. Miller, Multidimensional Gaussian Distributions, New York: Wiley, 1964. 14. O.V. Sarmanov, Maximum Correlation Coefficient (Symmetrical Case), DAN, Vol. 120, 1958, pp. 715–718. 15. C. Nikias and A Petropulu, Higher-Order Spectra Analysis, New Jersey: Prentice Hall, 1993. 16. D. L. Wallace, Asymptotic Approximations to Distributions. Ann. Math. Stat. 29, 635–654, 1958. 17. H. Bateman and A. Erdelyi, Higher Transcendental Functions, New York: McGraw-Hill, 1953– 1987. 18. O.V. Sarmanov, Maximum Correlation Coefficient (Non-Symmetrical Case), DAN, Vol. 121, 1958, pp. 52–55.

3 Random Processes 3.1

GENERAL REMARKS

A random process ðuÞ describes a physical quantity which depends on a parameter u. A specific trace obtained as the result of a simple observation (umin < u < umax ) is called a realization of this random process, a trajectory or a sample path. In future these terms are used interchangeably. In communication theory, the parameter u usually represents time t or spacial coordinates x; y; z, or both, depending on the application. For now, discussions will be concentrated on the case of temporal random processes, i.e. u t. As an example of a random process, one can consider the variation of voltage (current) across electric circuit elements in the presence of noise. In this case, the notation ðtÞ is used to emphasize that the random process is a function of time. For every fixed moment of time, t ¼ ti , the value of the random process i ¼ ðti Þ is a random variable, since this value may differ from experiment to experiment, even if they are conducted under the same conditions. The random value i is called a sample of the random process ðtÞ taken at the moment of time t ¼ ti . Depending on the shape of the trajectories and the method of description, random processes can be divided into three classes: impulse, fluctuations and a special form. An impulse process can be visualized as a sequence of single pulses, in general of different shapes, which follow each other in time with random intervals between consecutive pulses. Usually, impulse processes are described by piece-wise continuous functions of time. Manmade noise, so-called Electromagnetic Interference (EMI), and a number of natural phenomena, such as lightning, can be described as impulse random processes. A fluctuation type of random process is usually a result of a combination of a large number of contributing sources, often impulses, which overlap in time. The trajectory of such processes resembles a continuous function of time. Thermal and cosmic noise are the prominent representatives of fluctuations. The special type of random processes usually involve a combination of impulse or fluctuation random processes mixed with a deterministic signal or a signal of a known deterministic shape and random parameters. As an example one can consider a signal of the following form ðtÞ ¼ A cosð!t þ Þ þ nðtÞ


ð3:1Þ

60

RANDOM PROCESSES

Here, nðtÞ can be a fluctuation or an impulse process and A; !; can be deterministic or random variables. The next few sections are dedicated to general methods of description of random processes.

3.2

PROBABILITY DENSITY FUNCTION (PDF)

Let one imagine that there is a large number N of completely identical systems (Fig. 3.1), constituting an ensemble system. All the systems work in identical conditions. The output of each system is a sample path of a random process ðtÞ under consideration. If identical registering devices are attached to the systems, we can record each system state at any moment of time t. For a fixed moment of time, t ¼ t1 , readings x1 ðt1 Þ; x2 ðt1 Þ; . . . ; xN ðt1 Þ from different systems are different from each other since systems are assumed to be described by a random process. In general, values of xi ðti Þ cover all possible regions of variation of ðtÞ if N is sufficiently large. Let n1 be a number of samples whose values belong to a small interval ½x; x þ x. If N is large enough, a relative number of observations inside this interval is proportional to x and depends on t1 as a parameter. In other words, n1 ¼ p1 ðx; t1 Þx N!1 N lim

ð3:2Þ

The function p1 ðx; t1 Þ is called the marginal probability density of the process ðtÞ considered at the moment of time t ¼ t1 . The marginal probability density is a very important, but not a sufficient, description of a random process. It completely describes the statistical properties only at one fixed moment

Figure 3.1

Illustration of the imaginary experiment with a large number of identical sources.

PROBABILITY DENSITY FUNCTION (PDF)

61

of time. However, it does not describe at all how values of the process ðtÞ are related at two different time instants t ¼ t1 and t ¼ t2 . It is possible to say that the marginal PDF gives a local view of the random process while ignoring its properties on a whole (even small) time interval. A more complete description can be achieved if a second order (two time instant) PDF p2 ðx1 ; x2 ; t1 ; t2 Þ is considered. It describes relations between any two values of the random process ðtÞ considered at any two separate moments of time t1 and t2 . Construction of the second order PDF is similar to that of the marginal PDF. One can observe the state of each of N devices, considered above, at two separate moments of time t1 and t2 (see Fig. 3.1) and let 2N random variables x1 ðt1 Þ; x2 ðt1 Þ; . . . ; xN ðt1 Þ;

x1 ðt2 Þ; x2 ðt2 Þ; . . . ; xN ðt2 Þ

ð3:3Þ

be the result of these observations. The next step is to count observations which fell into a small interval ½x1 ; x1 þ x1 at the moment of time t ¼ t1 and into a small interval ½x2 ; x2 þ x2 at the moment of time t2 . Once again it is assumed that if N is large enough, a relative number n12 of such events is proportional to the length of each interval and depends on both t1 and t2 as parameters. Mathematically speaking n12 ¼ p2 ðx1 ; x2 ; t1 ; t2 Þx1 x2 N!1 N lim

ð3:4Þ

The function p2 ðx1 ; x2 ; t1 ; t2 Þ is called the joint two-dimensional PDF of the process ðtÞ. In general, the joint second order PDF p2 ðx1 ; x2 ; t1 ; t2 Þ also does not provide a complete description of the random process ðtÞ. It describes only the relation between two time instants of the process. A more detailed description can be obtained by considering a series of probability density functions of an arbitrary order M pðx1 ; x2 ; . . . ; xM ; t1 ; t2 ; . . . ; tM Þ constructed as follows: for M fixed moments of time t1 ; t2 ; . . . ; tM , let xi ðtj Þ be a sample of the random variable ðtÞ at the output of the i-th system at the time instant t ¼ tj . For small intervals, ½xi ; xi þ xi , let n be the number of outcomes such that at ti the observed quantity fits in the interval ½xi ; xi þ xi for every i ¼ 1; . . . ; M. In this case n ¼ pM ðx1 ; x2 ; . . . ; xM ; t1 ; t2 ; . . . ; tM Þx1 x2 xM N!1 N lim

ð3:5Þ

A PDF of order M allows conclusions to be drawn about properties of a random process ðtÞ at M separate moments of time. A higher order of PDF provides additional information about the process with respect to the lower order PDF. An M-dimensional PDF must satisfy the following conditions 1. Positivity: pM ðx1 ; x2 ; . . . ; xM ; t1 ; t2 ; . . . ; tM Þ 0 for any xi and tj This follows from the definition (3.5) since n 0

ð3:6Þ

62

RANDOM PROCESSES

2. Normalization: 1 ð

1 ð

1

pM ðx1 ; x2 ; . . . ; xM ; t1 ; t2 ; . . . ; tM Þ dx1 dx2 dxM ¼ 1

ð3:7Þ

1

3. Symmetry: pM does not change under any permutation of its arguments x1 ; x2 ; . . . ; xM . 4. Matching condition: for any m < M ð ð ð pm ðx1 ; x2 ; . . . ; xm ; t1 ; t2 ; . . . ; tm Þ ¼

pM ðx1 ; x2 ; . . . ; xM ; t1 ; t2 ; . . . ; tM Þdxmþ1 dxM RMm

ð3:8Þ This equation shows that if the M-dimensional PDF is known, the PDF of a smaller order m can be found by integrating the former over ‘‘additional’’ variables. In other words, if the PDF of order M is known, then a PDF of order m < M can be found by integrating as in eq. (3.8). Since t is continuous, there is no finite order M which allows for a complete description of the process (except a few particular cases such as Gaussian or Markov processes). However, it can be shown that an infinite but countable sequence of PDFs describes a random process completely [1,2]. If a value of the process under consideration is known and fixed at a moment of time t ¼ t2 , the conditional PDF p1 ðx1 ; t1 jx2 ; t2 Þ can be defined as follows p1 ðx1 ; t1 Þ ¼

p2 ðx1 ; x2 ; t1 ; t2 Þ ¼ p1 ðx1 ; t1 jx2 ; t2 Þ p1 ðx2 ; t2 Þ

ð3:9Þ

Here, of course, 1 ð

p1 ðx2 ; t2 Þ ¼

p2 ðx1 ; x2 ; t1 ; t2 Þdx1

ð3:10Þ

1

The probability density p1 ðx1 ; t1 jx2 ; t2 Þ is known as the conditional PDF for a given x2 . It can be shown that p1 ðx1 ; t1 jx2 ; t2 Þ is a proper PDF. Indeed, since p2 0 and p1 0 then their ratio is also a positive function, thus satisfying one of the conditions. The normalization condition is also satisfied since 1 ð

1

1 p1 ðx1 ; t1 jx2 ; t2 Þdx1 ¼ p1 ðx2 ; t2 Þ

1 ð

p2 ðx1 ; x2 ; t1 ; t2 Þdx2 ¼ 1

1 p1 ðx2 ; t2 Þ ¼ 1 p1 ðx2 ; t2 Þ ð3:11Þ

It is important to realize that the conditional PDF contains more (or at least as much) information about the process ðtÞ at the time instant t ¼ t1 than the marginal PDF p1 ðx1 ; t1 Þ.

THE CHARACTERISTIC FUNCTIONS AND CUMULATIVE DISTRIBUTION FUNCTION

63

Indeed, the value of the marginal PDF can be restored from the conditional PDF as follows. If t2 ! 1, then the knowledge of x2 ¼ ðt2 Þ does not produce any new information about ðt1 Þ since they are independent, i.e. lim p1 ðx1 ; t1 jx2 ; t2 Þ ¼ p1 ð1 ; t1 Þ

t2 !1

ð3:12Þ

However, it is not possible to restore the conditional PDF from the marginal one. The amount of additional information carried by the conditional PDF is quite different in different cases. If ðt1 Þ and ðt2 Þ are independent samples of a random processes ðtÞ, then1 p2 ðx1 ; x2 ; t1 ; t2 Þ ¼ p1 ðx1 ; t1 Þp1 ðx2 ; t2 Þ

ð3:13Þ

The conditional PDF is then defined as p1 ðx1 ; t1 jx2 ; t2 Þ ¼

p2 ðx1 ; x2 ; t1 ; t2 Þ ¼ p1 ðx1 ; t2 Þ p1 ðx2 ; t2 Þ

ð3:14Þ

thus stating that the knowledge of x2 does not contribute any additional knowledge about the value of the process at the previous time. In the other extreme case, two samples ðt1 Þ and ðt2 Þ can be related by a deterministic function dependence, i.e. ðt1 Þ ¼ g½ðt2 Þ. In this case p2 ðx1 ; x2 ; t1 ; t2 Þ ¼ ðx1 gðx2 ÞÞp1 ðx2 ; t2 Þ

ð3:15Þ

and the conditional PDF is reduced to a function. Most of the realistic situations fall in between these two extremes. Eqs. (3.9)–(3.15) can be generalized to the case of a few variables.

3.3

THE CHARACTERISTIC FUNCTIONS AND CUMULATIVE DISTRIBUTION FUNCTION

Instead of probability densities of an arbitrary order, one can consider corresponding characteristic functions of the same order. The characteristic function n ðu1 ; u2 ; . . . ; un ; t1 ; t2 ; . . . ; tn Þ of order n is defined as the n-dimensional Fourier transform of the PDF of order n, i.e.

n ðu1 ; u2 ; . . . ; un ; t1 ; t2 ; . . . ; tn Þ 1 1 ð ð pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; ; tn Þ ¼ 1

1

exp½ jðu1 x1 þ u2 x2 þ þ un xn Þdx1 dx2 dxn

1

ð3:16Þ

p1 ðx1 ; t1 Þ and p1 ðx2 ; t2 Þ are different functions in general. However, since they represent the same process, the notation is kept the same.

64

RANDOM PROCESSES

It follows from eq. (3.16) that the characteristic function is an average value of the complex exponent

n ðu1 ; u2 ; . . . ; un ; t1 ; t2 ; . . . ; tn Þ ¼ hexp½ jðu1 x1 þ u2 x2 þ þ un xn Þi

ð3:17Þ

The corner brackets here are used as a short-cut for averaging over an assembly. It follows from eqs. (3.16) and (3.17) that

n ð0; 0; . . . ; 0; t1 ; t2 ; . . . ; tn Þ 1 1 ð ð ... pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ exp½ j0dx1 dx2 . . . dxn 1 ¼ 1

ð3:18Þ

1

and

n ðu1 ; u2 ; . . . ; un ; t1 ; t2 ; . . . ; tn Þ ¼ ðnþmÞ ðu1 ; u2 ; . . . ; 0; . . . ; 0; t1 ; t2 ; . . . ; tn Þ

ð3:19Þ

Cumulative distribution functions (CDF) Pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ of any order can be defined as @ n Pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ ¼ pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ @x1 @x2 . . . @xn

ð3:20Þ

thus leading to Pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ xð1 xð2 xðn ¼ ... pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; . . . ; tn Þdx1 dx2 . . . dxn 1 1

ð3:21Þ

1

Since pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ 0 then Pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ is a nondecreasing function in all arguments, which satisfies the following normalization condition Pn ð1; 1; . . . ; 1; t1 ; t2 ; . . . ; tn Þ ¼ 1

ð3:22Þ

Considering that there is one-to-one correspondence between pn ðxÞ, Pn ðxÞ and n ðxÞ, either of these functions provides the same description of the random process.

3.4

MOMENT FUNCTIONS AND CORRELATION FUNCTIONS

As was pointed out in Section 3.2, a complete description of a random process is accomplished by a series of PDFs (CDFs, characteristic functions) of increasing order. However, in practice, it is important to find a simpler (though in most cases incomplete) description of random processes for the following reasons:


65

1. In many cases, one needs to consider a transformation of a random process by a linear or a non-linear system with memory. Even if a complete description of the input of such systems is achieved (i.e. PDF of an arbitrary order is known), it is impossible to provide a general method of deriving PDFs at the output of the system under consideration. The problem can be solved partially by calculating a sufficient number of moments (or cumulants) of the output process instead. 2. Often, a process cannot be described statistically based on some physical (theoretical) model due to its complexity or lack of precise description. In this case, some statistical properties of the process can be obtained from experimental measurements. For example, low order moments of the process can be measured in many cases; however, a complete description in terms of PDF of an arbitrary order is impossible. It is usually possible to determine the marginal PDF and correlation function, but not the higher order PDFs. 3. Some processes are defined by a small number of parameters. For example, knowledge of the mean value and the covariance function completely describe a Gaussian process. Another interesting example is so-called ‘‘Spherically Invariant Processes,’’ which are also defined by their marginal PDF and the covariance function. In addition, a widely used parametric technique allows the fitting of a model distribution based on a few lower order moments: the Pearson’s class of PDF is widely used in applications and requires only knowledge of the first four moments [3]. 4. Answers to many practical questions can be obtained by considering certain particular characteristics of random processes (such as mean, variance, correlation, etc.). For example, analysis of properties of the output of a receiver with an amplitude detector driven by a linearly amplitude modulated signal Að1 þ m cos tÞ sinð!t þ Þ, usually ignores statistical properties of a random phase . In this case, only statistical properties of A have impact on the output [4]. 5. It is often desired to obtain rough estimates of the output process, rather than its complete and accurate description. Moment functions, Mi1 ðtÞ; Mi1 i2 ðt1 ; t2 Þ; . . . ; of a random process ðtÞ are defined as follows. Let I ¼ ½i1 ; i2 ; . . . ; in be a set of integer numbers greater than or equal to zero. Then for different moments of time, t1 ; t2 ; . . . ; tn , one can define an n-dimensional moment n P ik as function of order ¼ i¼1

Mi1 i2 ...in ðt1 ; t2 ; . . . ; tn Þ ¼ hi1 ðt1 Þi2 ðt2 Þ . . . in ðtn Þi ð1 ð1 ... xi11 xi22 . . . xinn pn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þdx1 . . . dxn ¼ 1

1

ð3:23Þ

For example, the one-dimensional moment of order i1 is nothing but the statistical average of the i1 -th power of the random process ðtÞ itself Mi1 ðtÞ ¼ h ðtÞi ¼

ð1

i1

1

xi1 p1 ðxÞdx

ð3:24Þ

66

RANDOM PROCESSES

while the two-dimensional moment Mi1 i2 ðt1 ; t2 Þ of order i1 þ i2 is defined as Mi1 i2 ðt1 ; t2 Þ ¼

ð1 ð1 1

1

xi11 xi22 p2 ðx1 ; x2 Þdx1 dx2

ð3:25Þ

and so on. It is important to note that in order to consider moment functions of order n, one needs to know a PDF of order n. Instead of moment functions Mi1 in ðt1 ; . . . ; tn Þ defined by eq. (3.23), one can consider socalled central moment functions of the same order and dimension, which are defined as follows i1 ;i2 ;...;in ðt1 ; t2 ; . . . ; tn Þ ¼ h½ðt1 Þ M1 ðt1 Þi1 . . . ½ðtn Þ Mn ðtn Þin i ð1 ð1 ¼ ... ðx1 M1 ðt1 ÞÞi1 ðx2 M2 ðt2 ÞÞi2 . . . ðxn Mn ðtn ÞÞin 1

1

p ðx1 ; x2 ; . . . ; xn Þdx1 . . . dxn

ð3:26Þ

There is an obvious relation between the moments and centred moments of the same random process. Indeed, expanding the expression ðx1 M1 ðt1 ÞÞi1 ðx2 M2 ðt2 ÞÞi2 . . . ðxn Mn ðtn ÞÞin into a power series one can obtain ðx1 M1 ðt1 ÞÞi1 ðx2 M2 ðt2 ÞÞi2 . . . ðxn Mn ðtn ÞÞin ¼

XX k1

k2

...

X

Ck1 k2 ...kn xk11 xk22 . . . xknn

kn

ð3:27Þ and, thus i1 ;i2 ;...;in ðt1 ; t2 ; . . . ; tn Þ ð1 X X X ð1 ¼ ... ... Ck1 k2 ...kn xk11 xk22 . . . xknn pn ðx1 ; x2 ; . . . ; xn Þdx1 . . . dxn 1

¼

XX k1

¼

1 k1

k2

XX k1

...

X kn

...

k2

X

k2

Ck1 k2 ...kn

kn

ð1

1

...

ð1 1

xk11 xk22 . . . xknn pn ðx1 ; x2 ; . . . ; xn Þdx1 . . . dxn

Ck1 k2 ...kn Mi1 ;i2 ;...;in ðt1 ; t2 ; . . . ; tn Þ

ð3:28Þ

kn

Here, Ck1 k2 ...kn ¼ Ck1 k2 ...kn ðt1 ; t2 ; . . . ; tn Þ is a function of time only and it can be obtained by collecting terms with the same power of xi . For example, the central moment of order k, considered at a single moment of time t, is related to the moments of order up to k as k ðtÞ ¼

ð1 1

ðx m1 ðtÞÞk p1 ðx; tÞdx ¼

k X

k! ð1Þi mi1 ðtÞMki ðtÞ i!ðk iÞ! i¼1

ð3:29Þ

67


Similarly 1;1 ðt1 ; t2 Þ ¼

ð1 ð1 1

1

½x1 m1 ðt1 Þ½x2 m1 ðt2 Þp2 ðx1 ; x2 ; t1 ; t2 Þdx1 dx2

¼ M1;1 ðt1 ; t2 Þ m1 ðt1 Þm1 ðt2 Þ

ð3:30Þ

and so on. It is interesting to note that in order to find an expression for the central moment of order , moments of the order up to (but not higher) must be considered. The reverse statement is also easy to verify. This can be accomplished by considering the following identity xi11 xi22 . . . xinn ¼ ð½x1 m1 ðt1 Þ þ m1 ðt1 ÞÞi1 ð½x2 m1 ðt2 Þ þ m1 ðt2 ÞÞi2 . . . ð½xn m1 ðtn Þ þ m1 ðtn ÞÞin

ð3:31Þ

Definition (3.23) can be extended to produce generalized averages. In particular, instead of powers of xi in eq. (3.23), one can use a general function gðx1 ; x2 ; . . . ; xn Þ of n variables: Mg ðt1 ; t2 ; . . . ; tn Þ ¼ hgðx1 ; x2 ; . . . ; xn Þi ¼

ð1 1

...

ð1 1

gðx1 ; x2 ; . . . ; xn Þpn ðx1 ; x2 ; . . . ; xn ; t1 ; t2 ; . . . ; tn Þ

dx1 dxn

ð3:32Þ

Obviously, this definition includes the definition of moments, central moments and fractional moments (obtained when indices i1 are not necessary integer). It also covers the characteristic function, absolute and factorial moments and many other quantities used in statistics and signal processing [4,5]. The moment functions can be obtained by differentiating the corresponding characteristic function, rather than by integration of the PDF. Indeed, using the Taylor series of the exponent, one can obtain ! ð1 ð1 1 X ðjuxÞn

1 ðu; tÞ ¼ p1 ðx; tÞ exp½ juxdx ¼ p1 ðx; tÞ dx n! 1 1 n¼0 !ð n 1 1 1 n X j Mn ðtÞ X ð juÞ un xn p1 ðx; tÞdx ¼ ð3:33Þ ¼ n! n! 1 n¼0 n¼0 Since the above eq. (3.33) is the Taylor expansion of 1 ðu; tÞ, its coefficients can be obtained through the values of its derivatives at u ¼ 0 as nd

Mn ðtÞ ¼ ðjÞ

n

1 ðu; tÞ dun u¼0

ð3:34Þ

It is possible to show that similar expressions are valid for k-dimensional moment functions [5]. It is possible to select a narrow group of moments which provide significant information about the random process under consideration, so called covariance functions. Cumulant

68

RANDOM PROCESSES

functions K1 ðt1 Þ, K2 ðt1 ; t2 Þ, K3 ðt1 ; t2 ; t3 Þ are defined through expansion of ln½ n ðu1 ; u2 ; . . . ; un ; t1 ; . . . ; tn Þ into the MacLaurin series. All of them are symmetric functions of their arguments. In the one-dimensional case, cumulant functions are known as cumulants, or semi-invariants [5]. It is also possible to express covariance functions through related moment functions of similar order. In order to show this, one can use the following expansion of the lnð1 þ zÞ into MacLaurin series [6] 1 1 1 ð1Þn n z lnð1 þ zÞ ¼ z z2 þ z3 z4 þ þ 2 3 4 n

ð3:35Þ

z ¼ 1 ðuÞ 1

ð3:36Þ

Setting

and using the expansion of 1 ðuÞ in terms of the moment functions Mu ðtÞ, one obtains 1 ln 1 ðuÞ ¼ ð 1 1Þ ð 1 1Þ2 2 !2 !3 1 1 1 X X X M 1 M 1 M ð juÞ ð juÞ þ ð juÞ þ þ 2 3 ! ! ! ¼1 ¼1 ¼1

ð3:37Þ

since

1 ð0Þ ¼

ð1 1

p1 ðxÞ exp½ ju0dx ¼

ð1 1

p1 ðxÞdx ¼ 1

ð3:38Þ

The right-hand side of expression (3.37) represents a polynomial with respect to ð juÞ. Rearranging terms in eq. (3.37), this equality can be rewritten as ln 1 ðuÞ ¼

1 X ¼1

!

ð juÞ

ð3:39Þ

or

1 ðuÞ ¼ exp

1 X ¼1

!

! ð juÞ

ð3:40Þ

Expansion (3.39) defines cumulants of order , or semi-invariants of order . It can be seen from eq. (3.37) that is a polynomial of n variables M1 ; M2 ; . . . ; Mn and vice versa. An implicit expression relating cumulant and moment functions can be obtained by equating terms of the same order of ð juÞ in eqs. (3.39) and (3.37). As a result 1 ¼ M 1 2 ¼ M2 M12 ¼ 2 ¼ 2 3 ¼ M3 3M1 M2 þ 2M12 ¼ 3 4 ¼ M4 3M22 4M1 M3 þ 12M12 M2 6M14 ¼ 4 322 and so on. Expressions for higher order cumulants can be found in [5,7].

ð3:41Þ


69

It is important to mention that the first cumulant coincides with the average of the random process: 1 ðtÞ ¼ M1 ðtÞ ¼ hðtÞi

ð3:42Þ

while the second cumulant is its variance 2 ðtÞ ¼ M2 ðtÞ M12 ðtÞ ¼ hðtÞ M1 ðtÞ2 i ¼ 2 ðtÞ

ð3:43Þ

Higher order cumulants are also very important in describing the shape of the marginal PDF p1 ðxÞ. Let us just mention that the normalized third and fourth cumulants are known as normalized skewness and excess [7,8]

1 ¼

3 3=2 2

;

2 ¼

4 ; 22

¼

4 22

ð3:44Þ

It is important to notice that higher order cumulants do not coincide with corresponding central moments. The difference starts with 4 , as can be seen from eq. (3.41). Higher order covariance (cumulant) functions K1 ðt1 Þ; K2 ðt1 ; t2 Þ; K3 ðt1 ; t2 ; t3 Þ are defined by expanding ln n ðu1 ; . . . ; un Þ into the MacLaurin series. The first three covariance functions are listed below: K1 ðtÞ ¼ 1 ¼ M1 ðtÞ ¼ hðtÞi K2 ðt1 ; t2 Þ ¼ h½ðt1 Þ M1 ðt1 Þ½ðt2 Þ M1 ðt2 Þi ¼ M2 ðt1 ; t2 Þ M1 ðt1 ÞM2 ðt2 Þ ¼ 1;1 ðt1 ; t2 Þ

ð3:45Þ

K3 ðt1 ; t2 ; t3 Þ ¼ M3 ðt1 ; t2 ; t3 Þ M1 ðt1 ÞK2 ðt2 ; t3 Þ M1 ðt2 ÞK2 ðt1 ; t3 Þ M1 ðt3 ÞK2 ðt1 ; t2 Þ þ 2M1 ðt1 ÞM1 ðt2 ÞM1 ðt3 Þ ¼ 1;1;1 ðt1 ; t2 ; t3 Þ It is easy to see that t1 ¼ t2 ¼ t3 , and eq. (3.45) becomes eq. (3.41). Knowledge of covariance functions allows one to restore characteristic functions and thus PDFs. This means that an adequate description of the random process can be achieved by a series of PDFs, a series of characteristic functions, moments or cumulants, as indicated in Fig. 3.2. It is important to note that the decision of which characteristic of a random process to use heavily depends on the type of random process. For a Gaussian process, which is

Figure 3.2

Relations between PDF, CF, moments and cumulants.

70

RANDOM PROCESSES

completely described by its mean and covariance function, all descriptions can be applied with the same complexity. In future, the first two cumulant functions M1 ðtÞ and K2 ðt1 ; t2 Þ will play a central role in the analysis of random processes and their transformation by linear and non-linear systems. A branch of the theory of random processes based on M1 ðtÞ and K2 ðt1 ; t2 Þ is called ‘‘the correlation theory’’ of random processes. Detailed investigation of cumulants can be found in [8]. Some discussions, closely following the approach taken in [8] can also be found in Section 3.8.

3.5

STATIONARY AND NON-STATIONARY PROCESSES

One of the important classes of random processes is so-called stationary random processes. A random process ðtÞ is called stationary in a strict sense if the PDF pn ðx1 ; x2 ; . . . ; xn ; t1 ; . . . ; tn Þ of any arbitrary order n does not change by a time translation of the time instants t1 ; t2 ; . . . ; tn along the time axes. In other words, for any t0 and n pn ðx1 ; . . . ; xn ; t1 ; . . . ; tn Þ ¼ pn ðx1 ; . . . ; xn ; t1 t0 ; t2 t0 ; . . . ; tn t0 Þ

ð3:46Þ

This means that the stationary process is homogeneous in time. Of course, eq. (3.46) implies that the same property is true for characteristic functions, moments and cumulants. Stationary processes are similar to steady-state processes in deterministic systems. All other processes are called non-stationary. As an example of a non-stationary process, one can consider a process of switching load of the power system (random number of users) before it reaches a steady state. It can be seen from eq. (3.46) that the marginal PDF p1 ðx; tÞ of a stationary process does not depend on time at all, i.e. p1 ðx; tÞ ¼ p1 ðx; t tÞ ¼ p1 ðx; 0Þ ¼ p1 ðxÞ

ð3:47Þ

At the same time the PDF of the second order depends only on the time difference, ¼ t2 t1 , between two instances of time p2 ðx1 ; x2 ; t1 ; t2 Þ ¼ p2 ðx1 ; x2 ; t1 t2 t1 Þ ¼ p2 ðx1 ; x2 ; 0; Þ ¼ p2 ðx1 ; x2 ; Þ

ð3:48Þ

In general, the PDF of order n depends only on ðn 1Þ time intervals k ¼ tkþ1 tk between t1 ; t2 ; . . . ; tn . Since the marginal PDF of a stationary process does not depend on time, neither do the characteristic function, moments, and cumulants of the first order. In particular, changing of a time scale also does not change these quantities. In a sense, a description using only the marginal PDF is similar to a description of a harmonic signal sðtÞ ¼ A cosð!t þ ’Þ, using only its magnitude A. It provides knowledge about the range of the process (from A to A) but does not tell how fast the process changes.

COVARIANCE FUNCTIONS AND THEIR PROPERTIES

71

Since the probability density of the second order p2 ðx1 ; x2 ; t1 ; t2 Þ depends only on the time difference ¼ t2 t1 (time lag ) the covariance2 function Cðt1 ; t2 Þ ¼ K2 ðt1 ; t2 Þ also depends only on the time lag CðÞ ¼ h½ðtÞ m½ðt2 Þ mi ¼ hðt1 Þðt1 þ Þi m2 ð1 ð1 ¼ ðx1 mÞðx2 mÞp2 ðx1 ; x2 ; Þdx1 dx2 1

1

ð3:49Þ

Variance of the random process 2 then coincides with the value of the covariance function at zero lag: ð1 2 ¼ Cð0Þ ¼ ðx mÞ2 p1 ðxÞdx ð3:50Þ 1

In many practical cases, one is limited to considering probability densities of order two. Moreover, only the mean value and the variance of the process are often sufficient to obtain an answer or a rough estimate. Due to this fact, a more relaxed definition of stationarity is usually used. A random process ðtÞ is called a wide sense stationary random process if its mean does not depend on time ðm ¼ M1 ðtÞ ¼ const:Þ and its covariance function C ðt1 ; t2 Þ depends only on the time lag ¼ t2 t1 , i.e. C ðÞ ¼ C ðt2 t1 Þ. A strict sense stationary process is also stationary in the wide sense. The reverse is not always true. It is important to mention that there is an important class of random processes for which these two definitions are equivalent: any wide sense Gaussian process is a strict sense stationary process as well. This follows from the fact that knowledge of the mean and covariance function completely describes any Gaussian process [1,4]. Since Gaussian processes are often encountered in applications, it is important to learn how to calculate or measure the mean value and the covariance function. On the other hand, no further information is required to describe such a process. This fact emphasizes the importance of the correlation theory for applications.

3.6


A covariance function between samples of the same process at two moments of time is called the autocovariance function. The general definition of autocovariance is given by eq. (3.45) C ðt1 ; t2 Þ ¼ h½ðt1 Þ m1 ðt1 Þ½ðt2 Þ m1 ðt2 Þi

ð3:51Þ

C ðÞ ¼ hðtÞðt þ Þi m2

ð3:52Þ

which is reduced to

in the case of a stationary process ðtÞ. 2

Here and in the future, the following special notations are used: C ðt1 ; t2 Þ ¼ h½ðt1 Þ m1 ðt1 Þ½ðt2 Þ m1 ðt2 Þi for the covariance function and R ðt1 ; t2 Þ ¼ C ðt1 ; t2 Þ þ m1 ðt1 Þm1 ðt2 Þ ¼ hðt1 Þðt2 Þi for the correlation function. Subindex indicating the random process will often be omitted if it does not lead to confusion.

72

RANDOM PROCESSES

The definition can be extended to consider covariance between two or more random processes. For simplicity, a case of two stationary random processes ðtÞ and ðtÞ with averages m and m respectively is considered. It is possible to define a mutual (joint) covariance function between these two processes as 4

C ðt1 ; t2 Þ ¼ h½ðt1 Þ m ½ðt2 Þ m i 4

C ðt1 ; t2 Þ ¼ h½ðt1 Þ m ½ðt2 Þ m i

ð3:53Þ

If both C and C depend only on the time lag ¼ t2 t1 , i.e. C ðt1 ; t2 Þ ¼ C ðÞ C ðt1 ; t2 Þ ¼ C ðÞ

ð3:54Þ

then ðtÞ and ðtÞ are called stationary related processes. For a stationary related process, one can obtain that C ðÞ ¼ h½ðtÞ m ½ðt þ Þ m i ¼ h½ðt Þ m ½ðtÞ m i ¼ C ðÞ ð3:55Þ Equations (3.52)–(3.53) can be generalized to the case of complex random processes as follows C ðÞ ¼ h½ðtÞ m ½ ðt1 þ Þ m i

ð3:56Þ

C ðÞ ¼ h½ðtÞ m ½ ðt1 þ Þ m i

ð3:57Þ

where stands for the complex conjugation. The importance of the mutual covariance function in describing the relation between two random processes can be explained by considering two extreme cases of dependence between two processes: (a) ðtÞ and ðtÞ are independent, and (b) ðtÞ and ðtÞ are linearly related, i.e. ðtÞ ¼ ðtÞ þ ; > 0. In case (a), the joint probability density is just a product of marginal PDFs p2 ðx1 ; x2 Þ ¼ p1 ðx1 Þp1 ðx2 Þ

ð3:58Þ

where p1 ðx1 Þ is the PDF of ðtÞ and p1 ðx2 Þ is the PDF of ðt þ Þ. In this case C ¼

ð1 ð1 1

1

ðx1 m Þðx2 m Þp1 ðx1 Þp1 ðx2 Þdx1 dx2 ¼ 0

ð3:59Þ

Thus, the mutual covariance function between two independent processes is equal to zero (i.e. there is no correlation between ðtÞ and ðtÞ). On the contrary, under the linear dependence as specified in case (b), one can write C ðt1 ; t2 Þ ¼ h½ðt1 Þ m ½ðt2 Þ m i ¼ C ðÞ

ð3:60Þ

73


since hðtÞi ¼ h ðtÞ þ i ¼ hðtÞi þ hi ¼ m þ

ð3:61Þ

Setting t2 ¼ t1 ð ¼ 0Þ, one obtains C ðt1 ; t1 Þ ¼ C ð0Þ ¼ 2 ¼

ð3:62Þ

It will be shown later that eq. (3.62) is the maximum correlation which can be achieved. In other words, the covariance function shows how much knowledge about the process ðtÞ can be obtained by observing the process ðtÞ. The following are some properties of autocovariance functions which will be often used 1. The covariance function CðÞ of a stationary process is an even function, i.e. CðÞ ¼ CðÞ

ð3:63Þ

Indeed, since the process is stationary, one can shift the time origin by to obtain CðÞ ¼ hðtÞðt þ Þi m2 ¼ hðt ÞðtÞi m2 ¼ CðÞ

ð3:64Þ

2. The absolute value of the covariance function does not exceed its value at ¼ 0, i.e. the variance of the process jCðÞj Cð0Þ ¼ 2

ð3:65Þ

It is clear that the average of a non-negative function is a non-negative quantity, i.e. Ef½ðtÞ m ½ðt þ Þ m g2 0

ð3:66Þ

The last inequality implies that hððtÞ m Þ2 2½ðtÞ m ½ðt þ Þ m þ ððt þ Þ m Þ2 i ¼ 2½2 CðÞ 0 ð3:67Þ and thus jCðÞj 2

ð3:68Þ

For many practical random processes, the following condition is also satisfied lim CðÞ ¼ 0

!1

ð3:69Þ

74

RANDOM PROCESSES

From a physical point of view, this result can be explained by the fact that most of the physically possible systems have an impulse response which approaches zero (stable systems). As a consequence, the effect of the current value of the system state on the future is limited by the time when the impulse response is significant and disappears when the time lag approaches infinity. To summarize, the covariance function CðtÞ of a stationary process is an even function, having a maximum at ¼ 0. This maximum is equal to the variance 2 , and the covariance function vanishes as ! 1. Some commonly used covariance functions are listed in Table 3.1. Not every function which satisfies the conditions 1–3 can be a proper covariance function, an autocovariance function must satisfy an additional condition. 3. The covariance function is a positively defined function; i.e. for any ai and aj and any N moments of time N X

ai aj Cðti ; tj Þ 0

ð3:70Þ

i;k ¼ 1

This is an obvious consequence of the fact that an average of a non-negative random variable is a non-negative quantity 2 + * + * X N N N X X ai ðti Þ ¼ ai aj ðti Þ ðtj Þ ¼ ai aj Cðti ; tj Þ 0 i ¼1 i¼1 i;k ¼ 1

ð3:71Þ

An equivalent form of this definition is the fact that the cosine transform of the covariance function is also a non-negative function: ð1

CðÞ cos ! d 0

ð3:72Þ

0

It is worth mentioning that mutual covariance functions do not satisfy the conditions above. Some of the properties of autocovariance can be generalized to a case of nonstationary processes. In this case, eqs. (3.63) and (3.65) become Cðt1 ; t2 Þ ¼ Cðt2 ; t1 Þ jCðt1 ; t2 Þj ðt1 Þðt2 Þ

3.7

ð3:73Þ

CORRELATION COEFFICIENT

Equations (3.53) and (3.62) show that the correlation between functions describes not only a level of dependence between two random processes, but it also takes into account their variance. Indeed if one of the functions ðtÞ or ðtÞ deviates a little from its mean, the absolute value of the covariance function would be small, no matter how closely related ðtÞ

75

CORRELATION COEFFICIENT

Table 3.1

Common covariance functions and their Fourier transforms [20]

Normalized covariance function CðÞ=2 ¼ RðÞ

Normalized PSD Sð!Þ ¼ Ð1 j! d 1 RðÞe

N0 ðÞ 2

N0 2

exp½ ð1 þ jjÞ expð Þ

exp½ 2 2

exp½ jj cosð!0 Þ

exp½ jj½cosð!0 Þ þ

Description or form-filter

sin !0 jj !0

sin ! !

Correlation interval ðcorr Þ corr ¼ Ð1 0 jRðÞjd

1

1

1

1

1.571

2

2

1.221

pffiffiffi 2

2 2

1.065

Resonant contour

1

1

1.571

Resonant contour

1

!20 þ 2

1.571

Ideal low-pass filter

2!

ð!Þ2 3

1

First order low-pass filter

4 3

Second order low-pass filter

ð 2 þ !2 Þ2 rffiffiffiffiffiffi !2 exp 2 2 4 " 1 2 þ ð! !0 Þ2 # 1 þ 2 þ ð! þ !0 Þ2 4 ð!20 þ 2 Þ 1 2 þ ð! !0 Þ2

1 þ 2 þ ð! þ !0 Þ2 8 < if j!j ! ! : 0 if j!j > !

feff f

0

White noise

2 2 þ !2

R00 ð0Þ

Gaussian filter

J0 ð2fD jjÞ

2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð2fD Þ2 !2

Jake’s fading

1 2fD

22 fD2

1

A20 cos !0 2

2 A ½ð! !0 Þ 2 0 þð! þ !0 Þ

Sinusoid with

1

A20 !20 2

1

a random phase

8 < if j! !0 j ! 2 ! : 0 if j! !0 j > ! 2

Ideal bandpass filter

rffiffiffiffiffiffiffiffi 2 4

Gaussian bandpass filter

! 2 ! 2

sin

expð 2 2 Þ cosð!0 Þ

e

ð!!0 Þ2 4 2

þ e

2

ð!þ!0 Þ 4 2

! pffiffiffi 2

!20 þ

ð!Þ2 12

!20 þ 2 2

1

1.065

76

RANDOM PROCESSES

and ðtÞ are.3 For a quantitative description of linear4 correlation, it is reasonable to introduce a normalized function, called the correlation coefficient RðÞ RðÞ ¼

C ðt1 ; t2 Þ CðÞ ; R ðt1 ; t2 Þ ¼ 2 ðt1 Þ ðt2 Þ

ð3:74Þ

Functions R and R are called the mutual correlation coefficient and autocorrelation coefficient respectively. Alternative notation ðÞ for the correlation coefficient will also be used here. It follows from eq. (3.62) that if two random functions are related by a deterministic function, then the coefficient of mutual correlation between them in any instant of time is equal to 1; if these functions are independent, the correlation coefficient becomes zero. Thus, the correlation coefficient describes linear (but not an arbitrary [10,11]) dependence between two processes. Two stationary processes ðtÞ and ðt þ Þ with the correlation coefficient between ðtÞ and ðt þ Þ equal to zero for all values of > 0, (i.e. R ðÞ ¼ 0) are called uncorrelated. It was shown above that any two independent processes are also uncorrelated. The reverse statement is not generally true. Indeed, RðÞ ¼ 0 does not generally provide enough grounds to state that any higher order moments (non-linear covariance functions) hi1 ðtÞi2 ðt2 Þi all equal zero in the case of non-Gaussian processes. One of the most important examples which shows that uncorrelation does not imply independence can be constructed as follows. Let the joint PDF of the second order be given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ps x21 þ x22 ð3:75Þ p2 ðx1 ; x2 Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi x21 þ x22 where ps ðxÞ is a PDF of an arbitrary positive random variable. It can be seen that, due to symmetry, the cross-correlation function is equal to zero, i.e. ðtÞ and ðtÞ are uncorrelated. However, p2 ðx1 ; x2 Þ can be represented as a product of p1 ðx1 Þ and p1 ðx2 Þ only in the case of Rayleigh PDF p2 ðsÞ. It follows from eqs. (3.63)–(3.72) that the autocorrelation coefficient has the following properties ð1 RðÞ ¼ RðÞ; jRðÞj Rð0Þ ¼ 1; lim RðÞ ¼ 0; RðÞ cos ! d 0 ð3:76Þ !1

0

In the following, a notion of the correlation interval corr will be often used for both the cases of practically important correlation coefficients: autocorrelation coefficient with a monotonic decay RðÞ ¼ ðÞ and fast oscillating decaying correlation coefficient, for example in the form RðÞ ¼ ðÞ cos !0 . In both cases, correlation interval corr is defined as ð ð1 1 1 corr ¼ ð3:77Þ j ðÞjd ¼ j ðÞjd 2 1 0 If one sets ðtÞ ¼ ðtÞ then C ðÞ ¼ C ðtÞ 2 . Arbitrarily small would make C ðÞ arbitrarily small. For non-linear correlation see [10,11]

3 4

77

ERGODICITY

Geometrically, the correlation interval corr is equal to the base of a rectangle of height 1 and having the same area as under the curve j ðÞj (see Table 3.1). The quantity corr gives a rough estimate of the maximum time lag between two samples of the random process which exhibit a significant correlation between them. It is also important to mention that the definition (3.77) is not unique and an alternative definition can be introduced, for example based on the average square spread [12], or at 3 dB level.

3.8

CUMULANT FUNCTIONS ½n

½n

Cumulant functions x;...;z ðt1 1 ; . . . ; tN N Þ of a random process, or a group of random processes, can be introduced using the definition of cumulants of the random variable or a group of random variables and by sampling the random processes under consideration at the moments of time t1 ; . . . ; tN . First, the definitions are given for a case of a single random process ðtÞ and then extended to the case of a group of processes. The cumulant and moment brackets technique is applied below to the case of random processes. As a result the cumulants become functions of moments of time t1 ; . . . ; tN . Let ðtÞ be a scalar random process considered at N moments of time t1 ; . . . ; tN . Then fðt1 Þ; ðt2 Þ; . . . ; ðtN Þg is a random vector which, in turn, can be described using the joint PDF, characteristic function, joint moments, or joint cumulants as discussed in Chapter 2. For example, in the case of marginal PDF p ðx; t1 Þ depending on time t1 , the characteristic function ðu; t1 Þ also depends on the time instant t1 . As a result so do the cumulants

ðu; t1 Þ ¼ he jðtÞ i ¼ exp

1 X ðt1 Þ ¼1

!

! ð juÞ

ð3:78Þ

In a similar way, higher order cumulants may depend on more than one moment of time. The following notations are adopted here. Let a random process ðtÞ be sampled at N moments P i , of time t ¼ ti , 1 i N, then ;;...; 1 ; 2 ;...; N ðt1 ; t2 ; . . . ; tN Þ is the cumulant of order which, in turn, is the coefficient of expansion of the log-characteristic function

N ð ju1 ; ju2 ; . . . ; juN Þ into a power series ln N ð ju1 ; ju2 ; . . . ; juN Þ ¼

1 X 1 X i1 ¼ 0 i2 ¼ 0

...

1 X

ðjÞði1 þi2 þ þiN Þ

iN ¼ 0

;;...; 1 ; 2 ;...; N ðt1 ; t2 ; . . . ; tN Þ i1 i2 u1 u2 . . . uiNN i1 !i2 ! . . . iN !

ð3:79Þ

3.9

ERGODICITY

Until now the description of a random process has been achieved through corresponding statistical averages over an ensemble of realizations, i.e. over responses of a large number of identical systems. In many cases, for some strict sense stationary processes, it is possible to obtain the same characteristics by averaging a single realization over time. This possibility is justified by the fact that a stationary process is homogeneous in time. Thus, a single

78

RANDOM PROCESSES

realization which is long enough would contain all the information about the process. A process which can be described based only on a single realization is called an ergodic process. All ergodic processes are obviously stationary but the reverse statement is not true. Leaving detailed mathematical considerations outside the scope of this book, it can be noted that sufficient and necessary conditions for a stationary process to be ergodic is the fact that its covariance function vanishes at ¼ 1, i.e. lim CðÞ ¼ 0

ð3:80Þ

!1

A few particular results can be derived from the ergodicity of a random process ðtÞ. First of all, let ZðtÞ be a random function of an ergodic process ðtÞ. It is assumed that ZðtÞ is also stationary and ergodic. Then, with the probability 1 1 hZðtÞi ¼ lim T!1 T

ðT

ð3:81Þ

ZðtÞdt 0

In order to prove this formula, one can consider a random variable Z T , defined as an average of ZðtÞ over the time interval ½0; T5 1 ZT ¼ T

ðT

ð3:82Þ

ZðtÞdt 0

Taking the statistical average of both sides, changing the order of integration and statistical averaging, it can be seen that the average of Z coincides with that of the original process: 1 hZ T i ¼ T

ðT

hZðtÞi hZðtÞidt ¼ T 0

ðT

dt ¼ hZðtÞi

ð3:83Þ

0

Here the fact that hZðtÞi does not depend on time for a stationary process has been used. The next step is to show that the variance 2 ðTÞ of Z T approaches zero when T approaches infinity. It can be done by subtracting hZ T i from both sides of eq. (3.82) and squaring the difference ðT

2 ð ð 1 1 T T fZðtÞ hZðtÞigdt ¼ 2 fZðt1 Þ hZðtÞigfZðt2 Þ hZðtÞigdt1 dt2 ðZ T hZ T iÞ ¼ T 0 T 0 0 2

ð3:84Þ Taking the statistical average of both sides of eq. (3.84), one obtains 1 ðTÞ ¼ hðZ T hZ T iÞ i ¼ 2 T 2

2

ðT ðT 0

CZ ðt2 t1 Þdt1 dt2

In the following, an overbar over a function indicates averaging over time, i.e. xðtÞ ¼ T1

5

ð3:85Þ

0

ÐT 0

xðtÞdt

79

ERGODICITY

Here, CZ ðt2 t1 Þ ¼ 2Z RZ ðt2 t1 Þ ¼ EðfZðt1 Þ hZðtÞigfZðt2 Þ hZðtÞigÞ

ð3:86Þ

is the covariance function of the process ZðtÞ and Rz ðt2 t1 Þ is its correlation coefficient. By assumption ZðtÞ is stationary process and thus both of these functions depend only on t2 t1 . Changing variables to ¼ t2 t1 t2 ¼ t0 þ 2

t1 þ t2 2 t1 ¼ t0 2

t0 ¼

ð3:87Þ

and taking into account that Cz ðÞ is an even function Cz ðÞ ¼ Cz ðÞ, one obtains 2 ðÞ ¼ T 2

ðT ð 22z T 1 CZ ðÞd ¼ 1 RZ ðÞd T T T 0 0

ð3:88Þ

The last equation shows that in order to calculate 2 ðTÞ one needs to know the correlation coefficient of ZðtÞ. In two particular cases (small and large T), a good approximation of these values can be obtained. Let corr be a correlation interval of the process ZðtÞ. Then, for T K we can assume that RZ 1 and thus 2 ðTÞ 2Z

ð3:89Þ

If T corr , then eq. (3.88) can be simplified to produce the following estimate 22 ðTÞ Z T

ð1

2

jRZ ðÞjd ¼

0

22Z corr T

ð3:90Þ

Thus, if a stationary process ZðtÞ has a finite variance, correlation interval and, in addition, its covariance function approaches zero, lim!1 CZ ðÞ ! 0, then 2 ðTÞ vanishes. This means that Z T approaches a deterministic quantity which coincides with the average value of the process hZðtÞi, according to eq. (3.83) 1 hZðtÞi ¼ lim T!1 T

ðT ZðtÞdt

ð3:91Þ

0

At the same time, one can estimate how fast expression (3.91) converges. Indeed, it follows from eq. (3.90) that ðT 1 2corr 1=2 ð3:92Þ T ZðtÞdt hZðtÞi Z T 0 This means that averaging over time converges at least as fast as the arithmetic mean N 1X ZT ¼ ZðiÞ N i¼1

ð3:93Þ

80

RANDOM PROCESSES

of identically distributed and independent random variables ZðiÞ if N ¼ T=ð2corr Þ. Thus, to simplify evaluation of the average hZðtÞi, it is reasonable to use the sum (eq. 3.93) instead of the integral (eq. 3.82), assuming that the discretization step is greater than 2corr ! In a very similar way, one can obtain different moments by averaging an ergodic process over time.

3.10

POWER SPECTRAL DENSITY (PSD)

The power spectral density (PSD) of a random process ðtÞ is defined as the Fourier transform of its covariance function CðÞ, i.e. ð1 4 Sð!Þ ¼ CðÞ exp½j!d ð3:94Þ 1

The inverse Fourier transform restores the covariance function from its PSD ð 1 1 CðÞ ¼ Sð!Þ exp½ j!d! 2 1

ð3:95Þ

Equations (3.94)–(3.95) are known as the Wiener–Khinchin theorem. Setting ¼ 0 in eq. (3.95), one can obtain that the variance of the process is equal to the area under the Sð!Þ curve, i.e. ð ð 1 1 1 1 2 ¼ Cð0Þ ¼ Sð!Þ exp½j!0d! ¼ Sð!Þd! ð3:96Þ 2 1 2 1 Since CðÞ is a symmetric positively definite function, its Fourier transform Sð!Þ is always a positive (and thus real) symmetric function. If one considers ðtÞ to be a random voltage (current) then 2 can be considered as the power dissipated by a resistor of R ¼ 1 . 1 Sð!Þd! can be attributed to the spectral components located between A differential power 2 ! and ! þ d!. This is why Sð!Þ is called the power spectral density or the energy spectrum. The covariance function CðÞ and PSD Sð!Þ of a stationary process ðtÞ ‘‘inherit’’ all the properties of a Fourier pair. In particular, if Sð!Þ is ‘‘wider’’ than some test function S0 ð!Þ then CðÞ must be ‘‘narrower’’ than the corresponding covariance function C0 ðÞ and vice versa. This is dictated by the so-called uncertainty principle of the Fourier transform [13]. It also imposes conditions between smoothness of the covariance function and the rate of roll off of the PSD [14]. The effective spectral width !eff of the PDF can be defined according to the following equation ð 1 1 !eff ¼ Sð!Þd! ð3:97Þ S0 0 where S0 is the value of the PSD at a selected characteristic frequency !0. It is common to choose !0 such that Sð!Þ achieves its maximum Smax ¼ Sð!0 Þ at !0 . In this case corr !corr const:

ð3:98Þ

81

POWER SPECTRAL DENSITY (PSD)

Alternative definitions are also possible [12]; for example spectral width !0:5 can be defined on the level of 0:5Smax . Of course, !eff > !0:5 . Sometimes instead of PSD, defined by eq. (3.94), normalized PSD sð!Þ is considered 1 sð!Þ ¼ Sð!Þ ¼ 2 2

ð1 CðÞ exp½j!d 1

ð3:99Þ

It follows from eq. (3.99) that sð!Þ ¼

ð1

1 RðÞ ¼ 2

RðÞ exp½j!d; 1

ð1 sð!Þ exp½ j!d! 1

ð3:100Þ

Using the fact that the covariance function and the correlation coefficient are even functions, one obtains the equivalent form of the Wiener–Khinchin theorem: Sð!Þ ¼ ¼

ð1 ð1 1

CðÞ exp½j!d ¼

ð0

ð1

CðÞ exp½j!d þ CðÞ exp½j!d 0 ð1 ð1 CðÞ exp½j!dðÞ þ CðÞ exp½j!d ¼ 2 CðÞ cos !d 1

0

0

0

ð3:101Þ and, conversely CðÞ ¼ 2

ð1

Sð!Þ cos !

0

d! 2

ð3:102Þ

It is important to note that the PSD Sð!Þ in eqs. (3.94), (3.95), (3.101), and (3.102) is defined for both positive and negative frequencies. In addition, Sð!Þ ¼ Sð!Þ. Instead of a two-sided ‘‘mathematical’’ spectrum, it is possible to consider a one-sided ‘‘physical’’ spectrum Sþ ð!Þ defined as þ

S ð!Þ ¼

0 Sð!Þ þ Sð!Þ ¼ 2Sð!Þ

! corr and the impulse response hðtÞ produces an output ðtÞ which approaches a Gaussian process. As the time constant s increases, the output processes becomes closer to a Gaussian process. This is due to the fact that the input process can be considered as a large sum of samples of the input processes which are almost independent: ðtÞ ¼

ð1 1

hð tÞðÞd

N X k ¼ N

hðk tÞðkÞ

ð3:176Þ

WHITE GAUSSIAN NOISE (WGN)

95

If s corr then one can choose corr < < c such that ðkÞ are almost independent while hðk tÞ does not change much, i.e. the weights are approximately equal. The greater corr, the more uniformity in weights one can observe and a larger number N of terms is needed to approximate the integral in eq. (3.176).

3.15

WHITE GAUSSIAN NOISE (WGN)

One of the most important processes used in applications is a stationary Gaussian process nðtÞ, whose covariance function is a delta function scaled by a positive constant N0 =2 CðÞ ¼

N0 ðÞ 2

ð3:177Þ

Since the delta function is equal to zero everywhere except ¼ 0, the process defined by the covariance function (3.177) is uncorrelated at any two separate instants of time. Such a process is called absolutely random. The expression for the PSD of this process can be easily obtained from the sifting property of the delta function ð1 N0 N0 Sð!Þ ¼ ¼ const; Sþ ð!Þ ¼ 2 ¼ N0 ð3:178Þ CðÞ exp½j!d ¼ CðÞj¼0 ¼ 2 2 1 Thus, the spectral density of an absolutely random process has a spectral density which is a constant at all frequencies. Such a process is also known as a ‘‘white’’ process, similar to the white light which is composed of all possible chromatic lines with approximately equal power (see Fig. 3.3). For the white Gaussian noise, eq. (3.96) produces a singular result ð 1 1 2 Sð!Þd! ¼ 1 ð3:179Þ ¼ 2 1 In other words, the variance of a white noise is equal to infinity, 2 ¼ 1. This can be explained by the fact that a white noise is rather a mathematical abstraction than a real physical process. In real life, PSD usually decays with frequency, thus providing for a finite correlation interval, corr 6¼ 0, and a limited variance. However, white noise is a useful abstraction, which can be used in a case where the correlation interval of the process corr is

Figure 3.3

Covariance function and PSD of a white process.

96

RANDOM PROCESSES

much smaller than a characteristic time constant of the system under consideration. In an equivalent statement, one can use white noise if the bandwidth of the system under consideration is much smaller than the bandwidth of the noise. A simple rule of thumb can be provided for substitution of a real process by a white noise nðtÞ in analysis of systems under random excitation. Let a system with a characteristic time constant s be considered. This system is excited by a random process with covariance function CðÞ ¼ 2 RðÞ which has a relatively wide spectrum, and thus corr s . In this case, one can assume that a white noise model is a suitable model. An equivalent spectral density of noise N0 =2 can be chosen such that Sð! ¼ 0Þ ¼ Sð0Þ. In turn, this can be calculated from the covariance function CðÞ as ð1 ð1 N0 ¼ Sð0Þ ¼ CðÞd ¼ 2 CðÞd ¼ 22 corr ð3:180Þ 2 1 0 One of the prominent examples of white noise is so-called thermal noise, caused by random movement of a large number of electrons inside metals. It is known that the spectral density of thermal noise of a resistor R is defined by the so-called Nyquist formula Sþ a ð!Þ ¼ 4kTR;

N0 ¼ 4kTR

ð3:181Þ

where k ¼ 1:38 1023 J/K is the Boltzman constant (for T ¼ 200, kT ¼ 4 1021 W/Hz). It can be seen from the last equation that the PSD of thermal noise is constant and it may appear that thermal noise is an ideal example of white noise. However, one has to take into account that eq. (3.181) is valid only for low frequencies. It is obtained from a quantum formula

1 hf hf exp 1 Su ð f Þ ¼ 4kTR kT kT

ð3:182Þ

where h ¼ 6:62 1034 J s is the Planck constant. If hf =kT 1, i.e. kT 4 1021 W=Hz

6 1012 Hz f 34 h 6:62 10 J s

ð3:183Þ

(i.e. eq. (3.181) works well for frequencies up to 100 GHz and normal temperature). However, in order to calculate variance of the thermal noise, one must use the exact expression (3.182) in order to obtain a finite result

2u

¼ 4kTR

ð1 0

1 ð hf hf kT 1 xdx 22 exp ¼ ðkTÞ2 R 1 dt ¼ 4kTR kT kT h 0 exp x 1 3h

ð3:184Þ

As another example, one can consider the covariance function of the random process sðtÞ ¼ nðtÞ cosð!0 t þ ’Þ

ð3:185Þ

REFERENCES

97

obtained by modulation of a harmonic signal with a random uniformly distributed phase by white noise, which can be obtained from eq. (3.138) and is given by the equation 1 A2 N0 2 A ðÞ Cs ðÞ ¼ N0 ðÞ 0 cos ! ¼ 2 2 4 0

ð3:186Þ

That is, such a process is also white.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

16. 17.

18. 19. 20.

J. Doob, Stochastic Processes, New York: Wiley, 1990, 1953. A. Kolmogorov, Foundations of the Theory of Probability, New York: Chelsea Pub. Co., 1956. J.K. Ord, Families of Frequency Distributions, London: Griffin, 1972. D. Middleton, An Introduction to Statistical Communication Theory, New York: McGraw-Hill, 1960. A. Stuart and K. Ord, Kendall’s Advanced Theory of Statistics, London: Charles Griffin & Co., 1987. A. Abramowitz and I. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, New York: Dover, 1972. C. Nikias and A. Petropulu, Higher-Order Spectra Analysis: A Non-Linear Processing Framework, Upper Sadle River: Prentice Hall, 1993. A.N. Malakhov, Cumulant Analysis of Non-Gaussian Random Processes and Their Transformations, Moscow: Sovetskoe Radio, 1978 (in Russian). A. Papoulis, Probability, Random Variables, and Stochastic Processes, New York: McGraw-Hill, 3rd edition, 1991. O.V. Sarmanov, Maximum Correlation Coefficient (Symmetrical Case), DAN, Vol. 120, 1958, pp. 715–718. O.V. Sarmanov, Maximum Correlation Coefficient (Non-Symmetrical Case), DAN, Vol. 121, 1958, pp. 52–55. P. Bello, Characterization of Randomly Time-Variant Linear Channels, IEEE Trans. Communications, vol. 11, no. 4, December, 1963, pp. 360–393. W. Siebert, Circuits, Signals and Systems, Boston: MIT Press, 1985. S. Mallat, A Wavelet Tour of Signal Processing, San Diego: Academic Press, 1999. L. Hanzo, W. Webb and T. Keller, Single- and Multi-Carrier Quadrature Amplitude Modulation: Principles and Applications for Personal Communications, WLANs and Broadcasting, New York: John Wiley & Sons, 2000. J. Proakis, Digital Communications, Boston: McGraw-Hill, 2001. H. Sakai, Theory of Cyclostationary Processes and Its Applications, in: Tohru Katayama, Sueo Sugimoto, Eds., Statistical Methods in Control and Signal Processing, New York: Marcel Dekker, 1997. F. Riesz and B. Sz.-Nagy, Functional Analysis, New York: Ungar, 1955. K. Miller, Multidimensional Gaussian Distributions, New York: Wiley, 1964. V. Tikhonov, Statistical Radiotechnique, Moscow: 1966 (in Russian).

4 Advanced Topics in Random Processes 4.1

CONTINUITY, DIFFERENTIABILITY AND INTEGRABILITY OF A RANDOM PROCESS

Analysis and design of systems is often based on their representation by a system of differential or difference equations with random forcing functions. In order to provide a proper mathematical foundation for such analysis one has to operate with properties such as continuity, differentiability and integrability in a way similar to the deterministic case. Some basic definitions are presented below. More detailed considerations can be found in the literature [1–3]. 4.1.1

Convergence and Continuity

Let fn g ¼ 1 ; 2 ; . . . ; n ; . . . be a sequence of random variables. In order to define a notion of the limit of a sequence one has to define a proper measure of the difference n . Based on the chosen measure, the following types of convergence can be defined: 1. Convergence with probability 1. In this case Probfn ! g ¼ 1;

n!1

ð4:1Þ

2. Convergence in probability. In this case for an arbitrary " > 0, the probability of deviations larger than " is equal to zero Probfjn j > "g ¼ 0;

n!1

ð4:2Þ

3. Mean square convergence. In this case it is required that the mean square deviation is zero lim Efjn j2 g ¼ 0

n!1

ð4:3Þ

These definitions are related: convergence in probability follows from both convergence in the mean square sense and from the convergence with probability 1. Stochastic Methods and Their Applications to Communications. S. Primak, V. Kontorovich, V. Lyandres # 2004 John Wiley & Sons, Ltd ISBN: 0-470-84741-7

100

ADVANCED TOPICS IN RANDOM PROCESSES

A similar definition can be extended to the case of a random process ðtÞ by considering samples of the sequence of random processes fn ðtÞg at a given moment of time t. In addition, the notion of continuity could be introduced by considering values of the random process at two closely spaced moments of time t and t þ h. In particular, it is said that a random process ðtÞ is continuous with probability 1 at the moment of time t, if ðt þ hÞ ! ðtÞ with probability 1 as h ! 0. It is important to mention though that continuity with probability 1 does not imply that each realization of the random process ðtÞ is continuous at the same point itself1. In the following, mainly convergence (and continuity) in the mean square sense is used. Such an approach is applicable to so-called random processes of the second order, i.e. to processes whose second moment is finite m2 ðtÞ ¼ Ef2 ðtÞg < 1 at any given moment of time t. A random process of the second order is continuous in the mean square sense at the moment of time t if lim Ef½ðt þ hÞ ðtÞ2 g ¼ 0

h!0

ð4:4Þ

In order to check the validity of this condition, it is necessary and sufficient to know the covariance function C ðt1 ; t2 Þ of the random process ðtÞ: a random process ðtÞ is continuous in the mean square sense if and only if its mean is continuous for the moment of time t and the covariance function C ðt1 ; t2 Þ is continuous at t1 ¼ t2 ¼ t. Indeed, if the conditions above are satisfied then lim Ef½ðt þ hÞ ðtÞ2 g

h!0

¼ lim Ef½ððt þ hÞ m ðt þ hÞÞ ððtÞ m ðtÞÞ2 g h!0

¼ lim½C ðt þ h; t þ hÞ 2C ðt þ h; tÞ þ C ðt; tÞ þ ðmðt þ hÞ mðtÞÞ2 ¼ 0 h!0

ð4:5Þ

and, thus, these conditions are necessary. Sufficiency follows from a simple estimate Ef½ðt þ hÞ mðt þ hÞ ðtÞ þ mðtÞ2 g ¼ C ðt þ h; t þ hÞ 2C ðt þ h; tÞ þ C ðt; tÞ 0 ð4:6Þ If the left-hand side converges to zero, each term in this expression must converge to zero, thus proving sufficiency of the conditions. 4.1.2

Differentiability

The random process ðtÞ of the second order is called a differentiable in the mean square sense random process at the moment of time t ¼ t0 if the limit ðt0 þ hÞ ðt0 Þ h!0 h

ðt0 Þ ¼ lim 1

ð4:7Þ

The definition based on the probability measure allows a certain number of realizations to have discontinuity, as long as the number of such realizations is ‘‘small’’, or more accurately, the number of deviations has measure zero.

CONTINUITY, DIFFERENTIABILITY AND INTEGRABILITY OF A RANDOM PROCESS

101

exists in the mean square sense. The new random process ðtÞ is called the derivative of the random process ðtÞ in the mean square sense if the derivative exists for all moments of time t considered. The necessary and sufficient conditions for differentiability of a process ðtÞ can be formulated as follows: a second order random process ðtÞ is differentiable at the moment of time t in the mean square sense if and only if its mean m ðtÞ is differentiable at the moment of time t and the second mixed derivative @2 C ðt1 ; t2 Þ ð4:8Þ @t1 @t2 t1 ¼ t2 ¼ t exists and is finite. In order to prove necessity of these conditions, one can consider the average of the following quantity

ðt0 þ h1 Þ ðt0 Þ ðt0 þ h2 Þ ðt0 Þ 2 h1 h2

2

ðt0 þ h1 Þ ðt0 Þ ðt0 þ h1 Þ ðt0 Þ ðt0 þ h2 Þ ðt0 Þ ðt0 þ h2 Þ ðt0 Þ 2 ¼ 2 þ h1 h1 h2 h2 ð4:9Þ Averaging of the middle term on the right-hand side results in

ðt0 þ h1 Þ ðt0 Þ ðt0 þ h2 Þ ðt0 Þ m ðt0 þ h1 Þ m ðt0 Þ m ðt0 þ h2 Þ m ðt0 Þ E ¼ h1 h2 h1 h2 C ðt0 þ h1 ; t0 þ h2 Þ C ðt0 þ h1 ; t0 Þ C ðt0 ; t0 þ h2 Þ þ C ðt0 ; t0 Þ þ ð4:10Þ h1 h2 Taking the limit on the both sides of eq. (4.10), one obtains (

) ðt0 þ h1 Þ ðt0 Þ ðt0 þ h2 Þ ðt0 Þ 2 lim E h1 ; h2 !0 h1 h2

2 @ @2 m ðtÞ þ ¼ C ðt1 ; t1 Þ @t @t1 @t2

ð4:11Þ

t1 ¼ t2 ¼ t

thus indicating that the average of the right-hand side of eq. (4.9) is finite. After calculating all the terms and taking the limit for h1 ; h2 ! 0, one verifies that (

) ðt0 þ h1 Þ ðt0 Þ ðt0 þ h2 Þ ðt0 Þ 2 ¼0 ð4:12Þ lim E h1 ; h2 !0 h1 h2 Thus, the derivative of the random process ðtÞ exists in the mean square sense. Necessity also can be easily proven by observing that the right-hand side of eq. (4.9) is a sum of two non-negative terms. If the left-hand side approaches zero, then each of these terms approaches zero separately which is equivalent to the requirements of existence of the derivatives of the mean and covariance functions.

102


4.1.3

Integrability

Finally, an integral of a random process in the mean square sense can be defined as follows. Let a time interval ½a; b be divided into sub-intervals by points a ¼ t0 < t1 < < tn ¼ b, and f ðtÞ be a deterministic function defined on the same interval ½a; b. A partial sum n ¼

n X

ðti Þf ðti Þðti ti1 Þ

ð4:13Þ

i¼1

is a sequence of random variables fn g. If this sequence has a finite limit J in the mean square sense when maxðtj tj1 Þ ! 0 J¼

n X

lim

maxðtj tj1 Þ!0


ð4:14Þ

i¼1

this is called the Rieman mean square integral of the random process ðtÞ over the time interval ½a; b. J ¼ J½a; b ¼

ðb

f ðtÞðtÞdt ¼

a

n X

lim

maxðtj tj1 Þ!0


ð4:15Þ

i¼1

Improper integrals also can be defined by considering the mean square limit of the integral over a finite but increasing time interval ½a; b. For example ð1

f ðtÞðtÞdt ¼ lim

ðb

b!1 a

a

f ðtÞðtÞdt

ð4:16Þ

The mean square integral of the Rieman type exists if the following two integrals I1 ¼ I2 ¼

ðb a ðb

f ðtÞm ðtÞdt ðb f ðt1 Þf ðt2 ÞC ðt1 ; t2 Þdt1 dt2

a

ð4:17Þ

a

exist. In this case ðb E

f ðtÞðtÞdt

a

¼

ðb

f ðtÞm ðtÞdt

ð4:18Þ

a

and ðb ðb E a

a

f ðt1 Þf ðt2 Þðt1 Þðt2 Þdt1 dt2

¼

ðb ðb a

a

f ðt1 Þf ðt2 ÞC ðt1 ; t2 Þdt1 dt2 þ

ðb

2 f ðtÞm ðtÞdt

a

ð4:19Þ

ELEMENTS OF SYSTEM THEORY

4.2 4.2.1

103

ELEMENTS OF SYSTEM THEORY General Remarks

Analytical investigation of any realistic system requires an adequate mathematical model. There are three main principles which guide choice of the model: (a) it must reflect real properties of the system; (b) it must be reasonably simple; and (3) it must allow well developed mathematical techniques to be applied to the analysis and the design. In order to build such a model, certain a priori information must be available. The amount of such information defines two significantly different approaches to modeling: (a) in some cases there is enough information to model a system’s behaviour quite accurately; (b) only minimal information is available. In the former case, it is usually assumed that the mathematical model of the system is known (analysis) or its parameters must be identified (estimation), while the latter requires that such a model be synthesized. One of the most prominent examples of such synthesis is phenomenological modeling of the communication channel, considered in this book. From the mathematical perspective, operation of any system can be thought of in terms of the input–output relationships, i.e. for an arbitrary input ðtÞ the output ðtÞ is defined as some function of the input ðtÞ ¼ T½ðtÞ

ð4:20Þ

If such description is possible the mathematical description of the system is achieved by providing a proper functional T. The function ðtÞ is referred to as the reaction to the stimulus ðtÞ. Depending on the nature of the input stimulus and the reaction, a great variety of systems can be considered. Of course, the stimulus can depend not only on time but on other variables such as frequency, space coordinates, etc. It is important to note that the structure of the operator T does not necessarily reflect physical processes taking place inside the real system. Such a description only allows accurate reproduction of the output of the system once the input is specified. There are a great variety of possible structures of the system and the corresponding operator T which allow input–output description. The input signal (stimulus) and the reaction can be vectors of dimension n and m respectively, thus describing multiple-input, multiple-output (MIMO) systems, as shown in Fig. 4.1(A). Such description incorporates a simple case of single-input single-output (SISO) systems m ¼ n ¼ 1 and autonomous systems ðn ¼ 0; m ¼ 1Þ. The operator T can be deterministic or stochastic. The deterministic operator describes systems which produce the same response to the same stimulus if the same initial conditions

Figure 4.1

(A) MIMO systems, (B) SISO systems.

104


are enforced. In other words, the randomness in the response ðtÞ may appear only due to the randomness in the input signal ðtÞ. On the contrary, the operator T is called stochastic if the randomness in the response ðtÞ appears also due to the internal structure of the system. For example, it can be attributed to the random initial conditions, changing structure of the system, etc. If all physical phenomena are to be taken into account, most realistic systems are described by stochastic operators due to such internal factors as thermal noise, fluctuating external conditions (temperature, humidity), etc. However, in many cases the influence of these random factors is negligible and the system can be sufficiently described by a deterministic operator with randomness attributed only to the input stimuli. In many cases, internal randomness can be also properly represented by an equivalent random input, etc. The majority of the discussion in this book deals with deterministic systems with random, inputs, with the notable exception of systems with random structure, described in Chapter 6. Further classification of operators can be achieved based on the type of output process ðtÞ. In particular, one can distinguish between the analog and discrete and digital systems, continuous systems with jumps, etc. It is also possible to distinguish between zero-memory non-linear (ZMNL) systems, linear systems, non-linear systems with memory, causal and non-causal2, etc. The system is called a zero-memory non-linear system if the input–output relationship can be represented in the following form ðtÞ ¼ g½ðtÞ; t

ð4:21Þ

where ½ðtÞ; t is a non-linear deterministic vector function. In other words, the output of a ZMNL system at the moment of time t can be predicted based on the value of the input signal ðtÞ at the same moment of time and does not depend on the values of the input and the output signals at any other previous moment of time t0 < t. A casual system is a system which transforms current and past values of the input signals to the current value of the system output: the future values of the system input have no effect on the current output value. In other words, the output always lags the excitation. While the causal systems are the only systems which can be physically implemented, non-causal systems are often a useful tool in theoretical analysis. As an example, one can mention an ideal low-pass filter [14]. It is often possible to represent a general non-linear system with memory as a cascaded connection of ZMNL and linear sub-systems as shown in Fig. 4.2. In this case, non-linear effects of the original system are reproduced by ZMNL blocks while the memory effects are attributed to the impact of the linear block.

Figure 4.2

2

Representation of a non-linear system with memory.

It will be seen from the following consideration that there is a group of methods well tailored for each separate class of systems, thus justifying such classification.

105


Strictly speaking, an operator T ¼ L is called a linear operator if the so-called superposition principle is valid for this operator, i.e. the following equation must be satisfied for any set of the input stimuli k ðtÞ " # n n X X ck k ðtÞ ¼ ck L½k ðtÞ ð4:22Þ ðtÞ ¼ L k¼1

k¼1

Here ck are arbitrary constants, which are independent of time. All other operators are treated as non-linear. 4.2.2

Continuous SISO Systems

The deterministic linear operator L for a SISO system can be given in the form of a differential or difference equation with corresponding initial conditions. For a continuous system with constant parameters, the differential equation can be written in the form m X

n X dk dk ak k ðtÞ ¼ bk k ðtÞ; dt dt k¼0 k¼0

n<m

ð4:23Þ

or, in the symbolic form Am ðsÞðtÞ ¼ Bn ðsÞðtÞ;

ð4:24Þ

where s¼

m n X X d ; Am ðsÞ ¼ ak sk ; Bn ðsÞ ¼ bk s k dt k¼0 k¼0

ð4:25Þ

are polynomials in s defined by the coefficients of differential equation (4.23). Equation (4.23) must be augmented by a set of initial conditions in the form d dm1 ðm1Þ 0 ¼ m1 ðtÞ ð4:26Þ 0 ¼ ðt0 Þ; 0 ¼ ðtÞ ; . . . ; 0 dt dt t¼t0 t¼t0 Instead of differential equations one can use the following alternative but equivalent descriptions of the same linear system and its operator L. 1. The impulse response of the system hðt; Þ is the reaction of a linear system to a delta pulse at time t ¼ : hðt; Þ ¼ L½ðt Þ

ð4:27Þ

The response to an arbitrary stimulus ðtÞ can be then found as ðtÞ ¼ L½ðtÞ ¼ L

ð 1 1

ðÞðt Þd ¼

ð1 1

ðÞL½ðt Þd ¼

ð1 1

ðÞhðt; Þd ð4:28Þ

106


Here the sifting property of the delta function is used in the third term and the superposition principle is used to exchange the order of integration and the operator L action. For a causal system, no response appears before the excitation, i.e. hðt; Þ ¼ 0;

for t <

ð4:29Þ

thus transforming eq. (4.28) to ðtÞ ¼

ðt 1

ðÞhðt; Þd

ð4:30Þ

Linear systems can be further divided into time-variant and time-invariant systems. For a time-invariant system, the impulse response is a function of difference of the arguments, i.e. hðt; Þ ¼ hðt Þ and, thus ðtÞ ¼

ðt 1

ðÞhðt Þd

ð4:31Þ

In other words, a time-invariant system is invariant to a shift of the origin of time: ðt t0 Þ ¼ L½ðt t0 Þ

ð4:32Þ

All other systems are called time-variant. The duration of the impulse response defines the memory of a system since it reflects for how long the perfectly localized input affects the output. A linear system is called stable if its impulse response is absolutely integrable, i.e. ð1

jhðtÞjdt < 1

ð4:33Þ

0

In this case a bounded input would always produce a bounded response. Stability is widely investigated in the control theory with numerous criteria of stability [5,6]. In particular, a system is stable if all roots of the characteristic polynomial m X

ak k ¼ 0

ð4:34Þ

k¼0

have negative real parts. 2. An equivalent description of a stationary linear system can be achieved by means of the frequency response, or the transfer function, Hð j!Þ, defined as the Fourier transform of the impulse response Hð j!Þ ¼

ð1 1

hðÞ expðj! Þd ¼ jHð j!Þj exp½ jð j!Þ

ð4:35Þ


107

or, equivalently, 1 hðtÞ ¼ 2

ð1 Hð j!Þ expð j!Þd 1

ð4:36Þ

The reaction of a linear system to a harmonic signal of frequency ! is the harmonic signal of the same frequency scaled by the value of the transfer function of the system, i.e. ðtÞ ¼ L½expð j!tÞ ¼

ðt 1

ðÞ exp½ j!ðt Þd ¼ Hð j!Þ expð j!tÞ

ð4:37Þ

Instead of the Fourier transform, one can use the Laplace transform to include signals which are not properly described by the Fourier transform. In this case HðsÞ ¼

ð1 hðÞ expðsÞd

ð4:38Þ

0

A detailed discussion of the advantages of both methods can be found in [7].

