detection algorithms for wireless communications with applications to wired and storage systems ferrari, colavolpe raheli 2004 10 08 Cấu trúc dữ liệu và giải thuật

Preface xi Acknowledgements xiii List of Figures xv List of Tables xxix 1 Wireless Communication Systems 1 1.1 Introduction 11.2 Overview of Wireless Communication Systems 41.3 Wireless

Detection Algorithms

for

Wireless Communications

To my wife Laura

Giulio Colavolpe

To Annapaola, Enrica and Alberto

Riccardo Raheli

Detection Algorithms

for

Wireless Communications

With Applications to Wired and Storage Systems

Gianluigi Ferrari, Giulio Colavolpe and Riccardo Raheli

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Preface xi Acknowledgements xiii List of Figures xv List of Tables xxix

1 Wireless Communication Systems 1

1.1 Introduction 11.2 Overview of Wireless Communication Systems 41.3 Wireless Channel Models 51.3.1 Additive White Gaussian Noise Channel 61.3.2 Frequency Nonselective Fading Channel 61.3.3 Frequency Selective Fading Channel 81.3.4 Phase-Uncertain Channel: Channel with Phase and

Frequency Instabilities 91.4 Demodulation, Detection, and Parameter Estimation 101.5 Information Theoretic Limits 121.5.1 Additive White Gaussian Noise Channel 121.5.2 Frequency Nonselective Fading Channel 121.5.3 Phase-Uncertain Channel 141.6 Coding and Modulation 151.6.1 Block and Convolutional Coding 151.6.2 Linear Modulation without Memory 161.6.3 Combined Coding and Modulation 171.7 Approaching Shannon Limits: Turbo Codes and Low Density ParityCheck Codes 19

v

1.8 Space Time Coding 201.9 Summary 211.10 Problems 21

2 A General Approach to Statistical Detection for Channels with Memory 25

2.1 Introduction 252.2 Statistical Detection Theory 262.3 Transmission Systems with Memory 322.3.1 Causality and Finite Memory 352.3.2 Stochastic Channels: Channels with Infinite Memory 382.4 Overview of Detection Algorithms for Stochastic Channels 402.5 Summary 432.6 Problems 43

3 Sequence Detection: Algorithms and Applications 49

3.1 Introduction 493.2 MAP Sequence Detection Principle 503.3 Viterbi Algorithm 513.4 Soft-Output Viterbi Algorithm 543.5 Finite Memory Sequence Detection 543.5.1 Inter-Symbol Interference Channel 573.5.2 Flat Slow Fading Channel 583.6 Estimation-Detection Decomposition 593.7 Data-Aided Parameter Estimation 633.8 Joint Detection and Estimation 663.8.1 Phase-Uncertain Channel 673.8.2 Dispersive Slow Fading Channel 693.9 Per-Survivor Processing 713.9.1 Phase-Uncertain Channel 753.9.2 Dispersive Slow Fading Channel 753.9.3 Remarks 753.10 Complexity Reduction Techniques for VA-Based Detection Algorithms 763.10.1 State Reduction by Memory Truncation 773.10.2 State Reduction by Set Partitioning 803.10.3 A Case Study: TCM on an ISI Channel 833.10.4 Reduced-Search Algorithms 873.11 Applications to Wireless Communications 883.11.1 Adaptive Sequence Detection: Preliminaries and Least

Mean Squares Estimation 89

Contents vii

3.11.2 Noncoherent Sequence Detection for Phase-Uncertain

Channels 953.11.3 Noncoherent Sequence Detection for Slowly Varying

Frequency Nonselective Fading Channels Ill3.11.4 Linear Predictive Sequence Detection for Phase-Uncertain

Channels 1243.11.5 Linear Predictive Sequence Detection for Frequency Flat

Fading Channels 1343.11.6 Linear Predictive Sequence Detection for Frequency

Selective Fading Channels 1413.12 Summary 1463.13 Problems 148

4 Symbol Detection: Algorithms and Applications 155

4.1 Introduction 1554.2 MAP Symbol Detection Principle 1564.3 Forward Backward Algorithm 1574.4 Iterative Decoding and Detection 1624.5 Extrinsic Information in Iterative Decoding: a Unified View 1684.5.1 A Review of the Use of the Extrinsic Information 1694.5.2 Forward Backward Algorithm 1724.5.3 Soft-Output Viterbi Algorithm 1784.6 Finite Memory Symbol Detection 1854.7 An Alternative Approach to Finite Memory Symbol Detection 1914.8 State Reduction Techniques for Forward Backward Algorithms 2004.8.1 Forward-Only RS-FB Algorithms 2014.8.2 Examples of Application of Fwd-Only RS-FB Algorithms 2044.8.3 Forward-Only RS FB-type Algorithms 2134.8.4 Examples of Application of Fwd-Only RS FB-type

Algorithms 2164.8.5 Generalized RS-FB Algorithms 2224.8.6 Examples of Application of Generalized RS-FB Algorithms 2374.9 Applications to Wireless Communications 2464.9.1 Noncoherent Iterative Detection of Binary Linear Coded

Modulation 2464.9.2 Noncoherent Iterative Detection of Spectrally Efficient

Linear Coded Modulation 260

4.9.3 Pilot Symbol-Assisted Iterative Detection for

Phase-Uncertain Channels 2724.9.4 Linear Predictive Iterative Detection for Phase-Uncertain

Channels 2854.9.5 Noncoherent Iterative Detection for Slow Frequency

Nonselective Fading Channels 2924.9.6 Linear Predictive Iterative Detection for Fading Channels 2944.10 Summary 2964.11 Problems 297

5 Graph-Based Detection: Algorithms and Applications 301

5.1 Introduction 3015.2 Factor Graphs and the Sum-Product Algorithm 3035.3 Finite Memory Graph-Based Detection 3075.4 Complexity Reduction for Graph-Based Detection Algorithms 3125.5 Strictly Finite Memory: Inter-Symbol Interference Channels 3135.5.1 Factor Graph 3145.5.2 Modified Graph 3185.6 Applications to Wireless Communications 3225.6.1 Noncoherent Graph-Based Detection 3235.6.2 Linear Predictive Graph-Based Detection for Phase-

Uncertain Channels 3235.6.3 Linear Predictive Graph-Based Detection for Frequency Flat

