Digital Communication over Fading Channels, Second Edition 9780471649533, 9780471715221

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Digital Communication over Fading Channels

Digital Communication over Fading Channels Second Edition

Marvin K. Simon Mohamed-Slim Alouini

A JOHN WILEY & SONS, INC., PUBLICATION

Copyright  2005 by John Wiley & Sons, Inc. All rights reserved. Published by John Wiley & Sons, Inc., Hoboken, New Jersey. Published simultaneously in Canada. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording, scanning, or otherwise, except as permitted under Section 107 or 108 of the 1976 United States Copyright Act, without either the prior written permission of the Publisher, or authorization through payment of the appropriate per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, 978-750-8400, fax 978-646-8600, or on the web at www.copyright.com. Requests to the Publisher for permission should be addressed to the Permissions Department, John Wiley & Sons, Inc., 111 River Street, Hoboken, NJ 07030, (201) 748-6011, fax (201) 748-6008. Limit of Liability/Disclaimer of Warranty: While the publisher and author have used their best efforts in preparing this book, they make no representations or warranties with respect to the accuracy or completeness of the contents of this book and specifically disclaim any implied warranties of merchantability or fitness for a particular purpose. No warranty may be created or extended by sales representatives or written sales materials. The advice and strategies contained herein may not be suitable for your situation. You should consult with a professional where appropriate. Neither the publisher nor author shall be liable for any loss of profit or any other commercial damages, including but not limited to special, incidental, consequential, or other damages. For general information on our other products and services please contact our Customer Care Department within the U.S. at 877-762-2974, outside the U.S. at 317-572-3993 or fax 317-572-4002. Wiley also publishes its books in a variety of electronic formats. Some content that appears in print, however, may not be available in electronic format. 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 and transmission–Fading. I. Alouini, Mohamed-Slim. II. Title. III. Series. TK5103.7.S523 2004 621.382–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 1.1 System Performance Measures 1.1.1 Average Signal-to-Noise Ratio (SNR) 1.1.2 Outage Probability 1.1.3 Average Bit Error Probability (BEP) 1.1.4 Amount of Fading 1.1.5 Average Outage Duration 1.2 Conclusions References

CHAPTER 2

Fading Channel Characterization and Modeling 2.1 Main Characteristics of Fading Channels 2.1.1 Envelope and Phase Fluctuations 2.1.2 Slow and Fast Fading 2.1.3 Frequency-Flat and Frequency-Selective Fading 2.2 Modeling of Flat-Fading Channels 2.2.1 Multipath Fading 2.2.1.1 Rayleigh 2.2.1.2 Nakagami-q (Hoyt) 2.2.1.3 Nakagami-n (Rice) 2.2.1.4 Nakagami-m

3 4 4 5 6 12 13 14 14 17 17 17 18 18 19 20 20 22 23 24 vii

viii

CONTENTS

2.2.1.5 2.2.1.6 2.2.1.7

2.3

CHAPTER 3

Weibull Beckmann Spherically-Invariant Random Process Model 2.2.2 Log-Normal Shadowing 2.2.3 Composite Multipath/Shadowing 2.2.3.1 Composite Gamma/Log-Normal Distribution 2.2.3.2 Suzuki Distribution 2.2.3.3 K Distribution 2.2.3.4 Rician Shadowed Distributions 2.2.4 Combined (Time-Shared) Shadowed/Unshadowed Fading Modeling of Frequency-Selective Fading Channels

25 28

References

39

30 32 33 33 34 34 36 37 37

Types of Communication 3.1 Ideal Coherent Detection 3.1.1 Multiple Amplitude-Shift-Keying (M-ASK) or Multiple Amplitude Modulation (M-AM) 3.1.2 Quadrature Amplitude-Shift-Keying (QASK) or Quadrature Amplitude Modulation (QAM) 3.1.3 M-ary Phase-Shift-Keying (M-PSK) 3.1.4 Differentially Encoded M-ary Phase-Shift-Keying (M-PSK) 3.1.4.1 π /4-QPSK 3.1.5 Offset QPSK (OQPSK) or Staggered QPSK (SQPSK) 3.1.6 M-ary Frequency-Shift-Keying (M-FSK) 3.1.7 Minimum-Shift-Keying (MSK) 3.2 Nonideal Coherent Detection

45 45

3.3

Noncoherent Detection

66

3.4

Partially Coherent Detection 3.4.1 Conventional Detection 3.4.1.1 One-Symbol Observation 3.4.1.2 Multiple-Symbol Observation 3.4.2 Differentially Coherent Detection 3.4.2.1 M-ary Differential Phase-Shift-Keying (M-DPSK) 3.4.2.2 Conventional Detection (Two-Symbol Observation) 3.4.2.3 Multiple-Symbol Detection

68 68 68 69 71

47

48 50 53 54 55 56 58 62

71 73 76

CONTENTS

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

ix

78 78

PART 2 MATHEMATICAL TOOLS CHAPTER 4

CHAPTER 5

Alternative Representations of Classical Functions 4.1 Gaussian Q-Function 4.1.1 One-Dimensional Case 4.1.2 Two-Dimensional Case 4.1.3 Other Forms for One- and Two-Dimensional Cases 4.1.4 Alternative Representations of Higher Powers of the Gaussian Q-Function 4.2 Marcum Q-Function 4.2.1 First-Order Marcum Q-Function 4.2.1.1 Upper and Lower Bounds 4.2.2 Generalized (mth-Order) Marcum Q-Function 4.2.2.1 Upper and Lower Bounds 4.3 The Nuttall Q-Function

100 105 113

4.4

Other Functions

117

References

119

Appendix 4A. Derivation of Eq. (4.2)

120

Useful Expressions for Evaluating Average Error Probability Performance 5.1 Integrals Involving the Gaussian Q-Function 5.1.1 Rayleigh Fading Channel 5.1.2 Nakagami-q (Hoyt) Fading Channel 5.1.3 Nakagami-n (Rice) Fading Channel 5.1.4 Nakagami-m Fading Channel 5.1.5 Log-Normal Shadowing Channel 5.1.6 Composite Log-Normal Shadowing/Nakagami-m Fading Channel 5.2 Integrals Involving the Marcum Q-Function 5.2.1 Rayleigh Fading Channel 5.2.2 Nakagami-q (Hoyt) Fading Channel 5.2.3 Nakagami-n (Rice) Fading Channel 5.2.4 Nakagami-m Fading Channel 5.2.5 Log-Normal Shadowing Channel

83 84 84 86 88 90 93 93 97

123 123 125 125 126 126 128 128 131 132 133 133 133 133

x

CONTENTS

5.2.6

5.3

5.4

Composite Log-Normal Shadowing/Nakagami-m Fading Channel 5.2.7 Some Alternative Closed-Form Expressions Integrals Involving the Incomplete Gamma Function 5.3.1 Rayleigh Fading Channel 5.3.2 Nakagami-q (Hoyt) Fading Channel 5.3.3 Nakagami-n (Rice) Fading Channel 5.3.4 Nakagami-m Fading Channel 5.3.5 Log-Normal Shadowing Channel 5.3.6 Composite Log-Normal Shadowing/Nakagami-m Fading Channel Integrals Involving Other Functions 5.4.1 The M -PSK Error Probability Integral 5.4.1.1 Rayleigh Fading Channel 5.4.1.2 Nakagami-m Fading Channel 5.4.2 Arbitrary Two-Dimensional Signal Constellation Error Probability Integral 5.4.3 Higher-Order Integer Powers of the Gaussian Q-Function 5.4.3.1 Rayleigh Fading Channel 5.4.3.2 Nakagami-m Fading Channel 5.4.4 Integer Powers of M -PSK Error Probability Integrals 5.4.4.1 Rayleigh Fading Channel References Appendix 5A. Evaluation of Definite Integrals Associated with Rayleigh and Nakagami-m Fading 5A.1 Exact Closed-Form Results 5A.2 Upper and Lower Bounds

CHAPTER 6

New Representations of Some Probability Density and Cumulative Distribution Functions for Correlative Fading Applications 6.1 Bivariate Rayleigh PDF and CDF 6.2 6.3 6.4

134 135 137 138 139 139 140 140 140 141 141 142 142 142 144 144 145 145 146 148 149 149 165

169 170

PDF and CDF for Maximum of Two Rayleigh Random Variables

175

PDF and CDF for Maximum of Two Nakagami-m Random Variables

177

PDF and CDF for Maximum and Minimum of Two Log-Normal Random Variables 6.4.1 The Maximum of Two Log-Normal Random Variables

180 180

CONTENTS

The Minimum of Two Log-Normal Random Variables References

xi

6.4.2

183 185

PART 3 OPTIMUM RECEPTION AND PERFORMANCE EVALUATION CHAPTER 7

Optimum Receivers for Fading Channels 7.1 The Case of Known Amplitudes, Phases, and Delays—Coherent Detection 7.2

7.3 7.4

7.5

CHAPTER 8

The Case of Known Phases and Delays but Unknown Amplitudes 7.2.1 Rayleigh Fading 7.2.2 Nakagami-m Fading The Case of Known Amplitudes and Delays but Unknown Phases The Case of Known Delays but Unknown Amplitudes and Phases 7.4.1 One-Symbol Observation—Noncoherent Detection 7.4.1.1 Rayleigh Fading 7.4.1.2 Nakagami-m Fading 7.4.2 Two-Symbol Observation—Conventional Differentially Coherent Detection 7.4.2.1 Rayleigh Fading 7.4.2.2 Nakagami-m Fading 7.4.3 Ns -Symbol Observation—Multiple Differentially Coherent Detection 7.4.3.1 Rayleigh Fading 7.4.3.2 Nakagami-m Fading The Case of Unknown Amplitudes, Phases, and Delays 7.5.1 One-Symbol Observation—Noncoherent Detection 7.5.1.1 Rayleigh Fading 7.5.1.2 Nakagami-m Fading 7.5.2 Two-Symbol Observation—Conventional Differentially Coherent Detection References

Performance of Single-Channel Receivers 8.1 Performance Over the AWGN Channel

189 191 195 195 196 198 199 199 201 206 211 214 217 217 218 218 219 219 220 221 221 222 223 223

xii

CONTENTS

8.1.1

8.1.2 8.1.3 8.1.4

8.1.5

8.1.6 8.2

Ideal Coherent Detection 8.1.1.1 Multiple Amplitude-Shift-Keying (M-ASK) or Multiple Amplitude Modulation (M-AM) 8.1.1.2 Quadrature Amplitude-ShiftKeying (QASK) or Quadrature Amplitude Modulation (QAM) 8.1.1.3 M-ary Phase-Shift-Keying (M-PSK) 8.1.1.4 Differentially Encoded M-ary Phase-Shift-Keying (M-PSK) and π /4-QPSK 8.1.1.5 Offset QPSK (OQPSK) or Staggered QPSK (SQPSK) 8.1.1.6 M-ary Frequency-Shift-Keying (M-FSK) 8.1.1.7 Minimum-Shift-Keying (MSK) Nonideal Coherent Detection Noncoherent Detection Partially Coherent Detection 8.1.4.1 Conventional Detection (One-Symbol Observation) 8.1.4.2 Multiple-Symbol Detection Differentially Coherent Detection 8.1.5.1 M-ary Differential Phase-Shift-Keying (M-DPSK) 8.1.5.2 M-DPSK with Multiple-Symbol Detection 8.1.5.3 π /4-Differential QPSK (π /4-DQPSK) Generic Results for Binary Signaling

224

224

225 228

234 235 236 237 237 242 242 242 244 245 245 249 250 251

Performance Over Fading Channels

252

8.2.1

252

Ideal Coherent Detection 8.2.1.1 Multiple Amplitude-Shift-Keying (M-ASK) or Multiple Amplitude Modulation (M-AM) 8.2.1.2 Quadrature Amplitude-ShiftKeying (QASK) or Quadrature Amplitude Modulation (QAM) 8.2.1.3 M-ary Phase-Shift-Keying (M-PSK) 8.2.1.4 Differentially Encoded M-ary Phase-Shift-Keying (M-PSK) and π /4-QPSK

253

254 256

258

CONTENTS

Offset QPSK (OQPSK) or Staggered QPSK (SQPSK) 8.2.1.6 M-ary Frequency-Shift-Keying (M-FSK) 8.2.1.7 Minimum-Shift-Keying (MSK) 8.2.2 Nonideal Coherent Detection 8.2.2.1 Simplified Noisy Reference Loss Evaluation 8.2.3 Noncoherent Detection 8.2.4 Partially Coherent Detection 8.2.5 Differentially Coherent Detection 8.2.5.1 M-ary Differential Phase-ShiftKeying (M-DPSK)—Slow Fading 8.2.5.2 M-ary Differential Phase-ShiftKeying (M-DPSK)—Fast Fading 8.2.5.3 π /4-Differential QPSK (π /4-DQPSK) 8.2.6 Performance in the Presence of Imperfect Channel Estimation 8.2.6.1 Signal Model and Symbol Error Probability Evaluation for Rayleigh Fading 8.2.6.2 Special Cases References

xiii

8.2.1.5

Appendix 8A. Stein’s Unified Analysis of the Error Probability Performance of Certain Communication Systems CHAPTER 9

Performance of Multichannel Receivers 9.1 Diversity Combining 9.1.1 Diversity Concept 9.1.2 Mathematical Modeling 9.1.3 Brief Survey of Diversity Combining Techniques 9.1.3.1 Pure Combining Techniques 9.1.3.2 Hybrid Combining Techniques 9.1.4 Complexity–Performance Tradeoffs 9.2 Maximal-Ratio Combining (MRC) 9.2.1 Receiver Structure 9.2.2 PDF-Based Approach 9.2.3 MGF-Based Approach 9.2.3.1 Average Bit Error Rate of Binary Signals

262 262 267 267 273 281 282 284 285 290 294 294

295 297 301

304 311 312 312 312 313 313 315 316 316 317 319 320 320

xiv

CONTENTS

9.2.3.2

9.3

9.4

9.5

9.6

Average Symbol Error Rate of M-PSK Signals 9.2.3.3 Average Symbol Error Rate of M-AM Signals 9.2.3.4 Average Symbol Error Rate of Square M-QAM Signals 9.2.4 Bounds and Asymptotic SER Expressions Coherent Equal Gain Combining 9.3.1 Receiver Structure 9.3.2 Average Output SNR 9.3.3 Exact Error Rate Analysis 9.3.3.1 Binary Signals 9.3.3.2 Extension to M-PSK Signals 9.3.4 Approximate Error Rate Analysis 9.3.5 Asymptotic Error Rate Analysis Noncoherent and Differentially Coherent Equal Gain Combining 9.4.1 DPSK, DQPSK, and BFSK Performance (Exact and with Bounds) 9.4.1.1 Receiver Structures 9.4.1.2 Exact Analysis of Average Bit Error Probability 9.4.1.3 Bounds on Average Bit Error Probability 9.4.2 M-ary Orthogonal FSK 9.4.2.1 Exact Analysis of Average Bit Error Probability 9.4.2.2 Numerical Examples 9.4.3 Multiple-Symbol Differential Detection with Diversity Combining 9.4.3.1 Decision Metrics 9.4.3.2 Average Bit Error Rate Performance 9.4.3.3 Asymptotic (Large Ns ) Behavior 9.4.3.4 Numerical Results Optimum Diversity Combining of Noncoherent FSK 9.5.1 Comparison with the Noncoherent Equal Gain Combining Receiver 9.5.2 Extension to the M-ary Orthogonal FSK Case Outage Probability Performance 9.6.1 MRC and Noncoherent EGC 9.6.2 Coherent EGC

322 323 324 326 331 331 332 333 333 339 340 342 342 343 343 346 352 353 356 364 367 367 368 371 372 375 377 378 379 379 380

CONTENTS

9.7

9.8

9.9

9.6.3 Numerical Examples Impact of Fading Correlation 9.7.1 Model A: Two Correlated Branches with Nonidentical Fading 9.7.1.1 PDF 9.7.1.2 MGF 9.7.2 Model B: D Identically Distributed Branches with Constant Correlation 9.7.2.1 PDF 9.7.2.2 MGF 9.7.3 Model C: D Identically Distributed Branches with Exponential Correlation 9.7.3.1 PDF 9.7.3.2 MGF 9.7.4 Model D: D Nonidentically Distributed Branches with Arbitrary Correlation 9.7.4.1 MGF 9.7.4.2 Special Cases of Interest 9.7.4.3 Proof that Correlation Degrades Performance 9.7.5 Numerical Examples Selection Combining 9.8.1 MGF of Output SNR 9.8.2 Average Output SNR 9.8.3 Outage Probability 9.8.3.1 Analysis 9.8.3.2 Numerical Example 9.8.4 Average Probability of Error 9.8.4.1 BDPSK and Noncoherent BFSK 9.8.4.2 Coherent BPSK and BFSK 9.8.4.3 Numerical Example Switched Diversity 9.9.1 Dual-Branch Switch-and-Stay Combining 9.9.1.1 Performance of SSC over Independent Identically Distributed Branches 9.9.1.2 Effect of Branch Unbalance 9.9.1.3 Effect of Branch Correlation 9.9.2 Multibranch Switch-and-Examine Combining 9.9.2.1 Classical Multibranch SEC 9.9.2.2 Multibranch SEC with Post-selection 9.9.2.3 Scan-and-Wait Combining

xv

381 389 390 390 392 392 393 393 394 394 394 395 395 396 397 399 404 405 406 409 409 410 411 411 413 415 417 419

419 433 436 439 440 443 446

xvi

CONTENTS

9.10 Performance in the Presence of Outdated or Imperfect Channel Estimates 9.10.1 Maximal-Ratio Combining 9.10.2 Noncoherent EGC over Rician Fast Fading 9.10.3 Selection Combining 9.10.4 Switched Diversity 9.10.4.1 SSC Output Statistics 9.10.4.2 Average SNR 9.10.4.3 Average Probability of Error 9.10.5 Numerical Results 9.11 Combining in Diversity-Rich Environments 9.11.1 Two-Dimensional Diversity Schemes 9.11.1.1 Performance Analysis 9.11.1.2 Numerical Examples 9.11.2 Generalized Selection Combining 9.11.2.1 I.I.D. Rayleigh Case 9.11.2.2 Non-I.I.D. Rayleigh Case 9.11.2.3 I.I.D. Nakagami-m Case 9.11.2.4 Partial-MGF Approach 9.11.2.5 I.I.D. Weibull Case 9.11.3 Generalized Selection Combining with Threshold Test per Branch (T-GSC) 9.11.3.1 Average Error Probability Performance 9.11.3.2 Outage Probability Performance 9.11.3.3 Performance Comparisons 9.11.4 Generalized Switched Diversity (GSSC) 9.11.4.1 GSSC Output Statistics 9.11.4.2 Average Probability of Error 9.11.5 Generalized Selection Combining Based on the Log-Likelihood Ratio 9.11.5.1 Optimum (LLR-Based) GSC for Equiprobable BPSK 9.11.5.2 Envelope-Based GSC 9.11.5.3 Optimum GSC for Noncoherently Detected Equiprobable Orthogonal BFSK 9.12 Post-detection Combining 9.12.1 System and Channel Models 9.12.1.1 Overall System Description 9.12.1.2 Channel Model 9.12.1.3 Receiver 9.12.2 Post-detection Switched Combining Operation 9.12.2.1 Switching Strategy and Mechanism

456 457 458 461 462 462 463 463 464 466 466 468 469 469 472 492 497 502 510 512 515 520 524 531 531 532 532 533 536

536 537 537 537 537 539 539 539

CONTENTS

9.12.2.2 9.12.3 Average 9.12.3.1 9.12.3.2

Switching Threshold BER Analysis Identically Distributed Branches Nonidentically Distributed Branches 9.12.4 Rayleigh Fading 9.12.4.1 Identically Distributed Branches 9.12.4.2 Nonidentically Distributed Branches 9.12.5 Impact of the Severity of Fading 9.12.5.1 Average BER 9.12.5.2 Numerical Examples and Discussion 9.12.6 Extension to Orthogonal M-FSK 9.12.6.1 System Model and Switching Operation 9.12.6.2 Average Probability of Error 9.12.6.3 Numerical Examples 9.13 Performance of Dual-Branch Diversity Combining Schemes over Log-Normal Channels 9.13.1 System and Channel Models 9.13.2 Maximal-Ratio Combining 9.13.2.1 Moments of the Output SNR 9.13.2.2 Outage Probability 9.13.2.3 Extension to Equal Gain Combining 9.13.3 Selection Combining 9.13.3.1 Moments of the Output SNR 9.13.3.2 Outage Probability 9.13.4 Switched Combining 9.13.4.1 Moments of the Output SNR 9.13.4.2 Outage Probability 9.14 Average Outage Duration 9.14.1 System and Channel Models 9.14.1.1 Fading Channel Models 9.14.1.2 GSC Mode of Operation 9.14.2 Average Outage Duration and Average Level Crossing Rate 9.14.2.1 Problem Formulation 9.14.2.2 General Formula for the Average LCR of GSC 9.14.3 I.I.D. Rayleigh Fading 9.14.3.1 Generic Expressions for GSC 9.14.3.2 Special Cases: SC and MRC 9.14.4 Numerical Examples

xvii

540 540 542 542 543 544 547 548 550 552 552 552 555 562 566 566 568 568 570 571 571 572 575 575 576 581 584 585 585 585 586 586 586 589 589 590 591

xviii

CONTENTS

9.15 Multiple-Input/Multiple-Output (MIMO) Antenna Diversity Systems 9.15.1 System, Channel, and Signal Models 9.15.2 Optimum Weight Vectors and Output SNR 9.15.3 Distributions of the Largest Eigenvalue of Noncentral Complex Wishart Matrices 9.15.3.1 CDF of S 9.15.3.2 PDF of S 9.15.3.3 PDF of Output SNR and Outage Probability 9.15.3.4 Special Cases 9.15.3.5 Numerical Results and Discussion References Appendix 9A. Alternative Forms of the Bit Error Probability for a Decision Statistic that Is a Quadratic Form of Complex Gaussian Random Variables

594 594 595 596 596 598 599 600 601 604

619

Appendix 9B. Simple Numerical Techniques for Inversion of Laplace Transform of Cumulative Distribution Functions 9B.1 Euler Summation-Based Technique 9B.2 Gauss–Chebyshev Quadrature-Based Technique Appendix 9C. The Relation between the Power Correlation Coefficient of Correlated Rician Random Variables and the Correlation Coefficient of Their Underlying Complex Gaussian Random Variables

626

Appendix 9D. Proof of Theorem 9.1

631

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

632

Appendix 9F. Special Definite Integrals

634

625 625

627

PART 4 MULTIUSER COMMUNICATION SYSTEMS CHAPTER 10 Outage Performance of Multiuser Communication Systems 10.1 Outage Probability in Interference-Limited Systems 10.1.1 A Probability Related to the CDF of the Difference of Two Chi-Square Variates with Different Degrees of Freedom

639 640

640

CONTENTS

10.1.2 Fading and System Models 10.1.2.1 Channel Fading Models 10.1.2.2 Desired and Interference Signals Model 10.1.3 A Generic Formula for the Outage Probability 10.1.3.1 Nakagami/Nakagami Scenario 10.1.3.2 Rice/Rice Scenario 10.1.3.3 Rice/Nakagami Scenario 10.1.3.4 Nakagami/Rice Scenario 10.2 Outage Probability with a Minimum Desired Signal Power Constraint 10.2.1 Models and Problem Formulation 10.2.1.1 Fading and System Models 10.2.1.2 Outage Probability Definition 10.2.2 Rice/I.I.D. Nakagami Scenario 10.2.2.1 Rice/I.I.D. Rayleigh Scenario 10.2.2.2 Extension to Rice/I.I.D. Nakagami Scenario 10.2.2.3 Numerical Examples 10.2.3 Nakagami/I.I.D. Rice Scenario 10.2.3.1 Rayleigh/I.I.D. Rice Scenario 10.2.3.2 Extension to Nakagami/I.I.D. Rice Scenario 10.2.3.3 Numerical Examples 10.3 Outage Probability with Dual-Branch SC and SSC Diversity 10.3.1 Fading and System Models 10.3.2 Outage Performance with Minimum Signal Power Constraint 10.3.2.1 Selection Combining 10.3.2.2 Switch-and-Stay Combining 10.3.2.3 Numerical Examples 10.4 Outage Rate and Average Outage Duration of Multiuser Communication Systems

xix

643 643 644 644 645 646 647 647 648 648 648 648 649 649 652 652 654 654 656 657 659 661 661 662 663 664 667

References

671

Appendix 10A. A Probability Related to the CDF of the Difference of Two Chi-Square Variates with Different Degrees of Freedom

674

Appendix 10B. Outage Probability in the Nakagami/Nakagami Interference-Limited Scenario

678

xx

CONTENTS

CHAPTER 11 Optimum Combining—a Diversity Technique for Communication over Fading Channels in the Presence of Interference 11.1 Performance of Diversity Combining Receivers 11.1.1 Single Interferer; Independent, Identically Distributed Fading 11.1.1.1 Rayleigh Fading—Exact Evaluation of Average Bit Error Probability 11.1.1.2 Rayleigh Fading—Approximate Evaluation of Average Bit Error Probability 11.1.1.3 Extension to Other Modulations 11.1.1.4 Rician Fading—Evaluation of Average Bit Error Probability 11.1.1.5 Nakagami-m Fading—Evaluation of Average Bit Error Probability 11.1.2 Multiple Equal Power Interferers; Independent, Identically Distributed Fading 11.1.2.1 Number of Interferers Less than Number of Array Elements 11.1.2.2 Number of Interferers Equal to or Greater than Number of Array Elements 11.1.3 Comparison with Results for MRC in the Presence of Interference 11.1.4 Multiple Arbitrary Power Interferers; Independent, Identically Distributed Fading 11.1.4.1 Average SEP of M-PSK 11.1.4.2 Numerical Results 11.1.5 Multiple-Symbol Differential Detection in the Presence of Interference 11.1.5.1 Decision Metric 11.1.5.2 Average BEP 11.2 Optimum Combining with Multiple Transmit and Receive Antennas 11.2.1 System, Channel, and Signals Models 11.2.2 Optimum Weight Vectors and Output SIR 11.2.3 PDF of Output SIR and Outage Probability 11.2.3.1 PDF of Output SIR 11.2.3.2 Outage Probability 11.2.3.3 Special Case When Lt = 1 11.2.4 Key Observations 11.2.4.1 Distribution of Antenna Elements

681 682 682

686

689 692 693 695 697 700

706 710 715 715 716 718 718 718 721 721 723 723 724 724 725 726 726

CONTENTS

xxi

11.2.4.2 Effects of Correlation between Receiver Antenna Pairs 11.2.5 Numerical Examples References

726 727 729

Appendix 11A. Distributions of the Largest Eigenvalue of Certain Quadratic Forms in Complex Gaussian Vectors 11A.1 General Result 11A.2 Special Case

732 732 733

CHAPTER 12 Direct-Sequence Code-Division Multiple Access (DS-CDMA) 12.1 Single-Carrier DS-CDMA Systems 12.1.1 System and Channel Models 12.1.1.1 Transmitted Signal 12.1.1.2 Channel Model 12.1.1.3 Receiver 12.1.2 Performance Analysis 12.1.2.1 General Case 12.1.2.2 Application to Nakagami-m Fading Channels 12.2 Multicarrier DS-CDMA Systems 12.2.1 System and Channel Models 12.2.1.1 Transmitter 12.2.1.2 Channel 12.2.1.3 Receiver 12.2.1.4 Notations 12.2.2 Performance Analysis 12.2.2.1 Conditional SNR 12.2.2.2 Average BER 12.2.3 Numerical Examples References

735 736 736 736 737 738 739 740 740 741 742 742 743 743 744 745 745 749 750 754

PART 5 CODED COMMUNICATION SYSTEMS CHAPTER 13 Coded Communication over Fading Channels 13.1 Coherent Detection 13.1.1 System Model 13.1.2 Evaluation of Pairwise Error Probability 13.1.2.1 Known Channel State Information 13.1.2.2 Unknown Channel State Information

759 761 761 763 764 768

xxii

CONTENTS

13.1.3 Transfer Function Bound on Average Bit Error Probability 13.1.3.1 Known Channel State Information 13.1.3.2 Unknown Channel State Information 13.1.4 An Alternative Formulation of the Transfer Function Bound 13.1.5 An Example 13.2 Differentially Coherent Detection 13.2.1 System Model 13.2.2 Performance Evaluation 13.2.2.1 Unknown Channel State Information 13.2.2.2 Known Channel State Information 13.2.3 An Example 13.3 Numerical Results—Comparison between the True Upper Bounds and Union–Chernoff Bounds

772 774 774 774 775 781 781 783 783 785 785 787

References

792

Appendix 13A. Evaluation of a Moment Generating Function Associated with Differential Detection of M-PSK Sequences

793

CHAPTER 14 Multichannel Transmission—Transmit Diversity and Space-Time Coding 14.1 A Historical Perspective

797 799

14.2 Transmit versus Receive Diversity—Basic Concepts

800

14.3 Alamouti’s Diversity Technique—a Simple Transmit Diversity Scheme Using Two Transmit Antennas

803

14.4 Generalization of Alamouti’s Diversity Technique to Orthogonal Space-Time Block Code Designs

809

14.5 Alamouti’s Diversity Technique Combined with Multidimensional Trellis-Coded Modulation 14.5.1 Evaluation of Pairwise Error Probability Performance on Fast Rician Fading Channels 14.5.2 Evaluation of Pairwise Error Probability Performance on Slow Rician Fading Channels 14.6 Space-Time Trellis-Coded Modulation

812

814

817 818

CONTENTS

14.6.1 Evaluation of Pairwise Error Probability Performance on Fast Rician Fading Channels 14.6.2 Evaluation of Pairwise Error Probability Performance on Slow Rician Fading Channels 14.6.3 An Example 14.6.4 Approximate Evaluation of Average Bit Error Probability 14.6.4.1 Fast-Fading Channel Model 14.6.4.2 Slow-Fading Channel Model 14.6.5 Evaluation of the Transfer Function Upper Bound on Average Bit Error Probability 14.6.5.1 Fast-Fading Channel Model 14.6.5.2 Slow-Fading Channel Model 14.7 Other Combinations of Space-Time Block Codes and Space-Time Trellis Codes 14.7.1 Super-Orthogonal Space-Time Trellis Codes 14.7.1.1 The Parameterized Class of Space-Time Block Codes and System Model 14.7.1.2 Evaluation of the Pairwise Error Probability 14.7.1.3 Extension of the Results to Super-Orthogonal Codes with More than Two Transmit Antennas 14.7.1.4 Approximate Evaluation of Average Bit Error Probability 14.7.1.5 Evaluation of the Transfer Function Upper Bound on the Average Bit Error Probability 14.7.1.6 Numerical Results 14.7.2 Super-Quasi-Orthogonal Space-Time Trellis Codes 14.7.2.1 Signal Model 14.7.2.2 Evaluation of Pairwise Error Probability 14.7.2.3 Examples 14.7.2.4 Numerical Results 14.8 Disclaimer References CHAPTER 15 Capacity of Fading Channels 15.1 Channel and System Model

xxiii

820

821 824 827 827 829 831 831 833 833 834

834 836

844 845

846 848 850 850 852 853 857 858 859 863 863

xxiv

Index

CONTENTS

15.2 Optimum Simultaneous Power and Rate Adaptation 15.2.1 No Diversity 15.2.2 Maximal-Ratio Combining 15.3 Optimum Rate Adaptation with Constant Transmit Power 15.3.1 No Diversity 15.3.2 Maximal-Ratio Combining 15.4 Channel Inversion with Fixed Rate 15.4.1 No Diversity 15.4.2 Maximal-Ratio Combining 15.5 Numerical Examples

865 865 866

15.6 Capacity of MIMO Fading Channels

876

867 868 869 869 870 870 871

References

877

Appendix 15A. Evaluation of Jn (µ)

878

Appendix 15B. Evaluation of In (µ)

880 883

PREFACE Regardless of the branch of science or engineering, theoreticians have always been enamored with the notion of expressing their results in the form of closed-form expressions. Quite often the elegance of the closed-form solution is overshadowed by the complexity of its form and the difficulty in evaluating it numerically. In such instances, one becomes motivated to search instead for a solution that is simple in form and likewise simple to evaluate. A further motivation is that the method used to derive these alternative simple forms should also be applicable in situations where closed-form solutions are ordinarily unobtainable. The search for and ability to find such a unified approach for problems dealing with the evaluation of the performance of digital communication over generalized fading channels is what provided the impetus to write this textbook, the result of which represents the backbone for the material contained within its pages. For at least four decades, researchers have studied problems of this type and system engineers have used the theoretical and numerical results reported in the literature to guide the design of their systems. While the results from the earlier years dealt mainly with simple channel models, such as Rayleigh or Rician multipath fading, the applications in more recent years have become increasingly sophisticated, thereby requiring more complex models and improved diversity techniques. Along with the complexity of the channel model comes the complexity of the analytical solution that enables one to assess performance. With the mathematical tools that were previously available, the solutions to such problems when possible had to be expressed in complicated mathematical form that provided little insight into the dependence of the performance on the system parameters. Surprisingly enough, not until 1998 had anyone demonstrated a unified approach that not only allows previously obtained complicated results to be simplified both analytically and computationally but also permits new results to be obtained for special cases that heretofore resisted solution in a simple form. This approach was first introduced to the public by the authors in a tutorial-style article that appeared in the September 1998 issue of the IEEE Proceedings. Since that time, it has spawned a large wave of publications on the subject in the technical journal and conference literature, by both the authors and many others and, based on the variety of applications 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 alternative representations of classic functions arising in the error probability analysis of digital communication systems (e.g., the Gaussian Q-function1 and the Marcum Q-function) in such a manner that the resulting expressions for various performance measures such as average bit or symbol error rate are in a form that is rarely more complicated than a single integral with finite limits and an integrand composed of elementary (e.g., exponential and trigonometric) functions. By virtue of replacing the conventional forms of the above-mentioned functions by their alternative representations, the integrand will contain the moment generating function (MGF) of the instantaneous fading SNR, and as such the unified approach is referred to as the MGF-based approach. The first edition of this book was aimed at collecting and documenting the huge compendium of results contained in the myriad of contributions developed from the MGF-based approach that had been reported until that time and, by virtue of its unified notation and collocation in a single publication, would thereby be useful to both students and researchers in the field. In 1999 the manuscript for the first edition was submitted to the publisher. Since that time, a great deal of additional significant 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 extremely beneficial to these same researchers and students in bringing them up to date on these new developments. Perhaps the most significant of these new developments is the explosion of interest and research that has taken place in the area of transmit diversity and spacetime coding and the associated multiple-input/multiple-output (MIMO) channel, a subject 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 this all-important subject that, in keeping with the main theme of the book, deals with the performance evaluation aspects of such systems. The performance of MIMO systems is also treated from other perspectives elsewhere in the text. Aside from these developments, many new and exciting results have been developed by the authors as well as other researchers that (1) have led to new and improved diversity schemes and (2) allow for the performance analysis of previously known schemes operating in new and different fading scenarios not discussed in the first edition. A few of these developments are (1) new alternative forms for classic mathematical functions such as the second-order Gaussian Q-function and also higher powers of the first-order Gaussian Q-function; (2) improved diversity schemes such as threshold and postdetection generalized selection combining, switch-and-examine combining, and switch-and-wait combining; (3) new channel fading models of interest in wireless and mobile applications; (4) new bounds on system performance in the presence of fading; and (5) new mathematical results 1 The

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

PREFACE

xxvii

related to quadratic forms in Gaussian random variables and the difference in chisquare random variables with different degrees of freedom, allowing for the analysis of practical communication performance measures such as the outage probability of 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 sample of the voluminous amount of material (on the order of several hundred pages) that has 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, differentially coherent, partially coherent, and noncoherent communication systems are all handled as well as a large variety of fading channel models typical of communication links of practical interest. Both single- and multichannel reception are discussed, and in the case of the latter, a large variety of diversity types are considered. In fact, the chapter on multichannel reception (Chapter 9) is by itself now over 325 manuscript pages long and, in reality, could stand alone as its own textbook. For each combination of communication (modulation/detection) type, channel fading model, and diversity type, expressions for various system performance measures are obtained in a form that can be readily evaluated.2 All cases considered correspond to real practical channels, and in many instances the closed-form expressions obtained can be evaluated numerically on a handheld calculator. In writing this book, our intent was to spend as little space as possible duplicating material dealing with basic digital communication theory and system performance evaluation 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 most value to those desiring to extend their knowledge beyond what ordinarily might be covered in the classroom. In this regard, the book should have a strong appeal to graduate students doing research in the field of digital communications over fading channels as well as practicing engineers who are responsible for the design and performance evaluation of such systems. With regard to the latter, the book contains copious numerical evaluations that are illustrated in the form of parametric performance curves (e.g., average error probability versus average signal-to-noise ratio). The applications chosen for the numerical illustrations correspond to practical systems and as such the performance curves provided will have far more than academic value. The availability of such a large collection of system performance curves in a single compilation allows researchers and system designers to perform tradeoff studies among the various communication type/fading channel combinations so as to 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 express purpose. The first part contains an introduction to the subject of communication system performance evaluation followed by discussions of the various types of fading channel models and modulation/detection schemes that together form the 2 The terms bit error probability (BEP) and symbol error probability (SEP) are quite often used as alternatives to bit error rate (BER) and symbol error rate (SER). With no loss in generality, we shall employ both usages in this text.

xxviii

PREFACE

overall system. Part 2 starts by introducing the alternative forms of the classic functions mentioned above and then proceeds to show how these forms can be used to (1) evaluate certain integrals characteristic of communication system error probability performance and (2) find new representations for certain probability density and distribution functions typical of correlated fading applications. Part 3 is the “heart and soul” of the book since, in keeping with its title, the primary focus of this part is on performance evaluation of the various types of fading channel models and modulation/detection schemes introduced in Part 1 both for single and multichannel (diversity) reception. Before presenting this comprehensive performance evaluation study, however, Part 3 begins by deriving the optimum receiver structures corresponding to a variety of combinations concerning the knowledge or lack thereof of the fading parameters: amplitude, phase, and delay. Several of these structures might be deemed as too complex to implement in practice; nevertheless, their performance serves as a benchmark against which many suboptimum but practical structures discussed in the remainder of the chapter might be compared. Part 4, which deals with multiuser communications, considers first the problem of outage probability evaluation followed by optimum combining (diversity) in the presence of cochannel interference. The unified approach is then applied to studying the performance of single- and multiple-carrier direct-sequence code-division multiple-access (DS-CDMA) systems typical of the current digital cellular wireless standard. Part 5 extends the theory developed in the previous parts for uncoded communication to error-correction-coded systems and then space-time-coded systems and concludes with a discussion of the capacity of fading channels. Whereas the first edition has already established itself as the classic reference text on the subject with no apparent competition in sight, it is a safe bet that the second edition will continue to maintain that reputation for years to come. The authors know of no other textbook currently on the market that addresses the subject of digital communication over fading channels in as comprehensive and unified a manner 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 the reprint 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 other textbooks [3–11] devote part of their contents3 to fading channel performance evaluation, by comparison with our book, the treatment is brief and as such is incomplete. Because of this, we believe that our textbook is unique in the field. By way of acknowledgment, the authors wish to express their personal thanks to Dr. Payman Arabshahi of the Jet Propulsion Laboratory, Pasadena, CA for providing his invaluable help and consultation in preparing the submitted electronic version of the manuscript. Mohamed-Slim Alouini would also like to also like to express his sincere acknowledgment and gratitude to his PhD advisor Prof. Andrea J. Goldsmith of Stanford University, Palo Alto, CA for her guidance, support, and constant encouragement. Some of the material presented in Chapters 9, 3 Although Ref: 11 is a book that is entirely devoted to digital communication over fading channels, the focus is on error-correction-coded modulation and therefore would relate primarily only to Chapter 13 of our book.

PREFACE

xxix

12, and 15 are the result of joint work with Prof. Goldsmith. Mohamed-Slim Alouini would also like to thank his past and current doctoral students Mr. Ming Kang, Prof. Hong-Chuan Yang, Dr. Young-Chai Ko, and Ms. Lin Yang for their major 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. Gang Huo, and Dr. Yan Xin to this book and more generally to the research efforts of Mohamed-Slim Alouini are also particularly noteworthy. Finally, Mohamed-Slim Alouini would like to acknowledge the support of the National Science Foundation, the Office of the Vice President for Research of the University of Minnesota, and the University of Minnesota’s McKnight Land-Grant Professorship program for sustaining his research activities in the performance analysis of wireless communication systems since 1999. Marvin K. Simon Mohamed-Slim Alouini Jet Propulsion Laboratory, Pasadena, California University of Minnesota, Minneapolis, Minnesota

REFERENCES 1. R. S. Kennedy, Fading Dispersive Communication Channels. New York, NY: WileyInterscience, 1969. 2. K. Brayer, ed., Data Communications via Fading Channels. Piscataway, NJ: IEEE Press, 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: McGrawHill, 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: CRC Press, 1993. 7. W. C. Jakes, Microwave Mobile Communication. 2nd ed. Piscataway, NJ: IEEE Press, 1994. 8. K. Pahlavan and A. H. Levesque, Wireless Information Networks, Wiley Series in Telecommunications and Signal Processing. New York, NY: Wiley-Interscience, 1995. 9. G. St¨uber, Principles of Mobile Communication. Norwell, MA: Kluwer Academic Publishers, 1996. 10. T. S. Rappaport, Wireless Communications: Principles and Practice. Upper Saddle River, NJ: PTR Prentice-Hall, 1996. 11. S. H. Jamali and T. Le-Ngoc, Coded-Modulation Techniques for Fading Channels. Norwell, MA: Kluwer Academic Publishers, 1994.

NOMENCLATURE

Notation

Description or Name of Function and Reference Citation for Its Definition

AF AGC AM AOD ASK AT AWGN Bx (p, q) BEP BER BFSK BPSK BRGC BRS CCDF CCI CDF CDMA CF CIR CNR CPFSK CSI DPSK DS-CDMA EGC FSK

Amount of fading Automatic gain control Amplitude modulation Average outage duration Amplitude-shift-keying Absolute threshold Additive white Gaussian noise Incomplete beta function [1, Eq. (8.391)] Bit error probability Bit error rate Binary frequency-shift-keying Binary phase-shift-keying Binary reflected Gray code Branch relative strength Complementary cumulative distribution function Cochannel interference Cumulative distribution function Code-division multiple access Characteristic Function Carrier-to-interference ratio Carrier-to-noise ratio Continuous-phase frequency-shift-keying Channel state information Differential phase-shift-keying Direct-sequence code-division multiple access Equal gain combining Frequency-shift-keying xxxi

xxxii

NOMENCLATURE

1 F1

(α, γ ; z)

2 F1

(α, β; γ ; z)

p Fq (α1 , α2 , . . . , αp ;

β1 , β2 , . . . , βq ; z) GaAs
   a1 , . . . , ap m,n  Gp,q x  b1 , . . . , bq GSC γ γ  (x) γ (a, x)  (a, x) Hn (x) Iν (x) I&D i.i.d. ISI Jν (x) K LCR LOS MA MAI M-AM MAP M-ASK M-DPSK M-FSK M-PSK MC-CDMA MGF Mγ (s) MIMO MIP MISO ML MLSE MMSE MRC MSDD

Kummer confluent hypergeometric function [1, Eq. (9.210.1)] Gaussian hypergeometric function [1, Eq. (9.14.2)] Generalized hypergeometric function [1, Eq. (9.14.1)] Gallium Arsenide Meijer’s G-function [1, Eq. (9.301)] Generalized selection combining Instantaneous fading SNR Average fading SNR Gamma function [1, Eq. (8.310.1)] Incomplete gamma function [1, Eq. (8.350.1)] Complementary incomplete gamma function [1, Eq. (8.350.2)] nth-order Hermite polynomial [1, Eq. (8.950.1)] νth-order modified Bessel function of the first kind [1, Eq. (8.431)] Integrate-and-dump Independent, identically distributed Intersymbol interference νth-order Bessel function of the first kind [1, Eq. (8.411.1)] Rician fading parameter Level crossing rate Line of sight Multiple access Multiple-access interference Multiple amplitude modulation Maximum a posteriori Multiple amplitude-shift-keying Multiple differential phase-shift-keying Multiple frequency-shift-keying Multiple phase-shift-keying Multicarrier code-division multiple access Moment generating function Moment generating function of γ Multiple input/multiple output Multipath intensity profile Multiple input/single output Maximum likelihood Maximum-likelihood sequence estimation Minimum mean-square error Maximal-ratio combining Multiple-symbol differential detection

NOMENCLATURE

MSK MTCM NC NSD NT OC OPRA OQPSK ORA Pb (E) PBI PDF PDP PEP PG PLL PMF PN PSAM Ps (E) QAM QASK Q (x) Q (x, y; ρ) Qm (α, β) Qm,n (α, β) QOSTBC QPSK ROC RV SC SC-CDMA SEC SECps SEO SEP SER SIR SINR SIMO SIRP SIRV SISO SNR SOSTTC

xxxiii

Minimum-shift-keying Multiple trellis-coded modulation Noncoherent Normalized standard deviation Normalized threshold Optimum combining Optimum power and rate adaptation Offset (staggered) QPSK Optimum rate adaptation Bit error probability Partial-band interference Probability density function Power delay profile Pairwise error probability Processing gain Phase-locked loop Probability mass function Pseudonoise Pilot-symbol-assisted modulation Symbol error probability Quadrature amplitude modulation Quadrature amplitude-shift-keying First-order Gaussian Q-function [2, Eq. (26.2.3)] Second-order Gaussian Q-function [2, Eq. (26.3.3)] mth-order Marcum Q-function [3] Nuttall Q-function [5,6] Quasi-orthogonal space-time block code Quadriphase-shift-keying Region of convergence Random variable Selection combining Single-carrier code-division multiple access Switch-and-examine combining Postselection SEC Symbol error outage Symbol error probability Symbol error rate Signal-to-interference ratio Signal-to-interference plus noise ratio Single input/multiple output Spherically invariant random process Spherically invariant random variable Single input/single output Signal-to-noise ratio Super-orthogonal space-time trellis code

xxxiv

NOMENCLATURE

SQOSTTC SQPSK SS SSC STBC STC STTC SWC TCM TDMA T-GSC TUB TB (m, n, r) UEP Wk.m (z)

Super-quasi-orthogonal space-time trellis code Staggered quadriphase-shift-keying Spread spectrum Switch-and-stay combining Space-time block code Space-time code Space-time trellis code Scan-and-wait combining Trellis-coded modulation Time-division multiple access Generalized selection combining with threshold test per branch True upper (union) bound Incomplete Toronto function [4] Uniform error probability Whittaker function [1, Eq. (9.222)]

REFERENCES 1. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, 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 Memorandum M-339, ASTIA Document AD 11 65 451, Rand Corporation, Santa Monica, CA, January 1, 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 Underwater Systems Center, New London, CT, April 17, 1972. 6. M. K. Simon, Probability Distributions Involving Gaussian Random Variables: A Handbook 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 his parents and his family.

PART 1 FUNDAMENTALS

1 INTRODUCTION As we continue to step forward into the new millennium with wireless technologies leading the way in which we communicate, it becomes increasingly clear that the dominant consideration in the design of systems employing such technologies will be their ability to perform with adequate margin over a channel perturbed by a host of impairments, not the least of which is multipath fading. This is not to imply that 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 to serve 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 the fading channel scenarios that occurred well prior to the wireless revolution since indeed 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 more sophisticated wireless applications will no doubt also be useful for dealing with the less complicated fading environments of the past. With the above in mind, what better opportunity is there than now to write a comprehensive book that will provide simple and intuitive solutions to problems dealing with communication system performance evaluation over fading channels? Indeed, as mentioned in the preface, the primary goal of this book is to present a unified method for arriving at a set of tools that will allow the system designer to compute the performance of a host of different digital communication systems characterized 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 allow easy yet accurate performance evaluation but at the same time provide insight into the manner in which this performance depends on the key system parameters. To emphasize what was stated above, the set of tools that will be developed in this book are useful not only for the wireless applications that are rapidly filling our current 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 Alouini ISBN 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 purpose of this introductory chapter, namely, to provide several measures of performance related to practical communication system design and to begin exploring the analytical methods by which they may be evaluated. While the deeper meaning of these measures will be truly understood only after their more formal definitions are presented in the chapters that follow, the introduction of these terms here serves to illustrate the various possibilities that exist depending on both need and relative ease of evaluation.

1.1 1.1.1

SYSTEM PERFORMANCE MEASURES Average Signal-to-Noise Ratio (SNR)

Probably the most common and well understood performance measure characteristic of a digital communication system is signal-to-noise ratio (SNR). Most often this is measured at the output of the receiver and is thus directly related to the data detection process itself. Of the several possible performance measures that exist, it is typically the easiest to evaluate and most often serves as an excellent indicator of the overall fidelity of the system. While traditionally the term “noise” in signalto-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 more appropriate performance measure is average SNR, where the term “average” refers to statistical averaging over the probability distribution of the fading. In simple mathematical terms, if γ denotes the instantaneous SNR [a random variable (RV)] at the receiver output that includes the effect of fading, then
∞  γ = γpγ (γ ) dγ (1.1) 0

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 unified approach to performance evaluation, we first rewrite (1.1) in terms of the moment generating function (MGF) associated with γ :
∞ pγ (γ ) esγ dγ (1.2) Mγ (s) = 0

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

dMγ (s) |s=0 ds

(1.3)

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

SYSTEM PERFORMANCE MEASURES

5

To gain further insight into the power of the statement above, we note that in many systems, particularly those dealing with a form of diversity (multichannel) reception known as maximal-ratio combining (MRC) (to be discussed in great detail in Chapter 9), the output SNR, γ , is expressed  as a sum (combination) of the individual branch (channel) SNRs, namely, γ = L l=1 γl , where L denotes the number of channels combined. In addition, it is often reasonable in practice to assume that the channels are independent of each other, that is, that the RVs γl L l=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) = L l=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, even with the assumption of channel independence, the computation of the PDF pγ (γ ),  that characterize the which requires convolutional of the various PDFs pγl (γl ) L l=1 L channels, can still be a monumental task. Even in the case where these individual channel PDFs are of the same functional form but are characterized by different average SNRs, γ l , the evaluation of pγ (γ ) can still be quite tedious. Such is the power of the MGF-based approach; namely, it circumvents the need for finding the first-order PDF of the output SNR, provided that one is interested in a performance measure that can be expressed in terms of the MGF. Of course, for the case of  γ average SNR, the solution is extremely simple, namely, γ = L l=1 l regardless of whether the channels are independent, and in fact, one never needs to find the MGF at all. However, for other performance measures and also the average SNR of other combining statistics, such as the sum of an ordered set of random variables typical of generalized selection combining (GSC) (to be discussed in Chapter 9), matters are not quite this simple and the points made above for justifying an MGF-based approach are, as we shall see, especially significant. 1.1.2

Outage Probability

Another standard performance criterion characteristic of diversity systems operating over fading channels is the so-called outage probability-denoted by Pout and defined as the probability that the instantaneous error probability exceeds a specified value or equivalently the probability that the output SNR, γ , falls below a certain specified threshold, γth . Mathematically speaking, we have
γth Pout = pγ (γ ) dγ (1.4) 0

which is the cumulative distribution function (CDF) of γ , namely, Pγ (γ ), evaluated at γ = γth . Since the PDF and the CDF are related by pγ (γ ) = dPγ (γ ) /dγ 1 Note that the existence of the product form for the MGF M γ (s) does not necessarily imply that the channels are identically distributed; thus, each MGF Mγl (s) is allowed to maintain its own identity independent of the others. Furthermore, even if the channels are not assumed to be independent, the relation in (1.3) is nevertheless valid and in many instances the MGF of the (combined) output can still be obtained in closed form.

6

INTRODUCTION

and since Pγ (0) = 0, then the Laplace transforms of these two functions are related by2 pˆ γ (s) Pˆγ (s) = s

(1.5)

Furthermore, since the MGF is just the Laplace transform of the PDF with argument reversed in sign [i.e., pˆ γ (s) = Mγ (−s)], then the outage probability can be found from the inverse Laplace transform of the ratio Mγ (−s) /s evaluated at γ = γth Pout

1 = 2πj




σ +j ∞ σ −j ∞

Mγ (−s) sγth e ds s

(1.6)

where σ is chosen in the region of convergence of the integral in the complex s plane. Methods for evaluating inverse Laplace transforms have received widespread attention in the literature. (A good summary of these can be found in the paper by Abate and Whitt [1]). One such numerical technique that is particularly useful for CDFs of positive RVs (such as instantaneous SNR) is discussed in Appendix 9B and applied therein in Chapter 9. For our purpose here, it is sufficient to recognize once again that the evaluation of outage probability can be performed based entirely on the knowledge of the MGF of the output SNR without ever having to compute its PDF. 1.1.3

Average Bit Error Probability (BEP)

The third performance criterion and undoubtedly the most difficult of the three to compute is average bit error probability (BEP).3 On the other hand, it is the one that is most revealing about the nature of the system behavior and the one most often illustrated in documents containing system performance evaluations; thus, it is of primary interest to have a method for its evaluation that reduces the degree of difficulty as much as possible. The primary reason for the difficulty in evaluating average BEP lies in the fact that the conditional (on the fading) BEP is, in general, a nonlinear function of the instantaneous SNR, as the nature of the nonlinearity is a function of the modulation/detection scheme employed by the system. Thus, for example, in the multichannel case, the average of the conditional BEP over the fading statistics symbol “ˆ·” above a function denotes its Laplace transform. 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 with system performance. Furthermore, the terms bit error rate (BER) and symbol error rate (SER) are often used in the literature as alternatives to BEP and SEP. Rather than choose a preference, in this text we shall use these terms interchangeably. 2 The

3 The

SYSTEM PERFORMANCE MEASURES

7

is not a simple average of the per channel performance measure as was true for average SNR. Nevertheless, we shall see momentarily that an MGF-based approach is still quite useful in simplifying the analysis and in a large variety of cases allows unification under a common framework. Suppose first that the conditional BEP is of the form Pb (E |γ ) = C1 exp (−a1 γ )

(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) (see Chapter 8). Then, the average BEP can be written as 

Pb (E) =




∞




∞

Pb (E |γ )pγ (γ )dγ =

0

C1 exp (−a1 γ ) pγ (γ ) dγ = C1 Mγ (−a1 )

0

(1.8) where again Mγ (s) is the MGF of the instantaneous fading SNR and depends only on the fading channel model assumed. Suppose next that the nonlinear functional relationship between Pb (E |γ ) and γ is such that it can be expressed as an integral whose integrand has an exponential dependence on γ in the form of (1.7),4
Pb (E |γ ) =

ξ2

C2 h (ξ ) exp (−a2 g (ξ ) γ ) dξ

(1.9)

ξ1

where for our purpose here h (ξ ) and g (ξ ) are arbitrary functions of the integration variable and typically both ξ1 and ξ2 are finite (although this is not an absolute 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 employing alternative forms of such classic nonlinear functions as the Gaussian Q-function and Marcum Q-function (see Chapter 4), which are characteristic of the relationship between Pb (E |γ ) and γ corresponding to, for example, coherent detection of PSK and noncoherent detection of quadriphase-shift-keying (QPSK), respectively. Still another possibility is that the nonlinear functional relationship between Pb (E |γ ) and γ is inherently in the form of (1.9); thus, no alternative representation need be employed. An example of such occurs for the conditional symbol error probability (SEP) associated with coherent and differentially coherent detection of M-ary PSK (M-PSK) (see Chapter 8). Regardless of the particular case at hand, 4 In the more general case, the conditional BEP might be expressed as a sum of integrals of the type in (1.9). 5 In principle, (1.9) includes (1.7) as a special case if h (ξ ) is allowed to assume the form of a Dirac delta function located within the interval ξ1 ≤ ξ ≤ ξ2 .

8

INTRODUCTION

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


∞

Pb (E) = 0







= C2




ξ2 ξ1

C2 h (ξ ) exp(−a2 g (ξ ) γ )dξ pγ (γ )dγ 0

ξ1

∞

exp (−a2 g (ξ ) γ )pγ (γ ) dγ dξ

h (ξ )


= C2

∞
ξ2

Pb (E |γ )pγ (γ )dγ =

(1.10)

0

ξ2

h (ξ ) Mγ (−a2 g (ξ )) dξ ξ1

As we shall see later on in the text, integrals of the form in (1.10) can, for many special cases, be obtained in closed form. At the very worst, with rare exception, the resulting expression will be a single integral with finite limits and an integrand composed of elementary functions.6 Since (1.8) and (1.10) cover a wide variety of different modulation/detection types and fading channel models, we refer to this approach for evaluating average error probability as the unified MGF-based approach and the associated forms of the conditional error probability as the desired forms. The first notion of such a unified approach was discussed in Ref. 2 and laid the groundwork for much of the material that follows in this text. It goes without saying that not every fading channel communication problem fits this description; thus, alternative, but still simple and accurate, techniques are desirable for evaluating system error probability in such circumstances. One class of problems for which a different form of MGF-based approach is possible relates to communication with symmetric binary modulations wherein the decision mechanism constitutes a comparison of a decision variable with a zero threshold. Aside from the obvious uncoded applications, the above-mentioned class also includes the evaluation of pairwise error probability in error-correction-coded systems as discussed in Chapter 12. In mathematical terms, letting D |γ denote the decision variable,7 then the corresponding conditional BEP is of the form (assuming arbitrarily 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 both positive and negative values whereas the instantaneous fading SNR, γ , is restricted to only positive values, there is a strong resemblance between the binary probability 6 As

we shall see in Chapter 4, the h (ξ ) and g (ξ ) that result from the alternative representations of the Gaussian and Marcum Q-functions are composed of simple trigonometric functions. 7 The notation “D |γ ” is not meant to imply that the decision variable explicitly depends on the fading SNR. Rather, it is merely intended to indicate the dependence of this variable on the fading statistics of 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 1 Pb (E |γ ) = 2πj




σ +j ∞ σ −j ∞

MD|γ (−s) ds s

(1.12)

where MD|γ (−s) now denotes the MGF of the decision variable D |γ , that is, the bilateral Laplace transform of pD|γ (D) with argument reversed. To see how MD|γ (−s) might explicitly depend on γ , we now consider the subclass of problems where the conditional decision variable D |γ corresponds to a quadratic form of independent complex Gaussian RVs, such as a sum of the squared magnitudes of, say, L independent complex Gaussian RVs—a chi-square RV with 2L degrees of freedom. Such a form occurs for multiple-(L)-channel reception of binary modulations with differentially coherent or noncoherent detection (see Chapter 9). In this instance, the MGF MD|γ (s) happens to be exponential in γ and has the generic form MD|γ (s) = f1 (s) exp (γf2 (s))

(1.13)

 If as before we let γ = L l=1 γl , then substituting (1.13) into (1.12) and averaging over the fading results in the average BEP8 1 Pb (E) = 2πj




σ +j ∞

σ −j ∞

MD (−s) ds s

(1.14)

where 

MD (s) =




∞




∞

MD|γ (s) pγ (γ ) dγ = f1 (s)

0

exp (γf2 (s)) pγ (γ ) dγ 0

(1.15)

= f1 (s) Mγ (f2 (s)) is the unconditional MGF of the decision variable, which also has the product form MD (s) = f1 (s)

L 

Mγl (f2 (s))

(1.16)

l=1

Finally, by virtue of the fact that the MGF of the decision variable can be expressed in 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 on the basis of knowledge of the latter MGF. It is not immediately obvious how to extend the inverse Laplace transform technique discussed in Appendix 9B to CDFs of bilateral RVs; thus other methods 8 The approach for computing average BEP as described by (1.13) was also described by Biglieri et 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 contour integration using residues, saddle point integration, and numerical integration by Gauss–Chebyshev quadrature rules, are discussed in the literature [3–6] and will be covered later on in the text. Although the methods dictated by (1.14) and (1.8) or (1.10) cover a wide variety of problems dealing with the performance of digital communication systems over fading channels, there are still some situations that don’t lend themselves to either of these two unifying methods. An example of such is the evaluation of the bit error probability performance of an M-ary noncoherent orthogonal system operating over an L-path diversity channel (see Chapter 9). However, even in this case there exists an MGF-based approach that greatly simplifies the problem and allows for a result [7] more general than that previously reported by Weng and Leung [8]. We now briefly outline the method, leaving the more detailed treatment to Chapter 9. Consider an M-ary communication system where, rather than comparing a single decision variable with a threshold, one decision variable U1 |γ is compared with the remaining M − 1 decision variables Um , m = 2, 3, . . . , M, all of which do not depend on the fading statistics.9 Specifically, a correct symbol decision is made if U1 |γ is greater than Um , m = 2, 3, . . . , M. Assuming that the M decision variables are independent, then, in mathematical terms, the probability of correct decision is given by Ps (C |γ ; u1 ) = Pr {U2 < u1 , U3 < u1 , . . . , UM < u1 |U1 |γ = u1 } M−1 
u1  M−1 = Pr {U2 < u1 |U1 |γ = u1 } = pU2 (u2 ) du2 



= 1 − 1 − PU2 (u1 )

0

M−1

(1.17)

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

M−1 i=1



i  M −1 (−1)i+1 1 − PU2 (u1 ) = g (u1 ) i

(1.18)

Averaging over u1 and using the Fourier transform relationship between the PDF pU1 |γ (u1 ) and the MGF MU1 |γ (j ω), we obtain


∞

Ps (E |γ ) =


∞

= 0 9 Again

g (u1 ) pU1 |γ (u1 ) du1

0

1 2π




(1.19)

∞

−∞

MU1 |γ (j ω) e

−j ωu1

g (u1 ) dω du1

the conditional notation on γ for U1 is not meant to imply that this decision variable is explicitly a 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 ) the conditional MGF MU1 |γ (j ω) is of the form in (1.13), then averaging (1.19) over γ transforms MU1 |γ (j ω) into MU1 (j ω) of the form in (1.15), which, when substituted in (1.19) and reversing the order of integration, produces


∞ 
∞ 1 −j ωu1 (1.20) f1 (j ω) Mγ (f2 (j ω)) e g (u1 ) du1 dω Ps (E) = 2π −∞ 0 Finally, because the CDF PU2 (u1 ) in (1.18) is that of a central chi-square RV with 2L degrees of freedom, the resulting form of g (u1 ) is such that the integral on u1 in (1.20) can be obtained in closed form. Thus, as promised, what remains again is an expression for average SEP (which for M-ary orthogonal signaling can be related to the average BEP by a simple scale factor) whose dependence on the fading statistics is solely through the MGF of the fading SNR. All the techniques considered thus far for evaluating average error probability performance rely on the ability to evaluate the MGF of the instantaneous fading SNR γ . In dealing with a form of diversity reception referred to as equal gain combining (EGC) (to be discussed in great detail in Chapter 9), √ the instantaneous  √ 2 fading SNR at the output of the combiner takes the form γ = [(1/ L) L l=1 γl ] . In this case, it is more convenient to deal with the MGF of the square root of the √  √ √ L  √ instantaneous fading SNR x = γ = (1/ L) L l=1 γl = (1/ L) l=1 xl since if the channels are again assumed independent then again this MGF takes on a  √  L product form, namely, Mx (s) = l=1 Mxl s/ L . Since the average BER can alternatively be computed from
∞ Pb (E |x ) px (x) dx Pb (E) = (1.21) 0

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

ξ2



C2 h (ξ ) exp −a2 g (ξ ) x 2 dξ

(1.22)

ξ1

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 (j ω) = F {Pb (E |x )} denote the Fourier transform of Pb (E |x ), then independent of the form of Pb (E |x ), we obtain10
∞ 1 Pb (E) = G (j ω) Mx (j ω) dω 2π −∞ (1.23)
1 ∞ Re {G (j ω) Mx (j ω)} dω = π 0 10 A

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

12

INTRODUCTION

where we have recognized that the imaginary part of the integral must be equal to zero since Pb (E) is real and that the even part of the integrand is an even function of ω. Making the change of variables θ = tan−1 ω, (1.23) can be written in the form of an integral with finite limits: 1 Pb (E) = π 2 = π





π/2

1 Re {G (j tan θ ) Mx (j tan θ )} dθ cos2 θ

π/2

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

0

0

(1.24)

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




ξ2

∞

C2 h (ξ ) ξ1

exp(−a2 g (ξ ) x 2 + j ωx) dx dξ

(1.25)

0

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


∞

exp(−a2 g (ξ ) x 2 + j ωx) dx

0

      1 3 (j ω)2 (j ω)2 = π a2 g (ξ ) exp + j ω 1 F1 1, ; 2a2 g (ξ ) 4a2 g (ξ ) 2 4a2 g (ξ ) (1.26) where 1 F1 (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 on ξ can also be evaluated in closed form; thus, in these instances, Pb (E) can be obtained as a single integral with finite limits and an integrand involving the MGF of the fading. Methods of error probability evaluation based on the type of MGF approach described above have been considered in the literature [11–13] and will be 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 most commonly employed to describe the behavior of digital communication systems in the presence of fading. Although not as descriptive as the other two, average SNR had the advantage that it was simple to compute in that it required knowledge of only the first statistical moment of the instantaneous SNR. However, in the context of diversity combining, this performance criterion does not capture all the diversity benefits. Indeed, if the diversity advantage were limited to an average SNR gain, then this could be achieved by simply increasing the transmitter power. Of more importance is the aptitude of diversity systems to reduce the fading-induced fluctuations 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 performance measures that take into account higher moments of the combiner output SNR. Following along this train of thought, another performance measure that is most often simple to compute and requires knowledge of only the first and second moments of the instantaneous SNR was introduced by the authors [15] when describing the behavior of dual-diversity combining systems over correlated log-normal fading channels. The measure, which is referred to as “amount of fading” (AF), is associated with the output of the combiner and is modeled after a criterion bearing the same name that was originally introduced by Charash [16, p. 29] as a measure of the severity of the fading channel by itself (see Chapter 2, Section 2.2). It is the authors’ suggestion that this same AF measure is often appropriate in the more general context of describing the behavior of systems with arbitrary combining techniques and channel statistics and thus can be used as an alternative performance criterion whenever convenient.11 Specifically, letting γt denote the total instantaneous SNR at the combiner output, we define AF by

  2 E γt2 − E γt var γt AF =  2 =

 2 E γt E γt

(1.27)

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

d 2 Mγt (s) |s=0 ds 2



−



2 dMγt (s) |s=0 ds

2 dMγt (s) |s=0 ds

(1.28)

Because the AF defined in (1.27) is computed at the output of the combiner, its evaluation will reflect the behavior of the particular diversity combining technique as well as the statistics of the fading channel and thus, as mentioned above, is a measure of the performance of the entire system. Closed-form expressions for a variety of such evaluations will be presented in Chapter 9. 1.1.5

Average Outage Duration

In certain communication system applications such as adaptive transmission schemes, the performance metrics discussed above do not provide enough information for the overall system design and configuration. In that case, in addition to these performance measures, the frequency of outages and the average outage duration (AOD) (also known as the “average fade duration”) are important performance criteria for the proper selection of the transmission symbol rate, interleaver depth, packet length, and/or time slot duration. 11 Perhaps

the earliest indication of such a performance measure appears in a paper by Win and Winters [17], who described the behavior of hybrid selection combining/maximal-ratio combining in the presence of Rayleigh fading in terms of the normalized standard deviation (NSD) of the diversity combiner 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 whenever 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 system remains in the outage state. Mathematically speaking, the AOD is well known to be given by [18] T (γth ) =

Pout N (γth )

(1.29)

where Pout was defined and discussed in Section 1.1.2 and N (γth ) is the frequency of outages or equivalently the average level crossing rate (LCR) of the output SNR γ at level γth , which can be obtained from the well-known Rice formula [19]
∞ γ˙ fγ ,γ˙ (γth , γ˙ ) d γ˙ , (1.30) N (γth ) = 0

with fγ ,γ˙ (γ , γ˙ ) the joint PDF of γ and its time derivative γ˙ . Methods and analytical expressions for the evaluation of the average LCR, and thus AOD, for various diversity combining schemes will be presented in Chapter 9.

1.2

CONCLUSIONS

Without regard to the specific application or performance measure, we have briefly demonstrated in this chapter that for a wide variety of digital communication systems covering virtually all known modulation/detection techniques and practical fading channel models, there exists an MGF-based approach that simplifies the evaluation of this performance. In the biggest number of these instances, the MGFbased approach is encompassed in a unified framework that allows the development of a set of generic tools to replace the case-by-case analyses typical of previous contributions in the literature. It is the authors’ hope that by the time the readers reach the end of this book and have experienced the exhaustive set of practical circumstances where these tools are useful, they will fully appreciate the power behind the MGF-based approach and as such will generate for themselves an insight into finding new and exciting applications.

REFERENCES 1. J. Abate and W. Whitt, “Numerical inversion of Laplace transforms of probability distributions,” 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 of digital 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 probabilities 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 evaluating error 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 modulations in 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 performance 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 orthogonal modulation 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 under Nakagami 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. San Diego, CA: Academic Press, 1994. 11. M.-S. Alouini and M. K. Simon, “Error rate analysis of M-PSK with equal-gain combining 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-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun., vol. 47, no. 9, September 1999, pp. 1335–1344. 13. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Unified analysis of equal-gain diversity on Rician and Nakagami fading channels,” Proc. IEEE Wireless Commun. and Networking Conf. (WCNC’99 ), New Orleans, LA, September 1999. 14. A. Annamalai, C. Tellambura, and V. K. Bhargava, “A general method for calculating error 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 in Rayleigh 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. St¨uber, Principles of Mobile Communications, 2nd ed. Boston, MA: Kluwer Academic 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 CHANNEL CHARACTERIZATION AND MODELING Radiowave propagation through wireless channels is a complicated phenomenon characterized by various effects such as multipath and shadowing. A precise mathematical description of this phenomenon is either unknown or too complex for tractable communication systems analyses. However, considerable efforts have been devoted to the statistical modeling and characterization of these different effects. The result is a range of relatively simple and accurate statistical models for fading channels that depend on the particular propagation environment and the underlying communication scenario. The primary purpose of this chapter is to briefly review the principal characteristics and models for fading channels. A more detailed treatment of this subject can be found in standard textbooks such as those by Proakis, Rappaport, and St¨uber [1–3]. This chapter also introduces terminology and notation that will be used throughout the book. The chapter is organized as follows. A brief qualitative description of the main characteristics of fading channels is presented in the next section. Models for frequency-flat fading channels, corresponding to narrowband transmission, are described in Section 2.2. Models for frequency-selective fading channels that characterize fading in wideband channels are described in Section 2.3.

2.1 2.1.1

MAIN CHARACTERISTICS OF FADING CHANNELS Envelope and Phase Fluctuations

When a received signal experiences fading during transmission, both its envelope and 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 Alouini ISBN 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 compensate for them at the receiver. Most often, analyses of systems employing such modulations assume that the phase effects due to fading are perfectly corrected at the receiver resulting in what is referred to as “ideal” coherent demodulation. For noncoherent modulations, phase information is not needed at the receiver and therefore the phase variation due to fading does not affect the performance. Hence, performance analyses for both ideal coherent and noncoherent modulations over fading channels requires knowledge of only the fading envelope statistics and will be the case most often considered in this text. Furthermore, for so-called slow fading (to be discussed next), wherein the fading is at least constant over the duration of a symbol time, the fading envelope random process can be represented by a random variable (RV) over the symbol time. 2.1.2

Slow and Fast Fading

The distinction between slow and fast fading is important for the mathematical modeling of fading channels and for the performance evaluation of communication systems operating over these channels. This notion is related to the coherence time Tc of the channel, which measures the period of time over which the fading process is correlated (or equivalently, the period of time after which the correlation function of two samples of the channel response taken at the same frequency but different time instants drops below a certain predetermined threshold). The coherence time is also related to the channel Doppler spread fd by Tc


1 fd

(2.1)

The fading is said to be slow if the symbol time duration Ts is smaller than the channel’s coherence time Tc ; otherwise it is considered to be fast. In slow fading a particular fade level will affect many successive symbols, which leads to burst errors, whereas in fast fading the fading decorrelates from symbol to symbol. In this latter case and when the communication receiver decisions are based on an observation of the received signal over two or more symbol times (such as differentially coherent or coded communications), it becomes necessary to consider the variation of the fading channel from one symbol interval to the next. This is done through a range of correlation models that depend essentially on the particular propagation environment and the underlying communication scenario. These various autocorrelation models and their corresponding power spectral density are tabulated in Table 2.1 [4], in which for convenience the variance of the fast-fading process is normalized to unity. 2.1.3

Frequency-Flat and Frequency-Selective Fading

Frequency selectivity is also an important characteristic of fading channels. If all the spectral components of the transmitted signal are affected in a similar manner, the fading is said to be frequency-nonselective or equivalently frequency-flat. This

19

MODELING OF FLAT-FADING CHANNELS

TABLE 2.1 Correlation and Spectral Properties of Various Types of Fading Processes of Practical Interest Type of Fading Spectrum

Fading Autocorrelation ρ

Rectangular

sin(2π fd Ts )/(2π fd Ts )

Gaussian


 2  exp − π fd Ts

Land mobile

  J0 2π fd Ts

First-order Butterworth

  exp −2π fd Ts 

Second-order Butterworth

   π fd Ts  exp − √

Normalized PSD (2fd )−1 ; |f| ≤ fd     −1 f 2 √ exp − π fd fd 2 2  −1/2   π f − fd2 ; f  ≤ fd 



π fd



f 1+ fd

 1 + 16

2

    

π fd Ts  √ × cos π f√d Ts + sin



f fd

2 −1

4 −1



2

2

Key: PSD—power spectral density; fd —Doppler spread; Ts —symbol time.

is the case for narrowband systems, in which the transmitted signal bandwidth is much smaller than the channel’s coherence bandwidth fc . This bandwidth measures the frequency range over which the fading process is correlated and is defined as the frequency bandwidth over which the correlation function of two samples of the channel response taken at the same time but different frequencies falls below a suitable value. In addition the coherence bandwidth is related to the maximum delay spread τmax by fc


1 τmax

(2.2)

On the other hand, if the spectral components of the transmitted signal are affected by different amplitude gains and phase shifts, the fading is said to be frequencyselective. This applies to wideband systems in which the transmitted bandwidth is bigger than the channel’s coherence bandwidth.

2.2

MODELING OF FLAT-FADING CHANNELS

When fading affects narrowband systems, the received carrier amplitude is modulated by the fading amplitude α, where α is a RV with mean-square value  = α 2 and probability density function (PDF) pα (α), which is dependent on the nature of the radio propagation environment. After passing through the fading channel, the signal is perturbed at the receiver by additive white Gaussian noise (AWGN), which is typically assumed to be statistically independent of the fading amplitude α, and which is characterized by a one-sided power spectral density N0

20

FADING CHANNEL CHARACTERIZATION AND MODELING

Watts/Hertz. Equivalently, the received instantaneous signal power is modulated by α 2 . Thus, we define the instantaneous signal-to-noise power ratio (SNR) per symbol by γ = α 2 Es /N0 and the average SNR per symbol by γ = Es /N0 , where Es is the energy per symbol.1 In addition, the PDF of γ is obtained by introducing a change of variables in the expression for the fading PDF pα (α) of α, yielding

  γ pα γ (2.3) pγ (γ) =  2 γγ The moment generating function (MGF) Mγ (s) associated with the fading PDF pγ (γ ) and defined by  ∞ Mγ (s) = pγ (γ ) esγ dγ (2.4) 0

is another important statistical characteristic of fading channels, particularly in the context of this book. In addition, consistent with Section 1.1.4 of Chapter 1, the amount of fading (AF), or “fading figure,” associated with the fading PDF is defined as

E (α 2 − )2 E[γ 2 ] − (E[γ ])2 var[α 2 ] = = (2.5) AF = (E[α 2 ])2 2 (E[γ ])2 where E[·] denotes statistical average and var[·] denotes variance. This figure was introduced by Charash [5, p. 29; 6] as a unified measure of the severity of the fading, and is typically independent of the average fading power . We now present the different radio propagation effects involved in fading channels, their corresponding PDFs, MGFs, AFs, and their relation to physical channels. These properties are summarized in Table 2.2. 2.2.1

Multipath Fading

Multipath fading is due to the constructive and destructive combination of randomly delayed, reflected, scattered, and diffracted signal components. This type of fading is relatively fast and is therefore responsible for the short-term signal variations. Depending on the nature of the radio propagation environment, there are different models describing the statistical behavior of the multipath fading envelope. 2.2.1.1 Rayleigh The Rayleigh distribution is frequently used to model multipath fading with no direct line-of-sight (LOS) path. In this case, the channel fading amplitude α is distributed according to   α2 2α exp − , α≥0 (2.6) pα (α) =   1 Our

performance evaluation of digital communications over fading channels will generally be a function of the average SNR per symbol γ .