4.2.3

Discrete Linear Systems

Similar characteristics can be obtained for discrete systems. It is also important to mention an important property of linear systems: characteristics of a discrete system can be obtained from similar characteristics of an equivalent continuous system, while the inverse is not always possible [8]. However, the analysis of discrete systems is important in itself, without relation to the underlying continuous systems. Let n ¼ ðt ntÞ be samples of a random process ðtÞ at the time instants separated by a fixed discretization interval t, and let be the shift (delay) operator n ¼ ðnþ1Þ ; m n ¼ nm

ð4:39Þ

The difference operator r can be defined as rn ¼ n ðnþ1Þ ¼ ð1 Þn

ð4:40Þ

Using this notation, a formal equivalent to eq. (4.23) is then given by ð1 þ 1 r þ þ p rp Þ ¼ ð0 þ 1 r þ þ q rq Þ

ð4:41Þ

In applied problems, a few particular forms of eq. (4.41) are used: (1) autoregressive (AR); (2) moving average (MA); (3) autoregressive with moving average (ARMA); and (4) autoregressive integrated moving average (ARIMA).

108


An AR model of the order M, or ARðMÞ, produces the current sample 0 of the centred output process3 as a linear combination of M samples of the output process and the sample of the input process 0 , i.e. 0 ¼

M X

m m þ 0

ð4:42Þ

m¼1

This model has M parameters m plus freedom in choosing the distribution of the input signal. Due to analytical difficulties with considering higher order difference equations, it is common to limit considerations only to ARð1Þ and ARð2Þ models. Despite such limitations, a large number of useful results can be obtained [9–12]. An MA model of order N, or MAðNÞ, produces the current value of a centred output signal as a weighted sum of the current and N past values of the input signal, i.e. 0 ¼ 0 þ

N X

n n

ð4:43Þ

n¼1

This model is also defined by values of N coefficients n and by choice of the distribution of the input process . Both AR and MA models can be treated as discrete filters. The AR model is described by an Infinite Impulse Response (IIR) filter, while the MA model is described by a Finite Impulse Response (FIR) filter. The MA model is well suited for description of relatively smooth processes, while the AR model is used when a certain impulse behaviour has to be properly represented. The ARMAðM; NÞ model is a generalization which has features of both the MA and AR models defined above. In this case, the current sample of the output signal is obtained as a sum of weighted averages of the past N samples of the input process, past M samples of the output process and the current input process. Mathematically, the equation describing ARMAðM; NÞ is thus 0 ¼

M X

m m þ 0 þ

m¼1

N X

n n

ð4:44Þ

n¼1

ARMA models also can be treated as discrete linear filters. Finally, the ARIMAðM; N; KÞ model is a further generalization of the ARMAðM; NÞ with AR part driven by the increments n ¼ rK n ¼ ð1 ÞK n

ð4:45Þ

of the output process: 0 ¼

M X m¼1

m m þ 0 þ

N X

n n

ð4:46Þ

n¼1

A non-zero mean value m can be added later. The variance of the sequence n can also be adjusted in such a way that the sample of the external excitation has the weight 1 in eq. (4.42).

3

109


Analysis of a discrete system is most conveniently performed using either the discrete Fourier transform (DFT) [13] 1 X

ðe Þ ¼ j!

n ej!n

ð4:47Þ

n zn

ð4:48Þ

n¼1

or the so-called z transform method [13] 1 X

ðzÞ ¼

n¼1

ðe j! Þ ¼ ðzÞjz¼expð j!Þ

ð4:49Þ

Both methods transform difference equations of the form (4.42)–(4.44) to an algebraic equation in the ! or z domain. Indeed, if n ¼ nk , its z transform is given by 1 X

ðzÞ ¼

n¼1

n ¼ nz

1 X

n ¼ zk nk z

n¼1

1 X

nk zðnkÞ ¼ zk ðzÞ; ðe j! Þ ¼ ej!k ðe j! Þ

n¼1

ð4:50Þ which allows the ARMA equation to be rewritten as " ðzÞ 1 þ

M X

#

"

m zm ¼ ðzÞ 1 þ

m¼1

N X

# n zn

ð4:51Þ

n¼1

and thus P 1 þ Nn¼1 n zn

ðzÞ ðzÞ ¼ P m 1þ M m¼1 m z

ð4:52Þ

Here ðzÞ is the z transform of the sequence n .

4.2.4

MIMO Systems

Analysis of MIMO systems can be conducted within the same framework considered in the previous section. In general, it is impossible to provide an expression for joint probability densities of the output vector processes but it is feasible to obtain expressions for moments and cumulants. Let n ðtÞ, 1 n N, be a random process acting at the n-th input of an N inputs, M outputs MIMO system. If the impulse response between the n-th input and m-th output is denoted by hnm ðtÞ, the response n ðtÞ of the system at the m-th output is then given by m ðtÞ ¼

N ðt X n¼1

0

hnm ðn Þn ðt n Þdn

ð4:53Þ

110


Thus, the average value mm ðtÞ of the m-th output and the mutual correlation function Rm1 ;m2 ðt1 ; t2 Þ between two outputs can be calculated as follows mm ðtÞ ¼

N ðt X n¼1

hnm ðn Þmn ðt n Þdn

Rm1 m2 ðt1 ; t2 ¼ Efm1 ðtÞm2 ðtÞg N X N ð t1 ð t2 X hm1 n1 ðn1 Þhm2 n2 ðn2 ÞRn1 n2 ðt1 n1 ; t2 n2 Þdn1 dn2 ¼ n2 ¼1 n1 ¼1 0

ð4:54Þ

0

ð4:55Þ

0

The same equations hold for the centred processes and thus for the mutual covariance functions Cm1 m2 ðt1 ; t2 Þ: Cm1 m2 ðt1 ; t2 Þ ¼

N X N ð t1 ð t2 X n2 ¼1 n1 ¼1 0

0

hm1 n1 ðn1 Þhm2 n2 ðn2 ÞCn1 n2 ðt1 n1 ; t2 n2 Þdn1 dn2 ð4:56Þ

If the input signals are stationary related, the expression for the mutual covariance function is somewhat simplified to read Cm1 m2 ðt1 ; t2 Þ ¼

N X N ð t1 þ ð t1 X n2 ¼1 n1 ¼1 0

0

hm1 n1 ðn1 Þhm2 n2 ðn2 ÞCn1 n2 ð n1 þ n2 Þdn1 dn2 ð4:57Þ

Finally, if t1 ! 1, the expression for the asymptotically stationary mutual covariance function becomes Cm1 m2 ðÞ ¼

4.2.5

N X N ð1 ð1 X n2 ¼1 n1 ¼1 0

0

hm1 n1 ðn1 Þhm2 n2 ðn2 ÞCn1 n2 ð n1 þ n2 Þdn1 dn2 :

ð4:58Þ

Description of Non-Linear Systems

As was mentioned earlier, two classes of non-linear system can be distinguished: zero memory (non-inertial) and non-zero memory (inertial) systems. Zero memory non-linearities are described by an operator of the type ðtÞ ¼ T½ðtÞ ¼ gððtÞÞ

ð4:59Þ

where y ¼ gðxÞ is a deterministic (or random) function which does not depend on time. Often a polynomial yðxÞ

N X n¼0

an ðx x0 Þn ;

ð4:60Þ


111

yðxÞ ai x þ bi ; x 2 ½xi ; xiþ1

ð4:61Þ

piece-wise linear

or transcendental approximations are used. In the latter case, the non-linearity gðxÞ is approximated by some simple and well studied transcendental functions such as exponent, tangent, logarithm, etc. Each of these approximations has its advantages and disadvantages. Often it is difficult to balance accuracy of approximation and simplicity of calculations. Mathematical models of non-linear systems with memory vary greatly. Often, they are described by a system of non-linear differential equations (state–space equations), written in the Cauchy form d ðtÞ ¼ f ½ðtÞ; t þ g½ðtÞ; t; ðtÞ dt

ð4:62Þ

where f ðÞ and gðÞ are deterministic functions, which may depend on time. In contrast to linear systems, when the general structure of the solution is known, the form of solution for a non-linear equation (4.62) depends on the order, type, excitation and many other factors. In most cases, it is impossible to obtain an exact solution even for the autonomous case ðtÞ ¼ 0. This is why the analysis of non-linear systems with memory is fundamentally different from analysis of linear systems and zero memory non-linearity: each non-linear system must be analyzed separately. However, it will be seen that for some classes of non-linear system it is possible to obtain significant information in a very general form. Instead of using differential equations, one can rewrite the state–space equations of type (4.62) using the so-called Volterra representation [14] ðtÞ ¼ h0 ðtÞ þ

1 ð X i¼1

hi ðt; 1 ; . . . ; i Þð1 Þ ði Þd1 di

ð4:63Þ

Ri

Such representation is legitimate if the function f ðxÞ is well approximated by the Taylor series and the function gðxÞ does not depend on ðtÞ. Furthermore, if functions f ðxÞ and gðxÞ do not depend explicitly on time, the eq. (4.63) can be recast as ðtÞ ¼ h0 ðtÞ þ

1 ð X i¼1

hi ð1 ; . . . ; i Þðt 1 Þ ðt i Þd1 di

ð4:64Þ

Ri

Functions hi ð1 ; . . . ; i Þ are called the Volterra kernels of the system. The first term of the sum h0 ðtÞ can be interpreted as the internal dynamics of the system since it can be obtained by setting the forcing function ðtÞ to zero. The second term is the reaction of a linear system with the impulse response h1 ðtÞ to the external signal ðtÞ. The third term is more complicated and can be viewed as a two-dimensional convolution. It also can be treated as a combination of a linear system and a ZMNL system. The same is true for higher order terms in eq. (4.64) [14]. Thus, in some cases, a non-linear system with memory can be represented by a combination of linear blocks and a ZMNL block as shown in Fig. 4.3. As a result, analysis, and sometimes synthesis, of a non-linear system can be reduced to analysis of linear systems and ZMNL systems.

112


Figure 4.3

4.3

Representation of non-linear systems using the Volterra series.

ZERO MEMORY NON-LINEAR TRANSFORMATION OF RANDOM PROCESSES

In this section, dependence of moments and cumulants of the marginal PDF p ðxÞ and the joint PDF p ðx; yÞ of the second order of a random process ¼ f ðÞ on the characteristic of the input process ðtÞ is studied. A zero memory non-linear transformation y ¼ f ðxÞ

ð4:65Þ

is assumed to be known. Transformation of a random process ðtÞ by a ZMNL system can be reduced to a problem of transformation of a random vector as considered in section 2.3. Indeed, if the joint PDF p ðy1 ; . . . ; yn ; t1 ; . . . ; tn Þ of the output process ðtÞ ¼ f ½ðtÞ considered at n moments of time is to be found, one can approach the problem in the following manner. Let ¼ ½ðt1 Þ; ðt2 Þ; . . . ; ðtn ÞT be a vector formed by the samples of the input random process ðtÞ and ¼ ½ðt1 Þ; ðt2 Þ; . . . ; ðtn ÞT be the vector formed by the samples of the output process ðtÞ. Since the transformation has no memory, each component is transformed independently and according to the same rule, i.e. y ¼ ½y1 ; y2 ; . . . ; yn T ¼ f ðxÞ ¼ ½ f ðx1 Þ; f ðx2 Þ; . . . ; f ðxn ÞT

ð4:66Þ

Thus, if the joint PDF p ðxÞ is known one can calculate the joint PDF p ðyÞ using rules of transformation of a random vector.

4.3.1

Transformation of Moments and Cumulants

At a fixed moment of time t values of the random processes ðtÞ and ðtÞ are random variables, and, thus, the relation for the moments and the central moments of the output


113

process are just given by mn ðtÞ

¼

ð1

y p ðy; tÞdy ¼

ð1

f n ðxÞp ðx; tÞdx 1 ð1 ð1 1 n n ðtÞ ¼ ½ y m1 ðtÞ p ðy; tÞdy ¼ ½ f ðxÞ m1 ðtÞn p ðx; tÞdx n

1

1

ð4:67Þ ð4:68Þ

In principle, eqs. (4.67) and (4.68) solve the problem of finding the moments of the marginal PDF of the output process ðtÞ at any given instant of time. Of course, for a stationary process, all these quantities do not depend on time. In general, a complete knowledge of the marginal PDF p ðx; tÞ is required. Further simplification can be achieved for a polynomial non-linear transformation. In this case, the transformation is a sum of powers of the input, i.e. f ðxÞ ¼

N X

ð4:69Þ

ai xi

i¼0

Using this expression in (4.67) and setting n ¼ 1, one obtains the expression for the average m1 ðt1 Þ of the output process m1 ðtÞ ¼

ð1

N X

1

! ai xi p ðx; tÞdx ¼

i¼0

N X

ai mi ðtÞ

ð4:70Þ

i¼0

Similarly, the expression for the second moment can be obtained by noting that f ðxÞ ¼ 2

N X

!2 ai x

i

¼

i¼0

N X

a2i x2i

þ2

N X

ai aj x iþj

ð4:71Þ

j¼0; j6¼i

i¼0

and, thus m2 ðtÞ

¼

N X

a2i m2i ðtÞ

þ2

N X

ai aj miþj ðtÞ

ð4:72Þ

j¼0; j6¼i

i¼0

In general, the expression f k ðxÞ would be a polynomial of the order kN and, thus, only the moments mn ðt1 Þ of order up to kN are needed to calculate the k-th moment of the output: f ðxÞ ¼ k

kN X i¼0

bi x

i

; mk ðt1 Þ

¼

kN X

bi mi ðt1 Þ

ð4:73Þ

i¼0

This approach works well for calculation of low order moments or computer simulations. However, sometimes a simpler way of calculating moments can be found based on application of eqs. (2.255)–(2.257) which allow differential equations for h f n ðxÞi in terms

114


of the cumulants of the input process ðtÞ to be obtained. Indeed, plugging f n ðxÞ into eqs. (2.257), one obtains @ 1 ds k mk ðtÞ ¼ f ðxÞ s! dxs @s ðtÞ ð4:74Þ sþp 2 @ 1 d k mk ðtÞ ¼ f ðxÞ s!p! dxsþp @s @p Example [15]: As an example an exponential transformation is considered here, i.e. ðtÞ ¼ exp½aðtÞ

ð4:75Þ

In this case @

heax i ¼

@s

as ax he i s!

ð4:76Þ

Integrating this equation, one obtains s a s he i ¼ As exp s!

ð4:77Þ

ax

where As is a constant to be determined. Since s in eq. (4.77) is an arbitrary integer, a constant A should exist such that ! 1 s X a s ð4:78Þ heax i ¼ A exp s! s¼1 Furthermore, if the random variable is identically zero, then s ¼ 0 for any s and thus A ¼ he0 i ¼ 1. Thus, finally ! 1 s X a s heax i ¼ exp ð4:79Þ s! s¼1 This result also allows one to calculate moments mn . Indeed, ! 1 s s X a k s mk ¼ hekax i ¼ exp s! s¼1

ð4:80Þ

If the input process ðtÞ is a Gaussian process then the calculation of the moments can be further simplified since the higher order cumulants vanish4. In this case it can be shown [15] that

s 1 X 1 d2s 2 D f ðxÞ ð4:81Þ m2 ðtÞ ¼ 2s s! dx 2 x¼m s¼0 1

4

While the cumulants vanish, even moments do not. Thus, the application of the approach given by eqs. (4.67) and (4.68) is not ‘‘optimal’’.


115

Example [15]: Let a Gaussian random process be transformed by a square law detector. In this case ¼ f ðÞ ¼ 2 . Using eq. (4.81), one can easily obtain expressions for the first and second moments m1 ðtÞ ¼ m21 ðtÞ þ d ðtÞ ¼ m2 ðtÞ

ð4:82Þ

m2 ðtÞ ¼ m41 ðtÞ þ 6m21 ðtÞD ðtÞ þ 3D2 ðtÞ

ð4:83Þ

Finally, the variance of the output process is just D ðtÞ ¼ m2 ðtÞ m21 ðtÞ ¼ 4m21 ðtÞD ðtÞ þ 2D2 ðtÞ

ð4:84Þ

The next task is to find an expression for the correlation function R ðt1 ; t2 Þ of the transformed process ðtÞ ð1 ð1 R ðt1 ; t2 Þ ¼ y1 y2 p ðy1 ; y2 ; t1 ; t2 Þdy1 dy2 1 1 ð1 ð1 ¼ f ðx1 Þ f ðx2 Þp ðx1 ; x2 ; t1 ; t2 Þdx1 dx2 ð4:85Þ 1

1

If both processes ðtÞ and ðtÞ are stationary processes then eq. (4.85) is simplified to ð1 ð1 f ðxÞ f ðx Þp ðx; x ; Þdx dx ð4:86Þ R ðÞ ¼ 1 1

Thus, in order to evaluate the correlation function at the output of a non-linear zero memory device, it is sufficient to know either the joint PDF p ðx1 ; x2 ; t1 ; t2 Þ at the input or its characteristic function ðju1 ; ju2 ; t1 ; t2 Þ. As a result, there are two main methods developed for the calculation of the correlation function [5]: (a) direct method based on the known PDF and (b) the Rice method, based on the orthogonal expansion of the characteristic function. Here both methods are briefly summarized. 4.3.1.1

Direct Method

In this section, the following orthogonal expansion of the joint PDF p ðx; x ; Þ, studied in section 2.2, is widely used p ðx; x ; Þ ¼ pðxÞpðx Þ

1 X

n ðÞQn ðxÞQn ðx Þ

ð4:87Þ

n¼0 5 Here fQn ðxÞg1 1 is a set of orthogonal polynomials generated by the PDF p ðxÞ and the coefficients Cn ðÞ can be found as ð1 ð1 Qn ðxÞQn ðx Þp ðx; x ; Þdx dx ð4:88Þ n ðÞ ¼

1 1

5

While it is always possible to build a system of orthogonal polynomials with a given weight p ðxÞ, this system is not always complete. It can be shown that for the Pearson system of PDFs, such a system is indeed complete. For more details see [16].

116


Using expansion (4.87), the expression for the correlation function can be rewritten as p ðx; x ; Þ ¼

1 X

n ðÞ

ð1

n¼0

1

f ðxÞQn ðxÞpðxÞdx

ð1 1

f ðx ÞQn ðx Þpðx Þdx ¼

1 X

n ðÞa2n

n¼0

ð4:89Þ In this expansion, the coefficients n can be calculated through the marginal PDF pðxÞ as an ¼

ð1 1

f ðxÞQn ðxÞpðxÞdx

ð4:90Þ

It is important to mention here that both n ðÞ and an depend on a particular choice of pðxÞ, which, in turn, defines the set of the orthogonal polynomials Qn ðxÞ. The most popular choice is a Gaussian PDF and the Gamma PDF, as discussed in section 2.2. As an alternative, pðxÞ can be chosen as the marginal PDF of the input process pðxÞ ¼ p ðxÞ.

4.3.1.2

The Rice Method

Let ð1

Fð juÞ ¼

1

f ðxÞejux dx

ð4:91Þ

be the Fourier transform of the non-linearity f ðxÞ under consideration 1 f ðxÞ ¼ 2

ð1

ð4:92Þ

Fð juÞe jux du 1

As a result, the expression for the correlation function R ðÞ can be rewritten as 1 R ðÞ ¼ 2 4 1 ¼ 2 4

ð 1 ð 1 ð 1 Fð juÞe 1

1

1

1

ð1 ð1

1

jux

du

ð1 1

Fð ju Þe

ju x

du p ðx; x ; Þdx dx

ð 1 ð 1

Fð juÞFð ju Þ p ðx; x ; Þe jðuxþu x Þ dx dx du du 1

1

ð4:93Þ Since the expression in the square brackets is the characteristic function of the second order ð ju; ju ; Þ, eq. (4.93) becomes 1 R ðÞ ¼ 2 4

ð1 ð1 1

1

Fð juÞFð ju Þ ð ju; ju ; Þdu du

ð4:94Þ


117

Calculations using eq. (4.94) can be simplified if ðtÞ is a zero mean Gaussian process. In this case 2 2 2 ðu; u Þ ¼ exp ½u1 þ 2 ðÞu1 u2 þ u2 2 1 2 2 2 2 X ð1Þn 2 ¼ exp u1 exp u2 ð ðÞu1 u2 Þn 2 2 n! n¼0

ð4:95Þ

As a result, eq. (4.94) becomes 1 1 X ð1Þn 2n n ðÞ C ðÞ ¼ 2 4 n¼0 n!

ð 1

2 X 1 2 u2 ð1Þn 2 n dn ðÞ Fð juÞu exp du ¼ 2 n! 1 n¼0 2n

ð4:96Þ where n dn ¼ 2

2 u2 Fð juÞu exp du 2 1

ð1

2n

ð4:97Þ

Relations (4.89)–(4.90) and (4.96)–(4.97) in principle allow calculation of the correlation function of the output of a non-linear device. They are especially helpful for numerical simulations. However, for the purpose of analytical analysis, these equations may become intractable. Especially, it is difficult to trace the effect of the input correlation function to the shape of the output correlation function. An alternative approach, based on the cumulant equations is considered in the following section.

4.3.2

Cumulant Method

The cumulant method is based on the application of the following identity, known as the Price formula [4,14,15,17,18] @n ðnÞ ðxÞf ðnÞ ðx Þi n R ðÞ ¼ h f @½R ðÞ

ð4:98Þ

This equation, which is valid for an arbitrary marginal PDF of the input process6, is most useful for analysis of transformation of Gaussian random processes. Without loss of generality, it is assumed that both the input and the output processes have zero mean, i.e. m1 ¼ m1 ¼ 0. Also, for convenience, the following notation is used gn ðÞ h f ðnÞ ðxÞ f ðnÞ ðx Þi 6

Originally it was derived only for the case of Gaussian processes.

ð4:99Þ

118


In the general case, the output correlation function depends on all cumulants of the input joint density p ðx; x ; Þ, i.e. gn ðÞ ¼ Fn ½2 ð0; Þ; 3 ð0; ; Þ pþq ð0½ p ; ½q Þ

ð4:100Þ

Combining the Price formula (4.98) with (4.100), one obtains a differential equation written now in terms of the cumulants @n ½2 n R ðÞ ¼ Fn ½R ðÞ; 3 ð0; Þ; . . . @½R ðÞ

ð4:101Þ

Let gn ðÞ be the value of the function Fn ðÞ at zero value of R ðÞ, i.e. gn ðÞ ¼ Fn ½R ðÞ; 3 ð0; ½2 Þ; . . . R ðÞ¼0

ð4:102Þ

As was shown in [17], the correlation function R ðÞ can be represented as a power series in terms of R ðÞ. In particular 1 1 X X 1 dn ðR ðÞÞ 1 ðkÞ n h f ðxÞ f ðkÞ ðx ÞiR ðÞ¼0 Rn ðÞ þ g0 ðÞ R ðÞ ¼ R ðÞ ¼ n dðR k! k! ðÞÞ R ð Þ¼0 k¼1 k¼1 ð4:103Þ

The unknown function g0 ðÞ can be defined from the condition R ðÞR ¼0 0

ð4:104Þ

Equation (4.103) represents a general form of dependence of the output correlation function on the input correlation function. This series is significantly simplified if the non-linear transformation is a polynomial function. In this case, an infinite series (4.103) is reduced to a sum of a finite number of terms. Equation (4.103) also allows an asymptotic value of the correlation function R ðÞ for large time lags to be obtained. Indeed, for a sufficiently large , the value of the correlation function is small, thus only the first term in expansion (4.103) could be kept to allow for an appropriate accuracy. In this case lim R ðÞ ffi h f ðxÞi2 R ðÞ

!1

4.4 4.4.1

ð4:105Þ

CUMULANT ANALYSIS OF NON-LINEAR TRANSFORMATION OF RANDOM PROCESSES Cumulants of the Marginal PDF

The cumulants of the marginal PDF of the output process can be found using a technique considered in section 2.8. A few examples are considered below.

CUMULANT ANALYSIS OF NON-LINEAR TRANSFORMATION OF RANDOM PROCESSES 119

Example 1: Let ¼ f ðÞ be given a deterministic ZMNL transformation. It is required to find the dependence of the variance D ¼ 2 of the output process on the input cumulants s . It follows from eq. (4.98) that @D 1 ds 2 ½f ðxÞ 2h f ðxÞif ðxÞ ¼ ð4:106Þ s! dx @s Setting the order s sequentially to 1; 2; . . . , one obtains the following system of equations @D d ¼ 2 f ðxÞ; f ðxÞ dx @m1 2 @D d d d2 d f ðxÞ; f ðxÞ þ f ðxÞ; 2 f ðxÞ þ f ðxÞ ¼ dx dx dx @D dx ð4:107Þ @D d d 1 d3 f ðxÞ; 2 f ðxÞ þ f ðxÞ; 3 f ðxÞ ¼ dx dx 3 dx @3 2 @D 1 d 1 d3 1 d4 f ðxÞ; 3 f ðxÞ þ f ðxÞ; 4 f ðxÞ ¼ f ðxÞ þ 3 12 dx dx @4 4 dx2 Example 2: Let ðtÞ ¼ A2 ðtÞ. In this case, equations for derivatives of the variance D with respect to low order cumulants can be easily obtained from eq. (4.107) by noticing that f 0 ðxÞ ¼ 2Ax, f 00 ðxÞ ¼ 2A and all the higher order derivatives are equal to zero. Using the rules of manipulating cumulant brackets, listed in the appendix of chapter 2, one obtains @D

@m1

¼ 2hAx2 ; 2Axi ¼ ð43 þ 81 2 ÞA2

@D ¼ h4a2 x2 i þ hAx2 ; 2Ai ¼ 4A2 ½2 þ ð1 Þ2 @D @D 1 ¼ h2Ax; 2Ai þ hAx2 ; 0i ¼ 4A2 1 3 ¼ 0 3 @2

ð4:108Þ

@D

1 1 1 d4 A2 2 h2Ai þ hAx h f ðxÞ; ¼ ; 0i þ f ðxÞi ¼ 3 12 dx4 2 4 @4 4

Solving this system of equations starting from the equation which responds to the fourth cumulant, one obtains D ¼ A2 ½4 þ 41 3 þ 22 þ 4ð1 Þ2 2

ð4:109Þ

It is worth noting that the expansion (4.109) does not depend on the particular form of the PDF p ðxÞ of the input process, which emphasizes the power of the cumulant method. 4.4.2

Cumulant Method of Analysis of Non-Gaussian Random Processes

In the previous section, cumulant equations were used to obtain relations between the cumulants of the marginal PDF at the input and the output of a zero memory non-linear

120


transformer. Application of the Price formula allows calculation of the correlation function. It is possible to generalize this equation [4,14,15,17,18] to the case of higher order correlation (cumulant) functions. For example, it can be shown [15] that Dn ½2 ðÞ

Dpþq ð0½p ; ½q Þn

¼

1 h f ðpnÞ ðxÞf ðqnÞ ðx Þi; p!q!

p 6¼ 0; q 6¼ 0

ð4:110Þ

It can be seen that the Price equation (4.98) is a particular case of eq. (4.110) with p ¼ q ¼ 1. However, its application to the case of a non-Gaussian driving process ðtÞ is limited since it does not provide a relation with all the summands of the input process ðtÞ. In order to illustrate this statement, the example of a square law transformation considered above can be used. Once again the transformation ðtÞ ¼ A2 ðtÞ is considered. Since the right-hand side of eq. (4.110) differs from zero only for p ¼ q ¼ 1, and n ¼ 2 or p; q ¼ 1 and n ¼ 1, then, eq. (4.110) produces the following five differential equations with respect to cumulants 2 2 D2 ½C ðÞ A2 d x dx2 ¼ ¼ 4A2 2 2 2 dx2 dx ð1!1!Þ D½C ðÞ D½C ðÞ 1 ¼ h4A2 i ¼ A2 2 ½2 ½2 ð2!Þ D½4 ð0 ; Þ D½C ðÞ 1 ð4:111Þ h2Ax 2Ax i ¼ 4A2 ½Cx ðÞ þ m21 ¼ D½C ðÞ 1!1! D½C ðÞ 1 h2Ax 2Ai ¼ 2A2 m1 ¼ ½2 D½3 ð0; Þ 2!1! D½C ðÞ

D½3 ð0½2 ; Þ

¼ 2A2 m1

If all cumulants s , s 1 are equal to zero, then the random processes ðtÞ and ðtÞ are equal to zero ðtÞ ¼ ðtÞ ¼ 0. Having this in mind, the following expression for the covariance function can be obtained from eq. (4.111) C ðÞ ¼ A2 f4 ð0½2 ; ½2 Þ þ 2m1 ½3 ð0½2 ; Þ þ 3 ð0; ½2 Þ þ 4m1 C ðÞ þ 2½C ðÞ2 g ð4:112Þ If the input process is a Gaussian process then all the cumulants of order s 3 are equal to zero and eq. (4.112) can be further simplified to R ðÞ ¼ A2 f2½R ðÞ2 þ 4m1 R ðÞg

ð4:113Þ

Of course, the same result can be obtained from the Price formula. Equation (4.113) remains valid if some non-Gaussian process has the skewness and the curtosis equal to zero, i.e. 3 ¼ 4 ¼ 0 and all other higher order cumulants are non-zero. Example: Square law detection of a telegraph signal. Let process ðtÞ be a random telegraph signal with the average bit rate Rb and equal probability of 1s and 0s, encoded by voltage values a. This signal has a symmetric PDF and the covariance function of the

121

LINEAR TRANSFORMATION OF RANDOM PROCESSES

exponential type C ðÞ ¼ a2 e2Rb jj

ð4:114Þ

The cumulant function of the fourth order is obtained in [17] as 4 ð0½2 ; ½2 Þ ¼ 2a4 e4Rb jj

ð4:115Þ

Application of eq. (4.112) results in the following expression for the output covariance function: C ðÞ ¼ A2 f2a4 e4Rb jj þ 2a4 e4Rb jj g 0

ð4:116Þ

This result is easy to understand, since the output process is a constant process ðtÞ ¼ Aa2 . Thus, the compensation of cumulant functions is possible only for non-Gaussian processes. For an input Gaussian process, the output process is always random and, thus, C ðÞ 0. Example: exponential detector. In this case the transformation under consideration is ðtÞ ¼ exp½ðtÞ;

f ðxÞ ¼ expðxÞ:

ð4:117Þ

pþq C ðÞ p!q!

ð4:118Þ

It follows from eq. (4.110) that D½C ðÞ

D½pþq ð0½p ; ½q Þ

¼

and its solution is given by ( C ðÞ ¼ exp

1 s X s X s¼1

s!

i¼0

) Csi s ð0½si ; ½i Þ

ð4:119Þ

Thus, in contrast to a square law detector, the correlation function of the output process ðtÞ depends on all cumulant functions s ð0½si ; ½i Þ of the input process ðtÞ. This can be explained by the fact that the exponential function has an infinite number of terms in its Taylor series; as a result, all the moments of the input process contribute to the resulting covariance function. It can be concluded that the method of cumulant equations is a very effective tool for investigation of non-linear transformations of random processes.

4.5 4.5.1

LINEAR TRANSFORMATION OF RANDOM PROCESSES General Expression for Moment and Cumulant Functions at the Output of a Linear System

The following quite general problem is to be solved in the analysis of transformation of random processes by a linear (time-invariant) system: given the system impulse response

122


hðtÞ and the statistical characteristics of the input process ðtÞ, such as joint PDF p ðx1 ; x2 ; . . . ; xn Þ of order n, find statistical characteristics of the output process ðtÞ ¼ L½ðtÞ, such as joint PDF p ðy1 ; y2 ; . . . ; yk Þ of order k < n. There is no known general method to solve this problem, unless the transformation of a Gaussian, a Spherically Invariant or a Markov process is considered7. Instead, the problem of evaluation of moment and cumulant functions could be studied.

4.5.1.1

Transformation of Moment and Cumulant Functions

Solution of the general problem for a Gaussian driving process ðtÞ is significantly simplified since it is known that the output process ðtÞ is also Gaussian and one has to find only the corresponding average m ðtÞ and the output covariance function C ðt1 ; t2 Þ. Let ðtÞ be the input of an LTI system8 with the impulse response hðtÞ and zero initial conditions. Then the output random process is given by the following integral ðtÞ ¼

ðt

hðt sÞðsÞds ¼

ðt

0

hðsÞðt sÞds

ð4:120Þ

0

This integral can be approximated by a partial sum as ðtÞ

K X

hðt sk Þðsk Þsk

k¼1

¼

K X

ð4:121Þ

hðsk Þðt sk Þsk ;

k¼1

¼ t; sk ¼ sk sk1

0 < s1 < s2 < < sk

Averaging both sides and taking into account that the average of a sum is a sum of the averages of the summands, one obtains m ðtÞ

K X

hðt sk Þm ðsk Þsk ¼

k¼1

K X

hðsk Þm ðt sk Þsk

ð4:122Þ

k¼1

Since in this relation only deterministic functions are present, one can easily take a limit of both sides assuming that all sk ! 0 m ðtÞ ¼

ðt 0

hðt sÞm ðsÞds ¼

ðt

hðsÞm ðt sÞds

ð4:123Þ

0

This property reflects the fact that operations of statistical averaging and time integration are interchangeable. 7

There are also a number of particular examples which can be solved completely. It is assumed here that ðtÞ ¼ 0 if t < 0.

8

123


Eq. (4.120) can be written for moments of time, thus producing the following chain of equations Efðt1 Þðt2 Þ ðtn Þg ð t 1

ð t2 ð tn ¼E hðt1 s1 Þðs1 Þds1 hðt2 s2 Þðs2 Þds2 hðtn sn Þðsn Þdsn 0

ð t1 ð t2

¼

0

0

ð tn

0

0

hðt1 s1 Þhðt2 s2 Þ hðtn sn ÞEfðs1 Þðs2 Þ ðsn Þgds1 ds2 dsn

0

ð4:124Þ or, equivalently ; ;...; m1; 1;...;1 ðt1 ; t2 ; . . . ; tn Þ ð t1 ð t2 ð tn ;...; ¼ hðt1 s1 Þhðt2 s2 Þ hðtn sn Þm;1;1;...;1 ðs1 ; s2 ; . . . ; sn Þds1 ds2 dsn 0

¼

0

ð t1 ð t2 0

0

0

ð tn 0

;...; hðs1 Þhðs2 Þ hðsn Þm;1;1;...;1 ðt1 s1 ; t2 s2 ; . . . ; tn sn Þds1 ds2 dsn

ð4:125Þ If all the moments of time are the same, t1 ¼ t2 ¼ ¼ tn ¼ t, an expression for the n-th moment mn ðtÞ of the output process ðtÞ can be obtained as mn ðtÞ

¼

ðt ðt

0 0

¼

ðt ðt

ðt 0

0 0

ðt

0

; ;...; hðt s1 Þhðt s2 Þ hðt sn Þm1; 1;...; 1 ðs1 ; s2 ; . . . ; sn Þds1 ds2 dsn

hðs1 Þhðs2 Þ hðsn Þm;;...; 1;1;...;1 ðt s1 ; t s2 ; . . . ; t sn Þds1 ds2 . . . dsn ð4:126Þ

Characteristically, this moment depends on the mixed moment function m;1; ;...; 1;...; 1 ðs1 ; s2 ; . . . ; sn Þ considered at n moments of time rather than on the moment mn ðtÞ, thus illustrating the effect of memory. The effect of linearity is emphasized by the fact that the output moment still depends only on a single moment function, rather than a combination of moments of different orders. Equation (4.126) allows one to understand the level of complexity of the problem of evaluation of the moments of the output process. Indeed, it indicates that if the moment function of order n is to be found, one has to measure or derive a joint moment function of the order n and perform n-fold integration. In other words, the complexity of this problem is very high unless one limits considerations only to a few lower order moments. The expression for the correlation function can be obtained from eq. (4.126) by setting n ¼ 2, thus producing R ðt1 ; t2 Þ ¼

m; 1;1 ðt1 ; t2 Þ

¼

ð t1 ð t2 0

0

hðs1 Þhðs2 ÞR ðt1 s1 ; t2 s2 Þds1 ds2

ð4:127Þ

124


In a very similar manner, equations for mutual moments between the input and the output processes can be defined as ;...; 0 0 m;1; ...;; ...;1; 1;...;1 ðt1 ; . . . ; tm ; t1 ; . . . ; tn Þ

¼ Efðt10 Þ ðtm0 Þðt1 Þ ðtn Þg ð t1 ð t2 ð tn ; ;...; 0 0 hðs1 Þhðs2 Þ hðsn Þm1; ¼ 1;...;1 ðt1 . . . ; tm ; t1 s1 ; t2 s2 ; . . . ; tn sn Þds1 ds2 dsn 0

0

0

ð4:128Þ

In particular, the mutual correlation function R ðt1 ; t2 Þ can be expressed as R ðt1 ; t2 Þ ¼

m; 1;1 ðt1 ; t2 Þ

¼

ð t2

hðsÞR ðt1 ; t2 sÞds

ð4:129Þ

0

It is also possible to obtain expressions for centred moments. The first step is to subtract eq. (4.123) from eq. (4.120) to obtain processes 0 ðtÞ ¼ ðtÞ m ðtÞ and 0 ðtÞ ¼ ðtÞ m ðtÞ with zero mean 0 ðtÞ ¼ ¼

ðt

hðt sÞðsÞds

0 ðt

ðt

hðt sÞm ðsÞds ¼

0

ðt

hðt sÞ½ðsÞ m ðsÞds ð4:130Þ

0

hðt sÞ0 ðsÞds

0

Thus, the relation between centred processes obeys the same equation as the non-centred processes. This mean that all derivations for moments and mutual moments remain valid for centred moments. The only difference is that centred moments must be used on the righthand side of eqs. (4.124)–(4.129). In particular, the covariance C ðt1 ; t2 Þ and the mutual covariance C ðt1 ; t2 Þ can be obtained as follows C ðt1 ; t2 Þ ¼ ¼ C ðt1 ; t2 Þ ¼

ð t1 ð t2 0 ð t1

0 ð t2

0 ð t1

0

hðt1 s1 Þhðt2 s2 ÞC ðs1 ; s2 Þds1 ds2

hðs1 Þhðs2 ÞC ðt1 s1 ; t2 s2 Þds1 ds2 ð t1 hðt2 sÞC ðt1 ; sÞds ¼ hðsÞC ðt1 ; t2 sÞds

0

ð4:131Þ ð4:132Þ

0

Thus, the variance of the output process is given by D ðtÞ ¼

ðt ðt 0 0

hðt s1 Þhðt s2 ÞC ðs1 ; s2 Þds1 ds2 ¼

ðt ðt

hðs1 Þhðs2 ÞC ðt s1 ; t s2 Þds1 ds2

0 0

ð4:133Þ If a linear time-variant system is considered, the above remains valid if the impulse response hðtÞ is substituted by the system function hðt; Þ.


4.5.1.2

125

Linear Time-Invariant System Driven by a Stationary Process

The results obtained in the previous section can be further simplified if the input process ðtÞ is a stationary process. In this case, expressions for the mean, covariance function, mutual covariance and the variance functions can be written as ðt

m ðtÞ ¼ m hðtÞdt 0 ð t1 ð t2 hðs1 Þhðs2 ÞC ðt2 t1 s2 þ s1 Þds1 ds2 C ðt1 ; t2 Þ ¼ 0 0 ð t2 hðsÞC ðt2 t1 sÞds C ðt1 ; t2 Þ ¼ 0 ðt ðt D ðtÞ ¼ hðs1 Þhðs2 ÞC ðs1 s2 Þds1 ds2

ð4:134Þ ð4:135Þ ð4:136Þ ð4:137Þ

0 0

These equations show that even if the input process is stationary, the output process could be non-stationary. This is very similar to the deterministic case when a constant input, applied at t ¼ 0, produces a transient process and the steady state is achieved only at t ¼ 1. However, in most realistic applications the impulse response hðtÞ decays fast enough to justify the steady state and the stationarity achieved after a time greater than the time scale of the impulse response. For example, if the impulse response has an exponential form hðtÞ ¼ expðt=corr Þ, t 0, it is reasonable to assume that at the moment of time t corr the transient process has been completed and the steady state, or stationary regime, is in place. Indeed, in this case, the finite limit of integration in eq. (4.134) can be substituted by infinite limits, thus producing m m

ð1 0

hðsÞds ¼ m Hðj!Þj! ¼ 0

ð4:138Þ

That is, the mean value does not depend on time. In order to obtain the expression for the covariance function, one can introduce a time lag ¼ t2 t1 and assume that t1 corr . Thus, changing the variable of integration and assuming infinite limits of integration, one obtains C ðt1 ; t2 Þ

¼

ð1 ð1

hðs1 Þhðs2 ÞC ð s1 þ s2 Þds1 ds2 ðs þ hðsÞds hð þ s tÞC ðtÞdt ¼ C ðÞ

0 0 ð1 0

ð4:139Þ

0

Therefore it can be said that after a transitional period, the output of a linear system driven by a wide sense stationary random process also becomes a wide sense stationary process. In other words, the output process is an asymptotically wide sense stationary process. Equation (4.139) also allows the relation between power spectral densities of the input and the output processes to be obtained. Indeed, taking the Fourier transform of both sides one

126


obtains the required relation S ð!Þ ¼ ¼

ð1 ð1 1

C ðÞ expðj!Þdt expðj!Þd

1 ð1

exp½j!ðs1 s2 Þhðs1 Þhðs2 ÞC ð s1 þ s2 Þ exp½j!ðs2 s1 Þds1 ds2 ð1 ¼ exp½j!s1 hðs1 Þds1 expðj!s2 Þhðs2 Þds2 0 0 ð1 C ð s1 þ s2 Þ exp½j!ð s1 þ s2 Þds ð01

1

ð4:140Þ

It also can be rewritten in the equivalent form in terms of the transfer function Hð j!Þ S ð!Þ ¼ Hð j!ÞHðj!ÞS ð!Þ ¼ jHð j!Þj2 S ð!Þ

ð4:141Þ

Thus, the spectral density of the output process is that of the input multiplied by the squared absolute value of the transfer function. This fundamental result allows one to effectively use the PSD for calculation of the parameters of the output process in linear systems. It is interesting to note that the transformation of the spectral densities is invariant with respect to the phase characteristic of the linear system. In other words, it is impossible, in general, to restore the transfer function of the system by measuring the PSD of the input and the output. Such limitation is related to the fact that the PSD is defined through the averaging of the Fourier transform of the input process and does not completely represent the property of the linear system. However, in many cases the phase characteristic can also be restored if there is some additional information about the properties of the system [13]. Equations (4.136) and (4.137) can be simplified to produce the following expression for the variance and mutual covariance function in the stationary regime: D ¼ C ðÞ ¼

ð1 ð1 ð01 0

hðs1 Þhðs2 ÞC ðs1 s2 Þds1 ds2

hðsÞC ð sÞds

ð4:142Þ ð4:143Þ

0

The last equation indicates that the input and the output process are stationary related. Example: filtering of WGN. One of the most often considered problems is filtering of WGN by a filter with the impulse response hðtÞ and the transfer function Hð j!Þ. In this case the output process is a zero mean Gaussian process, since m ¼ 0 in eq. (4.138). The PSD of the output process can be obtained from eq. (4.141) by noting that the PSD of WGN is a constant N0 =2: S ð!Þ ¼ jHð j!Þj2 S ð!Þ ¼

N0 jHð j!Þj2 2

ð4:144Þ


127

The covariance function of the output process ðtÞ can be obtained either by taking the inverse Fourier transform of S ð!Þ or by applying eq. (4.139) and taking into account the fact that the WGN is delta correlated ð ð1 ð sþ N0 N0 1 D ¼ hðsÞds hð þ s tÞ ðtÞdt ¼ hðsÞhð þ sÞds ð4:145Þ 2 2 0 0 0 Example: sliding averaging of a stationary random process. Let the output of a linear system be given as an integral of the input process over a finite interval of time, so-called sliding averaging of the process 1 ðtÞ ¼ 2

ð tþ

ðsÞds

ð4:146Þ

t

The random process ðtÞ is assumed to be wide sense stationary with the average m and the covariance function C ðÞ. The mean value of the process ðtÞ is thus 1 m ¼ 2

ð tþ

EfðsÞgds ¼ m

ð4:147Þ

t

The expression for the covariance function can be found to be 1 C ðÞ ¼ EfððtÞ m Þððt þ Þ m Þg ¼ 42

ð tþ ð tþþ t

C ðs1 s2 Þds1 ds2 ð4:148Þ

tþ

Using the expression of the covariance function in terms of the PSD 1 C ðs1 s2 Þ ¼ 2

ð1 1

S ð!Þ exp½j!ðs1 s2 Þd!

ð4:149Þ

Equation (4.148) can be rewritten as ð ð tþþ ð tþ 1 1 1 S ð!Þd! exp½j!s1 ds1 exp½ j!s2 ds2 C ðÞ ¼ 2 42 1 t tþ ð 1 1 sin ! 2 ¼ s ð!Þ expð j!Þd! 2 1 !

ð4:150Þ

This also implies that the PSD s ð!Þ of the output signal is equal to sin ! 2 S ð!Þ ¼ S ð!Þ !

ð4:151Þ

Since the function sin !=! is concentrated around small values of the frequency ! < , the sliding averaging removes high frequency components of the signal spectrum, thus eliminating fast, noise-like components.

128


The same results can be obtained by noting that the impulse response of the system described by eq. (4.147) is 8 < 1 jtj < hðtÞ ¼ 2 : 0 jtj >

ð4:152Þ

and the corresponding transfer function is thus Hð j!Þ ¼

sin ! !

ð4:153Þ

Having this in mind, eq. (4.139) can be now written as 1 C ðÞ ¼ 2

ð 2 jsj 1 C ð sÞds 2 2

ð4:154Þ

Setting ¼ 0, the expression for the variance of the output process can be obtained as 1 D ¼ C ð0Þ ¼ 2

ð 2 2

jsj 1 C ðsÞds 2

ð4:155Þ

since C ðÞ is an even function. A slight modification of eq. (4.147) shows that the variance of the integrated process 1 ðtÞ ¼ T

ðT

ðsÞds

ð4:156Þ

0

is given by 2 D ðTÞ ¼ T

ðT 0

2D 1 C ðÞd ¼ T T

ðT 1 r ðÞd T 0

ð4:157Þ

This equation is instrumental in the ergodic theory of random processes. In particular, it allows one to find if the variance of the average process approaches zero, i.e. the values of the average converge to the value of the quantity under consideration. Equation (4.157) indicates that if the covariance function decays fast enough, the resulting process is ergodic in the wide sense. Example: reaction of an RC circuit to the WGN. Let WGN drive an RC circuit shown in Fig. 4.4. The WGN ðtÞ, applied at the moment of time t0 ¼ 0, has constant PSD N0 =2 and the covariance function given by C ðÞ ¼

N0 ðÞ 2

ð4:158Þ


Figure 4.4

129

Reaction of RC circuit.

The output voltage ðtÞ is measured across the capacitor C and it obeys the following differential equation @ ðtÞ þ ðtÞ ¼ ðtÞ; @t

¼

1 RC

ð4:159Þ

The general solution, given the initial condition 0 ¼ ðt0 Þ is well known and is given by the following equation ðt ðtÞ ¼ 0 expð tÞ þ expð tÞ expð sÞðsÞds ð4:160Þ 0

Since the output process ðtÞ is a Gaussian process for a fixed initial condition, it is only necessary to calculate its mean and covariance functions to completely describe properties of the process ðtÞ. In order to account for the random initial condition, ðt0 Þ ¼ 0 , it is assumed that the distribution of the random variable 0 is given by a known PDF p0 ðÞ. A deterministic initial condition can be considered as a particular case with p0 ðÞ ¼ ð 0 Þ. The first step is to solve the problem for a deterministic 0 . In this case, taking the expectation of both sides of the solution (4.160), one obtains an expression for the mean value m ðt j 0 Þ of the output process ðtÞ as m ðt j 0 Þ ¼ 0 expð tÞ

ð4:161Þ

since the average of the WGN is zero. Using the definition of the covariance function, one calculates C ðt1 ; t2 Þ ¼ Ef½ðt1 Þ m ðt1 j 0 Þ½ðt2 Þ m ðt2 j 0 Þg ð t1 ð t2 2 N 0 exp½ ðt1 þ t2 Þ exp½ ðs1 þ s2 Þðs2 s1 Þds1 ds2 ¼ 2 0 0

ð4:162Þ

Setting t2 ¼ t1 þ ¼ t þ and using the sifting properties of the delta function, expression (4.162) becomes C ðt; t þ Þ ¼

N0 expð jjÞ½1 expð2 tÞ 4

ð4:163Þ

130


This immediately produces an expression for the variance of the output process as a function of time D ðtÞ ¼

N0 ½1 expð2 tÞ 4

ð4:164Þ

For the asymptotically stationary regime, i.e. for t ! 1, eqs. (4.161) and (4.163)–(4.164) are reduced to m ðt j 0 Þ ¼ 0;

D ¼

N0 ; 4

N0 expð jjÞ 4

C ðÞ ¼

ð4:165Þ

Thus, it can be seen that the stationary mean and variance do not depend on the value of the initial condition. However, the transient characteristics do depend on 0 . In order to account for the effect of random initial conditions, one can split solution (4.160) into two parts ðtÞ ¼ 1 ðtÞ þ 2 ðtÞ where 1 ðtÞ ¼ 0 expð tÞ; and 2 ðtÞ ¼ expð tÞ

ðt expð sÞðsÞds

ð4:166Þ

0

The process 2 ðtÞ is a linear transformation of a Gaussian process and thus is a Gaussian process itself. It is easy to see from the analysis above that this process has zero mean and the covariance function defined by eq. (4.165). The process 1 has the PDF defined by p1 ðxÞ ¼ expð tÞp0 ½expð tÞx

ð4:167Þ

which follows the shape of the PDF of the initial condition but has a different scale, which varies in time. The mean and the variance are given by m1 ðtÞ ¼ m0 expð tÞ; D1 ðtÞ ¼ D0 expð2 tÞ

ð4:168Þ

respectively. Here m0 and D0 are the mean and the variance of the initial condition 0 . The resulting distribution can be found as a composition of the distributions of 1 ðtÞ and 2 ðtÞ. It is important to mention here that the situation described above is common for all linear systems driven by a Gaussian process: the stationary solution does not depend on the initial state of the system and it is a Gaussian process. Example: derivative of a Gaussian process. Properties of the derivative ðtÞ ¼ 0 ðtÞ of a random process ðtÞ have been considered above in section 4.1.2. To complete the considerations, the spectral domain analysis is presented here. Since the transfer function of the ideal differentiator is given by Hð j!Þ ¼ j!

ð4:169Þ

the expression for the PSD S ð!Þ of the output process can be written as S ð!Þ ¼ jHð j!Þj2 S ð!Þ ¼ !2 S ð!Þ ¼ ð j!Þ2 S ð!Þ

ð4:170Þ

It is easy to see that this condition is equivalent to eq. (4.10) derived by different means.

131


4.5.2

Analysis of Linear MIMO Systems

Analysis of MIMO systems can be conducted within the same framework which was used in the previous section. In general, it is impossible to provide an expression for joint probability densities of the output vector processes, but it is feasible to obtain expressions for moments and cumulants. Let n ðtÞ1 n N be a random process acting at the n-th input of an N inputs, M outputs MIMO system. If the impulse response between the n-th input and m-th output is denoted by hnm ðtÞ, the response n ðtÞ of the system at the m-th output is then given by m ðtÞ ¼

N ðt X n¼1

hnm ðn Þn ðt n Þdn

ð4:171Þ

0

Thus, the average value mm ðtÞ of the m-th output and the mutual correlation function Rm1 ;m2 ðt1 ; t2 Þ between two outputs can be calculated as follows mm ðtÞ ¼

N ðt X n¼1

hnm ðn Þmn ðt n Þdn

ð4:172Þ

0

and Rm1 m2 ðt1 ; t2 Þ ¼ Efm1 ðtÞm2 ðtÞg N X N ð t1 ð t2 X ¼ hm1 n1 ðn1 Þhm2 n2 ðn2 ÞRn1 n2 ðt1 n1 ; t2 n2 Þdn1 dn2 n2 ¼1 n1 ¼1 0

0

ð4:173Þ The same equation will hold for the centred processes and, thus, for the mutual covariance functions Cm1 m2 ðt1 ; t2 Þ: Cm1 m2 ðt1 ; t2 Þ ¼

N X N ð t1 ð t2 X n2 ¼1 n1 ¼1 0

0

hm1 n1 ðn1 Þhm2 n2 ðn2 ÞCn1 n2 ðt1 n1 ; t2 n2 Þdn1 dn2 ð4:174Þ

If the input signals are stationary related, the expression for the mutual covariance function is somewhat simplified to read Cm1 m2 ðt1 ; t2 Þ ¼

N X N ð t1 þ ð t1 X n2 ¼1 n1 ¼1 0

0

hm1 n1 ðn1 Þhm2 n2 ðn2 ÞCn1 n2 ð n1 þ n2 Þdn1 dn2 ð4:175Þ

Finally, if t1 ! 1, the expression for the asymptotically stationary mutual covariance function becomes Cm1 m2 ðÞ ¼

N X N ð1 ð1 X n2 ¼1 n1 ¼1 0

0

hm1 n1 ðn1 Þhm2 n2 ðn2 ÞCn1 n2 ð n1 þ n2 Þdn1 dn2

ð4:176Þ

132

4.5.3


Cumulant Method of Analysis of Linear Transformations

It has been shown in section 4.2.2 that a transformation of a random process ðtÞ by a linear system with constant parameters can be described by linear differential equations of the form NðpÞðtÞ ¼ MðpÞðtÞ;

Nð pÞ ¼

n X

ak p ; k

Mð pÞ ¼

k¼0

m X

bk pk

ð4:177Þ

k¼0

Here Nð pÞ and Mð pÞ are polynomials of order n and m of the differential operators. If yðtÞ and xðtÞ are deterministic signals then an auxiliary variable zðtÞ, defined as m X dk dk ak k yðtÞ ¼ bk k xðtÞ zðtÞ ¼ Nð pÞyðtÞ ¼ Mð pÞxðtÞ ¼ dt dt k¼0 k¼0 n X

ð4:178Þ

is also a deterministic signal. It is known from the theory of ordinary differential equations that the solution of eq. (4.177) is a sum of a general solution y0 ðtÞ of the homogeneous equation zðtÞ ¼ 0 and a particular (forced) solution of the non-homogeneous equation yp ðtÞ: yðtÞ ¼ y0 ðtÞ þ yp ðtÞ

ð4:179Þ

If the linear system is stable9, then the part of the response y0 ðtÞ induced by a non-zero initial condition y0 ¼ yðt0 Þ vanishes after a period of time greater than the characteristic time constant s of the system. Thus, for t t0 s, the output of the system consists only of the forced solution ð1 zðt ÞhðÞd ð4:180Þ yðtÞ ¼ yp ðtÞ ¼ 0

Here hðtÞ is the impulse response of the system zðtÞ ¼ Nð pÞ yðtÞ. The function hðtÞ is also Green’s function of the operator NðpÞ, since it satisfies the condition ðtÞ ¼ Nð pÞ yðtÞ. Thus, the operator (4.180) describes a stationary regime of a process transformed by a linear system. By the same argument, the stationary solution of the operator (4.177) can be written as yðtÞ ¼

ð1

hM ðÞxðt Þd

ð4:181Þ

0

where m X d dk bk k hðtÞ hM ðtÞ M hðtÞ ¼ dt dt k¼0

ð4:182Þ

if m < n. At this moment, it is important to mention that introduction of a auxiliary function zðtÞ does not play a significant role in the derivation of eqs. (4.181)–( 4.182). However, it will be extremely useful in the calculation of cumulants of the transformed process. 9

See [5] for discussion of different criteria of stability.


The correlation function of the systems (4.177)–( 4.178) can be defined as ð1 Rh ðÞ ¼ hðtÞhðt þ Þd 1 ð1 hM ðtÞhM ðt þ Þdt RhM ðÞ ¼

133

ð4:183Þ

1

respectively. These and similar characteristics are often used to describe communication channels [4,17]. Similarly, moment functions mhk ð1 ; 2 ; . . . s Þ of linear systems can be defined as ð1 h mk ð1 ; 2 ; . . . k Þ hðtÞhðt þ 1 Þhðt þ 2 Þ hðt þ s Þdt ð4:184Þ 1

The next step is to quantitatively define the characteristic time s . While this definition is not unique, here the energy-based definition is considered: ð 1 1 2 s ¼ 2 h ðtÞdt ð4:185Þ h0 0 where h0 ¼ max½hðtÞ. Similarly, the correlation interval h can be defined as ð1 1 R2h ðÞd h ¼ 2 Rh ð0Þ 0

ð4:186Þ

Instead of considering a time-domain description of a linear system, one can use its description in the frequency domain. In this case, the system is completely described by its transfer function Hð j!Þ, which, in turn, is the Fourier transform of the impulse response hðtÞ ð1 Hð j!Þ ¼ hðtÞ expðj!tÞdt ð4:187Þ 1

The transfer function completely describes the system response in the steady state; however, some additional information (such as initial conditions) is needed to describe the reaction to non-zero initial conditions. Assuming that the transfer function is known, the next task is to consider calculation of the moment and cumulant functions. At this point, it is assumed that the moment and cumulant functions ms ðt1 ; t2 ; . . . ts Þ, s ðt1 ; t2 ; . . . ts Þ of the auxiliary process ðtÞ ¼ NðpÞðtÞ are also known. If the system has deterministic non-zero initial conditions, then it will produce a deterministic relaxation process10. The random component is thus only a result of the external driving force. Thus, in both stationary and non-stationary cases, the random output signal ðtÞ can be represented as ðt ðtÞ ¼ hðÞðt Þd ð4:188Þ 0

10

This deterministic process would contribute only to the first moment (cumulant) m1 ðtÞ ¼ 1 ðtÞ.

134


The latter allows one to immediately write expressions for the moment and cumulant functions, ms ðtÞ and s ðtÞ, of the marginal distribution. Indeed, using the definition of the moments it is possible to write ms ðt1 ; t2 ; . . . ; ts Þ ¼ Efðt1 Þðt2 Þ ðts Þg ð t1

ðt ðt ¼E hð1 Þðt 1 Þd1 hð2 Þðt 2 Þd2 hðs Þðt s Þds ¼

ð1 0

0

ð1 0

0

0

hð1 Þhð2 Þ hðs Þms ðt1 1 ; t2 2 ; . . . ; ts s Þd1 d2 ds

ð4:189Þ

The cumulant functions can be found in many ways. First of all one can calculate them from the moments, obtained through the use of eq. (4.189), using relations between cumulants and moments. This works well for the lower order cumulants. In the general case, it is more convenient to use the relation between moment and cumulant brackets, as discussed in [15]. This approach produces the following set of equations ð t1 1 ðt1 Þ ¼ hð1 Þ1 ðt1 1 Þd1 ð4:190Þ 0

2 ðt1 ; t2 Þ ¼ s ðt1 ; t2 ; . . . ; ts Þ

¼

ð t1 ð t2 0

ð t1 0

0

hð1 Þhð2 Þ2 ðt1 1 ; t2 2 Þd1 d2 ð ts 0

ð4:191Þ

hð1 Þhð2 Þ . . . hðs Þs ðt1 1 ; t2 2 ; . . . ; ts s Þd1 d2 . . . ds ð4:192Þ

It follows from eqs. (4.189) and (4.192) that the relationship between moment and cumulant functions at the input and the output of a linear system are linear and inertial. This means that the values of the cumulants s of the marginal PDF depend only on the cumulant of the same order but considered in separate time instants. Indeed, setting all the limits of integration to a fixed moment of time tk ¼ t, one obtains ðt ðt ð4:193Þ s ðtÞ ¼ hð1 Þhð2 Þ hðs Þs ðt 1 ; t 2 ; . . . ; t s Þd1 d2 ds 0

0

thus indicating non-zero memory of linear systems. This leads to a significant change of the PDF of the process ðtÞ compared to the PDF of the input signal ðtÞ with only one remarkable exception11: linear transformation of a Gaussian random process remains Gaussian! This follows from the fact that all higher order cumulant functions of the input process are zero. In general, if some cumulant function is zero for the process ðtÞ, the cumulant function of the same order of the process ðtÞ is also equal to zero. This constitutes so-called cumulant invariance of linear transformation [15] and emphasizes the importance of the model approximation: if only a first few cumulants are sufficiently non-zero (two for 11

This is also true for so-called Spherical Invariant Processes considered in Section 4.8.


135

the Gaussian approximation and four for the model approximation), the same would be valid for the output process, i.e. both these models are sufficient to describe both the input and the output process. However, the cumulant functions themselves can be changed significantly, as will be seen from the example below. A stationary regime is obtained from eq. (4.189) by setting t ! 1. In this case s ð0; 2 ; 3 ; . . . ; s Þ ð1 ð1 ¼ hðu1 Þ hðus Þs ð0; 2 u2 þ u1 ; . . . ; s us þ u1 Þdu1 dus 0

ð4:194Þ

0

A similar expression can be derived for the moment functions ms ð0; 2 ; 3 ; . . . ; s Þ. In order to obtain particular results, one has to specify the structure of the input random process. This will be considered below. It was assumed in eqs. (4.192)–(4.194) that the characteristics of the auxiliary process ðtÞ are known. However, it is worth recalling that m X d dk ðtÞ ¼ M bk k ðtÞ ð4:195Þ ðtÞ ¼ dt dt k¼1 That is, the properties of the processes ðtÞ and ðtÞ should be expressed in terms of cumulant and moment functions of the input process ðtÞ. This can be easily done if one recalls that the order of operations of statistical averaging and differentiation with respect to time is interchangeable. Having this in mind, the following relation can be derived ms ðt1 ; t2 ; . . . ; ts Þ ¼ hðt1 Þ ðt2 Þ ðts Þi @ @ @ ¼ M ðt1 ÞM ðt1 Þ M ðts Þ @t1 @t2 @ts @ @ ¼M M hðt1 Þðt2 Þ ðts Þi @t1 @ts s Y @ ¼ M ms ðt1 ; . . . ; ts Þ @ti i¼1

ð4:196Þ

Thus, the moments of the auxiliary process ðtÞ are the partial derivatives of the corresponding moments of the input random process. The same can be derived for the cumulants s Y @ s ðt1 ; t2 ; . . . ; ts Þ ¼ M ð4:197Þ s ðt1 ; . . . ; ts Þ @t i i¼1 Finally, substituting eqs. (4.196) and (4.197) into eqs. (4.189) and (4.192) one obtains the desired expressions for the moments and the cumulants of the output process ðtÞ in terms of the moments and cumulants of the input process ðtÞ. It can be seen that if a cumulant of the input process ðtÞ is equal to zero, so will be the corresponding cumulant of the auxiliary process ðtÞ. Thus the cumulant invariance is preserved for the process ðtÞ. If the input process ðtÞ is Gaussian, then the output process is a Gaussian process as well and it is sufficient to find only second order cumulants.

136


Example: response of a linear circuit to a Poisson pulse. Poisson pulse flow is described by a flow of pulses of a deterministic or random shape. Time instants of the impulse appearance obey the Poisson distribution: the probability that there are exactly N pulses on the interval of duration t is given by Pn ¼

ðtÞN expðtÞ N!

ð4:198Þ

Here, is the average number of pulses per unit interval. Thus, the Poisson pulse flow can be represented as ðtÞ ¼

1 X

ai Fi ðt ti Þ

ð4:199Þ

i¼1

where Fi ðtÞ is the so-called elementary pulse (random or deterministic) and ai is the pulse magnitude (random or deterministic). It is shown in [15] that the cumulants of this process are given by ð1 s s ð0; 2 ; . . . ; s Þ ¼ ha i FðuÞ Fðu þ 2 Þ . . . Fðu þ s Þdu ð4:200Þ 1

Now, the problem is reduced to calculation of the cumulants of the output process using the relation (4.193). If the linear system has the impulse response hðtÞ, then its response to an elementary pulse is ð1 IðtÞ ¼ hðÞ Fðt Þd ð4:201Þ 1

It can be clearly seen that, using this notation, eq. (4.193) for the output cumulants can be rewritten as ð1 s IðuÞIðu þ 1 Þ Iðu þ s Þdu ð4:202Þ s ð0; 1 ; . . . ; s Þ ¼ ha i 1

Thus, it can be deducted by comparing eqs. (4.200) and (4.202), that these equations are identical in structure and the only difference is that instead of the pulse shape function FðtÞ, the response waveform IðtÞ of the linear system to the pulse function FðtÞ is used. This implies that the output process is also a Poisson process defined as ðtÞ ¼

1 X

ai Iðt ti Þ

ð4:203Þ

i¼1

The cumulant functions s of the output process ðtÞ are thus given by eq. (4.202). Setting all the lag variables to zero, i ¼ 0, one obtains ð1 s F s ðtÞdt s ¼ ha i ð1 ð4:204Þ 1 s s s ¼ ha i I ðtÞdt 1


137

A characteristic time constant for the cumulant function s ðÞ can be introduced in the very same manner as it was introduced for the second order cumulant function. Specifically, they can be defined by the following equations ð 1 1 s ¼ 2 F ðtÞdt F0 1 ð 1 1 s I ðtÞdt Ts ¼ 2 I0 1 Ts

ð4:205Þ

where F0 ¼ maxfFðtÞg and I0 ¼ maxfIðtÞg. Using this notation, the corresponding cumulants can be written as s ¼ has iF0s Ts ;

s ¼ has iI0s Ts

ð4:206Þ

It is clear that eq. (4.206) can be used to define normalized cumulant coefficients s , similarly to the definition introduced in section 2.2. It also can be shown that the relation between the input cumulant coefficients sin and the output cumulant coefficient sout under the assumption that hai ¼ 0 Ts ½T2 2 s

sout

¼

sin

½T2 2 Ts s

ð4:207Þ

It is shown in [15] that " out

2in

2s

T2 T2

#s1 ð4:208Þ

and

out

2s1

" #s12 T in 1 T2

2s1 T1 T2

ð4:209Þ

It can be seen by inspection of expression (4.207) that the main factor contributing to change of the PDF of the output process is the ratio between duration of pulses FðtÞ and IðtÞ. Since this ratio is defined by the length of the impulse response of the linear system under consideration, one can conclude that the systems with long memory (or, equivalently, with long response time) would significantly change the PDF of the output process.

4.5.4

Normalization of the Output Process by a Linear System

In this section, it is shown that a linear system tends to normalize the output process, independently of the distribution of the process at the input. For simplicity, the case of a

138


symmetric distribution of pulses is considered. This implies that all odd moments of ai are equal to zero, i.e. ha2sþ1 i ¼ 0

ð4:210Þ

In turn, eq. (4.210) implies that all odd moments of the output process are also equal to zero: h2sþ1 i ¼ 0 and the output PDF is also a symmetric curve. In this case, it is possible to rewrite eqs. (4.208)–(4.209) to obtain the following expression for the even cumulant coefficients ! s1 T out in 2 ¼ 2s ð4:211Þ

2s T2 out ¼ 0. Since, for any linear keeping in mind that the odd coefficients are equal to zero: 2sþ1 system with memory, the impulse response has non-zero duration, then T2 < T2 , which immediately implies that the relative weight of the higher order cumulants decreases with number: ! 2 out in in

2s T2

2s out in 2s in ð4:212Þ

2s < 2s ; out ¼ in

2sþ2 2sþ2 T2

2sþ2

This fact shows that the linear system with memory and T2 > T2 suppresses higher order cumulants12, i.e. it normalizes the output. If the memory of the linear system is long, i.e. T2 T2 , the output process is certainly very close to a Gaussian random process. The normalization property is clearly related to the scale of the time constant s of the linear system with respect to the correlation interval of the input signal if s T2 . Then, during the correlation interval of the input signal the impulse response of a linear system remains virtually unchanged, and, thus ð1 ð1 IðtÞ ¼ hðÞFðt Þd ¼ hðtÞ FðuÞdu ¼ hðtÞA ð4:213Þ 0

0

and T2

1 ¼ 2 2 E h0

ð1 1

A2 h2 ðtÞdt ¼ s T2

ð4:214Þ

Ð1 Here E ¼ 0 F 2 ðtÞdt is the energy of a single pulse. Thus, the bigger s is with respect to T2 the bigger the normalization of the output process is as well. This statement can also be recast in terms of the effective bandwidth of the linear system Ð1 feff ¼

12

0

jHð j!Þj2 K02

d! 2

ð4:215Þ

Since the duration of pulses is defined in an equal area then it is possible that a linear system with memory would shorten the duration of the pulse defined in an equal area sense.


139

Since in all of the physical systems the following relationship is valid [19] s

1 feff

ð4:216Þ

it follows from eq. (4.214) that a deeper normalization is achieved by a system with a narrower bandwidth. Usually the normalization property of a linear system is explained in terms of the Central Limit Theorem [1]. Indeed, the number of impulses exciting the system during a time interval T2 is on average equal to T2 . During the same period of time, the input process can be represented as a sum of T2 terms on average. If T2 T2 , and T2 1, then the conditions of the Central Limit Theorem are satisfied and the output process approaches a Gaussian one. Example: normalization of a process by a linear RC circuit. The impulse response of a linear RC circuit is an exponential function hðtÞ ¼ e t ;

t0

ð4:217Þ

1 RC

ð4:218Þ

where ¼ feff ¼

is the effective bandwidth of the filter. It is assumed that han i ¼ 0 and the excitation pulses in eq. (4.199) are the delta functions, i.e. FðtÞ ¼ ðtÞ

ð4:219Þ

It is known that in this case, the cumulants of the input process are delta functions as well [15] 1 ðtÞ ¼ 0; s ðt1 ; t2 ; . . . ; ts Þ ¼ Ds ðt2 t1 Þ ðt2 t1 Þ

ð4:220Þ

Ds ¼ has i

ð4:221Þ

where

Taking into account that all the cumulants are either zero or delta functions, the integration of eq. (4.202) results in 1 ¼ 0;

and

s ¼

Ds ; 2 s

s ¼ 2; 3; . . .

ð4:222Þ

It seems that there is no normalization, since all the cumulants are still non-zero, while the time constant s ¼ 1= is much greater than T2 ¼ 0. However, the cumulants of the filtered process are finite, while the cumulants of the original process are equal to infinity. It is also

140


characteristic that the normalized magnitude of cumulants decays with order. Thus, it could be claimed that the filtered process ðtÞ is indeed closer to Gaussian than the original one

s

4.6 4.6.1

¼

ð2 Þs=2 Ds s=2

s Dd

;

s3

ð4:223Þ

OUTAGES OF RANDOM PROCESSES General Considerations

The problem of a boundary attainment existed for a relatively long period of time. One of the first results in this area was the work of A. Pontryagin [20], who considered calculations of the distribution of the first passage time by a Markov random process, and the pioneering work of S.O. Rice [21,22]. Essentially, the latter laid a foundation for the applied analysis of the characteristics of random processes related to a level crossing problem. A comprehensive review of the recent advances in the theory can be found in [20,23]. In particular, level crossing problems have a great importance in communication problems. The following are just a few examples from a long list: calculation of the average duration of the outage of the fading carrier in channels with fading [24], loss of synchronization in phase-lock loops [25], the average level crossing rate and fade duration in channels with fading [26], evaluation of the average length of bursts of errors in discrete communication channels [27], etc. Let 1 ðtÞ be a realization of a random continuous process ðtÞ defined on a finite time interval ½0; T and min < H < max be a fixed (for now) level, as shown in Fig. 4.5.

Figure 4.5 Definition of the first passage time and the average level crossing rate.

Let be the duration of the segment of the realization 1 ðtÞ which exceeds the chosen level H. This interval of time is called a duration of the upward excursion, and the portion of the trajectory above the threshold H is called a positive pulse. Similarly, the duration downward excursion is defined as the duration of the portion of the trajectory below the threshold H (negative pulse). In addition, let tm be the moment of time when the trajectory achieves its maximum value max . Finally, let n be the number of pulses (both negative and positive). Distributions of the number of extrema and related statistics are studied in extreme value theory [26] and are not considered in this book. Here, the main focus of discussion is kept on

OUTAGES OF RANDOM PROCESSES

141

the evaluation of some statistical characteristics of the random variables , and n due to their applied importance.

4.6.2

Average Level Crossing Rate and the Average Duration of the Upward Excursions

Let random process ðtÞ be differentiated in a mean square sense (see Section 4.1.2 for more details). It is also assumed for now that the joint probability density p_ ðx; x_ ; tÞ of the process ðtÞ and its mean square sense derivative _ðtÞ ¼ dðtÞ=dt at the same moment of time are also known. Details of its evaluation from the joint PDF are considered below. The process ðtÞ will cross the level H in the vicinity ½ti t=2; ti þ t=2 of the moment of time ti if the following condition is met

t t H ti þ H 0. This is also equivalent to the condition that ðt t=2Þ < H and ðt þ t=2Þ > H. Let the interval ½0; T be split into m non-intersecting sub-intervals of equal duration t. The number m is to be chosen in such a way that there is no more than one upward excursion on each of the intervals. In other words, the length of the interval t is so short that the probability of more than one intersection is much smaller than the probability of a single upward excursion [21]. Inside this small interval, a continuous process ðtÞ can be approximated as ðiÞ ðt0 Þ ðiÞ ðtÞ þ _ðiÞ ðtÞðt0 tÞ;

t < t0 < t þ t

ð4:225Þ

Here ðiÞ ðtÞ is the i-th realization of the random process ðtÞ. Thus, on the interval ½t; t þ t there are two possibilities: there is, with the probability q0, a level crossing inside this interval,

142


or, with the probability p0 ¼ 1 q0, there is none. Having this in mind, a binary sequence of ðiÞ ðiÞ ðiÞ ðiÞ m pulses 1 ; 2 ; . . . ; m , where j ¼ 1 if _ðtj Þ > 0 and ji ¼ 0 otherwise, can be constructed. Thus, the number of upward excursions on the i-th realization can be calculated as nþ ðH; TÞ ¼

m X

ðiÞ

j

ð4:226Þ

i¼1

Since this number varies from realization to realization, it is more appropriate to calculate the average number of level crossings as N þ ¼ hnþ ðH; TÞi ¼

m X ðiÞ hj i

ð4:227Þ

i¼1

The average value of an individual pulse can be calculated using the probability of each pulse. This produces ðiÞ

hj i ¼ 0 Probfji ¼ 0g þ 1 Probfji ¼ 1g ¼ q0

ð4:228Þ

Thus, the problem of finding an average number of upward excursions is reduced to finding the probability q0. It can be seen by inspection of Fig. 4.6 and eq. (4.224) that the conditions of the appearance of the positive pulse can be rewritten on the interval ½tj ; tj þ tj as H j < ðtj Þ H; and _ðtj Þ > 0

ð4:229Þ

j ¼ _ðtj Þtj

ð4:230Þ

where

This immediately implies that q0 ¼ ProbfH j < ðtj Þ H;

_ðtj Þ > 0g

ð4:231Þ

Assuming that tj ; j ! 0 and the derivative of the process varies in a narrow interval d_x around a fixed value x_ , it is possible to approximate the right-hand side of eq. (4.231) by ProbðH j < j H; x_ < _ðtj Þ x_ þ d_xÞ p_ ðH; x_ ; tj Þj d_x p_ ðH; x_ ; tj Þtj x_ dx ð4:232Þ where p_ ðH; x_ ; tj Þ is the joint PDF of the process ðtÞ and its time derivative. Integrating eq. (4.232) over all possible positive slopes (only such trajectories will be able to achieve the boundary), one concludes that q0 ¼ t

ð1 0

x_ p_ ðH; x_ ; tÞd_x

ð4:233Þ


143

Finally, the average number of positive pulses N þ can be obtained by setting the number of intervals m ! 1 and adding contributions from each interval: þ

N ¼

ðT

ð1 dt

0

0


ð4:234Þ

In the very same manner, the average number N of negative pulses is

N ¼

ðT

ð0 dt 1

0


ð4:235Þ

Ultimately, the average number of level crossings is then þ

N ¼N þN ¼

ðT

ð1 dt 1

0

jx_ jp_ ðH; x_ ; tÞd_x

ð4:236Þ

Furthermore, expressing the joint probability density p_ ðH; x_ ; tÞ in terms of conditional density p_jH ð_x; t j H; tÞ as p_ ðH; x_ ; tÞ ¼ p_jH ð_x; t j H; tÞp ðH; tÞ

ð4:237Þ

expression (4.236) can be rewritten as N¼

ðT

p ðH; tÞdt

ð1

0

1

j_xjp_jH ð_x; t j H; tÞd_x

ð4:238Þ

These equations were first obtained in [21,22]. If the process under consideration is a stationary process then its joint PDF does not depend on time, i.e. p_ ðx; x_ ; tÞ ¼ p_ ðx; x_ Þ ¼ p_jH ð_x j xÞp ðxÞ

ð4:239Þ

In this case, eq. (4.238) can be further simplified to N¼

ðT 0

p ðHÞdt

ð1 1

j_xjp_jH ð_x j xÞd_x ¼ Tp ðHÞ

ð1 1

j_xjp_jH ð_x j xÞd_x

ð4:240Þ

Dividing both parts of the latter by the duration of the observation interval T, the average number n of crossings per unit time can be obtained as N n ¼ ¼ p ðHÞ T

ð1 1

j_xjp_jH ð_x j xÞd_x

ð4:241Þ

144


Similarly, the average number of upward nþ and downward n excursions are given by ð1 Nþ þ ¼ p ðHÞ n ¼ x_ p_jH ð_x j xÞd_x ð4:242Þ T 0 ð0 N ¼ p ðHÞ x_ p_jH ð_x j xÞd_x ð4:243Þ n ¼ T 1 It was shown earlier that the values of a random process ðtÞ and its derivative _ðtÞ are uncorrelated at the same moment of time. This means that the values of the process and its velocity at the same level are only loosely related. In many cases, an even stronger statement is true: the process and its derivative are independent, i.e. p_jH ð_x j xÞ ¼ p_ ð_xÞ and, thus, eqs. (4.240)–(4.243) can be further simplified to produce ð1 N n ¼ ¼ p ðHÞ j_xj p_ ð_xÞd_x ¼ Cp ðHÞ T 1 ð1 Nþ þ ¼ p ðHÞ x_ p_ ð_xÞd_x ¼ C þ p ðHÞ n ¼ T 0 ð0 N ¼ p ðHÞ x_ p_ ð_xÞd_x ¼ C p ðHÞ n ¼ T 1

ð4:244Þ

ð4:245Þ

Thus, for the case of an independent derivative the level crossing rates (LCR) of a level H n, nþ , and n are proportional to the values of the PDF of the process considered at the level H. This fact can be used in order to experimentally measure the PDF of the process by counting a number of crossings at different levels of the process. The next step is to calculate an average duration of upward (downward) excursions. Without loss of generality, only the case of upward excursions is considered. For a given time interval ½0; T, the average time spent by the process above a given threshold H is approximately given by ð1 p p ðxÞdx ð4:246Þ T ¼T H

Since, during the same time interval, there are approximately N þ ¼ nþ T positive pulses, the average duration T þ of a positive pulse is Ð1 p p ðxÞdx T ÐH1 Tþ ¼ þ ¼ N p ðHÞ 0 x_ p_jH ð_x j xÞd_x

ð4:247Þ

In the very same manner, the average duration of a negative pulse is ÐH n T 1 p ðxÞdx T ¼ ¼ Ð0 N p ðHÞ 1 x_ p_jH ð_x j xÞd_x

ð4:248Þ


145

It is common to refer to the quantities defined by eqs. (4.245) and (4.248) as an average level crossing rate (LCR) and an average fade duration (AFD). These statistics are often misleadingly called higher order statistics (HOS) of signals. However, it is commonly agreed in statistical signal processing that the higher order statistics refer to the cumulants, moments and their spectra of order higher than two [28]. The calculations above assume that the expression for the joint PDF p_ ðx; x_ Þ of the process and its derivative is known. In the case of a stationary process, it can be shown that this function can be calculated from the joint PDF p ðx1 ; x2 ; Þ of the process at two moments of time. Indeed, choosing t1 ¼ t

t ; 2

t2 ¼ t þ

t ; 2

t ¼ t2 t1

and using continuity of the process ðtÞ, one can write t t _ t x_ tþ ¼ ðtÞ þ ðtÞ; x2 ¼ x þ 2 2 2 t t _ t x_ t ¼ ðtÞ ðtÞ; x1 ¼ x 2 2 2 This transformation allows the joint PDF of ðtÞ and _ðtÞ to be obtained as t t p_ ðx; x_ Þ ¼ lim tp x x_ ; x þ x_ ; t t!0 2 2

ð4:249Þ

ð4:250Þ

ð4:251Þ

Since the joint PDF p ðx1 ; x2 Þ is a symmetric function of its arguments, eq. (4.251) implies that p_ ðx; x_ Þ ¼ p_ ðx; _xÞ

ð4:252Þ

Thus, if a process ðtÞ is stationary and has a derivative, then the distribution of the velocity _ðtÞ is symmetric with respect to zero. If it were not so, the bias in velocity distribution would lead to non-stationarity, since the process will tend either to drift ‘‘upward’’ or ‘‘downward,’’ thus causing a change in the mean value of the process in time. 4.6.3

Level Crossing Rate of a Gaussian Random Process

The problem of level crossing can be solved completely for a Gaussian case. Since ðtÞ and _ðtÞ are related by a linear operation of differentiation, the Gaussian nature of the process ðtÞ would imply that its derivative is also a Gaussian process. Due to the symmetry prescribed by eq. (4.252), the mean value of the process _ðtÞ is equal to zero and it is only required to find its correlation function13 R__ ðÞ. Using eqs. (4.131)–(4.132), it is easy to show that R_ ðÞ ¼

@ R ðÞ; @

R__ ðÞ ¼

@2 R ðÞ @ 2

ð4:253Þ

Since the process ðtÞ has zero mean, the correlation function and the covariance function are equal: R ðÞ ¼ C ðÞ.

