DigitalCommunication over

Fading Channels

DigitalCommunication over

Fading ChannelsSecond Edition

Marvin K. SimonMohamed-Slim Alouini

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright 2005 by John Wiley & Sons, Inc. All rights reserved.

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Library of Congress Cataloging-in-Publication Data:

Simon, Marvin Kenneth, 1939–Digital communication over fading channels/Marvin K. Simon and Mohamed-Slim

Alouini.—2nd ed.p. cm.—(Wiley series in telecommunications and signal processing)

“A Wiley-Interscience publication.”Includes bibliographical references and index.ISBN 0-471-64953-8 (cloth : acid-free-paper)

1. Digital communications–Reliability–Mathematics. 2. Radio–Transmitters andtransmission–Fading. I. Alouini, Mohamed-Slim. II. Title. III. Series.

TK5103.7.S523 2004621.382–dc22

2005042040

Printed in the United States of America.

10 9 8 7 6 5 4 3 2 1

CONTENTS

Preface xxv

Nomenclature xxxi

PART 1 FUNDAMENTALS

CHAPTER 1 Introduction 31.1 System Performance Measures 4

1.1.1 Average Signal-to-Noise Ratio (SNR) 41.1.2 Outage Probability 51.1.3 Average Bit Error Probability (BEP) 61.1.4 Amount of Fading 121.1.5 Average Outage Duration 13

1.2 Conclusions 14

References 14

CHAPTER 2 Fading Channel Characterization and Modeling 172.1 Main Characteristics of Fading Channels 17

2.1.1 Envelope and Phase Fluctuations 172.1.2 Slow and Fast Fading 182.1.3 Frequency-Flat and Frequency-Selective

Fading 182.2 Modeling of Flat-Fading Channels 19

2.2.1 Multipath Fading 202.2.1.1 Rayleigh 202.2.1.2 Nakagami-q (Hoyt) 222.2.1.3 Nakagami-n (Rice) 232.2.1.4 Nakagami-m 24

vii

viii CONTENTS

2.2.1.5 Weibull 252.2.1.6 Beckmann 282.2.1.7 Spherically-Invariant Random

Process Model 302.2.2 Log-Normal Shadowing 322.2.3 Composite Multipath/Shadowing 33

2.2.3.1 Composite Gamma/Log-NormalDistribution 33

2.2.3.2 Suzuki Distribution 342.2.3.3 K Distribution 342.2.3.4 Rician Shadowed Distributions 36

2.2.4 Combined (Time-Shared)Shadowed/Unshadowed Fading 37

2.3 Modeling of Frequency-Selective Fading Channels 37

References 39

CHAPTER 3 Types of Communication 453.1 Ideal Coherent Detection 45

3.1.1 Multiple Amplitude-Shift-Keying (M-ASK)or Multiple Amplitude Modulation (M-AM) 47

3.1.2 Quadrature Amplitude-Shift-Keying(QASK) or Quadrature AmplitudeModulation (QAM) 48

3.1.3 M-ary Phase-Shift-Keying (M-PSK) 503.1.4 Differentially Encoded M-ary

Phase-Shift-Keying (M-PSK) 533.1.4.1 π /4-QPSK 54

3.1.5 Offset QPSK (OQPSK) or Staggered QPSK(SQPSK) 55

3.1.6 M-ary Frequency-Shift-Keying (M-FSK) 563.1.7 Minimum-Shift-Keying (MSK) 58

3.2 Nonideal Coherent Detection 62

3.3 Noncoherent Detection 66

3.4 Partially Coherent Detection 68

3.4.1 Conventional Detection 683.4.1.1 One-Symbol Observation 683.4.1.2 Multiple-Symbol Observation 69

3.4.2 Differentially Coherent Detection 713.4.2.1 M-ary Differential

Phase-Shift-Keying (M-DPSK) 713.4.2.2 Conventional Detection

(Two-Symbol Observation) 733.4.2.3 Multiple-Symbol Detection 76

CONTENTS ix

3.4.3 π /4-Differential QPSK (π /4-DQPSK) 78

References 78

PART 2 MATHEMATICAL TOOLS

CHAPTER 4 Alternative Representations of ClassicalFunctions 834.1 Gaussian Q-Function 84

4.1.1 One-Dimensional Case 844.1.2 Two-Dimensional Case 864.1.3 Other Forms for One- and Two-Dimensional

Cases 884.1.4 Alternative Representations of Higher

Powers of the Gaussian Q-Function 90

4.2 Marcum Q-Function 93

4.2.1 First-Order Marcum Q-Function 934.2.1.1 Upper and Lower Bounds 97

4.2.2 Generalized (mth-Order) MarcumQ-Function 1004.2.2.1 Upper and Lower Bounds 105

4.3 The Nuttall Q-Function 113

4.4 Other Functions 117

References 119

Appendix 4A. Derivation of Eq. (4.2) 120

CHAPTER 5 Useful Expressions for Evaluating AverageError Probability Performance 1235.1 Integrals Involving the Gaussian Q-Function 123

5.1.1 Rayleigh Fading Channel 1255.1.2 Nakagami-q (Hoyt) Fading Channel 1255.1.3 Nakagami-n (Rice) Fading Channel 1265.1.4 Nakagami-m Fading Channel 1265.1.5 Log-Normal Shadowing Channel 1285.1.6 Composite Log-Normal

Shadowing/Nakagami-m Fading Channel 128

5.2 Integrals Involving the Marcum Q-Function 131

5.2.1 Rayleigh Fading Channel 1325.2.2 Nakagami-q (Hoyt) Fading Channel 1335.2.3 Nakagami-n (Rice) Fading Channel 1335.2.4 Nakagami-m Fading Channel 1335.2.5 Log-Normal Shadowing Channel 133

x CONTENTS

5.2.6 Composite Log-NormalShadowing/Nakagami-m Fading Channel 134

5.2.7 Some Alternative Closed-Form Expressions 135

5.3 Integrals Involving the Incomplete Gamma Function 137

5.3.1 Rayleigh Fading Channel 1385.3.2 Nakagami-q (Hoyt) Fading Channel 1395.3.3 Nakagami-n (Rice) Fading Channel 1395.3.4 Nakagami-m Fading Channel 1405.3.5 Log-Normal Shadowing Channel 1405.3.6 Composite Log-Normal

Shadowing/Nakagami-m Fading Channel 140

5.4 Integrals Involving Other Functions 141

5.4.1 The M -PSK Error Probability Integral 1415.4.1.1 Rayleigh Fading Channel 1425.4.1.2 Nakagami-m Fading Channel 142

5.4.2 Arbitrary Two-Dimensional SignalConstellation Error Probability Integral 142

5.4.3 Higher-Order Integer Powers of theGaussian Q-Function 1445.4.3.1 Rayleigh Fading Channel 1445.4.3.2 Nakagami-m Fading Channel 145

5.4.4 Integer Powers of M -PSK Error ProbabilityIntegrals 1455.4.4.1 Rayleigh Fading Channel 146

References 148

Appendix 5A. Evaluation of Definite IntegralsAssociated with Rayleigh and Nakagami-m Fading 149

5A.1 Exact Closed-Form Results 1495A.2 Upper and Lower Bounds 165

CHAPTER 6 New Representations of Some Probability Densityand Cumulative Distribution Functions forCorrelative Fading Applications 1696.1 Bivariate Rayleigh PDF and CDF 170

6.2 PDF and CDF for Maximum of Two RayleighRandom Variables 175

6.3 PDF and CDF for Maximum of Two Nakagami-mRandom Variables 177

6.4 PDF and CDF for Maximum and Minimum of TwoLog-Normal Random Variables 180

6.4.1 The Maximum of Two Log-NormalRandom Variables 180

CONTENTS xi

6.4.2 The Minimum of Two Log-Normal RandomVariables 183

References 185

PART 3 OPTIMUM RECEPTION AND PERFORMANCEEVALUATION

CHAPTER 7 Optimum Receivers for Fading Channels 1897.1 The Case of Known Amplitudes, Phases, and

Delays—Coherent Detection 191

7.2 The Case of Known Phases and Delays butUnknown Amplitudes 195

7.2.1 Rayleigh Fading 1957.2.2 Nakagami-m Fading 196

7.3 The Case of Known Amplitudes and Delays butUnknown Phases 198

7.4 The Case of Known Delays but UnknownAmplitudes and Phases 199

7.4.1 One-Symbol Observation—NoncoherentDetection 1997.4.1.1 Rayleigh Fading 2017.4.1.2 Nakagami-m Fading 206

7.4.2 Two-Symbol Observation—ConventionalDifferentially Coherent Detection 2117.4.2.1 Rayleigh Fading 2147.4.2.2 Nakagami-m Fading 217

7.4.3 Ns-Symbol Observation—MultipleDifferentially Coherent Detection 2177.4.3.1 Rayleigh Fading 2187.4.3.2 Nakagami-m Fading 218

7.5 The Case of Unknown Amplitudes, Phases, andDelays 219

7.5.1 One-Symbol Observation—NoncoherentDetection 2197.5.1.1 Rayleigh Fading 2207.5.1.2 Nakagami-m Fading 221

7.5.2 Two-Symbol Observation—ConventionalDifferentially Coherent Detection 221

References 222

CHAPTER 8 Performance of Single-Channel Receivers 2238.1 Performance Over the AWGN Channel 223

xii CONTENTS

8.1.1 Ideal Coherent Detection 2248.1.1.1 Multiple Amplitude-Shift-Keying

(M-ASK) or Multiple AmplitudeModulation (M-AM) 224

8.1.1.2 Quadrature Amplitude-Shift-Keying (QASK) or QuadratureAmplitude Modulation (QAM) 225

8.1.1.3 M-ary Phase-Shift-Keying(M-PSK) 228

8.1.1.4 Differentially Encoded M-aryPhase-Shift-Keying (M-PSK)and π /4-QPSK 234

8.1.1.5 Offset QPSK (OQPSK) orStaggered QPSK (SQPSK) 235

8.1.1.6 M-ary Frequency-Shift-Keying(M-FSK) 236

8.1.1.7 Minimum-Shift-Keying (MSK) 2378.1.2 Nonideal Coherent Detection 2378.1.3 Noncoherent Detection 2428.1.4 Partially Coherent Detection 242