21

1 ≤m 2

Nakagami-m

Composite gamma/log-normal

m and 0 ≤ σ

σ

0≤n

Nakagami-n (Rice)

Log-normal shadowing

0≤q≤1

Fading Parameter

Nakagami-q (Hoyt)

Rayleigh

Type of Fading exp −

γ γ



PDF pγ (γ )

∞

mm γ m−1

1 − 2sγ +

(2sγ )2 q2 (1 + q2 )2

−1/2

sγ m

n=1

1−

−m

Np

√  1  Hxn exp 10( 2σ xn +µ)/10 s √ π



  (1 + n2 ) n2 sγ exp (1 + n2 ) − sγ (1 + n2 ) − sγ



(1 − sγ )−1

MGF Mγ (s)

Np

−m
mγ  √ 1  Hxn 1 − 10( 2σ xn +µ)/10 s/m exp − √ m w (m) w π 0 n=1  2 ξ (10 log10 w − µ) dw × √ exp − 2σ 2 2πσw



 (10 log10 γ − µ)2 4.34 exp − √ 2 2σ 2πσγ

  2 (1 + n2 )e−n (1 + n2 )γ exp − γ γ    (1 + n2 )γ × I0 2n γ   m m−1 m γ mγ exp − m γ (m) γ

  (1 + q2 ) (1 + q2 )2 γ exp − 2 2qγ 4q γ   4 (1 − q )γ × I0 4q2 γ

1 γ



TABLE 2.2 Probability Density Function (PDF) and Moment Generating Function (MGF) of the SNR per Symbol γ for Some Common Fading Channels

22

FADING CHANNEL CHARACTERIZATION AND MODELING

and hence, following (2.3), the instantaneous SNR per symbol of the channel γ is distributed according to an exponential distribution given by   γ 1 , pγ (γ) = exp − γ γ

γ≥0

(2.7)

The MGF corresponding to this fading model is given by Mγ (s) = (1 − sγ )−1

(2.8)

In addition, the moments associated with this fading model can be expressed by E(γ k ) = (1 + k) γ k

(2.9)

where (.) is the gamma function. The Rayleigh fading model therefore has an AF equal to 1, and typically agrees very well with experimental data for mobile systems where no LOS path exists between the transmitter and receiver antennas [3]. It also applies to the propagation of reflected and refracted paths through the troposphere [7] and ionosphere [8,9], and to ship-to-ship [10] radio links. 2.2.1.2 Nakagami-q (Hoyt) The Nakagami-q distribution, also referred to as the Hoyt distribution [11], is given in Eq. (52) of Ref. 12 by pα (α) =

    (1 − q 4 ) α 2 (1 + q 2 )2 α 2 (1 + q 2 ) α I , exp − 0 q 4q 2  4q 2 

α≥0

(2.10) where I0 (.) is the zeroth-order modified Bessel function of the first kind and q is the Nakagami-q fading parameter, which ranges from 0 to 1. Using (2.3), it can be shown that the SNR per symbol of the channel (γ) is distributed according to pγ (γ) =

    (1 − q 4 ) γ (1 + q 2 ) (1 + q 2 )2 γ exp − I , 0 2qγ 4q 2 γ 4q 2 γ

γ≥0

(2.11)

It can be shown that the MGF corresponding to (2.11) is given by −1/2  (2sγ )2 q 2 Mγ (s) = 1 − 2sγ + (1 + q 2 )2

(2.12)

Also the moments associated with this model are given by [12, Eq. (52)]  E(γ ) = (1 + k) 2 F1 k

2   1 − q2 k−1 k γk − , − ; 1; 2 2 1 + q2

(2.13)

MODELING OF FLAT-FADING CHANNELS

23

where 2 F1 (·, ·; ·, ·) is the Gauss hypergeometric function, and the AF of the Nakagami-q distribution is therefore given by AFq =

2 (1 + q 4 ) , (1 + q 2 )2

0≤q≤1

(2.14)

and hence ranges between 1 (q = 1) and 2 (q = 0). The Nakagami-q distribution spans the range from one-sided Gaussian fading (q = 0) to Rayleigh fading (q = 1). It is typically observed on satellite links subject to strong ionospheric scintillation [13,14]. Note that one-sided Gaussian fading corresponds to the worst-case fading or equivalently, the largest AF for all multipath distributions considered in our analyses. 2.2.1.3 Nakagami-n (Rice) The Nakagami-n distribution is also known as the Rice distribution [15]. It is often used to model propagation paths consisting of one strong direct LOS component and many random weaker components. Here the channel fading amplitude follows the distribution [12, Eq. (50)]      2 2 −n 2 2 2 (1 + n ) α 1+n  2(1 + n )e α I0 2nα exp − , α≥0 pα (α) =    (2.15) where n is the Nakagami-n fading parameter, which ranges from 0 to ∞. This parameter is related to the Rician K factor by K = n2 which corresponds to the ratio of the power of the LOS (specular) component to the average power of the scattered component. Applying (2.3) shows that the SNR per symbol of the channel, γ, is distributed according to a noncentral chi-square distribution given by      2 (1 + n2 )γ (1 + n2 )γ (1 + n2 )e−n I0 2n exp − , γ≥0 pγ (γ) = γ γ γ (2.16) It can also be shown that the MGF associated with this fading model is given by   n2 sγ (1 + n2 ) exp (2.17) Mγ (s) = (1 + n2 ) − sγ (1 + n2 ) − sγ and that the moments are given by [12, Eq. (50)] E(γ k ) =

(1 + k) 2 k 1 F1 (−k, 1; −n ) γ (1 + n2 )k

(2.18)

where 1 F1 (·, ·; ·) is the Kummer confluent hypergeometric function. The AF of the Nakagami-n distribution is given by AFn =

1 + 2 n2 , (1 + n2 )2

n≥0

(2.19)

24

FADING CHANNEL CHARACTERIZATION AND MODELING

and hence ranges between 0 (n = ∞) and 1 (n = 0). The Nakagami-n distribution spans the range from Rayleigh fading (n = 0) to no fading (constant amplitude) (n = ∞). This type of fading is typically observed in the first resolvable LOS paths of microcellular urban and suburban land–mobile [16], picocellular indoor [17], and factory [18] environments. It also applies to the dominant LOS path of satellite [19,20] and ship-to-ship [10] radio links. 2.2.1.4 Nakagami-m The Nakagami-m PDF is in essence a central chi-square distribution given by [12, Eq. (11)]   2mm α 2m−1 mα 2 exp − , α≥0 (2.20) pα (α) = m (m)  where m is the Nakagami-m fading parameter, which ranges from 12 to ∞. Figure 2.1 shows the Nakagami-m PDF for  = 1 and various values of the m parameter. Applying (2.3) shows that the SNR per symbol γ is distributed according to a gamma distribution given by   mγ mm γ m−1 , γ≥0 (2.21) exp − pγ (γ) = m γ (m) γ 2 1.8

m = 1/2

Probability Density Function Pa(a)

1.6

m=1 m=2

1.4

m=4 1.2 1 0.8 0.6 0.4 0.2 0 0

0.5

1

1.5

2

Channel Fade Amplitude a

Figure 2.1

Nakagami PDF for  = 1 and various values of the fading parameter m.

2.5

MODELING OF FLAT-FADING CHANNELS

It can also be shown that the MGF is given in this case by   sγ −m Mγ (s) = 1 − m

25

(2.22)

and that the moments are given by [12, Eq. (65)] E[γ k ] =

(m + k) k γ (m) mk

(2.23)

1 2

(2.24)

which yields an AF of AFm =

1 , m

m≥

Hence, the Nakagami-m distribution spans via the m parameter the widest range of AF (from 0 to 2) among all the multipath distributions considered in this book. For instance, it includes the one-sided Gaussian distribution (m = 12 ) and the Rayleigh distribution (m = 1) as special cases. In the limit as m −→ +∞, the Nakagami-m fading channel converges to a nonfading AWGN channel. Furthermore, when m < 1, equating (2.14) and (2.24) we obtain a one-to-one mapping between the m parameter and the q parameter allowing the Nakagami-m distribution to closely approximate the Nakagami-q (Hoyt) distribution, and this mapping is given by m=

(1 + q 2 )2 , 2(1 + 2q 4 )

m≤1

(2.25)

Similarly, when m > 1, equating (2.19) and (2.24), we obtain another one-toone mapping between the m parameter and the n parameter (or equivalently the Rician K factor) allowing the Nakagami-m distribution to closely approximate the Nakagami-n (Rice) distribution, and this mapping is given by (1 + n2 )2 , n≥0 1 + 2n2  √ m2 − m n= , m≥1 √ m − m2 − m

m=

(2.26)

Finally, the Nakagami-m distribution often gives the best fit to land–mobile [21–23] and indoor–mobile [24] multipath propagation, as well as scintillating ionospheric radio links [9,25–28]. 2.2.1.5 Weibull The Weibull distribution [29] is yet another mathematical description of a probability model for characterizing amplitude fading in a multipath environment, particularly that associated with mobile radio systems operating in the 800/900 MHz frequency range [30–32]. The Weibull PDF is given by      c/2 c/2  2   1 + 2c 2 α  1+ α c−1 exp − , α≥0 pα (α) = c   c (2.27)

26

FADING CHANNEL CHARACTERIZATION AND MODELING

where c is a parameter that is chosen to yield a best fit to measurement results and as such affords the shape flexibility of the Nakagami distributions. The corresponding CDF is    c/2  2 α2  1+ , α≥0 (2.28) Pα (α) = 1 − exp −  c For the special case of c = 2, (2.27) describes a Rayleigh distribution, and when c = 1, it describes an exponential distribution. Applying (2.3) to (2.27), the SNR per symbol (γ ) has PDF and CDF c pγ (γ ) = 2

     c/2  c/2   1 + 2c 2 γ  1+ γ c/2−1 exp − , γ γ c

γ ≥0 (2.29)

and      2 c/2 γ  1+ , Pγ (γ ) = 1 − exp − γ c

γ ≥0

(2.30)

A comparison of (2.29) with (2.27) reveals that γ is also a Weibull RV but with parameter c/2 instead of c. This observation can be generalized in the form of a theorem relating to a property of Weibull RVs as follows. Theorem 2.1 If R is a sample of a Weibull distribution with parameter c, then R β is also a sample of a Weibull distribution with parameter c/β. The theorem is most easily proved by dealing with the normalized form of Weibull RVs that have the PDF   pR (R) = cR c−1 exp −R c ,

R≥0

Now let y = R β . Then the PDF of y is given in terms of the PDF of R by     cy (c−1)/β exp −y c/β pR y 1/β py (y) =   =  dy  βy (β−1)/β  dR    c = y (c/β)−1 exp −y c/β , β

(2.31)

(2.32)

y≥0

which completes the proof. Theorem 2.1 is particularly useful in finding all higherorder moments of Weibull RVs from knowledge of only their first moment. Specifically, it is known [29] that the first moment of R is given by   1 E {R} =  1 + (2.33) c

MODELING OF FLAT-FADING CHANNELS

27

Then, it immediately follows that     β E Rβ =  1 + c

(2.34)

Until just recently, a tractable and useful expression for the MGF of the Weibull distribution in terms of tabulated functions was considered to be unknown despite the fact that this MGF was known to exist for c ≥ 1 [33]. Recognizing that the form of the MGF of the Weibull distribution is related to Faxen’s integral [34, p. 332], then applying a Mellin transform approach [35], Cheng et al. [36] have been able to derive a closed-form expression for the MGF as follows:    c/2  1 + 2c 1 s −c Mα (s) = c (2π )(1−c)/2 √ −  c c     −c/2   2 s c  1 c,1   1 + c  − × G1,c  c  1, 1 + 1/c, . . . , 1 + (c − 1) /c (2.35) where Gc,1 is the Meijer’s G-function [37] which is available in standard scien(·) 1,c tific software packages such as Mathematica and Maple. In view of Theorem 2.1, the MGF of the instantaneous SNR, Mγ (s), would then be obtained from (2.35) by replacing c by c/2 and  by γ . Although the availability of (2.35), in principle, allows application of the MGFbased approach for analytically evaluating the average error probability of systems operating over a Weibull fading channel, because of the difficulty of evaluating integrals whose integrand involves Meijer’s G-function, obtaining closed-form expressions for such performance is still not possible for most modulation/detection schemes. By comparison, the amount of fading for the Weibull channel model is easily obtained from the results presented above for the moments of this distribution. Specifically, applying the definition of AF in (2.5) to the normalized version of α, namely, R, from (2.34) we obtain     2   E R2 − E R2 var R 2 AF =   2 =   2 E R2 E R2        1 + 4c −  2 1 + 2c  1 + 4c = =     −1  2 1 + 2c  2 1 + 2c

(2.36)

which varies between 0 and ∞. For the Rayleigh channel (c = 2), (2.36) yields the previously given result AF = 1. Figure 2.2 is a plot of the mapping between the Weibull fading parameter c and the Nakagami-m and Rician fading parameters. The curves in this plot are obtained by first finding the amount of fading for a given value of n2 or m using (2.19) or (2.24), respectively, and then finding the corresponding value of c by numerically inverting (2.36). The analogous

28

FADING CHANNEL CHARACTERIZATION AND MODELING

7

m parameter Rician factor K = n 2

Weibull Fading Parameter c

6

5

4

3

2

1 0

1

2

3

4

5

6

7

8

9

10

Fading parameter Figure 2.2 Mapping between the Nakagami-m and the Nakagami-n (Rician) fading parameters with the Weibull fading parameter.

mapping between the Weibull, Nakagami-m, and Nakagami-q (Hoyt) distributions is illustrated in Fig. 2.3. 2.2.1.6 Beckmann The Beckmann distribution [38] is a four-parameter distribution corresponding to the envelope of two independent Gaussian RVs, each with their own mean and variance and as such includes the Rayleigh, Rician, Nakagami-q, and single-sided Gaussian distributions as special cases. Specifically,  X and Y be indepen letting dent Gaussian √ RVs with parameters (µx , σx ) and µy , σy , respectively, then the envelope α = X2 + Y 2 has a Beckmann distribution with PDF α pα (α) = 2π σx σy



2π 0



 2  α sin θ − µy (α cos θ − µx )2 exp − − dθ , 2σx2 2σy2

α≥0

(2.37) Other forms of the Beckmann PDF exist in terms of doubly infinite series of products of modified Bessel functions of the first kind; however, for our purposes in this book, the form in (2.37) is the most useful.

MODELING OF FLAT-FADING CHANNELS

29

2

q parameter m parameter

1.95

Weibull Fading Parameter c

1.9 1.85 1.8 1.75 1.7 1.65 1.6 1.55 1.5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Fading parameter

Figure 2.3 Mapping between the Nakagami-m and Nakagami-q (Hoyt) fading parameters and the Weibull fading parameter.

Corresponding to the PDF in (2.37), the MGF of the instantaneous SNR, γ = α 2 Es /N0 , is given by   Es Es µ2x N s s µ2y N 1 0 0 Mγ (s) =    exp 1 − 2σ 2 Es s + 1 − 2σ 2 Es s Es x N0 y N0 2 Es s 1 − 2σx2 N s 1 − 2σ y N0 0 (2.38) Defining

q2 =

σx2 , σy2



r2 =

µ2x , µ2y

K=

µ2x + µ2y σx2 + σy2

(2.39)

where q and K are consistent with their definitions for the Nakagami-q and Rician models, the average fading SNR is given by  Es    Es γ = σx2 + σy2 + µ2x + µ2y = σy2 1 + q 2 (1 + K) N0 N0   2 1+q Es = σx2 (1 + K) q2 N0

(2.40)

30

FADING CHANNEL CHARACTERIZATION AND MODELING

whereupon the MGF can be expressed in terms of γ as 

 1 + q 2 (1 + K) Mγ (s) =  



  1 + q 2 (1 + K) − 2q 2 γ s 1 + q 2 (1 + K) − 2γ s

2       r 1 K 1+r K 1+r 1 + q2 γ s 1 + q2 γ s 2 2    + × exp   1 + q 2 (1 + K) − 2q 2 γ s 1 + q 2 (1 + K) − 2γ s (2.41) The following special cases immediately follow:  µ2x = µ2y = 0, σx2 = σy2 r 2  µ2x = µ2y = 0, σx2 = σy2 r 2  µ2x = µ2y = 0, σx2 = σy2 r 2  µ2x = µ2y = 0, σx2 = 0, σy2 = 0 r 2

 = 1, q 2 = 1, K = 0  = 1, q 2 = 1, K = 0  = 1, q 2 = 1, K = 0  = 1, q 2 = 0, K = 0

(Rayleigh) (Rician) (Nakagami-q) (single-sided Gaussian) (2.42)

2.2.1.7 Spherically-Invariant Random Process Model Yet another statistical characterization of the fading channel is the sphericallyinvariant random process (SIRP) [39], which is a generalization of the Gaussian random process and, similar to the Beckmann distribution, provides a unification of a variety of statistical models. Our interest here is in applying an SIRP to the complex fading amplitude;2 thus, we shall define a complex spherically-invariant RV (SIRV) X = X1 + j X2 [equivalently a two-dimensional random vector X = (X1 , X2 )] derived from two samples X (t1 ) , X (t2 ) of an SIRP process X (t). Then, by the representation theorem [40], the joint PDF of this vector is given by pX (x1 , x2 ) =

1 ! 2π σ1 σ2 1 − ρ 2  2  × h2 

x1 −µ1 σ1

+

x2 −µ2 σ2

2

− 2ρ

x1 −µ1 σ1



x2 −µ2 σ2



1 − ρ2

 

(2.43)

where h2 (r) is a positive-valued function satisfying the Kolmogorov consistency $∞ $∞ condition [i.e., pX1 (x1 ) = −∞ pX (x1 , x2 ) dx2 and pX2 (x2 ) = −∞ pX (x1 , x2 ) dx1 must both be PDFs] and is defined by the one-dimensional integral 

∞

h2 (r) = 0

r  v −2 exp − 2 pV (v) dv, 2v

0 1, the first (noncoherent) term in (3.46) in the absence of noise is not identical for all phase sequences and thus contributes to the decisionmaking process. This term, however, does have a phase ambiguity associated with it in that the multiplication of each term in the sum by e−j θa , where θa is an arbitrary fixed phase does not change the value of the term. Hence, on the basis of the first term alone (i.e., for ρc = 0), the decision on the transmitted phase sequence would be ambiguous by θa radians where θa could certainly assume the value of one of the transmitted information phases. The second term in (3.46) does not have such a phase ambiguity associated with it, and thus for ρc = 0 the decision rule would be unique. To guarantee a unique decision rule for the ρc = 0 case, one can employ differential phase encoding of the information phase symbols as was discussed in Section 3.1.4. The specific details of how such differential encoding provides for a unique decision rule in this special case will be discussed in Section 3.5 in connection with differential detection of M-PSK with multiple symbol observation. Figure 3.15 illustrates a partially coherent receiver for M-PSK based on the decision statistics of (3.46). The performance of this receiver will be presented in Chapter 8. 3.4.2

Differentially Coherent Detection

3.4.2.1 M-ary Differential Phase-Shift-Keying (M-DPSK) Suppose once again that one does not specifically attempt to reconstruct a local carrier at the receiver from an estimate of the received carrier phase. We saw

72

*

Received Carrier Oscillator

c˜ r (t) = e j(2pfct)

˜ r(t)

e jb1

(•)dt

Figure 3.15 the AWGN.

. . . (•)dt

e jbM

Delay Ts

e jb2

Delay Ts

e jb1

Delay Ts

∑

Delay Ts

∑

Delay Ts

∑

Delay Ts

...

...

...

e jbM

Delay Ts

e jb2

Delay Ts

e jb1

Delay Ts

•

Re{•}

•

Re{•}

•

Re{•}

2

2

2

. . .

zn2

zn1

zn,MNs

rc

rc

rc

i

Choose Phase Sequence Corresponding to max zni

Complex form of optimum receiver for multiple-symbol partially coherent detection over

e jbM

1 ∫ N0 nTs

(n+1)Ts

e jb2

1 (n+1)Ts (•)dt ∫ N0 nTs

1 ∫ N0 nTs

(n+1)Ts

{qˆn}

PARTIALLY COHERENT DETECTION

73

in Section 3.3. that for an observation interval corresponding to a single transmitted symbol, the optimum noncoherent receiver could not be used to detect M-PSK modulation. Instead, let us now reconsider the noncoherent detection problem assuming an observation interval greater than one symbol in duration. This problem is akin to the partially coherent detection problem considered in the last section except that the memory that is introduced into the modulation now comes directly from the received carrier phase θc (assumed to be constant over, say, Ns symbols) rather than the phase error φc that results from its attempted estimation. As such, the maximum-likelihood solution to the problem would involve averaging the conditional likelihood function based on an Ns -symbol observation over a uniformly distributed phase, namely, θc rather than a Tikhonov-distributed phase (i.e., φc ). Receivers designed according to these principles are referred to as differential detectors and clearly represent an extension of noncoherent reception to the case of multiple-symbol observation. The term “differential” came about primarily because, in the conventional technique, a two-symbol observation is used (Ns = 2), and thus as we shall see that the decision is based on the difference between two successive matched-filter outputs. However, Divsalar and Simon [16] showed that by using an observation greater than two symbols in duration, one could obtain a receiver structure that provided further improvement in performance in the limit as Ns → ∞ approaching that of differentially encoded M-PSK (see Section 3.1.4). Practically speaking, it is necessary to have Ns on the order of only 3 to achieve most of the performance gain. With a little bit of thought, it should also be clear that the Tikhonov PDF of (3.37) with ρφ = 0 becomes a uniform PDF and thus from the above-mentioned analogy the solution to the multiple-symbol (including Ns = 2) differential detection problem can be directly obtained as a special case of the results obtained for the multiple-symbol partially coherent detection problem. 3.4.2.2 Conventional Detection (Two-Symbol Observation) We begin our discussion of differential detection of M-PSK by considering the conventional case of a two-symbol observation. According to the preceding discussion, the decision variables can be obtained from the first term of (3.46) with Ns = 2. Substituting (3.44) together with (3.45) in this term gives   2  (n+1)Ts 2 1 2  Ac  y˜n,k0 + y˜n−1,k1  = R˜ (t) e−jβk0 dt N0 N0  nTs 2  nTs  −jβk1 ˜ dt  , k0 , k1 = 1, 2, . . . , M (3.47) R (t) e + 

znk =

(n−1)Ts

where βk0 represents the assumed value for the information phase θ0 transmitted in the nth symbol interval and βk1 represents the assumed value for the information phase θ−1 transmitted in the n − 1st symbol interval. As mentioned above, multiplying each of the two matched-filter outputs in (3.47) by e−j θa with θa arbitrary does not change the decision variables. To resolve this phase ambiguity we employ differential phase encoding at the transmitter as discussed in Section 3.1.4.

74

TYPES OF COMMUNICATION

In particular, the transmitted information phases, now denoted by {θn } are first converted (differentially encoded) to the set of phases {θn } in accordance with the relation θn = θn−1 + θn modulo 2π

(3.48)

where βk0 , βk1 in (3.47) now represent the assumed values for the differentially encoded phases in the nth and n − 1st symbol intervals, respectively. Note that for θn and θn−1 to both range over the set βk = (2k − 1) π /M, k = 1, 2, . . . , M we must now restrict the information phase θn to range over the set βk = kπ /M, k = 0, 1, 2, . . . , M − 1. If we now choose the arbitrary phase equal to the negative of the information phase in the n − 1st interval (i.e., θa = −βk0 ), then, multiplying each matched-filter output term in (3.47) by e−j θa = ejβk0 , we can rewrite (3.47) as (ignoring the (Ac /N0 )2 scaling term)  (n+1)Ts  2   nTs   −j βk1 −βk0  ˜ ˜ znk =  dt  R (t) dt + R (t) e (n−1)Ts

nTs

  = 

(n+1)Ts

R˜ (t) dt +



nTs

(n−1)Ts

nTs

2  −j βk  ˜ dt  , R (t) e

k = 0, 1, . . . , M − 1

(3.49) Choosing the largest of the znk values in (3.49) then directly gives an unambiguous decision on the information phase θn . Expanding the squared magnitude in (3.49) as 2  (n+1)Ts  nTs    ˜ (t) dt + ˜ (t) e−j βk dt  R R   (n−1)Ts

nTs

  = 

(n+1)Ts nTs

2    R˜ (t) dt  + 



+ 2Re

(n+1)Ts

nTs

R˜ (t) e

−j βk

(n−1)Ts

R˜ (t) dt

∗ 

nTs

2  dt 

R˜ (t) e−j βk dt

 (3.50)

(n−1)Ts

nTs

and noting that the first two terms of (3.50) are independent of the decision index k, then an equivalent decision rule is to choose the largest of   ∗  nTs (n+1)Ts −j β k dt R˜ (t) dt R˜ (t) e znk = Re = Re e

(n−1)Ts

nTs −j βk



(n+1)Ts

nTs

k = 0, 1, . . . , M − 1

R˜ (t) dt

∗ 

nTs

R˜ (t) dt

 ,

(n−1)Ts

(3.51)

A receiver that implements this decision rule is illustrated in Fig. 3.16 and is the optimum receiver under the constraint of a two-symbol observation. For binary

75

* ˜ R(t) (n+1)Ts

∫nTs (•)dt

Ts

Delay

*

e−j∆bM−1

. . . y˜ n,M−1

y˜n1

y˜ n0

i

max z˜ni

i

max Re{y˜ni} =

Choose Data Phase Corresponding to

Figure 3.16 Complex form of optimum receiver for conventional (two-symbol observation) differentially coherent detection of M-PSK over the AWGN.

Received Carrier Oscillator

c˜ r(t) = e j(2pfct)

˜ r(t)

e−j∆b1

e−j∆b0

n

Data Phase Decision ∆qˆ

76

TYPES OF COMMUNICATION

˜ r(t)

c˜ r(t) = e

*

˜ R(t)

(n+1)Ts

∫nTb

*

(•)dt

1 e jDqˆn

Re{•}

−1

j(2pfct)

Delay Received Carrier Oscillator

Tb

Figure 3.17 Complex form of optimum receiver for conventional (two-symbol observation) differentially coherent detection of DPSK over the AWGN.

DPSK, the decision rule simplifies to   ∗  (n+1)Tb j θˆn e = sgn Re R˜ (t) dt

nTb

R˜ (t) dt

 (3.52)

(n−1)Tb

nTb

and is implemented by the receiver illustrated in Fig. 3.17. Note that the structure of the receiver in Fig. 3.16 and its special case in Fig. 3.17 is such that the previous matched-filter output acts as the effective baseband demodulation reference for the current matched-filter output. In this context, the differentially coherent receiver behaves like the nonideal coherent receiver discussed in Section 3.2 with a reference signal as in (3.39) having a gain G = 1 and an additive noise independent of that associated with the received signal. 3.4.2.3 Multiple-Symbol Detection Analogous to what was true for partially coherent detection, the performance of the differentially coherent detection system can be improved by optimally basing the design of the receiver on an observation of the received signal for more than two symbol intervals [16]. The appropriate decision variables are now obtained from the first term of (3.46) with now Ns > 2. Once again using differential phase encoding to resolve the phase ambiguity inherent in this term, in particular, setting the arbitrary phase θa = −θn−Ns +1 and using the differential encoding algorithm of (3.48), we obtain, analogous to (3.49), the decision variables  (n+1)Ts  nTs  R˜ (t) dt + R˜ (t) e−j βk1 dt znk =  (n−1)Ts

nTs

 +···+

(n−Ns +2)Ts (n−Ns +1)Ts

2

 −j βkN −1 s dt  R˜ (t) e

 ,

ki = 0, 1, . . . , M − 1, i = 1, 2, . . . , Ns − 1

(3.53)

from which a decision on the information sequence θn−Ns +2 , θn−Ns +3 , . . . , θn−1 , θn is made corresponding to the largest of the znk values. Note that an Ns -symbol observation results in a simultaneous decision on Ns − 1 information phase symbols. The squared magnitude in (3.53) can be expanded analogous to

77

c˜ r(t) =

˜ * R(t) (n+1)Ts

∫nTs (•)dt

Delay Ts

*

. . .

Delay Ts

e−jDbM−1

e−jDb1

e−jDb0

*

. . .

. . .

e−jDbM−1

e−jDb1

e−jDb0

e−j2DbM−1

y˜ n,M2−1

y˜ n1

y˜ n0

i

i

max ˜zni

max Re{y˜ ni} =

Choose Data Phases Corresponding to

Figure 3.18 Complex form of optimum receiver for three-symbol differentially coherent detection of M-PSK over the AWGN.

Received Carrier Oscillator

e j(2pfct)

˜ r(t)

*

e−j(Db0 +Db1)

e−j2Db0

qˆn,qˆn−1

78

TYPES OF COMMUNICATION

(3.50) to simplify the decision rule. For example, for Ns = 3 the decision rule is to choose the pair of information phases θn−1 , θn corresponding to the maximum over k1 and k2 of ∗  nTs   (n+1)Ts −j βk1 ˜ ˜ R (t) dt R (t) dt znk = Re e (n−1)Ts

nTs

+e

−j βk2



nTs

R˜ (t) dt

∗ 

(n−1)Ts

+e

  −j βk1 +βk2



(n−1)Ts

R˜ (t) dt



(n−2)Ts (n+1)Ts

R˜ (t) dt

nTs

k1 , k2 = 0, 1, . . . , M − 1

∗ 

(n−1)Ts

R˜ (t) dt

 ,

(n−2)Ts

(3.54)

A receiver that implements this decision rule is illustrated in Fig. 3.18. We conclude this section by mentioning that although it appears that the complexity of the receiver implementation grows exponentially with the observation block size Ns [1, Ch. 7, Sect. 7.2.3], Mackenthun [17] has developed algorithms for implementing multiple symbol differential detection of M-PSK that considerably reduce this complexity, thus making it a viable alternative to coherent detection of differentially encoded M-PSK. These algorithms and their complexity in terms of the number of operations per Ns -symbol block being processed are also discussed in Ref. 1. 3.4.3

π /4-Differential QPSK (π /4-DQPSK)

The π /4-QPSK introduced in Sect. 3.1.4.1 in combination with coherent detection as a means of reducing the regeneration of spectral sidelobes in bandpass filtered/nonlinear systems can also be used for that same purpose when combined with differential detection. The resulting scheme, called π /4-differential QPSK (π /4DQPSK), behaves quite similarly to ordinary differential detection of QPSK as discussed in Section 3.5.1 with the following exception. Since the set of phases {βk } used to represent the information phases {θn } is now βk = (2k − 1) π /4, k = 1, 2, 3, 4, this set must be used in place of the set βk = kπ /4, k = 0, 1, 2, 3 in the phase comparison portion of Fig. 3.16.

REFERENCES 1. M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques: Signal Design and Detection, Englewood Cliffs, NJ: PTR Prentice-Hall, 1995. 2. W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering, Englewood Cliffs, NJ: PTR Prentice-Hall, 1973. 3. M. K. Simon and D. Divsalar, “On the optimality of classical coherent receivers of differentially encoded M-PSK,” IEEE Commun. Lett., vol. 1, no. 3, May 1997, pp. 67–70.

REFERENCES

79

4. P. A. Baker, “Phase-modulation data sets for serial transmission at 2000 and 2400 bits per second, part I,” AIEE Trans. Commun. Electron., July 1962. 5. J. B. Anderson, T. Aulin, and C.-E. Sundberg, Digital Phase Modulation, New York, NY and London, U.K.: Plenum Press, 1986. 6. M. L. Doelz and E. T. Heald, Minimum-Shift Data Communication System, U.S. Patent 2,977,417, March 28, 1961. 7. S. Pasupathy, “Minimum shift keying: A spectrally efficient modulation,” IEEE Commun. Mag., vol. 17, no. 4, July 1979, pp. 14–22. 8. A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK modulation carrier phase with application to burst digital transmission,” IEEE Trans. Inform. Theory, vol, IT-32, July 1983, pp. 543–551. 9. M. P. Fitz, Open loop techniques for carrier synchronization, PhD dissertation, Univ. Southern California, Los Angeles, CA, June 1989. 10. V. I. Tikhonov, “The effect of noise on phase-locked oscillator operation,” Automation and Remote Control, vol. 20, 1959, pp. 1160–1168; translated from Automatika i Telemekhaniki, Akademya Nauk, SSSR, vol. 20, September 1959. 11. A. J. Viterbi, “Phase-locked loop dynamics in the presence of noise by Fokker-Planck techniques,” Proc. IEEE, vol. 51, no. 12, December 1963, pp. 1737–1753. 12. M. P. Fitz, “Further results in the unified analysis of digital communication systems,” IEEE Trans. Commun., vol. 40, no. 3, March 1992, pp. 521–532. 13. S. Stein, “Unified analysis of certain coherent and noncoherent binary communication systems,” IEEE Trans. Inform. Theory, vol. IT-10, no. 1, January 1964, pp. 43–51. 14. A. J. Viterbi, “Optimum detection and signal selection for partially coherent binary communication,” IEEE Trans. Inform. Theory, vol, IT-11, April 1965, pp. 239–246. 15. M. K. Simon and D. Divsalar, “Multiple symbol partially coherent detection of MPSK,” IEEE Trans. Commun., vol. 42, no. 2/3/4, February/March/April 1994, pp. 430–439. 16. D. Divsalar and M. K. Simon, “Multiple-symbol differential detection of MPSK,” IEEE Trans. Commun., vol. COM-38, no. 3, March 1990, pp. 300–308. 17. K. M. Mackenthun, Jr., “A fast algorithm for multiple-symbol differential detection of MPSK,” IEEE Trans. Commun., vol. 42, no. 2/3/4, February/March/April 1994, pp. 1471–1474.

PART 2 MATHEMATICAL TOOLS

4 ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

Having characterized and classified the various types of fading channels and modulation/detection combinations that can be communicated over these channels, the next logical consideration is evaluation of the average error probability performance of the receivers of such signals. Before moving on in the next part of the book to a description of these receivers and the details of their performance on the generalized fading channel, we divert our attention to developing a set of mathematical tools that will unify and greatly simplify these evaluations. The key to such a unified approach is the development of alternative representations of several mathematical functions, the most classical of which are the Gaussian Q-function and the Marcum Q-function, which characterize the error probability performance of digital signals communicated over the AWGN channel in a form that is analytically more desirable for the fading channel. The specific nature and properties of this desired form will become clear shortly. For the moment, suffice it to say that the canonical forms of the Gaussian and Marcum Q-functions that have been around for many decades and to this day still dominate the literature dealing with error performance evaluation have an intrinsic value in their own right with respect to their relation to well-known probability distributions. What we aim to show, however, is that aside from this intrinsic value, these canonical forms suffer a major disadvantage in situations where the arguments of the functions depend on random parameters that require further statistical averaging. Such is the case when evaluating average error probability on the fading channel as well as on many other channels with random disturbances. Herein lies the most significant value of the alternative representations of these functions: namely, their ability to enable simple and in many cases closed-form evaluation of such statistical averages. Digital Communication over Fading Channels, Second Edition. By Marvin K. Simon and Mohamed-Slim Alouini ISBN 0-471-64953-8 Copyright  2005 John Wiley & Sons, Inc.

83

84

4.1 4.1.1

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

GAUSSIAN Q-FUNCTION One-Dimensional Case

The one-dimensional Gaussian Q-function (often referred to as the Gaussian probability integral ), Q (x), is defined as the complement (with respect to unity) of the cumulative distribution function (CDF) corresponding to the normalized (zero mean, unit variance) Gaussian random variable (RV) X. The canonical representation of this function is in the form of a semi-infinite integral of the corresponding probability density function (PDF):
Q(x) = x

∞

 2 1 y dy √ exp − 2 2π

(4.1)

In principle, the representation of (4.1) suffers from two disadvantages. From a computational standpoint, this relation requires truncation of the upper infinite limit when using numerical integral evaluation or algorithmic techniques. More important, however, the presence of the argument of the function as the lower limit of the integral poses analytical difficulties when this argument depends on other random parameters that ultimately require statistical averaging over their probability distributions. For the pure AWGN channel, only the first of the two disadvantages comes into play that ordinarily poses little difficulty and therefore accounts for the popularity of this form of the Gaussian Q-function in the performance evaluation literature. However, for channels perturbed by other disturbances, in particular the fading channel, the second disadvantage plays an important role since, as we shall see later, the argument of the Q-function depends, among other parameters, on the random fading amplitudes of the various received signal components. Thus, to evaluate the average error probability in the presence of fading, one must average the Q-function over the fading amplitude distributions. It is primarily this second disadvantage, namely, the inability to average analytically over one or more random variables when they appear in the lower limit of an integral, that serves as the primary motivation for seeking alternative representations of this and similar functions. Clearly, then, what would be more desirable in such evaluations would be to have a form for Q (x) wherein the argument of the function is in neither the upper nor the lower limit of the integral and furthermore appears in the integrand as the argument of an elementary function (e.g., an exponential). Still more desirable would be a form wherein the argument-independent limits are finite. In what follows, any function that has the two properties above will be said to be in the desired form. A number of years ago, Craig [1] cleverly showed that evaluation of the average probability of error for the two-dimensional AWGN channel could be considerably simplified by choosing the origin of coordinates for each decision region as that defined by the signal vector as opposed to using a fixed coordinate system origin for all decision regions derived from the received vector. This shift in vector space coordinate systems allowed the integrand of the two-dimensional integral describing the conditional (on the transmitted signal) probability of error to be independent of the transmitted signal. A byproduct of Craig’s work was a definite integral form

GAUSSIAN Q-FUNCTION

85

for the Gaussian Q-function, which was in the desired form.1 In particular, Q(x) of (4.1) could also now be defined (but only for x ≥ 0) by 1 Q(x) = π


0

π/2

 exp −

x2 2 sin2 θ

 dθ

(4.2)

The form in (4.2) is not readily obtainable by a change of variables directly in (4.1). However, by first extending (4.1) to two dimensions (x and y), where one of the dimensions (y) is integrated over the half-plane, a change of variables from rectangular to polar coordinates readily produces (4.2). Furthermore, (4.2) can be obtained directly by a straightforward change of variables of a standard known integral involving Q(x), in particular, Eq. (3.363.2) in Ref 5. Both of these techniques for arriving at (4.2) are described in Appendix 4A. Another derivation of (4.2) is given in Ref. 6 and is based on the fact that since the product of two independent random variables, one of which is a Rayleigh and the other a sinusoidal random process with random phase, is a Gaussian random variable, determining the CDF of this product variable is equivalent to evaluating the Gaussian Q-function. Finally, Tellambura and Annamalai [25] offer other derivations of the Craig representation in (4.2), some of which bear similarity to those discussed above. On the basis of our previous discussion, it is clear that Q(x) of (4.2) is in the desired form, that is, in addition to the advantage of having finite integration limits independent of the argument of the function x, it has the further advantage that the integrand now has a Gaussian form with respect to x ! We shall see in Chapter 5 that this exponential dependence of the integrand on the argument of the Q-function will play a very important role in simplifying the evaluation of performance results for coherent communication over generalized fading channels. Before exploiting this property of (4.2) in great detail, however, we wish to give further insight into the alternative definition of the Gaussian Q-function with regard to how it relates to the well-known Chernoff bound. Note that the maximum of the integrand in (4.2) occurs when θ = π /2 [i.e., the integrand achieves its maximum value, viz., exp −x 2 /2 at the upper limit]. Thus, replacing the integrand by its maximum value, we immediately get the well known upper bound on Q(x), namely, Q(x) ≤ 12 exp −x 2 /2 , which is the Chernoff bound. As we shall see on many occasions later in the book, the advantage of this observation is that the form of Q(x) in (4.2) allows manipulations akin to those afforded by the Chernoff bound but without the necessity of invoking a bound ! In principle, one simply operates on the integrand in the same fashion as if the Qfunction had been replaced by the Chernoff bound, and then at the end performs a single integration over the variable θ . For example, many problems dealing with 1 This form of the Gaussian Q-function was earlier implied in the work of Pawula et al. [2] and Weinstein [3]. The earliest reference to this form of the Gaussian Q-function found by the authors appeared in a classified report (which has since become unclassified) by Nuttall [4]. The relation given there is actually for the complementary error function, which is related to the Gaussian Q-function by √  2x . erfc (x) = 2Q

86

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

sequence detection whose error probability performance was heretofore characterized by a combined union–Chernoff bound can now be described by just a union bound, thereby improving its tightness. This behavior is discussed in more detail in Chapter 13. 4.1.2

Two-Dimensional Case

The normalized two-dimensional Gaussian probability integral is defined by


∞
∞ 1 x 2 + y 2 − 2ρxy   exp − dx dy (4.3) Q(x1 , y1 ; ρ) =  2 1 − ρ2 2π 1 − ρ 2 x1 y1 Rewriting (4.3) as 1  2π 1 − ρ 2
∞
∞  (x + x1 )2 +(y + y1 )2 − 2ρ(x + x1 )(y + y1 ) dx dy × exp − 2(1 − ρ 2 ) 0 0 (4.4) we see that we can interpret this double integral as the probability that a signal vector s = (−x1 , −y1 ) received in correlated unit variance Gaussian noise falls in the upper right quadrant of the (x, y) plane. Defining

y1 S = x12 + y12 , φs = tan−1 (4.5) x1 Q(x1 , y1 ; ρ) =

then, using the geometry of Fig. 4.1, it is straightforward to show that Q(x1 , y1 ; ρ) can be expressed as 2


π/2−φs  1 − ρ2 S 1 − ρ sin 2θ cos2 φs 1   exp − dθ Q(x1 , y1 ; ρ) = 2π 0 1 − ρ sin 2θ 2 1 − ρ2 sin2 θ 2


φs  1 − ρ2 S 1 − ρ sin 2θ sin2 φs 1   exp − dθ + 2π 0 1 − ρ sin 2θ 2 1 − ρ2 sin2 θ (4.6) which using (4.5) simplifies still further to 1 Q(x1 , y1 ; ρ) = 2π +




1 2π



x12 1 − ρ sin 2θ 1 − ρ2 exp −  dθ  1 − ρ sin 2θ 2 1 − ρ 2 sin2 θ

 y12 1 − ρ sin 2θ 1 − ρ2 exp −   dθ , 1 − ρ sin 2θ 2 1 − ρ 2 sin2 θ

π/2−tan−1 y1 /x1

0




tan−1 y1 /x1 0

x1 ≥ 0, y1 ≥ 0



(4.7)

87

GAUSSIAN Q-FUNCTION

Ne jΘ = (x + x1) + j(y + y1) (x + x1)2 + (y + y1)2 − 2r(x + x1)(y + y1) = N 2(1 − rsin2Θ) q = p −Θ 2 fs ≤ Θ ≤ p 2

0 ≤ q ≤ p − fs 2

N

y

R

fs cos S cos Θ

x

−x1

fs Θ

S

−y1

Ne jq= (x + x1) + j(y + y1) (x + x1)2 + (y + y1)2 − 2r(x + x1)(y + y1) = N 2(1 − r sin2q) y

0 ≤ q ≤ fs

R

−x1

fs

x

N

S sin fs S sin q

Figure 4.1

q

−y1

Geometry for Eq. (4.6).

For the special case of ρ = 0, (4.7) simplifies to

Q(x1 , y1 ; 0) = Q (x1 ) Q (y1 ) = 1 + 2π


0

tan−1 y1 /x1

1 2π




π/2−tan−1 y1 /x1 0

 exp −

y12 2 sin2 θ

 exp −

 dθ

x12 2 sin2 θ

 dθ (4.8)

88

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

In addition, when x1 = y1 = x, we have Q(x, x; 0) = Q2 (x) =

1 π




π/4

 exp −

0

x2 2 sin2 θ

 dθ

(4.9)

which is a single-integral form for the square of the Gaussian Q-function.2 The form of the result in (4.9) can also be obtained directly from (4.1) by squaring the latter, rewriting it as a double integral of a two-dimensional Gaussian PDF, and then converting from rectangular to polar coordinates (see Appendix 4A). Comparing (4.9) with (4.2), we see that to compute the square of the one-dimensional Gaussian probability integral, one integrates the same integrand but only over the first half of the domain. 4.1.3

Other Forms for One- and Two-Dimensional Cases

In the previous section, following the geometric approach used by Craig to obtain the form of Q(x) given in (4.2), we derived a representation of the two-dimensional joint Gaussian Q-function [see (4.7)] which has similar desirable properties; however, the integrand is not purely exponential in the integration variable, and furthermore the argument of the exponential in the integrand is not as simple as in (4.2). Applying a clever transformation to (4.2), Simon [26] was able to arrive at a simpler form for Q(x1 , y1 ; ρ) that dispenses with the trigonometric factor that precedes the exponentials in its integrands and furthermore results in an exponential argument that is precisely in the same simple form as that in the Craig representation of Q(x) of (4.2). As such, the entire dependence of Q(x1 , y1 ; ρ) on ρ now appears only in the limits of integration. We now present a brief summary of the results in Ref. 26. Consider applying to the integrals in (4.7) the change of variables   −1 tan  ± ρ  (4.10) θ = tan 1 − ρ2 It can easily be shown that 

dθ =

1 − ρ2 d 1 ± ρ sin 2

(4.11)

Furthermore, after some trigonometric manipulation of (4.10), we obtain   [1 ∓ ρ sin 2] 1 1 + tan2 θ 1 =  2 =  2 2 2 1−ρ sin  2 1 − ρ 2 tan θ ± ρ sin θ 1 − ρ 2 ± ρ cot θ (4.12) 2 This result can also be obtained from Lebedev [7, Ch. 2, Prob. 6] after making the change of variables θ = π /2 − tan−1 t.

89

GAUSSIAN Q-FUNCTION

Finally, substituting (4.11) and (4.12) into (4.7) gives     x1 /y1 −ρ
tan−1 √ 2 x1 1 1 1−ρ 2     exp − Q(x1 , y1 ; ρ) =  2  dθ 2 2π − tan−1 √ ρ 2 sin θ 1 − ρ 2 + ρ cot θ 1−ρ 2

+

1 2π




 tan−1

y1 /x1 −ρ √

 − tan−1

1−ρ 2

√

ρ







y12 2



 exp −

2 sin θ

1−ρ 2



1 1−

ρ2

+ ρ cot θ

 2  dθ (4.13)

Next, letting α = tan−1 

ρ

(4.14)

1 − ρ2

then 

sin2 θ sin2 θ 2 =  2 = 2 sin (θ + α) 1 − ρ 2 + ρ cot θ 1 − ρ 2 sin θ + ρ cos θ 1

(4.15)

and (4.13) simply becomes Q(x1 , y1 ; ρ) =

1 2π +




1 2π

 tan−1

√x1 /y1

1−ρ 2

−α




 tan−1

√

 −tan α

 exp − 

y1 /x1 1−ρ 2

−tan α

−α



x12 2 sin2 (θ + α)

 exp −

dθ (4.16)



y12 2 sin (θ + α) 2

dθ

Finally, performing the change of variables  = θ + α and using the trigonometric identity for tan−1 A + tan−1 B gives the desired result in the Craig form as3 √  1−ρ 2 x1 /y1  
tan−1 1−ρx1 /y1 x12 1 d Q(x1 , y1 ; ρ) = exp − 2π 0 2 sin2  (4.17) √  1−ρ 2 y1 /x1  
tan−1 2 1−ρy1 /x1 y1 1 + d exp − 2π 0 2 sin2  which, as we shall see in Chapter 9, has application in computing outage probability for dual-diversity selection combining over correlated nonidentically distributed channels. Also note that for ρ = 0 but y1 = x1 = x, (4.17) simplifies to 1 Q(x, x; ρ) = π




tan−1 0



1+ρ 1−ρ



 exp −

x2 2 sin2 

 d

(4.18)

3 Since 1 − ρx /y and 1 − ρy /x can take on positive or negative values, the arctangents in the upper 1 1 1 1 limits of the integrals in (4.17) are defined by tan−1 X/Y = π (1 − sgn Y ) /2 + (sgn Y ) tan−1 X/ |Y |.

90

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

which has the same application as above but for identically distributed log-normal channels. Also, for ρ = 0, we immediately get the Craig form for Q2 (x) as given by (4.9) whereas for ρ = 1, (4.18) becomes the Craig form for Q(x) as in (4.2). Although not derived from (4.7), a parametric (in ρ) form of the one-dimensional Gaussian Q-function was also proposed [27, Eq. (74)] that bore a striking resemblance to the above two-dimensional representation, namely   2
π/2  x 1 1 − ρ2 1 (1 − ρ sin 2) d exp − Q(x) = 2π 0 1 − ρ sin 2 2 1 − ρ2 sin2    2
π/2  1 x 1 1 − ρ2 (1 + ρ sin 2) + d, x ≥ 0 exp − 2π 0 1 + ρ sin 2 2 1 − ρ2 sin2  (4.19) where now 0 ≤ ρ ≤ 1 is simply a parameter (not to be interpreted as correlation) that characterizes the integrand. Applying the identical change of variables as in (4.10) to the parametric form of Q(x) in (4.19), we immediately obtain an expression analogous to (4.13):  
π/2 2 x 1 1   Q(x) =  √  exp −  2  dθ 2 2π − tan−1 ρ/ 1−ρ 2 2 sin θ 1 − ρ 2 + ρ cot θ (4.20)  
π/2 2 1 x 1   +  √  exp −  2  dθ 2 2π tan−1 ρ/ 1−ρ 2 2 sin θ 1 − ρ 2 − ρ cot θ Replacing θ by −θ in the second integral of (4.20), the two integrands become identical and the two integrals combine into  
π/2 2 x 1 1   (4.21) Q(x) = exp −  2  dθ 2 2π −π/2 2 sin θ 1 − ρ 2 + ρ cot θ Finally, using (4.14) and (4.15) and performing the change of variables  = θ + α, (4.21) becomes  
π/2+α 1 x2 Q(x) = d (4.22) exp − 2π −π/2+α 2 sin2  which is easily shown to be equivalent to the Craig representation of (4.2) by recognizing the periodic symmetry of the sin2  function. 4.1.4 Alternative Representations of Higher Powers of the Gaussian Q-Function In Section 4.1.2 we showed [see Eq. (4.9)] that the square of the Gaussian Q-function could be expressed in the Craig form of the Gaussian Q-function given in (4.2) with a mere change of the upper limit from π /2 to π /4. Thus, it is natural to ask

91

GAUSSIAN Q-FUNCTION

whether a similar alternative form can be found for higher powers of the Gaussian Q-function. In Ref. 28, Simon demonstrated that it is indeed possible to obtain Craigtype forms for the third and fourth powers of the Gaussian Q-function, namely, single integrals with finite limits independent of the argument of the function and an integrand whose dependence on the argument is purely Gaussian. The motivation for finding such forms comes from the desire to analytically evaluate the performance over the generalized fading channel of certain modulation/detection techniques whose error probability expressions involve higher powers of the Gaussian Q-function such as coherent detection of 4-ary orthogonal signaling and differentially encoded QPSK. These applications will be discussed in detail in Chapter 8. Consider the generic double integral 
ξu   
φu x2 x2 π  1 = dξ , φu , ξ u ≤ exp − exp − f (x) 2 2 π2 0 2 2 sin φ 2 sin ξ 0 (4.23) Letting X=

1 1 , Y = sin φ sin ξ

⇒R=



β = tan−1

X2 + Y 2 ,

Y X

(4.24)

and performing the transformation of variables from rectangular to polar coordinates, we can write (4.23) as f (x) =

where  βu

sin−1 (Yu


∞
cos−1 (Xu /R) 4 1

√   2 π Xu2 +Yu2 sin−1 (Yu /R) (R sin 2β) R 4 sin2 2β − 4 R 2 − 1  2 2 x R dβ dR × exp − 2 

(4.25)



Xu = 1/ sin φu , Yu = 1/ sin ξu . Partitioning the inner integral into  cos−1 (Xu /R)  (·) dβ + βu (·) dβ with βu = tan−1 (Yu /Xu ), and making the /R)

change of variables y = 1/ sin2 2β, we obtain after a good bit of manipulation 




√2Xu
1/ sin2 2 sin−1 (Yu /R) 1 dy f (x) =    √ 2 2     2 −1 2π 2 2 Xu +Yu 1/ sin 2 sin (Xu /R) −y + 1 + R 4 / 4 R 2 − 1 y    −R 4 / 4 R 2 − 1  2 2 1 x R dR × √ exp − 2 R R2 − 1 1 + 2π 2




∞ √

2Xu


1

  1/ sin2 2 sin−1 (Yu /R)



−y 2



dy

   + 1 + R4/ 4 R2 − 1 y    −R 4 / 4 R 2 − 1

92

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

×

 2 2 x R dR exp − 2 R R2 − 1

1 + 2π 2

×

1

√







∞ √

2Xu

  1/ sin2 2 sin−1 (Xu /R)

1



dy

   −y + 1 + R 4 / 4 R 2 − 1 y    −R 4 / 4 R 2 − 1 2



 2 2 x R dR exp − 2 2 R R −1 1

√

(4.26)

Interestingly enough, the integral on y in (4.26) can be evaluated in closed form. In particular, from Eq. (2.261) of Ref. 5, we obtain


1 sin−1 = −√ −c cy 2 + by + a dy



 2cy + b , √ −

c < 0,

 = 4ac − b2 < 0

(4.27)       Letting a = −R 4 / 4 R 2 − 1 , b = 1 + R 4 / 4 R 2 − 1 , and c = −1, then  = − (a + 1)2 and the conditions for (4.27) are satisfied. Finally, applying (4.27) to (4.26) and making one more change of variables, namely, R = 1/ sin θ , we obtain after considerable simplification the desired result 1 f (x) = 2π 2




sin−1



sin2 φu sin2 ξu sin2 φu +sin2 ξu

 √  sin−1 sin φu / 2

 −1  cos g (θ , ξu ) − cos−1 g (θ , φu ) 




sin−1 sin φu /√2 1 dθ + [cos−1 g (θ , ξu ) 2π 2 0 2 sin2 θ   x2 −1 dθ + cos g (θ , φu )] exp − 2 sin2 θ  × exp −

x2



(4.28)

where 

g (θ , ζ ) = −1 + 2

sin2 ζ − sin2 θ − sin4 ζ cos2 θ  2  sin ζ − sin2 θ cos2 2θ

(4.29)

A simplification of (4.28) occurs for the case φu = ξu : 1 f (x) = 2 π

 √  sin−1 sin φu / 2




cos 0

−1

 g (θ , φu ) exp −

x2 2 sin2 θ

 dθ

(4.30)

We are now in a position to apply (4.29) and (4.30) to the evaluation of Q3 (x) and Q4 (x) in single integral form. Specifically, noting that Q3 (x) = Q (x) Q2 (x) and

MARCUM Q-FUNCTION

93

Q4 (x) = Q2 (x) Q2 (x), then, using (4.2) and (4.9) and letting φu = π /4, ξu = π /2 in (4.28), gives after some simplification   
π/6 x2 1 3 −1 3 cos 2θ − 1 − 1 exp − cos dθ Q (x) = 2 π 0 2 cos3 2θ 2 sin2 θ      
sin−1 1/√3  x2 1 −1 3 cos 2θ − 1 dθ π − cos − 1 exp − + 2π 2 0 2 cos3 2θ 2 sin2 θ (4.31) Similarly, letting φu = π /4 in (4.30) and simplifying gives   
π/6 3 cos 2θ − 1 x2 1 − 1 exp − dθ cos−1 Q4 (x) = 2 π 0 2 cos3 2θ 2 sin2 θ    
sin−1 1/√3  1 3 −1 3 cos 2θ − 1 (4.32) π − cos −1 = Q (x) − 2π 2 0 2 cos3 2θ   x2 dθ × exp − 2 sin2 θ 4.2

MARCUM Q-FUNCTION

Motivated by the form of the alternative Gaussian Q-function in (4.2), one questions whether a similar form is possible for the generalized Marcum Q-function [8], which, as we shall see in later chapters, is common in performance results for communication problems dealing with partially coherent, differentially coherent, and noncoherent detection. We now present the steps leading up to this desirable form and then show how it offers the same advantages as the alternative representation of the Gaussian Q-function. For simplicity of the presentation, we shall first demonstrate the approach for the first-order (m = 1) Marcum Q-function and then generalize to the mth-order function, where in general m can be noninteger as well as integer. The derivations and specific forms that will be derived can be found in Ref. 9 with similar derivations and forms found in Ref. 10. 4.2.1

First-Order Marcum Q-Function  √  The first-order Marcum Q-function, Q1 s, y , is defined as the complement (with respect to unity) of the  CDF corresponding to the normalized noncentral chi-square random variable, Y = 2k=1 Xk2 , whose canonical representation is in the form of a semi-infinite integral of the corresponding probability density function (PDF), namely4    2
∞  √  x + s2 I0 (sx) dx Q1 s, y = √ x exp − (4.33) 2 y 4 It is common in the literature to omit the “1” subscript on the Marcum Q-function when referring to the first-order function. For the purpose of clarity and distinction from the generalized (mth-order) Marcum Q-function to be introduced shortly, we shall maintain the subscript notation.

94

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

where s 2 is referred to as the noncentrality parameter. Also, for simplicity of nota√ tion, we shall replace the arguments s and y in (4.33) by α and β, respectively, in which case (4.33) is rewritten in the more common form5    2
∞ x + α2 I0 (αx) dx Q1 (α, β) = x exp − (4.34) 2 β Using integration by parts, it has also been shown [12,13] that the first-order Marcum Q-function has the series form  ∞   ∞  2  2     α + β2  α k β  1+ζ 2 Ik (αβ) = exp − ζ k Ik β 2 ζ Q1 (α, β) = exp − 2 β 2 k=0

k=0

(4.35) 

where ζ = α/β. The reason for introducing the parameter ζ to represent the ratio of the arguments of the Marcum Q-function is in the same sense that the definition in (4.33) has one argument that represents the true argument of the function √ (i.e., y), whereas the second argument (i.e., s) is a parameter. More insight into the significance of ζ in the digital communications application and its dependence on the modulation/detection form is given in Chapter 5. Suffice it to say at the moment, that in terms of the analogy with Craig’s result, we are attempting to express the Marcum Q-function as an integral with finite limits and an integrand that is a Gaussian function of β. The modified Bessel function of k th order can be expressed as the integral [5, Eqs. (8.406.3) and (8.411.1)]
π  k 1 (4.36) −j e−j θ e−z sin θ dθ Ik (z) = 2π −π 5 It is interesting to note that the complement (with respect to unity) of the first-order Marcum Q-function can be regarded as a special case of the incomplete Toronto function [11, pp. 227–228] which finds its roots in the radar literature and is defined by

TB (m, n, r) = 2r n−m+1 e−r

2




B

2

t m−n e−t In (2rt) dt

0

In particular, we have   α Tβ/√2 1, 0, √ = 1 − Q1 (α, β) 2 Furthermore, as β → ∞, Q1 (α, β) can be related to the Gaussian Q-function as follows. Using the asymptotic (for large argument) form of the zeroth-order modified Bessel function of the first kind, we get [4, Eq. (A-27)]   2 x + α 2 exp (αx) dx x exp − √ 2 2π αx β

 
∞ β 1 β (x − α)2 ∼ Q (β − α) exp − dx = √ = α 2π β 2 α

Q1 (α, β) ∼ =




∞

MARCUM Q-FUNCTION

95

√ where j = −1 and it is clear that the imaginary part of the right-hand side of (4.36) must be equal to zero [since Ik (z) is a real function of the real argument z ]. Although (4.36) is not restricted to values of ζ less than unity, to arrive at the alternative representation of the Marcum Q-function, it will be convenient to make this assumption. (Shortly we shall give an alternative series form from which an alternative representation can be derived for the case where the ratio α/β is greater than unity.) Thus, assuming in (4.36) that 0 ≤ ζ < 1, after substitution in (4.35), we obtain
π ∞  1   k β2  2 2 Q1 (α, β) = exp − 1+ζ ζ −j e−j θ e−β ζ sin θ dθ 2 2π −π k=0 (4.37) 
π 2   1 1 β 2 −β 2 ζ sin θ  e 1+ζ dθ = exp − 2 2π −π 1 + ζ j e−j θ 

Simplifying the complex factor of the integrand as 1 1 + ζ (sin θ − j cos θ ) 1 =  = 1 + ζ (sin θ + j cos θ ) 1 + ζ j e−j θ (1 + ζ sin θ)2 + (ζ cos θ)2 1 + ζ (sin θ − j cos θ ) = 1 + 2ζ sin θ + ζ 2

(4.38)

and recognizing again that the imaginary part of (4.38) must result in a zero integral [since Q1 (α, β) is real], substituting (4.38) into (4.37) gives the final result
π 1 + ζ sin θ 1 Q1 (α, β) = Q1 (βζ , β) = 2π −π 1 + 2ζ sin θ + ζ 2   β2  β>α≥0 × exp − 1 + 2ζ sin θ + ζ 2 dθ , 2

(0 ≤ ζ < 1)

(4.39) which is in the desired form of a single integral with finite limits and an integrand that is bounded and well behaved over the interval −π ≤ θ ≤ π and is Gaussian in the argument β. We observe from (4.39) that ζ is restricted to be less than unity (i.e., α = β). The reason for this stems from the closed form used for the geometric series in (4.37), which, strictly speaking, is valid only when ζ < 1. This special case, which has limited interest in communication performance applications, has been evaluated [14, Eq. (A-3-2)] and has the closed-form result     1 + exp −α 2 I0 α 2 Q1 (α, α) = 2

(4.40)

96

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

For the case α > β ≥ 0, the appropriate series form is [12,13]6  ∞    α2 + β 2  β k Q1 (α, β) = 1 − exp − Ik (αβ) 2 α  = 1 − exp −

k=1

α2 2

∞      2 ζ k Ik α 2 ζ 1+ζ

(4.41)

k=1

whereupon an analogous development to that leading up to (4.39) would yield the result7
π ζ 2 + ζ sin θ 1 Q1 (α, β) = Q1 (α, αζ ) = 1 + 2π −π 1 + 2ζ sin θ + ζ 2   α2  2 1 + 2ζ sin θ + ζ dθ , α>β≥0 × exp − 2

(0 ≤ ζ < 1) (4.42)



where now ζ = β/α < 1. Once again the expression in (4.42) is a single integral with finite limits and an integrand that is bounded and well behaved over the interval −π ≤ θ ≤ π and is Gaussian in one of the arguments, in this case, α. Aside from its analytical desirability in the applications discussed in later chapters, the form of (4.39) and (4.42) is also computationally desirable relative to other methods suggested previously by Parl [16] and Cantrell and Ojha [17] for numerical evaluation of the Marcum Q-function. The results in (4.39) and (4.42) can be put in a form with a more reduced integration interval. In particular, using the symmetry properties of the trigonometric functions over the intervals (−π , 0) and (0, π), we obtain the alternative forms




1 ± ζ cos θ 1 ± 2ζ cos θ + ζ 2 0   β2  2 × exp − dθ , 1 ± 2ζ cos θ + ζ 2

Q1 (α, β) = Q1 (βζ , β) =

1 π

π



β>α≥0

(0 ≤ ζ < 1) (4.43)

6 We

note that (4.41) is valid even if α < β, but for our purpose the series form given in (4.35) is more convenient for this case. 7 At first glance it might appear from (4.42) that the Marcum-Q function can exceed unity. However, the integral in (4.42) is always less than or equal to zero. It should also be noted that the results in (4.39) and (4.42) can also be obtained from the work of Pawula [15] dealing with the relation between the Rice Ie-function and the Marcum Q-function. In particular, equating Eqs. (2a) and (2c) of Ref. 15 and using the integral representation of the zeroth-order Bessel function obtained from (4.36) with k = 0 in the latter of the two equations, one can, with an appropriate change of variables, arrive at (4.39) and (4.42).

MARCUM Q-FUNCTION

97

and
 ζ 2 ± ζ cos θ 1 π Q1 (α, β) = Q1 (α, αζ ) = 1 + π 0 1 ± 2ζ cos θ + ζ 2   α2  × exp − 1 ± 2ζ cos θ + ζ 2 dθ , α>β≥0 2

(0 ≤ ζ < 1)

(4.44) Since, as we shall soon see, for the generalized (mth-order) Marcum Q-function the reduced integration interval form is considerably more complex than the form between symmetric (−π , π) limits, we shall tend to use (4.39) and (4.42) when dealing with the applications. As a simple check on the validity of (4.39) and (4.42), we examine the limiting cases Q1 (0, β) and Q1 (α, 0). Letting ζ = 0 in (4.39), we immediately have the well-known result   β2 Q1 (0, β) = exp − 2

(4.45)

Similarly, letting ζ = 0 in (4.42) gives Q1 (α, 0) = 1

(4.46)

4.2.1.1 Upper and Lower Bounds Simple upper and lower bounds on Q1 (α, β) can be obtained in the same manner that the Chernoff bound on the Gaussian Q-function was obtained from (4.2). In particular, for β > α ≥ 0, we observe that the maximum and minimum of the integrand in (4.39) occurs for θ = −π /2 and θ = π /2, respectively. Thus, replacing the integrand by its maximum and minimum values leads to the upper and lower “Chernoff-type” bounds



β 2 (1 + ζ )2 β 2 (1 − ζ )2 1 1 exp − exp − ≤ Q1 (βζ , β) ≤ (4.47) 1+ζ 2 1−ζ 2 or equivalently



β β (β + α)2 (β − α)2 exp − exp − ≤ Q1 (α, β) ≤ β+α 2 β −α 2

(4.48)

which, in view of (4.45), are asymptotically tight as α → 0. For α > β ≥ 0, the integrand in (4.42) has a minimum at θ = −π /2 and  a maximum at θ = π /2. Since the maximum of the integrand, [ζ /(1 + ζ )] exp −α 2 (1 + ζ )2 /2 , is always positive, the upper bound obtained by replacing the integrand by this value would exceed unity and hence be useless. On the other hand, the

98

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

  minimum of the integrand, − [ζ /(1 − ζ )] exp −α 2 (1 − ζ )2 /2 , is always negative. Hence a lower Chernoff-type bound on Q1 (α, β) is given by8

α 2 (1 − ζ )2 ζ exp − 1− (4.49) ≤ Q1 (α, αζ ) 1−ζ 2 or equivalently



α (α − β)2 exp − 1− ≤ Q1 (α, β) α−β 2

(4.50)

Another alternative and in some sense simpler form of the first-order Marcum Q-function was disclosed in Ref. 18. This form dispenses with the trigonometric factor that precedes the exponential in the integrands of (4.39) and (4.42) in favor of the sum of two purely exponential integrands each still having the desired dependence on β or α as appropriate. In particular, with a change in notation suitable to that used previously in this chapter, the results obtained in Ref. 18 can be expressed as follows: 
π  β2  1 exp − 1 + 2ζ sin θ + ζ 2 Q1 (α, β) = Q1 (βζ , β) = 4π −π 2      2 1 − ζ2 β2 dθ , β ≥ α ≥ 0 (0 ≤ ζ ≤ 1) + exp − 2 1 + 2ζ sin θ + ζ 2 (4.51) 
π  α2  1 exp − 1 + 2ζ sin θ + ζ 2 Q1 (α, β) = Q1 (α, αζ ) = 1 + 4π −π 2     2 1 − ζ2 α2 dθ , α ≥ β ≥ 0 (0 ≤ ζ ≤ 1) − exp − 2 1 + 2ζ sin θ + ζ 2 (4.52) or equivalently in the reduced forms analogous to (4.43) and (4.44) 
π  β2  1 2 exp − 1 ± 2ζ cos θ + ζ Q1 (α, β) = Q1 (βζ , β) = 2π 0 2    2  1 − ζ2 β2 dθ , β ≥ α ≥ 0 (0 ≤ ζ ≤ 1) + exp − 2 1 ± 2ζ cos θ + ζ 2 (4.53) 
π 2   1 α Q1 (α, β) = Q1 (α, αζ ) = 1 + exp − 1 ± 2ζ cos θ + ζ 2 2π 0 2     2 1 − ζ2 α2 dθ , α ≥ β ≥ 0 (0 ≤ ζ ≤ 1) − exp − 2 1 ± 2ζ cos θ + ζ 2 (4.54) 8

Clearly, since Q1 (α, β) can never be negative, the lower bound of (4.49) or (4.50) is useful only for values of the arguments that result in a nonnegative value.

MARCUM Q-FUNCTION

99

Since the first exponential integrand in each of (4.51) through (4.54) is identical to the exponential integrand in the corresponding equations (4.39), (4.42), (4.43), and (4.44), we can consider the second exponential in the integrands of the former group of equations as compensating for the lack of the trigonometric multiplying factor in the integrands of the latter equation group. The forms of the Marcum Q-function in (4.51) and (4.52) [or (4.53) and (4.54)] immediately allow us to obtain tighter upper and lower bounds of this function than those in (4.47) and (4.48). In particular, once again recognizing that for β > α ≥ 0 the maximum and minimum of the first exponential integrand in (4.51) occurs for θ = −π /2 and θ = π /2, respectively, and vice versa for the second exponential integrand, then we immediately obtain9



β 2 (1 + ζ )2 β 2 (1 − ζ )2 exp − ≤ Q1 (βζ , β) ≤ exp − (4.55) 2 2 or equivalently

(β + α)2 exp − 2





(β − α)2 ≤ Q1 (α, β) ≤ exp − 2

(4.56)

Making a similar recognition in (4.52), then, for α > β ≥ 0, we obtain the lower bound



 1 α 2 (1 − ζ )2 α 2 (1 + ζ )2 (4.57) 1− exp − − exp − ≤ Q1 (α, αζ ) 2 2 2 or equivalently10



 1 (α − β)2 (α + β)2 1− exp − − exp − ≤ Q1 (α, β) 2 2 2

(4.58)

We note that the bounds in (4.57) and (4.58) cannot be obtained directly from (4.42) by lower-bounding the exponential in the integrand since the factor that precedes it is not positive over the entire domain of the integral. We also note that although 9 It has been pointed out to the authors by W. F. McGee of Ottawa, Canada that the same tighter bounds can be obtained from (4.39) by upper- and lower-bounding only the exponential factor in the integrand (thus making it independent of the integration variable θ) and then recognizing that the integral of the remaining factor of the integrand can be obtained in closed form and evaluates to unity. We point out to the reader that this procedure of only upper and lower bounding the exponential is valid when the remaining factor is positive over the entire domain of the integral as is the case in (4.39). 10 Note that the upper bound in this case would become

Q1 (α, β) ≤ 1 +

1 2

exp −

which exceeds unity and is thus not useful.

(α − β)2 2



− exp −

(α + β)2 2



100

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

tighter bounds on the first-order Marcum Q-function have been obtained by Chiani [19], they are not in the desired form and thus are not helpful in applying the MGF-based approach to upper-bound the average BEP performance of noncoherent and differentially coherent communication systems perturbed by slow fading. Before concluding this section, we alert the reader to the inclusion of the endpoint α = β (ζ = 1) in the alternative representations of (4.51)–(4.54), all of which yield the value of Q1 (α, α) in (4.40). This is in contrast to the alternative representation pairs (4.39), (4.42) or (4.43), (4.44), which yield different limits as α approaches β (ζ approaches 1) from the left and right, respectively. The reason for these different left and right limits [the arithmetic average of which does in fact produce the result in (4.40)] is again tied to the fact that these representations rely on the convergence of a geometric series that, strictly speaking, is not convergent at the point ζ = 1. On the other hand, the derivation of the representations in (4.51) through (4.54) is based on a different approach [18] and as such are continuous across the point ζ = 1. Thus, even in the neighborhood of ζ = 1, one would anticipate better behavior from these representations. 4.2.2

Generalized (mth-Order) Marcum Q-Function

The generalized Marcum Q-function is defined analogous to (4.33) by   2
∞  √  1 x + s2 Im−1 (sx) dx Qm s, y = m−1 √ x m exp − s 2 y

(4.59)

or equivalently11 Qm (α, β) =




1 α m−1

∞ β

  2 x + α2 Im−1 (αx) dx x m exp − 2

(4.60)

where for m integer, the canonical form in (4.59) has the significance of being the complement (with respect to unity) of the CDF corresponding to the normalized 11 The

complement of the generalized Marcum Q-function can also be viewed as a special case of the incomplete Toronto function. In particular   α = 1 − Qm (α, β) Tβ/√2 2m − 1, m − 1, √ 2 Furthermore, as β → ∞, Qm (α, β) can be related to the Gaussian Q-function in the same manner as was done for the first-order Marcum Q-function. Specifically, since the asymptotic (for large argument) form of the k th-order modified Bessel function of the first kind is independent of the order, then   2 x + α 2 exp (αx) dx exp − √ α 2 2π αx β

 m−1/2
∞ β 1 (x − α)2 ∼ √ exp − dx = α 2 2π β

Qm (α, β) ∼ =

=




∞

x

 x m−1

 m−1/2 β Q (β − α) α

MARCUM Q-FUNCTION

101

 2 noncentral chi-square random variable, Y = m+1 k=1 Xk . It would be desirable to obtain integral forms analogous to (4.39) and (4.42) to represent the generalized Marcum Q-function regardless of whether m is integer or noninteger. Unfortunately, this has been shown to be possible only for the case of m integer, at least in the sense of an exact representation [9,10]. As we shall see from the derivation of these forms, however, the ones derived for m integer are also applicable in an approximate sense to the case of m noninteger in certain regions of the function’s arguments. Thus, we begin by proceeding with an approach analogous to that taken in arriving at (4.39) and (4.42) without restricting m to be integer, applying this restriction only when it becomes necessary. The details are as follows. m−1 Im−1 (αx)u = x m−1 Im−1 (αx) Applying integration  to (4.60) with u = x   2 by parts 2 and dv = x exp − x + α /2 dx and using the Bessel function recursion relation Im−1 (x) − Im+1 (x) = (2m/x) Im (x) [20, Eq. (9.6.26)], it is straightforward to show that the generalized Marcum Q-function satisfies the recursion relation    2  m−1 α + β2 β Im−1 (αβ) + Qm−1 (α, β) Qm (α, β) = exp − α 2

(4.61)

Recognizing that regardless of the values of α and β, Q−∞ (α, β) = 0 and Q∞ (α, β) = 1, then, iterating (4.61) in both the forward and backward directions gives the series forms  ∞    α2 + β 2  α r I−r (αβ) Qm (α, β) = exp − 2 β

(4.62)

 ∞  r β Ir (αβ) α r=m

(4.63)

r=1−m

and 

α2 + β 2 Qm (α, β) = 1 − exp − 2

Note that when m is integer, the values of the summation index r are also integer, and since in this case I−r (x) = Ir (x), we can rewrite (4.62) as  ∞    α2 + β 2  α r Ir (αβ) Qm (α, β) = exp − 2 β

(4.64)

r=1−m

Equations (4.63) and (4.64) are the series forms of the generalized Marcum Q-function that are found in the literature and apply when m is integer. When m is noninteger, the values of the summation index r are also noninteger, and since in this case I−r (x) = Ir (x), then (4.64) is no longer valid; instead one must use (4.62). Note that (4.63) is valid for m integer or m noninteger and together with (4.64) reduce to (4.41) and (4.35), respectively, for m = 1. Although the discussion above appears to make a mute point, it is important in the approach taken in Ref. 9 since certain trigonometric manipulations applied

102

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

there when deriving the alternative representation of the Marcum Q-function from the series representation hold only for m integer. Despite this fact, however, if the Ir (x) function r could still be represented exactly by the integral Ir (x) = π (1/2π) −π −j e−j θ e−x sin θ dθ [which is the same as (4.36) with r substituted for k ], then, even though the summation indices in (4.63) and (4.64) are noninteger, adjacent values are separated by unity and the same geometric series manipulations could be performed as were done previously for the first-order Marcum Q-function. Unfortunately, however, the integral representation of Ir (x) above is approximately valid only when its argument x is large irrespective of the value of r, and thus the steps that follow and the results that ensue are only approximate when m, the order of the Marcum Q-function, is noninteger. In what follows, however, we shall proceed as though this integral representation is exact (which it is for r integer, or equivalently m integer) with the understanding that the final integral representations obtained for the mth-order Marcum Q-function will be exact for m integer and approximate (for large values of the argument β or α as appropriate) for m noninteger. As discussed previously with regard to application of the alternative representation, it is convenient to introduce the parameter ζ < 1 to represent the ratio of the smaller to the larger of the two variables of the Marcum Q-function. We can therefore rewrite (4.62) and (4.63) as  2  ∞ β Qm (βζ , β) = exp − (1 + ζ 2 ) ζ r I−r (β 2 ζ ), 2

0+ ≤ ζ =

α < 1 (4.65) β



β < 1 (4.66) α



r=1−m

and ∞      α2  ζ r Ir α 2 ζ , Qm (α, αζ ) = 1 − exp − 1 + ζ2 2 r=m

0≤ζ =

Letting N < m < N + 1 (i.e., N is the largest integer less than or equal to m), substituting the integral form of the modified Bessel function in (4.65) gives 
π  ∞  1  −r β2  2 2 1+ζ Qm (βζ , β) = exp − ζ r −j e−j θ e−β ζ sin θ dθ 2 2π −π r=1−m N−m 
π   r  1 β2  (4.67) ej (θ +π/2) ζ = exp − 1 + ζ2 2 2π −π r=1−m

∞   j (θ +π/2) r −β 2 ζ sin θ e ζ dθ + e r=N−m+1

Recognizing, as mentioned above, that the sums in (4.67) are still geometric series despite the fact that the summation index r does not take on integer values,

MARCUM Q-FUNCTION

103

we obtain 
π  1 β2  Qm (βζ , β) = exp − ζ −(m−1) e−j (m−1)(θ +π/2) 1 + ζ2 2 2π −π   1 − ζ N ej N(θ+π/2) (4.68) × 1 − ζ ej (θ +π/2)   1 2 N+1−m j (N+1−m)(θ+π/2) e−β ζ sin θ dθ +ζ e 1 − ζ ej (θ +π/2) Since Qm (α, β) is a real function of its arguments, then, taking the real part of the right-hand side of (4.68) and simplifying results in the desired expression !      " ζ −(m−1) cos (m − 1) θ + π2 − ζ cos m θ + π2 1 + 2ζ sin θ + ζ 2 −π   β2  α 2 1 + 2ζ sin θ + ζ dθ , 0+ ≤ ζ = < 1 × exp − 2 β (4.69) Note that the limit of Qm (βζ , β) as ζ → 0 is difficult to evaluate directly from the form in (4.69), which explains the restriction on its region of validity. However, this limit can be evaluated starting with the integral form of (4.60) and using the small argument form of the modified Bessel function: 1 Qm (βζ , β) = 2π




π

Iv (z) ∼ =

(z/2)v  (ν + 1)

(4.70)

When this is done, the following results Qm (0, β) =

   m, β 2 /2  (m)

(4.71)

where  (α, x) is the complementary Gauss incomplete gamma function [5, Eq. (8.350.2)]. Using a particular integral representation of  (α, x), [21, Eq. (11.10)], then, after some changes of variables, Qm (0, β) can be expressed in the desired form  
π/2 cos θ β2 β 2m dθ (4.72) exp − Qm (0, β) = m−1 2  (m) 0 2 sin2 θ (sin θ )1+2m For m integer, the gamma function can be evaluated in closed form [5, Eq. (8.352.2)] and (4.71) reduces to Qm (0, β) =

m−1  n=0

  2 n  β /2 β2 exp − 2 n!

(4.73)

104

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

which is a special case of another form of the Marcum Q-function proposed by Dillard [22]: Qm (α, β) =

∞  n=0

    2 n n+m−1   2 k  α /2 β /2 α2 β2 exp − exp − 2 n! 2 k!

(4.74)

k=0

In a similar fashion, substituting the integral form of the modified Bessel function in (4.66) gives
π  ∞  1  −r α2  2 2 Qm (α, αζ ) = 1 − exp − 1+ζ ζ r −j e−j θ e−β ζ sin θ dθ 2 2π −π r=m 


π  ∞  1  j (θ +π/2) r −β 2 ζ sin θ β2  2 ζ e dθ e = 1 − exp − 1+ζ 2 2π −π r=m 

(4.75) whereupon, recognizing the sum as a geometric series, we get  2
π  1 1 α  2 m+1 j (m+1)(θ +π/2) ζ 1+ζ Qm (α, αζ ) = 1−exp − e 2 2π −π 1 − ζ ej (θ +π/2) × e−β

2ζ

sin θ

dθ

(4.76)

Finally, taking the real part of the right-hand side of (4.76) and simplifying gives the complementary expression to (4.69):    "
π m!   ζ cos m θ + π2 − ζ cos (m − 1) θ + π2 1 Qm (α, αζ ) = 1 − 2π −π 1 + 2ζ sin θ + ζ 2  2 α β 2 × exp − (1 + 2ζ sin θ + ζ ) dθ , 0 ≤ ζ = < 1 (4.77) 2 α For m integer, (4.69) and (4.77) simplify slightly to


π




π

(−1)m−1/2 ζ −(m−1) [cos(m − 1)θ + ζ sin mθ] 1 + 2ζ sin θ + ζ 2 −π   β2  α 2 × exp − dθ , 0+ < ζ = < 1, m odd 1 + 2ζ sin θ + ζ 2 β

1 Qm (βζ , β) = 2π

Qm (βζ , β) =

(−1)m/2 ζ −(m−1) [sin(m − 1)θ − ζ cos mθ] 1 + 2ζ sin θ + ζ 2 −π   β2  α 0+ < ζ = < 1, m even × exp − 1 + 2ζ sin θ + ζ 2 dθ , 2 β (4.78) 1 2π

MARCUM Q-FUNCTION

105


π 1 (−1)m−1/2 ζ m [sin mθ + ζ cos(m − 1)θ ] 2π −π 1 + 2ζ sin θ + ζ 2    β α2  2 1 + 2ζ sin θ + ζ dθ , 0 ≤ ζ = < 1, m odd × exp − α
π2 1 (−1)m/2 ζ m [cos mθ − ζ sin(m − 1)θ ] Qm (α, αζ ) = 1 − 2π −π 1 + 2ζ sin θ + ζ 2    α2  β × exp − 0 ≤ ζ = < 1, m even 1 + 2ζ sin θ + ζ 2 dθ , 2 α (4.79) which are the forms reported by Simon [9, Eqs. (7) and (10)]. Finally, the limit of (4.77) as ζ → 0 is easily seen to be Qm (α, 0) = 1, which is in agreement with the similar result in (4.46) for the first-order Marcum Q-function. As before, we observe from (4.69) and (4.77) that ζ is restricted to be less than unity (i.e., α = β) for the reason mentioned previously relative to the alternative representations of the first-order Marcum Q-function. For m integer, this special case has the closed-form result [10]    m−1      2  I0 α 2 1 (4.80) + Ik α 2 Qm (α, α) = + exp −α 2 2 Qm (α, αζ ) = 1 +

k=1

For m noninteger, the authors have been unable to arrive at an approximate closedform result. Finally, we note that the approach taken in Ref. 18 for arriving at the alternative forms for the first-order Marcum Q-function given in (4.51) through (4.54) unfortunately does not produce an equivalent simplification in the case of the mth-order Marcum Q-function. Similarly, upper and lower bounds on the mth-order Marcum Q-function are not readily obtainable by upper and lower bounding the exponential in the integrands of (4.69) and (4.77) since the first factor of these integrands is not positive over the domain of the integral. Thus, throughout the remainder of the book, unless the forms in (4.39) through (4.42) produce a specific analytical advantage, we shall tend to use the alternative forms of the first-order Marcum Q-function given in (4.39) and (4.42) because of their synergy with the equivalent forms in (4.69) and (4.77) for the mth-order Marcum Q-function. 4.2.2.1 Upper and Lower Bounds Despite the fact that upper and lower bounds on the mth-order Marcum Q-function are not readily obtainable from (4.69) and (4.77), it is nevertheless possible [23] (see also Figs. 4.2–4.4) for m integer to obtain such bounds by using the upper and lower bounds on the first-order Marcum Q-function given in (4.55) and (4.56) together with the recursive relation of (4.61).12 In particular, (4.61) can first be 12 We

emphasize that we are again looking for simple (exponential-type) bounds recognizing that although these may not be the tightest bounds achievable over all ranges of their arguments, relative to others previously reported in the literature [24], they are particularly useful in the context of evaluating error probability performance over fading channels.