13

146


Since the correlation function R ðÞ has maximum at ¼ 0, this means that R_ ð0Þ ¼ 0

ð4:254Þ

For a Gaussian process, this implies that the process and its derivatives are also independent and, thus, p_ ðx; x_ Þ ¼ p ðxÞp_ ð_xÞ

ð4:255Þ

The variance of the derivative _ðtÞ can be found as the values of its correlation function at zero @2 D_ ¼ 2 R ðÞ ¼ R00 ð0Þ ¼ D 00 ð0Þ ð4:256Þ @ ¼0 Here ðÞ is the normalized covariance function (coefficient) of the process ðtÞ and D is the variance of this process. Combining eqs. (4.255) and (4.256), one obtains an expression for the joint PDF of a Gaussian process and its derivative. ( " #) 2 _ x 1 1 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð4:257Þ x2 00 p_ ðx; x_ Þ ¼ p ðxÞp_ ð_xÞ ¼ 2D 00 ð0Þ 2D ð0Þ

Finally, the level crossing rate of a Gaussian process with the covariance function R ðÞ can be obtained from eq. (4.245) qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 ð0Þ H 2 ð 1 s2 e 2D LCRðHÞ ¼ se 2 ds; ð4:258Þ 2 0 where s¼

x_ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi D 00 ð0Þ

ð4:259Þ

Since the last integral in eq. (4.258) is equal to unity, one obtains

LCRðHÞ ¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 ð0Þ 2

H2 2D e

¼


H2 2D e

¼


e

ðH=

pffiffiffiffi 2 D Þ

2

ð4:260Þ

The values of 00 ð0Þ for some important correlation functions are summarized in Table 3.1. Some of these values are infinite, thus indicating that the process with such a correlation function has a derivative with infinite variance. The expression for the level crossing rate (4.260) can now be used to obtain an expression for an average duration of the positive outage. Indeed, using the Laplace function ðxÞ [29] ð1 H

! H p ðxÞdx ¼ 1 pffiffiffiffiffiffi D

ð4:261Þ

147


expression (4.260) can be transformed to produce

2 Hffiffiffiffi H p 2 1 exp 2D D þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T ¼ 00 ð0Þ

ð4:262Þ

It can be seen from eqs. (4.260) and (4.262) that the expressions for the LCR and AFD are more conveniently expressed in terms of the normalized threshold H h ¼ pffiffiffiffiffiffi D

ð4:263Þ

Dependence of the LCR and AFD on the normalized threshold h is shown in Fig. 4.7 for two cases of covariance function ðÞ ¼ J0 ð2fD Þ;

00 ð0Þ ¼ 22 fD2

ð4:264Þ

2

ð4:265Þ

and 2 Þ; ðÞ ¼ expð 2 =corr

00 ð0Þ ¼

2 corr

It can be seen that the LCR and AFD are expressed in terms of the time characteristic of the random process, in particular both quantities are functions of the values of the second

Figure 4.7 Dependence of LCR (a) and AFD (b) on the normalized threshold and the type of correlation function with the same correlation interval.

148


Figure 4.7

(Continued )

derivative of the correlation function 00 ð0Þ. The same calculation could be conducted in the frequency domain as well. Indeed, using the Parseval theorem and the properties of the Fourier transform one obtains

00 ð0Þ

¼

1 2

ð1

ð j!Þ2 Gð!Þd!

0

2

4 ¼ 2

ð1

f 2 Gð f Þdf

ð4:266Þ

0

Example: Bandlimited (pink) noise. In this case of an ideal baseband filter with the cut-off frequency F, the one-sided PSD of filtered WGN is given by PSDð f Þ ¼

0 f F f > F

N0 0

ð4:267Þ

The value of 00 ð0Þ for this process is thus Ð1 2 Ð F 2 4 f Gð f Þd f 4 f N0 df 4N0 ðFÞ3 4ðFÞ2 00 0 0 ¼ ¼ ¼ Ð F ð0Þ ¼ Ð 1 N0 F 3 3 N0 df 0 Gð f Þd f

ð4:268Þ

0

Example: A Gaussian process with a Gaussian PSD. zero mean be described by a Gaussian PSD:

Let a Gaussian process ðtÞ with

"

2 # f PSDð f Þ ¼ N0 exp 4 feff

ð4:269Þ

149


This PSD corresponds to a Gaussian correlation function 2 2 Þ ðÞ ¼ expðfeff

ð4:270Þ

It is easy to check that both time-domain and frequency-domain methods produce the same answer: 00 ð0Þ ¼ 2ð feff Þ2

ð4:271Þ

Example: Clark’s PSD. It is commonly accepted that the one-sided PSD of the signal received by a moving receiver is provided by the following expression [30] PSDð f Þ ¼

2 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; fD 1 f 2 =fD2

0 f fD

ð4:272Þ

where fD ¼ c f cos is the Doppler frequency of the signal moving with velocity and the angle of arrival of the signal at the receiver’s antenna is ; c is the speed of light in the air. Calculations using both eqs. (4.253) and (4.266) produce 00 ð0Þ ¼ 2fD2 4.6.4

ð4:273Þ

Level Crossing Rate of the Nakagami Process

The Nakagami process is often used to describe non-Rayleigh fading [31]. As was pointed out in section 4.6.2, the LCR can be found using the joint PDF of the process at two separate moments of time. In contrast to a Gaussian random process and the Rayleigh process as an envelope of a complex Gaussian process, the Nakagami process in general cannot be described completely by the joint PDF of the second order. In addition, a different joint PDF can be derived for the same marginal PDF and the correlation function. However, the following model is the most popular in practical applications [31]: p ðx1 ; x2 ; Þ

4ðx1 x2 Þm mmþ1 mx21 þ mx22 2m 0 ¼ exp Im1 x1 x2 ; ðmÞð1 20 Þ m1 mþ1 ð1 20 Þ ð1 20 Þ 0

m

1 2

ð4:274Þ Here 0 ðÞ is the correlation function of the underlying Gaussian process [31], ðÞ 20 ðÞ is the normalized covariance function of the squared process 2 ðtÞ (square of the envelope) and In ðxÞ is the modified Bessel function of the first type of the order n [29]. Thus, eq. (4.274) can be rewritten as

pffiffiffi 2m 4ðx1 x2 Þm mx21 þ mx22 x1 x2 p ðx1 ; x2 ; Þ ¼ exp Im1 ð1 Þ ð1 Þ ðmÞð1 Þ ðm1Þ=2 mþ1 ð4:275Þ

150


The next step is calculation of the joint PDF of the process ðtÞ and its derivative _ðtÞ by means of eq. (4.251). At this stage, the time lag can be assumed to be a small quantity, thus allowing for the following approximate relations, derived from the Taylor expansion of the correlation coefficient and the fact that 0 ð0Þ ¼ 0:

ðÞ ffi 1 þ

00 ð0Þ 2 þ Oð 3 Þ; 1 ðÞ ffi 00 ð0Þ 2 þ Oð 3 Þ; 2 pffiffiffiffiffiffiffiffiffi 00 ð0Þ 2 þ Oð 3 Þ ðÞ ffi 1 þ 4

ð4:276Þ

Furthermore, it can be seen that the argument of the Bessel function approaches infinity since 1 approaches zero. In this case, the following asymptotic values of the Bessel function can be used [29] ex Im1 ðxÞ pffiffiffiffiffiffiffiffi 2x

ð4:277Þ

Having this in mind and using eq. (4.251), one easily obtains

p_ ðx; x_ Þ 8 > > >
4 > > :

2 2 3 39 2 1 00 ð0Þ 2 1 1 > x x_ þ x þ x_ 7 2m 1 þ x þ x_ > 6 7= 2 4 2 2 7 6 7 7Im1 4 5> 00 ð0Þ 2 00 ð0Þ 2 5 > ; 2 2

1 x x_ 2

39 2 00 ð0Þ 2 1 2 2 > 2 x x_ 7> > 62m 1 þ 4 4 > 3 exp6 7> > 2 > 00 5 4 > 1 ð0Þ 2 2 > 2 2 > _ 2m x þ x = 6 7 4 2 7 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi v exp6 lim 4 5 u !0> 00 ð0Þ > > > u 2mpffiffi ffi x 12 x_ x þ 12 x_ > > > > t > > 2 2 > > 00 ð0Þ 2 > > > > > > 2 > > : ; 8 > > > > > > > > > > 0 implies that the expression under the square root is positive14. Finally, the LCR and AFD can be expressed as pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 00 ð0Þ 2m1 C2 C N1þ ¼ e ðmÞ

ð4:280Þ

T1þ ¼

ðm; C 2 Þ 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi eC 2m1 00 C ð0Þ

ð4:281Þ

N1þ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi C2 00 ð0ÞCe

ð4:282Þ

where pffiffiffiffi m C ¼ pffiffiffiffi H

ð4:283Þ

It can be noted that for m ¼ 1 these equations coincide with eq. (4.257) derived for the Rayleigh process. It has been noted already that the Nakagami m distribution is widely used to describe fading in communication channels. This also poses a problem to the investigation of LCR and AFD in systems with combining. The combining methods can be viewed as techniques allowing formation of a continuous communication channel which allows much smaller variance of the PDF, describing the received signal. This question was investigated as far back as in [31]. It was shown there that if all L branches of the combining scheme are

14

It is interesting to note that eq. (4.279) was probably first published in [32,33] the latter of which unfortunately is not widely available in English. Since then this equation has been rediscovered in a number of publications.

152


identical, the resulting process is also approximately a Nakagami m process with the following parameters mL ¼ mL;

L ¼ L2

ð4:284Þ

This fact can be used to obtain rough estimates of the performance of the combining scheme under consideration. More detailed and accurate results are described in [35].

4.6.5

Concluding Remarks

All the attention in the previous section has been devoted to calculation of only the first moments of the number of crossings, duration of fades and overshoots, etc. However, such description does not provide complete information about the mentioned quantities, which can be provided only by a marginal PDF. Rice [23] outlined an approach allowing calculation of the marginal PDF pF ðÞ of the fade duration. However, a general approach to this problem is still unknown. Even calculation of the variance D appears to be a difficult problem. Often this quantity is estimated from experiments or numerical simulations. However, for the case of rare fades15 ðC 1Þ, it can be assumed [20,22] that the outages obey the Poisson law and, thus pT ðÞ ¼

1 exp þ ; Tþ T

>0

ð4:285Þ

In the other limiting case C ! 0 of the Gaussian process, it can be shown that the fade duration obeys the Rayleigh law [22]. A number of authors suggest a similar representation for other distributions of the process ðtÞ, however, there are no convincing arguments to support such approximation. Good approximations for a sum of Gaussian noise and a harmonic signal are obtained in [20,23]. However, these expressions are quite complicated and difficult to use in analysis and synthesis. A situation when such equations can be directly used is rarely found.

4.7

NARROW BAND RANDOM PROCESSES

Most communication systems work at a very high frequency (the carrier frequency f0 ) and utilize bandwidth f (measured at 3 dB level) which is usually much smaller than the carrier16 f 1 f

15

ð4:286Þ

That is, the outage process is an ordinary process. However, recently great advances have been achieved in the area of so-called ultra-wide band systems, which utilize a relatively wide spectral band. For more detailed discussion see [36].

16


153

If a stationary wide band excitation such as WGN acts on the input of such systems, the asymptotically stationary output process has spectral density which is concentrated in a narrow band around the carrier frequency f0, according to eq. (4.144). It will be shown below that the covariance function of a narrow band random process ðtÞ can be written in the following form C ðÞ ¼ D ðÞ cos½!0 þ ðÞ

ð4:287Þ

Here !0 ¼ 2f0 and ðÞ and ðÞ are functions which vary slowly compared to cos !0 . In order to justify representation (4.288), one can introduce a new variable fD , which is sometimes referred to as the Doppler shift, by the rule fD ¼ f f0

ð4:288Þ

Then, using the one-sided PSD Sþ ðf Þ, one can write C ðÞ ¼

ð1

þ

S ðf Þ cosð2f Þdf ¼

0

ð1 f0

Sþ ðf0 þ fD Þ cos½2ðf0 þ fD ÞdfD

ð4:289Þ

Using the following notation D c ðÞ ¼ D s ðÞ ¼

ð1 f

ð 10 f0

Sþ ðf0 þ fD Þ cos½2fD dfD ð4:290Þ þ

S ðf0 þ fD Þ sin½2fD dfD

one can rewrite eq. (4.289) in the following form C ðÞ ¼ D ½ c ðÞ cosð2f0 Þ s ðÞ sinð2f0 Þ ¼ D ðÞ cos½!0 þ ðÞ

ð4:291Þ

where qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s ðÞ ðÞ ¼ 2c ðÞ þ 2s ðÞ; tan ðÞ ¼ c ðÞ

ð4:292Þ

Since c ðÞ is an even function and s ðÞ is an odd function, ðÞ is always an even function and ðÞ is always an odd function of . If the spectral density Sþ ð f Þ is a symmetric function around f0 , the lower limit of integration in eq. (4.290) can be substituted by 1, thus producing s ðÞ ¼ 0 and ðÞ ¼ 0. C ðÞ ¼ D ðÞ cos !0 ¼ cos !0

ð1 f0

Sþ ð f0 þ Þ cosð2fD Þd

ð4:293Þ

Thus, the covariance function of a random narrow band process with symmetric PSD can be represented by a modulated cosine function (4.293). Since the spectral density Sþ ðf0 þ Þ is

154


almost completely located at the low frequency band (compared to f0 ), functions c ðÞ, and s ðÞ, and as a result ðÞ and ðÞ, are slow functions of time-shift compared to cos !0 .

4.7.1

Definition of the Envelope and Phase of Narrow Band Processes

It could be also said that a realization of a random process looks like a modulated harmonic signal, that is why it is often referred to as a quasiharmonic signal or modulated signal. It is convenient to write a quasiharmonic process ðtÞ as ðtÞ ¼ AðtÞ cos½!0 t þ ðtÞ;

AðtÞ 0

ð4:294Þ

where AðtÞ and ðtÞ are random slowly varying functions with respect to cos !0 t. The random process AðtÞ is called the envelope and the process ðtÞ is called the random phase. The representation (4.294) is not unique, unless additional conditions are specified. This stems from the fact that for a very short observation interval, it is impossible to estimate the central frequency !0 of the random signal, thus causing uncertainties in the value of the phase and the magnitude. Moreover, the form of the equation will not be changed if the phase shift is 2n. In order to eliminate such uncertainties, one has to apply some meaningful restrictions on the phase ðtÞ and the magnitude. First of all, it will be assumed that the phase is confined to an interval ½0; 2 (or, equivalently ½; ). The restriction on the magnitude is more complex and requires definition of the analytical signal. In this case, the original process ðtÞ is augmented by a conjugate process ðtÞ defined by the so-called Hilbert transform [4] ð1

ðsÞ ds 1 t s ð 1 1 ðsÞ ðtÞ ¼ ds 1 t s

1 ðtÞ ¼

ð4:295Þ

The Hilbert transform of a signal ðtÞ can be considered as the reaction of a filter with the impulse response hðtÞ ¼ 1=t to the signal ðtÞ. In the frequency domain, this transform is described by the following transfer function 1 Hð j!Þ ¼

ð1

1 expðj!tÞdt ¼ jsgn! 1 t

ð4:296Þ

Since jHð j!Þj ¼ 1, this filter shifts the phase of all negative frequencies up by =2 and shifts the phase of all positive frequencies down by =2. Equivalently, any component of the form cos !t is transformed into sin !t. The Hilbert filter is often called the quadrature filter, since it transforms in-phase components of signals into quadrature components of the same signal. The Hilbert transform filter is also an all-pass system, since the magnitude spectrum is unchanged at the output of the system with respect to one at the input of the system. This immediately implies that the PSDs of both ðtÞ and ðtÞ are the same S ð!Þ ¼ jHð j!Þj2 S ð!Þ ¼ S ð!Þ

ð4:297Þ


155

so are the covariance functions: C ðÞ ¼ C ðÞ

ð4:298Þ

The mutual PSD and the covariance function can be directly calculated from eq. (4.143) S ð!Þ ¼ Hðj!ÞS ð!Þ ¼ jS ð!Þsgn! ð ð 1 1 1 1 C ðÞ ¼ C ðÞ ¼ jS ð!Þsgn ! exp½ j!d! ¼ S ð!Þ sin ! d! 2 1 0

ð4:299Þ ð4:300Þ

The latter shows that the processes ðÞ and ðÞ are uncorrelated if considered at the same moment of time, since EfðtÞðtÞg ¼ C ð0Þ ¼ 0

ð4:301Þ

For a Gaussian process, this also means independence, however, for non-Gaussian processes it is not so. The in-phase component ðtÞ and the quadrature component ðtÞ obtained through the Hilbert transform can be combined in a single complex signal called a pre-envelope [4] ðtÞ ¼ ðtÞ þ jðtÞ ¼ AðtÞ exp½jðtÞ

ð4:302Þ

Thus, the envelope and the full phase can be defined as AðtÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðtÞ ¼ !0 t þ ðtÞ 2 ðtÞ þ 2 ðtÞ; ðtÞ ¼ atan ðtÞ

ð4:303Þ

It can be seen that AðtÞ jðtÞj 0 and AðtÞ ¼ jðtÞj if ðtÞ ¼ 0 and, thus, the process AðtÞ envelopes the process ðtÞ. Equation (4.248) also allows definition of a low frequency equivalent #ðtÞ of the complex pre-envelope as ðtÞ ¼ AðtÞ exp½ jð!0 t þ ðtÞÞ ¼ AðtÞ exp½ jðtÞ expðj!0 tÞ ¼ #ðtÞ expðj!0 tÞ

ð4:304Þ

The frequency content of the process #ðtÞ is concentrated around zero frequency if the original process ðtÞ is narrow band and is concentrated around the frequency f0. Furthermore, the low frequency equivalent can be obtained through a demodulation process as #ðtÞ ¼ ðtÞ expðj!0 tÞ ¼ ½ðtÞ þ jðtÞ½cos !0 t þ j sin !0 t ¼ Ac ðtÞ þ jAs ðtÞ

ð4:305Þ

and, thus Ac ðtÞ ¼ AðtÞ cos ðtÞ ¼ ðtÞ cos !0 t þ ðtÞ sin !0 t As ðtÞ ¼ AðtÞ sin ðtÞ ¼ ðtÞ sin !0 t þ ðtÞ cos !0 t

ð4:306Þ

In order to find characteristics of the envelope and the phase of a narrow band process, it is necessary to find a joint distribution of the in-phase and quadrature components p ðx1 ; x2 Þ.

156


Such calculations are relatively simple to perform for a Gaussian process, as shown in the example below. An alternative definition of the conjugate process and the envelope can be obtained following [37]. In this case sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 d 1 _2 ðtÞ ðtÞ

ð4:307Þ ðtÞ ¼ _ðtÞ; AðtÞ ¼ 2 ðtÞ þ 2 !0 dt !0 !0 It can be seen that, for a deterministic harmonic signal xðtÞ ¼ A cos !0 t, eq. (4.307) produces an exact result yðtÞ ¼ A sin !0 t. However, for a finite bandwidth of the signal ðtÞ this definition is an approximate one. Both definitions will be used in the considerations below. 4.7.2

The Envelope and the Phase Characteristics

One of the most important questions in the theory of narrow band processes is the relation between the characteristics of the random process itself and the characteristics of its envelope, especially relations between the marginal densities and the covariance functions. 4.7.2.1

Blanc-Lapierre Transformation

Let a narrow band process be represented in the following form ðtÞ ¼ R ðtÞ ¼ AðtÞ cos½!0 t þ ðtÞ ¼ AðtÞ cos ðtÞ;

ð4:308Þ

where the envelope AðtÞ and the phase ðtÞ are slowly varying random functions with respect to cosð!0 tÞ and are defined by eqs. (4.303). Jointly, the envelope AðtÞ and the total phase ðtÞ can be described by the joint PDF pA; ða; Þ. The marginal PDF pA ðaÞ of the envelope and the marginal PDF p ð Þ of the phase can be obtained by integration of the joint PDF over the auxiliary variable. The process R ðtÞ is described by its PDF pR ðxÞ. The goal of this section is to derive the relationship between pA ðaÞ and pR ðxÞ assuming that the process R ðtÞ is a stationary process. As a by-product of this analysis, it will be shown that the phase distribution must be uniform to obtain a stationary process: p ð Þ ¼

1 2

ð4:309Þ

Following the technique discussed in section 4.7.1, the conjugate I ðtÞ and the preenvelope ðtÞ processes can be defined as I ðtÞ ¼ AðtÞ sin½!0 t þ ðtÞ ¼ AðtÞ sin ðtÞ

ð4:310Þ

ðtÞ ¼ R ðtÞ þ jI ðtÞ

ð4:311Þ

The complex process ðtÞ can be described by means of the joint PDF of R ðtÞ and I ðtÞ, as discussed in section 2.1 @2 ProbfR ðtÞ x1 ; I ðtÞ x2 g p ðx1 ; x2 Þ ¼ @x1 @x2

ð4:312Þ

157


Conversantly, both pR ðxÞ and pI ðxÞ can be obtained from p ðx1 ; x2 Þ by integration, that is ð1 pR ðxÞ ¼ p ðx; x2 Þdx2 ð4:313Þ 1 ð1 p ðx1 ; xÞdx1 ð4:314Þ pI ðxÞ ¼ 1

An alternative description of ðtÞ can be achieved through its characteristic function ðu; Þ ¼

ð1 ð1 1

1

p ðx1 ; xÞ exp½ jðux1 þ x2 Þdx1 dx2

ð4:315Þ

As a particular case of eq. (4.315), one can see that ðu; 0Þ ¼ ¼

ð1 ð1 ð1 1 1

p ðx1 ; x2 Þ expðjux1 Þdx1 dx2 ð1 ð1 expðjux1 Þdx1 p ðx1 ; x2 Þdx2 ¼ expðjux1 ÞpR ðx1 Þdx1 ¼ R ðuÞ 1

1

1

ð4:316Þ In the very same way, the characteristic function of the quadrature is given by ð0; Þ ¼ I ðÞ;

ð4:317Þ

These last two equations allow one to determine the PDFs of R ðtÞ and I ðtÞ through ðu; Þ. In fact 1 pR ðx1 Þ ¼ 2

ð1

1 R ðuÞ exp½jux1 du ¼ 2 1

ð1 1

ðu; 0Þ exp½jux1 du

ð4:318Þ

ð0; Þ exp½jx2 d

ð4:319Þ

and 1 pI ðx2 Þ ¼ 2

ð1

1 I ðÞ exp½jx2 d ¼ 2 1

ð1 1

As the next step, one can express the joint characteristic function ðu; Þ, given by eq. (4.315), in terms of the joint PDF pA; ða; Þ of the magnitude and phase. In order to do so, the following change of variables can be used in eq. (4.315) x1 ¼ a cos ð4:320Þ x2 ¼ a sin The Jacobian J of this transformation is just cos J ¼ a sin

sin ¼a a cos

ð4:321Þ

158


and, thus, the relationship between PA; ða; Þ and p ðx1 ; x2 Þ has the following form (see eq. (2.163) in section 2.6) pA; ða; Þ ¼ p ða cos ; a sin Þa

ð4:322Þ

pA; ða; Þ a

ð4:323Þ

or p ða cos ; a sin Þ ¼

After changing the variables in eq. (4.315), using eqs. (4.320) and (4.323), one obtains ð1 ð1 R ðuÞ ¼ ðu; 0Þ ¼ p ðx1 ; x2 Þ exp½ jux1 dx1 dx2 1 1 ð 1 ð 2 ð4:324Þ ¼ p ða cos ; a sin Þ exp½ju cos aa da d 0 0 ð 1 ð 2 PA; ða; Þ exp½ jua cos da d ¼ 0

0

The exponential term in the last equation can be expressed as a sum of Bessel functions of the first kind (eqs. (9.1.44–9.1.45) on page 361 in [29]) exp½ jua cos ¼

1 X

jn Jn ðuaÞ exp½ jn

ð4:325Þ

n¼1

so that eq. (4.324) becomes R ðuÞ ¼

ð 1 ð ð2Þ X 1 0

¼ ¼

0

1 X n¼1 1 X n¼1

jn Jn ðuaÞ exp½ jn pA; ða; Þda d

n¼1

ð1 j

n 0

ð1 j

n 0

Jn ðuaÞ

ð 2

exp½jnð!0 t þ ’ÞPA; ða; ’Þd’ da

0

Jn ðuaÞpA ðaÞda

ð 2 0

exp½ jnð!0 t þ ’ÞpjA ð’ j aÞd’

ð4:326Þ

Here, pjA ð’ j aÞ, stands for PDF of the phase, ðtÞ, given a value of a for the magnitude pjA ð’ j aÞ ¼

pA; ða; ’Þ pA ðaÞ

ð4:327Þ

One can obtain R ðuÞ which is independent of time (i.e. R ðtÞ will be a stationary process) if and only if17 pjA ð’ j aÞ ¼

1 2

ð4:328Þ

Recall that the expression for the total phase ðtÞ contains the term !0 t. This time dependence is gone if eq. (4.328) holds.

17

159


This means that the distribution of the phase does not depend on the value of a, i.e. the magnitude and the phase are statistically independent18. Also, taking into account that ð 2

exp½ jnð!0 t þ ’Þd’ ¼ 0

ð4:329Þ

0

for n > 0, eq. (4.326) can be rewritten as R ðuÞ ¼

ð1

J0 ðuaÞpA ðaÞda

ð4:330Þ

0

or, taking the inverse Fourier transform 1 pR ðx1 Þ ¼ 2

ð1

1 R ðuÞ exp½jx1 udu ¼ 2 1

ð1 ð1 1

J0 ðuaÞpA ðaÞ exp½jx1 uda du:

0

ð4:331Þ In order to express pA ðaÞ in terms of pR ðx1 Þ, it could be observed that R ðuÞ is just the Hankel transform19 (see [38]) of pA ðaÞ=a and thus can be easily inverted to produce pA ðaÞ ¼ a

ð1

J0 ðuaÞR ðuÞu du ¼

0

ð1 0

uJ0 ðuaÞ

ð1 1

pR ðx1 Þ exp½ jx1 udx1 du

ð4:332Þ

The transform pair given by eqs. (4.331) and (4.332) is known as the Blanc-Lapierre transform [19]. An alternative expression can be obtained by rewriting eq. (4.331) as ð ð 1 1 1 pR ðx1 Þ ¼ J0 ðuaÞpA ðaÞ exp½jx1 uda du 2 1 0 ð ð1 1 1 pA ðaÞda J0 ðuaÞ exp½jx1 udu ¼ 2 0 1 Since the inner integral is equal to (equation in [39]) 8 2 ð1 < pffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi if 2 x2 J0 ðuaÞ exp½jx1 udu ¼ a 1 : 1 0 if one can write 1 pR ðx1 Þ ¼

18

ð1

pa ðaÞ 1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi da ¼ 2 2 a x1 jx1 j

ð1

a jx1 j

ð4:334Þ

a < jx1 j

pA ðjx1 j cosh aÞda

0

This condition can be relaxed if only wide sense stationarity is required. Ð1 Ð1 A pair of Hankel transforms is defined by FðuÞ ¼ 0 f ðxÞJ0 ðxuÞx dx; f ðxÞ ¼ 0 FðuÞJ0 ðxuÞu du [38].

19

ð4:333Þ

ð4:335Þ

160

4.7.2.2


Kluyver Equation

Very often a narrow band random process ðtÞ on a physical level is obtained as a sum of a number of narrow band processes with known characteristics. For example, multipath propagation of radio waves results in a weighted sum of randomly amplified and delayed replicas of the transmitted signal. A similar situation is encountered with sea radar when the reflected wave is a combination of reflections from numerous irregularities of the sea surface. A great number of similar examples can be found elsewhere. In such problems, it is beneficial to find a relationship between the PDF of each summand and the PDF of the net narrow band process. This problem is treated in this section. Let a complex random variable be a sum of N independent complex random variables n , n ¼ 1; 2; . . . ; N: ¼ þ j ¼

N X

n ¼

n¼1

N X

n þ j

n¼1

N X

ð4:336Þ

Yn

n¼1

where n ¼ Reðn Þ, n ¼ Imðn Þ. It is assumed that the joint PDF pn ;n ðxn ; yn Þ is known for each individual n . This can be recast in terms of a joint PDF of magnitude An and phase n as in eq. (4.323) n ¼ An expð jn Þ;

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi An ¼ n2 þ n2 ;

pAn ;n ðan ; ’n Þ ¼ an pn ;n ðan cos ’n ; an sin ’n Þ ¼

n ¼ atan

n n

1 pA ðan Þ 2 n

ð4:337Þ ð4:338Þ

The process ðtÞ itself can also be described by the joint PDF of its real and imaginary parts p ðx; yÞ or, equivalently, by its characteristic function ðu; Þ ¼

ð1 ð1 1

1

p ðx; yÞ exp½jðux þ yÞdx dy

ð4:339Þ

Since is a sum of N independent random variables its characteristic function is a product of the characteristic functions of each single term [18], i.e. ðuÞ ¼

N Y

n ðuÞ ¼

N ð1 Y

N Y J0 ðuan ÞpAn ðan Þdan ¼ hJ0 ðuan Þi

n¼1 0

n¼1

ð4:340Þ

n¼1

A similar equation holds for ðuÞ. In turn, according to the Blanc-Lapierre transformation (eq 4.332), pA ðaÞ can be expressed in terms of ðuÞ as pA ðaÞ ¼ a

ð1

J0 ðuaÞX ðuÞu du ¼ a

0

¼a

ð1 0

" J0 ðuaÞ

N ð1 Y n¼1 0

ð1 0

J0 ðuaÞ

N Y hJ0 ðuan Þiu du n¼1

#

J0 ðuan ÞpAn ðan Þdan u du

ð4:341Þ


161

This equation relates the distribution of the magnitude of the sum through distributions of the magnitude of each individual component. In practice, it is the value of magnitude squared, I ¼ a2 , which is available for measurement. In this case, the PDF of the intensity I can be easily found to be pffiffi Ð 1 pffiffi Q Ð 1 pffiffi I 0 J0 ðu I Þ Nn¼1 0 J0 ðuan ÞpAn ðan Þdan u du pA ð I Þ pffiffi pi ðIÞ ¼ pffiffi ¼ 2 I 2 I ð N pffiffi Y 1 1 ¼ J0 ðu I Þ hJ0 ðuan Þiu du ð4:342Þ 2 0 n¼1 The last equation is known as the Kluyver equation.

4.7.2.3

Relations Between Moments of pAn ðan Þ and pi ðIÞ

The Kluyver equation (4.342) can rarely be used to produce analytical results. However, a useful relation between the moments of the sum and those of its components can be analytically obtained. In order to do so, one can make use of the moment generating function ðÞ as discussed in Section 2.1.1. N N @ ðÞ ð4:343Þ mkI ¼ ð1Þ @N ¼0 A Taylor-type expansion for ðÞ can be obtained directly from the Kluyver equation (4.342) ðÞ ¼

ð1 e

I

pi ðIÞdI ¼

0

ð1 e 0

I

1 2

ð1

N pffiffi Y J0 ðu I Þ hJ0 ðuan Þiu du dI

0

n¼1

ð1 Y ð N pffiffi 1 1 I hJ0 ðuan Þiu du e J0 ðu I ÞdI ¼ 2 0 0 n¼1

ð4:344Þ

The first step is to calculate the inner integral in eq. (4.344). This can be achieved by using eq. (6.614-1) on page 709 in [39] rffiffiffiffiffi 2 2

ð1 pffiffiffi 2 x þ1 exp e J ð xÞdx ¼ I1 I ð4:345Þ 3 2 2 4 8 8 8 0 and setting ¼ , ¼ u, ¼ 0, x ¼ I, which implies that 2 u2 1 1 þ 1 1 ¼ ; ¼ ¼ ; 2 2 2 8 8 2

ð4:346Þ

and thus ð1 e 0

I

pffiffi u J0 ðu I ÞdI ¼ 4

rffiffiffiffiffi 2 2

u2 u u 1 1 exp I I 2 2 3 8 y 8

ð4:347Þ

162


This last equation can be further simplified by observing that (using 10.2.13–14 on page 443 in [29]) rffiffiffiffiffi rffiffiffiffiffi 2z sinh z 2 I1=2 ðzÞ ¼ ¼ sinh z ð4:348Þ z z rffiffiffiffiffi rffiffiffiffiffi 2z cosh z 2 I1=2 ðzÞ ¼ ¼ cosh z ð4:349Þ z z and thus 2 rffiffiffiffiffiffiffiffi u2 u 16 u2 exp I12 ¼ u2 8 8 8

I12

ð4:350Þ

Taking eq. (4.350) into account, eq. (4.347) becomes ð1 e 0

I

pffiffi u J0 ðu I ÞdI ¼ 4

rffiffiffiffiffi rffiffiffiffiffiffiffiffi u2 16 u2 1 u2 exp exp ¼ exp ð4:351Þ 3 u2 8 8 4

Finally, the expression for ðÞ becomes 1 ðÞ ¼ 2

ð1 Y N 1 u2 hJ0 ðuan Þiu du exp 4 0 n¼1

ð4:352Þ

As the next step, let us use the following substitution of variables in eq. (4.352) x ¼ u2 =4; u ¼

pffiffiffiffiffiffiffiffi 4x;

du ¼

pffiffiffi dx x

ð4:353Þ

so that eq. (4.352) becomes 1 ðÞ ¼ 2

ð1 Y N

pffiffiffiffiffiffiffiffi hJ0 ð 4xan Þi expðxÞdx

ð4:354Þ

0 n¼1

The last integral cannot be calculated analytically in closed form, therefore one can use pffiffiffiffiffiffiffi ffi the Taylor expansion of J0 ð 4xan Þ with respect to variable . Using eq 9.1.10 in [29] J ðzÞ ¼ with ¼ 0 and z ¼

1 z X

2

ðz2 =4Þk k!ð þ k þ 1Þ k¼0

ð4:355Þ

pffiffiffiffiffiffiffiffi 4xan one can obtain 1 n X pffiffiffiffiffiffiffiffi ð1Þkn kn xkn a2k 2 J0 ð 4xan Þ ¼ ; kn !kn ! k ¼0 n

ð4:356Þ

163


and, thus 1 X n pffiffiffiffiffiffiffiffi ð1Þkn kn xkn ha2k n i hJ0 ð 4xan Þi ¼ kn !kn ! k ¼0

ð4:357Þ

n

Furthermore N N X 1 Y Y n pffiffiffiffiffiffiffiffi ð1Þkn kn xkn ha2k n i hJ0 ð 4xan Þi ¼ kn !kn ! n¼1 n¼1 k ¼0 n

¼ ¼

1 X 1 X

1 2kN1 2k1 N X ð1Þm m xm ha2k N1 ihaN1 i ha1 i kN !kN !kN1 !KN k1 !k1 ! k ¼0

kN ¼0 kN1 ¼0 1 X m

1

ð1Þ cm m xm

ð4:358Þ

m¼0

where m¼

N X

kn

ð4:359Þ

1 2kN1 2k1 N X ha2k iha i ha N1 N1 1 k !k !k !k k1 !k1 !PN ¼0 k ¼0 N N N1 N1

ð4:360Þ

n¼1

and cm ¼

1 X 1 X kN ¼0 kN1

k ¼m n¼1 n

1

Finally, integrating eq. (4.354) with respect to x and taking advantage of eq. (4.359) one obtains ðÞ ¼

ð1 X 1 0

! m

ð1Þ cm x

m m

1 X expðxÞdx ¼ ð1Þm cm m m!

m¼0

ð4:361Þ

m¼0

Comparing this to eq. (4.358), the following expression can be derived mkI ¼ k!k!ck ¼ k!k!

kn ¼0 kN1

4.7.2.4

1 2kN 2kN1 2k1 X haN1 ihaN1 i ha1 i k !k !k !k ! k1 !k1 !PN ¼0 k ¼0 N N N1 N1

1 X 1 X

1

n¼1

ð4:362Þ kn ¼k

The Gram–Charlier Series for pR ðxÞ and pi ðIÞ

As has been mentioned above, it is the distribution of the intensity pi ðIÞ which is usually available for measurement. At the same time, it is important for numerical simulations to

164


know and reproduce the statistic of the field amplitude pR ðxÞ. The exact analytical form, given by eqs. (4.330)–(4.331) is rarely achievable (however, a number of results can be found in [19]). Important relations like (4.362) give insight into how the distribution behaves but do not really allow its numerical simulation. In this section, useful results allowing the reconstruction of the PDF pR ðxÞ from the PDF of the intensity pi ðIÞ are derived. As the first step, the Blanc-Lapierre transformation (4.331) can be rewritten in terms of pi ðIÞ, using the fact that pffiffi pffiffi pffiffi pA ð I Þ ð4:363Þ pi ðIÞ ¼ pffiffi ) pA ð I Þ ¼ 2 I pi ðIÞ 2 I and thus ð ð 1 1 1 pR ðxÞ ¼ J0 ðuaÞpa ðaÞda exp½jxudu 2 1 0 ð ð pffiffi pffiffi pffiffi 1 1 1 ¼ J0 ðu I ÞpA ð I Þ exp½jxudu d I 2 1 0 ð ð pffiffi pffiffi 1 1 1 dI J0 ðu I Þ2 I pi ðIÞ pffiffi exp½jxudu ¼ 2 1 0 2 I ð1 ð1 p ffiffi 1 ¼ exp½jxudu J0 ðu I Þpi ðIÞdI 2 1 0

ð4:364Þ

Let now the PDF pi ðIÞ be represented by its expansion through the Laguerre polynomials Lk ðIÞ as discussed in section 2.2, i.e. pi ðIÞ ¼ exp½I

1 X

k Lk ðIÞ

ð4:365Þ

k¼0

where k ¼

ð1

pi ðIÞLk ðIÞdI

ð4:366Þ

0

In this case, eq. (4.364) becomes 1 pR ðxÞ ¼ 2 ¼ ¼

ð1

ð1 exp½jxudu

1 1 X

k 2 k¼0 1 X k 2 k¼0

1 X pffiffi J0 ðu I Þ exp½I k Lk ðIÞdI

0

ð1

ð1

exp½jxudu 1

ð1

1

k¼0

pffiffi J0 ðu I ÞLk ðIÞ exp½IdI

0

Fk ðuÞ exp½jxudu

ð4:367Þ

where Fk ðuÞ ¼

ð1 0

pffiffi J0 ðu I ÞLk ðIÞ exp½IdI

ð4:368Þ

165


In order to simplify eq. (4.368), one can use the following integral (2.19.12-7 on page 474 in [40]) ð1 pffiffiffi xffi Þ =2 px J ðbpffiffi x e ð4:369Þ Ln ðcxÞdx ¼ Jðp; cÞ ðb x Þ I 0 where 2 b ðp cÞn b b2 c Jðp; cÞ ¼ exp ; L 2 pþnþ1 4pc 4p2 4p n 2 ð1Þn b þ2n b exp Jðc; cÞ ¼ þnþ1 2 n!c 4c

p 6¼ c

ð4:370Þ ð4:371Þ

Using the last equation with ¼ 0, p ¼ c ¼ , n ¼ k and b ¼ u, one can obtain 1 u2k u2 exp Fk ðuÞ ¼ k!kþ1 2 4

ð4:372Þ

Rewriting eq. (4.367) by taking into account eq. (4.372), one can obtain with u ¼ I 1 X k pR ðxÞ ¼ 2 k¼0

ð1 1

Fk ðuÞ exp½jxudu

ð1 1 X 1 u2k u2 k exp ¼ exp½jxudu kþ1 2 2 k¼0 4 1 k! pffiffiffi X pffiffiffi X ð p ffiffiffi 1 k 1 2k 1 k 2 Gk ðxÞ t expðt Þ exp½jð2 xÞtdt ¼ ¼ k¼0 k! 1 k¼0 k!

ð4:373Þ

where the following notation is used Gk ðxÞ ¼

ð1

ð1 pffiffiffi pffiffiffi 2 t expðt Þ exp½jð2 xÞtdt ¼ exp½x t2k exp½ðt þ j xÞ2 dt 2k

1

2

1

ð4:374Þ Comparing the standard integral 3.462-4 on page 338 in [39] ð1 1

pffiffiffi xn exp½ðx Þ2 dx ¼ ð2jÞn Hn ðjÞ

ð4:375Þ

one can obtain Gk ðxÞ ¼ exp½x

ð1

2

1

t2k exp½ðt þ j

pffiffiffi 2 pffiffiffi pffiffiffi xÞ dt ¼ 22k ð1Þk exp½x2 H2k ð xÞ ð4:376Þ

166


and thus pffiffiffi X pffiffiffi 1 X pffiffiffi 1 k ð1Þk k pR ðxÞ ¼ Gk ðxÞ ¼ pffiffiffi exp½x2 H2k ð xÞ 2k k¼0 k! 2 k! k¼0

ð4:377Þ

Here Hn ðxÞ is the Hermitian polynomial of order n as discussed in section 2.2. The last expression should not come as a surprise. If pi ðIÞ is an exponential distribution (2 of two degrees of freedom) then 0 ¼ 1 and k ¼ 0 for k > 0. This reduces eq. (4.377) to the Gaussian PDF as it must. At the same time, eq. (4.377) is nothing but the Gram–Charlier expansion of the PDF pR ðx1 Þ. The odd order terms are missing due to the fact that pR ðx1 Þ is a symmetrical PDF. Treatment of non-stationary processes becomes very complicated since independence between the phase and the magnitude does not hold. It is possible to obtain a number of useful results for the Gaussian process using direct calculations of the related statistics. This approach is described in some detail in section 4.7.3 below. A number of examples for the non-Gaussian case is considered in section 4.7.4.

4.7.3 4.7.3.1

Gaussian Narrow Band Process First Order Statistics

Let I ðtÞ and Q ðtÞ be two independent Gaussian processes with equal variance 2 and different mean values mI and mQ respectively. These can be considered as components of a complex random process ðtÞ ¼ I ðtÞ þ jQ ðtÞ ¼ AðtÞ exp½ jðtÞ

ð4:378Þ

The goal in this section is to derive first order statistics of the envelope AðtÞ and phase ðtÞ of this complex Gaussian process. Since in-phase and quadrature components are independent, their joint PDF is a product of the marginal PDF of each component, i.e. " # 1 ðx mI Þ2 þ ðy mQ Þ2 exp pI Q ðx; yÞ ¼ pI ðxÞpQ ðyÞ ¼ ð4:379Þ 22 22 Making a transformation of variables according to x ¼ A cos

ð4:380Þ

y ¼ A sin the Jacobian of this transformation is @x @y @A @A cos ¼ J¼ @x @y A sin @ @

sin ¼A A cos

ð4:381Þ


167

and, thus, the joint PDF of the envelope and phase is pA; ðA; Þ ¼ pI Q ðA cos ; A sin ÞA " # A ðA cos mI Þ2 þ ðA sin mQ Þ2 ¼ exp 22 22 " # A2 þ m2I þ m2Q 2AðmI cos þ mQ sin Þ A exp 22 22

ð4:382Þ

Further simplification can be achieved by introducing two new parameters m¼

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2I þ m2Q ;

¼ a tan

mQ mI

ð4:383Þ

and thus mi ¼ m cos ;

mQ ¼ m sin

ð4:384Þ

As a result, eq. (4.382) can be rewritten as

A A2 þ m2 2Amðcos cos þ sin sin Þ pA; ðA; Þ ¼ exp 22 22

A A2 þ m2 2Am cosð Þ ¼ exp 22 22

ð4:385Þ

This in hand, one can calculate the PDF of the amplitude by integrating expression (4.385) over :

A A2 þ m2 2Am cosð Þ pA; ðA; Þd ¼ exp d pA ðAÞ ¼ 2 22 2

ð

A A2 þ m2 Am cosð Þ A A2 þ m 2 mA exp exp exp d ¼ I ¼ 0 22 2 2 2 22 22 ð

ð

ð4:386Þ Here the following standard integral has been used

ð Am cosð Þ Am cosðÞ Am exp exp d ¼ d ¼ 2I0 2 2 2

ð

ð4:387Þ

Thus, the distribution of the envelope is Rician. In the very same manner, the distribution of the phase can be found to be p ðÞ ¼

ð

pA; ðA; ÞdA ¼

ð1 0

A A2 þ m2 2Am cosð Þ exp dA 22 22

ð4:388Þ

168


Using substitutions z¼

A m cosð Þ ;

dA ¼ dz;

A ¼ z þ m cosð Þ

ð4:389Þ

eq. (4.388) becomes " # z þ m cosð Þ ðzÞ2 þ m2 þ m2 cos2 ð Þ exp p ðÞ ¼ dz 22 22 m cosð Þ

1 m2 m cosð Þ m cosð Þ m2 sin2 ð Þ pffiffiffiffiffiffi exp 2 þ ¼ exp 2 22 2 2 ð4:390Þ ð1

where ðsÞ is the CDF of the standard Gaussian PDF 2

ðs 1 x pffiffiffiffiffiffi exp ðsÞ ¼ dx 2 1 2

ð4:391Þ

For a particular case of m ¼ mI ¼ mQ ¼ 0

A A2 pA ðAÞ ¼ 2 exp 2 2 1 p ðÞ ¼ 2

4.7.3.2

ð4:392Þ ð4:393Þ

Correlation Function of the In-phase and Quadrature Components

Second order statistics of the in-phase and quadrature components can be easily found using the fact that SI ð!Þ ¼ jHI ð!Þj2

N0 ; 2

2 N0 SQ ð!Þ ¼ HQ ð!Þ 2

ð4:394Þ

where SI ð!Þ and SQ ð!Þ are power spectral densities (PSD) of the in-phase and quadrature components. According to the Wiener–Khinchin theorem, we have then 1 RI ðÞ ¼ 2

ð1 1

SI ð!Þ exp½ j!d!

ð4:395Þ

Alternatively, the correlation function can be obtained through factorization, as described below. In general, the transfer function of a filter can be represented as a ratio of two polynomials HðsÞ ¼

BðsÞ bn sn þ bn1 sn1 þ þ b0 ¼ AðsÞ an sn þ an1 sn1 þ þ a0

ð4:396Þ

169


One can expand HðsÞ using the method of partial fraction expansion: HðsÞ ¼

a1 a2 an þ þ þ þ KðsÞ s p1 s p2 s pn

ð4:397Þ

where ak are the residues, pk are the poles and KðsÞ is the remaining direct term. Assuming that KðsÞ ¼ 0, one can see that the time-domain response of the filter to an impulse is given by: ð1 ð1 n X a1 an Hð!Þ exp½j!td! ¼ þ þ ak exp½pk t hðtÞ ¼ exp½j!td! ¼ s pn 1 1 s p1 k¼1 ð4:398Þ Now that the expression for hðtÞ is factorized, one can determine the autocorrelation function: ) ð1 ð 1 (X n X n RðÞ ¼ hðtÞhðt þ Þdt ¼ ak al exp½pk exp½ðpk þ pl Þt dt n X n X k¼1 l¼1

ak al exp½pk

ð1

0

0

exp½ðpk þ pl Þtdt ¼

o

¼

k¼1 l¼1 n n XX

ak al

k¼1 l¼1 n X

exp½pk pk þ pl

ak exp½pk t

k¼1

n X l¼1

al pk þ pl ð4:399Þ

Finally, the expression for RðÞ has the following form RðÞ ¼

n X

ck exp½pk

ð4:400Þ

k¼1

where coefficients ck are given by: ck ¼ ak

4.7.3.3

n X

al p þ pl l¼1 k

ð4:401Þ

Second Order Statistics of the Envelope

In order to find second order statistics of the envelope and the phase, one ought to derive the fourth order joint probability density function of I ðtÞ, I ðt þ Þ, Q ðtÞ and Q ðt þ Þ. Since I ðtÞ and Q ðt þ Þ are independent in any moment of time, one obtains pI Q I Q ðx; y; x ; y Þ ¼ pl ðx; x ÞpQ Q ðy; y Þ 2

1 x 2 ðÞxx þ x2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ¼ 22 ð1 2 ðÞÞ 22 1 2 ðÞ 2

1 y 2 ðÞyy þ y2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 22 ð1 2 ðÞÞ 22 1 2 ðÞ

ð4:402Þ

170


or 2

1 x þ y2 2 ðÞðxx þ yy Þ þ x2 þ y2 exp pI Q I Q ðx; y; x ; y Þ ¼ 2 4 4 ð1 2 ðÞÞ 22 ð1 2 ðÞÞ ð4:403Þ Here, ðÞ is the normalized correlation function, such that ð0Þ ¼ 1. Changing variables according to x ¼ A cos ; x ¼ A cos ;

y ¼ A sin y ¼ A sin

ð4:404Þ

and noticing that the Jacobian of the transformation (4.404) is cos A sin J¼ 0 0

sin

0

A cos 0

0 cos

0

A sin

0 ¼ AA sin A cos 0

ð4:405Þ

the following joint PDF of magnitude and phase in two separate moments of time can be obtained pAA ðA; A ; ; Þ

AA 1 2 2 exp 2 ½A þ A 2 ðÞAA cosð Þ ¼ 2 4 2 ð1 2 ðÞÞ 4 ð1 2 ðÞÞ ð4:406Þ

A joint probability density function of amplitude in two separate moments of time can be calculated by integrating (4.406) over phase variables, i.e. pAA ðA; A Þ ¼

ð ð

pAA ; ðA; A ; ; Þd d

AA A2 þ A2 exp 2 ¼ 2 4 4 ð1 2 ðÞÞ 2 ð1 2 ðÞÞ

ð ð ðÞAA cosð Þ d d exp 2 ð1 2 ðÞÞ

AA A2 þ A2 ðÞ AA exp 2 I0 1 2 ðÞ 2 4 ð1 2 ðÞÞ 2 ð1 2 ðÞÞ

ð4:407Þ

Here, once again, the integral (4.387) has been used. The last expression allows one to calculate the correlation function of the envelope. The first step is to calculate an average value of the magnitude. Since it is already known that the marginal PDF is given by the


171

Rayleigh distribution (4.392), the mean and variance can be easily found to be rffiffiffi

A2 A2 mA ¼ ApA ðAÞdA ¼ exp dA ¼ 2 2 22 0 0 rffiffiffi 2

ð1 ð1 2 A A 2 2 2 ðA mA Þ pA ðAÞdA ¼ A exp 2 dA ¼ 2 A ¼ 2 2 2 2 0 0 ð1

ð1

ð4:408Þ ð4:409Þ

Further simplification can be achieved by normalizing the magnitude variables by variance z1 ¼

A ;

z2 ¼

A ;

J ¼ 2

ð4:410Þ

and, thus

z1 z2 z21 þ z22 ðÞ exp z1 z2 pAA ðz1 ; z2 Þ ¼ I0 1 2 ðÞ 1 2 ðÞ 2ð1 2 ðÞÞ

ð4:411Þ

Using a Bessel function expansion in terms of the Laguerre polynomial I0 ðaz1 z2 Þ ¼

1 X n¼0

Ln ðz21 =2ÞLn ðz22 =2Þan

ð4:412Þ

the expression (4.411) can be transformed into

1 2 2 2n z21 þ z22 X z z Ln 1 Ln 1 pAA ðz1 ; z2 Þ ¼ z1 z2 exp 2 2 2 n!n! n¼0

ð4:413Þ

The expansion (4.413) can be now used to calculate the second moment of the envelope hAA i ¼ hz1 z2 i ¼ 2

ð1 ð1 2 1 1

z1 z2 pAA ðz1 ; z2 Þdz1 dz2

2

2 2 ð ð 1 X 2n 1 1 z1 þ z22 z z ¼ z1 z2 exp Ln 1 Ln 1 dz1 dz2 n!n! 1 1 2 2 2 n¼0 2

ð4:414Þ

ð 2 2 2 1 X 2n 1 z z ¼ z1 exp 1 Ln 1 dz1 n!n! 1 2 2 n¼0 2

Finally

CAA ðÞ ¼ hAA i

m2A

1 1 2 X 1 2 2n X 1 2 2n 2 ¼ ðÞ ¼ A ðÞ 4 n ¼ 1 2n!! 2 n ¼ 1 2n!!

ð4:415Þ

172


If only two terms are taken into account then CAA ðÞ ¼ 2A ð0:921 2 ðÞ þ 0:058 4 ðÞ þ Þ 2A 2 ðÞ

ð4:416Þ

Introducing a new variable ¼ and integrating with reference to eq. (4.406) over A, A and , we can obtain the expression for the PDF of the phase difference in two moments of time separated by " # þ a sin x 1 2 ðÞ 1 ; þx2 p ðÞ ¼ 2 1 x2 ð1 x2 Þ3=2

4.7.3.4

x ¼ ðÞ cos

ð4:417Þ

Level Crossing Rate

The level crossing rate (LCR) of the random process is defined as the number of crossings of a fixed threshold R per unit time in the positive direction (i.e. from a level below R to a level above R). In order to find the LCR, one needs to calculate the joint probability density of the random process and its derivative p_ ðx; x_ Þ. With this in hand, the LCR can be calculated as LCRðRÞ ¼

ð1 0

xp_ ðx; x_ Þd_x

ð4:418Þ

For a random process with Gaussian components pAA_ ðA; A_ Þ ¼ pA ðAÞpA_ ðA_ Þ

ð4:419Þ

A A2 þ m 2 mA pA ðAÞ ¼ 2 exp I 0 2 22

ð4:420Þ

_2 1 A ffi exp 2 pA_ ðA_ Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð 000 Þ 2ð 000 Þ

ð4:421Þ

where

and

The following short notation is also used 000

@2 ¼ 2 ðÞ @ ¼0

ð4:422Þ

173


Thus, the LCR can be calculated as LCRðRÞ ¼

ð1 0

pA ðRÞpA_ ðA_ ÞA_ dA_ ¼ pA ðRÞ

ð1 0

pA_ ðA_ ÞA_ dA_

A_ A_ 2 ffi exp 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ pA ðRÞ dA_ ð4:423Þ 00 2 ð 000 Þ 0 2ð 0 Þ

ð1 pA ðRÞ A_ 2 A_ 2 2 00 ffi d ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð Þ exp 0 22 ð 000 Þ 22 ð 000 Þ 2ð 000 Þ 0 rffiffiffiffiffiffiffiffiffi

rffiffiffiffiffiffiffiffi00ffi ð pa ðRÞ2 ð 000 Þ 1 000 R R2 þ m2 mR 0 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi exp ¼ exp½tdt ¼ p ðRÞ I A 0 2 2 2 2 2 2ð 000 Þ 0 ð1

In the particular case of m ¼ 0, one can obtain

rffiffiffiffiffiffiffiffi00ffi R R2 0 LCRðRÞ ¼ exp 2 2 2

ð4:424Þ

In many cases, it is easier to calculate the coefficient 000 through the power spectral density. Indeed, using properties of the Fourier transform, one can obtain 000

4.7.4 4.7.4.1

ð1 @2 1 ¼ 2 ðÞ ¼ !2 Sð!Þ d! 2 2 @ 0 ¼0

ð4:425Þ

Examples of Non-Gaussian Narrow Band Random Processes K Distribution

The K distribution 2b ba K1 ðbaÞ; pA ðaÞ ¼ ðÞ 2

> 0; x 0

ð4:426Þ

is a rare case when the Blanc-Lapierre transformation can be calculated analytically. In order to do so, the characteristic function of the PDF (4.426) using the Bessel transform [19,38] can be found first

R ðsÞ ¼

ð1 0

bþ1 pA ðaÞJ0 ðsaÞda ¼ 1 2 ðÞ

ð1

A K1 ðbaÞJ0 ðsaÞda

ð4:427Þ

0

It can be calculated using the standard integral (6.576–7 on page 694 in [39]) ð1 0

xþþ1 J ðaxÞK ðbxÞdx ¼ 2þ a b

ð þ þ 1Þ ða2 þ b2 Þþþ1

ð4:428Þ

174


where the following should be used: ¼ 0;

a ¼ s;

x ¼ A;

¼1

ð4:429Þ

This yields

R ðsÞ ¼

bþ1 ðÞ b2 1 1 2 b ¼ ða2 þ b2 Þ ðs2 þ b2 Þ 21 ðÞ

ð4:430Þ

The PDF px ðxÞ itself can be found by using an inverse Fourier transform of its characteristic function (4.430) [40]: ð ð 1 1 b2 1 expðisxÞ pR ðxÞ ¼

ðsÞ expðisxÞds ¼ ds 2 1 R 2 1 ðs2 þ b2 Þ ð1 expðisxÞ ¼ ds 1 ðis þ bÞ ðis þ bÞ ð b2 1 ¼ ðis þ bÞ2 2 ðis þ bÞ2 2 expðisxÞds 2 1

ð4:431Þ

The last integral is a table integral (3.384–9 on page 321 in [39]): ð1 1

ð þ jxÞ2 ð jxÞ2 ejpx dx ¼ 2ð2Þ2

jpj21 W 1 ð2jpjÞ ð2Þ 0;22

ð4:432Þ

Thus, setting ¼ ; 2

¼ b;

p ! x;

x!s

ð4:433Þ

the expression for the PDF becomes pR ðxÞ ¼

b jxj1 22vþ1 b ð2bjxÞv1 W0;12 ð2bjxjÞ 2ð2bÞ W0;12 ð2bjxjÞ ¼ ðÞ 2 ðÞ

ð4:434Þ

This equation can be further simplified using the following expression for the Whittaker function W0;1=2 ð Þ, first in terms of the Kummer function Uða; b; zÞ and then in terms of the modified Bessel function K of the second kind (eq. 13.1–33 on page 505 and eq. 13.6–21 in Table 3.6 on page 510 in [29]): z z z1=2 W0; 12 ðzÞ ¼ exp z Uð1 ; 2 2; zÞ ¼ pffiffiffi K12 2 2

ð4:435Þ

1 2þ2 b pffiffiffi ðbjxjÞ2 K1 ðbjxjÞ pR ðxÞ ¼ 2 ðÞ

ð4:436Þ

and thus 1


175

Therefore, the distribution of instantaneous values corresponding to the K distribution of the magnitude is again expressed in terms of the modified Bessel function K. 4.7.4.2

Gamma Distribution

Let the PDF of the intensity be the Gamma distribution pi ðIÞ ¼

þ1 I expðIÞ ð þ 1Þ

Coefficients m of the expansion (4.365) can be found as ð1 ð 1 þ1 I expðIÞ Lm ðIÞdI pi ðIÞLm ðIÞdI ¼ m ¼ ð þ 1Þ 0 0 ð1 ð1 1 1 ¼ ðIÞ exp½ILm ðIÞd I ¼ x exp½xLm ðxÞdx ð þ 1Þ 0 ð þ 1Þ 0

ð4:437Þ

ð4:438Þ

The last integral can be calculated using the table integral 7.414 –11 on page 845 in [39] ð1 ð ðð1 þ þ m Þ ; Ref g > 0 ð4:439Þ x 1 exp½xLm ðxÞdx ¼ m!ð1 þ Þ 0 with ¼ þ 1, and ¼ 0: ð1 1 1 ð þ 1Þð1 þ m 1Þ ðm Þ ¼ m ¼ x exp½xLm ðxÞdx ¼ ð þ 1Þ 0 ð þ 1Þ m!ð1 1Þ m!ð Þ ð4:440Þ Substituting eq. (4.440) into eq. (4.377) results in the following expression for the PDF of the amplitude of the distribution pffiffiffi 1 X pffiffiffi ð1Þk ðk Þ 2 pR ðxÞ ¼ pffiffiffi exp½x H ð xÞ 2k 22k K! K!ð Þ k¼0

ð4:441Þ

A few examples showing the accuracy of the approximation of the Gamma PDF can be found in Fig. 4.8. Since the coefficients of the expansion (4.441) decay as fast as k , one can expect that the approximation (4.441) converges as fast to the exact PDF pR ðxÞ as the corresponding approximation of pi ðIÞ converges to the exact Gamma PDF. 4.7.4.3

Log-Normal Distribution

The log-normal distribution is also frequently used in modeling of random EM fields. In this case " # 1 ðlnðIÞ aÞ2 pi ðIÞ ¼ pffiffiffiffiffiffi exp ð4:442Þ 22 I 2

176


Figure 4.8

Approximation of the Gamma distribution by its Gram–Charlier series.

with moments given by mLn

n2 2 ¼ exp an þ 2

ð4:443Þ

It is impossible to obtain an expression similar to eq. (4.440) in this case. However, the fact that the moments of this distribution are analytically known can be used as follows. Let Lk ðIÞ be rewritten as a power series of ðIÞ: Lk ðIÞ ¼

k X

ðkÞ

al ðIÞl

ð4:444Þ

l¼0 ðkÞ

where the coefficients al can be recursively calculated using eq. 22.3.9 on page 775 in [29] 1 k! 1 k ðkÞ l ¼ ð1Þl ð4:445Þ l ¼ ð1Þ k l l! ðk lÞ! l!l! Plugging eq. (4.444) into eq. (4.366), one can obtain a recursive algorithm for the calculation of coefficients in expansion (4.377) ! ð1 k k k X X X l 2 2 ðkÞ ðkÞ l ðkÞ l l k ¼ ð4:446Þ pi ðIÞ al ðIÞ dI ¼ al m l ¼ al exp al þ 2 0 l¼0 l¼0 l¼0 thus leading to the final expression for pR ðxÞ in the form pffiffiffi 1 k X pffiffiffi X 1 ð1Þkl l l 2 2 2 exp al þ H ð xÞ pR ðxÞ ¼ pffiffiffi exp½x 2k 2k 2 ðk lÞ!l!l! 2 k¼0 l¼0

ð4:447Þ

177


Figure 4.9 Approximation of log-normal PDF by its Gram–Charlier series.

A few examples showing the accuracy of approximation of the log-normal PDF can be found in Fig. 4.9.

4.7.4.4

A Narrow Band Process with Nakagami Distributed Envelope

In this case, the distribution of the envelope is described by the Nakagami PDF 2 mm 2m1 ma2 1 a exp ; m ;a 0 pA ðaÞ ¼ ðmÞ 2

ð4:448Þ

The characteristic function can be found by first finding the characteristic function of the PDF (4.448) using the Bessel transform [19,38] ð1 ð 2 mm 1 2m1 ma2

R ðsÞ ¼ pA ðaÞJ0 ðsaÞda ¼ a exp ð4:449Þ J0 ðsaÞda ðmÞ 0 0 This integral can be calculated using standard integral (6.631–1 on page 698 in [39])

ð1 0

þþ1 2

þþ1 2 ; þ 1; x expð x ÞJ ðxÞdx ¼ 1 F1 þþ1 2 4 2þ1 2 ð þ 1Þ

2

ð4:450Þ

by setting ¼ 0;

¼ s;

¼

m

and

¼ 2m 1

ð4:451Þ

178


Here 1 F1 ða; b; zÞ is the so-called confluential hypergeometric function20 [39]. As a result s2

R ðsÞ ¼ 1F1 m; 1; 4m

ð4:452Þ

The distribution of the in-phase component can be found through the Fourier transform of the characteristic function, i.e. 1 pR ðxÞ ¼ 2

ð s2 1 1 s2 expðisxÞds ¼ cosðsxÞdx 1 F1 m; 1; 1 F1 m; 1; 4m 0 4m 1

ð1

ð4:453Þ Using the substitution z ¼ 12

pffiffimffi s, the last integral is converted to rffiffiffiffi ð 1 rffiffiffiffi m m 2 z dz 1 F1 ðm; 1; z Þ cos 2x 0

2 pR ðxÞ ¼

ð4:454Þ

which, in turn, can be calculated with the help of the standard integral 7.642 on page 822 in [39] ð1

pffiffiffi ðcÞ 2a1 1 1 2 2 y cosð2xyÞ1 F1 ða; c; x Þdx ¼ expðy Þ c ; a þ ; y 2 ðaÞ 2 2 2

0

ð4:455Þ

by setting c ¼ 1;

rffiffiffiffi m y¼x and

a¼m

ð4:456Þ

Thus pR ðjxjÞ ¼

mm

m 1 1 1 x2 m 2m1 2 pffiffiffi jxj exp x ; m þ ; 2 2 ðmÞ

ð4:457Þ

Here ða; b; zÞ is the second confluent hypergeometric function, also known as the Kummer Uða; b; zÞ function [39]. Since the Kummer function ða; b; zÞ can also be expressed in terms of the Whittaker function W ; ðzÞ as z 1 1 ð4:458Þ þ ; 1 þ 2; z ¼ zð2þÞ exp W; ðzÞ 2 2 Equation (4.457) can be rewritten in the form suggested in [41] m 1 pffiffiffi pR ðjxjÞ ¼ ðmÞ

2m1 4

3 jxjm 2

2 m 2 mx x W2m1 ; 2m1 exp 2 4 4

The same function is known as the Kummer M function Mða; b; zÞ or ða; b; zÞ [39].

20

ð4:459Þ

179


Figure 4.10 PDF of the in-phase (quadrature) component of a stationary random process with the Nakagami distribution for different values of the severity parameter m.

The shape of the PDF of the in-phase component is shown in Fig. 4.10 for different values of parameter m. Alternatively, one can obtain a series expansion for pR ðxÞ in terms of Hermitian polynomials by following the procedure established in section 2.2. Indeed, using eqs. (2.65) and (4.366) # " 1 x X 1 m1 I 1 I I exp k L k ¼ exp 1þ pi ðIÞ ¼ ðmÞ k¼1

ð4:460Þ

with k ¼

ð1 0

ð1 I 1 1 1 ð1 þ k mÞ m m1 dI ¼ Lk dI ¼ pi ðIÞLk I exp ðmÞ 0 ð1 mÞk! ð4:461Þ

one obtains that the series (4.377) for the PDF of the in-phase component becomes 2 X 1 x 1 ð1Þk ð1 þ k mÞ x H2k pffiffiffiffi pR ðxÞ ¼ pffiffiffiffiffiffiffi exp k¼0 22k k!ð1 mÞk!

ð4:462Þ

For integer values of m ¼ n, n ¼ 1; 2; . . ., expressions (4.452) and (4.457) can be further simplified since 2 2 2 2 k ! n1 X s2 s s s s ck ¼ exp Ln1 ¼ exp 1þ 1 F1 n; 1; 4n 4n 4n 4n 4n k¼1 ð4:463Þ

180


and, thus 2 " 2 k # n1 X s s expðisxÞdx exp ck 1þ 4n 4n 1 k¼1 k 2 ð 2 ð n1 X 1 1 s 1 1 s2 s ¼ expðisxÞds þ expðisxÞds ck exp 2 1 4n 2 1 4n 4n k¼1 2 x " # 2 exp k n1 X ð1Þ x x pffiffiffiffiffiffiffi ð4:464Þ ¼ ck 2n exp 1þ H2n pffiffiffiffi 2 k¼1

1 pR ðxÞ ¼ 2

ð1

Knowledge of the characteristic function (4.452) of the in-phase process also allows one to derive an expression for the PDF of the envelope of the mixture of the narrow band process with the Nakagami envelope and a narrow band Gaussian noise ðtÞ with the variance 2 ¼ N0 f . Indeed, since in this case R ðtÞ

¼ R ðtÞ þ ðtÞ

ð4:465Þ

and, using independence of the noise and the signal s2 2 s 2 exp ð jsÞ ¼ R ð jsÞ ð jsÞ ¼ 1 F1 m; 1; 4m 2

ð4:466Þ

Taking the inverse Bessel transform (4.332), one obtains the PDF p ðaÞ of the envelope of the process ðtÞ ð1 ð1 s2 2 s2 exp J0 ðasÞs ds ðjsÞJ0 ðasÞs ds ¼ a pA ðaÞ ¼ a 1 F1 m; 1; 4m 2 0 0 ð4:467Þ This integral can be expressed in terms of Meijer functions [39] using the table integral (7.663-1 on page 826 of [39]) 0 1 ð1 2 2 1;b 2 @y A x2 1 F1 ða; b; x2n ÞJ ðxyÞdx ¼ G21 23 1 1 1 1 2þ1 ðaÞy 4 0 þ þ ; a; þ 2 2 2 2 ð4:468Þ if the exponent exp½2 s2 =2 is represented by its Taylor series. Setting a ¼ m;

b ¼ 1;

¼

; 4m

¼ 0;

¼nþ

1 2

ð4:469Þ

one obtains

ð1 s2nþ1 1 F1 0

! 2 1;1 s2 22n ma m; b; G21 J0 ðasÞdx ¼ 4m ðmÞa2nþ1 23 nþ1;m;nþ1

ð4:470Þ

SPHERICALLY INVARIANT PROCESSES

181

and, thus 1 2k ð 1 X s2 2kþ1 k ð1Þ k J0 ðasÞs ds pA ðaÞ ¼ a s 1 F1 m; 1; 4m 2 k! 0 k¼0 ! 1 2k k 2 1;1 1 X 2 ma ð1Þk G21 ¼ ðmÞ k¼0 k! a2k 23 kþ1; m; kþ1

ð4:471Þ

Although this equation is convenient in calculations of the PDF itself21 it is not useful in analysis.

4.8 4.8.1

SPHERICALLY INVARIANT PROCESSES Definitions

A formal definition of a Spherically Invariant Vector (SIRV) is as follows: a random vector ¼ ½1 ; 2 ; . . . ; N T is called a spherically invariant vector if its PDF is uniquely defined by the specification of a mean vector m , a covariance matrix C , and the characteristic PDF p ðsÞ. It is also known as an elliptically contoured distribution [42–68]. This definition does not provide any specific details about the general form of the PDF of a SIRV. However, the answer is given by the following. Representation theorem [42]: If a random vector is a SIRV, then there exists a nonnegative random variable such that the PDF of the random vector conditioned on is a multivariate Gaussian PDF: ¼

ð4:472Þ

A Spherically Invariant Random Process (SIRP) ðtÞ is a random process such that every random vector n obtained by means of sampling of the random process ðtÞ is a SIRV. It is important to distinguish between a SIRP and a compound process, obtained as a product of two random processes ðtÞ ¼ ðtÞðtÞ

ð4:473Þ

which will be extensively used in the following derivations. It is clear from the definition of the SIRV that any particular realization of a SIRV is a Gaussian process with variance changing from one realization to another. Thus, while the process ðtÞ is non-Gaussian and stationary, it is not an ergodic process, since the distribution of the variance cannot be estimated by observing a single realization of the process. On the contrary, a compound process defined by a similar equation, eq. (4.473), defines an ergodic process. However, the structure of the compound process is more complicated since every realization is not Gaussian and it will not be possible to use general expressions for the joint probability density of an arbitrary order. Nevertheless, it will be shown that a significant gain can be obtained from the The Meijer functions are implemented in such products as Maple# or Mathematica#

21

182


representation of a compound process as a product of a Gaussian process and an exponentially correlated Markov process.

4.8.2 4.8.2.1

Properties Joint PDF of a SIRV

It is often convenient to assume that the modulating random variable is independent on the underlying zero mean Gaussian vector n. In this case, it is relatively easy to obtain the joint PDF p ðxÞ of the vector g in terms of the covariance matrix C of the Gaussian vector n and the characteristic PDF p ðsÞ of the modulating variable . Indeed, the PDF p ðxÞ is given by " # h pi xT C1 x 1 1 qffiffiffiffiffiffiffiffiffiffi exp ¼ p ðpÞ; p ¼ xT C1 ¼ p ðxÞ ¼ exp pffiffiffiffiffiffiffiffi x N=2 N=2 2 2 jC j ð2Þ C ð2Þ ð4:474Þ Here p is a scalar parameter, defined as a quadratic form of the vector x. Equation (4.474) indicates a well-known property of a Gaussian vector. The conditional PDF is thus given by " # T 1 h pi x C x 1 sN N ffiffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffiffiffi q exp pj ðy j sÞ ¼ ¼ s exp 2 ds 2s2 ð2ÞN=2 C ð2ÞN=2 x

ð4:475Þ

since multiplication by a fixed number scales the covariance matrix by a factor s2 (see section 2.3). Finally, the unconditional PDF p ðxÞ can be obtained by integration of the PDF (4.475) over the range of and the weight p ðsÞ: p ð y; NÞ ¼

ð1 0

¼

1 qffiffiffiffiffiffiffiffiffiffi pj ð y j sÞp ðsÞds ¼ N=2 C ð2Þ

ð1 0

h pi sN exp 2 p ðsÞds 2s

1 qffiffiffiffiffiffiffiffiffiffi hN ðpÞ ð2ÞN=2 C

ð4:476Þ

where hN ðpÞ ¼

ð1 0

h pi sN exp 2 p ðsÞds 2s

ð4:477Þ

Equations (4.476)–( 4.477) show that the PDF of a SIRV is a function of a quadratic form p, as is the Gaussian distribution. However, the form of this dependence is more complicated than a simple exponent. Thus, a SIRV can be considered as a natural extension of a Gaussian random vector. It is shown in a number of publications [47–58] that a SIRV inherits a number of properties of a Gaussian vector such as invariance under a deterministic linear transformation, properties of an optimal estimator, etc.

SPHERICALLY INVARIANT PROCESSES

183

It follows from eq. (4.472) that the covariance matrices M of a SIRV and the covariance matrix of the underlying Gaussian process M are M ¼ Ef2 gM ¼ m2 M

ð4:478Þ

Without loss of generality, it is possible to assume that the second moment m2 is unity: m2 ¼ 1. In this case, the correlation matrix of the SIRV is the same as that of the underlying Gaussian process. Setting N ¼ 1 in eq. (4.476), the marginal PDF of the resulting process p ðyÞ is related to the marginal PDF p ðsÞ of the modulation process as

ð1 1 y2 ds ð4:479Þ exp 2 2 p ðsÞ p ðyÞ ¼ pffiffiffiffiffiffiffiffi s 2s 2 0 Taking the Fourier transform of both sides, the following equation for the characteristic function of the output process can be easily derived

ð1 ð1 ð 1 ds 1 y2 ðj!Þ ¼ p ðyÞ expð j!yÞdy ¼ pffiffiffiffiffiffiffiffi p ðsÞ exp 2 2 expð j!yÞdy s 1 2s 2 0 1 " # 2 2 2 ð1 ð1 1 2k 2k 2k X s ! ks ! p ðsÞ exp p ðsÞ ð1Þ ds ¼ ds ¼ 2 2k k! 0 0 k¼0 ¼1þ

1 X 2k !2k ð1Þk k mu2k 2 k! k¼1

ð4:480Þ

Comparing this expansion with the Taylor series of the characteristic function , ð j!Þ, the following relations between the moments of the modulating PDF and the resulting process can be easily deduced ð j!Þ ¼

1 X ðjÞl m l

l¼0

l!

!l ¼ 1 þ

1 X k¼1

ð1Þk

2k !2k m dk k! 2k

ð4:481Þ

or, equivalently m2k1 ¼ 0; m2k ¼

ð2kÞ!2k m2k ¼ m2k m2k 2k k!

ð4:482Þ

The last equation can be of course directly obtained by manipulating moment brackets, as in section 2.8. Interestingly, only even moments of the modulating PDF matter, since the odd moments are cancelled by zero odd moments of the Gaussian distribution.