Fading Channels 3265.7 Strong Phase Noise: An Alternative Approach to Graph-BasedDetection 3295.7.1 System Model and Exact Sum-Product Algorithm 3295.7.2 Proposed Algorithms 3335.7.3 Numerical Results 3435.8 Summary 3475.9 Problems 349

A Discretization by Sampling 353

A.I Introduction 353A.2 Continuous-Time Signal Model 353A.2.1 Power Spectrum of a Rayleigh Faded Signal 355A.2.2 Signal Oversampling 359A.2.3 Signal Symbol-Rate Sampling 363A.3 Discrete-Time Signal Model 364

Contents ix

References 367 List of Acronyms 387 Index 391

This book presents, in a unitary and novel perspective, some of the research workthe authors have carried out over the last decade, along with several collaborators andstudents The roots of this book can be traced back to the design of adaptive sequencedetection algorithms for channels with parametric uncertainty The explosion of turbocodes and iterative decoding around the middle of the Nineties has motivated thedesign of iterative (turbo and graph-based) detection algorithms

This book aims at providing the reader with a unified perspective on the design of

detection algorithms for wireless communications What does this statement reallymean? It has become clear to us, in recent years, that most of the proposed detection

algorithms evolve from a simple idea, which can be described as finite-memory tection and synthesized by a simple metric This unique metric is the key ingredient

de-to derive:

• sequence detection algorithms based on the Viterbi algorithm;

• symbol detection algorithms based on the forward-backward algorithm;

• graph-based detection algorithms based on the sum-product algorithm

Although simple, and probably familiar to several researchers working in thisarea, to the best of our knowledge a unified approach to the design of detection al-gorithms, based on a single metric, has never been proposed clearly in the literature.This book tries to address this lack, by giving a comprehensive treatment, with sev-eral examples of application

This book should, however, be interpreted by the reader as a starting point, ratherthan a purely tutorial work In fact, we believe that the proposed simple unifyingidea can find many applications beyond those explored in this book We would like

to mention a single (and significant) example In current and future wireless munication systems, it will be more and more important to support high data-ratetransmissions Multiple-input multiple-output systems, based on the use of multipleantennas, have received significant interest from the research community over the

com-xi

last years All the detection algorithms presented in this book apply to single-inputsingle-output systems The reader is therefore invited to entertain herself/himself bytrying to extend these algorithms to multiple-input multiple-output communicationscenarios.

A final comment is related to the subtitle: "With Applications to Wired and age Systems." As the reader will see, most of the examples presented in this book arerelated to wireless communication systems However, several of the proposed com-munication scenarios apply also to storage and wired systems: for example, properinter-symbol interference channels may characterize several storage systems More-over, the proposed approach is general and, therefore, suitable for application to sce-narios different from those considered explicitly Again, the reader is invited to usethe tools proposed in this book and apply them to solve her/his own communicationproblems

Stor-As an extra resource we have set up a companion website for our book containing

a solutions manual and a sample chapter Also, for those wishing to use this materialfor lecturing purposes, electronic versions of most of the figures from our book areavailable Please go to the following URL and take a look: ftp://ftp.wiley.co.uk/pub/books/ferrari

This book would have never been possible without the help of several people acrossthe years of research, pain and happiness at the University of Parma (and not only).Since it is impossible to thank explicitly all of them, we would like to take this op-portunity to "unitarily" thank all of them, guaranteeing that their help has never beenforgotten, but it is well remembered In particular, we first wish to thank many stu-dents at the University of Parma, who have indirectly contributed to this book withtheir thesis works

Although explicit and comprehensive acknowledgments are impossible, severalresearchers must be explicitly mentioned, for particularly significant and importantcontributions We would like to thank Prof Achilleas Anastasopoulos (University ofMichigan, Ann Arbor, USA), whose collaboration led to some results presented inChapter 4 Moreover, we would also like to thank him for kindly proof-reading theentire manuscript Many thanks go to Prof Keith M Chugg (University of SouthernCalifornia, Los Angeles, USA) and Dr Phunsak Thiennviboon (TrellisWare Tech-nologies Inc., San Diego, USA), for the collaboration (while Gianluigi Ferrari wasvisiting the University of Southern California in 2000-2001) from which some resultspresented in Chapter 4 come from The invaluable collaboration of Prof GiuseppeCaire (Institut Eurecom, Sophia Antipolis, France) in part of Chapter 5 is acknowl-edged This collaboration started while Giulio Colavolpe was visiting the InstitutEurecom in 2000 and has continued and consolidated in the following years Thanks

to Dr Alberto Ginesi and Dr Riccardo De Gaudenzi (ESA-ESTEC, Noordwijk, TheNetherlands) for their appreciation and encouragement on some results in Chapter 5.Alan Barbieri (PhD student, University of Parma, Italy) and Gianpietro Germi, arealso acknowledged for their contribution to part of Chapter 5 Going backward intime, we would like to acknowledge the contribution of Prof Andreas Polydoros(University of Athens, Greece) for a long research collaboration which lead to per-survivor processing, described in Chapter 3 We would also like to acknowledgethe contribution of Prof Piero Castoldi (Scuola Superiore Sant'Anna, Pisa, Italy) tosome of the results described in Chapter 3 and the Appendix Finally, we wish to ac-

xiii

knowledge the encouragement and support through the years of our senior colleaguesProf Giorgio Picchi (University of Parma, Italy) and Prof Giancarlo Prati (ScuolaSuperiore Sant'Anna, Pisa, Italy), who first appreciated our scientific achievements.Besides research collaborators, several people have helped in the process of edit-ing the manuscript Among them, we would like to thank Alessandra De Conti, whoread very carefully the entire manuscript, providing extremely valuable comments

on the English style Luca Consolini (PhD student, University of Parma, Italy) isalso thanked for reading the entire manuscript and providing comments We wishalso to thank Annuccia Babayan, for proof-reading parts of the manuscript, andMichele Franceschini (PhD student, University of Parma, Italy), for providing usefulcomments and helping significantly in the editing process Prof Enrico Forestieri(Scuola Superiore Sant'Anna, Pisa, Italy) is finally acknowledged for using his vastknowledge of ETgX to solve a few (unsolvable to us) editing problems and make themanuscript more compliant with the requests of the Publisher