106

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

Exponential Bounds on the First Order Marcum Q-function

Q1(1,b)

100

10−5

10−10

10−15

1

2

3

4

5

6

7

8

9

10

b Exponential Bounds on the Second Order Marcum Q-function

Q2(1,b)

100

10−5

10−10

10−15

1

2

3

4

5

6

7

8

9

10

9

10

b Exponential Bounds on the Fourth Order Marcum Q-function

Q4(1,b)

100

10−5

10−10

10−15

1

2

3

4

5

6

7

8

b Figure 4.2 Plots of Q1 (1, β), Q2 (1, β), Q4 (1, β), and their bounds versus β [(−) exact; (∗) upper bound (4.84); (×) Chernoff upper bound from Ref. 23; ( ) Chernoff lower bound from Ref. 23; and () lower bound of (4.62)].

MARCUM Q-FUNCTION

107

Exponential Bounds on the First Order Marcum Q-function

Q1(5,b)

100

10−10

10−20

10−30

0

2

4

6

8

10

12

14

16

18

20

18

20

18

20

b Exponential Bounds on the Second Order Marcum Q-function

Q2(5,b)

100

10−10

10−20

10−30

0

2

4

6

8

10

12

14

16

b Exponential Bounds on the Fourth Order Marcum Q-function

Q4(5,b)

100

10−10

10−20

10−30

0

2

4

6

8

10

12

14

16

b Figure 4.3 Plots of Q1 (5, β), Q2 (5, β), Q4 (5, β), and their bounds versus β [(−) exact; (∗) upper bound of (4.84); (×) Chernoff upper bound from Ref. 23; ( ) Chernoff lower bound from Ref. 23; () lower bound of (4.90)].

108

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

Exponential Bounds on the First Order Marcum Q-function 100

Q1(10,b)

10−5 10−10 10−15 10−20

0

2

4

6

8

10 b

12

14

16

18

20

18

20

18

20

Exponential Bounds on the Second Order Marcum Q-function 100

Q2(10,b)

10−5 10−10 10−15 10−20 10−25 0

2

4

6

8

10 b

12

14

16

Exponential Bounds on the Fourth Order Marcum Q-function 100

Q1(10,b)

10−5 10−10 10−15 10−20

0

2

4

6

8

10 b

12

14

16

Figure 4.4 Plots of Q1 (10, β), Q2 (10, β), Q4 (10, β), and their bounds versus β [(−) exact; (∗) upper bound of (4.84); (×) Chernoff upper bound from Ref. 23; ( ) Chernoff lower bound from Ref. 23; () lower bound of (4.90)]. Note that the lower bounds given by (4.90) and the Chernoff upper bound from Ref. 23 (m = 4) are out of the range in this case of α = 10.

MARCUM Q-FUNCTION

109

rewritten as  m−1    α2 + β 2  β n Qm (α, β) = exp − In (αβ) + Q1 (α, β) 2 α

(4.81)

n=1

Now expressing In (z) in its integral form analogous to (4.36), that is
1 π z cos θ e cos nθ dθ In (z) = π 0

(4.82)

and recognizing that the exponential part of the integrand has maximum and minimum values of ez and e−z , respectively, then because of the n-fold periodicity of cos nθ and the equally spaced (by π /n) regions where cos nθ is alternately positive and negative within the interval 0 ≤ θ ≤ π , we can upper bound In (z) by13 

3π/2n
2π/n n z 1 π/2n −z 1 z1 e In (z) ≤ cos nθ dθ + e cos nθ dθ + e cos nθ dθ 2 π 0 π π/2n π 3π/2n =

ez − e−z , π

z≥0

(4.83)

which is independent of n for n ≥ 1. This allows the series in (4.81) to be summed as a geometric series that has a closed-form result. Finally, using (4.83) in (4.81) together with the upper bound on Q1 (α, β) for 0+ ≤ ζ = α/β < 1 as given by (4.56), we obtain after some manipulation



1 (β − α)2 (β − α)2 Qm (α, β) ≤ exp − + exp − 2 π 2 (4.84)

  

β m−1 1 − (α/β)m−1 (β + α)2 − exp − 2 α 1 − α/β or equivalently



1 β 2 (1 − ζ )2 β 2 (1 − ζ )2 + exp − Qm (βζ , β) ≤ exp − 2 π 2     1 1 − ζ m−1 β 2 (1 + ζ )2 − exp − 2 ζ m−1 1−ζ

(4.85)

The first term of (4.84) or (4.85) represents the upper bound on the first-order Marcum Q-function, and thus, as would be expected, for m = 1 the remaining terms in these equations evaluate to zero. 13 Note

that (4.83) is valid for n odd as well as n even. It has also recently been pointed out to the authors that tighter bound than that given in (4.83) is In (z) ≤ (1/2) sinh z or equivalently,   a slightly In (z) ≤ ez − e−z /4. Thus, the upper bounds on the Marcum Q-function in (4.84) and (4.85) can likewise be slightly improved.

110

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

To obtain the lower bound on Qm (α, β) for 0+ ≤ ζ = α/β < 1, we can again use the lower bound on Q1 (α, β) [as given by (4.53b)] in (4.81); however, the procedure used to obtain the upper bound on In (z) that led to (4.83) would now yield the lower bound In (z) ≥

e−z − ez π

(4.86)

which for z ≥ 0 is always less than or equal to zero and therefore not useful relative to the simpler lower bound In (z) ≥ 0, n ≥ 1. Thus, to get a useful lower bound on In (z), we must employ an alternative form of its integral definition, namely [20, p. 376, Eq. (9.6.18)]
π (z/2)n ez cos θ sin2n θ dθ (4.87) In (z) = √   π n + 12 0 Once again replacing the exponential factor of the integrand by its minimum value, e−z , we obtain the lower bound
π (z/2)n −z sin2n θ dθ (4.88) In (z) ≥ √ e   π n + 12 0 which using [5, Eqs. (3.621.3) and (8.339.2)] yields14 In (z) ≥

zn −z e (2n)!!

(4.89)

Finally, substituting (4.89) in (4.81) and using the lower bound on Q1 (α, β) as given by (4.56) results after some simplification in   m−1 (β + α)2  (β/2)n ≤ Qm (α, β), 0≤α β (i.e., it is always extremely tight). In the case of (4.92), the lower bound was examined both with and without the additional term involving the summation; the latter is equivalent to (4.58). Over the range of values considered, the numerical results that take into account the presence of the extra series term are indistinguishable (when plotted) from those without it. Hence we can conclude that this series term can be dropped without losing tightness on the overall result. This observation will be important in the application discussions that follow in later chapters. Another approach [29] to upper and lower bounding of the mth-order Marcum Q-function (again only for m integer) is to apply the Cauchy–Schwarz inequality, 15 It is to be noted that whereas these upper and lower bounds of Ref. 24 are of interest on their own, their regions of validity do not share a common boundary in the α versus β plane, thus prohibiting their use in evaluating upper bounds on expressions containing the difference of two Marcum Q-functions with reversed arguments [i.e., Qm (α, β) − Qm (β, α)]. We shall see later in the book that expressions of this type are characteristic of many types of error probability evaluations over fading channels, and thus upper bounding such error probabilities requires an upper bound on the first Q-function and a lower bound on the second, with a boundary between their regions of validity given by α = β. The bounds presented in this chapter clearly satisfy this requirement, and thus with regard to the primary subject matter of this book they are the bounds of interest.

112

ALTERNATIVE REPRESENTATIONS OF CLASSICAL FUNCTIONS

namely #
# # #

b a

#2
# g1 (θ )g2 (θ ) dθ ## ≤

b




b

|g1 (θ )| dθ 2

a

|g2 (θ )|2 dθ

(4.94)

a

to the integral forms in Eqs. (4.78) and (4.79). For this purpose, it is convenient to combine the m even and m odd cases into a single integral that can be obtained by shifting the integration variable in (4.78) and (4.79) by π /2. Specifically, we have the more compact forms Qm (βζ , β) =




ζ −(m−1) [cos(m − 1)θ − ζ cos mθ] 1 − 2ζ cos θ ! +ζ 2 0 (4.95)   β2  α 2 + 1 − 2ζ cos θ + ζ × exp − dθ , 0 0 (1 + K) sin2 θ + a 2 γ /2 5.1.4

Nakagami-m Fading Channel

For the Nakagami-m distribution with instantaneous SNR per bit PDF given by [see (2.21)] pγ (γ ) =

  mγ mm γ m−1 exp − , γ m  (m) γ

γ ≥0

(5.14)

with Laplace transform [2, Eq. (3)]   sγ −m Mγ (−s) = 1 + , m

s>0

(5.15)

4 This particular Laplace transform is not tabulated directly in Ref. 2 but can be evaluated from a definite integral in the same reference, in particular, Eq. (6.631.4).

127

INTEGRALS INVOLVING THE GAUSSIAN Q-FUNCTION

the integral in (5.3) evaluates to 

I = Im (a, m, γ ) =

1 π




π/2 0

 1+

a2γ 2m sin2 θ

−m dθ

(5.16)

which can be evaluated in closed form using the definite integral derived in Appendix 5A:5 1 π




π/2 0

 1+

c sin2 θ

−m dθ

    m−1   2k   1 − µ2 (c) k  1 c    = , , µ (c) 1 − µ (c)   k  2 4 1 + c   k=0   m integer   = √    1  m + 12 1 1 c    , √ 2 F1 1, m + ; m + 1;   2 1+c 2 π (1 + c)m+(1/2)  (m + 1)     m noninteger

(5.17) where 2 F1 (·, ·; ·; ·) is the Gauss hypergeometric function [1, Eq. (15.1.1)]. Thus, using (5.17) in (5.16) gives Im (a, m, γ )     k    2  m−1   1 − µ2 a 2 γ     2m 1 a γ 2k      1 − µ ,   k 2m 4  2  k=0      2    a 2 γ /2 a γ    =  , µ   2m m + a 2 γ /2   m integer =  "       m + 12 a 2 γ /2m 1    m+(1/2)  Im (a, m, γ ) = 2√π   (m + 1)  1 + a 2 γ /2m        m 1    , × 2 F1 1, m + ; m + 1;   2 m + a 2 γ /2    m noninteger

(5.18)

Note that for m = 1, (5.18) reduces to the result for the Rayleigh case as given by (5.6). 5 This

definite integral does not appear to be available in standard integral tables such as Ref. 2.

128

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

5.1.5

Log-Normal Shadowing Channel

For the log-normal shadowing distribution with instantaneous SNR per bit PDF given by [see (2.25)]   2  10 log10 γ − µ 10/ ln 10 pγ (γ ) = √ ,γ ≥ 0 exp − 2σ 2 2π σ 2 γ µ (in dB) = 10 log10 γ

(5.19)

σ (in dB) = logarithmic standard deviation of shadowing the Laplace transform cannot be obtained in closed form. Instead, we substitute directly (5.19)  √ into (5.2) and then make a change of variables, namely, x = 10 log10 γ − µ / 2σ , which results in 

I = Iln (a, µ, σ ) =

1 π




π/2 0



1 √ π




 exp −

∞ −∞

a2 2 sin2 θ

10(x

√

2σ +µ)/10

  2 e−x dx dθ

(5.20) The inner integral can be efficiently computed using a Gauss–Hermite quadrature integration [1, Eq. (25.4.46)], that is 1 √ π




∞

−∞

1 =√ π

 exp − n  i=1

a2 2

2 sin θ  wi exp −

√ (x 2σ +µ)/10

10

a2 2 sin2 θ



e−x dx 2

√ (xi 2σ +µ)/10



(5.21)

10

where {xi } , i = 1, 2, . . . , n are the zeros of the nth-order Hermite polynomial H en (x) and {wi } ; i = 1, 2, . . . , n are weight factors tabulated in Table 25.10 of Ref. 1 for values of n from 2 to 20. Since the xi and wi terms are independent of θ , then, substituting (5.21) in (5.20) and making use of the desired form of the Gaussian Q-function as given in (4.2), we get n   " √ 1  Iln (a, µ, σ ) = √ wi Q a 10(xi 2σ +µ)/10 π i=1

(5.22)

where the value of n is chosen depending on the desired degree of accuracy. 5.1.6 Composite Log-Normal Shadowing/Nakagami-m Fading Channel The class of composite shadowing/fading channels is discussed in Section 2.2.3. A popular example of this class that is characteristic of congested downtown areas with a large number of slow-moving pedestrians and vehicles is the composite lognormal shadowing/Nakagami-m fading channel. For this channel, pγ (γ ) is obtained

INTEGRALS INVOLVING THE GAUSSIAN Q-FUNCTION

129

by averaging the instantaneous Nakagami-m fading average power (treated now as a random variable) over the conditional PDF of the log-normal shadowing, which from (5.14) and (5.19) results in the composite gamma/log-normal PDF


pγ (γ ) =

 mγ  mm γ m−1 exp − m  (m) 0 #   2 $ 10 log10 − µ 10/ ln 10 × √ d , exp − 2σ 2 2π σ 2 ∞

(5.23) γ ≥0

Since the Laplace transform of the Nakagami-m fading portion of (5.23) is known in closed form [see (5.15)], then the Laplace transform of the composite PDF in (5.23) can be obtained as the single integral ∞


Mγ (−s) = 0

s 1+ m

  2 $ −m # 10 log10 − µ 10/ ln 10 exp − d , √ 2σ 2 2π σ 2

s>0

(5.24) Substituting (5.24)  √into (5.2) and then making a change of variables, namely, x =  10 log10 − µ / 2σ , results in 

I = Ig/ ln (a, µ, σ , m)   −m
∞
√ a2 1 π/2 1 2 1+ 10(x 2σ +µ)/10 e−x dx dθ = √ π 0 π −∞ 2m sin2 θ

(5.25)

Once again the inner integral can be computed efficiently using a Gauss–Hermite quadrature integration [1, Eq. (25.4.46)]: 1 √ π




∞



a2

√

10(x

2σ +µ)/10

−m

e−x dx 2m sin2 θ  −m n √ a2 1  (xi 2σ +µ)/10 wi 1 + = √ 10 π i=1 2m sin2 θ −∞

1+

2

(5.26)

Since, as mentioned previously, the xi s and wi s terms are independent of θ , then, substituting (5.26) in (5.25) and making use of the closed-form integral in (5.17), we get   n m−1   2k   1 − µ2 (ci ) k 1  Ig/ ln (a, µ, σ , m) = √ wi 1 − µ (ci ) , k 4 2 π i=1 k=0  2 √ ci   a = 10(xi 2σ +µ)/10 , ci = (5.27) µ (ci ) 1 + ci 2m

130

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

Before moving on to a consideration of integrals involving the Marcum Q-function, we briefly discuss integrals involving the square of the Gaussian Q-function, since these will be found useful when we discuss evaluating average symbol error probability of coherently detected square QAM over generalized fading channels. Integrals involving higher-order powers of the Gaussian Q-function will be considered in Section 5.4.3. Analogous to (5.1), then, it is of interest to evaluate
∞  √  Q2 a γ pγ (γ ) dγ (5.28) I= 0

for the various fading channel PDFs. Using the classical definition of the Gaussian Q-function, such  √  integrals would be extremely difficult to obtain in closed form √ since Q2 a γ would be written as a double integral, each of which has γ in its lower limit. However, in view of the similarity between the desired forms of the Gaussian Q-function and the square of the Gaussian Q-function [compare Eqs. (4.2) and (4.9)], in principle it becomes a simple matter to evaluate I of (5.28)—in particular, one merely need replace the π /2 upper limit in the integration on θ in the evaluations of I of (5.1) with π /4 to arrive at the desired results. Although this may seem like a simple generalization, depending on the channel, the foregoing replacement of the upper limit can lead to closed-form expressions that are significantly more complicated. For the Rayleigh fading channel, the result, analogous to (5.6), is straightforward in view of the fact that the indefinite integral form of this equation has a closed-form result [see Eq. (5A.11) in Appendix 5A]. Thus, using (5A.13) we arrive at 

I = Ir(2) (a, γ ) =

1 π




π/4

 1+

a2γ

−1

dθ 2 sin2 θ   

2 γ /2 1 a 2 γ /2  4 1 + a  = tan−1 1− 4 1 + a 2 γ /2 π a 2 γ /2 0

(5.29)

For the Nakagami-m channel with m integer, the result is considerably more complex than (5.18). However, using (5A.17) with M = 4, we obtain 

I = Im(2) (a, m, γ ) =

1 π

π/4 




1+

a2γ

−m

dθ 2m sin2 θ # π  m−1   2k  1 1 1 − tan−1 α = − α k 4 π 2 (4 (1 + c))k 0

k=0



−1

− sin tan

α

k   m−1 k=1 i=1

Tik %  −1 &2(k−i)+1 cos tan α (1 + c)k

(5.30) $

INTEGRALS INVOLVING THE MARCUM Q-FUNCTION

131

where a2γ , c= 2m





α=µ =

c = 1+c

a 2 γ /2 m + a 2 γ /2

(5.31)

and 

2k



k



Tik =   2(k−i) 4i [2 (k − i) + 1]

(5.32)

(k−i)

5.2

INTEGRALS INVOLVING THE MARCUM Q-FUNCTION

When characterizing the performance of differentially coherent and noncoherent digital communications, the generic form of the expression for the error probability typically involves the generalized Marcum Q-function, both of whose arguments are proportional to the square root of the instantaneous SNR of the received signal. To compute the average error probability over a slowly fading channel, one must evaluate an integral whose integrand consists of the product of the above-mentioned Marcum Q-function and the PDF of the instantaneous SNR per bit. Thus, analogous to (5.1), we wish to investigate integrals having the generic form


∞

I=

 √ √  Ql a γ , b γ pγ (γ ) dγ

(5.33)

0

where a and b are constants that depend on the specific modulation/detection combination; l, the order of the Marcum Q-function; and pγ (γ ) again depends on the type of fading as discussed in Chapter 2. As was true for the Gaussian Q-function, if one were to use the classical definition of the Marcum Q-function given by Eq. (4.60) in (5.33), then, in general, evaluation of (5.33) would be diffi√ cult because of the presence of γ in the lower limit of the Marcum Q-function integral. If instead we were to use the desired form of the Marcum Q-function of (4.69) or (4.77) in (5.33), the result of this substitution would be %  & %  & 
π  −(l−1)  cos (l − 1) θ + π2 − ζ cos l θ + π2 ζ 1 I= 2π −π 1 + 2ζ sin θ + ζ 2  
∞  2  & b γ % a 2 pγ (γ ) dγ dθ , × exp − 0+ ≤ ζ = < 1 1 + 2ζ sin θ + ζ 2 b 0 (5.34)

132

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

or %  & %  & 
π  l ζ cos l θ + π2 − ζ cos (l − 1) θ + π2 1 I =1− 2π −π 1 + 2ζ sin θ + ζ 2  
∞  2  & a γ % b 1 + 2ζ sin θ + ζ 2 pγ (γ ) dγ dθ , 0+ ≤ ζ = < 1 × exp − 2 a 0 (5.35) where the inner integral is again in the form of a Laplace transform with respect  ∞ to the variable γ ; that is, if, as in Section 5.1, Mγ (s) = 0 esγ pγ (γ )dγ denotes the MGF of γ , then (5.34) and (5.35) can be rewritten as  %  & %  &  ζ −(l−1) cos (l − 1) θ + π2 − ζ cos l θ + π2 1 + 2ζ sin θ + ζ 2 −π   2 & b % a 0+ ≤ ζ = < 1 × Mγ − 1 + 2ζ sin θ + ζ 2 dθ , 2 b

1 I= 2π




π



(5.36)

or %  & %  & 
π  l ζ cos l θ + π2 − ζ cos (l − 1) θ + π2 1 I =1− 2π −π 1 + 2ζ sin θ + ζ 2  2  & a % b × Mγ − 0+ ≤ ζ = < 1 1 + 2ζ sin θ + ζ 2 dθ , 2 a

(5.37)

We now evaluate I of (5.36) for the variety of fading channel PDFs derived in Chapter 2, where, for simplicity of notation, we introduce the functions 

g (θ ; ζ ) = 1 + 2ζ sin θ + ζ 2 '   '  π ( π (  − ζ cos l θ + h (θ ; ζ , l) = ζ −(l−1) cos (l − 1) θ + 2 2

(5.38)

Also, the corresponding results for I of (5.37) can then be obtained by inspection. 5.2.1

Rayleigh Fading Channel

For the Rayleigh channel with a Laplace transform of the instantaneous SNR per bit PDF given by (5.5), the integral I of (5.36) [or equivalently (5.33) for a < b] evaluates to 

I = Jr (b, ζ , γ , l) =

1 2π




π

−π



h (θ ; ζ , l) g (θ ; ζ )

−1  b2 γ 1+ g (θ ; ζ ) dθ 2

(5.39)

INTEGRALS INVOLVING THE MARCUM Q-FUNCTION

5.2.2

133

Nakagami-q (Hoyt) Fading Channel

For the Nakagami-q (Hoyt) distribution with a Laplace transform of the instantaneous SNR per bit PDF given by (5.7), the integral I of (5.36) evaluates to 

I = Jq (b, ζ , q, γ , l) 1 = 2π 5.2.3




π



−π

h (θ ; ζ , l) g (θ ; ζ )

−1/2  q 2 b4 γ 2 g 2 (θ ; ζ ) 2 dθ 1 + b γ g (θ ; ζ ) +  2 1 + q2

(5.40)

Nakagami-n (Rice) Fading Channel

For the Nakagami-n (Rice) distribution with a Laplace transform of the instantaneous SNR per bit PDF given by (5.11), the integral I of (5.36) evaluates to  
π 1 + n2 h (θ ; ζ , l)  1  I = Jn (b, ζ , n, γ , l) = 2 2π −π g (θ ; ζ ) 1 + n2 + b 2γ g (θ ; ζ ) (5.41)   2 n2 b 2γ g (θ ; ζ )  dθ × exp − 2 1 + n2 + b 2γ g (θ ; ζ ) or equivalently in terms of the Rician parameter 1 I = Jn (b, ζ , K, γ , l) = 2π 

 × exp −

5.2.4

K

b2 γ 2

1+K +




π



−π

g (θ ; ζ ) b2 γ 2

h (θ ; ζ , l) g (θ ; ζ ) 



1+K 1+K +

b2 γ 2

g (θ ; ζ ) (5.42)

 dθ

g (θ ; ζ )

Nakagami-m Fading Channel

For the Nakagami-m distribution with a Laplace transform of the instantaneous SNR per bit PDF given by (5.15), the integral I of (5.36) evaluates to 1 I = Jm (b, ζ , m, γ , l) = 2π 




π −π



h (θ ; ζ , l) g (θ ; ζ )



−m b2 γ 1+ g (θ ; ζ ) dθ 2m

(5.43)

which reduces to (5.39) for the Rayleigh (m = 1) case. 5.2.5

Log-Normal Shadowing Channel

As discussed in Section 5.1.5, the Laplace transform of the instantaneous SNR per bit PDF for the log-normal shadowing distribution cannot be obtained in closed

134

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

form. Thus, we proceed as before and substitute (5.19) directly   √ into (5.34) and then make a change of variables, namely, x = 10 log10 γ − µ / 2σ , which results in 

I = Jln (b, ζ , µ, σ , l)    2 
∞
π 1 h (θ ; ζ , l) b g (θ ; ζ ) (x √2σ +µ)/10 −x 2 1 10 e dx dθ √ exp − = 2π −π g (θ ; ζ ) 2 π −∞ (5.44) The inner integral can be efficiently computed using a Gauss–Hermite quadrature integration [1, Eq. (25.4.46)]:   2
∞ b g (θ ; ζ ) (x √2σ +µ)/10 −x 2 1 e dx 10 exp − √ 2 π −∞ (5.45)   2 n b g (θ ; ζ ) (xi √2σ +µ)/10 1  10 wi exp − =√ 2 π i=1

Substituting (5.45) into (5.44) and making use of the desired from of the generalized Marcum Q-function as given in (4.69), we get n "  "  √ √ 1  wi Ql bζ 10(xi 2σ +µ)/10 , b 10(xi 2σ +µ)/10 (5.46) Jln (b, ζ , µ, σ , l) = √ π i=1

5.2.6 Composite Log-Normal Shadowing/Nakagami-m Fading Channel Finally, we consider the composite log-normal shadowing/Nakagami-m fading channel treated in Section 5.1.6. For this channel, we again make use of the singleintegral form of the Laplace transform of pγ (γ ) as given in (5.24), which, on  substitution into (5.36) together with the change of variables x = 10 log10 − µ / √ 2σ , results in 
π h (θ ; ζ , l) 1  I = Jg/ ln (b, ζ , µ, σ , m, l) = 2π −π g (θ ; ζ )   (5.47) −m
∞ b2 g (θ ; ζ ) (x √2σ +µ)/10 1 2 −x 1+ 10 e dx dθ × √ 2m π −∞ Once again the inner integral can be computed efficiently using a Gauss–Hermite quadrature integration [1, Eq. (25.4.46)]:  −m b2 g (θ ; ζ ) (x √2σ +µ)/10 2 e−x dx 1+ 10 2m −∞  −m n b2 g (θ ; ζ ) (xi √2σ +µ)/10 1  10 wi 1 + = √ 2m π

1 √ π




∞

i=1

(5.48)

INTEGRALS INVOLVING THE MARCUM Q-FUNCTION

135

Substituting (5.48) in (5.47) and making use of the closed-form integral in (5.17), we get  
π n 1 h (θ ; ζ , l) 1  wi Jg/ ln (b, ζ , µ, σ , m, l) = √ 2π −π g (θ ; ζ ) π i=1 (5.49) −m   b2 g (θ ; ζ ) (xi √2σ +µ)/10 10 × 1+ dθ 2m Unfortunately, because a closed-form result was not obtainable for (5.43), we cannot similarly obtain a closed-form result for (5.49). 5.2.7

Some Alternative Closed-Form Expressions

In the case of the Rayleigh channel, there exists another approach that leads to a closed-form alternative to (5.39). In particular, starting with a relation between Qm (α, β) and Q1 (α, β) derived from the recursion relation in (4.61) and then making use of the integrals in Eq. (55) of Ref. 10 and Eq. (20) of Ref. 11, it is shown in Ref. 12 that6   '  ( Z−Z l 2 2 2  2 
∞ p 1 + 2 + a − b  √ 1−Z p y 1  √   Ql a y, b y dy = 2 1 + ) exp −  2 2 p 0 2 2 2 2 2 p + a + b − 4a b (5.50) where 

Z=

p 2 + a 2 + b2 −

)

p 2 + a 2 + b2

2

− 4a 2 b2

2a 2

Thus, letting p2 /2 = 1/γ and y = γ , we obtain after some simplification   l + a 2 γ2 (1 − Z) 1 + Z−Z 1−Z I =    2 1 + a 2 + b2 γ2 − a 2 b2 γ 2

(5.51)

(5.52)

where 



Z=

1+ a +b 2

2

γ 2

−



  2 1 + a 2 + b2 γ2 − a 2 b2 γ 2 a2γ

(5.53)

 √ √  interpreting the integral in (5.50) as the Laplace transform of Ql a γ , b γ , the closed-form expression on the right-hand side of this equation can be shown to be consistent with the ninth entry in Table 2 of Ref. 13.

6 By

136

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

Alternatively, in terms of the ratio ζ = a/b, we have 1+ I = 



1+

Z−Z l 1−Z

b2





+ b2 ζ 2 γ2 (1 − Z)

1+

ζ2

 γ 2 2

−

(5.54)

b4 ζ 2 γ 2

where



Z=

1+

b2



1+

ζ2

γ 2

−



  2 1 + b2 1 + ζ 2 γ2 − b4 ζ 2 γ 2 b2 ζ 2 γ

(5.55)

In the case of the Nakagami-m channel, a generalization of (5.50) allows us to find a closed-form alternative to (5.43). In particular, using an integration-by-parts approach similar to that used by Nuttall [see [10], Appendix A relating to Eq. (55)], it is shown in Ref. 12 that7
∞    √ √  y m−1 exp −p2 y/2 Ql a y, b y dy 0

#    2 i m−1  2 m p b2l  l + i = − 1)! 1 + (m i p2 sl s i=0    2 l+i+1 l+i+2 4a 2 b2 a , ; l + 1; × 2 F1 s 2 2 s2  $   l + i l + i + 1 4a 2 b2 l , ; l; − 2 F1 l+i 2 2 s2 

(5.56)

where s = p2 + a 2 + b2 and 2 F1 (a, b; c; z) is the Gaussian hypergeometric function [1, Eq. (15.1.1)]. Thus, letting p2 /2 = m/γ and y = γ , we obtain   m−1  b2l  l + i 2m I =1+ l i s sγ i=0



 i a2 l+i+1 l+i+2 4a 2 b2 , ; l + 1; F 2 1 s 2 2 s2     l + i l + i + 1 4a 2 b2 l , ; l; − F 2 1 l+i 2 2 s2

×

(5.57)

7 In principle, a closed-form expression for the integral in (5.56) could be obtained by differentiating the right-hand side of (5.50) m − 1 times with respect to p 2 .

INTEGRALS INVOLVING THE INCOMPLETE GAMMA FUNCTION

137

where now s = 2m/γ + a 2 + b2 . Alternatively, in terms of the ratio ζ = a/b, we have   i  2 2 m−1  2m l+i+1 l+i+2 b ζ 4b4 ζ 2 b2l  l + i , ; l + 1; F 2 1 i sl sγ s 2 2 s2 i=0     l + i l + i + 1 4b4 ζ 2 l (5.58) , ; l; − 2 F1 l+i 2 2 s2

I =1+

  with s = 2m/γ + b2 1 + ζ 2 . In certain applications (to be discussed in Chapter 9), rather than attempt to find a closed form for the integral in (5.33) by itself, it is of interest to evaluate a particular sum that involves the difference of two such integrals. This sum takes the form S=


∞ L   1 2L − 1 y L−1 exp (−y) L−l 22L−1 − 1)! (L 0 l=1 %  √  √ √  √ & × 1 − Ql b y, a y + Ql a y, b y dy 1

(5.59)

As a limiting case of results obtained by Lao and Haimovich [14] for multiplesymbol differential detection of M -PSK with L-diversity reception in the presence of interference, it can be shown that L

  1  S =  1 − ) 2  

1  ×  1 + ) 2

5.3

L−1 

  2    k=0 b2 − a 2 + 4 a 2 + b2 + 1 k b2

−

a2

L−1+k k



(5.60)

b2 − a 2     2 b2 − a 2 + 4 a 2 + b2 + 1

INTEGRALS INVOLVING THE INCOMPLETE GAMMA FUNCTION

In the preceding section, we considered integrals involving the Marcum Q-function where the desired (finite integral) form of this function as in (4.69) was used to simplify the evaluations. A special case of the Marcum Q-function corresponding to its first argument equal to zero is expressible as a ratio of complementary Gauss incomplete gamma functions [see Eq. (4.67)]. As we shall see in Chapter 8, integrals involving such a ratio are appropriate to the unification of the error probability performance of coherent, differentially coherent, and noncoherent binary PSK and FSK systems over generalized fading channels. However, since the desired form

138

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

of the Marcum Q-function of (4.69) requires that the first argument be greater than zero, the specific results derived in Section 5.2 cannot be used in this instance. Fortunately, however, the special case Qm (0, β) can be put in a separate desired form8 as given by (4.72). In this section, we derive the analogous results to those in Section 5.2 using this special desired form of Qm (0, β). On the basis of the discussion above, then, we are interested in evaluating  
∞  l, b2 γ
∞  √  2 pγ (γ ) dγ Ql 0, b γ pγ (γ ) dγ = (5.61) I=  (l) 0 0 for the various characterizations of pγ (γ ) or substituting the form of (4.72) in (5.61), we are equivalently interested in evaluating  
∞  √ 2l
π/2 b γ cos θ b2 γ dθ pγ (γ ) dγ exp − (5.62) I= 2l−1  (l) 0 2 sin2 θ (sin θ )1+2l 0 Reversing the order of integration and grouping together like variables, we can rewrite (5.62) as  
π/2
∞ cos θ b2 γ b2l l pγ (γ ) dγ dθ γ exp − (5.63) I = l−1 2  (l) 0 2 sin2 θ (sin θ)1+2l 0 where the integral on γ is in the form of a Laplace transform that is similar to but slightly more complicated than the MGF of γ . 5.3.1

Rayleigh Fading Channel

Substituting (5.4) in (5.63) and making use of Eq. (3.381.4) of Ref. 2, we obtain −l−1   2 l
π/2 b2 γ cos θ b γ  1+ dθ (5.64) I = Jr (b, γ , l) = 2l 2 2 sin2 θ (sin θ )1+2l 0 Making the change of variables t = [1 + b2 γ /(2 sin2 θ )]−1 , after some manipulation we arrive at the equivalent compact result  −1 1+b2 γ /2


Jr (b, γ , l) = l 0

(1 − t)l−1 dt = l B(1+b2 γ /2)−1 (1, l)

where 

Bx (p, q) =




x

t p−1 (1 − t)q−1 dt

(5.65)

(5.66)

0

is the incomplete beta function [2, p. 960, Eq. (8.391)]. 8 The

desired form of the integral for Qm (0, β) is slightly less desirable than that for Qm (α, β) , 0 < α < β, in that the integrand contains a term β 2m in addition to the usual Gaussian dependence on β. Nevertheless, it is still useful in carrying out integrals involving the statistics of the fading channel by using Laplace transform manipulations.

INTEGRALS INVOLVING THE INCOMPLETE GAMMA FUNCTION

5.3.2

139

Nakagami-q (Hoyt) Fading Channel

Substituting (5.7) in (5.63) and making use of the Laplace transform found in the Erdelyi et al. table [3, p. 196, Eq. (8)], then, recognizing the relation between the associated Legendre function and the Gaussian hypergeometric function [2, Eq. (8.771.1)], we obtain 

I = Jq (b, q, γ , l)  2 l  
π/2 b γ 1 + q2 cos θ =l 2 q (sin θ )1+2l 0    2   2 −[(l+1)/2] 2 2 2 4 1 + q b γ 1−q  × − + 4q 2 4q 2 2 sin2 θ    1 1 × 2 F1  −l, l + 1; 1; 2 − 2  

5.3.3

 b2 γ 2 sin2 θ b2 γ 2 sin2 θ

+

1+q 2

2 1+q 2 + ( 4q 2 )

2

4q 2

2

−

(5.67) 

   dθ  4 2   1−q 4q 2

Nakagami-n (Rice) Fading Channel

Substituting (5.11) in (5.63) and making use of the Laplace transform found in Erdelyi et al. [3, Eq. (20)], then, recognizing the relation between the Whittaker function and the confluent hypergeometric function [2, Eq. (9.220.2)], we obtain 

−l−1  l
π/2  cos θ b2 γ  b2 γ 2 2 1+n + 1 + n2 e−n 1+2l 2 2 sin2 θ 0 (sin θ )     n2 1 + n2  dθ (5.68) × 1 F1 1 + l, 1; b2 γ 1 + n2 + 2 sin 2θ 

I = Jn (b, n, γ , l) = 2l

or equivalently in terms of the Rician parameter 

−l−1   2 l
π/2 b2 γ cos θ b γ 1 + K + (1 + K) e−K 1+2l 2 2 sin2 θ 0 (sin θ)   K (1 + K)  dθ (5.69) × 1 F1 1 + l, 1; b2 γ 1+K + 2 sin 2θ

I = Jn (b, K, γ , l) = 2l

where 1 F1 (·, ·; ·) is the confluent hypergeometric function [2, Sect. 9.20].

140

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

5.3.4

Nakagami-m Fading Channel

Substituting (5.15) in (5.63) and making use of Eq. (3.381.4) of Ref. 2, we obtain 

I = Jm (b, m, γ , l) =

2 B (m, l)



b2 γ 2m

l


π/2

cos θ (sin θ )1+2l

0

 1+

b2 γ

−l−m

2m sin2 θ

dθ (5.70)

where 

B (m, l) = B (l, m) =

 (m)  (l)  (m + l)

(5.71)

is the beta function [2, Eq. (8.384.1)]. Making the change of variables % &−1 t = 1 + b2 γ /(2m sin2 θ ) , after some manipulation we arrive at the equivalent compact result 1 Jm (b, γ , l) = B (m, l)

 −1 1+b2 γ /2m




t m−1 (1 − t)l−1 dt =

B(1+b2 γ /2m)−1 (m, l)

0

B (m, l) (5.72)

or in terms of the incomplete beta function ratio [2, Eq. (8.392)] 

Ix (p, q) =

Bx (p, q) B (p, q)

(5.73)

the still simpler form Jm (b, γ , l) = I(1+b2 γ /2m)−1 (m, l)

(5.74)

For the Rayleigh (m = 1) case, (5.72) clearly reduces to (5.65) since B (1, l) = l −1 . 5.3.5

Log-Normal Shadowing Channel

Substituting the PDF of (5.19) into (5.63) and making the change of variables,   √ x = 10 log10 γ − µ / 2σ results after much simplification in   2 n √  b 1 (x 2σ +µ)/10 i I = Jln (b, µ, σ , l) = √  l, 10 2 π (l) i=1 

(5.75)

where again {xi } , i = 1, 2, . . . , n, are the zeros of the nth order Hermite polynomial H en (x) as discussed in Section 5.1.5. 5.3.6 Composite Log-Normal Shadowing/Nakagami-m Fading Channel Finally, for the composite log-normal shadowing/Nakagami-m fading channel treated in Section 5.1.6, we substitute the PDF of (5.23) into (5.63) together with

INTEGRALS INVOLVING OTHER FUNCTIONS

141

  √ the change of variables x = 10 log10 − µ / 2σ , resulting in n 1   −1 (m, l) √ wi I I = Jg/ ln (b, µ, σ , m, l) = √ 1+(b2 /2m)10(xi 2σ +µ)/10 π i=1

(5.76)

where now, in addition, {wi } , i = 1, 2, . . . , n are the Gauss quadrature weights as discussed in Section 5.1.5.

5.4

INTEGRALS INVOLVING OTHER FUNCTIONS

When studying the error probability performance of certain modulation schemes over generalized fading channels, we shall have reason to evaluate integrals involving special functions other than the three considered previously in this chapter. In this section we consider integrals involving two such special functions corresponding to well-known modulation schemes. 5.4.1

The M-PSK Error Probability Integral

When studying the average error probability performance of M -PSK over generalized fading channels, we shall have reason to evaluate integrals of the form   a2γ K= dθ pγ (γ ) dγ exp − 2 sin2 θ 0 0  
∞  
a2γ 1 (M−1)π/M p exp − dγ dθ = (γ ) γ π 0 2 sin2 θ 0


∞

1 π




(M−1)π/M

(5.77)

where specifically a 2 = 2 sin2 π /M. The integral in (5.77) is a generalization of the one in (5.2) in the sense that the latter is a special case of the form corresponding to M = 2. Thus, (5.77) follows directly from (5.3) and is given by 1 K= π




(M−1)π/M 0

 Mγ −

a2 2 sin2 θ

 dθ

(5.78)

Although this may seem like a simple generalization, unfortunately the replacement of the π /2 upper limit in (5.3) by (M − 1)π /M results wherever possible in closed-form expressions for (5.78) that, in general, are significantly more complicated. Without further ado, we present the results for the evaluation of (5.78) corresponding to the various types of fading channels, where closed-form results can be obtained. The results corresponding to the remainder of the fading channels can be obtained by the same upper limit replacement as mentioned above in the corresponding expressions of Section 5.1.

142

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

5.4.1.1 Rayleigh Fading Channel Substituting (5.5) in (5.78) and making use of (5A.15), we obtain  −1
a2γ 1 (M−1)π/M 1+ dθ π 0 2 sin2 θ 

    M −1 a 2 γ /2 M = 1−  M 1 + a 2 γ /2 (M − 1) π     2 γ /2 π π a  cot ×  + tan−1   2 1 + a 2 γ /2 M 

K = Kr (a, γ , M) =

(5.79)

which reduces to (5.6) when M = 2. 5.4.1.2 Nakagami-m Fading Channel Here we need to substitute the Laplace transform of (5.15) into (5.78). After this is done, then making use of (5A.17), we obtain 

K = Km (a, γ , m, M)  −m
a2γ 1 (M−1)π/M 1+ = dθ π 0 2m sin2 θ

 m−1   2k  1 a 2 γ /2m / π M −1 1 −1 − + tan α =   k k M π 1 + a 2 γ /2m 2 4 1 + a 2 γ /2m k=0 $ k  %  −1 &2(k−i)+1  −1  m−1 Tik + sin tan α  k cos tan α 2 k=1 i=1 1 + a γ /2m (5.80) where



α=

π a 2 γ /2m cot 1 + a 2 γ /2m M

(5.81)

and Tik is again given by (5.32). 5.4.2 Arbitrary Two-Dimensional Signal Constellation Error Probability Integral As a generalization of QAM, Craig [4] showed that the evaluation of the average error probability performance of an arbitrary two-dimensional (2D) signal constellation with polygon-shaped decision regions over the AWGN channel can be

INTEGRALS INVOLVING OTHER FUNCTIONS

expressed as a summation of integrals of the form9  
θi ai2 sin2 ψi 1 Pi = exp − dθ 2π 0 2 sin2 (θ + ψi )

143

(5.82)

where ai2 is a signal-to-noise ratio parameter associated with the i th signal in the set and θi and ψi are angles associated with the correct decision region corresponding to that same signal. Thus, when studying the average error probability performance of these 2D signal constellations over generalized fading channels, we shall have reason to evaluate integrals of the form  
∞
θi ai2 γ sin2 ψi 1 L= exp − dθ pγ (γ ) dγ 2π 0 2 sin2 (θ + ψi ) 0 (5.83)   
θi 
∞ ai2 γ sin2 ψi 1 pγ (γ ) dγ dθ = exp − 2π 0 2 sin2 (θ + ψi ) 0 By comparison with (5.77), we observe that (5.83) can be expressed in the form of (5.78), namely  
θi ai2 sin2 ψi 1 Mγ − dθ (5.84) L= 2π 0 2 sin2 (θ + ψi ) where again Mγ (s) is the MGF of γ . Evaluation of the Laplace transform integrand in (5.84) for the various types of fading channels follows exactly along the lines of the previous results and hence is not repeated here. Unfortunately, however, for arbitrary θi it is not always possible now to obtain closed-form expressions for Li even when the integrand is obtainable in closed form. However, for the Rayleigh channel, using (5.5) for Mγ (−s) and the indefinite form of the integral in (5A.11), it is straightforward to obtain the following closed-form solution 

L = Lr (ai , γ , θi , ψi )

  0 ci 1 1 + ci 1 0θi −ψi −1 tan tan θ 0−ψ = − i 4 2π ci (1 + ci ) ci

ci 1 1 = − 4 2π ci (1 + ci )      1 + ci 1 + ci −1 −1 tan (θi − ψi ) + tan tan ψi × tan ci ci 9 Equation (5.82) appears as Eq. (13) in Ref. 4 but with an error of a factor of that premultiplies the integral there should be 1/2π , as shown here in (5.82)].

1 2

(5.85)

[i.e., the factor 1/π

144

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

where 

ci =

ai2 γ sin2 ψi 2

(5.86)

For Nakagami-m fading, using the Laplace transform in (5.15), we obtain m  
θi −ψi 1 sin2 φ  L = Lm (ai , γ , m, θi , ψi ) = dφ 2π sin2 φ + ci /m 0 (5.87) m 
ψi  sin2 φ + dφ sin2 φ + ci /m 0 with ci still as defined in (5.86). If, depending on the signal constellation, θi and θi − ψi both turn out to be either in the form (M − 1) π /M or π /M for M = 2m , m integer, the closed-form results of (5A.16) and (5A.21) can be used to obtain (5.87) in closed form. Otherwise, the single-integral form of (5.87) must be used. The results for the other fading channel types will, in general, be expressed as a single integral with finite limits (0, θi ) in accordance with (5.84) and the various closed-form expressions obtained previously for Mγ (−s). 5.4.3

Higher-Order Integer Powers of the Gaussian Q-Function

Associated with the study of the average error probability performance of coherent communication systems using differentially encoded QPSK and M -ary orthogonal signals in the presence of slow fading, we shall have need to evaluate integrals of the form
∞  √   Qk a γ pγ (γ ) dγ (5.88) Ik = 0

where k is assumed to be integer. In general, for arbitrary integer values of k, Ik cannot be obtained in the desired form. However, certain special cases, namely, k = 1, 2, 3, 4, do exist either in closed form or in the form of a single integral with finite limits and an integrand composed of elementary functions. For k = 1, 2, the results were presented in Section 5.1. The specific results corresponding to k = 3, 4 for Rayleigh and Nakagami-m fading are presented in what follows. 5.4.3.1 Rayleigh Fading Channel For k = 3, 4, we use the alternative forms for Q3 (x) and Q4 (x) as given in (4.31) and (4.32), respectively. Substituting these expressions in (5.88), we obtain    
π/6 3 cos 2θ − 1 1 a2  dθ cos−1 − − 1 M I3 = I3,r (a, γ ) = 2 γ π 0 2 cos3 2θ 2 sin2 θ      2
sin−1 1/√3 1 a2 1 −1 3 cos 2θ − 1 dθ π − cos − − 1 M + γ 2π 2 0 2 cos3 2θ 2 sin2 θ (5.89)

INTEGRALS INVOLVING OTHER FUNCTIONS

145

and 

I4 = I4,r (a, γ ) =

1 π2


0

π/6

cos−1



   3 cos 2θ − 1 a2 dθ (5.90) − − 1 M γ 2 cos3 2θ 2 sin2 θ

which, using (5.5) for Mγ (−s), results in the desired forms 

I3 = I3,r (a, γ )   
π/6  1 sin2 θ −1 3 cos 2θ − 1 − 1 dθ = 2 cos π 0 2 cos3 2θ sin2 θ + a 2 γ /2   1  2
sin−1 1/√3  sin2 θ 1 −1 3 cos 2θ − 1 π − cos − 1 dθ + 2π 2 0 2 cos3 2θ sin2 θ + a 2 γ /2 (5.91) and   
π/6  1 sin2 θ  −1 3 cos 2θ − 1 dθ − 1 I4 = I4,r (a, γ ) = 2 cos π 0 2 cos3 2θ sin2 θ + a 2 γ /2 (5.92) 5.4.3.2 Nakagami-m Fading Channel Following the same procedure as for the Rayleigh fading channel, we can evaluate (5.88) for the Nakagami-m fading channel by using (5.15) for Mγ (−s) in (5.89) and (5.90), resulting in 

I3 = I3,r (a, γ ) m  
π/6  1 sin2 θ −1 3 cos 2θ − 1 dθ − 1 cos = 2 π 0 2 cos3 2θ sin2 θ + a 2 γ /2m   m 1  2
sin−1 1/√3  sin2 θ 1 −1 3 cos 2θ − 1 π − cos −1 dθ + 2π 2 0 2 cos3 2θ sin2 θ + a 2 γ /2m (5.93) and m  
π/6  1 sin2 θ  −1 3 cos 2θ − 1 = − 1 dθ cos I4 I4,r (a, γ ) = 2 π 0 2 cos3 2θ sin2 θ + a 2 γ /2m (5.94) 5.4.4

Integer Powers of M-PSK Error Probability Integrals

Associated with the study of the average error probability performance of coherently detected differentially encoded M -PSK in the presence of slow fading, we

146

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

shall have need to evaluate integrals of the form 

K2 (a, γ ) =

∞


0

1 π




(M−1)π/M

0

 2  a2γ dθ pγ (γ ) dγ exp − 2 sin2 θ

(5.95)

and    a12 γ L2 (θu1 , θu2 , a1 , a2 , γ ) = dθ exp − 2 sin2 θ 0 0     a22 γ 1 θu2 × dθ pγ (γ ) dγ exp − π 0 2 sin2 θ 




∞

1 π





θu1

(5.96)

where, as was the case in Section 5.4.1, a 2 = 2 sin2 π /M, and now in addition a12 and a22 assume the possible values 2 sin2 (2k ± 1) π /M, k = 0, 1, 2, . . . , M − 1, and θu1 and θu2 assume the possible values π [1 − (2k ± 1) /M]. While (5.95) can be evaluated in the desired form for both Rayleigh and Nakagami-m fading, unfortunately, (5.96) can be obtained in such a form only for the Rayleigh case. Thus, we shall present specific results only for this single fading case. 5.4.4.1 Rayleigh Fading Channel To evaluate (5.95), we make use of the generic integral of (4.30) with φu = (M − 1)π /M, resulting in   /M] √ sin−1 sin[(M−1)π

    a2 (M − 1) π Mγ − dθ cos g θ , M 2 sin2 θ 0 (5.97) where g (θ , ζ ) is as defined in (4.29). Again using (5.5) for Mγ (−s) results in the desired form    /M]   
sin−1 sin[(M−1)π √ 1 sin2 θ (M − 1) π 2 −1 dθ g θ , cos K2 (a, γ ) = 2 π 0 M sin2 θ + a 2 γ /2 (5.98) For the Nakagami-m case one would simply use (5.15) instead of (5.5) for Mγ (−s) in (5.97). To evaluate (5.96), we proceed as follows   2 

a22 a1 1 1 θu1 1 θu2 dθ dφ L2 (θu1 , θu2 , a1 , a2 ) = Mγ − + π 0 π 0 2 sin2 θ sin2 φ  −1

a12 γ a22 γ 1 θu1 1 θu2 = 1+ + dθ dφ π 0 π 0 2 sin2 θ 2 sin2 φ   
θu2 
1 2 sin2 θ 1 θu1 c12 (φ) = dθ dφ π 0 π 0 sin2 θ + c12 (φ) a12 γ (5.99) 1 K2 (a, γ ) = 2 π




2

−1

INTEGRALS INVOLVING OTHER FUNCTIONS

147

where c12 (φ) is defined as a2γ c12 (φ) = 1 2





sin2 φ



(5.100)

sin2 φ + a22 γ /2

Rewriting the integral in brackets as
0




sin2 θ

θu2

sin θ + c12 (φ) 2

θu2

dθ = θu2 − 0

c12 (φ) dθ sin2 θ + c12 (φ)

(5.101)

and then making use of Eq. (2.562.1) of Ref. 2, we obtain


θu2 0

sin2 θ

1 dθ = 2 π sin θ + c12 (φ)

 θu2 −

c12 (φ) tan−1 1 + c12 (φ)



 1 + c12 (φ) tan θu2 c12 (φ) (5.102)

and hence  2  
θu1 1 2 c12 (φ) π a12 γ 0

   c12 (φ) 1 + c12 (φ) −1 tan tan θu2 dφ × θu2 − 1 + c12 (φ) c12 (φ) (5.103) Note that it is also possible to obtain a closed-form expression for K2 (a, γ ) as a special case of (5.103). In particular, since K2 (a, γ ) = L2 ((M − 1)π /M, (M − 1)π / M, a, a, γ ), then this special case evaluates as L2 (θu1 , θu2 , a1 , a2 , γ ) =


(M−1)π/M  2  2 1 K2 (a, γ ) = c (φ) π a2γ 0

   c (φ) 1 + c (φ) (M − 1) π (M − 1) π −1 − tan tan × dφ M 1 + c (φ) c (φ) M (5.104) where c (φ) is defined as a2γ c (φ) = 2 



sin2 φ sin2 φ + a 2 γ /2

 (5.105)

The other special cases that will be of interest in later chapters dealing with differentially encoded, coherently detected M -PSK are L2 (θ+ , θ+ , a+ , a+ ), L2 (θ− , θ− ,

148

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE





2 = 2 sin2 a− , a− ), and L2 (θ+ , θ− , a+ , a− ), where θ± = π (1 − (2k ± 1) /M) , a± ± 1) π /M, k = 0, 1, 2, . . . , M − 1. These special cases of (5.103) evaluate as (2k

L2 (θ± , θ± , a± , a± )    
π(1−(2k±1)/M)  2  2 c± (φ) 2k ± 1 1 − c± (φ) π 1 − = 2 π M 1 + c± (φ) a± γ 0      1 + c± (φ) 2k ± 1 × tan−1 tan π 1 − dφ (5.106) c± (φ) M and L2 (θ+ , θ− , a+ , a− )    
π(1−(2k+1)/M)  2  2 c+− (φ) 2k − 1 1 − c+− (φ) π 1 − = 2 π M 1 + c+− (φ) a+ γ 0      2k − 1 1 + c+− (φ) tan π 1 − × tan−1 dφ (5.107) c+− (φ) M where a2 γ c± (φ) = ± 2 



sin2 φ 2 sin2 φ + a± γ /2

 ,

a2 γ c+− (φ) = + 2 



sin2 φ



2 sin2 φ + a− γ /2 (5.108)

REFERENCES 1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York, NY: Dover Press, 1972. 2. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1994. 3. A. Erdelyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Table of Integral Transforms, vol. 1. New York, NY: McGraw-Hill, 1954. 4. J. W. Craig, “A new, simple and exact result for calculating the probability of error for two-dimensional signal constellations,” IEEE MILCOM’91 Conf. Rec., Boston, MA, pp. 25.5.1–25.5.5. 5. T. Eng and L. B. Milstein, “Coherent DS-CDMA performance in Nakagami multipath fading,” IEEE Trans. Commun., vol. 43, no. 2/3/4, February/March/April 1995, pp. 1134–1143. 6. J. Proakis, Digital Communications, 3rd ed. New York, NY: McGraw-Hill, 1995. 7. S. Chennakeshu and J. B. Anderson, “Error rates for Rayleigh fading multichannel reception of MPSK signals,” IEEE Trans. Commun., vol. 43, February/March/April 1995, pp. 338–346.

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

149

8. J. Edwards, A Treatise on the Integral Calculus,Vol. II. London, U.K.: Macmillan, 1922. 9. E. Villier, “Performance analysis of optimum combining with multiple interferers in flat Rayleigh fading,” IEEE Trans. Commun., vol. 47, October 1999, pp. 1503–1510. 10. A. Nuttall, Some Integrals Involving the Q-Function, Technical Report 4297, Naval Underwater Systems Center, New London, CT, April 17, 1972. 11. A. Nuttall, Some Integrals Involving the Q M -Function, Technical Report 4755, Naval Underwater Systems Center, New London, CT, May 15, 1974. 12. M. K. Simon and M.-S. Alouini, “Some new results for integrals involving the generalized Marcum Q-function and their application to performance evaluation over fading channels,” IEEE Trans. Wireless Commun., vol. 2, no. 4, July 2003, pp. 611–615. 13. A. Annamalai, C. Tellambura, and V. K. Bhargava, “A general method for calculating error probabilities over fading channels,” Proc. IEEE Int. Conf. Commun. (ICC ‘00 ), New Orleans, LA, June 2000. 14. D. Lao and A. Haimovich, “Multiple-symbol differential detection with interference suppression,” Proc. 2001 Conf. Inform. Sciences and Systems, Johns Hopkins Univ., Baltimore, MD, March 2001, vol. 2, pp. 850–855.

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS ASSOCIATED WITH RAYLEIGH AND NAKAGAMI-m FADING 5A.1

Exact Closed-Form Results

1. 1 π






π/2

sin2 θ

m

sin2 θ + c

0

dθ

We wish to consider evaluating the integral 1 Im = π




π/2

0



sin2 θ sin2 θ + c

m dθ

(5A.1)

for m both integer and noninteger. To do this, we shall make an equivalence with another definite integral for which closed-form results have been reported in the literature. In particular, it has been shown [5, App. A, Eq. (A8)] that the integral
∞ √  am  Jm (a, b) = e−at t m−1 Q bt dt, n≥0 (5A.2)  (m) 0 has the closed-form result √ 

Jm (a, b) = Jm (c) = 

c=

b , 2a

c/π 1

2 (1 + c)m+ 2 m noninteger

     m + 12 1 1 1, m + ; m + 1; , F 2 1  (m + 1) 2 1+c (5A.3)

150

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

When m is restricted to positive integer values, it has been further shown [5, App. A, Eq. (A13)] that (5A.3) simplifies to   m−1   2k   1 − µ2 (c) k 1  , Jm (a, b) = Jm (c) = 1 − µ (c) k 2 4 k=0  c  , m integer (5A.4a) µ (c) = 1+c which was also previously obtained by Eq. (14-4-15) in Ref. 6 in the form  Jm (c) =

1 − µ (c) 2

m m−1  k=0

m−1+k k



1 + µ (c) 2

k ,

m integer

(5A.4b) Using the alternative representation of the Gaussian Q-function [as given in Eq. (4.1.2) of Chapter 4] in (5A.2) gives 
π/2 
∞ 1 am − bt2 −at m−1 2 sin θ e t e dθ dt Jm (a, b) =  (m) 0 π 0 (5A.5)  
π/2
∞ b am m−1 − a+ 2 sin2 θ t t e dt dθ = π  (m) 0 0 The inner integral on t can be expressed in terms of the integral definition of the gamma function [1, Eq. (6.1.1)]:
∞ m t m−1 e−αt dt (5A.6)  (m) = α 0

Thus, using (5A.6) in (5A.5), we obtain
π/2
(m) 1 1 π/2 am Jm (a, b) =  m dθ =  m dθ (5A.7) π (m) 0 π 0 b a + 2 sinb 2 θ 1 + 2a sin 2θ Finally, letting c = b/2a, we can rewrite (5A.7) as m 
1 π/2 sin2 θ  Jm (a, b) = Jm (c) = dθ π 0 sin2 θ + c

(5A.8)

which is identical with Im of (5A.1). Thus, equating (5A.8) with (5A.3) and (5A.4) establishes the desired results for m noninteger and m integer, respectively. One final note is to observe from (5A.4) that J1 (c) = (1 − µ(c)) /2. Thus, a special case of (5A.8) that is of interest on Rayleigh channels is   
c 1 1 π/2 sin2 θ 1 − (5A.9) dθ = π 0 2 1+c sin2 θ + c

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

151

which could also be obtained directly as follows: 1 π




π/2

sin2 θ sin2 θ + c

0

dθ =

1 π




π/2

0

1 1 = − 2 π

 1−




dθ

sin2 θ + c

π/2

c sin2 θ + c

0



c

(5A.10)

dθ

Making use of the definite integral of Ref. 2 [p. 185, Eq. (2.562.1)], we arrive at 1 π




π/2

0

0  0 c sin2 θ 1 1+c 1 −1 tan tan θ 00π/2 dθ = − 0 2 2 π c (1 + c) c sin θ + c    c 1  = P (c) (5A.11) 1− = 2 1+c

This alternative derivation has been included because it is useful in deriving closedform results for two other integrals of interest related to evaluating the performance of QAM and M-PSK over Rayleigh channels. In particular, for QAM we will have a need to evaluate 1 π




π/4 0

sin2 θ sin2 θ + c

dθ =

1 π




π/4

0

1 1 = − 4 π

 1−




sin2 θ + c

π/4

c sin2 θ + c

0



c

dθ (5A.12)

dθ

Making use of the same indefinite integral as used in (5A.11), we immediately arrive at the desired result:

0 
0 1 π/4 sin2 θ c 1 1 + c 1 tan−1 tan θ 00π/4 dθ = − 0 2 π 0 4 π c + c) c (1 sin θ + c      c 1+c 1 4 −1 tan = 1− (5A.13) 4 1+c π c 2. 1 π




(M−1)π/M

sin2 θ sin2 θ + c

0

dθ

For M-PSK, we will need to evaluate 1 π




(M−1)π/M 0

sin2 θ sin θ + c 2

dθ =

1 M −1 − M π


0

(M−1)π/M

c sin2 θ + c

dθ

(5A.14)

152

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

Making use of the same indefinite integral as used in (5A.11), we immediately arrive at the desired result: 1 π




(M−1)π/M

sin2 θ

dθ sin2 θ + c         M −1 M c 1+c (M − 1) π −1 = tan tan 1− M 1 + c (M − 1) π c M        M π c c π M −1 1− + tan−1 cot = M 1 + c (M − 1) π 2 1+c M (5A.15) 0

3. 1 π




(M−1)π/M



sin2 θ

m

sin2 θ + c

0

dθ

For evaluation of symbol error probability corresponding to single-channel reception of M-PSK on Nakagami-m fading channels and also for multichannel reception of M-PSK on Rayleigh fading channels, we must evaluate 1 Km = π




(M−1)π/M



sin2 θ sin2 θ + c

0

m dθ

(5A.16)

Using a result [viz., Eq. (21)] from Ref. 7 for the symbol error probability performance of M-PSK over a Rayleigh channel with multichannel reception, it is straightforward to show that for m integer 1 π




(M−1)π/M 0



sin2 θ sin2 θ + c

m dθ

# π  m−1   2k  c 1 + tan−1 α k 1+c 2 (4 (1 + c))k k=0 $ k   −1  m−1 Tik %  −1 &2(k−i)+1 cos tan α + sin tan α (1 + c)k k=1 i=1

1 M −1 − = M π



(5A.17)

where 

α=



π c cot 1+c M

(5A.18)

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

153

and  

Tik = 

2(k−i)



2k



k

(5A.19)

4i [2 (k − i) + 1]

k−i

For m = 1, (5A.17) reduces to (5A.15). 4. 1 π




π/4



sin2 θ

m

sin2 θ + c

0

dθ

For evaluation of symbol error probability corresponding to single-channel reception of QAM on Nakagami-m fading channels and also for multichannel reception of QAM on Rayleigh fading channels, we need to evaluate 1 Lm = π




π/4 0



sin2 θ sin2 θ + c

m dθ

(5A.20)

Using a result [viz., Eq. (18)] from Ref. 7 with θU = (M + 1)π /M and θL = (M − 1)π /M, it is straightforward to show that for m integer 1 π




π/M 0

=



sin2 θ sin2 θ + c

m dθ



 m−1   2k  c / π 1 − tan−1 α k 1+c 2 (4 (1 + c))k k=0 $ k   −1  m−1 Tik %  −1 &2(k−i)+1 cos tan α − sin tan α (1 + c)k k=1 i=1 1 1 − M π

(5A.21)

where α and Tik are as evaluated in (5A.18) and (5A.19), respectively. Let√ ting M = 4 in (5A.21) whereupon α = c/ (1 + c) gives the desired result in (5A.20). Finally, for exact evaluation of bit error probability corresponding to singlechannel reception of M-PSK on Nakagami-m fading channels and also for multichannel reception of M-PSK on Rayleigh fading channels, we have to evaluate integrals of the form in (5A.17) or (5A.21) but with upper limits given by π (1− (2k ± 1)/M) for k = 1, 2, . . . , M − 1. What is needed to evaluate the

154

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

above-mentioned bit error probabilities is the difference of specific pairs of these integrals that can be related to the generic closed-form result given in Eq. (18) of Ref. 7. Specifically, it can be shown that Im (θU , θL ; K) 1 = 2π




π−θL



sin θ + 2

0

m

sin2 θ µ2L

1 dθ − 2π




π−θU



sin θ + 2

0

m

sin2 θ

dθ =

µ2U

/ π  m−1   2k  1 1 −1 + βU + tan αU   k k 2π 2 4 1 + µ2U k=0 

−1

+ sin tan

αU

k   m−1



k=1 i=1

Tik

%  −1 &2(k−i)+1  cos tan αU 2 k



+ sin tan−1 αL

k   m−1



k=1 i=1

$

1 + µU

/ π  m−1   2k  1 1 −1 βL + tan αL −   k k 2π 2 4 1 + µ2L k=0 %

Tik 1 + µ2L

(5A.22)

$



&2(k−i)+1

,

αL = βL cot θL ,

,

αU = βU cot θU

−1 k cos tan αL

θU − θL 2π

where  

µL =  

µU =

µL

K sin θL , m

βL = )

K sin θU , m

βU = )







1 + µ2L µU 1+



(5A.23)

µ2U

with K a constant. Our interest will be in the case where θU = (2k + 1) π /M, θL = (2k − 1) π /M, and K is related to signal-to-noise ratio. Alternatively for  √ θ)U = (M + 1)π /M, θL = (M − 1)π /M, then µL = −µU = c, βL = −βU =    c/ 1 + c2 , and αL = αU = α in which case (5A.22) immediately simplifies to (5A.21). 5. 1 π




φ 0



sin2 θ sin2 θ + c

m dθ

Interestingly enough, a closed-form expression for the integral in (5A.16) or (5A.21) with an arbitrary upper limit, say, φ, can be obtained from (5A.22). In particular,

155

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

setting θL = π − φ and θU = π , whereupon the second integral in (5A.22) disappears, we arrive at the result # m
 π  1 φ 1 φ sin2 θ −1 − β + tan Im (φ; c) = dθ = α π 0 π π 2 sin2 θ + c m−1 



  1 + sin tan−1 α k (4 (1 + c)) k=0 $ m−1 k  Tik %  −1 &2(k−i)+1 × cos tan α , (1 + c)k k=1 i=1

×

2k k

where 

β=



c sgn φ, 1+c



α = −β cot φ

(5A.24)

−π ≤ φ ≤ π

(5A.25)

Clearly (5A.24) reduces to (5A.16) and (5A.21) when φ = (M − 1) π /M and φ = π /M, respectively. Another closed form for the integral in (5A.24) has been suggested to the authors by R. F. Pawula and is readily derived using a clever change of variables due to Euler and Legendre [8, p. 316]. Although this alternative closed form is quite similar in structure to (5A.24) and therefore does not offer a significant computational advantage, it is nevertheless worth documenting because of the elegance associated with its derivation and the simplicity with which the final result is obtained relative to that employed in arriving at (5A.24). To begin, we first employ simple trigonometry to convert the integral to a slightly different form as follows m m

 1 − cos 2θ 1 φ 1 φ sin2 θ dθ = dθ Im (φ; c) = π 0 π 0 1 + 2c − cos 2θ sin2 θ + c (5A.26) 
2φ  1 − cos ξ m 1 dξ = 2π (1 + 2c)m 0 1 − d cos ξ 

where d = 1/ (1 + 2c). Next, employing the Euler–Legendre change of variables √

1 − d2 , 1 − d cos ξ = 1 + d cos x

dξ =

1 − d2 dx 1 + d cos x

(5A.27)

then, after some algebraic and trigonometric manipulation, we obtain the form Im (φ; c) =

√ d c 2m π (1 + c)m−1/2


0

xmax

(1 − cos x)m dx 1 + d cos x

(5A.28)

156

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

where tan xmax

√ √ 2 c (1 + c) sin 2φ 1 − d 2 sin 2φ = = cos 2φ − d (1 + 2c) cos 2φ − 1

(5A.29)

Finally, letting x = 2t and taking care to ensure that xmax as derived from (5A.29) is interpreted in the four-quadrant arctangent sense, we get the simpler integral form √ Im (φ; c) =




sin2m t dt c + cos2 t

(5A.30)

   π 1+D + 1−N 2 2

(5A.31)

c

π (1 + c)

m−1/2

T 0

where 

xmax 1 T = = tan−1 2 2

N D



with " N = 2 c (1 + c) sin 2φ,

D = (1 + 2c) cos 2φ − 1

(5A.32)

The integral form of (5A.30) is valid for m integer as well as m noninteger but is restricted to values of φ [the upper limit in the integral of (5A.26)] between zero and π . Later on, after obtaining the desired closed-form result, we will show how to remove this restriction. To obtain the closed form of (5A.30), we use the well-known geometric series 3m−1 k m k=0 x = (1 − x ) / (1 − x) to rewrite this equation as 1 Im (φ; c) = π 1 = π

 

c 1+c c 1+c




T

  1 − 1 − a 2m sin2m t 1 − a 2 sin2 t

0




T

0

1

1 dt − 2 2 π 1 − a sin t

dt




m−1 c  2k T a sin2k t dt 1+c 0 k=0

(5A.33) 1/ (1 + c). The first term is the original integral when m = 0 and thus where from (5A.26) must be equal to φ/π . The second integral is available in Eq. [2.513 (1)] of Ref. 2:  a2 =




T 0

sin2k t dt =

T 22k



2k k



% &   k−1 (−1)k  2k sin (2k − 2j ) T j + 2k−1 (−1) j 2 2k − 2j j =0

(5A.34)

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

157

Combining these two results and simplifying gives the alternative closed-form result φ T Im (φ; c) = − π π 2 − π





m−1   1 c  2k k [4 (1 + c)]k 1+c k=0

% & m−1 k−1   c   2k (−1)j +k sin (2k − 2j ) T , 0≤φ≤π j [4 (1 + c)]k 1+c 2k − 2j k=0 j =0

(5A.35) To extend this result so as to apply for upper integration limits in the region π ≤ φ ≤ 2π , we proceed as follows. First we partition the integral in (5A.26) as m m

 1 π 1 φ sin2 θ sin2 θ Im (φ; c) = dθ = dθ π 0 π 0 c + sin2 θ c + sin2 θ (5A.36) m
 sin2 θ 1 φ + dθ π π c + sin2 θ In the second integral we make the change of variables θ  = θ − π . Then m  m

sin2 θ sin2 θ  1 φ−π 1 π Im (φ; c) = dθ + dθ  (5A.37) 2 2  π 0 π c + sin θ c + sin θ 0 The second integral in (5A.37) can be evaluated using (5A.35) with φ replaced φ − π . For the first integral we have to first evaluate T in the limit when φ = π and then use (5A.35). Since φ approaches π from below, it is straightforward to show that the first term of (5A.31) will be zero and the second term will approach π . Thus, limφ→π T = π . Using this value of T in (5A.35), the double sum evaluates to zero and hence the first integral above becomes m  
 m−1  sin2 θ 1 1 π c  2k dθ = 1 − (5A.38) k π 0 1+c [4 + c)]k (1 c + sin2 θ k=0 Thus, when π ≤ φ ≤ 2π , the final result can be written as  Im (φ; c) = 1 −

 m−1  c  2k 1 k 1+c [4 + c)]k (1 k=0 

 m−1  c  2k 1 (5A.39) k 1+c [4 (1 + c)]k k=0 & %   m−1 k−1  c   2k 2 (−1)j +k sin (2k − 2j ) T  − j π 1+c 2k − 2j [4 (1 + c)]k T φ−π − + π π

k=0 j =0

158

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

where T  is T evaluated with φ replaced by φ − π . However, because of the periodicity of T with respect to the 2φ process, we have T  = T . Thus, the final result is     m−1  T c  2k c 2 φ 1 Im (φ; c) = − 1 + − k k π π 1+c π 1 + c [4 (1 + c)] k=0 ×

m−1 k−1  



2k j

k=0 j =0

% & (−1)j +k sin (2k − 2j ) T , 2k − 2j [4 (1 + c)]k

π ≤ φ ≤ 2π (5A.40)

or combining this with (5A.35), we obtain 

 m−1  c  2k 1 k 1+c [4 (1 + c)]k k=0 % &   m−1 k−1  c   2k 2 (−1)j +k sin (2k − 2j ) T , − j π 1+c 2k − 2j [4 (1 + c)]k

φ Im (φ; c) = − π



1 + (φ − π) T + 2 π

k=0 j =0

0 ≤ φ ≤ 2π

6. 1 π




φ



(5A.41)

sin2 θ

m 

sin2 θ + c1

0

sin2 θ



sin2 θ + c2

dθ

In the study of generalized diversity selection combining to be discussed in Chapter 9, we will need to evaluate an extension of the integral in (5A.24), 1 Im (φ; c1 , c2 ) = π




φ 0



m 

sin2 θ sin2 θ + c1

sin2 θ sin2 θ + c2

 dθ

(5A.42)

where, in general c1 = c2 . Since a closed form for such an integral cannot be obtained from the results of Ref. 7 or, for that matter, from any other reported contributions, we once again turn to the method suggested by Pawula for arriving at the alternative closed form for Im (φ; c) given in (5A.35) but instead apply it now to (5A.42). In particular, following steps analogous to (5A.26)–(5A.30), it is straightforward to show that Im (φ; c1 , c2 ) =

√ c1 (1 − d1 ) d2 π (1 + c1 )m−1/2 (d1 − d2 )




T1 0



sin2(m+1) t c1 + cos2 t



 1 dt D + cos2 t (5A.43)

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

159



where, as before, di = 1/ (1 + 2ci ) , i = 1, 2 and now also 

D=

1 − d1 d2 − d1 + d2 2 (d1 − d2 )

(5A.44)

In addition, T1 corresponds to T of (5A.31) with c replaced by c1 . Now, using the same geometric series manipulation as in (5A.33), we can rewrite (5A.43) as   2(m+1) √ 2(m+1) 2
T1 1 − 1 − a1 sin t c1 (1 + c1 ) (1 − d1 ) d2 b1    dt Im (φ; c1 , c2 ) = π (d1 − d2 ) 1 − a12 sin2 t 1 − b12 sin2 t 0 (5A.45) 

where, as before a12 = 1/ (1 + c1 ) and now in addition 

b12 =

1 2 (d1 − d2 ) c2 − c1 = = 1+D 1 + d1 − d2 − d1 d2 c2 (1 + c1 )

(5A.46)

Expanding the integrand of (5A.45) into a partial fraction expansion and evaluating the fractional coefficient in front of the integral purely in terms of c1 and c2 , we obtain after considerable algebraic simplification   
T1 1 − 1 − a12m sin2m t c1 1 Im (φ; c1 , c2 ) = dt π 1 + c1 0 1 − a12 sin2 t (5A.47)    m
T1  1 − 1 − b12m sin2m t c2 c1 1 dt − π 1 + c1 c2 − c1 1 − b12 sin2 t 0 Comparing the first term of (5A.47) with (5A.33), we immediately see that    m
T1  1 − 1 − b12m sin2m t c2 c1 1 Im (φ; c1 , c2 ) = Im (φ; c1 ) − dt π 1 + c1 c2 − c1 1 − b12 sin2 t 0 (5A.48) which indicates that the second term in (5A.48) accounts for the additional factor in the integrand of Im (φ; c1 , c2 ) that is not present in the integrand of Im (φ; c1 ). Since for c1 = c2 , we have from (5A.46) that b12 = 0, then, writing the second term of (5A.48) as 1 π

 m
T1 2m 2m b1 sin t c2 c1 dt 1 + c1 c2 − c1 1 − b12 sin2 t 0 
T1 c1 1 sin2m t 1 dt = m π 1 + c1 (1 + c1 ) 0 1 − b12 sin2 t 
T1 1 c1 1 = sin2m t dt π 1 + c1 (1 + c1 )m 0



(5A.49)

160

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

and using (5A.34), we obtain  m
T1 2m 2m  b1 sin t c2 c1 1 dt π 1 + c1 c2 − c1 1 − b12 sin2 t 0    c1 T1 1 2m = m [4 (1 + c1 )]m π 1 + c1 & %   m−1  c1  2m 2 (−1)j +m sin (2m − 2j ) T1 − j [4 (1 + c1 )]m π 1 + c1 2m − 2j

(5A.50)

j =0

Substituting (5A.50) into (5A.48) and recognizing the form of Im (φ; c) in (5A.35), we immediately see that for c1 = c2 Im (φ; c1 , c1 ) = Im+1 (φ; c1 )

(5A.51)

as it should from the definition of Im (φ; c1 , c2 ) in (5A.42). For the case c1 = c2 , we return to the form in (5A.48) and analogous to (5A.33) partition it into two integrals: m   
T1 1 c1 1 c2 Im (φ; c1 , c2 ) = Im (φ; c1 ) − dt c2 − c1 π 1 + c1 0 1 − b12 sin2 t  
m−1 c1  2k T1 2k 1 b1 sin t dt (5A.52) − π 1 + c1 0 k=0

The first integral in (5A.52) can be evaluated by first noting from (5A.47) that 
 sin2 θ 1 φ I0 (φ; c1 , c2 ) = dθ = I1 (φ; c2 ) π 0 sin2 θ + c2  
T1
T1 1 c1 1 c1 1 1 = dt − dt π 1 + c1 0 1 − a12 sin2 t π 1 + c1 0 1 − b12 sin2 t 
T1 φ c1 1 1 = − dt (5A.53) π π 1 + c1 0 1 − b12 sin2 t Evaluating I1 (φ; c2 ) from (5A.35) as T2 φ I1 (φ; c2 ) = − π π



c2 1 + c2

(5A.54)

where T2 now corresponds to T of (5A.31) with c replaced by c2 , then, combining (5A.53) and (5A.54), we get  
T1 c1 1 c2 T2 1 (5A.55) dt = 2 2 π 1 + c1 0 1 − b1 sin t π 1 + c2

161

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

The second integral of (5A.52) is evaluated as before using (5A.34). Without further ado we present the desired closed-form result for Im (φ; c1 , c2 ):  m c2 c2 1 + c2 c2 − c1 m−k    m−1  c2 c1  T1 1 2k + k π 1 + c1 c2 − c1 [4 (1 + c1 )]k k=0

Im (φ; c1 , c2 ) = Im (φ; c1 ) −

2 + π



T2 π



m−k   m−1 k−1  c2 c1   2k j 1 + c1 c2 − c1

(5A.56)

k=0 j =0

% & (−1)j +k sin (2k − 2j ) T1 , × 2k − 2j [4 (1 + c1 )]k

0≤φ≤π

To extend the range of coverage of the upper integration limit from 0 ≤ φ ≤ π to 0 ≤ φ ≤ 2π , we proceed as before and arrive at the final desired result  m  c2 c2 1 + (φ − π) T2 + 2 π 1 + c2 c2 − c1 m−k   m−1  c2 c1  1 + (φ − π) T1 + + 2 π 1 + c1 c2 − c1 

Im (φ; c1 , c2 ) = Im (φ; c1 ) −

k=0

m−k m−1 k−1  c2 c1   1 2 × + π 1 + c1 c2 − c1 [4 (1 + c1 )]k k=0 j =0 % &   (−1)j +k sin (2k − 2j ) T1 2k , 0 ≤ φ ≤ 2π × j 2k − 2j [4 (1 + c1 )]k (5A.57) where now Im (φ; c1 ) is evaluated from (5A.41). 

7. 1 π

2k k


0

π/2







sin2 θ sin2 θ + c1

m1 

sin2 θ sin2 θ + c2

m2 dθ

An extension of the previous integral wherein each of the two factors in the integrand is raised to an arbitrary power is of interest in the study of diversity (optimum) combining in the presence of interference (see Chapter 10 for a complete discussion of this topic). Unfortunately, it appears difficult to apply the previous derivation approaches to obtain a result for the most generic form of this integral where the powers are not necessarily restricted to be integer and the upper limit of the integral is arbitrary. However, for the case where the upper limit is equal to π /2 and the

162

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

powers are restricted to be integer, which is of interest in evaluating the average error probability performance of PSK with optimum combining over a Rayleigh fading channel, then, making an association with a closed-form result obtained by Villier [9], we present (without derivation) the following result 1 π




π/2



m1 

sin2 θ sin2 θ + c1  m2 −1

0

c1 c2

=

 2 1−

c1 c2

m2

sin2 θ

dθ

sin2 θ + c2

m1 +m2 −1

#m −1 2  c2

$ k  m1 −1 c1 k c1  − 1 Bk Ik (c2 ) − Ck Ik (c1 ) 1− c1 c2 c2

k=0

k=0

(5A.58) where10  

Bk = 

Ak m1 +m2 −1



Ck =

,



n=0

k



m2 −1

k n



m1 +m2 −1

 An ,

n

(5A.59)

 m2 4

k



Ak = (−1)m2 −1+k

m 2 −1

(m2 − 1)!