8.1.4.1 Conventional Detection(One-Symbol Observation) 242

8.1.4.2 Multiple-Symbol Detection 2448.1.5 Differentially Coherent Detection 245

8.1.5.1 M-ary DifferentialPhase-Shift-Keying (M-DPSK) 245

8.1.5.2 M-DPSK with Multiple-SymbolDetection 249

8.1.5.3 π /4-Differential QPSK(π /4-DQPSK) 250

8.1.6 Generic Results for Binary Signaling 251

8.2 Performance Over Fading Channels 252

8.2.1 Ideal Coherent Detection 2528.2.1.1 Multiple Amplitude-Shift-Keying

(M-ASK) or Multiple AmplitudeModulation (M-AM) 253

8.2.1.2 Quadrature Amplitude-Shift-Keying (QASK) or QuadratureAmplitude Modulation (QAM) 254

8.2.1.3 M-ary Phase-Shift-Keying(M-PSK) 256

8.2.1.4 Differentially Encoded M-aryPhase-Shift-Keying (M-PSK) andπ /4-QPSK 258

CONTENTS xiii

8.2.1.5 Offset QPSK (OQPSK) orStaggered QPSK (SQPSK) 262

8.2.1.6 M-ary Frequency-Shift-Keying(M-FSK) 262

8.2.1.7 Minimum-Shift-Keying (MSK) 2678.2.2 Nonideal Coherent Detection 267

8.2.2.1 Simplified Noisy Reference LossEvaluation 273

8.2.3 Noncoherent Detection 2818.2.4 Partially Coherent Detection 2828.2.5 Differentially Coherent Detection 284

8.2.5.1 M-ary Differential Phase-Shift-Keying (M-DPSK)—Slow Fading 285

8.2.5.2 M-ary Differential Phase-Shift-Keying (M-DPSK)—Fast Fading 290

8.2.5.3 π /4-Differential QPSK(π /4-DQPSK) 294

8.2.6 Performance in the Presence of ImperfectChannel Estimation 2948.2.6.1 Signal Model and Symbol Error

Probability Evaluation forRayleigh Fading 295

8.2.6.2 Special Cases 297

References 301

Appendix 8A. Stein’s Unified Analysis of the ErrorProbability Performance of Certain CommunicationSystems 304

CHAPTER 9 Performance of Multichannel Receivers 311

9.1 Diversity Combining 312

9.1.1 Diversity Concept 3129.1.2 Mathematical Modeling 3129.1.3 Brief Survey of Diversity Combining

Techniques 3139.1.3.1 Pure Combining Techniques 3139.1.3.2 Hybrid Combining Techniques 315

9.1.4 Complexity–Performance Tradeoffs 316

9.2 Maximal-Ratio Combining (MRC) 316

9.2.1 Receiver Structure 3179.2.2 PDF-Based Approach 3199.2.3 MGF-Based Approach 320

9.2.3.1 Average Bit Error Rate of BinarySignals 320

xiv CONTENTS

9.2.3.2 Average Symbol Error Rate ofM-PSK Signals 322

9.2.3.3 Average Symbol Error Rate ofM-AM Signals 323

9.2.3.4 Average Symbol Error Rate ofSquare M-QAM Signals 324

9.2.4 Bounds and Asymptotic SER Expressions 326

9.3 Coherent Equal Gain Combining 331

9.3.1 Receiver Structure 3319.3.2 Average Output SNR 3329.3.3 Exact Error Rate Analysis 333

9.3.3.1 Binary Signals 3339.3.3.2 Extension to M-PSK Signals 339

9.3.4 Approximate Error Rate Analysis 3409.3.5 Asymptotic Error Rate Analysis 342

9.4 Noncoherent and Differentially Coherent EqualGain Combining 342

9.4.1 DPSK, DQPSK, and BFSK Performance(Exact and with Bounds) 3439.4.1.1 Receiver Structures 3439.4.1.2 Exact Analysis of Average Bit

Error Probability 3469.4.1.3 Bounds on Average Bit Error

Probability 3529.4.2 M-ary Orthogonal FSK 353

9.4.2.1 Exact Analysis of Average BitError Probability 356

9.4.2.2 Numerical Examples 3649.4.3 Multiple-Symbol Differential Detection with

Diversity Combining 3679.4.3.1 Decision Metrics 3679.4.3.2 Average Bit Error Rate

Performance 3689.4.3.3 Asymptotic (Large Ns) Behavior 3719.4.3.4 Numerical Results 372

9.5 Optimum Diversity Combining of NoncoherentFSK 375

9.5.1 Comparison with the Noncoherent EqualGain Combining Receiver 377

9.5.2 Extension to the M-ary Orthogonal FSKCase 378

9.6 Outage Probability Performance 379

9.6.1 MRC and Noncoherent EGC 3799.6.2 Coherent EGC 380

CONTENTS xv

9.6.3 Numerical Examples 381

9.7 Impact of Fading Correlation 389

9.7.1 Model A: Two Correlated Branches withNonidentical Fading 3909.7.1.1 PDF 3909.7.1.2 MGF 392

9.7.2 Model B: D Identically DistributedBranches with Constant Correlation 3929.7.2.1 PDF 3939.7.2.2 MGF 393

9.7.3 Model C: D Identically DistributedBranches with Exponential Correlation 3949.7.3.1 PDF 3949.7.3.2 MGF 394

9.7.4 Model D: D Nonidentically DistributedBranches with Arbitrary Correlation 3959.7.4.1 MGF 3959.7.4.2 Special Cases of Interest 3969.7.4.3 Proof that Correlation Degrades

Performance 3979.7.5 Numerical Examples 399

9.8 Selection Combining 404

9.8.1 MGF of Output SNR 4059.8.2 Average Output SNR 4069.8.3 Outage Probability 409

9.8.3.1 Analysis 4099.8.3.2 Numerical Example 410

9.8.4 Average Probability of Error 4119.8.4.1 BDPSK and Noncoherent BFSK 4119.8.4.2 Coherent BPSK and BFSK 4139.8.4.3 Numerical Example 415

9.9 Switched Diversity 417

9.9.1 Dual-Branch Switch-and-Stay Combining 4199.9.1.1 Performance of SSC over

Independent Identically DistributedBranches 419

9.9.1.2 Effect of Branch Unbalance 4339.9.1.3 Effect of Branch Correlation 436

9.9.2 Multibranch Switch-and-ExamineCombining 4399.9.2.1 Classical Multibranch SEC 4409.9.2.2 Multibranch SEC with

Post-selection 4439.9.2.3 Scan-and-Wait Combining 446

xvi CONTENTS

9.10 Performance in the Presence of Outdated orImperfect Channel Estimates 4569.10.1 Maximal-Ratio Combining 4579.10.2 Noncoherent EGC over Rician Fast Fading 4589.10.3 Selection Combining 4619.10.4 Switched Diversity 462

9.10.4.1 SSC Output Statistics 4629.10.4.2 Average SNR 4639.10.4.3 Average Probability of Error 463

9.10.5 Numerical Results 4649.11 Combining in Diversity-Rich Environments 466

9.11.1 Two-Dimensional Diversity Schemes 4669.11.1.1 Performance Analysis 4689.11.1.2 Numerical Examples 469

9.11.2 Generalized Selection Combining 4699.11.2.1 I.I.D. Rayleigh Case 4729.11.2.2 Non-I.I.D. Rayleigh Case 4929.11.2.3 I.I.D. Nakagami-m Case 4979.11.2.4 Partial-MGF Approach 5029.11.2.5 I.I.D. Weibull Case 510

9.11.3 Generalized Selection Combining withThreshold Test per Branch (T-GSC) 5129.11.3.1 Average Error Probability

Performance 5159.11.3.2 Outage Probability Performance 5209.11.3.3 Performance Comparisons 524

9.11.4 Generalized Switched Diversity (GSSC) 5319.11.4.1 GSSC Output Statistics 5319.11.4.2 Average Probability of Error 532

9.11.5 Generalized Selection Combining Based onthe Log-Likelihood Ratio 5329.11.5.1 Optimum (LLR-Based) GSC for

Equiprobable BPSK 5339.11.5.2 Envelope-Based GSC 5369.11.5.3 Optimum GSC for Noncoherently

Detected Equiprobable OrthogonalBFSK 536

9.12 Post-detection Combining 5379.12.1 System and Channel Models 537

9.12.1.1 Overall System Description 5379.12.1.2 Channel Model 5379.12.1.3 Receiver 539

9.12.2 Post-detection Switched CombiningOperation 5399.12.2.1 Switching Strategy and Mechanism 539

CONTENTS xvii

9.12.2.2 Switching Threshold 5409.12.3 Average BER Analysis 540

9.12.3.1 Identically Distributed Branches 5429.12.3.2 Nonidentically Distributed

Branches 5429.12.4 Rayleigh Fading 543

9.12.4.1 Identically Distributed Branches 5449.12.4.2 Nonidentically Distributed

Branches 5479.12.5 Impact of the Severity of Fading 548

9.12.5.1 Average BER 5509.12.5.2 Numerical Examples and

Discussion 5529.12.6 Extension to Orthogonal M-FSK 552

9.12.6.1 System Model and SwitchingOperation 552

9.12.6.2 Average Probability of Error 5559.12.6.3 Numerical Examples 562

9.13 Performance of Dual-Branch Diversity CombiningSchemes over Log-Normal Channels 5669.13.1 System and Channel Models 5669.13.2 Maximal-Ratio Combining 568

9.13.2.1 Moments of the Output SNR 5689.13.2.2 Outage Probability 5709.13.2.3 Extension to Equal Gain

Combining 5719.13.3 Selection Combining 571

9.13.3.1 Moments of the Output SNR 5729.13.3.2 Outage Probability 575

9.13.4 Switched Combining 5759.13.4.1 Moments of the Output SNR 5769.13.4.2 Outage Probability 581

9.14 Average Outage Duration 5849.14.1 System and Channel Models 585

9.14.1.1 Fading Channel Models 5859.14.1.2 GSC Mode of Operation 585

9.14.2 Average Outage Duration and AverageLevel Crossing Rate 5869.14.2.1 Problem Formulation 5869.14.2.2 General Formula for the Average

LCR of GSC 5869.14.3 I.I.D. Rayleigh Fading 589

9.14.3.1 Generic Expressions for GSC 5899.14.3.2 Special Cases: SC and MRC 590

9.14.4 Numerical Examples 591

xviii CONTENTS

9.15 Multiple-Input/Multiple-Output (MIMO) AntennaDiversity Systems 594

9.15.1 System, Channel, and Signal Models 5949.15.2 Optimum Weight Vectors and Output SNR 5959.15.3 Distributions of the Largest Eigenvalue of

Noncentral Complex Wishart Matrices 5969.15.3.1 CDF of S 5969.15.3.2 PDF of S 5989.15.3.3 PDF of Output SNR and Outage

Probability 5999.15.3.4 Special Cases 6009.15.3.5 Numerical Results and Discussion 601

References 604

Appendix 9A. Alternative Forms of the Bit ErrorProbability for a Decision Statistic that Is aQuadratic Form of Complex Gaussian RandomVariables 619