4.8.2.2

Narrow Band SIRVs

If the underlying Gaussian process ðtÞ in eq. (4.472) is a narrow band process, then the resulting SIRV (or compound process) will also be a narrow band process. In turn, the real

184


Gaussian process ðtÞ can be considered as a real part of a complex process ðtÞ ¼ R ðtÞ þ jI ðtÞ; ðtÞ ¼ RefðtÞg

ð4:483Þ

where the in-phase component R ðtÞ and the quadrature component I ðtÞ are defined using the Hilbert transformation, as described in section 4.7.1. As a result, the process ðtÞ can be thought of as the real part of a complex process ðtÞ ¼ ðtÞðtÞ ¼ ðtÞR ðtÞ þ jðtÞI ðtÞ

ð4:484Þ

with the envelope defined as A ðtÞ ¼ jðtÞj ¼ ðtÞjðtÞj ¼ ðtÞA ðtÞ

ð4:485Þ

Thus, the envelope of the resulting process is an envelope of a complex Gaussian process modulated by a random variable or a random process ðtÞ. Since the envelope of a narrow band Gaussian process is the Rayleigh process a a2 pA ðaÞ ¼ 2 exp 2 2

ð4:486Þ

the PDF of the envelope pA ðaÞ of the process ðtÞ can be expressed as pA ðaÞ ¼

4.8.3

ð1 0

a a2 ds p ðsÞ 2 exp 2 2 2 2s s

ð4:487Þ

Examples

A number of analytically tractable examples are summarized in Table 4.1.

Table 4.1 Marginal PDF Gaussian

p ðyÞ 2 y exp 2 2 pffiffiffiffiffiffiffiffiffiffi 22

p ðsÞ; s 0 ðs 1Þ

Two-side Laplace

b 2 expðbjxjÞ

2 2 b2 s exp b 2s

Cauchy

b 2 ðb þ x2 Þ

b2 b2 exp 2 s3 2s

K-distribution

2b bx K1 ðbxÞ ðÞ 2

No analytical expression

h2N ðpÞ p exp 2 2N b pffiffiffi pffiffiffi N1 kN1 ðb pÞ ðb pÞ 2N b N þ 12 1 pffiffiffi 2 ðb þ pÞNþ2 pffiffiffi b2N ðb pÞvN pffiffiffi KNv ðb pÞ v1 ðÞ2

REFERENCES

185

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5 Markov Processes and Their Description 5.1

DEFINITIONS

Let ðtÞ be a random process which depends on only one parameter—time1. Conditional on the range of values which the random process assumes and the nature of the parameter t, one can distinguish five major types of random process: 1. A discrete random sequence m ¼ ðtm Þ or a random chain (discrete process in discrete time). In this case, the random process attains a fixed and countable set of values xk , k ¼ 1; . . . ; K, and the time parameter is discrete as well, i.e. t belongs to a discrete countable set tm , m ¼ 1; . . . ; M. It is important to note that both K and M can be infinite. Processes of this type are often encountered in practice: a random tossing of an object, random telegraph signals and remote sensing are just a few examples. Random discrete sequences are often obtained as a result of sampling and quantization of a continuous random process. Most of the modern communication systems use such signals and thus can be described by random chains. 2. A random sequence m (continuous process in discrete time). This kind of process is different from the discrete chain only in the range of attainable values. In the case of the random sequence it is a continuum of values rather than a discrete set as in the former case. Quite often a random sequence is obtained as the result of sampling of a continuous random process. 3. A discrete random process or continuous time chain (discrete process with continuous time). In this case the random process ðtÞ assumes values from a countable set xk , k ¼ 1; . . . ; K, while the parameter t continuously changes from t ¼ t0 to t0 þ T. It is also common to assume that t0 ¼ 0. The length of a serving queue and the number of users of a computer network are two examples of processes of this type. Often, a continuous time chain can be considered as an output of a quantizer driven by a continuous random process. 4. A continuous random process ðtÞ. In this case, the random process attains a value from a continuum of values with the time parameters also changing continuously. It is also often 1

Among the other possible choices the spatial parameter is often used to model space–time variation of signals.


190

MARKOV PROCESSES AND THEIR DESCRIPTION

assumed that the trajectory of the continuous process does not experience large vertical jumps. Noise is one of quite a few examples of this kind of random process. 5. Mixed random processes. In this case, a random process is a mixture of the different types listed above. In particular, continuous processes with jumps can be considered as mixed. In this case, the random process resembles a continuous random process most of the time. However, from time to time, a rapid variation in the value (jump) appears, often due to changes in the system structure. Noise in communication systems using power lines and channels with impulsive and continuous noise are systems whose random structure can be described as a mixed process. Often, it is possible to describe a mixed process using a vector random process nðtÞ ¼ ½ðtÞ; ðtÞT , where ðtÞ is a continuous random process (type 4) and ðtÞ is a discrete chain2. If, in addition, a process poses a Markov property as defined below, one can distinguish Markov chains, Markov sequences, continuous time Markov chains, continuous Markov processes and mixed Markov processes. A few typical trajectories are shown in Figure 5.1. The classification can be extended to a case of a random vector nðtÞ ¼ ½1 ðtÞ; 2 ðtÞ; . . . ; r ðtÞT . In this case the vector process is called r-dimensional. In general, each component belongs to a different type of process and thus classification should be done on a component basis. Extensive literature exists on the theory of Markov processes and their applications [1–6 to name a few]. Only a brief review is included in this book to facilitate further considerations.

5.1.1

Markov Chains

Let a physical system which has K possible states #ð1Þ ; #ð2Þ ; . . . ; #ðKÞ be considered. The system can change its state randomly, depending on the external factors. These changes appear in a sudden manner at time instants t1 < t2 < t3 < . . ., i.e. transitions

1 ! 2 ! 3 ! . . . take place only at these specified moments of time. Here m ¼ ðtm Þ stands for the state of the system after m 1 transitions, and 1 ¼ ðt1 Þ is the initial state of the system (see Fig. 5.1(a)). Of course all k are chosen from a finite set f#ðlÞ g; l ¼ 1; . . . ; K. A complete description of such a system can be achieved if the joint probability Pð 1 ; 2 ; . . . ; m Þ can be calculated for an arbitrary m. Using the Bayes formula one can write Pð k1 ; k2 ; . . . ; km Þ ¼ Pð 1 ¼ #k1 ; 2 ¼ #k2 ; . . . ; m ¼ #km Þ ¼ Pð 1 ¼ #k1 ; 2 ¼ #k2 ; . . . ; m1 ¼ #km1 Þ Pð m ¼ #km j 1 ¼ #k1 ; 2 ¼ #k2 ; . . . ; m1 ¼ #km1 Þ

ð5:1Þ

The notation ðtÞ for a continuous random process and ðtÞ for a discrete random process or chain is reserved for the presentation in this chapter.

2

191

DEFINITIONS

Figure 5.1 Classification of Markov processes: (a) Markov chain; (b) Markov sequence; (c) continuous time Markov chain; (d) continuous Markov process.

where #k is one of the possible states #ðÞ . In the very same manner, one can further expand Pð 1 ¼ #k1 ; 2 ¼ #k2 ; . . . ; m1 ¼ #km1 Þ in terms of conditional probabilities to obtain Pð k1 ; k2 ; . . . ; km Þ ¼ P1 ð 1 Þ

m Y

ð j 1 ; . . . ; 1 Þ

ð5:2Þ

¼2

where P1 ð 1 Þ is the probability of the initial state 1 ¼ ðt1 Þ and ð#k j #k1 ; . . . ; #k1 Þ ¼ Pð ¼ #k j 1 ¼ #k1 ; 2 ¼ #k2 ; . . . ; 1 ¼ #k1 Þ;

2m

ð5:3Þ

192


is the conditional probability to reach state #k at the time instant t if the previous states are

1 ; 2 ; . . . ; 1 . The probability ð#k j #k1 ; . . . ; #k1 Þ is also known as the transitional probability. Thus, in order to calculate the joint probabilities Pð 1 ; 2 ; . . . ; m Þ, it is sufficient to know the initial state of the system and all the transitional probabilities , or the methodology of their calculation. There are a few specific cases when such calculations are relatively simple: 1. Changes in all states are independent. In this case, the transition to the next state does not depend on the previous state of the system, i.e. m ð m j 1 ; . . . ; m1 Þ ¼ Pm ð m Þ

ð5:4Þ

and eq. (5.2) is significantly simplified to become

Pð 1 ; 2 ; . . . ; m Þ ¼

m Y

P ð Þ

ð5:5Þ

¼1

As can be seen, this case coincides with the case of independent trials [4,5]. It is also a degenerate case of a Markov chain considered below. 2. A more general model can be obtained if the transitional probabilities depend only on the current state m1 ¼ ðtm1 Þ of the system and do not depend on its past states at the time instances t1 ; t2 ; . . . ; tm2 . This property is general to all types of Markov process. For this reason Markov processes are often called processes without memory.3 Mathematically speaking, the following equality holds for any index m m ð m j 1 ; . . . ; m1 Þ ¼ m ð m j m1 Þ

ð5:6Þ

Conditional probabilities m ð m j m1 Þ are also called transition probabilities. Equation (5.6) defines the so-called Markov property of a random chain (process). Using the chain rule, one can easily transform eq. (5.2) into

Pð 1 ; 2 ; . . . ; m Þ ¼ P1 ð 1 Þ

m Y

ð j 1 Þ

ð5:7Þ

¼2

As a result, the complete description of a Markov chain can be achieved by providing the probability of the initial state P1 ð 1 Þ and the transitional probabilities ð j 1 Þ. It is important to emphasize that eq. (5.7) holds true for the case of two moments of time ðm ¼ 2Þ for any random process. However, for the case of three or more moments, this equation does not describe a general situation, i.e. not all random processes are Markov. 3

In the case (2), the process under consideration does not remember its past only, in contrast to a process considered in the case (1) which does not remember its past or its present.

193

DEFINITIONS

3. In a similar manner, one could define a Markov process of order nðn 1Þ: if the transitional PDF depends only on n previous states (including the current state) of the system, and does not depend on a more distant past. Mathematically speaking ( m ð m j 1 ; . . . ; m1 Þ ¼

m ð m j mn ; . . . ; m1 Þ;

m>n

m ð m j 1 ; . . . ; m1 Þ;

2mn

ð5:8Þ

A regular Markov chain, defined in Section 2, corresponds to the case n ¼ 1. A Markov chain of order n can be reduced to a simple vector Markov chain for a vector of dimension n. In order to do so, one can consider n sequential states of the system as components of a single n-dimensional vector h defined as hm ¼ f i; m ; i ¼ 1; . . . ; ng ¼ f mnþ1 ; . . . ; m g;

i;m ¼ mnþi

ð5:9Þ

On the previous time step the vector hm1 is expressed as f i; m1 ; i ¼ 1; . . . ; ng ¼ f mn ; . . . ; m1 g

ð5:10Þ

and the following relations between components of the vector are satisfied

i; m ¼ iþ1; m1 ¼ mnþi ;

1; m1 ¼ mn ;

n<m

ð5:11Þ

Thus, a Markov chain of the order n is a particular case of a simple Markov vector of dimension n. In this case, the transitional PDF is defined as m ðhm j hm1 Þ ¼ m ð n; m j 1; m1 ; . . . ; n; m1 Þ

n1 Y

ð i;m ; iþ1; m1 Þ

ð5:12Þ

i¼1

where ð; Þ ¼ is the Kronecker symbol. This relation follows from the definition (5.11) of the components of the vector hm. It is often necessary to evaluate the probabilities Pm ðhm Þ and Pm1 ðhm1 Þ of the system state at two sequential moments of time. For a simple Markov chain, the necessary relation can be obtained directly from eq. (5.7) Pm ðhm Þ ¼

#k X

m ðhm j hm1 ÞPm1 ðhm1 Þ;

m ¼ 2; . . . ; M

ð5:13Þ

m1 ¼ #1

Indeed, it follows from eq. (5.7) that Pðh1 ; . . . ; hm Þ ¼ m ðhm j hm1 ÞPðh1 ; . . . ; hm1 Þ

ð5:14Þ

Adding both sides of eq. (5.14) over h1 ; . . . ; hm1, one immediately obtains eq. (5.13).

194


If a simple vector Markov chain of dimension r is considered, hm ¼ f i; m ; i ¼ 1; . . . ; rg, then the recurrent relation (similar to eq. (5.13)) has the following form Pm ð 1; m ; . . . ; r; m Þ #1;K1

¼

X

# r; Kr X

1; m1 ¼#1; 1

m ð 1; m ; . . . ; r; m j 1; m1 ; . . . ; r; m1 ÞPð 1; m1 ; . . . ; r; m1 Þ

r; m1 ¼ #r; 1

ð5:15Þ or Pm ð m Þ ¼

X

m ðhm j hm1 ÞPm1 ðhm1 Þ

ð5:16Þ

m1 ¼ #

and the summation is conducted over all possible states 0 ¼ f#i;ki ; ki ¼ 1; . . . ; Ki g of each component i; m ; i ¼ 1; . . . ; r. Here Ki is the number of states of the i-th component. For a Markov chain of order n, one can write an equation similar to eq. (5.7), based on eqs. (5.1) and (5.8) Pm ð mnþ1 ; . . . ; m Þ ¼

#K X

m ð m j mn ; . . . ; m1 Þ Pm1 ð mn ; . . . ; m1 Þ

ð5:17Þ

mn ¼ #1

where m > n. At the initial steps 2 m n, we have to set n ¼ m 1. In this case the following recurrent relation holds Pm ð 2 ; . . . ; m Þ ¼

#K X

m ð m j 1 ; . . . ; m1 Þ Pm1 ð mn ; . . . ; m1 Þ

ð5:18Þ

m ¼ #1

Expressions (5.17) and (5.18) relate joint probabilities of the last n states of the discrete random variable f m g. It can also be noted that the same relations can be obtained from the recurrent relation (5.16) for the probabilities of states of the Markov chain of order n if one sets r ¼ n and uses eqs. (5.10) and (5.12). In this case the state space of each component is identical and thus Pm ðhm Þ ¼

#K X

m ð n; m j 1; m1 ; 1; m ; . . . ; n1; m Þ Pm1 ð 1; m1 ; 1; m ; . . . ; n1; m Þ

1; m1 ¼ #1

¼

#K X

m ð m j mn ; mnþ1 ; . . . ; m1 Þ Pm1 ð mn ; mnþ1 ; . . . ; m1 Þ

mn ¼ #1

ð5:19Þ where n < mðm 2Þ. The obtained relation coincides with eq. (5.17). As an example of the application of a simple Markov chain, one can consider the following problem. Let the initial state of a system be known at the initial moment of time t1

195

DEFINITIONS

and let the transition probabilities be specified as pk ðmÞ ¼ Pð m ¼ #k Þ;

jk ð; mÞ ¼ Pf m ¼ #k j ¼ #j g;

j; k ¼ 1; . . . ; K; 1 m ð5:20Þ

The interpretation of this notation is as follows: the probability pk ðmÞ describes the (unconditional) probability of finding the system at the state #k at the ðm 1Þ-th step (i.e. at the moment of time t ¼ tm ); the probability pk ð0Þ ¼ p0k describes the initial distribution of the states of the system. Conditional probabilities jk ð; mÞ define the probability of the system to reside at the state #k at the moment of time tm , given that the system has resided at the state # at the moment of time t ¼ t < tm . This probability is often referred to as the transitional probability from the state #j to the state #k over ðm Þ steps. It is required to find the probabilities of states of the system at the moment of time tm > t1 , and to find their limit when m approaches infinity. It is clear that all the quantities defined by eq. (5.20) are non-negative and satisfy the following normalization conditions K X

pk ðmÞ ¼ 1;

pk ðmÞ 0;

m ¼ 1; . . . ; M

ð5:21Þ

k¼1 K X

jk ð; mÞ ¼ 1;

jk ð; mÞ 0;

j ¼ 1; . . . ; K

ð5:22Þ

k¼1

Using the Bayes theorem one can immediately write pk ðmÞ ¼

K X

pj ðÞjk ð; mÞ;

k ¼ 1; . . . ; K;

1m

ð5:23Þ

j¼1

Transition of the system from the state #j to the state #k over a few steps can be completed in a number of ways. In other words, the system can reside in different intermediate states #i . Using the Bayes rule and the definition of a Markov process one deduces jk ð; mÞ ¼

K X

ji ð; nÞik ðn; mÞ; j; k ¼ 1; . . . ; K;

ð5:28Þ

All other chains are non-homogeneous. For a homogeneous Markov chain, one can consider a one-step transitional matrix, which does not depend on any time parameter P ¼ Pð1Þ;

jk ¼ jk ð1Þ

ð5:29Þ

There is a simple relation between the matrix P and the transitional probability matrix Pð; mÞ ¼ Pðm Þ between the -th and the m-th time step. Indeed, using property (5.27) one can sequentially obtain Pðm Þ ¼ Pðm 1ÞPð1Þ ¼ Pðm 2ÞPð1ÞPð1Þ ¼ Pm

ð5:30Þ

Thus, the k-step transitional matrix is just the one-step transitional matrix raised to the power k PðkÞ ¼ Pk

ð5:31Þ

Furthermore, using a property of matrix transposition, it is possible to calculate probabilities Pðm þ kÞ of states of the system after m steps if the initial probabilities PðkÞ are known. Indeed, using eq. (5.30) in eq. (5.26) one readily concludes that Pðm þ kÞ ¼ PT ðk; mÞPðkÞ ¼ ðPm ÞT PðkÞ ¼ ðPT Þm PðkÞ

ð5:32Þ

A class of homogeneous Markov chains can be further restricted to stationary chains. For a stationary chain, the probability PðmÞ of the states at each step of the system does not depend on the index m, i.e. they are equal at each step. All other chains are non-stationary. It follows from the definition of a stationary chain that PðmÞ ¼ Pð1Þ ¼ P ¼k pk k

ð5:33Þ

This, combined with eq. (5.26), shows that P ¼ PT P

ð5:34Þ

DEFINITIONS

197

i.e. the vector of stationary probabilities is the eigenvector of the transitional matrix P corresponding to the eigenvalue ¼ 1 n. It is important in many applications to decide if there is a limiting stationary distribution P ¼ limm!1 PðmÞ, and, if it exists, how to find it. The following theorems provide an answer to this question. A detailed proof can be found in the literature [7]. It is said that a state #k is reachable from a state #j , if, after a certain (finite) number of steps, there is a positive probability that the system can reach the state #k . In other words, there is a finite number m such that kj ðmÞ > 0. Theorem 1. For any Markov chain with a finite number of states and with all states reachable from all other states, i.e. for any j and k there is an integer mjk such that jk ðmjk Þ > 0

ð5:35Þ

pk ¼ lim pk ðMÞ

ð5:36Þ

there are final probabilities, such that M!1

These final probabilities do not depend on the initial probabilities pk ð1Þ at the first step. A Markov chain for which the final probabilities exist is called an ergodic Markov chain. Theorem 2. If the condition (5.35) is satisfied, then the final probabilities pk , k ¼ 1; . . . ; K are the unique solution of the system of linear equations pk ¼

K X

pj jk ;

k ¼ 1; . . . ; K

ð5:37Þ

j¼1

subject to an additional condition (normalization) K X

pj ¼ 1

ð5:38Þ

j¼1

These final probabilities form a stationary probability vector (distribution) P ¼k pk k. A Markov chain is stationary if its initial distribution Pð1Þ coincides with the final probabilities. It is easy to see that the matrix form of eq. (5.37) coincides with that of eq. (5.34). It can be seen from eqs. (5.37) and (5.38) that there are K þ 1 equations for determining only K unknown final probabilities. However, the first K equations in eq. (5.37) are linearly dependent. Indeed, by summing both sides of the equation over all possible values of the index k one obtains an identity, thus indicating that only ðK 1Þ equations are really independent. As a result, the unknown probabilities pk could be defined from ðK 1Þ eq. (5.37) and an additional eq. (5.38). In order to illustrate this fact, the following two important examples are considered. Example: A symmetric Markov chain with two states (Figure 5.2). Let m , m ¼ 1; . . . ; M be a Markov chain with two possible states #1 and #2 . In the initial state the probability of the state #1 is p1 ð0Þ ¼ p01 ¼ , while the probability of the second

198


Figure 5.2 A symmetric Markov chain with two states.

state #2 is p2 ð0Þ ¼ p02 ¼ , where þ ¼ 1. One-step transitional probabilities are described by the following equalities: 11 ¼ 22 ¼ p and 12 ¼ 21 ¼ q, where p þ q ¼ 1, p > 0. Since its one-step transitional matrix P is symmetric, the corresponding Markov chain is called a symmetric Markov chain. The goal is to determine the transitional probabilities jk ðm 1Þ after ðm 1Þ steps, probabilities pk ðmÞ of each state on the m-th step, and the final probabilities pk . One also requires to determine the reverse probabilities kj ðm 1Þ ¼ Pf 1 ¼ #j j m ¼ #k g. In particular, the probability 11 ðm 1Þ ¼ Pf 1 ¼ #1 j m ¼ #1 g is of interest. This symmetric Markov chain is homogeneous, since the transitional probabilities do not depend on the step m. Setting ¼ 1 and n ¼ 0 in the Markov equation (5.27), one immediately obtains jk ðm 1Þ ¼

2 X

ji ik ðm 2Þ

ð5:39Þ

i¼1

At the same time, the equation for the probabilities of the states on the m-th step can be found from the expression (5.26): pk ðmÞ ¼

2 X

pj ð1Þjk ðm1 Þ

ð5:40Þ

j¼1

The probabilities jk ðm 1Þ can be calculated using two methods: either by using eq. (5.31) and matrix calculus or by calculating these probabilities step by step, using eq. (5.39). The second technique is investigated here in more detail. The normalization condition (5.22) for the case of ¼ 0 can be written as 11 ðm 1Þ þ 12 ðm 1Þ ¼ 1 22 ðm 1Þ þ 21 ðm 1Þ ¼ 1

ð5:41Þ

Since there is a complete symmetry in this problem, one can justify that 12 ðm 1Þ ¼ 21 ðm 1Þ, and, consequently, 11 ðm 1Þ ¼ 22 ðm 1Þ. Thus one needs to calculate only one of these probabilities in order to find all the others. For example, one can concentrate on calculating 11 ðm 1Þ. In order to do that, relation (5.22) can be exploited to produce 11 ðmÞ ¼ 11 11 ðm 1Þ þ 12 12 ðm 1Þ ¼ p11 ðm 1Þ þ q½1 11 ðm 1Þ 1 ðp qÞ 1 ðp qÞ ¼ þ ½211 ðm 1Þ 1 ¼ ðp qÞ11 ðm 1Þ þ 2 2 2

ð5:42Þ

199

DEFINITIONS

Since, for m ¼ 1 the system is in the initial state, one deduces that 11 ð0Þ ¼ 22 ð0Þ ¼ 1;

12 ð0Þ ¼ 21 ð0Þ ¼ 0

ð5:43Þ

Now it is possible to derive the expression for all the transitional probabilities using eqs. (5.41) and (5.42) in a sequential manner. Indeed, it follows from eqs. (5.42) and (5.43) that 1 ðp qÞ 1 ðp qÞ ½211 ð0Þ 1 ¼ þ 11 ð1Þ ¼ þ 2 2 2 2 1 ðp qÞ 1 ðp qÞ2 ½211 ð1Þ 1 ¼ þ 11 ð2Þ ¼ þ 2 2 2 2

ð5:44Þ

1 ðp qÞ 1 ðp qÞm ½211 ðm 1Þ 1 ¼ þ 11 ðmÞ ¼ þ 2 2 2 2 Once 11 ðmÞ is found, the rest of the transitional probabilities can be found from eq. (5.41): 1 ðp qÞm 22 ðmÞ ¼ 11 ðmÞ ¼ þ 2 2

ð5:45Þ

1 ðp qÞm 12 ðmÞ ¼ 21 ðmÞ ¼ 1 11 ðmÞ ¼ 2 2

ð5:46Þ

Finally, probabilities of each state can be found through the probabilities in the initial state and the transitional probabilities (5.44)–(5.45) p1 ðmÞ ¼ p1 ð1Þ11 ðm 1Þ þ p2 ð1Þ21 ðm 1Þ

1 ðp qÞm 1 ðp qÞm 1 ð Þ ðp qÞm1 ¼ þ þ ¼ þ 2 2 2 2 2 2 p2 ðmÞ ¼ p2 ð1Þ22 ðm 1Þ þ p1 ð1Þ12 ðm 1Þ

1 ðp qÞm 1 ðp qÞm 1 ð Þ ðp qÞm1 þ ¼ þ ¼ þ 2 2 2 2 2 2

ð5:47Þ

ð5:48Þ

Since p > 0 and all the states are reachable, then ðp qÞ < 1 and lim p1 ðmÞ ¼ lim p2 ðmÞ ¼

m!1

m!1

1 2

ð5:49Þ

This indicates that this chain is an ergodic Markov chain and the final probabilities p1 and p2 of the states can be found. The latter can be accomplished by solving a system of linear equations, obtained directly from eqs. (5.37) and (5.38) p1 ¼ pp1 þ qp2

p2 ¼ qp1 þ pp2

or

p1 þ p2 ¼ 1

ð5:50Þ

The only solution of either of these systems is p1 ¼ p2 ¼ 0:5. Further analysis of expressions of probabilities of states shows that there is a transient process before a stationary (final) distribution of states is achieved. It also becomes obvious

200


that the approach to stationarity has an exponential nature, since the variable part of p1 ðmÞ is proportional to ðp qÞm . On the other hand if p ¼ q ¼ 0:5, the final probabilities are immediately achieved and the process is stationary. In order to find the reversed probabilities kj ðm 1Þ ¼ Pf 1 ¼ #j j m ¼ #k g, the Bayes theorem can be used. Indeed, according to this theorem kj ðm 1Þ ¼ Pf 1 ¼ #j j m ¼ #k g ¼

Pf 1 ¼ #j ; m ¼ #k g jk ðmÞpj ð1Þ ¼ pk ðmÞ pk ðmÞ

ð5:51Þ

In particular, this leads to the following expression for the reversed transition probability 11 ðm 1Þ 11 ðmÞp1 ð1Þ 11 ðmÞp1 ð1Þ þ ðp qÞm ¼ ¼ 11 ðm 1Þ ¼ p1 ðmÞ p1 ðmÞ 1 þ ð Þð p qÞm

ð5:52Þ

This clearly shows that, in general, the reversed transitional probability is not equal to the forward transitional probability. Example: A quasi-random telegraph signal. Let ðtÞ be a random process with two states #0 , which changes states only at the fixed moments of time tn ¼ nT0 , where T0 is a constant, n ¼ 0; 1; 2; . . . ; and is a random variable uniformly distributed on the interval ½0; T0 and independent of ðtÞ. It is assumed that the conditions of the previous example also hold. In particular, #1 ¼ #0 and #2 ¼ #0 . Such a signal is known as a quasi-random telegraph signal. It is important to note that the change of state is only allowed in certain instants of time, in contrast to the example considered later in Section 5.1.3. It is required to find the correlation function and the power spectrum density of such a signal in the stationary case, i.e. assuming that p1 ¼ p2 ¼ 0:5. The expectation h ðtÞi of the process is equal to zero since h ðtÞi ¼ p1 #0 þ p2 ð#0 Þ ¼ 0

ð5:53Þ

Thus, its covariance function CðÞ is just an average of the product of two values of the process at separate moments of time, i.e. CðÞ ¼ h ðtÞ ðt þ Þi

ð5:54Þ

The covariance function can be first evaluated assuming that the value of is fixed. Let the interval of length contain m points where the process can change its value, as shown in Fig. 5.3. Since the signal can assume only two values #0 , the product ð0Þ ðÞ assumes only two possible values as well. This value is #20 if both ends of the interval lie on the same side of the real axis, i.e. if either ð0Þ ¼ ðÞ ¼ #0 or ð0Þ ¼ ðÞ ¼ #0. If the ends of these intervals lie on pulses of different polarity then ð0Þ ðÞ ¼ #20 . Thus, one can write Cm ð j Þ ¼ #20 Pf ð0Þ ¼ #0 ; ðÞ ¼ #0 g þ #20 Pf ð0Þ ¼ #0 ; ðÞ ¼ #0 g #20 Pf ð0Þ ¼ #0 ; ðÞ ¼ #0 g #20 Pf ð0Þ ¼ #0 ; ðÞ ¼ #0 g ¼ #20 ½p1 11 ðmÞ þ p2 22 ðmÞ #20 ½p1 12 ðmÞ þ p2 21 ðmÞ

ð5:55Þ

DEFINITIONS

Figure 5.3

201

Quasi-random telegraph signal.

Since for a stationary case p1 ¼ p2 ¼ 0:5 and, as has been shown in the example above, 1 þ ðp qÞm ; 11 ðmÞ ¼ 22 ðmÞ ¼ 2

1 ðp qÞm 12 ðmÞ ¼ 21 ðmÞ ¼ 2

ð5:56Þ

then the expression for the covariance function becomes Cm ð j Þ ¼ #20 ðp qÞm

ð5:57Þ

If the values of and m are such that ðm 1ÞT0 m T0 then there are two possibilities: 1. If < mT0 , then the interval contains ðm 1Þ points where the system is able to change its state. As a result, the covariance function for these values of and is Cm1 ð j Þ ¼ #20 ðp qÞm1

ð5:58Þ

2. If > mT0 then there are m points where the system is able to change its state. As a result, the correlation function for these values of and is Cm ð j Þ ¼ #20 ðp qÞm

ð5:59Þ

Since the random variable is uniformly distributed on the interval ½0; T0 , its probability density function is a constant p ðÞ ¼

1 ; T0

2 ½0; T0

ð5:60Þ

202


and the unconditional covariance function CðÞ can be obtained by averaging the conditional correlation function Cð j Þ ð

ð T0 ð T0 1 mT0 CðÞ ¼ Cð j Þp ðÞd ¼ Cm1 ð j Þd þ Cm ð j Þd T0 0 0 mT0

m1 2 2 ¼ #0 m þ #0 ðm 1Þ ðp qÞm ð5:61Þ ðp qÞ T0 T0 Given that CðÞ is an even function of , the final expression for the correlation function takes the following form

jj m1 2 2 jj þ #0 ðm 1Þ ðp qÞm ð5:62Þ CðÞ ¼ #0 m ðp qÞ T0 T0 The power spectral density of the quasi-random telegraph signal can be then found by applying the Wiener–Khinchin theorem ð1 CðÞ cos !d Sð!Þ ¼ 2 0

ð mT0 1 X m1 2 2#0 ðp qÞ m ðm 1Þ ð p qÞ cos !d ¼ þ T0 T0 ðm1ÞT0 m¼0 12 0 !T0 sin B #20 t0 ½1 ðp qÞ2 2 C C B ð5:63Þ 1 2ðp qÞ cos !T0 þ ðp qÞ2 @ !T0 A 2 For a particular case of p ¼ q ¼ 0:5 this expression can be further simplified to produce 12 0 8 !T0 sin jj < 2 B 2 C #0 1 ; jj < T0 2 B C ; Sð!Þ ¼ #0 T0 @ ð5:64Þ kðÞ ¼ T0 !T0 A : 0; jj > T0 2 It is important to note that if is deterministic and equal to zero, ¼ 0, then the quasitelegraph signal becomes non-stationary. The easiest way to show this is to consider a case of p ¼ q ¼ 0:5. For this signal the covariance function is4 Cðt1 ; t2 Þ ¼ h ðt1 Þ ðt2 Þi ¼ #20 Pf ðt1 Þ ¼ #0 ; ðt2 Þ ¼ #0 g þ #20 Pf ðt1 Þ ¼ #0 ; ðt2 Þ ¼ #0 g #20 Pf ðt1 Þ ¼ #0 ; ðt2 Þ ¼ #0 g #20 Pf ðt1 Þ ¼ #0 ; ðt2 Þ ¼ #0 g ( 2 #0 ðm 1ÞT0 < t1 ; t2 < mT0 ¼ 0; otherwise

4

This signal still has zero mean at any moment of time.

ð5:65Þ

203

DEFINITIONS

Thus, the convariance function depends separately on the values of t1 and t2 and the process

ðtÞ is non-stationary. 5.1.2

Markov Sequences

A Markov sequence is a sequence of random variables m ¼ ðtm Þ which attains a continuum of values at discrete moments of time t1 < t2 < < tM . A sampled continuous Markov random process is an example of a Markov sequence. A Markov sequence is a good way to describe coherent digital communication systems, since they change their state only at certain fixed moments of time. At the same time a number of possible states is so large that the description in terms of a discrete number of states is rather complicated. A sequence of random variables m is called a Markov sequence if for any m the conditional distribution function (or, equivalently, the conditional probability density) satisfies the following conditions Pm ðxm j x1 ; . . . ; xm1 Þ ¼ Pm ðxm j xm1 Þ

ð5:66Þ

m ðxm j x1 ; . . . ; xm1 Þ ¼ m ðxm j xm1 Þ

ð5:67Þ

These relations reflect the fact that, for a Markov sequence, conditional probabilities and distributions for the future moment of time tm depend only on the state of the system at the current moment of time tm1 , not on the past state of the system. It follows from eq. (5.67) that the joint probability density function pM ðx1 ; . . . ; xM Þ can be expressed in terms of one-step transitional probability densities m ðxm j xm1 Þ, m ¼ 2; . . . ; M and the probability of the initial state p1 ðx1 Þ: pm ðx1 ; . . . ; xm Þ ¼ p1 ðx1 Þ

m Y

ðx j x 1 Þ

ð5:68Þ

¼2

The last equation can also be considered as a definition of a Markov sequence, since it follows from eq. (5.68) that m ðxm j x1 ; . . . ; xm1 Þ ¼

pm ðx1 ; . . . ; xm Þ ¼ m ðxm j xm1 Þ pm1 ðx1 ; . . . ; xm1 Þ

ð5:69Þ

The following is a brief summary of properties of a Markov sequence. 1. If the state of a Markov sequence is known at the current moment of time, its future state does not depend on any past state. This means that if the state xn is known, then for any m > n > random variables xm and x are independent: pðxm ; x j xn Þ ¼ mn ðxm j xn Þn ðx j xn Þ

ð5:70Þ

Indeed, using eq. (5.67) one can write pðxm ; x j xn Þ ¼

pðx ; xn ; xm Þ pðx ; xn Þmn ðxm j xn Þ ¼ ¼ mn ðxm j xn Þn ðx j xn Þ pðxn Þ pðxn Þ

ð5:71Þ

A similar statement can be derived for a few moments in the past and in the future.

204


2. Any sub-sequence of a Markov sequence is also a Markov sequence, i.e. for a given tm and moments of time tm1 < tm2 < < tmn the following equation is valid: ðxmn j xm1 ; . . . ; xmn1 Þ ¼ ðxmn j xmn1 Þ

ð5:72Þ

3. A Markov sequence remains Markov if the time direction is reversed, i.e. ðxm j xmþ1 ; . . . ; xmþn Þ ¼ ðxm j xmþ1 Þ

ð5:73Þ

This can be proven by expanding eq. (5.73) using the Bayes formula and the property (5.67) ðxm j xmþ1 ; . . . ; xmþn Þ ¼ ¼

pðxm ; xmþ1 ; . . . ; xmþn Þ pðxmþ1 ; . . . ; xmþn Þ pðxm ; xmþ1 Þ pðxmþ2 ; . . . ; xmþn j xmþ1 Þ ¼ ðxm j xmþ1 Þ pðxmþ1 Þ pðxmþ2 ; . . . ; xmþn j xmþ1 Þ

ð5:74Þ

4. The transitional probability density satisfies the following equation ðxm j x Þ ¼

ð1 1

ðxm j xn Þðxn j x Þdxn ;

m>n>

ð5:75Þ

which is a particular case of the Kolmogorov–Chapman equation. Indeed, for any Markov sequence, the joint probability density at three moments of time can be expressed as pðx ; xn ; xm Þ ¼ pðx Þðxn j x Þðxm j xn Þ

ð5:76Þ

Integrating the latter over xn one obtains pðx ; xm Þ ¼

ð1 1

pðx ; xn ; xm Þdxn ¼ pðx Þ

ð1 1

ðxn j x Þðxm j xn Þdxn

ð5:77Þ

which is equivalent to eq. (5.75). A Markov sequence is called a homogeneous Markov sequence if the transitional probability density m ðxm j xm1 Þ does not depend on m. A Markov sequence is called a stationary Markov sequence if it is a homogeneous sequence and the probabilities of states do not depend on the instant of time pm ðxm Þ ¼ pðxm Þ ¼ const:

ð5:78Þ

Similar to a Markov chain of n-th order, one can consider a Markov sequence of order n 1, defined as m ðxm j x1 ; . . . ; xm1 Þ ¼

m ðxm j xmn ; . . . ; xm1 Þ; m ðxm j x1 ; . . . ; xm1 Þ;

m>n 2mn

ð5:79Þ

205

DEFINITIONS

For n ¼ 1 the Markov chain of order n coincides with a simple Markov chain. For m n one has to formally assume n ¼ m 1 ðm 2Þ. It is possible to introduce an n-dimensional random vector nm ¼ fi; m ; i ¼ 1; . . . ; ng

ð5:80Þ

with components on two sequential steps defined as i; m ¼ iþ1; m1 ¼ mnþ1 ; 1; m1 ¼ mn ðn < mÞ

ð5:81Þ

It is easy to see that this new vector is a simple Markov vector. Its transitional probability density can be expressed as m ðXm j X m1 Þ ¼ m ðxn;m j x1; m1 ; . . . ; xn;m1 Þ

n1 Y

ðxi; m xiþ1; m1 Þ

ð5:82Þ

i¼1

The probability density pm ðxmnþ1 ; . . . ; xm Þ obeys the following recurrence equations, similar to eq. (5.13): pm ðxmnþ1 ; . . . ; xm Þ ¼

ð1 1

m ðxm j xmn ; . . . ; xm1 Þpm1 ðxmn ; . . . ; xm1 Þdxmn

ð5:83Þ

where m > n. On the steps 2 m n the following equality holds for a Markov vector pm ðx2 ; . . . ; xm Þ ¼

ð1 1

m ðxm j x1 ; . . . ; xm1 Þpm1 ðx1 ; . . . ; xm1 Þdx1

ð5:84Þ

In the case of a simple scalar Markov chain eq. (5.84) becomes pm ðxm Þ ¼

ð1 1

m ðxm j xm1 Þpm1 ðxm1 Þdxm1

Similarly, for a vector case eq. (5.84) becomes ð1 pm ðXm Þ ¼ m ðX m j X m1 Þpm1 ðXm1 ÞdX m1 1

ð5:85Þ

ð5:86Þ

where m ¼ 2; . . . ; M and the integration is performed over all the components xi; m1 ði ¼ 1; . . . ; rÞ. Proofs of eqs. (5.82)–(5.86) are similar to those for the case of a Markov chain. Example. Let independent random variables 1 ; 2 ; . . . ; m ; . . . have probability densities pm ðxÞ, respectively. It is possible to prove that the following sequence 1 ¼ 1 ; is a Markov sequence.

2 ¼ 1 þ 2 . . . ;

m ¼ 1 þ 2 þ þ m ; . . .

ð5:87Þ

206


Since differences kþ1 k ¼ k are independent, one can obtain an expression of joint probability density function of the first m random variables k as pðx1 ; . . . ; xm Þ ¼ p1 ðx1 Þp2 ðx2 x1 Þ . . . pm ðxm xm1 Þ

ð5:88Þ

Using this relation, the expression for transitional probability becomes m ðxm j x1 ; . . . ; xm1 Þ ¼

pðx1 ; . . . ; xm Þ ¼ pm ðxm xm1 Þ pðx1 ; . . . ; xm1 Þ

ð5:89Þ

Since this transitional density does not depend on x1 ; . . . ; xm2 the considered sequence is a Markov sequence. If, in addition, all the random variables m have zero mean, hm i ¼ 0, then hm i ¼ 0. Furthermore, given that m ¼ m þ m1 , then hm j m1 i ¼ hm j m1 i þ hm1 j m1 i ¼ m1

ð5:90Þ

since m does not depend on m1 and hm i ¼ 0. Finally, using a Markov property of the sequence it is possible to conclude that hxm j x1 ; . . . ; xm1 i ¼ xm1

ð5:91Þ

A sequence of random variables which has the property described by eq. (5.91) is called a martingale [8]. Example. Let 1 ; . . . ; M be a sequence of independent random variables, described by the probability density function pm ðxm Þ; m ¼ 1; . . . ; M. A new sequence fm ; m ¼ 1; . . . ; Mg is defined such that it satisfies the following equation m X ¼ mn

am ¼ m ;

m>n

ð5:92Þ

where fam g are known coefficients, a0 6¼ 0 and an 6¼ 0. It is clear that the sequence m is a sequence of independent variables if n ¼ 0 and Markov for n ¼ 1. For n > 1 one has a case of a Markov chain of order n, which is described by the transitional probability density ! m X m ðxm j xmn ; . . . ; xm1 ¼ a0 pm ð5:93Þ am x ¼ mn

A conditional expectation is then " hm j 1 ; . . . ; m1 i ¼ hm j mn ; . . . ; m1 i ¼ a1 hxm i 0

m X ¼ mn

# am

ð5:94Þ

It can be seen from this equation that the conditional mean depends on n previous values of the process mn ; . . . ; m1 .

207

DEFINITIONS

5.1.3

A Discrete Markov Process

Let process ðtÞ attain values from a finite set #1 ; . . . ; #K and transitions from state to a state appear at random moments of time. The transitional probability ij ðt0 ; tÞ ¼ Pf ðtÞ ¼ #j j ð ðt0 Þ ¼ #i Þg;

t > t0

ð5:95Þ

describes the probability that the system is in state #j at the moment of time t, given that the system has been in the state #i at the moment of time t0 . It is clear that K X

ij ðt0 ; tÞ ¼ 1;

ij ðt0 ; tÞ 0

ð5:96Þ

j¼1

and ij ðt0 ; t0 Þ ¼ ij ¼

1; 0;

i¼j i 6¼ j

ð5:97Þ

The main problem in studying Markov processes is to calculate transitional probabilities and unconditional probabilities of the states given the initial distribution of the states and one-step transitional probability densities. The same considerations used to derive eq. (5.24) can be used to obtain the Kolmogorov– Chapman equation in the following form ij ðt0 ; t þ tÞ ¼

K X

ik ðt0 ; tÞkj ðt; t þ tÞ;

t > t0 ;

t > 0

ð5:98Þ

k¼1

In the case of discrete Markov processes and a small time interval t, transitional probabilities can be written as kk ðt; t þ tÞ ¼ Pf ðt þ tÞ ¼ #k j ðtÞ ¼ #k g ¼ 1 akk ðtÞt þ oðtÞ kj ðt; t þ tÞ ¼ Pf ðt þ tÞ ¼ #j j ðtÞ ¼ #k g ¼ akj ðtÞt þ oðtÞ

ð5:99Þ

where oðtÞ are terms of the higher order with respect to t. It can be seen from eq. (5.99) that it is a characteristic property of a Markov process that for small increments of time t, the probability of kk remaining in the current state is much higher than the probability of kj changing state. The first of the two equations (5.99) is consistent with eq. (5.97) and reflects two facts: (1) for t ¼ 0 the system is certainly in the state #k ; (2) the probability of a transition from the state #k to any other state in the future depends on the moment of time to be considered, and, for a small interval t, these probabilities are proportional to the duration of this time interval. Since the probability of transition from one state to another is non-negative, then akj ðtÞ 0. Furthermore, it follows from eq. (5.96) that the following normalization condition must also be satisfied X akk ðtÞ ¼ akj ðtÞ 0; akj ðtÞ 0 ð5:100Þ j6¼k

208


Substituting eq. (5.99) into the right-hand side of eq. (5.98) one obtains a system of differential equations for transitional probabilities X @ ij ðt0 ; tÞ ¼ ajj ðtÞij ðt0 ; tÞ þ akj ðtÞik ðt0 ; tÞ @t k 6¼ j

ð5:101Þ

where akj ðtÞ satisfies the condition (5.100). The solution of this system of differential equations with initial conditions (5.97) provides the dependence of the transitional probabilities on time t. If the number of possible states of the system is finite, then for any continuous functions aij ðtÞ, satisfying the conditions (5.100), the system of differential equations (5.101) has a unique non-negative solution which defines a discrete Markov process. Similarly to a Markov sequence, a Markov discrete process remains Markov when the direction of time is reversed. It can be shown that in addition to the system of equations (5.101), an alternative system of equations can be written. In order to derive this new system of equations one ought to consider a point in time which is close to the initial moment of time t0 and rewrite eq. (5.98) in the following form X ij ðt0 ; tÞ ¼ ik ðt0 ; t0 þ tÞkj ðt0 þ t; tÞ; t > t0 þ t > t0 ð5:102Þ k

The next step is to use the approximation of probabilities for a small time interval t, similar to eq. (5.99) ii ðt0 ; t0 þ tÞ ¼ 1 aii ðt0 Þt þ oðtÞ

ð5:103Þ

Substituting approximation (5.103) into eq. (5.102) and taking a limit for t ! 0 one obtains the so-called reversed5 Kolmogorov equation for a discrete Markov process X @ ij ðt0 ; tÞ ¼ aii ðt0 Þij ðt0 ; tÞ aik ðt0 Þkj ðt0 ; tÞ; @t0 k 6¼ j

t > t0

ð5:104Þ

Equation (5.101) is satisfied not only by transitional probabilities ij ðt0 ; tÞ but also by unconditional probabilities pj ðtÞ of states at any moment of time t. Indeed, once the initial probabilities pj ð0Þ ¼ pj ðt0 Þ are known then, for a discrete Markov process, the following equations are satisfied X X pj ðtÞ ¼ pi ðt0 Þij ðt0 ; tÞ; pj ðt þ tÞ ¼ pi ðtÞij ðt; t þ tÞ ð5:105Þ i

i

After multiplication of both sides of eq. (5.101) by pi ðt0 Þ and summing over all possible states one obtains X @ pj ðtÞ ¼ ajj ðtÞpj ðtÞ þ akj ðtÞpk ðtÞ @t k 6¼ j 5

Equation (5.101) is called the ‘‘forward’’ Kolmogorov equation.

ð5:106Þ

209

DEFINITIONS

A unique solution of eq. (5.106) is obtained using the initial conditions pj ðtÞ ¼ pj ðt0 Þ ¼ p0j

ð5:107Þ

A discrete Markov process is called a homogeneous Markov process if its transitional probabilities ij ðt0 ; tÞ depend only on time difference ¼ t t0 , i.e. ij ðt0 ; tÞ ¼ ij ðÞ;

¼ t t0

ð5:108Þ

It follows from eq. (5.106) that, for a homogeneous Markov process, coefficients akj ðtÞ ¼ akj are constants (as functions of time) and eqs. (5.99)–(5.103) are simplified to a system of linear differential equations with constant coefficients X d ij ðÞ ¼ ajj ij ðÞ þ akj ik ðÞ d k 6¼ j ij ð0Þ ¼ ij

ð5:109Þ ð5:110Þ

A homogeneous discrete Markov process is called ergodic if there is a limit lim ij ðÞ ¼ pj

ð5:111Þ

!1

which does not depend on the initial state #i . Probabilities pj define an equilibrium, or final, distribution of the states of the system. These probabilities can be defined from algebraic equations, which are the limiting case of eq. (5.109) ajj pj þ

X k 6¼ j

akj pk ¼ 0;

K X

pk ¼ 1

ð5:112Þ

k¼1

The concept of the final probabilities is illustrated by the following examples. Example: A discrete binary Markov process (signal). Let ðtÞ be a discrete Markov process with two possible states #1 ¼ 1 and #2 ¼ 1. The probability of transition from the state #1 ¼ 1 to the state #2 ¼ 1 for a short time interval t is equal to t and the probability of the transition from the state #2 ¼ 1 to the state #1 ¼ 1 is equal to t. Initially, the distribution of these two states is described by the following probabilities ¼ Pf ðt0 Þ ¼ #1 g and ¼ Pf ðt0 Þ ¼ #2 g ¼ 1 . It is required to determine the probabilities of transitions ij ðt0 ; tÞ, the final stationary probabilities of each state, as well as the mean values of the process ðtÞ and its correlation function. It is worth noting that if a binary Markov process ðtÞ has two states #1 ¼ ’1 and #2 ¼ ’2 , then it could be represented using a linear transformation of a binary process ðtÞ with two states 1: ðtÞ ¼

’1 þ ’2 ’1 ’2 þ

ðtÞ 2 2

ð5:113Þ

210


Thus, the statistical properties of the process ðtÞ can be easily expressed in terms of statistics of the process ðtÞ. It follows from the definition of ðtÞ that the following are the values of the constants in eq. (5.109): a12 ¼ , a22 ¼ . Since all the coefficients aij are constants, the process ðtÞ is homogeneous6. The next step is to rewrite eq. (5.109) as @ i1 ðt0 ; tÞ @t

9 > ¼ i1 ðt0 ; tÞ þ i2 ðt0 ; tÞ > =

@ i2 ðt0 ; tÞ @t

> > ¼ i2 ðt0 ; tÞ þ i1 ðt0 ; tÞ ;

;

i ¼ 1; 2

ð5:114Þ

Using property (5.96) one obtains 12 ðt0 ; tÞ ¼ 1 11 ðt0 ; tÞ, which, in turn, implies that the first of the eqs. (5.114) can be rewritten as @ 11 ðt0 ; tÞ ¼ ð þ Þ11 ðt0 ; tÞ þ ; @t

t > t0

ð5:115Þ

A general solution of this equation which satisfies the condition 11 ðt0 ; t0 Þ is given by 11 ðt0 ; tÞ ¼

ðt

exp½ð þ Þðt sÞds þ exp½ð þ Þðt t0 Þ

t0

¼

þ exp½ð þ Þ þ þ

ð5:116Þ

In the very same way a solution can be obtained for all transitional probabilities. Here, only the final results are presented: 11 ðÞ ¼

þ exp½ð þ Þ; 12 ðÞ ¼ ½1 exp½ð þ Þ þ þ þ

ð5:117Þ

22 ðÞ ¼

þ exp½ð þ Þ; 21 ðÞ ¼ ½1 exp½ð þ Þ þ þ þ

ð5:118Þ

These equations indicate that the process considered is homogeneous since all the transitional probabilities depend only on the time difference ¼ t t0 . In addition, this process is ergodic since the transitional probabilities approach a limit at t ! 1 thus defining stationary probabilities p1 ¼

6

See derivations below.

; þ

p2 ¼

þ

ð5:119Þ

211

DEFINITIONS

Given an initial distribution of the probabilities and the transitional probabilities it is possible to find unconditional probabilities of each state: þ exp½ð þ Þs p1 ðt0 þ sÞ ¼ 11 ðsÞ þ ð1 Þ21 ðsÞ ¼ þ þ ð5:120Þ p2 ðt0 þ sÞ ¼ 22 ðsÞ þ ð1 Þ12 ðsÞ ¼ exp½ð þ Þs þ þ The average h ðtÞi of the process ðtÞ can be evaluated directly from the definition of average and the expression (5.120) for probabilities of the states at any moment of time m ðtÞ ¼ h ðtÞi ¼ 1 p1 ðtÞ þ ð1Þ p2 ðtÞ ¼ 11 ðÞ þ ð1 Þ21 ðÞ þ2 22 ðÞ ð1 Þ12 ðÞ ¼ exp½ð þ Þ þ þ

ð5:121Þ

Here ¼ t t0 . The next step is to find the covariance function C ðs; Þ ¼ h ðt0 þ sÞ ðt0 þ s þ Þi h ðt0 þ sÞih ðt0 þ s þ Þi;

s > 0; > 0

ð5:122Þ

Once again, using a definition of the average one obtains h ðt0 þ sÞ ðt0 þ s þ Þi ¼ 1 p1 ðt0 þ sÞ11 ðÞ þ p2 ðt0 þ sÞ22 ðÞ þ ð1Þ p1 ðt0 þ sÞ12 ðÞ p2 ðt0 þ sÞ21 ðÞ;

s > 0; > 0 ð5:123Þ

Finally, using eqs. (5.117), (5.118) and (5.123) the expression for the covariance function is reduced to (

4 þ exp½ð þ Þs 4 C ðs; Þ ¼ þ þ ð þ Þ2 ð5:124Þ

exp½ð þ Þs exp½ð þ Þ þ If the process starts from the stationary distribution of the states, i.e. ¼ p1 ¼ =ð þ Þ, then the process is a stationary process from the initial moment of time t0 . Its mean does not depend on time m ðtÞ ¼

¼ const: þ

ð5:125Þ

and its covariance function depends only on C ðÞ ¼

4 ð þ Þ2

exp½ð þ Þ;

>0

ð5:126Þ

212


Finally, using symmetry of the covariance function of a stationary process one obtains C ðÞ ¼

4 ð þ Þ2

exp½ð þ Þjj

ð5:127Þ

If ¼ the binary Markov process is called a symmetric binary Markov process or a random telegraph signal. Using this condition, eqs. (5.117)–(5.127) can be significantly simplified to produce 1 11 ðÞ ¼ 22 ðÞ ¼ ð1 þ e2 Þ; 2 1 p1 ¼ p2 ¼ ; m ¼ 0; K ðÞ ¼ exp½2 2 1 12 ðÞ ¼ 21 ðÞ ¼ ð1 e2 Þ 2

5.1.4

ð5:128Þ

Continuous Markov Processes

In this section a continuous Markov process is the subject of discussion. In this case the process can attain a continuum of values and is a function of a continuous parameter—time t. Let ðt0 Þ; ðt1 Þ; . . . ; ðtm1 Þ, and ðtm Þ be values of the random process ðtÞ at the moments of time t0 < t1 < < tm1 < tm . The process ðtÞ is called a continuous Markov process if its conditional transitional probability density m ðxm ; tm j xm1 ; tm1 ; xm2 ; tm2 ; . . . ; x0 ; t0 Þ ¼

pmþ1 ðx0 ; x1 ; . . . ; xm ; t0 ; t1 ; . . . ; tm Þ pm ðx0 ; x1 ; . . . ; xm1 ; t0 ; t1 ; . . . ; tm1 Þ ð5:129Þ

depends only on the values of the process at the last moment of time ðtm1 Þ, i.e. m ðxm ; tm j xm1 ; tm1 ; xm2 ; tm2 ; . . . ; x0 ; t0 Þ ¼ m ðxm ; tm j xm1 ; tm1 Þ;

m>1

ð5:130Þ

As usual @m Probfðtm1 Þ < xm1 ; . . . ; ðt0 Þ < x0 g pm ðx0 ; x1 ; . . . ; xm1 ; t0 ; t1 ; . . . ; tm1 Þ ¼ @xm1 . . . @x0 ð5:131Þ In other words, the future of a Markov process does not depend on its past if its value at the present is known. It follows from eqs. (5.129) and (5.130) that pmþ1 ðx0 ; x1 ; . . . ; xm ; t0 ; t1 ; . . . ; tm Þ ¼ m ðxm ; tm j xm1 ; tm1 Þpm ðx0 ; x1 ; . . . ; xm1 ; t0 ; t1 ; . . . ; tm1 Þ ð5:132Þ

213

DEFINITIONS

Integration of the last equation over x0 produces pm ðx1 ; . . . ; xm ; t1 ; . . . ; tm Þ ¼ m ðxm ; tm j xm1 ; tm1 Þpm1 ðx1 ; . . . ; xm1 ; t1 ; . . . ; tm1 Þ ð5:133Þ This indicates that for any m > 1, the expression for the joint probability function contains the same transitional probability density ðx; t j x0 ; t0 Þ; t > t0, (from the state x0 to the state x). Sequential application of integration to eq. (5.130) leads to the following important representation of a joint probability density function, considered at any m þ 1 moments of time tm < tm1 < < t1 < t0 pmþ1 ðx0 ; x1 ; . . . ; xm ; t0 ; t1 ; . . . ; tm Þ ¼

m 1 Y

ðxmk ; tmk j xmk1 ; tmk1 Þpðx0 ; t0 Þ

ð5:134Þ

k¼0

In particular p2 ðx0 ; x1 ; t0 ; t1 Þ ¼ ðx1 ; t1 j x0 ; t0 Þpðx0 ; t0 Þ

ð5:135Þ

This shows that a complete description of a continuous Markov process can be achieved if its marginal and transitional probability densities are known. This explains why Markov processes are so important in applications—a limited amount of information (only second order joint probability densities) is needed to obtain a complete description. Using expression (5.130) and the Bayes formula one can show that a continuous valued Markov process remains Markov if time is reversed. In other words ðx0 ; t0 j x1 ; t1 ; x2 ; t2 ; . . . ; xm ; tm Þ ¼ ðx0 ; t0 j x1 ; t1 Þ;

t0 < t1 < < tm

ð5:136Þ

Using the definition of the transitional probability density ðx; t j x0 ; t0 Þ one can obtain the following properties of the transitional density. 1. The transitional probability density is non-negative and is normalized to one after integration over the destination state x: ð1 ðx; t j x0 ; t0 Þdx ¼ 1 ð5:137Þ ðx; t j x0 ; t0 Þ 0 and 1

2. The transitional probability density reduces to a delta function for coinciding moments of time ðx; t j x0 ; tÞ ¼ ðx x0 Þ

ð5:138Þ

3. The transitional probability density satisfies the Kolmogorov–Chapman equation [1,2] ðx; t j x0 ; t0 Þ ¼

ð1 1

ðx; t j x0 ; t0 Þðx0 ; t0 j x0 ; t0 Þd0

ð5:139Þ

This equation can be easily derived using eq. (5.134) for m ¼ 2 p3 ðx0 ; x1 ; x; t0 ; t1 ; tÞ ¼ ðx; t j x1 ; t1 Þðx1 ; t1 j x0 ; t0 Þpðx0 ; t0 Þ Integration over x1 produces the required relation (5.139).

ð5:140Þ

214


A continuous Markov process is called a homogeneous Markov process if its transitional probability density depends only on a difference of time instances, i.e. ðx; t j x1 ; t1 Þ ¼ ðx; x1 j t t1 Þ ¼ ðx; x1 j Þ;

¼ t t1

ð5:141Þ

Furthermore, if ðx; x1 j Þ approaches a limit pst ðxÞ when ! 1 lim ðx; x1 j Þ ¼ pst ðÞ

!1

ð5:142Þ

and this limit is independent of the initial state x0 then this Markov process is called ergodic. It is worth noting that the Kolmogorov–Chapman equation in the form (5.139) describes all types of Markov process considered above if the transitional density ðx; t j x0 ; t0 Þ can be a generalized function and the parameter t allows change on a discrete or continuous set. For example, for a Markov chain one can represent the transitional density as ðx; t j x0 ; t0 Þ ¼ ðx; kt j x0 ; ðk 1ÞtÞ ¼

M X M X

Pmn ðx xm Þðx xn Þ

ð5:143Þ

m¼1 n¼1

5.1.5

Differential Form of the Kolmogorov–Chapman Equation

The Kolmogorov–Chapman equation (5.139) is an integral equation and under some conditions it can be converted to a differential equation with a more revealing structure. The following section follows closely Section 3.4 in [1]. It is assumed for now that for all " > 0 1. The following limit exists uniformly in x; x0 , and t such that jx x0 j > " ðx; t þ t; x0 ; tÞ ¼ Wðx j x0 ; tÞ 0 t!0 t lim

ð5:144Þ

2. The following two limits are uniform in x0 , ", and t 1 lim t!0 t 1 lim t!0 t

ð ð

jxx0 j" t

ð jxx0 j>"

Wðx j x0 ; tÞdx

ð5:148Þ

is the probability that a jump in magnitude of at least " may appear at the moment of time t. The former is called a diffusion Markov process and the latter is a continuous process with jumps. If a random process ðtÞ is treated as a coordinate of some imaginary particle, the quantity K1 ðx; tÞ describes the average value of the velocity of the process at the moment of time t. Large deviations of the particle position are excluded in eq. (5.145) to avoid contributions from jumps: the uniformity in " would insure that arbitrary small deviations can be considered, and, thus, contributions of larger velocities can be neglected. Using mechanical analogy the coefficient K1 ðx; tÞ is often called the drift of the Markov process and it is attributed to an external force (potential). Finally, the coefficient K2 ðx; tÞ describes the average energy of fluctuations of the particle around a fixed position x. Such deviations could be caused by external noise and in general do not pull the particle in a certain direction as an external force. It is common to call K2 ðxÞ the diffusion to reflect the fact that a particle can randomly change its position without external force. Thus, the motion of the particle can be thought of as a superposition of the drift, diffusion, and jumps. Let f ðzÞ be a deterministic function which is at least twice differentiable and ðtÞ be a continuous Markov process. In this case the rate of change in the average value of f ðÞ, given that ðt0 Þ ¼ y, is defined by the following equation @ @ h f ðÞ j yi ¼ @t @t

ð1

Ð1

f ðxÞ½ðx; t þ t j y; t0 Þ ðx; t j y; t0 Þdx t!0 t Ð1 Ð1 f ðnÞ 1 ½ðx; t þ t j z; tÞðz; t j y; t0 Þdzdx 1 f ðzÞðz; t j y; t0 Þdz t ð5:149Þ

f ðxÞðx; t j y; t0 Þdx ¼ lim

1 Ð1 1

¼ lim

t!0

1

If " > 0 is fixed then the region of integration can be divided into two intervals: jx zj " and jx zj < ", while the function f ðxÞ can be represented by its Taylor series d ðx zÞ2 d2 f ðzÞ þ jx zj2 Rðx; zÞ f ðxÞ ¼ f ðzÞ þ ðx zÞ f ðzÞ þ 2 dz 2 dz

ð5:150Þ

where limjxzj!0 jRðx; zÞ ¼ 0. As a result, the Kolmogorov–Chapman equation (5.149) can be written as [2]

2 @f 1 2@ f ðx zÞ þ ðx zÞ ðx; t þ t j z; tÞðz; t j y; t0 Þdxdz 2 @z 2 @z jxzj 0. The probability for this process to remain in the original state during the same time interval is then 1 t þ oðtÞ. It is also assumed that the probability of more than one change of state during an arbitrary short interval of time is of the order oðtÞ. Mathematically speaking Probf ðt þ tÞ ¼ j j ðtÞ ¼ j 1g ¼ t þ oðtÞ Probf ðt þ tÞ ¼ j j ðtÞ ¼ jg ¼ 1 t oðtÞ

ð5:176Þ

It is also reasonable to assume that the initial state of the process is zero, i.e. p0j

pj ðt0 Þ ¼ Probf ðt0 Þ ¼ jg ¼

1; 0;

j¼0 j ¼ 1; 2; . . .

ð5:177Þ

The first task is to calculate unconditional probabilities of each state, pj ðtÞ ¼ Probf ðtÞ ¼ jg. This can be accomplished by using eq. (5.106) with ajj ¼ and aj; j1 ¼ , thus leading to @ pj ðtÞ ¼ pj ðtÞ þ pj1 ðtÞ; j 1 @t @ p0 ðtÞ ¼ p0 ðtÞ; j¼0 @t

ð5:178Þ

Once again, a generation function can be used to solve this equation. Indeed, let

Gðs; tÞ ¼

1 X j¼0

sj pj ðtÞ

ð5:179Þ

222


be the generating function of the Poisson process. Multiplying both sides of eq. (5.178) by sj and summing over j one obtains 1 1 X X @ @ j @ p0 ðtÞ þ s pj ðtÞ ¼ Gðs; tÞ ¼ p0 ðtÞ ðsj pj ðtÞ sj pj1 ðtÞÞ @t @t @t j¼1 j¼1

ð5:180Þ

¼ ð1 þ sÞGðs; tÞ The initial condition can be obtained from eq. (5.177) as Gðs; 0Þ ¼

1 X

s j pj ð0Þ ¼ 1

ð5:181Þ

j¼0

The solution of eq. (5.180) with initial condition (5.181) is then Gðs; tÞ ¼ Gðs; 0Þ exp½ð1 þ sÞt ¼ exp½ð1 þ sÞt

ð5:182Þ

Expanding the latter into the Taylor series and equating coefficients with equal powers of s one obtains Gðs; tÞ ¼

1 X n¼0

s pn ðtÞ ¼ expðtÞ n

1 X ðstÞn n¼0

n!

¼ expðtÞ

1 X ðtÞn n¼0

n!

Sn

ð5:183Þ

and thus ðtÞn t pn ðtÞ ¼ e n!

ð5:184Þ

Since the parameter does not depend on time or the state, the Poisson process is homogeneous. Using eqs. (5.101) and (5.109) one obtains the following forward and backward differential equations for transitional probabilities d ij ðÞ ¼ ij ðÞ þ i; j1 ðÞ d d ij ðÞ ¼ ij ðÞ þ iþ1; j1 ðÞ d

ð5:185Þ ð5:186Þ

It can be shown that the solution of these equations with initial conditions ij ð0Þ ¼ ij is given by the following equation 8 ji > < ðÞ exp½; ij ðÞ ¼ ð j iÞ! > : 0;

ji j

ð5:187Þ

SOME IMPORTANT MARKOV RANDOM PROCESSES

223

It can be pointed out that, for a non-homogeneous Poisson process, parameter ¼ ðtÞ changes in time and the expression for probabilities of states becomes ð t n ðt

1 ðÞd exp ðÞd ð5:188Þ pn ðt0 ; tÞ ¼ n! t0 t0 5.2.2.2

A Birth Process

The next step is to consider a process ðtÞ which also attains only integer values 0; 1; 2; . . . but in contrast to the Poisson process, the probability of change of state during a short interval ðt; t þ tÞ depends on the current state. The process considered is known as a birth process. A birth process is defined as follows. If the system is at the state ðtÞ ¼ j; j ¼ 0; 1; 2; . . ., at a moment of time, then the probability of transition j ! j þ 1 during a short interval of time ðt; t þ tÞ is equal to j t þ oðtÞ; the probability of remaining in the same state is 1 j oðtÞ, where j > 0. The probability of transition to any other state is of the order oðtÞ, and thus is negligible. Applying eq. (5.104) to the birth process one obtains a system of linear differential equations d pj ðtÞ ¼ j pj ðtÞ þ j1 pj1 ðtÞ; dt d p0 ðtÞ ¼ 0 p0 ðtÞ dt

j ¼ 1; 2; . . . ð5:189Þ

The last equation must be supplemented by the initial conditions which have the following form p 0j ¼ pj ð0Þ ¼ j;1

ð5:190Þ

The simplest case of the birth process is the case where the rates j are proportional to the current state j: j ¼ j;

¼ const > 0;

j>1

ð5:191Þ

which reflects the fact that the probability of transition is proportional to the number of specimens in the total population.8 Under this assumption eq. (5.189) can be rewritten as d pj ðtÞ ¼ jpj ðtÞ þ ðj 1Þpj1 ðtÞ; dt

j1

ð5:192Þ

The last system of equations can be solved using sequential integration, thus leading to the following Yule–Farri distribution: ( pj ðtÞ ¼

et ð1 et Þ 0;

8

This is approximately true for a case of newborn species [10]

j1

;

j1 j¼0

ð5:193Þ

224


The mean value and the variance of this process exponentially increase in time: m ðtÞ ¼ et ;

2 ðtÞ ¼ et ðet 1Þ

ð5:194Þ

Since coefficients j do not explicitly depend on time, the process itself is a homogeneous process, forward and backward Kolmogorov equations (5.109) and (5.110) can be thus written as d ij ðÞ ¼ j ij ðÞ þ j1 i; j1 ðÞ d d ij ðÞ ¼ i ij ðÞ þ j iþ1; j ðÞ d

ð5:195Þ

It can be shown [3] that the solution of these equations is given by ij ðÞ ¼

5.2.2.3

j1 i Cj1 e ð1 e Þj1 ; 0;

ji j

ð5:196Þ

A Death Process

A death process ðtÞ attains only integer values and is a non-increasing process and is defined by the following conditions: 1. At the initial moment of time a system under consideration is in a state j0 1, i.e.

ð0Þ ¼ j0 . 2. If in the moment of time t the system resides at a state j, i.e. ðtÞ ¼ j, then the probability of transition j ! j 1 during a short interval of time is equal to j t þ oðtÞ and the probability of remaining at the same state is 1 j t oðtÞ. 3. The probabilities of other transitions are of order oðtÞ2 or less. Similar to the case of the birth process it is easy to obtain a differential equation for probabilities of each state d pj ðtÞ ¼ j pj ðtÞ þ jþ1 pjþ1 ðtÞ dt

ð5:197Þ

The solution of the death equation can be obtained in the very same way as the solution of the birth equation.

5.2.2.4

A Death and Birth Process

A birth and death process ðtÞ attains integer values and is allowed to increase or decrease its value according to the following rules: 1. If the system is at the state ðtÞ ¼ j at the moment of time t, then the probability of transition j ! j þ 1 during a short time interval ðt; t þ tÞ is equal to j t þ oðtÞ.

225

SOME IMPORTANT MARKOV RANDOM PROCESSES

2. If the system is at the state ðtÞ ¼ j at the moment of time t, then the probability of transition j ! j 1 during a short time interval ðt; t þ tÞ is equal to j t þ oðtÞ. 3. The probability of remaining in the current state ðtÞ ¼ j is 1 ðj þ j Þt oðtÞ. 4. The probability of change into an adjacent state is of an order of oðtÞ. 5. The state ðtÞ ¼ 0 is an absorbing state, i.e. if the system reaches this state it remains in this state forever. Once again resorting to eq. (5.104) one can obtain a system of differential equations describing a death and birth random process: d pj ðtÞ ¼ j1 pj1 ðtÞ ðj þ j Þpj ðtÞ þ jþ1 pjþ1 ðtÞ dt

ð5:198Þ

In general, for j ¼ 0 one can write d p0 ðtÞ ¼ 1 p1 ðtÞ dt

ð5:199Þ

Since j ¼ 0 is absorbing, only downward trajectories are allowed. It is assumed that at the initial moment of time the system resides in some non-zero initial state 0 < j0 < 1. As a result, the initial conditions are simply pj ð0Þ ¼ jj0 ¼

1; 0;

j ¼ j0 j 6¼ j0

ð5:200Þ

In turn, eq. (5.109) for transitional probabilities becomes d ij ðtÞ ¼ j1 i; i1 ðtÞ ðj þ j Þij ðtÞ þ jþ1 j; jþ1 ðtÞ dt

ð5:201Þ

In general, the solution of eqs. (5.199)–(5.201) is difficult to obtain. However, in a linear case j ¼ j and j ¼ j, ¼ const > 0, ð ¼ const > 0Þ and the initial condition ðtÞ ¼ j0 of the process can be treated analytically. It is shown in [3] that the probability of the states can be found as p0 ðtÞ ¼ ðtÞ pj ðtÞ ¼ ½1 ðtÞ½1 ðtÞ j1 ðtÞ; ij ðtÞ ¼

i X

j ¼ 1; 2; . . .

iþin1 Cni Cn1 ½ ðtÞin ½ðtÞ jn ½1 ðtÞ ðtÞn ;

ð5:202Þ ij

ð5:203Þ

n¼0

where ðtÞ ¼

ðeðÞt 1Þ ; eðÞt

ðtÞ ¼

ðeðÞt 1Þ eðÞt

ð5:204Þ

226


The mean and variance of the linear death and birth process are given by m ðtÞ ¼ exp½ð Þt;

2 ðtÞ ¼

þ ðÞ ðÞt e ½e 1

ð5:205Þ

In particular, if ¼ and ð0Þ ¼ j0 ¼ 1, the average growth rate is equal to zero, m ðtÞ ¼ 1, and the variance grows linearly, 2 ðtÞ ¼ 2t. It follows from eqs. (5.202) and (5.204) that the probability that the system resides at the final state ðtÞ ¼ 0 at the moment of time t is given by eðÞt 1 p0 ðtÞ ¼ ðÞt e

ð5:206Þ

This expression allows one to obtain the condition defining survivability of the system in the long run. Indeed, if < , then lim p0 ðtÞ ¼ 1;

then lim p0 ðtÞ ¼

t!1

;

>

ð5:208Þ

and the system would have a non-zero probability p ¼ ð Þ= to never reach the absorbing state. This result can be easily visualized if and represent the birth and death rate of the population . If the birth rate is less than the death rate, then, eventually, the population becomes extinct. If the birth rate exceeds the death rate then it is possible that the population would never be extinct. It is possible to show [3] that eqs. (5.202) and (5.203) remain valid for a more general case where coefficients and depend on time. In this case one has to use the following expressions for ðtÞ and ðtÞ ðtÞ ¼ 1

1 exp½ ðtÞ; "ðtÞ

ðtÞ ¼ 1

1 "ðtÞ

ð5:209Þ

where

ðtÞ ¼

ðt

ðt ½ðÞ ðÞd and "ðtÞ ¼ exp½ ðtÞ 1 þ ðÞ exp½ ðÞd

0

ð5:210Þ

0

For such non-homogeneous processes, the mean and variance can be expressed as [3] m ðtÞ ¼

exp½ ðtÞ; 2 ðtÞ

¼ exp½2 ðtÞ

ðt 0

½ðÞ þ ðÞ exp½ ðÞd

ð5:211Þ

THE FOKKER–PLANCK EQUATION

227

The probability p0 ðtÞ that a system reaches its absorbing state ðtÞ ¼ 0 is given by Ðt

p0 ðtÞ ¼

ðÞ exp½ ðÞd Ðt 1 þ 0 ðÞ exp½ ðÞd 0

ð5:212Þ

Ðt It approaches unity every time when the integral 0 ðÞ exp½ ðÞd diverges for t ! 1. All the results shown are valid for j0 ¼ 1. If j0 > 1 all equations for the mean value and the variance can be obtained by multiplying the right-hand side by j0. The probability of reaching the absorbing state at ðtÞ ¼ 0 is still one if < and equals ð=Þ j0 if > . If and depend on time, then the probability p0 ðtÞ is obtained by raising expression (5.212) to power j0.

5.3 5.3.1

THE FOKKER–PLANCK EQUATION Preliminary Remarks

The transitional probability density ðx; t j x0 ; t0 Þ; t > t0 of a continuous diffusion Markov process satisfies the following partial differential equations (PDE) @ @ 1 @2 ðx; t j x0 ; t0 Þ ¼ ½K1 ðx; tÞðx; t j x0 ; t0 Þ þ ½K2 ðx; tÞðx; t j x0 ; t0 Þ @t @x 2 @x2 @ ð@Þ 1 @2 ðx; t j x0 ; t0 Þ ¼ K1 ðx0 ; t0 Þ ðx; t j x0 ; t0 Þ þ K2 ðx0 ; t0 Þ 2 ðx; t j x0 ; t0 Þ @t0 @x0 2 @x0

ð5:213Þ ð5:214Þ

The first of these two equations is known as the Fokker–Planck (FPE), or direct, equation. The second equation is called the Kolmogorov equation or the backward equation. It is important to mention that the direct equation involves a time derivative with respect to t (forward time), while the second one involves a time derivative with respect to t0 (backward time). It will be shown later that these two equations form a conjugate pair. 5.3.2

Derivation of the Fokker–Planck Equation

The direct Fokker–Planck equation can be obtained directly from the differential Kolmogorov–Chapman equation (5.160) by setting wðx j y; tÞ ¼ 0. However, in this section an alternative proof is provided. It is assumed for a moment that the transitional probability density ðx; t j x0 ; t0 Þ has all the necessary derivatives needed to formally write eqs. (5.213) and (5.214). Both equations can be derived from the Smoluchowski equation (5.139) ðx; t j x0 ; t0 Þ ¼

ð1 1

ðx; t j x0 ; t0 Þðx0 ; t0 j x0 ; t0 Þdx0

ð5:215Þ

The direct equation can be obtained by choosing t0 close to t, while the reversed equation can be obtained by choosing t0 close to t0 .

228


In order to obtain the Fokker–Planck equation, the Smoluchowski equation (5.139) can be rewritten in the following form ðx; t þ t j x0 ; t0 Þ ¼

ð1 1

ðx; t þ t j x0 ; tÞðx0 ; t j x0 ; t0 Þdx0 ;

t þ t > t > t0 ð5:216Þ

where the time interval t is very short. Let ð ; t þ j x0 ; tÞ represent a conditional characteristic function of the random increment x ¼ x x0 over a short interval t given x0 . Using the definition of the characteristic function one can write 0

0

0

ð ; t þ j x ; tÞ ¼ hexp½j ðx x Þ j x ; ti ¼

ð1

0

1

e j ðxx Þ ðx; t þ t j x0 ; tÞdx ð5:217Þ

Using the inverse Fourier transform one obtains 1 ðx; t þ t j x ; tÞ ¼ 2 0

ð1

0

1

ej ðxx Þ ð ; t þ j x0 ; tÞd

ð5:218Þ

The conditional characteristic function can be expanded into the Taylor series as ð ; t þ j x0 ; tÞ ¼

1 X mn ðx0 ; tÞ n¼0

n!

ðj Þn

ð5:219Þ

where 0

n

mn ðx ; tÞ ¼ hfðt þ tÞ ðtÞg j ðtÞi ¼

ð1 1

ðx x0 Þn ðx; t þ t j x0 ; tÞdx

ð5:220Þ

are conditional moments of the increment x ¼ x x0 . Using eq. (5.220) in expression (5.219) one can derive that ðx; t þ t j x0 ; tÞ 1 ¼ 2

ð1

j ðxx Þ

e 1

0

1 X mn ðx0 ; tÞ n¼0

n!

