Sample Rate Conversion in Software Configurable Radios (Artech House Mobile Communications Series)

Sample Rate Conversion in Software Configurable Radios For a listing of recent titles in the Artech House Mobile Comm...

Author: Tim Hentschel

29 downloads 455 Views 2MB Size Report

This content was uploaded by our users and we assume good faith they have the permission to share this book. If you own the copyright to this book and it is wrongfully on our website, we offer a simple DMCA procedure to remove your content from our site. Start by pressing the button below!
Report copyright / DMCA form

DOWNLOAD PDF

Sample Rate Conversion in Software Configurable Radios

For a listing of recent titles in the Artech House Mobile Communications Series, turn to the back of this book.


Tim Hentschel

Artech House Boston London www.artechhouse.com *

iv


Library of Congress Cataloging-in-Publication Data Hentschel, Tim. Sample rate conversion in software configurable radios / Tim Hentschel. p. cm.—(Artech House mobile communications series) Includes bibliographical references and index. ISBN 1-58053-336-1 (alk. paper) 1. Software radio. 2. Telecommunication—Traffic—Management. 3. Sampling (Statistics). I. Title. II. Series. TK5103.4875 .H45 2002 621.384—dc21 2002074697 British Library Cataloguing in Publication Data Hentschel, Tim Sample rate conversion in software configurable radios.—(Artech House mobile communications series) 1. Signal processing—Digital techniques 2. Digital filters (Mathematical) 3. Sampling (Statistics) 4. Mobile communication systems 5. Radio—Receivers and reception I. Title 621.3’822 ISBN 1-58053-336-1 Cover design by Gary Ragaglia Ó 2002 ARTECH HOUSE, INC. 685 Canton Street Norwood, MA 02062 All rights reserved. Printed and bound in the United States of America. No part of this book may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording, or by any information storage and retrieval system without permission in writing from the publisher. All terms mentioned in this book that are known to be trademarks or service marks have been appropriately capitalized. Artech House cannot attest to the accuracy of this information. Use of a term in this book should not be regarded as affecting the validity of any trademark or service mark. International Standard Book Number: 1-58053-336-1 Library of Congress Catalog Card Number: 2002074697 10

9 8 7 6 5

4 3 2 1

Contents

v

Contents Preface 1

xi

Sample Rate Conversion in the Context of Software Radio

1

1.1

Introduction

1

1.2

Software Radio

1

1.3

Critical Functionality of a Software Radio Terminal Analog-to-Digital Conversion SRC Channelization Application of SRC Summary and Conclusion

7 7 20 22 25 25

References

26

Basic Principles of SRC and Multirate Systems

29

2.1

Introduction

29

2.2 2.2.1 2.2.2 2.2.3

Basic Considerations of SRC Resampling After Reconstruction Irrational Factor SRC Rational Factor SRC

30 30 38 43

1.3.1 1.3.2 1.3.3 1.3.4 1.3.5

2

v

vi


2.2.4 2.2.5 2.2.6 2.2.7 2.2.8

Integer Factor SRC Up- and Downsampling with Phase Offset Polyphase Decomposition Modulation Representation Summary

2.3

Linear Periodically Time-Varying Systems and Block Filtering Introductory Remarks The Raising Procedure Block-Filtering and Multirate Systems Frequency Behavior of Multirate Systems: The Bifrequency Approach

2.3.1 2.3.2 2.3.3 2.3.4 2.3.5

45 52 55 60 61 61 61 62 71 76

Summary

82

References

82

A Concept for SRC in Software Radio Terminals

85

3.1

Introduction

85

3.2

Review of the OSR of a Signal

85

3.3

The Conversion Factor: Integer Factor and Fractional SRC

86

3.4

Processing Issues

89

3.5

Summary

95

References

96

4

Systems for Integer Factor SRC

99

4.1

Introduction: The Direct Approach

99

4.2 4.2.1 4.2.2

Parallelization: Polyphase Filters Introductory Remarks Increasing the Sample Rate

3

100 100 100

Contents

4.2.3 4.2.4

Reducing the Sample Rate Reducing the Size of the Coefficient Memory

4.3

Efficient Realization of FIR Filters on Periodically Time-Varying Systems Computing All Coefficients from One Polyphase Component Storing More Than One Polyphase Component Alternative Interpretation of Polynomial Polyphase Filters

4.3.1 4.3.2 4.3.3 4.4

vii

106 110 113 113 124 133

Conclusion

141

References

141

Comb Filters: Special Systems for Integer Factor SRC

143

5.1

Introduction

143

5.2

Basic Considerations of Comb Filters

143

5.3

Cascaded Comb Filters

145

5.4 5.4.1 5.4.2 5.4.3 5.4.4

147 147 149 152

5.4.5

CIC Filters The Structure of CIC Filters Noise Shaping with CIC Decimators CIC Interpolators The Frequency-Sampling Structure of FIR-Filters Raising a CIC Decimator

5.5 5.5.1 5.5.2 5.5.3 5.5.4 5.5.5

Recursive Multirate Filters The Idea of Recursive Multirate Filters A Second-Order Decimator A Second-Order Interpolator Higher-Order Structures Higher-Order Integrate and Dump

155 155 156 158 161 164

5

152 153

viii


5.6 5.6.1 5.6.2 5.6.3

Generalizing Recursive Multirate Filters Motivation Arbitrary Poles of the Filter G ðzÞ Designing Generalized Recursive Multirate Filters

165 165 165

Conclusion

185

References

185

6

Systems for Fractional SRC

187

6.1

Introduction: The Direct Approach

187

6.2 6.2.1

188

6.2.2

Polyphase and Block Realization Block Filtering and Conventional Polyphase Structures Time-Varying Polyphase Structures

188 191

6.3 6.3.1 6.3.2 6.3.3

Time-Varying CIC Filters CIC Filters for Fractional SRC State-Space Equations Simplification of the State Equation

197 197 198 204

6.4

A Note on Irrational Factor SRC

208

6.5

Summary

209

References

210

7

Applications of SRC

211

7.1

Introduction

211

7.2

Scenario of the Application

212

7.3

Requirements on SRC in the Context of Mobile Communications

213

Conventional Filters Direct Implementation Direct-Block Implementation

216 216 217

5.7

7.4 7.4.1 7.4.2

172

Contents

7.4.3 7.4.4 7.4.5 7.5 7.5.1 7.5.2

Time-Varying Polyphase/Block Implementation Filter Characteristics Conclusion

ix

217 218 219 219 219

7.5.5

Polynomial Filters Review The Conventional and Transposed Farrow Structures The Generalized (Time-Varying) Conventional Farrow Structure and the Transposed Farrow Structure Antialiasing with the Generalized Conventional Farrow Structure Conclusion

7.6

Time-Varying CIC Filters

229

7.7

Comparison of Methods for SRC

230

7.8

Summary

232

References

232

8

Conclusions

235

8.1

Summary

235

8.2

SRC Questionnaire

237

Appendix: Derivations and Calculations

239

Polyphase Decomposition of the Transfer Function

239

A.2

The Modulation Representation

241

A.3 A.3.1

The Raising Procedure The State-Space Description of Raised Systems

243

7.5.3

7.5.4

A.1

220

224 226 229

243

x


A.3.2

The Transfer Matrix of Raised Time-Invariant Systems

244

A.4 A.4.1 A.4.2 A.4.3

Time-Varying Comb Filters Time-Varying Differentiator Second-Order Case R th-Order Case

245 245 246 247

A.5

Simplification of the State Equation of Time-Varying CIC Filters

250

References

252

Symbols and Abbreviations

253

About the Author

259

Index

261

Preface A whole book on sample rate conversion? When working in the field of software radio, I quickly became aware of a new problem in communications terminal design. One main property of software radio terminals is the independence from existing standards defining an air interface. Thus, software radio terminals have the ability to process many various communications standards. These standards are generally based on different master clock rates and therefore employ different bit/chip rates. A solution to cope with the diversity of master clock rates in one terminal is to have a dedicated master clock for each standard of operation. This kind of solution is not only costly, it limits the applicability of a realized terminal to what is defined by the implemented clock rates. Hence, it is much more elegant to run the terminal on a fixed clock rate and perform digital sample rate conversion controlled by software. But is sample rate conversion not a well-investigated field with lots of solutions? Indeed, there are many solutions. However, none of them serves the requirements of an application in communications terminals. Complexity and power consumption are the main issues in terminal design. On the other hand, often only one channel out of a wide-receive bandwidth needs to be processed. Hence, I sought solutions to perform sample rate conversion where certain errors are allowed. This led to an investigation of the topic of sample rate conversion from its beginning. Sample rate conversion can be tackled with the idea of resampling after reconstruction. This is a theoretical approach based on the ideal reconstruction of the discrete-time input signal and on resampling of the reconstructed continuous-time signal. From this idea, a mathematical description can be derived that enables a realization by means of digital signal processing. xi

xii


The idea of resampling after reconstruction shows that sample rate conversion is a process of sampling. Hence, the two fundamental effects of sampling, imaging and aliasing, must be expected. Because it is aliasing that destroys possibly interesting signal components, we can conclude that antialiasing is the most important property to be obeyed by any sample rate conversion system. Antialiasing can be achieved by means of filtering. In order to have a flexible hardware platform for sample rate conversion, it is sensible to implement a continuous time impulse response rather than samples of the impulse response of the antialiasing filter (i.e., a conventional digital filter). This enables us to sample the impulse response arbitrarily at run time and, thus, accommodate any sample rate or master clock rate. Polynomial filters have a continuous time impulse response whose samples can be calculated with moderate effort. Therefore, they are a candidate for realizing the necessary antialiasing behavior of a flexible sample rate conversion system. Systems for sample rate conversion belong to the class of linear periodically time-varying systems. Due to this fact, it is not possible to directly apply the well-known methods of system analysis and synthesis of linear time-invariant systems. The nature of systems for sample rate conversion is even better described by the name (L,M )-shift-invariant system. This name means that a shift of the input signal by M samples causes a shift of the output signal by L samples. On this basis, a time-invariant multiple-input multiple-output (MIMO) system with M inputs and L outputs can be constructed that has exactly the same transfer characteristics as the single-input single-output (L,M )-shift-invariant system (given the proper construction of the input and output signals of the MIMO system). Hence, it is possible to analyze the (L,M )-shift-invariant system by means of analyzing the equivalent timeinvariant MIMO system. Signal processing with such MIMO systems is called block-signal processing. There is a strong relationship between block-processing systems and the polyphase representation of systems and signals. Combining the ideas of polynomial filtering and block processing yields the well-known Farrow structure for sample rate conversion. It turns out that the Farrow structure does not have the capability to perform sufficient antialiasing filtering. Still, the transposed Farrow structure provides a means for antialiasing filtering. It is presented as a novel structure in this book. The power of the block-processing approach is further demonstrated by deriving novel time-varying structures for fractional sample rate conversion

Preface

xiii

on the basis of the well-known cascaded-integrator comb filters for integer factor sample rate conversion. Moreover, this book illustrates how the blockprocessing framework can be used to design recursive filters with time-varying feedback loops for integer factor sample rate conversion. The presented solutions have been derived in the context of software radio but are not limited to this application. This book provides a general overview on the problem of sample rate conversion. The book is based on the Ph.D. thesis that I wrote at the Mannesmann Mobifunk Chair for Mobile Communications Systems of Technische Universita¨t Dresden. I would like to thank Professor Gerhard Fettweis for his supervision and my colleagues Matthias Henker and Michael Lo¨hning for our numerous fruitful discussions and their invaluable critical comments. Moreover, I would like to thank Professor Hans-Joachim Jentschel of Technische Universita¨t Dresden and Professor Jeffrey H. Reed of Virginia Polytechnic Institute, Blacksburg, Virginia, and State University for their opinions and comments. I would like to thank the people at Artech House and the anonymous reviewer whose comments helped to improve the book. Finally, I cannot forget the support provided by my wife.

1 Sample Rate Conversion in the Context of Software Radio 1.1 Introduction This chapter briefly presents some concepts related to software radio to introduce the reader to particular aspects of sample rate conversion (SRC) in the context of software radio. This chapter does not attempt to investigate receiver structures and the respective digital signal processing algorithms in any depth. The chapter is organized as follows. First, the idea of software radio is introduced. After that, the main part of this chapter is dedicated to the critical functionality of software radio. It is the very part of a software radio transceiver that is difficult to realize with current technology (e.g., for reasons of power consumption or dynamic range). Analog-to-digital conversion, SRC (in brief ), and channelization are covered. The chapter is completed with a note on applications of SRC that are different from software radio.

1.2 Software Radio Software radio is a notion that comes from applying the computer industry’s qualitative step from calculator to computer to communications systems business. Initiated by the rapid development of digital signal processing, and specifically its enabling technologies (above all, semiconductor technology), the idea of defining a radio by means of software has arisen. It is as simple as: 1

2


Receiver—the radio frequency (RF) signal is digitized close to the antenna and fed to a software-programmable digital signal processor, which will do the rest. Transmitter—the transmit signal is created in the signal processor and constructed (digital-to-analog converted) directly before the antenna (or a power amplifier). The aim is to have a terminal that has multipersonal capabilities. Any combination of RF band, air interface, protocol, coding, and related functions [1] must be realized on a common software-programmable hardware platform. Thus, the hardware platform is an enabling component of the software radio concept. In many publications, only the receiver side of the platform is investigated, as the receiver is generally the more demanding design problem. In a similar way, this book concentrates on investigations into the receiver only. However, most of the solutions to SRC are just as applicable to the transmitter. Due to the success and high penetration of mobile communications systems, the idea of software radio is strongly related to mobile terminals and base stations of mobile communications systems. The ideal software radio approach is shown in Figure 1.1. It is the most rigorous approach. However, restrictions with respect to bandwidth and dynamic range render this approach infeasible, at least with today’s technologies. Therefore trade-offs between the ideal software radio and the available technology must be made. In most cases, the trade-offs are made with a special focus. A network operator is concerned about the downloadability of new software, while an equipment manufacturer is interested in multimode terminals that could reduce the product diversity and thus reduce cost. A semiconductor manufacturer might be interested in the software configurability itself, which would allow him to extensively use digital signal processors instead of application-specific integrated circuits (ASICs).

Figure 1.1 Ideal software radio receiver.


3

It is important to note that software radio is qualitatively more than just a digital radio. Shifting the analog-to-digital interface towards the antenna is certainly a main issue and a requirement of software radio. Reconfigurability and programmability of the processing platform are just as important. There are several stages in a software radio receiver where reconfigurability can be introduced. Ultimately, in such an evolution, the whole receiver should be reconfigurable. Finally, this evolution could yield a design similar to the ideal software radio. To start with, one could concentrate on the very part of a receiver that is realized by digital signal processing already today (i.e., the baseband processing). By increasing the computational power of the baseband processor, it becomes possible to run different algorithms on it and thus serve different mobile communications standards. New fields are opened for the application of digital signal processing and the design of a reconfigurable hardware platform as the analog-to-digital interface is shifted towards the antenna. In this context, the requirements on the analog-to-digital converter (ADC) are analyzed. In the ideal software radio concept of Figure 1.1, a system filter selects a bandwidth covering all of the services the software radio should support. Within this bandwidth, there are signals of different air interfaces (standards) possibly used by several operators. This is, in principle, shown in Figure 1.2, where three different services are suggested. These are the global system for telecommunications (GSM) as a typical single carrier system

Figure 1.2 Software radio–supported services sketched over frequency, adapted from [2]. Ó 1999 by Kluwer Academic Publishers.

4


based on frequency division multiple access (FDMA), the Universal Mobile Telecommunications System (UMTS) as a representative of a pure spreadspectrum system, and HIPERLAN/2 as a multicarrier system based on orthogonal frequency division multiplex (OFDM). The bandwidth of each service is split again into several channels, in the case of FDMA or multicarrier transmission, or is wholly occupied by each channel of the service, in the case of pure spread-spectrum systems. Due to the typical characteristics of mobile communications signals (defined by [3]), the power levels of different frequency-divided channels in a particular air interface can vary considerably. Thus, mobile communications signals comprising several channels (i.e., multichannel signals) or multiple services can have a very high dynamic range. This, and the very large bandwidth that a software radio terminal might have to support, makes it impossible to receive and process (especially analog-to-digital conversion) the whole bandwidth of the software radio service at once. In realizable software radio receivers, the signal bandwidth must be reduced. Thus, two digitization concepts of a software radio receiver can be distinguished. The concept of the ideal software radio of Figure 1.1 is termed full-band digitization. A more practical concept employing bandwidth reduction is termed partial-band digitization. Both concepts are illustrated in Figure 1.3.

Figure 1.3 (a) Full-band and (b) partial-band digitization principle, adapted from [2]. Ó 1999 by Kluwer Academic Publishers.


5

The reduction of bandwidth in the case of partial-band digitization can be realized with conventional receiver techniques (i.e., with frequency conversion to a low or high intermediate frequency and filtering). Two example architectures are shown in Figure 1.4. The well-known heterodyne receiver is shown in Figure 1.4(a). Its main feature is the intermediate frequency (IF), to where the signal of interest is mixed, and at which it is then filtered. The reason for mixing the signal to a fixed intermediate frequency is that the IF filter can have a fixed center frequency. This simplifies the realization of the filter considerably. Depending on the current frequency of the local oscillator (LO), different partial bands (e.g., channels or channel bundles) are passed through the IF filter. Due to the nature of mixing with a real LO (i.e., multiplication with a cosine or sine signal), an image-rejection filter is required before mixing (see [4, 5]). This filter can be avoided with the low-IF receiver where the signal is mixed with a complex LO frequency, as shown in Figure 1.4(b). Saving the image-rejection filter must be paid for with a complex mixer that poses its own realization problems. A consequence of partial-band digitization is that an analog front end is necessary. Although it might have less functionality than in conventional terminals (where, more or less, the complete channel selection is done in

Figure 1.4 (a) Heterodyne receiver concept and (b) low-IF receiver concept.

6


the analog front end), the requirements are heavily increased due to the higher bandwidth and dynamic range of the multichannel signal a software radio receiver must process. In both full-band and partial-band digitization, the signal is processed digitally, after digitization. Before baseband processing can be performed, the channel of interest must be selected from the digitized multichannel signal and shifted to baseband. To perform baseband processing, the sample rate of the selected signal must be in accordance with the air interface. Hence, there is some digital signal processing necessary before baseband processing. This digital signal processing is done traditionally through analog techniques in the analog front end. Due to the shifting of the ADC towards the antenna, this functionality is realized digitally. It is termed the digital front end (DFE) [6, 7]. The functionality of the DFE is channelization (i.e., digital downconversion and channel filtering) and possibly SRC (necessary to provide the correct symbol or chip rate). The DFE is usually not implemented on programmable digital signal processor (DSP) devices. This is due to the very high sample rates resulting from the high dynamic range and bandwidth of the signals. In general, it can be stated that today’s DSPs cannot provide the processing capacity needed for processing wideband and high-bit-rate signals [8]. To illustrate this statement, the functionality of digital downconversion (see Section 1.3.3) is examined. As with analog mixing (see Figure 1.4), digital downconversion is realized by multiplying the signal with an LO signal. If both the signal and the LO are complex, digital downconversion requires four real multiplications per signal sample. With a sample rate of 80 million samples per second (MSps), which is state of the art for high resolution ADCs, this results in 320 million multiplications per second (Mmps). There are certainly DSPs that can serve this demand. However, digital downconversion would occupy a considerable portion of the processing power of a DSP without really requiring its flexiblity. The only thing that must be tuned with digital downconversion is the frequency of the LO (i.e., the signal at one of the inputs of each of the four multipliers). Hence, it is more sensible to implement digital downconversion on a dedicated hardware platform. Yet this does not violate the software radio idea. Such considerations are the motivation for regarding the DFE as a separate entity. The challenge of designing this DFE lies in the requirement of reconfigurability and/or programmability in the presence of very high data rates. Hence, the DFE can be regarded as an extremely critical part of a software radio receiver. Other critical parts are the wideband analog front


7

end and the ADC. All of these components are enabling components of a software radio terminal. Given all of these considerations, the architecture of today’s feasible software radio receiver can be derived. It is shown in Figure 1.5 and comprises a minimal analog front end, the ADC, a DFE, and the digital signal processor. The next section is dedicated to the critical parts analog-digital-analog (A/D/A) interface (specifically the ADC) and the DFE with its functionality channelization and SRC.