Appendix 9B. Simple Numerical Techniques forInversion of Laplace Transform of CumulativeDistribution Functions 625

9B.1 Euler Summation-Based Technique 6259B.2 Gauss–Chebyshev Quadrature-Based

Technique 626

Appendix 9C. The Relation between the PowerCorrelation Coefficient of Correlated RicianRandom Variables and the Correlation Coefficientof Their Underlying Complex Gaussian RandomVariables 627

Appendix 9D. Proof of Theorem 9.1 631

Appendix 9E. Direct Proof of Eq. (9.438) 632

Appendix 9F. Special Definite Integrals 634

PART 4 MULTIUSER COMMUNICATION SYSTEMS

CHAPTER 10 Outage Performance of Multiuser CommunicationSystems 63910.1 Outage Probability in Interference-Limited Systems 640

10.1.1 A Probability Related to the CDF of theDifference of Two Chi-Square Variates withDifferent Degrees of Freedom 640

CONTENTS xix

10.1.2 Fading and System Models 64310.1.2.1 Channel Fading Models 64310.1.2.2 Desired and Interference Signals

Model 64410.1.3 A Generic Formula for the Outage

Probability 64410.1.3.1 Nakagami/Nakagami Scenario 64510.1.3.2 Rice/Rice Scenario 64610.1.3.3 Rice/Nakagami Scenario 64710.1.3.4 Nakagami/Rice Scenario 647

10.2 Outage Probability with a Minimum Desired SignalPower Constraint 648

10.2.1 Models and Problem Formulation 64810.2.1.1 Fading and System Models 64810.2.1.2 Outage Probability Definition 648

10.2.2 Rice/I.I.D. Nakagami Scenario 64910.2.2.1 Rice/I.I.D. Rayleigh Scenario 64910.2.2.2 Extension to Rice/I.I.D. Nakagami

Scenario 65210.2.2.3 Numerical Examples 652

10.2.3 Nakagami/I.I.D. Rice Scenario 65410.2.3.1 Rayleigh/I.I.D. Rice Scenario 65410.2.3.2 Extension to Nakagami/I.I.D. Rice

Scenario 65610.2.3.3 Numerical Examples 657

10.3 Outage Probability with Dual-Branch SC and SSCDiversity 659

10.3.1 Fading and System Models 66110.3.2 Outage Performance with Minimum Signal

Power Constraint 66110.3.2.1 Selection Combining 66210.3.2.2 Switch-and-Stay Combining 66310.3.2.3 Numerical Examples 664

10.4 Outage Rate and Average Outage Duration ofMultiuser Communication Systems 667

References 671

Appendix 10A. A Probability Related to the CDFof the Difference of Two Chi-Square Variates withDifferent Degrees of Freedom 674

Appendix 10B. Outage Probability in theNakagami/Nakagami Interference-LimitedScenario 678

xx CONTENTS

CHAPTER 11 Optimum Combining—a Diversity Technique forCommunication over Fading Channels in thePresence of Interference 681

11.1 Performance of Diversity Combining Receivers 682

11.1.1 Single Interferer; Independent, IdenticallyDistributed Fading 68211.1.1.1 Rayleigh Fading—Exact

Evaluation of Average Bit ErrorProbability 686

11.1.1.2 Rayleigh Fading—ApproximateEvaluation of Average Bit ErrorProbability 689

11.1.1.3 Extension to Other Modulations 69211.1.1.4 Rician Fading—Evaluation of

Average Bit Error Probability 69311.1.1.5 Nakagami-m Fading—Evaluation

of Average Bit Error Probability 69511.1.2 Multiple Equal Power Interferers;

Independent, Identically Distributed Fading 69711.1.2.1 Number of Interferers Less than

Number of Array Elements 70011.1.2.2 Number of Interferers Equal to or

Greater than Number of ArrayElements 706

11.1.3 Comparison with Results for MRC in thePresence of Interference 710

11.1.4 Multiple Arbitrary Power Interferers;Independent, Identically Distributed Fading 71511.1.4.1 Average SEP of M-PSK 71511.1.4.2 Numerical Results 716

11.1.5 Multiple-Symbol Differential Detection inthe Presence of Interference 71811.1.5.1 Decision Metric 71811.1.5.2 Average BEP 718

11.2 Optimum Combining with Multiple Transmit andReceive Antennas 721

11.2.1 System, Channel, and Signals Models 72111.2.2 Optimum Weight Vectors and Output SIR 72311.2.3 PDF of Output SIR and Outage Probability 723

11.2.3.1 PDF of Output SIR 72411.2.3.2 Outage Probability 72411.2.3.3 Special Case When Lt = 1 725

11.2.4 Key Observations 72611.2.4.1 Distribution of Antenna Elements 726

CONTENTS xxi

11.2.4.2 Effects of Correlation betweenReceiver Antenna Pairs 726

11.2.5 Numerical Examples 727

References 729

Appendix 11A. Distributions of the LargestEigenvalue of Certain Quadratic Forms inComplex Gaussian Vectors 732

11A.1 General Result 73211A.2 Special Case 733

CHAPTER 12 Direct-Sequence Code-Division Multiple Access(DS-CDMA) 73512.1 Single-Carrier DS-CDMA Systems 736

12.1.1 System and Channel Models 73612.1.1.1 Transmitted Signal 73612.1.1.2 Channel Model 73712.1.1.3 Receiver 738

12.1.2 Performance Analysis 73912.1.2.1 General Case 74012.1.2.2 Application to Nakagami-m

Fading Channels 740

12.2 Multicarrier DS-CDMA Systems 741

12.2.1 System and Channel Models 74212.2.1.1 Transmitter 74212.2.1.2 Channel 74312.2.1.3 Receiver 74312.2.1.4 Notations 744

12.2.2 Performance Analysis 74512.2.2.1 Conditional SNR 74512.2.2.2 Average BER 749

12.2.3 Numerical Examples 750

References 754

PART 5 CODED COMMUNICATION SYSTEMS

CHAPTER 13 Coded Communication over Fading Channels 75913.1 Coherent Detection 761

13.1.1 System Model 76113.1.2 Evaluation of Pairwise Error Probability 763

13.1.2.1 Known Channel State Information 76413.1.2.2 Unknown Channel State

Information 768

xxii CONTENTS

13.1.3 Transfer Function Bound on Average BitError Probability 77213.1.3.1 Known Channel State Information 77413.1.3.2 Unknown Channel State

Information 77413.1.4 An Alternative Formulation of the Transfer

Function Bound 77413.1.5 An Example 775

13.2 Differentially Coherent Detection 781

13.2.1 System Model 78113.2.2 Performance Evaluation 783

13.2.2.1 Unknown Channel StateInformation 783

13.2.2.2 Known Channel State Information 78513.2.3 An Example 785

13.3 Numerical Results—Comparison between the TrueUpper Bounds and Union–Chernoff Bounds 787

References 792

Appendix 13A. Evaluation of a Moment GeneratingFunction Associated with Differential Detection ofM-PSK Sequences 793

CHAPTER 14 Multichannel Transmission—Transmit Diversityand Space-Time Coding 79714.1 A Historical Perspective 799

14.2 Transmit versus Receive Diversity—BasicConcepts 800

14.3 Alamouti’s Diversity Technique—a SimpleTransmit Diversity Scheme Using TwoTransmit Antennas 803

14.4 Generalization of Alamouti’s Diversity Techniqueto Orthogonal Space-Time Block Code Designs 809

14.5 Alamouti’s Diversity Technique Combined withMultidimensional Trellis-Coded Modulation 812

14.5.1 Evaluation of Pairwise Error ProbabilityPerformance on Fast Rician FadingChannels 814

14.5.2 Evaluation of Pairwise Error ProbabilityPerformance on Slow Rician FadingChannels 817

14.6 Space-Time Trellis-Coded Modulation 818

CONTENTS xxiii

14.6.1 Evaluation of Pairwise Error ProbabilityPerformance on Fast Rician FadingChannels 820

14.6.2 Evaluation of Pairwise Error ProbabilityPerformance on Slow Rician FadingChannels 821

14.6.3 An Example 82414.6.4 Approximate Evaluation of Average Bit

Error Probability 82714.6.4.1 Fast-Fading Channel Model 82714.6.4.2 Slow-Fading Channel Model 829

14.6.5 Evaluation of the Transfer Function UpperBound on Average Bit Error Probability 83114.6.5.1 Fast-Fading Channel Model 83114.6.5.2 Slow-Fading Channel Model 833

14.7 Other Combinations of Space-Time Block Codesand Space-Time Trellis Codes 833

14.7.1 Super-Orthogonal Space-Time Trellis Codes 83414.7.1.1 The Parameterized Class of

Space-Time Block Codes andSystem Model 834

14.7.1.2 Evaluation of the Pairwise ErrorProbability 836

14.7.1.3 Extension of the Results toSuper-Orthogonal Codes withMore than Two Transmit Antennas 844

14.7.1.4 Approximate Evaluation ofAverage Bit Error Probability 845

14.7.1.5 Evaluation of the TransferFunction Upper Bound on theAverage Bit Error Probability 846

14.7.1.6 Numerical Results 84814.7.2 Super-Quasi-Orthogonal Space-Time Trellis

Codes 85014.7.2.1 Signal Model 85014.7.2.2 Evaluation of Pairwise Error

Probability 85214.7.2.3 Examples 85314.7.2.4 Numerical Results 857

14.8 Disclaimer 858

References 859

CHAPTER 15 Capacity of Fading Channels 86315.1 Channel and System Model 863

xxiv CONTENTS

15.2 Optimum Simultaneous Power and Rate Adaptation 865

15.2.1 No Diversity 86515.2.2 Maximal-Ratio Combining 866

15.3 Optimum Rate Adaptation with Constant TransmitPower 867

15.3.1 No Diversity 86815.3.2 Maximal-Ratio Combining 869

15.4 Channel Inversion with Fixed Rate 869

15.4.1 No Diversity 87015.4.2 Maximal-Ratio Combining 870

15.5 Numerical Examples 871

15.6 Capacity of MIMO Fading Channels 876

References 877

Appendix 15A. Evaluation of Jn(µ) 878Appendix 15B. Evaluation of In(µ) 880

Index 883

PREFACE

Regardless of the branch of science or engineering, theoreticians have always beenenamored with the notion of expressing their results in the form of closed-formexpressions. Quite often the elegance of the closed-form solution is overshadowedby the complexity of its form and the difficulty in evaluating it numerically. Insuch instances, one becomes motivated to search instead for a solution that issimple in form and likewise simple to evaluate. A further motivation is that themethod used to derive these alternative simple forms should also be applicable insituations where closed-form solutions are ordinarily unobtainable. The search forand ability to find such a unified approach for problems dealing with the evaluationof the performance of digital communication over generalized fading channels iswhat provided the impetus to write this textbook, the result of which represents thebackbone for the material contained within its pages.