(m1 + m2 − n)

n=1 n=k+1

and  Ik (c) = 1 −

  k  c (2n − 1)!! 1+ 1+c n!2n (1 + c)n

(5A.60)

n=1

with the double factorial notation denoting the product of only odd integers from 1 to 2k − 1. It is straightforward (requiring, however, some tedious manipulations) to show that (5A.58) reduces to (5A.56) when m1 = m and m2 = 1. Also, by symmetry it can be shown that (5A.58) reduces to (5A.56) with c1 and c2 switched when m1 = 1 and m2 = m. 8. 1 π




φ 0

sin2m θ c + sin2 θ

dθ



10 Note

 k = 0 for n > k. Also, for m2 = 1, by convention the product n (m1 + m2 − n) = 1 and the only non-zero-valued coefficients are A0 = B0 = Ck = 1. For

that by convention

5m2 n=1 n=k+1

m2 > 1, the coefficients Ak , Bk and Ck clearly depend on both m1 and m2 .

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

163

Yet another integral that arises in the study of generalized diversity selection combining to be discussed in Chapter 9 is
φ sin2m θ  1 dθ (5A.61) Jm (φ; c) = π 0 c + sin2 θ This integral is similar in form to that in (5A.30) and can be evaluated by following an approach analogous to that used in arriving at the closed form in (5A.35). The procedure is as follows. Let a 2 = 1/c. Then
φ 2m 2m a sin θ 1 a2 dθ Jm (φ; c) = π a 2m 0 1 + a 2 sin2 θ  
φ 1 − 1 − a 2m sin2m θ 1 = dθ (5A.62) π a 2(m−1) 0 1 + a 2 sin2 θ  

φ φ 1 1 − a 2m sin2m θ 1 dθ − dθ = π a 2(m−1) 0 1 + a 2 sin2 θ 1 + a 2 sin2 θ 0   3 For l odd, li=0 (−1)i x i = 1 − x l+1 / (1 + x). Thus, letting x = a 2 sin2 φ, then for m even, we get  

φ m−1 φ  1 1 i 2i 2i sin θ dθ (5A.63) dθ − Jm (φ; c) = (−1) a π a 2(m−1) 0 1 + a 2 sin2 θ 0 i=0

Finally, using Eq. (2.562) due to Gradshteyn and Ryzhik [2] to evaluate the first integral,
φ "  1 1 −1 2 tan φ tan 1 + a (5A.64) dθ = √ 2 2 1 + a2 0 1 + a sin θ and (5A.34) for the second integral, we arrive at the desired result (for m even)  m−1  #  c 1 + c 1 cm−1 tan−1 tan φ − Jm (φ; c) = (−1)i i π 1+c c c i=0  % &      i−1  i  sin − 2j φ (2i ) φ (−1) 2i 2i  ×  2i + 2i−1 (−1)j i j  2 2 2i − 2j j =0

(5A.65) For m odd we slightly change the procedure. First rewriting (5A.62) as  
φ −1 + 1 + a 2m sin2m θ 1 dθ Jm (φ; c) = π a 2(m−1) 0 1 + a 2 sin2 θ  

φ φ 1 1 + a 2m sin2m θ 1 dθ + dθ − = 2 2 π a 2(m−1) 1 + a 2 sin2 θ 0 1 + a sin θ 0 (5A.66)

164

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

3l

then, noting that for l even Jm (φ; c) =

1 π a 2(m−1)

i=0 (−1)


−

φ

0

i

  x i = 1 + x l+1 / (1 + x), we obtain

1 1 + a 2 sin2 θ

dθ +

m−1 


i

(−1) a

i=0



φ

2i

2i

sin θ dθ 0

(5A.67) which is the negative of (5A.63). Thus, for arbitrary integer m, we have  m−1  # m−1  c 1+c 1 m c −1 Jm (φ; c) = (−1) tan tan φ − (−1)i i π 1+c c c i=0  % &      i−1  i  sin − 2j φ (2i ) φ (−1) 2i 2i  + 2i−1 ×  2i (−1)j i j  2 2 2i − 2j j =0

(5A.68) A special case of interest is when φ = π /2, in which case (5A.68) simplifies to    m−1 π  m−1  c c 1 2i (5A.69) Jm ; c = (−1)m − (−1)i 2i i i 2 2 1+c 2 c i=0

which reduces to (5A.9) when m = 1, as it should. 9. 1 π




L π/2 4 0

l=1



ml

sin2 θ sin2 θ + cl

dθ

Finally, we consider the generalization of the integral in paragraphs 7 of this appendix [(5A.58)–(5A.60) and related text] to an L-fold product of factors in the integrand where again the powers of each factor are restricted to be integer. The approach taken to obtain a closed-form result is to first express the product form of the integrand in a partial-fraction expansion whereupon the resulting terms can each be integrated using the result in (5A.4a). Applying the residue theorem, it is straightforward to show that  ml  k ml L L  4  sin2 θ sin2 θ = Akl (5A.70) sin2 θ + cl sin2 θ + cl l=1 l=1 k=1 where

  

L 5 d ml −k ml −k dx  n=1 

Akl =

n= l



 0 mn  0 1 0x=−c−1 1+cn x l   ml −k

(ml − k)!cl

(5A.71)

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

165

For the special case of m1 = m2 = · · · = mL = 1, (5A.70) and (5A.71) simplify to L 4





sin2 θ

=

sin2 θ + cl

l=1

L 

 A1l

l=1

sin2 θ

 (5A.72)

sin2 θ + cl

and A1l =

L  4

cl cl − cn

 (5A.73)

n=1 n=l

Now using the partial-fraction form of the integrand in (5A.70) together with (5A.4a), we obtain the desired result 1 π




L π/2 4



sin2 θ

ml

dθ sin2 θ + cl     L ml k   c 1  1 2j l  = Akl 1 − j 2 1 + cl [4 (1 + cl )]j l=1 k=1 j =0 0

l=1

(5A.74)

or for the special case of m1 = m2 = · · · = mL = 1 1 π




L π/2 4 0

l=1





sin2 θ

dθ =

sin2 θ + cl

  4  L  L  cl cl 1 1− 2 1 + cl cl − cn l=1

(5A.75)

n=1 n=l

Further extension of the results in (5A.74) in (5A.75) to integrals of the same form but with different upper limits can be obtained the same way but instead using the results in paragraphs 3–5 of this appendix [(5A.16)–(5A.41) and related text] to evaluate the resulting integrals in the partial-fraction expansion terms. 5A.2

Upper and Lower Bounds

Before concluding, we present some simple  upper and lower bounds on the integrals considered in this appendix. Since sin2 θ / sin2 θ + c is a monotonically increasing function of θ in the interval (0, π /2), an upper bound on its integral in the interval (0, φ) , φ ≤ π /2 can be found by setting sin θ = sin φ. Thus 1 π


0

φ



sin2 θ sin2 θ + c

m

φ dθ ≤ π



sin2 φ sin2 φ + c

m (5A.76)

166

USEFUL EXPRESSIONS FOR EVALUATING AVERAGE ERROR PROBABILITY PERFORMANCE

which for φ = π /2 becomes 1 π




π/2



m

sin2 θ

dθ ≤

sin θ + c 2

0

1 2 (1 + c)m

(5A.77)

Another upper bound can be obtained by setting sin θ = 0 in the denominator of the integrand, resulting in m  
 sin2 θ 1 φ 1 2m φ dθ ≤ 2m m m π 0 2 πc sin2 θ + c  (5A.78)   m−1  2m sin [(2m − 2k) φ] m+k +2 (−1) k 2m − 2k k=0

which for φ = π /2 simplifies to 1 π




φ



0

2

sin θ sin2 θ + c



m dθ ≤

2m m

 (5A.79)

22m+1 cm

To obtain a lower bound, we can set sin θ = 1 in the denominator of the integrand, resulting in   1 2m φ m 22m π (1 + c)m m   
φ m−1 2  [(2m sin sin − 2k) φ] θ 1 2m dθ ≤ +2 (−1)m+k k 2m − 2k π 0 sin2 θ + c k=0

(5A.80)

which for φ = π /2 simplifies to 

2m m



22m+1 (1 + c)m

1 ≤ π




φ 0



sin2 θ sin2 θ + c

m dθ

(5A.81)

For large c, the upper and lower bounds of (5A.78) and (5A.80) [or (5A.79) and (5A.81)], respectively, approach each other and thus are asymptotically tight whereas, for example, the bound in (5A.76) is asymptotically in error by a factor 

22m 2m m



For small values of c, the upper bound in (5A.76) is tighter than that in (5A.78) since the former approaches a finite value of φ/π in the limit of c → 0 whereas

APPENDIX 5A. EVALUATION OF DEFINITE INTEGRALS

167

the latter becomes unbounded. If desired one could use a composite of (5A.76) and (5A.78) to obtain the tightest upper bound. For example, for φ = π /2, the upper bounds of (5A.77) and (5A.79) cross each other at  c0 =

4−

2m m



1/m

2m m

1/m

(5A.82)

Thus, the composite upper bound would consist of (5A.77) for c ≤ c0 and (5A.79) for c > c0 . Finally, using the results found here together with the partial fraction expansion of (5A.70), it is straightforward to find upper and lower bounds on integrals having a product of integrands of the type in (5A.76).

6 NEW REPRESENTATIONS OF SOME PROBABILITY DENSITY AND CUMULATIVE DISTRIBUTION FUNCTIONS FOR CORRELATIVE FADING APPLICATIONS Later on in the text we shall have reason to study the performance of digital communication systems over correlative fading channels. Such channels occur, for example, in small-size terminals equipped with space antenna diversity where the antenna spacing is insufficient to provide independent fading among the various signal paths. In such instances, the received signal will consist of two or more replicas of the transmitted signal with fading amplitudes that are correlated random variables. In order to assess the performance of receivers of such signals, it is therefore necessary to study the joint statistics of correlated random variables with probability distributions characterized by the various fading channel models of Chapter 2. One important application of this scenario pertains to a system wherein the channel is assumed to be modeled by two paths and as such the receiver implements a diversity combiner with two branches. Evaluation of the performance of such a dual-diversity combining receiver (to be discussed in great detail in Chapter 9) requires, in general, knowledge of the two-dimensional (bivariate) fading amplitude PDF and CDF. For the specific case of selection combining (SC) [1, Sect. 10-4, p. 432], the combiner chooses the branch with the highest signal-to-noise ratio (or equivalently with the strongest signal assuming equal noise power among the branches) and outputs this signal to the threshold decision device. To evaluate performance in this instance, it is sufficient to obtain the one-dimensional PDF and CDF of the SC output which is tantamount to finding the PDF and CDF of the maximum of two correlated fading random variables. The SC output CDF is Digital Communication over Fading Channels, Second Edition. By Marvin K. Simon and Mohamed-Slim Alouini ISBN 0-471-64953-8 Copyright  2005 John Wiley & Sons, Inc.

169

170

NEW REPRESENTATIONS OF SOME PDF’s & CDF’s FOR CORRELATIVE FADING APPLICATIONS

used to evaluate so-called outage probability (the probability that neither SC input exceeds the detection threshold, or equivalently, the probability that the SC output falls below this threshold), while the SC output PDF is used to evaluate average error probability. In what follows, we shall focus on the Rayleigh, Nakagami-m, and log-normal fading channels since they are the most commonly used in digital communication system analyses and, as discussed previously, are typical of many wireless environments.

6.1

BIVARIATE RAYLEIGH PDF AND CDF

From a purely mathematical standpoint, the bivariate Rayleigh and Nakagami-m distributions can viewed as the joint statistics of the envelopes, R1 , R2 , of two correlated chi-square random variables of degrees 2 and 2m, respectively. Specifically, the bivariate Nakagami-m PDF is given by [1, Eq. (126); 3, Eq. (1)] pR1 ,R2 (r1 ,
1 ; r2 ,
2 |m, ρ ) =

  2  r1 r22 4mm+1 (r1 r2 )m m exp − + m−1
√ 1 − ρ
1
2  (m)
1
2 (1 − ρ)
1
2 ρ   √ 2m ρr1 r2 (6.1) × Im−1 √ , r1 , r2 ≥ 0
1
2 (1 − ρ)


 

 where
i = ri2 , i = 1, 2, and ρ = cov r12 , r22 / var r12 var r22 is the power correlation coefficient (0 ≤ ρ < 1).1 The special case of the bivariate Rayleigh PDF is given by [2, Eq. (122); 4, Eq. (3.7-13)] pR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ ) =

  2  r1 r2 4r1 r2 1 exp − + 2
1
2 (1 − ρ) 1 − ρ
1
2   √ 2 ρr1 r2 √ × I0 , r1 , r2 ≥ 0 (1 − ρ)
1
2

1 The

(6.2)

power correlation coefficient, ρ, of the two Rayleigh-distributed envelopes, R1 and R2 is related to the correlation coefficient, ρX1 X2 , of the underlying complex Gaussian RVs, X1 and X2 by  2 ρ = ρX1 X2 

where Ri = |Xi | , i = 1, 2 and

ρX1 X2 =

1 2E





∗

X1 − X 1 X2 − X 2 √ , varX1 varX2



varXi =

2

1  E Xi − X i  , 2

In Appendix 10B we shall generalize this relationship for correlated Rician RVs.

i = 1, 2

BIVARIATE RAYLEIGH PDF AND CDF

171

Tan and Beaulieu [3] were successful in finding infinite series representations of the CDFs corresponding to (6.1) and (6.2), in particular PR1 ,R2 (r1 ,
1 ; r2 ,
2 |m, ρ )     mr12 mr22 γ m + k, γ m + k, ∞
1 (1−ρ)
2 (1−ρ) (1 − ρ)m  k ρ =  (m) k! (m + k)

(6.3)

k=0



where γ (α, x) = Eq. (6.5.2)] and

x 0

e−t t α−1 dt, Re {α} > 0 is the incomplete gamma function [5,

PR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ ) = (1 − ρ)

∞  k=0

ρk

 × P k + 1,

   r22 r12 P k + 1,
1 (1 − ρ)
2 (1 − ρ)

(6.4) > 0 is another common form of where P (α, x) = (1/ (α)) 0 the incomplete gamma function [5, Eq. (6.5.3)]. Although (6.3) and (6.4) appear to have a simple structure, they have the drawback that because they are infinite series of the product of pairs of integrals, their computation requires truncation of the series. Bounds on the error resulting from this truncation along with empirical results for indicating the rate of convergence and tightness of the ensuing bounds are discussed by Tan and Beaulieu [3], who further point out that the complementary Rayleigh bivariate CDF (and thus also the Rayleigh bivariate CDF itself) had been previously expressed in terms of the Marcum Q-function [1, App. A]: x

α−1 e−t t dt, Re {α}

PR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ ) = 1 − Pr {R1 > r1 } − Pr {R2 > r2 } + Pr {R1 > r1 , R2 > r2 } 

    r12 r2 r1 2 2ρ Q1 = 1 − exp − √ , √
1 1 − ρ
2 1 − ρ
1 

    r22 r2 r1 2ρ 2 − exp − 1 − Q1 √ , √
2 1 − ρ
2 1 − ρ
1

(6.5)

Although Tan and Beaulieu [3] abandoned this result because of the lack of availability of the Marcum Q-function in standard distributions of such mathematical software packages as Maple V, MATLAB, and Mathematica, Simon and Alouini [6] recognized the value of (6.5) in terms of the desired form of the Marcum Q-function as described by (4.39) and (4.42). Indeed, as we shall soon see, this desired form of the Marcum Q-function allows the bivariate Rayleigh CDF to be similarly expressed as a single integral with finite limits and an integrand that includes a type of bivariate Gaussian PDF. This resulting form is simple and exact, and requires no special function evaluations (i.e., the integrand is entirely composed of elementary functions such as exponentials and trigonometrics).

172

NEW REPRESENTATIONS OF SOME PDF’s & CDF’s FOR CORRELATIVE FADING APPLICATIONS

Since the Marcum Q-function as represented by (4.39) and (4.42) depends on the relative values of its arguments, we must consider its use in (6.5) separately for different regions of the arguments r1 and r2 . For simplicity of notation, we shall also introduce the normalized (by the square root of the average power) envelope √  random variables Yi = ri /
i , i = 1, 2. √ r such that 2/[
2 (1 − ρ)]r2 < √ Consider first the region of r1 and √ 2 2ρ/[
1 (1 − ρ)]r1 or equivalently Y2 < ρY1 , which corresponds to the fact that the first argument is less than the second argument in the first Marcum Q-function √ in (6.5). Since in this region we would also have ρY2 < Y1 , then in the second Marcum Q-function in (6.5), the first argument is also less than the second argument. As such, we now substitute (4.39) in both of these two terms. After much simplification, one arrives at the desired result: PR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ )

  √  π Y12 + Y22 + 2 ρY1 Y2 sin θ 1 = 1 − exp + exp − (6.6) 2π −π 1−ρ  

 √ 1 − ρ 2 Y12 Y22 + ρ (1 − ρ) Y1 Y2 Y12 + Y22 sin θ 
 dθ ×
2 √ √ ρY1 + 2 ρY1 Y2 sin θ + Y22 Y12 + 2 ρY1 Y2 sin θ + ρY22 √ The complement of the region just considered is where Y2 > ρY1 or equivalently √ √ √ ρY2 > ρY1 . Here, however, we can have either ρY2 > Y1 or ρY1 < ρY2 < Y . Thus, two separate subcases must be considered. For the first subcase where √ √1 ρY2 > Y1 , we would certainly also have ρY2 > ρY1 , and thus for both Marcum Q-function terms in (6.5), the second argument is greater than the first argument. Thus, substituting Eq. (4.42) in both of these terms, we obtain after much simplifi
cation the identical
result of (6.6) except that the second term, namely, exp −Y22 ,  √ now becomes exp −Y12 . Finally, for the second subcase where ρY1 <
ρY2 < Y1 , 2 once again (6.6)
is appropriate 
with,  however, the second term, exp −Y2 , now 2 2 replaced by exp −Y1 + exp −Y2 . What remains is to evaluate the bivariate Rayleigh CDF at the endpoints between the regions where one must make use of the relation in (4.40). When this is done, √ the following results
are obtained for the second term in (6.6). When Y2 =
ρY1 , use    

√ 1 2 2 ρY2 , use 12 exp −Y22 + exp −ρY22 . 2 exp −Y1 + exp −ρY1 , and when Y1 = Summarizing, the bivariate Rayleigh can be expressed in the form of a single integral with finite limits and an integrand composed of elementary functions as follows:


−Y22



PR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ )

  √  π Y12 + Y22 + 2 ρY1 Y2 sin θ 1 = 1 − g (Y1 , Y2 |ρ ) + exp − 2π −π 1−ρ   

√ 1 − ρ 2 Y12 Y22 + ρ (1 − ρ) Y1 Y2 Y12 + Y22 sin θ 
 dθ , (6.7) ×
2 √ √ ρY1 + 2 ρY1 Y2 sin θ + Y22 Y12 + 2 ρY1 Y2 sin θ + ρY22 ri  Yi = √
i

BIVARIATE RAYLEIGH PDF AND CDF

173

where    exp −Y22 ,       2  1  −Y + exp −ρY12 ,   2 exp  2  2 1 exp −Y1 + exp −Y2 , g (Y1 , Y2 |ρ ) =    1 exp −Y 2  + exp −ρY 2  ,   2 2  2   exp −Y12 ,

√ 0 ≤ Y2 < ρY1 √ Y2 = ρY1 √ √ ρY1 < Y2 < Y1 / ρ √ Y2 = Y1 / ρ √ Y1 / ρ < Y2

(6.8)

At first glance, one might conclude from (6.8) that the bivariate CDF as given √ √ by (6.7) is discontinuous at the boundaries Y2 = ρY1 and Y2 = Y1 / ρ. Clearly this cannot be true since the Marcum Q-function itself is continuous over the entire range of both of its arguments and thus from the form in (6.5), the CDF must also be continuous over these same ranges. The explanation for this apparent discontinuity is that the integral portion of (6.7) is also discontinuous at these same boundaries but in such a way as to completely compensate for the discontinuities in g (Y1 , Y2 |ρ ) and thus produce a CDF that is continuous for all positive Y1 , Y2 . The bivariate Rayleigh CDF of Eq. (6.7) has been evaluated numerically using Mathematica and compared with the double-integral representation [3, Eqs. (1) and (2)], the infinite series representation [3, Eq. (4)] and Eq. (6.5) using direct evaluation of the Marcum Q-function. Both the infinite sum and the proposed integral representation have a significant speedup factor compared to the other two methods (double-integral approach and the one where Marcum-Q is evaluated numerically). Furthermore, the proposed approach always gives the exact result (up to the precision/accuracy allowed by the platform) whereas the infinite series representations (when programmed with the available Mathematica routines and setting the upper limit to infinity as allowed by Mathematica) loses its accuracy for high values of ρ such as 0.8 and 0.9 and a truncation of the series is required.2 Note that the number of terms for the truncation must be determined for each set of values of r1 , r2 , ρ. Tan and Beaulieu [3] derived a bound on the error resulting from truncation of the infinite series but reported that this bound becomes loose as ρ approaches 1, which we have verified is the case. An alternative simple form of the bivariate Rayleigh CDF can be obtained by substituting the representations of the first-order Marcum Q-function of (4.51) and (4.52) in (6.5). When this is done, then after considerable algebraic manipulation the following result is obtained
 √ PR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ ) = 1 − g (Y1 , Y2 |ρ ) + sgn Y2 − ρY1 I (Y1 , Y2 |ρ )
 √ + sgn Y1 − ρY2 I (Y2 , Y1 |ρ ) (6.9) 2 Note that the infinite series representation itself converges to the correct result for all values of ρ between zero and one. It is the limitation of the numerical evaluation of this series caused by the software used to make this evaluation that results in the loss of accuracy for large ρ.

174

NEW REPRESENTATIONS OF SOME PDF’s & CDF’s FOR CORRELATIVE FADING APPLICATIONS

where, analogous to (6.8), we obtain    √  exp −Y22 , 0 ≤ Y2 < ρY1    2  2 √ √ g (Y1 , Y2 |ρ ) = exp −Y1 + exp −Y2 , ρY1 ≤ Y2 < Y1 / ρ    exp −Y 2  , Y /√ρ ≤ Y 1 2 1

(6.10)

and I (Y1 , Y2 |ρ) =

     π 1 1 exp − Y12 + 4π −π 1−ρ


2  2 ρY1 − Y22 × √ Y22 + 2 ρY1 Y2 sin θ + ρY12

(6.11)

Note that the compensation for the discontinuities in g (Y1 , Y2 |ρ ) at the boundaries √ √ Y2 = ρY1 and Y2 = Y1 / ρ is now immediately obvious from the form of the last two terms in (6.9). Moreover, the values of the CDF at these endpoints are given as PR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ )     1 exp −Y12 − exp −Y22 2  
     π 2 2 2 1 − ρ Y 1 1 1 + exp − Y22 + , (6.12) 4π −π 1−ρ 1 + 2ρ sin θ + ρ 2 √ Y2 = ρY1 =1−

and PR1 ,R2 (r1 ,
1 ; r2 ,
2 |ρ )     1 exp −Y22 − exp −Y12 2  
2     π Y22 1 − ρ 2 1 1 2 + exp − Y1 + , (6.13) 4π −π 1−ρ 1 + 2ρ sin θ + ρ 2 √ Y1 = ρY2 =1−

One might anticipate that the bivariate Nakagami-m CDF could be expressed in a form analogous to (6.5) depending instead on the mth-order Marcum Q-function. If this were possible, then, using the desired form of the generalized Marcum Q-function as in (4.69) and (4.77), one could express the bivariate Nakagami-m CDF also in the desired form. Unfortunately, to the author’s knowledge an expression analogous to (6.5) has not been reported in the literature, and the authors have themselves been unable to arrive at one.

175

PDF AND CDF FOR MAXIMUM OF TWO RAYLEIGH RANDOM VARIABLES

6.2 PDF AND CDF FOR MAXIMUM OF TWO RAYLEIGH RANDOM VARIABLES In this section we consider the distributions of the random variable R = max (R1 , R2 ) where R1 , R2 are correlated Rayleigh random variables with joint PDF as in (6.2). As mentioned previously, the random variable R characterizes the output of an SC whose inputs are R1 and R2 . Since Pr {R ≤ R ∗ } = Pr{R1 ≤ R ∗ , R2 ≤ R ∗ }, then the CDF of R is immediately obtained from the joint CDF of R1 , R2 by equating its two arguments. Since we are ultimately interested in the PDF of the  instantaneous SNR per bit,3 γ = r 2 Eb /N0 with mean γ = r 2 Eb /N0 =
Eb /N0 , it is convenient for the Rayleigh case to start by renormalizing the bivariate CDF of  (6.7).4 Thus, noting that Yi2 = ri2 /
i = γi /γ i , i = 1, 2, then the joint CDF of γ1 , γ2 is given by 
Pγ1 ,γ2 γ1 , γ 1 ; γ2 , γ 2 |ρ


  = 1 − G H γ1 , γ 1 , H γ2 , γ 2 |ρ       γ1 γ2 γ1 γ2      π sin θ + + 2 ρ   γ1 γ2 γ1 γ2 1 + exp −   2π −π 1−ρ     

       √ γ1 γ2 γ1 γ2 1−ρ   γ1 γ 2 + ρ (1 − ρ) γ1 γ2 γ 1 + γ 2 sin θ   ×           dθ  γ1 γ1 γ2 γ2 γ1 γ1 γ2 γ2  ρ γ +2 ρ γ sin θ + γ sin θ +ρ γ γ γ +2 ρ γ γ


2

1

  γ1   γ2 

1

2

2

1

1

2

2

(6.14) where 
  H γ2 , γ 2 ,       1 H
γ , γ  +H
γ , γ  ,  1 2  1 2  2   

 
 

G H γ1 , γ 1 , H γ2 , γ 2 |ρ = H γ1 , γ 1 + H γ2 , γ 2 ,    
 1
  H γ2 , γ 2 +H γ1 , γ 1 ,    2   
   H γ1 , γ 1 ,

3 As

γ1 γ2 0 (6.22)

with


 −H γ , γ i , m = 

mm  (m) γ i



γ γi

m−1

  mγ , exp − γi

i = 1, 2

(6.23)

Also h1 (θ |ρ ) is still given by (6.17), which is independent of m and 

h (θ |ρ ) =

  1 γ 1 (m−1)/2 γm ργ 2 1  )
)
* ( *  −γ 1 cos (m − 1) θ + π2 + ργ 1 γ 2 cos m θ + π2 ( × ργ 2 + 2 ργ 1 γ 2 sin θ + γ 1 (6.24)   1 ργ 1 −(m−1)/2 + m γ2 γ2  * ( *  )
)
γ 2 cos (m − 1) θ + π2 − ργ 1 γ 2 cos m θ + π2 × ( γ 2 + 2 ργ 1 γ 2 sin θ + ργ 1

6 Note that the alternative representation of the generalized Marcum Q-function (m = 1) is valid only for ρ = 0.

PDF AND CDF FOR MAXIMUM OF TWO NAKAGAMI-m RANDOM VARIABLES

179

Note that for m = 1, (6.24) simplifies to     ( ( γ 1 + ργ 1 γ 2 sin θ γ 2 + ργ 1 γ 2 sin θ 1 1 h (θ |ρ ) = + ( ( γ 1 ργ 2 + 2 ργ 1 γ 2 sin θ + γ 1 γ 2 γ 2 + 2 ργ 1 γ 2 sin θ + ργ 1 (6.25) which can be shown to be equal to h
1 (θ |ρ ) h2 (θ
|ρ ) with h
2 (θ |ρ ) obtained from (6.17). Thus, also noting that −H  γ , γ i , 1 = 1/γ i exp −γ /γ i , i = 1, 2, the PDF of (6.24) reduces to (6.18) as it should. Note here that the dependence on γ of pγ (γ ) in (6.22) resembles the behavior of the instantaneous SNR per bit corresponding to a single Rayleigh RV, namely, pγ (γ ) = [mm γ m−1 / γ m  (m)] exp (−mγ / γ ). Because of this similarity, it is possible to draw an analogy with results for the average error probability performance of single-channel (no diversity) digital modulations transmitted over a Nakagami-m fading channel (see Chapter 8) that make use of the integrals developed in Sections 5.1.4 and 5.2.4 of Chapter 5 on the basis of the desired forms of the Gaussian and Marcum Q-functions. However, because of the additional integration on θ required by the second term in (6.18), the functional form of the results will be somewhat different. The CDF of the SC output can now be found directly by integration of (6.22) with the result (for ρ = 0) 
 

Pγ (γ ) = G −H γ , γ 1 , m , −H γ , γ 2 , m |ρ   π  γ mm 1 m−1 − y exp {−myh1 (θ |ρ )} dy h (θ |ρ ) dθ  (m) 2π −π 0


  = G −H γ , γ 1 , m , −H γ , γ 2 , m |ρ  π +  , mm 1 |ρ − , m h (θ |ρ ) dθ (h1 (θ |ρ ))−m −H γ , h−1 (θ ) 1  (m) 2π −π (6.26) where now  k mγ  m−1   
 γ 
γi mγ , i = 1, 2 −H  y, γ i , m dy = 1 − exp − −H γ , γ i , m = γ k! 0 i k=0

(6.27) For ρ = 0, the PDF of γ can be obtained from [7, Eq. (20)], which after some changes of variables becomes    mγ  m−1    m, m γ2 γ mγ  m  1− pγ (γ ) = exp −  (m) γ 1 γ 1 γ1  (m) +

mm  (m) γ 2



γ γ2

m−1

   mγ    m, γ1 mγ  , exp − 1− γ2  (m)

γ ≥0 (6.28)

180

NEW REPRESENTATIONS OF SOME PDF’s & CDF’s FOR CORRELATIVE FADING APPLICATIONS

∞ m−1 where  (m, x) = x e−t t dt is the complementary incomplete gamma function [5, Eq. (6.5.3)]. For m integer,  (m, x)has a closed-form expression [9, Eq. (8.352.2)] and (6.28) simplifies to pγ (γ ) =

mm (m − 1)!γ 1



mm + (m − 1)!γ 2

 
* mγ ) −H γ , γ 2 , m exp − γ1 m−1  
* γ mγ ) −H γ , γ 1 , m , exp − γ2 γ2

γ γ1 

m−1

(6.29) γ ≥0

The corresponding CDFs are obtained by integration of (6.28) and (6.29) between 0 and γ . For m noninteger, integration of (6.28) does not produce a closed-form result, whereas for m integer, integration of (6.29) results in 

Pγ (γ ) = −H γ , γ 1 , m − H γ , γ 2 , m 



  m−1  (n + m − 1)! γ 1 n γ 2 m + γ 1 m γ 2 n −
n+m n! (m − 1)! γ1 + γ2 n=0    γ 1γ 2 × −Hn γ , ,m γ1 + γ2

(6.30)

where analogous to (6.27)   m+n−1  mγ  −Hn (γ , γ , m) = 1 − exp − γ



k=0

mγ γ

k

k!

(6.31)

Note that −H0 (γ , γ , m) is equal to −H (γ , γ , m) of (6.27).

6.4 PDF AND CDF FOR MAXIMUM AND MINIMUM OF TWO LOG-NORMAL RANDOM VARIABLES Later on in the book, when discussing the bounds on and exact performance of dual-diversity combining over correlated log-normal channels in Chapter 9, we shall have need for the PDF and CDF of the maximum and minimum of two log-normal random variables. We consider the maximum case first. 6.4.1

The Maximum of Two Log-Normal Random Variables

Consider a pair of correlated log-normal RVs, γ1 and γ2 , whose bivariate PDF is the extension of the one-dimensional log-normal PDF in (2.43), specifically

PDF AND CDF FOR MAXIMUM AND MINIMUM OF TWO LOG-NORMAL RANDOM VARIABLES

181

   10 log10 γ1 − µ1 2 1 ξ2  pγ1 ,γ2 (γ1 , γ2 ) = exp −
( σ1 2 1 − ρ2 2π σ1 σ2 1 − ρ 2 γ1 γ2 2    10 log10 γ2 − µ2 10 log10 γ1 − µ1 + − 2ρ σ2 σ1  10 log10 γ2 − µ2 × (6.32) σ2 where, as before, ξ = 10/ ln 10 = 4.3429. Letting γ = max (γ1 , γ2 ), then the PDF of γ is directly evaluated from  γ  γ pγ1 ,γ2 (γ , γ2 ) dγ2 + pγ1 ,γ2 (γ1 , γ ) dγ1 (6.33) pγ (γ ) = 0

0

the value of γ in decibels
and introducing the Letting  = 10 log10 γ denote
changes of variables U1 = 10 log10 γ1 − µ1 /σ1 and U2 = 10 log10 γ2 − µ2 /σ2 , we can rewrite (6.33) as    (−µ2 )/σ2  − µ1 pU1 ,U2 , U2 dU2 pγ (γ ) = σ1 0 (6.34)    (−µ1 )/σ1  − µ2 dU1 + pU1 ,U2 U1 , σ2 0 where pU1 ,U2 (U1 , U2 ) is a normalized bivariate Gaussian PDF:   U12 + U22 − 2ρU1 U2 1 
exp − pU1 ,U2 (U1 , U2 ) = ( 2 1 − ρ2 2π 1 − ρ 2

(6.35)

Finally, substituting (6.35) into (6.34) and carrying out the integrations produces the desired result for the PDF of the maximum of two log-normal RVs:    −µ2 − ρ −µ1   1  − µ1 2 σ σ1 Q − 2( pγ (γ ) = exp − 2 2 σ1 1−ρ (6.36)     −µ1 − ρ −µ2  1  − µ2 2 σ1 σ2 + exp − Q − ( 2 σ2 1 − ρ2 The CDF, Pγ (γ ), for the maximum of two log-normal RVs can be obtained by integrating (6.36). However, a more direct approach to obtaining this result is to recognize that Pγ (γ ) = Pr {γ1 ≤ γ , γ2 ≤ γ } = 1 − Pr {γ1 ≥ γ } − Pr {γ2 ≥ γ } + Pr {γ1 ≥ γ , γ2 ≥ γ }

(6.37)

182

NEW REPRESENTATIONS OF SOME PDF’s & CDF’s FOR CORRELATIVE FADING APPLICATIONS

or equivalently Pγ (γ ) = Pr {1 ≤ , 2 ≤ } = 1 − Pr {1 ≥ } − Pr {2 ≥ } + Pr {1 ≥ , 2 ≥ }

(6.38)

where 1 = 10 log10 γ1 , 2 = 10 log10 γ2 are the values of γ1 , γ2 , respectively, in decibels. Since 1 and 2 are Gaussian RVs with means µ1 , µ2 and variances σ12 , σ22 , respectively, then the probabilities in (6.37) can be expressed in terms of the one- and two-dimensional Gaussian Q-functions, which gives        − µ2  − µ1  − µ2  − µ1 −Q +Q , ;ρ (6.39) Pγ (γ ) = 1 − Q σ1 σ2 σ1 σ2 Using an equivalent alternative form of the Craig representation of the onedimensional Gaussian Q-function in (4.2), namely    π x2 1 dθ , x≥0 (6.40) exp − Q (x) = 2π 0 2 sin2 θ and the Craig form of the two-dimensional Gaussian Q-function in (4.17), the CDF of (6.39) can be expressed as      π  − µ1 2 1 1

 √ Pγ (γ ) = 1 − exp − dθ 1−ρ 2 (−µ1 )/σ1 2π tan−1 σ1 2 sin2 θ (−µ2 )/σ2 −ρ (−µ1 )/σ1

1 − 2π



π



tan−1

√

1−ρ 2 (−µ2 )/σ2 (−µ1 )/σ1 −ρ (−µ2 )/σ2

 exp

−

1 2 sin2 θ



 − µ2 σ2

 ≥ µ1 ,  ≥ µ2

2  dθ , (6.41)

Using the properties Q (−x) = 1 − Q (x) Q (−x, y; ρ) = Q (y) − Q (x, y; −ρ) Q (x, −y; ρ) = Q (x) − Q (x, y; −ρ)

(6.42)

Q (−x, −y; ρ) = 1 − Q (x) − Q (y) + Q (x, y; ρ) the other three cases of (6.41) become

√   1−ρ 2 (−µ1 )/σ1     tan−1 (µ2 − )/σ2 +ρ (−µ1 )/σ1  − µ1 2 1 1 Pγ (γ ) = − exp − dθ 2π 0 σ1 2 sin2 θ      π µ2 −  2 1 1

 √ dθ , + exp − 1−ρ 2 (µ2 − )/σ2 2π tan−1 σ2 2 sin2 θ (−µ1 )/σ1 +ρ (µ2 − )/σ2

 ≥ µ1 ,  ≤ µ2

(6.43)

183

PDF AND CDF FOR MAXIMUM AND MINIMUM OF TWO LOG-NORMAL RANDOM VARIABLES

1 Pγ (γ ) = 2π





π

tan−1

1−ρ 2 (µ1 − )/σ1 (−µ2 )/σ2 +ρ (µ1 −)/σ1



1 − 2π

√

tan−1

 exp

√

1−ρ 2 (−µ2 )/σ2 (µ1 − )/σ1 +ρ (−µ2 )/σ2

−





1 2 sin2 θ

 exp −

0

µ1 −  σ1 

1 2 sin2 θ

2  dθ

 − µ2 σ2

2  dθ ,

 ≤ µ1 ,  ≥ µ2

1 Pγ (γ ) = 2π



tan−1

(6.44) √

1−ρ 2 (µ1 − )/σ1 (µ2 −)/σ2 −ρ (µ1 −)/σ1

0

1 + 2π



tan−1



 exp −

√

1−ρ 2 (µ2 − )/σ2 (µ1 − )/σ1 −ρ (µ2 − )/σ2





1 2 sin2 θ

 exp −

0

µ1 −  σ1 

1 2 sin2 θ

2 

µ2 −  σ2

dθ 2 

 ≤ µ1 ,  ≤ µ2

dθ , (6.45)

For the special case of µ1 = µ2 = µ and σ1 = σ2 = σ , (6.41) and (6.45) simplify to 1 Pγ (γ ) = 1 − π





π tan−1



1+ρ 1−ρ

 exp

−



1 2 sin2 θ

−µ σ

2   ≥ µ (6.46)

dθ ,

and 1 Pγ (γ ) = π 6.4.2



tan−1 0



1+ρ 1−ρ



 exp −

1 2 sin2 θ



µ− σ

2  dθ ,

≤µ

(6.47)

The Minimum of Two Log-Normal Random Variables

For the case where γ = min (γ1 , γ2 ), then, analogous to (6.37), the CDF is now given by Pγ (γ ) = Pr {γ1 or γ2 ≤ γ } = 1 − Pr {γ1 ≥ γ , γ2 ≥ γ }

(6.48)

or equivalently Pγ (γ ) = Pr {1 or 2 ≤ } = 1 − Pr {1 ≥ , 2 ≥ }

(6.49)

Once again using the alternative Craig form for the two-dimensional Gaussian Q-function, then the CDF corresponding to the four cases considered in (6.41) and

184

NEW REPRESENTATIONS OF SOME PDF’s & CDF’s FOR CORRELATIVE FADING APPLICATIONS

(6.43) through (6.45) now become 1 Pγ (γ ) = 1 − 2π 1 − 2π



tan−1

√

1−ρ 2 (−µ1 )/σ1 (−µ2 )/σ2 −ρ (−µ1 )/σ1



 exp −

0

√  1−ρ 2 (−µ2 )/σ2 tan−1 (−µ )/σ −ρ (−µ 1 1 2 )/σ2



 exp −

0



1 2 sin2 θ 

1

 − µ1 σ1

 − µ2 σ2

2 sin2 θ

2  dθ

2  dθ ,

 ≥ µ1 ,  ≥ µ2

−

1 2π



1 Pγ (γ ) = 1 + 2π

(6.50)

tan−1

√

0



π

tan−1

√

1−ρ 2 (µ2 − )/σ2 (−µ1 )/σ1 +ρ (µ2 − )/σ2





   − µ1 2 exp − dθ σ1 2 sin2 θ     µ2 −  2 1  exp − dθ , σ2 2 sin2 θ

1−ρ 2 (−µ1 )/σ1 (µ2 − )/σ2 +ρ (−µ1 )/σ1



1

 ≥ µ1 ,  ≤ µ2

−

1 2π



1 Pγ (γ ) = 1 + 2π

(6.51)

tan−1

√

0



π

tan−1

√

1−ρ 2 (−µ2 )/σ2 (µ1 − )/σ1 +ρ (−µ2 )/σ2





  µ1 −  2 exp − dθ σ1 2 sin2 θ      − µ2 2 1  exp − dθ , σ2 2 sin2 θ

1−ρ 2 (µ1 − )/σ1 (−µ2 )/σ2 +ρ (µ1 − )/σ1



1

 ≤ µ1 ,  ≥ µ2 1 Pγ (γ ) = 2π



1 + 2π



π

tan−1



(6.52) √

1−ρ 2 (µ1 − )/σ1 (µ2 −)/σ2 −ρ (µ1 −)/σ1

 exp

−

2 sin2 θ



π

tan−1

√

1−ρ 2 (µ2 − )/σ2 (µ1 − )/σ1 −ρ (µ2 − )/σ2

 exp



1

−

µ1 −  σ1 

1 2 sin2 θ

2 

µ2 −  σ2

 ≤ µ1 ,  ≤ µ2

dθ 2  dθ , (6.53)

For the special case of µ1 = µ2 = µ and σ1 = σ2 = σ , (6.50) and (6.53) simplify to 1 Pγ (γ ) = 1 − π

 0

tan−1



1+ρ 1−ρ



 exp −

1 2 sin2 θ



−µ σ

2  dθ ,

≥µ (6.54)

REFERENCES

185

and 1 Pγ (γ ) = π





π tan−1



1+ρ 1−ρ

 exp

−

1 2 sin2 θ



−µ σ

2  dθ ,

≤µ

(6.55)

REFERENCES 1. M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques, New York, NY: McGraw-Hill, 1966. 2. M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation, Oxford, U.K.: Pergamon Press, 1960, pp. 3–36. 3. C. C. Tan and N. C. Beaulieu, “Infinite series representation of the bivariate Rayleigh and Nakagami-m distributions,” IEEE Trans. Commun., vol. 45, no. 10, October 1997, pp. 1159–1161. 4. S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J., vol. 23, 1944, pp. 282–332; vol. 24, 1945, pp. 46–156. 5. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York, NY: Dover Press, 1972. 6. M. K. Simon and M.-S. Alouini, “A simple single integral representation of the bivariate Rayleigh distribution,” IEEE Commun. Lett., vol. 2, no. 5, May 1998, pp. 128–130. 7. G. Fedele, I. Izzo, and M. Tanda, “Dual diversity reception of M -ary DPSK signals over Nakagami fading channels,” IEEE Int. Symp. Personal, Indoor, and Mobile Radio Commun. Toronto, Canada, September 1995, pp. 1195–1201. 8. M. K. Simon and M.-S. Alouini, “A unified performance analysis of digital communication with dual selection combining diversity over correlated Rayleigh and Nakagami-m fading channels,” IEEE Trans. Commun., vol. 47, no. 1, January 1999, pp. 33–43; also presented in part in the IEEE Global Commun. Conf. (GLOBECOM’98 ), Sydney, Australia, November 8–12, 1998. 9. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1994. 10. M.-S. Alouini and M. K. Simon, “Dual diversity reception over correlated lognormal fading channels,” IEEE Trans. Commun., vol. 50, no. 12, December 2002, pp. 1946–1959; also presented in part in the IEEE Int. Conf. Commun. (ICC‘01 ), Helsinki, Finland, June 2001.

PART 3 OPTIMUM RECEPTION AND PERFORMANCE EVALUATION

7 OPTIMUM RECEIVERS FOR FADING CHANNELS

As far back as the 1950s, researchers and communication engineers recognized the need for investigating the form of receivers that would provide optimum detection of digital modulations transmitted over a channel composed of a combination of AWGN and multiplicative fading. For the most part, most of these contributions dealt with only the simplest of modulation/detection schemes and fading channels, namely, BPSK with coherent detection and Rayleigh or Rician fading. In some instances, the work pertained to single-channel reception, while in others multichannel reception was considered. Our goal in this chapter is to present the work of the past under a unified framework based on the maximum-likelihood approach and also to consider a larger number of situations corresponding to more sophisticated modulations, detection schemes, and fading channels. In addition, we shall treat a variety of combinations of channel state knowledge relating to the amplitude, phase, and delay parameter vectors associated with the fading channels. In many instances, implementation of the optimum structure may not be simple or even feasible, and thus a suboptimum solution is preferable and will be discussed. Also, evaluating the error probability performance of these optimum receivers may not always be possible to accomplish using the analytical tools previously discussed in this book or anywhere else, for that matter. Nevertheless, it is of interest to determine in each case the optimum receiver since it serves as a benchmark against which to measure the suboptimum but simpler-to-implement and- analyze structure. We begin our discussion by reviewing the mathematical models for the transmitted signal and generalized fading channel as introduced in previous chapters. In particular, consider that during a symbol period of Ts seconds the transmitter Digital Communication over Fading Channels, Second Edition. By Marvin K. Simon and Mohamed-Slim Alouini ISBN 0-471-64953-8 Copyright  2005 John Wiley & Sons, Inc.

189

190

OPTIMUM RECEIVERS FOR FADING CHANNELS

sends the real bandpass signal1
 sk (t) = Re {˜sk (t)} = Re S˜k (t) ej 2πfc t

(7.1)

where s˜k (t) is the kth complex bandpass signal and S˜k (t) is the corresponding kth complex baseband signal chosen from the set of M equiprobable message waveforms representing  the transmitted information. At this point, we do not restrict
˜ the signal set Sk (t) in any way, for instance, we do not require that the signals have equal energy, and thus we are able to handle all of the various modulation types discussed in Chapter 3. The signal of (7.1) is transmitted over the generalized fading channel, which is characterized by Lp independent paths, each of which is a slowly varying channel that attenuates, delays, and phase-shifts the signal and adds an AWGN noise source. Thus, the received signal is a set of noisy replicas of the transmitted signal,2
  rl (t) = Re αl s˜k (t − τl ) ej θl + n˜ l (t) = Re αl S˜k (t − τl ) ej (2πfc t+θl )  +N˜ l (t) ej 2πfc t (7.2)
 l = 1, 2, . . . , Lp = Re {˜rl (t)} = Re R˜ l (t) ej 2πfc t , Lp where {N˜ l (t)}l=1 is a set of statistically hertz independent3 complex AWGN proLp Lp Lp cesses, each with PSD 2Nl Watts/Hertz. The sets {αl }l=1 , {θl }l=1 , and {τl }l=1 are the random channel amplitudes, phases, and delays, respectively, which, because of the slow-fading assumption, are assumed to be constant over the transmission (symbol) interval Ts . Also, without loss of generality, we take the first channel to be the reference channel whose delay τ1 = 0 and further assume that the delays are ordered: τ1 < τ2 < · · · < τLp . The optimum receiver computes the set of a posteriori probabilities Lp  , k = 1, 2, . . . , M and chooses as its decision that message p sk (t) {rl (t)}l=1 1

Without any loss in generality, we shall assume that the carrier phase θc is arbitrarily set equal to zero since the various paths that compose the channel will each introduce their own random phase into the transmission. 2 In deriving the various optimum receiver configurations, we shall assume a so-called “one shot” approach, namely, a single transmission, wherein intersymbol interference (ISI) that would be produced by the presence of the path delays on continuous transmission is ignored. 3 It should be noted that Turin [1] originally considered optimal diversity reception for the more general case where the link noises (as well as the link fades) could be mutually correlated; however, the noises and fades were statistically independent. Later on, however, Turin [2] restricted his considerations to link noises that were white Gaussian and statistically independent (the link fades, however, were still allowed to be correlated—statistically independent and exponentially correlated fades were considered as special cases.)

THE CASE OF KNOWN AMPLITUDES, PHASES, AND DELAYS—COHERENT DETECTION

191

whose signal sk (t) corresponds to the largest of these probabilities.4 Since the messages (signals) are assumed to be equiprobable, then, by Bayes’ rule, the equivalent decision rule is to choosesk (t) corresponding  to the largest of the conLp |sk (t) , k = 1, 2, . . . , M, which is ditional probabilities (likelihoods) p {rl (t)}l=1 the maximum-likelihood (ML) decision rule. Using the law of conditional probability, each of these conditional probabilities can be expressed as5 

   Lp  Lp Lp Lp p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1

  Lp Lp Lp Lp Lp Lp d {αl }l=1 , {θl }l=1 , {τl }l=1 d {θl }l=1 d {τl }l=1 × p {αl }l=1

(7.3)

and as such depends on the degree of knowledge [amount of channel state informaLp Lp Lp , {θl }l=1 , and {τl }l=1 . For instance, tion (CSI) available on the parameter sets {αl }l=1 if any of the three parameter sets are assumed to be known (e.g., through channel measurement), then the statistical averages on that set of parameters need not be performed. In the limiting case (to be considered shortly) where all parameters are assumed to be known to the receiver, none of the statistical averages in (7.3) need beperformedand hence the ML decision rule  simplifies to choosing the largest of Lp  Lp Lp Lp p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1 , k = 1, 2, . . . , M. Receivers that make use of CSI have been termed self-adaptive [3] in that the estimates of the system parameters are utilized to adjust the decision structure, thereby improving system performance by adaptation to slowly varying channel changes. We start our detailed discussion of optimum receivers with the most general case of all parameters known since the decision rule is independent of the statistics of the channel parameters and leads to a well-known classic structure whose performance is better than all others that are based on less than complete parameter knowledge. Also, since detection schemes are typically classified based on the degree of knowledge related to the phase(s) of the received signal, ideal coherent detection implying perfect knowledge, the assumption of complete parameter knowledge falls into this category.

7.1 THE CASE OF KNOWN AMPLITUDES, PHASES, AND DELAYS—COHERENT DETECTION Conditioned on perfect  of the amplitudes, phases, and  knowledge  delays, the condiLp  Lp Lp Lp tional probability p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1 is a joint Gaussian 4 The receiver is assumed to be time-synchronized to the transmitted signal; that is, it knows the time epoch of the beginning of the transmission. 5 Each integral in (7.3) is in fact an L -fold integral. p

192

OPTIMUM RECEIVERS FOR FADING CHANNELS

PDF that, because of the independence assumption on the additive noise components, can be written as    Lp  Lp Lp Lp p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1 =

Lp l=1

=

Lp l=1

  Ts +τl   1 j θl  2  r˜l (t) − αl s˜k (t − τl ) e Kl exp − dt 2Nl τl

(7.4)

 Ts +τl  2  1 ˜ j θl  ˜ Kl exp − Rl (t) − αl Sk (t − τl ) e  dt 2Nl τl

where Kl is an integration constant. Substituting (7.2) into (7.4) and simplifying yields    Lp  Lp Lp Lp p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1

  αl2 Ek αl −j θl =K exp Re e ykl (τl ) − Nl Nl l=1    

Lp Lp 2  α E α k l l −j θ  Re e l ykl (τl ) − = K exp  Nl Nl Lp

l=1

(7.5)

l=1

where 

ykl (τl ) =



Ts +τl τl

R˜ l (t) S˜k∗ (t − τl ) dt =



Ts 0

R˜ l (t + τl ) S˜k∗ (t) dt

(7.6)

is the complex crosscorrelation of the lth received signal and the kth signal waveform and   2 2 1 Ts  ˜ 1 Ts +τl  ˜   (7.7) Ek = Sk (t) dt = Sk (t − τl ) dt 2 0 2 τl is the energy of the kth signal sk (t). Also, the 2 constant K absorbs all the Kl s as   Lp  well as the factor exp( l=1 (1/2Nl ) R˜ l (t) dt), which is independent of k and thus has no bearing on the decision. Since the natural logarithm is a monotonic function of its argument, we can equivalently maximize (with respect to k)     Lp  Lp Lp Lp k = ln p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1 =

Lp  l=1



  αl2 Ek αl −j θl e ykl (τl ) − Re Nl Nl

(7.8)

THE CASE OF KNOWN AMPLITUDES, PHASES, AND DELAYS—COHERENT DETECTION

193

where we have ignored the ln K term since it is independent of k.6 The first bracketed term in the summation of (7.8) requires a complex weight (αl ej θl ) to be applied to the lth crosscorrelator output (scaled by the noise PSD Nl ), and the second bracketed term is a bias dependent on the signal energy-to-noise ratio in the lth path. For constant envelope signal sets (i.e., Ek = E; l = 1, 2, . . . , M), the bias can be omitted from the decision making process. A receiver that implements (7.8) as its decision statistic is illustrated in Fig. 7.1 and is generically referred to as a RAKE receiver [4,5] because of its structural similarity to the teeth on a garden rake.7 Note that this receiver is, for the CSI conditions specified (i.e., perfect knowledge of all channel parameters), optimum regardless of the statistics of these parameters. We shall see shortly that as soon as we deviate from this ideal condition, that is, when one or more sets of parameters are unknown, then the receiver structure will immediately depend on the channel parameter statistics. We conclude this subsection by noting that if instead of the generalized fading channel model consisting of Lp independently received noisy replicas of the transmitted signal, we had assumed the random multipath channel model suggested by Turin [6] wherein the received signal would instead be of the form r (t) =

Lp 

  Re αl s˜k (t − τl ) ej θl + Re {n˜ (t)}

l=1

=

Lp 



Re αl S˜k (t − τl ) ej (2πfc t+θl ) + Re N˜ (t) ej 2πfc t

(7.9)

l=1


= Re R˜ (t) ej 2πfc t with N˜ (t) a complex AWGN processes with PSD 2N0 Watts/Hertz, then the decision metric analogous to (7.8) would be 

k =

Lp 



l=1

  αl2 Ek αl −j θl e ykl (τl ) − Re N0 N0

(7.10)

which is in agreement with [6]. Since N0 is now a constant independent of l, we can eliminate it from (7.10) in so far as the decision is concerned and rewrite the decision metric as 

k =

Lp    −j θ   Re αl e l ykl (τl ) − αl2 Ek

(7.11)

l=1 6 For

convenience, in what follows we shall use the notation k for all decision metrics associated with the kth signal regardless of any constants that will be ignored because they do not depend on k. 7 Such a receiver is also considered to implement the maximum-ratio combining (MRC) form of diversity and will be discussed further in Chapter 9, which deals with the performance of multichannel receivers.

194

S˜k(t)

R˜ Lp(t)

R˜ 2(t)

R˜ 1(t)

p

∫tL (•)dt

(•)dt

(•)dt

Ts +tLp

∫t2

Ts +t2

Ts +t1

∫t1

aLp e jqLp NLp

a2 jq e 2 N2

. . . Re{•}

Re{•}

Re{•}

+

N Lp

aL2pEk

+

a22Ek N2

+

a12Ek N1

−

−

−

ΛM

. . .

Λk

Λ2

Λ1

k

Choose Signal Corresponding to max Λk

Complex form of optimum receiver for known amplitudes, phases, and delays: coherent

(The asterisk on the multiplier denotes complex conjugate multiplication)

*

*

Figure 7.1 detection.

Delay tLp

Delay t2

Delay t1

*

a1 jq e 1 N1

Decision ˆ s(t)

THE CASE OF KNOWN PHASES AND DELAYS BUT UNKNOWN AMPLITUDES

195

For single-channel reception (i.e., Lp = 1), (7.8) or (7.11) simplifies to    k = Re αe−j θ ykl (τ ) − α 2 Ek

(7.12)

which is identical to the decision metric for a purely AWGN channel except for the scaling of the first term by the known fading amplitude α and the second (bias) term by α 2 . For the special case of constant envelope signal sets, the second term becomes independent of k and can therefore be ignored leaving as a decision  metric k = αRe e−j θ ykl (τ ) . Since α now appears strictly as a multiplicative constant that is independent of k, it has no bearing on the decision and thus can also be eliminated from the decision metric. Hence, for single-channel reception of constant envelope signal sets, the decision metric is identical to that for the pure AWGN channel and knowledge of the fading amplitude does not aid in improving the performance. It should be emphasized, however, that despite the lack of dependence of the optimum decision metric on knowledge of the channel fading amplitude, the error probability performance of this receiver does indeed depend on the fading amplitude statistics and will, of course, be worse for the fading channel than for the pure AWGN channel. On the other hand, for nonconstant envelope signal sets (e.g., M-QAM), the second term in (7.12) cannot be ignored and optimum performance requires perfect knowledge of the channel fading amplitude (typically provided by an AGC). Finally, note that if in the generalized fading channel model all paths have equal noise PSD (i.e., Nl = N0 , l = 1, 2, . . . , Lp ), then, the decision metric of (7.8) reduces to that of (7.10). 7.2 THE CASE OF KNOWN PHASES AND DELAYS BUT UNKNOWN AMPLITUDES When the amplitudes are unknown, then the conditional probability of (7.5) must be averaged over their joint PDF to arrive at the decision metric. Assuming inde Lp , we obtain pendent amplitudes with first-order PDFs pal (αl ) l=1    Lp  Lp Lp p {rl (t)}l=1 sk (t) , {θl }l=1 , {τl }l=1 =K

Lp  l=1

0

∞



  α 2 Ek αl exp Re e−j θl ykl (τl ) − l Nl Nl



(7.13) pal (αl ) dαl

We now consider the evaluation of (7.13) for Rayleigh and Nakagami-m fading. 7.2.1

Rayleigh Fading

For Rayleigh fading with channel PDFs  αl2 2αl exp − pal (αl ) = , l l

αl ≥ 0

(7.14)

196

OPTIMUM RECEIVERS FOR FADING CHANNELS

   and l = E αl2 , the integrals of (7.13) can be evaluated in closed form. In particular, using [7, Eq. (3.462.5), p. 382], we obtain    Lp  Lp Lp p {rl (t)}l=1 sk (t) , {θl }l=1 , {τl }l=1 =K

Lp 

1 + γ kl

−1



 √ Ukl2 Ukl 1 + π Ukl exp 1−Q √ 4 2



l=1

(7.15)



where Q (x) is the Gaussian Q-function (see Chapter 4), γ kl = l Ek /Nl is the average SNR of the kth signal over the lth path, and  

Ukl =



Ek Nl

γ kl 1 + γ kl



  −j θ  1 l Re e ykl (τl ) Ek

(7.16)

The combination of (7.15) and (7.16) agrees, after a number of corrections, with the results of Eq. (28) in Ref. 3 using a different notation. The decision metric analogous to (7.8) is obtained by taking the natural logarithm of (7.15) and ignoring the ln K term, which results in k = −

Lp 





ln 1 + γ kl +

l=1

Lp  l=1





 √ Ukl2 Ukl 1−Q √ ln 1 + π Ukl exp 4 2

(7.17) The first summation in (7.17) is a bias, and the second summation is the decision variable that depends on the observation. For large average SNR (i.e., γ kl  1), √ the decision metric above simplifies to (ignoring the ln π term)

k = −

Lp 

ln γ kl +

l=1

Lp   l=1

1 ln Ukl + Ukl2 4

 (7.18)

A receiver that implements the decision rule based on the high-SNR decision shown above metric is illustrated in Fig. 7.2. 7.2.2

Nakagami-m Fading

For Nakagami-m fading with channel PDFs 2 pal (αl ) = (ml )



ml l

ml

2m −1 αl l exp

ml αl2 − l

 ,

αl ≥ 0

(7.19)

197

S˜k(t)

R˜ Lp(t)

R˜ 2(t)

R˜ 1(t)

Delay tLp

Delay t2

Delay t1

p

∫tL (•)dt

(•)dt

(•)dt

Ts +tLp

∫t2

Ts +t2

Ts +t1

∫t1

. . . xLpe jqLp

x2e jq2

Re{•}

Re{•}

Re{•}

(•)2 In(•)+ 4

(•)2 ln(•)+ 4

(•)2 ln(•)+ 4

−

−

xl =

1

−

k l

√E N

+

gkl 1 + gkl

ln(1 + gkLp )

+

ln(1 + gk2)

+

ln(1 + gk1)

ΛM

. . .

Λk

Λ2

Λ1

k

Choose Signal Corresponding to max Λk

Figure 7.2 Complex form of optimum receiver for known amplitudes, phases and delays: Rayleigh fading, high average SNR (γ kl  1).

*

*

*

x1e jq1

Decision ˆ s(t)

198

OPTIMUM RECEIVERS FOR FADING CHANNELS

the integrals of (7.13) can be evaluated in closed form using Eq. (3.462.1) from Ref. 7 (p. 382) with the result    Lp  Lp Lp p {rl (t)}l=1 sk (t) , {θl }l=1 , {τl }l=1 =K

Lp

l=1

(2ml ) 2ml −1 (ml )



ml ml + γ ml

ml



Vkl2 exp 8



D−2ml

−Vkl √ 2



(7.20)

where 



Ek Nl



Vkl =

γ kl ml + γ kl



   1 Re e−j θl ykl (τl ) Ek

(7.21)

and Dp (x) is the parabolic cylinder function [7, Eq. (3.462.1), Sect. 9.24, p. 1092].

7.3 THE CASE OF KNOWN AMPLITUDES AND DELAYS BUT UNKNOWN PHASES When the phases are unknown, then the conditional probability of (7.5) must be averaged over their joint PDF to arrive at the decision metric. Assuming independent phases with PDFs specified over the interval (0, 2π), we obtain    Lp  Lp Lp p {rl (t)}l=1 sk (t) , {θl }l=1 , {τl }l=1 =K

Lp  l=1

2π 0



  α 2 Ek αl exp Re e−j θl ykl (τl ) − l Nl Nl



(7.22) pθl (θl ) dθl

For uniformly distributed phases as is typical of Rayleigh and Nakagami-m fading, (7.22) becomes    Lp  Lp Lp p {rl (t)}l=1 sk (t) , {θl }l=1 , {τl }l=1 =K

Lp l=1

=K

Lp l=1

=K

Lp l=1













α 2 Ek exp − l Nl α 2 Ek exp − l Nl α 2 Ek exp − l Nl

1 2π 1 2π

 0



I0

2π

2π 0



  −j θ  αl l exp Re e ykl (τl ) dθl Nl

   αl |ykl (τl )| cos θl − arg (ykl (τl )) dθl exp Nl

αl |ykl (τl )| Nl

 (7.23)

199

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

Taking the natural logarithm of (7.23) and ignoring the ln K term, we obtain the decision metric k =

Lp 

ln I0

l=1

  Lp αl2 Ek αl |ykl (τl )| − Nl Nl

(7.24)

l=1

which for constant envelope signal sets simplifies to (ignoring the bias term) k =

Lp 

ln I0

l=1

 αl |ykl (τl )| Nl

(7.25)

An implementation of a receiver that bases its decisions on the metric of (7.24) is illustrated in Fig. 7.3. For large arguments, the function ln I0 (x) is approximated by a scaled version of |x|, and thus for high SNR, the decision metric is likewise approximated by k =

Lp  αl |ykl (τl )| Nl

(7.26)

l=1

7.4 THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES When only the delays are known, then the conditional probability of (7.5) must be averaged over both the unknown amplitudes and phases to arrive at the decision. Assuming as was done in Section 7.3 the case of i.i.d. uniformly distributed phases, then the conditional probability needed to compute the decision statistic is obtained by averaging (7.23) over the PDFs of the independent amplitudes, resulting in    Lp  Lp p {rl (t)}l=1 sk (t) , {τl }l=1 =K

Lp  l=1

7.4.1

∞ 0



α 2 Ek exp − l Nl



I0

 αl |ykl (τl )| pαl (αl ) dαl Nl

(7.27)

One-Symbol Observation—Noncoherent Detection

In this subsection, we consider the case where the observation interval of the received signal is one symbol in duration. Receivers that implement their decision rules on the basis of statistics formed from one-symbol duration correlations are referred to as noncoherent receivers. This is in direct contrast to the cases that will be considered next wherein the observation of the received signal extends over two or more symbols resulting in so-called differentially coherent receivers. The abovementioned distinction in this terminology regarding the method of detection (i.e., noncoherent versus differentially coherent) employed by the receiver and its relation to the observation interval is discussed in Ref. 8 (App. 7A) for AWGN channels.

200

S˜k(t)

R˜ Lp(t)

R˜ 2(t)

R˜ 1(t)

p

∫tL (•)dt

(•)dt

(•)dt

Ts +tLp

∫t2

Ts +t2

Ts +t1

∫t1

aLp NLp

a1 N2

. . . ln I0 | • |

ln I0 | • |

ln I0 | • |

+

NLp

aL2pEk

+

a22Ek N2

+

a12Ek N1

−

−

−

ΛM

. . .

Λk

Λ2

Λ1

k

Choose Signal Corresponding to max Λk

Complex form of optimum receiver for known amplitudes and delays but unknown

(The asterisk on the multiplier denotes complex conjugate multiplication)

*

*

Figure 7.3 phases.

Delay tLp

Delay t2

Delay t1

*

a1 N1

Decision ˆ s(t)

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

201

7.4.1.1 Rayleigh Fading For the Rayleigh fading PDF of (7.14), the conditional probability of (7.27) can be evaluated in closed form. In particular, using Eq. (6.633.4) of Ref. 7 (on p. 739), we obtain after some manipulation  p



Lp  {rl (t)}l=1 s k

Lp (t) , {τl }l=1



=K

Lp 

1 + γ kl

−1

 exp

l=1

Ukl 4

2 

where, analogous to (7.16) for the coherent case  

 1 Ek γ kl   = |ykl (τl )| Ukl Nl 1 + γ kl Ek

(7.28)

(7.29)

Once again taking the natural logarithm of the likelihood of (7.28) and ignoring the ln K term, we obtain the decision metric k = −

Lp  l=1



ln 1 + γ kl



2

 Lp  1 Ek γ kl |ykl (τl )| + 4Nl 1 + γ kl Ek

(7.30)

l=1

A receiver that implements a decision rule based on the metric of (7.30) is illustrated in Fig. 7.4. For the special case of constant envelope signal sets, wherein the bias [first term of (7.30)] becomes independent of k and can be ignored, the decision metric becomes (ignoring the scaling by the energy E)  Lp

 |ykl (τl )|2 γl k = 1 + γl Nl

(7.31)

l=1



where γ l = l E/Nl . If we further assume Nl = N0 ; l = 1, 2, . . . , Lp , then (7.31) simplifies still further to (ignoring the scaling by N0 ) k =

 Lp

 γl |ykl (τl )|2 1 + γl

(7.32)

l=1

Finally, for a flat power delay profile (PDP), l = ; l = 1, 2, . . . , Lp , then ignoring the scaling by γ / (1 + γ ), the decision metric is simply k =

Lp 

|ykl (τl )|2

(7.33)

l=1

which is identical in structure to the optimum receiver for a pure AWGN multichannel, that is, each finger implements a complex crosscorrelator matched to the

202

S˜k(t)

R˜ Lp(t)

R˜ 2(t)

R˜ 1(t)

*

*

p

∫tL (•)dt

(•)dt

(•)dt

Ts +tLp

∫t2

Ts +t2

∫t1

Ts +t1

xLp

x2

. . .

xl =

1 | • |2 4

1 | • |2 4

1 | • |2 4 In(•)+

−

−

1

k l

√E N

+

gkl 1 + gkl

−

ln(1 + gkLp )

+

ln(1 + gk2)

+

ln(1 + gk1)

ΛM

. . .

Λk

Λ2

Λ1

k

Choose Signal Corresponding to max Λk

Figure 7.4 Complex form of optimum receiver for known delays, unknown amplitudes and phases: Rayleigh fading, one-symbol observation (noncoherent detection).

Delay tLp

Delay t2

Delay t1

*

x1

Decision ˆ s(t)

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

203

delayed signal for that path followed by a square-law envelope detector with no postdetection weighting. Methods for analytically evaluating the average bit error probability (BEP) performance of multichannel receivers with square-law detection are discussed in Chapter 9. Analyses are provided there for the performance of both the optimum receiver that implements the decision metric of (7.32) and one that implements the unweighted decision metric of (7.33), which for other than a uniform PDP would be suboptimum. In the case of the latter, the performance is equivalent to that obtained from analysis of equal gain combining (EGC) diversity reception, which is studied as an independent topic in Chapter 9. In what follows, we examine and compare the performances of both the optimum and suboptimum receivers for the case of binary FSK and an exponential PDP described by γ l = γ 1 e−δ(l−1) , l = 1, 2, . . . , Lp . Figures 7.5a–e and 7.6a–e respectively illustrate the average BEP performances of the optimum and suboptimum receivers as a function of the fading power decay factor δ and the number of paths Lp . Specifically, for values of δ = 0, 0.1, 0.5, 1.0, and 2.0, the average SNR/bit of the first path, γ 1 , is allowed to vary over a range of 0–16 dB, and the number of paths Lp is varied from 1 to 4. For δ = 0 (i.e., a uniform PDP), the corresponding curves of Figs. 7.5a and 7.6a are seen to agree exactly since in this case the suboptimum receiver corresponding to the decision metric of (7.33) is indeed optimum, as mentioned previously. For δ > 0 the optimum receiver clearly outperforms (has a smaller BEP than) the suboptimum receiver, as it should. We further observe from the curves of Fig. 7.5 that for fixed δ the performance of the optimum receiver always improves monotonically with increasing Lp over the entire range of γ 1 considered. By contrast, the curves in Fig. 7.6 illustrate that for large δ, the performance of the suboptimum receiver can in fact degrade with increasing Lp as a result of the “noncoherent combining loss,” which is more prevalent at low SNRs. Comparing the various groups of curves within each set of figures also reveals that the improvement in BEP obtained by increasing Lp is larger when the fading power decay factor δ is smaller; thus, a uniform PDP stands to gain more from an increase in the number of combined paths than one with an exponentially decaying multipath and the same average SNR/bit of the first path. To compare the behavior of the optimum and suboptimum receivers, Fig. 7.7a,b illustrates their performance for two different combinations of δ and Lp , namely, δ = 1, Lp = 5 and δ = 2, Lp = 4. Also illustrated in these figures are the corresponding results for Lp = 1, in which case the two receivers once again yield  identical performance since the single scaling factor γ 1 / 1 + γ 1 in (7.32) is now inconsequential. We observe from these figures that the suboptimum receiver performs quite well with respect to its optimum counterpart but does in fact exhibit a noncoherent combining loss at sufficiently low SNR as mentioned previously. As a further comparison of the behavior of the optimum and suboptimum BFSK receivers, Fig. 7.8 illustrates their performance with Lp = 4 and varying δ. Finally,

204

OPTIMUM RECEIVERS FOR FADING CHANNELS

100

Lp = 1 Lp = 2 Lp = 3 Lp = 4

Average Bit Error Rate Pb(E)

10−1

10−2

10−3

10−4

10−5

0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(a)

100

Lp = 1 Lp = 2 Lp = 3 Lp = 4

Average Bit Error Rate Pb(E )

10−1

10−2

10−3

10−4

10−5

0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(b)

Figure 7.5 Average BEP performance for optimum reception of noncoherently detected binary FSK over Rayleigh fading with an exponential PDP; m = 1, M = 2: (a) δ = 0; (b) δ = 0.1; (c) δ = 0.5; (d) δ = 1.0; (e) δ = 2.0.

205

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

100

Lp = 1 Lp = 2 Lp = 3 Lp = 4

Average Bit Error Rate Pb(E )

10−1

10−2

10−3

10−4 0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(c)

100

Average Bit Error Rate Pb(E )

Lp = 1 Lp = 2 Lp = 3 Lp = 4

10−1

10−2

10−3

0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB] (d)

Figure 7.5 (Continued)

14

16

206

OPTIMUM RECEIVERS FOR FADING CHANNELS

100

Average Bit Error Rate Pb(E )

Lp = 1 Lp = 2 Lp = 3 Lp = 4

10−1

10−2 0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(e)

Figure 7.5 (Continued)

Fig. 7.9 gives an analogous performance comparison for 4-ary FSK with δ = 1.0 and varying Lp . 7.4.1.2 Nakagami-m Fading For the Nakagami-m fading PDF of (7.19), the conditional probability of (7.27) can also be evaluated in closed form. In particular, using [7, Eq. (6.631.1), p. 737], we obtain [9]  p



Lp  {rl (t)}l=1 sk

Lp (t) , {τl }l=1



  2   Lp

Vkl γ kl −ml 1+ =K (7.34) 1 F1 ml , 1; ml 4 l=1

where analogous to (7.21) for the coherent case  

 1 Ek γ kl   = |ykl (τl )| Vkl Nl ml + γ kl Ek

(7.35)

and 1 F1 (a, b; x) is Kummer’s confluent hypergeometric function [7, Sect. 9.210, p. 1085] which has the property that for x > 0, a > 0, 1 F1 (a, 1; x) is a monotonically increasing function of x. Also, the larger a is, the greater the rate of increase.

207

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

100

Average Bit Error Rate Pb(E )

10−1

Lp = 1 10−2

Lp = 2 10−3

Lp = 3

10−4

Lp = 4

10−5

0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(a)

100

Average Bit Error Rate Pb(E )

10−1

Lp = 1 10−2

Lp = 2 10−3

Lp = 3

10−4

10−5

Lp = 4

0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(b)

Figure 7.6 Average BEP performance for suboptimum reception of noncoherently detected binary FSK over Rayleigh fading with an exponential PDP; m = 1, M = 2: (a) δ = 0; (b) δ = 0.1; (c) δ = 0.5; (d) δ = 1.0; (e) δ = 2.0.

208

OPTIMUM RECEIVERS FOR FADING CHANNELS

100

Average Bit Error Rate Pb(E )

10−1

Lp = 1 10−2

Lp = 2 Lp = 3 10−3

Lp = 4

10−4 0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(c)

Average Bit Error Rate Pb(E )

100

10−1

Lp = 1

10−2

Lp = 2 Lp = 3 Lp = 4 10−3

0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB] (d)

Figure 7.6 (Continued)

14

16

209

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

Average Bit Error Rate Pb(E )

100

10−1

Lp = 2 Lp = 3 Lp = 1

Lp = 4 10−2 0

2

4

6 8 10 12 Average SNR per Bit of First Path [dB]

14

16

(e)

Figure 7.6 (Continued)

Finally, since 1 F1 (1, 1; x) = ex , then, for ml = 1, l = 1, 2, . . . , Lp , the conditional probability of (7.34) reduces to (7.28), as it should. The decision metric for this case is obtained by taking the natural logarithm of (7.34) with the result (ignoring the ln K term)

k = −

Lp  l=1

  2   

Lp V γ kl + ml ln 1 + ln 1 F1 ml , 1; kl ml 4

(7.36)

l=1

Once again the first summation in (7.36) is a bias term, whereas the second summation has a typical term that is a nonlinearly processed sample (at time τl ) of the crosscorrelation modulus |ykl (τl )|. A receiver that implements a decision rule based on (7.36) would be similar to Fig. 7.4, where, however, the square-law nonlinearity is replaced by the ln 1 F1 (·, ·; ·) nonlinearity and the bias is modified accordingly. To compare the behavior of the optimum and suboptimum receivers, Fig. 7.10 illustrates their performance as a function of the m parameter for δ = 2 and Lp = 4. Here we observe that the difference between the suboptimum and optimum performances increases with m, that is, as the severity of the fading decreases.

210

OPTIMUM RECEIVERS FOR FADING CHANNELS

Average Bit Error Rate Pb(E )

100

Lp = 1 Optimum Suboptimum

10−1

10−2

10−3

0

2

4

6

8

10

12

14

16

Average SNR per Bit of First Path [dB] (a)

Average Bit Error Rate Pb(E )

Lp = 1 Optimum Suboptimum

100

10−2

Average SNR per Bit of First Path [dB] (b)

Figure 7.7 Comparison of the average BEP performance for optimum and suboptimum reception of noncoherently detected binary FSK over Rayleigh fading with an exponential PDP; m = 1, M = 2: (a) δ = 1.0, Lp = 5; (b) δ = 2.0, Lp = 4.

211

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

10−1

Average Bit Error Rate Pb(E )

Optimum Suboptimum

d = 2.0 d = 1.0

d = 2.0

d=0 10−2

d = 1.0

10−3

10−4

10−5

0

2

4

6

8

10

12

14

16

Average SNR per Bit of First Path [dB]

Figure 7.8 Comparison of the average BEP performance for optimum and suboptimum reception of noncoherently detected binary FSK over Rayleigh fading with an exponential PDP; Lp = 4, varying δ, m = 1, M = 2.

7.4.2 Two-Symbol Observation—Conventional Differentially Coherent Detection We assume here that in addition to the channel phases and amplitudes being unknown, the channel is sufficiently slow-varying that these parameters can be considered to be constant over a time interval that is at least two symbols in duration. Furthermore, we consider only constant envelope modulations, namely, M-PSK. For a purely AWGN channel, the optimum receiver has been shown [8, App. 7A] to implement differentially coherent detection that for M-PSK results in so-called M-DPSK. What we seek here is the analogous optimum receiver when, in addition to AWGN, fading with unknown amplitude is present on the received signal. The derivation of this optimum receiver to be presented here follows the development found in Ref. 8 (App. 7A). We begin by rewriting (7.4) with integration limits corresponding to a 2Ts -sec observation:    Lp  Lp Lp Lp p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1 =

Lp l=1

 2Ts +τl  2  1 ˜ j θl  ˜ Kl exp − Rl (t) − αl Sk (t − τl ) e  dt 2Nl τl

(7.37)

212

OPTIMUM RECEIVERS FOR FADING CHANNELS

Optimum Suboptimum

Average Bit Error Rate Pb(E )

Lp = 1 Lp = 4 Lp = 2

Lp = 1

Lp = 2 Lp = 4

10−1

0

1

2

3

4

5

6

7

8

Average SNR per Symbol of First Path [dB]

Figure 7.9 Comparison of the average BEP performance for optimum and suboptimum reception of noncoherently detected 4-ary FSK over Rayleigh fading with an exponential PDP; δ = 1.0, varying Lp , m = 1, M = 4.

Defining the individual symbol energies of the kth signal as Eki =

1 2



(i+1)Ts iTs

  2 2 1 (i+1)T +τl  ˜ ˜   S dt = (t)  k  Sk (t − τl ) dt, 2 iTs +τl

i = 0, 1 (7.38)

we obtain, analogous to (7.5)    Lp  Lp Lp Lp p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1 =K

Lp l=1



  αl2 (Ek0 + Ek1 ) αl −j θl exp Re e ykl (τl ) − Nl Nl 

= K exp 

Lp  l=1

   Lp αl2 (Ek0 + Ek1 ) αl −j θl  Re e ykl (τl ) − Nl Nl

l=1

(7.39)

213

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

Optimum Suboptimum

Average Bit Error Rate Pb(E )

m = 0.5 m = 0.5

m=1 m=2 10−1

m=1

m=4 m=2 m=4

10−2

0

1

2

3

4

5

6

7

8

9

10

Average SNR per Bit of First Path [dB]

Figure 7.10 Comparison of the average BEP performance for optimum and suboptimum reception of noncoherently detected binary FSK over Nakagami-m fading with an exponential PDP; δ = 2.0, Lp = 4, varying m. Note that the BEP performance of the optimum receiver was obtained by Monte Carlo simulations.

where now 

ykl (τl ) =



2Ts +τl

τl

R˜ l (t) S˜k∗ (t − τl ) dt =

 0

2Ts

R˜ l (t + τl ) S˜k∗ (t) dt

(7.40)

Since we have assumed constant envelope M-PSK modulation, the kth complex baseband signal can be expressed as8  Es j φ (i) e k , iTs ≤ t ≤ (i + 1) Ts , i = 0, 1 (7.41) S˜k (t) = Ts where Ek0 = Ek1 = Es (the energy per symbol) and φk(i) denotes the information phase transmitted in the ith symbol interval of the kth signal and ranges over the set βk = (2k − 1) π /M, k = 1, 2, . . . , M. Substituting (7.41) into (7.40), we can 8 To

avoid notational confusion with the channel fading phases, we use φ (as opposed to θ from Chapter 3) to denote the transmitted phases.