1.3 Critical Functionality of a Software Radio Terminal 1.3.1 Analog-to-Digital Conversion The placement of the A/D/A interface in a software radio is the key [9]. This means that shifting the A/D/A interface closer and closer to the antenna theoretically enables the realization of more and more functionality by means of software, thus approaching the ideal software radio. On the receiver side, the shift of the ADC towards the antenna implies an increase of the digitization bandwidth and eventually a shift from partial-band digitization to full-band digitization. However, technology limits the performance of ADCs and digital-to-analog converters (DACs), thus slowing down their ‘‘move’’ towards the antenna.

Figure 1.5 Feasible software radio receiver.

8


As this book deals with sampled signals, it is necessary to understand how these signals are obtained. Therefore, some fundamentals on analog-todigital conversion are given. There are several classifications of ADCs. When regarding the ADC as a black box and having interest only for different concepts that have fundamentally different performance, it is opportune to distinguish between memoryless ADCs and sigma-delta (SD) ADCs. The analog-to-digital conversion process takes place in two steps: quantization (i.e., the allocation of the signal amplitude to discrete levels) and sampling (i.e., capturing a signal at an instant in time). When designing a system, it is vital to have models of all of the signal processing steps. The mathematical models of quantization and sampling are briefly introduced. For a detailed treatment on analog-to-digital conversion, the reader is referred to [10, 11]. 1.3.1.1

Sampling

Sampling means taking amplitude values of the signal xðt Þ at certain timeinstances (e.g., sampling at t0 , t1 , t2 , etc., yields xðt0 Þ, xðt1 Þ, xðt2 Þ, etc.). We only want to consider the case when the time difference between two sample times is constant (i.e., t1 t0 ¼ t2 t1 ¼ ¼: T ). This type of sampling is called equidistant sampling, and T is the sampling period. The mathematical model of sampling is the multiplication of the signal with a Dirac impulse train. The Dirac impulse dðt Þ is a function (also called Dirac’s delta function) introduced by the physicist Paul Dirac. It has the following properties: dðt Þ ¼ Z1

0 1

t¼ 6 0 t ¼0

dðt Þdt ¼ 1

ð1:1Þ ð1:2Þ

1

dðt Þ ¼ dðt Þ

ð1:3Þ

Since there is no classical function with the property of (1.1) it is—strictly speaking—mathematically intractable. However, physicists and engineers have been successfully working with this function. Therefore, mathematicians have redefined Dirac’s delta function as the limit of a function series. This definition belongs to the mathematical apparatus of distribution theory. Still, even without this theory the engineer can work with the Dirac impulse on the basis of (1.1) to (1.3). An important property of the Dirac impulse is xðt Þ dðt T Þ ¼ xðT Þ dðt T Þ

ð1:4Þ


9

where xðt Þ is a function (e.g., describing a signal). From (1.4) the sifting property of the Dirac impulse can be derived by using (1.2): Z1

xðt Þ dðt T Þdt ¼ xðT Þ

ð1:5Þ

1

The sum of an infinite number of T -shifted Dirac impulses is called a Dirac impulse train. Multiplying a signal with this Dirac impulse train yields the mathematical description of a sampled signal: xa ðt Þ ¼ xðt Þ T

1 X

dðt kT Þ

ð1:6Þ

xðkT Þdðt kT Þ

ð1:7Þ

k¼1

¼T

1 X k¼1

From (1.7) it can be seen that besides the Dirac impulse, only the desired samples of xðt Þ are used to describe the sampled signal. Due to (1.2) the dimension of dðt Þ must be time1 . Therefore the coefficient T is introduced in (1.6), which ensures the same dimension for xðt Þ and xa ðt Þ. One question remains: how to choose T ? The sampling theorem states that the signal xðt Þ can be perfectly reconstructed from its samples xðkT Þ; k ¼ 0; 1; 2; . . . if the sample rate 1=T is higher than the so-called Nyquist rate (i.e., twice the highest frequency component of the signal to be sampled). Otherwise, irreversible distortions result known as aliasing. This can easily be seen from frequency domain considerations. The spectrum Xa ð f Þ of xa ðt Þ can be obtained by employing the Fourier-transform1 Xa ð f Þ ¼ FT fxa ðt Þg ¼

Z1

xa ðt Þ e j2pft dt

ð1:8Þ

1

Before substituting (1.6) into (1.8), it is useful to replace the Dirac impulse train in (1.6) by its Fourier-series representation. 1 X k¼1

dðt kT Þ ¼

1 1 X k e j2pT t T k¼1

ð1:9Þ

1. It is assumed that xa ðt Þ fulfills the necessary conditions to ensure convergence of the integral.

10


Now, (1.6) is substituted into (1.8), thereby using (1.9) for the Diracimpulse train. Xa ð f Þ ¼ FT fxa ðt Þg ¼

Z1 X 1 1

¼

Z1 1 X k¼1

¼

1 X

k

k¼1

f

k T

ð1:10Þ

k¼1

xðt Þ e j2pð f T Þt dt

1

X

k

xðt Þ e j2pT t e j2pft dt

ð1:11Þ

ð1:12Þ

Equation (1.12) has been obtained from (1.11) by using (1.8) with xa ðt Þ replaced by xðt Þ, and thus Xa ð f Þ replaced by X ð f Þ . It can be concluded that the introduction of the scaling factor T in (1.6) is necessary to maintain the original spectrum X ð f Þ as part of Xa ð f Þ for k ¼ 0 in (1.12). From (1.12) the typical characteristics of sampling can be identified: Sampling the signal xðt Þ with sampling period T causes a repetition of the spectrum X ð f Þ at integer multiples of 1=T . This effect is called imaging, the spectral copies are called images. The previously mentioned sampling theorem can easily be understood when inspecting Figure 1.6. As long as the sample rate is larger than twice the highest frequency component of X ð f Þ, there is no overlap of the images. Thus the spectrum X ð f Þ can be separated from Xa ð f Þ (which is done by lowpass filtering). If the sampling theorem is not obeyed, the images overlap. This irreversible effect is called aliasing. It can be concluded that in general, aliasing must be avoided. This will be discussed in Section 1.3.2 and in Chapter 2. 1.3.1.2

Quantization

Having discussed sampling, the second part of analog-to-digital conversion, namely quantization, is examined. Quantization is a nonlinear operation. It is a mapping of the signal values (from a possibly continuous set) to quantized signal values (of a limited set). There are uniform and nonuniform quantizer characteristics. Quantizer characteristics are uniform if the quantization step DV is constant independent of the current signal value: DV ¼

Vpp Vpp ¼ b q 2

with q ¼ 2b

ð1:13Þ


11

Figure 1.6 Spectral characteristics of sampling, with two different sampling periods TB > TA :

where Vpp is the peak-to-peak voltage of the quantizer, and q is the number of quantization steps, here regarded as being the bth power of two. Thus, the amplitudes of the quantized signal samples can be coded with b bits. Two typical uniform quantization characteristics are shown in Figure 1.7. Nonuniform quantization characteristics are also used. They are not treated here. The quantization process causes distortions of the sampled signal. These distortions are described by the error signal that is the difference between the quantized signal xQ ðnT Þ and the signal xðnT Þ eðnT Þ ¼ xQ ðnT Þ xðnT Þ

ð1:14Þ

Due to the nonlinear character of quantization, ‘‘new’’ frequency components are created ði.e., xQ ðnT Þ consists of frequency components that were not necessarily part of x (nT )Þ. These new frequency components are harmonics and intermodulation products of the frequency components of xðnT Þ. In case of a so-called busy signal at the input of the quantizer

12


Figure 1.7 Quantization characteristics: (a) midtread quantizer with q ¼ 9, and (b) midrise quantizer with q ¼ 8 quantization steps.

(meaning that the signal has sufficiently high amplitudes and comprises a sufficiently large number of different frequencies), the quantization error is modeled as additive white noise that is uncorrelated with the signal xðnT Þ. Thus, eðnT Þ represents white noise, called quantization noise. Obviously, there are always cases where this model is not sufficient. Nonetheless, it helps very much to estimate the theoretical performance of a quantizer in a very simple way. For a system designer, the most interesting figure resulting from modeling the quantization error as additive white noise is the signal-toquantization-noise ratio (SNR) of a quantizer and, thus, of an ADC. To determine the SNR, the signal power and the quantization-noise power must be known. Assuming a busy signal at the input of the quantizer, the quantization error eðnT Þ is a stochastic process e with constant probability density of the amplitude, as shown in Figure 1.8. With Figure 1.8, the quantization noise power Pe can be calculated as the variance s2e of the error process e (assuming an impedence level of 1O):


13

Figure 1.8 Probability density function of the quantization error e.

Pe ¼ s2e ¼

Zemax

DV

e 2 pðeÞde ¼

emin

Z2

e2

1 1 de ¼ ðDV Þ2 DV 12

ð1:15Þ

DV 2

The signal power Px depends on the signal and on the exitation of the ADC. In general, the signal power is equal to the variance of the signal: Px ¼ s2x

ð1:16Þ

With this knowledge, we can write down the logarithmic ratio of noise power and signal power (i.e., the SNR): Px Pe 12 s2x ¼ 10 log ðDV Þ2 12 22b 2 ¼ 10 log sx Vpp2 s2 ¼ 10 log x2 þ 10 log 12 þ 10 log 22b Vpp sx ¼ 20 log þ 10:8 þ 6:02b Vpp

SNR ¼ 10 log

ð1:17Þ ð1:18Þ ð1:19Þ ð1:20Þ ð1:21Þ

Thus, the SNR rises with approximately 6 dB per additional bit quantizer resolution. sx =Vpp is the ratio between the standard deviation of the signal amplitude and the peak-to-peak voltage of the quantizer. Assuming maximum exitation of the quantizer (i.e., the maximum amplitude of the signal is Vpp =2), the ratio sx =Vpp can be interpreted as a characteristic of the signal—namely a relation between its power and its highest amplitude.

14


pffiffiffiffiffi For a signal with equiprobable amplitudes, it is sx =Vpp ¼ 1= 12; for a pffiffiffi pure sine wave, it is sx =Vpp ¼ 1= 8. sx =Vpp results in a signal-dependent offset of the SNR, which does not influence the fundamental relation between SNR and the number of bits b. Because the quantization noise is assumed to be white, the noise power is uniformly distributed in the frequency interval ½fS =2; fS =2, where fS is the sample rate. If the quantized signal covers only a fraction of this interval, namely a bandwidth B < fS , then only the noise power within this bandwidth B really distorts the signal. The remaining noise components can be removed by filtering (see Figure 1.9). This is called oversampling (i.e., the sample rate is higher than twice the highest frequency component of the signal). The oversampling ratio (OSR) is defined as: OSR ¼

fS B

ð1:22Þ

Usually the bandwidth B is located symmetrically at frequency f ¼ 0. Because only the noise power inside the bandwidth B is of interest, the relevant noise power is reduced by OSR if the power spectral density (PSD) of the noise is constant (Figure 1.9). It is Pe ðOSRÞ ¼

1 Pe OSR

ð1:23Þ

Thus, the SNR in the relevant frequency interval is SNROSR

Px ¼ 10 log OSR Pe

SNROSR ¼ 20 log

sx þ 10:8 þ 6:02b þ 10 log OSR Vpp

ð1:24Þ ð1:25Þ

Figure 1.9 Illustrating the effect of oversampling. (Note that PSD refers to power spectral density.)


15

When referring to memoryless ADCs, it is this SNR that is referred. From (1.25) it can be concluded that 1. The SNR rises with 6.02 dB per additional bit of quantizer resolution. 2. The SNR rises with 3.01 dB per doubling the oversampling ratio, which is equivalent to 1/2-bit quantizer resolution. Thus, oversampling by a factor OSR leads to a gain of quantizer resolution of 1 b ¼ log2 OSR 2

bit

ð1:26Þ

It can be concluded that the resolution gain obtained from oversampling is comparatively low. Before leaving the subject of quantization, a generalized type of oversampling is introduced. It is called generalized oversampling. The idea behind this is that the signal at the input of the quantizer can be a multichannel mobile communications signal (i.e., the available bandwidth fS =2 < f fS =2 is fully occupied by the multichannel signal). There is no oversampling in the conventional way. However, a certain channel of bandwidth b (i.e., the channel of interest) covers only a fraction of the available bandwidth (see Figure 1.10, which illustrates the generalized oversampling ratio (GOSR) of a signal). Thus, the channel of interest is oversampled by the factor

Figure 1.10 Illustrating the GOSR of a signal.

16


GOSR ¼

fS b

ð1:27Þ

The difference between conventional and generalized oversampling is that in the case of generalized oversampling, the power of the channel of interest is lower than the total signal power. The total signal power is the one that determines peak-to-peak excitation of the ADC, while the power of the channel of interest equals the signal power in the SNR calculations. A very rough estimate is to assume that the signal power Px ðGOSRÞ (i.e., the power of the channel of interest) is GOSR times lower than the total signal power Px Px ðGOSRÞ ¼

1 Px GOSR

ð1:28Þ

and the quantization-noise power inside the bandwidth b is Pe ðGOSRÞ ¼

1 Pe GOSR

ð1:29Þ

Thus, we can conclude that the SNR with generalized oversampling is Px =GOSR SNRGOSR ¼ 10 log ð1:30Þ Pe =GOSR SNRGOSR ¼ SNR

ð1:31Þ

Hence, there is no gain from generalized oversampling if the quantization error can be regarded as additive white noise, as introduced earlier. 1.3.1.3

Sigma-Delta Modulation

Noise-shaping techniques can be employed to increase the resolution gain from oversampling. Sigma-delta modulators (SDMs) are such noise shapers. They are part of sigma-delta ADCs, which are commonly regarded as a combination of an SDM and a decimator. The latter is a digital filter that attenuates aliasing components before the sample rate of the highly oversampled signal is reduced (this is covered comprehensively in Chapter 4). This combined view might be good for a chip manufacturer, but in the context of signal processing it is not advantageous. Instead, a separated view will be discussed here. The SDM is the analog-to-digital interface and is therefore a separate unit that itself can be called an ADC. Everything that follows the modulator is digital signal processing and should therefore be looked at in conjunction with the other digital signal processing functionality.


17

The SDM usually comprises a noise-shaping filter and a quantizer, together in a feedback loop. More generally, it can be modeled as shown in Figure 1.11 [12, 13]. It should be noted that this is a discrete-time model, which is a relevant model because SDMs can be realized by discrete-time switched capacitor technology. The quantizer in the feedback loop is nonlinear. Therefore, an SDM is a nonlinear system. In order to estimate its behavior in terms of SNR, a linearized model is obtained by replacing the quantizer by an additive white noise source with noise characteristics as described earlier. This leads to a system with two inputs, namely signal X ðzÞ and noise E ðzÞ, and one output Y ðzÞ. Thus, two transfer functions of the discrete-time system can be derived: the signal-transfer function STF ðzÞ and the noise-transfer function NTF ðzÞ: Y ðzÞ AðzÞ ¼ X ðzÞ 1 BðzÞ Y ðzÞ 1 ¼ NTF ðzÞ ¼ E ðzÞ 1 BðzÞ STF ðzÞ ¼

ð1:32Þ ð1:33Þ

By properly selecting AðzÞ and BðzÞ, it becomes, in principle, possible to separate signal and noise. AðzÞ and BðzÞ can be merged to form the socalled loop filter. Thus, designing the loop filter is the key. For our short excursion to SDMs, the classical double-integrating SDM as shown in Figure 1.12 is chosen, resulting in

Figure 1.11 Generic SDM, adapted from [2]. Ó 1999 by Kluwer Academic Publishers.

Figure 1.12 Second-order lowpass SDM with double-integration (quantizer modeled as additive noise source), adapted from [2]. Ó 1999 by Kluwer Academic Publishers.

18


STF ðzÞ ¼ z 1

ð1:34Þ

NTF ðzÞ ¼ ð1 z

1 2

Þ

ð1:35Þ

Evaluating the NTF on the unit circle yields the frequency characteristics:

f 2 j2pf S ð1:36Þ NTF e j2pf =fS ¼ 1 e 2 f pf j2pf S sin ¼ 4e ð1:37Þ fS This represents highpass filtering. Hence, the quantization noise is highpass filtered. If the signal is highly oversampled (i.e., it covers only a small bandwidth B around f ¼ 0), not much quantization noise can distort the signal. The resulting noise power within the bandwidth B is equal to the integral over the power spectral density of the quantization noise multiplied with the squared magnitude of the frequency characteristics of (1.37): B

Pe jSDM

Z 2 4 Pe pf ¼ 16 sin df fS fS B2

3 1 1 p sin ¼ Pe 16 8 OSR 2p OSR Pe

p4 5 OSR 5

ð1:38Þ !

! 1 2p sin þ 16p OSR

for OSR 4

ð1:39Þ ð1:40Þ

Equation (1.40) has been obtained by approximating the sine function of (1.39) with a third-order Taylor series that holds if the OSR is larger than 4. Finally, the SNR of the second-order SDM of Figure 1.12 is SNRSDM ¼ 20 log

sx 5 þ 10:8 þ 6:02b þ 10 log 4 þ 50 log OSR p Vpp

for OSR 4 sx ¼ 20 log 2:1 þ 6:02b þ 50 log OSR Vpp for OSR 4

ð1:41Þ

ð1:42Þ

The presented technique can be extended to Lth order SDMs employing integrators as loop filters [14]:


SNROSR ¼ 20 log

19

sx 2L þ 1 þ 10:8 þ 6:02b þ 10 log 2L p Vpp

þ ð2L þ 1Þ10 log OSR

for OSR 4

ð1:43Þ

From (1.43) it can be concluded that: 1. The SNR rises with 6.02 dB per additional bit of quantizer resolution. 2. The SNR rises with 6ðL þ 1=2Þ dB per doubling of the OSR, equivalent to L þ 1=2 bit quantizer resolution. Hence, oversampling by a factor OSR with an Lth order SDM leads to a resolution gain equivalent to 1 ð1:44Þ log2 OSR bit b¼ Lþ 2 In the case of generalized oversampling, (1.27) and (1.28) can be employed leading to SNRGOSR ¼ 20 log

sx 2L þ 1 þ 10:8 þ 6:02b þ 10 log 2L p Vpp

þ ð2LÞ10 log GOSR

for GOSR 4

ð1:45Þ

The resulting gain from generalized oversampling is b ¼ L log2 GOSR

bit

ð1:46Þ

The main advantage of sigma-delta modulation is that quantizer resolution (number of bits b) can be traded off against the oversampling ratio, with much greater effect than in the case of memoryless ADCs. By choosing a higher-order loop filter in the modulator or by choosing a very high OSR, the resolution gain from oversampling becomes so high that the quantizer can have a small number of bits, typically only 1 bit. However, it should be mentioned that loop filters of an order higher than 2 cannot analytically be designed to be stable. So far, only simulations have lead to ‘‘practically stable’’ higher-order modulators. The possibility of using simpler quantizers increases linearity and reduces cost and power consumption. A drawback of SDMs is that the high SNR is only achievable in the stopband of the noise-shaping filter. In the context of mobile communications, this means that only one (or very few) channel of interest can be digitized. As long as parameterizable loop filters

20


are not available, this channel of interest must be centered in the stopband of the noise-shaping filter that can be a lowpass filter, as in Figure 1.12, or a bandpass filter. Concluding the remarks on sigma-delta analog-to-digital conversion, it should be mentioned that a comprehensive overview on this topic can be found in [14, 15]. The application to digitization in mobile communications terminals has been proposed in [2, 16]. 1.3.1.4

Conclusion

Concluding the discussion on ADCs, it should be mentioned that the quantizer resolution and the sampling frequency are limited by technology. Beside quantization noise, the following technology-dependent mechanisms limit the achievable SNR [17]: 1. Input-referred circuit noise (equivalent thermal noise); 2. Aperture uncertainty (sampling jitter); 3. Comparator ambiguity. As claimed in [17], aperture uncertainty is the most limiting factor, with only small improvements in terms of SNR over the last few years.