! ð j Þ

n

ð 1 X mn ðx0 ; tÞ 1 1 j ðxx0 Þ d ¼ e ð j Þn d n! 2 1 n¼0

ð 1 1 0 X X mn ðx0 ; tÞ ð1Þn @ n 1 j ðxx0 Þ n mn ðx ; tÞ @ e d ¼ ð1Þ ðx x0 Þ ¼ n n n! n! @x 2 @ 1 n¼0 n¼0 ð5:221Þ Finally, substituting series (5.221) into the Smoluchowski equation (5.216) one obtains

ðx; t þ t j x0 ; t0 Þ ¼

1 X ð1Þn @ n ½mn ðx; tÞðx; t j x0 ; t0 Þ n! @xn n¼0

ð5:222Þ

229


or, equivalently, 1 X ð1Þn @ n ½mn ðx; tÞðx; t j x0 ; t0 Þ ðx; t þ t j x0 ; t0 Þ ðx; t j x0 ; t0 Þ ¼ n! @xn n¼1

ð5:223Þ

Dividing both parts by t and taking the limit when t ! 0 the following partial differential equation is obtained 1 X @ ð1Þn @ n ðx; t j x0 ; t0 Þ ¼ ½Kn ðx; tÞðx; t j x0 ; t0 Þ @t n! @xn n¼1

ð5:224Þ

Here 1 1 hfðt þ tÞ ðtÞgn jðtÞi ¼ lim Kn ð; tÞ ¼ lim t!0 t t!0 t

ð1 1

ðx x0 Þn ðx; t þ t j x0 ; tÞdx ð5:225Þ

It is worth noting here that since only the Chapman equation is used to derive eq. (5.224), it is valid for any Markov process as long as quantities Kn ðx; tÞ are finite. Most attention in this book is paid to Markov processes with finite coefficients K1 ð; tÞ and K2 ð; tÞ, and zero higher order coefficients Kn ðx; tÞ: Kn ðx; tÞ 6¼ 0;

n ¼ 1; 2;

Kn ðx; tÞ ¼ 0;

n>2

ð5:226Þ

It follows from eq. (5.225) that condition (5.226) describes the rate of decay in the probability of large deviations with decreasing t. Large deviations of the process ðtÞ are permitted, but they must appear in opposite directions with approximately the same probability. Thus, thepaverage increment of the random process over a short time interval ffiffiffiffiffiffi t is of the order of t [4]. Finite jumps of this process appear with zero probability; all the trajectories are continuous in a regular sense with probability one. It can be proven that if, for a Markov process with finite coefficients Kn ðx; tÞ < 1 for all n, some even coefficient is zero, i.e. K2n ðx; tÞ ¼ 0, then all the higher order coefficients n 3 are equal to zero. This statement is known as the Pawula theorem [2,11]. Thus one may just require K4 ðx; tÞ ¼ 0. Thus, for a Markov process with continuous range to be a continuous Markov process it is necessary and sufficient that Kn ðx; tÞ ¼ 0 for all n 3. For continuous Markov processes, the Fokker–Planck equation becomes @ @ 1 @2 ðx; t j x0 ; t0 Þ ¼ ½K1 ðx; tÞðx; t j x0 ; t0 Þ þ ½K2 ðx; tÞðx; t j x0 ; t0 Þ @t @x 2 @x2

ð5:227Þ

Historically, the Fokker–Planck equation in the form of eq. (5.227) was obtained to study Brownian motion and is sometimes called a diffusion equation [1,2]. The coefficient K1 ðx; tÞ describes the average value of the particle displacement and is also known as the drift. Likewise, K2 ðx; tÞ can be interpreted as a variance of a particle’s displacement from the average value and is called the diffusion coefficient.

230


Partial differential equation (5.227) is a linear partial differential equation of a parabolic type and a variety of methods can be used to obtain its solution. A few of these methods will be reviewed later. For a detailed analysis the interested reader can refer to [1,2]. Any solution ðx; t j x0 ; t0 Þ of the Fokker–Planck equation must satisfy the following conditions: a transitional probability density should be positive for all values of the arguments and be normalized to one after integration over the destination state, i.e. ð1 ðx; t j x0 ; t0 Þdx ¼ 1 ð5:228Þ ðx; t j x0 ; t0 Þ 0; 1

In addition, this density must be reduced to a function if t ¼ t0 , i.e. ðx; t0 j x0 ; t0 Þ ¼ ðx x0 Þ

ð5:229Þ

A solution of the Fokker–Planck equation for unbounded space x 2 ð1; 1Þ and the initial condition (5.229) is called the fundamental solution of the Fokker–Planck equation. If the initial value of the process x0 is not fixed but distributed according to probability density function pðx; t0 Þ ¼ p0 ðxÞ

ð5:230Þ

then this probability density must be considered as a part of the initial conditions. An unconditional marginal probability density function can be calculated in two ways. 1. Using the property of a joint probability function to normalize to a marginal probability density one readily obtains pðx; tÞ ¼

ð1 1

p2 ðx; t; x0 ; t0 Þdx0 ¼

ð1 1

pðx0 ; t0 Þðx; t j x0 ; t0 Þdx0

ð5:231Þ

This implies that in order to find the marginal probability density it is necessary to know the initial condition (5.229) and the fundamental solution ðx; t j x0 ; t0 Þ of the Fokker– Planck. 2. It is possible to calculate pðx; tÞ directly from the Fokker–Planck equation. Indeed, multiplying both sides of eq. (5.227) by pðx0 ; t0 Þ and integrating over x0 one obtains @ @ 1 @2 pðx; tÞ ¼ ½K1 ðx; tÞpðx; tÞ þ ½K2 ðx; tÞpðx; tÞ @t @x 2 @x2

ð5:232Þ

This indicates that the marginal probability density also satisfies the same Fokker–Planck equation as the transitional probability density. If the initial condition is expressed by eq. (5.229) then the marginal probability density coincides with the fundamental solution of the Fokker–Planck equation. This statement follows directly from eq. (5.231). Taking the latter property into account, as well as the fact that it is the marginal probability density which is usually of interest for applications, it is possible to concentrate on studying the Fokker– Planck equation in the form (5.232). This equation must be solved with initial conditions


231

(5.230). The solution must also satisfy the positivity and normalization conditions given by pðx; tÞ 0;

ð1 1

pðx; tÞdx ¼ 1

ð5:233Þ

Furthermore, if the covariance function C ðt; t0 Þ, t > t0 of the Markov random process ðtÞ is to be found, one has to find both the transitional probability density ðx; t j x0 ; t0 Þ and the marginal probability density pðx0 ; t0 Þ. Once this is accomplished, the desired covariance function can be found as C ðt; t0 Þ ¼ h½ðtÞ hðtÞi½ðt0 Þ hðt0 Þii ðð ¼ ½x m ðtÞ½x0 m ðt0 Þðx; t j x0 ; t0 Þpðx0 ; t0 Þdxdx0

5.3.3

ð5:234Þ

Boundary Conditions

In order to obtain the solution of the Fokker–Planck equation (5.232) with the initial conditions (5.230) one must also consider appropriate boundary conditions, which depend on a particular problem and can vary from case to case. In order to choose proper boundary conditions the following interpretation of the Fokker– Planck equation can be helpful. In particular, one can associate the value of probability density with some quantity proportional to a number of particles in a flow of an abstract substance [2]. In this case the probability density pðx; tÞ reflects concentration of this substance at the point x at the moment of time t. The flow (or current) of the probability Gðx; tÞ (flow of substance) along the axis x consists of two parts: a local average flow K1 ðx; tÞpðx; tÞ of particles moving in the field of a potential and a random (diffusion) flow 0:5 @t@ ½K2 ðx; tÞpðx; tÞ, i.e. Gðx; tÞ ¼ K1 ðx; tÞpðx; tÞ

1@ ½K2 ðx; tÞpðx; tÞ 2 @t

ð5:235Þ

Using the notion of probability current Gðx; tÞ, the Fokker–Planck equation can be rewritten as @ @ pðx; tÞ þ Gðx; tÞ ¼ 0 @t @x

ð5:236Þ

which can be now interpreted as a law of conservation of the total probability. The conservation equation (5.236) can be rewritten using small increments x and t instead of derivatives as pðx; t þ tÞ pðx; tÞ Gðx þ x; tÞ Gðx; tÞ þ

0 t x

ð5:237Þ

232


or ½ pðx; t þ tÞ pðx; tÞx ½Gðx; tÞ Gðx þ x; tÞt

ð5:238Þ

Thus, it can be said that the increment of the probability (substance) during a small interval of time t at the location x is equal to a difference between the incoming flow Gðx; tÞ, entering on the left of the interval ½x; x þ x and the outcoming flow Gðx þ x; tÞ, leaving it, as shown in Fig. 5.7. The expression (5.238) can be interpreted as a law of conservation of probability, or continuity of probability. This also allows one to write similar expressions for the case of discontinuous Markov processes, both in continuous and discrete time.

Figure 5.7

Illustration of the concept of probability current.

If a random process ðtÞ attains all values on the real axis, i.e. x 2 ð1; 1Þ, then eq. (5.236) is valid on the whole real axis. As the boundary condition one must specify the values of the process at x ¼ 1. Integrating eq. (5.236) over x from 1 to 1 and using the normalization condition (5.233) one deduces that the following conditions must be satisfied Gð1; tÞ ¼ Gð1; tÞ

ð5:239Þ

However, in many practical problems, the latter condition is substituted by somewhat stronger conditions Gð1; tÞ ¼ Gð1; tÞ ¼ 0; pð1; tÞ ¼ pð1; tÞ ¼ 0

ð5:240Þ

which are known as zero boundary conditions or natural conditions. In a case when the random process ðtÞ attains values from a finite interval ½c; d, eq. (5.236) should be considered only on this interval, with the solution vanishing outside this interval. However, in this case a number of different boundary conditions can be considered, similar to those used in section 5.2.1. In particular, one can distinguish between the absorbing and reflecting wall. For some problems periodic boundary conditions can be imposed [2]. Detailed discussions of the boundary conditions can be found in [1,2]. An absorbing boundary condition at x ¼ c or x ¼ d requires that the value of the probability density vanishes on this boundary at all times, i.e. pðc; tÞ ¼ 0

or pðd; tÞ ¼ 0

ð5:241Þ

while a reflecting boundary condition requires that the probability current vanishes on the boundary Gðc; tÞ ¼ 0

or Gðd; tÞ ¼ 0

ð5:242Þ

233


Many problems can be described by an absorbing boundary on one end and a reflecting boundary on the other end. If one of the boundaries is at infinity, natural boundary conditions must be applied on that boundary. The periodic boundary conditions pðx; tÞ ¼ pðx þ L; tÞ; Gðx; tÞ ¼ Gðx þ L; tÞ

ð5:243Þ

can be found in such applications as the theory of matter, crystals, phase lock loops, etc. [1,2,12]. For a sub-class of stationary processes, the transitional probability density ðx; t j x0 ; t0 Þ depends only on the difference of the time arguments ¼ t t0 , i.e. ðx; t j x0 ; t0 Þ ¼ ðx; x0 ; Þ

ð5:244Þ

while the drift and the diffusion do not explicitly depend on time, i.e. K1 ðx; tÞ ¼ K1 ðxÞ;

K2 ðx; tÞ ¼ K2 ðxÞ

ð5:245Þ

The marginal stationary probability density function pst ðxÞ, if it exists, depends neither on time nor on the initial distribution p0 ðxÞ ¼ pðx; t0 Þ. Thus, for a stationary distribution @ pst ðxÞ ¼ 0 @t

ð5:246Þ

and, due to the conservation property of the Fokker–Planck equation @ GðxÞ ¼ 0; @x

GðxÞ ¼ G ¼ const:

ð5:247Þ

As a result, the Fokker–Planck equation is reduced to the following ordinary differential equation d ½K2 ðxÞpst ðxÞ 2K1 ðxÞpst ðxÞ ¼ 2G dx

ð5:248Þ

The general solution of this equation can be found elsewhere (see [13] for example) ðx

ðy

ð C K1 ðsÞ 2G x K1 ðsÞ exp 2 ds ds dy exp 2 pst ðxÞ ¼ K2 ðxÞ K2 ðxÞ x0 x0 K2 ðsÞ y0 K2 ðsÞ

ð5:249Þ

Here, a constant C can be defined from the normalization property of the probability density function. The lower limit of integration is an arbitrary point on the real axis where the process ðtÞ is defined. For zero boundary conditions on the probability flow ðG ¼ 0Þ eq (5.248) becomes d ½K2 ðxÞpst ðxÞ 2K1 ðxÞpst ðxÞ ¼ 0 dx

ð5:250Þ

234


The general solution of this equation is ðx

C K1 ðsÞ exp 2 pst ðxÞ ¼ ds bðxÞ x 0 K2 ðsÞ

ð5:251Þ

and the constant C can be obtained to normalize the probability density function pst ðxÞ to unity. The latter results illustrate the importance of the Fokker–Planck equation for applications. Indeed, once the drift K1 ðxÞ and diffusion K2 ðxÞ coefficients are known, it is possible to write an expression for the stationary probability density. In many cases integral (5.251) can be evaluated in quadratures. Unfortunately, investigation of the non-stationary solution is much more complicated. However, as will be shown later, a number of useful results can be obtained. 5.3.4

Discrete Model of a Continuous Homogeneous Markov Process

In order to visualize the nature of time variation of a continuous homogeneous Markov process ðtÞ for small time increments t, it is useful to consider an approach of a discrete Markov process to its continuous limit. In order to do so, a discrete Markov chain with transitions only to one of the neighbouring states is considered. In other words, if the chain is in the state j at the moment of time t, then at the moment of time t þ t the chain can be found in states j 1, j, or j þ 1. Let the probabilities of these transitions be j , 1 j j and j , respectively. In addition, let ij ðmÞ be the probability of transition from the state i to the state j in m steps. Then ij ðmÞ ¼ j1 i; j1 ðm 1Þ þ ð1 j j Þi; j ðm 1Þ þ j i;jþ1 ðm 1Þ

ð5:252Þ

In a similar way the continuous process can be approximated as a discrete process with increments x, 0, and x with probabilities ðxÞ, 1 ðxÞ ðxÞ and ðxÞ, respectively. If ðx; t j x0 ; t0 Þ stands for the conditional probability to find the trajectory of the process inside an interval ðx; x þ xÞ at the moment of time t, given that this trajectory was at x0 at the moment of time t0 , then the equation for evolution of ðx; t j x0 ; t0 Þ in time can be written as ðx; t j x0 ; t0 Þx ¼ ðx x; t t j x0 ; t0 Þx ðx xÞ þ ðx; t t j x0 ; t0 Þx½1 ðxÞ ðxÞ þ ðx þ x; t t j x0 ; t0 Þxðx þ xÞ

ð5:253Þ

or ðx; t j x0 ; t0 Þ ¼ ðx x; t t j x0 ; t0 Þ ðx xÞ þ ðx; t t j x0 ; t0 Þ ½1 ðxÞ ðxÞ þ ðx þ x; t t j x0 ; t0 Þðx þ xÞ

ð5:254Þ

In order for the approximation to approach a continuous process, the time increment t has to approach zero, t ! 0. It will be shown below that certain restrictions must be applied on probabilities ðxÞ and ðxÞ. Since the particle located at x at the moment of time t is allowed only to remain in the current position or move to the closest location separated by x, average displacement and


235

its variance during a small time interval t are thus mðtÞ ¼ ½ ðxÞ ðxÞx

ð5:255Þ

2 ðtÞ ¼ ½ ðxÞ þ ðxÞ ½ ðxÞ ðxÞ2 ðxÞ2

ð5:256Þ

The drift and diffusion coefficients now can be obtained as the limit of mðtÞ and 2 ðtÞ 2 x 2 ðxÞ ; K2 ðxÞ ¼ lim ½ ðxÞ þ ðxÞ ½ ðxÞ ðxÞ K1 ðxÞ ¼ lim ½ ðxÞ ðxÞ x; t!0 x; t!0 t t ð5:257Þ

In order to obtain a finite drift and diffusion, certain restrictions must be imposed on how ðxÞ and ðxÞ behave for small t and x. Assuming that the diffusion coefficient K2 ðxÞ is limited K2 ðxÞ < C ¼ const:

ð5:258Þ

ðxÞ2 ¼ Ct

ð5:259Þ

and choosing x such that

one can easily verify that ðxÞ ¼

1 ½K2 ðxÞ þ K1 ðxÞx; 2C

ðxÞ ¼

1 ½K2 ðxÞ K1 ðxÞx 2C

ð5:260Þ

satisfies eq. (5.257). Thus, in a discrete model pffiffiffiffiffiffi of a continuous Markov process, spacial increments x must be of the order of Oð tÞ and the probabilities ðxÞ and ðxÞ differ only by a quantity of the same order pffiffiffiffiffiffi K1 ðxÞ pffiffiffiffiffiffi ðxÞ ðxÞ pffiffiffiffiffiffiffiffiffiffiffi t Oð tÞ K2 ðxÞ 5.3.5

ð5:261Þ

On the Forward and Backward Kolmogorov Equations

Since the forward and the backward Kolmogorov equations (5.213) and (5.214) define the same transitional probability density ðx; t j x0 ; t0 Þ, these two equations are not independent. In particular, it can be shown that these two equations define two adjoint operations. According to [14], two differential functionals

d d d d kðxÞ uðxÞ lðxÞ uðxÞ þ ½ðxÞuðxÞ ðxÞuðxÞ ð5:262Þ L½uðxÞ ¼ dx dx dx dx and

d d d d kðxÞ ðxÞ þ lðxÞ ðxÞ ðxÞ ðxÞ ðxÞ M½ðxÞ ¼ dx dx dx dx

ð5:263Þ

236


where kðxÞ, lðxÞ, ðxÞ and ðxÞ are given functions of x; uðxÞ and ðxÞ are two functions which satisfy the following condition ð

d d ððxÞL½uðxÞ uðxÞM½ðxÞÞdx ¼ kðxÞ ðxÞ uðxÞ uðxÞ ðxÞ dx dx

þ ððxÞ lðxÞÞuðxÞðxÞ

ð5:264Þ

and are called conjugate. In this case a differential operator M½ is uniquely defined by the differential operator L½. It is easy to show that if uðxÞ ¼ ðx; t j x0 ; t0 Þ in eq. (5.213) and ðxÞ ¼ ðx; t j x0 ; t0 Þ in eq. (5.214) then direct and inverse Fokker–Planck operators are conjugate with kðxÞ ¼ 0:5K2 ðx; tÞ, lðxÞ ¼ 0, ðxÞ ¼ 0:5K2 ðx; tÞ K1 ðx; tÞ and ðxÞ ¼ 0. Depending on a particular problem it is wise to use either the direct equation or the reversed equation. For example, if the initial state x0 ¼ xðt0 Þ is given, it is more appropriate to use the direct equation (5.213). However, if it is required to calculate the probability of the first level crossing it is better to use the reversed equation (5.214). Furthermore, since the final value of the process is known, it can be found that if certain limitations are imposed on the solution xðtÞ, the direct equation cannot be used, while the reversed equation is still valid. For example, one can consider movement of a particle on a limited interval of its range. When the particle reaches one of the two boundaries of this interval it remains on the boundary for a certain random time , distributed exponentially. At the end of the time interval the particle instantaneously returns to a point x inside the specified interval described by a known probability density. Further motion of the particle is described by a regular Fokker–Planck equation. The described process is a Markov process. However, the direct equation cannot be immediately applied since the transitions are not local.9 Meanwhile the reversed equation would not change and can be applied [1,2].

5.3.6

Methods of Solution of the Fokker–Planck Equation

Since the Fokker–Planck equation is an equation of a parabolic type, one can apply all the known methods [14]. Only four methods are considered below: (a) method of separation of variables, (b) the Laplace transform, (c) method of characteristic function, and (d) changing the independent variables.

5.3.6.1

Method of Separation of Variables

This method is reasonable to apply when both the drift and the diffusion coefficients do not depend on time (homogeneous Markov process). In this case eq. (5.232) becomes @ @ 1 @2 pðx; tÞ ¼ ½K1 ðxÞpðx; tÞ þ ½K2 ðxÞpðx; tÞ @t @x 2 @x2

9

Since jumps are allowed.

ð5:265Þ


237

The solution of this equation can be thought of in the form of an anzatz [2,6] pðx; tÞ ¼ ðxÞTðtÞ

ð5:266Þ

where ðxÞ and TðtÞ are functions of x and t only. Substituting eq. (5.266) into eq. (5.265) and dividing both sides by pðx; tÞ ¼ ðxÞTðtÞ one obtains 1 @ TðtÞ ¼ TðtÞ @t

@ 1 @2 ½K1 ðxÞðxÞ þ ½K2 ðxÞðxÞ @x 2 @x2 ðxÞ

ð5:267Þ

The left-hand side of this equation depends only on t, while the right-hand side depends only on x. This implies that both sides of this equation are equal to some constant, say 2. Thus, the partial differential equation can be substituted by an equivalent system of two ordinary differential equations 1 d TðtÞ ¼ 2 ¼ ; TðtÞ dt

0

1 d2 d ½K2 ðxÞðxÞ ½K1 ðxÞðxÞ þ 2 ðxÞ ¼ 0 2 2 dx dx

ð5:268Þ ð5:269Þ

Differential equation (5.268) has an exponential solution10 TðtÞ ¼ expð 2 tÞ

ð5:270Þ

and the second order differential equation (5.269) can be solved using traditional methods [13,14]. This solution ðx; A; B; Þ depends on two arbitrary constants A and B as well as on the separation parameter . Since the Fokker–Planck equation is a linear equation, its general solution is a linear combination of particular solutions, i.e. pðx; tÞ ¼

1 X n¼0

ðx; An ; Bn ; n Þ expð n2 tÞ

ð5:271Þ

where constants An , Bn and n are defined by the corresponding boundary and initial conditions, imposed by a particular physical problem. It will be seen that it is often possible to satisfy the boundary conditions only for particular values of , called the eigenvalues, while the corresponding function n ðx; A; BÞ ¼ ðx; A; B; n Þ is called the eigenfunction corresponding to the eigenvalue n . A set of all possible eigenvalues is also known as the spectrum of the Fokker–Planck equation. If there is a countable number of isolated eigenvalues, the spectrum is called a discrete spectrum; if there is a continuum of eigenvalues then the spectrum is called a continuous spectrum. Finally, if there are both isolated eigenvalues and a region of continuous eigenvalues, the spectrum is called a mixed

10

The weight of the exponent is arbitrarily chosen to be unity with a proper constant absorbed by the function .

238


spectrum. A comprehensive treatment of the so-called eigenvalue problem can be found in [15]. It can be shown that if the flow Gðx; tÞ across the boundaries is zero and the spectrum of the Fokker–Planck operator is discrete, the solution can be further simplified to pðx; tÞ ¼ P0 pst ðxÞ þ

1 X n¼1

n ðxÞTn exp½ n2 ðt t0 Þ;

0 ðxÞ ¼ pst ðxÞ

ð5:272Þ

where n ðxÞ are eigenfunctions of eq. (5.269) corresponding to the eigenvalues n2 . P0 and Tn are some constant coefficients to be defined from the initial conditions. The eigenfunction which corresponds to the smallest eigenvalue 0 ¼ 0 is the stationary solution pst ðxÞ of the stationary Fokker–Planck equation as given by eqs. (5.250) and (5.251). Functions n ðxÞ form an orthonormal set with weight 1=pst ðxÞ. Mathematically speaking ð

n ðxÞn ðxÞ dx ¼ mn ¼ pst ðxÞ

1; 0;

n¼m n 6¼ m

ð5:273Þ

In particular ð

ð 0 ðxÞn ðxÞ dx ¼ n ðxÞdx ¼ 0 for n > 0 pst ðxÞ

ð5:274Þ

If the initial probability density is specified, pðx; t0 Þ ¼ p0 ðxÞ then the coefficients Tn and P0 can be obtained simply by setting t ¼ t0 in eq. (5.272) and by multiplying both parts of the latter by m ðxÞ followed by integration over x. Due to the orthogonality property (5.272) one obtains that the coefficients P0 and Tn are uniquely defined by ð P0 ¼ 1 and Tn ¼

n ðxÞp0 ðxÞ dx; pst ðxÞ

n>0

ð5:275Þ

If the initial value of the process is fixed at x0 , i.e. p0 ðxÞ ¼ ðx x0 Þ then the transitional density ðx; t j x0 ; t0 Þ is recovered pðx; tÞ ¼ ðx; t j x0 ; t0 Þ ¼

1 X n ðxÞn ðx0 Þ n¼0

pst ðx0 Þ

exp½ n2 ðt t0 Þ

It is sometimes beneficial to consider normalized eigenfunctions following rule11 n ðxÞ

11

Another useful normalization is

n ðxÞ

¼

pffiffiffiffiffiffiffiffiffiffiffi ¼ n ðxÞ= pst ðxÞ.

ðxÞ pst ðxÞ

n ðxÞ,

ð5:276Þ

defined by the

ð5:277Þ

239


In this case eqs. (5.272)–(5.276) can be rewritten as " pðx; tÞ ¼ pst ðxÞ 1 þ

ð n ðxÞ n ðxÞpst ðxÞdx

1 X

# n ðxÞTn

n¼1

1; n ¼ m

exp½ n2 ðt t0 Þ ;

0 ðxÞ

¼1

ð5:278Þ

¼ mn ¼ 0; n 6¼ m ð Tn ¼ n>0 n ðxÞp0 ðxÞdx;

ð5:279Þ ð5:280Þ

and pðx; tÞ ¼ ðx; t j x0 ; t0 Þ ¼ pst ðxÞ

1 X

n n ðxÞ n ðx0 Þ exp½ 2 ðt

n¼0

t0 Þ;

t t0

ð5:281Þ

Once the transitional density is known, one is able to calculate all parameters of the corresponding Markov process. For example, the joint PDF p2 ðx; x0 ; t; t0 Þ is thus p2 ðx; x0 ; t; t0 Þ ¼ ðx; t j x0 ; t0 Þpðx0 ; t0 Þ ¼ pst ðxÞpðx0 ; t0 Þ

1 X n¼0

2 n ðxÞ n ðx0 Þ exp½ n ðt

t0 Þ ð5:282Þ

and the correlation function is given by ðð R ðt; t0 Þ ¼

xx0 p2 ðx; x0 ; t; t0 Þdxdx0 ð ð 1 X 2 ¼ exp½ n ðt t0 Þ x n ðxÞpst ðxÞdx x0

ð5:283Þ

n ðx0 Þpðx0 ; t0 Þ

n¼0

In the stationary case, pðx0 ; t0 Þ ¼ pst ðx0 Þ and the correlation function depends only on the time difference ¼ t t0 R ðÞ ¼

1 X n¼0

h2n

exp½ n2 jj

¼

h20

þ

1 X n¼1

h2n exp½ n2 jj

ð5:284Þ

where ð hn ¼ x

n ðxÞpst ðxÞdx

ð5:285Þ

Since 0 ðxÞ ¼ 1 one can obtain from eq. (5.285) that h0 is just the mean value m of the process and, thus, the expression for the covariance function C ðÞ becomes simply C ðÞ ¼ R ðÞ m2 ¼

1 X n¼1

h2n exp½ n2 jj

ð5:286Þ

240


Finally, taking the Fourier transform of both sides of eq. (5.286) one obtains the expression for the power spectral density of the Markov process ðtÞ ð1

1 X 4 n2 h2n C ðÞ expðj!Þd ¼ Sð!Þ ¼ !2 þ n4 1 n¼1

ð5:287Þ

Higher order statistics are derived in section 5.6. Example: Gaussian process [6]. Let the following Fokker–Planck equation be considered @ 1 @ 2 @ 2 pðx; tÞ ¼ ð½xpðx; tÞÞ þ pðx; tÞ @t c @x c @x2

ð5:288Þ

with natural boundary conditions at x ¼ 1, i.e. Gð1; tÞ ¼ 0. In this case eq. (5.269) becomes 2

@2 @ ½xðxÞ þ 2 c ðxÞ ¼ 0 ðxÞ þ @x @x2

ð5:289Þ

This problem has a discrete spectrum given by [15,16]

n2 ¼

n ; c

n ¼ 0; 1; 2; . . .

ð5:290Þ

and the eigenfunctions x 1 1 n ðxÞ ¼ pffiffiffiffi F ðnþ1Þ n!

ð5:291Þ

where function F ðnþ1Þ ðzÞ can be easily expressed in terms of the Hermitian polynomials using the Rodrigues formula [9] F

ðnþ1Þ

1 dn z2 1 p ðzÞ ¼ ffiffiffiffiffiffi n e 2 ¼ pffiffiffiffiffiffi 2 dz 2

ð5:292Þ

The stationary solution corresponds to 0 ðxÞ and is thus 1 ð1Þ x 1 x2 ¼ pffiffiffiffiffiffiffiffiffiffi exp 2 pst ðxÞ ¼ 0 ðxÞ ¼ F 2 22

ð5:293Þ

i.e. the stationary distribution is a Gaussian one with zero mean and variance 2 . Having this in mind, the normalized eigenfunctions n ðxÞ are given by n ðxÞ ð1Þn 2n=2 x pffiffiffiffi ¼ Hn pffiffiffi n ðxÞ ¼ pst ðxÞ 2 n!

ð5:294Þ


241

This, in turn, provides the following expression for the transitional density 1

1 x2 X 1 x x0 Hn pffiffiffi Hn pffiffiffi exp ðx; t j x0 ; t0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi exp 2 ð5:295Þ c 2 n ¼ 0 2n n! x 2 22 pffiffiffi Taking into account that H1 ðxÞ ¼ 2x one obtains that the expression for the covariance function is reduced to a single exponent

C ðÞ ¼ exp c

ð5:296Þ

2

Example: one-sided Gaussian distribution [6]. The Fokker–Planck equation (5.288), considered with different boundary conditions, leads to a different expression for the eigenvalues and eigenfunctions. Let there be a reflecting wall at x ¼ 0 and the natural boundary conditions at x ¼ 1, i.e. Gð0; tÞ ¼ Gð1; tÞ ¼ 0

ð5:297Þ

It is easy to verify that in this case one can keep only eigenvalues (5.290) and eigenfunctions (5.291) with even indices, i.e.

n2 ¼

2n c

ð5:298Þ

thus resulting in the following expression for the stationary PDF rffiffiffiffiffiffiffiffi 2 x2 pst ðxÞ ¼ exp 2 ; 2 2

x0

ð5:299Þ

and the transitional density rffiffiffiffiffiffiffiffi 1

2 x2 X 1 x x0 2 H2n pffiffiffi H2n pffiffiffi exp ðx; t j x0 ; t0 Þ ¼ exp 2 2 c 2 n¼0 22n ð2nÞ! 2 2 ð5:300Þ Example: Nakagami distributed process [6]. Let the Fokker–Planck equation have the following form @ 1 @ pðx; tÞ ¼ @t 2c @x

ð2m 1Þ @2 x pðx; tÞ þ pðx; tÞ 2mx 4c m @x2

ð5:301Þ

with the boundary conditions Gð0; tÞ ¼ Gð1; tÞ ¼ 0

ð5:302Þ

242


Separation of variables produces the following eigenvalue problem d2 @ ðxÞ þ 2m dx2 @x

ð2m 1Þ x ðxÞ þ 2c 2 ðxÞ ¼ 0 2mx

ð5:303Þ

Using the following substitution 2mx2 and uðzÞ ¼ zðm1Þ=2 z¼

rffiffiffiffiffiffi! z 2m

ð5:304Þ

Eq. (5.303) can be transformed to a more standard form [6] d2 d uðzÞ þ uðzÞ þ 2 dz dz

"

m 2

2 # þ 2 c 14 m1 2 þ uðzÞ ¼ 0 z z2

ð5:305Þ

Once again, using [9], one can obtain that the eigenvalues of eq. (5.305) are given by

n2 ¼

n c

ð5:306Þ

with corresponding eigenfunctions expressed in terms of the generalized Laguerre polynomials [9] un ðzÞ ¼ zm=2 expðzÞLm1 ðzÞ n

ð5:307Þ

or, returning to the original variables and normalizing the eigenfunctions, one obtains [6] 1 mx2 m1 mx2 2m1 exp n ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x Ln n!ðn þ mÞðmÞ

ð5:308Þ

and the joint stationary PDF is given by 2

X m1 mx2 m1 mx0 1 2 2 Ln Ln 4 mxx0 2m1 mðx þ x0 Þ njj exp p2 ðx; x0 ; Þ ¼ exp c n!ðn þ mÞðmÞ n¼0 ð5:309Þ Thus, the stationary distribution described by the Fokker–Planck equation (5.301) is the Nakagami distribution 2 mm 2m1 mx2 pst ðxÞ ¼ x exp ðmÞ

ð5:310Þ


5.3.6.2

243

The Laplace Transform Method

The effectiveness and popularity of the Laplace transform is explained by the fact that it eliminates the dependence of the solution on time t and the partial differential equation becomes an ordinary differential equation. This approach can be directly applied to the Fokker–Planck equation. In order to do so, it can be written as

@ 1 @2 @ @ pðx; tÞ ¼ K2 ðxÞ 2 pðx; tÞ þ K2 ðxÞ K1 ðxÞ pðx; tÞ @t 2 @x @x @x

1 @2 @ þ K2 ðxÞ K1 ðxÞ pðx; tÞ 2 @x2 @x

ð5:311Þ

Next, let Pðx; sÞ stand for the Laplace transform of the probability density pðx; tÞ, i.e. Pðx; sÞ ¼

ð1

pðx; tÞest dt

ð5:312Þ

0

Taking the Laplace transformation of both sides of the Fokker–Planck equation one obtains the following ordinary differential equation

1 d2 d 1 00 0 0 K2 ðxÞ 2 Pðx; sÞ þ ½K2 ðxÞ K1 ðxÞ Pðx; sÞ þ K2 ðxÞ K1 ðxÞ s Pðx; sÞ þ p0 ðxÞ ¼ 0 2 dx 2 dx ð5:313Þ where p0 ðxÞ is the initial PDF of the process under consideration, pðx; tÞ ¼ p0 ðxÞ. The corresponding boundary conditions are also obtained by taking the Laplace transform of these conditions. Example: thermal driven motion in a well with an absorbing wall. If a particle is moving in a limited region with a reflecting wall at x ¼ 0 and an absorbing wall under the influence of the thermal force, the corresponding Fokker–Planck equation can be written as @ 2 @ 2 pðx; tÞ ¼ pðx; tÞ @t c @x2

ð5:314Þ

@ pðx; tÞ ¼ pðL; tÞ ¼ 0 @x x¼0

ð5:315Þ

with the boundary conditions

Assuming that the initial position of the particle is uniformly distributed over the interval ½0; L the Laplace transform of eq. (5.314) becomes 1 2 @ 2 @ sPðx; sÞ ¼ Pðx; sÞ; Pðx; sÞ ¼ PðL; sÞ ¼ 0 2 L c @x @x x¼0

ð5:316Þ

244


After simple manipulation the solution of the problem (5.316) is given by pffiffiffiffiffi s cosh 1 x pffiffiffiffiffi Pðx; sÞ ¼ sL s L sL cosh

ð5:317Þ

Since there is an absorbing wall, the particle eventually will be removed from the system. This conclusion is confirmed by the fact that12 lim pðx; tÞ ¼ lim sPðx; sÞ ¼ 0

!1

s!0

ð5:318Þ

Furthermore, one can define a residency (survival) distribution d Pr ðtÞ ¼ dt

ðL

ð5:319Þ

pðx; tÞdx 0

Since it is difficult to find the exact inverse Laplace transform of eq. (5.317), instead of finding the exact distribution pr ðtÞ one can settle for finding the average residency time Tc which can be expressed in terms of Pðx; sÞ as Tc ¼

ð1

tpr ðtÞdt ¼

0

ð1 ðL 0

0

ÐL pðx; tÞdxdt ¼ lim s!0

0

Pðx; sÞdx c L2 ¼ 2 2 s

ð5:320Þ

The latter equation is derived using the finite limit theorem for the Laplace transform. As expected, the average residency time depends on the width of the well and the intensity of noise. Higher order moments of the distribution pr ðtÞ can also be calculated in a similar manner. 5.3.6.3

Transformation to the Schro¨dinger Equations

Let the process considered have a constant diffusion coefficient K2 ðx; tÞ ¼ K2 ¼ const. In this case the eigenvalue problem for the Fokker–Planck equation becomes

d K2 d2 ½K1 ðxÞ ðxÞ þ ðxÞ ¼ ðxÞ dx 2 dx2

ð5:321Þ

or K2 d2 d d ðxÞ þ K1 ðxÞ ðxÞ ¼ 0 ðxÞ K1 ðxÞ dx dx 2 dx2 12

Here the initial and the final value theorem for the Laplace transform is used [17].

ð5:322Þ

STOCHASTIC DIFFERENTIAL EQUATIONS

245

It is easy to see that substitution [18] ð ðxÞ ¼ uðxÞ exp

K1 ðxÞ dx K2

ð5:323Þ

reduces eq. (5.322) to the canonical form 3 d K1 ðxÞ K 2 ðxÞ d2 d2 7 6 1 dx uðxÞ ¼ uðxÞ þ 2 uðxÞ þ ½E VðxÞuðxÞ ¼ 0 5 4 dx2 dx2 K2 K22 2

ð5:324Þ

where E¼

2 K2

ð5:325Þ

and d K1 ðxÞ K 2 ðxÞ VðxÞ ¼ 2 dx þ 12 K2 K2

ð5:326Þ

The importance of such a representation is obvious when approximate methods of solution of the Fokker–Planck are to be found: there is a wealth of approximate methods developed for the Schro¨dinger equation in the form (5.324). Example: Gaussian process and Hermitian polynomials. Let the drift of K1 ðxÞ be a linear function of its argument K1 ðxÞ ¼ x

ð5:327Þ

and let the natural boundary conditions be used for the corresponding Fokker–Planck equation. In this case eq. (5.326) is reduced to

d2 þ 2 x uðxÞ þ 2 2 uðxÞ ¼ 0 K2 dx2 K2

ð5:328Þ

which is well studied [15].

5.4


Let a deterministic system be described by a differential equation known as the Liouville equation d xðtÞ ¼ f ðx; tÞ þ gðx; tÞuðtÞ; xðt0 Þ ¼ x0 dt

ð5:329Þ

246


where xðtÞ is the state of the system, x0 is the initial state of the system, f ðx; tÞ and gðx; tÞ are some deterministic functions, and uðtÞ is external force, for a moment considered deterministic. In this case the state evolution is purely deterministic and can be found through a closed form solution of eq. (5.329). Of course, such a solution is deterministic as well.13 Equation (5.329) can be rewritten in equivalent differential and integral forms as dxðtÞ ¼ f ðxðtÞ; tÞdt þ gðxðtÞ; tÞdðtÞ; xðt0 Þ ¼ x0

ð5:330Þ

and xðtÞ ¼ xðt0 Þ þ

ðt

f ðxðÞ; Þd þ

ðt

t0

gðxðÞ; ÞdðÞ

ð5:331Þ

t0

where dðÞ ¼ uðÞd

ð5:332Þ

Randomness can be brought in by a number of ways: one can consider initial conditions xðt0 Þ ¼ , with functions f ðx; tÞ and gðx; tÞ being random. However, one of the most common situations is the case where the external excitation uðtÞ is itself a random process. In particular, if uðtÞ is a white Gaussian noise ðtÞ or a Poisson flow ðtÞ, the corresponding equation (5.329) is called a Stochastic Differential Equation (SDE), expression (5.331) is called a Stochastic Differential, and eq. (5.331) is called a Stochastic Integral. Treatment of stochastic integrals is somewhat different from regular integrals and is considered in some detail below. Comprehensive discussions can be found in [6,8].

5.4.1

Stochastic Integrals

Let f ðx; tÞ and gðx; tÞ be smooth functions of their arguments. In the case when both t and x are deterministic variables the Riemann integral and the Stielties integrals are defined through the limits of the following sums ðt ðt t0

t0

f ðxðÞ; Þd ¼ lim

!0

gðxðÞ; ÞdðÞ ¼ lim

!0

m1 X

f ðxði Þ; i Þðtiþ1 ti Þ

ð5:333Þ

gðxði Þ; i Þ½ðtiþ1 Þ ðti Þ

ð5:334Þ

i¼0 m1 X i¼0

Here t0 < t1 < < tm ¼ t; ¼ maxðtiþ1 ti Þ, and i are some points belonging to intervals ½ti ; tiþ1 , i.e. ti i tiþ1 . For any piece-wise continuous function f ðxðtÞ; tÞ and continuous function gðxðtÞ; tÞ limits in eqs. (5.333) and (5.334) do not depend on the choice of points i inside each interval. 13

In chaotic systems such a statement is not valid if one allows for small errors in knowledge of the initial conditions [19].


247

It can be shown [8] that integrals involving random functions can be defined using eqs. (5.333) and (5.334) if the following conditions are met. 1. Random functions f ðx; tÞ and gðx; tÞ are uniformly continuous in the mean square sense on the time interval ½t0 ; t, i.e. Ef½ f ðxð þ Þ; þ Þ f ðxðÞ; Þ2 g ! 0 as ! 0 uniformly with respect to 2 ½t0 ; t. A similar equality must hold for gðx; tÞ. 2. The following integrals exist and are finite ð t

ð t

2 2 E f ðxðtÞ; tÞdt < 1; E g ðxðtÞ; tÞdt < 1 t0

ð5:335Þ

t0

3. Limits in eqs. (5.333) and (5.334) are understood in the mean square sense. The definition of an integral of a random function does not differ much from the definition of the integrals of deterministic functions. However, in a striking contrast, the value of the integral (5.334) depends on the way in which points i are chosen. Definition (5.334) implies that a continuous function gðxðtÞ; tÞ is substituted on the interval ½t0 ; t by a piece-wise constant function g ðtÞ such that g ði Þ ¼ gðxði Þ; i Þ;

i 2 ½ti ; tiþ1

ð5:336Þ

and i are chosen according to the following rule i ¼ i ¼ ð1 Þti þ tjþ1 ;

01

ð5:337Þ

Since it is assumed that xðtÞ is a solution of an SDE in the form (5.331) its trajectory can be approximated by a straight line on a small time interval xðÞ ¼ xðti Þ þ

xðtiþ1 Þ xðti Þ ð ti Þ þ oðÞ tiþ1 ti

ð5:338Þ

Using expression (5.337) in eq. (5.338) one obtains xðÞ ¼ ð1 Þxðti Þ þ xðtiþ1 Þ þ oðÞ

ð5:339Þ

If one further assumes that the function gðxðtÞ; tÞ is continuously differentiable, then the following approximation can be easily obtained from eqs. (5.337) and (5.339) gðxði Þ; i Þ ¼ gðð1 Þxðti Þ þ xðtiþ1 Þ; ti Þxðti Þ þ xðtiþ1 Þ þ oðÞ

ð5:340Þ

Finally, eq. (5.340) allows one to approximate the stochastic integral (5.334) as J ¼

ðt

gðxðÞ; Þd ðÞ ¼ lim

!0

t0

¼ lim

!0

m1 X i¼0

m1 X

gðxði Þ; i Þ½ðtiþ1 Þ ðti Þ

i¼0

gðð1 Þxðti Þ þ xðtiþ1 Þ; ð1 Þti þ tiþ1 Þ½ðtiþ1 Þ ðti Þ

ð5:341Þ

248


Here the sub-index emphasizes the choice of the time i inside the interval ½ti ; tiþ1 . If ¼ 0, eq. (5.341) defines the so-called Ito stochastic integral [1,2] J0 ¼ lim

!0

m1 X

gðxðti Þ; ti Þ½ðtiþ1 Þ ðti Þ

ð5:342Þ

i¼0

which is widely used in the theory of Markov processes. The next step is to evaluate the difference J J0 for an arbitrary value of from the interval ½0; 1. The assumption of differentiability of function gðx; tÞ allows one to represent it by the first two terms of the corresponding Taylor expansion, i.e. gðð1 Þxðti Þ þ xðtiþ1 Þ; ti Þ " # @ gðx; ti Þ ½xðtiþ1 Þ xðti Þ þ o½xðtiþ1 Þ xðti Þ

gðxðti Þ; ti Þ þ @x x ¼ xðti Þ

ð5:343Þ

Since the trajectory xðtÞ is the solution of a differential equation (5.329), one can substitute the corresponding differential (5.330) by a finite difference for a small time interval tiþ1 ti xðtiþ1 Þ xðti Þ ¼ f ðxði Þ; i Þðtiþ1 ti Þ þ gðxði Þ; i Þ½ðtiþ1 Þ ðti Þ þ oðÞ

ð5:344Þ

The latter can be used to transform eq. (5.341) to (

m1 X

@ gðxðti Þ; ti Þ gðxðti Þ; ti Þ½ðtiþ1 Þ ðti Þ2 þ oðÞ J J0 ¼ lim lim !0 !0 @x i¼0

) ð5:345Þ

If the external force ðtÞ is white Gaussian noise, this implies that (see section 3.15 or [2]) Ef½ðtiþ1 Þ ðti Þ2 g ¼

N0 ðtiþ1 ti Þ 2

ð5:346Þ

@ ðgðxðÞ; ÞÞd @x

ð5:347Þ

and eq. (5.346) approaches in the limit to N0 J J0 ¼ 2

ðt t0

gðxðÞ; Þ

Thus, the relationship between J and the Ito stochastic integral J0 is given by ðt

ðt

N0 J ¼ gðxðÞ; Þd ðÞ ¼ gðxðÞ; Þd0 ðÞ þ 2 t0 t0

ðt t0

gðxðÞ; Þ

@ gðxðÞ; Þd @x ð5:348Þ

249


In more general settings it can be proven that if xðtÞ is solution of a stochastic differential equation (5.330), ðxðtÞ; tÞ is [8] ðt

ðt

N0 ðxðÞ; Þd ðÞ ¼ ðxðÞ; Þd0 ðÞ þ 2 t0 t0

ðt

gðxðÞ; Þ

t0

@ ðxðÞ; Þd @x

ð5:349Þ

It is important to note that if the function gðx; tÞ does not explicitly depend on the current value of the random process xðtÞ then J ¼ J0 for all , i.e. all definitions of stochastic integrals are identical. While the Ito integrals are mostly used in theoretical analysis due to the fact that they are martingales [4,8], other definitions (with 6¼ 0) are very important. In particular, the case of ¼ 0:5 is known as the Stratonovich form of stochastic integrals [6] m1 X xðti Þ þ xðtiþ1 Þ ti þ tiþ1 ; ¼ gðxðÞ; Þd0:5 ðÞ ¼ lim g ½ðtiþ1 Þ ðti Þ !0 2 2 t0 i¼0 ðt

J0:5

ð5:350Þ The formal difference between Ito and Stratonovich integrals lies in the fact that in order to evaluate values of the function gðx; tÞ in the pre-limit sum (5.334) one picks points at the beginning of the time intervals (Ito form) or in the middle of the time intervals (Stratonovich form). Ito and Stratonovich integrals lead to different results for integration of random functions. For example, integrals of a Wiener process can be obtained using a straightforward equality 1 wðti Þ½wðtiþ1 Þ wðti Þ ¼ fw2 ðtiþ1 Þ w2 ðti Þ ½wðtiþ1 Þ wðti Þ2 g 2

ð5:351Þ

Thus

1 2 N 2 J0 ¼ wðÞd0 wðÞ ¼ w ðtÞ w ðt0 Þ ðt t0 Þ 2 2 t0 ðt

ð5:352Þ

while the Stratonovich definition will produce J0:5 ¼ J0 þ

N 1 ðt t0 Þ ¼ ½w2 ðtÞ w2 ðt0 Þ 4 2

ð5:353Þ

It is possible to prove that for any ‘‘good’’ function ðxðtÞ; tÞ [8] ðt

ðt

1 ðxðÞ; Þd0:5 xðÞ ¼ ðxðÞ; Þd0:5 xðÞ þ 2 t0 t0

ðt t0

g2 ðxðÞ; Þ

@ ðxðÞ; Þd @x

ð5:354Þ

It follows from the discussion above that different definitions of the stochastic integrals would lead to a different solution of the stochastic differential equation, eqs. (5.329)– (5.331). That is why it is always necessary to indicate which form of stochastic integrals are

250


used. Sometimes, different notations are used for Stratonovich and Ito forms, as discussed in Appendix C on the web. While the Ito form is convenient in theoretical studies, it can be seen from the example below that the usual rules of calculus of integrals do not apply to the Ito form of stochastic integrals. The rules of differentiation also become more complicated than those of classical calculus. In particular, it can be proven that

@ @ 1 2 @2 ðxðtÞ; tÞ þ f ðxðtÞ; tÞ ðxðtÞ; tÞ þ g ðxðtÞ; tÞ 2 ðxðtÞ; tÞ dt dðxðtÞ; tÞ ¼ @t @x 2 @x 1 @ þ gðxðtÞ; tÞ gðxðtÞ; tÞd0 ðtÞ 2 @x

ð5:355Þ

while the usual rules of calculus result in

@ @ 1 @ ðxðtÞ; tÞ þ f ðxðtÞ; tÞ ðxðtÞ; tÞ dt þ gðxðtÞ; tÞ ðxðtÞ; tÞdðtÞ dðxðtÞ; tÞ ¼ @t @x 2 @x ð5:356Þ This fact makes Ito integrals unattractive for engineering calculations. Interestingly enough, the Stratonovich form of stochastic calculus preserves the rules of classical calculus for derivatives and substitution of variables. Further discussion will be provided later in this section. The next step is to realize that a random process described by a stochastic differential equation (5.329) defines a diffusion Markov process if the excitation is white Gaussian noise. This fact is known as Doob’s first theorem [8]. Indeed, let t3 > t2 > t1 t0 be three distinct moments of time. Using the stochastic integral (5.331) one can write xðt3 Þ ¼ xðt2 Þ þ

ð t3

f ðxðÞ; Þd þ

t2

ð t3

gðxðÞ; Þd wðtÞ

ð5:357Þ

t2

Since increments of the Wiener process on non-intersecting intervals are independent, the value xðt3 Þ [given value of xðt2 Þ] is independent of the value xðt1 Þ, i.e. the process xðtÞ is indeed a Markov process. Since both the SDE and the Fokker–Planck describe a diffusion Markov process it is possible to relate the coefficients of SDE (5.329) with those of the Fokker–Planck equation. It will be assumed that the Ito form of stochastic differentials is used, i.e. dxðtÞ ¼ f ðxðtÞ; tÞdt þ gðxðtÞ; tÞd0 wðtÞ

ð5:358Þ

Integrating both sides of eq. (5.358) over a short interval ½t; t þ t one obtains xðt þ tÞ xðtÞ ¼

ð tþt t

f ðxðÞ; Þd þ

ð tþt gðxðtÞ; tÞd0 wðtÞ t

ð5:359Þ


251

For a short interval t, the value of the first integral in eq. (5.359) is just ð tþt E t

f ðxðÞ; Þd xðtÞ ¼ x ¼ f ðx; tÞt

ð5:360Þ

while from the definition of the Ito integral it follows that ð tþt E t

f ðxðÞ; Þd0 wðtÞ xðtÞ ¼ x ¼ 0

ð5:361Þ

The latter is the consequence of the fact that the increment of the Wiener process over the time interval ½t; t þ t does not depend on the value of the function xðtÞ ¼ x at the moment of time t. This property also allows one to write the expression for the mean root square value of the integral of gðx; tÞ ( ð )

2 tþt N0 2 E f ðxðÞ; Þd0 wðtÞ xðtÞ ¼ x ¼ ð5:362Þ g ðx; tÞt 2 t Thus, the drift and diffusion of the solution of SDE (5.358) are given by

xðt þ tÞ xðtÞ K10 ðx; tÞ ¼ lim E ¼ f ðx; tÞ t!0 t ( ) ½xðt þ tÞ xðtÞ2 N0 2 g ðx; tÞ ¼ K20 ðx; tÞ ¼ lim E t!0 t 2

ð5:363Þ ð5:364Þ

In case of general the relation between J and J0 allows one to write the drift and diffusion as @ @ b ðx; tÞ ¼ K10 ðx; tÞ þ b ðx; tÞ K1 ðx; tÞ ¼ f ðx; tÞ þ 2 @x ( 2 @x ) ½xðt þ tÞ xðtÞ2 N0 2 g ðx; tÞ ¼ K20 ðx; tÞ ¼ K2 ðx; tÞ ¼ lim E t!0 t 2

ð5:365Þ ð5:366Þ

One-to-one correspondence between the drift and the diffusion and functions f ðx; tÞ and gðx; tÞ allows one to rewrite SDE (5.358) directly in terms of parameters of a Markov process:

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ K2 ðx; tÞ dt þ K2 ðx; tÞd wðtÞ dxðtÞ ¼ K1 ðx; tÞ 2 @x

ð5:367Þ

In particular, the Ito and the Stratonovich forms of the SDE thus become pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dxðtÞ ¼ K1 ðx; tÞdt þ K2 ðx; tÞd0 wðtÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1@ K2 ðx; tÞ dt þ K2 ðx; tÞd0:5 wðtÞ dxðtÞ ¼ K1 ðx; tÞ 4 @x

ð5:368Þ ð5:369Þ

252


Formula (5.367) also shows that equations with different can be transformed into each other simply by changing the value of function f ðx; tÞ. However, it also shows that the same Markov process can be described by various SDEs and such ambiguity creates some uncertainty in using SDEs as models for realistic physical processes. As has been pointed out, the Ito form is attractive in theoretical investigations since it defines the solutions of SDEs as martingales. However, rules of differentiation and changes of variables are more complicated than those of deterministic calculus. The latter explains the popularity of the Stratonovich form among engineers and physicists: the rules of calculus are the same as in the deterministic case. It is often important to evaluate the adequacy of the model (5.329) which uses white Gaussian noise as an excitation, since none of the realistic waveforms can be WGN. In principle it is possible to avoid such a problem by considering excitation to be a correlated Gaussian noise which is produced by a linear SDE. In this case the Langevin equation will become a system of two SDEs d xðtÞ ¼ f ðx; tÞ ¼ gðx; tÞuðtÞ dt sffiffiffiffiffiffiffiffi d 1 uðtÞ ¼ uðtÞ þ dt c

22 ðtÞ c

ð5:370aÞ ð5:370bÞ

It can be shown that the random vector ½xðtÞ; uðtÞT is a Markov vector and can be described in a similar way as a scalar Markov process (see section 5.7 for more details). As a result, it has been shown that if c ! 0 the solution of SDE (5.370a) approaches the Stratonovich form of idealized SDE (5.329). Thus, the Stratonovich form of integrals is probably closer to a physical model of the system excited by a very wide band noise than the Ito form. This fact is an additional reason why the Stratonovich form is often used in applied research—one can use deterministic steady-state equations of systems under investigation and then simply assume that the excitation is WGN. No additional changes of non-linear function f ðx; tÞ are then necessary. Numerical simulation of SDEs is considered in more detail in Appendix C on the web and in much greater detail in a number of monographs [20,21]. It is important to mention here that a different numerical scheme would converge to a different form of the solution of SDE: thus the second and the fourth order Runge–Kutta scheme converges to the Stratonovich form of the solution, while the Euler scheme converges to the Ito form of the solution. In general, on each step of simulation ½ti ; tiþ1 WGN is substituted by a Gaussian random variable ni ð 1 tiþ1 ni ¼ ðtÞdt; ¼ tiþ1 ti ð5:371Þ ti It is easy to see from the definition that this random variable has zero mean and variance N0 =2. Samples of the Gaussian random variable are independent of each other. For uniform sampling WGN is substituted by an independent identically distributed (iid) Gaussian sequence with variance N0 =2t. Example: log-normal process. Let the following equation be considered dxðtÞ ¼ xðtÞduðtÞ

ð5:372Þ


253

Its solution satisfying the initial condition xðt0 Þ ¼ x0 is given by [13,18] xðtÞ ¼ x0 exp½uðtÞ uðt0 Þ

ð5:373Þ

For simplicity it is also assumed that x0 > 0. If uðtÞ ¼ ðtÞ is white Gaussian noise, the process wðtÞ ¼ uðtÞ uðt0 Þ is the Wiener process, and its PDF is then

1 x2 pw ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp ð5:374Þ N0 ðt t0 Þ N0 ðt t0 Þ Using rules of zero memory non-linear transformation of densities one obtains the following expression for the PDF of the solution of SDE (5.373)

1 1 ln2 ðx=x0 Þ ð5:375Þ pðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp x N0 ðt t0 Þ N0 ðt t0 Þ Thus, the resulting process is distributed according to a log-normal law. It is interesting to find out which form of SDE produces the same solution. In order to do that one can obtain solutions in both Ito and Stratonovich forms and compare the results with eq. (5.375). Stratonovich form of the solution. Let time interval ½t0 ; t be partitioned by points t0 < t1 < < tm < t. For the i-th sub-interval one can write xðti Þ ¼ xðti1 Þ þ ½xðti Þ þ xðti1 Þ

½wðti Þ wðti1 Þ 2

ð5:376Þ

or, equivalently wðti Þ wðti1 Þ 2 xðti Þ ¼ xðti1 Þ wðti Þ wðti1 Þ 1 2 1þ

ð5:377Þ

The last equality, recursively applied i times, leads to wðti Þ wðti1 Þ 2 xðti Þ ¼ xðt0 Þ wðti Þ wðti1 Þ i¼1 1 2 m 1þ Y

ð5:378Þ

The limit in eq. (5.378) can be calculated by noting that if maxjti ti1 j ! 0 and m ! 1 then wðti Þ wðti1 Þ ! 0. Thus, taking the logarithm of both sides of eq. (5.378) and using approximation lnð1 þ xÞ x x2 =2 for small x one obtains

m X wðti Þ wðti1 Þ wðti Þ wðti1 Þ ln 1 þ ln xðtÞ ¼ ln x0 þ ln 1 2 2 i¼1

ln x0 þ

m X i¼1

½wðti Þ wðti1 Þ ¼ ln x0 þ wðtÞ wðt0 Þ

ð5:379Þ

254


and thus xðtÞ ¼ xðt0 Þ exp½wðtÞ wðt0 Þ

ð5:380Þ

The latter coincides with eq. (5.374). Using expressions for univariate and bivariate characteristic functions of the Wiener process one can obtain expressions for the mean value and the covariance function of non-stationary process xðtÞ:

N0 mx ðtÞ ¼ x0 Efexp½wðtÞ wðt0 Þg ¼ x0 exp ðt t0 Þ ð5:381Þ 4 Cxx ðt1 ; t2 Þ ¼ x20 Efexp½ðwðt1 Þ wðt0 Þ þ ðwðt2 Þ wðt0 ÞÞÞg

N0 2 ½t1 þ t2 2t0 þ 2minðt1 t0 ; t2 t0 Þ ¼ x0 exp 4

ð5:382Þ

Ito solution: Since in this case the point is taken on the left of each sub-interval, one obtains that xðti Þ ¼ xðti1 Þ þ xðti1 Þ½wðti Þ wðti1 Þ ¼ xðti1 Þ½1 þ wðti Þ wðti1 Þ

ð5:383Þ

(Euler scheme) and, after recursive application of the latter xðtÞ ¼ xðt0 Þ

m Y

ð1 þ wðti Þ wðti1 ÞÞ

ð5:384Þ

i¼1

Taking the logarithm of both sides and using approximation lnð1 þ xÞ x x2 =2 once again one easily concludes that ln xðtÞ ¼ ln x0 þ

m X

lnð1 þ wðti Þ wðti1 ÞÞ ¼ ln x0 þ wðtÞ wðt0 Þ þ

i¼1

m X ½wðti Þ wðti1 Þ2 i¼1

2 ð5:385Þ

If m ! 1 and maxjti ti1 j ! 0, then the last term approaches N0 ðt t0 Þ=2 thus leading to

N0 xðtÞ ¼ x0 exp wðtÞ wðt0 Þ ðt t0 Þ 4

ð5:386Þ

It is easy to confirm that this solution is described by the following PDF, average and correlation function (

2 ) 1 1 1 x N0 ln pðx; tÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp þ ðt t0 Þ x N0 ðt t0 Þ N0 ðt t0 Þ x0 4 mx ðtÞ ¼ x0 Cxx ðt1 ; t2 Þ ¼ x20 exp

N0 minðt1 t0 ; t2 t0 Þ 4

ð5:387Þ ð5:388Þ ð5:389Þ


255

The same result can be obtained by changing variables in the original eq. (5.372). Since conventional rules of calculus do not apply for the Ito form of SDE one has to use a special rule for substitution: if one considers the SDE (Ito form) d xðtÞ ¼ f ðx; tÞ þ gðx; tÞðtÞ dt

ð5:390Þ

and the following substitution yðtÞ ¼ ðxðtÞÞ is required, then the Ito form SDE for yðtÞ can be written as d N0 yðtÞ ¼ f ðx; tÞ0 ðxðtÞÞ þ gðx; tÞ0 ðxðtÞÞðtÞ þ g2 ½ðxðtÞÞ00 ðxðtÞÞ dt 4

ð5:391Þ

Applying this rule to eq. (5.373) and choosing yðtÞ ¼ ln½xðtÞ one obtains d N0 yðtÞ ¼ þ ðtÞ dt 4

ð5:392Þ

This, in turn, has an exact solution N0 yðtÞ ¼ yðt0 Þ ðt t0 Þ þ 4

ðt

ðÞd

ð5:393Þ

t0

which immediately produces the result obtained earlier (5.386)

N0 xðtÞ ¼ xðt0 Þ exp wðtÞ wðt0 Þ ðt t0 Þ 4

ð5:394Þ

In a general case, one can write xðti Þ ¼ xðti1 Þ þ ½ð1 Þxðti1 Þ þ xðti Þ½wðti Þ wðti1 Þ which leads to a solution in the form

1 N0 ðt t0 Þ xðtÞ ¼ xðt0 Þ exp wðtÞ wðt0 Þ þ 2 2

ð5:395Þ

ð5:396Þ

Thus, solutions of the SDE (5.372) obtained by methods applied to regular differential equations correspond to the Stratonovich form of SDE. Physically, this can be explained by the fact that the white Gaussian noise term ðtÞ in the Stratonovich form of stochastic integrals can be treated as a zero mean Gaussian process ðtÞ with covariance function CðÞ with a very small correlation interval c (much smaller than the characteristic time of the system under consideration, s c ). The impact of such wide band noise has approximately the same effect as pure WGN if one substitutes CðÞ ! ðN0 =2ÞðÞ, where N0 ¼ 2

ð1 CðÞd 1

ð5:397Þ

256


However, substitution of WGN by a correlated process allows one to treat an SDE as a regular differential equation, thus explaining why the Stratonovich solution is obtained as a result of application of the regular rules of calculus. It is beneficial to illustrate this fact on the example considered above. Let the covariance function of ðtÞ be represented in the form CðÞ where is a time scale factor. For example, the exponential covariance function can be written as CðÞ ¼ N0 f1 expð f1 jjÞ

ð5:398Þ

where ¼ f =f1 and f1 is the ‘‘bandwidth’’ corresponding to ¼ 1. On a fixed subinterval ½ti1 ; ti eq. (5.373) now can be rewritten as d ~xðtÞ ¼ ~xðtÞ~ðtÞ dt Its solution is given by a simple formula ð t1 ~xðti Þ ¼ ~xðti1 Þ ~ ðti1 Þ ðtÞdt ¼ ~xðti1 Þ½~ wðti Þ w

ð5:399Þ

ð5:400Þ

ti1

~ ðtÞ approaches the Wiener process and the solution If approaches infinity, the process w (5.400) approaches that of the Stratonovich solution (5.380). Indeed, the variance of the ~ ðtÞ over time interval T is given by increment of the process w ~ ðtÞ g ¼ Ef½~ wðt þ TÞ w 2

ð tþT ð tþT t

¼

Ef~ð1 Þ~ð2 Þgd1 d2

t

ð tþT ð tþT

Rðð1 2 ÞÞd1 d2 ðT jj ¼T 1 RðjjÞdðÞ T T ð1 ð T jsj N0 RðsÞdðsÞ ! T T 1 RðsÞdðsÞ ¼ ¼T T 2 T 1 t

t

ð5:401Þ

if ! 1. Thus the variance of the process ~ðtÞ approaches that of the Wiener process implying that ~ðtÞ itself approaches the Wiener process. The same conclusion can be reached for a general case of SDE: if WGN is substituted by wide band noise, the solution of the corresponding SDE approaches the Stratonovich form of the solution as bandwidth of the noise increases. Finally, it is possible to summarize the feasibility of modelling of real physical processes using stochastic differential equations into the following three key points. 1. Such modelling is possible if the bandwidth of the driving noise is much wider than the bandwidth of the system, or equivalently c s, where s is the characteristic time constant of the system and c is the correlation interval of the noise. If a continuous time system without delays is described by a system of differential state equations under deterministic input, the same equations can be used if the input is substituted by WGN. In this case one has to treat the obtained SDE as the Stratonovich form of SDE.

257

TEMPORAL SYMMETRY OF THE DIFFUSION MARKOV PROCESS

2. If the discrete system is described by a difference equation under deterministic input, such as xiþ1 ¼ Fi ðxi Þ þ gi ðxi Þi

ð5:402Þ

where i is an iid Gaussian process with variance D ¼ 2 and discretization step t, such a system can be described by a continuous SDE dx xiþ1 xi Fi ðxi Þ xi

¼ þ gi ðxi Þi ¼ f ðx; tÞ þ gðx; tÞðtÞ dt t t

ð5:403Þ

In this case SDE (5.403) must be treated as the Ito form of SDE since the Euler numerical scheme is used (see Appendix C on the web). However, if the input sequence i has correlation interval c exceeding t, c > t one has to treat the approximate continuous SDE as the Stratonovich form of SDE. 3. If a system under consideration is driven by wide band noise (i.e. again c s ) but contains delay d one has to carefully examine the amount of delay. If such delays are small compared to the time constant of the system d s they can be omitted from the differential equations describing the model. However, the delay must also be compared to the correlation interval of the noise. If d c then the system of SDE inferred from the state model must be treated as the Stratonovich SDE, while in the opposite case the Ito form must be used.

5.5

TEMPORAL SYMMETRY OF THE DIFFUSION MARKOV PROCESS

In this section the so-called temporally symmetric property of diffusion Markov processes is considered. It has been found [22–26] that not all actual world processes have this property. Fortunately, stationary processes in communication channels can be thought of as temporally symmetric. This enables one to apply diffusion Markov processes as models of the real ones. Temporal symmetry in a strict sense stationary stochastic process is defined in terms of its joint distributions. A stationary process ðtÞ, t 2 T is temporally symmetric if, for every n and every t1 ; . . . ; tn in T, the random vectors fðt1 Þ; . . . ; ðtn Þg and fðt1 Þ; . . . ; ðtn Þg have the same probability distributions [23]. The set of time indices, T, is assumed to be the entire real line for continuous-time processes, and the set of all integers for discrete-time processes. Thus, whenever t0 is in T, so is t0 . Although this facilitates the definition of temporal symmetry given above, it should be noted that if T does not have this property, the definition can be suitably modified, essentially retaining the same ideas. It is said that a stationary process is second order temporally symmetric if the joint distributions of fðt1 Þ; ðt2 Þg and fðt1 Þ; ðt2 Þg are identical for every t1 and t2 . The

258


relationship of second order temporal symmetry to the more general symmetry is similar to that of a wide sense stationary process to a strict sense stationary process. By analogy with [24] the following proposition is proven Proposition 1. A stationary process is second order temporally symmetric if and only if its bivariate distributions are symmetric. Proof: If ðtÞ is second order temporally symmetric, by definition Pððt2 Þ; ðt1 ÞÞ ¼ Pððt2 Þ; ðt1 ÞÞ

ð5:404Þ

for every t1 and t2 , where Pððt2 Þ; ðt1 ÞÞ denotes a bivariate distribution function. Shifting the time origin by t1 þ t2 and using the stationary condition, one can write Pððt1 Þ; ðt2 ÞÞ ¼ Pððt2 Þ; ðt1 ÞÞ

ð5:405Þ

Combining eqs. (5.404) and (5.405) one obtains Pððt2 Þ; ðt1 ÞÞ ¼ Pððt1 Þ; ðt2 ÞÞ

ð5:406Þ

The converse can be proved by retracing the above steps in reverse order. All stationary Gaussian processes are temporally symmetric [24]. This follows from the fact that all joint distributions of these processes depend only on the covariance matrices and that due to stationarity, those matrices are Hermitian. It is possible to prove the following. Proposition 2. A stationary Markov process ðtÞ, generated by the Fokker–Planck equation is temporally symmetric. Proof: We prove this proposition for the case of a discrete spectrum for a one-dimensional process. The proof in the case of a continuous spectrum, as well as a vector case, repeats the same steps. Taking into account the expression for bivariate PDF (5.282) p ðx; x0 Þ ¼

1 X

m ðxÞm ðx0 Þ exp½m

ð5:407Þ

m¼0

one can conclude that this bivariate PDF is symmetric. Then, following Proposition 1, it is possible to conclude that the Markov process generated by the corresponding SDE is temporally symmetric. This means that this approach does not allow one to model temporally asymmetric random processes. One possible approach to modelling asymmetricin-time processes, staying within the SDE framework, is the use of the Poisson flow of delta pulses as the standard excitation.

5.6

HIGH ORDER SPECTRA OF MARKOV DIFFUSION PROCESSES

The high order spectra of random processes became a topic of great interest in recent decades [25]. Higher order statistical analysis involves high order distributions which, in


259

the case of Markov diffusion processes generated by proper SDEs, are defined by transient density 1 X m ðxÞm ðx0 Þ

ðx; x0 Þ ¼

m¼0

p st ðx0 Þ

exp½m jj

ð5:408Þ

Let t0 , t0 þ 1 , and t0 þ 1 þ 2 be three sequential moments of time and the corresponding values of the Markov diffusion process ðtÞ are x0 ¼ ðt0 Þ, x1 ¼ ðt0 þ 1 Þ, and x2 ¼ ðt0 þ 1 þ 2 Þ. In this case, expression (5.408), written for two intervals 1 and 2 , has the following form 1 X m ðx1 Þm ðx0 Þ

1 ðx1 ; x0 Þ ¼

m¼0

p st ðx0 Þ

exp½m j1 j

ð5:409Þ

exp½n j2 j

ð5:410Þ

and 2 ðx2 ; x1 Þ ¼

1 X n ðx2 Þn ðx1 Þ n¼0

p st ðx1 Þ

respectively. The trivariate density can be then obtained by using the Markov property of the process ðtÞ, i.e. p2 1 ðx2 ; x1 ; x0 Þ ¼ 2 ðx2 ; x1 Þ1 ðx1 ; x0 Þp st ðx0 Þ 1 X 1 X n ðx2 Þn ðx1 Þm ðx1 Þm ðx0 Þ exp½n j2 j m j1 j ¼ p st ðx1 Þ m¼0 n¼0

ð5:411Þ

The knowledge of the trivariate density allows one to calculate moment mk2 k1 k0 ð2 ; 1 Þ of order k2 ; k1 ; k0 ð mk2 k1 k0 ð2 ; 1 Þ ¼ xk22 xk11 xk00 p2 1 ðx2 ; x1 ; x0 Þdx2 dx1 dx0 3

ð ¼

1 X 1 X

3 m ¼ 0 n ¼ 0

xk22 xk11 xk00

n ðx2 Þn ðx1 Þm ðx1 Þm ðx0 Þ p st ðx1 Þ

exp½n j2 j m j1 jdx2 dx1 dx0 ¼

1 X 1 X m¼0 n¼0

xk22 xk11 xk00 exp½n j2 j m j1 j

ð ð ð

n ðx2 Þn ðx1 Þm ðx1 Þm ðx0 Þ dx2 dx1 dx0 P st ðx1 Þ ð ð 1 X 1 X n ðx1 Þm ðx1 Þ k2 ¼ exp½n j2 j m j1 j x n ðx2 Þdx2 xk11 dx1 px st ðx1 Þ m¼0 n¼0 ð 1 X 1 X xk2 n ðx2 Þdx2 ¼ hn;k2 hm;k0 hn;m;k1 exp½n j2 j m j1 j

m¼0 n¼0

ð5:412Þ

260


where is the support of the marginal PDF p st ðxÞ, and the following notation is used ð hn;k ¼ xk n ðxÞdx ð5:413Þ ð n ðxÞm ðxÞ dx ð5:414Þ hn;m;k ¼ xk p st ðxÞ Thus the following proposition is proven. Proposition 3. The bivariate moment function of order k2, k1 , k0 of the Markov diffusion process has the form mk2 k1 k0 ð2 ; 1 Þ ¼

1 X 1 X

m; n;k2 k1 k0 exp½n j2 j m j1 j

ð5:415Þ

m¼0 n¼0

Corollary. The third order moment of the diffusion Markov process is given by m3 ð1 ; 2 Þ ¼ m111 ð1 ; 2 Þ ¼

1 X 1 X

hn hm hnm exp½n j2 j m j1 j

ð5:416Þ

m¼0 n¼0

where

ð hn ¼

ð

xn ðxÞdx and

hnm ¼

n ðxÞm ðxÞ dx p st ðxÞ

ð5:417Þ

xk n ðxÞdx ¼ hn;k

ð5:418Þ

x

Proposition 4. hn;k ¼ hn;0;k Proof: Since 0 ðxÞ ¼ p st ðxÞ, one obtains ð

hn;0;k

n ðxÞ0 ðxÞ dx ¼ ¼ x pst ðxÞ

ð

k

The two-dimensional moment spectra Mk2 k1 k0 ð!2 ; !1 Þ can be found as a two-dimensional Fourier transform of mk2 k1 k0 ð2 ; 1 Þ Mk2 k1 k0 ð!2 ; !1 Þ ¼

1

ð1 ð1

mk2 k1 k0 ð2 ; 1 Þ exp½j!2 2 j!1 1 ð2Þ2 1 1 1 X 1 X 16n m hn;k2 hm;k0 hn;m;k1 ¼ ð!22 þ 2n Þð!21 þ 2m Þ m¼0 n¼0

ð5:419Þ

thus proving the following proposition. Proposition 5. Mk2 k1 k0 ð!2 ; !1 Þ ¼

1 X 1 X 16n m hn;k2 hm;k0 hn; m;k1 ð!22 þ 2n Þð!21 þ 2m Þ m ¼ 0 n¼0

ð5:420Þ


261

Corollary. For the third order moment, two-dimensional spectra have the form 1 X 1 X 16n m hn hm hnm M3 ð!2 ; !1 Þ ¼ ð!22 þ 2n Þð!21 þ 2n Þ m¼0 n¼0

ð5:421Þ

A third order cumulant can be expressed in terms of moments of order up to 3 as [25,26] c3 ð2 ; 1 Þ ¼ m3 ð2 ; 1 Þ m1 ½m2 ð2 Þ þ m2 ð1 Þ þ m2 ð2 1 Þ þ 2m31

ð5:422Þ

or, after substitution of eq. (5.416) into eq. (5.422), c3 ð2 ; 1 Þ ¼

1 X 1 X

hm hn hmn exp½n j2 j m j1 j m1

m¼1 n¼1

1 X m¼1

h2m exp½m ðj2 j j1 jÞ ð5:423Þ

and consequently the following proposition is proved. Proposition 6. Bispectrum C3 ð!2 ; !1 Þ [25] of the diffusion Markov process is given by the following formula 1 X 1 1 X X 16n m hm hn hmn 162n h2m C3 ð!2 ; !1 Þ ¼ m 1 ð!22 þ 2n Þð!21 þ 2m Þ ð!22 þ 2m Þð!21 þ 2m Þ m¼1 n¼1 m¼1

ð5:424Þ

The coefficients hn;m;k and eigenvalues m play a significant role in third order statistics of the diffusion Markov process. We now show that all spectral properties of any order are defined by the same coefficients. Proposition 7. All spectral properties of a diffusion Markov process are defined by spectra of eigenvalues m and the coefficients hnm . Proof: Consider n samples of the process ðtÞ at the moments t0 ; t0 þ 1 ; . . . ; t0 þ 1 þ . . . þ n1 . Using Markov properties of the process, the n-variate density is given by p1 ;...;n1 ðxn1 ; xn2 ; . . . ; x0 Þ ¼ p st ðx0 Þ

n1 Y

i ðxi ; xi1 Þ

i¼1

¼

1 X 1 X kn1 kn2

1 X kn1 ðxn1 Þkn1 ðxn2 Þkn2 ðxn2 Þkn2 ðxn3 Þ . . . k1 ðx0 Þ p st ðxn2 Þp st ðxn3 Þ . . . p st ðx1 Þ k ¼0 0

exp½kn1 jn1 j kn3 jn3 j k1 j1 j

ð5:425Þ

262


To obtain the expression for the moment of order n we should integrate eq. (5.425) n times over real axis R mn ðn1 ; n2 ; . . . ; 1 Þ ð ¼ xn1 xn2 . . . x0 p1 ;...;n1 ðxn1 ; . . . ; x0 Þdxn1 . . . dx0 Rn

¼

1 X 1 X

...

kn1 kn2

ð

1 X

exp½kn1 jn1 j kn3 jn3 j k1 j1 j

k0 ¼0

ð

kn2 ðxn2 Þkn3 ðxn2 Þ dxn2 . . . px st ðxn2 Þ R R ð ð ð kj ðxj Þkj1 ðxj Þ k1 ðx1 Þk0 ðx1 Þ dxj x1 xj dx1 x0 k0 ðx0 Þdx0 px st ðxj Þ px stðx1 Þ R R R

xn1 kn1 ðxn1 Þdxn1

xn2

ð5:426Þ

thus mn ðn1 ; n2 ; . . . ; 1 Þ 1 X 1 1 X X ¼ ... hkn1 hkn2 kn3 . . . hk2 k1 hk0 exp½kn1 jn1 j kn3 jn3 j k1 j1 j kn 1 kn2

k0 ¼0

ð5:427Þ Since mkn1 ...k1 k0 ðn1 ; . . . ; 2 ; 1 Þ ¼ mkn1 þ þ k1 þ k0 ðn1 ; 0; . . . ; 0; n2 ; . . . ; 0; 1 ; . . . ; 0Þ ð5:428Þ an n-variate moment can be expressed in terms of kn1 þ þ k1 þ k0 variate moment with some values of time equal to zero. The last statement together with eq. (5.427) proves the proposition. Example: The random process with Rayleigh distribution is generated by the following SDE K2 1 x x_ ¼ þ KðtÞ ð5:429Þ 2 x 2 In this case [6] m ¼

2Km 2

ð5:430Þ

and

2 x x2 x m ðxÞ ¼ exp 2 Lm 2 n! 2 22

ð5:431Þ

263

VECTOR MARKOV PROCESSES

Table 5.1 Coefficients hmn = for Rayleigh process (5.429) m=n

0

1

2

3

4

0 1 2 3 4

1.2533 0.6267 0.0783 0.0131 0.0020

0.6267 2.1933 0.4308 0.0326 0.0039

0.0783 0.4308 0.7099 0.0873 0.0048

0.0131 0.0326 0.0873 0.0934 0.0084

0.0020 0.0039 0.0048 0.0084 0.0066

Then, using eq. (5.417) one obtains hmn

2 2 ð1 1 x2 x x 2 ¼ 2 x exp 2 Ln Lm dx 2 n!m! 0 2 2 22 pffiffiffi ð 1 2 x1=2 exp½xLn ðxÞLm ðxÞdx ¼ n!m! 0

ð5:432Þ

Values of coefficients hmn in eq. (5.432) for some integers n and m are presented in Table 5.1. It can be seen from this table that the most important terms hmn are those whose index does not exceed 3. Thus, even for higher order statistics, it is possible to take into account only a few first terms in expansion (5.427).14

5.7 5.7.1

VECTOR MARKOV PROCESSES Definitions

The notion of a scalar Markov process can be extended to vector processes. In particular, let nðtÞ ¼ f1 ðtÞ; 2 ðtÞ; . . . ; M ðtÞg be a random vector of dimension M which can be described by a series of probability density functions pnþ1 ðx0 ; . . . ; xn ; t0 ; . . . ; tn Þ of order n, as defined in Chapter 2, or in terms of a series of conditional probability densities n ðxn ; tn j xn1 ; tn1 ; . . . ; x0 ; t0 Þ ¼

pnþ1 ðx0 ; . . . ; xn ; t0 ; . . . ; tn Þ pn ðx0 ; . . . ; xn1 ; t0 ; . . . ; tn1 Þ

ð5:433Þ

The Markov property is defined by the fact that the transitional density, defined by eq. (5.429), depends only on the value of the process at the moment of time tn1 , i.e. n ðxn ; tn j xn1 ; tn1 ; . . . ; xn1 ; tn1 Þ ¼ ðxn ; tn j xn1 ; tn1 Þ

ð5:434Þ

All the properties of a scalar Markov process discussed in sections 5.1–5.5 can be readily extended to the vector case. In particular, one can show that the transitional density 14

For the case of the correlation function this result was considered in [26,29].