214

OPTIMUM RECEIVERS FOR FADING CHANNELS

rewrite (7.39) as    Lp  Lp Lp Lp p {rl (t)}l=1 sk (t) , {αl }l=1 , {θl }l=1 , {τl }l=1 =K

Lp l=1

=K

Lp l=1

 αl −j θl  (0) (1) ykl (τl ) + ykl (τl ) exp Re e Nl

 !  αl  (0)  (1) (0) (1) exp y (τl ) + ykl (τl ) cos θl − arg ykl (τl ) + ykl (τl ) Nl kl

(7.42)  Lp 2  where we have absorbed the constant term exp −2Es l=1 αl /Nl in K and   (i) ykl (τl ) =

Es Ts



(i+1)Ts +τl iTs +τl

(i)

R˜ l (t) e−j φk dt,

i = 0, 1

(7.43)

As in Section 7.3, we first need to average (7.42) over the uniformly distributed statistics of the unknown channel phases. Proceeding as we did in (7.23), we arrive at the result    Lp  Lp Lp p {rl (t)}l=1 sk (t) , {θl }l=1 , {τl }l=1 =K

Lp l=1



α 2 Ek exp − l Nl



I0

αl Nl

   (0)  (1) + y y (τ ) (τ )  kl l l  kl

(7.44)

Next, we must average over the statistics of the unknown amplitudes. 7.4.2.1 Rayleigh Fading Following analogous steps to those taken in Section 7.4.1.1, we obtain   2    (0) (1)  Lp   l ykl   + y (τ ) (τ )   l l kl Lp  Lp exp p {rl (t)}l=1 sk (t) , {τl }l=1 = K   4Nl   l=1

(7.45)

with the equivalent decision metric (ignoring the ln K term) Lp 2  l  (0)  (1) k = ykl (τl ) + ykl (τl ) 4Nl l=1

 2 Lp  2Ts +τl   (0) (1) γ l  Ts +τl ˜ −j φk −j φk  ˜ = dt + dt e e R R l (t) l (t)   4Ts τl Ts +τl l=1

(7.46)

215

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

The decision rule based on the decision metric in (7.46) is to choose as the transmitted signal that pair of phases φk(0) = βj0 , φk(1) = βj1 that results in the largest k . We note that adding an arbitrary phase, say, β, to both φk(0) and φk(1) does not affect the decision metric, and thus the joint decision on φk(0) and φk(1) in accordance with the above-mentioned decision rule will be completely ambiguous. To resolve this phase ambiguity, we observe that although the decisions on φk(0) and φk(1) can each be ambiguous with an arbitrary phase β, the difference of these two decisions is not ambiguous at all. Thus, an appropriate solution is to encode the phase information as the difference between two successive transmitted phases, that is, employ differential phase encoding at the transmitter. This is exactly the solution discussed in Section 3.5 for phase ambiguity resolution on the pure AWGN channel (also see Ref. 8, App. 7A). Mathematically speaking, we can set the arbitrary phase β = −φk(0) , in which case (7.46) becomes  Lp     2  2Ts +τl (0) (1)   γ l  Ts +τl ˜ −j φk +β −j φk +β dt + dt  Rl (t) e R˜ l (t) e k =  4Ts τl Ts +τl l=1

 Lp  2   2Ts +τl (1) (0)   γ l  Ts +τl ˜ −j φk −φk = dt  Rl (t) dt + R˜ l (t) e  4Ts τl Ts +τl

(7.47)

l=1

 2 Lp  2Ts +τl   (1) γ l  Ts +τl ˜ −j φk ˜ = Rl (t) dt + e Rl (t) dt   4Ts τl Ts +τl l=1



where φk(i) = φk(i) − φk(i−1) represents the information phase corresponding to the ith transmission interval that ranges over the set of values βk = 2kπ /M, k = 0, 1, . . . , M − 1. Expanding the squared magnitude in (7.47) and retaining only terms that depend on the information phase φk(1) , we obtain (ignoring other multiplicative constants) k =

Lp 


(1) γ l Re V˜0l V˜1l∗ ej φk

(7.48)

l=1

where  V˜il =



(i+1)Ts +τl

iTs +τl

R˜ l (t) dt,

i = 0, 1

(7.49)

A receiver that bases its decision rule on the decision metric of (7.48) is illustrated in Fig. 7.11. For a flat power delay profile and equal channel noise PSDs, the metric of (7.48) reduces to that corresponding to optimum reception in a pure AWGN environment (Section 3.5).

216 R˜ Lp(t)

2Ts +t2

R˜ 2(t)

(•)dt Delay Ts

Delay Ts

Delay Ts

*

(1) e j∆fk

(1)

g2e j∆fk

gLp

*

*

Re{•}

Re{•}

Re{•}

ΛM

. . .

Λk

Λ2

Λ1

k

Choose Signal Corresponding to max Λk

Figure 7.11 Complex form of optimum receiver for known phases, but unknown amplitudes and delays: Rayleigh fading, two-symbol observation (conventional differentially coherent detection).

p

2Ts +tLp

∫Ts +tL

. . .

∫Ts +t2 (•)dt

2Ts +t1

∫Ts +t1 (•)dt

R˜ 1(t)

(1)

g1e j∆fk

Decision ∆fˆ (1)

THE CASE OF KNOWN DELAYS BUT UNKNOWN AMPLITUDES AND PHASES

217

7.4.2.2 Nakagami-m Fading By comparison of the conditional probabilities of (7.23) and (7.44) corresponding, respectively, to noncoherent and differentially coherent detection, it is straightforward to show that for Nakagami-m fading the decision metric becomes k =

Lp  l=1

ln 1 F1

W2 ml , 1; kl 4

 (7.50)

where  

Wkl = 

E Nl



γl ml



 1  (0)  (1) y (τl ) + ykl (τl ) E kl

   2Ts +τl  (0) (1) 1  Ts +τl ˜ −j φk −j φk ˜ dt + dt  Rl (t) e Rl (t) e = √  ETs τl Ts +τl (7.51) As for the Rayleigh case, the decision metric of (7.50) in combination with (7.51) is ambiguous to an arbitrary phase shift β. With differential phase encoding employed at the transmitter, the unambiguous decision metric is still given by (7.50) with now Wkl now defined as E Nl



 

Wkl =  =

E Nl E Nl

γl ml





γl ml γl ml

 √ 

 (1) 1  ˜  V0l + e−j φk V˜1l  ETs

1 ETs

    1/2
(1)  ˜ 2  ˜ 2 ∗ j φk ˜ ˜ V V + + 2Re e V V  0l   1l  0l 1l

(7.52)

Note that because of the nonlinear  2postdetection processing via the ln 1 F1 (·, ·; ·)    ˜ 2 function, the terms V0l  and V˜1l  cannot be ignored, nor can the other multiplicative factors in (7.52) despite the fact that they are all independent of k. 7.4.3 Ns -Symbol Observation—Multiple Differentially Coherent Detection Divsalar and Simon [10], considered differential detection of M-PSK over an AWGN channel based on an Ns -symbol (Ns > 2) observation of the received signal. The optimum receiver (see Fig. 3.18) was derived and shown to yield improved (monotonically with increasing Ns ) performance relative to that attainable with the conventional (two-symbol observation) M-DPSK receiver. Our intent here is to generalize the results of Divsalar and Simon [10] (also see Ref. 8, Ch. 7, Sect. 7.2) to the fading multichannel with unknown amplitudes. Clearly, for Ns > 2, the decision metric and associated receiver derived here will reduce to those obtained in Section 7.4.2.

218

OPTIMUM RECEIVERS FOR FADING CHANNELS

Without going into great detail, it should be immediately obvious that for an Ns -symbol observation, the conditional probability of (7.44) generalizes to    Lp  Lp Lp p {rl (t)}l=1 sk (t) , {θl }l=1 , {τl }l=1 =K

Lp l=1



 αl2 Ek αl exp − I0 Nl Nl

N −1  s     (n) ykl (τl )   

(7.53)

n=0

7.4.3.1 Rayleigh Fading Averaging (7.53) over Rayleigh statistics for the unknown amplitudes results in the generalization of the decision metric in (7.46): N −1  2 Lp s  (n+1)Ts +τl  (n) γ l    −j φk ˜ dt  Rl (t) e k =   4Ts  nTs +τl l=1

(7.54)

n=0

Using the same differential phase encoding rule as for the two-symbol observation case to resolve the phase ambiguity in (7.54), then the unambiguous form of this decision metric becomes Lp  γl k = 4Ts l=1

N −1 2  (n+1)Ts +τl s   n (i)   e−j i=0 φk R˜ l (t) dt     nTs +τl

(7.55)

n=0

where by definition φk(0) = 0. As before, expanding the squared magnitude and retaining only terms that depend on the information phases, we obtain (ignoring other multiplicative constants)  Lp

k =

l=1

γ l Re

   s −1 N s −1 N     i=0

j =0 i 2, Pawula [20, p. 93, Eq. (2.29)] had previously found an upper bound on this performance given by   π 1 + cos M π 2Es  1 − cos Ps (E) ≤ 2.06 (8.95) π Q 2 cos M N0 M which, applying the Chernoff bound to the Gaussian Q-function, results in  
π 1 + cos M π Es  1 − cos exp − Ps (E) ≤ 1.03 π 2 cos M N0 M (8.96)  
π 1 + cos M π 2Es exp − sin2 = 1.03 π 2 cos M N0 2M Figure 8.2 illustrates a comparison of the exact evaluation of Ps (E) from (8.84) or (8.91) with upper bounds obtained from (8.93), (8.95), and (8.96). As can be observed, the two exponential bounds, (8.93) and (8.96), are reasonably tight at high SNR whereas the Q-function bound of (8.95) is virtually a perfect match to the exact result over the entire range of SNRs illustrated. 8.1.5.2 M-DPSK with Multiple-Symbol Detection In Section 3.5.1.2 we discussed the notion of multiple-symbol differential detection of M-PSK and developed the associated decision variables and optimum receiver (see Fig. 3.18 for a three-symbol observation, i.e., Ns = 3). The error probability performance of this receiver was first reported by Divsalar and Simon [21] and later included in Ref. 5 (Sect. 7.2). Since for differential detection a block of Ns symbols (phases) is observed in making a decision on Ns − 1 information symbols, then, following the procedure developed for partially coherent detection, an upper bound on average BEP can be obtained analogous to (8.83), namely Pb (E) ≤



*  ) 1 w u, uˆ Pr zˆ nk > znk (Ns − 1) log2 M β=βˆ

(8.97)

where now (Ns − 1) log2 M represents the number of bits corresponding to the information symbol sequence, β and βˆ now refer to the correct and incorrect sequences associated with the information (prior to differential encoding) phases,

250

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

100

10−1

Symbol Error Rate (SER)

10−2

10−3

10−4

10−5

Exact (8.84) or (8.91) Bound (8.95) Exponential Bound (8.93) Bound (8.96)

10−6

10−7

0

5

10

15 20 SNR per Symbol [dB]

25

30

Figure 8.2 A comparison of exact evaluation and upper bounds on the symbol error probability of coherent 16-DPSK.

* ) and Pr zˆ nk > znk is determined from the decision variables in (3.53) in the form of (8.80) with now 

b a

( =

 ( 0 Eb log2 M Ns ± Ns2 − |δ|2 2N0

(8.98)

and δ now defined analogous (because of the differential encoding) to (8.82) by 

δ=

N s −1 i=0

. exp j

Ns −i−2 

βki−m − βˆki−m



 (8.99)

m=0

8.1.5.3 π /4-Differential QPSK (π /4-DQPSK) As discussed in Section 3.5.2, the only conceptual difference between π /4-DQPSK and conventional DQPSK is that the set of phases {βk } used to represent the information phases {θn } is βk = (2k − 1) π /4, k = 1, 2, 3, 4 for the former and βk = kπ /4, k = 0, 1, 2, 3 for the latter. Since the performance of the M-DPSK receiver of Fig. 3.16 is independent of the choice of the information symbol set, then we can immediately conclude that π /4-DQPSK has a behavior identical to

PERFORMANCE OVER THE AWGN CHANNEL

251

that of DQPSK on the ideal linear AWGN channel and hence is characterized by (8.84) and (8.86) with M = 4. 8.1.6

Generic Results for Binary Signaling

Although specific results for the BEP of binary signals transmitted over the AWGN have been given in the previous sections, an interesting unification of some of these results into a single BEP expression is possible as discussed by Wojnar [22], who, in particular, cites a result privately communicated to him by Lindner (see footnote 2 of Ref. 22) stating that the BEP of coherent, differentially coherent, and noncoherent detection of binary signals transmitted over the AWGN is given by the generic expression [see also Eq. (4.71)]

Pb (E) =

  Eb  b, a N 0 2 (b)

 1 Eb = Qb 0, 2a 2 N0

(8.100)

where  (•, •) is the complementary incomplete gamma function [23, p. 949, Eq. (8.350.2)], which for convenience, is provided here as 

 (α, x) =

∞

e −t t α−1 d t

(8.101)

x

The parameters a and b depend on the particular form of modulation and detection and are presented in Table 8.1. We have also indicated in this table the specific equations to which (8.100) reduces in each instance. Although the result in (8.100) does not provide any new results relative to those indicated in Table 8.1, it does offer a nice unification of five different BEP expressions into a single one that can easily be programmed using standard mathematical software packages such as Mathematica. Furthermore, when evaluating the average BEP performance of these very same binary communication systems over the generalized fading channel, the form in (8.100) will also be helpful in unifying these results. This will be discussed in Section 8.2 of this chapter making use of the special integrals given in Section 5.3.

TABLE 8.1 b

a 1 2

1 0≤g≤1

Parameters a and b for Various Modulation/Detection Combinations 1 2

Orthogonal coherent BFSK [Eq. (8.43)] Antipodal coherent BPSK [Eq. (8.19)] Correlated coherent binary signaling [Chapter 8, footnote 6]

1 Orthogonal noncoherent BFSK [Eq. (8.69)] Antipodal differentially coherent BPSK (DPSK) [Eq. (8.85)] —

252

8.2

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

PERFORMANCE OVER FADING CHANNELS

In this section of the chapter, we apply the special integrals evaluated in Chapter 5 to the AWGN error probability results presented in Section 8.1 to determine the performance of these same communication systems over generalized fading channels. Wherever possible, we shall again make use of the desired forms rather than the classical representations of the mathematical functions introduced in Chapter 4. By comparison with the level of detail presented in Section 8.1, the treatment here will be quite brief since, indeed, the entire machinery that allows us to determine the desired results has by this time been completely developed. Thus, for the most part we shall merely present the final results except for the few situations where further development is warranted. When fading is present, the received carrier amplitude Ac is attenuated by the fading amplitude, α, which is a random variable (RV) with mean square value α 2 =  and probability density function (PDF) dependent on the nature of the fading channel. Equivalently, the received instantaneous signal power is attenuated by α 2 ,  and thus it is appropriate to define the instantaneous SNR per bit by γ = α 2 Eb /N0  and the average SNR per bit by γ = α 2 Eb /N0 = Eb /N0 . As such, conditioned on the fading, the BEP of any of the modulations considered in Section 8.1 is obtained by replacing Eb /N0 by γ in the expression for AWGN performance. Denoting this conditional BEP by Pb (E; γ ), the average BEP in the presence of fading is obtained from

Pb (E) =

∞

Pb (E; γ ) pγ (γ ) d γ

(8.102)

0

where pγ (γ ) is the PDF of the instantaneous SNR. On the other hand, if one is interested in the average SEP, then the same relation as (8.102) applies using instead the conditional SEP in the integrand, which is obtained from the AWGN result with Es /N0 replaced by γ log2 M. Our goal in the remainder of this chapter is to evaluate (8.102) for the various modulation/detection schemes considered in Section 8.1 and the various fading channel models characterized in previous chapters. Because of the multitude of different signal/channel combinations, however, we shall give explicit results for only one or two of the fading channel models and then indicate how to obtain the rest of the results. 8.2.1

Ideal Coherent Detection

In this section, we evaluate the average BEP of the various modulations considered in Section 8.1.1 when transmitted over the generalized fading channel and detected with an ideal phase coherent reference signal. The results will be obtained by applying the integrals presented in Section 5.1.1 to the appropriate expressions for BEP over the AWGN with the abovementioned replacement of Eb /N0 by γ .

PERFORMANCE OVER FADING CHANNELS

253

8.2.1.1 Multiple Amplitude-Shift-Keying (M-ASK) or Multiple Amplitude Modulation (M-AM) For M-AM the SEP over the AWGN channel is given by (8.3). To obtain the average SEP of M-AM over a Rayleigh fading channel, one first obtains the conditional SEP by replacing Es /N0 with γ log2 M in (8.3) and then evaluates (8.102) for the Rayleigh PDF of (5.4). This type of evaluation was carried out in Chapter 5, in particular, comparing (8.102) with (5.1), and making use of (5.6), we obtain  
3γ s M −1 (8.103) 1− Ps (E) = M M 2 − 1 + 3γ s 

where γ s = γ log2 M denotes the average SNR per symbol. For the binary case, (8.103) becomes  γ 1 Pb (E) = 1− (8.104) 2 1+γ To obtain the remainder of the results for average SEP, one finds the particular integral in Section 5.1 corresponding to (5.1) for  4 channel  of interest,

the fading M 2 − 1 for a 2 . For multiplies it by 2 (M − 1) /M, and substitutes 6 log2 M example, for Nakagami-m fading, the appropriate integrals to use are (5.18). Thus, the average SEP of M-AM over a Nakagami-m fading channel is given by 
 m−1 
2k 
1 − µ2 k M Ps (E) = , 1−µ k M −1 4 k=0

 

µ=

(8.105)

3γ s  , 2 m M − 1 + 3γ s

m integer which clearly reduces to (8.103) for m = 1 and 0 3γ s 


 m + 12 1 M −1 m(M 2 −1) Ps (E) = √   M π m(M 2 −1)+3γ s m+1/2  (m + 1) 2 m(M −1)   × 2 F1 1, m +

1 2; m

m M 2 −1

+ 1; m M 2 −1 +3γ ( ) s

(8.106)

,

m noninteger It is tempting to try to evaluate the average BEP over the fading channel by using the asymptotic (large SNR) relation between the AWGN BEP and SEP as given in (8.7) to determine the conditional BEP needed in (8.102). Unfortunately,

254

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

this procedure is inappropriate since, as mentioned earlier in this chapter, on the fading channel the symbol SNR of the AWGN SEP is replaced by log2 M times the instantaneous SNR per bit, γ , which is a RV varying between zero and infinity. Rather, one needs to compute the exact relation between AWGN BEP and SEP substitute γ log2 M for Es /N0 and then average over the PDF of γ . As mentioned in Section 8.1.1.1, this relation, that is, the conditional BEP on γ , can be computed for any given M and a Gray code bit-to-symbol mapping. 8.2.1.2 Quadrature Amplitude-Shift-Keying (QASK) or Quadrature Amplitude Modulation (QAM) For QAM, the SEP over the AWGN channel is given by (8.10). To obtain the average SEP of M-AM over a Rayleigh fading channel, one proceeds as for the M-AM case by first obtaining the conditional SEP, that is, replacing Es /N0 with γ log2 M in (8.10), and then evaluating an integral such as (8.102) for the Rayleigh PDF of (5.4). This type of evaluation involves two integrals that were developed in Chapter 5. In particular, comparing the terms (5.28) and (8.102) combined with (5.1), and making use of (5.6) and (5.29), we obtain  √ M −1 1.5γ s Ps (E) = 2 1− √ M − 1 + 1.5γ s M   √  2  M −1 1.5γ s M − 1 + 1.5γ s 4 −1 − tan 1− √ M − 1 + 1.5γ s π 1.5γ s M (8.107) which for 4-QAM reduces to      γ γ 1+γ 4 1 −1 tan Ps (E) = 1 − − 1− (8.108) 1+γ 4 1+γ π γ To obtain the remainder of the results for average SEP, one finds the particular integrals in Section 5.1 corresponding √ to (5.1) and √ (5.28) for the fading channel of interest, multiplies the first by 4 M − 1 / M and the second by  √ ,2

 + √ M − 1 / M , and substitutes 3 log2 M / (M − 1) for a 2 . For example, for 4 Nakagami-m fading with m integer, the appropriate integrals to use are (5.18) and (5.30). Thus, the average SEP of QAM over a Nakagami-m fading channel is given by √   m−1 
2k 
1 − µ2 k M Ps (E) = 2 √ 1−µ k 4 M −1 k=0 √ 2   m−1 
2k  M 4 2 π 1 − tan−1 µ − √ 1− µ k π 2 (4 (1 + c))k M −1 k=0  k 

−1  m−1 Tik + −1 ,2(k−i)+1 − sin tan µ cos tan µ (8.109) (1 + c)k k=1 i=1

255

PERFORMANCE OVER FADING CHANNELS

100

Average Symbol Error Probability PS(E )

10−1

m = 0.5

m=1

10−2

m=2

10−3

10−4

m=4 10−5

10−6

0

5

10

15

20

25

30

Average SNR per Symbol [dB] Figure 8.3 Average SEP of 16-QAM over a Nakagami-m channel versus the average SNR per symbol.

where

1.5γ s , c= m (M − 1)



µ=

-

c 1+c

(8.110)

and Tik is as defined in (5.32). Figure 8.3 is an illustration of the average SEP of 16-QAM as computed from (8.109) with m as a parameter. To compute the average BEP performance, again one should not use the approximate asymptotic form of (8.7) but rather use the exact AWGN BEP of (8.14), substituting γ log2 M for Es /N0 and then average over the PDF of γ . Instead, one can use the approximate BEP expression obtained by Lu et al. [8] for the AWGN, which is accurate for a wide range of SNRs, again making the substitution γ log2 M for Es /N0 followed by averaging over the PDF of γ . Using the alternative form of the Gaussian Q-function of (4.2), it is straightforward to show that the result of this evaluation is given by Pb (E) ∼ =4

√


 √

M/2 1 M −1 1 π/2 (2i − 1)2 3Eb log2 M Mγ − dθ √ log2 M π 0 2 sin2 θ N0 (M − 1) M i=1 (8.111)

256

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

where Mγ (s) is again the MGF of the instantaneous fading power γ . For example, for a Rayleigh fading channel, we obtain the following analogous to (8.107):    √
 √ M/2 2 1 M − 1 1.5 − 1) γ log M (2i 2  1 − Pb (E) ∼ √ =2 2 log2 M M − 1 + 1.5 − 1) γ log2 M (2i M i=1 (8.112) 8.2.1.3 M-ary Phase-Shift-Keying (M-PSK) For M-PSK, the classical form of the SEP over the AWGN channel is given by (8.18) and the desired form is given by (8.23). To obtain the average SEP of M-PSK over a Rayleigh fading channel, one first obtains the conditional SEP by replacing Es /N0 with γ log2 M in (8.23) and then evaluates (8.102) for the Rayleigh PDF of (5.1.4). In particular, comparing (8.102) with (5.77) and making use of (5.79), we obtain
Ps (E) =

M −1 M 

×


.  M gPSK γ s 1− 1 + gPSK γ s (M − 1) π

π + tan−1 2



gPSK γ s π cot 1 + gPSK γ s M

(8.113)





where gPSK = sin2 π /M. For M = 2, (8.113) reduces to (8.104) since binary PSK and binary AM are identical. For Rician fading, the average SEP is obtained from (5.78) together with (5.11), or equivalently (5.13) [with the upper limit changed from π /2 to (M − 1) π /M] with a 2 = 2gPSK and γ s substituted for γ , resulting in

Ps (E) =

1 π

(M−1)π/M 0


× exp −

(1 + K) sin2 θ (1 + K) sin2 θ + gPSK γ s KgPSK γ s

(1 + K) sin2 θ + gPSK γ s

(8.114)

 dθ

An equivalent result was reported by Sun and Reed [24, Eq. (2.38)].10 10 It should be noted that an error occurs in Eqs. (10)–(12) of [24] in that the upper limit of their integrals should be π /2 − π /M rather than π /2.

257

PERFORMANCE OVER FADING CHANNELS

For Nakagami-m fading with m integer, the average SEP is obtained from (5.80) with the same substitutions for a 2 and γ resulting in the closed-form solution 

4 gPSK γ s m 4

1 + gPSK γ s m 2 π  m−1 
2k  1 −1 × + tan α +

 4 ,k k 2 4 1 + gPSK γ s m k=0

1 M −1 − Ps (E) = M π



−1

+ sin tan

α

k   m−1 k=1 i=1

Tik

+



−1

+

 4 ,k cos tan 1 + gPSK γ s m

α

(8.115) ,2(k−i)+1



where from (5.81) 



α=

4 gPSK γ s m π  4 cot

M 1 + gPSK γ s m

(8.116)

and again Tik is as defined in (5.32). Figure 8.4 illustrates the average SEP as computed from (8.115) with m as a parameter. Exact results for average BEP of 4-PSK, 8-PSK, and 16-PSK over Rayleigh fading channels can be obtained by averaging (8.31) over the fading PDF in (5.4). In particular, using a generalization of (5A.15) when the upper limit of the integral is π (1 − (2k ± 1) /M), we obtain

∞  Pk = Pk pγ (γ ) d γ = K+ − K− , k = 0, 1, 2, . . . , M − 1 (8.117) 0

where 
  M 2k ± 1 gPSK (k ± ) γ s 1− M 1 + gPSK (k ± ) γ s (2k ± 1) π   1 + gPSK (k ± ) γ s (2k ± 1) π −1 tan × tan , (8.118) gPSK (k ± ) γ s M

1 K± = 2




 (2k ± 1) π gPSK k ± = sin2 M Using P k of (8.117) for Pk in (8.31) gives the desired results for M = 4, 8, 16. Similarly for Nakagami-m fading, P k can be computed from (5A.22) as
 (2k + 1) π (2k − 1) π k = 0, 1, 2, . . . , M − 1 (8.119) , ;γs , P k = Im M M which again should be used in place of Pk in (8.31) to obtain average BEP.

258

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

100

Average Symbol Error Probability PS (E )

10−1

m = 0.5

10−2

m=1

10−3

m=2 10−4

m=4

10−5

10−6

0

5

10

15

20

25

30

Average SNR per Symbol [dB]

Figure 8.4 Average SEP of 8-PSK over a Nakagami-m channel versus the average SNR per symbol.

For other values of M, one can again use the approximate AWGN result of Lu et al. [8] as given in (8.32) substituting γ log2 M for Es /N0 followed by averaging over the PDF of γ . Using the alternative form of the Gaussian Q-function of (4.2), the end result of this evaluation is Pb (E) ∼ =

2  max log2 M, 2


π/2 Mγ −

× 0

max(M/4,1)  i=1

1 π

Eb log2 M (2i − 1) π sin2 N M sin θ 0 1

2

(8.120)

 dθ

Specific results for the variety of fading channels considered are easily worked out using the results of Chapter 5 and are left as exercises for the reader. 8.2.1.4 Differentially Encoded M-ary Phase-Shift-Keying (M-PSK) and π /4-QPSK Consider first the case of differentially encoded QPSK for which the classical form of the SEP over the AWGN channel is given by (8.39). To apply this result to

259

PERFORMANCE OVER FADING CHANNELS

the evaluation of the average SEP on the slow-fading channel, we merely replace  Es /N0 by γs = α 2 Es /N0 and then average over the PDF of the fading distribution pγs (γs ). It should be noted that, whereas for the AWGN channel the first two terms of (8.39) are typically sufficient for a good approximation of performance in high-SNR applications, on the fading channel all four terms are important because of the need to average over γs between zero and infinity. Since the four Q-function forms in (8.39) are all expressible as single integrals with Gaussian integrands in γs , after the replacement mentioned above, the MGF-based approach is now applicable. Specifically, the average SEP is given by 4 π

Ps (E) =

+

+

π/2


Mγs −

0

4 π2 4 π2

π/6

1 2 sin2 θ

cos−1

0

sin−1 √1

3



#

 dθ −

8 π

π/4


Mγs −

0

1 2 sin2 θ

 dθ


$  3 cos 2θ − 1 1 − M − 1 dθ γs 2 cos3 2θ 2 sin2 θ

π − cos−1

#

0


$( 3 cos 2θ − 1 1 dθ − M − 1 γ s 2 cos3 2θ 2 sin2 θ

(8.121) For the special case of Rayleigh fading, with the aid of (5.6), (5.29), (5.91), and (5.92), letting a = 1 and replacing γ by γ s , one obtains the result Ps (E) = 4I1 − 8I2 + 4I3 + 4I4

(8.122)

where  γ s /2 1 I1 = 1− , 2 1 + γ s /2 1 I3 = 2 π

1 I4 = 2 π

π/6 0



0



sin2 θ

−1

sin2 θ + γ s /2  √  sin−1 1/ 3

    4 γ s /2 1 + γ s /2 1 −1 tan I2 = 1− 4 1 + γ s /2 π γ s /2



cos

sin2 θ sin2 θ + γ s /2

#

$ 3 cos 2θ − 1 − 1 dθ 2 cos3 2θ



π − cos−1

#

(8.123)

$( 3 cos 2θ − 1 − 1 dθ 2 cos3 2θ

For Nakagami-m fading with m integer, the average SEP can similarly be obtained from (8.122) with the aid of (5.18), (5.30), (5.93), and (5.94), letting

260

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

a = 1 and replacing γ by γ s . Specifically, the Ik values needed in (8.122) are now given by     

 1 − µ2 γ s
 m−1    2m γs γ γ s /2 1  2k  s k , = µ 1−µ I1 = k 2 2m 4 2m m + γ s /2 k=0



 m−1
  2k γ /2 1 π s − tan−1



k k 2 1 + γ s /2 4 1 + γ s /2 k=0 m−1 k  γ s /2 Tik (8.124)

k 1 + γ s /2 k=1 i=1 1 + γ s /2   2(k−i)+1   γ /2

1 γ s /2 1 I2 = − 4 π 1 + γ s /2  − sin tan−1 



× cos tan−1

1 I3 = 2 π

π/6



.

sin2 θ

s

 

1 + γ s /2 m −1

#

$ 3 cos 2θ − 1 − 1 dθ 2 cos3 2θ

cos sin2 θ + γ s /2m   m  # $(

sin−1 1/√3 sin2 θ 1 −1 3 cos 2θ − 1 I4 = 2 π − cos − 1 dθ π 0 2 cos3 2θ sin2 θ + γ s /2m 0

For the more general case of differentially encoded M-PSK, we need to evaluate the average of (8.36) over the fading PDF. Here we can obtain the result in only the simple desired form for Rayleigh fading. The average of the first term of (8.36) is given by (8.113) multiplied by 2: 
. 


∞ M gPSK γ s M −1 2Ps (E) |M -PSK pγ (γ ) d γ = 2 1− M 1 + gPSK γ s (M − 1) π 0    gPSK γ s π π + tan−1 cot × (8.125) 2 1 + gPSK γ s M The corresponding average of the second term is obtained from (5.104) with a 2 = 2gPSK = 2 sin2 π /M and γ replaced by γ s

∞ (Ps (E) |M -PSK )2 pγ (γ ) d γ 0

=

#  (M−1)π/M
2
1 1 (M − 1) π c (φ) π gPSK γ s M 0    c (φ) 1 + c (φ) (M − 1) π −1 tan tan − dφ 1 + c (φ) c (φ) M

(8.126)

PERFORMANCE OVER FADING CHANNELS

where now



c (φ) = gPSK γ

261



sin2 φ

(8.127)

sin2 φ + gPSK γ

For the average of the third term we must first square Pk of (8.30) and then make   2 = 2 sin2 (2k ± 1) π /M and θ± = π (1− use of (5.106) through (5.108) with a± (2k ± 1) /M) for k = 0, 1, 2, . . . , M − 1 . The result is

∞ Pk2 pγ (γ ) d γ = L+ + L− − 2L+− , k = 0, 1, 2, . . . , M − 1 (8.128) 0

where

 π(1−(2k±1)/M) 1 2 1 L± =

2 c± (φ)  2π sin (2k ± 1) π /M γ s 0
  c± (φ) 2k ± 1 − × π 1− M 1 + c± (φ) . $ #
2k ± 1 1 + c± (φ) −1 × tan tan π 1 − dφ c± (φ) M

and




(8.129)

 π(1−(2k+1)/M) 1 1 2 =

2 c+− (φ)  2π sin (2k + 1) π /M γ s 0
  2k − 1 c+− (φ) − × π 1− M 1 + c+− (φ) . #
$ 1 + c+− (φ) 2k − 1 −1 tan π 1 − × tan dφ c+− (φ) M


L+−

with 



2





2





c± (φ) = sin (2k ± 1) π /M γ s 

c+− (φ) = sin (2k + 1) π /M γ s



sin2 φ

 sin2 φ + sin2 (2k ± 1) π /M γ s sin2 φ

(8.130)



(8.131)

 sin2 φ + sin2 (2k − 1) π /M γ s

Finally, since, as was pointed in Section 8.1.1.4, the performance of coherently detected π /4-QPSK transmitted over a linear AWGN channel is identical to that of differentially encoded QPSK, the same conclusion can be made for the fading channel. Hence, the SEP performance of coherently detected π /4-QPSK over the Rayleigh and Nakagami-m fading channels is also given by (8.122) together with (8.123) or (8.124), respectively.

262

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

8.2.1.5 Offset QPSK (OQPSK) or Staggered QPSK (SQPSK) In Section 8.1.1.5 it was concluded that because of the similarity between the conventional and offset QPSK receivers and the fact that time offset of the I and Q channels has no effect on the decisions made on the I and Q data bits, the BEP performances of these two modulation techniques on a linear AWGN channel with ideal coherent detection are identical. Thus, without further ado, we conclude that the same is true on the fading channel and hence the error probability performance results of Sections 8.1.2.3 and 8.1.2.4 apply. 8.2.1.6 M-ary Frequency-Shift-Keying (M-FSK) In Section 8.1.1.6 we observed that the expression [see (8.41)] for the average SEP of orthogonal M-FSK involves the (M-1)st power of the Gaussian Q-function. Since for arbitrary M an alternative form [analogous to (4.2)] is not available for QM−1 (x), (8.41) cannot be put in the desired form to allow simple evaluation of the average SEP on the generalized fading channel. However, for the special case of binary FSK (M = 2), the conditional error probability is in the desired form [see (8.44) for orthogonal signals or (8.45) for nonorthogonal signals], and thus simple exact evaluation of average BEP on the generalized fading channel is possible. Furthermore, using a different approach based on an M-dimensional extension of Craig’s method [10], Dong and Beaulieu [25] have been able to obtain exact closed-form results for average BEP and SEP of 3-ary and 4-ary (i.e., M = 3 and M = 4) orthogonal signaling in slow Rayleigh fading. Also shown there is the fact that the results obtained for M = 4 can be used as close approximations to the exact results for values of M > 4. In addition, because the results in Ref. 25 for conditional error probability can be expressed in what we have previously referred to as the “desired form,” then, by applying the MGF-based approach, it is possible to extend them to the generalized fading channel. Finally, in an effort to simplify matters still further, we shall derive easy-to-evaluate, asymptotically tight upper bounds on the average BEP and SEP of 4-ary FSK. Before moving on to the more difficult 4-ary case, we first quickly dispense with the results for binary FSK since these follow immediately from the integrals developed in Chapter 5 or equivalently from the results previously obtained for binary AM and BPSK replacing γ by γ /2 for orthogonal BFSK and by (γ /2) (1 − (sin 2π h) /2π h) for nonorthogonal BFSK. For example, for Rayleigh fading the average BEP of orthogonal BFSK is given by  γ /2 1 Pb (E) = 1− 2 1 + γ /2

(8.132)

whereas for Nakagami-m fading the analogous results are  Pb (E) = 2 1 − µ

m−1 
k=0

m integer

2k k




1 − µ2 4



k  ,



µ=

γ /2 , m + γ /2

(8.133)

PERFORMANCE OVER FADING CHANNELS

and 1 Pb (E) = √ 2 π

√ γ /2m (1 + γ /2m)m+

m noninteger

1 2

263




 m + 12 1 1 ; m + 1; , 1, m + F 2 1  (m + 1) 2 1 + γ /2m (8.134)

Although Dong and Beaulieu [25] explicitly pursue results for the Rayleigh fading channel, along the way they implicitly arrive at expressions for the conditional SEP of 3-ary and 4-ary orthogonal FSK that, interestingly enough, are either in or can be put in the desired form to allow averaging over the statistics of the generalized fading channel. Specifically, it is shown in Eq. (17) of Ref. 25 that for 3-ary orthogonal FSK 


γ 1 2π/3 dθ (8.135) exp − Ps (E; γ ) = π 0 2 sin2 θ Note the resemblance of (8.135) to the Craig representation for Q (x) in (4.2) and Q2 (x) in (4.9). Thus, we immediately the conditional SEP of 3-ary is

√  see that √ upper- and lower-bounded by Q γ and Q2 γ , respectively. Furthermore, performing the average required in (8.102), we immediately obtain the average SEP as 


1 1 2π/3 dθ (8.136) Mγ − Ps (E) = π 0 2 sin2 θ where, as before, Mγ (s) is the MGF of the fading SNR. While one might think that the analogous result for 4-ary orthogonal FSK would still be in the simple Craig-type form of (8.135) with a change in only the upper limit of the integral, unfortunately such is not the case. Instead, the result is given by Eq. (22) of Ref. 25: -
 5π/6 3γ 3 sin θ 3γ ' exp − + Ps (E; γ ) = Q 2 4 + 2 sin2 θ π/6 2π 2 + sin2 θ (8.137)  √  3γ sin θ × 1−Q ' dθ 4 + 2 sin2 θ However, using the Craig form of the Gaussian Q-function given in (4.2), it is still possible to express (8.137) in the desired form as 


3γ 1 π/2 dθ exp − Ps (E; γ ) = π 0 4 sin2 θ #


5π/6 3γ 3 sin θ ' exp − + (8.138) 4 + 2 sin2 θ π/6 2π 2 + sin2 θ   

1 π/2 3γ sin2 θ − exp − 1+ dφ dθ π 0 4 + 2 sin2 θ 2 sin2 φ

264

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

whereupon, performing the average required in (8.102), we obtain 1 π

Ps (E) =

π/2

Mγ


−

0


# × Mγ − 1 − π

π/2



5π/6 3 3 sin θ ' d θ + 4 sin2 θ π/6 2π 2 + sin2 θ  3

4 + 2 sin2 θ

Mγ 0

3 − 4 + 2 sin2 θ



sin2 θ 1+ 2 sin2 φ



(8.139)

 dφ dθ

For the Rayleigh fading case where Mγ (−s) is specifically given by (5.5), the average SEPs of (8.136) and (8.139) become, respectively 

 1 γ + 2+γ π

2 Ps (E) = − 3

γ tan−1 2+γ



3 (2 + γ ) γ

(8.140)

and     γ 3 (2 + γ ) 2+γ −1 −1 tan + tan 2+γ γ 4 + 3γ (8.141) For M-ary orthogonal FSK, the average SEP on the AWGN can be obtained from (8.42) as 

3 3 Ps (E) = − 4 2



3 γ + 2 + γ 2π







M−1


2 q 1 dq Ps (E) = 1 − √ exp − 2 2π −∞  M−1 
2

∞ 1 q 2E s   = dq 1− 1−Q q + √ exp − N0 2 2π −∞   M−1  

∞ √ 

1 Es 1 − 1 − Q  exp −u2 d u = √ 2 u+ N0 π −∞

∞

2Es Q −q − N0  

(8.142) and the corresponding BEP is obtained from (8.142) using (8.42). The most straightforward way of numerically evaluating (8.142) (and therefore the BEP derived from it) is to apply Gauss–Hermite quadrature [26, p. 890, Eq. (25.4.46)] resulting in   M−1   Np    √ Es 1 Ps (E) ∼ wn 1 − 1 − Q 2 xn + =√   N0 π n=1

(8.143)

265

PERFORMANCE OVER FADING CHANNELS

* ) where xn ; n = 1, 2, . . . , Np are the zeros of the Hermite polynomial of order Np and wn are the associated weight factors [26, Table 25.10, p. 924]. A value of Np = 20 is typically sufficient for excellent accuracy. When slow fading is present, the average symbol error probability is obtained from (8.142) or (8.143) by first replacing Es /N0 with γ = α 2 Es /N0 and then averaging over the PDF of γ , specifically

√ 2

∞ + ,M−1  ∞ y − 2γ 1 1 − 1 − Q (y) exp − Ps (E) = √ pγ (γ ) d γ d y 2 2π −∞ 0 (8.144) or approximately  ( Np

∞% √

1  √ &M−1 ∼ 1−Q wn 1 − 2 xn + γ pγ (γ ) d γ Ps (E) = √ π n=1 0 (8.145) Numerical evaluation of (8.144) and the associated bit error probability using (8.34) for Rayleigh and Nakagami-m fading channels is computationally intensive. Equation (8.145) does yield numerical values; however, its evaluation is very timeconsuming, especially for large values of m. Thus, tight upper bounds on the result in (8.144) that are simple to use and evaluate numerically are highly desirable. Using Jensen’s inequality [27], Hughes [28] derived a simple bound on the AWGN performance in (8.132). In particular, it was shown that 

 Ps (E) ≤ 1 − 1 − Q

Es N0

M−1 (8.146)

which is tighter than the more common union upper bound [5, Eq. (4.97)]  Es Ps (E) ≤ (M − 1) Q (8.147) N0 Evaluation of an upper bound on average error probability for the fading channel by averaging the right-hand side of (8.146) (with Es /N0 replaced by γs ) over the PDF of γs and using the conventional form for the Gaussian probability integral as in (4.1) is still computationally intensive. Using the alternative forms of the Gaussian Q-function and its square as in (4.2) and (4.9), respectively, it is possible to simplify the evaluation of this upper bound on performance. The details are as follows. We begin by applying a binomial expansion to the Hughes bound of (8.146), which when averaged over the fading PDF, results in Ps (E) ≤

M−1  k=1


(−1)

k+1

M −1 k

 Ik

(8.148)

266

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

where 

Ik =

∞

Qk

√  γs pγs (γs ) d γs ,

k = 1, 2, . . . , M − 1

(8.149)

0

[Note that the result based on the union upper bound would simply be the first term (k = 1) of (8.148).] Using (4.2), (4.9), (4.31), and (4.32), the integral in (8.149) can be evaluated for M = 4 (k = 1, 2, 3) using the MGF-based approach. Using these integral results in (8.148), we obtain the upper bound on average SEP given by 3 Ps (E) ≤ π

π/2


Mγs −

1



3 dθ − π

π/4


Mγs −

1



dθ sin2 θ sin2 θ 0  #
$

π/6 1 1 −1 3 cos 2θ − 1 dθ (8.150) + 2 − 1 Mγs − 2 cos π 0 2 cos3 2θ sin θ #
$( 

sin−1 √1  1 1 3 −1 3 cos 2θ − 1 + π − cos − M − 1 dθ γ s 2π 2 0 2 cos3 2θ sin2 θ 0

where the first term is what would be obtained from the union upper bound of (8.147) and the remaining terms correspond to the improved tightness of the Hughes bound. Specific results for the Rayleigh and Nakagami-m fading channels can also be directly obtained using integrals from Sections 5.4.3.1 and 5.4.3.2. Specifically, using (4.2) and (4.9) and assuming a Nakagami-m channel with instantaneous PDF given by (5.14), the integral in (8.149) can be evaluated for M = 4 (k = 1, 2, 3) either in closed form or in the form of a single integral with finite limits and an integrand composed of elementary functions, specifically, exponentials and trigonometrics. The results are summarized here as follows: I1 = (P (c))

m

m−1 
k=0

m−1+k k

 (1 − P (c))k ,

(8.151)
 c  1  γ 1− , c= s P (c) = 2 1+c 2m

 m−1 
2k  1 π c c 1 1 −1 − tan I2 = − k 4 π 1+c 2 1+c (4 (1 + c))k k=0 
#

m−1 $2(k−i)+1
k  c T c ik cos tan−1 − sin tan−1 ,  1+c 1+c (1 + c)k k=1 i=1

2k 



Tik = 2(k−i) k−i

k

4i

[2 (k − i) + 1]

,



c=

γs 2m

(8.152)

PERFORMANCE OVER FADING CHANNELS

267

and 1 I3 = π 

c (φ) =

γ 2

π/4 0






2 c (φ) γs sin2 φ

sin2 φ +

m m

[P (c (φ))]

γs 2

m−1 
k=0

m−1+k k

 [1 − P (c (φ))]k d φ,

(8.153)

Illustrated in Fig. 8.5 are curves for average bit error probability versus average bit SNR for 4-ary orthogonal signaling over the Nakagami-m fading channel, the special case of m = 1 corresponding to the Rayleigh channel. For each value of m, three curves are calculated. The first is the exact result obtained (with much computational power and time) by averaging (8.145) over the PDF in (5.14). The second is the Hughes upper bound obtained from (8.148) together with the three integrals in (8.151)–(8.153). Finally, the third is the union upper bound obtained from the first term of (8.145) together with the integral in (8.151). The curves labeled m = ∞ correspond to the nonfading (AWGN-only) results. We observe, not surprisingly, that as m increases (the amount of fading decreases), the three results are asymptotically equal to each other. For Rayleigh fading (the smallest integer value of m) we see the most disparity between the three with the Hughes bound falling approximately midway between the exact result and the union upper bound. More specifically, the “averaged” Hughes bound is 1 dB tighter than the union bound for high average bit SNR values. As m increases the difference between the Hughes bound and the exact results is at worst less than a few tenths of a decibel over a wide range of average bit SNRs. Hence, for high values of m we can conclude that it is accurate to use the former as a prediction of true system performance with the advantage that the numerical results can be obtained instantaneously. Note also that for high values of m a slightly less accurate result can be obtained by using the union bound. 8.2.1.7 Minimum-Shift-Keying (MSK) Following the same line of reasoning as discussed in Section 8.1.1.7 for the AWGN channel, we conclude here for the fading channel that the average BEP performance of the MSK receiver implemented as that is optimum for half-sinusoidal pulse-shaped OQPSK is identical to that of AM, BPSK, QPSK, and conventional (rectangular pulse-shaped) OQPSK. As a result of this observation, no further discussion is necessary. 8.2.2

Nonideal Coherent Detection

To compute the average error probability performance of nonideal coherent receivers of BPSK, QPSK, OQPSK, and MSK modulations transmitted over a fading channel, we again follow the approach taken by Fitz [16] wherein the randomness of the demodulation reference signal is modeled as an additive Gaussian noise independent of the AWGN associated with the received signal. In the absence

268

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

100

Average Bit Error Rate

10−1

10−2

m = 1 (Rayleigh)

m=2 a b

10−3

m=4

c

10−4

m = ∞ (AWGN)

10−5 −5

0

5

10

15

20

25

30

Average SNR per Bit [dB] Figure 8.5 Average BEP of 4-ary orthogonal signals over a Nakagami-m channel versus the average SNR per bit (a) Union bound; (b) Hughes bound; and (c) exact result.

of fading, this model was introduced in Section 3.2 and the performance of the receiver based on this model was given in Section 8.1.2. When Rician fading is present, Fitz [16] proposes a suitable modification of the Gaussian noise reference signal model as follows.

PERFORMANCE OVER FADING CHANNELS

269

Let ηn = ηI n + j ηQn denote a complex Gaussian RV that represents the fading associated with the received signal in the nth symbol interval. In the most general case when ηI n , ηQn are nonzero mean, αn = |ηn | is a Rician RV, which is the case considered by Fitz. With reference to (3.38), the k th matched-filter output in this symbol interval ynk , k = 1, 2, . . . , M now becomes specular component

random component

6

78 78  9  9 6

y˜nk = s˜k ηn e + N˜ nk = s˜k ηI n + j ηQn e j θc + s˜k ξI n + j ξQn e j θc +N˜ nk (8.154) The reference signal is also assumed to be degraded by the channel fading. As such, the additive Gaussian noise model for this signal given in (3.39) is now modified to j θc

specular component

random component

6 ' 78 9 6 ' 78  9

 c˜r = Ar Gs ηI n + j ηQn e j θc + Ar Gr ξI n + j ξQn e j θc +N˜ r

(8.155)

where Gs and Gr denote the SNR gains associated with its specular and random  components, respectively11 and ξin = ηin − ηin , i = I , Q. In view of the preceding complex Gaussian fading models for the received signal and * the reference signal, ) the decision statistic for the nth symbol, namely, Re y˜nk c˜r∗ is, as was the case for the fading-free channel, in the form of the real part of the product of two nonzero mean complex Gaussian random variables; hence, the error probability analysis discussed in Appendix 8A is once again applicable. To apply Stein’s analysis [29], we need to specify the first and second moments of y˜nk and c˜r . These are computed as follows. Assume that the real and imaginary components of the complex fading RV ηn have first and second moments η I = mI ,

η Q = mQ ,

var ηI = var ηQ = σ 2

(8.156)

2 2 m2I + m2Q ηI + ηQ specular power = K= = random power var ηI + var ηQ 2σ 2

(8.157)

Then the Rician factor K is given by

and the total power of ηn is given by ) * * ) 2 = 2σ 2 + m2I + m2Q = 2σ 2 (1 + K) E |ηn |2 =  = E ηI2 + ηQ

(8.158)

11 Later on, we shall specifically consider (as does Fitz [16]) the slow-fading case, which implies that the fading changes slowly in comparison to the memory length of the phase estimator. This implies that Gs = Gr = G. For the moment, however, we shall allow the specular and random gains to maintain their individual identity.

270

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

For BPSK signaling, s˜k = Ac Tb an (an = ±1 represents the binary data) and Ar = Ac = A. Thus, from (8.154) and (8.155)

  y˜nk

0 0  

2 2 K   2 2 A2 Tb2 y˜ nk  = ATb ηI n + ηI n = ATb mI + mQ = 1+K 2 3 2 1  A2 Tb2 + N0 Tb − y˜ nk  = (ATb )2 2σ 2 + var N˜ nk = (8.159) 1+K

and 0 0   '

2 2 ' Gs K   2 2 A2 Tb2 c˜r  = Gs ATb ηI n + ηI n = Gs ATb mI +mQ = 1+K  2 3 2 Gr   A2 Tb2 + N0 Tb (8.160) c˜r − c˜r  = Gr (ATb )2 2σ 2 + var N˜ nk = 1+K Letting z1p = c˜r , z2p = y˜nk , and A2 Tb = Eb , then, relating these moments to the parameters defined in (8A.3) and (8A.4), we get 1  2 1  2 1 Gs K Eb Tb , z1p = c˜r  = 2 2 21+K 1  2 1  2 1 K Eb Tb = z2p  = y˜ nk  = 2 2 21+K
 2 2 Gr Eb 1  1  N0 Tb   z1p − z1p = c˜r − c˜r  = = +1 2 2 2 1 + K N0
 2 2 1 Eb 1 1  N0 Tb  = z2p − z2p  = y˜nk − y˜ nk  = +1 2 2 2 1 + K N0

S1p = S2p N1p N2p

∗

 1 z1p − z1p z2p − z2p ρp = ρcp + jρsp = ' 2 N1p N2p  ∗   1 = ' c˜r − c˜r y˜nk − y˜ nk 2 N1p N2p = -

θ1p = θ2p ,

√ Gr Eb 1+K N0

Gr Eb 1+K N0

  1 Eb + 1 1+K + 1 N0

φ=0

(8.161)

PERFORMANCE OVER FADING CHANNELS

271

Using these parameters in (8A.6a) gives the arguments of the Marcum Q-function in (8A.5) as [16, Eqs. (8a), (8b)] 

a b

(

   2  S1p S1p S2p S1p S2p S2p 1 + ∓2 ∓ = N1p N2p N1p N2p 2 N1p N2p :  : 2 ; ; K K ; 1+K γ 1 ;  Eb 1+K Gs γ < <  , = γ = ∓ 1 1 2 N0 1 + 1+K Gr γ 1 + 1+K γ 1 = 2



(8.162)

where once again we have elected to express the result in terms of the average fading SNR per bit γ . Also since from (8.161) ρp is real, then from (8A.6a), we obtain A= 0

ρcp 2 1 − ρsp

= ρp = 0

√ Gr 1+K γ

Gr 1+K γ

 1  + 1 1+K γ +1

(8.163)

Finally, the average BEP for nonideal coherent detection of BPSK in a Rician fading environment is given by (8A.5), namely
  √ √  √ √ & A √  a+b 1% I0 1 − Q1 b, a + Q1 a, b − exp − ab 2 2 2 (8.164) where a, b, and A are as defined above. As mentioned in footnote 11, we will be interested in the case of slow fading, which implies Gs = Gr = G. Making this substitution in (8.162) and (8.163) gives the simplified results Pb (E) =



a b

(

: 2 : ; ; K K ; ; Gγ γ 1 ∓ < 1+K1  = < 1+K1 2 1 + 1+K Gγ 1 + 1+K γ

(8.165)

and A = 0

√ G 1+K γ

G 1+K γ

 1  + 1 1+K γ +1

(8.166)

which agrees with an unnumbered equation [between (8c) and (9)] in [16]. Figure 8.6 is an illustration of average BEP as computed from (8.164) together with (8.165) and (8.166) for K = 0 and K = 10 and three different nonideal coherence parameter values, namely, G = 3, 10, and 20 dB. We observe from these numerical results that over a wide range of average SNRs the BEP is rather insensitive to the value of G, particularly for the higher value of K.

272

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

100

Average Bit Error Probability Pb (E )

10−1

K =0 10−2

a K = 10

b

c

10−3

10−4

b a c

10−5

10−6

0

5

10

15

20

25

30

Average SNR per Bit [dB] Figure 8.6 Average BEP for nonideal coherent detection of BPSK over a Rician channel versus the average SNR per bit (a) G = 3 dB; (b) G = 10 dB; and (c) G = 20 dB.

As a check on previous results, the no-fading case which corresponds to K → ∞, γ → Eb /N0 , would result in 

a b

(

1 = 2



 Eb N0

Eb G ∓ N0

2 =

2 Eb √ G∓1 , 2N0

A=0

(8.167)

which agrees with (8.62). For Rayleigh fading (K = 0), the corresponding results are 

a b

(

 =

0 0

√

( ,

A= √

Gγ (Gγ + 1) (γ + 1)

(8.168)

Since Q1 (0, 0) = 1, then, from (8.164), we obtain   √ Gγ 1 Pb (E) = 1− √ 2 (Gγ + 1) (γ + 1)

(8.169)

PERFORMANCE OVER FADING CHANNELS

For a perfect phase reference (i.e., G → ∞), (8.169) simplifies to  γ 1 Pb (E) = 1− 2 1+γ

273

(8.170)

which is consistent with the result given in (8.104) for ideal coherent detection. To extend the results presented above to other quadrature modulation schemes with I and Q carrier components that are independently modulated, such as QPSK, OQPSK, MSK, and QAM, one merely recognizes that for such schemes the average BEP in the presence of fading can be expressed as Pb (E) = 12 PbI (E) + 12 PbQ (E)

(8.171)

where PbI (E) , PbQ (E) are respectively the average BEPs for the I and Q data streams. Thus, Stein’s analysis technique [29] of Appendix 8A can again be applied to separately evaluate PbI (E) and PbQ (E) and thereby arrive at the generalization of the AWGN BEP results given by (8.63) through (8.66) to the fading channel case. For example, for QPSK the analogous results to (8.165) and (8.166) are obtained by replacing G by 2G whereupon the former reduces to (8.63) when K → ∞. The specific details for the remaining quadrature modulation schemes are left to the reader. 8.2.2.1 Simplified Noisy Reference Loss Evaluation In this section, we take a different approach [44] for evaluating performance in the presence of slow fading and carrier phase error based on expanding the conditional (on the carrier phase error) BEP in a Maclaurin series followed by averaging the leading terms of this result over both the statistics of the carrier phase error and the channel fading. This approach is valid for small phase errors and allows closed-form expressions for the average BEP to be derived in forms that lend themselves toward obtaining simple formulas for the associated noisy reference loss, that is, the additional loss in SNR at a given BEP caused by the lack of perfect carrier synchronization. Numerical evaluation of this loss based on the use of these formulas provides excellent accuracy when compared with that obtained from exact BEP evaluation that requires twofold numerical integration. We shall illustrate the method for BPSK and QPSK in the presence of Rayleigh fading; however, it is easily extended to other forms of modulation whose conditional bit error probability is known in closed form as well as other fading channel models. As pointed out in Section 3.2, for a broad class of carrier synchronization loops, such as a PLL in the discrete carrier case or a Costas or decision feedback datastripping loop (data-aided loop), in the suppressed carrier case, the PDF of the phase error is typically modeled by a Tikhonov distribution of the form in (3.37). Although Section 8.2 explicitly considers the Costas loop as the carrier synchronization means, for simplicity of our discussion here, we shall assume that a pilot tone (unmodulated carrier) is transmitted and recovered at the receiver by a PLL to provide carrier synchronization. In this case, the PDF of φ is explicitly given by  (3.37), where ρc = Pc /N0 BL is the loop SNR with Pc denoting the power in the

274

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

discrete carrier (pilot) and BL the single-sided loop bandwidth.12 For a given available total power Pt , a fixed fraction of it η = Pc /Pt is allocated to the pilot and the remaining fraction 1 − η = (Pt − Pc ) /Pt = Pd /Pt is available for data detection. Since Pd = Eb /Tb , then the loop SNR can be linearly related to Eb /N0 by ρc =

1 Eb  Eb Pc Pc /Pt Pd Tb η =C = = N 0 BL 1 − η BL T b N 0 N0 (Pd /Pt ) N0 BL Tb

(8.172)

where C is a constant of proportionality that, when expressed in decibels (dB), represents the amount by which the loop SNR (in dB) exceeds the energy-per-bit to noise power spectral density ratio (in dB). Note that a plot of error probability versus Eb /N0 for the scenario described above wherein the ratio of carrier-to-data power is fixed does not exhibit an irreducible error probability, that is, a finite BEP in the limit of infinite Eb /N0 , since ρc increases (linearly) as Eb /N0 increases. For the slow-fading channel and under the further assumption that the fading bandwidth is much smaller than the loop bandwidth,13 the average BEP can be evaluated from the approach discussed in Section 8.1.2 modified by replacing Eb /N0 by the instantaneous fading SNR γ . Specifically, using the conditional BEP of BPSK in (8.49), the Tikhonov phase error PDF of (3.37) and the Rayleigh fading SNR PDF in (2.7), we get14 ∞ π

Pb (E) = 0

−π

Pb (E; φc , γ ) pφc (φc |γ ) pγ (γ ) d φc d γ


  exp (Cγ cos φ ) 1 γ c d φc d γ exp − 2π I0 (Cγ ) γ γ 0 −π (8.173) to a plot A plot of Pb (E) versus γ obtained from (8.173) for fixed C relative  of BEP

√ for ideal coherent detection in AWGN, namely, Pb (E) = Q 2Eb /N0 , reveals the amount of degradation in this performance measure caused by the combination of fading and carrier phase error. Alternatively, this degradation is specified in terms of the amount of additional SNR that must be provided at a fixed value of

=

∞ π

Q

'

2γ cos φc

12 Extension to the case of suppressed carrier synchronization loops is straightforward; however, because of the squaring loss associated with such loops, the effective loop SNR (which includes the squaring loss) will not be linearly related to Eb /N0 [see (8.51) together with (8.52)]. Thus, although the technique to be described is, in principle, still applicable, the ability to obtain closed-form results for the SNR loss due to the noisy reference will depend on the particular fading statistics assumed. 13 This is the same assumption as that made by Eng and Milstein [45]. The implication of this assumption in the mobile communication application where Doppler spread is a consideration that the loop bandwidth should be at least on the order of 5 times the Doppler spread. This, in turn, together with the fixed carrier-to-data power ratio assumed, has a direct impact on the values of the constant C that characterize this application. Since the theory is most accurate when C is large, the numerical results that follow are more typical of the low-mobility (small-Doppler-spread) case. 14 Eng and Milstein [45] also consider an approximation to (8.173) obtained by replacing the average of 

√

√ 

√  Pb (E; φc , γ ) over φc by Q 2γ cos φc = Q 2γ I1 (ρc ) /I0 (ρc ) = Q 2γ I1 (Cγ ) /I0 (Cγ ) , thus eliminating one of the integrations.

PERFORMANCE OVER FADING CHANNELS

275

BEP, specifically, the noisy reference loss. To compute noisy reference loss, one must translate the vertical degradation (increase in BEP at a fixed SNR) obtained from the plot of Pb (E) versus SNR to a horizontal degradation (increase in required SNR at a fixed BEP). A simple method to accomplish this goal was discussed in Ref. 45 for the AWGN channel and relied on first simplifying (approximating) the evaluation of Pb (E) by expanding Pb (E; φc ) in a Maclaurin series in φc prior to averaging over pφc (φc ) and then maintaining only the first two terms. This approach will now be extended to the fading channel to arrive at a similar simple rule-of-thumb formula for evaluating noisy reference loss. For BPSK modulation, adapting the expansion of (8.49) into a Maclaurin series to the fading channel and averaging over the PDF of φc , we obtain the following equation, analogous to Eq. (26) of Ref. 46: '  1 - γ exp (−γ ) σφ2c + higher-order terms 2γ + (8.174) Pb (E; γ ) = Q 2 π For the AWGN case, if one makes the assumption of large loop SNR, then the loop is said to perform in its linear region of operation and the Tikhonov PDF of (3.37) can be approximated by a Gaussian PDF for which σφ2c ∼ = 1/ρc = −1 [C (Eb /N0 )] . For the fading channel, we shall (as was done by Eng and Milstein [45]) make the analogous assumption that σφ2c ∼ = (Cγ )−1 . Strictly speaking, the assumption of linear region behavior for the fading channel case is not totally valid since here the SNR, γ , is a RV that ranges from zero to infinity. However, for large γ the PDF of γ will be skewed to the right, and thus the contribution to the average BEP from small values of γ (where the linear region assumption would be violated) will likewise be small. The true degree of validity of this linear region approximation over the entire range of γ can be justified only by comparison with the exact results computed by using (8.173) for evaluation of average BEP and will be demonstrated shortly. In the meantime, proceeding with the inverse linear relation given above between σφ2c and γ and neglecting higher-order terms, we obtain from (8.174) Pb (E; γ ) = Q

'

 2γ +

1 exp (−γ ) √ 2C π γ

(8.175)

where the first term represents the ideal (perfect carrier phase reference) performance. Averaging (8.175) over the PDF of γ gives the desired average BEP, which is tabulated below in closed form (with the help of tabulated integrals in Ref. 23) for several different fading channel models whose PDFs are found in Chapter 2: Nakagami-m Fading


  m + 12 1 mm m 1 γ ∼ 1, m + , m + 1; F Pb (E) = 2 1 2 π (m + γ )m+1/2  (m + 1) 2 m+γ

  m − 12 mm 1 (8.176) , m> + √ 2 2 π γ C (m) (m + γ )m−1/2

276

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

Rician Fading

2  a + b2 p 1 1+ exp − I0 (ab) Pb (E) ∼ = Q1 (a, b) − 2 1+p 2 
2γ + 1 + K (1 + K) (8.177) + exp −K ' 2 (γ + 1 + K) 2C γ 2 + (1 + K) γ
 K (1 + K) × I0 2 (γ + 1 + K) where  a=

K




1 + 2p − 2 (1 + p)

-

 p , 1+p

 b=

K




1 + 2p + 2 (1 + p)

-

 p 1+p (8.178)

with p = γ / (1 + K). Rayleigh Fading  γ 1 1 ∼ Pb (E) = 1− + √ 2 1+γ 2C γ (1 + γ )

(8.179)

Figures 8.7 and 8.8 illustrate the average BEP performances for Nakagami-m and Nakagami-n (Rician) fading with a loop SNR of 10 dB above Eb /N0 . In each figure the exact BEP together with the appropriate fading PDF is plotted along with the approximate result obtained from (8.176) or (8.177). We observe that over a wide range of values for the m and K fading parameters the approximate and exact results are in virtually perfect agreement. Further numerical experiments show that these approximate expressions can be used safely over a wide range of fading conditions as long as the loop parameter C is above approximately 8 dB (i.e., loop SNR is about 8 dB above Eb /N0 ). Because of the linear relation between Eb /N0 and loop SNR in (8.172), a plot of Pb (E) on a logarithmic scale versus average SNR in dB would be asymptotically parallel to the analogous curve for ideal coherent detection. Thus, the noisy reference loss at a given BEP can be evaluated by dividing the degradation in BEP from the ideal by the slope of the BEP performance curve. In mathematical terms, the noisy reference loss L (in dB) is evaluated as L=

Pb (E) PbI (E) d log10 PbI (E) − d

log10

,

 = 10 log10 γ

(8.180)

where PbI (E) denotes the ideal BEP as determined from the first term in the Maclaurin series expansion of Pb (E).

PERFORMANCE OVER FADING CHANNELS

277

100 Perfect Sync. Imperfect Sync. (Exact) Imperfect Sync. (Approxim. (8.176))

Average Bit Error Probability Pb(E )

10−1

m = 0.75

10−2

m = 1 (Rayleigh)

10−3

m=2

10−4

10−6

m=4

(AWGN)

10−5

0

5

10

15

20

25

30

Average Received SNR per Bit [dB] Figure 8.7 Combined effect of Nakagami-m fading and imperfect synchronization on the average bit error probability of BPSK.

As an example of the evaluation of (8.180), consider the case of Rayleigh fading where from (8.179)  
1 1 γ 1 1− √ log10 PbI (E) = log10 1− = log10 2 1+γ 2 10−/10 + 1 (8.181) Differentiating (8.181) with respect to  and then substituting back in terms of γ gives 1 d log10 PbI (E) =−  0  1 d 20γ 1 + γ 1+

1 γ

 −1

(8.182)

Finally, since from (8.179) 1 Pb (E) = 1 + √ √ √  PbI (E) C γ 1+γ − γ

(8.183)

278

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

100 Perfect Sync. Imperfect Sync. (Exact) Imperfect Sync. (Approxim. (8.177))

Average Bit Error Probability Pb(E )

10−1

10−2

K = 0 (Rayleigh)

10−3

K = 3 dB K = 6 dB

10−4

K = 10 dB

(AWGN) 10−5

10−6

0

5

10

15

20

25

30

Average Received SNR per Bit [dB] Figure 8.8 Combined effect of Nakagami-n (Rice) fading and imperfect synchronization on the average bit error probability of BPSK.

then the noisy reference loss (in dB) is [from (8.180)] given by the desired simpleto-evaluate formula  
1 1 1 L = 20γ 1 + 1 + − 1 log10 1 + √ √ √  (8.184) γ γ C γ 1+γ − γ where from the first term of (8.179) γ is related to the ideal BEP at which one desires to evaluate this loss by γ =

(1 − 2PbI (E))2 1 − (1 − 2PbI (E))2

(8.185)

For sufficiently large γ , the noisy reference loss of (8.184) asymptotically becomes
 2 L = 10 log10 1 + C

(8.186)

PERFORMANCE OVER FADING CHANNELS

279

TABLE 8.2 Noisy Reference Loss Lr (in dB) for BPSK over Rayleigh Fading Channels Average BEP PbI (E)

Loop Parameter C = 10 (10 dB)

Loop Parameter C = 20 (13 dB)

10−2 10−3 10−4 10−5

0.824 0.795 0.792 0.791

0.431 0.415 0.414 0.413

independent of γ . This behavior is exhibited by the numerical results of Fig. 8.7 for m = 1 and Fig. 8.8 for K = 0. Table 8.2 shows for values of C = 10 and 20 the evaluation of the noisy reference loss for BPSK as given by (8.184) as a function of the average BEP at which one desires to operate and also confirms the convergence of this loss to the asymptotic result given in (8.186). As would be expected, the noisy reference loss decreases as BEP decreases since for smaller BEP, Eb /N0 is larger and hence from the linear relationship in (8.172), ρc is proportionally larger resulting in a smaller loss. These results are easily extended to QPSK by using the appropriate Maclaurin series expansion of the conditional BEP for that modulation as in (8.57), which when adapted to the fading channel, becomes [analogous to Eq. (26) in Ref. 31] '



1 2γ + 2

-

γ (1 + 2γ ) exp (−γ ) σφ2c + higher-order terms π (8.187) Then, the exact average BEP is computed by substituting γ for Eb /N0 in (8.56) and averaging over the fading PDF, whereas the approximation to the average BEP would be obtained by averaging (8.187) over this same PDF substituting (Cγ )−1 for σφ2c . With regard to the latter, the following, expressed in terms of (8.176) and (8.179), are obtained: Pb (E; γ ) = Q

Nakagami-m Fading  1 Pb (E) QPSK ∼ = Pb (E) |BPSK + C

-



 m + 12 mm γ , π (m + γ )m+1/2  (m)

1 2 (8.188) m>

Rayleigh Fading  1 ' 1 Pb (E) QPSK ∼ γ = Pb (E) |BPSK + 2C (1 + γ )3/2

(8.189)

Unfortunately, a closed-form result for the Rician case is not obtainable. The results in (8.188) and (8.189) clearly identify the added BER degradation attributed to QPSK relative to BPSK. Figure 8.9 illustrates the analogous curve to Fig. 8.7 now for the QPSK case. As would be expected, because of the crosstalk between

280

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

Average Bit Error Probability Pb(E ) of QPSK

100 Perfect Sync. Imperfect Sync. (Exact) Imperfect Sync. (Approxim. (8.188))

10−1

10−2

m = 0.75

10−3

m=2

m = 1 (Rayleigh)

10−4

m=4

(AWGN) 10−5

0

5

10

15

20

25

30

Average Received SNR per Bit [dB] Figure 8.9 Combined effect of Nakagami-m fading and imperfect synchronization on the average bit error probability of QPSK.

the I and Q channels the noisy reference loss is larger for QPSK than it is for BPSK; however, the relative accuracy of the approximate evaluation is still quite good. Thus, we can still use the result in (8.187) to come up with a simple expression for the noisy reference loss in the QPSK case. Following the same procedure as for BPSK we eventually arrive at the noisy reference loss formula (in dB)    
1 1 1 + 2γ L = 20γ 1 + 1 + − 1 log10 1 + √ √ √  γ γ C γ 1 + γ − γ (1 + γ ) (8.190) with γ still given by (8.185). For sufficiently large γ , the noisy reference loss of (8.190) asymptotically becomes
 4 L = 10 log10 1 + C

(8.191)

which is again independent of γ . Again, Table 8.3 shows for values of C = 10, 20 the evaluation of the noisy reference loss for QPSK as given by (8.190) as a

PERFORMANCE OVER FADING CHANNELS

281

TABLE 8.3 Noisy Reference Loss Lr (in dB) for QPSK over Rayleigh Fading Channels Average BEP PbI (E)

Loop Parameter C = 10 (10 dB)

Loop Parameter C = 20 (13 dB)

10−2 10−3 10−4 10−5

1.494 1.4664 1.4616 1.4613

0.8088 0.7935 0.7919 0.7918

function of the average BEP at which one desires to operate and also confirms the convergence of this loss to the asymptotic result given in (8.191). 8.2.3

Noncoherent Detection

As has previously been alluded to, in a multipath environment it is often difficult in practice to achieve good carrier synchronization; hence, in such instances, it is necessary to employ a modulation for which noncoherent detection is possible. The most popular choice of such a modulation in fading channel applications is orthogonal M-FSK whose error probability performance in AWGN was considered in Section 8.1.3. It is a simple matter now to extend these results to the fading channel. In particular, since each term of (8.67) is purely an exponential of the SNR, then, applying the MGF-based approach to this equation, we obtain the average SEP Ps (E) =

M−1 


m+1

(−1)

m=1

M −1 m




 m 1 Mγ − m+1 s m+1

(8.192)

where the moment generating function Mγs (−s) is obtained from any of the results in Section 5.1 with γ replaced by the average symbol SNR γ s . Thus, for Rayleigh fading, using (5.5), we have Ps (E) =

M−1 


(−1)m+1

m=1

M −1 m



1

 1 + m 1 + γs

(8.193)

which for the special case of binary FSK simplifies to Pb (E) = 1/ (2 + γ ) in agreement with [6, Eq. (14-3-12)]. For Rician fading, using (5.11) gives Ps (E) =

M−1  m=1


(−1)

m+1

M −1 m



1+K

 1 + K + m 1 + K + γs

Kmγ s 

× exp − 1 + K + m 1 + K + γs

(8.194)

282

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

which agrees with Sun and Reed [24, Eq. (8)] and reduces to (8.193) when K = 0. Finally, for Nakagami-m fading, using (5.15), we obtain Ps (E) =

M−1  l=1


(−1)l+1

M −1 l

 %

(l + 1)m−1  &m γ 1 + l 1 + ms

(8.195)

where we have changed the summation index to avoid confusion with the Nakagamim fading parameter. As expected, (8.195) reduces to (8.193) when m = 1. Before concluding this section, we note that the results for average BEP over a fading channel can be obtained, as was the case for the AWGN channel, by applying the relation between bit and symbol error probability given in (8.68) to the preceding results. We furthermore note that although we have specifically addressed M-FSK, these results apply equally well to any M-ary orthogonal signaling scheme transmitted over a slow, flat-fading channel and noncoherently detected at the receiver. For nonorthogonal M-FSK, we observed in Section 8.1.3 that a simple analytical result for average BEP over the AWGN was possible for the binary case, namely, (8.70). To extend this result to the fading channel, we first rewrite it in the alternative form [see Eq. (9A.14) of Appendix 9A] % √ √  √ √ & b, a + Q1 a, b (8.196) Pb (E) = 12 1 − Q1 and then make use of the alternative representation of the Marcum Q-function in (4.39) and (4.42) to allow application of the MGF-based approach [see Eq. (8A.12)]. Using the definitions of a and b in (8.71), the result of this application produces




'  1 1 − ζ2 1 π 2 1+2ζ sin θ + ζ 2 1 + − dθ, M 1−ρ γ 4π −π 1 + 2ζ sin θ + ζ 2 4 : ' ; 1 − 1 − ρ2  ; < ' ζ= (8.197) 1 + 1 − ρ2

Pb (E) =

where ρ is the correlation coefficient of the two signals. To obtain specific results for the various fading channels, one merely substitutes the appropriate MGF from Section 5.1 in (8.197) analogous to what was done previously for the orthogonal signaling case. The specific analytical results are left as an exercise for the reader. As an illustration of the numerical results that can be obtained from (8.197) after making the aforementioned substitutions, Figs. 8.10, 8.11, and 8.12 illustrate the average BEP performance for Nakagami-q (Hoyt), Nakagami-n (Rice), and Nakagami-m channels, respectively. 8.2.4

Partially Coherent Detection

In this section, we apply the MGF-based approach to the AWGN results of Section 8.1.4 to predict the performance of partially coherent detection systems in the

PERFORMANCE OVER FADING CHANNELS

283

Average Bit-Error-Rate Pb(E )

100

10−1

q=0 c d ab 10−2

q = 0.3

q=1 Rayleigh

10−3

0

5

10

15

20

25

d c b da b a c

30

Average SNR per Bit [dB] Figure 8.10 Average BEP of correlated BFSK over a Nakagami-q (Hoyt) channel (a) ρ = 0; (b) ρ = 0.2; (c) ρ = 0.4; and (d) ρ = 0.6.

presence of fading. The steps to be followed parallel those of the previous sections and as such the presentation will be brief. For BPSK with conventional (one-symbol observation) detection, the conditional (on a fixed phase error φc ) BEP is in the form of a Gaussian Q-function as described by (8.72). Thus, first performing the averaging over the fading takes the form of (5.1), which is expressed in terms of the MGF of the fading as in (5.3). Finally, performing the averaging over the Tikhonov phase error PDF gives the desired result: 


π π/2 exp (ρc cos φc ) cos2 φc 1 dθ d φc Mγ − (8.198) Pb (E) = 2 π 2π I0 (ρc ) sin θ −π 0 For orthogonal and nonorthogonal BFSK, the results [see (8.75) together with (8.70)] are expressed in terms of the first-order Marcum Q-function. However, in these cases the ratio of the two arguments of this function [see (8.76) and (8.77)] are not independent of SNR, and thus the MGF-based approach is not useful here in allowing an easy evaluation of average BEP. Instead, one must resort to the brute-force approach of replacing Eb /N0 by γ in the a and b parameters and then performing the average over the PDF of γ as appropriate for the type of fading

284

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

100 10−1

n = 0 (Rayleigh)

Average Bit-Error-Rate Pb(E )

10−2

n =2 (K = 6.02 dB)d bc

10−3

c d ab a

10−4 10−5

n =4 (K = 12.04 dB)

10−6

d a

n > ∞ (AWGN)

10−7

c b

d c

10−8 10−9

a b 0

5

10

15

20

25

30

Average SNR per Bit [dB] Figure 8.11 Average BEP of correlated BFSK over a Nakagami-n (Rice) channel (a) ρ = 0; (b) ρ = 0.2; (c) ρ = 0.4; and (d) ρ = 0.6.

channel under consideration. A similar statement is made for the multiple-symbol detection case since again the ratio of the two arguments of the Marcum Q-function [see (8.82)] is not independent of SNR. 8.2.5

Differentially Coherent Detection

In this section of the chapter, we consider the characterization of the error probability performance of M-DPSK when transmitted over a fading channel. This modulation/detection combination has received a lot of attention in the literature, particularly the M = 4 case (DQPSK), which has been adopted in the most recent North American and Japanese digital cellular system standards. For instance, Tjhung et al. [30] and Tanda [31] analyzed the average BEP of DQPSK over slow Rician and Nakagami-m fading channels, respectively. Later, Tellambura and Bhargava [32] presented an alternative unified BEP analysis of DQPSK over Rician and Nakagami-m fading channels. In keeping with the unifying theme of this book, our purpose in this section is to once again unify and add to the previous contributions by obtaining results for arbitrary values of M as well as for a broad class of fading channels. As in Section 8.1.5, we first focus on conventional (two-symbol

PERFORMANCE OVER FADING CHANNELS

285

100 10−1

m = 1/2

b d a m = 1 (Rayleigh) bd a

Average Bit-Error-Rate Pb(E )

10−2 10−3

m=2 d c a b

10−4 10−5

m =4 d c ab

10−6

m > ∞ (AWGN) 10−7

a

10−8 10−9 0

5

10

b c

d

15

20

25

30

Average SNR per Bit [dB] Figure 8.12 Average BEP of correlated BFSK over a Nakagami-m channel (a) ρ = 0; (b) ρ = 0.2; (c) ρ = 0.4; and (d) ρ = 0.6.

observation) detection of M-PSK for which, as noted there, the SEP is already in the desired form, namely, one that lends itself to immediate application of the MGF-based approach. 8.2.5.1 M-ary Differential Phase-Shift-Keying (M-DPSK)—Slow Fading Conventional Detection (Two-Symbol Observation) With reference to (8.84), which gives the SEP of M-DPSK for the AWGN channel, we observe that the integrand is already an exponential function of the symbol SNR. Thus, unlike the cases where the integrand’s dependence on SNR is through the Gaussian and Marcum Q-functions, no alternative form is necessary here to allow averaging over the fading statistics of the channel. All that needs to be done is to replace Es /N0 by γs in the argument of the exponential and then average over the PDF of γs resulting in the MGF-based expression

+ , √ √

gPSK π/2 Mγs − 1 − 1 − gPSK cos θ π  dθ, gPSK = sin2 Ps (E) = √ 2π M 1 − 1 − gPSK cos θ −π/2 (8.199)

286

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

or the simpler form derived from (8.90): 1 Ps (E) = π

(M−1)π/M


Mγs −

0

gPSK √ 1 + 1 − gPSK cos θ

 dθ

(8.200)

The special case of binary DPSK wherein gPSK = 1 simplifies to the closed-form result Pb (E) = 12 Mγ (−1)

(8.201)

Comparing (8.201) with the special case of (8.192) corresponding to M = 2, namely, Pb (E) = 12 Mγ − 12 , then, since, independent of the type of fading, the MGF of the fading SNR Mγ (−s) is only a function of the product sγ [see, e.g., Eqs. (5.5), (5.8), (5.11), (5.15)], we conclude that the BEP of noncoherent orthogonal FSK is 3 dB worse in average fading SNR than that of DPSK. We remind the reader that this is the same conclusion reached when comparing these two modulation/detection schemes over the AWGN. To obtain the average BEP corresponding to values of M > 2, we make use of the AWGN results in (8.86), which correspond to a Gray code bit-to-symbol mapping. Since each of the BEP results in (8.86) is expressed in terms of the function F (ψ) defined in (8.87), which, analogous to (8.84), has an integrand with exponential dependence on symbol SNR, then clearly the average BEP over the fading channel can be obtained from (8.86) by replacing F (ψ) with sin ψ F (ψ) = − 4π

π/2

−π/2



  Mγ − log2 M (1 − cos ψ cos t) dt 1 − cos ψ cos t

(8.202)

or the simpler form [see Eq. (4.123)] 1 F (ψ) = − 4π



π−ψ −(π−ψ)

Mγ



− log2 M

sin2 ψ 1 + cos ψ cos t

dt

(8.203)

The average BEP for the special case of DQPSK can, of course, be obtained from the first relation in (8.86) together with (8.202) or (8.203) with M = 4. In view of (8.89) for the AWGN channel, it can also be obtained in a form analogous to (8.197):

 

π  1

1 − ζ2 1 2 − 1 + 1 + 2ζ sin θ + ζ M dθ, √ γ 4π −π 1 + 2ζ sin θ + ζ 2 2  √ 2− 2  (8.204) ζ= √ 2+ 2

Pb (E) =

PERFORMANCE OVER FADING CHANNELS

287

Using instead the alternative forms of the first-order Marcum Q-functions given in (4.20) and (4.21) corresponding to only positive values of the integration variable, a form equivalent to (8.204) can be obtained from Eq. (3) of Ref. 32: Pb (E) =

1 2π

π 0

√

1 2 − cos θ

   √ Mγ − 2 − 2 cos θ d θ

(8.205)

Without further ado, we now give the specific results from above corresponding to Rayleigh, Rician, and Nakagami-m channels. These results as well as those for the other fading channels previously discussed are taken from Ref. 33. Rayleigh Fading given by Ps (E) =

π sin M 2π

From (8.199) and (5.5), the average SEP of M-DPSK is

π/2 −π/2



1 − cos

π M

cos θ



1

 d θ (8.206) π 1 + γ s 1 − cos M cos θ

which is in agreement with Eq. (6) of Ref. 24. The corresponding binary DPSK result is Pb (E) =

1 2 (1 + γ )

(8.207)

which agrees with Eq. (14-3-10) of Ref. 6. For DQPSK, the average BEP is evaluated from (8.204) in closed form as  Pb (E) =



1 1 1 −  2 (1+2γ )2 2γ 2

−1

  

(8.208)

which agrees with an equivalent result obtained by Eqs. (18) and (13) of Refs. 30 and 31, respectively, namely  √  √ ' 2 2γ + 2 − 1 1 + 2γ − 1 + 4γ + 2γ 1 √  √  Pb (E) = ' ' 2 1 + 4γ + 2γ 2 2γ − 2 − 1 1 + 2γ − 1 + 4γ + 2γ 2 (8.209) or the one reported by Tellambura and Bhargava [32, Eq. (8)]:   √ 2γ 1 1− ' Pb (E) = 2 1 + 4γ + 2γ 2

(8.210)

288

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

For other values of M, the average BEP is computed from (8.86) using

F (ψ) = −

sin ψ 4π

π/2 −π/2

1

  dt

(1 − cos ψ cos t) 1 + γ log2 M (1 − cos ψ cos t) (8.211)

in place of F (ψ). Rician Fading Ps (E) =

From (8.199) and (5.11), the average SEP of M-DPSK is given by

π sin M 2π

1+K + ,

π cos θ 1 + K + γ s 1 − cos M cos θ −π/2 1 − cos   

π Kγ s 1 − cos M cos θ

 dθ (8.212) × exp − π 1 + K + γ s 1 − cos M cos θ π/2



π M

which is in agreement with Sun and Reed [24, Eq. (5)]. The corresponding binary DPSK result is Pb (E) =

1 2




1+K 1+K +γ




exp −

Kγ 1+K +γ

 (8.213)

For DQPSK, the average BEP is most easily evaluated from (8.205), which produces e −K Pb (E) = 2π

1+K √  % & √ 0 2 − cos θ 1 + K + γ s 2 − 2 cos θ   K + K) (1   dθ × exp − √ 1 + K + γ s 2 − 2 cos θ π

(8.214)

in agreement with Ref. 32, Eq. (6). For other values of M, the average BEP is computed from (8.86) using

1+K

 , −π/2 (1 − cos ψ cos t) 1 + K + γ log2 M (1 − cos ψ cos t)

 Kγ log2 M (1 − cos ψ cos t)

 × exp − dt (8.215) 1 + K + γ log2 M (1 − cos ψ cos t)

sin ψ F (ψ) = − 4π

in place of F (ψ).