For at least four decades, researchers have studied problems of this type andsystem engineers have used the theoretical and numerical results reported in theliterature to guide the design of their systems. While the results from the ear-lier years dealt mainly with simple channel models, such as Rayleigh or Ricianmultipath fading, the applications in more recent years have become increasinglysophisticated, thereby requiring more complex models and improved diversity tech-niques. Along with the complexity of the channel model comes the complexity ofthe analytical solution that enables one to assess performance. With the mathe-matical tools that were previously available, the solutions to such problems whenpossible had to be expressed in complicated mathematical form that provided littleinsight into the dependence of the performance on the system parameters. Surpris-ingly enough, not until 1998 had anyone demonstrated a unified approach that notonly allows previously obtained complicated results to be simplified both analyti-cally and computationally but also permits new results to be obtained for specialcases that heretofore resisted solution in a simple form. This approach was firstintroduced to the public by the authors in a tutorial-style article that appeared inthe September 1998 issue of the IEEE Proceedings. Since that time, it has spawneda large wave of publications on the subject in the technical journal and conferenceliterature, by both the authors and many others and, based on the variety of appli-cations to which it has already been applied, will no doubt continue well into the

xxv

xxvi PREFACE

new millennium. The key to the success of this approach relies on employing alter-native representations of classic functions arising in the error probability analysisof digital communication systems (e.g., the Gaussian Q-function1 and the MarcumQ-function) in such a manner that the resulting expressions for various performancemeasures such as average bit or symbol error rate are in a form that is rarely morecomplicated than a single integral with finite limits and an integrand composed ofelementary (e.g., exponential and trigonometric) functions. By virtue of replacingthe conventional forms of the above-mentioned functions by their alternative rep-resentations, the integrand will contain the moment generating function (MGF) ofthe instantaneous fading SNR, and as such the unified approach is referred to asthe MGF-based approach.

The first edition of this book was aimed at collecting and documenting the hugecompendium of results contained in the myriad of contributions developed fromthe MGF-based approach that had been reported until that time and, by virtue ofits unified notation and collocation in a single publication, would thereby be usefulto both students and researchers in the field. In 1999 the manuscript for the firstedition was submitted to the publisher. Since that time, a great deal of additionalsignificant work on the subject has been performed and reported on in the literature,so much so that a second edition of the book is warranted and will be extremelybeneficial to these same researchers and students in bringing them up to date onthese new developments.

Perhaps the most significant of these new developments is the explosion ofinterest and research that has taken place in the area of transmit diversity and space-time coding and the associated multiple-input/multiple-output (MIMO) channel, asubject that was briefly alluded to but not discussed in any detail in the first edition.One of the key elements of the second edition is a comprehensive chapter on thisall-important subject that, in keeping with the main theme of the book, deals withthe performance evaluation aspects of such systems. The performance of MIMOsystems is also treated from other perspectives elsewhere in the text.

Aside from these developments, many new and exciting results have been devel-oped by the authors as well as other researchers that (1) have led to new andimproved diversity schemes and (2) allow for the performance analysis of previ-ously known schemes operating in new and different fading scenarios not discussedin the first edition. A few of these developments are (1) new alternative forms forclassic mathematical functions such as the second-order Gaussian Q-function andalso higher powers of the first-order Gaussian Q-function; (2) improved diver-sity schemes such as threshold and postdetection generalized selection combining,switch-and-examine combining, and switch-and-wait combining; (3) new channelfading models of interest in wireless and mobile applications; (4) new bounds onsystem performance in the presence of fading; and (5) new mathematical results

1The Gaussian Q-function Q (x) has a one-to-one mapping with the complementary error function [i.e.,

Q (x) = 12 erfc(x/

√2)

] commonly found in standard mathematical tabulations. In much of the engi-

neering literature, however, the two functions are used interchangeably, and as a matter of conveniencewe shall do the same in this book.

PREFACE xxvii

related to quadratic forms in Gaussian random variables and the difference in chi-square random variables with different degrees of freedom, allowing for the analysisof practical communication performance measures such as the outage probabilityof digital communication systems in the presence of multiple interferers. In fact,because of the importance of the latter issue in multiuser communication systems,a new chapter has been added on this subject. The list above is only a small sampleof the voluminous amount of material (on the order of several hundred pages) thathas been added to the second edition.

As in the first edition, in dealing with the application of the MGF-based approach,the coverage in this edition of the book is extremely broad in that coherent, differ-entially coherent, partially coherent, and noncoherent communication systems areall handled as well as a large variety of fading channel models typical of com-munication links of practical interest. Both single- and multichannel reception arediscussed, and in the case of the latter, a large variety of diversity types are consid-ered. In fact, the chapter on multichannel reception (Chapter 9) is by itself now over325 manuscript pages long and, in reality, could stand alone as its own textbook.For each combination of communication (modulation/detection) type, channel fad-ing model, and diversity type, expressions for various system performance measuresare obtained in a form that can be readily evaluated.2 All cases considered corre-spond to real practical channels, and in many instances the closed-form expressionsobtained can be evaluated numerically on a handheld calculator.

In writing this book, our intent was to spend as little space as possible duplicatingmaterial dealing with basic digital communication theory and system performanceevaluation that is well documented in many fine textbooks on the subject. Rather,this book serves to advance the material found in these texts and as such is of mostvalue to those desiring to extend their knowledge beyond what ordinarily might becovered in the classroom. In this regard, the book should have a strong appeal tograduate students doing research in the field of digital communications over fadingchannels as well as practicing engineers who are responsible for the design andperformance evaluation of such systems. With regard to the latter, the book containscopious numerical evaluations that are illustrated in the form of parametric perfor-mance curves (e.g., average error probability versus average signal-to-noise ratio).The applications chosen for the numerical illustrations correspond to practical sys-tems and as such the performance curves provided will have far more than academicvalue. The availability of such a large collection of system performance curves ina single compilation allows researchers and system designers to perform tradeoffstudies among the various communication type/fading channel combinations so asto determine the optimum choice in the face of their available constraints.

The structure of the book is composed of five parts, each with its own expresspurpose. The first part contains an introduction to the subject of communicationsystem performance evaluation followed by discussions of the various types offading channel models and modulation/detection schemes that together form the

2The terms bit error probability (BEP) and symbol error probability (SEP) are quite often used asalternatives to bit error rate (BER) and symbol error rate (SER). With no loss in generality, we shallemploy both usages in this text.

xxviii PREFACE

overall system. Part 2 starts by introducing the alternative forms of the classicfunctions mentioned above and then proceeds to show how these forms can beused to (1) evaluate certain integrals characteristic of communication system errorprobability performance and (2) find new representations for certain probabilitydensity and distribution functions typical of correlated fading applications. Part 3is the “heart and soul” of the book since, in keeping with its title, the primary focusof this part is on performance evaluation of the various types of fading channelmodels and modulation/detection schemes introduced in Part 1 both for single andmultichannel (diversity) reception. Before presenting this comprehensive perfor-mance evaluation study, however, Part 3 begins by deriving the optimum receiverstructures corresponding to a variety of combinations concerning the knowledge orlack thereof of the fading parameters: amplitude, phase, and delay. Several of thesestructures might be deemed as too complex to implement in practice; nevertheless,their performance serves as a benchmark against which many suboptimum butpractical structures discussed in the remainder of the chapter might be compared.Part 4, which deals with multiuser communications, considers first the problem ofoutage probability evaluation followed by optimum combining (diversity) in thepresence of cochannel interference. The unified approach is then applied to study-ing the performance of single- and multiple-carrier direct-sequence code-divisionmultiple-access (DS-CDMA) systems typical of the current digital cellular wire-less standard. Part 5 extends the theory developed in the previous parts for uncodedcommunication to error-correction-coded systems and then space-time-coded sys-tems and concludes with a discussion of the capacity of fading channels.

Whereas the first edition has already established itself as the classic referencetext on the subject with no apparent competition in sight, it is a safe bet thatthe second edition will continue to maintain that reputation for years to come. Theauthors know of no other textbook currently on the market that addresses the subjectof digital communication over fading channels in as comprehensive and unified amanner as is done herein. In fact, prior to the publication of this book, to the authors’best knowledge there existed only two works (the textbook by Kennedy [1] and thereprint book by Brayer [2]) that, like our book, are totally dedicated to this subject,and both them are more than a quarter of a century old. While a number of othertextbooks [3–11] devote part of their contents3 to fading channel performanceevaluation, by comparison with our book, the treatment is brief and as such isincomplete. Because of this, we believe that our textbook is unique in the field.

By way of acknowledgment, the authors wish to express their personal thanksto Dr. Payman Arabshahi of the Jet Propulsion Laboratory, Pasadena, CA for pro-viding his invaluable help and consultation in preparing the submitted electronicversion of the manuscript. Mohamed-Slim Alouini would also like to also liketo express his sincere acknowledgment and gratitude to his PhD advisor Prof.Andrea J. Goldsmith of Stanford University, Palo Alto, CA for her guidance, sup-port, and constant encouragement. Some of the material presented in Chapters 9,

3Although Ref: 11 is a book that is entirely devoted to digital communication over fading channels, thefocus is on error-correction-coded modulation and therefore would relate primarily only to Chapter 13of our book.

PREFACE xxix

12, and 15 are the result of joint work with Prof. Goldsmith. Mohamed-SlimAlouini would also like to thank his past and current doctoral students Mr. MingKang, Prof. Hong-Chuan Yang, Dr. Young-Chai Ko, and Ms. Lin Yang for theirmajor contributions to some of the results presented in Chapters 9, 10, and 11.The contributions of Prof. Mazen O. Hasna, Mr. Fadel F. Digham, Mr. Pavan K.Vitthaladevuni, Prof. Ali Abdi, Dr. Henrik Holm, Mr. Wing C. Lau, Mr. GangHuo, and Dr. Yan Xin to this book and more generally to the research efforts ofMohamed-Slim Alouini are also particularly noteworthy. Finally, Mohamed-SlimAlouini would like to acknowledge the support of the National Science Founda-tion, the Office of the Vice President for Research of the University of Minnesota,and the University of Minnesota’s McKnight Land-Grant Professorship programfor sustaining his research activities in the performance analysis of wireless com-munication systems since 1999.