1.3.2 SRC Having discussed the issues with analog-to-digital conversion, we now know that the sample rate is an important concern that is vital to properly interpret and reconstruct the analog signal from its samples. Moreover, we have learned that as long as the sampling theorem is obeyed, the actual known sample rate can be varied to meet other system requirements. Therefore, it is possible that two digital signals with different sample rates represent the same continuous-time signal (i.e., both signals carry the same information). Audio signals are used as an example. If the audio-signal spectrum is limited to an upper frequency of 20 kHz, it does not matter if the music is recorded on a compact disc (CD) recorder with a sample rate of 44.1 kHz or on a digital audio tape (DAT) recorder with a sample rate of 48 kHz. However, what happens if the signal is recorded with 44.1 kHz on a CD recorder and we want to copy it to the DAT recorder that works with a sample rate of 48 kHz? This question can be formulated more generally: How can a digital signal be represented at a different sample rate than that at which it was first acquired? The answer is SRC.


21

In software radio terminals, the problem of SRC emerges if the sample rate of the ADC is different from the symbol, chip, or bit rate (i.e., the target rate) of the air interface whose signals are processed. This is a problem because baseband processing is usually carried out at the target rate and not at an arbitrary sample rate. Because a software radio terminal should support a great variety of air interfaces, many different target rates must be supported. This can, in principle, be realized by clocking the ADC at a rate specific to the actual air interface, which means that a tunable oscillator of high quality (low jitter) must be designed to the terminal. Complex analog components must be avoided in mass production. Therefore, a tunable sample rate of the ADC is not the best choice. Alternatively, a digital signal with the target sample rate can be obtained directly from the first digital signal provided by the ADC. SRC is the solution. In the case of equidistant sampling with two different sampling periods T1 ; T2 2 R, and k; m 2 Z, SRC can be defined as follows: A discrete-time signal xðmT2 Þ shall be derived from the signal xðkT1 Þ, which is the result of sampling a continuous-time signal xðt Þ with period T1 , where the signal xðmT2 Þ shall equal the result of hypothetically sampling xðt Þ with period T2 .

This target can only be achieved under certain conditions. However, it must be noted that in some cases it might not be necessary to absolutely achieve this target. Instead it can be sufficient to merely maintain certain characteristics of the signal xðt Þ. This will be investigated in Chapter 2. Throughout this book, only equidistant sampling periods are considered. Hence, SRC can be characterized by a factor relating both the sample rate of the original (input) signal and that of the new (output) signal. Generally, this factor is a real number. Considering certain subsets of the real numbers, special types of SRC result (e.g., integer factor SRC that can be used for increasing or reducing the sample rate of a signal by integer factors). As will be seen, SRC is a special process of sampling. Therefore, all characteristics and rules known from sampling theory also apply to SRC, especially the two fundamental effects imaging and aliasing. 1.3.2.1

Notation

Discrete-time signals are defined as sequences. Sampling a signal xðt Þ with a period T1 yields the sequence s½k j s½k ¼ xðkT1 Þ ; k 2 Z

22


The relationship between the elements of s½k and the sampling period T1 is lost with such a definition. However, in the context of SRC and multirate signal processing, this timing relationship is vital for understanding, as one sees from the earlier definition of SRC. Therefore, instead of writing s½k in this book, all signals are described as samples xðkT1 Þ of a continuous-time signal xðt Þ. In cases where the relationship to a continuous-time signal is not obvious, or where the timing relationship is evident from the context, the sampling period is omitted. 1.3.3 Channelization First a definition of what will be understood by channelization is given: Channelization is the function that comprises all necessary tasks to extract one or more user channels for further processing at baseband, thus involving downconversion, channel filtering, and possibly despreading. 1.3.3.1

Digital Downconversion

The point of downconversion is to shift a signal by fIF in the frequency domain. A channel of interest centered around the intermediate frequency f IF is thus shifted to baseband. Downconversion of a signal to baseband is realized by multiplying the IF (or RF) signal with a rotating complex phasor. xBB ðt Þ ¼ xðt Þ e j2pfIF t

ð1:47Þ

Digital downconversion (DDC) is the digital equivalent of (1.47). xBB ðnT Þ ¼ xðnT Þ e j2pfIF nT

ð1:48Þ

Obviously, each sample xBB ðnT Þ of the downconverted signal is the result of a complex multiplication. The complex samples e j2pfIF nT of the rotating phasor can be: Stored in memory; Calculated on-line (e.g., by using the coordinate rotation digital computer (CORDIC) algorithm [18], which has the great advantage of providing the samples of the respective sine and cosine functions simultaneously). An alternative to implementing the complex multiplication is to use the CORDIC algorithm as a vector rotator [19]. In this case, the CORDIC algorithm provides the complex samples of the rotating phasor and performs the multiplication all at once.


23

DDC can be simplified considerably if the center frequency of the signal equals a certain fraction of the sample rate. If the center frequency of the channel of interest before digitization equals an odd multiple of a quarter of the sample rate, at a quarter of the sample rate the channel is centered after digitization i.e., fIF ¼ 1=ð4T Þ . Hence, p

xBB ðnT Þ ¼ xðnT Þ e j 2n with e

j p2n

¼ f1; j; 1; j; . . .g

ð1:49Þ ð1:50Þ

Thus, in this case, downconversion can be accomplished by simply multiplying the digital IF signal with the sequences f1; 0; 1; 0g and f0; 1; 0; 1g, representing the real and imaginary part of the complex rotating phasor at a quarter of the sample rate. This technique is well documented [20, 21]. Other simplifications (e.g., downconversion of signals centered about 1/8 of the sample rate [22]) are also possible. All of these solutions are based on multiplying the signal with sequences comprising the factors f0; 1; 1g and scaling the result with a fixed factor if required [23]. For more details on DDC, the reader is referred to [7]. 1.3.3.2

Channel Filtering

Apart from downconversion, channel filtering is required to extract frequency divided channels from a wideband signal. The channel filter must attenuate adjacent-channel interferers and meet the so-called blocking characteristic requirements of the particular air interface. Dynamic range leads to requirements similar to those that the ADC has to meet. In software radio terminals where the channel filter must be tunable to the application requirements, it is possible to follow two fundamentally different approaches: 1. The filters can be implemented on reconfigurable hardware (e.g., on field programmable gate arrays (FPGAs) that enable a complete reconfiguration for different air interfaces). Thus, the filters can be optimized for each air interface individually. 2. Another approach is to find a common hardware platform that can serve all of the filter requirements in the context of channel filtering in software radio terminals. This hardware platform is not reconfigurable but parameterizable. It can be realized as an ASIC. While the first approach does not bear challenges for the designer (as to the filter algorithms themselves), the latter requires sophisticated solutions due

24


to the possibly great diversity of filter characteristics that must be supported. However, this is rewarded with a much more efficient realization than the FPGA-based one. Conventional solutions (e.g., direct finite-length impules response (FIR) filters or polyphase filters), are not well suited for the second approach. Because the filter characteristics depend not only on the coefficient values but also on the number of coefficients, it is difficult to decide how many coefficients must be foreseen. A possible solution to overcome this problem is the application of multirate filters, which can be parameterized easily by choosing the ratechange factor [24]. 1.3.3.3

Despreading

In spread-spectrum systems (e.g., wideband code division multiple access (W-CDMA)) the user-channel selection is done by despreading. The rake receiver is a prominent solution in a typical multipath environment. It employs several identical parallel rake fingers, which themselves are realized by a correlator and a decimator. Each rake finger can be regarded as a symbol-matched filter. The different fingers ‘‘collect’’ the same symbol arriving from different propagation paths. A final combination of the results of all the rake fingers yields the received symbol sample. The correlator is usually realized by a multiplier for multiplying the spreading code with the signal, followed by an integrate-and-dump circuit. Because the spreading code is binary or ternary, the multiplier can often be realized by a simple switch. 1.3.3.4

Multiple-Channel Versus Single-Channel Reception

There are two possible areas of application for channelization. In a mobile terminal, only one channel must be selected, while a base station needs to select many channels. As to the first case, downconversion and filtering/ despreading can be applied, as briefly described earlier. The result is a one-channel channelizer. If several channels must be selected, it is possible to use the per-channel approach, which essentially is a parallel realization of M one-channel channelizers, if M is the number of channels. An alternative approach is the filter bank channelizer. Among others, Zangi and Koilpillai [25] have favored this solution, which is based on the discrete Fourier transform (DFT) filter bank and polyphase filter framework [24, 26]. A characteristic of these DFT filter banks is that the sample rate of the incoming signal must be an integer multiple of the channel spacing. Simultaneously, the output sample rate is an integer multiple of the channel


25

spacing. Consequently, there is a dependency between input sample rate, channel spacing, and output sample rate. The number of channels is determined by the number of polyphase branches of the filter, which, moreover, is a power of 2 if the DFT is implemented by means of a fast Fourier transform (FFT). In a fixed hardware platform with a fixed number of polyphase branches, SRC must be performed before and after channelization. While the latter is no problem, SRC before channelization is difficult to realize due to the high dynamic range and the wideband character of the signal (see also Section 7.3). For details on multiple-channel reception, see [27]. 1.3.4 Application of SRC At the end of this chapter it should be stressed that SRC is used in many applications. It is not limited to digital radio. Whenever two digital systems with different sample rates are connected, SRC is required. As mentioned in Section 1.3.2, digital audio signal processing is an area in which SRC is widely used. Digital control is another field of application for SRC [28]. Although SRC is treated with respect to digital radio (specifically, software radio) in this book, it should be stressed that any of the presented techniques and methods can also be used in other applications. The material in Chapter 2 prepares the reader both for the remainder of this book and to exploit other areas of application of SRC.

1.3.5 Summary and Conclusion Analog-to-digital conversion, channelization, and SRC have been identified as critical functions of a software radio receiver. The critical functions of analog-to-digital conversion and channelization are part of every conventional digital communications receiver. Although problems arise with their realization for software-defined radio, especially with analog-to-digital conversion, their theoretical foundation has been well investigated. However, there are still many outstanding research issues concerned with the efficient implementation of SRC. The necessity for SRC in communications receivers arises with the need to implement different air interfaces (e.g., in a software radio terminal). It is well known that readily available solutions to SRC are based on interpolation, which is a well-investigated mathematical problem. Interpolation is not a generally applicable solution to SRC. A considerable part

26


of this book is dedicated to prove this claim, and moreover, to suggest solutions that perform better than interpolation. Chapter 2 deals with the theoretical background of sampling and SRC. Moreover, the idea of block processing is introduced, which greatly simplifies and harmonizes the derivations of solutions for SRC. Besides very few conference publications, this is the first time that SRC is comprehensively tackled on the basis of block processing. Before dealing with systems for SRC, the concept for SRC in software radio terminals is introduced in Chapter 3. In conjunction with the theoretical background, the treatment of systems for integer factor SRC in Chapters 4 and 5 represents a main part of this book. Well-known concepts for SRC are repeated in the context of the block-processing framework. Moreover, several novel concepts are presented. Based on the concepts for integer factor SRC, systems for fractional SRC are investigated in Chapter 6. Finally, selected solutions are compared in the context of an application.

References [1]

Mitola III, J., ‘‘Software Radio Architecture: A Mathematical Perspective,’’ IEEE Journal on Selected Areas in Communications, Vol. 17, April 1999, pp. 514–538.

[2]

Hentschel, T., and G. P. Fettweis, ‘‘Software Radio Receivers,’’ in CDMA Techniques for Third Generation Mobile Systems, Swarts, Pieter van Rooyen et. al. (eds.), Boston, MA: Kluwer Academic Publishers, 1999, pp. 257–283.

[3]

‘‘Digital Cellular Telecommunications System (Phase 2+),’’ Radio Transmission and Reception (GSM 05.05 version 5.9.0 Release 1996), European Telecommunication Standard pr ETS 300 910, July 1998. DRAFT.

[4]

Beach, M., ‘‘Radio Frequency Design for SDR,’’ in Software Radio: Enabling Technologies, Walter Tuttlebee (ed.), Chichester, West Sussex, England: John Wiley & Sons Limited, 2002.

[5]

Mirabbasi, S., and K. Martin, ‘‘Classical and Modern Receiver Architectures,’’ IEEE Communications Magazine, Vol. 38, No. 11, November 2000, pp. 132–139.

[6]

Hentschel, T., M. Henker, and G. P. Fettweis, ‘‘The Digital Front-End of Software Radio Terminals,’’ IEEE Personal Communications, Vol. 6, No. 4, August 1999, pp. 40–46.

[7]

Hentschel, T., and G. Fettweis, ‘‘The Digital Front-End — Bridge Between RF- and Baseband-Processing,’’ in Software Radio: Enabling Technologies, Walter Tuttlebee (ed.), Chichester, West Sussex, England: John Wiley & Sons Limited, 2002.

[8]

Gunn, J. E., K. S. Barron, and W. Ruczczyk, ‘‘A Low-Power DSP Core-Based Software Radio Architecture,’’ IEEE Journal on Selected Areas in Communications, Vol. 17, No. 4, April 1999, pp. 574–590.


27

[9]

Mitola III, J., ‘‘The Software Radio Architecture,’’ IEEE Communications Magazine, May 1995, pp. 26–38.

[10]

Oppenheim, A. V., and R. W. Schafer, Discrete-Time Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1989.

[11]

Proakis, J. G., and D. G. Manolakis, Digital Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, Inc., 1996.

[12]

Jantzi, S. A., W. M. Snelgrove, and P. F. Ferguson, Jr., ‘‘A Fourth-Order Bandpass Sigma-Delta Modulator,’’ IEEE Jounal of Solid State Circuits, Vol. 28, No. 3, March 1993, pp. 282–291.

[13]

Shoaei, O., and W. M. Snelgrove, ‘‘Optimal (Bandpass) Continuous-Time SD Modulator,’’ in Proceedings of the ISCAS, London, England, May 30–June 2, 1994, pp. 489–492.

[14]

Aziz, P. M., H. V. Sorensen, and J. van der Spiegel, ‘‘An Overview of Sigma-Delta Converters,’’ IEEE Signal Processing Magazine, January 1996, pp. 61–84.

[15]

Norsworthy, S. R., R. Schreier, and G. C. Temes, Delta-Sigma Data Converters — Theory, Design and Simulation, Piscataway, NJ: IEEE Press, 1996.

[16]

Hentschel, T., and G. P. Fettweis, ‘‘AD Conversion and Channelization for MultiMode Terminals,’’ in MTT-S European Wireless 98, Amsterdam, The Netherlands, October 8–9, 1998, pp. 169–174.

[17]

Walden, R. H., ‘‘Analog-to-Digital Converter Survey and Analysis,’’ IEEE Jounal on Selected Areas in Communications, Vol. 17, No. 4, April 1999, pp. 539–550.

[18]

Edwards, B. H., R. T. Jackson, and J. M. Underwood. ‘‘How Do Calculators Calculate?’’ in Proceedings of the 9th Annual International Conference on Technology in Collegiate Mathematics, Reno, NV, November 1996.

[19]

Lo¨hning, M., T. Hentschel, and G. P. Fettweis, ‘‘Digital Down Conversion in Software Radio Terminals,’’ in Proceedings of the 10th European Signal Processing Conference (EUSIPCO), Vol. 3, Tampere, Finland, September 5–8, 2000, pp. 1517–1520.

[20]

Boehm, K., et. al., ‘‘An IF Digitizing Receiver for a Combined GPS/GSM Terminal,’’ in IEEE Radio and Wireless Conference (RAWCON), August 1998, pp. 39–42.

[21]

Mu¨ller, T., K. Boehm, and T. Hentschel, ‘‘A GSM/GPS Receiver With a Bandpass Sigma-Delta Analog to Digital Converter,’’ in Proceedings of the European Microwave Conference, Munich, Germany, Vol. 2, October 1999, pp. 295–298. Also published in Proc. of the European Wireless 1999, Munich, Germany, pp. 27–30.

[22]

Schreier, R., and W. Martin Snelgrove, ‘‘Decimation for Bandpass Sigma-Delta Analog-to-Digital Conversion,’’ in IEEE International Symposium on Circuits and Systems (ISCAS), May 1990, pp. 1801–1804.

[23]

Fettweis, A., ‘‘Method and Apparatus for Frequency Division Multiplex System,’’ U.S.-patent US4326288, April 1982.

[24]

Crochiere, R. E., and L. R. Rabiner, Multirate Digital Signal Processing, Englewood Cliffs, NJ: Prentice-Hall, 1983.

28


[25]

Zangi, K. C., and R. D. Koilpillai, ‘‘Software Radio Issues in Cellular Base Stations,’’ IEEE Jounal on Selected Areas in Communications, Vol. 17, No. 4, April 1999, pp. 561–573.

[26]

Vaidyanathan, P. P., Multirate Systems and Filter Banks, Englewood Cliffs, NJ: Prentice Hall 1993.

[27]

Hentschel, T., ‘‘Channelization for Software Defined Base-Stations,’’ Annales des Te´lećommunications, special issue on software radio, May/June 2002.

[28]

Feuer, A., and G. C. Goodwin, Sampling in Digital Signal Processing and Control, Boston, MA: Birkhaüser, 1996.

2 Basic Principles of SRC and Multirate Systems 2.1 Introduction As discussed in Section 1.3.2, the problem of SRC arises when a discrete-time signal whose samples are located at well-defined positions in time has to be processed in a way that a new discrete-time signal is generated with its samples located at different positions. This process should maintain the key information of the original signal. In this chapter, the necessary fundamentals are discussed. Well-known approaches to SRC are discussed. The discussion starts with an analog interpretation of SRC. From this, it is possible to derive the equations describing digital SRC. The obtained general result will be used to derive SRC by a rational factor as well as integer factor SRC (i.e., decimation and interpolation). A second part of this chapter deals with signal processing concepts that will be used in Chapters 4, 5, and 6. It will be seen that systems for SRC are linear periodically time-varying systems. The analysis of such systems can be simplified when transforming them into linear time-invariant block systems. This approach is strongly related to the polyphase concept. To fully exploit this, a polyphase description of general transfer functions and impulse responses (i.e., not limited to finite-length impulse responses) is given based on the state-space description of linear time-invariant systems.