π/2

+

PERFORMANCE OVER FADING CHANNELS

289

Nakagami-m Fading From (8.199) and (5.15), the average SEP of M-DPSK is given by π π/2 sin M 1 Ps (E) = % & dθ 

2π −π/2 1 − cos π cos θ 1 + γ s 1 − cos π cos θ  m M m M (8.216) and is illustrated in Fig. 8.13 as a function of average symbol SNR and parameterized by m. The corresponding binary DPSK result is
m m 1 Pb (E) = (8.217) 2 m+γ which agrees with the expression attributed to Barrow [34] and later reported by Wojnar [22, Eq. (2.38)] and Crepeau [35, Eq. (B1)]. For DQPSK, the average BEP is evaluated from (8.205) as
m π m 1 1 Pb (E) = √  m d θ (8.218) √ 2γ 2π m + 2γ 0 2 − cos θ 1 − m+2γ cos θ which agrees with Eq. (2.33) of Ref. 32.

100

m = 0.5

Average Symbol Error Probability PS (E )

10−1

m =1 10−2

m =2 10−3

m =4 10−4

10−5

10−6

0

5

10

15

20

25

30

Average SNR per Symbol [dB]

Figure 8.13 Average SEP of 8-DPSK over a Nakagami-m channel versus the average SNR per symbol.

290

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

Finally, the function necessary to compute average BEP for other values of M is given by m 

1 sin ψ π/2 m

 F (ψ) = − dt 4π −π/2 (1 − cos ψ cos t) m + γ log2 M (1 − cos ψ cos t) (8.219) Multiple-Symbol Detection The upper bound on the BEP for the AWGN channel as given in (8.97) is easily extended to the*fading channel case by recognizing ) that the form of the probability Pr zˆ nk > znk as described by (8.80) is identical to (8.196), which characterizes noncoherent detection of orthogonal FSK. As such, one can make use of the results in Section 8.2.3 to express each term in the sum of (8.97) in an MGF-based form analogous to (8.197), namely



π 0 ) * log2 M 1 − ζ2 1 2 |δ| Ns + Ns − Pr zˆ nk > znk = Mγ − 4π −π 1 + 2ζ sin θ + ζ 2 4 : ' ; 

Ns − Ns − |δ|2  ; 2 < dθ, ζ= ' × 1 + 2ζ sin θ + ζ Ns + Ns − |δ|2 (8.220) with δ as defined in (8.99). Substituting (8.220) into (8.97) gives the desired upper bound on BEP. It is left as an exercise for the reader to evaluate (8.220) for the various fading channels based on the same procedure as that stated at the end of Section 8.2.3. 8.2.5.2 M-ary Differential Phase-Shift-Keying (M-DPSK)—Fast Fading Conventional Detection (Two-Symbol Observation) Until now in this chapter we have focused entirely on the performance of digital communication systems operating over slow-fading channels. For conventional differentially coherent detection of M-PSK the assumption of slow fading is tantamount to assuming that the fading amplitude is constant over a duration of at least two symbol intervals. A suitable modification of this model for the case of fast fading is to assume that the fading amplitude is constant within the duration of a single symbol but varies from symbol to symbol. In other words, each symbol interval is characterized by its own fading amplitude that, relative to that of another symbol interval, satisfies a given discrete correlation function that is related to the nature of the fading channel (more about this later on). Such a discrete fast-fading model is an approximation to the true channel behavior wherein the fading varies continuously with time. To fully understand the method used to evaluate average error probability for this scenario, we must first review the system model discussed in Section 3.5, making the necessary modifications in notation to account for the presence of fast fading on the signal. We shall focus all our attention on the Rician channel (with results for the Rayleigh channel obtained as a special case) and develop only the binary DPSK case.

291

PERFORMANCE OVER FADING CHANNELS

Consider a binary DPSK system transmitting information bits over an AWGN channel that is also perturbed by fast Rician fading. The normalized k th information bit at the input to the system is given by xk = e j θk

(8.221)

where for binary transmission θk takes on values of 0 and π corresponding, respectively, to values of 1 and −1 for xk . The input information bits are differentially encoded, resulting in the transmitted bit vk =

' ' 2Eb e j θk = 2Eb e j (θk−1 +θk ) = vk−1 xk

(8.222)

After passing through the fast-fading channel, the received information bit in the k th transmission interval is wk = Gk vk + Nk

(8.223)

where Gk is the complex Gaussian fading amplitude associated with the k th received bit and) Nk is*a zero mean complex Gaussian noise RV with correlation function E Nk∗ Nm = 2N)0 δ (k − m). * Denoting the mean and variance of Gk by η = E {Gk } and σ 2 = 12 E |Gk − η|2 (both assumed to be independent of k ), 

then, for the assumed Rician channel, the magnitude of Gk , namely, αk = |Gk |, has PDF
 αk2 (1 + K) 2αk ' 2 (1 + K) exp −K − K (1 + K) (8.224) p (αk ) = αk I0    ) * where  = E αk2 = 2σ 2 (1 + K). Furthermore, the adjacent complex fading amplitudes have correlation 1 2E

) * (Gk−1 − η)∗ (Gk − η) = ρσ 2 ,

0≤ρ≤1

(8.225)

where ρ is the fading correlation coefficient whose value depends on the fast-fading channel model that is assumed. At the receiver, the received signal wk for the current bit interval is complex conjugate multiplied by the same signal corresponding to the previous bit interval and the real part of the resulting product forms the decision variable (which is multiplied by 2 for mathematical convenience) ) *  ∗ zk = 2Re wk∗ wk−1 = wk∗ wk−1 + wk wk−1

(8.226)

Comparison of zk with a zero threshold results in the final decision on the transˆ mitted bit xk , namely, xˆk = ej θk = sgn zk , which is consistent in form with the decision rule given in (3.52).

292

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

We note that conditioned on the information bit xk , the components wk−1 and wk are complex Gaussian RVs (since both the fading amplitude and additive noise RVs are complex Gaussian). Thus, (8.226) represents a Hermitian quadratic form of complex variables. While it is possible to use an MGF-based approach based on the conditional MGF of such a quadratic form first considered by Turin [36] and later reported in Appendix B of Schwartz et al. [37], in this particular case there is an easier way to proceed. Specifically, letting D = zk xk =1 denote the decision variable corresponding to transmission of a +1 information bit, then, based on the abovementioned decision rule, the average BEP is given by Pb (E) = Pr {D < 0}

(8.227)

The solution to (8.227) is a special case of the problem considered in Ref. 6, App. B, which has also been reconsidered in alternative forms in Appendix 9A of this text. Specifically, letting A = B = 0, C = 1, Xk = wk−1 , Yk = wk in (9.A.2), then the decision variable (8.226) is identical to that in (9.A.1) when L = 1. Evaluating the various coefficients required in (9A.10) produces after much simplification the following results v2 1 + K + γ (1 + ρ) , η= = v1 1 + K + γ (1 − ρ)

 a = 0,

b=

2Kγ 1+K +γ

(8.228)

where γ = Eb /N0 is as before the average fading SNR. Finally, substituting 2 (8.228) in (9A.10) and recalling that Q1 (0, b) = e−b /2 , we obtain the desired average BEP Pb (E) =

# $
 Kγ 1 1 + K + γ (1 − ρ) exp − 2 1+K +γ 1+K +γ

(8.229)

The corresponding result for the Rayleigh (K = 0) channel is # $ 1 1 + γ (1 − ρ) Pb (E) = 2 1+γ

(8.230)

As a check, the previously presented results for slow fading can be obtained by letting ρ = 1 in (8.229) and (8.230), which respectively results in (8.213) and (8.207) as expected. What is different about the fast-fading case in comparison with the slow-fading case is the limiting behavior of Pb (E) as the average fading SNR approaches infinity. Letting γ → ∞ in (8.229) and (8.230) gives lim Pb (E) =

γ →∞

1−ρ exp (−K) 2

(8.231)

PERFORMANCE OVER FADING CHANNELS

293

and lim Pb (E) =

γ →∞

1−ρ 2

(8.232)

that is, an irreducible bit error probability exists for any ρ = 1. The amount of this irreducible error probability can be related (through the parameter ρ) to the ratio of the Doppler spread (fading bandwidth) of the channel to the data rate. The specific functional relationship between these parameters depends on the choice of the fading channel correlation model. Mason [39] has tabulated such relationships for various types of fast-fading processes of interest. These results are summarized in Table 2.1, where fd Tb denotes the Doppler spread/data rate ratio and in addition the variance of the fading process has, for convenience, been normalized to unity. For example, for the land–mobile channel where ρ = J0 (2πfd Tb ), Fig. 8.14 illustrates the average BEP as computed from (8.229) as a function of average bit SNR for Rician factors K = 0 (Rayleigh channel) and K = 10 with fd Tb as a parameter. As one would expect, as fd Tb diminishes, the irreducible error becomes smaller. Nevertheless, depending on the value of Rician factor, a Doppler spread of only 1% of the data rate can still cause a significant error floor.

100

K=0

Average Bit Error Probability Pb(E )

10−1

fdTb = 0.1 fdTb = 0.05

10−2

K=0 10−3

fdTb = 0.01

10−4 10−5

fdTb = 0.1 10−6

fdTb = 0.05

10−7

fdTb = 0.01

10−8

0

5

10

15

20

25

30

35

40

45

50

Average SNR per Bit [dB]

Figure 8.14 Average BEP of binary DPSK over a fast fading Rician channel versus the average SNR per bit; land–mobile channel.

294

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

8.2.5.3 π /4-Differential QPSK (π /4-DQPSK) From the conclusion drawn in Section 8.1.5.2 relative to the equivalence in behavior between DQPSK and π /4-DQPSK on the ideal linear AWGN channel, it is clear that the same statement can be made for the fading channel. Thus, without any additional detail, we immediately conclude that the error probability performance of π /4-DQPSK on the fading channel is characterized by the results of Section 8.2.5.1, namely, the generic BEP of (8.204) [or (8.205)] or the more specific results that followed these equations. 8.2.6

Performance in the Presence of Imperfect Channel Estimation

In Section 8.2.2 we considered a form of nonideal coherent detection in which the carrier demodulation reference was assumed to be imperfect with a randomness modeled as an additive Gaussian noise source independent of (but with power equal to) the AWGN associated with the received signal. Since the primary function of this reference signal was to remove the unknown phase of the received signal prior to detection, no explicit attempt was made to estimate the channel fading. Rather, the reference signal was assumed to be degraded by fading identical to that induced by the channel on the transmitted signal and thus for Rayleigh or Rician fading, the decision statistic took the form of the product of two highly correlated complex Gaussian RVs. In an effort to improve on the above, one might consider estimating the channel and using this estimate to remove the fading from the received signal prior to detection. The most common way of accomplishing this is to divide the received faded signal plus noise decision statistic by the channel estimate. Several different channel estimation techniques and their associated performances for a variety of modulations have been discussed in the literature, including pilot-symbol-assisted modulation (PSAM) [50–53] and minimum mean-squared error (MMSE) estimation [54–56]. Assuming a complex Gaussian fading channel, these estimation techniques have been shown to generate a channel estimate that is also complex Gaussian and correlated with the true fading. Thus, the decision statistic now takes the form of the ratio of two complex Gaussian RVs. The brute-force method of evaluating the performance of such a system is to compute the conditional (on the channel and its estimate) error probability as would be done for the AWGN case and then average over the joint distribution of the channel and the channel estimate, involving, in general, a four-dimensional nested integration. Several of the above-referenced works follow this approach, although the simplification of the final result depends to a large extent on the particular modulation being transmitted. For example, in the general case of arbitrary 2D constellations with polygonal decision regions, this approach will most often lead to an intractable analysis. Most recently, a new approach to solving this problem was presented in Ref. 57 for the Rayleigh fading case. The heart of the solution relies on finding the joint

PERFORMANCE OVER FADING CHANNELS

295

PDF (in polar form) for the ratio of two correlated zero-mean complex Gaussian RVs in terms of their individual and joint second-order moments and then applying it to Craig’s method for analyzing the SEP for arbitrary 2D signaling on the AWGN [10], which was later extended to the perfectly known slow-fading channel in Refs. 58 and 59. Since the generality of the approach in Ref. 57 allows one to obtain results that also include the special cases previously treated in the literature, we consider only this approach in what follows and present a brief summary of the key results. 8.2.6.1 Signal Model and Symbol Error Probability Evaluation for Rayleigh Fading For an arbitrary M-ary modulation, the complex baseband equivalent (with time suppressed) of the received signal in any symbol interval15 given that s˜i = s˜I + j s˜Q was transmitted is given by y˜ = η˜si + n˜

(8.233)

where n˜ = n˜ I + j n˜ Q is the complex AWGN with variance N0 for both its real and imaginary parts and η = ηI + j ηQ is again the complex fading applied to the signal which for the Rayleigh case is a zero-mean complex Gaussian RV with parts. Furthermore, the average signal variance s in both its real and imaginary= si |2 , is equal to Es . Using one of energy of the constellation, namely, (1/M) M i=1 |˜ the channel estimation techniques, an estimate ηˆ is produced that is also a complex Gaussian RV that is correlated with the true fading η. The received observation y˜ is divided by η, ˆ the result of which can be expressed as 

D=

y˜ ˜ = s˜i + N; ηˆ

η − ηˆ n˜ N˜ = s˜i + ηˆ ηˆ

(8.234)

where the additive “noise” term N˜ now includes the effect of the imperfect channel estimation as well as the AWGN. Note that even in the case of perfect channel estimation (i.e., ηˆ = η), the additive “noise” is no longer complex Gaussian but rather the ratio of two independent complex Gaussian RVs (assuming the fading and AWGN are independent as is typically the case). Nevertheless, for the perfect channel estimation case, if D is used as a decision statistic, the optimum (maximumlikelihood) decision rule is equivalent to the minimum Euclidean distance decision rule as in the AWGN case. However, for the imperfect channel estimation case, this equivalence is no longer true. Nevertheless, for simplicity of receiver design, we shall pursue only the case where the decision boundaries are determined from an AWGN model with the understanding that the resulting performance will be somewhat suboptimum when the channel estimation is imperfect. 15 Since

we will be dealing only with the slow-fading case, for simplicity of notation, we drop the n dependence, which previously referred to the nth symbol interval.

296

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

The first step in determining the average SEP performance of the receiver is to evaluate the statistics of the decision variable D. Since, from (8.233) and (8.234), D is defined as the ratio of two correlated zero-mean complex Gaussian RVs, then, expressing it in polar form (i.e., D = rd ej θd ), it is shown in Ref. 57 that the joint PDF of rd and θd is given by  2 µyy µηˆ ηˆ − µy ηˆ  rd pD (rd , θd ) =

*2 ) π µηˆ ηˆ rd2 + µyy − 2Re µy ηˆ rd e −j θd where 3 3 2 2 ) * µyy = E |y| ˜ 2 = E |η|2 |˜si |2 + E |n| ˜2 2  3 2 µηˆ ηˆ = E ηˆ 

(8.235)

(8.236)

* ) * ) µy ηˆ = E y˜ ηˆ ∗ = E ηηˆ ∗ s˜i Since Craig’s method for analyzing the SEP of arbitrary 2D constellations rests on shifting the origin of coordinates to the signal, thereby redefining the decision regions in terms of the equivalent additive noise vector, the next step is to transform the joint PDF of D into the joint PDF of N˜ = rej θ . From (8.234), this merely entails a translation by the signal s˜i . Thus, first transforming the PDF in (8.235) into rectangular coordinates, then shifting by s˜i = s˜I + j s˜Q and retransforming the result back to polar coordinates, the following result is obtained [57]  2 µyy µηˆ ηˆ − µy ηˆ  r pN˜ (r, θ) =

2 π cr 2 + b (θ ) r + a

(8.237)

where ) * a = µηˆ ηˆ |˜si |2 + µyy − 2Re µy ηˆ s˜i∗

* ) * ) b (θ ) = 2 µηˆ ηˆ Re s˜i e −j θ − Re µy ηˆ e −j θ

(8.238)

c = µηˆ ηˆ Next, following the approach taken in Ref. 59 for the perfectly known fading channel case, the erroneous decision region for signal s˜i is partitioned into a set of distinct subregions (some of which are closed and others of which are open) and the probability of error associated with the j th erroneous subregion Rj is given by

pN˜ (r, θ) d r d θ Ps,j (E) = (8.239) Rj

297

PERFORMANCE OVER FADING CHANNELS

Finally, the average SEP is obtained from a weighted sum of the Pj (E) terms in (8.239), namely Ps (E) =

N 

(8.240)

wj Ps,j (E)

j =1

where N is the total number of erroneous subregions for all the signal points in the constellation and wj is the a priori probability of the symbol to which the j th subregion corresponds. Since most constellations of practical interest have a certain degree of symmetry, the number of distinct subregions that have distinct geometry requiring individual evaluation of Pj (E) is usually much smaller than N. 8.2.6.2

Special Cases

M-PSK As an example of the evaluation of the above for a particular modulation, consider the case of M-PSK. As we have discussed previously, for this constellation, the correct decision region for each signal point is a wedge of angular dimension 2π /M symmetrically located around the point itself. In fact, for this modulation, it is actually simpler to work with the coordinate system based on the signal-plus“noise” vector rather √ than shifting it to the location of the signal. Specifically, assuming that s˜i = Es (the signal point located on the circle at zero degrees) was transmitted, then based on the previous assumption of AWGN decision boundaries, the erroneous decision region is the area outside a wedge of angular width 2π /M centered around the positive horizontal axis. As such, the SEP is given by

Ps (E) =

=

π π/M π π/M

∞

pD (rd , θd ) d rd d θd +

0

pθd (θd ) d θd +

−π/M

−π

∞

pD (rd , θd ) d rd d θd 0

−π/M −π

pθd (θd ) d θd

(8.241)

where pθd (θd ) is the marginal PDF of the phase difference θd = θy − θˆ with θy the phase of the received signal y˜ and θˆ the phase of the channel estimate η. ˆ Thus, from (8.241) we see that the SEP performance (still assuming AWGN decision boundaries) equivalently corresponds to a decision based on the statistic obtained by multiplying y˜ by e−j θ , which is the complex conjugate of the reference signal determined from only the phase of the channel estimate η. ˆ In other words, for M-PSK, the receiver need not make use of the amplitude of the channel estimate in forming its decision statistic but rather can perform phase compensation in the conventional manner employed by systems that obtain their phase estimate from a synchronization loop or by other means. Finally, it should be noted that because of the symmetry of the constellation, the SEP in (8.241) is also the average SEP over all symbols.

298

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

If instead one were to shift the origin of coordinates to the signal as discussed previously and work instead with the joint PDF of N˜ as given in (8.237), then it is shown in Ref. 57 that, for this special case, (8.240) evaluates to  2 µyy µηˆ ηˆ − µy ηˆ  Ps (E) = π   2a sin2 θ − π  + √E b (θ ) sin π  sin θ − π  s  M × +

π 

M π √

M π  2 π/M   (θ ) a sin θ − M + Es b (θ ) sin M sin θ − M

π , +Es c sin2 M

π   √ π π b (θ ) 2 −1 b (θ ) sin θ − M + 2 Es c sin M 

− √ 1 − tan π π  (θ ) sin θ − M  (θ )3/2

 √



π  π π 2a sin2 θ − M + Es b (−θ) sin M sin θ − M +

 √

 +

π  π π + Es b (−θ) sin M sin θ − M  (−θ ) a sin2 θ − M

π , +Es c sin2 M 

π   √ π π b (−θ) 2 −1 b (−θ) sin θ − M + 2 Es c sin M 

− 1 − tan dθ √ π π  (−θ) sin θ − M  (−θ )3/2 (8.242) √ where a, b (θ ), and c are evaluated from (8.238) using s˜i = Es , specifically

π

' * * ) ) a = µηˆ ηˆ Es + µyy − 2 Es Re µy ηˆ = µηˆ ηˆ Es + 2 (s Es + N0 )−2Es Re µηηˆ ' '

* * ) ) b (θ ) = 2 Es µηˆ ηˆ cos θ − Re µy ηˆ e −j θ = 2 Es µηˆ ηˆ cos θ − Re µηηˆ e −j θ c = µηˆ ηˆ

(8.243)

and in addition # $  2  ' *2 ) −j θ    (θ ) = 4ac−b (θ ) = 4 µyy µηˆ ηˆ − µy ηˆ + Es µηˆ ηˆ sin θ + Im µy ηˆ e 2

%  2

*2 & ) = 4 2 (s Es + N0 ) µηˆ ηˆ − Es µηηˆ  + Es µηˆ ηˆ sin θ + Im µηηˆ e −j θ (8.244) Note from (8.243) that if µy ηˆ is real (as turns out to be the case for the PSAM and MMSE channel estimation techniques [57]), then b (θ ) = b (−θ), in which case the third and fourth terms of the integrand in (8.242) become identical to the first and second terms, and thus the SEP simplifies to

PERFORMANCE OVER FADING CHANNELS

Ps (E) =

299

  2  2 µyy µηˆ ηˆ − µy ηˆ  

π

 2a sin2 θ − π  + √E b (θ ) sin π  sin θ − π  s  M ×

M π √

M π +

π   2  sin θ − M  (θ ) a sin θ − M + Es b (θ ) sin M π/M

π , +Es c sin2 M 

π   √ π π b (θ ) 2 −1 b (θ ) sin θ − M + 2 Es c sin M 

− 1 − tan dθ √ π π  (θ ) sin θ − M  (θ )3/2

π

(8.245) We now consider the specific evaluation of (8.245) for minimum mean-squared error estimation of the channel. In the case of MMSE of a RV, say, η, it is well known that the estimate ηˆ and the estimation error e = η − ηˆ are uncorrelated. Moreover, since in our application η and ηˆ are both assumed to be complex Gaussian (and thus so is η − η), ˆ the RVs ηˆ and e are independent. Denote the power of the channel estimation error by 2e . Thus, in view of the statistical properties presented above, the second moments of (8.236) become µyy = 2 (s Es + N0 ) * ) * ) µηˆ ηˆ = E |η|2 − E |e|2 = 2 (s − e ) µy ηˆ =

(8.246)

' ' )

 * ' Es µηηˆ = Es E e + ηˆ ηˆ ∗ = Es µηˆ ηˆ

Using (8.246) to evaluate (8.243) and (8.244), we find that b (θ ) = 0 and  (θ ) = 4aµηˆ ηˆ , which, after substitution in (8.245) and evaluation of the integral in closed form, gives [57] 1 M −1 − Ps (E) = M π 1 M −1 − = M π





gPSK gPSK +



−1

× π − tan

 a/Es g + π c π − tan−1  PSK tan  gPSK M

 gPSK γ s − γ e

 1 + γ e + gPSK γ s − γ e





a/Es c





 1 + γ e + gPSK γ s − γ e π

 tan M gPSK γ s − γ e

(8.247)

300

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

where γ s = s Es /N0 is the true faded average SNR, γ e = e Es /N0 is the average SNR of the channel estimation error, and as before gPSK = sin2 π /M. Clearly, for perfect channel estimation (i.e., γ e = 0), (8.247) becomes 1 M −1 − Ps (E) = M π

 gPSK γ s 1 + gPSK γ s



 −1

π − tan

1 + gPSK γ s π tan gPSK γ s M



(8.248) which is equivalent to (8.113), as it should be. M-QAM As a second example of the evaluation of (8.239) and (8.240) for a particular modulation, consider the case of M-QAM. Analogous to (8.247), for MMSE the SEP for the j th decision subregion is given by [57] :   ; θ2,j − θ1,j  u2j 1 ; < − Ps,j (E) = s 2π 2π u2j + a/E c

 : ; 2 a/Es ; uj + c 

  × tan−1 < tan θ2,j − θ1,j  + ψj  u2j 

:  ; 2 a/Es ; uj + c − tan−1 < tan ψj  u2j :

   ; θ2,j − θ1,j  u2j γ s − γ e 1 ; <

 − = 2π 2π 1 + γ e + u2j γ s − γ e

TABLE 8.4 Geometric Parameters of Rectangular 16-QAM for Calculating its SER, 1 wj = 12 √ √ xj ψj θ1,j θ2,j xj ψj θ1,j θ2,j s˜ i / Es s˜ i / Es √ √ √ √ √ √ π π π 3 10 π 3π 10 10 5 10 5 , − , 0 10 10 5 4 4 4 10 10 5 4 4 √ √ π π 3π π 5 5 3π − 0 4 4 4 4 4 √5 √5 π 3π 5π π 3π 5π 5 5 5 4 4 4 5 4 4 4 √ √ √ √ π π π 5 10 3 10 5 3π π π − − , − 5 4 4 4 10 10 5 4 4 2 √ √ √ √ π 5π π π 5π π 3 10 3 10 5 5 , 10 10 5 4 4 2 5 4 4 2 √ √ π π 5 5 3π 3π π − 0 − − 5 4 4 5 4 4 4

301

REFERENCES

 :

 ; ; 1 + γ e + u2j γ s − γ e  



 tan θ2,j − θ1,j  + ψj  × tan−1 < u2j γ s − γ e 

: 

 ; ; 1 + γ e + u2j γ s − γ e

 − tan−1 < tan ψj  u2j γ s − γ e

(8.249)

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35. P. J. Crepeau, “Uncoded and coded performance of MFSK and DPSK in Nakagami fading channels,” IEEE Trans. Commun., vol. 40, no. 3, March 1992, pp. 487–493. 36. G. L. Turin, “The characteristic function of Hermitian quadratic forms in complex normal variables,” Biometrika, vol. 47, June 1960, pp. 199–201. 37. M. Schwartz, W. R. Bennett, and S. Stein, Communication Systems and Techniques. New York, NY: McGraw-Hill, 1966. 38. R. F. Pawula, “Relations between the Rice Ie-function and Marcum Q-function with applications to error rate calculations,” Electron. Lett., vol. 31, no. 20, September 28, 1995, pp. 1717–1719. 39. L. J. Mason, “Error probability evaluation of systems employing differential detection in a Rician fast fading environment and Gaussian noise,” IEEE Trans. Commun., vol. 35, no. 1, January 1987, pp. 39–46. 40. M. O. Fitz and J. P. Seymour, “On the bit error probability of QAM modulation,” Int. J. Wireless Inform. Networks, vol. 1, no. 2, 1994, pp. 131–139. 41. D. Yoon, K. Cho, and J. Lee, “Bit error probability of M-ary quadrature amplitude modulation,” Proc. IEEE Veh. Technol. Conf. (VTC’2000-Fall ), Boston, MA, vol. 5, September 2000, pp. 2422–2427; see also full journal paper in IEEE Trans. Commun., vol. 50, no. 7, July 2002, pp. 1074–1080. 42. K. Cho, D. Yoon, W. Jeong, and M. Kavehrad, “Bit error probability analysis of arbitrary rectangular QAM,” 35th Asilomar Conf. Signals, Systems and Computers, vol. 2, 2001, pp. 1056–1059; see also full journal paper in IEEE Trans. Commun., vol. 50, no. 7, July 2002, pp. 1074–1080. 43. L.-L. Yang and L. Hanzo, “A recursive algorithm for the error probability evaluation of M-QAM,” IEEE Commun. Lett., vol. 4, no. 10, October 2000, pp. 304–306. 44. M. K. Simon and M.-S. Alouini, “Simplified noisy reference loss evaluation for digital communication in the presence of slow fading and carrier phase error,” IEEE Trans. Veh. Technol., vol. 50, no. 2, March 2001, pp. 480–486. 45. T. Eng and L. B. Milstein, “Partially coherent DS-SS performance in frequency selective multipath fading,” IEEE Trans. Commun., vol. 45, no. 1, January 1997, pp. 110–118. 46. M. K. Simon, “Error probability performance of unbalanced QPSK receivers,” IEEE Trans. Commun., vol. COM-26, no. 9, September 1978, pp. 1390–1397. 47. J. Lassing, E. Strom, E. Agrell, and T. Ottoson, “Computation of the exact bit error rate of coherent M-ary PSK with Gray code bit mapping,” IEEE Trans. Commun., vol. 51, no. 11, November 2003, pp. 1758–1760. 48. P. K. Vitthaladevuni and M.-S. Alouini, “Exact BER computation of generalized hierarchical PSK constellations,” IEEE Trans. Commun., vol. 51, no. 12, December 2003, pp. 2030–2037. 49. S. Park and D. Yoon, “Effect of the phase error, quadrature error, and I-Q gain mismatch on symbol error probability for an M-PSK system in fading channels,” IEEE International Symposium of Personal, Indoor and Mobile Radio Communications (PIMRC 2004), Barcelona, Spain, September 5-8, 2004. 50. J. K. Cavers, “An analysis of pilot symbol assisted modulation for Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 40, November 1991, pp. 686–693. 51. Y. S. Kim, C. J. Kim, G. Y Jeong, Y. J. Bang, H. K. Park, and S. S. Choi, “New Rayleigh fading channel estimator based on PSAM channel sounding technique,” Proc. IEEE Int. Conf. Communications (ICC’97 ), June 1997, pp. 1518–1520.

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52. X. Tang, M. S. Alouini, and A. Goldsmith, “Effect of channel estimation error on MQAM BER performance in Rayleigh fading,” IEEE Trans. Commun., vol. 47, no. 12, December 1999, pp. 1856–1864. 53. K. Yu, J. Evans, and I. Collings, “Performance analysis of pilot symbol aided QAM for Rayleigh fading channels,” Proc. IEEE Int. Conf. Communications (ICC’02 ), April 2002. 54. A. Aghamohammadi and H. Meyr, “On the error probability of linearly modulated signals on Rayleigh frequency-flat fading channels,” IEEE Trans. Commun., vol. 38, no. 11, November 1990, pp. 1966–1970. 55. M. G. Shayesteh and A. Aghamohammadi, “On the error probability of linearly modulated signals on frequency-flat Ricean, Rayleigh and AWGN channels,” IEEE Trans. Commun., vol. 43, February/March/April 1995, pp. 1454–1466. 56. S. K. Wilson and J. M Cioffi, “Probability density function for analyzing multi-amplitude constellations in Rayleigh and Ricean channels,” IEEE Trans. Commun., vol. 47, no. 3, March 1999, pp. 380–386. 57. X. Dong and N. C. Beaulieu, “A new method for calculating symbol error probabilities of two-dimensional signalings in Rayleigh fading with channel estimation errors,” IEEE Trans. Commun. (in press). Available at http://www.exe.ualberta.ca/∼xdong/channelerr.pdf 58. X. Dong, N. C. Beaulieu, and P. H. Wittke, “Error probabilities of two-dimensional M-ary signaling in fading,” IEEE Trans. Commun., vol. 47, no. 3, March 1999, pp. 352–355. 59. X. Dong, N. C. Beaulieu, and P. H. Wittke, “Signal constellations for fading channels,” IEEE Trans. Commun., vol. 47, no. 5, May 1999, pp. 703–714.

APPENDIX 8A. STEIN’S UNIFIED ANALYSIS OF THE ERROR PROBABILITY PERFORMANCE OF CERTAIN COMMUNICATION SYSTEMS The analysis of the error probability performance of differential and noncoherent detection as well as certain nonideal coherent detection systems on an AWGN channel is characterized by a decision statistic that is in the form of either the product of two complex Gaussian random variables or the difference of the squares of such variables. In what has now become a classic paper in the annals of communication theory literature, Stein [29] showed how, using a simple algebraic relation between the product and difference of square forms of the decision variable, the error probability of certain such binary systems could be analyzed by a unified approach. Our intent in this appendix is to summarize (without proof) the results found in Stein’s original paper in a generic form that can be easily referenced in the main text where it is applied to specific communication scenarios. This generic form will also be useful when extending Stein’s results to M-ary communication systems [21] and fading channels [6] as well as certain nonideal coherent detection systems   [16]. We start by considering two complex Gaussian variables z1p = z1p  ej 1p and z2p = z2p  ej 2p that are in general correlated and whose sum and difference

305

APPENDIX 8A. STEIN’S UNIFIED ANALYSIS OF THE ERROR PROBABILITY PERFORMANCE

 



z1f = z1p + z2p /2 and z2f = z1p − z2p /2 are also correlated complex Gaussian random variables.16 A simple algebraic manipulation shows that 
 


 2  2 z1p + z2p ∗ z1p − z2p ∗ z1p − z2p z1f  − z2f  = z1p + z2p − 2 2 2 2 
∗ ∗ z1p z2p + z1p z2p ) ∗ * = Re z1p =2 z2p (8A.1) 4 3 2  2  2 ∗ Hence, a test of z1f  − z2f  or Re z1p z2p against a zero threshold, which are respectively typical of noncoherent FSK and differentially coherent PSK systems, would produce equivalent error probability performance expressions, i.e., * * ) ) ∗ (8A.2a) z2p < 0 P = Pr Re z1p or

2  3 2   2  2 3 )   * 2 2 P = Pr z1f  − z2f  < 0 = Pr z1f  < z2f  = Pr z1f  < z2f 

(8A.2b) To evaluate the error probability P , Stein used a succession of linear transformations to transform both the FSK and PSK models to a canonical problem that had a convenient solution. In particular, he showed that the solution to (8A.2a) or (8A.2b) could be expressed in terms of an equivalent noncoherent FSK problem based on two nonzero-mean but uncorrelated complex Gaussian variables, ) t1 and t*2 , wherein the desired error probability could be stated as P = Pr |t1 |2 < |t2 |2 = Pr {|t1 | < |t2 |}. By relating t1 and t2 to z1p , z2p and z1f , z2f , Stein arrived at the following generic results. Define the first- and second-order moments of z1p , z2p by (using Stein’s notation)    zip = mip + j µip = zip  e θip , i = 1, 2    2  2   Sip = 12 zip  = 12 m2ip + µ2ip , Nip = 12 zip − zip  ,

i = 1, 2

' ∗

 

ρp N1p N2p = 12 z1p − z1p z2p − z2p , ρp = ρcp + jρsp



 1 2 z1p − z1p z2p − z2p = 0

(8A.3)

and likewise for z1f , z2f . Finally, define the phase angle φ by & % ' φ = arg N1p − N2p − j 2ρsp N1p N2p

(8A.4a)

or + , φ = arg ρcf + jρsf

(8A.4b)

16 The f and p subscript notations refer, respectively, to FSK and PSK modulations, as will become clear shortly.

306

PERFORMANCE OF SINGLE-CHANNEL RECEIVERS

for the problems characterized by (8A.2a) and (8A.2b), respectively. Then
  √ √  √ √ & A √  a+b 1% 1 − Q1 I0 b, a + Q1 a, b − exp − ab 2 2 2 (8A.5) where for the definition of P as in (8A.2a) we have   



'  ( 1  S1p + S2p + S1p − S2p cos φ + 2 S1p S2p sin θ1p − θ2p sin φ a 0

=  2 b 2 2 N N N1p + N2p + N1p − N2p + 4ρsp 1p 2p  



' S1p + S2p − S1p − S2p cos φ − 2 S1p S2p sin θ1p − θ2p sin φ + 0

2 2 N N N1p + N2p − N1p − N2p + 4ρsp 1p 2p  

' 2 S1p S2p cos θ1p − θ2p  ∓ 0

  2 N N 1 − ρsp 1p 2p P =

ρcp A= 0 2 1 − ρsp

(8A.6a)

and for the definition of P as in (8A.2b) we have 

 '  ( 1 a  S1f + S2f + 2 S1f S2f cos θ1f − θ2f + φ 0 =   2 b 2 N1f + N2f + 2 N1f N2f ρf 

 ' S1f + S2f − 2 S1f S2f cos θ1f − θ2f + φ + 0  2 N1f + N2f − 2 N1f N2f ρf   

2 S1f − S2f  ∓ 0

 2  2 N1f + N2f − 4N1f N2f ρf  A = 0

N1f

(8A.6b)

N1f − N2f  2 2 + N2f − 4N1f N2f ρf 

Several special cases of (8A.6a) and (8A.6b) are of interest. First, if z1p and  z2p are uncorrelated (i.e., ρp  = 0), then φ = 0 or π (depending, respectively, on whether N1p > N2p or N1p < N2p . In either event, (8A.6a) simplifies to 

a b

(

1 = 2



  

S2p S1p S1p S2p + ∓2 cos θ1p − θ2p , N1p N2p N1p N2p

A=0

(8A.7)

APPENDIX 8A. STEIN’S UNIFIED ANALYSIS OF THE ERROR PROBABILITY PERFORMANCE



307 

A further special case of (8A.8) corresponds to S1p = S2p = Sp , N1p = N2p = Np , θ1p = θ2p , in which case we obtain 

a b

( =

    

0

  

, 2Sp   Np

A=0

(8A.8)



If for (8A.6b) z1f and z2f have equal noise power (i.e., N1f = N2f = Nf ), then (8A.6b) simplifies to 

a b

(



 '

 1  S1f + S2f − 2 ρf  S1f S2f cos θ1f − θ2f + φ =   2 2Nf 1 − ρf   S1f − S2f  ∓0  2  , 1 − ρf 

A=0

(8A.9)

  If, in addition, z1f and z2f are uncorrelated (i.e., ρf  = 0), then (8A.9) further simplifies to 

a b

(

 S2f    Nf = S1f    Nf

      

,

A=0

(8A.10)

The generic result in (8A.5) can be simplified by using some of the alternative representations of classical functions given in Chapter 4. In particular, substituting (4.16), (4.19), and (4.65) in (8A.5) and combining terms, we arrive at the result $
 , 1−A + 2ζ A sin θ −ζ 2 (1 + A) b+ 2 1 + 2ζ sin θ + ζ exp − d θ, 1 + 2ζ sin θ + ζ 2 2 −π a  0 and that it becomes more accentuated as δ increases. 9.3.3

Exact Error Rate Analysis

9.3.3.1 Binary Signals We begin our discussion by considering the performance of an EGC receiver when coherent binary BPSK or binary BFSK modulation is transmitted over a multilink

334

PERFORMANCE OF MULTICHANNEL RECEIVERS

channel with L paths. Conditioned on the fading amplitudes {αl }L l=1 , the BER  Pb E|{αl }L l=1 , of an EGC receiver is given by     2gγEGC Pb E|{αl }L l=1 = Q  * +  L 2 +
 + 2gEb  , = Q α l   L l=1 Nl l=1

(9.59)

where, as for the MRC case, g is a modulation dependent parameter such that g = 1 for BPSK, g = 12 for orthogonal BFSK, and g = 0.715 for BFSK with minimum correlation [6]. The average BER Pb (E) is obtained by averaging (9.59) over the joint PDF of the channel fading amplitudes pα1 ,α2 ,...,αL (α1 , α2 , . . . , αL ):  * +  L 2 + ∞ ∞
 + 2gEb , Pb (E) = ··· Q αl    L N 0 0 



l=1

l

l=1

×pα1 ,α2 ,...,αL (α1 , α2 , . . . , αL ) d α1 d α2 · · · d αL

(9.60)

The L-fold integral in (9.60) can be collapsed to a single integral, namely 



∞

Pb (E) =

Q 0

2gEb αt2 L N l l=1

 pαt (αt ) dαt

(9.61)

 where αt = L l=1 αl denotes the sum of the fading amplitudes after combining. In general, there are two difficulties associated with analytically evaluating the average BER as expressed in (9.61). The first relates to the requirement of obtaining the PDF of the total fading RV αt . When the fading amplitudes can be assumed independent (the case to be considered in this section), finding this PDF requires a convolution of the PDFs of the αl terms and can often be quite difficult to evaluate. The second difficulty has to do with the fact that the argument of the classical definition of the Gaussian Q-function in (4.1) appears in the lower limit of the integral, which is undesirable when trying to perform the average over αt . To circumvent these difficulties, we now propose a new method of solution based on the alternative representation of the Gaussian Q-function as given by (4.2): Q(x) =

1 π

 0

π/2

exp −

x2 2 sin2 φ

dφ;

x≥0

(9.62)

COHERENT EQUAL GAIN COMBINING

335

First, using (9.62) in (9.60) gives 

∞

Pb (E) = 0

 ··· 0

∞

1 π





π/2 0

 2  L gEb α l=1 l   exp − L  2 N sin φ l=1 l

×pα1 (α1 ) · · · pαL (αL ) d φ d α1 d α2 · · · d αL

(9.63)

Unfortunately, we cannot represent the exponential in (9.63) as a product of exponentials each involving only a single αl because of the presence of the αk αl crossproduct terms. Hence, we cannot partition the L-fold integral into a product of one-dimensional integrals as is possible for MRC [21], and thus we must abandon this approach. Instead, we use the alternative representation of the Gaussian Q-function in (9.61), which gives after switching the order of integration

 π/2  ∞ 1 A2 2 Pb (E) = αt pαt (αt ) dαt dφ, exp − (9.64) π 0 2 sin2 φ 0  where A = (2gEb )/( L l=1 Nl ). While this maneuver cures the second difficulty by getting the total fading RV αt out of the lower limit of the integral and into the integrand, it appears that we are still faced with the problem of determining the PDF of αt . To get around this difficulty, we represent pαt (αt ) in terms of its characteristic function αt (j v), which, because of the independence assumption on the fading channel amplitudes, becomes  ∞ 1 pαt (αt ) = α (j v) e −j vαt d v 2π −∞ t   ∞  L 1 αl (j v) e −j vαt d v (9.65) = 2π −∞ l=1

where αl (j v) is the characteristic function of the fading amplitude αl corresponding to the lth path. Substituting (9.65) into (9.64) gives   π/2  ∞  L 1 αl (j v) Pb (E) = 2π 2 0 −∞ l=1

 ∞ A2 αt2 × d αt d v d φ − j vα exp − (9.66) t 2 sin2 φ 0    J (v,φ)

The integral J (v, φ) can be obtained in terms of the complementary error function erfc(·) as   .

π sin φ sin φ sin2 φ 2 (9.67) v erfc j √ v exp − J (v, φ) = 2 A 2A2 2A

336

PERFORMANCE OF MULTICHANNEL RECEIVERS

or alternatively by separately evaluating its real and imaginary parts, namely [36, Eqs. (3.896.4) and (3.896.3)]    # π sin2 φ A2 sin2 φ 2 2 αt cos(vαt ) d αt = exp − v 2A2 2A2 2 sin2 φ 0   # "  ∞ A2 v sin2 φ sin2 φ 2 2 exp − exp − v αt sin(vαt ) d αt = A2 2A2 2 sin2 φ 0   1 3 sin2 φ 2 , ; ×1 F1 v (9.68) 2 2 2A2



∞

" exp −

where 1 F1 (·, ·; ·) is the Kummer confluent hypergeometric function [53, p. 504, Eq. (13.1.2)]. Thus, letting . X(φ) =

π sin φ 2 A

v sin2 φ Y (v, φ) = − 1 F1 A2



1 3 sin2 φ 2 , ; v 2 2 2A2

 (9.69)

we can write the integral J (v, φ) in the form 

 sin2 φ 2 J (v, φ) = (X(φ) + j Y (v, φ)) exp − v 2A2

# "  Y (v, φ) = X2 (φ) + Y 2 (v, φ) exp j tan−1 X(φ)   sin2 φ 2 v × exp − 2A2

(9.70)

In general, the characteristic function of a PDF will be a complex quantity, and hence the product of characteristic functions in (9.66) will also be complex. However, since the average BER is real, it is sufficient to consider only the real part of the right-hand side of (9.66), which yields  L    π/2  ∞ 1 Re αl (j v) J (v, φ) dv dφ (9.71) Pb (E) = 2π 2 0 −∞ l=1

Expressing the characteristic function of each fading path PDF by αl (j v) = Ul (v) + j Vl (v)

# " −1 Vl (v) 2 2 = Ul (v) + Vl (v) exp j tan Ul (v)

(9.72)

COHERENT EQUAL GAIN COMBINING

and then substituting (9.70) and (9.72) into (9.71) gives    π/2  ∞ 1 sin2 φ 2 Pb (E) = F(v, φ) exp − v dv dφ 2π 2 0 2A2 −∞

337

(9.73)

where F(v, φ) = R(v, φ) cos (v, φ) R(v, φ) =

L  X2 (φ) + Y 2 (v, φ) Ul2 (v) + Vl2 (v)

(v, φ) = tan−1



Y (v, φ) X(φ)

l=1

+

L


tan−1

l=1



Vl (v) Ul (v)




π [L + 1 − sgn(Y (v, φ)) − sgn(Vl (v))] 2 L

+

(9.74)

l=1

where sgn(·) denotes the sign function and the arctangent function is defined with respect to the standard principal value as available, for example, in the Mathematica routine for that function. The characteristic function corresponding to the Nakagami-m fading PDF can be evaluated with the help of Eq. 9.240 of Ref. 36 in terms of the parabolic cylinder function D−v (·) [36, Sect. 9.24–9.25]    # " l 1 (2ml ) l 2 D−2ml −j v αl (j v) = m −1 v exp − 2 l (ml ) 2ml 8ml or alternatively by separately evaluating its real and imaginary parts by using sine and cosine Fourier transforms found in [36] with the results [see Eq. (9.72)]

l 2 v Ul (v) = Al (v) exp − 4ml

l 2 Vl (v) = Bl (v) exp − v (9.75) 4ml where

1 1 v 2 l − ml , ; Al (v) = 1 F1 2 2 4ml   

ml + 12 l 3 v 2 l Bl (v) = (9.76) v 1 F1 1 − m l , ; (ml ) ml 2 4ml Thus, the functions defined in (9.74) become R(v, φ) =



X2 (φ)

+

Y 2 (v, φ)

L l=1



L
l 2 × exp − v 4ml l=1



A2l (v) + Bl2 (v)

338

PERFORMANCE OF MULTICHANNEL RECEIVERS

(v, φ) = tan−1



Y (v, φ) X(φ)

+

L


tan−1



l=1

Bl (v) Al (v)




π [L + 1 − sgn(Y (v, φ)) − sgn(Bl (v))] 2 L

+

(9.77)

l=1

with X(φ) and Y (v, φ) as defined in (9.69) and Al (v) and Bl (v) as defined in (9.76). It is convenient in this case to absorb the exponential factor in R(v, θ ) into the exponential factor in the integrand of (9.73). Hence, we can write the average BER of (9.73) as      π/2  ∞ L 1 sin2 φ
l F0 (v, φ) exp − + v 2 dv dφ (9.78) Pb (E) = 2 2π 2 0 2A 4m l −∞ l=1

where F0 (v, φ) is a normalized version of F(v, φ) defined by F0 (v, φ) = R0 (v, φ) cos (v, φ)

(9.79)

with R0 (v, φ) =



X2 (φ) + Y 2 (v, φ)

L A2l (v) + Bl2 (v)

(9.80)

l=1

and (ν, φ) still as defined in (9.77). Finally, letting sin2 φ
l + 2A2 4ml L

η(φ) =

(9.81)

l=1

√ and making the change of variables x = η(φ)v, the inner doubly infinite integral is of the form

 ∞ x 2 F0 √ (9.82) , φ e−x dx η(φ) −∞ which can be readily evaluated by the Gauss–Hermite quadrature formula [53, p. 890, Eq. (25.4.46)], yielding the desired final result in the form of a single finite-range integral on φ, namely Pb (E) =

1 2π 2



π/2 0



Np
xn 1 Hxn F0 √ , φ dφ √ η(φ) n=1 η(φ)

(9.83)

where Np is the order of the Hermite polynomial, HNp (.). Setting Np to 20 is typically sufficient for excellent accuracy. In (9.83) xn are the zeros of the Np order Hermite polynomial and Hxn are the weight factors of the Np -order Hermite polynomial and are given by [53, Table 25.10, p. 924] √ 2Np −1 Np ! π (9.84) Hxn = 2 2 Np HNp −1 (xn )

339

COHERENT EQUAL GAIN COMBINING

100 EGC MRC

Average Bit Error Rate Pb (E )

10−1

10−2

m = 1/2 10−3

m =1

10−4

m =2 10−5

m =4 10−6 −10

−8

−6

−4

−2

0

2

4

6

8

10

12

Average Received SNR per Bit and per Branch g¯ [dB] Figure 9.5 Average BER of BPSK over Nakagami-m fading channels with MRC and coherent EGC (L = 4).

Abramowitz and Stegun tabulated both the zeros and the weight factors of the Hermite polynomial [53, Table (25.10), p. 924] for various polynomial orders Np . Note that by substituting (9.79), (9.80), and (9.81) in (9.83), it can be shown that if Nl = N0 (l = 1, 2, . . . , L) then the average BER in (9.83) is solely a function of the various average SNR/bit/path γ l = (l Eb )/N0 . As a numerical example, Fig. 9.5 compares the BER performance of BPSK with MRC and EGC over i.i.d. Nakagami-m fading channels. Note that EGC approaches the performance of MRC as m increases. 9.3.3.2 Extension to M-PSK Signals Recall that the SER for M-PSK over an AWGN is given by the integral expression (8.22), namely Ps (E|αt ) =

1 π

 0

(M−1)π/M

gPSK Es dφ exp − N0 sin2 φ

(9.85)

where gPSK = sin2 (π /M) and Es /N0 is the received symbol SNR. For EGC reception in the presence of fading, the conditional SER is obtained from (9.83) by

340

PERFORMANCE OF MULTICHANNEL RECEIVERS

replacing Es /N0 by γEGC , which represents the instantaneous SNR per symbol after combining. Following the same steps as in Section 9.3.3.1, it is straightforward to show that the average SER is given by an equation analogous to (9.83), namely Ps (E) =

1 2π 2



(M−1)π/M 0


1 Hx F0 √ ηPSK (φ) n=1 n Np

√

xn ηPSK (φ)

, φ dφ

(9.86)

where L
l sin2 φ + 4ml 2A2PSK l=1  2 gPSK Es = L l=1 Nl

ηPSK (φ) =

APSK

(9.87)

and all other parameters and functions remain the same as for the binary signal case. It can also be shown that if Nl = N0 (l = 1, 2, . . . , L) then (9.86) is solely a function of the various average SNR/symbol/path γ l = (l Es )/N0 . 9.3.4

Approximate Error Rate Analysis

As mentioned previously, one of the difficulties in evaluating (9.61) is the requirement of obtaining the PDF of the total fading RV αt . Even the use of the alternative representation of the Gaussian Q-function which leads to (9.64) does not alleviate this problem. Although there is no known closed-form exact expression for the PDF of the sum of L i.i.d. Nakagami-m RVs, Nakagami [54] showed after rather complex calculations that such a sum can be accurately approximated by another Nakagami-m distribution [54, p. 22] with a parameter 0 m and average power 0  given by 0m

= mL



2 (m + 12 )  = L 1 + (L − 1) 0 m 2 (m)



1  L2  1 − 5m

(9.88)

Specifically, modeling the PDF pαt (αt ) by a Nakagami-m PDF with parameters 0 m and 0 , the average BER for binary signals as given by (9.61) can be evaluated in closed form as [55, App. A] 

Lm   Lm + 12 m π (m + gγ eq ) (Lm + 1) m + gγ eq   1 m × 2 F1 1, Lm + , Lm + 1; 2 m + gγ eq

1 Pb (E)  2

gγ eq

(9.89)

COHERENT EQUAL GAIN COMBINING

341

where 2 F1 (·, ·; ·; ·) is the Gauss hypergeometric function [53, p. 556, Eq. (15.1.1)], γ is the SNR/symbol/path, and γ eq

 

2 (m + 12 ) γ 1 = γ 1 − 1 + (L − 1)  L m 2 (m) 5m

(9.90)

For the special case where m is integer, it can be shown (using Ref. [55, App. A]) that (9.89) simplifies to  Pb (E) 

1 1 − 2



gγ eq

Lm−1


m + gγ eq

l=0



2l  l

  4 1+

gγ eq m

 l 

(9.91)

For M-PSK, the same approximate modeling of αt can be used together with the conditional SER obtained from (9.85) to compute an approximate expression for the average SER. For an arbitrary noninteger m, no closed-form solution is available and the final average has to be computed numerically. However, when m is restricted to integer values, using the expression (5A.35), the average SER can be expressed in closed form as M −1 T − Ps (E)  M π 2 − π ×





gPSK γ eq

Lm−1


m + gPSK γ eq

l=0

gPSK γ eq m + gPSK γ eq

l−1 Lm−1

l=0 j =0

2l j

2l    4 1+

l l gPSK γ eq m

(−1)l+j   l g γ 4 1 + PSKm eq

sin(2(l − j )T ) , 2(l − j )

(9.92)

where 

T =

"



#

2g γ π 1 + 1 1 + sgn 1 + PSK eq cos 2π − 1 2 2 m M  .    2π   gPSK γ eq gPSK γ eq 2 1 + sin m m M     − tan−1   2π     2gPSK γ eq 1+ cos − 1 m M

(9.93)

For M = 2, it can be shown that T as given in (9.93) becomes equal to π /2 and therefore (9.92) reduces to (9.91).

342

PERFORMANCE OF MULTICHANNEL RECEIVERS

9.3.5

Asymptotic Error Rate Analysis

In this section we continue the discussion started in Section 9.2.4 and consider the asymptotic SER performance of M-PSK with coherent EGC reception. In this case, the conditional combined SNR depends on the γl s as given in (9.51), namely γt = γEGC

2  L 1
√ = γl L

(9.94)

l=1

assuming that all the branches have the same AWGN power spectral density (i.e., Nl = N0 , l = 1, 2, . . . , L). Then, Abdel-Ghaffar and Pasupathy [44] showed that C (L, M) for EGC is computed from C (L, M)|EGC =

1 (2L)L L! (2L)! (L − 1)!



∞ 0

γtL−1 Ps (E |γt ) dγt

(9.95)

It is of interest to compare the asymptotic performance of coherent MRC and EGC receivers. Since the conditional SER of these two receivers would be equal when γEGC = γMRC , then comparing (9.95) with (9.42) we observe that C (L, M) |EGC (2L)L L! ∼ 1  e L = , =√ C (L, M) |MRC (2L)! 2 2

L1

(9.96)

where the latter approximation is obtained using Stirling’s formula x! ∼ = √ −x x+(1/2) 2π e x , x  1. Thus, we conclude that the asymptotic SER degradation of EGC relative to MRC is given by (9.96). Hence, combining (9.96) and (9.47) immediately gives the desired expression for C (L, M) for M-PSK with EGC reception: C (L, M) =

1 (L/2)L L! (2L)! (sin(π /M))2L  



L 2L M − 1
2L l sin (2π l/M) − × (−1) L L−l M πl

(9.97)

l=1

9.4 NONCOHERENT AND DIFFERENTIALLY COHERENT EQUAL GAIN COMBINING In this section, we consider the performance of several differentially coherent and noncoherent modulations when used in conjunction with post-detection EGC [7, Sect. 12-1, p. 680; 3, Sect. 5.5.6, p. 253]. In Section 9.4.1 we present an approach based on the alternative representation of the generalized Marcum Q-function that applies to binary DPSK and FSK as well as DQPSK [56]. We then present in section 9.4.2 another approach which applies to noncoherent orthogonal M-FSK [57].

NONCOHERENT AND DIFFERENTIALLY COHERENT EQUAL GAIN COMBINING

9.4.1

343

DPSK, DQPSK, and BFSK Performance (Exact and with Bounds)

A large number of papers deal with the performance of noncoherent and differentially coherent communication and detection systems when used in conjunction with post-detection EGC over AWGN as well as fading channels. For example, Proakis [58] developed a generic expression for evaluating the BER for multichannel noncoherent and differentially coherent reception of binary signals over L independent AWGN channels. Proakis further [7, Sect. 14-4, p. 777] provides closed-form expressions for the average BER of binary orthogonal square-law detected FSK and binary DPSK with multichannel reception over L independent, identically distributed (i.i.d.) Rayleigh fading channels. Lindsey [59], derived a general expression for the average BER of binary correlated FSK with multichannel communication over L independent Rician fading channels in which the strength of the scattered component is assumed to be constant for all the channels. Charash [60] analyzed the average BER performance of binary orthogonal FSK with multichannel reception over L i.i.d. Nakagami-m fading channels. More recently, Weng and Leung [61] derived a closed-form expression for the average BER of binary DPSK with multichannel reception over L i.i.d. Nakagami-m fading channels. Patenaude et al. [62] extended the results of Charash [60] and Weng and Leung [61] by providing a closed-form expression for the average BER performance of binary orthogonal square-law detected FSK and binary DPSK with multichannel reception over L independent but not necessarily identically distributed Nakagami-m fading channels. Their derivation is based on the characteristic function method and the resulting expression contains L − 1 order derivatives, which can be found for small L but that become more complicated to find as L increases. In this section, we present two unified approaches for the performance evaluation of such systems over generalized fading channels. The first approach, which is described in Section 9.4.1.2, exploits the alternative integral form of the Marcum Q-function as presented in Section 4.2 and the resulting alternative integral representation of the conditional BER as well as the Laplace transforms and/or Gauss–Hermite quadrature integration derived in Chapter 5 to independently average over the PDF of each channel that fades. In all cases, this approach leads to exact expressions of the average BER that involve a single finite-range integral whose integrand contains only elementary functions and that can therefore be easily computed numerically. The second approach, which is presented in Section 9.4.1.3, relies on the bounds on the generalized Marcum Q-function developed in Section 4.2 to derive tight closed-form bounds on the average BER of the systems under consideration. 9.4.1.1 Receiver Structures We consider L-branch (finger) post-detection EGC receivers, as shown in Figs. 9.6 and 9.7 for differentially coherent and noncoherent detection, respectively. Both receivers utilize M correlators to detect the maximum a priori transmitted symbol. Without loss of generality, let us consider the mth symbol correlator. Each of the L received signals rl (t) is first delayed by τL − τl and then appropriately demodulated [symbol correlation followed by integration and dump (I&D), then

344 A exp {−j fm}

A exp {−j fm}

A exp {−j fm}

A exp {−j fm}

∫tL

tL + Ts

∫tL

tL + Ts

∫tL

tL + Ts

∫tL

tL + Ts

Figure 9.6

L Received Replicas of the Signal

rL(t )

Delay 0

r3(t )

Delay tL − t3

r2(t )

Delay tL − t2

r1(t )

Delay tL − t1

Sampler

Sampler

Sampler

Sampler

Delay Ts

Delay Ts

Delay Ts

*

*

*

*

From Other Correlators

e−j∆bm

From Other Correlators

Conjugate Operation

Differentially coherent multichannel receiver structure.

(.)dt

(.)dt

(.)dt

(.)dt

Delay Ts

P A R T

R E A L

L A R G E S T

C H O O S E

Decision

345

tL + Ts

∫ tL

A exp {−j (2pfm(t − tL))}

∫tL

tL + Ts

A exp {−j (2pfm(t − tL))}

∫tL

tL + Ts

A exp {−j (2pfm(t − tL))}

∫tL

tL + Ts

A exp {−j (2pfm(t − tL))}

(.)dt

(.)dt

(.)dt

(.)dt

Sampler

Sampler

Sampler

Sampler

.

.

.

.

2

2

2

2

From Other Correlators

From Other Correlators

Figure 9.7 Noncoherent multichannel receiver structure.

L Received Replicas of the Signal

rL(t )

Delay 0

r3(t )

Delay tL − t3

r2(t )

Delay tL − t2

r1(t )

Delay tL − t1

L A R G E S T

C H O O S E

Decision

346

PERFORMANCE OF MULTICHANNEL RECEIVERS

baud-rate sampling]. These operations assume that the receiver is correctly time synchronized at every branch (i.e., perfect time delay {τl }L l=1 estimates). For differentially coherent detection (see Fig. 9.6) the receiver takes, at every branch l, the difference of two adjacent transmitted phases to arrive at the decision rm,l . For noncoherent detection (see Fig. 9.7) no attempt is made to estimate the phase and the receiver yields the decision rm,l based on the squared envelope (i.e., square-law detection). Using EGC the L decision outputs {rm,l }L l=1 are summed to form the final decision variable rm rm =

L


rm,l ;

m = 1, 2, . . . , M

(9.98)

l=1

Last, the receiver selects the symbol corresponding to the maximum decision variable, as shown in Figs. 9.6 and 9.7. For equally likely transmitted symbols, the total conditional SNR per bit γt at the output of the post-detection EGC combiner is given [7, Sect. 12-1] as γt =

L


γl

(9.99)

l=1

9.4.1.2 Exact Analysis of Average Bit Error Probability Many problems dealing with the BER performance of multichannel reception of differentially coherent and noncoherent detection of PSK and FSK signals in AWGN channels have a decision variable that is a quadratic form in complex-valued Gaussian random variables. In 1968 Proakis [58] developed a general expression for evaluating the BER when the decision variable is in that particular form. Indeed, the development and results originally obtained in [58] later appeared as App. B of Ref. [7] and have become classics in the annals of communication system performance literature. The most general form of the BER expression [7, Eq. (B-21)] obtained by Proakis was given in terms of the first-order Marcum Q-function and modified Bessel functions of the first kind. Although implied but not explicitly given in Refs. [58] and [7], this general form can be rewritten in terms of the generalized Marcum Q-function, Ql (., .) as (see Appendix 9A) Pb (L, γt ; a, b, η)

     L−1 2L−1 l   √ η a 2 + b2 γt √  l=0 l = Q1 a γt , b γt − 1 − exp − I0 (abγt ) (1 + η)2L−1 2   L
2L − 1

 1 √ √ √ √  L−l η + Ql (a γt , b γt ) − Q1 (a γt , b γt ) L−l (1 + η)2L−1 l=2   L
2L − 1

 1 √ √ √ √  L−1+l − η Ql (b γt , a γt ) − Q1 (b γt , a γt ) , L−l (1 + η)2L−1 l=2

(9.100)

347

NONCOHERENT AND DIFFERENTIALLY COHERENT EQUAL GAIN COMBINING

  where 2L−1 L−l = [(2L − 1)!]/[(L − l)! (L + l − 1)!] denotes the binomial coefficient, and where all the modulation-dependent parameters have already been defined previously. As a check for L = 1 and η = 1, the latter two summations in (9.100) do not contribute and hence one immediately obtains the single-channel result (8.196), as expected. Note that although the form in (9.100) does not give the appearance of being much simpler than Eq. (B-21) of Ref. [7], we shall see shortly that it does have particular advantage for obtaining the average BER performance over generalized fading channels. As in the single-channel reception case, the parameters a and b in (9.100) are typically independent of SNR and furthermore b > a. For instance, for noncoherent detection of equal energy, equiprobable, correlated binary signals, η = 1 and  a=  b=

1−

1+





1 − |λ|2 2 1 − |λ|2 2

1/2

1/2 (9.101)

where λ (0 ≤ |λ| ≤ 1) is the complex-valued crosscorrelation coefficient between the two signals. The special case λ = 0 corresponds to orthogonal noncoherent BFSK for which a = 0 and b = 1.Furthermore, inthe case of binary DPSK, a = 0, √ √ √ b = 2, and η = 1. Finally a = 2 − 2, b = 2 + 2, and η = 1 correspond to DQPSK with Gray coding. At this point let us introduce again a modulationdependent parameter ζ = (a/b), which is independent of SNR. With this in mind, we now show how the alternative integral representations of the generalized Marcum Q-function yields a desired product form representation of the conditional BER. In particular, it was shown in Section 4.2.2 that 

ζ −(l−1) (cos [(l − 1)(φ + π /2)] − ζ cos [l(φ + π /2)]) 1 + 2ζ sin φ + ζ 2 −π # "  w2  u 0+ ≤ ζ = 0, s > k

(9.211)

together with the other identity [53, Eq. (15.1.8), p. 556; 83, Eq. (A-5)] 2 F1 (a, b; b; z)

= (1 − z)−a

(9.212)

it can be shown that

−m √ √ γ (1 − ρ + D ρ) s Mb (s; γ ; m; ρ; D) = Mb (s) = 1 − m

−m(D−1) √ γ (1 − ρ) × 1− , s < 0. s m 

(9.213)

11 It should be noted at this point that in Eq. (18) of Ref. 83 [or equivalently Eq. (41) of Ref. 74] the symbol ρ is used to denote the correlation coefficient of the underlying Gaussian processes that produce the fading on the channels. This correlation coefficient is equal to the square root of the power correlation coefficient in Rayleigh fading (see Appendix 9C). Following a study by Lawson and Uhlenbeck [102, p. 62] Pierce and Stein have shown [82, App. V] that, for all practical purposes, the envelope correlation coefficient can be assumed to be equal to the power correlation coefficient that is denoted by ρ throughout this section so as to follow what seems to be the more conventional usage of this symbol.

394

PERFORMANCE OF MULTICHANNEL RECEIVERS

For D = 2, as a check, it can be easily shown that (9.213) agrees with (9.206) for γ 1 = γ 2. 9.7.3 Model C: D Identically Distributed Branches with Exponential Correlation Model C was also proposed by Aalo [83, Sect. II-B] for identically distributed Nakagami-m channels (i.e., all channels are assumed to have the same average SNR per symbol γ and the same fading parameter m). This model assumes an exponential power correlation coefficient ρdd between any pair of channels (d, d = 1, 2, . . . , D) as given by cov(rd2 , rd2 )

ρdd = = ρ |d−d | , var(rd2 )var(rd2 )

0≤ρ≤1

(9.214)

and may therefore correspond to the scenario of multichannel reception from equispaced diversity antennas in which the correlation between the pairs of combined signals decays as the spacing between the antennas increases. 9.7.3.1 PDF On the basis of the work of Kotz and Adams [103], Aalo showed that the PDF of γt can be very well approximated by a gamma distribution given by [83, Eq. (19)]12   t exp − mDγ rρ γ ,    2 /r ) (mD ρ 2

mD 2 /rρ −1

pc (γt ) =

γt



mD rρ

γt ≥ 0

(9.215)

rρ γ mD

where13 rρ = D +



√ 2 ρ 1 − ρ D/2 D− √ √ 1− ρ 1− ρ

(9.216)

9.7.3.2 MGF Substituting (9.215) in (9.193) and then using the Laplace transform [36, Eq. (3.381.4)], it can be shown that

−mD2 /rρ rρ γ Mc (s; γ ; m; ρ; D) = Mc (s) = 1 − , s mD 

s 0)

403

IMPACT OF FADING CORRELATION

(m = 1) Average Bit Error Probability Pb(E)

Average Bit Error Probability Pb(E)

(m = 0.5) 100

10−1

b a

r=0 r = 0.2 r = 0.4 10−2

0

c

5

10

15

100

10−1 r=0 r = 0.2 r = 0.4 10−2

0

Average SNR per Bit of First Path [dB]

5

10−1 10−2

c b r=0 r = 0.2 r = 0.4 0

a

5

15

(m = 4) Average Bit Error Probability Pb(E)

Average Bit Error Probability Pb(E)

(m = 2)

10−4

10

Average SNR per Bit of First Path [dB]

100

10−3

c

b

a

10

Average SNR per Bit of First Path [dB]

15

100 10−1 10−2 r=0 r = 0.2 r = 0.4

10−3

10−4

0

a

5

b

c

10

15

Average SNR per Bit of First Path [dB]

Figure 9.26 Average bit error probability Pb (E) of binary noncoherent orthogonal FSK with dual diversity (L = 2) square-law combining versus the average SNR per bit of the first path γ 1 over unbalanced correlated Nakagami-m channels with an exponentially decaying power delay profile [(a) δ = 0, (b) δ = 0.5, (c) δ = 1].

than for channels with a uniform power delay profile (δ = 0), which explains in part the relatively important performance degradation due to the exponentially decaying power delay profile. Figure 9.26 is a plot of Pb (E) for binary orthogonal FSK with dual diversity over correlated unbalanced (γ 2 = e−δ γ 1 ) Nakagami-m fading channels. On the other hand, Fig. 9.27 shows the average BEP performance of binary orthogonal FSK with square-law combining over a multilink channel with L = 5, an exponentially decaying power delay profile, and an exponential correlation profile (i.e., ρij = ρ |i−j | , 1 ≤ i < j ≤ L). In both figures, for the parameters of interest, the BEP degradation induced by the power delay profile is higher than the degradation due to the fading correlation profile, where the degradation here is with respect to a system operating over a uniform power delay profile with independent multipaths. Furthermore, comparing Figs. 9.26 and 9.27, we conclude that this deterioration is more noticeable as the number of combined paths increases.

404

PERFORMANCE OF MULTICHANNEL RECEIVERS

(m = 1) Average Bit Error Probability Pb(E)

Average Bit Error Probability Pb(E)

(m = 0.5) 100 10−1 10−2 10−3

b a

10−4

r=0 r = 0.2 r = 0.4

10−5 10−6

c

0

5

10

15

20

25

100 10−1 10−2 10−3

0

Average SNR per Bit of First Path [dB]

5

10

r=0 r = 0.2 r = 0.4

10−3

a

b

c

10−4 10−5 10−6 0

5

10

15

20

25

(m = 4) Average Bit Error Probability Pb(E)

Average Bit Error Probability Pb(E)

(m = 2)

10−2

15

Average SNR per Bit of First Path [dB]

100 10−1

c

r=0 r = 0.2 r = 0.4

10−5 10−6

b

a

10−4

20

Average SNR per Bit of First Path [dB]

25

100 r=0 r = 0.2 r = 0.4

10−1 10−2 10−3

a

10−4

b

c

10−5 10−6 0

5

10

15

20

25

Average SNR per Bit of First Path [dB]

Figure 9.27 Average bit error probability Pb (E) of binary noncoherent orthogonal FSK with square-law combining (L = 5) versus the average SNR per bit of the first path γ 1 over correlated Nakagami-m channels with an exponential fading correlation profile and an exponentially decaying power delay profile [(a) δ = 0, (b) δ = 0.5, (c) δ = 1].