Marvin K. SimonMohamed-Slim Alouini

Jet Propulsion Laboratory, Pasadena, CaliforniaUniversity of Minnesota, Minneapolis, Minnesota

REFERENCES

1. R. S. Kennedy, Fading Dispersive Communication Channels. New York, NY: Wiley-Interscience, 1969.

2. K. Brayer, ed., Data Communications via Fading Channels. Piscataway, NJ: IEEEPress, 1975.

3. M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques.New York, NY: McGraw-Hill, 1966.

4. W. C. Y. Lee, Mobile Communications Engineering. New York, NY: McGraw-Hill, 1982.

5. J. Proakis, Digital Communications, 4th ed. New York, NY: McGraw-Hill, 1998 (1st,2nd, and 3rd editions in 1983, 1989, and 1995 respectively).

6. M. D. Yacoub, Foundations of Mobile Radio Engineering. Boca Raton, FL: CRCPress, 1993.

7. W. C. Jakes, Microwave Mobile Communication. 2nd ed. Piscataway, NJ: IEEEPress, 1994.

8. K. Pahlavan and A. H. Levesque, Wireless Information Networks, Wiley Series inTelecommunications and Signal Processing. New York, NY: Wiley-Interscience, 1995.

9. G. Stüber, Principles of Mobile Communication. Norwell, MA: Kluwer Academic Pub-lishers, 1996.

10. T. S. Rappaport, Wireless Communications: Principles and Practice. Upper SaddleRiver, NJ: PTR Prentice-Hall, 1996.

11. S. H. Jamali and T. Le-Ngoc, Coded-Modulation Techniques for Fading Channels. Nor-well, MA: Kluwer Academic Publishers, 1994.

NOMENCLATURE

Notation Description or Name of Function and ReferenceCitation for Its Definition

AF Amount of fadingAGC Automatic gain controlAM Amplitude modulationAOD Average outage durationASK Amplitude-shift-keyingAT Absolute thresholdAWGN Additive white Gaussian noiseBx (p, q) Incomplete beta function [1, Eq. (8.391)]BEP Bit error probabilityBER Bit error rateBFSK Binary frequency-shift-keyingBPSK Binary phase-shift-keyingBRGC Binary reflected Gray codeBRS Branch relative strengthCCDF Complementary cumulative distribution functionCCI Cochannel interferenceCDF Cumulative distribution functionCDMA Code-division multiple accessCF Characteristic FunctionCIR Carrier-to-interference ratioCNR Carrier-to-noise ratioCPFSK Continuous-phase frequency-shift-keyingCSI Channel state informationDPSK Differential phase-shift-keyingDS-CDMA Direct-sequence code-division multiple accessEGC Equal gain combiningFSK Frequency-shift-keying

xxxi

xxxii NOMENCLATURE

1F1 (α, γ ; z) Kummer confluent hypergeometric function[1, Eq. (9.210.1)]

2F1 (α, β; γ ; z) Gaussian hypergeometric function [1, Eq. (9.14.2)]pFq (α1, α2, . . . , αp; Generalized hypergeometric function

β1, β2, . . . , βq ; z) [1, Eq. (9.14.1)]GaAs Gallium Arsenide

Gm,np,q

(x

∣∣∣∣a1, . . . , apb1, . . . , bq

)Meijer’s G-function [1, Eq. (9.301)]

GSC Generalized selection combiningγ Instantaneous fading SNRγ Average fading SNR� (x) Gamma function [1, Eq. (8.310.1)]γ (a, x) Incomplete gamma function [1, Eq. (8.350.1)]� (a, x) Complementary incomplete gamma function

[1, Eq. (8.350.2)]Hn (x) nth-order Hermite polynomial [1, Eq. (8.950.1)]Iν (x) νth-order modified Bessel function of the first kind

[1, Eq. (8.431)]I&D Integrate-and-dumpi.i.d. Independent, identically distributedISI Intersymbol interferenceJν (x) νth-order Bessel function of the first kind

[1, Eq. (8.411.1)]K Rician fading parameterLCR Level crossing rateLOS Line of sightMA Multiple accessMAI Multiple-access interferenceM-AM Multiple amplitude modulationMAP Maximum a posterioriM-ASK Multiple amplitude-shift-keyingM-DPSK Multiple differential phase-shift-keyingM-FSK Multiple frequency-shift-keyingM-PSK Multiple phase-shift-keyingMC-CDMA Multicarrier code-division multiple accessMGF Moment generating functionMγ (s) Moment generating function of γMIMO Multiple input/multiple outputMIP Multipath intensity profileMISO Multiple input/single outputML Maximum likelihoodMLSE Maximum-likelihood sequence estimationMMSE Minimum mean-square errorMRC Maximal-ratio combiningMSDD Multiple-symbol differential detection

NOMENCLATURE xxxiii

MSK Minimum-shift-keyingMTCM Multiple trellis-coded modulationNC NoncoherentNSD Normalized standard deviationNT Normalized thresholdOC Optimum combiningOPRA Optimum power and rate adaptationOQPSK Offset (staggered) QPSKORA Optimum rate adaptationPb (E) Bit error probabilityPBI Partial-band interferencePDF Probability density functionPDP Power delay profilePEP Pairwise error probabilityPG Processing gainPLL Phase-locked loopPMF Probability mass functionPN PseudonoisePSAM Pilot-symbol-assisted modulationPs (E) Symbol error probabilityQAM Quadrature amplitude modulationQASK Quadrature amplitude-shift-keyingQ (x) First-order Gaussian Q-function [2, Eq. (26.2.3)]Q (x, y; ρ) Second-order Gaussian Q-function [2, Eq. (26.3.3)]Qm (α, β) mth-order Marcum Q-function [3]Qm,n (α, β) Nuttall Q-function [5,6]QOSTBC Quasi-orthogonal space-time block codeQPSK Quadriphase-shift-keyingROC Region of convergenceRV Random variableSC Selection combiningSC-CDMA Single-carrier code-division multiple accessSEC Switch-and-examine combiningSECps Postselection SECSEO Symbol error outageSEP Symbol error probabilitySER Symbol error rateSIR Signal-to-interference ratioSINR Signal-to-interference plus noise ratioSIMO Single input/multiple outputSIRP Spherically invariant random processSIRV Spherically invariant random variableSISO Single input/single outputSNR Signal-to-noise ratioSOSTTC Super-orthogonal space-time trellis code

xxxiv NOMENCLATURE

SQOSTTC Super-quasi-orthogonal space-time trellis codeSQPSK Staggered quadriphase-shift-keyingSS Spread spectrumSSC Switch-and-stay combiningSTBC Space-time block codeSTC Space-time codeSTTC Space-time trellis codeSWC Scan-and-wait combiningTCM Trellis-coded modulationTDMA Time-division multiple accessT-GSC Generalized selection combining with threshold test

per branchTUB True upper (union) boundTB (m, n, r) Incomplete Toronto function [4]UEP Uniform error probabilityWk.m (z) Whittaker function [1, Eq. (9.222)]

REFERENCES

1. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. SanDiego, CA: Academic Press, 1994.

2. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas,Graphs, and Mathematical Tables, 9th ed. New York, NY: Dover Press, 1972.

3. J. I. Marcum, Table of Q Functions, U.S. Air Force Project RAND Research MemorandumM-339, ASTIA Document AD 11 65 451, Rand Corporation, Santa Monica, CA, January1, 1950.

4. J. I. Marcum and P. Swerling, “Studies of target detection by pulsed radar,” IEEE Trans.Inform. Theory, vol. IT-6, April 1960.

5. A. Nuttall, Some Integrals Involving the Q-Function, Technical Report 4297, Naval Under-water Systems Center, New London, CT, April 17, 1972.

6. M. K. Simon, Probability Distributions Involving Gaussian Random Variables: A Hand-book for Engineers and Scientists, Boston, MA: Kluwer Academic Publishers, 2002.

Marvin K. Simon dedicates this book to his wife, Anita,whose devotion to him and this project never once faded

during its preparation.

Mohamed-Slim Alouini dedicates this book to hisparents and his family.

PART 1FUNDAMENTALS

1

INTRODUCTION

As we continue to step forward into the new millennium with wireless technologiesleading the way in which we communicate, it becomes increasingly clear that thedominant consideration in the design of systems employing such technologies willbe their ability to perform with adequate margin over a channel perturbed by a hostof impairments, not the least of which is multipath fading. This is not to implythat multipath fading channels are something new to be reckoned with; indeed,they have plagued many a system designer for well over 40 years, but rather toserve as a motivation for their ever-increasing significance in the years to come.At the same time, we do not in any way wish to diminish the importance of thefading channel scenarios that occurred well prior to the wireless revolution sinceindeed many of them still exist and will continue to exist in the future. In fact,it is safe to say that whatever means are developed for dealing with the moresophisticated wireless applications will no doubt also be useful for dealing withthe less complicated fading environments of the past.

With the above in mind, what better opportunity is there than now to write acomprehensive book that will provide simple and intuitive solutions to problemsdealing with communication system performance evaluation over fading channels?Indeed, as mentioned in the preface, the primary goal of this book is to presenta unified method for arriving at a set of tools that will allow the system designerto compute the performance of a host of different digital communication systemscharacterized by a variety of modulation/detection types and fading channel models.By “set of tools” we mean a compendium of analytical results that not only alloweasy yet accurate performance evaluation but at the same time provide insight intothe manner in which this performance depends on the key system parameters. Toemphasize what was stated above, the set of tools that will be developed in thisbook are useful not only for the wireless applications that are rapidly filling ourcurrent technical journals but also to a host of others involving satellite, terrestrial,and maritime communications.

Digital Communication over Fading Channels, Second Edition.By Marvin K. Simon and Mohamed-Slim AlouiniISBN 0-471-64953-8 Copyright 2005 John Wiley & Sons, Inc.

3

4 INTRODUCTION

Our repetitive use of the word “performance” thus far brings us to the purposeof this introductory chapter, namely, to provide several measures of performancerelated to practical communication system design and to begin exploring the ana-lytical methods by which they may be evaluated. While the deeper meaning ofthese measures will be truly understood only after their more formal definitions arepresented in the chapters that follow, the introduction of these terms here servesto illustrate the various possibilities that exist depending on both need and relativeease of evaluation.