29

30


2.2 Basic Considerations of SRC 2.2.1 Resampling After Reconstruction 2.2.1.1

An Analog Interpretation

The starting point of SRC is a discrete-time signal. In order to perform SRC, some intersample values must be calculated. It is an open question how to do this. The sampling theorem may be applied. It states that a bandlimited signal of finite energy can be reconstructed from its samples as long as the sample rate is higher than two samples per period of the highest frequency of the signal. This condition can be formulated as: the sample rate must be higher than the ‘‘two-sided’’ bandwidth of the signal (see Section 1.3.1). Hence, if it is possible to reconstruct a continuous-time signal from the discrete-time signal, the reconstructed signal can be newly sampled with a new sample rate. A system realizing this approach, resampling after reconstruction, is sketched in Figure 2.1. Resampling after reconstruction is a model that helps us to analyze and understand SRC. However, it is not a pragmatic means of realizing SRC. The signal xa ðt Þ is defined according to (1.6) and (1.7). xa ðt Þ ¼ T1

1 X

x ðkT1 Þdðt kT1 Þ

k ¼ 1

¼ xðt Þ T1

1 X

ð2:1Þ dðt kT1 Þ

k ¼ 1

with x ðkT1 Þ representing samples of the analog signal xðt Þ. The spectrum Xa ðf Þ of xa ðt Þ can be calculated by applying the Fourier-transform (see (1.8) to (1.12)):

Figure 2.1 Resampling of a reconstructed discrete-time signal, adapted from [1]. Ó 1999 by IEEE.


Xa ð f Þ ¼ FTfxa ðt Þg ¼

1 X

X

k ¼ 1

k f T1

31

ð2:2Þ

where X ð f Þ is the Fourier transform of xðt Þ. Equation (2.2) describes the well-known spectral repetition due to sampling. It is assumed that the sampling period T1 is sufficiently short to cause no aliasing: jX ð f Þj ¼ jFTfxðt Þgj ¼ 0

for j f j

1 2T1

ð2:3Þ

In this case xðt Þ can be reconstructed from its samples xðkT1 Þ according to the Cardinal series [2] 1Þ 1 sin pðt kT X T1 x ðkT1 Þ pðt kT Þ ð2:4Þ xðt Þ ¼ 1

k ¼ 1

T1

This can also be interpreted as filtering with an ideal lowpass filter having its cut-off frequency at half the sample rate [3]. Equation (2.4) can be generalized by introducing a reconstruction filter with the impulse response hc ðt Þ. Hence, the output signal yðt Þ of this reconstruction filter is yðt Þ ¼

¼

Z1

xa ðtÞhc ðt tÞd t

t ¼ 1 Z1

1 X

T1 t ¼ 1 1 X

¼ T1

x ðkT1 Þdðt kT1 Þhc ðt tÞd t

k ¼ 1

x ðkT1 Þhc ðt kT1 Þ

ð2:5Þ

k ¼ 1

With (2.4) it can be concluded that yðt Þ ¼ xðt Þ for

hc ðt Þ ¼

sin Tpt1 pt

1 t ¼ sinc T1 T1

ð2:6Þ

Resampling the reconstructed signal yields: y ðmT2 Þ ¼ T1

1 X k ¼ 1

x ðkT1 Þ hc ðmT2 kT1 Þ

ð2:7Þ

32


For reasons of simplicity, the following normalization of the impulse response of the reconstruction filter is introduced: hðt Þ :¼ T1 hc ðt Þ

ð2:8Þ

Note, that hðt Þ has no dimension. Thus, (2.7) can be rewritten as y ðmT2 Þ ¼

1 X

x ðkT1 Þ h ðmT2 kT1 Þ

ð2:9Þ

k ¼ 1

Equation (2.9) is a convolution. It involves samples of the impulse response hðt Þ at positions ðmT2 kT1 Þ. The understanding of (2.9) can be facilitated when rearranging the arguments of x and h: mT2 x ðkT1 Þ h T1 k y ðmT2 Þ ¼ T1 k ¼ 1 1 X mT2 x ðkT1 Þ h T1 þ mm k ¼ T1 k ¼ 1 1 X

mT2 n h T1 ðn þ mm Þ x T1 ¼ T1 n ¼ 1 1 X

ð2:10Þ

ð2:11Þ

with mT2 mT2 mm ¼ T1 T1 mT2 n¼ k T1

ð2:12Þ ð2:13Þ

For mm the following inequality holds: 0 mm < 1

ð2:14Þ

The quantity mm reflects the position of the current sample within the sample period T1 . It is called the intersample position. From (2.11) some obvious points can be derived:

Equation (2.11) is a direct digital representation of SRC.

The impulse response hðt Þ is sampled with period T1 of the input signal, and then with different time offsets mm T1 , which depend on


33

the position of the output sample with respect to the input samples.

Thus, to calculate different output samples, different sets of samples of the filter impulse response hðt Þ are used.

For T1 ¼ T2 ¼ T , (2.11) reduces to the well-known discrete convolution 1 X

x T ðm nÞ hðnT Þ ð2:15Þ yðmT Þ ¼ n ¼ 1

Thus, we have obtained a description of SRC for arbitrary sample periods T1 and T2 . However, it is still open if the signal after SRC contains the same information as the signal before SRC. For the case

jX ð f Þj ¼ jFTfxðt Þgj ¼ 0

1 1 ; for j f j min 2T1 2T2

ð2:16Þ

and perfect reconstruction according to (2.6) it can be concluded that perfectly reconstructing xðkT1 Þ will yield the same result as perfectly reconstructing yðmT2 Þ. A deeper insight in the characteristics of SRC can be obtained when considering the spectra of the individual signals of Figure 2.1. For that purpose ya ðt Þ ¼ T2

1 X

y ðmT2 Þdðt mT2 Þ

ð2:17Þ

m¼1

is defined following (2.1). 2.2.1.2

Spectral Interpretation of Resampling

In Figure 2.2, the spectral interpretation of the different steps of signal processing, from sampling the signal xðt Þ to resampling yðt Þ, is sketched. The reconstruction filter hc cancels all signal components resulting from spectral repetition (images), thus yielding a band-limited signal and avoiding aliasing. However, a perfect reconstruction filter cannot be realized. For practical applications, the constraints on hc can be relaxed if certain amounts of aliasing and passband attenuation are allowed. For example, 8 > < ¼ 1 d 0 j f j < fc fc j f j < fstop ð2:18Þ jHc ð f Þj ¼ jFTfhc ðt Þgj 1 > : H 1 fstop j f j < T2 stop

34


Figure 2.2 Spectral interpretation of resampling of a reconstructed signal, adapted from [1]. Ó 1999 by IEEE.


35

with d being the passband ripple, fc the cut-off frequency, fstop the stopband frequency, and 1=Hstop the stopband attenuation. After resampling, only the frequency band 0 j f j < T12 fstop is free of aliasing, depending on the stopband attenuation 1=Hstop and the characteristics of the signal x ðkT1 Þ. However, especially when considering only one channel of a multichannel communications signal—if there is a GOSR > 1 for the channel of interest (see Section 1.3.1)—it is not necessary to reconstruct (interpolate) the complete multichannel signal. Instead, the amount of aliasing in the channel of interest must be controlled in order to preserve the interesting information. This is shown in Figure 2.3. Substituting this spectrum to Figure 2.2 immediately reveals that there is only a very small region that must be kept free from aliasing. This has direct influence on the design constraints of the filter hc . Concluding this short discussion leads to the following remark. 2.2.1.3

Key Note

Antialiasing is the most prominent constraint to be obeyed by an SRC system.

The reconstruction filter hc can be regarded as a filter that attenuates distorting components of xa ðt Þ. These distorting components are those signal components that potentially cause aliasing. With the applicationdependent definition of the frequency band that has to be kept free from aliasing, the antialiasing filter hc can be designed. All aliasing components

Figure 2.3 Wideband signal comprising several channels (multichannel signal) with channel of interest and region of allowed aliasing.

36


that do not fall in the frequency band that has to be kept aliasing-free can be removed by filtering after SRC. On the basis of this antialiasing constraint, the filter hc can be simplified considerably. 2.2.1.4

Some Notes on Interpolation

Very often, SRC is equated with interpolation. Historically, interpolation is a mathematical problem of calculating in-between values of tabulated functions. The interpolation problem can be formulated as follows: n þ 1 numbers a t0 < t1 < t2 < . . . < tn b are given in the interval ½a; b. Moreover, n þ 1 real numbers yi ði ¼ 0; 1; . . . ; nÞ (e.g., representing the samples of a function yðtÞ) are given. The interpolation problem is then generally defined as finding a polynomial In ðtÞ of maximum degree n, so that In ðti Þ ¼ yi for 0 i n, thus finding a polynomial that takes the originally given samples at the positions defined by the numbers ti ði ¼ 0; 1; . . . ; nÞ. Under these constraints, exactly one polynomial In ðtÞ exists [4].

The representation of this polynomial can vary. It can be based upon different interpolation formulas (e.g., the Lagrange, Newton, Stirling, or Gau interpolation formulas). Usually, signals are equidistantly sampled in the context of communications. Moreover, the number of samples ðn þ 1Þ of a signal is generally very large, which makes the application of interpolation (as defined earlier) infeasible. Therefore, piecewise polynomial interpolation can be applied to such equidistantly sampled signals. An alternative approach is spline interpolation [5]. Splines are piecewise polynomials that are smoothly connected together. Piecewise polynomial interpolation including spline interpolation can be interpreted as a filtering operation [5–7]. The following fundamental property of the impulse response of interpolation filters can be derived from this interpretation and the definition of interpolation [8]: 1 if k ¼ 0 ð2:19Þ h ðkT1 Þ ¼ 0 if k ¼ 1; 2; . . . Obviously, (2.19) does not limit the impulse response to be derived from polynomial interpolation of given samples. It is interesting to note that the ideal lowpass filter of (2.6) and any derivatives of it obtained by windowing are interpolation filters with respect to (2.19). Although polynomial filters are not the only filters fulfilling (2.19), interpolation in the digital domain is usually related to the application


37

of polynomial filters. This is because the sampled impulse response of polynomial filters can be calculated with relatively low effort in real time, thus providing a means to avoid large coefficient memories. Polynomial interpolation filters derived from Lagrange’s interpolation formula have zeros of the transfer function that tend to be clustered about integer multiples of 1=T1 [8]. The spectral interpretation of SRC with such polynomial filters is in principle shown in Figure 2.4. It can be seen that the polynomial filter achieves high attenuation at integer multiples of 1=T1 . This is also referred to as anti-imaging. However, the aliasing components at integer multiples of 1=T2 are only poorly attenuated, which results in severe

Figure 2.4 Spectral interpretation of resampling of a reconstructed signal with a polynomial interpolation filter.

38


aliasing. On the other hand, if there are no signal components beside the interesting ones, the only source for aliasing are the images. Therefore it can be concluded that these polynomial filters are only applicable to SRC of oversampled and band-limited signals, which means that the sample rate of the signals must be several times higher than the Nyquist rate. In this case, the relatively small notches of the transfer function of these interpolation filters attenuate the images sufficiently to avoid severe aliasing. Another area of application for such filters is the case T1 T2 , where the required antialiasing filter can be approximated by an interpolation filter. This is possible because the first few transfer zeros at integer multiples of 1=T1 are very close to integer multiples of 1=T2 (areas of potential aliasing components) in this case. Further inside the stopband, the attenuation is usually high enough to avoid aliasing, even outside the transfer zeros. Other types of interpolation filters that obey (2.19) might perform better than polynomial filters. However, if they are not built on the antialiasing constraint, it is not possible to easily control the amount of aliasing as early as during filter design. This is possible with true antialiasing filters, as sketched in Figure 2.5. At this point it should be reminded that the aim of SRC is a signal at a new sample rate with well-defined and limited distortions in the frequency band of interest. Restrictions are not acceptable regarding band limitation of the signal, oversampling, or conversion factor. However, interpolation filters provide sufficient aliasing attenuation only in the case of certain oversampling ratios and SRC factors as we have learned. Therefore, they cannot meet this requirement. Hence, it is certainly sensible to refrain from interpolation (antiimaging) and stick to the spectral interpretation given in Section 2.2.1.2. The spectral interpretation of SRC and the resulting antialiasing constraint provide a means to directly estimate the aliasing distortions in a certain frequency band. This cannot be achieved as easily with the interpolation approach, which is basically a time-domain approach. Other time-domain methods, such as minimizing the quadratic error between the impulse response of an ideal interpolation filter and its approximation as suggested in [9], also have to be disqualified for their lack of direct spectral interpretation of the error.

2.2.2 Irrational Factor SRC Irrational factor SRC is the case if T1 ¼ nT2

n 2 RnQ:

ð2:20Þ


39

Figure 2.5 Spectral interpretation of resampling of a reconstructed signal with a polynomial antialiasing filter.

This means that T1 and T2 are related by an irrational number. In reality, this type of SRC is rather theoretical. It is covered here for reason of completeness. The irrational relation between T1 and T2 requires a different set of samples of the analog impulse response hðt Þ for each calculation of an output sample (see Section 2.2.1). Systems with those characteristics have been named continuous-time digital filters [10]. Since generally an infinite number of different sets is used, a digital realization is only feasible if: 1. The impulse response of the analog filter (or an approximation of it) is available so that the samples can be calculated on demand.

40


2. The number of sets can be limited, which is equivalent to approximating n by a rational factor L=M . The first case can be met if the impulse response of the filter is composed of functions that enable a simple evaluation of their values (e.g., polynomial representations). This is the reason that polynomial interpolation is so popular, although it is not satisfactory unless the signals are sampled at rates considerably higher than the Nyquist rate, as explained in Section 2.2.1. If the OSR (see (1.22)) can be chosen to be very high, simple interpolation schemes like sample and hold or linear interpolation are sufficient. However, in many applications, the signals cannot be oversampled. This is the case, for example, if wideband signals must be processed at relatively high sample rates or in simulation environments where high oversampling is not possible due to limited memory size for storing the signals. In the case where n is approximated by a rational factor L=M , n¼

T1 L ¼ þe T2 M

ð2:21Þ

an approximation error e remains. Obviously, a desired accuracy can lead to very large values of L and M making the implementation expensive. Starting from a synchronized position, it can be written after m output samples m

M M T1 ¼ mT2 þ m T2 e L L

ð2:22Þ

Thus, after m¼

L 1 jej M

ð2:23Þ

times the output period, the accumulated error equals T2 and can theoretically be compensated by dropping or repeating one output sample, depending on whether e is negative or positive, respectively. Likewise, input samples can theoretically be dropped or repeated after k times the input sample period. In this case n1 could be approximated by a rational number n1 ¼ leading after k input samples to

T2 M ¼ þ e L T1

ð2:24Þ


k

L L T2 ¼ kT1 þ k T1e M M

41

ð2:25Þ

Thus, after k¼

M 1 jej L

ð2:26Þ

times the input sample period the accumulated error equals T1 and could be compensated by repeating or dropping one input sample, depending on whether e is positive or negative, respectively. Comparing (2.21) and (2.24) yields e ¼

M 2e L 2 þ MLe

ð2:27Þ

As both e and e are irrational numbers, (2.23) and (2.26) do not yield integer numbers. Hence, the exact time instance for dropping or repeating samples is not available. It lies in between the time instances defined by the input or output sample periods, respectively. Therefore, this time has to be approximated. However, the time instances defined by (2.23) and (2.26) must be taken into consideration as average values with respect to the output sample rate or the input sample rate, respectively. Exact timing is possible if both periods T1 and T2 are known, for example, if two digital systems fed from different clock sources are connected [3]. In this case, a digital continuous-time zero-order-hold circuit can be applied: The source system writes its samples to a register. Each new sample overwrites the old one. The sink system simply reads the contents of the register at the new clock rate. Hence, samples are dropped or repeated automatically at appropriate time instances. Due to the irrational nature of the ratio between T1 and T2 , the occurrence of dropping and repeating samples is generally nonperiodic. Having discussed how the error of approximating an irrational rate change factor by a rational one can be dealt with, it is necessary to discuss the effect of the sample drop or repeat. In [3] this error is estimated on the basis of the energy of the signal. However, this does not say anything about the spectral distribution of the error, which is important in many applications as we have learned in Section 2.2.1. Thus, the resulting distortions caused by dropping or repeating samples should be estimated in the frequency domain. The dropping and repeating process can be realized by a system equivalent to Figure 2.1 with the filter h being a zero-order-hold circuit—having the impulse response

42


hðt Þ ¼

n

1 0

0 t < T1 else

ð2:28Þ

Its transfer function is the sinc function having zeros of first order at integer multiples of the sample rate 1=T1 (see Figure 2.4). The zero-order-hold filter is an interpolation filter (it fulfills (2.19)). Its first-order zeros only poorly suppress image components. Moreover, it is clear that the filter h does not suppress potential aliasing components. Therefore, the dropping (or repeating) process causes aliasing and thus inband distortions to a channel of interest of a multichannel signal. The interpolation filter can be exchanged by an equivalent antialiasing filter (see Figure 2.5 for its basic spectral performance). According to (2.28) its impulse response would be hðt Þ ¼

n

1 0

0 t < T2 else

ð2:29Þ

It is a continuous-time integrate-and-dump circuit. The samples of the input signal are integrated until the output picks up the current content of the accumulator, thus clearing it. It should be noted that a filter according to (2.29) does not avoid aliasing. However, it minimizes aliasing in a certain frequency band, namely around f ¼ 0. From this discussion, we have learned that approximating an irrational rate change factor by a rational one shifts the task of irrational factor SRC to the zero-order-hold (or integrate-and-dump) circuit, which performs the residual irrational factor SRC determined by e or e, respectively. Thus, irrational factor SRC by n can be divided into two steps: rational factor SRC by n1 ¼ L=M and remaining irrational factor SRC by n2 . The sequential order of performing both steps does not matter in principle. L þe M L M 1þ e ¼ M L ¼ n1 n2

n¼

ð2:30Þ

As an alternative to performing the two steps separately, SRC by n2 can be put inside the rational factor SRC by first interpolating by L, then performing SRC by n2 , and finally decimating by M . A system realizing this approach is sketched in Figure 2.6 (details on rational and integer factor SRC are discussed in Sections 2.2.3 and 2.2.4). Depending on the value of L, this solution


43

Figure 2.6 Irrational factor SRC.

provides an oversampled signal, which ensures that the aliasing error due to SRC by n2 is small. The larger L is, the higher is the OSR of the signal, and thus, the smaller is the aliasing error, especially when using an antialiasing filter according to (2.29). It should be noted that the case T1 T2 might result in infeasibly large values for L. In that case, it is possible to interpolate by any L that ensures sufficient oversampling for the subsequent SRC by n 1. Finally, the signal is decimated by L. Thus, we have approximated n by n1 ¼ LL ¼ 1 and set n2 ¼ n. This is exactly what Crochiere and Rabiner [3] have suggested except that we do suggest to use an antialiasing filter instead of the zero-orderhold interpolation filter. Concluding this section, it should be stressed that the realization of irrational factor SRC requires the availability of two clock signals, with the period T1 of the input signal and the period T2 of the output signal. These clock signals control an asynchronous interface (e.g., continuous-time sample and hold or integrate and dump). Strictly speaking, such systems are not discrete time and thus not digital. They are two separate discrete-time systems with a continuous-time interface, which, however, can be implemented on digital hardware. 2.2.3 Rational Factor SRC Rational factor SRC is the case if the sample periods of the input signal and the output signal are related by two positive integers. T1 L ¼ T2 M

L; M 2 Nþ

First, (2.11) and (2.12) are rewritten as: 1 X

mM y ðmT2 Þ ¼ x T1 n h T1 ð n þ mm Þ L n ¼ 1 1 X mM x T1 n hm ðnT1 Þ ¼ L n ¼ 1

ð2:31Þ

ð2:32Þ ð2:33Þ

44


with

hm ðnT1 Þ ¼ h T1 ðn þ mm Þ mM mM ðmM Þðmod LÞ ¼ mm ¼ L L L

ð2:34Þ ð2:35Þ

For mm (2.14) still holds: 0 mm < 1 The intersample position mm can only take L distinct values. Thus, the number of different sets of samples of hðt Þ used for the calculation of y ðmT2 Þ is limited to L. Because, moreover, mm is periodic in L, mmþL ¼

ðmM þ LM Þðmod LÞ ðmM Þðmod LÞ ¼ ¼ mm L L

ð2:36Þ

the filter h can be implemented as a periodically time-varying filter with period LT2 ¼ MT1 . A different view can be obtained by using (2.31) and rewriting (2.10) to 1 X T1 x ðkT1 Þ h ðmM kL Þ ð2:37Þ y ðmT2 Þ ¼ L k ¼ 1 which is interpreted in the following. First the signal vðnT0 Þ ¼

1 X

w ðkT0 Þ h T0 ðn k Þ

ð2:38Þ

k ¼ 1

with T0 ¼ and

w ðnT0 Þ ¼

T1 T2 ¼ L M

x Ln T1 0

ð2:39Þ n ¼ 0; L; 2L; . . . else

is defined. Then, (2.38) can be rewritten by using (2.40): 1 X

kL T1 h T0 ðn kL Þ x v ðnT0 Þ ¼ L k ¼ 1 1 X T1 x ðkT1 Þ h ¼ ðn kL Þ L k ¼ 1

ð2:40Þ

ð2:41Þ ð2:42Þ


45

Comparing (2.42) with (2.37) leads to y ðmT2 Þ ¼ v ðmMT0 Þ

ð2:43Þ

Equation (2.38) reflects the implementation of rational factor SRC by a time-invariant digital filter with the impulse response h ðnT0 Þ, while (2.40) and (2.43) describe time-varying operations necessary before and after filtering. The discrete-time impulse response h ðnT0 Þ can be obtained by sampling the continuous-time impulse response of the analog reconstruction filter hðt Þ with the period T0 . It is interesting to note that the different filter sets hm ðnT1 Þ of (2.34) are the polyphase components (see Section 2.2.6) of h ðnT0 Þ. Finally, it should be mentioned that systems realizing (2.40) are called upsampler or sample rate expander [3]. Often they are called interpolator, which is very misleading, because an interpolator consists of an upsampler and a filter, as will be seen in the next section. The process of upsampling is realized by zero filling (also called zero padding). Likewise, systems realizing (2.43) are called downsampler or sample rate compressor [3]. Both the upsampler and downsampler, together with the filter h ðnT0 Þ, realize a system for rational factor SRC shown in Figure 2.7. It can also be obtained from Figure 2.6 by leaving out the integrate-and-dump circuit and combining hint and hdec into h. By applying the discrete-time Fourier transform (DTFT) to the different signals shown in Figure 2.7, their spectra can be estimated. This will be dealt with in the next section. A typical constellation of these spectra for L ¼ 3 and M ¼ 5 is shown in Figure 2.8.