9.8

SELECTION COMBINING

Of the three types of linear diversity combining MRC, EGC, and SC normally employed in receivers of digital signals transmitted over multipath fading channels, SC is the least complicated since it processes only one of the diversity branches. Specifically, the combiner chooses the branch with the highest signal-to-noise ratio (or equivalently with the strongest signal assuming equal noise power among the branches) [108, Sect. 10-4, p. 432]. In order to obtain significant diversity gain, independent fading in the channels should be achieved. However, as mentioned previously, this is not always realized in practice because, for example, of insufficient antenna spacing in small-size terminals equipped with space antenna diversity and as a result, the maximum theoretical diversity gain cannot be achieved. In addition, the diversity branches in a practical system may have unequal average SNRs due to different noise figures or feedline lengths [109,110]. Hence, it is important to assess the effect of correlation and average SNR unbalance on the outage probability and average error probability of an SC diversity

SELECTION COMBINING

405

receiver, in particular, a dual-diversity (two-branch) SC receiver, which is the specific case to be considered in this section. Some special cases of the performance of various modulation schemes with dual SC over independent and correlated Rayleigh and Nakagami-m slow-fading channels have been reported in the literature [111–115]. For instance, Blanco [111] studied the performance of noncoherent BFSK with dual-SC over independent identically distributed Nakagami-m fading channels. Al-Hussaini and Al-Bassiouni [116] analyzed the effect of fading correlation and average SNR imbalance on the BER performance of BFSK with dual SC over Nakagami-m fading channels. Adachi et al. [112] analyzed the performance of DQPSK over correlated unequal average power Rayleigh fading channels. Okui [113] studied the probability of cochannel interference for selection diversity reception in the Nakagami-m fading channel, whereas Wan and Chen [114] presented simulation results for DQPSK with dual SC over correlated Rayleigh fading channels. Fedele et al. [117] analyzed the performance of M-ary DPSK with dual SC over independent and correlated Nakagami-m fading channels. Finally, Ugweje and Aalo [115] analyzed the average BER performance of DPSK and BPSK with dual SC over correlated Nakagami-m fading channels. In this section, we use the unifying MGF-based approach combined with the results presented in Chapter 6 to assess the performance of dual SC over independent and correlated slow Rayleigh and Nakagami-m fading channels [118,119]. 9.8.1

MGF of Output SNR

Recall that the PDF of the output SNR, γSC , of a dual SC over correlated Nakagamim fading channels is given by (6.21), namely

    mγ  (m/ γ 1 )m γ m−1 1 − Qm A1 2ργ , A2 2γ exp − pγSC (γ ) = (m) γ1

  m m−1   mγ  (m/ γ 2 ) γ 1 − Qm A2 2ργ , A1 2γ (9.235) exp − + (m) γ2  where Al is given by Al = m/[γ l (1 − ρ)]. Hence, the MGF of the output SNR can be written as MγSC (s) = I1 (s; γ 1 , γ 2 , ρ, m) + I2 (s; γ 1 , γ 2 , ρ, m) where

"

#  m (m/ γ 1 )m ∞ m−1 I1 (s; γ 1 , γ 2 , ρ, m) = γ exp − −s γ (m) γ1 0      × 1 − Qm A1 2ργ , A2 2γ d γ "

#  (m/ γ 2 )m ∞ m−1 m γ exp − −s γ I2 (s; γ 1 , γ 2 , ρ, m) = (m) γ2 0      × 1 − Qm A2 2ργ , A1 2γ d γ

(9.236)

(9.237)

(9.238)

406

PERFORMANCE OF MULTICHANNEL RECEIVERS

Using the identity in Ref. 113 [Eq. (6)], a straightforward change of integration variables in (9.237) and (9.238) allows us to express I1 (s; γ 1 , γ 2 , ρ, m) and I2 (s; γ 1 , γ 2 , ρ, m) in closed form as I1 (s; γ 1 , γ 2 , ρ, m) =

I2 (s; γ 1 , γ 2 , ρ, m) =

23m (2m)m2m X1−2m [W1 (1 + W1 )]−m (m) (m + 1)(γ 1 γ 2 )m (1 − ρ)m "

# 1 1 1− ×2 F1 1 − m, m; 1 + m; 2 W1 23m (2m)m2m X2−2m [W2 (1 + W2 )]−m (m) (m + 1)(γ 1 γ 2 )m (1 − ρ)m "

# 1 1 1− ×2 F1 1 − m, m; 1 + m; 2 W2

(9.239)

(9.240)

where 2 F1 [ · , · ; · ; · ] is the Gauss hypergeometric function [53, Ch. 15] and Xi , Yi , and Wi are given by X1 = 2(a − s) X2 = −2(a + s) 1/2  Y1 = Y2 = 2 (b − s)2 − c2 

2 1/2 b−s 2 Y1 c W1 = = − X1 a−s a−s 

2 1/2 b−s 2 Y2 c W2 = =− − X2 a+s a+s

(9.241)

where a, b, and c are given by a=

9.8.2

m(γ 2 − γ 1 ) , γ 1 γ 2 (1 − ρ)

b=

m(γ 1 + γ 2 ) , γ 1 γ 2 (1 − ρ)

c= 

√ 2m ρ γ 1 γ 2 (1 − ρ)

(9.242)

Average Output SNR

General Case Using the well-known result that the first moment of γSC is equal to its statistical average [120, Eq. (5-67)] ! dMγSC (s) !! γ SC = (9.243) ! ds s=0 we obtain, after substituting (9.236) in (9.243) and using the differentiation formula given in Ref. 53, [Eq. (15.2.1)], and after much manipulation, the final desired closed-form result as γ SC = K (K1 + K2 )

(9.244)

SELECTION COMBINING

407

where 2m m2m+1 (2m) (m) (m + 1)(γ 1 γ 2 )m (1 − ρ)m   √ (b2 − c2 − ab)(b2 − c2 + a b2 − c2 )−m K1 = √ (b2 − c2 )( b2 − c2 − a) 

m−1

 1 1 a a × + √ − 2 F1 1 − m, m; 1 + m; − √ 2 2 b2 − c2 2 2 b2 − c2   √ (b2 − c2 + ab)(b2 − c2 − a b2 − c2 )−m + √ (b2 − c2 )( b2 − c2 + a) 

m−1

 1 1 a a − √ × − 2 F1 1 − m, m; 1 + m; + √ 2 2 b2 − c2 2 2 b2 − c2

 b2 − c2 + ab 2b + √ K2 = (b2 − c2 + a b2 − c2 )−m−1 b2 − c2

1 a × 2 F1 1 − m, m; 1 + m; − √ 2 2 b2 − c2

 b2 − c2 − ab 2 2 −m−1 2 2 2b + √ + (b − c − a b − c ) b2 − c2

1 a (9.245) × 2 F1 1 − m, m; 1 + m; + √ 2 2 b2 − c2 K=

Using the integral representation of the Gauss hypergeometric function as given by Abramowitz and Stegun [53, Eq. (15.1.25)] it can be easily shown that 2 F1 (0, 1; 2; x) = 1 for all x. Hence for the Rayleigh fading (m=1) case, (9.244) reduces to the much simpler formula   γ 21 + γ 22 − 2ργ 1 γ 2 1 γ SC = γ1 + γ2 +  (9.246) 2 (γ 1 + γ 2 )2 − 4ργ 1 γ 2 Special Cases For the equal average SNR (γ 1 = γ 2 = γ ) but correlated fading √ case (ρ = 0), we have from (9.242) a = 0, b = 2m/(γ (1 − ρ)), and c = 2m ρ/ (γ (1 − ρ)). This, combined with the identity in Ref. 53 [Eq. (15.1.26)] relating the Gauss hypergeometric function to the gamma function as well as the duplication formula [53, Eq. (6.1.18)] of the gamma function, leads to the average SNR in the greatly simplified form given by √ " # (2m) 1 − ρ γ SC = γ 1 + 2m−1 (9.247) 2 (m) (m + 1)

408

PERFORMANCE OF MULTICHANNEL RECEIVERS

which further reduces for the special Rayleigh fading case (m = 1) to √ " # 1−ρ γ SC = γ 1 + 2

(9.248)

For the unequal average SNR (γ 1 = γ 2 ) with uncorrelated fading (ρ=0) case, we have from (9.242) a = m(γ 2 − γ 1 )/(γ 1 γ 2 ), b = m(γ 1 + γ 2 )/(γ 1 γ 2 ), and c = 0. Substituting this in (9.245) and using the linear transformation formula [53, Eq. (15.3.3)], it can be shown after some algebraic manipulations that (9.244) reduces to # " (γ 1 γ 2 )m+1 (2m + 1) γ SC = γ 1 + γ 2 − (m + 1) (m) (m + 1)(γ 1 + γ 2 )2m+1 "

γ1 × 2 F1 1, 2m + 1; m + 2; γ1 + γ2

# γ2 + 2 F1 1, 2m + 1; m + 2; . (9.249) γ1 + γ2 Using the series definition of the Gauss hypergeometric function as given in Ref. 53 [Eq. (15.1.1)], it can be easily shown that 2 F1 (1, 3; 3; x) = 1/(1 − x). Hence for the Rayleigh fading case (m = 1), (9.249) reduces to γ SC = γ 1 + γ 2 −

γ 1γ 2 γ1 + γ2

(9.250)

Finally, for the equal average SNR (γ 1 = γ 2 = γ ) with uncorrelated fading (ρ = 0) case, we have in (9.242) a = 0, b = 2m/γ , and c = 0. This combined again with [53, Eq. (15.1.26) and Eq. (6.1.18)] leads to # " (2m) (9.251) γ SC = γ 1 + 2m−1 2 (m) (m + 1) which reduces for the Rayleigh case (m = 1) to γ SC = 1.5γ in agreement with Eq. (6.62) of Ref. 4 or equivalently Eq. (5.86) of Ref. 3. Numerical Examples As an example, Fig. 9.28 plots the first branch normalized average SNR, γ SC /γ 1 , versus the correlation coefficient ρ for an equal average dual SC receiver [(a) (γ 1 = γ 2 )] as well as for an unbalanced dual SC receiver [(b) (γ 1 = 2 γ 2 ), (c) (γ 1 = 5 γ 2 ), and (d) (γ 1 = 10 γ 2 )]. The average SNR degrades quite rapidly as the correlation coefficient ρ increases especially for the equal average SNR case and for low values of m. In particular, when ρ = 0 (uncorrelated fading), the average SNR is maximum, and in the limit of fully fading correlation (ρ = 1), the average output SNR approaches the average SNR of a single branch (i.e., without diversity), as expected. In addition for a fixed fading correlation the average SNR decreases as the severity of fading decreases (i.e., m decreases), which may seem to be surprising at first glance. However, as m increases, the distribution becomes more skewed, which reduces the effective area of integration and therefore explains this dependence of the average SNR on the fading parameter m.

SELECTION COMBINING

1.3

1.6

1.5

First Branch Normalized Average SNR

First Branch Normalized Average SNR

1.7

m = 0.5

1.4

m=1

1.3

m=2

1.2

m=4

1.1

1

1.25

1.15

m=1

1.1

m=2 1.05

m=4

0

0.2

0.4

0.6

0.8

1

1

0

0.2

0.4

0.6

0.8

1

Correlation Coefficient r

1.03

1.08 1.07

First Branch Normalized Average SNR

First Branch Normalized Average SNR

m = 0.5

1.2

Correlation Coefficient r

1.06

m = 0.5 1.05 1.04 1.03

m=1 1.02 1.01 1

409

m=2 m=4 0

0.2

0.4

0.6

0.8

Correlation Coefficient r

1

1.025

m = 0.5 1.02

1.015

1.01

m=1 1.005

1

m=2 m=4 0 0.2

0.4

0.6

0.8

1

Correlation Coefficient r

Figure 9.28 First-branch normalized average SNR (γ SC /γ 1 ) of SC versus correlation coefficient (ρ) for correlated Nakagami-m fading channels with (a) γ 1 = γ 2 , (b) γ 1 = 2γ 2 , (c) γ 1 = 5γ 2 , and (d) γ 1 = 10γ 2 .

9.8.3

Outage Probability

9.8.3.1 Analysis The outage probability Pout is defined as the probability that the SC output SNR γ = max(γ1 , γ2 ) falls below a given threshold, say, γth . Since this probability is simply the probability that neither γ1 nor γ2 exceeds the threshold γth , then, by

410

PERFORMANCE OF MULTICHANNEL RECEIVERS

inspection, the outage probability is obtained by replacing γ with γth in the CDF expression given in (6.16) yielding for Rayleigh fading: Case 1: Identical Channels (γ 1 = γ 2 = γ )

 1−ρ π 1 γth + Pout = 1 − 2 exp − √ γ 2π −π 1 + ρ + 2 ρ sin θ

# " √ 2γth 1 + ρ sin θ dθ × exp − γ 1−ρ

(9.252)

Note that this result is equivalent to Eq. (10-10-7) in Ref. 108, which is expressed in terms of the Marcum Q-function. Note that even for this simpler case of identical channels, Tan and Beaulieu’s result [121, Eq. (4)] (or equivalently Staras’ result [122]) does not simplify considerably since it is still an infinite series of squares of integrals. Furthermore, in the limiting case of uncorrelated branches (i.e., ρ = 0), (9.252) reduces to Pout = (1 − exp(−γth /γ ))2 , as expected. Case 2: Nonidentical Channels (γ 1
= γ 2 )   Pout = 1 − G H (γth , γ 1 ), H (γth , γ 2 )|ρ     π γ 1 + γ 2 + 2 ργ 1 γ 2 sin θ 1 exp −γth + 2π −π γ 1 γ 2 (1 − ρ)  √ (1 − ρ 2 )γ 1 γ 2 + ρ(1 − ρ) γ 1 γ 2 (γ 1 + γ 2 ) sin θ   dθ × (ργ 2 + 2 ργ 1 γ 2 sin θ + γ 1 )(γ 2 + 2 ργ 1 γ 2 sin θ + ργ 1 )

(9.253)

  where G H (γth , γ 1 ), H (γth , γ 2 )|ρ is as given in (6.15). Note that (9.253) is equivalent to Eq. (10-10-3) of Ref. 108, which is expressed in terms of the Marcum Q-function. Furthermore, in the limiting case of uncorrelated branches (i.e., ρ = 0), (9.253) together with (6.15), reduces to Pout = (1 − exp(−γth /γ 1 ))(1 − exp(−γth /γ 2 )), as expected. The outage probability expressions for Nakagami-m fading are immediately obtained from the CDF expressions (6.26) and (6.30) by replacing γ with γth . Since no further simplifications are possible, and in the interest of brevity, we shall not write down the specific results for the two cases of identical and nonidentical channels previously considered for Rayleigh fading. 9.8.3.2 Numerical Example Figure 9.29 compares the outage probability of dual-branch MRC and SC for various values of the fading parameter m, correlation coefficient ρ, and average SNR unbalance. In this figure, the SC outage probability results are based on (9.252), whereas the MRC outage probability results are obtained by substituting (9.206) in (9.186).

411

SELECTION COMBINING

100

Outage Probability Pout

10−1

m=1 10−2

m=4

10−3

a b

10−4 10−5 10−6 −5

a

d

d

b

c

c d bc d a

bc

Maximal Ratio Combining (MRC) Selection Combining (SC) 0

5

10

15

20

Normalized Average SNR per Path [dB] 100

m=1

Outage Probability Pout

10−1

d ab c

m=4

10−2

a

10−3 10−4

a

10−5

Maximal Ratio Combining (MRC) Selection Combining (SC)

10−6 −5

0

5

b c d b b

10

d bc

c d 15

20

Normalized Average SNR of First Path [dB] Figure 9.29 Comparison of outage probability with MRC and SC versus average SNR of the first branch for various values of the correlation coefficient [(a) ρ = 0, (b) ρ = 0.5, (c) ρ = 0.7, (d) ρ = 0.9] and for equal average branch SNRs (γ 1 = γ 2 ) (upper plot) and unequal average branch SNRs (γ 1 = 10 γ 2 ) (lower plot).

9.8.4

Average Probability of Error

9.8.4.1 BDPSK and Noncoherent BFSK Recall that the conditional BER for BDPSK and noncoherent BFSK is given by Eqs. (5-2-69) and (5-4-47) of Ref. 7 Pb (E|γ ) =

1 2

exp(−gγ )

(9.254)

where g is again a modulation constant: g = 1 for BDPSK and g = 12 for orthogonal BFSK. Averaging (9.254) over the PDF of the SC output (6.18), we obtain the

412

PERFORMANCE OF MULTICHANNEL RECEIVERS

following expression for the average BER: " #  π h1 (θ |ρ)h2 (θ |ρ) 1 1 G((1 + gγ 1 )−1 , (1 + gγ 2 )−1 |ρ) − dθ 2 2π −π g + h1 (θ |ρ) (9.255) The integral term in (9.255) can be evaluated in closed form by first expanding the integrand into a partial fraction expansion and then making use of a well-known definite integral. In particular, identifying h1 (θ |ρ) and h2 (θ |ρ) from (6.17), it is straightforward to show that Pb (E) =

 γ 1 + γ 2 + 2 ργ 1 γ 2 sin θ h1 (θ |ρ)h2 (θ |ρ) =  g + h1 (θ |ρ) g(1 − ρ)γ 1 γ 2 + γ 1 + γ 2 + 2 ργ 1 γ 2 sin θ  √ (1 − ρ 2 )γ 1 γ 2 + ρ(1 − ρ) γ 1 γ 2 (γ 1 + γ 2 ) sin θ , ×   (ργ 2 + 2 ργ 1 γ 2 sin θ + γ 1 )(γ 2 + 2 ργ 1 γ 2 sin θ + ργ 1 ) (9.256) which is in the form γ1 + γ2 h1 (θ |ρ)h2 (θ |ρ) , = g + h1 (θ |ρ) γ 1 + γ 2 + gγ 1 γ 2

ρ=0

h1 (θ |ρ)h2 (θ |ρ) A + B sin θ + C sin2 θ = g + h1 (θ |ρ) (a1 + b sin θ )(a2 + b sin θ )(a3 + b sin θ ) c1 c2 c3 = + + , a1 + b sin θ a2 + b sin θ a3 + b sin θ

(9.257)

ρ = 0 (9.258)

with a1 = g(1 − ρ)γ 1 γ 2 + γ 1 + γ 2 a2 = ργ 2 + γ 1 a3 = ργ 1 + γ 2  b = 2 ργ 1 γ 2 A = (1 − ρ 2 )γ 1 γ 2 (γ 1 + γ 2 )  √ √ B = 2 ρ(1 − ρ 2 )(γ 1 γ 2 )3/2 + ρ(1 − ρ) γ 1 γ 2 (γ 1 + γ 2 )2 C = 2ρ(1 − ρ)γ 1 γ 2 (γ 1 + γ 2 ). The coefficients of the partial-fraction expansion are readily determined as c1 =

b2 A − a1 bB + a12 C (a1 − a2 )(a1 − a3 )b2

(9.259)

413

SELECTION COMBINING

c2 = −

b2 A − a2 bB + a22 C (a1 − a2 )(a2 − a3 )b2

b2 A − a3 bB + a32 C . (9.260) (a1 − a3 )(a2 − a3 )b2 Finally, substituting (9.257) and (9.258) into (9.255) and making use of the definite integral [36, p. 425, Eq. (3.661.4)]  π 1 1 1 dθ = √ , a ≥ b, (9.261) 2 2π −π a + b sin θ a − b2 c3 =

we get the desired closed-form result15 1 1
Pb (E) = βi , 2 1 + gγ i 3



γ3 =

i=1

γ 1γ 2 , β1 = β2 = 1, β3 = −1, γ1 + γ2

ρ=0 (9.262)



Pb (E) =




ci 1  , G((1 + gγ 1 )−1 , (1 + gγ 2 )−1 |ρ) − 2 2 ai − b2 3

ρ = 0

i=1

(9.263) Note that for the special case of γ 1 = γ 2 and g = 1 (BDPSK), (9.262) is in agreement with St¨uber’s result [3, p. 242, Eq. (5.88)]. 9.8.4.2 Coherent BPSK and BFSK Recall that, on the basis of an alternative representation of the Gaussian Q-function as given in (4.2), the conditional BER of BPSK and BFSK can be written in the integral form

 1 π/2 gγ dθ (9.264) exp − 2 Pb (E|γ ) = π 0 sin θ where g = 1 for BPSK, g = 12 for orthogonal BFSK, and g = 0.715 for BFSK with minimum correlation. Recognizing the analogy between (9.264) and (9.254) insofar as its functional dependence on γ , we can immediately write the average BER as   3     3
1 gγ i 1 π/2
1 βi βi dθ = , ρ=0 Pb (E) = 1− π 0 1 + g(θ )γ i 2 1 + gγ i i=1

i=1

(9.265) 1 Pb (E) = π −



π/2 0

3
i=1

15





G((1 + g(θ )γ 1 )−1 , (1 + g(θ )γ 2 )−1 |ρ) ci (θ ) ai 2 (θ ) − b2

 dθ ,

ρ = 0

(9.266)

Note from (9.259) that it can easily be shown that ai ≥ b for i = 1, 2, 3. Hence, (9.261) applies.

414

PERFORMANCE OF MULTICHANNEL RECEIVERS

where now g(θ ) = g/ sin2 θ , a2 (θ ) = a2 , a3 (θ ) = a3 , and a1 (θ ), c1 (θ ), c2 (θ ), and c3 (θ ) are obtained by substituting g(θ ) for g in (9.259) and (9.260), respectively, and the same substitution is made in G((1 + gγ 1 )−1 , (1 + gγ 2 )−1 |ρ). For identical channels (γ 1 = γ 2 = γ ) using the Adachi et al. approximate expression of the PDF [112, Eq. (42)] pγ (γ ) ≈

 −γ / γ  2 e − e−[2γ /(1−ρ)γ ] (1 + ρ) γ

(9.267)

it can be shown that the average BER of BPSK can be written in closed form as 1 Pb (E) = (1 + ρ)



1+ρ − 2



gγ 1−ρ + 1+gγ 2



(1 − ρ) g γ 2 + (1 − ρ) g γ

 (9.268)

Since it would be useful to know the relative accuracy improvements of the exact expressions (9.265) and (9.266) over the approximation (9.268), we plot all of them in Fig. 9.30. Note that the approximation tightly upper-bounds the exact BER expression and the bound gets tighter as the average SNR increases and as 100 Exact Approximation

Average Bit Error Rate (BER)

10−1

10−2

10−3

10−4

b

a

c d

10−5

10−6

0

5

10

15

20

25

30

Average Signal-to-Noise-Ratio (SNR) g1 = g2 [dB] Figure 9.30 Exact and approximate average BER of BPSK versus average SNR of the first branch for equal average branch SNRs (γ 1 = γ 2 ) and for various values of the correlation coefficient [(a) ρ = 0.9, (b) ρ = 0.7, (c) ρ = 0.5, (d) ρ = 0].

415

SELECTION COMBINING

the correlation coefficient decreases. Extension of the average probability of error performance to Nakagami-m fading is omitted here but can be found in Ref. 118. 9.8.4.3 Numerical Example We present in this section various numerical examples to illustrate the effects of (1) the severity of fading, (2) branch correlation, and (3) branch average SNR imbalance on the performance of the system. Figures 9.31 and 9.32 plot the average BER of BDPSK versus the average SNR of the first branch for various values of the fading parameter m and correlation coefficient ρ for equal average branch SNRs and unequal average branch SNRs, respectively. Figures 9.33 and 9.34 plot the average BER of BPSK versus the average SNR of the first branch for various values of the fading parameter m and correlation coefficient ρ for equal average branch SNRs and unequal average branch SNRs, respectively. Note in all figures that the diversity gain decreases with the increase of the correlation coefficient, as expected. Note also that the effect of branch correlation is more important for channels with a lower amount of fading (higher m parameter). Finally, comparing the equal average branch SNR figures (Figs. 9.31 and 9.33) with the unequal average branch ones (Figs. 9.32 and 9.34), observe that (1) unbalance in the average branch SNRs always leads to lower overall system performance, and (2) the effect of correlation is more important for equal average branch SNRs.

100 10−1

Average Bit Error Rate (BER)

10−2 10−3

m=1

10−4

a b

10−5 10−6

m=2

c d

10−7

a

10−8

m=4

b

10−9 10−10

c b

d 0

5

10

15

20

a 25

c d 30

Average Signal-to-Noise-Ratio (SNR) g1 = g2 [dB]

Figure 9.31 Average BER of BDPSK versus average SNR of the first branch for equal average branch SNRs (γ 1 = γ 2 ) and for various values of the correlation coefficient [(a) ρ = 0.9, (b) ρ = 0.7, (c) ρ = 0.5, (d) ρ = 0].

416

PERFORMANCE OF MULTICHANNEL RECEIVERS

100 10−1

Average Bit Error Rate (BER)

10−2

m=1

10−3

a b c d

10−4

m=2

10−5 10−6

m=4

a b c d

10−7 10−8 10−9 10−10

b d c 0

5

10

15

20

a

25

30

Average Signal-to-Noise-Ratio (SNR) g1 = 10 g2 [dB]

Figure 9.32 Average BER of BDPSK versus average SNR of the first branch for unequal average branch SNRs (γ 1 = 10 γ 2 ) and for various values of the correlation coefficient [(a) ρ = 0.9, (b) ρ = 0.7, (c) ρ = 0.5, (d) ρ = 0]. 100 10−1

Average Bit Error Rate (BER)

10−2 10−3 10−4

m=1

10−5

a b

10−6

c d

m=2

10−7 10−8

a

m=4 10−9 10−10

c b 5 10 15 20 Average Signal-to-Noise-Ratio (SNR) g1 = g2 [dB] d

0

b d c

a 25

30

Figure 9.33 Average BER of BPSK versus average SNR of the first branch for equal average branch SNRs (γ 1 = γ 2 ) and for various values of the correlation coefficient [(a) ρ = 0.9, (b) ρ = 0.7, (c) ρ = 0.5, (d) ρ = 0].

SWITCHED DIVERSITY

417

100 10−1

Average Bit Error Rate (BER)

10−2 10−3

m=1 10−4

m=2

a b c d

10−5 10−6

a

10−7

m=4

b c

10−8

d

10−9 10−10

d 0

5

10

15

20

c b 25

a 30

Average Signal-to-Noise-Ratio (SNR) g1 = 10 g2 [dB]

Figure 9.34 Average BER of BPSK versus average SNR of the first branch for unequal average branch SNRs (γ 1 = 10 γ 2 ) and for various values of the correlation coefficient [(a) ρ = 0.9, (b) ρ = 0.7, (c) ρ = 0.5, (d) ρ = 0].

Figure 9.35 compares the average BER performance of BPSK with dual-branch MRC and SC diversity for equal and unequal average SNR. In this figure, the SC average BER results are based on (9.265) and (9.266), whereas the MRC average BER results are based on the MGF given by (9.206). On the other hand, in both of these figures note that MRC outperforms SC, as expected, and that the diversity gain of MRC compared to SC is more important for channels with a low amount of fading (high m) regardless of the correlation between the two branches. 9.9

SWITCHED DIVERSITY

Switched diversity is an attempt at simplifying the complexity of selection diversity systems still further with, of course, an attendant loss in performance. In this case, rather than continually connecting the diversity path with the best quality, the receiver selects a particular diversity path until its quality drops below a predetermined threshold. When this happens, the receiver switches to another diversity path. This results in a complexity reduction relative to selection combining in that the simultaneous and continuous monitoring of all diversity paths is no longer necessary.

418

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Bit Error Rate (BER) Pb (E )

100 Maximal Ratio Combining (MRC) Selection Combining (SC)

10−1 10−2

m=1

10−3

m=4

10−4

a

10−5 10−6

d a

0

5

bc b c d 10

b

d

15

bc d c

20

25

30

Average SNR per Bit [dB]

Average Bit Error Rate Pb (E )

100 Maximal Ratio Combining Selection Combining

10−1 10−2

m=1

10−3 10−4

m=4 a

10−5

d b c d 10 15 20 Average SNR per Bit of First Path [dB]

d b cd b c

a

10

−6

0

5

25

30

Figure 9.35 Comparison of average BER of BPSK with MRC and SC for various values of the correlation coefficient [(a) ρ = 0, (b) ρ = 0.5, (c) ρ = 0.7, (d) ρ = 0.9] and for equal average branch SNRs (γ 1 = γ 2 ) (upper plot) and unequal average branch SNRs (γ 1 = 10 γ 2 ) (lower plot).

For more than two decades, communication theorists have analyzed the performance of switched diversity systems [8–10,111,123,128–130,132–134] and communication engineers have used their theoretical and numerical results to guide the design and optimization of these systems. Whereas initial work focused on dualbranch switch-and-stay combining (SSC) [8–10,111,123,128–132], more recent applications (e.g., transmit diversity) have motivated studies of multibranch switchand-examine combining (SEC) [133–135]. In this section, we first study the performance of dual-branch SSC systems. We then look into the various variants of SEC systems.

SWITCHED DIVERSITY

control

Switch Logic Comparator

Receiver

419

Present Treshold

Channel Estimator Data

Figure 9.36 Mode of operation of dual-branch switch-and-stay combining (SSC) diversity.

9.9.1

Dual-Branch Switch-and-Stay Combining

In this subsection, we focus on the performance evaluation and optimization of some SSC systems over a wide variety of fading conditions in conjunction with several communication types of practical interest. In particular, we consider dualbranch diversity systems, for which, when the SNR of the currently connected branch falls below a predetermined threshold, the receiver switches to, and stays with, the other branch regardless of whether the SNR of that branch is above or below the predetermined threshold (see Fig. 9.36) [9,123,129,130,132]. Other SSC systems for which a switch is initiated only when a downward crossing of the predetermined threshold occurs are analyzed in Refs. 8 and 131. The setting of the predetermined threshold is an additional important system design issue for SSC. For instance, if this threshold level is chosen too high, the switching unit is almost continually switching between the two antennas, which results not only in a poor diversity gain but also in an undesirable increase in the rate of the switching transients on the transmitted data stream. On the other hand, if this threshold level is chosen too low, the switching unit is almost locked to one of the diversity branches, even when the SNR level is quite low, and again little diversity gain is achieved. Hence, another goal of this section is to determine the optimal switching threshold as a function of channel characteristic, performance measure, and modulation type. Work related to this kind of SSC systems can be found in the literature [8–10,111,124–128,130–132]. 9.9.1.1 Performance of SSC over Independent Identically Distributed Branches Let γSSC denote the SNR per symbol of the SSC combiner output, and let γT denote the predetermined switching threshold. Following the mode of operation of SSC as described earlier, we derive in this section, the CDF, PγSSC (γ ), PDF, pγSSC (γ ), and MGF, MγSSC (γ ), of γSSC assuming i.i.d. branches.

420

PERFORMANCE OF MULTICHANNEL RECEIVERS

TABLE 9.5 Statistics of the SNR per Symbol γ for the Three Multipath Fading Models under Consideration Model

PDF (pγ (γ ))

Rayleigh

1 −(γ / γ ) e γ

Nakagami-m

(1 + n2 )e−n −(1+n2 )γ / γ e γ    2 1 + n γ × I0 2n γ m m−1 (m γ ) γ

(m)

MGF (Mγ (s))

1 − e−γ / γ 2

Nakagami-n

CDF (Pγ (γ ))

−mγ / γ

e



√ 1 − Q1 n 2,

1−



(1 − sγ )−1

 2(1 + n2 )  γ γ

 m,

m γ γ



1 + n2 1 + n2 − sγ

sγ n2 × exp 1 + n2 − sγ 1−

(m)

sγ m

−m

SSC Output Statistics CDF The CDF of γSSC can be written as [9,10] 1 Pr[(γ1 ≤ γT ) and (γ2 ≤ γ )] PγSSC (γ ) = Pr[(γT ≤ γ1 ≤ γ ) or (γ1 ≤ γT and γ2 ≤ γ )]

γ < γT γ ≥ γT

(9.269)

which can be expressed in terms of the CDF of the individual branches Pγ (γ ) as 1 PγSSC (γ ) =

γ < γT Pγ (γT ) Pγ (γ ) Pγ (γ ) − Pγ (γT ) + Pγ (γ ) Pγ (γT ) γ ≥ γT

(9.270)

Using the one-branch CDFs given in Table 9.5, we can write the CDF of γSSC over Rayleigh channels as 1 1 − (e −γT / γ + e −γ / γ ) + e −(γT +γ )/ γ γ < γT (9.271) PγSSC (γ ) = 1 − 2e−γ / γ + e −(γT +γ )/ γ γ ≥ γT over Nakagami-m channels as     (m, m (m, m  γ γT ) γ γ)   1− 1−    (m) (m)      m m  (m, γT ) − (m, γ ) γ γ PγSSC (γ ) =  (m)          (m, m (m, m   γ γT ) γ γ)  1−   + 1 − (m) (m)

γ < γT

(9.272) γ ≥ γT

SWITCHED DIVERSITY

421

and over Nakagami-n (Rice) channels as     2 )γ  √  2(1 + n   1 − Q1  2n2 ,   γ             2 √  2(1 + n )γ T    × 1 − Q1  2n2 , γ < γT   γ               2 )γ 2 )γ √ √ 2(1 + n 2(1 + n T  −Q1  2n2 ,  PγSSC (γ ) = Q1  2n2 ,  γ γ            2 )γ  √ 2(1 + n     +1 − Q1  2n2 ,   γ            2 )γ  √ 2(1 + n  T    × 1 − Q1  2n2 , γ ≥ γT   γ (9.273) PDF Differentiating PγSSC (γ ) with respect to γ , we get the PDF of the SSC output in terms of the CDF [Pγ (γ )] and the PDF [pγ (γ )] of the individual branches as pγSSC (γ ) =

d PγSSC (γ ) = dγ

1

γ < γT Pγ (γT ) pγ (γ ) (1 + Pγ (γT )) pγ (γ ) γ ≥ γT

(9.274)

which can be written for Rayleigh fading as  1 −γ / γ −γ / γ    (1 − e T )e γ pγSSC (γ ) =   1 (2 − e γT / γ )e −γ / γ  γ

γ < γT (9.275) γ ≥ γT

for Nakagami-m fading as  m m m−1   (m, m (γ ) γ  γ γT )   1− e −(m/ γ )γ   (m) (m)   m m m−1 pγSSC (γ ) =  (γ ) γ (m, m  γ γT )   e −(m/ γ )γ  2− (m) (m)

γ < γT (9.276) γ ≥ γT

422

PERFORMANCE OF MULTICHANNEL RECEIVERS

and for Nakagami-n (Rice) fading as    

 2 2 2  1 + n 1 + n 1 + n   exp −n2 − γ I0 2n γ    γ γ γ           2)  √ 2(1 + n   × 1 − Q1 n 2, γT     γ    pγSSC (γ ) =  2 2

2  1 + n 1 + n 1 + n   exp −n2 − γ I0 2n γ    γ γ γ           2  √ 2(1 + n )   × 2 − Q1 n 2, γT     γ

γ < γT

γ ≥ γT (9.277)

MGF The MGF of γSSC can be expressed in terms of the individual branch MGFs as  ∞ MγSSC (s) = Pγ (γT ) Mγ (s) + pγ (γ ) esγ dγ (9.278) γT

For Rayleigh fading, (9.278) simplifies to   MγSSC (s) = (1 − sγ )−1 1 − e −γT / γ + e −(1−sγ )γT / γ

(9.279)

For Nakagami-m fading using Eq. (3.381.3) of Ref. 36, (9.278) can be expressed in terms of the complementary incomplete gamma function as

sγ MγSSC (s) = 1 − m

−m  (m, (1 − 1+

γ mγT m s) γ

) − (m, m γ γT )



(m)

(9.280)

For Nakagami-n (Rice) fading (9.278) can be expressed in terms of the first-order Marcum Q-function as

−1

sγ sγ n2 exp 1 + n2 1 + n2 − sγ     2 )γ √ 2(1 + n T  × 1 − Q1 n 2, γ

MγSSC (s) = 1 −

 

 

2 2(1 + 1+n , 2 − s γT  +Q1 n 1 + n2 − sγ γ n2 )

(9.281)

SWITCHED DIVERSITY

423

Average Output SNR Analysis The average SNR at the SSC output can be obtained by averaging γ over pγSSC (γ ) as given by (9.274), yielding  ∞  ∞ γ SSC = Pγ (γT ) γpγ (γ ) d γ + γpγ (γ ) d γ 0



= Pγ (γT ) γ +

γT ∞

γpγ (γ ) d γ

(9.282)

γT

Differentiating (9.282) with respect to γT and setting the result to zero, we can easily show that γ SSC is maximized when the switching threshold is set to γT∗ = γ . For Rayleigh fading, substituting the one-branch CDF and PDFs given in Table 9.5 in (9.282), we get

γT −γT / γ (9.283) e γ SSC = γ 1 + γ which reduces for the optimal threshold case to γ ∗SSC = γ (1 + e −1 ) = 1.368 γ

(9.284)

Similarly, for the Nakagami-m fading case, the average output SNR can be found as   m −mγT / γ (m γ γT ) e γ SSC = γ 1 + (9.285) (m + 1) with the simplification to γ ∗SSC

mm−1 e −m =γ 1+ (m)

(9.286)

when using the optimum switching threshold. For Nakagami-n (Rice) fading using the identity [36, Sect. 8.486] 2 I0 (z) = I2 (z) + I1 (z) (9.287) z the average output SNR can be expressed in closed form in terms of generalized Marcum Q-functions as     2 √ 2(n + 1)  γT γ SSC = γ 1 − Q1 n 2, γ     γ   √ 2(n2 + 1)  + 2 Q2 n 2, γT n +1 γ 

√ + n Q3 n 2, 2



 2(n2 + 1)  γT , γ

(9.288)

424

PERFORMANCE OF MULTICHANNEL RECEIVERS

which reduces to  √    γ ∗SSC = γ 1 − Q1 n 2, 2(n2 + 1) +

 √    γ   √  2 2 2 + 1) n n Q 2, 2(n + 1) + n Q 2, 2(n 2 3 n2 + 1

(9.289)

when the optimum switching threshold is used. Comparison with MRC and SC For comparison purposes recall that the average SNR at the output of a dual-branch MRC diversity system is given by (9.55) γ MRC = 2γ

(9.290)

regardless of the fading model. On the other hand, the CDF of a dual-branch SC output is given by [3, Sect. 5.5.2]  2 PγSC (γ ) = Pγ (γ ) (9.291) which, when differentiated with respect to γ , gives the PDF pγSC (γ ) = 2pγ (γ )Pγ (γ ) Hence the average output SNR of a dual-branch SC is given by  ∞ γ SC = 2 γpγ (γ )Pγ (γ ) d γ

(9.292)

(9.293)

0

Using the one-branch PDFs and CDFs given in Table 9.5 for Rayleigh fading, (9.293) reduces to [3, Eq. (5.86)] γ SC = 1.5 γ

(9.294)

for Rayleigh fading channels. Similarly, the average output SNR of SC can be obtained in closed form for Nakagami-m fading by using the one-branch PDFs and CDFs given in Table 9.5 as well as Eq. (6.455) of Ref. 36 and Eq. (15.1.25) of Ref. 53, yielding

(2m + 1) γ SC = γ 1 + 2m (9.295) 2 (m (m))2 As an example, Fig. 9.37 plots the normalized average output SNR, γ MRC / γ , γ SC / γ , and γ SSC / γ versus the Nakagami-m fading parameter for MRC, SC, and SSC with optimum threshold, respectively. Outage Probability As before, the outage probability Pout is defined as the probability that the combiner output SNR falls below a given threshold γth and is therefore obtained by replacing γ with γth in the CDF expressions given previously:   SSC Pout = Pr γSSC < γth = PγSSC (γth ) (9.296)

SWITCHED DIVERSITY

425

2.1

a

2

Normalized average SNR

1.9 1.8 1.7 1.6 1.5 1.4

b

1.3

c

1.2 1.1

1

2

3

4 5 6 7 Nakagami Fading Parameter m

8

9

10

Figure 9.37 Normalized average SNR of (a) MRC (γ MRC / γ ), (b) SC (γ SC / γ ), and (c) SSC (γ SSC / γ ) versus the Nakagami-m fading parameter m.

Similarly the outage probability of dual-branch SC systems can easily be deduced from (9.291) as  2 SC = Pγ (γth ) Pout

(9.297)

Note that if we substitute γT for γth in (9.296), then, using (9.270), Eq. (9.296) reduces to (9.297). Since SC can be viewed as an optimal implementation of any switched diversity system, we can conclude that the optimal switching threshold in the minimum outage probability sense is given by γT∗ = γth . Hence, using (9.271), (9.272), and (9.273), we can write the outage probability of SC and SSC systems with optimal switching thresholds as 2  SSC SC = Pout = 1 − e γth / γ Pout

(9.298)

for Rayleigh fading  SSC Pout

=

SC Pout

=

(m) − (m, m γ γth ) (m)

2 (9.299)

426

PERFORMANCE OF MULTICHANNEL RECEIVERS

for Nakagami-m fading, and 



SSC SC Pout = Pout

√ = 1 − Q1 n 2,



2(1 + γ

2 n2 )

γth 

(9.300)

for Nakagami-n (Rice) fading. For comparison purposes, we also derive the outage probability of dual-branch MRC receivers from known results for the central and noncentral chi-square distributions [6, App. 5A]. For Rayleigh fading the outage probability is given by

γth MRC e −(γth / γ ) =1− 1+ (9.301) Pout γ while for Nakagami-m it is given by MRC Pout

=

(2m) − (2m, m γ γth ) (m)

and finally for Nakagami-n (Rice) fading it can be expressed as    2) √ 2(1 + n MRC Pout γth  = 1 − Q2 n 2, γ

(9.302)

(9.303)

Figures 9.38 and 9.39 compare the outage probability of dual-branch MRC and SC/SSC with optimal switching thresholds versus the normalized threshold SNR γth / γ for Nakagami-m and Nakagami-n (Rice) fading channels, respectively. The difference of diversity gain between MRC and SSC is about 2 dB in Rayleigh fading, and it increases as the fading environment improves, that is, as m or n (or equivalently the Rice factor) increases. Average Probability of Error Analysis In the previous sections an MGF-based approach was taken for the evaluation of the average error rate over fading channels, which, although specifically explored for MRC, EGC, and SC, is also applicable for SSC. Indeed, it was shown that the key to evaluating the average error rate of systems operating over fading channels is to express the MGF of the combiner output in a form that is both simple and suitable for single integration. Since we have already derived the MGF of the SSC output SNR in Section 9.9.1.1, the evaluation of average error rate over the fading channel can be accomplished as before. As an example, and in view of the alternative conditional SER expressions presented in Chapter 8, the average SER of M-PSK signals is given by

 gPSK 1 (M−1)π/M dφ (9.304) MγSSC − 2 Ps (E) = π 0 sin φ where gPSK = sin2 (π /M).

SWITCHED DIVERSITY

427

100 MRC SSC

m = 0.5 10−2

m=1 Outage Probability

−4

10

m=2 10−6

m=4

10−8

10−10

10−12 −15

0 5 −10 −5 Normalized threshold SNR [dB] per symbol per branch

10

Figure 9.38 Outage probability of MRC and SSC (SC) versus normalized threshold SNR γth / γ for Nakagami-m fading channel. 100 MRC SSC

10−1 10−2

Outage Probability

10−3

n2 = 5dB

10−4 10−5 10−6

n2 = 10 dB

10−7 10−8 10−9 10−10 −15

0 5 −10 −5 Normalized threshold SNR [dB] per symbol per branch

10

Figure 9.39 Outage probability of MRC and SSC (SC) versus normalized threshold SNR γth / γ for Nakagami-n (Rice) fading channel.

428

PERFORMANCE OF MULTICHANNEL RECEIVERS

For the particular case of BPSK over Rayleigh fading, the average BER can in fact be found in closed form in terms of the Gaussian Q-function as we show now. Indeed, the average BER of BPSK using SSC can be written as  ∞  Q( 2γ ) pγSSC (γ ) d γ Pb (E) = 0



∞

= Pγ (γT ) 0

 ∞ 1 −(γ / γ )  1 −(γ / γ )  e e Q( 2γ ) d γ + Q( 2γ ) d γ γ γT γ

(9.305)

which can be found in closed form with the help of (5A.4) and Luke’s equation [136, p. 185, Eq. (24)] as       γ 1 −γT / γ 1−e Pb (E) = 1− 2γT + e −γT / γ Q 2 1+γ    γ − Q 2γT (1 + γ )/ γ (9.306) 1+γ However, the form given by (9.304) has the advantage of leading to a generic expression for the optimum switching threshold in a minimum average error rate sense as shown next. Optimum Threshold in (9.304), we obtain

Let us first focus on the binary case (M = 2). Using (9.278)

1 Pb (E) = π



π/2 0

1 + π

Pγ (γT ) Mγ −



π/2

"

sin2 φ

∞

pγ (γ )e 0



1

dφ

−γ / sin2 φ

# d γ d φ.

(9.307)

γT

For γT = 0, Pγ (γT ) = 0, and hence the first term of (9.307) vanishes, resulting in 1 Pb (E) = π



π/2

Mγ −

0

1 sin2 φ

dφ

(9.308)

which is the BER performance of a single-branch (no-diversity) receiver. On the other hand, as γT tends to infinity, the second term of (9.307) vanishes, and since Pγ (γT ) = 1 in the first term, (9.307) reduces again to the average BER performance of a single-branch (no-diversity) receiver. Since the average BER is a continuous function of γT , there exists an optimal value of γT for which the average BER is minimal. This optimal value γT∗ is a solution of the equation ! dPb (E) !! =0 (9.309) dγT !γT =γ ∗ T

SWITCHED DIVERSITY

429

Substituting (9.307) in (9.309), we get 1 π

 0

π/2

pγ (γT∗ )Mγ

−

1



2

sin φ

dφ −

1 π

 0

π/2

∗

pγ (γT∗ )e−(γT / sin

2 φ)

dφ = 0 (9.310)

which after simplification reduces to 1 π



π/2

Mγ −

0

1 sin2 φ

dφ − Q

-

2γT∗

=0

(9.311)

where we have used the alternative representation of the Gaussian Q-function in (4.2). Solving for γT∗ in (9.311) leads to γT∗

 π/2 "

#2 1 1 1 −1 Q dφ = Mγ − 2 2 π 0 sin φ

(9.312)

where Q−1 (·) denotes the inverse Gaussian Q-function. Substituting the singlebranch MGFs given in Table 9.5 for Rayleigh and Nakagami-m fading in (9.312), then, using the trigonometric integrals derived in Appendix 5A, we get the optimum threshold for BPSK over Rayleigh fading as γT∗

    2 γ 1 −1 1 = Q 1− 2 2 1+γ

(9.313)

and for Nakagami-m fading as 



-

1  γT∗ = Q−1 2 2 (1 +

γ πm γ m+1/2 m)



(m + 1 1 2 F1 1, m + ; m + 1; (m + 1) 2 1+ 1 2)

γ m

2   

(9.314) For Nakagami-n (Rice) fading, it can be shown using the integrals given in [137, Eqs. (6) and (7)] that the optimum threshold is given by " " #

# . p 1 1 2 2 Q−1 Q1 (a, b) − 1+ e −(a +b )/2 I0 (ab) (9.315) γT∗ = 2 2 1+p where γ n2 + 1 " #1/2 . 1 + 2p p a=n − 2(1 + p) 1+p #1/2 " . p 1 + 2p + b=n 2(1 + p) 1+p

p=

(9.316)

430

PERFORMANCE OF MULTICHANNEL RECEIVERS

The optimum threshold can be found in a similar fashion for other modulation scheme/fading model combinations. In general, this optimal threshold will be a solution of an integral equation similar to (9.311), but explicit closed-form solutions similar to the ones presented in (9.313), (9.314), and (9.316) will not be always possible to obtain. For example, the optimum threshold γT∗ with DQPSK is the solution of the integral equation −

b2 (1 − ζ 2 )2 dφ 2(1 + 2ζ sin φ + ζ 2 ) −π

 π 1 b2 (1 − ζ 2 )2 ∗ = dφ γ exp − 4π −π 2(1 + 2ζ sin φ + ζ 2 ) T

1 4π



π

Mγ

(9.317)

  √ √ where a = 2 − 2, b = 2 + 2, and ζ = a/b. In this case, one has to rely on numerical root finding techniques to get an accurate numerical solution for γT∗ in (9.317). Comparison with MRC and SC In Figs. 9.40 through 9.43 we present some numerical results comparing the average error rate performance of several modulation schemes with SSC, SC, and MRC. The SSC curves are generated as per the average error rate expressions given in Section 9.9.1.1 and with the optimum switching thresholds derived previously. MRC curves are based on the expressions

100 MRC SC SSC

10−1

Average BER

10−2

m = 0.5

10−3

m=1

10−4 10−5

m=2 10−6

m=4

10−7 10−8

0

5

10 15 20 Average SNR per bit per branch [dB]

25

30

Figure 9.40 Comparison of the average BER of BPSK with MRC, SC, and SSC versus average SNR per bit per branch γ for Nakagami-m fading channel.

SWITCHED DIVERSITY

431

100 10

−1

MRC SC SSC

m = 0.5

10−2

Average SER

m=1 10−3 10−4

m=2

10−5

m=4

10−6 10−7 10−8

0

5

10 15 20 Average SNR per symbol per branch [dB]

25

30

Figure 9.41 Comparison of the average SER of 8-PSK with MRC, SC, and SSC versus average SNR per symbol per branch γ for Nakagami-m fading channel.

100

10−2

Average SER

MRC SC SSC

m = 0.5

10−1

m=1

10−3

m=2

10−4 10−5

m=4

10−6 10−7 10−8

0

5

10 15 20 Average SNR per symbol per branch [dB]

25

30

Figure 9.42 Comparison of the average SER of 16-QAM with MRC, SC, and SSC versus average SNR per symbol per branch γ for Nakagami-m fading channels.

432

PERFORMANCE OF MULTICHANNEL RECEIVERS

100 SC SSC

Average BER

10−1 10−2

m = 0.5

10−3

m=1

10−4

m=2 10−5 10−6

m=4

10−7 10−8

0

5

10 15 20 Average SNR per bit per branch [dB]

25

30

Figure 9.43 Comparison of the average BER of DQPSK with SC and SSC versus average SNR per bit per branch γ for Nakagami-m fading channels.

derived in Section 9.2 on the performance of MRC receivers. For SC, the Rayleigh and Nakagami-m curves are based on the results presented in Section 9.8. For Nakagami-n (Rice) fading channels, we need the MGF at the SC output to be able to get average error rate expressions. Using the integral [138, App. 4.6, p. 577] this MGF can be found in closed form in terms of the Marcum Q-function as

−1

sγ n2 1 + n2 − γ s "

# 1 λv µ v2 ,√ 1 − Q1 √ × 1− 1 + v2 1 + v2 1 + v2

4 µ λv 1 Q1 √ ,√ − 2 1 + v2 1+v 1 + v2

MγSC (s) = 2 1 −

γs 1 + n2



exp

(9.318)

where µ=

√

λ= v=

2 n,

2n2 (1 + n2 ) 1 + n2 − sγ 1 + n2 1 + n2 − sγ

(9.319)

1/2

1/2 (9.320)

433

SWITCHED DIVERSITY

9.9.1.2 Effect of Branch Unbalance In Section 9.9.1.1 we analyzed SSC for the case of identically distributed branches. In this section we consider the performance of SSC systems in the more general case where the branches are still independent but not necessarily identically distributed. In particular, let us denote by pγ1 (γ1 ) and pγ2 (γ2 ) the PDFs of the two branches, by Pγ1 (γ1 ) and Pγ2 (γ2 ) their respective CDFs, and by γ 1 and γ 2 their respective average SNRs. SSC Output Statistics Assuming a discrete-time implementation of SSC let γ1(n) and γ2(n) be the instantaneous SNR of branch 1 and 2, respectively, at time t = nT , and let γn denote the SSC output SNR at time t = nT . According to the mode of operation of SSC described above, we have 1 γn−1 = γ1(n−1) γ1(n) ≥ γT (9.321) γn = γ1(n) iff γn−1 = γ2(n−1) γ2(n) < γT CDF The CDF of γn , PγSSC (γ ), can be written as PγSSC (γ ) = Pr[γn ≤ γ ] = Pr[γn = γ1(n) and γ1(n) ≤ γ ] + Pr[γn = γ2(n) and γ2(n) ≤ γ ] = Pr[γ1(n) ≥ γT and γ1(n) ≤ γ ] Pr[γn−1 = γ1(n−1) ] +Pr[γ2(n) < γT and γ1(n) ≤ γ ] Pr[γn−1 = γ2(n−1) ] +Pr[γ2(n) ≥ γT and γ2(n) ≤ γ ] Pr[γn−1 = γ2(n−1) ] +Pr[γ1(n) < γT and γ2(n) ≤ γ ] Pr[γn−1 = γ1(n−1) ]

(9.322)

which can be expressed in terms of the CDF of the individual branches as  Pr[γn−1 = γ1(n−1) ]Pγ1 (γT )Pγ2 (γ )     + Pr[γn−1 = γ2(n−1) ]Pγ2 (γT )Pγ1 (γ ) γ ≤ γT   PγSSC (γ ) =  Pr[γn−1 = γ1(n−1) ] Pγ1 (γ ) − Pγ1 (γT ) + Pγ1 (γT )Pγ2 (γ )      + Pr[γn−1 = γ2(n−1) ] Pγ2 (γT )Pγ1 (γ ) + Pγ2 (γ )−Pγ2 (γT ) γ > γT (9.323) In order to obtain the CDF of the SSC output, we need to find

p1 = Pr[γn−1 = γ1(n−1) ]  = Pr (γn−2 = γ1(n−2) and γ1(n−2) ≥ γT ) or (γn−2 = γ2(n−2)  and γ2(n−1) < γT )

p2 = Pr[γn−1 = γ2(n−1) ]  = Pr (γn−2 = γ2(n−2) and γ2(n−2) ≥ γT ) or (γn−2 = γ1(n−2)  and γ1(n−1) < γT ) .

(9.324)

434

PERFORMANCE OF MULTICHANNEL RECEIVERS

Assuming that the pairs of samples from each branch are i.i.d. [i.e.,γ1(n−1) and γ1(n) are i.i.d. and γ2(n−1) and γ2(n) are i.i.d.], we can rewrite (9.324) as p1 = p1 (1 − Pγ1 (γT )) + p2 Pγ2 (γT ) p2 = p2 (1 − Pγ2 (γT )) + p1 Pγ1 (γT )

(9.325)

Using the fact that the events γn = γ1(n) and γn = γ2(n) are mutually exclusive (i.e., p1 + p2 = 1), we can solve for p1 and p2 to get p1 =

Pγ2 (γT ) Pγ1 (γT ) + Pγ2 (γT )

p2 =

Pγ1 (γT ) Pγ1 (γT ) + Pγ2 (γT )

(9.326)

Substituting (9.326) in (9.323), the CDF of the SSC output can be written solely in terms of the individual branches CDFs as   Pγ1 (γT )Pγ2 (γT )    γ ≤ γT Pγ1 (γ ) + Pγ2 (γ )    P (γ ) + P (γ ) γ1 T γ2 T     P (γ )P (γ )   γ1 T γ2 T Pγ1 (γ ) + Pγ2 (γ ) − 2 PγSSC (γ ) = (9.327)  Pγ1 (γT ) + Pγ2 (γT )      Pγ (γ )Pγ2 (γT ) + Pγ1 (γT )Pγ2 (γ )   γ > γT + 1  Pγ1 (γT ) + Pγ2 (γT ) PDF Differentiating the expression of PγSSC (γ ) as given by (9.327) with respect to γ , we get the PDF at the SSC output as   Pγ1 (γT )Pγ2 (γT )    pγ1 (γ ) + pγ2 (γ )    P (γ ) + P (γ ) γ1 T γ2 T     P (γ )P (γ )   γ1 T γ2 T pγ1 (γ ) + pγ2 (γ ) pγSSC (γ ) =  Pγ1 (γT ) + Pγ2 (γT )      pγ (γ )Pγ2 (γT ) + Pγ1 (γT )pγ2 (γ )   + 1  Pγ1 (γT ) + Pγ2 (γT )

γ ≤ γT (9.328) γ > γT

MGF Taking the Laplace transform of the PDF as given by (9.328), it can be shown that the MGF of the SSC output MγSSC (s) when the branches are not necessarily identically distributed can be expressed as MγSSC (s) =

Pγ2 (γT ) M (1) (s) Pγ1 (γT ) + Pγ2 (γT ) γSSC +

Pγ1 (γT ) M (2) (s) Pγ1 (γT ) + Pγ2 (γT ) γSSC

(9.329)

435

SWITCHED DIVERSITY

where Mγ(i)SSC (s) is the MGF given by (9.279), (9.280), or (9.281). The superscript (i) in Mγ(i)SSC (s) refers to the fact that in this analysis the two combined branches are allowed to have different average SNRs γ 1 and γ 2 , different fading parameters such as m1 and m2 in the case of Nakagami-m fading, or (even) to be distributed according to two different families of distributions like Nakagami-m and Nakagami-n (Rice). As a check, note that (9.329) reduces to (9.279), (9.280), or (9.281) in the case of i.i.d. branches. Average Output SNR The closed-form expression of the PDF as given by (9.328) readily allows us to obtain the SSC output average SNR in the case of unbalanced branches. For instance, let us consider the average SNR at the SSC output of the Nakagami-m fading branches with the same fading parameter m but different average SNRs γ 1 and γ 2 . In this case, averaging γ over (9.328), it can be shown after some manipulations that the average output SNR is given by

γ SSC

  mm−1 ( γγT )m e −mγT / γ 1 Pγ2 (γT ) 1 = γ1 1+ Pγ1 (γT ) + Pγ2 (γT ) (m)   mm−1 ( γγT )m e −mγT / γ 2 Pγ1 (γT ) 2 + γ2 1+ Pγ1 (γT ) + Pγ2 (γT ) (m)

(9.330)

where Pγi (·) is as given in Table 9.5 for the Nakagami-m case. As a comparison, recall that the average SNR at the output of a dual-branch MRC receiver is given by Eq. (5.102) of Ref. 3, γ MRC = γ 1 + γ 2 , regardless of the fading model, whereas the average SNR at the output of a dual-branch SC over Nakagami-m fading is given by (9.249) as (γ 1 γ 2 )m+1 (2m + 1) m(m + 1) (m)2 (γ 1 + γ 2 )2m+1

" γ1 × 2 F1 1, 2m + 1; m + 2; γ1 + γ2

# γ2 + 2 F1 1, 2m + 1; m + 2; γ1 + γ2

γ SC = γ 1 + γ 2 −

(9.331)

where 2 F1 ( · , · ; · ; · ) is the Gauss hypergeometric function [53, Ch. 15]. Average Probability of Error Using the closed-form MGF expression (9.329), we are in a position to derive the average probability of error for a wide variety of modulation schemes as explained in Section 9.9.1.1. For example, the average BER of BPSK over Nakagami-m fading channels with unequal average SNRs is

436

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

m = 0.5 m=1 m=2 m=4

Average BER

10−1 10−2

a b

10−3

a b a

10−4

b 10−5

a

10−6

b

10−7 10−8

0

5

10 15 20 25 Average SNR of branch 1 per bit per branch [dB]

30

Figure 9.44 Average BER of BPSK versus average SNR of the first branch γ 1 for (a) unequal average branch SNRs (γ 1 = γ 2 /5) and (b) equal average branch SNRs (γ 1 = γ 2 ) over Nakagami-m fading channels.

given by Pγ2 (γT ) 1 Pb (E) = Pγ1 (γT ) + Pγ2 (γT ) π



π 2

0

Pγ1 (γT ) 1 + Pγ1 (γT ) + Pγ2 (γT ) π

Mγ(1) SSC



π 2

0

−

1



dφ sin2 φ

1 (2) dφ MγSSC − 2 sin φ

(9.332)

where Mγ(i)SSC (·) is given by (9.280). Figure 9.44 shows the effect of average SNR unbalance on the average BER of BPSK with SSC over Nakagami-m fading channels. We used the optimum switching threshold to generate these curves, and these optimum thresholds were found numerically by minimizing (9.332) with respect to γT . Note, for example, that in the case of m = 2 the unbalanced system under consideration suffers about 3-dB penalty for an average BER of 10−6 compared to a balanced system. 9.9.1.3 Effect of Branch Correlation Recall that in Section 9.9.1.1 we considered the performance of SSC over i.i.d. branches while in the previous section we addressed the problem of branch unbalance. We assess in this section the effect of fading correlation on the performance of SSC receivers.

SWITCHED DIVERSITY

437

SSC Output Statistics PDF In the case of correlated Nakagami-m fading envelopes, the joint PDF pγ1 ,γ2 (γ1 , γ2 ) of the instantaneous SNRs γ1 and γ2 is given by [9, Eq. (19)] pγ1 γ2 (γ1 , γ2 ) =

m+1

m(γ1 +γ2 ) γ1(m−1)/2 γ2(m−1)/2 − γ (1−ρ) e (m)ρ (m−1)/2 (1 − ρ)

√ 2m ρ √ γ1 ≥ 0, γ2 ≥ 0 (9.333) × Im−1 γ1 γ2 , (1 − ρ)γ

m γ

Under these conditions and following the mode of operation of SSC systems as described above, Abu-Dayya and Beaulieu [9] showed that the PDF of the SSC output is given by [9, Eq. (21a)]    A(γ ) m pγSSC (γ ) = m γ m−1   + A(γ ) e −mγ / γ γ (m)

γ ≤ γT (9.334) γ > γT

where A(γ ) can be written with the help of Eq. (11) of Ref. 139 as [123]  m A(γ ) =

m γ

γ m−1 e −mγ / γ



(m) 

× 1 − Qm

 2mργ , (1 − ρ)γ

2mγT (1 − ρ)γ

 (9.335)

MGF Taking the Laplace transform of (9.334), the MGF of the SSC output SNR can be expressed in closed form in terms of the incomplete gamma function, with the help of [139, Eq. (11)], as

γ −m MγSSC (γ ) = 1 − s m   m, mγγ T (1 − × 1 + (m)



γ m s)

−

 m, mγγ T

1−γ s/m 1−(1−ρ)γ s/m

(m)

  . (9.336)

Average Output SNR The average output SNR of SSC with correlated branches is obtained by averaging γ over the PDF of (9.334), yielding with the help of Eq. (28) of Ref. 140 and Eq. (8.356.2) of Ref. 36, and after some manipulations  γ SSC = γ

1+

m −mγT / γ (1 − ρ)( m γ γT ) e

m (m)

 (9.337)

438

PERFORMANCE OF MULTICHANNEL RECEIVERS

As a check, note that (9.337) reduces the average SNR of a single branch γ for fully correlated branches with ρ = 1. On the other hand, (9.337) reduces to the average output SNR over i.i.d. branches as given by (9.285) when ρ = 0. Differentiating (9.337) with respect to γT and setting the result equal to zero, it can be easily shown that the optimum threshold for maximum average output SNR is γT∗ = γ , which is the same as in the uncorrelated fading case. Hence the maximum average output SNR for SSC over correlated Nakagami-m fading channels is given by γ SSC = γ

(1 − ρ)mm−1 e −m 1+ (m)

(9.338)

For the purpose of comparison, recall that the average output SNR for dualbranch MRC is unaffected by fading correlation while we showed that the average output SNR for dual-branch SC over equal average SNR correlated Nakagamim fading paths is given by (9.247). Figure 9.45 compares the effect of fading correlation on the average output SNR of SSC and SC receivers. SC outperforms SSC, as expected, but has a slightly higher sensitivity to fading correlation. Average Probability of Error Using the MGF of the SSC output SNR given by (9.336), we can determine the average probability of error of several modulation

1.7 SC SSC

Normalized average SNR

1.6

1.5

a 1.4

a

1.3

b

b

c

c

1.2

d

d 1.1

1

0

0.1

0.2

0.3

0.4 0.5 0.6 0.7 Correlation coefficient (r)

0.8

0.9

1

Figure 9.45 Comparison of the normalized average SNR of SC (γ SC / γ ) and SSC (γ SSC / γ ) versus correlation coefficients(ρ) over Nakagami fading with parameters (a) m = 0.5, (b) m = 1, (c) m = 2, and (d) m = 4.

SWITCHED DIVERSITY

439

100 10−1

Average BER

m = 0.5

d

10−2

c

b

10−3

d

d 10−4

ab

c

c c b

10−5

a

10−6

d

a m=1

b a

m=2

m=4

10−7 10−8

0

5

10 15 20 Average SNR per bit per branch [dB]

25

30

Figure 9.46 Average BER of BPSK with SSC versus average SNR per branch γ for various values of the correlation coefficient [ρ = (a) 0, (b) 0.3, (c) 0.6, (d) 0.9] over Nakagami-m fading.

schemes, as explained in Section 9.9.1.1. For example, Fig. 9.46 depicts the effect of correlation on the BER performance of BPSK with SSC. From Fig. 9.46 one can conclude that correlation coefficients up to 0.6 do not seriously degrade BER performance. Note that the curves in Fig. 9.46 used the optimum threshold in the minimum BER sense. In general, this optimum threshold cannot be expressed in an explicit closed form but can be numerically found by solving an integral equation. For example, in the particular BPSK case substituting (9.308) in (9.309) [with Mγ (s) given by (9.336)] and differentiating with respect to γT [with the help of Eq. (8.356.4) of Ref. 36] leads to the following integral equation for γT∗ : "

#  π/2 1 m + exp − γT∗ d φ γ sin2 φ 0    γ

 π/2 1 + m sin 2φ mγT∗ γ (1 − ρ) −m  d φ.  1+ exp − = γ γ m sin2 φ 1 + (1 − ρ) m sin 0 2φ (9.339) 9.9.2

Multibranch Switch-and-Examine Combining

Because only two paths are involved at most in the diversity combining decision of SSC schemes, these schemes cannot benefit in diversity from additional paths when these paths are i.i.d. or equicorrelated and identically distributed. In this case, one should rather implement an SEC type of combining [133–135] for

440

PERFORMANCE OF MULTICHANNEL RECEIVERS

which it is assumed that if the current path is not of acceptable quality, then the combiner switches and examines the quality of the next available path. This switching–examining process is repeated until either an acceptable path is found or all available diversity paths have been examined. In the latter case, the combiner either settles on the last examined path [133] or connects to the receiver the path with the best quality among all examined paths [134]. Another possibility is just to wait for a certain period of time (on the order of the channel coherence time) and to restart after that period the switching–examining process on all the available diversity paths. This scanning, followed by waiting, can then be repeated indefinitely until a path with acceptable quality is found. This variant of multibranch SEC, termed “scan and wait” combining (SWC), is, for example, suitable for the transmission of information that has a minimum instantaneous BER requirement but that is tolerant to some time delay [135]. In what follows, we briefly summarize the performance of traditional multibranch SEC and then multibranch SEC with postselection over i.i.d. paths. More details and more results about the performance of these schemes in more general non-i.i.d. environments are available in Refs. 133 and 134, respectively. We finally conclude this section by covering in more detail the performance of SWC in a generalized fading environment [135]. 9.9.2.1

Classical Multibranch SEC

Analysis If diversity paths are i.i.d., then the CDF of the output SNR with traditional L-branch SEC is given by [133, Eq. (33)]  [Pγ (γT )]L−1 Pγ (γ ), γ < γT      L−1
(9.340) PγSEC (γ ) = [Pγ (γ ) − Pγ (γT )][Pγ (γT )]j    j =0   + [Pγ (γT )]L , γ ≥ γT where Pγ (·) is the common CDF of the SNR for every diversity path and is given in Table 9.5 for various fading scenarios and γT is again the common predetermined switching threshold. Differentiating (9.340) with respect to γ , we obtain the PDF of the output SNR as  [Pγ (γT )]L−1 pγ (γ ), γ < γT    L−1
(9.341) pγSEC (γ ) =  [Pγ (γT )]j pγ (γ ), γ ≥ γT   j =0

where pγ (·) is the common PDF of every diversity path, which is given in Table 9.5 for various fading scenarios. Finally, the MGF of the output SNR is given by MγSEC (s) = [Pγ (γT )]L−1 Mγ (s)  ∞ L−2
j + [Pγ (γT )] e sγ pγ (γ ) d γ j =0

γT

(9.342)

SWITCHED DIVERSITY

441

where Mγ (·) is the common MGF of the diversity paths, which is also given in Table 9.5 for various fading scenarios. Note that the integral in (9.342) is a “partial” MGF that can be evaluated in closed form for many fading scenarios of interest, as will be discussed in Section 9.11.2.4. With the MGF in hand and using the MGF-based approach, we can now evaluate the average bit and symbol error rate of different modulation schemes with SEC over various fading scenarios. As such, at most a single finite-interval integration suffices for direct computation of the desired average error rate. However, in some special cases, such as, for example, the BPSK over i.i.d. Rayleigh paths case, the average BER Pb (E) can be obtained in a compact closed-form expression given by [133, Eq. (48)]  

#j  L−1 " γ 1
γT Pb (E) = 1 − exp − 1− 2 γ 1+γ j =0

−



#j L−2 " γT 1
1 − exp − 2 γ /

j =0



 γT × 1 − 2 exp − Q( 2γT ) γ    0 γ 2γT (1 + γ ) − 1 − 2Q 1+γ γ

(9.343)

As a double check, it is easy to verify that when L = 2, (9.343) reduces to (9.306). Numerical Examples Figure 9.4716 shows the average BER performance of BPSK with SEC (with the switching threshold set to 10 dB) over L i.i.d. Rayleigh branches versus the average SNR per path. As we can see, for this particular choice of the switching threshold, increasing L leads to relatively important but diminishing performance gain for SEC in the medium average SNR region. On the other hand, as the average SNR becomes very small (below 5 dB) or sufficiently large, the BER improvement decreases, as all curves converge asymptotically to the performance of a dual-branch switched diversity system. This is due to the fact that, if the average SNR is very small in comparison to the switching threshold, all branches will be unacceptable most of the time. Thus, the additional branches will not be able to provide any improvement. On the other hand, if the average SNR is very high in comparison with the switching threshold, all branches will be acceptable and the combiner will use one branch most of the time. Thus, the additional branches will not contribute and again will not lead to any improvement. A similar conclusion can be draw from Fig. 9.48, which plots the average BER of BPSK with SEC 16 Note that the curves in Figs. 9.47, 9.48, and 9.49 correct the corresponding ones published in Ref. 133, which were generated on the basis of an error in programming the closed-form expression for the average BER.

442

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

L=2 L=3 L=4 L=5 L=6

10−1 10−2

Average BER, Pb(E )

10−3 10−4 10−5 10−6 10−7 10−8 10−9 10−10

0

5

10

15 20 25 Average SNR, g (dB)

30

35

40

Figure 9.47 Average BER of BPSK with SEC over L i.i.d. Rayleigh branches as a function of the average SNR γ with γT = 10 dB.

10−2

Average BER, Pb(E )

10−3

10−4

10−5

10−6

10−7

10−8 −10

L=2 L=3 L=4 L=5 L=6 −5

0

5 10 15 Switching Threshold, gT (dB)

20

25

30

Figure 9.48 Average BER of BPSK with SEC over L i.i.d. Rayleigh branches as a function of the switching threshold γT with γ = 20 dB.

443

SWITCHED DIVERSITY

100

L=2 L=3 L=4 L=5 L=6

10−1

Average BER, Pb(E )

10−2 10−3 10−4 10−5 10−6 10−7 10−8

0

5

10

15

20

25

30

Average SNR, g (dB)

Figure 9.49 Average BER of BPSK with SEC over L i.i.d. Rayleigh branches as a function of the average SNR γ with optimal γT .

over i.i.d. Rayleigh fading paths versus the switching threshold. We note that there exists again an optimal switching threshold which minimizes the average BER. Figure 9.49 shows the average BER of BPSK with SEC over L i.i.d. Rayleigh fading with optimal switching threshold. It is clearly observed that, with an optimal choice of the switching threshold γT , the BER performance benefits from increasing L over the whole average SNR range. As an additional numerical example, Fig. 9.50 compares the error performance of 8-PSK with SSC, SEC, SC, and MRC over i.i.d. Rayleigh fading branches when L = 2, 4. Note that the symbol error probabilities of SSC and SEC have been numerically minimized over the possible choice of switching threshold γT . It can be seen that both SSC and SEC lead to higher error probability than MRC and SC, as expected. It also should be noted that as L increases, the error performance of SEC improves while that of SSC remains the same since SSC does not benefit from additional diversity branches. 9.9.2.2 Multibranch SEC with Post-selection If the diversity paths are i.i.d., then the CDF of the output SNR when L-branch SEC with postselection (SECps) is used is given by [134] PγSECps (γ ) = 1 −

L−1


[Pγ (γT )]i [1 − Pγ (γ )],

γ ≥ γT

i=0

= [Pγ (γ )]L ,

γ < γT

(9.344)

444

PERFORMANCE OF MULTICHANNEL RECEIVERS

100 SSC SEC SC MRC

Average Error Probability

10−1

L=2

10−2

10−3

L=4

10−4

10−5

10−6

0

5

10

15 Average SNR, dB

20

25

30

Figure 9.50 Comparison of the average error rate performance of 8-PSK with SSC, SEC, SC, and MRC over i.i.d. Rayleigh fading branches when L = 2, 4.

Differentiating PγSECps (γ ) in (9.344) with respect to γ , we obtain a generic formula for the PDF of the combined SNR, namely, pγSECps (γ ), as pγSECps (γ ) =

L−1


[Pγ (γT )]i pγ (γ ),

γ ≥ γT

= L[Pγ (γ )]L−1 pγ (γ ),

γ < γT

i=0

(9.345)

For the Rayleigh fading special case, the CDF of the combined SNR with SECps can be written as  L  (γ −γ )/ γ  T e , γ ≥ γT PγSECps (γ ) = 1 − 1 − 1 − e −γT / γ  L = 1 − e −γ / γ ,

γ < γT

(9.346)

The corresponding PDF and MGF with L-branch SECps over i.i.d. Rayleigh fading is thus given by  L  −(γ −γ )/ γ 1 T e 1 − 1 − e −γT / γ , γ ≥ γT pγSECps (γ ) = γ  L−1 1 −γ / γ e = L 1 − e −γ / γ , γ < γT (9.347) γ

445

SWITCHED DIVERSITY

and MγSECps (s) = L

L−1
i=0

+

L−1 i



(−1)i i + 1 − sγ

(i+1)γ sγ − γ T 1−e T

 L  sγ 1  1 − 1 − e −γT / γ e T, 1 − sγ

(9.348)

respectively. With the MGF in hand and using the MGF-based approach, we can now evaluate the average bit and symbol error rate of different modulation schemes with SECps over i.i.d. Rayleigh fading. As an example, for binary DPSK modulation, the average BER with L-branch SECps over i.i.d. Rayleigh fading is given by L−1

L
Pb (E) = 2 i=0

+

L−1 i



(−1)i i+1+γ

(i+1)γ −γT − γ T 1−e

 L 1 − 1 − e −γT / γ e −γT 2(1 + γ )

(9.349)

Figure 9.51 shows the average BER of binary DPSK with L-branch SECps in an i.i.d. Rayleigh fading environment. As an example, the switching threshold

100 No Diversity Dual-branch SECps 4-branch SEC 4-branch SECps 4-branch SC

Average Bit Error Probability, Pb(E )

10−1

10−2

10−3

10−4

10−5

10−6 10−7

5

10

15 20 25 − in dB Average SNR per path, g,

30

35

Figure 9.51 Average error rate of binary DPSK with L-branch diversity as function of average SNR γ in comparison with traditional SEC and SC (γT = 8 dB).

446

PERFORMANCE OF MULTICHANNEL RECEIVERS

γT is fixed to 8 dB. As the number of available diversity paths increases from 2 to 4, the error performance of SECps improves, as expected. In addition, note that four-branch SECps has nearly the same BER performance as four-branch SC and much lower average error rate than four-branch SECps in the low average SNR region. However, as the average SNR increases, the average BER of SECps degrades considerably and becomes eventually the same as that of four-branch traditional SEC. 9.9.2.3 Scan-and-Wait Combining We now present the details behind the mode of operation of multibranch SWC and its performance analysis in terms of average probability of error, waiting time statistics before channel access, and number of paths examination/estimation per channel access. System and Channel Models We consider a diversity environment in which L diversity paths are available for processing different faded replicas of the information-bearing signal. We assume that the means of creating diversity paths are available at the receiver and/or transmitter and that information transmission is done on a time-slot-based fashion. Each time slot consists of a short guard period followed by a data burst. During the guard periods, the receiver performs a series of operations, including channel quality estimations and necessary comparisons to a predetermined quality threshold, before an acceptable diversity path is found for information transmission during the subsequent data burst period. We adopt a block fading channel model. More specifically, the data burst is assumed to experience roughly the same fading as that occurring in the preceding guard period. On the other hand, the fading conditions during different guard period and data burst pairs are assumed to be independent. However, we allow the fading to be correlated across diversity paths for a fixed guard period (or equivalently a fixed data burst). In addition, we assume that the fading on each diversity path follows any of the popular fading models such as Rayleigh, Rice, or Nakagami-m. Operating Assumptions Let γl (l = 1, 2, . . . , L) be the received SNR of the lth diversity path during a particular slot time.17 At the beginning of each guard period, the receiver first measures γ1 and compares it to a predetermined SNR threshold γT1 . If this first diversity path is acceptable (i.e., γ1 ≥ γT1 ), then this path is picked for information transmission/reception during the subsequent data burst. On the other hand, if this first diversity path is not acceptable (i.e., γ1 < γT1 ), then the receiver switches to the second path and compares its SNR γ2 to its predetermined SNR threshold γT2 .18 This process of diversity path switching followed by SNR examining and comparison to a predetermined SNR threshold is repeated until 17 Note that more generally, as we will see in Chapter 11, the receiver may choose to monitor other channel quality measures such as signal-to-interference ratio in interference-limited scenarios. 18 If the diversity paths are identically distributed, then γT1 = γT2 = · · · = γTL . However, if the paths are not identically distributed, these predetermined thresholds can be different and can be set/chosen as a function of the average SNR of their corresponding diversity paths.

SWITCHED DIVERSITY

447

either an acceptable path is found (and in that case information transmission in the subsequent data burst occurs on the first found diversity path whose SNR exceeds its predetermined SNR threshold) or all L available diversity paths have been examined without finding any acceptable path. In the latter case, we assume that the receiver informs the transmitter (through a feedback channel) not to transmit during the subsequent data burst and to buffer the input data for a certain waiting period of time (on the order of the channel coherence time). After that waiting period, the system restarts the channel probing and the sequential diversity path examining-and-switching process. This scanning through the available diversity paths followed by waiting can then be repeated indefinitely until a path with an acceptable SNR is found. Performance of SWC Statistics of The Output SNR To evaluate the performance of SWC, we need to first characterize the statistics of the SWC output SNR γSWC . Let P1 = Pγ1 (γT1 ), where Pγ1 (·) is the CDF of γ1 , and more generally, let Pl = Pγ1 ,γ2 ,...γl (γT1 , γT2 , . . . , γTl ), where Pγ1 ,γ2 ,...γl (·, ·, . . . , ·) is the joint CDF of γ1 , γ2 , . . ., γl , and which is available for the Rayleigh case and Nakagami case in Refs. 141 and 142, respectively. Define ξ1 = P [γ1 ≥ γT1 ] = 1 − P1 and more generally ξl = P [γ1 < γT1 , γ2 < γT2 , . . . , γl−1 < γTl−1 , γl ≥ γTl ] = Pl−1 − Pl for l = 2, . . . , L. According to the mode of operation of the SWC scheme described above, the PDF of the SWC output SNR can be written as pγSWC (γ ) =

∞


PLn (ξ1 pγT1 (γ ) + ξ2 pγT2 (γ ) + · · · + ξL pγTL (γ ))

n=0

=

ξ1 pγT1 (γ ) + ξ2 pγT2 (γ ) + · · · + ξL pγTL (γ ) 1 − PL

(9.350)

where pγTl (γ ) is the conditional PDF of the truncated (above γTl ) random variable (RV) γl given that γ1 < γT1 , γ2 < γT2 , . . . , γl−1 < γTl−1 . More specifically   1 p (γ ), γ ≥ γ γ T1 T ξ1 1 pγ1 (γ ) = (9.351)  0, otherwise  γT  1  1 pγ1 ,γ2 (γ1 , γ ) d γ1 , γ > γT2 T ξ2 0 pγ2 (γ ) = (9.352)  0, otherwise and more generally  γT   γT 1 l−1 1 ··· pγ1 ,...,γl (γ1 , . . . , γl−1 , γ ) d γ1 · · · d γl−1 , γ > γTl T pγl (γ ) = ξl 0 0  0, otherwise (9.353) where pγ1 ,...,γl (γ1 , . . . , γl ) is the joint PDF of γ1 , γ2 , . . ., γl .

448

PERFORMANCE OF MULTICHANNEL RECEIVERS

If the fading identically distributed,

paths are independent but not

necessarily l−1 then PL = L P (γ ), ξ = (1 − P (γ )) P (γ ), l γl Tl l=1 γl Tl n=1 γn Tn and  pγl (γ )  , γ ≥ γTl T pγl (γ ) = (9.354) 1 − Pγl (γTl )  0, otherwise Finally, if the fading paths are i.i.d. whereupon their SNR follow the same PDF pγ (γ ), γT1 = γT2 = · · · = γTL , then PL = (Pγ (γT ))L , ξl = (Pγ (γT ))l−1 (1 − Pγ (γT )) and (9.350) simplifies to  pγ (γ )  , γ ≥ γT (9.355) pγSWC (γ ) = 1 − Pγ (γT )  0, otherwise Average Probability of Error Assume that a digital modulation with a conditional BEP Pb (E|γ ) is adopted. The average BEP Pb (E) is obtained by averaging this conditional BEP over the PDF of the SWC output SNR presented in the previous subsection. Considering (for analytical tractability) the case where the paths are independent but not necessarily identically distributed. Then averaging over the SWC output SNR PDF given in (9.350) [along with (9.354)] yields an average BEP of L l−1 Pγ (γT )(1 − Pγl (γTl ))Pb (El ) (9.356) Pb (E) = l=1 n=1 n Ln 1 − l=1 Pγl (γTl ) where

∞ Pb (El ) =

γTl

Pb (E|γ ) pγl (γ ) d γ 1 − Pγl (γTl )

For i.i.d fading conditions, the average BEP in (9.356) reduces to ∞ γ Pb (E|γ ) pγ (γ ) d γ Pb (E) = T 1 − Pγ (γT )

(9.357)

(9.358)

The average BEP in (9.356) along with (9.357) (and similarly (9.358)) can be obtained in closed form for many modulations and channel models of interest. As an example, if we consider BPSK operating over Rayleigh fading paths with average SNRs γ l (l = 1, 2, . . . , L), then it is well known that Pγl (γTl ) = 1 − exp(−γTl / γ l ) while Pb (El ) can be shown, with the help of another Luke equation [136, p. 185, Eq. (24)], to be given by      γl 1 + γl 2γTl − Q 2γTl (9.359) Pb (El ) = Q e γTl / γ l 1 + γl γl

449

SWITCHED DIVERSITY

Delay Statistics Let Nc denote the number of coherence times that the system has to wait before an acceptable path is found and transmission occurs. It is easy to see that Nc is a discrete RV with probability mass function (PMF) given by Pr[Nc = n] = PLn (ξ1 + ξ2 + · · · + ξL ), =

PLn (1

n = 0, 1, . . .

− PL ),

n = 0, 1, . . .