Last, but not least, we heartly wish to thank several people at John Wiley andSons Ltd, who made the realization of this book possible First of all, we are greatlyindebted to our Development Editor, Sarah Hinton, who first believed in this projectand promoted it Our gratitude goes also to our Publishing Editor, Mark Hammond,who supported the project along its entire realization Finally, the Project Editor, DanGill, is thanked for managing very efficiently the editorial phase of the manuscriptand the Copyeditor, Helen Heyes, is also thanked for providing extremely detailedcorrections to the final version of the manuscript

List of Figures

1.1 Examples of wireless channel models 71.2 Classical model of a communication system 111.3 Lower bound on the noncoherent channel capacity for different val-

ues of N Reproduced from [44] with permission of John Wiley &

Sons 15

1.4 Rate-1/2 convolutional encoder with generators G\ = 5 and G^ = 7 22

2.1 M-ary signaling and detection 262.2 Discretization of the received signal 292.3 Decision regions 302.4 Transmission system 332.5 Constellation for 32-APSK 442.6 Possible communication systems Reproduced from [92] by permis-sion of John Wiley & Sons 453.1 Add-compare-select operation in a VA 533.2 Receiver based on estimation-detection decomposition 613.3 Training and tracking operational mode 673.4 PSP-based detection 733.5 Trellis evolution: universal and PSP-based estimation 743.6 Pictorial description of trellis folding 783.7 Set partitioning for 8-PSK constellation 813.8 TCM encoder and mapping for 16-QAM (the subsets are specified inFigure 3.9) 843.9 Set partition and mapping rule for 16-QAM constellation 843.10 Equivalent discrete-time channel response of an ISI channel 853.11 Performance of TCM with 16-QAM for transmission over the 4-tap ISI channel considered in Figure 3.10 Reproduced from [106],

©1996 IEEE, by permission of the IEEE 86

xv

3.12 SER performance of uncoded QPSK transmission over a 3-tap persive fading channel and LMS-based adaptive detection The nor-

©1995 IEEE, by permission of the IEEE 923.13 SER performance of uncoded QPSK transmission over a 3-tap dis-persive fading channel and LMS-based adaptive detection with dual

3.14 SER performance of PSP-based detection of a TCM with 8-PSK Forcomparison, the performance of a conventional data-aided receiver

(with d = 2) is also shown Reproduced from [34], ©1995 IEEE, by

permission of the IEEE 953.15 BER of NSD detection schemes for 16-DQAM with various degrees

of complexity Reproduced from [47], ©1999 IEEE, by permission

of the IEEE 1003.16 BER of NSD detection schemes for 8-state TC-16-QAM Repro-duced from [47], ©1999 IEEE, by permission of the IEEE 1013.17 BER of the proposed detection schemes for 16-DQAM on the two

considered ISI channels and various values of N The noncoherent

detectors search a trellis with £' = 256 states Reproduced from [47],

©1999 IEEE, by permission of the IEEE 102

3.18 BER of the proposed receiver for DQPSK with N = 5 and £' = 1

for various values of phase jitter standard deviation (white marks)and frequency offset (black marks) Reproduced from [47], ©1999IEEE, by permission of the IEEE 1033.19 System model in the case of a channel with phase and frequencyuncertainty 1043.20 Examples of indistinguishable sequences: (a) noncoherent receiver(zero-th order); (b) advanced receiver with frequency estimation (firstorder) Reproduced from [117], ©2002 IEEE, by permission of theIEEE 1073.21 BER of the receiver based on (3.137) (white marks) for DQPSK andcomparison with NSD (black marks) and coherent receivers The

frequency offset is v — 0 Reproduced from [117], ©2002 IEEE, by

permission of the IEEE 1093.22 BER of the receiver based on (3.137) (white marks) for DQPSK,

N = 6, L = 11, and f' = 16 The performance of an NSD receiver

(black marks) with N = 6 and (J = 16 is also shown for comparison.

Reproduced from [117], ©2002 IEEE, by permission of the IEEE 110

List of Figures xvii

3.23 BER at an SNR of 10 dB versus the normalized frequency offset of

the receiver based on (3.137) for DQPSK and N = 6, L = 6, and

£' = 16 Method 1 (white marks) based on a limitation of the tion interval and method 2 (black marks) based on DDE are consid-

estima-ered The performance of an NSD receiver with N = 6 and £' = 16 is

also shown for comparison Reproduced from [117], ©2002 IEEE,

by permission of the IEEE Ill3.24 System model for transmission over a frequency nonselective fadingchannel Reproduced from [120], ©2000 IEEE, by permission of theIEEE 1123.25 BER of the proposed receivers with and without CSI, 5 = 4 and

TV = 2, for differentially encoded 16-QAM and Rice fading with

KR = 10 dB The performance of an ideal coherent receiver is also

shown for comparison Reproduced from [120], ©2000 IEEE, bypermission of the IEEE 1223.26 BER of the proposed receivers with and without CSI, ("' = 4 and

N — 2, for differentially encoded 16-QAM and Rayleigh fading.

The performance of an ideal coherent receiver is also shown for parison Reproduced from [120], ©2000 IEEE, by permission of theIEEE 1233.27 BER of the proposed receiver without CSI, (' = 4 and TV = 3, for dif-ferentially encoded QPSK, slow Rayleigh fading and various values

com-of phase noise standard deviation (black marks) The performance com-of

a coherent receiver based on a decision-directed PLL (white marks) isalso shown for comparison Reproduced from [120], ©2000 IEEE,

by permission of the IEEE 124

equal to 10~3 of the proposed receiver without CSI, f' = 1 and N —

3, for differentially encoded QPSK, slow Rayleigh fading (black marks),and comparison with a coherent receiver based on a decision-directedPLL (white marks) Reproduced from [120], ©2000 IEEE, by per-mission of the IEEE 1253.29 System model for linear prediction-based receivers 1263.30 BER of a TCM scheme with 16-QAM Linear predictive receiverswith various complexity levels are considered For comparison, theperformance of the equivalent coherent receiver is also shown 130

3.31 Prediction coefficients as a function of the phase noise standard viation <JA, for an equal-energy modulation, prediction order TV — 4

de-and Eb/No — 4 dB Various values of the frequency offset intensity

a are considered Reproduced from [128], ©2003 IEEE, by

permis-sion of the IEEE 1323.32 BER as a function of the phase noise standard deviation CTA for DQPSK,symbol by symbol decision, and various values of the frequency off-

set intensity a Reproduced from [128], ©2003 IEEE, by permission

of the IEEE 1333.33 BER of a linear predictive receiver for transmission of QPSK over atime-varying flat Rayleigh fading channel Various values of the nor-malized maximum Doppler rate are considered Reproduced from [124],