264


ðx; t j x0 ; t0 Þ must obey the Smoluchowski–Chapman equation ðx; t j x0 ; t0 Þ ¼

ð1 1

ðx; t j x1 ; t1 Þðx1 ; t1 j x0 ; t0 Þdx1 ;

t0 < t1 < t

ð5:435Þ

Here the integration is taken over RM. It is also possible to show that if ðtÞ is a Markov vector process with M components and continuous trajectories, i.e. only the first two conditional moments of increments are non-zero 1 Ef½i ðt þ tÞ i ðtÞ j ðtÞ ¼ xg t!0 t 1 Ef½i ðt þ tÞ i ðtÞ½j ðt þ tÞ j ðtÞ j ðtÞ ¼ xg K2ij ðx; tÞ ¼ lim t!0 t K1i ðx; tÞ ¼ lim

ð5:436Þ ð5:437Þ

then the marginal PDF pðx; tÞ and the transitional density ðx; t j x0 ; t0 Þ must obey the Fokker–Planck equation @ pðx; tÞ ¼ Lfpðx; tÞg @t

ð5:438Þ

where the Fokker–Planck operator is defined as Lfpðx; tÞg ¼

M M X M X @ 1X @2 ½K1m ðx; tÞpðx; tÞ þ ½K2mk ðx; tÞpðx; tÞ @xm 2 m¼1 k¼1 @xm @xk m¼1

ð5:439Þ

In addition to continuity of K1m ðx; tÞ and K2mk ðx; tÞ and their derivatives, it is required that the quadratic form HðxÞ ¼

M X M X

xm xk K2mk ðx; tÞ

ð5:440Þ

m¼1 k¼1

is positively defined, i.e. for all xi and xj HðxÞ 0. Markov processes which satisfy the corresponding Fokker–Planck equations are called diffusion Markov processes. Partial differential equations (5.438) and (5.439) must be supplemented by a proper set of initial and boundary conditions and their solution, even in a stationary case, is much more complicated than the case of a scalar diffusion Markov process. In contrast to the one-dimensional case, the exact solution of the Fokker–Planck equation, even in a stationary case, is impossible in a general case [1,2]. In this section, we consider the solution for one particular case, called the potential isotropic case [6]. In this case the Fokker–Planck equation can be rewritten in the form [6] M M M X X @ @ 1X @2 @ px ðx; tÞ ¼ ½K1i ðxÞpx ðx; tÞ þ ½K ðxÞp ðx; tÞ ¼ Gi ðxÞ 2ij x 2 @t @xi 2 i; j¼1 @xj @xi i¼1 i¼1

ð5:441Þ


265

where the following notation is used Gi ðxÞ ¼ K1i ðxÞpx ðx; tÞ

M 1X @ ½K2ij ðxÞpx ðx; tÞ 2 j ¼ 1 @xj

ð5:442Þ

It can be seen from eq. (5.441) that the stationary PDF, if it exists, satisfies the equation M X @ Gi ðxÞ ¼ 0 @xi i¼1

ð5:443Þ

In the vector case the probability current G ¼ fG1 ; G2 ; . . . ; GM g does not have to vanish inside the region R under consideration, even if G satisfies zero boundary conditions [6] Gi ðxÞ ¼ 0;

i ¼ 1; . . . ; M

ð5:444Þ

since rotational probability flows can occur. In fact, the current G vanishes in the whole region R, i.e. M 1X @ Gi ðxÞ ¼ K1i ðxÞpx ðx; tÞ ½K2ij ðxÞpx ðx; tÞ 2 j ¼ 1 @xj

ð5:445Þ

only in the special case which is called the potential one [6] when @ K1j ðxÞ @ K1i ðxÞ ¼ @xi K2 ðxÞ @xj K2 ðxÞ

ð5:446Þ

If the conditions (5.446) are met, the stationary PDF can be represented in the form [6] " ð P # M x2 C K ðxÞdx i i¼1 1i exp 2 px st ðxÞ ¼ K2 ðxÞ K2 ðxÞ x1

ð5:447Þ

The non-stationary PDF px ðx; tÞ can sometimes be obtained using Fourier methods in the same manner as in a one-dimensional case.15 By using the representation pðx; tÞ ¼ TðtÞðxÞ one can separate eq. (5.441) into the system T_ ðtÞ ¼ TðtÞ

15

M M X @ 1X @2 K1i ðxÞðxÞ þ K ðxÞðxÞ þ ðxÞ ¼ 0 2 2ij @x 2 @x i j i¼1 i; j¼1

ð5:448Þ

In the case of a potential isotropic system the differential operator of the generating SDE is a symmetric one and this means that application of the Fourier method is possible [14].

266


The transition density, bivariate density, and cross-covariance function between two deterministic functions F1 ðxÞ and F2 ðxÞ of the process xðtÞ can be given in terms of the eigenvalues and eigenfunction of the boundary problem as [27,29]16 ðx; x0 Þ ¼ p ðx; x0 Þ ¼ CF1 F2 ðÞ ¼

1 X m ðxÞm ðx0 Þ m¼0 1 X m¼0 1 X

exp½m jj

ð5:449Þ

m ðxÞm ðx0 Þ exp½m jj

ð5:450Þ

hm1 hm2 exp½m

ð5:451Þ

px st ðx0 Þ

m¼0

where hmj ¼

ð x2

Fj ðxÞm ðxÞdx;

j ¼ 1; 2

ð5:452Þ

x1

Having the transient and stationary, or initial PDF, one can construct the PDF of any dimension using the Markov properties of the process xðtÞ. In addition to the Fourier method of solution of FPE (5.441), some other methods, such as the Laplace transform, transition to the Volterra integral equation, and different numerical schemes can be considered [2]. However, these methods do not give sufficient information for synthesis of the SDE and therefore they are not considered here. As was shown earlier, any diffusion Markov process can be considered as a solution of a properly defined stochastic differential equation as discussed in Section 5.4. A similar statement can be made about vector diffusion Markov processes. Indeed, Doob proved [8] that the following system of SDEs dxi ðtÞ ¼ fi ðx; tÞdt þ

M X

gim ðx; tÞd wk ðtÞ; xðt0 Þ ¼ x0 ;

01

ð5:453Þ

m¼1

defines a diffusion Markov process with the drift and diffusion defined as K1i ðx; tÞ ¼ fi ðx; tÞ þ

M X M X Nk k¼1 j ¼ 1

2

gjk ðx; tÞ

@ gjk ðx; tÞ @xj

M 1X K2ij ðx; tÞ ¼ Nk gik ðx; tÞgjk ðx; tÞ 2 k¼1

ð5:454Þ

Here wk ðtÞ is the Wiener process with zero mean and linear cross-covariance Efwk ðtÞg ¼ 0;

pffiffiffiffiffiffiffiffiffiffi Ef½wk ðt2 Þ wk ðt1 Þ½wl ðt2 Þ wl ðt1 Þg ¼ kl Nk Nl jt2 t1 j ¼ gkl Nk jt2 t1 j; 1 k M ð5:455Þ 16

The continuous spectrum case can also be easily obtained as a generalization of eqs. (5.449) and (5.450).

267


Equivalently, wk ðtÞ can be thought of as a time integral of a zero mean white Gaussian noise wk ðtÞ ¼

ðt 1

k ðtÞdt; Efk ðt1 Þk ðt2 Þg ¼ Nn ðt2 t1 Þ; Efk ðt1 Þl ðt2 Þg ¼ 0;

k 6¼ l

ð5:456Þ

Having this in mind, one can formally rewrite the system of SDE (5.453) as M X d xi ðtÞ ¼ fi ðx; tÞ þ gim ðx; tÞm ðtÞ; xðt0 Þ ¼ x0 ; dt m¼1

01

ð5:457Þ

with the additional specification of the form of stochastic integrals used. The latter form of SDE is particularly useful if a deterministic system excited by a narrow band random input is studied. In this case one can reuse the deterministic state equations to write SDE (5.457), of course keeping in mind that such a procedure would lead to the solution of SDE (5.457) in the Stratonovich form.17 In particular, two types of the state equation are often found in applications. In the first case the state equation is resolved with respect to the higher order derivative, i.e. it is written in the following form dm xðtÞ ¼ Fðx; x0 ; . . . ; xðm1Þ ; t; ðtÞÞ m dt

ð5:458Þ

Using change of variables x0 ðtÞ ¼ xðtÞ, xk ðtÞ ¼ ðdk =dtk ÞxðtÞ, one can rewrite eq. (5.458) as a system of m first order SDEs, thus converting it to the form (5.457). In the second case, the system is linear and thus it can be described by a system of linear SDEs N X d xj ðtÞ ¼ ajk xk ðtÞ þ j ðtÞ; dt k¼1

1 < k;

j 1 m m 2 1 > > 1 22 Hn pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > 2 < n! 1 2 1 22 ¼ ! 2 n=2 > > 1 2 m2 m1 > > > ffi jn Hn j pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi : n! 2 1 2 22 21 1

if

2 < 1

if

2 > 1

ð6:115Þ

If 2 ¼ 1 ¼ , one can use the following standard integral 7.374–6 in [18] ð1 1

exp½ðx yÞ2 Hn ðxÞdx ¼

pffiffiffi n n y 2

ð6:116Þ

300

MARKOV PROCESSES WITH RANDOM STRUCTURES

to obtain ð2Þ k;l

1 ¼ pffiffiffi n!

" # m2 m1 2 2n=2 m2 m1 n Hn ðzÞdz ¼ exp z pffiffiffi n! 2 1

ð1

ð6:117Þ

Finally, eq. (6.105) produces a series for the solution of the generalized Fokker–Planck equation h i2 3 ðxm1 Þ2 xm 1 ffiffi p 1 exp 2 2 X Hk 2 1 ð2Þ k;1 5 pð1Þ ðxÞ P1 pffiffiffiffiffiffiffiffiffiffiffi2ffi1 41 þ P2 s 1 k 2 1 k¼1 h i 2 3 2 xm 2Þ ffiffi 2 1 Hk p exp ðxm X 2 2 2 2 ð1Þ k;2 5 pð2Þ ðxÞ P2 pffiffiffiffiffiffiffiffiffiffiffi2ffi2 41 þ P1 s 2 k 2 2 k¼1 ð1Þ

ð6:118Þ

ð1Þ

k;2 by k;1 can be easily obtained from the expression for where an expression for permutation of indices. It is worth noting that the correction terms depend not only on the marginal distributions in the isolated states but also on correlation intervals of the partial ð1Þ ð2Þ k;2 and k;2 processes. Characteristically in the case 2 ¼ 1 and m2 ¼ m1 , coefficients become zero as follows from eq. (6.117), thus confirming an intuitive conclusion that switching between two processes with identical PDFs must preserve the marginal PDF and the difference will appear only in the second order statistics. Results of numerical simulations are shown in Fig. 6.6. Example: Switching between two Gamma distributed processes. Let each of two isolated processes be described by the Gamma PDF ð1Þ p0 st ðxÞ

l l l x expðl xÞ; ¼ ð 1 Þ

l ¼ 1; 2

ð6:119Þ

In this case, a system of SDEs and corresponding generalized Fokker–Planck equations can be written as 1 x 1 2 þ x 1 ðtÞ 1 s 1 1 s 1 2 x 2 2 x_ ¼ þ x 2 ðtÞ 2 s 2 2 s 2 x_ ¼

ð6:120Þ

and

d 1 x 1 ð1Þ d2 1 ð1Þ 0¼ p ðxÞ þ 2 xp ðxÞ P2 pð1Þ ðxÞ þ P1 pð2Þ ðxÞ dx 1 s 1 dx 1 s 1

d 2 x 2 ð2Þ d2 1 ð2Þ p ðxÞ þ 2 p ðxÞ P1 pð2Þ ðxÞ þ P2 pð1Þ ðxÞ 0¼ dx 2 s 2 dx 2 s 2 ð6:121Þ

APPROXIMATE SOLUTION OF THE GENERALIZED FOKKER–PLANCK EQUATIONS

301

Figure 6.6 Numerical simulation of two switched Gaussian processes. Parameters of the simulation: p1 ¼ 0:3, p2 ¼ 0:7, m1 ¼ 0, m2 ¼ 2, 1 ¼ 1, 2 ¼ 0:7, 1 ¼ 0:2, 2 ¼ 0:3. ¼ 0 *, ¼ 0:5, r, ¼ 1 . . ., ¼ 1 þ þ.

302


respectively. Each isolated FPE defines a set of eigenfunctions [17] rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð þ kÞ ð 1Þ expðl xÞx l 1 Lk l ðl xÞ k; l ðxÞ ¼ k!

ð6:122Þ

Application of eq. (6.104) produces ð2Þ k;l

2 2 ¼ ð 2 Þ

ð1 1

ð 1Þ expð2 xÞx 2 1 Lk 1 ð1

ðk þ 1 Þ 1 xÞdx ¼ 2 F1 2 ; k; 1 ; k!ð 1 Þ 2 ð6:123Þ

Here the following standard integral, 7.414–7 in [18], has been used ð1 0

est t Lð Þ n ðtÞdt

ð þ 1Þðn þ þ 1Þ 1 1 s ¼ F n; þ 1; þ 1; n!ð þ 1Þ s

ð6:124Þ

where Fða; b; c; zÞ ¼ 2 F1 ða; b; c; zÞ is the hypergeometric function [17]. Results of a numerical simulation are shown in Fig. 6.7. 6.3.1.3

Form of the Solution for Large Intensity

The case of large intensities of switching also provokes great interest from the point of view of applications. While the perturbation analysis of this case is conducted in some detail in section 6.3.3, here some conclusions are drawn based on the analysis of eq. (6.100): ðlÞ

k;l ¼

M X 1 Pn ðnÞ n;l k;l k;l þ l n6¼1 Pl

Let l be a large but finite number such that 1 l . Then, it is possible to find an index NðÞ 1 such that NðÞ . If 1 < k NðÞ then k l and, thus ðlÞ l Pl k;l

M X

ðnÞ

n;l Pn k;l

ð6:125Þ

n6¼l

Comparing eq. (6.125) with eq. (6.102), one concludes that, for small values of k, the coefficients of expansion (6.96) are equal for every state of the process with random structure, i.e. ðnÞ

ðlÞ

k;l ¼ k;l

ð6:126Þ

Once l approaches infinity, NðÞ also must approach infinity and thus the number of equal terms in expansion (6.96) approaches infinity as well. As a result, it can be contended that for a very large intensity of switching, the resulting distributions in each state are proportional to the same function.


303

Figure 6.7 Numerical simulation of two switched Gamma processes. Parameters of the simulation: p1 ¼ 0:3, p2 ¼ 0:7, 1 ¼ 0, 2 ¼ 3, 1 ¼ 1, 2 ¼ 0:5, 1 ¼ 0:2, 2 ¼ 0:3. ¼ 0 *, ¼ 0:5 r, ¼ 1 . . ., ¼ 1 þ þ.

304


6.3.2

Solution by the Perturbation Method for the Case of Low Intensities of Switching

In this section, a small perturbation solution of the time-variant system of two generalized Fokker–Planck equations is considered [19,20]. In order to obtain such an approximation, one has to find correction terms for the eigenvalues and eigenvectors in the isolated states. At this stage, it is assumed that the isolated Fokker–Planck operators are described by a discrete spectrum, i.e. there are two discrete sets of eigenvalues fn g, fn g and eigenfunctions f n ðxÞg and fn ðxÞg such that d ð1Þ 1 d2 ð1Þ L1 FP ½ n ðxÞ ¼ ½K1 ðxÞ n ðxÞ þ ½K ðxÞ n ðxÞ ¼ n n ðxÞ dx 2 dx2 2 d ð2Þ 1 d2 ð2Þ L2 FP ½n ðxÞ ¼ ½K1 ðxÞn ðxÞ þ ½K ðxÞn ðxÞ ¼ n n ðxÞ dx 2 dx2 2 and

ð1 1

m ðxÞ dx n ðxÞ ð1Þ pst ðxÞ

¼

ð1 1

n ðxÞ

m ðxÞ ð2Þ pst ðxÞ

dx ¼ mn ¼

ð1Þ

1 0

ð6:127Þ

if m ¼ n if m ¼ 6 n

ð6:128Þ

ð2Þ

The stationary solutions 0 ðxÞ ¼ pst ðxÞ and 0 ðxÞ ¼ pst ðxÞ correspond to the zero eigenvalue for each isolated equation, i.e. 0 ¼ 0 ¼ 0. 6.3.2.1

General Small Parameter Expansion of Eigenvalues and Eigenfunctions

The next step is to consider the eigenvalues and the eigenfunctions of the perturbed Fokker– Planck operators L1 ½ ðxÞ ¼ L1 FP ½ ðxÞ P2 ðxÞ þ P1 pð2Þ ðxÞ L2 ½ðxÞ ¼ L2 FP ½ðxÞ P1 ðxÞ þ P2 pð1Þ ðxÞ

ð6:129Þ

While both equations must be solved simultaneously, the assumption of a small allows one to treat these equations separately. Due to a similar structure of both operators, the discussion is limited to the operator L1 only. At this point, the exact value of the function p2 ðxÞ is not specified; however, it is assumed to be a known function. A further assumption is that the eigenvalues n and the eigenvectors n ðxÞ of the perturbed operator are holomorphic functions of the small parameter s ¼

P1 P2 s 1 s 2 P2 s 1 þ P1 s 2

ð6:130Þ

where s 1 and s 2 are the characteristic time constants of the isolated systems. The holomorphic property allows the following expansions for small values of s X 2 ð2Þ n ¼ n þ ð s Þð1Þ þ ð Þ þ ¼ þ ð s Þk ðkÞ s n n n n k¼1

n ðxÞ ¼

n ðxÞ þ ð s Þ

ð1Þ n ðxÞ

þ ð s Þ2

ð2Þ n ðxÞ

þ ¼

n ðxÞ þ

1 X ð s Þk

ðkÞ n ðxÞ

k¼1

ð6:131Þ

APPROXIMATE SOLUTION OF THE GENERALIZED FOKKER–PLANCK EQUATIONS ðkÞ

305

ðkÞ

where n and n ðxÞ are to be defined. Since n ðxÞ is the eigenfunction corresponding to the eigenvalue n the following equality must hold L1 ½n ðxÞ ¼ L1 FP ½n ðxÞ P2 n ðxÞ þ P1 pð2Þ ðxÞ 1 X ¼ n n ðxÞ þ ð s Þk L1 FP ½ ðkÞ n ðxÞ P2 k¼1

1 X ð s Þk n ðxÞ P2

ðkÞ n ðxÞ

þ P1 pð2Þ ðxÞ

k¼1

"

¼ n n ðxÞ ¼ n

n ðxÞ

þ s nð1Þ

n ðxÞ

þ s n

ð1Þ n ðxÞ

1 X

þ

# ð s Þmþl ðmÞ n

ðlÞ n ðxÞ

mþ1>1

ð6:132Þ Equating terms with equal powers of on both sides of this expansion, one obtains a system ðkÞ ðkÞ of equations for n and n ðxÞ: P2 ð1Þ þ ¼ n L1 FP ½ s X P2 ðk1Þ ðxÞ ðxÞ ¼ ðmÞ L1 FP ½ ðkÞ n n s n mþl¼k ð1Þ n ðxÞ

6.3.2.2

n nð1Þ ðxÞ

n ðxÞ

P1 ð2Þ p ðxÞ s

ðlÞ n ðxÞ

ð6:133Þ ð6:134Þ

Perturbation of 0 ðxÞ

In the following it is assumed that the solution of the system of the Fokker–Planck eq. (6.129) has a stationary solution. Thus one has to enforce that 0 ¼ 0 ¼ 0, while the corresponding eigenfunction n ðxÞ still can be expressed as 0 ðxÞ ¼

ð1Þ pst ðxÞ

þ

1 X

ð s Þk

ðkÞ 0 ðxÞ

ð6:135Þ

k¼1

A similar expansion must be valid for the perturbed stationary solution of the second Fokker–Planck equation, i.e. 0 ðxÞ ¼

ð2Þ pst ðxÞ

þ

1 X

ðkÞ

ð s Þk 0 ðxÞ

ð6:136Þ

k¼1

Taking this into account, one can rewrite the first of eqs. (6.129) as " 1 X ð1Þ L1 FP ½0 ðxÞ P2 pð1Þ ðxÞ þ 1 Pð2Þ ðxÞ ¼ L1 FP pst ðxÞ þ ð s Þk "

1 X ð1Þ P2 pst ðxÞ þ ðs Þk k¼1

# ðkÞ 0 ðxÞ

# ðkÞ 0 ðxÞ

k¼1

" ð2Þ

þ P1 pst ðxÞ þ

1 X k¼1

# ðkÞ

ð s Þk 0 ðxÞ ¼ 0

ð6:137Þ

306

MARKOV PROCESSES WITH RANDOM STRUCTURES ð1Þ

Since L1 FP ½pst ðxÞ ¼ 0, one obtains the following system of perturbation equations 1 X

L1 FP ½

ð1Þ 0 ðxÞ

P2 ð1Þ P1 ð2Þ pst ðxÞ þ pst ðxÞ ¼ 0 s s

ð6:138Þ

L1 FP ½

ðkÞ 0 ðxÞ

P2 s

P1 ðk1Þ ðxÞ ¼ 0 s 0

ð6:139Þ

n ðxÞ

ð6:140Þ

k¼1 1 X k¼1

Representing

ð1Þ 0 ðxÞ

on the basis of f

ðk1Þ 0

n ðxÞg

ð1Þ 0 ðxÞ

¼

þ

as

1 X

0;p

p¼0

and plugging it into eq. (6.138) one derives 1 X k¼1

L1 FP ½

ðkÞ 0 ðxÞ

¼

1 X

0;p L1 FP ½

n ðxÞ

¼

p¼0

1 X

0;p p

n ðxÞ

¼

p¼0

P2 ð1Þ P1 ð2Þ pst ðxÞ pst ðxÞ s s ð6:141Þ

ð1Þ

Furthermore, multiplying both parts of eq. (6.141) by m ðxÞ=pst ðxÞ and integrating over x, the expression for the coefficients 0;m for m > 0 can be deduced: 0;m

P1 ¼ s m

ð1 1

ð2Þ

pst ðxÞ

m ðxÞ dx; ð1Þ pst ðxÞ

m1

ð6:142Þ

with 0;0 still undefined. The latter can be obtained from the normalization condition, since ð1 p

ð1Þ

1

ð1

ðxÞdx ¼

ð1Þ

1

pst ðxÞdx ¼ P1

ð6:143Þ

which is equivalent to 0;0 ¼ 0. Thus, the first order approximation is p1 ðxÞ

ð1Þ pst ðxÞ

P1

1 X

n ðxÞ

m¼1

m

ð1Þ

ð1 1

ð2Þ

pst ðxÞ

m ðxÞ dx ð1Þ pst ðxÞ

ð6:144Þ

ð1Þ

Furthermore, approximation of n and n ðxÞ can be obtained directly from eq. (6.133). ð1Þ Since f n ðxÞg forms a complete orthogonal basis, the function n ðxÞ can be expanded in terms of this basis: ð1Þ n ðxÞ

¼

1 X p¼0

n;p

p ðxÞ

ð6:145Þ


307

In this case, eq. (6.133) can be rewritten as L1 FP ½

ð1Þ n ðxÞ

"

¼ L1 FP

ð1Þ n ðxÞ

þ n

1 X

n;p

#

p ðxÞ þ n

1 X

p¼0

P2 ð1Þ n ¼ s

n;p

p ðxÞ ¼

p¼0 n ðxÞ

1 X

n;p ðn p Þ

p ðxÞ

p¼0

P1 ð2Þ p ðxÞ s

ð6:146Þ ð1Þ n ðxÞ=pst ðxÞ,

Multiplying both sides of eq. (6.146) by orthogonality, one obtains P2 0¼ ð1Þ n s

integrating over x and using

ð1

P1 ð2Þ n ðxÞ p ðxÞ ð1Þ d x 1 s pst ðxÞ

ð6:147Þ

i.e. ð1Þ n

P2 ¼ s

ð1

P1 ð2Þ n ðxÞ p ðxÞ ð1Þ d x 1 s pst ðxÞ ð1Þ m ðxÞ=pst ðxÞ; m

Furthermore, multiplying both sides of eq. (6.146) by over x and using orthogonality one obtains P1 n;m ðm n Þ ¼ s

ð1 1

ð6:148Þ 6¼ n, integrating

pð2Þ ðxÞ


ð6:149Þ

pð2Þ ðxÞ


ð6:150Þ

and, thus n;m

P1 ¼ s ðm n Þ

ð1 1

Returning to eq. (6.131), one derives n ¼ n þ P2 P1 n ðxÞ ¼ ð1 n;n s Þ

n

ð1 1

pð2Þ ðxÞ

P1

1 X m¼0 m6¼n

n ðxÞ dx ð1Þ pst ðxÞ m ðxÞ

m n

ð1

1

pð2Þ ðxÞ


ð6:151Þ

Integrating the second equation in (6.151) over x and taking into account the normalization conditions, one obtains, for n ¼ 0, 1¼

ð1 1

n ðxÞd x ¼ ð1 0;0 s Þ

ð6:152Þ

308


which implies that 0;0 ¼ 0 and, thus 0 ðxÞ ¼

ð1Þ pst ðxÞ

P1

1 X m¼0 m6¼n

ð1

m ðxÞ

m n

pð2Þ ðxÞ

1


ð6:153Þ

In general n;n ¼ 0 and, therefore n ¼ n þ P2 P1 n ðxÞ ¼

n

P1

1 X

ð1

pð2Þ ðxÞ

1 m ðxÞ

m n

m¼0 m6¼n

ð1 p

n ðxÞ dx ð1Þ pst ðxÞ

ð2Þ

1

ð6:154Þ

m ðxÞ ðxÞ ð1Þ dx pst ðxÞ

It is interesting to note how intensity of switching changes the first smallest non-zero eigenvalue 1 which defines the correlation interval. Indeed, if switching is performed between two processes with the same PDF but different correlation properties, i.e. ð1Þ ð2Þ P2 pst ðxÞ ¼ P1 pst ðxÞ but 1 6¼ 1 , then it follows from eq. (6.154) that 1 ¼ 1 þ P2 ¼ 1 þ 12 > 1

ð6:155Þ

thus providing a shorter correlation interval. The second order approximation can be obtained from eq. (6.134) by setting k ¼ 2 L1 FP ½

ð2Þ n ðxÞ

P2 s

ð1Þ n ðxÞ

X

¼

nðmÞ

ðlÞ n ðxÞ

ð2Þ n ðxÞ

¼ n

ð1Þ n

ð1Þ n ðxÞ

nð2Þ

n ðxÞ

mþl¼2

ð6:156Þ or, equivalently L1 FP ½ Expanding

ð2Þ n ðxÞ

ð2Þ n ðxÞ

þ

n ð2Þ n ðxÞ

P2 ð1Þ ¼ n s

into a series over the basis f ð2Þ n ðxÞ

¼

1 X

ð1Þ n ðxÞ

ð2Þ n

n ðxÞ

ð6:157Þ

n ðxÞg

n;l l ðxÞ

ð6:158Þ

l¼0

and plugging it into eq. (6.157), one obtains L1 FP ½

ð2Þ n ðxÞ 1 X

þ n

ð2Þ n ðxÞ

n;l l l ðxÞ þ n

¼

l¼0

P2 ð1Þ ¼ n s

1 X

n;l l ðxÞ ¼

l¼0 ð1Þ n ðxÞ

nð2Þ

1 X

n;l ðn l Þ l ðxÞ

l¼0 n ðxÞ

ð6:159Þ


Multiplication by

ð1Þ m ðxÞ=pst ðxÞ

309

and integration over x produces, for m ¼ n, ð2Þ n ¼0

ð6:160Þ

P2 ð1Þ n;m n;m ðm n Þ ¼ n s

ð6:161Þ

P2 n;m ð1Þ ; ¼ n s ðm n Þ

ð6:162Þ

and

As a result n;m

m 6¼ n

Thus, it can be seen that the second order approximation does not produce any significant correction terms. This fact justifies application of the small perturbation analysis. Example: Two Gaussian processes. A case of commutation of two Gaussian processes is described by SDE (6.106). In this case n n ¼ ; s 1

" # 1 ðx m1 Þ2 x m1 ffi exp Hn pffiffiffi n ðxÞ ¼ P1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 21 2 1 n!2 21

ð6:163Þ

and "

n ; n ¼ s 2

#

1 ðx m2 Þ x m2 ffi exp Hn pffiffiffi n ðxÞ ¼ P2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 22 2 2 n!2 22 2

ð6:164Þ

Taking eq. (6.163) into account, the expression for the stationary PDF of the process in the first state becomes

2 P2 s 1 ð1Þ 0 ðxÞ ¼ pst ðxÞ41 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2ffi n!2 2

pffiffi 1 1 Hm xm X 21 m¼0 m6¼n

mn

3 ðx m2 Þ x m1 Hm pffiffiffi exp d x5 2 2 2 2 1 1

ð1

"

2

#

!3 pffiffi 1 1 Hm xm 2 n=2 X P2 s 1 m2 m1 21 ð1Þ 1 22 Hm pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 5 pst ðxÞ41 pffiffiffiffi 1 n! m¼0 m n 2 21 22 2

ð6:165Þ

m6¼n

which is very similar to that given by eq. (6.118). Here the following standard integral (7.374.8 in [18]) has been used

pffiffiffi y 2 n=2 exp½ðx yÞ Hn ð xÞdx ¼ ð1 Þ Hn pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 1

ð1

2

ð6:166Þ

310


The first approximation of the eigenvalues is given by eq. (6.151) as " !# n=2 n 1 22 m2 m1 ffi þ P2 1 pffiffiffiffi 1 2 Hm pffiffiffipffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi n ¼ s 1 1 n! 2 21 22

6.3.3

ð6:167Þ

High Intensity Solution

In this section, an approximation of the solution for high intensity of switching is considered. The first step is to obtain conditions for the infinite intensity of switching and then to convert the problem into a small perturbation problem.

6.3.3.1

Zero Average Current Condition

It is easy to see that the Fokker–Planck equations can be written in the form of conservation of probability as P1

@ @ p1 ðx; tÞ ¼ P1 J1 ðxÞ P2 pð1Þ ðx; tÞ þ P1 pð2Þ ðx; tÞ @t @x

@ @ J2 ðxÞ P1 pð2Þ ðx; tÞ þ P2 pð1Þ ðx; tÞ P2 p2 ðx; tÞ ¼ P2 @t @x

ð6:168Þ

where ð1Þ

J1 ðx; tÞ ¼ K1 ðxÞp1 ðx; tÞ J2 ðx; tÞ ¼

ð2Þ K1 ðxÞp2 ðx; tÞ

1 @ ð1Þ ½K2 ðxÞp1 ðx; tÞ 2@ x

1 @ ð2Þ ½K ðxÞp2 ðx; tÞ 2@ x 2

ð6:169Þ

and pð1Þ ðx; tÞ ¼ P1 p1 ðx; tÞ pð2Þ ðx; tÞ ¼ P2 p2 ðx; tÞ

ð6:170Þ

are the probability current densities, created by the drift and the diffusion of the continuous Markov process. Additional probability flows P2 p1 ðx; tÞ and P1 p2 ðx; tÞ are due to the absorption and regeneration of trajectories. probabilities P1 and P2 represent the proportion of time during which a given flow is observed. Adding both eqs. (6.168), one obtains @ @ pðx; tÞ ¼ Jðx; tÞ @t @x

ð6:171Þ


311

where the change in the average probability current Jðx; tÞ ¼ P1 J1 ðx; tÞ þ P2 J2 ðx; tÞ

ð6:172Þ

is now the only factor responsible for the variation of the probability density pðx; tÞ ¼ p1 ðx; tÞ þ p2 ðx; tÞ ¼ P1 p1 ðx; tÞ þ P2 p2 ðx; tÞ

ð6:173Þ

in time. For a stationary flow, this current must remain constant, i.e. JðxÞ ¼ P1 J1 ðxÞ þ P2 J2 ðxÞ 1 @ ð1Þ ½K ðxÞp1 ðxÞ 2 @x 2 P1 @ ð1Þ P2 @ ð2Þ ð1Þ ð2Þ ½K2 ðxÞp1 ðxÞ ½K ðxÞp2 ðxÞ ¼ P1 K1 ðxÞp1 ðxÞ þ P2 K1 ðxÞp2 ðxÞ 2 @x 2 @x 2 ð1Þ

ð2Þ

¼ P1 K1 ðxÞp1 ðxÞ þ P2 K1 ðxÞp2 ðxÞ

¼ J0 ¼ const:

ð6:174Þ

If p1 ðxÞ and p2 ðxÞ are defined on a semi-infinite or infinite interval, probability densities p1 ðxÞ and p2 ðxÞ, as well as their derivatives, must vanish at infinity, together with the probability current density JðxÞ. Thus, the constant J0 must be equal to zero: J0 ¼ 0

ð6:175Þ

This condition is known as the natural boundary condition in the theory of continuous Markov processes [13 and Section 5.3.3]. However, in the case of processes with random structure it is the average probability current density which vanishes; individual current densities J1 ðxÞ and J2 ðxÞ may remain functions of x in such a way that their sum is zero. 6.3.3.2

Asymptotic Solution P1 ðxÞ

For a very high intensity all the probability densities pðkÞ ðxÞ must approach a fixed distribution p1 ðxÞ pðkÞ ðxÞ ¼ ak p1 ðxÞ

ð6:176Þ

in order to satisfy the balance conditions L X

kl pðkÞ ðxÞ 0

ð6:177Þ

k¼1

for any l and x. The normalizing coefficients are to be chosen to satisfy the normalization condition Pk ¼

ð1 p 1

ðkÞ

ðxÞdx ¼ ak

ð1 1

p1 ðxÞdx ¼ ak

ð6:178Þ

312


As far as a particular form of the asymptotic distribution p1 ðxÞ is concerned, nothing can be deduced from the eq. (6.177) itself. In order to obtain the expression for p1 ðxÞ, one can take advantage of the fact that pðlÞ ðxÞ ¼ Pl p1 ðxÞ. Taking this into account, and adding all M generalized Fokker–Planck equations, one obtains M M @ X @ X pðlÞ ðx; tÞ ¼ p1 ðx; tÞ @ t l¼1 @ t l¼1 " # " # M M 2 X X d 1 d ðlÞ ðlÞ p1 ðx; tÞ ¼ Pl K1 ðx; tÞ þ p1 ðx; tÞ Pl K2 ðx; tÞ 2 dx 2 dx l¼1 l¼1

ð6:179Þ or M @ X d 1 d2 p1 ðx; tÞ ¼ ½K1 1 ðxÞp1 ðx; tÞ þ ½K2 1 ðxÞp1 ðx; tÞ @ t l¼1 dx 2 dx2

ð6:180Þ

Here the average drift K1 1 ðxÞ and the average diffusion are given by K1 1 ðxÞ ¼

L X

ðlÞ

Pl K1 ðxÞ; K2 1 ðxÞ ¼

l¼1

L X

ðlÞ

Pl K2 ðxÞ

ð6:181Þ

l¼1

The formal solution of eq. (6.180) is easy to obtain since it is a regular Fokker–Planck equation: " ð P # ðlÞ L x l¼1 Pl K1 ðsÞ ð6:182Þ ds p1 ðxÞ ¼ C exp 2 PL ðlÞ 1 l¼1 Pl K2 ðsÞ Example: Two Gaussian processes. Once again a system of two SDEs (6.106) is 2 21 2 22 ð1Þ ð2Þ ð1Þ ð2Þ xm2 1 considered. In this case, K1 ¼ xm s 1 ; K1 ¼ s 2 ; K2 ðxÞ ¼ s 1 and K2 ðxÞ ¼ s 2 . Thus, the averaged drift and diffusion are

ðx m1 Þ ðx m1 Þ x m1 1 þ P2 ¼ K1 1 ðxÞ ¼ P1 s 1 s 1 s 1

ð6:183Þ

and K2 1 ðxÞ ¼ P1

2 21 2 2 2 þ P2 2 ¼ 2 1 s 1 s 2 1

ð6:184Þ

respectively. Here m1 1

P1 m1 s 2 þ P2 m2 s 1 ¼ ; P1 s 2 þ P2 s 1

21

P1 21 s 2 þ P2 22 s 1 ¼ P1 s 2 þ P2 s 1

and

1 ¼

s 1 s 2 P1 s 2 þ P2 s 1 ð6:185Þ


313

are parameters of the limiting PDF. It can be seen that the limiting distribution p1 ðxÞ itself is Gaussian with parameters defined by eq. (6.185): " # 1 ðx m1 Þ2 ð6:186Þ p1 ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2 21 2 21 Example: Two Gamma distributed processes. In this case, the system of two SDEs ð1Þ ð2Þ ð1Þ 1 2 ; K1 ¼ 2 x ; K2 ðxÞ ¼ 12s 1 and (6.120) is analyzed. In this case, K1 ¼ 1 x 1 s 1 2 s 2 ð2Þ K2 ðxÞ ¼ 22s 2 . Thus, the averaged drift and diffusion are

1 x 1 2 x 2 1 x a1 K1 1 ðxÞ ¼ P1 þ P2 ¼ 1 s 1 2 s 2 1 s 1

ð6:187Þ

and K2 1 ðxÞ ¼ P1

2 2 2 þ P2 ¼ 1 s 1 2 s 2 1 1

ð6:188Þ

respectively. Here P1 11 s 2 þ P2 12 s 1 1 ¼ ; 1 P1 s 2 þ P2 s 1

P1 11 s 2 þ P2 22 s 1

1 ¼

P1 s 2 þ P2 s 1

and

1 ¼

s 1 s 2 P1 s 2 þ P2 s 1 ð6:189Þ

are the parameters of the limiting PDF. It can be seen that the limiting distribution p1 ðxÞ itself is a Gamma PDF with parameters defined by eq. (6.189). p1 ðxÞ ¼

a1 1 expð1 xÞ ð 1 Þ

ð6:190Þ

Taking into account that for the Gamma distribution m¼

and

¼

1

ð6:191Þ

one can rewrite eq. (6.189) as m1 ¼

P1 m1 s 2 þ P2 m2 s 1 ; P1 s 2 þ P2 s 1

1 ¼

P1 1 s 2 þ P2 2 s 1 P1 s 2 þ P2 s 1

and

1 ¼

s 1 s 2 P1 s 2 þ P2 s 1 ð6:192Þ

i.e. transformations of the mean value and standard deviation follow the same rules as for the Gaussian process considered above. Another important conclusion which can be drawn is the importance of correlation properties of the isolated system on the resulting parameters of the distribution. In the case when the distributions in the isolated states are identical, the resulting marginal PDF of

314


course would remain the same as in the isolated states. However, if there is a difference, even in scale (i.e. only in variance), the effect of correlation in each isolated case begins to play a prominent role in the final result. 6.3.3.3

Case of a Finite Intensity

Of course, for a finite but large , a certain correction term must be added. It is a goal of the considerations of this section to derive an appropriate expression for such a correction term. In order to keep the derivations compact the case of two equations is considered below: @ ð1Þ 1 @ 2 ð1Þ ½K1 ðxÞpð1Þ ðxÞ þ ½K ðxÞpð1Þ ðxÞ P2 pð1Þ ðxÞ þ P1 pð2Þ ðxÞ @x 2 @x2 2 @ ð2Þ 1 @ 2 ð2Þ 0 ¼ ½K1 ðxÞpð2Þ ðxÞ þ ½K2 ðxÞpð2Þ ðxÞ þ P2 pð1Þ ðxÞ þ P1 pð2Þ ðxÞ 2 @x 2 @x

0¼

ð6:193Þ

This can be rewritten in the vector form as

! ! ! @ 1 @2 ½K 1 ðxÞ p ðxÞ þ ½K 2 ðxÞ p ðxÞ Q p ðxÞ ¼ 0 2 @x 2 @x

ð6:194Þ

or !

!

LFP ½ p ðxÞ ¼ Q p ðxÞ

ð6:195Þ

where " K 1 ðxÞ ¼

ð1Þ

K1 ðxÞ 0

# 0 ; ð2Þ K1 ðxÞ

" K 2 ðxÞ ¼

ð1Þ

K2 ðxÞ 0

# 0 ; ð2Þ K2 ðxÞ

P2 Q¼ P2

P1 P1

ð6:196Þ and !

LFP ½ p ðxÞ ¼

! ! @ 1 @2 ½K 1 ðxÞ p ðxÞ þ ½K 2 ðxÞ p ðxÞ 2 @x 2 @x

ð6:197Þ

is the Fokker–Planck operator written in a matrix form. One may look for the solution of eq. (6.194) in the form of an infinite series !

!

p ðxÞ ¼ P1 p ðxÞ þ 1

1 X Pm m¼1

m

!

!

½ p m ðxÞ p 1 ðxÞ

ð6:198Þ

where

P1

P1 ¼ 0

0 ; P2

! p 1 ðxÞ

p ðxÞ ¼ 1 p1 ðxÞ

ð6:199Þ


315

Pm and ~ pm ðxÞ are the matrix and the vector function to be defined. Substituting a solution in the form (6.199) into eq. (6.194), and using the linearity of the Fokker–Planck equation, one obtains LFP ½~ p ðxÞ

1 X @ 1 @2 @ Pm ½K 1 ðxÞ~ K 1 ðxÞ m ½~ p1 ðxÞ þ ½K 2 ðxÞ~ p1 ðxÞ þ pm ðxÞ ~ p1 ðxÞ ¼ @x 2 @ x2 @x m¼1 " # !

1 1 X X 1 @2 Pm Pm Pm ~ ~ K 2 ðxÞ m ½~ pm ðxÞ ~ p1 ðxÞ ¼ Q P1 þ p1 ðxÞ þ p ðxÞ 2 @ x2 m m m m¼1 m¼1

¼

1 X QPm m¼1

m1

½~ pm ðxÞ ~ p1 ðxÞ

ð6:200Þ

p1 ¼ 0. By equating the terms with equal negative powers of , one can obtain a since QP1 ~ sequence of equations QP1 ½~ p1 ðxÞ ~ p1 ðxÞ ¼

@ 1 @2 ½K 1 ðxÞ~ p1 ðxÞ þ ½K 2 ðxÞ~ p1 ðxÞ; m ¼ 0 @x 2 @x2

ð6:201Þ

and pm ðxÞ ~ p1 ðxÞ ¼ QPm ½~

1 X

m¼1

þ

@ fK 1 ðxÞPmþ1 ½~ pmþ1 ðxÞ ~ p1 ðxÞg @x

1 @2 fK 2 ðxÞPmþ1 ½~ pmþ1 ðxÞ ~ p1 ðxÞg; 2 @x2

m>0

ð6:202Þ

It follows from eq. (6.182) that

@ ð1Þ 1 @ 2 ð1Þ P1 ½K1 ðxÞp1 ðxÞ þ ½K ðxÞp1 ðxÞ @x 2 @x2 2

@ ð2Þ 1 @ 2 ð2Þ ¼ P2 ½K1 ðxÞp1 ðxÞ þ ½K ðxÞp1 ðxÞ @x 2 @x2 2

ð6:203Þ

or, equivalently ð1Þ

ð2Þ

P1 LFP ½p1 ðxÞ ¼ P2 LFP ½p1 ðxÞ

ð6:204Þ

Thus, the set of eqs. (6.201)–(6.202) are just the same equation repeated twice, i.e. a unique solution cannot be obtained just by considering eq.(6.202). This is the consequence of the fact that the matrix Q is degenerate, i.e. detðQÞ ¼ 0. As a result, some additional conditions are required to obtain a unique solution of the original problem. This additional condition is a constant probability current condition (6.174), considered in section 6.3.3.1. To make proper use of this condition, eq. (6.202) can be written as ð1Þ

ð2Þ

P1 L1 FP ½p1 ðxÞ P1 P2 ½p1 ðxÞ p1 ðxÞ þ P1 P2 ½p1 ðxÞ p1 ðxÞ ¼ 0

ð6:205Þ

316


or, equivalently ð1Þ

1 1 ð2Þ ð1Þ L1 FP ½p1 ðxÞ; p1 ðxÞ ¼ p1 ðxÞ L1 FP ½p1 ðxÞ P2 P2

ð2Þ

p1 ðxÞ p1 ðxÞ ¼

ð6:206Þ

Using approximation (6.198), the zero current density conditions can now be rewritten as P1 @ P2 @ ð1Þ ð2Þ ½K ðxÞp1 ðxÞ ½K ðxÞp1 ðxÞ 2 @x 2 2 @x 2

1 P1 @ P2 @ ð1Þ ð1Þ ð2Þ ð2Þ ð1Þ ð1Þ ð2Þ ð2Þ P1 K1 ðxÞp1 ðxÞ þ P2 K1 ðxÞp1 ðxÞ ½K ðxÞp1 ðxÞ ½K ðxÞp1 ðxÞ ¼ 0 þ 2 @x 2 2 @x 2 ð1Þ

ð2Þ

P1 K1 ðxÞp1 ðxÞ þ P2 K1 ðxÞp1 ðxÞ

ð6:207Þ

Furthermore, since p1 ðxÞ satisfies the condition ð1Þ

ð2Þ

½P1 K1 ðxÞ þ P2 K1 ðxÞp1 ðxÞ

1 @ ð1Þ ð2Þ ½fP1 K2 ðxÞ þ P2 K2 ðxÞgp1 ðxÞ ¼ 0 2@ x

ð6:208Þ ð1Þ

eq. (6.207) can be simplified using expression (6.206) to produce an equation for p1 ðxÞ: ð1Þ

1 @ ð1Þ ð2Þ ð1Þ f½P1 K2 ðxÞ þ P2 K2 ðxÞp1 ðxÞg 2@ x 1 @ ð2Þ ð2Þ fK2 ðxÞL1 FP ½p1 ðxÞg ¼ K1 ðxÞL1 FP ½p1 ðxÞ 2@ x

ð2Þ

ð1Þ

½P1 K1 ðxÞ þ P2 K1 ðxÞp1 ðxÞ

ð6:209Þ

Alternatively, using the notation of eq. (6.181), one obtains ð1Þ

K1 1 ðxÞp1 ðxÞ

1 @ 1 @ ð1Þ ð2Þ ð2Þ ½K2 1 ðxÞp1 ðxÞ ¼ K1 ðxÞL1 FP ½p1 ðxÞ fK2 ðxÞL1 FP ½p1 ðxÞg 2@x 2@ x ð6:210Þ

which is an ordinary non-homogeneous differential equation of the first order. The solution of the homogeneous equation pðxÞ is once again proportional to p1 ðxÞ: ðx

1 K1 1 ðxÞ pðxÞ ¼ C p1 ðxÞ ¼ exp 2 dx ð6:211Þ K2 1 ðxÞ 1 K2 1 ðxÞ ð1Þ

The next step is to find solution p1 ðxÞ using the method of undefined coefficients, i.e. the solution is thought of in the form ðx

1 K1 1 ðxÞ ð1Þ exp 2 dx ð6:212Þ p1 ðxÞ ¼ CðxÞ K1 1 ðxÞ 1 K2 1 ðxÞ where function CðxÞ has to be determined. Substituting expression (6.212) into eq. (6.210), one obtains a differential equation for the unknown function CðxÞ ðx

1 K1 1 ðxÞ d 1 @ ð2Þ ð2Þ exp 2 CðxÞ ¼ K1 ðxÞL1 FP ½p1 ðxÞ fK2 ðxÞL1 FP ½p1 ðxÞg dx 2 dx 2@ x 1 K2 1 ðxÞ ð6:213Þ

317

CONCLUDING REMARKS

whose general solution is

K1 1 ðyÞ dy exp 2 CðxÞ ¼ C0 þ 1 1 K2 1 ðyÞ

@ ð2Þ ð2Þ fK ðsÞL1 FP ½p1 ðsÞg 2 K1 ðsÞL1 FP ½p1 ðsÞ d s @s 2 ðx

ð2Þ

ðs

ð2Þ

ð6:214Þ

ð2Þ

Since K1 ðxÞ and K2 ðxÞ are related through the probability density p0 st ðxÞ in the second isolated state as ð2Þ

ð2Þ

K1 ðsÞ ¼

ð2Þ

1 dds ½K2 ðsÞp0 st ðsÞ ð2Þ 2 p ðsÞ

ð6:215Þ

0 st

then the expression in braces can be further simplified to produce @ ð2Þ ð2Þ fK2 ðsÞL1 FP ½p1 ðsÞg 2 K1 ðsÞL1 FP ½p1 ðsÞ @s ð2Þ ð2Þ ð2Þ ð2Þ K ðsÞp20 ðsÞ dsd fK2 ðsÞL1 FP ½p1 ðsÞg K2 ðsÞL1 FP ½p1 ðsÞ dds ½K2 ðsÞp20 ðsÞ ð2Þ ½K2 ðsÞp20 ðsÞ ¼ 2 ð2Þ ½K2 ðsÞp20 ðsÞ2 d L1 FP ½p1 ðsÞ ð2Þ ð6:216Þ ¼ K2 ðsÞp20 ðsÞ d s p20 ðsÞ

Finally, eq. (6.214) can be written as ðx

ð2Þ

K2 ðsÞp20 ðsÞ d L1 FP ½p1 ðsÞ ds CðxÞ ¼ C0 þ C1 p20 ðsÞ 1 K2 1 ðsÞp1 ðsÞ d s " ð ! # ðx ð2Þ s K1 ðuÞ K1 1 ðuÞ L1 FP ½p1 ðsÞ ¼ C0 þ C1 exp 2 du d ð2Þ K2 1 ðuÞ p20 ðsÞ 1 1 K2 ðuÞ ( )x ! ðx ð2Þ ð2Þ ð2Þ K2 ðsÞðL1 FP ½p1 ðsÞÞ K1 ðsÞ K1 1 ðsÞ K2 ðsÞL1 FP ½p1 ðsÞ ds ¼ C0 þ C1 C1 ð2Þ K2 1 ðsÞp1 ðxÞ K2 1 ðsÞ K2 1 ðsÞp1 ðsÞ 1 K ðsÞ 1

2

ð6:217Þ

The values of these constants can be determined through the normalization conditions.

6.4

CONCLUDING REMARKS

The following remarks can be added to the results obtained for processes with random structure: 1. The general theory of Markov processes allows the same level of development as the theory of continuous Markov processes. In particular, these results can be extended to the

318


Pontryagin equations in attainment problems [21], narrow band processes with random structure, etc. 2. Analytical results in a closed form can only be obtained for a case of a very low or a very high intensity of switching. These limiting cases are especially attractive since they can be seen as perturbed solutions of isolated systems. 3. Despite the asymptotic nature of the results discussed, they can be readily applied to model processes with weak non-stationary, mixture models such as Middleton Class A noise, etc. [22]. 4 In general, the model of a process with random structure, with a proper choice of the absorption and restoration flows, allows for modelling of a great variety of relatively little studied processes which are very important in applications, such as branching processes, etc.

REFERENCES 1. I. Kazakov and V. Artem’ev, Optimization of Dynamic Systems with Random Structure, Moscow: Nauka, 1980 (in Russian). 2. D. Middleton, Non-Gaussian Noise Models in Signal Processing for Telecommunications: New Methods and Results for Class A and Class B Noise Models, IEEE Trans. Information Theory, vol. 45, no, 5, May, 1999, pp. 1129–1149. 3. E. Lutz, D. Cugan, M. Dippold, F. Dolaninsky, and W. Papke, The Land Mobile Satellite Communication Channel–Recording, Statistics and Channel Model, IEEE Trans. Vehicular Technology, vol. 40, no. 2, 1991, pp. 375–380. 4. A Brusentzov and V. Kontorovich, Radio Channel Modeling Based on Dynamic Systems with Random Structure, Proc. of the Int. Con. Commsphere-91, Hertzlia, Israel, December, 1991 5. D. Goodman and S. Wei, Efficiency of Packet Reservation Multiple Access, IEEE Trans. Vehicular Technology, vol. 40, no. 1, pp. 170–176, February, 1991. 6. L. R. Rabiner, A Tutorial on Hidden Markov Models and Selected Applications in Speech Recognition, Proc. of the IEEE, vol. 77, February, 1989, pp. 257–286. 7. J. Dellec, J. G. Proakis, and H. L. Hausen, Discrete Time Processing of Speech Signals, New York: MacMillan, 1993. 8. M. Zorzi, Outage and Error Events in Bursty Channels, IEEE Trans. Communications, vol. 46, March, 1998, pp. 349–356. 9. W. Turin and R. van Nobelen, Hidden Markov Modeling of Flat Fading Channels, IEEE Journal Selected Areas of Communications, vol. 16, no. 12, December, 1998, pp. 1809–1817. 10. F. Babich and G. Lombardi, A Measurement Based Markov Model for the Indoor Propagation Channel, Proc. IEEE Vehicular Technology Conferenece Proc, vol. 1, Phoenix, AZ, May, 1997, pp. 77–81. 11. F. Babich, G. Lombardi, S. Shiavon, and E. Valentinuzzi, Measurements and Analysis of the Digital DECT Propagation Channel, Proc. of the IEEE Conf. Wireless Personal Communications, Florence, Italy, October, 1998, pp. 593–597. 12. K. K. Fang, S. Kotz, and K. W. Ng, Symmetric Multivariate and Related Distributions, New York: Chapman and Hall, 1990. 13. R. Risken, The Fokker–Planck Equation: Methods of Solution and Applications, Berlin: Springer, 1996, 472 p. 14. V. Kontorovich and V. Lyandres, Dynamic Systems with Random Structure: an Approach to the Generation of non-Stationary Stochastic Processes, Journal of Franklin Institute, vol. 336, 1999, pp. 939–954.

REFERENCES

319

15. V. Ya. Kontorovich, S. Primak, and K. Almustafa, On Limits Of Applicability of Cumulant Method to Analysis of Dynamic Systems with Random Structure and Deterministic Systems With Chaotic Behaviour, Proc. DCDIS-2003, Guelph, Ontario, Canada, May, 2003. 16. J. Ord, Families of Frequency Distributions, London: Griffin, 1972. 17. M. Abramovitz and I. Stegun, Handbook of Mathematical Functions, Dover, 1965. 18. I. Gradsteyn and I. Ryzhik, Tables of Integrals, Series, and Products, New York: Academic Press, 2000. 19. Yu. A. Mitropolskii and N. van Dao, Applied Asymptotic Methods in Nonlinear Oscillations, Boston: Kluwer Academic Publishers, 1997 20. T. Kato, Perturbation Theory for Linear Operators, New York: Springer-Verlag, 1966. 21. V. Kontorovich, Pontryagin Equations for Non-Linear Dynamic Systems with Random Structure, Nonlinear Analysis, vol. 47, 2001, pp. 1501–1512. 22. H. Inoue, K. Sasajima, and M. Tanaka, A simulation of ‘‘Class A’’ Noise by P-CNG, Proc. IEEE Int Symp. EMC, 1999, vol. 1, pp. 520–525. 23. O. A. Ladyzhenskaia and N. N. Ural’tseva, Linear and quasilinear elliptic equation, Trans. Editor: L. Ehrenpreis, New York: Academic Press, 1968. 24. B. S. Everitt and D. J. Hand, Finite Mixture Distributions, New York: Chapman and Hall, 1981. 25. V. Kontorovich and V. Lyandres, Pulse Non-Stationary Process Generated by Dynamic Systems with Random Structure, Journal of Franklin Institute, 2003. 26. V. Kontorovich, V. Lyandres, and S. Primak, Non-Linear Methods in Information Processing: Modeling, Numerical Simulation and Applications, DCDIS, series B: Applications & Algorithms, vol. 10, October 2003, pp. 417–428. 27. A. N. Malakhov, Cumulant Analysis of Non-Gaussian Random Processes and Their Transformations, Moscow: Sovetskoe Radio, 1978 (in Russian).

7 Synthesis of Stochastic Differential Equations 7.1

INTRODUCTION

Modeling of signals and interference in the fields of communications, radar, sonar and speech processing is usually based on an assumption that these processes may be considered as stationary (or at least locally stationary) and that experimental estimations of their simplest statistical characteristics, autocovariance function (ACF) and marginal probability density function (PDF), are available. While the generation of stationary random processes (sequences) with either a specified PDF or a required valid ACF does not present any principal difficulty, the solution of the joint problem requires much more effort. One of the most often used methods was suggested in [1]. This approach utilizes sequential combinations of linear filtering and zero memory non-linear transformations of White Gaussian Noise (WGN). A different technique is based on treatment of a process with the prescribed characteristics as a stationary solution of a proper system of stochastic differential equations (SDE) with the WGN or Poisson process as an external driving force [2,3]. Such an interpretation seems attractive as it takes advantage of Markov process theory and appears to be efficient in the modeling of correlated non-Gaussian processes. Applications of this approach can be found useful in modeling of continuous and impulsive noise in communication systems [4], radar [5,6], biology [7], statistical electromagnetics [8], bursty internet traffic, Middleton Class A noise, intersymbol interference combined with additive noise [9,10–13], speech [14] and stochastic ratchets [15,16], etc. It is important to mention [2,64] that an SDE approach provides a unified framework for simulation of the above mentioned phenomena. For practical purposes it is important to distinguish between baseband processes with an exponential type covariance function and narrow band processes which often have exponential decaying cosine covariation functions. A class of processes with a certain PDF and exponent ACF was originally considered by E. Wong [18,21], and A. Haddad [22,25] in the sixties and has been extended to narrow band processes in [26].


322

SYNTHESIS OF STOCHASTIC DIFFERENTIAL EQUATIONS

The following two definitions are important in the further derivations Definition 7.1. A Markov scalar diffusion random stationary process ðtÞ is called a process if its autocovariance function is a pure exponent, i.e. C ðÞ ¼ 2 expðjjÞ

ð7:1Þ

Definition 7.2. A stationary random process ðtÞ is called a ð; !Þ process if it is a component of a diffusion Markov process and has the covariance function given by a decaying exponent Cxx ðÞ ¼ 2 exp½jj cos !

ð7:2Þ

Although processes are usually referred to as wide sense stationary Markov (WSSM) [27] or separable class [25] processes, the terminology of definitions used is more physically revealing.

7.2 7.2.1

MODELING OF A SCALAR RANDOM PROCESS USING A FIRST ORDER SDE General Synthesis Procedure for the First Order SDE

The goal of this section is to design a procedure which allows one to synthesize a first order SDE (Ito form) dx ¼ f ðxÞ þ gðxÞðtÞ dt

ð7:3Þ

such that its stationary solution would have a given PDF px st ðxÞ and a given correlation interval corr . pffiffiffiffiffiffiffiffiffiffiffi For a moment one can assume that a non-negative function gðxÞ ¼ K2 ðxÞ 0 is already chosen and the goal is to find a proper drift K1 ðxÞ ¼ f ðxÞ such that ðx

C K1 ðsÞ exp 2 ds ð7:4Þ px st ðxÞ ¼ K2 ðxÞ 1 K2 ðsÞ Multiplying both sides of eq. (7.4) by K2 ðxÞ ¼ g2 ðxÞ, taking the natural logarithm and differentiating with respect to x one obtains 2

K1 ðxÞ d ¼ ln½K2 ðxÞpx st ðxÞ K2 ðxÞ dx

ð7:5Þ

Solving eq. (7.5) with respect to K1 ðxÞ ¼ f ðxÞ the desired synthesis equation becomes [2] 2 3 d px;st ðxÞ d 1 d 16 7 þ K2 ðxÞ5 ð7:6Þ f ðxÞ ¼ K1 ðxÞ ¼ K2 ðxÞ ln½K2 ðxÞpx st ðxÞ ¼ 4K2 ðxÞ dx 2 dx 2 dx px st ðxÞ

MODELING OF A SCALAR RANDOM PROCESS USING A FIRST ORDER SDE

323

Recalling that K2 ðxÞ ¼ g2 ðxÞ, eq. (7.6) can be rewritten as 3 d p ðxÞ x st 16 d 7 þ 2gðxÞ gðxÞ5 f ðxÞ ¼ K1 ðxÞ ¼ 4g2 ðxÞ dx 2 dx px st ðxÞ 2

ð7:7Þ

Thus, there are many SDEs which have the same marginal PDF as [28]. It will be seen from future analysis that the choice of function gðxÞ defines higher order statistics, such as the covariance function, correlation interval, etc. Further pffiffiffiffisimplification can be achieved if the diffusion is assumed to be constant, i.e. gðxÞ ¼ K ¼ const. In this case K d px st ðxÞ K d ln px st ðxÞ f ðxÞ ¼ 2 dx ¼ 2 dx px st ðxÞ

ð7:8Þ

This was first suggested in [30] and first published in English in [2]. The only parameter yet to be determined is the scale parameter K, defining the correlation interval. The value of the correlation interval sometimes1 can be estimated using an approximate formula, derived in [2,31] (see also Section 5.8): corr

1 a1 þ a3 n0

ð7:9Þ

where k 1 dk1 K d ¼ K ðxÞ ln p ðxÞ ak ¼ 1 xst k! dxk1 2k! dxk x¼0 x¼0

ð7:10Þ

Ð1 4 x px st ðxÞdx m4 n0 ¼ ¼ Ð1 1 2 m2 1 x px st ðxÞdx

ð7:11Þ

and

As a result K

1

1

corr d ln px st ðxÞ dx

2 n0 d3 þ ln p ðxÞ x st 3 x ¼ 0 6 dx x¼0

ð7:12Þ

This approximation works well for the case when drift is a symmetric non-linear function well approximated by a cubic polynomial, i.e. for K1 ðxÞ b1 x þ b3 x3. In other cases alternative methods must be used. If the spectrum of the corresponding Fokker–Planck equation is discrete and known, it is appropriate to choose K such that 1 ¼ 1=corr .

324


Example: Gaussian Markov process. In this case the stationary marginal PDF px st ðxÞ is given by " # 1 ðx mÞ2 px st ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp 22 22

ð7:13Þ

d xm ln px st ðxÞ ¼ 2 dx

ð7:14Þ

and, thus

In turn, eq. (7.12) provides the value of K equal to K¼

22 corr

ð7:15Þ

Finally, plugging expressions (7.14) and (7.15) into eq. (7.8) one obtains the generating SDE for a Markov process with a Gaussian marginal PDF xm þ x_ ¼ corr

sffiffiffiffiffiffiffiffi 22 ðtÞ corr

ð7:16Þ

The corresponding Fokker–Planck equation for the transitional density ðx; t j x0 ; t0 Þ has been solved in section 5.3.6, only the final result is shown below

1 1 ðs m2 Þ X 1 xm x0 m kjj Hk pffiffiffi ðx; t j x0 ; t0 Þ ¼ pffiffiffiffiffiffiffiffiffiffi exp Hk pffiffiffi exp 22 k! corr 2 2 22 k¼0 ð7:17Þ Here Hk ðxÞ are Hermitian polynomials [32] and ¼ t t0 . It is easy to see that the transitional density (7.17) gives rise to an exponential covariance function Cxx ðÞ ¼

jj ðx mÞðx0 mÞðx; t j x0 ; t0 Þpx st ðx0 Þdx dx0 ¼ exp ð7:18Þ corr 1

ð1 ð1

2

1

i.e. approximation (7.12) produces the exact result. It is interesting to note that the Markov process defined by SDE (7.16) is also a Gaussian process, i.e. the joint PDF at any number of time instances remains Gaussian. However, choosing K2 ðxÞ ¼ x in eq. (7.7) one can obtain a non-Gaussian Markov process with a Gaussian marginal PDF. This process would also have a non-exponential covariance function.


325

Example: a random process with two-sided Laplace marginal PDF. The two-sided Laplace distribution is defined by the following PDF px st ðxÞ ¼ exp½2jxj

ð7:19Þ

where > 0 is a parameter of the distribution. PDF (7.19) is often used to model telephone noise [33] and other phenomena. It is easy to see from eq. (7.7) that the generating SDE has the following form pffiffiffiffi x_ ¼ K sgn x þ K ðtÞ ð7:20Þ with the constant diffusion coefficient K yet to be determined. Since further differentiation of the drift is impossible, the approximation (7.12) cannot be used here. It is also known [38] that the corresponding Fokker–Planck equation has only one discrete eigenvalue 0 ¼ 0 and the rest is a continuous spectrum. Thus it is not possible to use an approximation by a second smallest eigenvalue as well. A fair estimate can be obtained through the statistical linearization of the non-linear function f ðxÞ in the generating SDE [36]. In other words, one needs to find a slope such that the mean square error, ð1 2 ð f ðxÞ xÞ2 px st ðxÞdx ð7:21Þ Efð f ðxÞ xÞ g ¼ 1

is minimized. Taking the derivative of both sides with respect to and equating it to zero one obtains the optimal value of the slope to be opt

Ð1 x f ðxÞpx st ðxÞdx 1 Ð1 ¼ 2 1 ¼ 2 corr 1 x px st ðxÞdx

ð7:22Þ

In the case of SDE (7.20) this estimate becomes opt ¼ 2K ¼

1 corr

ð7:23Þ

Thus, in order to provide a proper correlation interval one has to choose K¼

1 2corr

ð7:24Þ

and the generating SDE becomes sffiffiffiffiffiffiffiffiffiffiffiffiffi 1 1 x_ ¼ ðtÞ sgnx þ 2corr 2corr

ð7:25Þ

A large number of further examples can be constructed and some of them are summarized in Appendix A on the web.

326

7.2.2


Synthesis of an SDE with PDF Defined on a Part of the Real Axis

It can be seen from eq. (7.8) that the synthesis of SDEs implicitly assumes existence of the logarithmic derivative on the whole real axis, i.e. px st ðxÞ 6¼ 0. However, many useful PDFs have non-zero values only on a part of the real axis. In this case the direct application of eq. (7.8) is not appropriate. Simply ignoring intervals where px st ðxÞ ¼ 0 leads to controversy. For example, a linear SDE x_ ¼ x þ ðtÞ

ð7:26Þ

describes both a Gaussian and a one-sided Gaussian process. Furthermore, numerical simulation of eq. (7.26), using for example the Euler scheme (see Appendix C on the web and [34]), would reproduce only the Gaussian PDF. Interestingly enough, the related Fokker– Planck equation uniquely defines the distributions properly since it is supplemented by a set of boundary conditions, which are different for the Gaussian (natural boundary conditions at x ¼ 1) and one-sided Gaussian (reflecting boundary at x ¼ 0 and natural boundary condition at x ¼ 1) [38]. Thus, certain modifications must be made to the synthesis eq. (7.8) to accommodate these boundary conditions. The following procedure has been suggested in [39]. Let px st ðxÞ be defined only on a finite interval x 2 ½a; b and ðx; "Þ be a function which satisfies the following conditions: ðx; "Þ > 0 on the real axis; ~ pðx; "Þ ¼ ½" þ px st ðxÞðx; "Þ is a PDF for " > 0; the logarithmic derivative of ~ pðx; "Þ is defined on the whole real axis and lim ~pðx; "Þ ¼ px st ðxÞ

"!0

One possible candidate for such a function is one defined in [39] ðx; "Þ ¼

Cð"Þ

2x a b 1þ ba

2="

ð7:27Þ

Here Cð"Þ is chosen in such a way that the conditions above are satisfied. It can be easily seen that the logarithmic derivative is defined by @ px st ðxÞ @ @ ~ pðx; "Þ ¼ @x þ ðx; "Þ @x px st ðxÞ þ " @x

ð7:28Þ

The limit of the second term in eq. (7.28) approaches a sum of delta functions as " approaches zero: @ ðx; "Þ ¼ 2ðx aÞ 2ðx þ bÞ "!0 @x lim

ð7:29Þ


327

thus constituting two barriers in the function f ðxÞ. Therefore, if the PDF px st ðxÞ is zero on some part of the real axis, one has to augment eq. (7.8) by a correction term (7.29) and, thus, the synthesis equation becomes

pffiffiffiffi K @ ln px st ðxÞ þ 2ðx aÞ 2ðx bÞ þ K ðtÞ x_ ¼ 2 @x

ð7:30Þ

If either a or b are infinite one has to omit the corresponding delta function from the synthesis equation. The suggested modification of SDE would also require modification of corresponding numerical schemes which are considered in Appendix C on the web. Example: A uniformly distributed correlated process. A uniformly distributed process with the marginal PDF px st ðxÞ ¼

1 ; ba

axb

ð7:31Þ

plays a significant role in the modeling of the phase of narrow band signals, as discussed in section 3.12.3. In this case the synthesis eq. (7.30) becomes x_ ¼

pffiffiffiffi K ½2ðx aÞ 2ðx bÞ þ K ðtÞ 2

ð7:32Þ

It is important to note that direct application of eq. (7.8) would result in the SDE for the Wiener process pffiffiffiffi x_ ¼ K ðtÞ ð7:33Þ thus, once again, indicating the importance of correction terms. The Fokker–Planck equation, corresponding to SDE (7.32) is well known to be @ K @2 pðx; tÞ ¼ pðx; tÞ; Gða; tÞ ¼ Gðb; tÞ ¼ 0 @t 2 @x2

ð7:34Þ

and has the following exact solution " # 1 x a x a 2 2 1 X k K 0 cos k exp cos k ðx; t j x0 ; t0 Þ ¼ b a k¼0 ba ba 2ðb aÞ2

ð7:35Þ

which implies that the covariance function is given by an infinite series of exponents " # 1 2 2 X k K h2k exp jj ð7:36Þ Cxx ðÞ ¼ 2ðb aÞ2 k¼1 where hk ¼ ½ð1Þkþ1 þ 1

ðb aÞ2 k 2 2

ð7:37Þ

328


Using the exponent with the slowest decay ðk ¼ 1Þ one can obtain the following approximation of the coefficient K 1 2ðb aÞ2 K

corr 2

ð7:38Þ

Thus, the generating SDE has the following form 1 2ðb aÞ2 ba ½ðx aÞ ðx bÞ þ x_ ¼ 2 corr

rffiffiffiffiffiffiffiffi 2 ðtÞ corr

ð7:39Þ

Example: A Nakagami distributed correlated process. A Nakagami distributed process is one of the most important models of the envelope of fading communication signals. Therefore its modeling is an important issue, especially for proper simulation of communication systems. Since the marginal PDF of the Nakagami process is given by 2 m2 2m1 mx2 px st ðxÞ ¼ x exp ðmÞ

ð7:40Þ

the generating SDE becomes

K 2m 1 2mx þ 2ðxÞ þ KðtÞ x_ ¼ 2 x

ð7:41Þ

The solution for the transitional PDF is also known to be [31]

2 mm 2m1 mx2 ðx; t j x0 ; t0 Þ ¼ x exp ðmÞ sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

2 1 X k!ðmÞ ðm1Þ x2 mkK ðm1Þ x0 Lk jj Lk exp ðm þ kÞ 2 2 k¼0

ð7:42Þ

and, thus

1 2 ðm þ 0:5Þ X 2 ðk 0:5Þ 2mkK exp jj Cxx ðÞ ¼ 4mðmÞ k ¼ 0 ðm þ kÞk!

ð7:43Þ

Once again estimating the value of the coefficient K through the exponent with the slowest rate of decay one obtains the generating SDE in the following form

x_ ¼

corr

sffiffiffiffiffiffiffiffi 2m 1 2mx 2 þ 2ðxÞ þ ðtÞ x corr

ð7:44Þ


329

If m ¼ 1 the Nakagami process coincides with the Rayleigh process whose generating SDE is then x_ ¼

7.2.3

corr

sffiffiffiffiffiffiffiffi 1 2x 2 þ 2ðxÞ þ ðtÞ x corr

ð7:45Þ

Synthesis of Processes

It has been seen from the examples above that the solution of the first order SDE with constant diffusion poses an exponentially decaying covariance function. However, approximation of the latter by a single exponent does not always provide good results. In this section a technique is developed to generate a random process with exact exponential correlation and arbitrary marginal PDF (i.e. a process as defined on page 322). The desired process is thought to be a solution of the following SDE with further zero memory monotonous non-linear transformation y ¼ FðxÞ: x_ ¼ f ðxÞ þ gðxÞðtÞ y ¼ FðxÞ

ð7:46Þ

At this stage it is assumed that the non-linear monotonous2 transformation y ¼ FðxÞ and its inverse x ¼ F 1 ðyÞ are given and both PDFs py st ðyÞ and px st ðxÞ are also known.3 Proposition 1. The autocovariance function Cxx ðÞ of the random process generated by system (7.46) is an exact exponent with damping factor , if and only if is the eigenvalue of the corresponding Fokker–Planck equation and the corresponding eigenfunction has the following form ðyÞ ¼ ðFðyÞ þ Þpy st ðyÞ

ð7:47Þ

where ¼

ð y2

FðyÞpy st ðyÞdy

ð7:48Þ

y1

Proof: Necessity. Without loss of generality it is possible to assume that corresponds to the first non-zero eigenvalue ¼ 1

2 3

ð7:49Þ

This requirement is needed to preserve the Markov nature of the transformed process. This is not a very restrictive assumption since px st ðxÞ and py st ðyÞ are related by px st ðxÞ ¼ py st ½FðxÞF 0 ðxÞ (see Section 2.3)

330


and the corresponding eigenfunction is 1 ðyÞ ¼ py st ðyÞðFðyÞ þ Þ

ð7:50Þ

Using expression (7.50) and property (5.273) of eigenfunctions one obtains hm ¼

ð y2

FðyÞm ðyÞdy ¼

y1

ð y2

ðFðyÞ þ Þm ðyÞdy ¼

ð y2

y1

y1

m ðyÞ1 ðyÞ dy ¼ 0 py st ðyÞ

ð7:51Þ

for any constant and m 6¼ 1. Consequently, the expression for the ACF consists of only one term Cxx ðÞ ¼ h21 exp½jj

ð7:52Þ

no matter what the other eigenvalues are. Sufficiency. By virtue of the fact that all the coefficients h2m in expansion (5.276) are nonnegative, and the assumption that the function ðFðyÞ þ Þpy st ðyÞ is not an eigenfunction, one obtains that expansion (5.276) contains more than one exponential term. The contradiction proves the sufficiency. Finally, it follows from eq. (5.273) that ð y2

1 ðyÞdy ¼

y1

ð y2

ðFðyÞ þ Þpy st ðyÞdy ¼

y1

ð y2

FðyÞpy st ðyÞdy

ð7:53Þ

y1

which, in turn, is equivalent to ¼

ð y2

py st ðyÞFðyÞdy

ð7:54Þ

y1

Proposition 2. formulae

If the drift K1 ðyÞ and the diffusion K2 ðyÞ are chosen according to the

2 K2 ðyÞ ¼ d py st ðyÞ FðyÞ dy 2K1 ðyÞ ¼ K2 ðyÞ

ðy

ðFðsÞ þ Þpy st ðsÞds

ð7:55Þ

y1

d d ln py st ðyÞ þ K2 ðyÞ dy dy

ð7:56Þ

then the process xðtÞ generated by the corresponding SDE is a process. Proof: It follows from the synthesis eq. (7.8), that the drift K1 ðyÞ can be expressed in terms of the diffusion K2 ðyÞ and the desired PDF py st ðyÞ as 2K1 ðyÞ ¼ K2 ðyÞ

d d ln py st ðyÞ þ K2 ðyÞ dy dy

ð7:57Þ


331

Substituting expression (7.57) into eq. (7.7) and taking equality (7.49) into account, one can derive a differential equation for K2 ðyÞ in the following form

p0y st ðyÞ d2 d þ K2 ðyÞ py st ðyÞ ¼ 2ðFðyÞ þ Þpy st ðyÞ ½K2 ðyÞðFðyÞ þ Þpy st ðyÞ K2 ðyÞ dy py st ðyÞ dy2 ð7:58Þ or, equivalently

d d K2 ðyÞ FðyÞpy st ðyÞ ¼ 2ðFðyÞ þ Þpy st ðyÞ dy dy The latter can be easily solved with respect to K2 ðyÞ to produce ðy 2 K2 ðyÞ ¼ ðFðsÞ þ Þpy st ðsÞds d y1 py st ðyÞ FðyÞ dy

ð7:59Þ

ð7:60Þ

Therefore the proposition has been proved. The next step is to prove that the function K2 ðyÞ defined by eq. (7.54), is non-negative, so there is no contradiction with the definition of the diffusion coefficient. Proposition 3. Proof:

K2 ðyÞ 0.

It is sufficient to show that if ðy ðyÞ ¼ ðFðsÞ þ Þpy st ðsÞds

ð7:61Þ

y1

then ðy1 Þ ¼ ðy2 Þ ¼ 0

ð7:62Þ

0 ðyÞ ¼ FðyÞ þ

ð7:63Þ

y ¼ F 1 ðÞ

ð7:64Þ

and the equation

has a unique solution

The derivative of the function ðyÞ increases monotonously, so it is less then zero when y < and greater than zero when y > . The last assertion is equal to the following inequalities FðyÞ < Fðy1 Þ;

y 2 ½y1 ; y

FðyÞ < Fðy2 Þ;

x 2 ½y ; y2

and therefore the proposition is proved.