1.1 SYSTEM PERFORMANCE MEASURES

1.1.1 Average Signal-to-Noise Ratio (SNR)

Probably the most common and well understood performance measure character-istic of a digital communication system is signal-to-noise ratio (SNR). Most oftenthis is measured at the output of the receiver and is thus directly related to the datadetection process itself. Of the several possible performance measures that exist, itis typically the easiest to evaluate and most often serves as an excellent indicatorof the overall fidelity of the system. While traditionally the term “noise” in signal-to-noise ratio refers to the ever-present thermal noise at the input to the receiver,in the context of a communication system subject to fading impairment, the moreappropriate performance measure is average SNR, where the term “average” refersto statistical averaging over the probability distribution of the fading. In simplemathematical terms, if γ denotes the instantaneous SNR [a random variable (RV)]at the receiver output that includes the effect of fading, then

γ�=

∫ ∞0

γpγ (γ ) dγ (1.1)

is the average SNR, where pγ (γ ) denotes the probability density function (PDF)of γ . In order to begin to get a feel for what we will shortly describe as a unifiedapproach to performance evaluation, we first rewrite (1.1) in terms of the momentgenerating function (MGF) associated with γ :

Mγ (s) =∫ ∞

0pγ (γ ) e

sγ dγ (1.2)

Taking the first derivative of (1.2) with respect to s and evaluating the result ats = 0, we immediately see from (1.1) that

γ = dMγ (s)ds

|s=0 (1.3)

that is, the ability to evaluate the MGF of the instantaneous SNR (perhaps in closedform) allows immediate evaluation of the average SNR via a simple mathematicaloperation, namely, differentiation.

SYSTEM PERFORMANCE MEASURES 5

To gain further insight into the power of the statement above, we note that inmany systems, particularly those dealing with a form of diversity (multichannel)reception known as maximal-ratio combining (MRC) (to be discussed in greatdetail in Chapter 9), the output SNR, γ , is expressed as a sum (combination) ofthe individual branch (channel) SNRs, namely, γ = ∑Ll=1 γl , where L denotesthe number of channels combined. In addition, it is often reasonable in practiceto assume that the channels are independent of each other, that is, that the RVsγl

∣∣Ll=1 are themselves independent. In such instances, the MGF Mγ (s) can be

expressed as the product of the MGFs associated with each channel [i.e., Mγ (s) =∏Ll=1 Mγl (s)], which as we shall later on in the text can, for a large variety of

fading channel statistical models, be computed in closed form.1 By contrast, evenwith the assumption of channel independence, the computation of the PDF pγ (γ ),which requires convolutional of the various PDFs pγl (γl)

∣∣Ll=1 that characterize the

L channels, can still be a monumental task. Even in the case where these individualchannel PDFs are of the same functional form but are characterized by differentaverage SNRs, γ l , the evaluation of pγ (γ ) can still be quite tedious. Such is thepower of the MGF-based approach; namely, it circumvents the need for finding thefirst-order PDF of the output SNR, provided that one is interested in a performancemeasure that can be expressed in terms of the MGF. Of course, for the case ofaverage SNR, the solution is extremely simple, namely, γ = ∑Ll=1 γ l regardless ofwhether the channels are independent, and in fact, one never needs to find the MGFat all. However, for other performance measures and also the average SNR of othercombining statistics, such as the sum of an ordered set of random variables typicalof generalized selection combining (GSC) (to be discussed in Chapter 9), mattersare not quite this simple and the points made above for justifying an MGF-basedapproach are, as we shall see, especially significant.

1.1.2 Outage Probability

Another standard performance criterion characteristic of diversity systems operatingover fading channels is the so-called outage probability-denoted by Pout and definedas the probability that the instantaneous error probability exceeds a specified valueor equivalently the probability that the output SNR, γ , falls below a certain specifiedthreshold, γth. Mathematically speaking, we have

Pout =∫ γth

0pγ (γ ) dγ (1.4)

which is the cumulative distribution function (CDF) of γ , namely, Pγ (γ ), evalu-ated at γ = γth. Since the PDF and the CDF are related by pγ (γ ) = dPγ (γ ) /dγ

1Note that the existence of the product form for the MGF Mγ (s) does not necessarily imply that thechannels are identically distributed; thus, each MGF Mγl (s) is allowed to maintain its own identityindependent of the others. Furthermore, even if the channels are not assumed to be independent, therelation in (1.3) is nevertheless valid and in many instances the MGF of the (combined) output can stillbe obtained in closed form.

6 INTRODUCTION

and since Pγ (0) = 0, then the Laplace transforms of these two functions arerelated by2

P̂γ (s) = p̂γ (s)s

(1.5)

Furthermore, since the MGF is just the Laplace transform of the PDF with argu-ment reversed in sign [i.e., p̂γ (s) = Mγ (−s)], then the outage probability canbe found from the inverse Laplace transform of the ratio Mγ (−s) /s evaluatedat γ = γth

Pout = 12πj

∫ σ+j∞σ−j∞

Mγ (−s)s

esγthds (1.6)

where σ is chosen in the region of convergence of the integral in the complex splane. Methods for evaluating inverse Laplace transforms have received widespreadattention in the literature. (A good summary of these can be found in the paper byAbate and Whitt [1]). One such numerical technique that is particularly useful forCDFs of positive RVs (such as instantaneous SNR) is discussed in Appendix 9Band applied therein in Chapter 9. For our purpose here, it is sufficient to recog-nize once again that the evaluation of outage probability can be performed basedentirely on the knowledge of the MGF of the output SNR without ever having tocompute its PDF.

1.1.3 Average Bit Error Probability (BEP)

The third performance criterion and undoubtedly the most difficult of the three tocompute is average bit error probability (BEP).3 On the other hand, it is the onethat is most revealing about the nature of the system behavior and the one mostoften illustrated in documents containing system performance evaluations; thus, itis of primary interest to have a method for its evaluation that reduces the degreeof difficulty as much as possible.

The primary reason for the difficulty in evaluating average BEP lies in thefact that the conditional (on the fading) BEP is, in general, a nonlinear functionof the instantaneous SNR, as the nature of the nonlinearity is a function of themodulation/detection scheme employed by the system. Thus, for example, in themultichannel case, the average of the conditional BEP over the fading statistics

2The symbol “·̂” above a function denotes its Laplace transform.3The discussion that follows applies, in principle, equally well to average symbol error probability(SEP). The specific differences between the two are explored in detail in the chapters dealing withsystem performance. Furthermore, the terms bit error rate (BER) and symbol error rate (SER) areoften used in the literature as alternatives to BEP and SEP. Rather than choose a preference, in thistext we shall use these terms interchangeably.

SYSTEM PERFORMANCE MEASURES 7

is not a simple average of the per channel performance measure as was true foraverage SNR. Nevertheless, we shall see momentarily that an MGF-based approachis still quite useful in simplifying the analysis and in a large variety of cases allowsunification under a common framework.

Suppose first that the conditional BEP is of the form

Pb (E |γ ) = C1 exp (−a1γ ) (1.7)

such as would be the case for differentially coherent detection of phase-shift-keying(PSK) or noncoherent detection of orthogonal frequency-shift-keying (FSK) (seeChapter 8). Then, the average BEP can be written as

Pb(E)�=

∫ ∞0

Pb(E |γ )pγ (γ )dγ =∫ ∞

0C1 exp (−a1γ ) pγ (γ ) dγ = C1Mγ (−a1)

(1.8)where again Mγ (s) is the MGF of the instantaneous fading SNR and depends onlyon the fading channel model assumed.

Suppose next that the nonlinear functional relationship between Pb (E |γ ) andγ is such that it can be expressed as an integral whose integrand has an exponentialdependence on γ in the form of (1.7),4

Pb (E |γ ) =∫ ξ2

ξ1

C2h (ξ) exp (−a2g (ξ) γ ) dξ (1.9)

where for our purpose here h (ξ) and g (ξ) are arbitrary functions of the inte-gration variable and typically both ξ1 and ξ2 are finite (although this is not anabsolute requirement for what follows).5 While not at all obvious at this point,suffice it to say that a relationship of the form in (1.9) can result from employingalternative forms of such classic nonlinear functions as the Gaussian Q-functionand Marcum Q-function (see Chapter 4), which are characteristic of the relation-ship between Pb (E |γ ) and γ corresponding to, for example, coherent detectionof PSK and noncoherent detection of quadriphase-shift-keying (QPSK), respec-tively. Still another possibility is that the nonlinear functional relationship betweenPb (E |γ ) and γ is inherently in the form of (1.9); thus, no alternative representa-tion need be employed. An example of such occurs for the conditional symbol errorprobability (SEP) associated with coherent and differentially coherent detection ofM-ary PSK (M-PSK) (see Chapter 8). Regardless of the particular case at hand,

4In the more general case, the conditional BEP might be expressed as a sum of integrals of the typein (1.9).5In principle, (1.9) includes (1.7) as a special case if h (ξ) is allowed to assume the form of a Diracdelta function located within the interval ξ1 ≤ ξ ≤ ξ2.

8 INTRODUCTION

once again averaging (1.9) over the fading gives (after interchanging the order ofintegration)

Pb (E) =∫ ∞

0Pb(E |γ )pγ (γ )dγ =

∫ ∞0

∫ ξ2ξ1

C2h (ξ) exp(−a2g (ξ) γ )dξpγ (γ )dγ

= C2∫ ξ2

ξ1

h (ξ)

∫ ∞0

exp (−a2g (ξ) γ )pγ (γ ) dγ dξ (1.10)

= C2∫ ξ2

ξ1

h (ξ) Mγ (−a2g (ξ)) dξ

As we shall see later on in the text, integrals of the form in (1.10) can, formany special cases, be obtained in closed form. At the very worst, with rareexception, the resulting expression will be a single integral with finite limits andan integrand composed of elementary functions.6 Since (1.8) and (1.10) cover awide variety of different modulation/detection types and fading channel models,we refer to this approach for evaluating average error probability as the unifiedMGF-based approach and the associated forms of the conditional error probabilityas the desired forms. The first notion of such a unified approach was discussedin Ref. 2 and laid the groundwork for much of the material that follows inthis text.

It goes without saying that not every fading channel communication problemfits this description; thus, alternative, but still simple and accurate, techniques aredesirable for evaluating system error probability in such circumstances. One classof problems for which a different form of MGF-based approach is possible relatesto communication with symmetric binary modulations wherein the decision mech-anism constitutes a comparison of a decision variable with a zero threshold. Asidefrom the obvious uncoded applications, the above-mentioned class also includesthe evaluation of pairwise error probability in error-correction-coded systems asdiscussed in Chapter 12. In mathematical terms, letting D |γ denote the decisionvariable,7 then the corresponding conditional BEP is of the form (assuming arbi-trarily that a positive data bit was transmitted)

Pb (E |γ ) = Pr {D |γ < 0} =∫ 0

−∞pD|γ (D) dD = PD|γ (0) (1.11)

where pD|γ (D) and PD|γ (D) are, respectively, the PDF and CDF of this variable.Aside from the fact that the decision variable D |γ can, in general, take on bothpositive and negative values whereas the instantaneous fading SNR, γ , is restrictedto only positive values, there is a strong resemblance between the binary probability

6As we shall see in Chapter 4, the h (ξ) and g (ξ) that result from the alternative representations of theGaussian and Marcum Q-functions are composed of simple trigonometric functions.7The notation “D |γ ” is not meant to imply that the decision variable explicitly depends on the fadingSNR. Rather, it is merely intended to indicate the dependence of this variable on the fading statisticsof the channel. More about this dependence shortly.