2.2.4 Integer Factor SRC By setting L or M to one, integer factor SRC as a special case of rational factor SRC can be achieved. Although all considerations made with respect to rational factor SRC can directly be applied to integer factor SRC, there

Figure 2.7 Rational factor SRC, adapted from [11]. Ó 2000 by IEEE.

46


Figure 2.8 Spectra of signals in Figure 2.7 for L ¼ 3 and M ¼ 5.


47

are some additional interesting characteristics that should be named in this section. 2.2.4.1

Sample Rate Increase by an Integer Factor L

Setting M ¼ 1 leads to an output signal of an SRC system with the reconstruction filter h according to (2.37) to (2.43): T1 ¼ T2 L

x Ln T1 n ¼ 0; L; 2L; . . . w ðnT2 Þ ¼ 0 else 1 X

yðnT2 Þ ¼ vðnT2 Þ ¼ w ðkT2 Þ h T2 ðn k Þ T0 ¼

k ¼ 1 1 X

T1 x ðkT1 Þ h ¼ ðn kLÞ L k ¼ 1

ð2:44Þ

ð2:45Þ

Clearly, the input signal x ðkT1 Þ is upsampled by zero filling, thereby creating the signal w ðnT2 Þ, which is filtered by a filter having the impulse response h ðnT2 Þ. The output signal y ðnT2 Þ equals the signal v ðnT2 Þ at the output of the filter h. A system performing integer factor sample rate increase, comprising the upsampler L " and the filter h, is sketched in Figure 2.9. Although the filter h does not have to be an interpolation filter (fulfilling (2.19)), such a system is usually called an interpolator. The best way to characterize the upsampler is to calculate the DTFT of an upsampled signal in order to estimate its spectral properties: 1 X

w ðnT2 Þe jn2pfT2 W e j2pfT2 ¼ n ¼ 1

By exploiting (2.44) it can be rewritten as:

Figure 2.9 Sample rate increase by an integer factor: interpolation.

ð2:46Þ

48


1 X j2pfT

2 W e x ðkT1 Þe jk2pfT1 ¼

ð2:47Þ

k ¼ 1

Hence

W e j2pfT2 ¼ X e j2pfT1 ¼ X e j2pfLT2

ð2:48Þ

From (2.48) it can be concluded that the spectrum of the signal is not changed by upsampling. Just the timing relation, specifically the sample rate, is changed. Graphically, this can be achieved by renaming the frequency axis, specifically by exchanging L=T1 with 1=T2 . The L 1 images of the original signal x ðkT1 Þ at multiples of the sample rate are part of the fundamental spectrum of w ðnT2 Þ. They can be removed by the filter h, which (in the context of upsampling) is called an anti-imaging filter. Removing these images (in the frequency domain) is equivalent to interpolating the signal (in the time domain). The anti-imaging filter has the same purpose as the reconstruction filter of a DAC, yet the anti-imaging filter has only to interpolate L 1 values in contrast to the continuous function (infinite number of values), which the reconstruction filter of a DAC has to interpolate. Thus, the spectrum of the signal after upsampling and filtering is

ð2:49Þ Y e j2pfT2 ¼ H e j2pfT2 W e j2pfT2 j2pfT j2pfLT

2 2 ¼H e X e ð2:50Þ Often a gain factor L is introduced to interpolation filters (e.g., [3]). This can be understood when considering the reconstruction of an upsampled signal according to the definition of (2.1) (e.g., wðnT2 Þ in Figure 2.9): wa ðt Þ ¼ T2

1 X

wðkT2 Þdðt kT2 Þ

k ¼ 1

With (2.44) it is wa ðt Þ ¼ T2

1 X

xðnT1 Þdðt nT1 Þ

n ¼ 1

¼

xa ðt Þ L

ð2:51Þ

In order to compensate the attenuation by L, the interpolation filter following the upsampling process can be given a gain of L. However, digital signal processing is determined by quantization, scaling, and normalization


49

of the signal. Therefore, this gain factor is of secondary importance. It is only of theoretical interest. It should be noted that by deriving rational or integer factor SRC as done in this book, the scaling factor is taken into account by (2.8). For reasons of completeness, it should be mentioned that the filter h can be designed as a bandpass filter, which selects one of the images centered at k=ðLT2 Þ with k ¼ 1; 2; . . . ; L 1. This version of interpolation can be named integer-band interpolation [3] or generalized interpolation [12]. Clearly, integer-band interpolation achieves upconversion of the signal. 2.2.4.2

Sample Rate Reduction by an Integer Factor M

Setting L ¼ 1 leads to an output signal of an SRC system according to (2.37) to (2.43): T0 ¼ T1 ¼ v ðnT1 Þ ¼ y ðmT2 Þ ¼ v ðmMT1 Þ ¼

1 X

T2 M

k ¼ 1 1 X

x ðkT1 Þ h T1 ðn kÞ

ð2:52Þ

x ðkT1 Þ h T1 ðmM kÞ

ð2:53Þ

k ¼ 1

Clearly, the input signal x ðnT1 Þ is filtered by a filter having the impulse response h ðnT1 Þ leading to the signal v ðnT1 Þ; whose sample rate is finally reduced by the factor M , eventually yielding the signal y ðmT2 Þ. The system performing integer factor sample rate reduction, comprising the filter h and the downsampler M #, is sketched in Figure 2.10. Such a system is usually called a decimator. When considering the delayed Kronecker impulse 1 n ¼ n0 ð2:54Þ dKr ðn n0 Þ ¼ 0 n 6¼ n0

Figure 2.10 Sample rate reduction by an integer factor: decimation.

50


as an input signal to the downsampler v ðnT1 Þ ¼ dKr ðn n0 Þ its output signal is ( y ðmT2 Þ ¼ dKr ðmM n0 Þ ¼

1 0

n0 2Z if m ¼ M else

ð2:55Þ

Thus, the output depends on the value n0 and, thus, on the delay n0 T1 of the input signal, which clearly indicates that the downsampler M # is a timevarying system (specifically, a periodically time-varying system). Delaying the input signal by Mn0 T1 causes the output to be delayed by n0 T2 . The periodically time-varying nature of the downsampler can also be observed if it is implemented by multiplying the input signal with the M -periodic Kronecker-impulse train ZKr ðn; M Þ, sketched in Figure 2.11. y ðnT1 Þ ¼ v ðnT1 Þ ZKr ðn; M Þ

ð2:56Þ

with ZKr ðn; M Þ :¼

1 X

dKr ðn kM Þ

ð2:57Þ

k ¼ 1

Equations (2.56) and (2.57) indicate that a downsampler is a resampler. It is the discrete-time representation of the second sampler in Figure 2.1. For further calculations it is useful to describe the Kronecker-impulse train by means of its Fourier series:

Figure 2.11 Kronecker-impulse train.


ZKr ðn; M Þ ¼

1 X 1 M k ck e j2pnM M k ¼0

51

ð2:58Þ

with ck ¼

M 1 X

1 X

n

dKr ðn lM Þ e j2pk M ¼ 1

ð2:59Þ

1 X 1 M k e j2pnM M k ¼0

ð2:60Þ

n ¼ 0 l ¼1

Thus, ZKr ðn; M Þ ¼

In order to estimate the spectrum of the decimated signal y ðmT2 Þ, its DTFT is calculated. 1 X

yðmT2 Þe jm2pfT2 Y e j2pfT2 ¼

¼

m ¼ 1 1 X

v ðkT1 Þ ZKr ðk; M Þe jk2pfT1

ð2:61Þ ð2:62Þ

k ¼ 1

¼

1 X k ¼ 1

v ðkT1 Þ

1 X 1 M n e j2pk M e jk2pfT1 M n¼0

ð2:63Þ

Equation (2.63) has been obtained by substituting (2.60) into (2.62). Finally, we get

1 1 X X

n 1 M Y e j2pfT2 ¼ v ðkT1 Þ e j2pkT1 f MT1 M n¼0 k ¼ 1

1 X n 1 M ¼ V e j2pT1 f MT1 M n¼0

ð2:64Þ ð2:65Þ

Equation (2.65) indicates the well-known aliasing property of sampling: M -times downsampling causes spectral repetition of the original spectrum at multiples of an M th of the original sample rate. Because the signal vðkT1 Þ is a discrete-time signal, its spectrum V e j2pT1 f is not band limited. Thus, spectral components are aliased into the frequency band between 1=ð2T1 Þ and 1=ð2T1 Þ. Irreversible distortions of the decimated signal occur if the bandwidth of the original (lowpass) signal extends 1=T2

52


(considering the region 1=ð2T1 Þ < f 1=ð2T1 Þ). Such distortions caused by aliasing can be avoided if the signal to be decimated is band limited to 1=T2 before decimation. The band limitation is realized by the antialiasing filter h. There is a decimation complement to integer-band interpolation. It can be called integer-band decimation [3] or bandpass decimation. In this case, the antialiasing filter h is designed as a bandpass filter, which selects a 1=T2 -wide frequency band centered at k=ðMT1 Þ with k ¼ 1; 2; . . . ; M 1. This frequency band is aliased to baseband by the decimation process. Thus, bandpass decimation achieves down conversion. Bandpass decimation is the discrete version of harmonic subsampling.

2.2.5 Up- and Downsampling with Phase Offset Generalizing (2.40), a signal w 0 ðnT0 Þ can be defined. nL

x L 0 T1 if n L0 ¼ 0; L; 2L; . . . w ðnT0 Þ ¼ 0 else with L0 2 f0; 1; . . . ; L 1g 0

ð2:66Þ ð2:67Þ

It reflects upsampling with a phase offset L0 —the sample xð0Þ is taken as w 0 ðL0 T0 Þ. In equivalence to (2.38) the signal v 0 ðnT0 Þ is defined. v 0 ðnT0 Þ ¼

1 X

w 0 ðkT0 Þ h T0 ðn kÞ

k ¼ 1

¼

1 X

w ðk L0 ÞT0 h T0 ðn kÞ

k ¼ 1

¼

1 X

wðkT0 Þ h T0 ðn L0 k Þ

k ¼ 1

¼ v ðn L0 ÞT0

ð2:68Þ

Likewise, the downsampling process can be the object of an offset M0 . Then, (2.43) can be rewritten (see also (2.82)).

yM0 ðmT2 Þ ¼ v ðmM þ M0 ÞT0

with M0 2 f0; 1; . . . ; M 1g ð2:69Þ


53

From (2.69) it can be concluded that yM0 ð0Þ ¼ vðM0 T0 Þ. Thus, every M th sample is taken from vðnT0 Þ starting with vðM0 T0 Þ. Still the time instance M0 T0 is in the future compared to t ¼ 0. Consequently, a system realizing (2.69) would be noncausal. Nevertheless (2.69) is very important as a mathematical model. For an implementation, causality can be obtained by replacing M0 by M0 M . Introducing both L0 and M0 to a system for SRC by a rational factor, (2.43) changes to

yM0 L0 ðmT2 Þ ¼ v 0 ðmM þ M0 ÞT0

¼ v ðmM þ M0 L0 ÞT0

ð2:70Þ

Thus, the L-times upsampled and filtered signal vðnT0 Þ is downsampled with a phase offset ðM0 L0 Þ. Causality is maintained for L0 M0 . The signal y ðmT2 Þ can be expressed with (2.41) and (2.70) T1 x ðkT1 Þ h ðmM þ M0 L0 kL Þ yM0 L0 ðmT2 Þ ¼ L k ¼ 1 1 X ðmM þ M0 L0 Þðmod LÞ ¼ x ðkT1 Þ h T1 L k ¼ 1 mM þ M0 L0 k þ L 1 X mM þ M0 L0 x ¼ n T1 L n ¼ 1 lm h T1 n þ ð2:71Þ L 1 X

with

mM þ M0 L0 k n¼ L mM þ M0 L0 L lm ¼ mM þ M0 L0 L ¼ ðmM þ M0 L0 Þðmod LÞ

ð2:72Þ

ð2:73Þ

The processes of upsampling, filtering, and downsampling with offsets L0 and M0 , respectively, are sketched in Figure 2.12.

54


Figure 2.12 Upsampling with offset L0 , filtering, and downsampling with offset M0 .


55

Although it seems as if L0 and M0 do not have great importance, it will be seen in Chapters 4, 5, and 6 that they are design parameters that can be used to simplify the implementation of filters for SRC.

2.2.6 Polyphase Decomposition Up- and downsampling of discrete-time signals are closely related to the polyphase concept. Multiplying the discrete-time signal x ðkT Þ with N impulse trains ZKr ðn; N Þ having different phase-offsets yields the so-called polyphase components of the signal x ðkT Þ:

½P xl ðkT Þ ¼ xðkT Þ ZKr ðk lÞ; N ;

l ¼ 0; 1; . . . ; N 1

ð2:74Þ

Although the polyphase components are always related to the number of phases N , this N is omitted in the notation because it should always be clear from the context. The original signal xðkT Þ can be composed from its polyphase components as follows: xðkT Þ ¼

N 1 X

½P

xl ðkT Þ

ð2:75Þ

xðkT Þ ZKr ðk lÞ; N

ð2:76Þ

l¼0

¼

N 1 X l¼0

¼

N 1 X

1 X

xðkT Þ

¼ xðkT Þ

dKr ðk nN lÞ

n ¼ 1

l¼0 1 X

dKr ðk nÞ ¼ xðkT Þ

n ¼ 1

Equations (2.75) and (2.76) are called the polyphase representation [12] or polyphase decomposition [13] of xðkT Þ. The z-transform of the polyphase components (or, the polyphase components of the z-transform) of xðkT Þ can be derived by inserting (2.76) into the z-transform of xðkT Þ, which yields (2.77). By definition (see (2.57)), ZKr ðn l; N Þ is different from zero only at defined points n ¼ kN þ l, which yields (2.78). In equivalence to (2.75), the addends of (2.79) are called the polyphase components of the z-transform of xðkT Þ.

56


X ðzÞ ¼

1 X

xðnT Þ z n

n ¼ 1

¼

1 N 1 X X

xðnT Þ ZKr ðn l; N Þ z n

ð2:77Þ

n ¼ 1 l ¼ 0

¼

N 1 X

1 X

x ðkN þ lÞT z ðkN þlÞ

ð2:78Þ

l ¼ 0 k ¼ 1

¼

N 1 X

z l Xl z N

ð2:79Þ

l¼0

¼

N 1 X

½P

Xl ðzÞ

ð2:80Þ

l¼0

with the polyphase components 1 X

½P Xl ðzÞ ¼ z l Xl z N ¼ z l x ðnN þ lÞT z nN

ð2:81Þ

n ¼ 1

The signals

½P xl ðk NT Þ :¼ x ðkN þ lÞT ¼ xl ðkN þ lÞT

ð2:82Þ

are the time-shifted and downsampled polyphase components of xðkT Þ. In order to clearly distinguish them from the ‘‘ordinary’’ polyphase components, ½P xl ðkT Þ of (2.74), they are termed the normalized polyphase components. Originally introduced in [14], the notion polyphase component is understood differently by later authors. Here the approach of [12] is used. In addition to that, in this book we strictly distinguish between polyphase components and normalized polyphase components. Many authors, including [3, 15], simply regard the normalized polyphase components as polyphase components. Having decomposed xðkT Þ into its (normalized) polyphase components, it is also useful to (re)compose a signal from polyphase components. To derive a specification of how to compose xðkT Þ from its polyphase components, it is useful to recall (2.35), which says k N þ ðkÞðmod N Þ k; N 2 Z k¼ N


57

Thus, with (2.82) the signal xðkT Þ can be described by its (normalized) polyphase components as k xðkT Þ ¼ x N þ ðkÞðmod N Þ T N k NT ¼ xðkÞðmod N Þ N

ð2:83Þ

½P

¼ xðkÞðmod N Þ ðkT Þ

ð2:84Þ

This directly reflects the behavior of a rotating switch (commutator). It is very often used for describing signal processing with polyphase components and will be employed in the following chapters. Typical network structures for polyphase decomposition (also known as analysis) and polyphase recomposition (also known as synthesis) are sketched in Figure 2.13. Different types of the polyphase decomposition are known from the literature. The one introduced here is commonly called the Type 1 polyphase decomposition. For further details, the reader is referred to [12, 16]. The following relationship between the polyphase components in time- and transform-domain can be concluded by inspection of (2.74), (2.77) and (2.79): n o

½P ½P Z xl ðkT Þ ¼ Xl ðzÞ ¼ z l Xl z N ;

l ¼ 0; 1; . . . ; N 1

ð2:85Þ

From (2.81) and (2.82) it follows: Zfxl ðk NT Þg ¼ Xl ðzÞ;

l ¼ 0; 1; . . . ; N 1

ð2:86Þ

The N polyphase components as well as the normalized polyphase components can be combined in a vector. By definition it is X ½P ðzÞ :¼ X0 ðz N Þ z 1 X1 ðz N Þ . . . XðzÞ :¼ ½ X0 ðzÞ

X1 ðzÞ

...

z ðN 1Þ XN 1 ðz N Þ

XN 1 ðzÞ T

T

ð2:87Þ ð2:88Þ

Especially, the latter will be of great importance in the context of linear periodically time-varying systems (see Section 2.3). The relationship between the two vectors is given by a simple matrix equation:

X z N ¼ LðzÞ X ½P ðzÞ

ð2:89Þ

58


Figure 2.13 Typical structures for (a) polyphase decomposition and (b) polyphase recomposition.

with 2

1 60 6 6 LðzÞ ¼ 6 0 6 .. 4. 0 2.2.6.1

0 0 z 0 0 z2 .. . 0

0

... ... ..