ð7:65Þ

332


Substituting eq. (7.55) into eq. (7.56) one obtains the expression for the drift K1 ðyÞ and consequently for the coefficients f ðyÞ and gðyÞ in SDE (7.3). It was shown in [25], that the simple exponential form of the ACF of a process is due to its linear drift. This property can be proved also using eqs. (7.55) and (7.56). Example: Exponentially correlated processes with the Pearson class PDF. As has been described in section 2.2, any Pearson PDF px st ðxÞ obeys the following differential equation d a1 x þ a0 AðxÞ ln px st ðxÞ ¼ ¼ 2 dx b2 x þ b1 x þ b0 BðxÞ

ð7:66Þ

It is easy to see that if y ¼ x then the stationary PDF px st ðxÞ satisfies eq. (7.66) and the desired drift is given by K2 ðxÞ ¼ ðb2 x2 þ b1 x þ b0 Þ

ð7:67Þ

Equation (7.54) is satisfied as well. In fact, expression (7.54) can be rewritten in the form K2 ðxÞ

d d px st ðxÞ þ K2 ðxÞ ¼ ðx þ Þ dx dx

ð7:68Þ

if FðxÞ ¼ x, using the definition of the Pearson PDF (7.66) and the fact that d K2 ðxÞ ¼ ð2b2 x þ b1 Þ dx

ð7:69Þ

ða1 þ 2b2 Þx þ a0 þ b1 ¼ ðx þ Þ

ð7:70Þ

Furthermore, eq. (7.68) becomes

which is an identity if and are defined as ¼

a1 þ 2b2

a0 þ b1 ¼ a1 þ 2b2

ð7:71Þ

It can be seen that for processes with a Pearson PDF, K2 ðxÞ is the polynomial of order not higher than 2. All processes with a non-Pearson PDF would have a non-polynomial diffusion. A number of such examples can be easily obtained Example: Symmetric (two-sided) Laplace distribution. It is often required to obtain an exponentially correlated process with a specified value of the two-sided Laplace PDF parameter . It is obvious that this distribution is not a Pearson one. Having x ¼ y in eq. (7.46), one can write out the PDF of xðtÞ as px st ðxÞ ¼

expð jxjÞ 2

ð7:72Þ


333

and consequently K2 ðxÞ ¼

2 ð1 þ jxjÞ 2

K1 ðxÞ ¼ x

ð7:73Þ ð7:74Þ

The corresponding generating SDE is rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ð1 þ jxjÞ dWðtÞ dx ¼ xdt þ 2

ð7:75Þ

Example: Generalized gamma distribution. Let the stationary PDF of the process be the generalized gamma PDF

= jxj 1 exp½jxj px st ðxÞ ¼ 2ð = Þ

ð7:76Þ

which is used to model radar clutter [40–42], telephone signals [43], and ocean noise [44]. As particular cases, this PDF includes the Rayleigh PDF ð ¼ 2; ¼ 2Þ, the Nakagami PDF ð ¼ 2m; ¼ 2Þ and the Weibull one ð ¼ Þ. Using formula (7.55) one obtains the expression for the diffusion in the form K2 ðxÞ ¼

ð1 Þ þ1

þ1

; jxj jxj1 exp½ðjxjÞ

ð7:77Þ

and the generating SDE is thus vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi u þ1 uð1 Þ ; jxj jxj1 expððjxjÞ Þ dWðtÞ dx ¼ xdt þ t þ1

ð7:78Þ

Example: Bimodal distribution. Let the stationary PDF of the process have now the following form px st ðxÞ ¼ Cðp; qÞ expðpx2 qx4 Þ

ð7:79Þ

where constant Cðp; qÞ is to be chosen to normalize px st ðxÞ. It can be shown that4 2 2 rffiffiffi q p p exp Cð p; qÞ ¼ 2 K1 p 8q 4 8q

4

See integral 3.469-1 in [68].

ð7:80Þ

334


Such a PDF has two modes when p < 0 and q > 0. As a particular case, it includes the Gaussian PDF when p > 0 and q ¼ 0. Using eq. (7.55) and setting FðxÞ ¼ x it is easy to obtain the expression for the diffusion in the form " 2 #

pffiffiffi p p p ffiffi ffi K2 ðxÞ pffiffiffi exp q x2 þ Erfc q x2 þ 2 q 2q 2q where 2 ErfcðzÞ ¼ pffiffiffi

ð1

expðt2 Þdt

ð7:81Þ

ð7:82Þ

z

is the complementary error function [32]. The corresponding generating SDE is thus vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi " u pffiffiffi 2 #

u p p pffiffiffi Erfc q x2 þ dx ¼ xdt þ t pffiffiffi exp q x2 þ dwðtÞ 2 q 2q 2q

ð7:83Þ

It is impossible to obtain the exact equation for the transitional probability function, except for a number of cases considered in [47]. However, for a small transitional time an approximate formula can be obtained. Indeed, since the solution of the SDE is a diffusion Markov process, it can be approximated with a Gaussian random process with the same local drift and local diffusion as in [47]. " # 1 ðx x0 K1 ðxÞÞ2 ðx j x0 ; Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp 2K2 ðxÞ 2K2 ðxÞ

ð7:84Þ

or, taking eqs. (7.55) and (7.56) into account " # pffiffiffiffiffiffiffiffiffiffi ps ðxÞ ðx x0 þ ðx mx ÞÞ2 ps ðxÞ ðx j x0 ; Þ ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ð7:85Þ Ð x ffi exp 4 Ð x ðm xÞp ðxÞdx x s 1 4 1 ðmx xÞps ðxÞdx This transitional probability can be used to numerically simulate the random process using the Markov chain method, as described in Appendix C on the web.

7.2.4

Non-Diffusion Markov Models of Non-Gaussian Exponentially Correlated Processes

As has been pointed out by Haddad [25] exponential correlation is due to linear drift of the generating SDE—a fact which can be directly seen from the following equation [49] d ðx mÞ 1 Cxx ðÞ ¼ hxK1 ðx Þi ¼ x Cxx ðÞ ¼ d corr corr

ð7:86Þ


thus leading to the unique solution in the form of an exponent 2 Cxx ðÞ ¼ exp corr

335

ð7:87Þ

Thus, in the case of linear drift, the correlation function is always exponential regardless of other kinetic coefficients Kn ðxÞ. This fact allows one to extend the synthesis of exponentially correlated processes to non-diffusion processes. In a general case, a non-linear system can be driven by a mixture of White Gaussian Noise ðtÞ and the Poisson flow of delta pulses ðtÞ dx ¼ f ðxÞ þ gðxÞðtÞ þ ðtÞ dt

ð7:88Þ

Statistical properties of the solution xðtÞ can be completely described by its transitional probability density ðx; t; x0 ; t0 Þ, which must obey the differential Chapman–Kolmogorov equation (see section 5.1.5 and [38]) @ @ 1 @2 ðx j x0 ; Þ ¼ ½K1 ðxÞðx; x0 ; t0 Þ þ ½K2 ðxÞðx; t; x0 ; t0 Þ @t @xð 2 @x2 þ

1

1

ð7:89Þ

½Wðx j z; tÞðz; t; x0 ; t0 Þ Wðz j x; tÞðz; t; x0 ; t0 Þdz

where drift K1 ðxÞ, diffusion K2 ðxÞ and probability of jumps Wðx j z; tÞ can be obtained from the corresponding parameters of eq. (7.88) as (Ito form of stochastic integrals) lim ðz; t; x0 ; t0 Þ ¼ Wðx j z; tÞ

tt0 !0 ð1 1

Wðx j z; tÞdx ¼ 1

ð7:90Þ ð7:91Þ

In the stationary case, if it exists, eq. (7.89) becomes an integro-differential equation of the following form ð1 ½Wðx j z; tÞðz j x0 ; Þ Wðz j x; tÞðx j x0 ; Þdz 1 ð7:92Þ @ 1 @2 ¼ ½K1 ðxÞðx j x0 ; Þ ½K2 ðxÞðx j x0 ; Þ @x 2 @x2 Different particular cases of SDE (7.88) have been considered in a number of publications [2,3,50,51]. The main goal here is to show how different Markov processes with the same non-Gaussian probability density and exponential covariance function can be obtained. 7.2.4.1

Exponentially Correlated Markov Chain—DAR(1) and Its Continuous Equivalent

In this section the discrete time scheme which generates a non-Gaussian Markov chain with an exponential correlation function and a given arbitrary PDF is considered.

336


Following [52], it is assumed that the stationary distribution of the Markov chain yn with N states

1 < 2 < < N

ð7:93Þ

is described by the following probabilities of individual states qk ¼ Probfyn ¼ k g

ð7:94Þ

Any Markov chain can be completely described by its transitional probability matrix T ¼ ½Tk;l where Tk;l ¼ Probfym ¼ k j ym1 ¼ l g;

k; l ¼ 1; 2; . . . ; N

ð7:95Þ

which is the probability of the event ym ¼ k when ym1 ¼ l and satisfies the following conditions Tk;l 0;

N X

Tk;l ¼ 1; K ¼ 1; 2; . . . ; N

ð7:96Þ

k¼1

The stationary probabilities qk are obtained as the eigenvectors of the transition probability matrix T corresponding to the eigenvalue ¼ 1 (see section 5.1.1) N X

Tk; i qi ¼ qk ;

k ¼ 1; 2; . . . ; N

ð7:97Þ

i¼1

The same matrix defines the power spectrum in the stationary case. However, one may consider the inverse problem, having defined only the stationary probabilities of the states. For a case where the correlation function is an exponential one, the solution has been obtained in [52]. It is possible to follow this procedure to obtain the chain approximation with an infinite number of states as a limit of a finite state Markov chain. To achieve the first goal, the following matrix Q is defined in terms of the probabilities of the states 3 q1 q1 . . . q1 6 q2 q2 . . . q2 7 7 Q¼6 4... ... ... ...5 qN qN . . . qN |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} 2

ð7:98Þ

N times

It is easy to check that Q2 ¼ Q

ð7:99Þ


337

and [52] det½Q I ¼ ð1 ÞðÞN1

ð7:100Þ

In terms of the matrix Q, the transition matrix {T} can be defined as [52] T ¼ Q þ dðI QÞ

ð7:101Þ

where 0 d < 1 will define the correlation properties (described below) and I is the identity matrix. At the same time T satisfies the condition (7.97). For any integer m one can obtain the following expression by making use of eq. (7.101) and eq. (7.99) T m ¼ Q þ dm ðI QÞ

ð7:102Þ

Since d is a positive number less than 1, one has lim T m ¼ Q

m!1

ð7:103Þ

which means that the Markov chain described by eq. (7.101) becomes ergodic [53] and has the stationary probability given by qk. The next step is to consider the correlation function Rm of the Markov chain ym . The average value and the average of the squared value can be obtained in terms of the stationary probability qk as hym i ¼

N X

k qk

ð7:104Þ

k2 qk

ð7:105Þ

i¼1

hy2m i

¼

N X i¼1

To calculate the covariance function, one has to consider the two-dimensional probability which may be obtained from eq. (7.102) as QðmÞ ðk; lÞ ¼ Probfym ¼ k ; y0 ¼ l g ¼ fqk þ d m ðk;l qk Þgql

ð7:106Þ

where m > 0 and QðmÞ ðk; lÞ stands for the m-step transitional probability. It is easy to find that eq. (7.106) fits the consistency relation N X k¼1

QðmÞ ðk; lÞ ¼

N X

QðmÞ ðl; kÞ ¼ ql

ð7:107Þ

k¼1

The covariance function Cm is an even function of m defined as Cm ¼ Cm ¼ hym y0 i hym i2

ð7:108Þ

338


Substitution of eqs. (7.104)–(7.107) into eq. (7.108) produces Cm ¼ Cm ¼ hym y0 i hym i2 ¼

N X

k l Probfym ¼ k ; y0 ¼ l g hym i2

k; l ¼ 1

¼

N X

k l fqk þ dm ðk; l qk Þgq1 hym i2

k; l ¼ 1

¼

N X

k l qk ql þ d m

k; l ¼ 1

N X

ð7:109Þ

k l ðk; l qk Þql hym i2

k; l ¼ 1

¼ hym i2 þ djmj ðhy2m i hym i2 Þ hym i2 ¼ d jmj ðhy2m i hym i2 Þ which is an exponential function with the correlation length Ncorr defined as Ncorr ¼ ð1Þ= ln d

ð7:110Þ

Formula (7.102) can be extended to the finite state continuous time Markov chain ym ðtÞ as in [52] to produce TðtÞ ¼ Q þ expðtÞðI QÞ

ð7:111Þ

The expression for the covariance function can be given in this case as Cyy ðÞ ¼ expðÞðhy2m i hym i2 Þ

ð7:112Þ

Before turning to the continuous time infinite state Markov chain (a Markov process with a continuum of states) let us point out an important property of the exponentially correlated Markov chain. It follows from the definition of the matrices T and TðtÞ as in eqs. (7.102) and (7.111) that the transition probability density does not depend on the current state. If N ! 1, then the Markov continuous time chain tends to a Markov process, which can be a non-diffusion one. In this case eq. (7.111) can be written as ðx j x0 ; Þ ¼ expðÞðx x0 Þ þ ð1 expðÞÞps ðxÞ

ð7:113Þ

Here ps ðxÞ is the stationary distribution of the limit process. It is important to validate that the latter expression indeed defines a proper PDF of a Markov process. Positivity is obvious, since both summands are positive numbers, thus ðx j x0 ; Þ 0

ð7:114Þ

When approaches infinity, the transitional PDF must correspond to the stationary PDF of the process since the values of the process far away from the observation points are independent on this observation: lim ðx j x0 ; Þ ¼ lim ½expðÞðx x0 Þ þ ð1 expðÞÞps ðxÞ ¼ ps ðxÞ

!1

!1

ð7:115Þ

339


At the same time the limit of the PDF when approaches zero must be the delta function ðx x0 Þ, since the process cannot assume two different values at the same moment. Taking the limit of eq. (7.113) one obtains that this condition is indeed satisfied: lim ðx j x0 ; Þ ¼ lim ½expðÞðx x0 Þ þ ð1 expðÞÞps ðxÞ ¼ ðx x0 Þ

!0

!0

ð7:116Þ

Finally, in order to represent a Markov process, PDF ðx j x0 ; Þ must obey the Smoluchowski equation [48] ðx j x0 ; Þ ¼

ð1 1

ðx j x1 ; 2 Þðx1 j x0 ; 1 Þdx1

ð7:117Þ

with ¼ 1 þ 2 . It is easy to check that this is the case for PDF given by eq. (7.113). Indeed ð1 1

ðx j x1 ; 2 Þðx1 j x0 ; 1 Þdx1 ¼

ð1 1

½expð2 Þðx x1 Þ þ ð1 expð2 ÞÞps ðxÞ

½expð1 Þðx1 x0 Þ þ ð1 expð1 ÞÞps ðx1 Þdx1 ¼ expðð1 þ 2 ÞÞðx x0 Þ þ ð1 expðð1 þ 2 ÞÞÞps ðxÞ ¼ ðx j x0 ; 1 þ 2 Þ

ð7:118Þ

Thus, in fact, the PDF (7.113) defines a Markov process. The covariance function of this process is indeed exponential, since Cxx ðÞ ¼ ¼

ð1 ð1 1 ð1

1 ð1

xx1 ðx j x1 ; Þps ðx1 Þdxdx1 xx1 ½expðÞðx1 xÞps ðx1 Þdxdx1

1 1 ð1 ð1

þ

1 1

½ð1 expðÞÞps ðxÞps ðx1 Þdxdx1

¼ expðÞ½2x þ m2x þ ð1 expðÞÞm2x ¼ 2x expðÞ þ m2x

ð7:119Þ

The next step is to understand if the Markov process defined by the PDF (7.113) represents a diffusion Markov process or a process with jumps. In order to do this, one must calculate the following limit, which describes the non-diffusion part of any general Markov process (see Section 5.1.5 and eq. 3.4.1 in [48]) Wðx j x0 ; Þ ¼

lim !0 jx x0 j > "

ðx j x0 ; Þ ¼

lim !0 j x x0 j > "

ð1 expðÞÞps ðxÞ ¼ ps ðxÞ ð7:120Þ

340


The last equation implies that the Markov process defined by the PDF (7.115) is a process with jumps, since Wðx j x0 ; Þ 6¼ 0. However, it is important to note that the probability of jumps Wðx j x0 ; Þ does not depend on the current state x0 . This property is inherited from the fact that the pre-limit Markov chain fyðtÞg has the same property. It is interesting to find out which SDE, if any, generates such a process. In order to accomplish this, one has to calculate the drift K1 ðxÞ and the diffusion K2 ðxÞ coefficients, which are defined as [48] 1 K1 ðxÞ þ Oð"Þ ¼ lim !0 1 ¼ lim !0

ð ð

j xzj

ð7:151Þ


345

It was shown in [41] that it is impossible to obtain a Gaussian process using an SDE of the form x_ ðtÞ ¼ x þ ðtÞ

ð7:152Þ

a Poisson flow of delta pulses. Indeed, in order to achieve a Gaussian process by a linear transformation, one requires the input to be Gaussian as well. However, the Poisson process is not a Gaussian one, unless its intensity is infinite. Example: Pearson class of PDFs. In this case the stationary PDF pst ðxÞ obeys the following equation d pst ðxÞ a1 x þ a0 dx ¼ 2 b2 x þ b1 x þ b0 pst ðxÞ

ð7:153Þ

It was shown in [32] that the following SDE (Ito form) allows one to generate all possible Pearson processes with exponential correlation function of the type

dx a0 þ b1 ¼ x dt a1 þ 2b2

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ðb2 x2 þ b1 x þ b0 Þ ðtÞ þ a1 þ 2b2

ð7:154Þ

In all cases the transition probability density can be expressed in terms of classical orthogonal polynomials [32] or in closed integral form as found in [47]. A few examples are given in Table 7.1. Table 7.1 Parameters of the continuous Markov process with exponential correlationa pst ðxÞ expðxÞ

x expðxÞ ð þ 1Þ

ð þ þ 2Þ ð þ 1Þð þ 1Þ

ð1 þ xÞ ð1 xÞ 2 þ þ1

d ln pst ðxÞ dx

ðx; x0 ; Þ

( " # 2 h xx i 1 ðx x Þ 0 0 pffiffiffiffiffiffi exp exp 1 2 4 2 4 " #) ð ðx þ x0 Þ2 1 x 1 2 þ pffiffiffi e þ exp ez dz ðx þ x Þ 0 4 pffiffiffi 2 =2 x 1 x x 1 e x0 expðÞ

=2 pffiffiffiffiffiffiffi 2e xx0 x þ x0 e I exp 1 e 1 e 1 ð Þ ð þ Þx ð1 þ xÞ ð1 xÞ X enðnþ þþ1Þ 2 þ þ1 1 x2 n¼0 ; An P; n ðx0 ÞPn ðxÞ

An ¼ a

ð2n þ þ þ 1Þðn þ þ þ 1Þ ðn þ þ 1Þðn þ þ 1Þn!

Here I ðxÞ is the modified Bessel function of the first kind and P; n ðxÞ are the Jacobi polynomials [32].

346


Table 7.2

Parameters of Markov processes with exponential correlation and jumps d ln pst ðxÞ dx

pst ðxÞ

h

i þ x expðxÞ h i x expðxÞ 1 ð xÞ ð þ 1Þ

ð Þx ð þ Þx2 1 1 x2

1

expðxÞ x expðxÞ ð þ 1Þ

x x

ð þ þ 2Þ ð þ 1Þð þ 1Þ

WðxÞ

ð Þ ð þ Þx 1 x2

ð1 þ xÞ ð1 xÞ 2 þ þ1

1

ð þ þ 2Þ ð1 þ xÞ ð1 xÞ ð þ 1Þð þ 1Þ 2 þ þ1

In order to obtain a convenient expression for the Pearson class of distributions, equation (7.134) can be rewritten as 3 @ @ @x pst ðxÞ7 6 WðxÞ ¼ 1 pst ðxÞ x pst ðxÞ ¼ 4 1 x 5pst ðxÞ @x pst ðxÞ 2

a1 x 2 þ a0 x ¼ 1 pst ðxÞ b2 x 2 þ b1 x þ b 0

ð7:155Þ

Table 7.2 contains some of the examples given in Table 7.1. Example: Khinchin PDF. As an example of a non-Pearson distribution one can consider the so-called Khinchin probability density pst ðxÞ ¼

1 cos x x2

ð7:156Þ

This distribution is interesting since none of its moments exist. Indeed, the characteristic function, corresponding to this distribution is ð!Þ ¼

0 j!j 1 1 j!j j!j < 1

ð7:157Þ

and it is not differentiable at ! ¼ 0. The last statement is equivalent to the fact that the PDF (7.156) does not have moments which converge. Nevertheless, it was shown in [41] that this distribution can be obtained if x y x y3 þ ðx yÞ sin expð Þ cos 1 5 4 A 1 C x y þ Wðx j yÞ ¼ 2 2 expð Þ þ ðx yÞ 2

ð7:158Þ

MODELING OF A ONE-DIMENSIONAL RANDOM PROCESS

347

where C ðwÞ ¼

ð1 w

cosðzÞ dz z

ð7:159Þ

ð7:160Þ

and

¼

7.3 7.3.1

MODELING OF A ONE-DIMENSIONAL RANDOM PROCESS ON THE BASIS OF A VECTOR SDE Preliminary Comments

It was shown in Section 7.2 that it is possible to synthesize a one-dimensional SDE with the solution being a wide band random process with a given PDF and correlation interval. At the same time many real applications require modeling of random processes with a more complicated correlation function (spectral density). Of a special importance is the problem of modeling narrow band random processes frequently encountered in radio communications and other applications [56–58]. To remain in the SDE framework one should consider the SDE of an order higher than one. In this case a one-dimensional random process can be considered as a component of some vector Markov process.5 The first part of this section is devoted to the generation of narrow band random processes (processes with an oscillating correlation function) using two simple forms of an SDE of second order, the stochastic analog of the Duffing and Van der Pol equations [31]. The following material shows how the concept of an exponentially correlated process, combined with the concept of compound processes [40,42], can be used to generate one-dimensional processes with a given PDF and rational spectrum. 7.3.2

Synthesis Procedure of a ð; !Þ Process

Since the ACF of the first order SDE solution is always monotonous, a random process with the ACF defined by eq. (7.2) can be generated only by a system of at least two coupling equations. Such a narrow band stationary process, or ð; !Þ process, may be written as ðtÞ ¼ AðtÞ cos ’ðtÞ sinð!tÞ þ AðtÞ sin ’ðtÞ cosð!tÞ

ð7:161Þ

with uncorrelated baseband components (considered at the same time instant) k ðtÞ ¼ AðtÞ cos ’ðtÞ ? ðtÞ ¼ AðtÞ sin ’ðtÞ hk ðtÞ? ðtÞi 0

5

It is important to point out here that in this case the one-dimensional process is not a Markov one.

ð7:162Þ

348


and I ðtÞ ¼ k ðtÞ sin !t R ðtÞ ¼ ? ðtÞ cos !t

ð7:163Þ

The marginal distribution p ðxÞ defines, according to the Blanc-Lapierre transform6 [59], the marginal distribution pA ðAÞ of the envelope qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 ðtÞ ð7:164Þ AðtÞ ¼ k2 ðtÞ þ ? ð1 ð1 p st ðxÞ!J0 ð!AÞ expð j!xÞdx d! ð7:165Þ pA ðAÞ ¼ A 1

1

where J0 ð Þ stands for a Bessel function of order zero of the first kind [32]. The phase

I ðtÞ ’ðtÞ ¼ atan ð7:166Þ R ðtÞ of ðtÞ does not depend on A and is uniformly distributed on ½0; 2. Thus pA’ ðA; ’Þ ¼

1 pA ðAÞ 2

ð7:167Þ

or, returning back to y and z7 variables, pffiffiffiffiffiffiffiffiffiffiffiffiffiffi p z2 þ y2 1 A p ffiffiffiffiffiffiffiffiffiffiffiffiffiffi pR I ðy; zÞ ¼ 2 z2 þ y 2

ð7:168Þ

Detailed derivation and discussion of the representation (7.161)–(7.168) is given in Section 4.7. Here it is just emphasized that the components R ðtÞ and I ðtÞ have identical distributions and autocovariance functions. Equation (7.168) allows one to obtain the bivariate PDF pR I ðy; zÞ corresponding to the given p ðxÞ. If one denotes ð1 2 ¼ x2 pI ðxÞdx ð7:169Þ 1

then the variance matrix of the vector process ¼ ðR ; I ÞT has the following form 2

0 ð7:170Þ ¼ 2 I C ¼ 0 2 The next step is to consider a system of two SDEs dR ¼ R dt þ !I dt þ g11 ðR ; I ÞdW1 ðtÞ þ g12 ðR ; I ÞdW2 ðtÞ dI ¼ !R dt I dt þ g21 ðR ; I ÞdW1 ðtÞ þ g22 ðR ; I ÞdW2 ðtÞ 6

See also Section 4.7. It is known that k ðtÞ, ? ðtÞ, R ðtÞ and I ðtÞ have the same PDF (see Section 4.7 for details).

7

ð7:171Þ


349

The stationary PDF should satisfy the following Fokker–Planck equation

@ @ 1 @2 ½K11 ðy; zÞpR I ðy; zÞ ½K12 ðy; zÞpR I ðy; zÞ þ ½K211 ðy; zÞpR I ðy; zÞ @y @z 2 @y2 1 @2 1 @2 ½K212 ðy; zÞpR I ðy; zÞ ¼ 0 þ ½K ðy; zÞp ðy; zÞ þ 222 R I 2 @z2 2 @z@y

ð7:172Þ

where the drifts K11 ðy; zÞ and K12 ðy; zÞ are given by K11 ðy; zÞ ¼ R þ !I and K12 ðy; zÞ ¼ !R I

ð7:173Þ

while the diffusion coefficients can be found according to eqs. (7.176) and (7.178) below. Since eqs. (7.171) form a system with linear drifts, the solution has the correlation matrix CðÞ which obeys the following differential equation _CðÞ ¼ !

! CðÞ

ð7:174Þ

Taking into account the initial conditions (7.170) one obtains the solution of eq. (7.174) as

exp½jj cos ! CðÞ ¼ exp½jj sin !

exp½jj sin ! Cð0Þ exp½jj cos !

ð7:175Þ

It can be checked by direct substitution that if the drift is chosen according to (7.173) and the diffusions as ðy K211 ðy; zÞ ¼ sp ðs; zÞds pR I ðy; zÞ 1 R I ðz K222 ðy; zÞ ¼ sp ðy; sÞds pR I ðy; zÞ 1 R I ð z ðy ! K212 ðy; zÞ ¼ K221 ðy; zÞ ¼ sp ðy; sÞds spR I ðs; zÞds pR I ðy; zÞ 1 R I 1

ð7:176Þ ð7:177Þ ð7:178Þ

then Fokker–Planck equation (7.172) is satisfied. Having these parameters defined by eq. (7.173) and eqs. (7.175)–(7.178), one can write the corresponding generating SDE8 as d

8

R I

¼

!

!

R I

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

dw1 ðtÞ K211 ðy; zÞK212 ðy; zÞ dt þ K212 ðy; zÞK222 ðy; zÞ dW2 ðtÞ

ð7:179Þ

In general one has to solve the equation gðy; zÞgH ðy; zÞ ¼ K 2 . Using the fact that K 2 is a symmetric matrix according to eq. (7.178) and the symmetry in the properties of y and z mentioned on page 348, one can choose the symmetrical solution gðy; zÞ which is unique in this case. A more general consideration can be the subject of a separate investigation.

350


Example: A Gaussian ð; !Þ process. In this case one can write

1 x2 px ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp 2 2 22

ð7:180Þ

A A2 pA ðAÞ ¼ 2 exp 2 2

ð7:181Þ

and

which results in the joint PDF of the in-phase and quadrature components 2

1 y þ z2 exp pR I ðy; zÞ ¼ 22 22

ð7:182Þ

and the corresponding vector generating SDE is, respectively, pffiffiffiffiffiffiffiffiffiffi 22 dW1 ðtÞ pffiffiffiffiffiffiffiffiffiffi dI ¼ !R dt I dt þ 22 dW2 ðtÞ

dR ¼ R dt þ !zdt þ

ð7:183Þ

Example: A ð; !Þ process with uniformly distributed envelope. In this case one has pA ðAÞ ¼

1 ; A0

0 A a0

qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 p ðxÞ ¼ ln A0 x þ A20 x2 þ 1 A0 pR I ðy; zÞ ¼

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2A0 y2 þ z2

ð7:184Þ ð7:185Þ ð7:186Þ

where the corresponding diffusions are 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 A0 þ y 1A K211 ðy; zÞ ¼ @ y 2 þ z2 0sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 A0 þ z 1A K222 ðy; zÞ ¼ @ y2 þ z2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi! A20 þ y2 A20 þ z2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi K211 ðy; zÞ ¼ y2 þ z2

ð7:187Þ

ð7:188Þ

ð7:189Þ


351

and the generating SDE is v2 ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi u sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 u 2 2 A20 þ y2 A20 þ z2 u6 A0 þ y 7" pffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 u6 " # " #" # # 7 2 þ z2 y 2 þ z2 y pffiffiffiu 6 7 ðtÞ dW R ! R 1 u 7 d ¼ dt þ u6 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 7 6 u p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! I I 7 dW2 ðtÞ 2 u6 2 A20 þ z2 A20 þ z2 5 t4 A0 þpy ffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 2 2 2 2 y þz y þz ð7:190Þ

7.3.3

Synthesis of a Narrow Band Process Using a Second Order SDE

General properties of a stationary narrow band random process have been discussed in some detail in Section 4.7. It was shown there that a narrow band process ðtÞ can be represented as ðtÞ ¼ AðtÞ cos½!0 t þ ’ðtÞ

ð7:191Þ

where AðtÞ and ’ðtÞ are random amplitude and phase, varying slowly [60–62] compared to the ‘‘central frequency’’ !0 . In this case the spectrum width of the process ðtÞ satisfies the following condition 1 !0

ð7:192Þ

For a narrow band stationary process, the joint distribution of the amplitude and phase has the following form [63] pðA; ’Þ ¼ pA ðAÞp’ ð’Þ

ð7:193Þ

where, p’ ð’Þ ¼ 1=ð2Þ, ’ 2 ½; and the distribution pA ðAÞ of the envelope AðtÞ and the distribution px ðxÞ of process xðtÞ are related by the Blanc-Lapierre transform pair [59] 1 p st ðxÞ ¼

ð1

pA ðAÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dA A2 x 2 jxj

A pA ðAÞ ¼ 2

ð1

d!J0 ð!AÞ!

0

ð1 1

px st ðxÞej!x dx

ð7:194Þ

The covariance function of a narrow band process ðtÞ can be represented in a similar form C ðÞ ¼ BðÞ cosð!0 Þ where BðÞ is a slow function compared to cosð!0 Þ.

ð7:195Þ

352


Assuming that the random process (7.191) is generated by the WGN-excited deterministic dynamic system of minimal order two, the general form of SDE can be written in the form [2] €x þ f1 ðx; x_ Þ_x þ f2 ðx; x_ Þ ¼ f3 ðx; x_ ÞðtÞ

ð7:196Þ

A full analytical investigation of eq. (7.196) is impossible. To make this problem more manageable only a few particular cases of this equation are considered below.

7.3.3.1

Synthesis of a Narrow Band Random Process Using a Duffing Type SDE

One of the simplest equations to consider is the so-called Duffing SDE [31] €x þ 2 !0 x_ þ !20 FðxÞ ¼

pffiffiffiffi K !0 ðtÞ

ð7:197Þ

Here the power spectral density of the WGN is 1. It can be taken that eq. (7.197) represents a non-linear resonant contour excited by WGN (narrow band form filter). The block diagram of this form filter is given in Fig. 7.1. Equation (7.197) can be written in vector form as x_ ¼ y y_ ¼ 2 !0 y !20 FðxÞ þ

pffiffiffiffi K !0 ðtÞ

ð7:198Þ

with the corresponding stationary Fokker–Planck equation 0¼

Figure 7.1

@ @ K @2 ½ypðx; yÞ þ ½ð2 !0 y þ FðxÞ!20 Þpðx; yÞ þ pðx; yÞ @x @y 2 @y2

ð7:199Þ

A block diagram of a form filter, corresponding to the Duffing equation (7.197).


353

It can be easily verified [31] that the stationary solution of the Fokker–Planck equation (7.199) has the following one-dimensional joint PDF ð 2 2 4 !0 x y FðsÞds ¼ px st ðxÞpy st ðyÞ pst ðx; yÞ ¼ C exp K!0 K x1

ð7:200Þ

From eq. (7.200) one can obtain the expression for the marginal PDF in the form ð 4 !0 x FðsÞds px st ðxÞ ¼ C exp K x1

ð7:201Þ

and, respectively, the unknown non-linearity FðxÞ can be expressed as FðxÞ ¼

K d ln px st ðxÞ 4 !0 dx

ð7:202Þ

Once the non-linearity FðxÞ is chosen, the parameters and K can be found from the linearized model of the type [17,66] pffiffiffiffi K !0 ðtÞ

ð7:203Þ

1 @ ðK; FÞ 2 FðxÞ ! !0 @x

ð7:204Þ

2 ðK; FÞ!20 x ¼ €x þ 2 !0 x_ þ ! where ¼ !;

2

is defined both by K and by the nonIt is important to point out here that the parameter ! linearity FðxÞ, and this fact complicates the application of the Duffing equation as a model of 2 ðK; FÞ ¼ 1 would ensure that the narrow band processes: a choice of K which satisfies ! average frequency is equal to the required central frequency !0 but it is difficult to define its characteristics in more detail. Since the PDF px ðxÞ must be an even function of its argument9 x, its log-derivative is an odd function of x. If the function FðxÞ is well behaved in the vicinity of zero it can be expanded into a Taylor series with only odd powers of x FðxÞ ¼

1 X

2kþ1 x

2kþ1

k¼0

Kmax X

2kþ1 x2kþ1

ð7:205Þ

k¼0

Thus, the average derivative of FðxÞ is then

9

d FðxÞ dx

Kmax X k¼0

ð2k þ 1Þ 2kþ1 hx2k i ¼

Kmax X

ð2k þ 1Þ 2kþ1 m2k

k¼0

A narrow band stationary process has a symmetric PDF—see Section 4.7.

ð7:206Þ

354


If such a derivative does not exist, one can approximate non-linearity in the mean square sense, i.e. minimize the values of the mean square error MSEðsÞ ¼

ð1

ð1

2

½FðxÞ sx px st ðxÞdx ¼ ½FðxÞ2 px st ðxÞdx 1 ð1 ð1 1 FðxÞxpx st ðxÞdx þ s2 x2 px st ðxÞdx 2s 1

ð7:207Þ

1

with the minimum achieved at Ð1 d Ð1 FðxÞxpx st ðxÞdx K 1 x dx px st ðxÞdx K Ð1 Ð1 sopt ðKÞ ¼ 1 ¼ !20 ¼ ¼ 2 2 4 ! 4 ! m x p ðxÞdx x p ðxÞdx 0 1 0 2x x st x st 1

ð7:208Þ

Solving for K one obtains K ¼ 4 !30 m2x ¼ 4 !30 2x

ð7:209Þ

The models which use the Duffing SDE also have an important property which is often assumed (or indeed, proved) that the derivative of a random process is independent of the value of the process itself at the same moment of time and is a Gaussian random variable. This fact makes models based on the Duffing equation attractive for use in fading modeling. Example: A Gaussian narrow band random process. In this case

1 x2 px st ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffi exp 2 2 22

2 A A pA ðAÞ ¼ 2 exp 2 2

ð7:210Þ ð7:211Þ

Using formula (7.201) one obtains the function FðxÞ as FðxÞ ¼

K x 4 !0 2

ð7:212Þ

If the central frequency !0 and bandwidth ! are desired, then the parameters K and should be chosen as10 K ¼ 4 2 !0 ;

¼ !

ð7:213Þ

and finally, the generating SDE is sffiffiffiffiffiffiffi ! €x þ 2!0 x_ þ !20 x ¼ 2!20 ðtÞ !0 10

The same result is obtained using the statistical linearization, (7.209).

ð7:214Þ


355

Example: A narrow band process with K0 -distributed envelope. Let the distribution of the envelope of a narrow band random process be given by pA ðAÞ ¼ 2 AK0 ðAÞ;

A0

ð7:215Þ

Then, substituting eq. (7.215) into eq. (7.194), one obtains that the distribution of instantaneous values obeys the two-sided Laplace distribution px ðxÞ ¼

exp½jxj 2

ð7:216Þ

and, thus, the corresponding non-linearity is FðxÞ ¼

K sgnx 8 !0

ð7:217Þ

Since the derivative of expression (7.217) does not exist, one has to statistically linearize this expression in the mean square sense. The last function, being statistically linearized using eq. (7.197) [63] produces K¼

4 !30 2

ð7:218Þ

Finally, the generating SDE is thus rffiffiffiffiffiffiffi ! 2 ! !0 2 €x þ 2!_x þ 0 sgnx ¼ 2 !0 ðtÞ 2

ð7:219Þ

Example: A bimodal distribution. It has been pointed out in Section 7.2.3 that the bimodal distribution px st ¼ C expðqx4 px2 Þ

ð7:220Þ

approximates well the distribution of instantaneous values of processes with the Nakagami envelope. In this case the non-linearity FðxÞ becomes a simple third order polynomial FðxÞ ¼

K d K ln px st ðxÞ ¼ ð4qx3 þ 2pxÞ 4 !0 dx 4 !0

ð7:221Þ

Estimation of the parameter K can be achieved either by using eq. (7.206) K1 ¼

4 !0 12qm2 þ 2p

ð7:222Þ

356


or by applying eq. (7.209) K2 ¼ 4 !0 m2

ð7:223Þ

where the second moment m2 is calculated to be 2

m2 ¼

p 1 p K3=4 8q

4q

K1=4 2

K1=4

2 p 8q

p 8q

ð7:224Þ

It can be seen from Fig. 7.2 that both methods provide a similar11 estimate for jpj > 2q.

7.3.3.2

An SDE of the Van Der Pol Type

The Duffing SDE is convenient only when the Blanc-Lapierre transformation can be expressed analytically in terms of the known functions. At the same time, as was mentioned before, the central frequency of the power spectral density is strongly dependent on (Fig. 7.3) the function FðxÞ and is varying when the parameters of the distribution (7.193) are changed.

Figure 7.2

11

Dependence of K2 =K1 ¼ m2 ð12qm2 þ 2pÞ on p and q.

Accuracy of a statistical linearization is often acceptable if it does not exceed 10–15 % [36].

357


Figure 7.3

Block diagram for a form filter, corresponding to the Duffing eq. (7.219).

The following simple second order equations often produce better results [64,66] €x þ f1 ðx; x_ Þ_x þ !20 x ¼ KðtÞ

ð7:225Þ

€x þ f2 ðx; x_ Þ_x þ !20 x ¼ K x_ ðtÞ

ð7:226Þ

where fi ðx; x_ Þ are some deterministic functions of x and x_ . The corresponding form filters are shown in Figs. 7.4 and 7.5. Once again the narrow band solutions of eqs. (7.225) and (7.226) can be represented in the form (7.191), xðtÞ ¼ AðtÞ cosð!0 t þ ’ðtÞÞ

Figure 7.4 Form filter corresponding to the Van der Pol SDE (7.225).

ð7:227Þ

358


Figure 7.5 Form filter corresponding to the Van der Pol SDE (7.225).

and the adjoined process ^xðtÞ can be approximated by the Tikhonov formula [67] ^xðtÞ ¼

x_ ðtÞ !0

ð7:228Þ

instead of the exact Hilbert transform (see Section 4.7.1). In this case the envelope can be defined as [67] AðtÞ ¼

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x2 ðtÞ þ ^x2 ðtÞ ¼ x2 ðtÞ þ x_ 2 ðtÞ

ð7:229Þ

Using approximation (7.228) and relationship (7.229) one can rewrite SDEs (7.225) and (7.226) in terms of the envelope AðtÞ and the process itself as [64,66] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 8 2 2 > < x_ ¼ !0 A x 2 2 > : A_ ¼ A x F1 ðx; AÞ A 8 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 > < x_ ¼ !0 A x

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A 2 x2 ðtÞ !0 A

2 2 2 2 > : A_ ¼ A x F2 ðx; AÞ þ A x ðtÞ A A

ð7:230Þ

ð7:231Þ

respectively. Here pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fi ðx; AÞ ¼ fi ðx; !0 A2 x2 Þ; i ¼ 1; 2

ð7:232Þ


359

The corresponding Fokker–Planck equations are then given by pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi @ @ p1 ðA; x; tÞ ¼ ½!0 A2 x2 p1 ðA; x; tÞ @t @x

@ A2 x2 K 2 x2 F1 ðx; AÞ þ 2 3 p1 ðA; x; tÞ @A A 2!0 A

2 2 1 @2 2A x þ K p1 ðA; x; tÞ 2 @A2 !20 A2

ð7:233Þ

and i @ @ h pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi p2 ðA; x; tÞ ¼ !0 A2 x2 p2 ðA; x; tÞ @t @x

@ A2 x 2 K 2 x2 F2 ðx; AÞ þ 2 3 p2 ðA; xtÞ @A A 2!0 A " # 2 2 2 1 @2 ðA x Þ þ K2 p2 ðA; x; tÞ 2 @A2 A2

ð7:234Þ

respectively. In the stationary case, when t ! 1, the joint PDF can be written as [64,66] pi ðAÞ pi ðx; AÞ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ; A2 x 2

i ¼ 1; 2

ð7:235Þ

taking into account the uniformly distributed phase and eq. (7.193). Here pi ðAÞ is the stationary distribution of amplitude. After substituting eq. (7.235) into eqs. (7.233)–(7.234) and integrating over interval ðA; AÞ with respect to x, one obtains the equation for the stationary PDF of amplitude ð A

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 @2 @ 1 K2 2 2 p1 ðAÞ þ F1 ðx; AÞ A x dx 2 p1 ðAÞ ¼ 0 @A A A 4!20 @A2 4!0 ð A

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K2 @2 @ 1 K p2 ðAÞ þ F2 ðx; AÞ A2 x2 dx 2 p2 ðAÞ ¼ 0 @A A A 4!20 @A2 4!0

ð7:236Þ ð7:237Þ

In turn, eqs. (7.236) and (7.237) can be rewritten to have the unknown functions Fi ðA; xÞ on the left hand side, i.e.

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K d F1 ðx; AÞ A2 x2 dx ¼ 2 1 A ln p1 ðAÞ ¼ F1 ðAÞ dA 4!0 A

ðA pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K 2 2 d 2 2 A 1 þ 3A ln p2 ðAÞ ¼ F2 ðAÞ F2 ðx; AÞ A x dx ¼ dA 16 A

ðA

ð7:238Þ ð7:239Þ

360


In order to simplify solution of the integral eqs. (7.238) and (7.239), additional requirements can be imposed on the form of unknown functions Fi ðx; AÞ. In the following, considerations are restricted to the case where Fi ðx; AÞ depends only on one variable. In particular, if it is assumed that they depend only on a single variable A, i.e. x_ 2 2 fi ðx; x_ Þ ¼ fi x þ ¼ Fi ðAÞ !0

ð7:240Þ

the left hand side of eqs. (7.238) and (7.239) can be transformed by noting that ðA

ðA pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A2 2 2 Fi ðx; AÞ A x dx ¼ Fi ðAÞ A2 x2 dx ¼ Fi ðAÞ 2 A A

ð7:241Þ

and, thus Fi ðAÞ ¼

2Fi ðAÞ A2

ð7:242Þ

If dependence only on the variable x is assumed in Fi ðx; AÞ, i.e. fi ðx; x_ Þ ¼ fi ðxÞ ¼ Fi ðxÞ

ð7:243Þ

eqs. (7.238) and (7.239) are reduced to

2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K d F1 ðxÞ A2 x2 dx ¼ 2 1 A ln P1 ðAÞ ¼ F1 ðAÞ dA 4!0 A

ðA 2 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K d 2 A 1 þ 3A ln P2 ðAÞ ¼ F2 ðAÞ F2 ðxÞ A2 x2 dx ¼ dA 16 A ðA

ð7:244Þ ð7:245Þ

It can be shown that the solution of eqs. (7.244) and (7.245) has the following form [64,66] 2 3 d d ð Fi ðAÞ x x d F ðAÞ 1 dA 6dA i 7 þ fi ðxÞ ¼ lim dA 4 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 A!0 0 dA A A x A2

ð7:246Þ

thus providing a convenient tool for modeling narrow band processes with the prescribed PDF of the envelope [64,66]. Example: A narrow band process with the Nakagami PDF [64,66]. For a narrow band process with the envelope distributed according to the Nakagami law with parameters m and , the marginal PDF of the envelope is given by 2 mm 2m1 mA2 A exp pA ðAÞ ¼ ðmÞ

ð7:247Þ

SYNTHESIS OF A ONE-DIMENSIONAL PROCESS WITH GAUSSIAN MARGINAL PDF

Function F2 ðAÞ according to eq. (7.239) is then

K 2 6mA4 2 F2 ðAÞ ¼ ð6m 3ÞA 16

361

ð7:248Þ

and eq. (7.242) becomes K2 6mA2 6m 3 f2 ðAÞ ¼ 8

ð7:249Þ

Finally, the generating SDE is then

K 2 6m 2 x_ 2 €x þ x þ 2 þ 3 6m x_ þ !20 x ¼ K x_ ðtÞ 8 !0 Example: given by

ð7:250Þ

A generalized Rayleigh distribution [64,66]. If the PDF of the envelope is pA ðAÞ ¼ CA exp½ A2 A4

ð7:251Þ

then, according to eq. (7.238), function F 1 ðAÞ is defined by F1 ðAÞ ¼

K 2 ð2 A2 þ 4A4 Þ 2 4!0

ð7:252Þ

The corresponding solution of eq. (7.246) is then K 2 f1 ðxÞ ¼ 2 ð þ 3x3 Þ !0

ð7:253Þ

which, in turn, results in the following generating SDE €x þ

K 2 ð þ 3x3 Þ_x þ !20 x ¼ KðtÞ !20

ð7:254Þ

It is easy to see that, when ¼ 0 (Rayleigh distribution), eq. (7.254) becomes a linear equation coinciding with eq. (7.213). These models were used in a hardware simulator of HF communication channels [64,66].

7.4

SYNTHESIS OF A ONE-DIMENSIONAL PROCESS WITH A GAUSSIAN MARGINAL PDF AND NON-EXPONENTIAL CORRELATION

In this section synthesis of a one-dimensional process with Gaussian marginal PDF and a given covariance function is considered. If the covariance function is a pure exponent with

362


decay , its mean and variance are m and 2 respectively, then the first order SDE pffiffiffiffiffiffi x_ ¼ ðx mÞ þ 2ðtÞ ð7:255Þ is the generating SDE for this problem. The problem becomes more difficult if the correlation function differs from the exact exponent. In this case one has to represent this process as a component12 of the vector Markov Gaussian process which, in turn, is the solution of a vector linear SDE x_ ¼ Ax þ BðtÞ

ð7:256Þ

where the state vector x is composed of components xðtÞ and its derivatives with an order not higher than M 1 2 3 xðtÞ 6 x0 ðtÞ 7 7 x¼6 ð7:257Þ 4 ... 5 xðM1Þ ðtÞ A and B are some constant matrices of dimension M M and M 1, respectively. It is well known that spectral density ð!Þ of the process xðtÞ defined by eq. (7.256) is a rational function and it can be written in the form [55] ð!Þ ¼

Nð!2 Þ Dð!2 Þ

ð7:258Þ

where Nð!2 Þ and Dð!2 Þ are two polynomials in !2 . Any given covariance function Cxx ðÞ of the process xðtÞ can be approximated by a finite series of exponents [55] as13 Cxx ðÞ ¼

M X

i expði Þ

ð7:259Þ

i¼1

The last equality produces a rational spectrum according to the Wiener–Khinchin theorem. If the non-rational power spectral density is needed it can be approximated with some accuracy by the rational spectrum of the form (7.258) [55]. Let hr , and hr be zeros of the spectral density ð!Þ, and r and r be its poles. It is assumed that hr and r are chosen in such a way that their real parts are negative. Then the spectral density can be written as Qm 2 2 r¼1 ð! þ 2Reðhr Þ! þ jhr j Þ ð7:260Þ ð!Þ ¼ C Qn 2 2 s¼1 ð! þ 2Reðr Þ! þ jr j Þ 12

The process itself is, thus, non-Markov. Note that here, in contrast to the one-dimensional SDE, both i and i can be complex but are encountered in complex-conjugate pairs.

13

SYNTHESIS OF A ONE-DIMENSIONAL PROCESS WITH GAUSSIAN MARGINAL PDF

363

where C is a constant. The last expression can be factorized to produce Fð j!Þ 2 Fð j!ÞFðj!Þ ¼ ð!Þ ¼ Gð j!Þ Gð j!ÞGðj!Þ

ð7:261Þ

where FðzÞ and GðzÞ are polynomials of degree m and n, respectively, with all their zeros in the left half-plane. In order for the process ðtÞ to have a finite variance, the degree m of the numerator must be less than the degree of the denominator, m < n. Without loss of generality, one can fix the leading coefficient of GðzÞ as equal to 1, so that GðzÞ ¼

n Y

ðz r Þ

r¼1 m Y FðzÞ ¼ f0 ðz hr Þ

ð7:262Þ

r¼1

where f0 is the leading coefficient of FðzÞ, f0 ¼ c1=2 .14 Finally the process xðtÞ can be derived from a WGN ðtÞ of unit spectral density by the equation of n-th degree d d g xðtÞ ¼ F ðtÞ dx dx

ð7:263Þ

dwr r wr ¼ Mr ðtÞ dt n X wr ðtÞ xðtÞ ¼

ð7:264Þ

or in the Causchy form

r¼1

where the coefficients Mr can be found as [55]: ðp r ÞFðpÞ p!r Gð pÞ

Mr ¼ lim

ð7:265Þ

Another, but equivalent, form of eq. (7.264) can be obtained if the coefficients of the polynomials GðzÞ and FðzÞ are known. Let GðzÞ ¼ FðzÞ ¼

n X r¼0 m X

gr z r

ð7:266Þ

f s zs

ð7:267Þ

s¼0

14

It is assumed that all zeros and poles are distinct. If this is not so, the appropriate formulas can be derived by allowing certain zeros or poles to coalesce [55].

364


Then the generating SDE has a form 2

fn1

6 6 fn2 16 ... w_ ¼ 6 fn 6 6 4 f2 f1 w1 x¼ fn

fn

0

0 ... 0 0

...

fn . . . ... ... 0 0

... ...

3

3 2 fn gm 7 0 7 7 7 16 6 fn1 gm1 7 7 . . . 7w þ 6 7ðtÞ fn 4 . . . 5 7 fn 5 fnm g0 0 0

ð7:268Þ

ð7:269Þ

where w is a vector of hidden states. If one wants to avoid the calculations, then the form (7.264) should be used if the correlation function is given (or approximated) as the sum of exponents, and the forms (7.268) and (7.269) should be used if the energetic spectrum is given (or approximated) as a rational function of the frequency. Example: a Gaussian process with a rational spectrum. Let the following power spectral density of a Gaussian process to be generated [36]: Sx ð!Þ ¼

N0 ða þ !0 Þb2 þ ða !0 Þ!2 b4 þ 2ða2 !20 Þ!2 þ !4

ð7:270Þ

where w20 ¼ b2 a2 . In this case the numerator has two roots w ¼ ib1 , b21 ¼ b2 ða þ !0 Þða !0 Þ1 , and the denominator has four roots ð!0 iaÞ. Selecting the roots disposed in the upper half-plane one obtains gðzÞ ¼ ðzÞ ¼ FðzÞ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2K 2 ða !0 Þðz þ b1 Þ z2 þ 2az þ b2

ð7:271Þ

The corresponding generating SDE is then w_ ¼

0

b2 x ¼ w1

where q1 ¼

7.5

q1 wþ ðtÞ 2a q2 1

ð7:272Þ

pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2K 2 ða !0 Þ and q2 ¼ 2K 2 ða !0 Þðb1 2aÞ

SYNTHESIS OF COMPOUND PROCESSES

This section builds on the theory of compound processes presented in Section 4.8. The results shown here are based on the synthesis of a process with exponential correlation and a Gaussian process with a rational spectrum, which have been discussed in Sections 7.2.3 and 7.4, respectively.


7.5.1

365

A Compound Process

In this section we show how to obtain the exponentially correlated zero mean random process to be a compound one, and, consequently, to be a component of the vector random process: xðtÞ ¼ x1 ðtÞx2 ðtÞ

ð7:273Þ

If processes x1 ðtÞ and x2 ðtÞ in eq. (7.273) have exponential covariance function Cx1 ðÞ ¼ hx1 ðtÞx1 ðt þ Þi ¼ 2x1 expð1 jjÞ

ð7:274Þ

Cx2 ðÞ ¼ hx2 ðtÞx2 ðt þ Þi ¼ 2x2 expð2 jjÞ

ð7:275Þ

then the correlation function of the process xðtÞ is Cx ðÞ ¼ hxðtÞxðt þ Þi ¼ hx1 ðtÞx1 ðt þ Þðx2 ðtÞx2 ðt þ ÞÞi ¼ 2x1 expð1 jjÞ2x2 expð2 jjÞ ¼ 2x expðjjÞ

ð7:276Þ

where ¼ 1 þ 2 , and, consequently, the random process xðtÞ is again a process. Being exponentially correlated, processes x1 ðtÞ and x2 ðtÞ are solutions of the following SDEs x_ 1 ¼ 1 x1 þ g1 ðx1 Þ1 ðtÞ

ð7:277Þ

x_ 2 ¼ 2 x2 þ g2 ðx2 Þ2 ðtÞ

ð7:278Þ

and

respectively. Here 1 ðtÞ and 2 ðtÞ are independent WGN and g1 ðxÞ and g2 ðxÞ are to be defined to provide proper distribution densities for x1 ðtÞ and x2 ðtÞ. Using the equality x_ ðtÞ ¼ x_ 1 ðtÞx2 ðtÞ þ x1 ðtÞ_x2 ðtÞ

ð7:279Þ

one can obtain the following vector SDE 8 > < x_ ¼ 1 x1 þ g1 ðx1 Þ1 ðtÞ x_ 2 ¼ 2 x2 þ g2 ðx2 Þ2 ðtÞ > : x_ ¼ 2 x2 x1 1 x1 x2 þ x1 g2 ðx2 Þ2 ðtÞ þ x2 g1 ðx1 Þ1 ðtÞ

ð7:280Þ

One of the SDEs (7.280) is dependent on two others, so it can be rewritten as a twodimensional SDE 8 > < x_ 1 ¼ 1 x1 þ g1 ðx1 Þ1 ðtÞ; ð7:281Þ x x > 2 ðtÞ þ g1 ðx1 Þ1 ðtÞ : x_ ¼ 2 x 1 x þ x1 g2 x1 x1

366

or


8 > < x_ 2 ¼ 2 x2 þ g2 ðx2 Þ2 ðtÞ; x x > 1 ðtÞ þ g2 ðx2 Þ2 ðtÞ : x_ ¼ 2 x 1 x þ x2 g1 x2 x2

ð7:282Þ

When representation (7.273) is found, the synthesis of the SDE (7.280)–(7.282) can be made by taking the following steps: choose the correlation time 1 ¼ 11 > x corr , where x corr is the desired correlation time of the process ðtÞ; find the generating SDE corresponding to p1 ðx1 Þ and 1 using eqs. (7.54)–(7.56); find the correlation time of the process x2 ðtÞ according to 2 ¼

1 1 ¼ 1 2

find the generating SDE corresponding to p2 ðx2 Þ and 2 using eqs. (7.54)–(7.56); write the generating SDE in the form of eq. (7.281) or eq. (7.282). Example: K0 -distributed process. The marginal PDF of K0 -distributed random processes is given by px ðxÞ ¼

pffiffiffiffiffiffiffiffi a2 jxj K0 ð ajxjÞ; 16

x>0

ð7:283Þ

The direct application of the methodology of Section 7.2.3 in synthesizing the corresponding process with the correlation time corr ¼ 1= leads to complicated calculations employing higher transcendental functions. At the same time, the product of two symmetric Gamma distributions 2 jx1 j expð jx1 jÞ 2 2 px2 ðxÞ ¼ jx2 j expð jx2 jÞ 2

px1 ðx1 Þ ¼

ð7:284Þ ð7:285Þ

gives the K0 -distributed random variable with PDF px ðxÞ ¼ 2 2 jxjK0 ð2

pffiffiffiffiffiffiffiffiffiffiffi jxjÞ

ð7:286Þ

The SDEs generating a process with PDFs (7.284) and (7.285) are the following pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 F1 ðx1 Þ1 ðtÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x_ 2 ¼ 2 x2 þ 2 F2 ðx2 Þ2 ðtÞ x_ ¼ 1 x1 þ

ð7:287Þ ð7:288Þ


Here

8 2 þ 2 x2 2 x > > ðx ; if x < 0 < 1 2 x F1 ðxÞ ¼ xpx ðxÞdx ¼ 2 2 > px1 ðxÞ 1 1 > : 2 þ x þ x ; if x > 0 3 x

367

ð7:289Þ

and

8 2 þ 2 x2 2x > > ; if x < 0 px2 ðxÞ 1 2 > : 2 þ x þ x ; if x > 0 3 x pffiffiffi By setting 1 ¼ 2 ¼ =2 and ¼ ¼ b one obtains the desired system of SDEs. ðx

7.5.2

Synthesis of a Compound Process With a Symmetrical PDF

In this section we show how the compound model can be used to obtain the symmetrical random process with a prescribed correlation function and marginal distribution. The following statement lays the foundation for such an approach: Proposition 4. Let ðtÞ be a zero mean stationary compound random process with a symmetrical PDF and the correlation (covariance) function C ðÞ which is bounded by some exponent C ðÞ 2 expðm j jÞ ð7:291Þ Then the process ðtÞ can be represented as a component of the random vector Markov process being the solution of the proper SDE if it can be represented as a product of a Gaussian process and some modulating process. Proof: By assumption it is possible to find a representation of the random process ðtÞ in the form of products of a normal distributed random process n ðtÞ, and a properly distributed and independent of n ðtÞ zero mean symmetric random process sðtÞ ðtÞ ¼ n ðtÞsðtÞ

ð7:292Þ

If ps ðsÞ is the PDF of the random process sðtÞ then the PDF of the process ðtÞ is [42]

ð1 ð1 1 x2 2 x2 1 1 jsj exp s exp p ðxÞ ¼ pffiffiffiffiffiffiffiffiffiffiffi ps ðjsjÞds ¼ pffiffiffiffiffiffiffiffiffiffiffi ps ðsÞds 2Dn s2 2Dn s2 2Dn 1 2Dn 0 ð7:293Þ The corresponding relation for the characteristic function is [42] ð’Þ ¼ 2’

ð1 1=4 0

pffiffiffiffi s ð!Þ!1=2 J1=2 ð! ’Þd!

ð7:294Þ

368


where s ð!Þ and ð!Þ are the characteristic functions of sðtÞ and ðtÞ, respectively. As has been shown above, the covariance function of the compound process ðtÞ is simply the product of the correlation functions of its components n ðtÞ and sðtÞ C ðÞ ¼ Cn ðÞCs ðÞ

ð7:295Þ

If sðtÞ is the process with decay s < m then xn ðtÞ is the normal process with correlation function Cn ðÞ ¼ C ðÞ expðs jjÞ

ð7:296Þ

It is easy to show that Cn ðÞ ! 0 when ! 1, so the random process n ðtÞ can be represented as a component of the random process generated by SDE w_ ¼ Aw þ Bn ðtÞ

ð7:297Þ

where w1 ðtÞ ¼ n ðtÞ, and A and B are the matrices obtained using the algorithm described in the previous section. If the generating SDE for process sðtÞ is x_ ¼ f ðsÞ þ gðsÞs ðtÞ

ð7:298Þ

then consequently the generating SDE for process ðtÞ is 8 w_ ¼ Aw þ B1 ðtÞ > < n x x X x x > a1i wi þ w1 g 2 ðtÞ þ b1 1 ðtÞ : x_ ¼ w1 f w þ w w1 w1 1 1 i¼1

ð7:299Þ

Example: A narrow band K-distributed random process. Let us to obtain the random process with marginal distribution p ðxÞ ¼ K0 ðaxÞ

ð7:300Þ

and the correlation (covariance) function of the following form K ðxÞ ¼ 2 expð0 jjÞ cosð!Þ

ð7:301Þ

In this case, the desired process ðtÞ can be represented as a product of a normal distributed random process n ðtÞ with correlation function

0 Cn n ðxÞ ¼ exp jj cosð!Þ 2 2

and an m-distributed random process SðtÞ with m ¼ 0:5 and ¼ 1.

ð7:302Þ

SYNTHESIS OF IMPULSE PROCESSES

369

In this case process xn ðtÞ is generated by SDE €xn þ

0 x_ n þ !2 xn ¼ n ðtÞ !

ð7:303Þ

and sðtÞ is generated by15 s_ ¼

0 2

rffiffiffi ! 2 þ FðsÞðtÞ s

ð7:304Þ

where FðsÞ is, according to equation (4.60), pffiffiffi 1 s2 0 2 ; 0; 2 2 FðsÞ ¼

ð7:305Þ

where ð; ; Þ is the generalized Gamma function [32].

7.6

SYNTHESIS OF IMPULSE PROCESSES

Let us now model a stationary impulse process with a given PDF p st ðxÞ. Synthesis of the corresponding generating SDE d xðtÞ ¼ f ðxÞ þ ðtÞ dt

ð7:306Þ

model is based on inversion of the stationary Kolmogorov–Feller equation (see Section 6.2.3) d ½K1 ðxÞp st ðxÞ þ dx

ð1 1

pA ðx AÞp st ðAÞdA p st ðxÞ ¼ 0

ð7:307Þ

or the generalized Fokker–Planck equation d 1 d2 ½K2 ðxÞp st ðxÞ þ ½K1 ðxÞp st ðxÞ þ dx 2 dx2

ð1 1

pA ðx AÞp st ðAÞdA p st ðxÞ ¼ 0 ð7:308Þ

if a mixture of Gaussian noise and Poisson impulses are present. For a moment the focus of discussions is on eq. (7.307). If it is also assumed that a particular form of the distribution of the excitation pulses pA ðAÞ is chosen, then the synthesis problem is reduced to a simple first order differential 15

This SDE is written in Ito form.

370


equation (7.307) with respect to K1 ðxÞ. Its solution is easy to obtain by simple integration, which produces K1 ðxÞp st ðxÞ ¼

ð x ð 1 1

1

pA ðs AÞp st ðAÞdA p st ðsÞ ds þ C0

ð7:309Þ

A constant C0 must be chosen in such a way that the net probability current vanishes, i.e. one has to choose C0 ¼ 0. This implies that K1 ðxÞ ¼ p st ðxÞ

7.6.1

ð x ð 1 1

1

pA ðs AÞp st ðAÞdA p st ðsÞ ds ¼ f ðxÞ

ð7:310Þ

Constant Magnitude Excitation

In this case all impulses have the same magnitude, i.e. pA ðAÞ ¼ ðA A0 Þ

ð7:311Þ

As a result, one can reduce eq. (7.311) using the sifting property of the delta function to a simpler equation f ðxÞ ¼ p st ðxÞ

ðx 1

½p st ðs A0 Þ p st ðsÞds ¼

P ðxÞ P ðx A0 Þ p st ðxÞ

ð7:312Þ

Example: An exponentially distributed process. We wish to synthesize an SDE with exponential distribution of the solution: p st ðxÞ ¼ expð xÞ;

x0

ð7:313Þ

In this case ( P ðxÞ ¼

0

if x < 0

1 expð xÞ if x 0

ð7:314Þ

and, thus, eq. (7.312) produces 8 > < ½expð xÞ 1 f ðxÞ ¼ K1 ðxÞ ¼ > : ½expð A0 Þ 1

if 0 x A0 ð7:315Þ x A0

371

SYNTHESIS OF AN SDE WITH RANDOM STRUCTURE

7.6.2

Exponentially Distributed Excitation

In this case all impulses have the same magnitude, i.e. pA ðAÞ ¼ expð AÞ;

A0

ð7:316Þ

As a result, one can reduce eq. (7.311) using the sifting property of the delta function to a simpler equation f ðxÞ ¼ p st ðxÞ

ðx 1

expð sÞ

ðs

expð AÞp st ðAÞda p st ðsÞ ds

ð7:317Þ

0

Example: A Gamma distributed process [64]. Let one desire to model an impulse process with Gamma PDF 1 x expðxÞ ð Þ

p st ðxÞ ¼

ð7:318Þ

Setting the parameter in eq. (7.316) equal to in eq. (7.318) one can easily see that eq. (7.317) becomes f ðxÞ ¼

x

ð7:319Þ

More examples can be found in Table D.2 in Appendix D on the web.

7.7

SYNTHESIS OF AN SDE WITH RANDOM STRUCTURE

In this section the inverse problem is considered for an SDE with random structure. In addition to the information required for synthesis of an ordinary SDE, such as marginal PDF of the process and the correlation function, one has to obtain a priori information about a number of states M (number of sub-structures) and intensity of transitions kl . Knowledge of intensities allows one to solve the Kolmogorov–Chapman equation to obtain stationary ðlÞ probabilities of states Pl . As the next step, the joint PDF pst ðxÞ can be obtained using the equation ðlÞ

pst ðxÞ ¼ Pl plst ðxÞ

ð7:320Þ

In the following, only a scalar case is considered. In order to further simplify discussion, ðlÞ ðlÞ systems with a constant drift are considered: K2 ðxÞ ¼ K2 . A stationary solution, if it exists, must satisfy the following system of equations: ðlÞ

pst ðxÞ

@ ðlÞ @ ðlÞ ðlÞ ½K1 ðxÞ þ K1 ðxÞ pst ðxÞ ¼ QðxÞ @x @x

ð7:321Þ

372


where ðlÞ

K2 @ 2 ðlÞ pst ðxÞ þ Vl ðxÞ Ul ðxÞ QðxÞ ¼ 2 2 @x

ð7:322Þ ðlÞ

The next step is to rewrite eq. (7.321) as a differential equation with respect to K1 ðxÞ, i.e. in the following form d ðlÞ d QðxÞ ðlÞ ðlÞ K1 ðxÞ þ K1 ðxÞ ln pst ðxÞ ¼ ðlÞ dx dx pst ðxÞ

ð7:323Þ

which can be easily solved using the method of undefined coefficients ðlÞ

K ðlÞ ðlÞ K1 ðxÞ ¼ 2 ln pst ðxÞ 2

l

ðx 1

ðlÞ

pst ðxÞdx þ

M X

kl

k¼1

!

ðx

ðkÞ

1

pst ðxÞdx

þC

ð7:324Þ

The constant of integration in eq. (7.324) can be rewritten for pðlÞ ðxÞ if the additional requirement ðlÞ

K1 ð1Þ ¼ 0

ð7:325Þ

ðlÞ

is imposed on K1 ðxÞ. In this case, the synthesis equation becomes aðlÞ ðxÞ ¼

M X bðlÞ d Fl ðxÞ Pk Fk ðxÞ ðlÞ ln pst ðxÞ þ l ðlÞ kl ðkÞ 2 dx pst ðxÞ k ¼ 1 Pl pst ðxÞ

ð7:326Þ

where Fl ðxÞ ¼

ðx 1

ðlÞ

pst ðxÞdx

ð7:327Þ ðlÞ

It can be seen that the expression for the drift coefficient K1 ðxÞ contains a term present in the synthesis equation for an ordinary SDE (the first term in eq. (7.326)), while the random structure is represented by additional (the second and the third) terms in eq. (7.326). These additional terms represent contributions of flows of absorption and regeneration of the trajectories of the process with random structure. ðlÞ The diffusion coefficient K2 , as in the case of a monostructure, should be chosen in accordance with the correlation interval of the random process ðtÞ. Quantitatively it can only be obtained from the expression for the covariance function, which can be obtained only for a case of high and low intensities of the switching process and under the assumption that such a correlation function is close to exponential. For the case of a low intensity of switching, this approach produces quite accurate results since the effect of switching can be neglected or accounted for only by first order correction terms. In this case the synthesis of the SDE with random structure is no different from the synthesis of an ordinary SDE.

REFERENCES

373

REFERENCES 1. B. Liu and D. Munson, Generation of a Random Sequence Having Jointly Specified Marginal Distribution and Autocovariance, IEEE Trans. Acoustics, Speech Signal Processing, Vol. 30, 1982, pp. 973–983. 2. V. Kontorovich and V. Lyandres, Stochastic Differential Equations: An Approach to the Generation of Continuous Non-Gaussian Processes, IEEE Trans. Signal Processing, Vol. 43, 1995, pp. 2372– 2385. 3. S. Primak and V. Lyandres, On the Generation of Baseband and Narrowband Non-Gaussian Processes, IEEE Trans. Signal Processing, Vol. 46, 1998, pp. 1229–1237. 4. D. Middleton, An Introduction to Statistical Communication Theory, New York: McGraw-Hill, 1960. 5. E. Conte, M. Longo, and M. Lops, Modeling and Simulation of Non-Rayleigh Radar Clutter, IEEE Proc. F 138, 1991, pp. 121–131. 6. M. Rangaswamy, D. Weiner, and A. Ozturk, Non-Gaussian Random Vector Identification Using Spherically Invariant Random Processes, IEEE Tran. Aeros. Elec. Systems, Vol. 29, 1993, pp. 111– 124. 7. A. Fulinski, Z. Grzywna, I. Mellor, Z. Siwy, and P.N.R. Usherwood, Non-Markovian Character of Ionic Current Fluctuations in Membrane Channels, Physical Review E, Vol. 58, 1998, pp. 919–924. 8. S. Primak and J. LoVetri, Statistical Modeling of External Coupling of Electromagnetic Waves to printed Circuit Boards Inside a Leaky Cavity, Technical Report, CRC, Ottawa, Canada. 9. J. Lee, R. Tafazolli, and B. Evans, Erlang Capacity of OC-CDMA with Imperfect Power Control, IEE Electronics Letters, Vol. 33, 1997, pp. 259–261. 10. R. J. Polge, E. M. Holliday, and B. K. Bhagavan, Generation of a Pseudorandom Set with Desired Correlation and Probability Distribution, Simulation, 1973, pp. 153–158. 11. W. J. Szajnowski, The Generation of Correlated Weibull Clutter for Signal Detection Problems, IEEE Trans. Aer. Electr. Syst., Vol. AES-13, 1977, pp. 536–540. 12. V. G. Gujar and R. J. Kavanagh, Generation of Random Signals with Specified Probability Density Functions and Power Density Spectra, IEEE Trans. Autom. Contr., Vol. AC-13, 1968, pp. 716–719. 13. B. Liu and D. C. Munson, Generation of a Random Sequence Having a Jointly Specified Marginal Distribution and Autocovariation, IEEE Trans. Acous., Speech, Signal Proc., Vol. ASSP-30, 1982, pp. 973–983. 14. R. Rosenfeld, Two Decades of Statistical Language Modeling: Where Do We Go From Here?, Proc. IEEE, Vol. 88, 2000, pp. 1270–1278. 15. J. Luczka, T. Czernik, and P. Hanggi, Symmetric White Noise Can Induce Directed Current in Ratchets, Physical Review E, Vol. 56, 1997, pp. 3968–3975. 16. M. Popescu, C. Arizmendi, A. Salas-Brito, and F. Family, Disorder Induced Diffusive Transport in Ratchets, Physical Review Letters, Vol. 85, 2000, pp. 3321–3324. 17. V. Z. Lyandres and M. Shahaf, Envelope Correlation Function of Narrow-Band Non-Gaussian Process, Int. J. Nonlinear Mechanics, Vol. 30, 1995, pp. 359–369. 18. E. Wong and J. B. Thomas, On Polynomial Expansion of Second-Order Distributions, J. Soc. Industr. Math., 1962, Vol. 10, pp. 507–516. 19. T. Ozaki, A Local Linearization Approach to Non-Linear Filtering, Int. Jour. Contr., Vol. 57, 1995, pp. 75–96. 20. S. Primak, V. Lyandres, O. Kaufman, and M. Kliger, On the Generation of Correlated Time Series with a Given PDF, Submitted to Signal Processing, February, 1997. 21. E. Wong, ‘‘The Construction of a Class of Stationary Markov Processes’’, Proc. Symp. Appl. Math., Vol. 16, 1964, pp. 264–275. 22. T. T. Soong, Random Differential Equations in Science and Engineering. New York: Academic Press, 1971.

374


23. K. Ito, Stochastic integrals, Proc. Imp. Acad., Tokyo, Vol. 40, 1944, pp. 519–524. 24. I. Stakgold, Boundary Value Problem of Mathematical Physics, New York: Macmillan, 1967. 25. A.H. Haddad, Dynamical Representation of Markov Processes of the Separable Class, IEEE Trans. Information Theory, Vol. 16, 1970, pp. 529–534. 26. S. Primak and V. Lyandres, On the Generation of the Baseband and Narrowband Non-Gaussian Processes, IEEE Trans. Signal Processing, Vol. 46, 1998, pp. 1229–1237. 27. J. L. Doob, Stochastic Processes, New York: Wiley, 1953. 28. V. Ya. Kontorovich and V. Z. Lyandres, Fluctuation equations generating a one-dimensional Markovian Process, Radio Engineering and Electronics Physics, July 1972, pp. 1093–1096 (in Russian). 29. M.G. Kendall and A. Stuart, The Advanced Theory of Statistics, London: Griffin, 1958. 30. R. Gut and V. Kontorovich, On Markov Representation of Random Processes, Voprosi Radiotechniki, Vol. 6, 1971, pp. 91–94 (in Russian). 31. R. L. Stratonovich, Topics in the Theory of Random Noise, Vol. 1, New York: Gordon and Breach, 1967. 32. M. Abramoviz and I.A. Stegun, Handbook of Mathematical Functions, New York: Dover Publications, 1971. 33. B. Stuck and B. Kleiner, A Statistical Analysis of Telephone Noise, Bell Technical J., Vol. 53, pp. 1236–1320. 34. R. Risken, The Fokker–Planck Equation: Methods of Solution and Applications, Berlin: Springer, 1996, 472 p. 35. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, New York: Springer-Verlag, 1992. 36. V. Pugachev and I. Sinitsin, Stochastic Differential Systems: Analysis and Filtering, New York: Wiley, 1987. 37. P. E. Kloeden and E. Platen, Numerical Solution of Stochastic Differential Equations, New York: Springer-Verlag, 1992. 38. C. W. Gardiner, Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences Berlin: Springer-Verlag, 1983. 39. V. Kontorovich, V. Lyandres, and S. Primak, The Generation of Diffusion Markovian Processes with Probability Density Function Defined on Part of the Real Axis, IEEE Signal Processing Letters, Vol. 3, 1996, pp. 19–21. 40. E. Conte and M. Longo, Characterization of Radar Clutter as a Spherically Invariant Random Process, IEE Proc., Part F., Vol. 134, 1987, pp. 191–201. 41. S. Primak, V. Lyandres, and V. Kontorovich, Markov Models of non-Gaussian Exponentially Correlated processes and Their applications, Physical Review E, Vol. 63, 2001, 061103. 42. M. Rangaswamy, D. Weiner, and A. Ozturk, Non-Gaussian Random Vector Identification Using Spherically Invariant Random Processes, IEEE Tran. Aeros. Elec. Syst., Vol. AES-29, 1993, pp. 111–201. 43. M. M. Sondhi, Random Processes with Specified Spectral Density and First Order Probability Density, Bell System Technical J., Vol. 62, 1983, pp. 679–701. 44. R. J. Webster, A Random Number Generator for Ocean Noise Statistics, IEEE J. Ocean Eng., Vol. 19, 1994, pp. 134–137. 45. S. Xu, Z. Huang, and Y. Yao, ‘‘An Analytically Tractable Model for Videoconferencing,’’ IEEE Trans. Circuits and Systems for Video Technology, Vol. 10, 2000, p. 63. 46. S. Xu and Z. Huang, A Gamma Autoregressive Video Model on ATM Networks, IEEE Trans. Circuits and Systems for Video Technology, Vol. 8, 1998, pp. 138–142. 47. E. Wong, The Construction of a Class of Stationary Markov Processes, Proc. Symp. Appl. Math., Vol. 16, pp. 264–275. 48. S. Primak, Generation of Compound Non-Gaussian Random Processes with a Given Correlation Function, Physical Review E, Vol. 61, 2000, p. 100.

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49. A. N. Malakhov, Cumulantnii Analiz Sluchainich Processov i Ich Preobrazovanii (Cumulant Analysis of Random Processes and Their Transformations), Moscow: Radio, 1978 (in Russian). 50. V. Kontorovich and V. Lyandres, Impulsive Noise: A Nontraditional Approach, Signal Processing, Vol. 51, 1996, pp. 121–132. 51. G. Kotler, V. Kontorovich, V. Lyandres, and S. Primak, On the Markov Model of Shot Noise, Signal Processing, 1999, pp. 79–88. 52. J. Nakayama, Generation of Stationary Random Signals with Arbitrary Probability Distribution and Exponential Correlation, IEICE Trans. Fundamentals, Vol. E77-A, 1994, pp. 917–922. 53. S. Karlin, A First Course in Stochastic Processes, New York: Academic Press, 1968. 54. N. S. Jayant and P. Noll, Digital Coding of Waveforms, New Jersey: Prentice-Hall, 1984. 55. C. W. Helstrom, Markov Processes and Application in Communication Theory, in Communication Theory, New York: McGraw-Hill, 1968, pp. 26–86. 56. T. Rappaport, Wireless Communications: Principles and Practice, Upper Saddle River, NJ: Prentice Hall PTR, 2002. 57. G. Stu¨ber, Principles of Mobile Communication, Boston: Kluwer Academic Publishers, 2001. 58. S. Primak, V. Lyandres, O. Kaufman, and M. Kliger, On the Generation of Correlated Time Series with a Given Probability Density Function, Signal Processing, Vol. 72, 1999, pp. 61–68. 59. A. Blanc-Lapierre, M. Savelli, and A. Tortrat, Etude de Modeles Statistiques Suggeres par la Consideration des Effets des Atmospheriques sur les Amplificateurs, Ann. Telecomm, Vol. 9, 1954, pp. 237–248. 60. T. Felbinger, S. Schiller, and J. Mlynek, Oscillation and Generation of Nonclassical States in ThreePhoton Down-Conversion, Physical Review Letters, Vol. 80, 1998, pp. 492–495. 61. A. C. Marley, M.J. Higgins, and S. Bhattacharya, Flux Flow Noise and Dynamical Transitions in a Flux Line Lattice, Physical Review Letters, Vol. 74, 1995, pp. 3029–3032. 62. P. Rao, D. Johnson, and D. Becker, Generation and Analysis of Non-Gaussian Markov Time Series, IEEE Trans. Signal Processing, Vol. 40, 1992, pp. 845–855. 63. S. M. Rytov, Yu. A. Kravtsov, and V. I. Tatarskii, Principles of Statistical Radiophysics, Vol. 2, Correlation Theory of Random Processes, New York: Springer-Verlag, 1988. 64. D. Klovsky, V. Kontorovich, and S. Shirokov, Models of Continuous Communications Channels Based on Stochastic Differential Equations, Moscow: Radio i sviaz, 1984 (in Russian). 65. I. Kazakov and V. Artem’ev, Optimization of Dynamic Systems with Random Structure, Moscow: Nauka, 1980 (in Russian). 66. A. Berdnikov, A. Brusentzov, V. Kontorovich, and V. Lyandres, Problems in the theory and design of Instruments for Analog Simulation of Multipath Channels, Telecomm. Radio. Eng., Vol. 33–34, 1978, pp. 60–64. 67. V. Tikhonov and M. Mironov, Markov Processes, Moscow: Sov. Radio, 1977 (in Russian). 68. I. S. Gradsteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, Academic Press, 1980.