SYSTEM PERFORMANCE MEASURES 9

of error in (1.11) and the outage probability in (1.4). Thus, by analogy with (1.6),the conditional BEP of (1.11) can be expressed as

Pb (E |γ ) = 12πj

∫ σ+j∞σ−j∞

MD|γ (−s)s

ds (1.12)

where MD|γ (−s) now denotes the MGF of the decision variable D |γ , that is, thebilateral Laplace transform of pD|γ (D) with argument reversed.

To see how MD|γ (−s) might explicitly depend on γ , we now consider thesubclass of problems where the conditional decision variable D |γ corresponds to aquadratic form of independent complex Gaussian RVs, such as a sum of the squaredmagnitudes of, say, L independent complex Gaussian RVs—a chi-square RV with2L degrees of freedom. Such a form occurs for multiple-(L)-channel receptionof binary modulations with differentially coherent or noncoherent detection (seeChapter 9). In this instance, the MGF MD|γ (s) happens to be exponential in γand has the generic form

MD|γ (s) = f1 (s) exp (γf2 (s)) (1.13)

If as before we let γ = ∑Ll=1 γl , then substituting (1.13) into (1.12) and averagingover the fading results in the average BEP8

Pb (E) = 12πj

∫ σ+j∞σ−j∞

MD (−s)s

ds (1.14)

where

MD (s)�=

∫ ∞0

MD|γ (s) pγ (γ ) dγ = f1 (s)∫ ∞

0exp (γf2 (s)) pγ (γ ) dγ

= f1 (s)Mγ (f2 (s))(1.15)

is the unconditional MGF of the decision variable, which also has the product form

MD (s) = f1 (s)L∏

l=1Mγl (f2 (s)) (1.16)

Finally, by virtue of the fact that the MGF of the decision variable can be expressedin terms of the MGF of the fading variable (SNR) as in (1.15) [or (1.16)], then,analogous to (1.10), we are once again able to evaluate the average BEP solely onthe basis of knowledge of the latter MGF.

It is not immediately obvious how to extend the inverse Laplace transformtechnique discussed in Appendix 9B to CDFs of bilateral RVs; thus other methods

8The approach for computing average BEP as described by (1.13) was also described by Biglieriet al. [3] as a unified approach to computing error probabilities over fading channels.

10 INTRODUCTION

for performing this inversion are required. A number of these, including contourintegration using residues, saddle point integration, and numerical integration byGauss–Chebyshev quadrature rules, are discussed in the literature [3–6] and willbe covered later on in the text.

Although the methods dictated by (1.14) and (1.8) or (1.10) cover a wide varietyof problems dealing with the performance of digital communication systems overfading channels, there are still some situations that don’t lend themselves to eitherof these two unifying methods. An example of such is the evaluation of the biterror probability performance of an M-ary noncoherent orthogonal system operatingover an L-path diversity channel (see Chapter 9). However, even in this case thereexists an MGF-based approach that greatly simplifies the problem and allows for aresult [7] more general than that previously reported by Weng and Leung [8]. Wenow briefly outline the method, leaving the more detailed treatment to Chapter 9.

Consider an M-ary communication system where, rather than comparing a singledecision variable with a threshold, one decision variable U1 |γ is compared withthe remaining M − 1 decision variables Um, m = 2, 3, . . . , M , all of which do notdepend on the fading statistics.9 Specifically, a correct symbol decision is made ifU1 |γ is greater than Um, m = 2, 3, . . . , M . Assuming that the M decision variablesare independent, then, in mathematical terms, the probability of correct decision isgiven by

Ps (C |γ ; u1 ) = Pr {U2 < u1, U3 < u1, . . . , UM < u1 |U1 |γ = u1 }

= [Pr {U2 < u1 |U1 |γ = u1 }]M−1 =

[∫ u10

pU2 (u2) du2

]M−1

= [1 − (1 − PU2 (u1))]M−1

(1.17)

Using the binomial expansion in (1.17), the conditional probability of errorPs (E |γ ; u1 ) = 1 − Ps (C |γ ; u1 ) can be written as

Ps (E |γ ; u1 ) =M−1∑i=1

(M − 1

i

)(−1)i+1 (1 − PU2 (u1)

)i �= g (u1) (1.18)

Averaging over u1 and using the Fourier transform relationship between the PDFpU1|γ (u1) and the MGF MU1|γ (jω), we obtain

Ps (E |γ ) =∫ ∞

0g (u1) pU1|γ (u1) du1

=∫ ∞

0

1

2π

∫ ∞−∞

MU1|γ (jω) e−jωu1g (u1) dω du1

(1.19)

9Again the conditional notation on γ for U1 is not meant to imply that this decision variable is explicitlya function of the fading SNR but rather to indicate its dependence on the fading statistics.

SYSTEM PERFORMANCE MEASURES 11

Again noting that for a noncentral chi-square RV (as is the case for U1 |γ ) theconditional MGF MU1|γ (jω) is of the form in (1.13), then averaging (1.19) overγ transforms MU1|γ (jω) into MU1 (jω) of the form in (1.15), which, when sub-stituted in (1.19) and reversing the order of integration, produces

Ps (E) = 12π

∫ ∞−∞

f1 (jω) Mγ (f2 (jω))

(∫ ∞0

e−jωu1g (u1) du1)

dω (1.20)

Finally, because the CDF PU2 (u1) in (1.18) is that of a central chi-square RV with2L degrees of freedom, the resulting form of g (u1) is such that the integral on u1in (1.20) can be obtained in closed form. Thus, as promised, what remains againis an expression for average SEP (which for M-ary orthogonal signaling can berelated to the average BEP by a simple scale factor) whose dependence on thefading statistics is solely through the MGF of the fading SNR.

All the techniques considered thus far for evaluating average error probabilityperformance rely on the ability to evaluate the MGF of the instantaneous fadingSNR γ . In dealing with a form of diversity reception referred to as equal gaincombining (EGC) (to be discussed in great detail in Chapter 9), the instantaneousfading SNR at the output of the combiner takes the form γ = [(1/√L) ∑Ll=1

√γl]2.

In this case, it is more convenient to deal with the MGF of the square root of the

instantaneous fading SNR x�= √γ = (1/√L) ∑Ll=1

√γl = (1/

√L)

∑Ll=1 xl since

if the channels are again assumed independent then again this MGF takes on a

product form, namely, Mx (s) =∏L

l=1 Mxl(s/

√L

). Since the average BER can

alternatively be computed from

Pb (E) =∫ ∞

0Pb (E |x ) px (x) dx (1.21)

then if, analogous to (1.9), Pb (E |x ) assumes the form

Pb (E |x ) =∫ ξ2

ξ1

C2h (ξ) exp(−a2g (ξ) x2

)dξ (1.22)

a variation of the procedure in (1.10) is needed to produce an expression for Pb (E)in terms of the MGF of x. First, applying Parseval’s theorem [9, p. 27] to (1.21)and letting G (jω) = F {Pb (E |x )} denote the Fourier transform of Pb (E |x ), thenindependent of the form of Pb (E |x ), we obtain10

Pb (E) = 12π

∫ ∞−∞

G (jω) Mx (jω) dω

= 1π

∫ ∞0

Re {G (jω)Mx (jω)} dω(1.23)

10A unified performance evaluation method based on the form of (1.23) and its further simplificationin (1.24) has been proposed by Annamalai et al. [14], who refer to their approach as the characteristicfunction (CHF) method based on Parseval’s theorem.

12 INTRODUCTION

where we have recognized that the imaginary part of the integral must be equal tozero since Pb (E) is real and that the even part of the integrand is an even functionof ω. Making the change of variables θ = tan−1 ω, (1.23) can be written in theform of an integral with finite limits:

Pb (E) = 1π

∫ π /20

1

cos2 θRe {G (j tan θ)Mx (j tan θ)} dθ

= 2π

∫ π /20

1

sin 2θRe {tan θ G (j tan θ) Mx (j tan θ)} dθ

(1.24)

Now, specifically for the form of Pb (E |x ) in (1.22), G (jω) becomes

G (jω) =∫ ξ2

ξ1

C2h (ξ)

∫ ∞0

exp(−a2g (ξ) x2 + jωx) dx dξ (1.25)

The inner integral on x can be evaluated in closed form as

∫ ∞0

exp(−a2g (ξ) x2 + jωx) dx

= 12a2g (ξ)

{√πa2g (ξ) exp

((jω)2

4a2g (ξ)

)+ jω 1F1

(1,

3

2;

(jω)2

4a2g (ξ)

)}

(1.26)where 1F1 (a, b; c) is the confluent hypergeometric function of the first kind [10,p. 1085, Eq. (9.210)]. Therefore, in general, the evaluation of the average BER of(1.24) requires a double integration. However, for a number of specific applications,specifically, particular forms of the functions h (ξ) and g (ξ), the outer integral onξ can also be evaluated in closed form; thus, in these instances, Pb (E) can beobtained as a single integral with finite limits and an integrand involving the MGFof the fading. Methods of error probability evaluation based on the type of MGFapproach described above have been considered in the literature [11–13] and willbe presented in detail in Chapter 9.

1.1.4 Amount of Fading

The performance measures discussed in Sections 1.1.1–1.1.3 are the ones mostcommonly employed to describe the behavior of digital communication systems inthe presence of fading. Although not as descriptive as the other two, average SNRhad the advantage that it was simple to compute in that it required knowledge ofonly the first statistical moment of the instantaneous SNR. However, in the contextof diversity combining, this performance criterion does not capture all the diversitybenefits. Indeed, if the diversity advantage were limited to an average SNR gain,then this could be achieved by simply increasing the transmitter power. Of moreimportance is the aptitude of diversity systems to reduce the fading-induced fluc-tuations or equivalently in statistical terms, to reduce the relative variance of the

SYSTEM PERFORMANCE MEASURES 13

signal envelope that cannot be achieved just by increasing the transmitter power.Thus, in order to capture this effect, we are motivated to look at other performancemeasures that take into account higher moments of the combiner output SNR. Fol-lowing along this train of thought, another performance measure that is most oftensimple to compute and requires knowledge of only the first and second momentsof the instantaneous SNR was introduced by the authors [15] when describing thebehavior of dual-diversity combining systems over correlated log-normal fadingchannels. The measure, which is referred to as “amount of fading” (AF), is asso-ciated with the output of the combiner and is modeled after a criterion bearing thesame name that was originally introduced by Charash [16, p. 29] as a measure ofthe severity of the fading channel by itself (see Chapter 2, Section 2.2). It is theauthors’ suggestion that this same AF measure is often appropriate in the moregeneral context of describing the behavior of systems with arbitrary combiningtechniques and channel statistics and thus can be used as an alternative perfor-mance criterion whenever convenient.11 Specifically, letting γt denote the totalinstantaneous SNR at the combiner output, we define AF by

AF = var γt(E

[γt

])2 =E

(γ 2t

) − (E [γt])2

(E

[γt

])2 (1.27)

which can be expressed in terms of the MGF of γt by

AF =d2Mγt (s)

ds2|s=0 −

(dMγt (s)

ds|s=0

)2(

dMγt (s)

ds|s=0

)2 (1.28)

Because the AF defined in (1.27) is computed at the output of the combiner, itsevaluation will reflect the behavior of the particular diversity combining techniqueas well as the statistics of the fading channel and thus, as mentioned above, is ameasure of the performance of the entire system. Closed-form expressions for avariety of such evaluations will be presented in Chapter 9.