©1995 IEEE, by permission of the IEEE 141

dB and BPSK as a function of the assumed memory N Reproduced

from [121] by permission of John Wiley & Sons 146

QPSK modulation and p — 0.998 Reproduced from [121] by

per-mission of John Wiley & Sons 147

3.36 BER versus E^/No of the blind recursive detector with (3 — 1 and

(3 = 2 for BPSK modulation and p = 0.99 Reproduced from [121]

by permission of John Wiley & Sons 1484.1 Transmission system and MAP symbol detection 156

4.2 Parallel concatenated convolutional code, or turbo code 163

4.3 Turbo decoder for a PCCC 1644.4 Typical BER performance curves of a turbo code for an increasingnumber of iterations Reprinted from [33], ©IEEE, by permission ofthe IEEE 1664.5 Decoder for a turbo code of rate 1/2 1704.6 BER of a turbo code and the FB algorithm The extrinsic infor-mation generated by each decoder is either modeled as a Gaussian-

distributed random variable (first method) or used to update the a

priori probabilities (second method) The considered numbers of

it-erations are 1, 3, 6 and 18 Reproduced from [147], ©2001 IEEE,

by permission of the IEEE 177

List of Figures xix

4.7 Average value of ratio r] z /a^ versus the number of iterations, for

vari-ous values of SNR and a turbo code The component decoders use the

FB algorithm The extrinsic information generated by each decoder

is modeled as a Gaussian-distributed random variable (first method).Reproduced from [147], ©2001 IEEE, by permission of the IEEE 1784.8 BER of a serially concatenated code and FB algorithm The consid-ered numbers of iterations are 1, 3, 6 and 18 Reproduced from [147],

©2001 IEEE, by permission of the IEEE 179

4.9 Average value of ratio TJ Z /(T^ versus number of iterations, for various

values of SNR and a serially concatenated code The component coders use the FB algorithm The extrinsic information generated byeach decoder is modeled as a Gaussian-distributed random variable(first method) Reproduced from [147], ©2001 IEEE, by permission

de-of the IEEE 1804.10 BER of the proposed detection schemes for a turbo code and SOVA.The extrinsic information generated by each decoder is either mod-eled as a Gaussian-distributed random variable (first method) or used

to update the a priori probabilities (second method) or heuristically

weighted (third method) The considered numbers of iterations are

1, 3 and 18 Reproduced from [147], ©2001 IEEE, by permission ofthe IEEE 1834.11 Average value of the ratio ry^/cr^ versus the number of iterations, forvarious values of SNR by considering a turbo code The componentdecoders use SOVA The extrinsic information generated by each de-coder is considered as a Gaussian-distributed random variable (firstmethod) 1844.12 BER of a serially concatenated code and SOVA The considered num-bers of iterations are 1, 3, 6 and 18 Reproduced from [147], ©2001IEEE, by permission of the IEEE 185

4.13 Average value of the ratio r) z /cr% versus the number of iterations, for

various values of SNR and a serially concatenated code The ponent decoders use SOVA The extrinsic information generated byeach decoder is considered as a Gaussian-distributed random variable(first method) 1864.14 Communication system 1864.15 Implicit phase estimation in the forward recursion, backward recur-sion and completion for the NCSOa algorithm 1984.16 Implicit phase estimation in the forward recursion, backward recur-sion and completion for the NCSOb algorithm 199

com-4.17 Noncoherent iterative decoding of a PCCC with BPSK For ison the performance of the corresponding coherent receiver is alsoshown In all cases 1,5, and 10 decoding iterations are considered 2004.18 Noncoherent iterative decoding of an SCCC with 8-PSK For com-parison the performance of the corresponding coherent receiver isalso shown In all cases 1 and 5 decoding iterations are considered 201

compar-4 19 Forward recursion for the computation of a& (sk) for a Fwd-only

RS-FB algorithm in the case of coherent detection for an ISI channel.Reproduced from [156], ©2001 IEEE, by permission of the IEEE 206

RS-FB algorithm in the case of coherent detection for an ISI channel.The survivor map is constructed during this recursion Reproducedfrom [156], ©2001 IEEE, by permission of the IEEE 2074.21 Application of the proposed technique to iterative decoding/detectionfor an ISI channel Receivers with various levels of complexity areconsidered and compared with the full-state receiver (£ = 16) Theconsidered numbers of iterations are 1 and 6 in all cases The per-formance in the case of coded transmission over an AWGN chan-nel, without ISI, is also shown (solid lines with circles) Reproducedfrom [156], ©2001 IEEE, by permission of the IEEE 2094.22 Application of the proposed Fwd-only state reduction technique toiterative decoding/detection, through linear prediction, for flat fading

com-plexity (in terms of prediction order TV and reduced-state parameter

Q) are shown The considered numbers of iterations are 1 and 6

in all cases The performance in the case of decoding with perfectknowledge of the fading coefficients is also shown (solid lines) Re-produced from [156], ©2001 IEEE, by permission of the IEEE 214

4.23 Forward recursion of the pdf &k(tk) for a general Fwd-only RS

FB-type algorithm Reproduced from [156], ©2001 IEEE, by sion of the IEEE 217

FB-type algorithm The metric c/>k is calculated using the survivor

map previously constructed in the forward recursion Reproducedfrom [156], ©2001 IEEE, by permission of the IEEE 217

List of Figures xxi

4.25 Application of the RS-NCSOb algorithm to noncoherent decoding

of an RSC code Receivers with various levels of complexity areconsidered and compared with a full-state receiver (TV = 2) and thecoherent receiver Reproduced from [156], ©2001 IEEE, by permis-sion of the IEEE 2224.26 Transmitter and receiver for the uncoded BPSK transmission overISI/AWGN channels 2394.27 Performance comparisons of various self-iterative detection algorithmsfor Channel A assuming perfect CSI The number of considered self-iterations / is 1 (solid lines) or 5 (dashed lines), and the number ofstates is indicated by £' in the case of state reduction For compar-ison, the performance of the full-state receiver (f = 2048) is alsoshown Reproduced from [166], ©2002 IEEE, by permission of theIEEE 2404.28 Performance comparisons of various self-iterative detection algorithmsfor Channel B assuming perfect CSI The number of considered self-iterations / is 1 (solid lines) or 5 (dashed lines), and the number ofstates, in the case of state reduction, is indicated by £' For compar-ison, the performance of the full-state receiver (£ = 2048) is alsoshown Reproduced from [166], ©2002 IEEE, by permission of theIEEE 2414.29 Performance comparisons of various self-iterative detection algorithmsfor Channel C assuming perfect CSI The number of considered self-iterations / is 1 (solid lines) or 5 (dashed lines), and the number ofstates is indicated, in the case of state reduction, by £' For compar-ison, the performance of the full-state receiver (£ = 2048) is alsoshown Reproduced from [166], ©2002 IEEE, by permission of theIEEE 242