0 0 0 .. .

. . . . z N 1

3 7 7 7 7 7 5

ð2:90Þ

A Note on Impulse Responses of Filters

From system theory, it is known that impulse responses of filters can be treated as signals. Consequently, impulse responses of filters also can be decomposed to their polyphase components. It does not matter if the impulse response is infinite length impulse response (IIR) or FIR. With respect to system analysis, the polyphase components of an impulse response and its z-transform can be expressed by means of the statespace matrices describing the linear time-invariant system S ðA; B; C ; D Þ with one input and one output


zðk þ 1Þ ¼ AzðkÞ þ BxðkT Þ

59

ð2:91Þ

yðkT Þ ¼ C zðkÞ þ DxðkT Þ

The normalized polyphase components hl ðkT Þ of the impulse response of a linear time-invariant system S are: n D if k ¼ 0 and l ¼ 0 hl ðk NT Þ ¼ ð2:92Þ kN þl1 B else CA while its z-transform is: l

Hl ðzÞ ¼ CA zI A

N 1 N 1 A

Bþ

D CA l1 B

l¼0 l ¼ 1;2;. . .; N 1 ð2:93Þ

Equation (2.93) gives the normalized polyphase components Hl ðzÞ of H ðzÞ. ½P In order to derive the polyphase components Hl ðzÞ, (2.85) can be used ½P resulting in Hl ðzÞ ¼ z l Hl ðz N Þ. For the derivation of (2.92) and (2.93), see the Appendix at the end of the book. Example 2.1

To illustrate (2.92) and (2.93), the N ¼ 4 polyphase components of the impulse response of the filter shown in Figure 2.14 are calculated. First, the state-space equations are derived by inspection: zðk þ 1Þ ¼ 0:5 zðkÞ þ xðkT Þ yðkT Þ ¼ zðkÞ The system has one state. The system matrices are A ¼ 0:5

B¼1

C ¼1

D¼0

Figure 2.14 A simple recursive filter.

60


With (2.92) it is

h0 ðk 4T Þ ¼

8 > >
> : h3 ðk 4T Þ ¼ ð0:5Þ4kþ3

and with (2.93) it is ð0:5Þ3 z 0:54 0:5 z H2 ðzÞ ¼ z 0:54

H0 ðzÞ ¼

z z 0:54 0:52 z H3 ðzÞ ¼ z 0:54 H1 ðzÞ ¼

The reader may substitute the Hl ðzÞ to (2.79) for a proof.

2.2.7 Modulation Representation The modulation representation of a signal [12] is introduced here and used in subsequent sections. By multiplying the independent variable z with the number k

wNk ¼ e j2p N

ð2:94Þ

the modulation of a z-transform X ðzÞ is accomplished leading to the modulated z-transform

½M X k ðzÞ :¼ X z wNk ;

k ¼ 0; 1; . . . ; N 1

ð2:95Þ

Considering the modulated z-transform in the frequency domain of the DTFT by substituting z ! e j2pfT reveals that (2.95) corresponds to a frequency shift of 2pk=N on the normalized frequency axis, which is what usually is understood by the notion modulation. All N modulation components of a signal can be combined in a vector that is defined as: h X ½M ðzÞ :¼ X0½M ðzÞ

½M

X1 ðzÞ

½M

. . . XN 1 ðzÞ

iT

ð2:96Þ

As in the case of the polyphase components, the N representing the number of components is omitted in the notation because it should always be clear from the context. The [M] indicates that these components are the modulation


61

components and should not be confused with the upsampling factor M which is widely used in this book. The importance of the modulation components stems from their relationship to the polyphase components. Thus, it is possible to relate the characteristics of individual polyphase components to the signal to which they belong and its modulated versions. This relationship is derived in the Appendix (see Section A.2). It is X ½P ðzÞ ¼

1 WN X ½M ðzÞ N

ð2:97Þ

where WN is the DFT matrix, which is also given in the Appendix (see Section A.2). Inspecting (2.97) and evaluating the first component of X ½P ðzÞ yields the well-known aliasing property of (2.65). Combining (2.89) and (2.97) yields a relationship between the normalized polyphase components and the modulation components of a signal:

1 X z N ¼ LðzÞ WN X ½M ðzÞ N

ð2:98Þ

2.2.8 Summary Based upon an analog interpretation, a discrete-time representation of resampling after filtering has been given by (2.11). Subsequently, the basic characteristics of this equation (and the filter h as a main part of it) have been derived for different types of SRC. While irrational factor SRC requires the complete continuous-time impulse response of h to be available, rational and integer factor SRC can be realized on periodically time-varying systems with only a limited set of samples of the impulse response of h required. Finally, the polyphase concept has been introduced. A representation of the polyphase components of the impulse response of an LTI system and its z-transform was derived.

2.3 Linear Periodically Time-Varying Systems and Block Filtering 2.3.1 Introductory Remarks In Sections 2.2.3 and 2.2.4 the periodically time-varying nature of systems for rational and integer factor SRC have been covered as outline only.

62


There are several ways of analyzing linear periodically time-varying (LPTV) systems. One particularly interesting approach is to transform the LPTV system into an equivalent linear time-invariant (LTI) system. The advantage of this method is that the comprehensive apparatus of LTI system theory can be applied to the transformed system. Different approaches have been introduced that transform LPTV systems to LTI systems [17–20]. The basic idea that connects all of these methods is to form blocks of data of the input and the output signals. The size of these blocks is directly related to the periodicity of the LPTV system. The ideas presented in [17] and [20] are based on the so-called lifting isomorphism or raising technique. Because there are strong relationships to the concepts of Section 2.2, the raising technique is introduced next. Although they were introduced to digital signal processing years ago, these techniques have only come to bloom in the field of control theory and its applications. Evident relationships to well-established signal-processing concepts (e.g., polyphase decomposition) have not been revealed. In [21] relationships between LPTV systems, polyphase filtering, and block filtering are explained, while [16] has introduced the relationships between polyphase and modulation descriptions of multirate systems on the basis of block signal processing. However, the relationships between block filtering and the raising technique are not covered. This might be because the basis of the raising technique is the state-space description of systems, which is not widely used in digital signal processing. Therefore, besides being an introduction to the raising technique, this section aims to highlight the relationships between these signal-processing concepts. It will be seen that these are not different concepts but are different formulations of a common signal processing framework. It will be used in subsequent chapters of this book. 2.3.2 The Raising Procedure The raising procedure has been widely applied to the design and analysis of N -periodic controllers. Its application to signal processing has been proposed in [17], but limited to an introduction and a short discussion of its potential. All applications are given with respect to digital control rather than signal processing. Because the raising technique bears great potential for analysis and synthesis of digital signal processing systems for SRC, it is introduced in this section. The raising technique transforms a linear periodically time-varying system of period N to a linear time-invariant system. This technique is generally applicable. However, if not explicitly stated, in this book it will


63

only be applied to single-input single-output (SISO) systems which are raised to MIMO systems, and more specifically N -input N -output systems. A discrete time N -periodically time-varying SISO system is considered. It can be expressed in state-space form: zðk þ 1Þ ¼ Ak zðkÞ þ Bk xðkT Þ yðkT Þ ¼ Ck zðkÞ þ Dk xðkT Þ

ð2:99Þ

where xðkT Þ, yðkT Þ, and zðkÞ are the input signal, the output signal, and the state-vector of the system, respectively, and Ak , Bk , Ck , and Dk are the N -periodically time-varying system matrices. Ak þ iN ¼ Ak Ck þ iN ¼ Ck

Bk þ iN ¼ Bk Dk þ iN ¼ Dk

for i 2 Z

ð2:100Þ

The following raised vectors are defined:1 xðk NT Þ :¼ xðkNT Þ

x ðkN þ 1ÞT

...

T x ðkN þ N 1ÞT ð2:101Þ

yðk NT Þ :¼ yðkNT Þ

y ðkN þ 1ÞT

T . . . y ðkN þ N 1ÞT ð2:102Þ

which are composed by the normalized polyphase components xl ðk NT Þ and yl ðk NT Þ of the signals xðkT Þ and yðkT Þ, respectively (see (2.82)). Thus, xðk NT Þ ¼ ½ x0 ðk NT Þ x1 ðk NT Þ

xN 1 ðk NT Þ T

ð2:103Þ

yðk NT Þ ¼ ½ y0 ðk NT Þ y1 ðk NT Þ yN 1 ðk NT Þ T ð2:104Þ

The response of the N -periodic system S Ak ; Bk ; Ck ; Dk can then be described by the raised system Sð A; B; C; DÞ which is clearly time invariant: zðk þ 1Þ ¼ A zðkÞ þ B xðk NT Þ yðk NT Þ ¼ C zðkÞ þ D xðk NT Þ

ð2:105Þ

where 1. All matrices and vectors related to the raised formulation of the system are printed in bold face.

64


zðkÞ :¼ zðkN Þ

ð2:106Þ

is the raised state vector. The raised system matrices are given by: A ¼ AN 1 AN 2 A0

ð2:107Þ

B ¼ ½AN 1 . . . A1 B0 ; AN 1 . . . A2 B1 ; . . . ; AN 1 BN 2 ; BN 1 ð2:108Þ 3 2 C0 7 6 C1 A0 7 6 7 6 C2 A1 A0 ð2:109Þ C¼6 7 7 6 .. 5 4 . CN 1 AN 2 . . . A0 3 2 D0 0 ... 0 6 D1 ... 0 7 C1 B0 7 6 6 C2 B1 0 7 C2 A1 B0 D¼6 7 6 .. 7 .. .. 4 . 5 . . CN 1 AN 2 . . . A1 B0 CN 1 AN 2 . . . A2 B1 . . . DN 1 ð2:110Þ An outline of a proof of (2.105) to (2.110) is given in the Appendix (Section A.3). The transfer-matrix HðzÞ establishes the well-known relationships between the z-transforms of the input and the output signal of an LTI system in the transform domain YðzÞ ¼ HðzÞ XðzÞ

ð2:111Þ

HðzÞ ¼ CðzI A Þ1 B þ D

ð2:112Þ

HðzÞ is given as

The components of the raised input and output signals of the system are the normalized polyphase components of the respective signals. Their z-transform is given by (2.88): XðzÞ ¼ ½ X0 ðzÞ

X1 ðzÞ

XN 1 ðzÞ T

YðzÞ ¼ ½ Y0 ðzÞ

Y1 ðzÞ

YN 1 ðzÞ T

ð2:113Þ


65

Linearity is not touched by the raising procedure. The great advantage is that the large apparatus of tools and methods available for time-invariant systems can be applied to the raised system and, thus, indirectly also to the LPTV system. In [17] it is mentioned that raised systems may behave ‘‘nicely’’ in the time domain. However, when implemented sequentially as an LPTV system, ‘‘intra-period oscillations and sensitivity to high frequency modeling errors’’ must be expected. As will be seen, the intra-period behavior is simply a result from aliasing being caused by the periodicity of the LPTV system. Detailed characteristics of the relation between LPTV systems and their LTI counterpart can be found in [17] and [22]. In particular, the latter deals with the problem of how to realize LPTV systems from a given transfer matrix of an LTI MIMO system. Example 2.2

To illustrate the raising concept, a simple periodically time-varying system is raised in the following. A digital mixer is a periodically timevarying system. It is just a multiplier that multiplies the signal with the sample sequence of a sine or cosine function. If the frequency of this sine function is exactly a quarter of the sample rate this sequence is f0; 1; 0; 1g. The period of the system is N ¼ 4. The system is shown in Figure 2.15. The system has no states. Thus, 8k : Ak ¼ 0; Bk ¼ 0; Ck ¼ 0 D0 ¼ 0; D1 ¼ 1; D2 ¼ 0; D3 ¼ 1 8k : Dk ¼ Dk þ 4 With (2.107) to (2.110) it is 2

A ¼ 0;

B ¼ 0;

Figure 2.15 Simple digital mixer.

C ¼ 0;

0 4 C¼ 0 0 0

0 1 0 0

0 0 0 0

3 0 0 5 0 1

66


and with (2.112) 2

0 4 HðzÞ ¼ D ¼ 0 0 0

0 1 0 0

0 0 0 0

3 0 0 5 0 1

This is a transfer matrix of a linear time-invariant system with four inputs and four outputs. 2.3.2.1

Raising a Linear Time-Invariant System

Of paramount importance is the fact that any time-invariant system is periodic with arbitrary period. This trivial fact enables us to raise time-invariant systems in addition to the periodically time-varying ones. The raised transfer matrix of a linear time-invariant system is (see the Appendix, Section A.3) HðzÞ ¼ CðzI A Þ1 B þ D 2 H0 ðzÞ z 1 HN 1 ðzÞ 6 H ðzÞ H0 ðzÞ 1 6 ¼6 .. .. 6 4 . . HN 1 ðzÞ

HN 2 ðzÞ

3

z 1 HN 2 ðzÞ

. . . z 1 H1 ðzÞ

z 1 HN 1 ðzÞ .. .

. . . z 1 H2 ðzÞ 7 7 7 .. 7 5 .

HN 3 ðzÞ

...

H0 ðzÞ ð2:114Þ

where Hl ðzÞ are the z-transforms of the normalized (type 1) polyphase components of the time-invariant transfer function H ðzÞ given in (2.93). The structure of HðzÞ in (2.114) is called (type 1) pseudocirculant.2 It is also called the polyphase matrix of the LTI SISO system S ðA; B; C ; D Þ having the transfer function H ðzÞ. Example 2.3

The system of Example 2.1 is linear time invariant. It should be raised to N ¼ 4. It should be noted that the system matrices are time invariant. Thus, 8k : Ak ¼ A ¼ 0:5; Bk ¼ B ¼ 1; Ck ¼ C ¼ 1;

Dk ¼ D ¼ 0

2. In [15], a pseudocirculant matrix is defined as a circulant matrix [23] with the additional feature that the elements below the main diagonal are multiplied by z 1 . However, this additional feature depends on the type of polyphase description. An exact description can be found in [16]. In type 1 polyphase pseudocirculant matrices, the elements above the main diagonal are multiplied by z 1 .


67

With (2.107) to (2.110) it is A ¼ A 4 ¼ 0:54 B ¼ ½A 3 B; A 2 B; AB; B ¼ ½0:53 ; 0:52 ; 0:5; 1 3 2 3 3 2 2 1 D 0 0 0 C 6 CA 7 6 0:5 7 6 CB D 0 07 7 6 7 7 6 6 D ¼ C ¼6 27¼ 6 7 7 6 4 CA 5 4 0:52 5 4 CAB CB D 05 2

CA 3

0:53 3 0 0 0 0 0 07 7 7 1 0 05

0 6 1 6 ¼6 4 0:5 0:52

0:5

CA 2 B

CAB

CB

D

1 0

and with (2.112) 2 HðzÞ ¼

3

1

6 0:5 7 1 7 6 7 ½0:53 ; 0:52 ; 0:5; 1 6 z 0:54 4 0:52 5 2

0:53 0 0

6 1 6 þ6 4 0:5 2

0:5 2

3

0

0

0

0

1

0

07 7 7 05

0:5 1

0:53 6 z 1 6 ¼ 6 4 z 0:5 4 0:5z 0:52 z

0

0:52 0:53 z 0:5z

0:5 0:52 0:53 z

3 1 0:5 7 7 2 7 0:5 5 0:53

This result can also be obtained by substituting the result of Example 2.1 into (2.114). In conclusion, it is noted that a (type 1) pseudocirculant transfer matrix can always be implemented as a SISO system having the following transfer function: H ðzÞ ¼

N 1 X l¼0

z l Hl z N

ð2:115Þ

68


Because any LTI SISO system results in a raised MIMO system with a pseudocirculant transfer matrix, it can further be concluded that a given transfer matrix HðzÞ can be reduced to an LTI SISO system if and only if HðzÞ is pseudocirculant. This has also been concluded in [21]. With the possibility of raising LTI systems to any period N , it is possible to analyze systems that comprise LPTV and LTI subsystems on the basis of these subsystems. The subsystems can be raised individually and be combined to yield the transfer characteristics of the complete system. An example will follow after the discussion of multirate systems in the context of block signal processing (Section 2.3.3). 2.3.2.2

Raising Multirate Systems

Generalizing the results of the previous sections, linear periodically timevarying systems with signals of different sample rates at the input and output are investigated. Initially, the system depicted in Figure 2.7 is considered. With respect to the intermediate period T0 , (2.37) can be rewritten as y ðm MT0 Þ ¼

1 X

x ðk LT0 Þ h T0 ðmM kLÞ

ð2:116Þ

k ¼ 1

Obviously, this system is not time invariant for L 6¼ M . A shift of the input signal by one sample does not cause a shift of the output signal by one sample. However, a shift of the input signal by M samples causes a shift of the output signal by L samples. More generally, such systems have been named ðL; M Þ shift invariant [24], or just ðL; M Þ systems [25], where L and M are relative prime (i.e., the greatest common divisor of L and M is one). Thus, only after a period of time of MT1 ¼ MLT0 ¼ LT2 , the behavior of the SRC system is the same again. It can be concluded that the system exhibits a generalized periodicity with the period N ¼ LM with respect to the intermediate sample rate 1=T0 . Thus, aiming for a timeinvariant description, multirate systems of the type shown in Figure 2.7 can be raised to the period N ¼ LM

ð2:117Þ

From this periodicity, it is obvious that systems for irrational factor SRC cannot be raised to a finite period. The description of the raised version of the filter h can be found by applying (2.105) to (2.110). Special care must be given to the specific


69

definition of the raised input and output signals of the filter. These signals are related by the upsampling and downsampling processes at the input and the output, respectively, of the complete system. Here, the generalized upand downsampling with a phase offset should be applied, as introduced in Section 2.2.5. The raised input and output signals of the filter can be defined as: 2 3 w 0 ðmLMT0 Þ

6 7 w 0 ðmLM þ 1ÞT0 6 7 ð2:118Þ w ðm LMT0 Þ ¼ 6 7 .. 4 5 .

w 0 ðmLM þ LM 1ÞT0 3 2 v 0 ðmLMT0 Þ

7 6 v 0 ðmLM þ 1ÞT0 7 6 ð2:119Þ vðm LMT0 Þ ¼ 6 7 .. 5 4 .

v 0 ðmLM þ LM 1ÞT0 with the elements of w ðmLMT0 Þ being defined by (2.66).

w 0 ðmLM þ iÞT0 ¼

x ðmM þ kÞT1 i ¼ L0 þ kL; k ¼ 0; 1; . . . ; M 1 0

ð2:120Þ

else

The output signal of the whole system is given by certain components of the raised output signal of the filter

y ðmL þ kÞT2 ¼ v 0 ðmLM þ M0 þ kM ÞT0 ; k ¼ 0; 1; . . . ; L 1 ð2:121Þ It should be noted that if the filter h is periodically time varying itself, the system might be ðLJ ; MJ Þ shift invariant, with J 2 Nþ relating to the periodicity of the filter processing the L-times upsampled input signal. This is dealt with in Section 2.3.3, where the somewhat complicated description of (2.120) and (2.121) will also be simplified. 2.3.2.3

Frequency Behavior of Raised Systems

The frequency behavior of a discrete-time system can be analyzed by evaluating the z-transform of the transfer matrix. This applies also to raised systems. Equation (2.111) can serve as a starting point. Because the frequency behavior is of interest, the modulation representation of the input and output signals

70


of the system is advantageous (see Section 2.2.7). A relationship between the modulation and the normalized polyphase representation of a signal has been given in (2.98). Exploiting this, the input–output relationship of the raised system with respect to the modulation components of the signals can be expressed by means of two equivalence transforms of Hðz N Þ. The latter is obtained by substituting z ! z N in HðzÞ.