1.1.5 Average Outage Duration

In certain communication system applications such as adaptive transmissionschemes, the performance metrics discussed above do not provide enough infor-mation for the overall system design and configuration. In that case, in additionto these performance measures, the frequency of outages and the average outageduration (AOD) (also known as the “average fade duration”) are important perfor-mance criteria for the proper selection of the transmission symbol rate, interleaverdepth, packet length, and/or time slot duration.

11Perhaps the earliest indication of such a performance measure appears in a paper by Win andWinters [17], who described the behavior of hybrid selection combining/maximal-ratio combining inthe presence of Rayleigh fading in terms of the normalized standard deviation (NSD) of the diversitycombiner output SNR, which coincidentally is equal to the square root of the amount of fading.

14 INTRODUCTION

As discussed above, in purely noise-limited systems, an outage is declared when-ever the output SNR, γ , falls below a predetermined threshold γth, (i.e., γ < γth).The AOD, T (γth) (in seconds) is a measure of how long, on the average, the systemremains in the outage state. Mathematically speaking, the AOD is well known tobe given by [18]

T (γth) = PoutN (γth)

(1.29)

where Pout was defined and discussed in Section 1.1.2 and N (γth) is the frequencyof outages or equivalently the average level crossing rate (LCR) of the output SNRγ at level γth, which can be obtained from the well-known Rice formula [19]

N (γth) =∫ ∞

0γ̇ fγ ,γ̇ (γth, γ̇ ) dγ̇ , (1.30)

with fγ ,γ̇ (γ , γ̇ ) the joint PDF of γ and its time derivative γ̇ . Methods and analyticalexpressions for the evaluation of the average LCR, and thus AOD, for variousdiversity combining schemes will be presented in Chapter 9.

1.2 CONCLUSIONS

Without regard to the specific application or performance measure, we have brieflydemonstrated in this chapter that for a wide variety of digital communication sys-tems covering virtually all known modulation/detection techniques and practicalfading channel models, there exists an MGF-based approach that simplifies theevaluation of this performance. In the biggest number of these instances, the MGF-based approach is encompassed in a unified framework that allows the developmentof a set of generic tools to replace the case-by-case analyses typical of previouscontributions in the literature. It is the authors’ hope that by the time the readersreach the end of this book and have experienced the exhaustive set of practicalcircumstances where these tools are useful, they will fully appreciate the powerbehind the MGF-based approach and as such will generate for themselves an insightinto finding new and exciting applications.

REFERENCES

1. J. Abate and W. Whitt, “Numerical inversion of Laplace transforms of probability dis-tributions,” ORSA J. Comput., vol. 7, no. 1, 1995, pp. 36–43.

2. M. K. Simon and M.-S. Alouini, “A unified approach to the performance analysis ofdigital communications over generalized fading channels,” IEEE Proc., vol. 86, no. 9,September 1998, pp. 1860–1877.

3. E. Biglieri, C. Caire, G. Taricco, and J. Ventura-Traveset, “Computing error probabili-ties over fading channels: A unified approach,” Eur. Trans. Telecommun., vol. 9, no. 1,February 1998, pp. 15–25.

REFERENCES 15

4. E. Biglieri, C. Caire, G. Taricco, and J. Ventura-Traveset, “Simple method for evaluatingerror probabilities,” Electron. Lett., vol. 32, February 1996, pp. 191–192.

5. J. K. Cavers and P. Ho, “Analysis of the error performance of trellis coded modulationsin Rayleigh fading channels,” IEEE Trans. Commun., vol. 40, no. 1, January 1992,pp. 74–80.

6. J. K. Cavers, J.-H. Kim, and P. Ho, “Exact calculation of the union bound on perfor-mance of trellis-coded modulation in fading channels,” IEEE Trans. Commun., vol. 46,no. 5, May 1998, pp. 576-579; see also Proc. IEEE ICUPC ’96, vol. 2, Cambridge, MA,September 1996, pp. 875–880.

7. M. K. Simon and M.-S. Alouini, “Bit error probability of noncoherent M-ary orthogonalmodulation over generalized fading channels,” Int. J. Commun. and Networks, vol. 1,no. 2, June 1999, pp. 111–117.

8. J. F. Weng and S. H. Leung, “Analysis of M-ary FSK square law combiner underNakagami fading channels,” Electron. Lett., vol. 33, September 1997, pp. 1671–1673.

9. A. Papoulis, The Fourier Integral and Its Application, New York, NY: McGraw-Hill,1962, p. 27.

10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. SanDiego, CA: Academic Press, 1994.

11. M.-S. Alouini and M. K. Simon, “Error rate analysis of M-PSK with equal-gain com-bining over Nakagami fading channels,” Proc. VTC’99, Houston, TX, pp. 2378–2382;see also IEEE Trans. Veh. Technol., vol. 50, no. 6, November 2001, pp. 1449–1463.

12. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratioand equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEETrans. Commun., vol. 47, no. 9, September 1999, pp. 1335–1344.

13. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Unified analysis of equal-gaindiversity on Rician and Nakagami fading channels,” Proc. IEEE Wireless Commun. andNetworking Conf. (WCNC’99 ), New Orleans, LA, September 1999.

14. A. Annamalai, C. Tellambura, and V. K. Bhargava, “A general method for calculatingerror probabilities over fading channels,” Proc. IEEE Int. Conf. Commun. (ICC’00 ),New Orleans, LA, June 2000.

15. M.-S. Alouini and M. K. Simon, “Dual diversity over log-normal fading channels,”IEEE Trans. Commun., vol. 50, no. 12, December 2002, pp. 1946–1959; see also Proc.IEEE Int. Conf. Commun. (ICC’01 ), Helsinki, Finland, June, 2001.

16. U. Charash, A Study of Multipath Reception with Unknown Delays, PhD dissertation,University of California, Berkeley, CA, January 1974.

17. M. Z. Win and J. H. Winters, “Analysis of hybrid selection/maximal-ratio combining inRayleigh fading,” IEEE Trans. Commun., vol. 47, no. 12, December 1999, pp. 1773–1776. See also Proc. IEEE Int. Conf. Commun. (ICC’99), Vancouver, British Columbia,Canada, June 1999, pp. 6–10.

18. G. L. Stüber, Principles of Mobile Communications, 2nd ed. Boston, MA: Kluwer Aca-demic Publishers, 2001.

19. S. Rice, “Statistical properties of a sine wave plus noise,” Bell Syst. Tech. J., vol. 27,January 1948, pp. 109–157.

2

FADING CHANNELCHARACTERIZATION

AND MODELING

Radiowave propagation through wireless channels is a complicated phenomenoncharacterized by various effects such as multipath and shadowing. A precise math-ematical description of this phenomenon is either unknown or too complex fortractable communication systems analyses. However, considerable efforts have beendevoted to the statistical modeling and characterization of these different effects.The result is a range of relatively simple and accurate statistical models for fadingchannels that depend on the particular propagation environment and the underlyingcommunication scenario.

The primary purpose of this chapter is to briefly review the principal characteris-tics and models for fading channels. A more detailed treatment of this subject can befound in standard textbooks such as those by Proakis, Rappaport, and Stüber [1–3].This chapter also introduces terminology and notation that will be used throughoutthe book. The chapter is organized as follows. A brief qualitative description ofthe main characteristics of fading channels is presented in the next section. Mod-els for frequency-flat fading channels, corresponding to narrowband transmission,are described in Section 2.2. Models for frequency-selective fading channels thatcharacterize fading in wideband channels are described in Section 2.3.

2.1 MAIN CHARACTERISTICS OF FADING CHANNELS

2.1.1 Envelope and Phase Fluctuations

When a received signal experiences fading during transmission, both its envelopeand phase fluctuate over time. For coherent modulations, the fading effects on the

Digital Communication over Fading Channels, Second Edition.By Marvin K. Simon and Mohamed-Slim AlouiniISBN 0-471-64953-8 Copyright 2005 John Wiley & Sons, Inc.

17

18 FADING CHANNEL CHARACTERIZATION AND MODELING

phase can severely degrade performance unless measures are taken to compensatefor them at the receiver. Most often, analyses of systems employing such modu-lations assume that the phase effects due to fading are perfectly corrected at thereceiver resulting in what is referred to as “ideal” coherent demodulation. For non-coherent modulations, phase information is not needed at the receiver and thereforethe phase variation due to fading does not affect the performance. Hence, perfor-mance analyses for both ideal coherent and noncoherent modulations over fadingchannels requires knowledge of only the fading envelope statistics and will be thecase most often considered in this text. Furthermore, for so-called slow fading (tobe discussed next), wherein the fading is at least constant over the duration of asymbol time, the fading envelope random process can be represented by a randomvariable (RV) over the symbol time.

2.1.2 Slow and Fast Fading

The distinction between slow and fast fading is important for the mathematicalmodeling of fading channels and for the performance evaluation of communicationsystems operating over these channels. This notion is related to the coherence timeTc of the channel, which measures the period of time over which the fading processis correlated (or equivalently, the period of time after which the correlation functionof two samples of the channel response taken at the same frequency but differenttime instants drops below a certain predetermined threshold). The coherence timeis also related to the channel Doppler spread fd by

Tc � 1fd

(2.1)

The fading is said to be slow if the symbol time duration Ts is smaller than thechannel’s coherence time Tc; otherwise it is considered to be fast. In slow fadinga particular fade level will affect many successive symbols, which leads to bursterrors, whereas in fast fading the fading decorrelates from symbol to symbol. Inthis latter case and when the communication receiver decisions are based on anobservation of the received signal over two or more symbol times (such as dif-ferentially coherent or coded communications), it becomes necessary to con