4.30 Performance of a (2,1,9) NRC code with generators G\ = 7604 and

G 2 = 4174 with BPSK on an AWGN channel / = 1 (solid lines)

and 7 = 5 (dashed lines) self-iterations are considered for different

numbers of reduced states £' and different packet length K The

performance in the full-state case (£ = 512 states) is also shown 243

4.31 Performance of a (2,1,12) NRC code with generators d = 42554

lines) and 7 = 5 (dashed lines) self-iterations are considered for

dif-ferent numbers of reduced states £' A packet length K = 128 is

considered in all cases, and for comparison, the performance of afull-state (C = 128) (2,1,7) convolutional code is also shown 244

4.32 Transmitter and receiver for the TCM system over an ISI/AWGNchannel 2454.33 Performance comparison of various iterative detection algorithms for

Opt 2: (/i}/0) = (3,5), (Mo) = (1,1); Opt 3: (/i,/0) = (2,6),

(0.625,0.375)) Reproduced from [166], ©2002 IEEE, by sion of the IEEE 2464.34 Communication system model for transmission over phase-uncertainchannels 2474.35 Schemes with separate detection and decoding using the proposedsoft-output noncoherent algorithms: (a) transmitter and (b) receiver 2504.36 Receiver with combined detection and decoding for a PCCC of rate 1/2.Reproduced from [162], ©2000 IEEE, by permission of the IEEE 2514.37 BER of the proposed iterative detection schemes using the NCSObalgorithm with predetection (dot-dashed curves), combined detectionand decoding (solid curves), coherent decoding (dashed curves), andcoherent predetection (dotted curves) The numbers of iterations are

permis-1, 3, 6, and 18 in all cases Reproduced from [162], ©2000 IEEE,

by permission of the IEEE 2524.38 BER of the proposed receivers using SO-NSD with combined de-tection and decoding for asymmetric (dotted curves) and symmetric(dashed curves) schemes The numbers of iterations are 1, 3, 6 and 18

in both cases The performance for coherent decoding (solid curve)and 18 iterations is also shown Reproduced from [162], ©2000IEEE, by permission of the IEEE 2544.39 Application of the Fwd-only RS NCSOb algorithm to noncoherentdecoding of a PCCC Receivers with various levels of complexity

are considered and compared with a full-state receiver (with N — 3)

and a coherent receiver The considered numbers of iterations are

1, 3 and 6 in all cases Reproduced from [156], ©2000 IEEE, bypermission of the IEEE 2554.40 BER of the proposed detection scheme using SO-NSD with com-bined detection and decoding for various levels of phase noise In allcases, the number of iterations is 6 Reproduced from [162], ©2000IEEE, by permission of the IEEE 2564.41 Iterative decoding of serially concatenated interleaved codes Repro-duced from [162], ©2000 IEEE, by permission of the IEEE 257

List of Figures xxiii

4.42 BER of the proposed detection scheme using the NCSOb algorithm

with N — 2 (dashed curves) and N = 4 (dotted curves) for the serial

concatenation of two interleaved convolutional codes For son, the performance of iterative coherent decoding (solid curves) isalso shown The numbers of iterations are 1, 3, 6 and 18 Reproducedfrom [162], ©2000 IEEE, by permission of the IEEE 2584.43 Transmission scheme relative to a concatenated code constituted by

compari-an outer rate-1/2 RSC code compari-and compari-an inner differential encoder 2594.44 BER of the proposed detection scheme using the NCSOb algorithm

with N = 3 (dashed curves) for the serial concatenation of a

convo-lutional code, an interleaver and a differential encoder For ison, the performance of iterative coherent decoding (dotted curves)and optimal coherent decoding of the single convolutional code (solidcurve) is also shown In the cases with iterative detection, the num-bers of iterations are 1, 3, and 10 Reproduced from [162], ©2000IEEE, by permission of the IEEE 2604.45 Berrou-type PCCC followed by differential encoding on the modu-lated symbols Reproduced from [163] by permission of GET/HermesScience 2614.46 Performance of the system shown in Figure 4.45 The considerednumbers of inner iterations are 1, 3 and 5 in all cases Reproducedfrom [163] by permission of GET/Hermes Science 2624.47 Benedetto-type PCCC followed by differential encoding on the mod-ulated symbols Reproduced from [163] by permission of GET/HermesScience 2634.48 Performance of the system shown in Figure 4.47 The considerednumbers of iterations are 1, 3 and 6 in all cases Reproduced from [163]

compar-by permission of GET/Hermes Science 2644.49 SCCC constituted by an outer convolutional code and an inner Unger-boeck code Reproduced from [163] by permission of GET/HermesScience 2654.50 Performance of the system shown in Figure 4.49 The outer code has

8 states and the number of iterations is 10 in all cases Reproducedfrom [163] by permission of GET/Hermes Science 2664.51 Performance of the system shown in Figure 4.49 The outer code has

16 states and the number of iterations is 10 in all cases Reproducedfrom [163] by permission of GET/Hermes Science 267

4.52 Performance of the system shown in Figure 4.49 The modulationformat is 16-QAM and the number of iterations is 1, 5 and 10 in allcases Reproduced from [163] by permission of GET/Hermes Science 268

4.53 Turbo trellis-coded scheme by Benedetto et al with 8-PSK

mod-ulation Puncturing may be embedded in the component boeck codes to take into account QPSK modulation Reproducedfrom [163] by permission of GET/Hermes Science 2694.54 Performance of the system proposed in Figure 4.53 The modulationformat is 8-PSK and the number of iterations is 1, 3 and 6 in all cases.Reproduced from [163] by permission of GET/Hermes Science 2714.55 Parallel concatenated coding scheme with separate detection and de-coding: transmitter, channel and iterative decoder constituted by theconcatenation of an A-SODEM and a coherent turbo decoder Re-produced from [129], ©2004 IEEE, by permission of the IEEE 2774.56 BER of the separate scheme with rate-1/2 PCCC and QPSK outputmodulation In all cases, 7e = 5 external iterations between the A-