Y ½M ðzÞ ¼ WN1 L1 ðzÞ H z N LðzÞ WN X ½M ðzÞ |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}

ð2:122Þ

H ½M ðzÞ

The DFT-matrix WN and its inverse are given in the Appendix (see Section A.2). LðzÞ is given in (2.90). Its inverse exists for z 6¼ 0. Equation (2.122) already suggests to write Y ½M ðzÞ ¼ H ½M ðzÞ X ½M ðzÞ

ð2:123Þ

with

H ½M ðzÞ ¼ WN1 L1 ðzÞ H z N LðzÞ WN

ð2:124Þ

The N N matrix H ½M ðzÞ is called the alias component matrix or modulation matrix. The notion alias component matrix should be clear from the following considerations. Because the frequency behavior is of interest, (2.123) can be evaluated on the unit circle:

Y ½M e j2pfT ¼ H ½M e j2pfT X ½M e j2pfT ð2:125Þ

½M Usually only one component of Y ½M e j2pfT , namely Y0 e j2pfT —the baseband component—is of interest. It is ½M j2pfT

Y0

e

¼

N 1 X

½M j2pfT

½M H0;i e j2pfT Xi e

ð2:126Þ

i ½M H0;i e j2pfT X e j2pðfT N Þ

ð2:127Þ

i¼0

¼

N 1 X i¼0

Comparison of (2.127) and (2.65) reveals that the spectrum of the output signal of an N -periodic linear time-varying system equals a sum of frequency-shifted spectra of the original signal, as in the case of decimation. A difference to decimation is that the sum comprises the frequency-shifted spectra of the input signal, where each can be regarded as being filtered by


71

a different filter. The transfer functions of these filters are the components of the alias component matrix. They determine the amount of aliasing due to the superposition of the different spectra. The derivation of the frequency characteristics of multirate systems— substituting the signals XðzÞ and YðzÞ by the signals defined in (2.118) to (2.121)—in conjunction with a further generalization of ðL; M Þ-shiftinvariant systems to ðLJ ; MJ Þ-shift-invariant systems ( J 2 Nþ ) will be dealt with in the next section. In case the underlying SISO system is LTI there is no source of aliasing. Therefore H ½M ðzÞ should be expected to have a diagonal structure. This is true because a pseudocirculant matrix (with z ! z N ) is similar to a circulant matrix with the similarity-transform matrix LðzÞ [21], with all matrices having the order N . Any circulant matrix of order N , however, is similar to a diagonal matrix with the similarity-transform matrix WN [23, 25]. Hence (2.124) yields a diagonal matrix if and only if HðzÞ is pseudocirculant. However, the pseudocirculant nature of HðzÞ has been derived to be a necessary and sufficient characteristic of the raised transfer matrix of an underlying LTI SISO system.

2.3.3 Block-Filtering and Multirate Systems The block-filtering concept [21, 24] is introduced here in order to unveil its strong relationships to multirate filtering and the raising technique. Special emphasis will be given to the block-filtering description of ðLJ ; MJ Þ-shiftinvariant filters that has been presented in [24]. Still, here it will be derived on the basis of the raised ðLJ ; MJ Þ-shift-invariant system. Block versions of the input and output signals of a SISO system (as given in (2.103) and (2.104)) are defined. While the so far scalar transfer function H ðzÞ has related the scalar input signal to the scalar output signal, the block versions are related by the transfer matrix H½B ðzÞ. However, the size of the block input and output vectors do not necessarily have to be the same. This is the case if an ðLJ ; MJ Þ-shift-invariant system ðJ 2 Nþ ) is described by its block version with the input signal block length MJ and the output signal block length LJ . Such a description is very graphical: The system generates LJ output samples from MJ input samples. This is exactly what an SRC system does. While L and M are the up- and downsampling factors, respectively, J 2 Nþ represents a possible periodicity of the filter h itself. It is introduced for reasons of generality. The system of Figure 2.7 employs a filter that is not time varying. Thus, for this system, J ¼ 1 holds. However, because any

72


ðL; M Þ-shift-invariant system is also ðLJ ; MJ Þ shift invariant, J can also be used as a design parameter for block implementations, if an integer increase of the block length is the issue. The raising procedure as defined here is only valid for LPTV single-rate systems (i.e., the input sample rate equals the output sample rate). Therefore, it was necessary to relate the signals with different sample rates to the raised single-rate system by means of (2.118) to (2.121). This can be simplified by generalizing to ðLJ ; MJ Þ-shift-invariant systems. A compact description can be achieved by means of rewriting (2.118) to (2.121) in matrix form. A downsampler will take LMJ samples and just pass every M th sample, thus letting LJ samples pass. In contrast to that, an upsampler takes MJ samples and introduces L 1 zero samples between each of the input samples, thus delivering LMJ samples at its output. Hence, the respective matrices describing these processes must have the dimensions LJ LMJ and LMJ MJ , respectively. Downsampling is achieved by taking a vector of length LMJ and multiplying it from the left with the downsampling matrix, which yields a vector of length LJ . Upsampling works equivalently. As both up- and downsamplers are memoryless, their transfer matrices describe a simple mapping of elements of the input vector to elements of the output vector. By inspection the up- and downsampling matrices H½up and H½down can be found: ½up HðLMJ ;MJ Þ ¼ li;k ð2:128Þ with li;k ¼

n

1 0

k ¼ 0; 1; . . . ; MJ 1; i ¼ L0 þ kL; 0 L0 < L else ð2:129Þ

where L is the upsampling factor, L0 the number of zeros inserted before the first signal element (i.e., the upsampling offset), and MJ the size of the input signal vector. h i ½down ð2:130Þ HðLJ ;LMJ Þ ¼ mi;k with mi;k ¼

n

1 0

i ¼ 0; 1; . . . ; LJ 1; k ¼ M0 þ iM ; 0 M0 < M else ð2:131Þ

where M is the downsampling factor, M0 the offset of the downsampling process, and LJ the size of the output signal vector. A similar approach was proposed in [16], still without the offsets L0 and M0 .


73

In order to describe the complete ðLJ ; MJ Þ-shift-invariant SISO system in its block form, the following MJ - and LJ -raised vectors are defined: 2 6 6 6 xðMJ Þ ðm MJT1 Þ :¼ 6 6 4

xðmMJT1 Þ

x ðmJM þ 1ÞT1 .. .

x ðmJM þ MJ 1ÞT1 3 2 yðmLJT2 Þ 6 y ðmLJ þ 1ÞT 7 7 6 2 7 6 y ðLJ Þ ðm LJT2 Þ :¼ 6 7 .. 7 6 . 5 4

y ðmLJ þ LJ 1ÞT2

3 7 7 7 7 7 5

ð2:132Þ

ð2:133Þ

which are the block input and output signals, respectively, of the MJ -input LJ -output block system. It should be noted that MT1 ¼ LT2 . Eventually, the ðLJ ; MJ Þ-shift-invariant SISO system can be described by the raised LMJ -input LMJ -output version of the system with the MJ - and LJ -raised vectors as input and output signals, respectively: zðk þ 1Þ ¼ A zðkÞ þ B H½up xðMJ Þ ðk MJT1 Þ h i y ðLJ Þ ðk LJT2 Þ ¼ H½down C zðkÞ þ D H½up xðMJ Þ ðk MJT1 Þ ð2:134Þ This is equivalent to

y ðLJ Þ

~ zðkÞ þ B ~ xðMJ Þ ðk MJT1 Þ zðk þ 1Þ ¼ A ~ zðkÞ þ D ~ xðMJ Þ ðk MJT1 Þ ðk LJT2 Þ ¼ C

ð2:135Þ

with ~¼A ~ ¼ B H½up A B ½down ~ ~ ¼ H½down D H½up C D C¼H

ð2:136Þ

Finally we get the block-transfer matrix of the system as H½B ðzÞ ¼ H½down HðzÞ H½up

ð2:137Þ

74


The description of an ðLJ ; MJ Þ-shift-invariant SISO system on the basis of block processing and the raising procedure is sketched in Figure 2.16. It should be noted that the signal flow is from right to left. This shows nicely how the ðLJ ; MJ Þ-shift-invariant SISO system is composed from an ðLJ ; 1Þshift-invariant system, a (1,1)-shift-invariant system, and a ð1; MJ Þ-shiftinvariant system. Moreover it can be realized that the concept of analyzing general ðLJ ; MJ Þ-shift-invariant systems on the basis of time-invariant blockprocessing systems is basically a shift of the time-varying characteristics into two well-known time-varying systems (polyphase decomposition and recomposition) at the input and the output of a time-invariant block system. Example 2.4

As an example, the decimator of Figure 2.10 should be analyzed using the block-filtering concept based on the raising technique. Obviously the system is (1,M )-shift-invariant (i.e., L ¼ J ¼ 1). First the raised transfer matrix of the filter is determined using (2.114): 3 2 H0 ðzÞ z 1 HM 1 ðzÞ z 1 HM 2 ðzÞ .. . z 1 H1 ðzÞ 6 H1 ðzÞ H0 ðzÞ z 1 HM 1 ðzÞ .. . z 1 H2 ðzÞ 7 7 6 HðzÞ ¼ 6 7 .. .. .. .. 5 4 . . . . HM 1 ðzÞ

HM 2 ðzÞ

HM 3 ðzÞ

.. .

H0 ðzÞ

Figure 2.16 Illustrating the block-filter concept with up- and downsampling matrices and raised filter (L0 ¼ 2, L ¼ 4, M0 ¼ 1; M ¼ 3).


75

As to the downsampler, we employ (2.130) to (2.131) with M0 ¼ M 1 to obtain the transfer matrix: ½down

Hð1;M Þ ¼ ½ 0

... 0

1

Finally we can combine the transfer matrices of the filter and the downsampler to obtain the transfer matrix of the decimator. Hdecimator ðzÞ ¼ H½down HðzÞ Because the matrix product is not commutative, the sequential order of the two matrices in the product must be observed. Hence, we have Hdecimator ð1;M Þ ðzÞ ¼ ½ HM 1 ðzÞ

HM 2 ðzÞ

HM 3 ðzÞ

...

H0 ðzÞ

Another example will be given in Section 6.2.1. From the example, the following conclusion can be drawn: The block implementation of a system for integer factor SRC that is given by (2.137), with either up- or downsampling (M ¼ 1 or L ¼ 1, respectively) and a linear time-invariant filter given by the pseudocirculant matrix HðzÞ of (2.114), is nothing but the well-known polyphase implementation [3, 12] (see also Chapter 4). Concluding this section, it should be mentioned that the raising technique has been introduced in the literature to simplify the analysis of periodically time-varying systems [17], while block filtering has been introduced for several reasons. Among those are the increased noise immunity [25]3 and reduction in computational effort [24]. Apparently, the raising technique can be used to easily find block descriptions of multirate and single rate filters. We shall employ the raising technique in subsequent sections to derive block-processing structures for SRC.

3. The noise immunity results from the fact that raising an LTI system leads to the eigenvalues of the system matrix A being raised to the same power to which the system, and thus the matrix A, is raised. However, these eigenvalues are the poles of the transfer matrix, which have to be inside the unit circle to ensure stability (jzpoles j < 1). Raising these poles causes them to move further to the center of the unit circle. Thus, quantization effects that originally could have caused the poles to move outside the unit circle, therefore making the system unstable, might not do so anymore.

76


2.3.4 Frequency Behavior of Multirate Systems: The Bifrequency Approach Extending the results obtained in Section 2.3.2, the frequency behavior of multirate systems, precisely ðLJ ; MJ Þ-shift-invariant systems, is investigated in the following section. With respect to SRC, these could be systems employing a J -periodic filter enclosed by an upsampler L " at the input and a downsampler M # at the output. In order to properly write down the equation for the frequency behavior of multirate systems, the vectors XðzÞ, YðzÞ, X ½M ðzÞ, and Y ½M ðzÞ, as well as the matrix LðzÞ are written with an index indicating their dimension. Equivalent to (2.123) and (2.124), the input-output relationship between the modulation components of the input signal and the modulation components of the output signal can be formulated as follows (see also Figure 2.16). With LMJ being the periodicity of the system, and thus LMJ being the block size of the raised version of the system, it is ½B

Y ðLJ Þ ðzÞ ¼ HðLJ ;MJ Þ ðzÞ X ðMJ Þ ðzÞ ½down

ð2:138Þ ½up

¼ HðLJ ;LMJ Þ HðLMJ ;LMJ Þ ðzÞ HðLMJ ;MJ Þ X ðMJ Þ ðzÞ Setting z ! z LMJ yields

½B Y ðLJ Þ z LMJ ¼ HðLJ ;MJ Þ z LMJ X ðMJ Þ z LMJ

ð2:139Þ

ð2:140Þ

which relates the LMJ -times upsampled normalized polyphase components4 of Y ðzÞ and X ðzÞ. With (2.98) it can be written

1 ½M

X ðMJ Þ z LMJ ¼ L z L WMJ X ðMJ Þ z L MJ LMJ

1 ½M L z M WLJ Y ðLJ Þ z M Y ðLJ Þ z ¼ LJ

ð2:141Þ ð2:142Þ

Equations (2.141) and (2.142) relate the upsampled modulation components to the upsampled normalized polyphase components of these signals. Substituting (2.141) and (2.142) to (2.140) yields

1

1 ½M ½B L z M WLJ Y ðLJ Þ z M ¼ HðLJ ;MJ Þ z LMJ L zL LJ MJ ½M

WMJ X ðMJ Þ z L 4. The upsampled polyphase components of a signal X ðzÞ are fundamentally different from the polyphase components of the upsampled signal.


L ½M ½B Y ðLJ Þ z M ¼ WLJ1 L1 z M HðLJ ;MJ Þ z LMJ L z L M ½M

WMJ X ðMJ Þ z L

77

ð2:143Þ

Hence,

½M ½M ½M

Y ðLJ Þ z M ¼ H ðLJ ;MJ Þ ðzÞ X ðMJ Þ z L

ð2:144Þ

with ½M

HðLJ ;MJ Þ ðzÞ ¼

L ½B WLJ1 L1 z M HðLJ ;MJ Þ z LMJ L z L WMJ M ð2:145Þ

and H½B ðzÞ being defined in (2.137). LJ (M -times upsampled) modulation components of Y ðzÞ are obtained from MJ (L-times upsampled) modulation components of X ðzÞ. They are related by H ½M ðzÞ. Note that H ½M ðzÞ is not a square matrix in general, though in the case of the input and the output of the system having the same sample rate (L ¼ M ¼ 1, and J representing the periodicity of the underlying time-varying SISO system), it is a square matrix (see Section 2.3.2). The spectrum of the output signal of the multirate system can be obtained by evaluating the first component of Y ½M ðz M Þ on the unit circle (see (2.126) and (2.127)). It should be noted ½B that by writing HðLJ ;MJ Þ ðz LMJ Þ in (2.145) the variable z is related to the period T0 at which the periodicity LMJ of the system can be observed. Thus,

½M ½M Y e j2pfT2 ¼ Y0 e j2pfT2 ¼ Y0 e j2pfMT0 ¼

MJ 1 X

i ½M H0;i e j2pfT0 X e j2pðfLT0 MJ Þ

ð2:146Þ

i¼0

Equation (2.146) describes the frequency behavior of ðLJ ; MJ Þ-shiftinvariant systems. The summation of different frequency components of the input signal after filtering reflects aliasing. Although (2.146) is not only sufficient but also very nicely describes the frequency behavior of multirate systems, we shall further elaborate this result to derive a relationship to the bifrequency approach that plays an important role in many publications [3, 25]. By exploiting the sifting property of the Dirac impulse (see (1.5))

78


xðt Þ ¼

Z

1

xðtÞdðt tÞd t

1

Equation (2.146) can be rewritten, which yields (2.147). The integral can be split then into an infinite sum of integrals over a finite region (Equation (2.148)). By changing variables F ¼ f 0 þ k=ðLT0 Þ, (2.149) results. By setting l ¼ i þ kMJ , with the equivalence i ¼ ði þ kMJ Þðmod MJ Þ ¼ ðl Þðmod MJ Þ, and by realizing that l covers the range from 1 to þ1, which enables us to combine the two sums into one, (2.150) can be obtained. Rearranging the terms and noting that T1 ¼ LT0 leads to (2.151) and (2.152). 1 X

MJ

½M Y e j2pfT2 ¼ H0;i e j2pfT0 i¼0

Z

1

j2pFLT

0 X e d f

1

¼

MJ 1 X

i F dF LMJT0

ð2:147Þ

½M H0;i e j2pfT0

i¼0 1 X

Z

k ¼ 1

kþ1 LT0

X e

j2pFLT0

k LT0

i F dF d f LMJT0 ð2:148Þ

¼

MJ 1 X

1 X

½M H0;i e j2pfT0

i¼0 k ¼ 1

Z

1 LT0

j2pLT0 f 0 þLTk

X e

0

0

¼

MJ 1 X

1 X

i þ kMJ 0 d f f df 0 LMJT0 ð2:149Þ

½M H0;i e j2pfT0

i¼0 k ¼ 1

Z

1 LT0

X e

j2pLT0 f

0

¼

1 X

0

df

0

¼1

1 LT0

0

!

i þ kMJ e|fflfflfflffl{zfflfflfflffl}0 d f f LMJT0 j2pLT0 LTk

½M H0;ðl Þðmod MJ Þ e j2pfT0

l ¼1

Z

0

X e

j2pf 0 LT0

d f

l f LMJT0

0

df

0

ð2:150Þ


79

Thus, with 0

K ðf ; f Þ ¼

1 X

½M H0;ðl Þðmod MJ Þ

l ¼1

e

j2pf

T1 L

l f d f MJT1

0

ð2:151Þ

it is

Y e j2pfT2 ¼

Z

1 T1

0 K ð f ; f 0 Þ X e j2pf T1 df

0

ð2:152Þ

0

K ð f ; f 0 Þ is the transmission function or the bifrequency system function [3]. It defines the mapping of frequency components of the input signal to frequency components of the output signal. In case of multirate systems with a rational SRC factor, this bifrequency system function has specific characteristics. Graphically, the addends of K ð f ; f 0 Þ form planes that are arranged mutually parallel and orthogonal to the plane being spread by the two frequency axes. The projections of these planes onto the P ðf ; f 0 Þ plane 1 are straight lines that are defined by the Dirac train l ¼1 dð f l =ðMJT1 Þ f 0 Þ. This property has been noted also in [25]. K ð f ; f 0 Þ also reflects the previously mentioned phenomena, aliasing and imaging. This is illustrated in the following example. Example 2.5

The frequency characteristics are derived of a system for SRC by L=M ¼ 2=3 as shown in Figure 2.7. The time-invariant filter h ( J ¼ 1) that is clocked at the high intermediate sample rate T0 is supposed to have the following transfer function: H ðzÞ ¼ 1 þ z 1 It is a simple first-order comb filter, which is used for interpolation often (see Chapter 5), also known as a sample-and-hold device. The four state-space matrices of this filter are: A¼0 C ¼1

B¼1 D¼1

The overall system is periodic with N ¼ LMJ ¼ 6 with respect to the intermediate sample rate T0 . Thus, the filter must be raised at least to N ¼ 6. Because the filter itself is time invariant, the transfer

80


matrix is composed by the polyphase components of H ðzÞ. With (2.93) the polyphase components of H ðzÞ are: H0 ðzÞ ¼ 1 H3 ðzÞ ¼ 0

H1 ðzÞ ¼ 1 H4 ðzÞ ¼ 0

H2 ðzÞ ¼ 0 H5 ðzÞ ¼ 0

Thus, HðzÞ can be derived (see (2.114)), as well as (2.128) and (2.129)) and H½down (see (2.130) and (2.131)) L0 ¼ M0 ¼ 0 : 3 2 1 0 0 0 0 z 1 61 1 0 0 0 0 7 7 6 60 1 1 0 0 0 7 7 6 HðzÞ ¼ 6 7 60 0 1 1 0 0 7 40 0 0 1 1 0 5 0 0 0 0 1 1 2 1 0 60 0 6 % & 60 1 1 0 0 0 0 0 ½down ½up H ¼ H ¼6 60 0 0 0 0 1 0 0 6 40 0 0 0

H½up (see by setting

3 0 07 7 07 7 07 7 15 0

With (2.137) the block-transfer matrix can be calculated: % & 1 0 0 H½B ðzÞ ¼ 0 1 0 Eventually, with (2.145) the alias-component matrix of the system is: #" " #" # 1 0 0 1 0 2 0:5 0:5 ½M H ðzÞ ¼ 3 0:5 0:5 0 z 3 0 1 0 2 32 3 1 1 1 1 0 0 2 4 7 6 76 4 0 z 2 0 54 1 e j 3p e j 3p 5 0

0

" 1 1 þ z 1 ¼ 3 1 z 1

z4

1

4

e j 3p

8

e j 3p

2

1 þ z 1 e j 3p

2

1 z 1 e j 3p

1 þ z 1 e j 3p 1 z 1 e j 3p

4 4

#


81

The components of the first row of H ½M ðzÞ determine the mapping of the input signal to the output signal as indicated in (2.146). Evaluating them on the unit circle reveals that different modulation components of the input signal are filtered by a filter with a cosine-like transfer function having its only transfer zero at half the sample rate (with respect to the intermediate sample rate). Hence, the image component resulting from two-times upsampling is attenuated by this transfer zero, which is the reason that this filter acts as an interpolation filter. Potential aliasing components are not explicitly suppressed. This behavior can be seen from Figure 2.17, where one period of the bifrequency system function (obtained by substituting H ½M ðzÞ to (2.151)) is shown inside the area spread by the input- and the output-signal frequency axis over the interval 0 f < 1=T0 .