8 Applications 8.1 8.1.1

CONTINUOUS COMMUNICATION CHANNELS A Mathematical Model of a Mobile Satellite Communication Channel

For some years many different organizations all over the world have been involved in activities of land mobile satellite communication services [1]. The quality of this kind of communication suffers from strong variations of the received signal power due to satellite signal shadowing and multipath fading. The shadowing of the satellite signal by obstacles in the propagation path (buildings, bridges, trees, etc.) results in attenuation over the entire signal bandwidth. This attenuation increases with carrier frequency, i.e. it is more pronounced in the L-band than in the UHF band. The shadowed areas are larger for low satellite elevations than for high elevations. Multipath fading occurs because the satellite signal is received not only via the direct path but also after having been reflected from objects in the surroundings. Due to their different propagation distances, multipath signals can add destructively, resulting in a deep fade. In analysing the scattered field characteristics the following assumptions are usually considered as valid: a large number of partial waves; identical partial wave amplitudes; no correlation between different partial waves; no correlation between the phase and amplitude of one partial wave; phase distribution is uniform on ½0; 2 . Physically, these assumptions are equivalent to a homogeneous diffuse scattered field resulting from randomly distributed point scatterers. With these assumptions, the central limit theorem leads to a complex stationary Gaussian process z ¼ x þ j y with zero mean, equal variance and uncorrelated components. The amplitude distribution thus obeys the Rayleigh PDF [2,3]

A A2 pA ðAÞ ¼ 2 exp 2 ð8:1Þ 2 Stochastic Methods and Their Applications to Communications. S. Primak, V. Kontorovich, V. Lyandres # 2004 John Wiley & Sons, Ltd ISBN: 0-470-84741-7

378

APPLICATIONS

which depends on a single parameter 2, equal to the variance of the in-phase and quadrature components. In the line of sight conditions, i.e. when the direct component is superimposed, pA ðAÞ is well described by the Rice distribution [2,3]

A A2 þ a 2 aA pA ðAÞ ¼ 2 exp I0 2 2 2

ð8:2Þ

where I0 ð Þ is the modified Bessel function of zero order, and a is a parameter describing the strength of the Line-of-Sight (LoS) component (see Section 2.6). The Rice factor K, defined as a2 K¼ 2 2

ð8:3Þ

not only defines the ratio of power of the LoS component to the power of the diffused component, but also can be used to compare different models as shown in Table 8.1. In general, it is possible that none of the aforementioned assumptions is valid due to correlation between a small number of partial waves. Numerous approximations have been suggested to describe the measurement results [4–17]. It was also found [3] that one of the most convenient and closely fitting distributions of the envelope is the Nakagami PDF

2 m 2 m1 m A2 exp pA ðAÞ ¼ A ðmÞ 2 2

ð8:4Þ

In scattering processes generating merely diffuse wave fields, m 1 and the Nakagami distribution is identical to the Rayleigh one (8.1). In the case of a line of sight, the distribution (8.4) with m > 1 approximates the Rice distribution. Another important characteristic of the communication channels is the power spectrum density of the fading signal [2]. Consider an unmodulated carrier being detected by a mobile receiver. Each component arrives at the receiver via different angles to the receiver velocity Table 8.1 Model Nakagami Suzuki

Loo

Different models of fading envelope and their comparison PDF

Rice factor 2 mm 2 m1 m A2 m2 m A exp ðmÞ A a2 pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 2 A 2 2 ! ðln s mÞ 3 ð1 þa 1 Aa 2 2 5 d s I0 4 3 exp s e 2 s s 0 A pffiffiffiffiffiffiffiffiffiffiffiffi 2 2 2 ðln s mÞ 3 ð1 2 2 1 A þ s A s 2 2 5 d s I0 e 4 exp 2 s 0

a2 2

References [6] [5]

[7]

379

CONTINUOUS COMMUNICATION CHANNELS

vector and thus each signal component acquires a certain Doppler shift that is related to the receiver speed and arrival angle i for the i-th ray according to [2] fi ¼

cos i ¼ fD cos i

ð8:5Þ

Due to the Doppler effect, a single frequency carrier results in multiple signals at the receiver having comparable amplitudes, random phases and a relative frequency shift confined to the Doppler spread about the carrier frequency. The theoretically obtained spectrum Sð f Þ (low pass equivalent) is given by [2] 1 Sð f Þ ¼ pffiffiffiffiffiffiffiffiffiffiffiffiffiffi fD2 f 2

ð8:6Þ

C ðÞ ¼ J0 ð2 fD jjÞ

ð8:7Þ

and the covariance function by

A number of alternative models have also been suggested, especially in the COST 207 report [4] which includes a combination of two Gaussian PDFs (see Table 3.2 in Chapter 3) and a combination of the Jakes PSD (8.6) and a delta function. These possibilities are summarized in Table 8.2. Finally, taking into account the multipath nature of the wave propagation one can represent the low pass equivalent impulse response of the channel as a sum of N rays embedded in white Gaussian noise rðtÞ ¼

N X

Ai ðtÞsðt i Þ þ ðtÞ

ð8:8Þ

k¼1

Table 8.2 Typical PSD of a fading signal [4] PSD Sð f Þ

Model Jakes

1 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi fD2 f 2

Gauss I

ð f þ 0:8 fD Þ2 0:909 exp 0:005 fD2

Gauss II

Normalized covariance function ðÞ J0 ð2 fD Þ !

ð f 0:4 fD Þ2 þ0:0909 exp 0:02 fD2 ! ð f 0:7 fD Þ2 0:968 exp 0:02 fD2

!

ð f þ 0:4 fD Þ2 þ0:0316 exp 0:045 fD2 Rice

!

0:681 pffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ 0:8319 ðf 0:7 fD Þ fD2 f 2

ð fD Þ2 0:909 exp 400

! expðj 1:6 fD Þ

! ð fD Þ2 expðj 0:8 fD Þ þ0:0909 exp 20 ! ð fD Þ2 expðj 1:4 fD Þ þ0:968 exp 200 ! ð fD Þ2 þ0:0316 exp expðj 0:8 fD Þ 40 0:1681 J0 ð2 fD Þ

380

APPLICATIONS

where N is the number of rays considered, Ai ðtÞ is the fading amplitude of the i-th ray and i is the delay of the i-th ray, and ðtÞ is the additive WGN. The number of rays, delay of each ray and their parameters vary greatly for different environments. COST 207 provides tables of corresponding parameters for 4, 6 and 12 path models in rural, bad urban, typical urban and hilly terrain environments. However, regardless of the particular environment, the general structure of the simulator of a mobile communication channel remains the same and is shown in Fig. 8.1.

Figure 8.1 General scheme of a mobile communication channel simulator.

8.1.2

Modeling of a Single-Path Propagation

From the diagram shown in Figure 8.1 one can conclude that the main task in the simulation of the multipath channel is to model a single ray propagation with the given characteristics. Two main schemes which allow one to maintain both the PDF of the envelope and the correlation function of the signal are suggested below. Numerous alternative methods can be found elsewhere [18–22].

8.1.2.1

A Process with a Given PDF of the Envelope and Given Correlation Interval

For the first approximation of the power spectrum (8.4) one can use a rational approximation with only one pole. Although this approximation is somewhat poor it is considered here


381

because this scheme facilitates modeling the non-Gaussian random process with any PDF of the envelope. Let the PDF pA ðAÞ of the envelope of a narrow band (with bandwidth fi ) signal be given. Then the bandwidth of the narrow band envelope is approximately 2 fi, and consequently, the correlation interval is corr i ¼

1 2 fi

ð8:9Þ

If the phase ’ðtÞ is uniformly distributed on the interval ½; and does not depend on AðtÞ then xðtÞ ¼ AðtÞ cos ’ðtÞ cos 2 f0 þ AðtÞ sin ’ðtÞ cos 2 f0

ð8:10Þ

is a stationary narrow band random process with the envelope AðtÞ and the power spectrum concentrated around the frequency f0 (see section 4.7). If AðtÞ is a process with certain PDF and correlation interval corr i , the problem is solved. The desired generating SDE can be obtained by applying the results of Chapter 7. The block diagram of the corresponding simulator is shown in Fig. 8.2.

Figure 8.2 Simulator of a narrow band process with given PDF of the envelope and bandwidth.

Example: Nakagami fading. In this case the distribution of the envelope is defined by eq. (8.4) and the corresponding SDE for the envelope is (see Section 7.2.2): K2 2 m 1 2 m A þ 2 ðxÞ þ K ðtÞ A¼ A 2

ð8:11Þ

The parameter K can be found from the correlation interval to be sffiffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K¼ fi ¼ f0 cos i ¼ m corr m mc

ð8:12Þ

382

APPLICATIONS

where c is the speed of light. Finally, the generating SDE of the envelope has the form rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi dA 2m 1 2mA ¼ f0 cos i þ 2 ðxÞ þ f0 cos i ðtÞ: dt 2m c A mc

ð8:13Þ

The results of the numerical simulation are shown in Figs. 8.3 and 8.4.

Figure 8.3 Estimated (solid line) and theoretical (dashed line) PDF of the process generated by SDE (8.13); m ¼ 1=2; ¼ 1.

Figure 8.4 Estimated (solid line) and theoretical (dashed line) correlation functions of the envelope generated by SDE (8.13); ¼ 0:1 s.


8.1.2.2

383

A Process with a Given Spectrum and Sub-Rayleigh PDF

A regular procedure to obtain the desired correlation function (power spectrum) is known only for the Gaussian random process. This synthesis procedure was considered earlier in Chapter 7. In the same chapter we showed that the compound representation of the random process could be used to obtain processes with a symmetrical PDF and any desired rational spectrum. In this case the complex (two-dimensional) Gaussian process xn ðtÞ is modulated by some positive process sðtÞ xðtÞ ¼ xn ðtÞsðtÞ

ð8:14Þ

Here the Gaussian process xn ðtÞ defines the spectral (correlation) properties of the process xðtÞ, and the modulating signal sðtÞ defines the distribution of xðtÞ. Let pAx ðAx Þ be a distribution of the envelope of the process xðtÞ. Then the distribution ps ðsÞ of the modulation is related to pAx ðAx Þ as Ax pAx ðAx Þ ¼ 2

ð1 0

A2x d exp 2 2 sðÞ 2 2

ð8:15Þ

where 2 is the parameter of the Rayleigh envelope of the process xn ðtÞ. If the modulation sðtÞ is Nakagami distributed, then the resulting process has PDF of the envelope in the form rffiffiffiffiffiffiffiffiffi ! m mþ1 2m 2 Am Ax x Km1 2 2 2 pAx ðAx Þ ¼ ð8:16Þ ðmÞ where Kv ð Þ is the modified Bessel function [23]. It is worth mentioning that the mean value of the modulation sðtÞ should be chosen to be 1, in order to keep the mean energy of the process xðtÞ the same as the mean energy of the Gaussian process xn ðtÞ. This leads to the following condition ð1 s ps ðsÞd s ¼ 1 ð8:17Þ 0

Which is satisfied if ¼m

2 ðmÞ 2 ðm þ 0:5Þ

ð8:18Þ

Finally, the PDF of the envelope is pAx ðAx Þ ¼

ðmÞ þ 0:5Þ2 2

2 2 ðm

mþ1 2 Am x Km1 ðmÞ

sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ! 2 2 ðmÞ Ax 2 ðm þ 0:5Þ2

ð8:19Þ

384

APPLICATIONS

pffiffiffi bþ1 þ1 Examples of K-PDF px ðxÞ ¼ 2þ1 K ðb xÞ. (1) ¼ 1; b ¼ 2—solid line; (2) ðÞ x pffiffiffi pffiffiffi ¼ 2; b ¼ 2 2—dashed line; (3) ¼ 3; b ¼ 3 2—dashed-dotted line; (4) Rayleigh distribution with 2 ¼ 1. Figure 8.5

The PDF (8.19) defines the K-distributed envelope, which is a suitable model for subRayleigh fading. When m ¼ 1, PDF (8.19) becomes a Rayleigh one.1 Some plots of the K distribution are given in Fig. 8.5. The desired power density spectrum (8.6) can be approximated by the third order FIR filter response yðnÞ ¼ b0 xðnÞ þ b1 xðn 1Þ þ b2 xðn 2Þ þ b3 xðn 3Þ

ð8:20Þ

where the coefficients fbi g can be calculated as 2 Fd Fs b0 ¼ 1 ¼ 0:43

!p ¼

ð8:22Þ ð8:23Þ

ð!p þ 2Þð2 !2p 8Þ þ ð!p 2Þð!2p þ 4 !p þ 4Þ ð!p þ 2Þð!2p þ 4 !p þ 4Þ

ð8:24Þ

ð!p 2Þð2 !2p 8Þ þ ð!p þ 2Þð!2p 4 !p þ 4Þ b2 ¼ ð!p þ 2Þð!2p þ 4 !p þ 4Þ

ð8:25Þ

b1 ¼

b3 ¼

ð!p 2Þð!2p 4 !p þ 4Þ ð!p þ 2Þð!2p þ 4 !p þ 4Þ

In this case the modulation sðtÞ becomes the deterministic function sðtÞ ¼ 1; ps ðsÞ ¼ ðs 1Þ.

1

ð8:21Þ

ð8:26Þ


385

Let z1 , z2 , z3 be the roots of the polynomial bðzÞ ¼ b3 z3 þ b2 z2 þ b1 z þ 1

ð8:27Þ

si ¼ i þ j !i ¼ Fs ln zi ; i ¼ 1; 2; 3

ð8:28Þ

and si and be defined as

¼

maxf i g 2

ð8:29Þ

This allows one to define a new form filter with poles sni ¼ si þ ¼ i þ þ j !i

zni ¼ zi exp Fs

ð8:30Þ ð8:31Þ

which is again a stable filter. Then, if process xn ðtÞ has the spectrum defined by the form filter with poles given by eq. (8.31), and sðtÞ is a process with the decay of the correlation function defined by eq. (8.29), the process xðtÞ ¼ xn ðtÞsðtÞ has the spectrum defined by eqs. (8.20)–(8.26), i.e. it has the desired spectral properties. Results of the simulation are shown in Figs. 8.6–8.8. Example: Nakagami sub-Rayleigh fading.2 A similar technique can be used for Nakagami sub-Rayleigh fading, i.e. for the case m < 1. In this case the modulation process

Figure 8.6

2

Estimated (solid line) and theoretical PDF, corresponding to eq. (8.14).

Material for these examples was prepared in collaboration with Ms Vanja Subotic.

386

APPLICATIONS

Figure 8.7 Correlation function corresponding to spectrum (8.6) (solid line), and those obtained according to eq. (8.31) (dashed line).

is a square root of a beta distributed random process [24–26] ym1 ð1 yÞm ; ps2 ðyÞ ¼ Bðm; 1 mÞ

0m1

ð8:32Þ

Results of the simulation are shown in Figs. 8.6–8.8. Samples of fading have been used to produce an error sequence (see section 8.2) and estimate the so-called Pðn; mÞ characteristic, i.e. the probability that there are exactly n errors in a block of m symbols (Figs. 8.9 and 8.10).

Figure 8.8 Estimated (solid line) and desired (dashed line) normalized correlation function, defined by eq. (8.14).


387

Figure 8.9 Nakagami cumulative distribution (CDF) and correlation functions for m ¼ 0:7, ¼ 1; fD ¼ 10 Hz and ¼ 1 103 s. Solid line—theory, dotted line—simulation.

Figure 8.10 Nakagami Pðm; nÞ characteristic for m ¼ 0:5 and m ¼ 0:7; SNR ¼ 5 dB; fD ¼ 10 Hz and ¼ 1 103 .

388

8.2

APPLICATIONS

AN ERROR FLOW SIMULATOR FOR DIGITAL COMMUNICATION CHANNELS

Most modern communications systems use discrete information sources and, consequently, discrete methods of modulation. A general scheme of a communication system of this type is given in Fig. 8.11 [3].

Figure 8.11

Digital communication system.

A large number of encoders–decoders (codecs) have been developed but their practical implementation sometimes leads to the results being far from the theoretical ones. The main reason for this failure is the fact that most scientific work in coding theory was applied to a binary symmetrical channel with independent errors—a model which cannot be considered as adequate for real communication conditions [27–29]. In recent years a number of works were devoted to the experimental investigation of real communication system behaviour and its description by different mathematical models [30– 97]. These models allow one to obtain statistical characteristics of the error flow, and then to calculate the efficiency of different correcting codes. It was found that simple mathematical models merely describe channel properties. The complexity of more accurate models arises from the fact that the discrete channel involves the transformation of signals from discrete to continuous, and back again. It can be seen from Fig. 8.11 that the discrete channel includes a continuous channel,3 and that the properties of the latter affect the properties of the discrete channel under consideration. The transition from continuous to discrete signals involves a modulator and demodulator and, consequently, a discrete communication channel should be defined by both blocks. Analytical investigation of such models is complicated, but the numerical simulation produces important information for channel designers.

3

This assertion allows the use of all models considered in the previous section as part of the discrete channel.


8.2.1

389

Error Flow in Digital Communication Systems

In the performance analysis of the modulator–demodulator (modem) where the average error rate is considered as the key parameter, the physical sources of errors are mathematically reasonably tractable and can be found from a mathematical model of the continuous part of the channel. For the coding of a discrete channel involving a modem and the propagation medium, any meaningful analytical description of channel memory in terms of the characteristics of individual physical causes of errors is difficult. It is more convenient to work with sample error sequences obtained from test runs of data transmitted on a given coding channel to obtain statistical descriptions of the error structure, and to use them in the design of error control systems. The required statistical parameters may be obtained from error sequences either by direct processing of the sequences or by forming a parameterized mathematical model that would ‘‘generate’’ similar sequences. The latter involves the following [30]: visualizing a model representative of real channel behaviour; developing methods to parameterize the model using error sequences; deriving the statistics required for code evaluation from the model. The error process on a digital communication link can be viewed as a binary discrete-time stochastic process, i.e. a family of binary random variables fXt ; t 2 Ig where I is the denumerable set of integers and t denotes time. The sequence x ¼ . . . ; x1 ; x0 ; x1 ; . . .

ð8:33Þ

representing a realization of the error process can be related to the channel input and output as follows. Let fan g be the digital sequence, and fbn g be the corresponding output sequence. The perturbation due to the channel between the input and output can be considered as an introduction of a noise sequence n originated from a hypothetical random noise source. This may be represented mathematically as the addition of a noise digit to an input digit, i.e. bn ¼ an þ x n

ð8:34Þ

where fan g, fbn g and fxn g are elements of the Galois field GFðqÞ and the addition is over the same field. For binary communications, q ¼ 2, and the addition is modulo 2. An error is said to occur whenever xn is different from zero. Accordingly, an error sequence fxn g is a stochastic mapping of the noise sequence fn g onto f0; 1g. 8.2.2

A Model of Error Flow in a Digital Channel with Fading

For the time being a communication channel without fading is considered. In this case the received signal ^sðtÞ is given by [3] ^sðtÞ ¼ sk ðt Þ þ Kch ðtÞ

ð8:35Þ

390

APPLICATIONS

where sk ðtÞ is the transmitted signal (logical 0 or 1 in the case of a binary channel), is a constant, describing the general attenuation of the signal in the medium of propagation, ðtÞ 2 is the intensity of WGN in the channel. After demodulation is WGN of unit intensity and Kch and decoding, the received signal ^sðtÞ is transformed to s^k ðtÞ. Error occurs if s^k ðtÞ does not coincide with sk ðtÞ. The error probability perr depends on the Signal-to-Noise Ratio (SNR), defined as SNR ¼

2 Eðsk Þ ¼ h2 ¼ 2 Kch

ð8:36Þ

and on the demodulation scheme. A great number of modulations is treated in [3]. The error probability perr does not depend on time and is equal to the average number of errors per bit transmitted. The error flow in this case can be modeled simply as the Poisson flow with intensity ¼ perr. If fading is present in the channel, the received signal should be written as ^sðtÞ ¼ ðtÞsk ðt Þ þ KðtÞ

ð8:37Þ

Now ðtÞ is a random process and, consequently, SNR and the error probability should be changed in time (i.e. they both are random processes). In the case of slow flat fading, it can be assumed that SNR and perr are slow functions of time, depending only on ðtÞ, defined as hðtÞ ¼

2 ðtÞEðsk Þ 2 Kch

perr ðtÞ ¼ FðhðtÞÞ

ð8:38Þ ð8:39Þ

where the structure of the function Fð Þ is defined by demodulation methods. Corresponding error flow can now be modeled as a Poison flow with slow varying intensity ðtÞ, equal to the instant error probability ðtÞ ¼ perr ðtÞ

ð8:40Þ

A corresponding scheme of the error flow simulator is shown in Fig. 8.12.

Figure 8.12

Error flow simulator.

It is worth noting that in many cases it is possible to calculate probability of error taking fading into account [3]. In this case the average probability of error would define a constant error rate over the duration of simulation. The model suggested here, while on a large scale leading to the same results, represents a finer structure of error flow since it follows local variations in the signal level. Consequently, it could be more appropriate for simulation of performance of relatively short codes or codes with memory.


Example: Error flow in a channel with Nakagami fading. channel experience Nakagami fading, described by PDF

391

Let a communication

2 mm 2 m1 m A2 A exp pA ðAÞ ¼ ðmÞ

ð8:41Þ

Thus, one can consider ðtÞ in eq. (8.37) as a random process generated by the following SDE _ ¼

2 m corr

sffiffiffiffiffiffiffiffiffiffiffiffi 2m 1 2m þ ðtÞ m corr

ð8:42Þ

Being proportional to the quantity zðtÞ ¼ 2 ðtÞ, the SNR ðtÞ can be thus generated by SDE

_ ¼

m corr

sffiffiffiffiffiffiffiffiffiffiffiffi 2m ðtÞ 2m 1 þ2 m corr

ð8:43Þ

where =m is the instantaneous SNR. The solution of this SDE ðtÞ can be used to drive the intensity of a Poisson flow of errors.

8.2.3

SDE Model of a Buoyant Antenna–Satellite Link

Small ships often employ a buoyant antenna array that extends into a ship’s wake to support satellite communications at sea [98]. Due to the changing surface environment, the instantaneous gain and phase of each array element is unknown because of multiple paths, changing interelement geometry and wash-over effects. The complex nature of communication systems based on this type of antenna system requires efficient numerical simulation to realistically assess performance during the communication system design process. The aim of this section is to present a mathematical model of the fading in the buoyant antenna array–satellite link which is useful both for numerical simulation and analytical calculations.4

8.2.3.1

Physical Model

A buoyant antenna array is often used for ship–satellite communications, especially by small boats and submerged submarines. Under the ideal conditions, such an array is represented by N equally spaced antenna elements, all transmitting at full power. In practice, some fraction of the elements are washed over by the sea due to wave motion on the sea surface. This results in random fluctuations in the antenna pattern and the signal received at the satellite. For a relatively slow bit rate it is possible to adjust directivity of the antenna in such a way that there is always a direct line-of-sight (LoS) link; however, the energy level of this link varies. Additionally, there are also some diffusion components present due to reflection from the sea surface and random scattering in the atmosphere. Under such conditions it is possible 4

This section is written in collaboration with Mr J. Weaver.

392

APPLICATIONS

to assume that for a short period of time (when there is no change in the number and position of the active elements), the fading is described either by the Rice law [3] A A 2 þ a2 aA pA ðaÞ ¼ 2 exp I ; 0 2 2 2

A0

ð8:44Þ

or by Nakagami fading [3]

2 mm 2 m1 m A2 A exp pA ðaÞ ¼ ; ðmÞ

m > 1;

A0

ð8:45Þ

with parameters a, , and m chosen to properly represent energy levels of the signal and noise. If only short messages are communicated, such as distress signals or very slow rate sensory information, a static model with constant parameters performs satisfactorily. If longterm communication is considered, these parameters cannot be treated as constants due to fluctuations of the geometry of the array which are themselves random processes. It is important to properly represent the effect of such fluctuations on the resulting quality of the communication channel. The following physical model of the channel is accepted here. The fluctuation of the signal level is described as a continuous random process with its parameters randomly changing over a finite number of possibilities or states. The latter is justified since there is a finite number (however large) of possible combinations of active and passive array elements [98]. Transitions from state to state are random and described by an underlying Markov chain as described in Chapter 6. At this stage it is assumed that the parameters of fading in each state are known, either from measurements or from detailed simulation, and the main engineering focus is the effect of state changes on the overall performance of communication systems. One of the most interesting questions that needs to be addressed is the effect of the intensity of state transitions. These transitions are defined by the roughness of the sea surface and the speed of the craft. While there is no exact formula which relates these parameters to the transitional probabilities of a Markov chain, it is reasonable to assume that faster transitions correspond to a rougher sea and a larger velocity. This chapter provides an analytical expression for the resulting PDF of the fading as a function of transitional intensity in the Markov chain. When experimental data becomes available it will be possible to trace a direct influence of the velocity and sea roughness on the bit error rate of the link. 8.2.3.2

Phenomenological Model

The mathematical model of the system is represented as M stochastic differential equations (SDEs), each representing a state of the antenna array ðnÞ

x_ ¼

K2 d ðnÞ ln pn ðxÞ þ K2 ðtÞ; 2 dx

n ¼ 1; . . . ; M

ð8:46Þ

These sequences are randomly switched, maintaining a single point continuity between adjacent sequences, according to the state of the underlying Markov chain. Here ðtÞ is the


393

white Gaussian noise with unit spectral density, pn ðxÞ is the desired marginal PDF of the ðnÞ solution and K2 is a constant defining the correlation interval of the resulting process. The probability density function of the solution can be described by a system of generalized Fokker–Planck equations of the following form (see Section 6.2.3) @ ðlÞ @ ðlÞ p ðx; tÞ ¼ ½K1 pðlÞ ðx; tÞ @t @x M X 1 @ 2 ðlÞ ðlÞ ðlÞ ½K p ðx; tÞ l p ðx; tÞ þ kl pðkÞ ðx; tÞ þ 2 @x2 2 k6¼l

ð8:47Þ

Here kl is the intensity of transition from the k-th state of the array to the l-th state of the array, and, as the total switching intensity from the l-th state l ¼

M X

kl

ð8:48Þ

k 6¼ l

in the steady state, the probability densities do not depend on time and so eq. (8.47) is further simplified to become an ordinary linear differential equation of the second order. Transitional intensities kl define the probability of each state of the antenna array. In particular, the vector of such probabilities P ¼ ½P1 ; . . . ; PM T is the eigenvector of the matrix of transitional intensity T ¼ fkl g TP ¼ P

ð8:49Þ

In general, the solution to eq. (8.47) can only be found numerically. However, for the two limiting cases, it is possible to obtain accurate and simple analytical approximations. Let s n be a characteristic (correlation) time of the system in the n-th state. Then the switching is considered to be slow if maxfkl sk g 1 and fast if maxfkl sk g 1. In the case of slow fading, the PDF of the process is just a weighted sum of the PDF in isolated states (see Section 6.3.2) pðxÞ ¼

M X

Pn pn ðxÞ

ð8:50Þ

n¼1

such that the switching effects last approximately sn seconds and are subsequently dominated by internal dynamics of the system in the same fashion as in an isolated state. On the contrary, if fast switching is predominant, then all states in the distribution approach a single distribution given by the following expression (see Section 6.3.3) ðx C K 1 ðsÞ exp 2 ds ð8:51Þ pðxÞ ¼ K 2 ðxÞ x0 K 2 ðsÞ where M X

ðnÞ

K d ½ln pn ðxÞ K 1 ðsÞ ¼ Pn 2 2 dx n¼1

ð8:52Þ

394

APPLICATIONS

and K 1 ðsÞ ¼

M X n¼1

ðnÞ

ð8:53Þ

Pn K2

To simplify the complexity of the model, a two-state Markov model is used to represent the possible states of the channel. The ‘‘Good’’ state represents the situation when almost all the elements of the array are active, and ‘‘Bad’’ represents situations with very few active arrays. In the former case, the fading could be described with the Nakagami law with a large value of the parameter mG > 1 and the average value of the signal level with respect to noise G . The latter case is described by a relatively small mB 1:5 and the average level of signal B < G . The probability of the Bad state is PB and the probability of the Good state is PG ¼ 1 PB . The transitional matrix T of the Markov chain is then defined as

ð1 ÞPG þ T¼ ð1 ÞPG

ð1 ÞPB ð1 ÞPB þ

ð8:54Þ

where the parameter 0 1 reflects correlation properties of the chain ( ¼ 0 corresponds to uncorrelated transitions). It is assumed that BPSK modulation [3] is used. In an isolated state (i.e. when no switching is present), it is possible to show that fading can be described by the following stochastic differential equation [29] K2G 2 mG 1 mG x þ K2G ðtÞ x_ ¼ x G 2 K2B 2 mB 1 mB x x_ ¼ þ K2B ðtÞ x B 2

ð8:55Þ

For a small intensity of fading, the distribution of the magnitude is therefore mG

mB

2 mG mG A 2 2 mB m B A2 2 mG1 2 mB1 A exp A exp pðAÞ ¼ PG þ PB ðmG Þ G ðmB Þ B G B ð8:56Þ For the fast switching it follows from eq. (8.52) that the distribution is again Nakagamidistributed m1

2 m1 m 1 A2 2 m1 1 pðAÞ ¼ A exp ðm1 Þ 1 1

ð8:57Þ

where ðGÞ

m1 ¼

PG mG K2

ðGÞ

PG K 2

ðBÞ

þ PB mB K2 ðBÞ

þ PB K2

ð8:58Þ


395

and ðGÞ

1

ðBÞ

PG mG K2 þ PB mB K2 ¼ mG ðGÞ mB ðBÞ PG K2 þ PB K G B 2

ð8:59Þ

Finally, since the expression for the bit error rate of BPSK in Nakagami fading is known in terms of the hypergeometric function [2,23] rffiffiffiffiffiffiffi 1 ðm þ 0:5Þ 2m þ 1 1 ; ; Perr ¼ ð8:60Þ 2 F1 2 m ðmÞ 2 2 m one can easily obtain the expression for the BER in fading with changing parameters. For the case of a slow switching, eq. (8.60) produces sffiffiffiffiffiffiffiffiffiffi 1 G ðmG þ 0:5Þ 2 mG þ 1 1 G ; ; P e ¼ PG 2 F1 2 2 2 mG mG ðmG Þ sffiffiffiffiffiffiffiffiffiffi B ðmB þ 0:5Þ 2 mB þ 1 1 B ; ; 2 F1 2 2 mB mB ðmB Þ

ð8:61Þ

For the fast switching, the following is produced 1 Pe ¼ 2

sffiffiffiffiffiffiffiffiffiffiffi 1 ðm1 þ 0:5Þ 2 m1 þ 1 1 1 ; ; 2 F1 2 2 m1 m1 ðm1 Þ

ð8:62Þ

Since parameters P1 , P2 and the intensity of transitions directly affect BER and are functions of the roughness of the sea and the velocity of the craft, the developed equations could be a useful tool for system designers.

8.2.3.3

Numerical Simulation

The modified Ozaki scheme (see Section C.3 in Appendix C on the web and [27–29]) could be used to simulate each non-linear SDE d ½ pst ðxn Þ dx

1 fn log 1 þ Kt ¼ ½expð fn tÞ 1 t xn fn sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi expð2 Kt tÞ 1 t ¼ 2 Kt fn ¼

xnþ1 ¼ j expðKt tÞxn þ t i j

ð8:63Þ ð8:64Þ ð8:65Þ ð8:66Þ

396

APPLICATIONS

where Kt is a factor inversely proportional to the correlation interval of the fading and i is a Gaussian iid process. The processes are then commutated by the two-state Markov process. Performance of a few representative signalling techniques is evaluated with the numerical model. Additionally, the two composite processes are compared on the basis of the Kolmogorov– Smirnov ‘‘Goodness-of-Fit’’ measure [99]. This measure maps the greatest absolute error from the empirical CDF to a reference CDF as shown below ^ obs ðxÞ Pref ðxÞj D ¼ supjP

ð8:67Þ

pffiffiffiffi 0:11 PðD > observedÞ ¼ Qks D N þ 0:12 þ pffiffiffiffi N

ð8:68Þ

and

where N is the number of bins in the empirical CDF and Qks is defined as Qks ðÞ ¼ 2

1 X

ð1Þj1 exp½2 j2 2

ð8:69Þ

j¼1

Using this measure, both fast and slow switching processes were highly likely to match their theoretical counterparts. In the slow switching case, the match was a virtual certainty ð < 0:4Þ and in the fast switching case the probability of a theoretical match was 98.5% ð 0:5Þ. Below, the results for the slow switching are reviewed first (PDF and K-S performance) followed by similar graphs for the fast switching case. Fig. 8.13 shows the results of numerical simulation of the suggested model.

Figure 8.13 Results of the numerical simulation of SDE (8.55) with random structure: (a) slow fading; (b) fast fading.

A SIMULATOR OF RADAR SEA CLUTTER WITH A NON-RAYLEIGH ENVELOPE

8.3

397


Clutter, being an unwanted phenomenon which usually accompanies target detection, has been the subject of numerous investigations related to the performance of sea microwave radar [100–153]. In many real situations (low grazing angles, requirement of high resolution) the probability density function of the clutter envelope (EPDF) does not fit the Rayleigh law [100–153], i.e. it exhibits higher tails and a larger standard deviation–to–mean ratio. Here two models of a simulator of radar clutter, utilizing a second order SDE considered earlier in Chapter 7, and generating clutter with the K or Weibull distribution of the envelope, are considered. Such simulators can be used in the testing of radar target discrimination schemes ([117–153] and many others).

8.3.1

Modeling and Simulation of the K-Distributed Clutter

The K-distribution of the clutter envelope, which is now widely applied both to the description of microwave scattering by a rough ocean surface and to the synthesis of the target detection algorithms [101–108], provides a better agreement with experimental data than the Rayleigh one. In this case the envelope of the signal is distributed according to the following PDF pA ðAÞ ¼

bþ1 A K1 ðb AÞ; 21 ðÞ

A0

ð8:70Þ

where ð Þ is the Euler Gamma function [23], is a positive shape parameter, Kv ð Þ is the modified second kind Bessel function of order [23] and b is related to the average power 2 by the formula b2 ¼ 2 =2

ð8:71Þ

which generalizes the Rayleigh law in the case ! 0. A non-Rayleigh, and in particular K-distributed, correlated clutter may be presented as a narrow band stationary process with a symmetrical non-Gaussian distribution of instantaneous values [101–108]. In addition to having its power spectrum concentrated around its central frequency, the corresponding generating system has to be at least of the second order, and the clutter itself can be considered as a component of the two-dimensional Markovian process, i.e. as a solution, for example, of the following Duffing stochastic SDE €x þ 2 _x þ FðxÞ ¼ K ðtÞ

ð8:72Þ

where ðtÞ is the white Gaussian noise (WGN) of a unit one-sided power spectral density. To find the parameters of eq. (8.72) the following steps have to be taken. 1. Choose a non-linear function FðxÞ which determines the desired distribution of the instantaneous values of the process xðtÞ.

398

APPLICATIONS

2. Estimate the damping factor and intensity K 2 of the WGN, which together define correlation properties of the process xðtÞ. The relation between the stationary distribution px st ðxÞ of the solution of SDE (8.72) and the non-linear term FðxÞ in its operator has been found to be (see Section 7.3.3, eq. (7.201))

px stðxÞ

4 ¼ C exp 2 k

ðx FðsÞd s

ð8:73Þ

x0

where C is the normalization condition constant. Function FðxÞ in SDE (8.72), generally non-linear, can be determined by inverting eq. (8.73) (see eq. (7.202)) FðxÞ ¼

K2 d ln px stðxÞ 4 dx

ð8:74Þ

In order to use the synthesis equation (8.74) the exact form of the PDF px st ðxÞ of the process xðtÞ must be evaluated from the known PDF of the envelope (8.70). Functions pA ðAÞ and px st ðxÞ are related by the Blanc-Lapierre transformation (4.331) for narrow band processes 1 px st ðxÞ ¼ 2

ð1

ð1 exp½i s xd s 1

pA ðAÞJ0 ðs AÞd A

ð8:75Þ

0

This has been accomplished in Section 4.7.4.1, eq. (4.436) 1

1 2þ2 b pffiffiffi ðbjxjÞ2 K 1 ðbjxjÞ px st ðxÞ ¼ 2 ðÞ

ð8:76Þ

To calculate the logarithmic derivative of the PDF (8.76) one can use the well known identity [23]

d K ðzÞ Kþ1 ðzÞ ð8:77Þ ¼ dz z z which leads to K3=2 ðbjxjÞ K3=2 ðbjxjÞ p0x st ðxÞ ¼ b sgnðxÞ ¼ b sgnðxÞ K1=2 ðbjxjÞ K1=2 ðbjxjÞ px st ðxÞ

ð8:78Þ

As a result, the generating SDE of a narrow band random process with K-distributed envelope is given by €x þ x_ þ sgnðxÞ

K 2 bK3=2 ðbjxjÞ ¼ KðtÞ 4 K1=2 ðbjxjÞ

ð8:79Þ

399


The next problem is to estimate the mean value of the central frequency of the narrow band process xðtÞ. For this purpose one can approximate the non-linear function FðxÞ in eq. (8.79) by a polynomial series as in Section 7.3.3: €x þ x_ þ

1 X

! a2 iþ1 x2 i !0 x ¼ KðtÞ

ð8:80Þ

i¼0

where a1 ¼ 1 and a2 iþ1 are the coefficients of FðxÞ’s power expansion. For this purpose one may use the following presentation of the modified Bessel function around zero [23]

ðÞ21 ð1 Þ 2 K ðxÞ

1þ x ; 4 ð2 Þ x

x>0

ð8:81Þ

Taking expansion (8.81) into account, the non-linear function FðxÞ can be rewritten as 3 2 ð1=2 Þ 2 x 1þ K 2 b2 ð 3=2Þ 6 4 ð1=2 Þ 7 7 6 x FðxÞ ¼ ð1=2 Þ 2 5 8 ð 1=2Þ 4 x 1þ 4 ð3=2 Þ

K 2 b2 ð 3=2Þ ð1=2 Þ ð1=2 Þ b2 x2

x 1þ 8 ð 1=2Þ ð1=2 Þ ð3=2 Þ 4

ð8:82Þ

Fig. 8.14 shows the satisfactory quality of such an approximation. It is was shown in Section 4.7 that a narrow band process can be represented as xðtÞ ¼ AðtÞ cosððtÞÞ ¼ AðtÞ cos½!0 t þ ’ðtÞ

Figure 8.14 line).

ð8:83Þ

Non-linear term in eq. (8.79) (solid line) and its approximation in eq. (8.82) (dashed

400

APPLICATIONS

where AðtÞ and ’ðtÞ are the slowly varying envelope and phase, respectively, defined via the Hilbert transform ^xðtÞ of the process xðtÞ. In this case A2 ðtÞ ¼ ^x2 ðtÞ þ x2 ðtÞ ðtÞ ¼ atan

ð8:84Þ

^xðtÞ xðtÞ

Since for a narrow band process its Hilbert transform can be approximated by a derivative of the original process (see Section 4.7) x_ ¼ !0 AðtÞ sin ðtÞ

ð8:85Þ

an equivalent first order SDE can be obtained [154] "

#

1 X

dA 1 sin ¼ 2 sin A sin a2 iþ1 A cos 2 iþ1 ðtÞ dt 2 !0 i ¼ 0 !0 " # 1 dA 1 X cos ¼ 2 cos A sin a2 iþ1 A cos 2 iþ1 ðtÞ dt A 2 !0 i ¼ 0 A !0

ð8:86Þ ð8:87Þ

which, in turn, can be simplified to dA K2 K ¼ 2 s þ þ

1 ðtÞ dt 4 A 2 !0

ð8:88Þ

d K2 K ¼ 2 c þ þ

1 ðtÞ ¼ !ðtÞ dt 4 A!0 2 !0

ð8:89Þ

where s and c are regular terms of eqs. (8.86) and (8.87), respectively [155], 1 ðtÞ, and

1 ðtÞ are independent white Gaussian noises with spectral density 0:5 K 2. Averaging both sides of eq. (8.89) over A, the mean angular frequency of the power spectrum of the process xðtÞ can be obtained as " ! ¼ !0 1 þ " ¼ !0 1 þ

ð 1 X a2 iþ1 ð2 iÞ! 1 i¼1

ði!Þ2 2i

1 X a2 iþ1 ð2 iÞ! i¼1

ði!Þ2 2i

1

# pA ðAÞA

2 iþ1

dA

#

m2 iþ1

ð8:90Þ

The initial moments of K-distribution are [101–103] ml ¼

ð1

bþ1 A pA ðAÞd A ¼ 1 2 ðÞ

ð1

l

0

A 0

lþ

2 þ 2l 1 þ 12 K1 ðbAÞdA ¼ b l ðÞ

ð8:91Þ


401

Using the latter expression and taking into account the fact that b2 ¼ 2 =2 , one can obtain the following expressions for the first three even moments (8.91) þ1 m2 ¼ 22 pffiffiffi 6 ð þ 3=2Þ m3 ¼ 3 b ðÞ 2 8 ð þ 2Þð þ 1Þ m4 ¼ 2

ð8:92Þ

Approximation (8.82) now can be rewritten as K 2 b2 ð 3=2Þ x½1 þ a3 m3 x2 8 ð 1=2Þ

ð8:93Þ

ð1=2 Þ ð1=2 Þ b2 a3 ¼ ð1=2 Þ ð3=2 Þ 4

ð8:94Þ

FðxÞ

where

and !20 ¼

K 2 b2 ð 3=2Þ 8 ð 1=2Þ

ð8:95Þ

Taking into account only terms of order 3 in eq. (8.80), one can arrive at the following expression for the central frequency of the solution of SDE (8.80) !c ¼ !0 ð1 þ a3 m3 Þ

ð8:96Þ

The next step is to show that if the parameter approaches infinity, the generating SDE (8.79) tends to a linear second order equation which generates a narrow band Gaussian process with the corresponding Rayleigh PDF. This can be accomplished by considering the following asymptotic expansion [23] rffiffiffiffiffiffi expð Þ K ð zÞ 2 ð1 þ z2 Þ1=4

ð8:97Þ

where ¼

pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ z2 þ ln

z pffiffiffiffiffiffiffiffiffiffiffiffi 1 þ 1 þ z2

ð8:98Þ

Using eq. (8.97), one can obtain that K ðbjxjÞ ¼ K ð zÞ; z ¼

bjxj

ð8:99Þ

402

APPLICATIONS

If x is fixed and z ! 0, eq. (8.98) can be simplified to 1 þ ln

z 2

ð8:100Þ

thus producing a simple approximation for K ð zÞ and Kv ðbjxjÞ in the following form rffiffiffiffiffiffi rffiffiffiffiffiffi z K ð zÞ

exp ln expð0:307 Þz ¼ 2 2 2 rffiffiffiffiffiffi K ðbjxjÞ ¼ expð0:307 ÞbÞjxj 2

ð8:101Þ ð8:102Þ

Finally, substituting expression (8.102) into eq. (8.78), one obtains that K3=2 ðbjxjÞ p0x st ðxÞ ¼ b sgnðxÞ K1=2 ðbjxjÞ px st ðxÞ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 =ð2 3Þ expð0:307ð 3=2ÞÞb2

b pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 3 =ð2 1Þ expð0:307ð 1=2ÞÞb2 sffiffiffiffiffiffiffiffiffiffiffi 12 ex ¼ !20 r x ¼ b2 2 32

ð8:103Þ

The last expression corresponds to the linear function FðxÞ in SDE (8.79) €x þ x_ þ !20 r x ¼ KðtÞ

ð8:104Þ

which generates a narrow band Gaussian process with a central frequency !0 r and with an envelope distributed by the Rayleigh law. The next step is to tune the correlation function of the envelope. It is reasonable to assume that a small deviation of the PDF of the envelope does not significantly influence the correlation time, and to use the approximation of the K-distribution by the Nakagami law (8.4). Clearly, the most feasible approximation condition is the equality of the second and fourth moments of both distributions. In the case of the Nakagami PDF, they are equal to [6] mN 2 ¼

ð8:105Þ

and m4 N ¼

mþ1 2 m

ð8:106Þ

It follows from eq. (8.92) that þ1 2 m þ 1 2 8 ð þ 2Þð þ 1Þ ¼ m 2 ¼ 2 2

ð8:107Þ ð8:108Þ


403

and, thus m¼

þ1 þ2

ð8:109Þ

The SDE which generates the Nakagami distribution is well known (7.41) 2 K 2 m 1 2 m A N A_ ¼ þ KN N ðtÞ A 2

ð8:110Þ

where WGN N ðtÞ has unit intensity and KN ¼ 0:5 K. The correlation function of the SDE (8.110) solution may be written as (see eq. (7.43)) 1 2 m þ 12 X 2 i 12 2mki jj exp CAA ðÞ ¼ 4 m ðmÞ i ¼ 1 ðm þ 1Þ

ð8:111Þ

Taking into account only the first term in eq. (8.92), one can obtain an approximation for the envelope covariance function and correlation interval, respectively 2 m þ 12 2 m KN2 CA ðÞ

exp jj 16 m ðmÞðm þ 1Þ corr ¼ 2 KN2 m

ð8:112Þ ð8:113Þ

The latter allows one to complete the set of equations required to design a simulator. If the parameters !c , corr , and 2 of sea clutter are given, then the steps to generate the SDE model are the following: 1. Calculate the spectral density Kn of the excitation from eq. (8.113) sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 2 2 ð þ 2Þ KN ¼ corr

ð8:114Þ

2. Using eq. (8.96) calculate the frequency !c 1 þ a3 m3

ð8:115Þ

ð þ 2Þð 3=2Þ 2 !20 ð 1=2Þ

ð8:116Þ

!0 ¼ 3. Calculation of the damping factor

¼

404

APPLICATIONS

Figure 8.15

Figure 8.16

Form filter corresponding to SDE (8.79).

Desired (dashed line) and estimated (solid line) PDF obtained from SDE (8.79).

The form filter, corresponding to model (8.79), is shown in Fig. 8.15 and results of the numerical simulation are given in Fig. 8.16.

8.3.2

Modeling and Simulation of Weibull Clutter

A different kind of radar clutter is the Weibull distributed radar clutter [112,122–153]. In this case the PDF of the envelope has the following form pA ðAÞ ¼ A 1 exp½ A

ð8:117Þ


405

where and are related to the average power 2 of the quadrature components as 2 2 2= 2 ¼ 1þ ð8:118Þ It is easy to see that, for the case of the Weibull distribution (8.118), eq. (7.244) can be rewritten in the form F1 ðAÞ ¼

K2 ½ A 1 2 4 !0

ð8:119Þ

and, thus, d K 2 2 2 F1 ðAÞ ¼ A dA 4 !20 3 2 d F ðAÞ d 6d A 1 7 K 2 2 ð 2Þ 3 A 5¼ 4 dA A 4 !20

ð8:120Þ

ð8:121Þ

The expression (8.120) has a zero limit when A ! 0 and > 2 (non-Rayleigh case) and is equal to K 2 2 =4 !20 in the Rayleigh case. The integral on the left side of eq. (7.244) can easily be calculated by using eq. (8.121) 2 3 d ð ðx d 6d A F1 ðAÞ7 d A K 2 2 ð 2Þ x A 3 d A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 5 pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ¼ A 4 !20 x 2 A2 x 2 A2 0 dA 0 2 pffiffiffi K 2 2 ð 2Þ 2 x 3 ð8:122Þ ¼ 2 1 4 !0 2 thus producing

2 pffiffiffi K 2 2 ð 2Þ 2 x 2 f ðxÞ ¼ 2 1 4 !0 2

and leading to a generating SDE of the van der Pol type (7.225) 2 pffiffiffi K 2 2 ð 2Þ 2 x 2 x_ þ !20 x ¼ !20 ðtÞ €x þ 2 1 4 !0 2

ð8:123Þ

ð8:124Þ

The next step is to find the correlation function of the solution xðtÞ. For this purpose one can rewrite eq. (8.124) in the Cauchy form x_ ¼ y y_ ¼ x 2 y !20 x þ !20 ðtÞ

ð8:125Þ

406

where

APPLICATIONS

2 pffiffiffi K 2 2 ð 2Þ 2

¼ 2 1 4 !0 2

ð8:126Þ

The drift coefficients K1 x and K1 y are thus defined as K1 x ðx; yÞ ¼ y K1 y ðx; yÞ ¼ x 2 y !20 x

ð8:127Þ

The latter produces the following Gaussian approximation for the covariance functions [156,157]: 3 2 @ @

K1 x ðx; yÞ 7 6 @ x K1 x ðx; yÞ d hx; x i hx; x i @y 7 6 ¼4 ð8:128Þ 5 hx; y i @ @ d hx; y i K1 y ðx; yÞ K1 y ðx; yÞ @x @y where h i denotes cumulant brackets (see section 4.4.2) and the initial conditions are hx; x ij¼0 ¼ 2 hx; y ij¼0 ¼ 0

ð8:129Þ

Using eq. (8.127) one can rewrite system of equation (8.128) in the following form

d hx; x i 0 1 hx; x i ¼ ð8:130Þ ð 2Þhx 3 yi !20 hx 2 i hx; y i d hx; y i The solution hx; x i of eq. (8.130) is given by hx; x i ¼

2 ð2 e1 1 e2 Þ 2 1

ð8:131Þ

where 1 and 2 are the eigenvalues of eq. (8.130). Defining the correlation interval corr as corr ¼

ð1

2

hx; x id

ð8:132Þ

0

one can obtain corr ¼

1 þ 2

hx 2 i ¼ 1 2

hx 3 yi !20

ð8:133Þ

We are now ready to present an algorithm for the synthesis of an SDE whose solution has the Weibull distributed envelope with parameters , , 2 related by formula (8.118) and


correlation interval corr . In fact, solving eq. (8.133) with respect to 2 pffiffiffi K 2 2 ð 2Þ !20 2

¼ 1 hx 2 i ð 2Þhx3 yi 4 !20 2

407

ð8:134Þ

therefore 1 1 8 !20 2 2 pffiffiffi 2 K ¼ 2 ð 2Þ hx 2 i ð 2Þhx 3 yi 2

ð8:135Þ

Thus, all the parameters in eq. (8.124) are now known. The form filter corresponding to the model (8.124) is given in Fig. 8.17 and the results of the numerical simulation are given in Fig. 8.18.

Figure 8.17

Figure 8.18

Form filter corresponding to SDE (8.124).

Desired (dashed line) and estimated PDF obtained from SDE (8.124).

408

APPLICATIONS

8.4

MARKOV CHAIN MODELS IN COMMUNICATIONS

In contrast to continuous models of fading channels, considered above, this section deals with the attempt to model the error flow directly, i.e. focusing on the discrete channel itself. Such models are powerful tools in the investigation of coding and network performance [38– 103,158–170]. Only a few topics are considered here. The reader is referred to [171] for more details and practical examples. 8.4.1

Two-State Markov Chain—Gilbert Model

Let a Markov chain with only two states, bad (B) and good (G) as shown in Fig. 8.19, be considered. This chain can be described by the following transitional matrix

1p p¼ r

Figure 8.19

p 1r

ð8:136Þ

Two-state Markov chain: (a) Zorzi notation [62], (b) original Gilbert paper notation.

In the good state there are no errors, while in the bad state error appears with probability 1 h. We assume that the probability of error, Perr , and the parameter h are given. Since errors appear only in the bad state, one has PB ¼ 1 P G ¼

Perr ; 1h

PG ¼ 1

Perr 1h

ð8:137Þ

Furthermore, any two-state transitional matrix can be decomposed into the following form

P T ¼ ð1 Þ G PG

PB 1 þ PB 0

ð1 ÞPB 0 ð1 ÞPG þ ¼ ð1 ÞPG ð1 ÞPB þ 1

ð8:138Þ

where parameter 0 1 reflects correlation properties of the Markov chain: ¼ 0 corresponds to independent error occurrence, while ¼ 1 results in completely deterministic error occurrence since the chain remains in only one of the states (see Table 8.3). Comparing eqs. (8.136) and (8.138) one obtains Perr r ¼ ð1 ÞPG ¼ ð1 Þ 1 ; 1h

p ¼ ð1 ÞPB ¼ ð1 Þ

Perr 1h

ð8:139Þ

409


Table 8.3 Correspondence between Markov chain parameters in different notations Quantity

Zorzi [62]

Perr

Gilbert [31]

ProbfG ! Gg

1p

Q

ProbfG ! Bg

p

P

ProbfB ! Gg

r

p

ProbfB ! Bg

1r

q

Probability of a correct digit in B state

h¼0

h

1 Perr 1h 1 Perr 1h 1 ð1 Perr hÞ 1h Perr þ ð1 Þ 1h 1

h

If the error-free run statistics are used to estimate parameters of the Gilbert–Elliot model, one has to fit the experimental data Pð0m j1Þ by a double exponential curve Pð0m j1Þ A J m þ ð1 AÞLm

ð8:140Þ

In this case the parameters of the Gilbert model can be defined as [31] LJ ; h¼ J AðJ LÞ

ð1 LÞð1 JÞ P¼ ; 1h

Lh p ¼ AðJ LÞ þ ð1 JÞ 1h

ð8:141Þ

For a more general Elliot model, the same approximation (8.140) of the error-free run produces the following parameters of the model Perr ½AðJ LÞ J JL qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Perr 1 þ JL ðPerr 1 þ JLÞ2 4 JL

ð8:142Þ

¼1

2 1 h Perr 1 h JL JþL h h p¼ kh JL P¼1 p hk k¼

8.4.2

h

Perr 1

ð8:143Þ ð8:144Þ

ð8:145Þ ð8:146Þ

Wang–Moayeri Model

One of the decisions which must be made by the simulator developer is to choose the proper order of Markov chain, N, and calculate elements of the transitional matrix. It was pointed out in [44] that the finite state models can produce unreasonable results if the model

410

APPLICATIONS

Figure 8.20

Wang–Moayeri model. Only transitions to a neighbouring state are allowed.

does not represent real physical characteristics of the continuous channel. One of the important features which must be represented is the so-called level crossing rate (LCR) of fading. Having this in mind, Wang and Moayeri have suggested a model [44] which is based on the assumption that transitions appear only between two neighbouring states and all other transitions are not allowed (see Fig. 8.20). Such an assumption is quite reasonable for the case of a very slow fading. Indeed, in this case the velocity A ðtÞ ¼

d AðtÞ dt

ð8:147Þ

of the process is relatively small and one can assume that the signal level AðtÞ is not able to change by an amount more than a quantization step jAðt þ TÞ AðtÞj jðtÞjT < minð Ai Þ

ð8:148Þ

Of course this condition will be violated for a faster fading. However, for a large number of systems with high bit rate, the duration of bit, T, without interleaving can be accurately described by the Wang–Moayeri model, described below. The range of magnitude variation can be divided into N regions by a set of thresholds Ak such that 0 ¼ A0 < A1 < < A K ¼ 1

ð8:149Þ

The fading is said to be at state sk if the magnitude of the signal resides inside interval ½Ak ; Akþ1 Þ. In each state, the probability of error ek is assigned according to the modulation scheme used. The probability pk of the channel being in state sk can be calculated if the fading PDF pA ðAÞ is known. The Wang–Moayeri model assumes that pk ¼

ð Akþ1

pA ðAÞd A

ð8:150Þ

Ak

Since only transitions into the neighbouring states are allowed, all transitional probabilities with indices differing by at least two are equal to zero: pij ¼ 0; for ji jj > 1;

0 < i;

j 1 by a Rice process with the Rice factor K defined as [3,172] pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m2 m pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi K¼ ð8:173Þ m m2 m thus resulting in the following expression rffiffiffiffiffiffiffiffiffiffiffi 000 LCRð Þ ¼ p ð Þ 2 sffiffiffiffiffiffiffiffiffiffi " !# 0 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi1 2 2

m m m2 m A pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi exp I 0 @2 ¼ fD

m m2 m m m2 m

ð8:174Þ

Another and more direct approach can be based on the fact that, for a certain class of Nakagami process, its derivative is indeed independent of the value of the process [3]. Thus, the general expression (see Section 4.6) can be used to produce rffiffiffiffiffiffiffiffiffiffiffi rffiffiffiffiffiffiffiffiffiffi m

000 2 1 m m m1 LCRð Þ ¼ p ð Þ ¼ fD

exp ð8:175Þ 2 ðmÞ 2

Of course, all three eqs. (8.172), (8.174) and (8.175) produce the same results for the case of Rayleigh fading, i.e. K ¼ 0 and m ¼ 1. It is suggested by Babich and Lombardi [65] that the distribution corresponding to the Rice fading power be approximated by means of the Weibull distribution " #

1

ð8:176Þ p ð Þ ¼ exp " # ð

ð8:177Þ p ð Þd ¼ 1 exp P ð Þ ¼ 1 Here the scale and the shape parameters and are related to the parameters of the Rice distribution K, 2 and A2s by the means of the following equalities, obtained by equalizing the first two moments of both distributions 1 þ 2 ðK þ 1Þ2 1 ¼ 2 1 þ 2K 1 þ 1 ð8:178Þ 2 þ A2s ¼ 1 þ 1 Here ðÞ is the Gamma function [23]. The advantage of using a Weibull PDF becomes apparent since a simple expression can be obtained for probabilities pk of states of the Markov chain if a set of thresholds k is chosen. The reverse procedure is also analytically treatable. Indeed, if one is given a set fpk g of the probabilities of the state of a Markov chain,

416

APPLICATIONS

then the corresponding thresholds can be found by noting that " # k X

k ¼ Pð k Þ pi ¼ Probf k g ¼ 1 exp i¼1 and thus

1

k ¼ ln 1 Pð k Þ

1=

" ¼ ln 1

k X

ð8:179Þ

!#1= pi

ð8:180Þ

i¼1

Using thresholds obtained through eq. (8.180), the level crossing rates can be estimated as sffiffiffiffiffiffiffiffiffiffiffiffiffiffi" !#121 ! rffiffiffiffiffiffiffiffiffiffiffi k k X X 000 000 ð8:181Þ pi 1 pi LCRð k Þ ¼ pA ðAÞ ¼ ln 1 2 2 i¼1 i¼1 This can be further used in order to obtain simple analytical expressions for the elements of the transitional matrix of the Wang–Moayeri model. Example: two-state model.5 Approximation (8.181) can be readily applied to the case of the Gilbert–Elliot model. In this case p1 ¼ " is the average probability of error in the channel and the transitional probabilities p ¼ pBG from the ‘‘Bad’’ to ‘‘Good’’ state and r ¼ pGB from the ‘‘Good’’ to the ‘‘Bad’’ state can now be approximated as [65] sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 00 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi T 0 ½ lnð1 p1 Þ12 ð1 p1 Þ 1 2 000 1 " ½ lnð1 "Þ12 r¼ ¼T ð8:182Þ 2 " p1 sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1 00 ð8:183Þ p ¼ 1 0 ½ lnð1 "Þ1 2 2 Similar equations were derived independently in [25] for the case of Nakagami and lognormal distributions and Clark’s correlation function: ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi s 2 m R 0:4439 fD NT 4 mm R2 m1 exp 8 m ð8:184Þ rN ¼ mb ðm; 0Þ m; "

# ðln R mÞ2 NT c exp 2 2 ! rL ¼ pffiffiffi 1 1 lnð bÞ m pffiffiffi þ erf 2 2 2

5

This example was developed in collaboration with Ms Vanja Subotic.

ð8:185Þ


417

Both studies [65] and [25] compared the average block failure rate predicted by eqs. (8.183)–(8.185) with those obtained from numerical simulation using the Jakes simulator and other simulators. It can be seen that there is a good agreement between the numerical simulation and prediction based on the two-state Markov model (Figs. 8.22–8.24).

Figure 8.22 Average block error burst length vs. the probability of the block error. Nakagami fading mean (left) and variance (right).

Figure 8.23 Average block error burst length vs. the probability of the block error. Log-normal fading mean values (left) and variance (right).

418

APPLICATIONS

Figure 8.24 Average block error burst length vs. the probability of the block error. Two vehicles, Nakagami PDF: mean (left) and variance (right).

8.4.3

Independence of the Channel State Model on the Actual Fading Distribution

An interesting comment can be made about the fact that after a Markov chain model is associated with a continuous fading process, i.e. a set of the state probabilities pk is assigned, it is impossible to restore the original distribution of the fading. Indeed, any two distributions can be discretized in such a way as to produce the same set of state probabilities of the corresponding Markov chain. This may be considered as a sort of a paradox. However, a proper model assigns a certain probability of error in each state6 which reflects the average SNR in this state and the modulation–demodulation techniques. Those probabilities will be different for different types of fading, since the same probability of the states of a Markov chain would require different quantization thresholds, thus producing different values of the average SNR in different states. Furthermore, using a state splitting technique [171] a hidden Markov model can be converted to a Markov model. However, due to different error probability assignments, the results of such a conversion will be different for different types of fading. 8.4.4

A Rayleigh Channel with Diversity

Since the only characteristic required to obtain the Wang–Moayeri model is to be able to calculate the marginal PDF of the signal and its level crossing rate, it is possible to adopt the same technique for a case of diversity channels with a single output signal. At this stage it can be realized that there would be a significant number of states which are described either by a very small probability of appearance or very small probability of errors. Both such states could be excluded from further consideration to reduce the complexity of the model and speed up simulation. 6

Thus leading to a Hidden Markov Model (HMM).

419


Following [91] one can consider a Rayleigh channel with N-branch pure selection combining with the following decision rule R ¼ maxðr1 ; r2 ; . . . ; rN Þ

ð8:186Þ

The simplest model assumes that the fading in different branches is independent and identically distributed. The cumulative distribution function of such a decision scheme is given by

N N Y Y R2 FR ðRÞ ¼ Probðr RÞ ¼ Probðri RÞ ¼ 1 exp 2 2 n n¼1 n¼1

N R2 ¼ 1 exp 2 ð8:187Þ 2 N

pffiffiffi F ð Þ ¼ ProbðSNR < Þ ¼ Probðr < Þ ¼ 1 exp ; ¼ 2 2 ð8:188Þ where 2n is the variance of the signal envelope in the n-th branch. Thus, stationary probabilities of the states of the approximating Markov chain can be obtained as

kþ1 N

k N 1 exp pk ¼ F ð kþ1 Þ F ð k Þ ¼ 1 exp

ð8:189Þ

It is shown in [91] that the level crossing rate of the combined signal is given by LCRN ð Þ ¼ N

N ð Y n¼2

pð 1 n Þd 1 n

N ð Y

pffiffi

n¼2 0

pðrn Þd rn

ð1

r_ 1 pð_r1 jr1 ¼

pffiffiffi

Þd r_ 1

0

sffiffiffiffiffiffiffiffiffiffi N1 2

fm exp ¼N 1 exp sffiffiffiffiffiffiffiffiffiffiffi

2 k

k

k N1 fm exp 1 exp LCRN ð k Þ ¼

ð8:190Þ

ð8:191Þ

Here 1 n represents the phase of the signal in the n-th element of the antenna array. Combining eqs. (8.189) and (8.191) with eqs. (8.153) and (8.154) derived earlier for the elements of the transitional matrix T one can obtain the desired Markov chain.

8.4.5

Fading Channel Models

Swarts and Ferreira carried out a comparison study [81] of Gilbert, Elliot and Fritchman models as applied to GSM systems with the band-time product BT ¼ 0:3, maximum Doppler shift fD ¼ 40 Hz and average SNR levels ¼ 5, 10, 15 and 20 dB. Parameters of the models can be estimated based on the error distribution statistics, obtained from the Jakes simulators and eqs. (8.141)–(8.146). Since only simple statistics were used, it appears that all three models perform almost identically.

420

APPLICATIONS

Guan and Turner [47] suggested modelling the fading envelope by approximating it by a general N-state Markov chain of the first order. The interval of variation of the envelope ½0; 1Þ is divided into N sub-intervals by a set of N 1 points n , 1 n N 1. It is suggested that elements of the transitional probability can be found from a joint probability density function pAA ðA; A ; Þ of the envelope considered in two separate moments of time pðAk ; Al Þ ¼ pðAk jAl Þ ¼ pðAl Þ

Ð k Ð l

k1 l1

pAA ðA; A ; Þd A d A Ð l l1 pA ðAÞd A

ð8:192Þ

Such an assignment ensures that not only the stationary probabilities of each state approximate the distribution of the continuous random process but the second order statistics at two sequential moments of time are also well approximated. As an example of the application of their model the authors of [47] consider modeling of the Nakagami distributed random envelope pðAk ; Al Þ ¼

4ðAA Þm ðmÞð1

m1 Þ 2

mmþ1 P

pffiffiffi 2 m A A m ðA2 þ A2 exp Im1 P 1 Pð1 Þ

ð8:193Þ

Here P ¼ EfA2 g is the average power of the envelope (proportional to the average SNR in the AWGN channel), 0:5 m 1 is the Nakagami parameter and is the correlation coefficient between two sequential samples of the envelope separated by a duration of bit J02 ð2 fD Þ

ð8:194Þ

While an analytical expression for integral (8.192) cannot be obtained in quadratures, a good approximation has recently been derived [3]. To validate the model, a comparison with the Wang–Moayeri approximation (see Section 8.4.2 for more details) has been conducted for the case of m ¼ 1, ¼ 105 s and fD ¼ 10; 100 Hz. A remarkable agreement between two models can be observed—most of the values differ by only the fifth decimal digit. The authors also consider an interleaved channel to find that the approximation of transitional probabilities calculated through the joint probability density (8.193) still produces good results, while the Wang–Moayeri approximation through the crossing rates becomes less accurate due to the fact that transitions to non-neighbouring states appear with greater probability. At the same time, larger time separation between samples, which is equivalent to larger Doppler spread, or a shorter correlation interval, also brings into consideration the non-Markov nature of Clark’s model of fading. This clearly shows limitations of simple Markov chain models for relatively fast fading. The question of choosing a proper order of Markov chain is treated in [65]. Modeling of a slow fading is considered in [72]. In this paper, the authors obtain measurements of average signal intensity and its variation based on 1.9 GHz measurements, collected in Austin, TX. Measurements were taken by a moving car. This data was later used to obtain fluctuating slow fading by means of numerical simulation of a random driving path, with the car turning randomly at each intersection. The value of the signal along each simulated path was then retrieved from the measurement database, filtered and decimated to produce a signal with 1 Hz sampling rate. The slow fading signal trace, obtained through this process, was used as a reference signal for modeling.


421

An 80-state Markov chain was chosen to approximate real data. Threshold levels were set in such a way that all states had equal probability pi ¼ 1=80 and spanned a region between 20 dB and 50 dB. The authors explained that such a choice would allow them to avoid situations when the chain resided in a few states for a long time. Transitional probabilities were estimated from the experimental data by counting a number of transitions between any two states and normalizing it by a total number of data points. Thus the constructed model was used to generate a sample of a Markov process and compare its characteristics with the one obtained from the measurements. It was found that the cumulative distribution function was well approximated, since a sufficient number of states was selected. The authors also point out that the model is able to approximate the level crossing rate with reasonable accuracy. This can be explained by the fact that the model has been constructed based on the transitional probabilities, i.e. level crossing rates. However, the model fails to approximate the power spectral density of the signal. The authors attribute this discrepancy to the fact that the actual channel does not show a Markov property. However, no further studies of the memory of measurements were conducted. It is clear that more accuracy in Markov models can be achieved by both increasing the number of states and increasing the memory of the Markov process.7 Such an increase in complexity is not desired for fast and approximate simulation. Rao and Chen [79] suggest a mechanism to reduce the number of states by aggregating a number of states into new superstates. The authors use the lumpability property of a Markov chain to derive a two-state model of the fading process and compare performance of such a simplified model with those produced by higher order models; in particular, one error state Fritchman model8 with four non-error states and different types of modulation schemes (FSK, DPSK, QPSK and 8-ary PSK), a hidden Markov model with six states (described in [173] and the Wang and Moayeri model9 with eight states. Many of the critics of Markov models point out that, for the correlation function, a simple Markov chain considered is a poor fit to realistic ones [170]. There is also ongoing discussion on applicability of modeling of fading as a Markov process [166–170,173]. While most of the researchers agree that the second order Markov models fit the canonical Jakes model of the envelope very well, there is no consensus on applicability of simple Markov models. In the view of the authors of the current book, such models are at least very helpful in discovering general trends and analysis performance of complicated codes.

8.4.6

Higher Order Models [170]

A number of methods allow one to extend Markov chain models to account for dependence between the future and the past of the process, given current observation. Such dependence leads to a non-Markov nature of the process under consideration. Zorzi et al. suggest [170] that both the values of the random process and its derivative (velocity) are approximated by a Markov chain. If there are M levels mi ð0 i M 1Þ of the random process ðtÞ to be considered, and there are S levels sj ð0 j S 1Þ of the velocity of the random process _ðtÞ, then the Markov chain consists of M S states obtained 7

That is, to consider Markov processes of order more than 1. See the chapter for more details. See Section 12.1.4. 9 See Section 8.4.2. 8

422

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as all possible pairs ðmi ; sj Þ. It is quite clear that the complexity of such a description grows very fast with the number of states. It is claimed in [170] that it is sufficient to choose only three levels of the velocity: S ¼ 3, s0 ¼ 0, s1 ¼ s2 ¼ ". Here velocity levels reflect a well known fact that for many useful processes such as Rayleigh, Nakagami, etc. the velocity is a zero mean Gaussian process. A particular value of " is chosen to compromise between the accuracy of approximation of the flat portion of the early time and the oscillatory portion in the late time of Clark’s model. The authors proceed further to show an example of an ARQ system simulation that provides a significant improvement when compared to the model which approximates only the states of the process ðtÞ.

8.5

MARKOV CHAIN FOR DIFFERENT CONDITIONS OF THE CHANNEL

A number of applications use a finite state Markov chain to describe changing conditions of the communication channel [16]. It is characteristic for a wireless link to change if at least one of the mobile receivers changes position. For example, a pedestrian can move from a residential area with relatively mild obstructions by two-storey wooden houses and trees to either an open area with virtually no obstruction or to downtown areas with high rises completely shadowing the signal. Such dramatic changes can be modeled as separate states of the Markov chain with certain non-deterministic transitions. One of the examples is considered in [56]. Here a land mobile satellite service (LMSS) channel is modeled as a three-state Markov chain. All states have contributions from weak scattered waves. In addition, the state A has a contribution from a strong, unattenuated line-of-sight ray; the state B is characterized by significant attenuation of the line-of-sight, and finally, the state C is described by fully blocked line-of-sight. As a result, fast fading statistics differ in each state. Probabilities of each state are pA , pB and pC , respectively (Fig. 8.25). Of course pA þ pB þ pC ¼ 1

ð8:195Þ

The three-state model presented in [56] is considered more general than the two-state models considered by Lutz [164] (only states B and C were considered, with a non-

Figure 8.25

Single satellite link model suggested in [56].

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Rayleigh model for the state C), and Barts and Stutzman [165] (only states A and B were considered). In all states, contribution from the weak diffused scatterers is approximated by Rayleigh fading. In the state A, the Nakagami–Rice model is used. In the state B, attenuation of the direct ray is described by a log-normal distribution. Superimposed with Rayleigh fast fading, this results in the well known Loo model [163]. Fast fading in the state C is represented just by Rayleigh fading. According to this model, the probability density function of the received signal is a weighted sum of the density functions at each state, with weights equal to the probabilities of the states pðrÞ ¼ PA pA ðr; Pr;A Þ þ Pb pB ðr; mL ; L ; Pr;B Þ þ PC pC ðr; Pr;C Þ

ð8:196Þ

Here 2r 1 þ r2 2r exp I0 pA ðr; Pr; A Þ ¼ Pr; A Pr; A Pr; A " # ð 6:930 r 1 1 f20 logðzÞ mL g2 r 2 þ z2 exp dz pB ðr; mL ; L ; Pr; B Þ ¼ L Pr; B 0 z Pr; B 2 2L 2r r2 pC ðr; Pr;C Þ ¼ exp Pr;C Pr;C

ð8:197Þ ð8:198Þ ð8:199Þ

and Pr; A ; Pr;B and Pr;C are the mean powers of the multipath components normalized to the direct ray power; ml and L are the mean and the variance of the signal fading in dB. In the original paper [56] these values were obtained from the measurements in different cities. Based on these measurements it is suggested that the following approximations are used for the state probabilities ð90 Þ2 ; PA ¼ 1

10 < < 90 ; ¼

7:0 103 1:66 104

for urban areas for suburban areas

ð8:200Þ

A similar idea has been suggested in [77]. In this paper the authors suggest that a satellite communication channel can be described by three significantly different conditions, described by the Rayleigh, Rician and log-normal distributions of the signal’s envelope. The transitions between states are described by a Markov chain of order 3. The main contribution of this paper is the fact that the model is implemented on a TMS320C30 microprocessor and can be expanded to model wide band channels by means of parallel computations. One of the drawbacks of this method is that the number of processors required is equal to the number of rays to be simulated, thus increasing the cost of such simulators [174–188].

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Index Analytical Signal 154 Average fade duration 145, 151 Birth process 223 Blanc-Lapierre transform 159, 348 Chapman equation 196, 204, 207, 214 Characteristic function 10, 15, 27, 63, 116, 228, 254, 343, 367 Clutter K-distributed 397 Weibull 404 Compound process 181 Conditional PDF 28, 62, 182, 287 Correlation coefficient 31, 74 Correlation interval 76–77, 133, 255, 290, 322, 380 Covariance 31 mutual 72 of a modulated carrier 87 of a periodic RP 85 of a sum and product of RV 84 of WGN 95 properties 73 relation to PSD 80, 83 Covariance function 67, 69, 71–72 Jakes’ 379 of process 322 of ! 322 of a Markov diffusion process 239 of Gaussian Markov process 241 of narrow band process 351 of telegraph signal 200 of the Wiener process 254 Covariance matrix 89, 181, 268 Cumulant brackets 48, 55 definition 11, 68 generating function 11 linear transformation 135

non-linear transformation 52, 112 of a random vector 30 of a sum of random variables 38 relation to moments 16, 48, 68 Cumulant equations 49, 51, 118, 289 Death process 224 Diffusion 229, 235, 312, 330 Diffusion process 217, 260 Drift 215, 235, 312, 322, 330, 335 Envelope 154, 156, 170, 177, 328, 348, 380 Error flow 388 Fading 43, 44, 140, 149, 377 Loo 378 Nakagami 378, 382, 385, 391 Rayleigh 41, 44, 76, 384 Rice 41 Suzuki 378 Fokker–Planck equation 217, 227, 393 boundary conditions 231 derivation 227 generalized 287 large intensity approximation 302, 310 methods of solution 236 small intensity approximation 297, 304 FPE, boundary conditions 219, 231, 288, 326 absorbing 219, 232–233 natural 232–233 periodic 232–233 reflecting 232 Gaussian process 71, 88, 115, 130, 146, 155, 166, 240, 270, 298 Gaussian RV 32–34 Hilbert transform 154, 358, 400 Independence 29, 31, 44, 87, 89, 155, 282, 419


434

INDEX

Joint Probability Density 26, 29, 32, 88, 146, 149, 239, 280 Kolmogorov equation 218, 227, 235 Kolmogorov–Feller equations 285–287, 369 curtosis 18, 49, 120 Level crossing rate 140, 144, 149, 172, 410, 411 Log-characteristic function 10, 30, 77, 343 Markov chain 190 ergodic 197 Fritchman 419 Gilbert–Elliot 408 homogeneous 196 models 408 of order n 193 simple 190, 287, 392 stationary 196 vector 193 Wang–Moayeri 409 Markov process 212–213 binary 209 classification 189–190 diffusion 215 ergodic 214 homogeneous 209 with jumps 221 with random structure 276 Markov sequence 190, 203 Master equation 217 Moment equations 51, 289 Moments 12, 64

Yule–Farri 224 Pearson PDF 10, 65, 115, 290, 322 Phase 25, 39, 65, 81, 126, 154, 168, 327 Poisson PDF 9 Poisson process 136, 221, 285, 345, 369, 390 Process with jump 215, 217, 221, 285, 296, 335, 339 Process with random structure 275 classification 279 examples 290, 292, 298, 300, 309, 312, 313, 391 Fokker–Planck equation 281, 288 high intensity approximation 302, 310 low intensity approximation 298, 304 moments equation 288 statistical description 280 Random telegraph process 120, 200 Schro¨dinger equation 244 SDE 245 Ito form 251 relation to Fokker–Planck equation 251, 266 relation to Kolmogorov–Feller 369 Stratonovich form 251 vector 266 Skewness 18 Smoluchowsky equation 195, 227, 339 Spherical invariant process 65, 181 Stochastic integrals 246 Ito 248 rules of change of variable 250 Stratonovich 249 Stratonovich 249, 252

Orthogonal polynomials 23, 115, 296, 345 PDF Cauchy 184 Gaussian, see Gaussian Process K 173, 184, 368, 383, 397 Nakagami 149, 177, 241, 328, 360, 378 Poisson 9, 136 Rayleigh 41, 262, 333, 361, 377 Rice 41, 378 Two-sided Laplace 184, 325 uniform 10, 25, 41, 85 Weibull 333, 397

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Temporal symmetry 257 Tikhonov 358 Transitional matrix 196, 278, 394, 408, 412, 414 Transitional PDF 199, 280, 204 Vector random variable and process 26, 112, 182, 205, 252, 263, 268 white Gaussian noise 88, 95, 246, 250, 252, 321, 379, 393 Wiener–Khinchin theorem 80

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