Unger-SODEM and the turbo decoder are considered Various numbers I\ of

inner decoding iterations are considered In the CL case, the adaptivealgorithm is characterized by TV = 0 for a A = 5 degrees and by

TV = 1 for <JA = 10 degrees In the OL case, the detection algorithm

is characterized by (TV, Q) = (7,3) Reproduced from [129], ©2004

IEEE, by permission of the IEEE 2794.57 Serially concatenated coding scheme with combined detection anddecoding: transmitter, channel and adaptive iterative decoder Re-produced from [129], ©2004 IEEE, by permission of the IEEE 2804.58 Parallel concatenated coding scheme with combined detection anddecoding: transmitter, channel and iterative decoder constituted bytwo adaptive component decoders Reproduced from [129], ©2004IEEE, by permission of the IEEE 2814.59 BER of an SCCC with OL and CL inner decoding algorithm, forphase jitter standard deviation <TA = 5 degrees and <JA = 10 degrees.The spectral efficiency is 1 bit/s/Hz For comparison, the perfor-mance of the equivalent coherent scheme is shown In all cases, 10decoding iterations are considered Reproduced from [129], ©2004IEEE, by permission of the IEEE 282

List of Figures xxv

4.60 BER of an SCCC with OL and CL inner decoding algorithms, forphase jitter standard deviation <JA = 5 degrees and <TA = 10 degrees.The spectral efficiency is 2 bits/s/Hz For comparison, the perfor-mance of the equivalent coherent scheme is shown In all cases, 10decoding iterations are considered Reproduced from [129], ©2004IEEE, by permission of the IEEE 284

4.61 E b fN 0 required to obtain a BER of 10~3 at 10 decoding iterationsversus the phase jitter standard deviation <JA- Both the OL and CLstrategies are considered Reproduced from [129], ©2004 IEEE, bypermission of the IEEE 285

4.62 BER of a PCCC with OL and CL component decoding algorithms,for phase jitter standard deviation <JA = 5 degrees and CTA = 10degrees For comparison, the performance of the equivalent coherentscheme is shown In all cases, 10 decoding iterations are considered

Nd = 16 in all the adaptive cases Reproduced from [129], ©2004

IEEE, by permission of the IEEE 286

4.63 FER of a PCCC with OL and CL component decoding algorithms forphase jitter standard deviation <JA = 5 degrees and <JA = 10 degrees.For comparison, the performance of the equivalent coherent scheme

is shown In all cases, 10 decoding iterations are considered N d =

16 in all adaptive cases Reproduced from [129], ©2004 IEEE, bypermission of the IEEE 287

4.64 BER of an SCCC with 8-PSK and inner linear prediction at the ceiver side Various receiver complexity levels are considered Forcomparison, the performance of the coherent system is also shown

re-In all cases, 5 decoding iterations are considered 289

4.65 BER as a function of the phase noise standard deviation a& for an

dB, various values of frequency offset intensity and levels of receivercomplexity In all cases, 5 decoding iterations are considered Re-produced from [128], ©2003 IEEE, by permission of the IEEE 2904.66 Sliding window linear prediction strategy in the case of oversamplingwith/? = 4 291

4.67 BER of a serially concatenated scheme given by an outer tional code and inner OMSK modulator At the receiver side there

convolu-is inner linear prediction with a sampling rate of (3 = 2 samples per

symbol Various receiver complexity levels are considered For parison, the performance of the coherent system is also shown In allcases, 5 decoding iterations are considered Reproduced from [128],

com-©2003 IEEE, by permission of the IEEE 2934.68 BER of an SCCC with inner noncoherent combined detection anddecoding over a Rayleigh flat fading channel with normalized band-width /DT = 0.01 In all cases, 5 decoding iterations are considered.Reproduced from [149], ©2003 IEEE, by permission of IEEE 2954.69 BER of an SCCC with inner linear prediction-based combined detec-tion and decoding over a Rayleigh flat fading channel with normal-ized bandwidth /DT = 0.01 In all cases 5 decoding iterations areconsidered Reproduced from [149], ©2003 IEEE, by permission ofIEEE 2975.1 The FG corresponding to the factorization (5.1) 3035.2 The FG corresponding to the factorization (5.2) 304

5.3 Factor graph corresponding to the factorization (5.14) for C = 2.

Reproduced from [201], ©2004 IEEE, by permission of the IEEE 3095.4 Factor graph corresponding to the factorization (5.12) 309

5.5 Factor graph corresponding to the factorization (5.17) for C = 2.

Reproduced from [201], ©2004 IEEE, by permission of the IEEE 3115.6 Complexity reduction in graph-based detection The messages insome of the incoming branches (dashed lines) are hard-quantized, soonly the remaining branches carry soft information to be used The

considered case corresponds to C — 3 and Q = 1 312

5.8 Performance for a sparse ISI channel 3185.9 Performance in the case of complexity reduction 3195.10 Part of the factor graph in Figure 5.7 after stretching 3205.11 Performance of the SP algorithm on a modified graph 3215.12 BER in the case of application to LDPC codes 3225.13 BER performance of an LDPC code transmitted over a noncoherentchannel Reproduced from [201], ©2004 IEEE, by permission of theIEEE 324

5.14 Simplified overall factor graph for PSK signals and C = 2

Repro-duced from [201], ©2004 IEEE, by permission of the IEEE 326

List of Figures xxvii

5.15 Performance in the case of a time-varying channel phase The value

of C is optimized in each individual case Reproduced from [201],

©2004 IEEE, by permission of the IEEE 3275.16 Performance of graph-based finite memory detection in the case of a

5.17 Factor graph corresponding to (5.40) 3325.18 Factor graph corresponding to (5.42) 3325.19 Performance of the algorithms based on discretization of channelparameters and Fourier parameterization BPSK and two differentphase models are considered 3435.20 Performance of the algorithms based on discretization of channel pa-rameters and Fourier parameterization QPSK and the Wiener modelwith <JA = 6 degrees are considered 3455.21 Performance of the algorithms based on Tikhonov and Gaussian pa-rameterizations BPSK and two different phase models are considered 3465.22 Performance of all the proposed algorithms and comparison withother algorithms proposed in the literature BPSK and the Wienerphase model with <JA = 6 degrees are considered 3475.23 Performance of the algorithms based on Tikhonov and Gaussian pa-rameterizations BPSK and two different pilot distributions are con-sidered 3485.24 Performance of the algorithms based on Tikhonov and Gaussian pa-rameterizations The ESA phase model is considered along with 8-PSK and 32-APSK modulations 349