Figure 2.17 Bifrequency system function magnitude of a sample rate converter L=M ¼ 2=3.

82


The transfer characteristics of the system of this example at a certain frequency (with respect to the output) equals a superposition of three different signal components of the input signal weighted with three different transfer functions. This is exactly what (2.152) describes. Somewhat simpler is the description of (2.146).

2.3.5 Summary Based on the raising technique, the block-processing concept has been introduced. It can serve as a basis for the understanding of LPTV systems, and thus ðLJ ; MJ Þ-shift-invariant systems, which exhibit a generalized periodicity with period LMJ . Multirate systems (whose different sample rates are related by rational factors) are ðLJ ; MJ Þ-shift-invariant. Hence, the block-processing concept enables the investigation and analysis of multirate systems. Since LTI systems belong to the class of LPTV systems with arbitrary period, the block-processing concept can also be applied to this class of systems. In that case, the transfer matrix has a specific structure called pseudocirculant. The well-known polyphase filter structures for interpolation and decimation can easily be derived from this pseudocirculant transfer matrix by introducing up- and downsampling matrices, which select one column or row of the pseudocirculant transfer matrix, respectively. This is a very plausible approach to polyphase interpolation and decimation filters. The frequency characteristics of block-processing systems can generally be estimated by means of the alias component matrix and the bifrequency system function, which inherently reveal the two phenomena of multirate systems, imaging and aliasing. Finally, it should be noted that the block-processing concept is extensively exploited in the context of filter banks. Some relationships between filter banks and block filtering have been unveiled in [21, 26].

References [1]

Hentschel, T., M. Henker, and G. P. Fettweis, ‘‘The Digital Front-End of Software Radio Terminals,’’ IEEE Personal Communications, Vol. 6, No. 4, August 1999, pp. 40–46.

[2]

Marks II, R. J., Introduction to Shannon Sampling and Interpolation Theory, New York: Springer-Verlag, 1991.

[3]

Crochiere, R. E., and L. R. Rabiner, Multirate Digital Signal processing, Englewood Cliffs, NJ: Prentice-Hall, 1983.


83

[4]

Bronstein, I. N., and K. A. Semendjajew, Teubner-Taschenbuch der Mathematik, Vol. 1, Leipzig, Germany: B.G. Teubner Verlagsgesellschaft, 1996.

[5]

Unser, M., ‘‘Splines, A Perfect Fit for Signal and Image Processing,’’ IEEE Signal Processing Magazine, Vol. 16, No. 6, November 1999, pp. 22–38.

[6]

Henker, M., ‘‘Abtastratenanpassung in Software-programmierbaren Mobilfunkempfa¨ngern,’’ Master’s thesis, Technische Universita¨t Dresden — Fakulta¨t Elektrotechnik — Institut fu¨r Nachrichtentechnik, October 1998.

[7]

Henker, M., and G. Fettweis, ‘‘Extended Algorithms for Sample Rate Conversion,’’ in 2nd Karlsruhe Workshop on Software Radios (WSR-2002), Karlsruhe, Germany, March 2002.

[8]

Schafer, R. W., and L. R. Rabiner, ‘‘A Digital Signal Processing Approach to Interpolation,’’ Proc. of the IEEE, Vol. 61, No. 6, June 1973, pp. 692–702.

[9]

Meyr, H., M. Moeneclaey, and S. A. Fechtel, ‘‘Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, New York: John Wiley & Sons, 1998.

[10]

Menkhoff, A., ‘‘Interpolationsverfahren fu¨r unbeschra¨nkte Abtastratenwandlungen,’’ Ph.D. thesis, Universita¨t Fridericiana Karlsruhe, Fakulta¨t Elektrotechnik, March 1996.

[11]

Hentschel, T., and G. P. Fettweis, ‘‘Sample Rate Conversion for Software Radio,’’ IEEE Communications Magazine, Vol. 38, No. 8, August 2000, pp. 142–150.

[12]

Fliege, N. J., Multirate Digital Signal Processing: Multirate Systems, Filter Banks, Wavelets, New York: John Wiley & Sons, 1994.

[13]

Vaidyanathan, P. P., ‘‘Multirate Digital Filters, Filter Banks, Polyphase Networks, and Applications: A Tutorial,’’ Proc. of the IEEE, Vol. 78, No. 1, January 1990, pp. 56–93.

[14]

Bellanger, M. G., G. Bonnerot, and M. Coudreuse, ‘‘Digital Filtering by Polyphase Network: Application to Sample-Rate Alteration and Filter Banks,’’ IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 24, No. 2, April 1976, pp. 109–114.

[15]

Vaidyanathan, P. P., Multirate Systems and Filter Banks, Englewood Cliifs, NJ: Prentice Hall, 1993.

[16]

Reng R. L., ‘‘Polyphase and Modulation Descriptions of Multirate Systems — A Systematic Approach,‘‘ in International Conference on Digital Signal Processing, Limassol, Cyprus, June 1995, pp. 212–217.

[17]

Feuer, A., and G. C. Goodwin, Sampling in Digital Signal Processing and Control, Boston, MA: Birkhaüser, 1996.

[18]

Misra, P., ‘‘Time-Invariant Representation of Discrete Periodic Systems,’’ Automatica, Vol. 32, No. 2, 1996, 267–272.

[19]

Van Dooren, P., and J. Sreedhar, ‘‘When Is a Periodic Discrete-Time System Equivalent to a Time-Invariant One?,’’ Linear Algebra and its Applications, No. 212/ 213, 1994, pp. 131–151.

84


[20]

Meyer, R. A., and C. S. Burrus, ‘‘A Unified Analysis of Multirate and Periodically Time-Varying Digital Filters,’’ IEEE Transactions on Circuits and Systems, Vol. 22, No. 3, March 1975, pp. 162–168.

[21]

Vaidyanathan, P. P., and S. K. Mitra, ‘‘Polyphase Networks, Block Digital Filtering, LPTV Systems, and Alias-Free QMF Banks: A Unified Approach Based on Pseudocirculants,‘‘ IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 36, No. 3, March 1988, pp. 381–391.

[22]

Colaneri, P., and S. Longhi, ‘‘The Realization Problem for Linear Periodic Systems, Automatica,’’ Vol. 31, No. 5, 1995, pp. 775–779.

[23]

Davis, P. J., Circulant Matrices, New York: Chelsea Publishing, 1994.

[24]

Miyawaki, T., and C. W. Barnes, ‘‘Multirate Recursive Digital Filters — A General Approach and Block Structures,’’ IEEE Transactions on Acoustics, Speech, and Signal Processing, Vol. 31, No. 5, October 1983, pp. 1148–1154.

[25]

Schu¨ler, H. W., Digitale Signalverarbeitung 1, Analyse diskreter Signale und Systeme, Berlin, Germany: Springer-Verlag, 1994.

[26]

Shenoy, R. G., ‘‘Multirate Specifications Via Alias-Component Matrices,’’ IEEE Transactions on Circuits and Systems-II: Analog and Digital Signal Processing, Vol. 45, No. 3, March 1998, pp. 314–320.

3 A Concept for SRC in Software Radio Terminals 3.1 Introduction To enable a comprehensive treatment using SRC, this chapter sets the scene further by bringing together the fundamentals of SRC addressed in Chapter 2 and the concept of software radio covered in Chapter 1. The intention of this chapter is to introduce the idea of separating integer factor and fractional SRC and to discuss briefly the ordering of these two in a cascaded structure realizing rational factor SRC.

3.2 Review of the OSR of a Signal The OSR and the GOSR of a signal have been introduced in Chapter 1. In Section 2.2.1, it has been suggested that with (generalized) oversampled signals, aliasing is acceptable in certain frequency bands. Thus, the OSR/ GOSR is very important in the context of SRC. The GOSR is the ratio between the sample rate fS of the signal and the bandwidth b of the signal of interest (i.e., the region to be kept free from aliasing). In the case where the signal of bandwidth b is ‘‘conventionally’’ oversampled (i.e., there are no signal components outside the bandwidth b), the GOSR equals the OSR. The GOSR after the SRC process directly determines the relative bandwidth (compared to the sample rate) of the potential aliasing components that have to be attenuated by the SRC filter (see Figure 2.5). The higher the GOSR is, the smaller the passband and the stopbands of the filter. Hence, a high GOSR (after SRC) relaxes the design constraints. Simpler filter structures result. A consequence of this is that SRC is 85

86


advantageously implemented on a cascaded multirate system (i.e., a cascade of filters with relaxed requirements at high sample rates, where the GOSR of the signal of interest is high, and strong requirements at low sample rates) [1, 2]. Example 3.1

The requirements on systems for rational factor SRC are compared. GSM is taken as an example air interface. In the first case, the output sample rate of the SRC system should be close to its input sample rate. However, it should be an integer multiple of the bit rate of GSM. In the second case, it should be twice the bit rate. The input sample rate is assumed to be 80 MSps. Table 3.1 lists all of the relevant parameters. Because the filter in a system for SRC should attenuate potential aliasing components, the stopbands of the respective filter are placed at integer multiples of 79:9 MHz or 541:6 kHz, respectively. This is sketched in Figure 3.1. The transition band is much wider in case 1 than in case 2. Thus, a filter fulfilling constraints as to aliasing attenuation in a certain application can be realized with considerably lower effort in case 1, where the GOSR after SRC is significantly higher than in case 2.

3.3 The Conversion Factor: Integer Factor and Fractional SRC In order to reduce the cost of analog components in mobile communications terminals, the ADC should be clocked at a fixed rather than a tunable rate (see Section 1.3.2). Therefore, the conversion between the digitization rate and the variable target rate (symbol/chip rate) should be realized digitally. This process is referred to as SRC. Table 3.1 Parameters of an SRC Example Case 1

Case 2 fS ¼ 80 MSps

Input sample rate

b ¼ 200 kHz

Signal bandwidth Target rate (after SRC)

767 768 0:9987 13295 48 79:9 MSps

1 147:7 541:6 kSps

GOSR after SRC

400

2:7

Conversion factor

13 1920

A Concept for SRC in Software Radio Terminals

87

Figure 3.1 Tolerance scheme of antialiasing filters for SRC according to Table 3.1.

Because both the digitization rate and the target rate can be expressed as an integer number of samples per time unit, the rate change (conversion) factor is a rational number. The ratio of the output sample rate f2 and the input sample rate f1 is f2 T1 L ¼ ¼ M f1 T2

ð3:1Þ

with L and M being relative prime (i.e., the greatest common divisor of L and M is one). It can be distinguished between effective reduction (L < M ) and effective increase (L > M ) of the sample rate. Although they are similar, integer factor SRC and fractional SRC also have differences, especially in implementation. Therefore, it is sensible to separate fractional SRC and integer factor SRC. To do this, the rate-change factor can be factorized to a fractional part ðL=M Þfrac and an integer part Lint (or Mint ): 8L < M frac Lint L ¼ M : L 1 M frac Mint

effective increase of the sample rate ð3:2Þ effective reduction of the sample rate

It should be noted that the fractional factor ðL=M Þfrac is limited to the interval

88


0:5
< P ck;i t k D h½i ðt Þ ¼ k¼0 > : 0

0t > ck;i cosk Dt 0t < k¼0

> > > :

else t o ¼ F cos n ; n ¼ 0; 1; . . . ; n 1 D n

ð4:41Þ

0

ð4:42Þ

Principally, other functions could be used for predistorting the argument t . However, two issues limit the variety of functions that can be used: computational complexity and the ability to approximate the desired impulse response. The cosine/sine functions can be realized with relatively low effort (e.g., by means of a CORDIC [8, 9], which simultaneously calculates the sine and the cosine of an argument). As well as the approximation of a function (impulse response) by polynomials, the approximation by sine and cosine functions is also a wellknown approach (i.e., the Fourier series). It should be noted that it is not necessary to approximate the complete impulse response by a Fourier series, but each of the N pieces can be dealt with separately. Equation (4.41) can be

Systems for Integer Factor SRC

113

regarded as a specific example of a Fourier series representation of one piece of a piecewise impulse response. Because a Fourier series is a periodic function, it is necessary to find a generalization of (4.38) and (4.39) that does not restrict the behavior of the individual pieces h½i ðt Þ outside the interval ½0; DÞ. The following can be used: jt k D ; 0t

Sample Rate Conversion in Software Configurable Radios (Artech House Mobile Communications Series)

Software Defined Radio for 3G (Artech House Mobile Communications Series)

Introduction to Mobile Communications Engineering (Artech House Mobile Communications Library)

Wireless Internet Telecommunications (Artech House Mobile Communications)

Advances in Mobile Information Systems (Artech House Mobile Communications Library)

GSM System Engineering (Artech House Mobile Communications Series)

New Horizons in Mobile and Wireless Communications: Radio Interfaces (Artech House Mobile Communication Series)

New Horizons in Mobile and Wireless Communications: Radio Interfaces (Artech House Mobile Communication Series)

New Horizons in Mobile and Wireless Communications: Reconfigurability (Artech House Mobile Communications)

Advances in 3G Enhanced Technologies for Wireless Communications (Artech House Mobile Communications Series)

Signal Processing Applications in CDMA Communications (Artech House Mobile Communications Series)

New Horizons in Mobile and Wireless Communications: Reconfigurability (Artech House Mobile Communications)

RDS: The Radio Data System (Artech House Mobile Communications Library)

Smart Antenna Engineering (Artech House Mobile Communications Library)

Wireless Technician's Handbook (Artech House Mobile Communications Library) 2nd edition

Wireless Technician's Handbook (Artech House Mobile Communications Library) 2nd edition

CDMA RF System Engineering (Artech House Mobile Communications Library)

Wireless Technician's Handbook 2nd Edition (Artech House Mobile Communications Library)

Wimax Rf Systems Engineering (Artech House Mobile Communications Library)

New Horizons in Mobile and Wireless Communications: Networks, Services and Applications (Artech House Universal Personal Communications)

New Horizons in Mobile and Wireless Communications: Networks, Services and Applications (Artech House Universal Personal Communications)

Radio Engineering for Wireless Communication and Sensor Applications (Artech House Mobile Communications Series)

Radio Engineering for Wireless Communication and Sensor Applications (Artech House Mobile Communications Series)

Radio Engineering for Wireless Communication and Sensor Applications (Artech House Mobile Communications Series)

Interference Analysis and Reduction for Wireless Systems (Artech House Mobile Communications Series.)

Interference Analysis and Reduction for Wireless Systems (Artech House Mobile Communications Series.)

Wireless Communications Security (Artech House Universal Personal Communications)

The Complete Wireless Communications Professional: A Guide for Engineers and Managers (Artech House Mobile Communications Library)

Simulation and Software Radio for Mobile Communications

Communications Receivers: DSP, Software Radios, and Design, 3rd Edition

Resource Allocation in Hierarchical Cellular Systems (Artech House Mobile Communications Library)

Sample Rate Conversion in Software Configurable Radios (Artech House Mobile Communications Series)

Software Defined Radio for 3G (Artech House Mobile Communications Series)

Introduction to Mobile Communications Engineering (Artech House Mobile Communications Library)

Wireless Internet Telecommunications (Artech House Mobile Communications)

Advances in Mobile Information Systems (Artech House Mobile Communications Library)

GSM System Engineering (Artech House Mobile Communications Series)

New Horizons in Mobile and Wireless Communications: Radio Interfaces (Artech House Mobile Communication Series)

New Horizons in Mobile and Wireless Communications: Radio Interfaces (Artech House Mobile Communication Series)

New Horizons in Mobile and Wireless Communications: Reconfigurability (Artech House Mobile Communications)

Advances in 3G Enhanced Technologies for Wireless Communications (Artech House Mobile Communications Series)

Signal Processing Applications in CDMA Communications (Artech House Mobile Communications Series)

New Horizons in Mobile and Wireless Communications: Reconfigurability (Artech House Mobile Communications)

RDS: The Radio Data System (Artech House Mobile Communications Library)

Smart Antenna Engineering (Artech House Mobile Communications Library)

Wireless Technician's Handbook (Artech House Mobile Communications Library) 2nd edition

Wireless Technician's Handbook (Artech House Mobile Communications Library) 2nd edition

CDMA RF System Engineering (Artech House Mobile Communications Library)

Wireless Technician's Handbook 2nd Edition (Artech House Mobile Communications Library)

Wimax Rf Systems Engineering (Artech House Mobile Communications Library)

New Horizons in Mobile and Wireless Communications: Networks, Services and Applications (Artech House Universal Personal Communications)

New Horizons in Mobile and Wireless Communications: Networks, Services and Applications (Artech House Universal Personal Communications)

Radio Engineering for Wireless Communication and Sensor Applications (Artech House Mobile Communications Series)

Radio Engineering for Wireless Communication and Sensor Applications (Artech House Mobile Communications Series)

Radio Engineering for Wireless Communication and Sensor Applications (Artech House Mobile Communications Series)

Interference Analysis and Reduction for Wireless Systems (Artech House Mobile Communications Series.)

Interference Analysis and Reduction for Wireless Systems (Artech House Mobile Communications Series.)

Wireless Communications Security (Artech House Universal Personal Communications)

The Complete Wireless Communications Professional: A Guide for Engineers and Managers (Artech House Mobile Communications Library)

Simulation and Software Radio for Mobile Communications

Communications Receivers: DSP, Software Radios, and Design, 3rd Edition

Resource Allocation in Hierarchical Cellular Systems (Artech House Mobile Communications Library)

Recommend Documents