A 1 Transmission system model 354A.2 Application of the concept of reversibility 360A.3 Sampling and filtering operations 362

List of Tables

2.1 A posteriori probabilities of the information messages 27

3.1 Summary of the derived branch metrics in the case of the absence ofCSI 1214.1 Different parameter settings for RS-FB algorithms 238

xxix

Wireless Communication Systems

1.1 Introduction

In the era of Information and Communication Technology, we are all familiar withthe concept of digital transmission of information The adjective "digital" encom-passes the fact that the transmitted message is one out of a finite number of possiblemessages In more practical terms, a digital message is composed of a sequence of

digits, or symbols, belonging to a finite alphabet This digital model of information arises for a number of reasons There exist information sources which inherently

generate digital messages A written text or a computer memory are simple ples of sources of digital information There also exist, however, information sources

exam-that inherently generate analog messages, such as voice and sounds These analog

messages can be approximated with the desired degree of accuracy in terms of digitalmessages by means of sampling and quantization

The goal of any digital transmission system is the delivery of a sequence of

dig-ital information symbols to a destination The transmission of a digdig-ital information symbol, or transmission act, takes place by selecting an analog signal, or waveform,

out of a finite set of possible signals and sending this waveform through a

transmis-sion medium, or channel, which connects the transmitter to a receiver located at the

destination

Wireless communications will have an increasing importance for future tions, where connectivity will be required everywhere and anyhow In particular, thelarge flow of information coming from the optical communication backbone callsfor wireless systems able to support large information transmission capacity It is

applica-Detection Algorithms for Wireless Communications- G Ferrari, G Colavolpe and R Raheli

©John Wiley & Sons, Ltd ISBN: 0-470-85828-1

1

1

becoming more and more important to push to the limit wireless access and tion, since larger and larger quantities of information need to be carried For instance,

distribu-in third generation (3G) mobile communication systems [1], multimedia data will betransferred and this will call for effective and efficient wireless communication sys-tems In particular, increased signal processing capabilities for lower cost will allow

the deployment of advanced detection algorithms in the receivers of wireless

termi-nals Several wireless communication systems are currently operational, and a fewsignificant examples will be summarized in the following

In passing through a wireless channel, the transmitted waveform is altered inseveral possible ways A primary alteration is due to thermal noise which is superim-posed on the transmitted waveform and effectively modeled as an additive Gaussianrandom process There is, however, a plethora of other possible modifications which

may be inflicted upon the signal by the transmission link In radio transmission, the

amplitude of the received signal, i.e., the signal at the output of the channel, is a tion of the link attenuation, which in turn may depend on the physical link geometry,such as the distance between transmitter and receiver antennas and their heights,but it may also depend on the simultaneous presence of several propagation paths,

func-or multipath, which may contribute constructively func-or destructively As a result, the

(possibly time-varying) signal amplitude may not be perfectly known by the receiver,which must adopt suitable countermeasures In bandpass transmission, similar con-siderations hold for the phase of the received signal, which is usually unknown at thereceiver When the amount of variability of multipath phenomena over space or time

is significant, the received signal may be characterized by unknown amplitude and

phase, which collectively manifest themselves as multiplicative noise termed fading.

In general terms, the transformation inflicted by a channel upon the transmittedwaveforms must be suitably modeled This modeling is needed in order to enable the

definition of an effective strategy to be undertaken by the receiver in order to decide

which message was actually transmitted on the basis of the signal actually received

This decision process, or detection, should obviously result in a minimal number

of decision errors By adopting a random model for both the information sourceand the transmission channel, the detection strategy can be optimized according to

a meaningful criterion, for example to minimize the probability of decision error A

receiver operating in accordance with such a strategy is said to be optimal.

There are several important applications in which random channel models areuseful for devising optimal receivers characterized by affordable implementationcomplexity, among which is the classical additive white Gaussian noise (AWGN)channel There are, however, applications where the implementation complexity of

an optimal receiver becomes prohibitive from a practical viewpoint The simple

pres-ence of a random time-invariant channel parameter, such as a phase rotation or a

tiplicative fading coefficient inflicted upon the received signal, may significantly

in-crease the complexity of the optimal receiver Time-varying random channel

parame-ters may also be responsible for the intractable complexity of an optimal receiver Inpresent-day communication systems, examples of such time-varying random chan-nel models arise to describe the effects of fading due to multiple transmission paths,phase noise due to the inherent instabilities of up- and down-conversion oscillators,

or Doppler shift encountered in low- and medium-earth satellite communications

It might be sometimes necessary to model some channel parameters as unknowndeterministic quantities This modeling assumption has the desirable consequence

of reducing the receiver complexity, by means of a decomposition approach In this

case, the receiver could be designed as the concatenation of a detection block, vised under the idealizing assumption of perfect knowledge of these deterministicchannel parameters, and one or more estimation blocks devoted to the acquisition ofthe necessary information about the channel parameters Nonetheless, this modelingassumption has the disadvantage that the concept of optimality, in terms of minimalerror probability, vanishes because an optimal detection strategy can only be definedfor specific known cases of deterministic channel parameters [2]

de-In the rest of the book, the focus will be mainly on channels characterized by

stochastic parameters In particular, detection algorithms will be designed taking

into account a statistical description of the parameters, e.g., by means of their jointprobability density function (pdf) This type of approach is usually referred to asBayesian Since taking into account the stochastic nature of the transmission chan-nel may lead to a significant increase of the complexity of the detection algorithm,suitable complexity reduction techniques will also be considered It is also important

to underline that, for the design of a detection algorithm, it is sometimes expedient

to consider a simplified channel model, with respect to the effective transmissionchannel

In this initial chapter, we will also present a brief overview of the concepts of

modulation and coding, providing a few significant examples for wireless

commu-nication scenarios The analysis is by no means complete, since it goes beyond thescope of this book However, the proposed set of examples should provide the readerwith a basic understanding of the concepts and schemes which will be considered inthe following chapters

Modulation can be, as a first instance, separated by coding It is somehow sible to state that modulation comes conceptually before coding, in the sense that it

pos-represents the way in which a signal needs to be shaped before undergoing channel transmission In particular, in the remainder of the book we will focus on digital

modulations, where the information to be transmitted belongs to a suitable set offinite cardinality In this case, considering a suitable set of orthonormal functions

3

it is possible to represent a modulated signal as a point in a constellation of finitecardinality.

After the breakthrough brought by the mathematical theory of communicationdeveloped by Shannon [3], it has been clear that coding the information to be trans-mitted can make the communication more efficient and reduce the probability of er-roneous reception at the receiver side In particular, the main application areas regard

source coding and channel coding [4-7] While in the first case the main principle

is that of compressing the source information still guaranteeing complete recovery,the main idea in the second case (of interest in this book) consists of adding redun-dancy to the generated information in order to increase the protection against channelimpairments

While modulation and coding can be considered separate as a first instance, inal work in the Seventies [8] culminated with the clear description of trellis codedmodulation (TCM) [9], which showed that modulation and coding should be regarded

sem-as two faces of the same coin This is intuitive, since once the information to betransmitted is suitably coded, the way in which this coded information is actuallytransmitted over the channel is strictly related to the considered modulation In theremainder of this chapter, we will review some basic concepts, relevant for the de-tection algorithms for wireless communications derived in the following chapters,

such as the capacity of a transmission channel and some interesting types of

modu-lation and coding formats Note that the focus, rather than on standard coding andmodulation techniques, will rather lie on interesting schemes to which the detectionalgorithms presented in the following chapters can be suitably applied

1.2 Overview of Wireless Communication Systems

There are various examples of wireless communication systems We consider here asimple summary of a few communication systems for which the detection algorithmsdiscussed in the remainder of this book might have significant impact

• Broadband radio access networks are gaining more and more interest, since

radio traffic is increasingly based on multimedia applications, requiring icant transmission capabilities Point to multipoint radio access is an importantexample of this radio communication paradigm

signif-• Cellular radio networks represent the current structure devoted to personal

wireless communications [10] They are based on the subdivision of territoryinto cells, with a base station placed in each cell and acting as a "reservoir" fortransmissions coming from mobile terminals in the cell

Wireless Channel Models

• Satellite communication systems are widely used to guarantee ubiquitous

con-nectivity and to allow transfer of information between distant locations [11]

In particular, mobile and satellite communication systems could merge, in thenear future, to guarantee ubiquitous coverage

• Wireless local area networks (such as IEEE 802.11 [12]) represent a new

com-munication paradigm which is under continuous development Its applicationsare currently limited to multiple access in limited areas, such as universitycampuses, airports, and buildings However, novel scenarios with extendedcoverage are emerging

• Personal area networks are currently becoming more and more popular, with

multiple diverse applications The network communication system nicknamed

"bluetooth" is a de facto standard which is developing to guarantee the communication between digital devices in very limited areas [13] This is thecase, for example, for communication among electronic appliances (comput-ers, video cameras, stereo systems) within a room

inter-• Mobile ad hoc wireless networks are emerging as a new communication paradigm

which could have numerous applications in the near future [14,15] In ular, the fundamental idea behind this type of network communication is thefact that radio communication should be supported by a nonhierarchical archi-tecture as long as the spatial distribution of the communicating terminals issufficiently dense

partic-1.3 Wireless Channel Models

In general, radio communications are affected by large (time) scale path loss (e.g., free space propagation model) and small scale phenomena [16] This book will fo-

cus on small scale phenomena In the following, we will consider a few significantexamples of communication channels which are relevant for the analysis of wirelesscommunication systems Various combinations of the proposed channel models mayalso be of interest The collection of examples is by no means complete, but theaim of the book is to develop a general framework for the design of detection algo-rithms to be used for transmissions over wireless channels The interested reader isinvited to use the tools presented in the following chapters to derive new detectionstrategies for other channel models not considered explicitly in this book Almost allconsidered models will correspond to the base-band equivalent of the effective trans-mission channels [17] Moreover, the derived detection algorithms will almost al-

5

ways stem from discrete-time equivalent versions of the transmission channels—theequivalence, in terms of statistical sufficiency, of the obtained discrete-time observ-ables will always be guaranteed, possibly by means of oversampling Appendix A isdevoted to a characterization of sampling as a means to derive sufficient statistics.

1.3.1 Additive White Gaussian Noise Channel

The basic communication channel is the AWGN channel, shown in Figure 1.1 (a) In

particular, the transmitted signal s(t) is corrupted by the additive noise n(t), yielding the received signal r(t), which can be written as follows:

The noise process n(t) is usually associated with the ubiquitous thermal noise at the

input of the receiver [16-18] In particular, the assumption of AWGN means that

n(t) is a circularly symmetric Gaussian process with constant power spectral density.

1.3.2 Frequency Nonselective Fading Channel

In this case, the transmitted signal is distorted by multiplicative noise f ( i ) modeled

as a circularly symmetric complex Gaussian random process, yielding the following

expression for the received signal:

We recall that a circularly symmetric Gaussian complex random variable is terized by independent and identically distributed (iid) Gaussian real and imaginary

charac-components [19] If f ( t ) is zero mean, then the fading amplitude \f(t)\ and phase

arg {/(£)} are Rayleigh and uniformly distributed, respectively In the frequency

do-main, this channel can be interpreted as a time-varying frequency nonselective (flat) linear filter Figure 1.1 (b) shows this channel model If the bandwidth of f ( t ) is

significantly narrower than that of the transmitted signal, its time variations can be

neglected and a slow fading model results As an extension of this model, in the case

of a Rice fading channel, f ( t ) has nonzero mean.

In order to describe the "memory" introduced by the channel it is expedient toconsider the autocorrelation function,1 usually characterized by the Clarke model [20,21] The Clarke model assumes a fixed transmitter with a vertically polarized andisotropic antenna [16], and uniform scattering around the mobile terminal—for this

lr This is meaningful in the case of circularly symmetric Gaussian fading with zero mean.

Wireless Channel Models

Figure 1.1: Examples of wireless channel models

reason, this model is usually referred to as the isotropic scattering model In

partic-ular, the statistical description of the received signal depends on (uniform) scatteringphenomena experienced by the transmitted waveform: under the assumption of theabsence of a direct line-of-sight component, the envelope of the received signal has aRayleigh distribution In [22], a spectral characterization of Clarke's model is stud-

7