(9.360)

The average waiting time (in terms of average number of coherence times N c ) required before transmission is given by Nc =

∞


nP [Nc = n]

(9.361)

n=0

Substituting (9.360) in (9.361) and using the identity 0 ≤ p < 1 leads to

∞

n=0 np

PL 1 − PL

Nc =

n−1

= 1/(1 − p)2 for

(9.362)

which reduces when the fading is independent across the diversity paths to

L Nc =

l=1

1−

L

Pγl (γTl )

l=1

Pγl (γTl )

(9.363)

and in the i.i.d. fading case can be written as (Pγ (γT ))L 1 − (Pγ (γT ))L

Nc =

(9.364)

The variance of the number of coherence times before var[Nc ] =  transmission, 2 p n = p(1 + p)/(1 − n E[Nc2 ] − (E[Nc ])2 , can be found using the identity ∞ n=0 p)3 for 0 ≤ p < 1 as var[Nc ] =

PL , (1 − PL )2

(9.365)

which, when the fading is independent across the diversity paths, reduces to

L var[Nc ] =

l=1

[1 −

L

Pγl (γTl )

l=1

Pγl (γTl )]2

(9.366)

and in the i.i.d. fading case further simplifies to var[Nc ] =

(Pγ (γT ))L [1 − (Pγ (γT ))L ]2

(9.367)

450

PERFORMANCE OF MULTICHANNEL RECEIVERS

For delay-sensitive data, it is of interest to determine the dropping probability Pd , which is the probability that the delay or waiting time before transmission exceeds a critical threshold nth . This threshold corresponds, for example, to the maximum delay tolerated in the transmission of the type of data of interest. In other words, Pd is the complementary CDF (CCDF) of Nc evaluated at nth and is then given by Pd = Pr[Nc > nth ] = 1 − Pr[Nc ≤ nth ]

(9.368)

Substituting (9.360) in (9.368) and after some simplifications, this leads to 1+nth

Pd = PL

(9.369)

which in the independent fading case reduces to Pd =

L

1+nth Pγl (γTl )

(9.370)

l=1

and  (1+nth )L Pd = Pγ (γT )

(9.371)

in the i.i.d. case. Number of Paths Estimation per Channel Access From complexity and processing power consumption perspectives, it is of interest to study the statistics of the number of paths estimated Ne before channel access. Given the mode of operation of SWC, this number Ne is a discrete RV whose PMF is given by Pr[Ne = nL + l] = PLn ξl , for n = 0, 1, . . . ,

and l = 1, 2, . . . , L

(9.372)

Hence the average number of estimates per channel access N e is given by Ne =

∞
L


(nL + l)Pr[Ne = nL + l]

(9.373)

n=0 l=1

Substituting (9.372) in (9.373) leads, after some manipulations and simplifications, to the final desired result as  1 + L−1 l=1 Pl Ne = (9.374) 1 − PL which can be written for the independent fading case as L−1 l Pγ (γT ) N e = l=0 Ln=1 n n 1 − l=1 Pγl (γTl )

(9.375)

SWITCHED DIVERSITY

451

and for the i.i.d fading scenario simplifies to 1 1 − Pγ (γT )

Ne =

(9.376)

The corresponding variance of the number of estimates var[Ne ] can be shown after some manipulations to be given by

var[Ne ] =

2LPL

L−1 l=0

L−1

Pl + (1 − PL )

l=0

(2l + 1)Pl −



L−1 l=0 Pl

(1 − PL )2

2 (9.377)

For the uncorrelated case, (9.377) can be used with Pl = ln=1 Pγn (γTn ). Finally, for  n L+1 − Lp L + p)/(1 − p)2 the i.i.d. case, using the identity L−1 n=0 np = ((L − 1)p for 0 ≤ p ≤ 1, we obtain, after much simplification, the following equation: var[Ne ] =

Pγ (γT ) (1 − Pγ (γT ))2

(9.378)

For certain applications, it is important that the number of estimations per channel access stays bounded with high probability below a critical value Nth . For instance, in the context of multiuser diversity, the number of channel estimations per channel access is related to the system feedback load since this number corresponds to the number of users that the base station has to probe before transmission. In this context, an excess feedback load probability Pe corresponds to the CCDF of Ne evaluated at Nth and is therefore given by Pe = Pr[Ne > Nth ] = 1 − Pr[Ne ≤ Nth = nth L + lth ] nth −1

=1−




PLn

n=1

L


n

ξl − PL th

lth


l=1

ξl

l=1

n

=1−

1 − PL th n (1 − PL ) − PL th (1 − Plth ) 1 − PL

n

n

= PL th Plth = PL th PNth −nth L For the uncorrelated case, using Pl = Pe =

L l=1

(9.379)

l

i=1 Pγi (γTi ),

nth Pγl (γTl )

Nth −nth L

(9.379) becomes

Pγl (γTl )

l=1

which in the i.i.d case simply reduces to Pe = (Pγ (γT ))Nth .

(9.380)

452

PERFORMANCE OF MULTICHANNEL RECEIVERS

100 SEC SWC

10−1

Average BEP

10−2

10−3

10−4

L= 3

10−5

L =4

L =6 10−6

0

5

10

15

20

25

Average SNR (dB) per Path Figure 9.52 Comparison of the average BEP of BPSK with SEC (using optimal switching threshold) and SWC (using a switching threshold yielding the same average number of path estimations as SEC for a fixed L) over i.i.d. Rayleigh fading paths and for various values of L.

Numerical Examples Figure 9.52 compares the average BEP of BPSK with traditional SEC [133] and the newly proposed SWC over L i.i.d. Rayleigh paths. In this figure, for a fixed average SNR, the switching threshold for traditional SEC is set to its optimal value γT∗ to guarantee minimum average BEP performance for SEC. Then a possible fair basis of comparison between SEC and SWC is equivalent processing complexity, which is tantamount to choosing the switching threshold for SWC such that the two schemes have the same average number of estimated diversity paths per channel access. For traditional SEC, it can be shown that this average number of estimates over L i.i.d. paths is given by L−1  1 − Pγ (γT ) Ne = 1 − Pγ (γT )

(9.381)

which for the Rayleigh case reduces to " L−1 # ∗  −γT∗ / γ e γT / γ Ne = 1 − 1 − e

(9.382)

SWITCHED DIVERSITY

453

0.1

L=3 L=4 L=6

0.09 0.08

Average Nc

0.07 0.06 0.05 0.04 0.03 0.02 0.01 0

0

5

10

15

20

25

Average SNR (dB) per Path Figure 9.53 Average number of coherence times required for SWC before channel access over i.i.d. Rayleigh fading paths as a function of the SNR per path and for various values of L.

Equating (9.382) and (9.376) for the i.i.d. Rayleigh case yields a switching threshold γT for SWC of " L−1 #  ∗ γT = γT∗ + γ ln 1 − 1 − e −γT / γ

(9.383)

From this figure it is clear that SWC outperforms SEC, but this gain seems to slowly diminish as L increases. The corresponding delay associated with SWC is illustrated in Figs. 9.53 and 9.54. More specifically, Fig. 9.53 shows the average number of coherence times needed before channel access, while Fig. 9.54 √ shows the dropping probability when the threshold n is set to N , N + var[Nc ], th c c √ and N c + 2 var[Nc ]. The interesting conclusion that one can draw from all these figures is that, at high SNR per path (say, above 10 dB), while SWC offers a certain gain over SEC (more than 1 dB), this gain comes at the expense of a negligible delay since the average number of coherence times is very close to zero and the probability that this average number is exceeded is below 10−2 . Figure 9.55 extends the comparison over L i.i.d. Rayleigh paths to MRC and SC diversity combining schemes. Since MRC and SC require L channel estimates per channel access, then again as one possible fair basis of comparison between SWC

454

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

nth = nc nth = nc + sc nth = nc + 2sc

10−1

Pr[Nc > nth]

10−2

10−3

L =3 10−4

L =4

10−5

L =6 10−6 0

5

10

15

20

25

Average SNR (dB) per branch

Figure 9.54 Dropping probability of SWC over i.i.d. Rayleigh fading paths as a function of the SNR per path and for various values of L.

and other combining schemes, the switching threshold for SWC γT is chosen such that the average number of estimates per channel access N e = L. More specifically, in the Rayleigh case, solving for γT in (9.376) yields γT = γ ln L. Comparing the performance of SWC, MRC, and SC for fixed L, one can observe (1) the tremendous gain offered by SWC over SC over the whole SNR range and (2) the considerable gain of SWC over MRC in the high-SNR region but with a diminishing range for this gain as L increases. It is interesting to note that this gain comes again at the expense of a negligible delay given by (9.364) with γT = γ ln L, which yields Nc =

(L − 1)L LL − (L − 1)L

(9.384)

More specifically, for (1) L = 2, N c = 13 = 0.333; (2) for L = 4, N c  0.463; and (3) for L = 6, N c  0.504. More generally, it can be shown that N c in (9.384) is an increasing function of L and converges to 1/(e − 1)  0.58 when L tends to infinity. Finally, Fig. 9.56 plots the average BEP of BPSK with SWC as a function of the total average SNR for L = 5. Both the i.i.d. and non-i.i.d. Rayleigh fading scenarios are addressed. For the i.i.d. scenario, the L paths are assumed to have the same average SNR per path γ , and the total average SNR is then γ t = Lγ .

SWITCHED DIVERSITY

455

100

10−1

SC MRC SWC

Average BEP

10−2 10−3

10−4

L=2 10−5

L=4 10−6

10−7

L=6

0

5

10

15

20

25

30

35

Average SNR (dB) per Path Figure 9.55 Comparison of the average BEP of BPSK with MRC, SC, and SWC (with a switching threshold set such that the average number of path estimations is equal to L) over i.i.d. Rayleigh fading paths as a function of the SNR per path and for various values of L.

On the hand, for the non-i.i.d. scenario, an exponentially decaying delay profile with power decay factor δ is assumed. In this case, the lth path average SNR γ l = γ 1 e−δ(l−1) , l = 1, . . . , L, where γ 1 is the average SNR    of the first path, and as such the total average SNR γ t = γ 1 1 − e−Lδ / 1 − e−δ . For the non-i.i.d scenario, three curves are plotted. The “best case” curve corresponds to a receiver that has average SNR path information and as such ensures that γ 1 ≥ γ 2 ≥ · · · γ L (i.e., the scanning of the L diversity paths starts from the strongest path on average). This curve yields a lower bound on the average BEP of SWC in a non-i.i.d. environment. In contrast, the “worst case” curve represents an upper bound on the BEP performance of SWC in a non-i.i.d. environment since it assumes that γ 1 ≤ γ 2 ≤ · · · γ L (i.e., the scanning of the L diversity paths starts from the weakest path on average). Hence, in the absence of average SNR information at the receiver, the BEP performance of SWC lies between these two bounds, as illustrated for example by the “random case” curve in Fig. 9.56. It is also interesting to note from this figure that when the total average SNR is below 8 dB (which corresponds to the switching threshold chosen in this particular example), then the three cases (i.e., best, worst, and random cases) yield the same average BEP performance. On the other hand, the gap between the best-case and the worst-case curves increases to about 5 dB in

456

PERFORMANCE OF MULTICHANNEL RECEIVERS

10−3 Non i.i.d-Worst Case Non i.i.d-Random Case I.I.D Non i.i.d-Best Case

Average BEP

10−4

10−5

10−6

0

5

10

15

20

25

Total Average SNR (dB) Figure 9.56 Comparison of the average BEP of BPSK with SWC (γT = 8 dB and L = 5) over an i.i.d. Rayleigh fading environment and a non-i.i.d Rayleigh fading environment (exponentially decaying power delay profile with δ = 0.3).

the high-total-SNR region. It should also be noted that while the average BEP, the ergodic capacity, and the average number of paths estimations are affected by the ordering of the average strength of the available diversity paths, the delay statistics are insensitive to this ordering, as one can conclude from Eqs. (9.363), (9.366), and (9.370). As a final remark, regarding this figure, one can conclude after comparing the i.i.d and non-i.i.d curves that the average BEP performance of the proposed SWC scheme benefits from non-i.i.d conditions, for all possible orderings in the low average SNR range and for the best-case ordering scheme (and sometimes for the random-case ordering) in the high average SNR range.

9.10 PERFORMANCE IN THE PRESENCE OF OUTDATED OR IMPERFECT CHANNEL ESTIMATES In general, diversity combining techniques rely, to a large extent, on accurate channel estimation. As a typical first step in performance analysis, perfect channel estimation is assumed as was done in the previous sections. However, in practice these estimates must be obtained in the presence of noise and time delay. For

PERFORMANCE IN THE PRESENCE OF OUTDATED OR IMPERFECT CHANNEL ESTIMATES

457

example, one common way to estimate the channel uses pilot-symbol-assisted modulation (PSAM) [143; 144, Sect. 10.3.2], which periodically inserts pilots into the stream of data symbols to extract the channel-induced fading. Estimation error (due to additive noise as well as wireless channel variation/decorrelation over time) will cause the channel fading extracted from the pilot symbols to differ from the actual fading affecting the data symbols, thereby inducing a performance degradation. The effects of channel estimation error or channel decorrelation on the performance of diversity systems has long been of interest. These previous studies focused on MRC receivers over Rayleigh fading channels [32,145–148], postdetection EGC receivers over fast Rician fading channels [149,150], SC receivers over Rayleigh [151] and Nakagami-m [152] fading channels, and SSC receivers over Nakagami-m fading channels [123]. In this section, we briefly summarize the work on the impact of channel estimation error or channel decorrelation on the performance of diversity systems. 9.10.1

Maximal-Ratio Combining

Gans [146] studied the effect of Gaussian distributed weighting errors on the performance of MRC receivers. In particular, he showed that if the combined branches are subject to i.i.d. Rayleigh fading, then the PDF of the combined SNR is given by L−1 " #l−1 L ργ (1 − ρ)L−1 exp(−γ / γ )
l−1 qγMRC (γ ) = γ (l − 1)! (1 − ρ)γ

(9.385)

l=1

where ρ ∈ [0, 1] is the power correlation coefficient between the estimated and actual fadings. This coefficient can be viewed as a measure of the channel’s rate of fluctuation and can be related solely to the time delay τ and to the maximum Doppler frequency shift fD [e.g., for land–mobile communication ρ = J02 (2πfD τ ), where J0 (·) is the zeroth-order Bessel function of the first kind]. The parameter ρ can also be viewed as a measure of the quality of the channel estimation and can be expressed, for example, in terms of the PSAM parameters such as the rate of pilot symbol insertion and SNR [153]. It is interesting to note that (9.385) can be rewritten as a weighted sum of L ideal MRC PDF’s [148, Eq. (7)] qγMRC (γ ) =

L
l=1

A(l)

1 γ l−1 e−γ / γ (l − 1)!γ l

(9.386)

where the weight coefficients

L−1 A(l) = (1 − ρ)L−l ρ l−1 l−1

(9.387)

are Bernstein polynomials. As a check, when ρ = 1 (perfect correlation between the pilot-extracted fading and the actual fading), perfect MRC combining is achieved

458

PERFORMANCE OF MULTICHANNEL RECEIVERS

and (9.385) or equivalently (9.386) reduce to (9.5), as expected. As ρ decreases, the correlation between the pilot-extracted fading and the actual fading diminishes and performance degrades. In the limit as ρ → 0, the fading estimate and its actual value are completely uncorrelated and (9.385) or equivalently (9.386) approach the PDF without diversity (i.e., L = 1) given by (9.4). Taking the Laplace transform of (9.386) and using Eq. (3.351.3) From Ref. 36, the corresponding MGF M(s) can be easily shown to be given by M(s) =

L
l=1

A(l) (1 − sγ )l

(9.388)

With (9.388) in hand and using the integrals derived in Appendix 5A, the average probability of error of several linear coherent modulations with imperfect MRC can be computed in closed form. The error probability analysis of diversity combining in the presence of Gaussian distributed weighting errors has been revisited by Ramesh and Milstein [154]. This new study shows that in this case the conditional probability of error is not a function of the output combiner SNR and that the result given in (9.388) actually yields a lower bound on the exact probability of error. These exact results were derived for i.i.d. Rayleigh fading [154] under the assumption that the pilot and data fades may have different powers denoted by p and d , respectively. More specifically, letting p denote the estimated Gaussian channel gain and letting h denote the Gaussian channel gain, then p = E[|p2 |], d = E[|h2 |], Rc = E[Re(p) Re(h)] = E[Im(p) Im(h)], and Rcs = E[Re(p) Im(h)] = −E[Im(p) Re(h)]. With this setup, Ramesh and Milstein [154] show that for variety of combining schemes the exact average probability of error under Gaussian weighting errors has the same form as the average probability error with ideal combining [as given in (9.6) for BPSK with ideal MRC] but with the average SNR per branch γ replaced by the effective SNR due to the weighting error γ ρ , defined by γρ =

γ ρc2 2 ) 1 + γ (1 − ρc2 − ρcs

(9.389)

2 /(p d ). where ρc = 4Rc2 /(p d ) and ρcs = 4Rcs

9.10.2

Noncoherent EGC over Rician Fast Fading

In this section, we extend the results presented in Section 8.2.5.2 and consider the effect of fast Rician fading on binary DPSK when used in conjunction with L-branch post-detection EGC. Using the same notation as in Sect. 8.2.5.2 the EGC output decision variable corresponding to transmission of a +1 information bit during the kth bit time becomes zk =

L
l=1

∗ wk∗l wk−1l +wkl wk−1 l

(9.390)

PERFORMANCE IN THE PRESENCE OF OUTDATED OR IMPERFECT CHANNEL ESTIMATES

459

where the subscript l refers to the lth branch. The probability of error is given by Pb (E) = Pr[zk < 0]

(9.391)

Because the decision variable is a quadratic form of complex Gaussian random variables, we rely again on App. B of Ref. 7. Specifically, letting A = B = 0, C = 1, D = zk , Xk = wk−1l , and Yk = wkl , we see that the decision variable (9.390) is identical to that in (9A.1) (or equivalently App. B in Ref. 7) for any arbitrary L. Evaluating the various coefficients required in (9A.10) produces after much simplification the following results η=

v2 1 + K + γ (1 + ρ) = v1 1 + K + γ (1 − ρ)

a=0  b=

2LKγ 1+K +γ

(9.392)

where K is the Rice factor and ρ is again the fading correlation whose value depends on the fast-fading channel model that is assumed, as mentioned earlier. Finally, substituting (9.392) in (9A.9) and recalling that Qm (b, 0) = 1 and (4.46), we obtain the desired average BER as



1 + K + γ (1 − ρ) 2L−1 LKγ 1+K +γ 2(1 + K + γ ) L−1
2L − 1 1 + K + γ (1 + ρ) l × l 1 + K + γ (1 − ρ)

Pb (E) = exp −

l=0

+



L−l−1

n  2L − 1 1 + K + γ (1 + ρ) l
1 LKγ l 1 + K + γ (1 − ρ) n! 1 + K + γ

L−2
l=0

n=1

(9.393) which can be shown to agree numerically with Eq. (76) of Ref. 150. As a check on consistency, letting ρ = 1 (i.e., perfect channel estimation) and replacing γ by γ /2 in (9.393), the result must reduce to (9.152), which corresponds to binary orthogonal FSK. Although by no means do the forms of these two equations appear to be the same, using appropriate combinatorial identities, however, these two equations can be analytically proven to be identical. The corresponding result of (9.393) for the Rayleigh (K = 0) channel is Pb (E) =

1 + γ (1 − ρ) 2(1 + γ )

2L−1 L−1
2L − 1 1 + γ (1 + ρ) l l 1 + γ (1 − ρ) l=0

in agreement with Kam [149, Eq. (3a)] and Chow et al. [150, Eq. (79)].

(9.394)

460

PERFORMANCE OF MULTICHANNEL RECEIVERS

Similar to the no-diversity case, these expressions exhibit an irreducible bit error probability floor for any ρ = 1. Letting γ approach infinity in (9.393) and (9.394) yields



2L−1 


L 2L − 1 1+ρ l Pb (E) = e l 1−ρ l=0  L−2
2L − 1 1 + ρ l L−l−1
1 n + (LK) l 1−ρ n! −LK



1−ρ 2

l=0

Pb (E) =

1−ρ 2

(9.395)

n=1

2L−1 L−1
2L − 1 1 + ρ l l 1−ρ

(9.396)

l=0

for Rician and Rayleigh channels, respectively. Figure 9.57 illustrates this bit error floor for a Rician factor K = 10 and assuming a correlation model with ρ = J02 (2πfd T ) and fd T = 0.04. Before concluding this section we should mention that Chow et al. [150] extend the analysis presented here to study the effect of fast Rician fading on M-DPSK (M ≤ 4) when used in conjunction with postdetection EGC.

100 10−2

Average Bit Error Rate Pb(E )

10−4

L=1

10−6 10−8 10−10

L=2

10−12 10−14 10−16

L=4

10−18 10−20

L =8 0

5

10

15

20

25

30

35

40

45

50

Total Average SNR per Bit [dB] Figure 9.57 Average BER of binary DPSK over fast Rician channels (K = 10, fd T = 0.04).

PERFORMANCE IN THE PRESENCE OF OUTDATED OR IMPERFECT CHANNEL ESTIMATES

9.10.3

461

Selection Combining

In this section, we adopt the approach introduced by Ritcey and Azizo˜glu [152] and study the impact of imperfect channel estimation or/and decorrelation on the performance of SC systems over Nakagami-m fading channels. This requires the second-order statistics of the channel variation, which are fortunately known for Nakagami-m fading. Let α and ατ denote the channel fading amplitudes at times t and t + τ , respectively.19 For a slowly varying channel we can assume that the average fading power remains constant over the time delay τ [i.e.,  = E(α 2 ) = E(ατ2 )]. Under these conditions the joint PDF pα,ατ (α, ατ ) of these two correlated Nakagami-m distributed channel fading amplitudes is given by [54, Eq. (126)] √

 m m+1 2m ρ αατ 4 (αατ )m Im−1 pα,ατ(α, ατ ) = (1 − ρ) (m) ρ (m−1)/2  (1 − ρ)

m (α 2 + ατ2 ) × exp − (9.397) (1 − ρ)  where ρ ∈ [0, 1] denotes again the power correlation factor between α and ατ . Denoting the instantaneous SNR per symbol at times t and t + τ by γ and γτ , respectively, the joint PDF of γ and γτ can be written as

m+1 γ m−1/2 γτm−1/2 m m(γ + γτ ) exp − pγ ,γτ (γ , γτ ) = γ (1 − ρ) (m)ρ m−1/2 (1 − ρ)γ √

2m ργ γτ (9.398) × Im−1 (1 − ρ)γ where γ is the average SNR per symbol over the time delay τ . Under these conditions it can be shown that for the dual-branch SC the MGF of the output of SC with an outdated or imperfect estimate can be obtained in closed form with the help of Eq. (6.455.2) from Ref. 36 as



γ −2m γ m 2 (2m) 2 − (2 − ρ) s s 1 − (1 − ρ) MγSC (s) = m 2 (m) m m   γ s 1 − (1 − ρ) m × 2 F1 1, 2m; m + 1; (9.399) γ 2 − (2 − ρ) m s Using the well-known result that the first moment of γSC is equal to its statistical average (9.243), we obtain the closed-form expression for the average output SNR of SC with an outdated or imperfect estimate as   1 2 (2m) 2 F1 (1, 2m; m + 1; 12 ) 2 F1 (2, 2m + 1; m + 2; 2 ) γ SC = +ρ γ m 2 (m) 22m 22m+1 (m + 1) (9.400) 19 Equivalently, as mentioned above, α and α can be viewed as the actual fading amplitude and the τ imperfectly estimated one, respectively.

462

PERFORMANCE OF MULTICHANNEL RECEIVERS

With the MGF (9.399) in hand, the average probability of error can be found for a wide variety of modulation schemes as explained in Chapter 8. 9.10.4

Switched Diversity

We now study the effect of channel decorrelation or imperfect channel estimation on the performance of dual-branch SSC systems operating over Nakagami-m fading channels [123]. 9.10.4.1 SSC Output Statistics The PDF of the SSC output at time t + τ , qγSSC (γτ ) can be expressed in terms of the PDF of the SSC output at time t, pγSSC (γ ), as  ∞ qγSSC (γτ ) = pγSSC (γ ) pγ (γτ |γ ) dγ (9.401) 0

where the PDF of γτ conditioned on γ , pγ (γτ |γ ), is given by pγτ |γ (γτ |γ ) =

pγ ,γτ (γ , γτ ) pγ (γ )

(9.402)

For simplicity let us consider the case of i.i.d. branches. Inserting (9.274) and (9.402) in (9.401), we can write qγSSC (γτ ) as  qγSSC (γτ ) = Pγ (γT ) 

γT

 pγ ,γτ (γ , γτ ) d γ + (1 + Pγ (γT ))

0 ∞

= Pγ (γT )

 pγ ,γτ (γ , γτ ) d γ +

0

= Pγ (γT )pγτ (γτ ) +



∞

pγ ,γτ (γ , γτ ) d γ γT

∞

pγ ,γτ (γ , γτ ) d γ γT

∞

pγ ,γτ (γ , γτ ) d γ

(9.403)

γT

As a check, when γ and γτ are fully correlated [i.e., pγ ,γτ (γ , γτ ) = δ(γ − γτ ) pγ (γ )], then it can be easily shown that qγSSC (γτ ) reduces to pγSSC (γτ ) as given in (9.274), and full SSC diversity gain is achieved. On the other hand, when γ and γτ are uncorrelated [i.e., pγ ,γτ (γ , γτ ) = pγ (γ )pγτ (γτ )], then it is straightforward to show that qγSSC (γτ ) reduces to pγτ (γτ ), which is the single-branch PDF, and hence no diversity gain is obtained. For Nakagami-m fading, inserting the single-branch PDF and CDF as given in Table 9.5 as well as (9.398) in (9.403), qγSSC (γτ ) can be expressed in closed form as



γτ m m γτm−1 exp −m γ (m) γ     2mργτ 2mγT (m, mγT / γ ) × 1− + Qm , (9.404) (m) (1 − ρ)γ (1 − ρ)γ

qγSSC (γτ ) =

463

PERFORMANCE IN THE PRESENCE OF OUTDATED OR IMPERFECT CHANNEL ESTIMATES

The MGF at time t + τ is given by  MγSSC (s) =

∞ 0

esγτ qγSSC (γτ ) dγτ

(9.405)

Substituting (9.404) in (9.405), and using the change of variable x = √ 2mργτ /[(1 − ρ)γ ], MγSSC (s) can be expressed in closed form with the help of Eq. (11) of Ref. 139 as  

−m m, mγγ T sγ 1 + MγSSC (s) = 1 − m

1−sγ /m 1−(1−ρ)sγ /m

(m)



  − m, mγγ T 

(9.406) When ρ = 1 and hence γ and γτ are perfectly correlated, it is easy to see that (9.406) reduces to the MGF with perfect SSC as given by (9.280). On the other hand, when γ and γτ are uncorrelated (i.e. ρ = 0), it is easy to see that (9.406) reduces to the MGF of single Nakagami-m channel reception as given in Table 9.5. 9.10.4.2 Average SNR Averaging γτ over the PDF qγSSC (γτ ) as given by (9.404) yields the average output SNR as    m ρ mγγ T e−(mγT / γ )   (9.407) γ SSC = γ 1 +  (m + 1) Differentiating (9.407) with respect to γT and setting the result to zero, we find that the optimal threshold γT∗ is given by γT∗ = γ and results in a maximum average output SNR γ ∗SSC given by γ ∗SSC = γ

mm−1 e−m 1+ρ (m)

(9.408)

9.10.4.3 Average Probability of Error Consider, as an example, the average BER of binary DPSK or noncoherent FSK. In this case the average BER is given by Pb (E) =

1 2

MγSSC (−g)

(9.409)

where g = 1 for DPSK and g = 12 for orthogonal FSK. Substituting (9.406) in (9.409) and then differentiating with respect to γT yields the optimal threshold as γT∗



m + gγ m + (1 − ρ)gγ ln = gρ m + (1 − ρ)gγ

(9.410)

464

PERFORMANCE OF MULTICHANNEL RECEIVERS

As a check, when ρ = 0, it is easy to see that (9.410) reduces to zero since no diversity gain can be achieved in this case and there is therefore no need to switch to the other branch. On the other hand, when ρ = 1 (9.410) reduces to the optimal threshold for perfect SSC as given by Abu-Dayya and Beaulieu [9, Eq. (14)], then γT∗



gγ m . = ln 1 + g m

(9.411)

Note that the optimum threshold γT∗ in (9.411) is dependent on the correlation coefficient ρ. In the case where ρ is viewed as a measure of the channel estimation quality, then γT∗ can be set to (9.411) once ρ is known for minimum BER performance. However, in the case of outdated estimates, ρ is a function of the time delay τ but γT∗ (which is set to a particular value during the whole slot time20 ) cannot be changed as a function of τ . In this case, to minimize the degradation due to channel decorrelation, one may want to find a “global” optimum threshold independent of the delay τ . This may be achieved by finding the value γT∗ that minimizes a cost function C similar to the one used in Eq. (29) of Ref. 9 over an average SNR range  C=

1 ρ1

Pb∗ (ρ) dρ log Pb∗ (ρ0 )

(9.412)

where Pb∗ (ρ) denotes the average BER, as given by (9.409), evaluated with the optimal threshold γT∗ , as given by (9.411). In (9.412), ρ1 denotes the minimal correlation coefficient experienced over the slot time whereas ρ0 can be chosen equal to ρ0 = (ρ1 + 1)/2. 9.10.5

Numerical Results

Figure 9.58 illustrates the analyses presented in the previous sections by showing the dependence of the average BER of BPSK on the correlation coefficient ρ ∈ [0, 1] between the estimated and actual fading for dual-branch MRC, SC, and SSC receivers and for Rayleigh type of fading. When ρ = 1 (i.e., perfect correlation between the pilot-extracted fading and the actual fading, or equivalently perfect channel estimation), MRC outperforms SC, which, in turn, outperforms SSC. As ρ decreases, the correlation between the pilot-extracted fading and the actual fading diminishes and performance degrades. In the limit as ρ → 0, the fading estimate and its actual value are uncorrelated and performance of all combining schemes approaches the performance without diversity. Figure 9.59 compares the effect of the correlation coefficient ρ on the average BER of binary DPSK with SC and 20 By

“slot time,” we mean the time interval T between two consecutive switching instants.

PERFORMANCE IN THE PRESENCE OF OUTDATED OR IMPERFECT CHANNEL ESTIMATES

465

10−1 MRC SC SSC

Average Bit Error Rate Pb(E )

SNR = 10 dB 10−2

SNR = 20 dB 10−3

10−4

10−5

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Correlation Coefficient (r) Figure 9.58 Average BER of BPSK with dual-branch MRC, SC, and SSC (with optimum threshold) versus the correlation coefficient (ρ).

SSC for various values of the Nakagami-m parameter. In this case, ρ is viewed as a measure of the channel estimation quality and the optimum switching threshold is set according to (9.410) for the SSC curves. We can see from these curves that the diversity gain offered by SC over SSC decreases as ρ decreases and tends eventually to zero as ρ tends to zero, as expected. Figure 9.60 shows the effect of channel decorrelation on the average BER of binary DPSK with SC and SSC. For the SSC curves the optimum switching threshold is fixed at the beginning of the slot time. For the dashed SSC curves the optimum switching threshold is fixed to the optimum value for ρ = 1 or equivalently for τ = 0. On the other hand, the solid SSC curves are generated using optimum thresholds that were optimized over the whole slot time (according to the cost function as explained above) where we assumed that ρ is bounded between 0.8 and 1. From these figures we can see that at low average SNRs (10 dB) the two procedures yield nearly indistinguishable performance results. However, for higher average SNR (20 dB), the first procedure clearly yields better performance for ρ close to 1 (or equivalently for small values of τ ) before the two curves cross and the “global” optimization procedure starts to pay off.

466

PERFORMANCE OF MULTICHANNEL RECEIVERS

m = 0.5

m=1

10−1

100

10−2

10 dB

10−1

Average BER

Average BER

10 dB

20 dB

10−2

10−3

20 dB

10−4 SSC SC 10−3

0

0.2

SSC SC 0.4

0.6

0.8

10−5

1

0

0.2

0.4

0.6

0.8

Correlation Coefficient (r)

Correlation Coefficient (r)

m=2

m=4

1

10−1

10−3

Average BER

Average BER

10−2

10 dB

10−2

10−4 20 dB

10−5

10−4

10−6 20 dB 10−8

10−6 10−7

10 dB

SSC SC 0

0.2

SSC SC 0.4

0.6

0.8

1

10−10

0

Correlation Coefficient (r)

0.2

0.4

0.6

0.8

1

Correlation Coefficient (r)

Figure 9.59 Average BER of binary DPSK with SC and SSC versus correlation coefficient (ρ) with (a) γ =10 dB and (b) γ =20 dB over Nakagami-m fading channel. The optimum threshold is set according to (9.410) and is thus a function of ρ and γ .

9.11 9.11.1

COMBINING IN DIVERSITY-RICH ENVIRONMENTS Two-Dimensional Diversity Schemes

In this section, we analyze the performance of the two-dimensional (2D) diversity systems that we described in Section 9.1.3.2. The aim is to accurately quantify the effect of the fading severity and correlation as well as the power delay profile on the error rate and outage probability performance. As in the previous section, we again rely on the MGF-based approach, which is going to be particularly handy

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

10−1

467

10−1 10 dB 10 dB 10−2 Average BER

Average BER

20 dB

10−2

20 dB 10−3

m = 0.5

m=1 10−4

SSC SSC SC 10−3 0.8

SSC SSC SC 10−5 0.8

0.85 0.9 0.95 1 Time Correlation Coefifcient (r)

10−2

10−2

10 dB

10 dB

10−3

10−4 20 dB

10−4

10−5

Average BER

Average BER

0.85 0.9 0.95 1 Time Correlation Coefficient (r)

m=4 20 dB

10−6

m=2 10−6

10−7 0.8

10−8 SSC SSC SC 0.85 0.9 0.95 1 Time Correlation Coefficient (r)

SSC SSC SC 10−10 0.8

0.85 0.9 0.95 1 Time Correlation Coefficient (r)

Figure 9.60 Average BER of binary DPSK with SC and SSC versus correlation coefficient (ρ) with (a) γ = 10 dB and (b) γ = 20 dB. The dashed line corresponds to the case where the optimum threshold is set for ρ = 1 or equivalently τ = 0. The solid line corresponds to the case where the optimum threshold is optimized over ρ = 0.8–1 range.

in this case. In particular, when MRC or post-detection EGC is used for both dimensions, finding the PDF of the combined SNR in a simple form (as required by the classical approach to tackle these problems) is particularly difficult in the presence of fading correlation, a nonuniform power delay profile and/or when the fading tends to follow other than Rayleigh statistics, whereas the MGF-based

468

PERFORMANCE OF MULTICHANNEL RECEIVERS

approach will circumvent much of the tedium and intractability in the classical approach, as we show next. 9.11.1.1 Performance Analysis Consider a 2D diversity system consisting, for example, of D antennas, each one followed by an Lc -finger RAKE receiver. As an example of practical channel c conditions of interest, let us assume that for a fixed antenna index d the {γl,d }L l=1 are independent but nonidentically distributed. On the other hand, let us assume that for a fixed multipath index l the {γl,d }D d=1 are correlated (in space) according to model A, B, C, or D (as described in Section 9.7). When MRC or post-detection EGC combining is applied for both space and multipath diversity, we have a conditional combined SNR/symbol given by

γt =

Lc D

d=1 l=1

=

D


=

 where γd =

γd

d=1 Lc


γl,d  γl,d

l=1

 γl

Lc


where γl =

D


l=1



γl,d

(9.413)

d=1

Finding the average error rate or outage probability performance of such systems with the classical PDF-based approach is difficult since the PDF of γt cannot be found in a simple form. However, using the MGF-based approach for the average BER of BPSK, for example, we have, after switching the order of integration 1 Pb (E) = π

 0

  L c l=1 γl dφ exp − sin2 φ



π/2

Eγ1 ,γ2 ,...,γLc

(9.414)

c Since the {γl }L l=1 are assumed to be independent, then

Pb (E) =

1 π

1 = π

 0

 0

Lc π/2 l=1 Lc π/2 l=1

# " γl dφ Eγl exp − 2 sin φ Mil −

1 sin2 φ

dφ

(9.415)

where Mil (s) is given by (9.206), (9.213), (9.217), or (9.219) depending on the space fading correlation model under consideration.

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

469

Application As an application, let us consider a 2D RAKE receiver operating over a Nakagami-m fading channel characterized by a spatial correlation coefficient ρl along the path of index l (l = 1, 2, . . . , Lc ) and the same exponential PDP for the D RAKE receivers γ l,d = γ 1,1 e−(l−1)δ

(l = 1, 2, . . . , Lc )

(9.416)

where δ denotes the average fading power decay factor. Substituting (9.213) in (9.415), we obtain the average BER for BPSK with a constant spatial fading correlation profile as Pb (E) =

1 π

Lc π/2



1+

0

l=1

γ l,d (1 −

√ γ l,d (1 − ρl ) × 1+ ml sin2 φ

√ √

ρl + D ρl ) −ml

ml sin2 φ

−ml (D−1) dφ

(9.417)

Similarly, substituting (9.228) in (9.415), we obtain the average BER for BPSK with a tridiagonal spatial fading correlation profile as Pb (E) =

1 π



Lc D 1 π/2

1−

0

l=1 d=1

"

#4−ml dπ √ 1 + 2 ρ cos dφ l D+1 ml sin2 φ (9.418) γ l,d

9.11.1.2 Numerical Examples As an example, the average BER performance curves of BPSK with 2D MRC RAKE reception over an exponentially decaying power delay profile and with constant or tridiagonal correlation between the antenna elements of the array [as given by (9.417) and (9.418), respectively] are shown in Fig. 9.61. The corresponding outage probability curves obtained by using the numerical technique presented in Section 9.6 are given in Fig. 9.62. Again notice the relatively important effect of the power delay profile. Also, diversity systems subject to tridiagonal correlation have a slightly better performance than do the ones subject to constant correlation in most cases. However, the opposite occurs at high average SNR for channels with a high amount of fading (m = 0.5) and a relatively strong correlation between the paths (ρ = 0.4). 9.11.2

Generalized Selection Combining

In the context of spread-spectrum communication with RAKE reception, the complexity of MRC and EGC receivers depends on the number of resolvable paths available, which can be quite high, especially for multipath diversity of wideband spread-spectrum signals. In addition, MRC is sensitive to channel estimation errors, and these errors tend to be more important when the instantaneous SNR

470

PERFORMANCE OF MULTICHANNEL RECEIVERS

(m = 0.5)

(m = 1)

100

100 Constant Correlation

10−2 10−3 10−4

Constant Correlation 10−1

Tridiagonal Correlation Average Bit Error Rate Pb(E)

Average Bit Error Rate Pb(E)

10−1

d=1

d = 0.5 d=0

10−5

a

a

a c

c

Tridiagonal Correlation

10−2 d=1 10−3 d = 0.5 10−4

d=0

10−5

c

10−6 −5 0 5 10 15 Average SNR per Bit of First Path [dB]

a c

a

(m = 2)

(m = 4) 100

Constant Correlation

Constant Correlation 10−1

Tridiagonal Correlation Average Bit Error Rate Pb(E)

Average Bit Error Rate Pb(E)

10−1 10−2

d=1

10−3

d = 0.5

10−5

c

10−6 −5 0 5 10 15 Average SNR per Bit of First Path [dB]

100

10−4

a c

d=0

a

a c

a c

c

10−6 −5 0 5 10 Average SNR per Bit of First Path [dB]

Tridiagonal Correlation

10−2 d=1 10−3 d = 0.5 10−4 d=0 10−5

a

c

a

c

a

c

10−6 −5 0 5 15 Average SNR per Bit of First Path [dB]

Figure 9.61 Average BER of BPSK with 2D MRC RAKE reception (Lc = 4 and D = 3) over an exponentially decaying power delay profile and constant or tridiagonal spatial correlation between the antennas for various values of the correlation coefficient [(a) ρ = 0, (b) ρ = 0.2, (c) ρ = 0.4].

is low. On the other hand, SC and SSC use only one path out of the L available (resolvable) multipaths [155] and hence do not fully exploit the amount of diversity offered by the channel. There has been an interest in bridging the gap between these two extremes (MRC/EGC and SC) by proposing generalized selection combining (GSC), which adaptively combines (as per the rules of MRC or EGC) the Lc

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

(m = 0.5) Constant Correlation

Constant Correlation

Tridiagonal Correlation

Tridiagonal Correlation

10−4

10−5

(m = 1)

10−3

d=1 d = 0.5 d=0

Outage Probability Pout

Outage Probability Pout

10−3

d=1

10−4 d = 0.5 d=0 10−5

a bc a b ca bc

a b c ab c a b c

10−6 −10 −5 0 5 Normalized Average SNR of First Path [dB]

10−6 −10 −5 0 5 Normalized Average SNR of First Path [dB]

(m = 2)

(m = 4)

100

100 Constant Correlation

Constant Correlation 10−1

Tridiagonal Correlation

10−2 d=1

10−3 d = 0.5 d=0 10−5

Outage Probability Pout

Outage Probability Pout

10−1

10−4

471

Tridiagonal Correlation

10−2 10−3 10−4

d=1 d = 0.5

d=0

10−5

a bc

abc abc

10−6 −10 −5 0 5 Normalized Average SNR of First Path [dB]

ab c

a bc a b c

10−6 −10 −5 0 5 Normalized Average SNR of First Path [dB]

Figure 9.62 Outage probability with 2D MRC or post-detection EGC RAKE reception (Lc = 4 and D = 3) over an exponentially decaying PDP and constant or tridiagonal spatial correlation between the antennas for various values of the correlation coefficient [(a) ρ = 0, (b) ρ = 0.2, (c) ρ = 0.4].

strongest (highest SNR) resolvable paths among the L available ones [11,157–160]. We denote such hybrid schemes as SC/MRC-Lc /L and SC/EGC-Lc /L. In the context of wideband spread-spectrum systems, these schemes offer less complex receivers than do the conventional MRC RAKE receivers since they have a fixed number of fingers independent of the number of multipaths. In addition, SC/MRC

472

PERFORMANCE OF MULTICHANNEL RECEIVERS

receivers are expected to be more robust toward channel estimation errors since the weakest SNR paths (and hence the ones that are most exposed to these errors) are excluded from the combining process. Finally, SC/MRC was shown to approach the performance of MRC [11], while SC/EGC was shown to outperform in certain cases conventional post-detection EGC since it is less sensitive to the “combining loss” of the very noisy (low-SNR) paths [11]. Kong, Eng, and Milstein [157,11] present an error rate analysis of binary signals with the GSC scheme over Rayleigh fading channels with both i.i.d. distributions and an exponentially decaying power delay profile for Lc = 2 and Lc = 3. Kong and Milstein [159], derived a simple neat closed-form expression for the average combined SNR at the output of GSC diversity systems operating over Rayleigh fading channels with a constant (uniform) power delay profile. Subsequently they extended their result to non-i.i.d. diversity paths [160]. In this section, we show that by starting with the MGF of the GSC output SNR it is possible to analyze the performance of GSC receivers over various i.i.d. and non-i.i.d. fading scenarios in terms of average combined SNR, outage probability, and average error rate for a wide variety of modulation schemes and for arbitrary Lc and L [161,162]. Finally, work on the performance analysis of GSC receivers using the “virtual branch” technique can also be found in papers by Win and colleagues [163–165]. 9.11.2.1

I.I.D. Rayleigh Case

GSC Input Joint PDF Let α1 , α2 , . . . , αL denote the set of i.i.d. Rayleigh random fading amplitudes associated with the SC inputs, each of which has average power . For RAKE reception with a matched-filter receiver for each diversity path, we define, as before, the instantaneous SNR per symbol of the lth path as γl = αl2 Es /N0 , l = 1, 2, . . . , L and the corresponding average SNR per symbol for each path as γ l = αl2 Es /N0 = Es /N0 . Let γ1:L ≥ γ2:L ≥ · · · γL:L ≥ 0 be the order statistics obtained by arranging the {γl }L l=1 in decreasing order of magnitude. Since Lc the {γl }L are i.i.d., the joint PDF p γ1:L ,...,γLc :L (γ1:L , . . . , γLc :L ) of the {γl:L }l=1 l=1 (Lc ≤ L) is given by [120, p. 185; 157, Eq. (9)] pγ1:L ,...,γLc :L (γ1:L , . . . , γLc :L )

Lc L−Lc L  Pγ (γLc :L ) = Lc ! pγ (γl:L ), Lc

γ1:L ≥ γ2:L ≥ · · · γLc :L (9.419)

l=1

  where LLc = L!/[Lc ! (L − Lc )!] denotes the binomial coefficient, pγ (γ ) is the PDF of the {γl }L l=1 such as pγ (γ ) = and Pγ (γ ) =

γ 0

1 −γ / γ e γ

(9.420)

pγ (y) dy is the corresponding CDF given by Pγ (γ ) = 1 − e−γ / γ

(9.421)

473

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

L It is important to note that although the {γl }L l=1 are independent, the {γl:L }l=1 are not, as can be seen from (9.419).

The MGF of the total combined SNR γGSC =

MGF of the Output SNR is defined by

 Lc  MγGSC (s) = EγGSC [esγGSC ] = Eγ1:L ,γ2:L ,···,γLc :L es l=1 γl:L

Lc

l=1 γl:L

(9.422)

where E[·] denotes the expectation operator. Substituting (9.419) in (9.422), we get  MγGSC (s) = 

∞ ∞ 0

 ···

γLc :L

γ2:L



Lc

l=1 γl:L

pγ1:L ,...,γLc :L (γ1:L , . . . , γLc :L )



Lc -fold

× es

∞

dγ1:L · · · dγLc −1:L dγLc :L

(9.423)

Although the integrand is in a desirable separable form in the γl:Lc terms, we cannot partition the Lc -fold integral into a product of one-dimensional integrals as was possible for MRC in Section 9.2 and post-detection EGC in Section 9.4 because of the γl:L terms in the lower limits of the semi-finite range (improper) integrals. To get around this difficulty, we take advantage of the following classical result, which is originally due to Sukhatme [166] and that subsequently played an important role in many order statistics problems [167,168], including radar detection analysis problems [169,170]. Theorem 9.1: Sukhatme [166] Consider the following transformation of random variables21 by defining the “spacings” 

xl = γl:L − γl+1:L ,

l = 1, 2, . . . , L − 1



xL = γL:L

(9.424)

Then it can be shown that the {xl }L l=1 are independent and distributed according to the exponential distribution pxl (xl ) given by pxl (xl ) =

l −(lxl / γ ) e , γ

xl ≥ 0,

l = 1, 2, . . . , L

(9.425)

A proof of this theorem is given in Appendix 9D. 21 It should be pointed out that Kong and Milstein considered a very similar transformation [159, App.] in their derivation of the combined average SNR of GSC and hence implicitly used Theorem 9.1.

474

PERFORMANCE OF MULTICHANNEL RECEIVERS

We now use Theorem 9.1 to derive a simple expression for the MGF of the total combined SNR γGSC , which can be expressed in terms of the xl values as γGSC =

Lc


γl:L =

Lc
L


xk

l=1 k=l

l=1

= x1 + 2x2 + · · · + Lc xLc + Lc xLc +1 + · · · + Lc xL

(9.426)

Hence the MGF of γGSC as defined in (9.422) can be expressed in terms of xl as  MγGSC (s) =

 ∞ . . . px1 ,...,xL (x1 ,. . . ,xL )  0  0  L-fold ∞

×es(x1 +2x2 +···+Lc xLc +Lc xLc +1 +···+Lc xL ) dx1 · · ·dxL Since the xl terms are independent [i.e., px1 ,...,xL (x1 , . . . , xL ) = can put the integrand in the desired product form resulting in

L

(9.427)

l=1 pxl (xl )],

we

  ∞  L MγGSC (s) = ··· pxl (xl )  0  0  l=1 L-fold 

∞

×esx1 e2sx2 · · · eLc sxLc eLc sxLc +1 · · · eLc sxL dx1 · · ·dxL

(9.428)

Grouping terms of index l, partitioning the L-fold integral of (9.428) into a product of L one-dimensional integrals, and then using the fact that the xl values are exponentially distributed, we get the desired closed-form result −Lc +1

MγGSC (s) = (1 − sγ )

L sγ Lc −1 1− l

(9.429)

l=Lc

Using a partial-fraction expansion of the product in (9.429), it can be shown that the MGF of γGSC can be rewritten in the following equivalent form: −Lc +1

MγGSC (s) = (1 − sγ )

L−L
c l=0

(−1)l 1+

 L L−Lc  Lc l Lc

l

− sγ

(9.430)

PDF of the Output SNR Having a simple expression for the MGF as given by (9.430), we are now in a position to derive the PDF of the GSC output combined SNR γGSC for an arbitrary Lc and L. Letting s = −p, the Laplace transform of the PDF of γGSC , namely, LγGSC (p), is related to the MGF of γGSC by LγGSC (p) = MγGSC (−p)

(9.431)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

475

which, using (9.430), can be written as L LγGSC (p) =

Lc Lc

γ

  

1 p+

1 γ

Lc +

L−L
c l=1

 p+

1 γ

  c (−1)l L−L l Lc −1  p+

 1+(l/Lc ) γ

 

(9.432)

Using the inverse Laplace transforms [171, p. 238, Eqs. (1) and (3)] as well as the identity Eq. (8.352.1) in Ref. 36, we obtain the PDF of γGSC in closed form as the inverse Laplace transform of (9.432):

"

γ Lc −1 e −γ / γ γ Lc (Lc − 1)!

Lc −1 L−L Lc 1
c Lc +l−1 L − Lc + (−1) e −(γ / γ ) γ l l l=1 

 L
c −2 1 −lγ m −lγ /(Lc γ ) × e − . m! Lc γ

L pγGSC (γ ) = Lc

(9.433)

m=0

As a check, note that (9.433) reduces to the well-known PDF of the SNR at an MRC (L = Lc ) output as given by (9.5) and SC (Lc = 1) output pγSC (γ ) =



" # L−1 L−1 (1 + l) γ L
exp − (−1)l l γ γ

(9.434)

l=0

Average Output SNR In this section, starting from the MGF of γGSC , we obtain the average combined SNR γ GSC at the GSC output. For this purpose we first introduce the “second” MGF of γGSC (using the terminology of Papoulis [120, Sect. 5.5]) or equivalently the cumulant generating function defined by γGSC (s) = ln(MγGSC (s))

(9.435)

which for the GSC after substituting of (9.429) in (9.435) is given as γGSC (s) = −Lc ln(1 − sγ ) −

L
l=Lc +1

sγ Lc ln 1 − l

(9.436)

We now use the well-known result that the first cumulant of γGSC is equal to its statistical average [120, (5-73), p. 117] γ GSC =

! dγGSC (s) !! ! ds s=0

(9.437)

476

PERFORMANCE OF MULTICHANNEL RECEIVERS

giving, after substituting (9.436) in (9.437) L
Lc γ l l=Lc +1   L
1 = 1 + Lc γ l

γ GSC = Lc γ +

(9.438)

l=Lc +1

which is the beautifully simple closed-form result originally obtained by Kong and Milstein [159, Eq. (7)]. In Appendix 9E we give an alternative simple and direct proof22 of the result in (9.438). It should be noted that since the MGF contains information about all the statistical moments of the underlying RV (and likewise for the cumulant generating function), it is then straightforward to obtain simple closedform expressions for the higher-order moments and cumulants directly from higherorder derivatives of (9.429) and (9.436), respectively. For example, the variance of γGSC , which is equal to the second cumulant of γGSC Eq. (5-73) in Ref.120 (p. 117), is given by   ! L
  d 2 γGSC (s) !! L c var γGSC = = 1 + Lc γ 2 ! ds 2 l2 s=0

(9.439)

l=Lc +1

in agreement with Eq. (18) of Ref. 164 independently derived by Win and Winters. We note, as pointed out in Ref. 159, that the result (9.438) generalizes the average SNR results for conventional SC and MRC. In particular, for the specific case of L = Lc (i.e., conventional MRC), it is easy to see that (9.438) reduces to the classical result given in (9.55). Similarly, for the particular case of Lc = 1 (i.e., conventional SC),  it is straightforward to see that (9.438) reduces to the well-known result γ SC = L l=1 (1/l)γ [4, Eq. (6.62), p. 327]. Figure 9.63 shows the normalized average combined SNR γ GSC / γ as a function of the number of available resolvable paths L for various values of the number of the Lc strongest combined paths. These results show that for a fixed number L of available diversity paths, diminishing diversity gain is obtained as the number of combined paths Lc increases. On the other hand, Fig 9.64 shows the normalized average combined SNR γ GSC / γ as a function of the number of strongest combined paths Lc for various values of the number of resolvable paths L. These curves indicate that for a fixed number of combined paths, a nonnegligible performance improvement can be gained by increasing the number of available diversity paths. Outage Probability The outage probability, Pout is defined as the probability that the GSC output SNR falls below a certain predetermined threshold SNR, γth , and 22 By “direct proof,” we mean that a proof that does not rely on Theorem 9.1 or equivalently on the transformation used in the appendix of Ref. 159.

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

477

5

4.5 Normalized average combined SNR per symbol [dB]

Lc = 4 4

Lc = 3 3.5

3

Lc = 2 2.5

2

Lc = 1 1.5

1

1

1.5

2 2.5 3 3.5 4 Number of available resolvable paths L

4.5

5

Figure 9.63 Normalized average combined SNR γ GSC / γ versus the number of available resolvable paths L for various values of the strongest combined paths Lc .

hence can be obtained by integrating the PDF of γGSC , which can be obtained in closed form with the help of [36, Eq. (3.351.1), p. 357] as

GSC Pout

L = Lc  ×

/

1 − e −γth / γ

L
c −1 l=0

1 − e −(1+l/Lc )(γth / γ ) 1 + l/Lc



Lc −1 L−L L − Lc Lc (γth / γ )l
c + (−1)Lc +l−1 l! l l l=1 0

 L
m c −2
−l m (γth / γ )k −γth / γ − 1−e . Lc k! m=0

k=0

(9.440) As a check, it is easy to see that when Lc = L, then (9.440) reduces to the well-known outage probability result for MRC [4, p. 329, Eq. (6.69)] MRC = 1 − e−(γth / γ ) Pout

L−1
l=0



γth γ

l!

l (9.441)

478

PERFORMANCE OF MULTICHANNEL RECEIVERS

5

L=5

Normalized average combined SNR per symbol [dB]

4.5

4

L=4 3.5

3

L=3

2.5

2

L=2 1.5 1

1.5 2 2.5 3 3.5 4 4.5 Number of strongest combined resolvable paths Lc

5

Figure 9.64 Normalized average combined SNR γ GSC / γ versus the number of the strongest combined paths Lc for various values of the number of the resolvable paths L.

In addition, for conventional SC (Lc = 1), (9.440) reduces to  

L−1 −(1+l)γth / γ
SC −γth / γ l L−1 1−e + (−1) Pout = L 1 − e l 1+l l=1

=L



L − 1 1 − e −(1+l)γth / γ (−1)l 1+l l

L−1
l=0

(9.442)

which can be easily shown using the identity in Ref. 172 (p. 171) and the binomial series expansion to be in agreement with the previously known result [4, Eq. (6.58), p. 326]  L SC = 1 − e−γth / γ (9.443) Pout as expected.

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

479

(L = 3)

Outage probability

100

10−2

(a) Lc = 1

10−4 (c) Lc = 3 −6

10

0

2

4

6

8

10

12

14

16

18

20

18

20

18

20

Normalized average SNR per symbol per path [dB] (L = 4) Average combined SNR per symbol [dB]

100

10−2 (a) Lc = 1 10−4 (d) Lc = 4 10−6

0

2

4

6

8

10

12

14

16

Normalized average SNR per symbol per path [dB] (L = 5)

Outage probability

100

10−2

10−4

10−6

(a) Lc = 1 (e) Lc = 5 0

2

4

6

8

10

12

14

16

Normalized average SNR per symbol per path [dB]

Figure 9.65 Outage probability PGSC out versus the normalized average SNR per path γ /γth [(a) Lc = 1 (SC), (b) Lc = 2, (c) Lc = 3, (d) Lc = 4, (e) Lc = 5]. GSC as function of the normalized Figure 9.65 shows the outage probability Pout average SNR per path γ /γth for various values of the available diversity paths L and strongest combined paths Lc . Notice again the diminishing returns as the GSC as function of γ /γth number of combined paths increases. Figure 9.66 shows Pout for a fixed Lc = 3 and L = 3, 4, 5. Clearly, these curves show that for fixed Lc a

480

PERFORMANCE OF MULTICHANNEL RECEIVERS

(Lc = 3) 100

10−1

10−2

Outage probability

10−3

10−4

(a) L = 3

10−5

10−6 (b) L = 4

10−7

10−8 (c) L = 5 10−9

10−10

0

5

10

15

20

25

30

Normalized average SNR per symbol per path [dB]

Figure 9.66 Outage probability PGSC out versus the normalized average SNR per path γ /γth for Lc = 3 [(a) L = 3, (b) L = 4, (c) L = 5].

significant decrease in the outage probability is obtained as the number of available diversity paths increases. Average Error Rate Using the closed-form expression for SNR MGF at the GSC output (9.430), we get the average BER as a single finite-range integral given by −1   L−Lc L  c π/2 (−1)l L−L 1 + Llc + singγ2 φ l=0 l L dφ (9.444) Pb (E) = c  Lc −1 π 0 1 + singγ2 φ

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

481

Switching the order of summation and integration, and defining the integral In (θ ; c1 , c2 ) as in (5A.42) where, in general, c1 and c2 are two constants (independent of φ) that might be different, we can rewrite the average BER as

L Pb (E) = Lc

L−L
c l=0

(−1)l



L−Lc 

1+

l l Lc

ILc −1

π gγ ; gγ , 2 1 + Ll

 (9.445)

c

Since the integrals In (θ ; c1 , c2 ) can be found in closed form (see Appendix 5A), (9.445) presents the final desired closed-form result. This result yields the same numerical results as in Eqs. (9) and (12) in Ref. 11 for the average BER of BPSK (g=1) with Lc = 2 and Lc = 3, respectively. Hence (9.445) [or equivalently (9.444)] is a generic expression valid for any Lc ≤ L. Similarly following the same steps as in (9.444)–(9.445), we obtain the average SER of M-PSK as   L−L  L−L L
c (−1)l l c gPSK γ (M − 1)π ; gPSK γ , ILc −1 Ps (E) = (9.446) M Lc 1 + Llc 1 + Llc l=0 The result (9.446) generalizes the M-PSK average SER results of Chennakeshu and Anderson [33] with MRC and conventional SC. For instance, for the particular case of L = Lc (i.e., MRC), it can be easily shown that (9.446) agrees with Eq. (21) of Ref. 33. Similarly, for the particular case of Lc = 1 (i.e., conventional SC), it can also be shown that (9.446) reduces to Eq. (26) of Ref. 33. Using the same steps as in (9.444)–(9.445), we obtain the average SER of square M-QAM as    



L−L
c (−1)l L−Lc π γ g 1 QAM l ILc −1 4 1− √ ; gQAM γ , 2 1 + Llc 1 + Llc M l=0   L−L 

L−L π 1 2
c (−1)l l c gQAM γ −4 1 − √ ; gQAM γ , ILc −1 4 1 + Llc 1 + Llc M l=0



L Ps (E) = Lc

(9.447) The result (9.447) generalizes the square M-QAM SER result given in (9.23) as well as those of Kim et al. [42,173,174] and Lu et al. [43] with conventional MRC and SC. For instance, for the particular case of L = Lc (i.e., MRC), it can be shown that (9.447) yields the same numerical results as Eq. (6) of Ref. 174, Eq. (15) of Ref. 42, or equivalently Eq. (12) of Ref. 43. Similarly, for the particular case of Lc = 1 (i.e., conventional SC), it can also be shown that (9.447) reduces to Eq. (23) of Ref. 43 or equivalently to Eq. (13) of Ref. 174. In Section 8.2.1.6, we considered the average SER performance of coherently detected 4-ary orthogonal signaling over a single Rayleigh fading channel. Here, as an additional example, we extend these results to the GSC case. More specifically, combining (9.430) with the MGF-based result for the SEP of coherent 4-ary

482

PERFORMANCE OF MULTICHANNEL RECEIVERS

orthogonal signaling as given in (8.138) with Mγt (s) replaced by MγGSC (s), namely Ps (E) =

1 π



π/2

MγGSC −



3

dθ 4 sin2 θ

"  5π/6 3 sin θ 3  (9.448) + MγGSC − 4 + 2 sin2 θ π/6 2π 2 + sin2 θ      1 π/2 sin2 θ 3 MγGSC − − 1+ dφ dθ π 0 4 + 2 sin2 θ 2 sin2 φ 0

we obtain a single-integral form of the average SEP for this modulation with GSC diversity. To evaluate this average SER of (9.448), we start by first evaluating the inner integral of the second term, namely 1 I (θ ) = π





π/2

MγGSC −

0



3 4 + 2 sin2 θ

1+

sin2 θ

 dφ

2 sin2 φ

(9.449)

for the MRC case and show that it can be obtained in closed form, thereby allowing the average SER to be obtained in single-integral form. For Lc = L, we can rewrite (9.449) in the form

I (θ ) =

1 π





L 

4 + 2 sin2 θ 4 + 2 sin θ + 3γ 2

L

π/2  0

 

sin2 φ sin2 φ +

3γ sin2 θ

  2 4+2 sin2 θ +3γ

  dφ (9.450) 

The integral in (9.450) can be evaluated in closed form with the help of Eq. (5A.4b) of Appendix 5A, resulting in  I (θ ) =

4 + 2 sin2 θ 4 + 2 sin2 θ + 3γ

×

L−1
k=0

where µ (c) =

√

L

L−1+k k



1 − µ (c1 (θ )) 2

1 + µ (c1 (θ )) 2

L

k (9.451)

c/ (1 + c) and c1 (θ ) =

3γ sin2 θ  2 4 + 2 sin2 θ + 3γ 

(9.452)

483

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

Following a similar procedure, we can also evaluate the inner integral of (9.449) for the more general GSC case in closed form. In particular, using (9.430) for the MGF we get 



3

sin2 θ



1+ 4 + 2 sin2 θ 2 sin2 φ Lc −1 L−L


c 4 + 2 sin2 θ L − Lc L k = (−1) k Lc 4 + 2 sin2 θ + 3γ

MγGSC −

k=0

 × 

4 + 2 sin θ  4 + 2 sin2 θ (1 + k/Lc ) + 3γ 2



  

Lc −1 2

sin φ sin2 φ +



  × 



sin φ +

 3γ  2 4+2 sin2 θ +3γ

  

  

(9.453)

3γ sin2 θ   2 4 + 2 sin2 θ (1 + k/Lc ) + 3γ

(9.454)

sin2 φ 2

sin2 θ

3γ  sin2 θ   2 4+2 sin2 θ (1+k/Lc )+3γ

Letting c2 (θ ; k) =



then each term of the sum in (9.453) will contribute an integral of the form 1 Ik (θ ) = π



π/2



L−1 

sin2 φ sin2 φ + c1 (θ )

0



sin2 φ

dφ

sin2 φ + c2 (θ ; k)

(9.455)

However, integrals of the form in (9.455) can be evaluated in closed form using the results in Eq. (5A.58) of Appendix 5A with m1 = L − 1, m2 = 1. Thus, the average SER for 4-ary orthogonal modulation can be expressed in single-integral form by combining all the above. The specific results are given below. Using (9.430), the first term of (9.448) becomes 1 I1 = π

 0

×

π/2

MγGSC −

1 1 + (k/Lc )



3



4 sin2 θ

1 dθ = π

sin2 θ sin2 θ + 3γ /4



π/2 0

Lc −1 



L Lc

L−L
c

(−1)

k=0

sin2 θ sin2 θ +

3γ 4(1+k/Lc )

k

L − Lc k



 dθ

(9.456)

484

PERFORMANCE OF MULTICHANNEL RECEIVERS

Applying Eq. (5A.58) for k > 0 with m = Lc − 1, c1 = 3γ /4, c2 = 3γ /[4(1 + k/N )] and Eq. (5A.35) for k = 0 with c1 = c2 = c = 3γ /4, (9.456) evaluates to 

/



Lc −1 L−L
c 1 3γ
1 2l L − Lc k + I1 = 1− (−1) k l 2 4 + 3γ (4 + 3γ )l l=0 k=1  



1 k 3γ 1 − 1 + × 1 − 1 + k/Lc (−k/Lc )Lc −1 4 (1 + k/Lc ) + 3γ Lc

0 L
c −2 k l 3γ − × Il (9.457) Lc 4

L Lc

l=0

Similarly, for the second term of (9.448), we get Lc −1

 4 + 2 sin2 θ 3 sin θ L I2 =  Lc 4 + 2 sin2 θ + 3γ π/6 2π 2 + sin2 θ 

 L−L
c 1 L − Lc k × (−1) k 1 + Lkc k=0   4 + 2 sin2 θ × dθ 3γ 4 + 2 sin2 θ + 1+k/L c 

5π/6

(9.458)

Finally, for the third term of (9.448), we get Lc −1

 1 3 sin θ 4 + 2 sin2 θ L I3 = −  Lc 2 4 + 2 sin2 θ + 3γ π/6 2π 2 + sin2 θ  /  

Lc −1 1 c1 (θ )
4 + 2 sin2 θ 2l × 1− l 1 + c1 (θ ) [4 (1 + c1 (θ ))]l 4 + 2 sin2 θ + 3γ l=0  

 L−L
c 4 + 2 sin2 θ 1 L − Lc k + (−1) 3γ k 1 + Lkc 4 + 2 sin2 θ + 1+k/L k=1 c   c1 (θ ) 1 c2 (θ ; k) − × Lc −1 1 − 1 + c2 (θ ; k) c2 (θ ; k) c1 (θ) 1 − c2 (θ;k) 0

L
c −2 c1 (θ ) 1− Il (c1 (θ )) d θ × (9.459) c2 (θ ; k) 

5π/6

l=0

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

485

For the MRC case, the results of (9.457), (9.458) and (9.459) simplify, respectively, to     L L−1 k  
L−1+k 1 3γ 3γ 1 I1 = 1− 1+ k 2 4 + 3γ 2 4 + 3γ k=0

(9.460) 

5π/6

I2 =

π/6



5π/6

=

3 sin θ  2π 2 + sin2 θ 3 sin θ  2π 2 + sin2 θ

MγMRC −



3

4 + 2 sin2 θ L 4 + 2 sin2 θ



dθ

dθ (9.461) 4 + 2 sin2 θ + 3γ     5π/6  3 sin θ 1 π/2 sin2 θ 3 I3 = − MγMRC − 1+ dφ dθ  4+2 sin2 θ 2 sin2 φ π/6 2π 2 + sin2 θ π 0 L 

 5π/6 1 − µ (c1 (θ )) L 3 sin θ 4 + 2 sin2 θ  =− 2 2 π/6 2π 2 + sin2 θ 4 + 2 sin θ + 3γ π/6

×

L−1
k=0

L−1+k k



1 + µ (c1 (θ )) 2

k dθ

(9.462)

Finally, using a somewhat different approach, Haghani and Beaulieu [175] arrive at an equivalent expression to (9.448), or equivalently, the sum of (9.457), (9.458), and (9.459), involving double integrals, which is given by    5π/6  L 3 sin θ 1 π/2 4 sin2 φ dφ + Ps (E) =  2 π 0 4 sin φ + 3bl π/6 2π 2 + sin2 θ l=1     L 3 sin θ 4 + 2 sin2 θ 1 5π/6 π/2 × d θ −  2 π 4 + 2 sin θ + 3b π/6 0 l 2π 2 + sin2 θ l=1     L 2 sin2 φ 4 + 2 sin2 θ ×     dφ dθ 2 sin2 φ 4 + 2 sin2 θ + 3 sin2 θ + 2 sin2 φ bl l=1 (9.463) where   γ, bl =  γ Lc , l

l = 1, 2, . . . , Lc l = Lc + 1, Lc + 2, . . , L

(9.464)

As some numerical examples, Figs. 9.67–9.72 show the effects of Lc and L on the average error rate of BPSK, 8-PSK, and 16-QAM. These curves confirm

486

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Bit Error Rate Pb(E)

(L = 3) 100

10−2 (a) Lc = 1 (c) Lc = 3

10−4

10−6

0

2

4

6

8

10

12

14

16

18

20

16

18

20

16

18

20

Average SNR per symbol per path [dB]

Average Bit Error Rate Pb(E)

(L = 4) 100 10−2 (a) Lc = 1 10−4

(d) Lc = 4

10−6 10−8 10−10

0

2

4

6

8

10

12

14

Average SNR per symbol per path [dB]

Average Bit Error Rate Pb(E)

(L = 5) 100

10−2 (a) Lc = 1

10−4 (e) Lc = 5 10−6

0

2

4

6

8

10

12

14

Average SNR per symbol per path [dB] Figure 9.67 Average BER of BPSK versus the average SNR per path γ [(a) Lc = 1 (SC), (b) Lc = 2, (c) Lc = 3, (d) Lc = 4, (e) Lc = 5].

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

487

100

10−1

Average Bit Error Rate Pb(E )

10−2

10−3 (a) L = 3

10−4 (b) L = 4

10−5

10−6

(c) L = 5

0

2

4

6 8 10 12 14 Average SNR per symbol per path [dB]

16

18

20

Figure 9.68 Average BER of BPSK versus the average SNR per path γ for Lc = 3 [(a) L = 3, (b) L = 4, (c) L = 5].

previous trends in the sense that diminishing returns are obtained as the number of strongest combined paths increases but a significant performance improvement can be gained by increasing the number of available diversity paths. SNR Penalty In this subsection, we assess the penalty of GSC with respect to MRC, which is defined as the increase in average fading SNR required for the former to achieve the same average SER as the latter. For the case of i.i.d. Rayleigh

488

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Symbol Error Rate Ps(E)

(L = 3) 100

10−2

(a) Lc = 1 (c) Lc = 3

10−4

10−6

0

2

4

6

8

10

12

14

16

18

20

Average SNR per symbol per path [dB]

Average Symbol Error Rate Ps(E)

(L = 4) 100

10−2 (a) Lc = 1 (d) Lc = 4

10−4

10−6

0

2

4

6

8

10

12

14

16

18

20

Average SNR per symbol per path [dB]

Average Symbol Error Rate Ps(E)

(L = 5) 100

10−2 (a) Lc = 1 (e) Lc = 5

10−4

10−6

0

2

4

6

8

10

12

14

16

18

20

Average SNR per symbol per path [dB] Figure 9.69 Average SER of 8-PSK versus the average SNR per symbol per path γ [(a) Lc = 1 (SC), (b) Lc = 2, (c) Lc = 3, (d) Lc = 4, (e) Lc = 5].

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

489

100

Average Symbol Error Rate Ps(E )

10−1

10−2 (a) L = 3

10−3

(b) L = 4 10−4

(c) L = 5

10−5

10−6

0

2

4

6 8 10 12 14 Average SNR per symbol per path [dB]

16

18

20

Figure 9.70 Average SER of 8-PSK versus the average SNR per symbol per path γ for Lc = 3 [(a) L = 3, (b) L = 4, (c) L = 5].

fading channels, this problem has been investigated in several papers [176–178]. In particular, both upper and lower bounds as well as low and high average SNR asymptotes on the SNR penalty were obtained initially for M -PSK modulation [176] and then later for arbitrary two-dimensional signaling constellations with polygonal decision regions [177,178]. We present here, without proof, the results of these investigations.

490

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Symbol Error Rate Ps(E )

(L = 3) 100 (a) Lc = 1

10−2

(c) Lc = 3 10−4

10−6

0

2

4

6

8

10

12

14

16

18

20

Average SNR per symbol per path [dB]

Average Symbol Error Rate Ps(E )

(L = 4) 100

10−2

(a) Lc = 1

10−4

10−6

(d) Lc = 4

0

2

4

6

8

10

12

14

16

18

20

Average SNR per symbol per path [dB]

Average Symbol Error Rate Ps(E )

(L = 5) 100

10−2

(a) Lc = 1

10−4 (e) Lc = 5 10−6

0

2

4

6

8

10

12

14

16

18

20

Average SNR per symbol per path [dB] Figure 9.71 Average SER of 16-QAM versus the average SNR per symbol per path γ [Lc = 1 (SC), (b) Lc = 2, (c) Lc = 3, (d) Lc = 4, (e) Lc = 5].

491

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

(Lc = 3) 100

Average Symbol Error Rate Ps(E )

10−1

10−2 (a) L = 3

10−3

(b) L = 4

(c) L = 5 10−4

10−5

10−6

0

2

4

6

8

10

12

14

16

18

20

Average SNR per symbol per path [dB]

Figure 9.72 Average SER of 16-QAM versus the average SNR per symbol per path γ for Lc = 3 [(a) L = 3, (b) L = 4, (c) L = 5].

In mathematical terms, the SNR penalty β, is defined as Ps (E; βγ ) |GSC = Ps (E; γ ) |MRC

(9.465)

where Ps (E; βγ ) |GSC is the SER of GSC at average SNR βγ and Ps (E; γ ) |MRC is the SER of MRC at average SNR γ . Solving (9.465) for β, we obtain the penalty β (γ ) =

1 −1 P (Ps (E; γ ) |MRC ) |GSC γ s

(9.466)

492

PERFORMANCE OF MULTICHANNEL RECEIVERS

which, in general, is a function of average SNR. For small SNR, it has been shown that the asymptotic penalty of (9.466) is given by "

βLA

K (L, L) = K (Lc , L)

#2 (9.467)

where 1 K (Lc , L) = 2π



∞





1 −

0

u2 1 + u2

Lc

L n=Lc +1



u2 Lc L

+ u2

  du

(9.468)

For large SNR, the analogous result for the SNR penalty is " βUA =

L!

#1/L (9.469)

c Lc !LL−L c

Note that the asymptotic penalties in (9.468) and (9.469) are independent of the average SNR, which implies that the horizontal separation between the SER performance curves of GSC and MRC approaches a fixed amount at these two extremes. 9.11.2.2

Non-I.I.D. Rayleigh Case

GSC Input Joint PDF The generalization of (9.419) to the case of independent but nonidentically distributed fading paths is given by   pγ1:L ,γ2:L ,...,γLc :L γ1:L , γ2:L , . . . , γLc :L L


=

pγi1 (γ1:L ) pγi2 (γ2:L ) . . . pγiL

 c

γLc :L



i1 ,i2 ...,iLc =1 i1 =i2 =...=iLc

×

L

  Pγij γLc :L

(9.470)

j =Lc +1 ij =i1 ,i2 ,...,iLc

where now 1 −γ /γ k e γk

(9.471)

Pγk (γ ) = 1 − e −γ /γ k

(9.472)

pγk (γ ) = and

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

493

In what follows, we shall be interested only in the joint PDF of (9.470) when Lc = L. Making this substitution, the product term in (9.470) becomes equal to unity and thus pγ1:L ,γ2:L ,...,γL:L (γ1:L , γ2:L , . . . , γL:L ) L


=

pγi1 (γ1:L ) pγi2 (γ2:L ) . . . pγiL (γL:L )

i1 ,i2 ...,iL =1 i1 =i2 =...=iL L


=

i1 ,i2 ...,iL =1 i1 =i2 =...=iL

  L 1 γl:L exp − γ il γ il

(9.473)

l=1

MGF of the Output SNR In treating the non-i.i.d. case, we still use the identical linear transformation as in (9.424); however, the xi values will now not be strictly independent. Nevertheless, each term of the joint PDF of the xi values can still be written as a product of exponentials similar to those in (9.425), and thus the MGF of the combiner output will still have a simple solution. Inverting the transformation in (9.424) and substituting the result together with (9.425) into (9D.3) gives px1 ,x2 ,...,xL (x1 , x2 , . . . , xL ) =

L
i1 ,i2 ...,iL =1 i1 =i2 =...=iL

=

L
i1 ,i2 ...,iL =1 i1 =i2 =...=iL

  L L 1 1
exp − xk γ il γ il k=l l=1

  L l
1 exp −xl γ −1 ik γ il l=1

(9.474)

k=1

Substituting (9.474) into (9.427), we are still able to obtain the MGF as a multiple sum of a product of separable integrals that is now given by23  MγGSC (s) =



∞

∞

... 0

0

px1 ,x2 ,...,xL (x1 , . . . , xL )

   × exp s x1 + 2x2 + · · · + Lc xLc + Lc xLc +1 + · · · + Lc xL d x1 · · · d xL

23 This equation corrects some minor typographic errors that appeared in Eq. (9) of Ref. 162 (without altering any of the subsequent results) as pointed out to the authors by P. R. Sahu and A. K. Chaturvedi of the Indian Institute of Technology, Kanpur, India.

494

PERFORMANCE OF MULTICHANNEL RECEIVERS Lc 

L


=

i1 ,i2 ...,iL =1 i1 =i2 =...=iL

 L

×

∞

n=1 0

∞

m=Lc +1 0

 γ −1 in exp

−xn

 n


 γ −1 ik

− ns

d xn

k=1

 γ −1 im exp −xm

 m


 γ −1 ik − Lc s

d xm

(9.475)

k=1

The integrals in (9.475) are easily evaluated, leading to the compact result L


L

i1 ,i2 ...,iL =1 i1 =i2 =...=iL

l=1

MγGSC (s) =

γ −1 il

 l


−1 γ −1 ik − min (l, Lc ) s

(9.476)

k=1



Note that, for γ 1 = γ 2 = . . . = γ L = γ , (9.476) simplifies to (9.429). Average Output SNR The average SNR of the combiner output can be easily obtained from the first derivative of MγGSC (s) evaluated at s = 0. Differentiating (9.476) with respect to s and evaluating the result at s = 0, we get γ GSC

! d MγGSC (s) !! = !s=0 ds =

L
i1 ,i2 ...,iL =1 i1 =i2 =···=iL

=

L
i1 ,i2 ...,iL =1 i1 =i2 =···=iL

! L γ −1 ! d il !s=0 l ! −1 ds k=1 γ i − min (l, Lc ) s l=1

k

  l  j −1  L −1 L


−1 −1  γ −1  γ min (l, Lc ) γ j =1

ij

im

m=1

ik

l=1

k=1

(9.477) which is equivalent to the more cumbersome form obtained by Kong and Milstein [160]. Average SER As an example, to obtain the average SER of M -PSK with GSC and non-i.i.d. Rayleigh fading, we start from 1 Ps (E) = π

 0

(M−1)π/M

gPSK dθ MγGSC − 2 sin θ

(9.478)

495

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

Substituting (9.475) in (9.478) and incorporating the multiplicative factor into the integrand gives the desired result L


Ps (E) =



1 π

i1 ,i2 ...,iL =1 i1 =i2 =···=iL

L (M−1)π/M 0

l

γ −1 il

−1 k=1 γ ik

l=1

(9.479)





 × gPSK min (l, Lc )

sin2 θ  l

−1 k=1 γ ik

−1

+ sin θ 2

  dθ

which agrees with a comparable form in [163, Eq. (23)] obtained from a modified version of the simple transformation in (9.424). For the non-i.i.d. MRC result, we let Lc = L in (9.479), resulting in Ps (E) =

L
i1 ,i2 ...,iL =1 i1 =i2 =...=iL

1 π



L (M−1)π/M 0

l

l=1

γ −1 il

−1 k=1 γ ik





2

 × gPSK l

 l

sin θ

−1 k=1 γ ik

−1

+ sin2 θ

 d θ

(9.480)

which is in agreement with the result obtained from (9.15), namely    L 1 (M−1)π/M sin2 θ Ps (E) = dθ π 0 gPSK γ i + sin2 θ l=1

(9.481)

although the analytical proof of this equivalence appears difficult to show for arbitrary L. The compact single integral expressions in (9.479) and (9.480) can also be obtained in closed form using the results of paragraphs 9 of Appendix 5A [Eqs. (5A.70)–(5A.75) and related text]. Specifically, we first rewrite (9.479) as L  L
γ −1 il Ps (E) = l −1 k=1 γ ik l=1 i1 ,i2 ...,iL =1 i1 =i2 =···=iL

×

1 π

 0





L (M−1)π/M l=1

2

  gPSK min (l, Lc )

sin θ  l

−1 k=1 γ ik

−1

+ sin θ 2

 d θ (9.482)

496

PERFORMANCE OF MULTICHANNEL RECEIVERS

Next we expand the product in the integrand into a partial-fraction expansion. In particular, letting

cl = gPSK min (l, Lc )

 l


−1 γ −1 ik

(9.483)

k=1

then, in view of the distinctness of the average branch SNRs, making use of (5A.72), we obtain L

L


Ps (E) =

i1 ,i2 ...,iL =1 i1 =i2 =···=iL

1 × π



l

L L


−1 k=1 γ ik

l=1

(M−1)π/M



γ −1 il



l=1



k=1 k=l



sin2 θ

dθ

sin2 θ + cl

0

cl cl − ck

(9.484)

Finally, using (5A.4a) to evaluate the integrals in (9.484) gives the desired result:  L

L


Ps (E) =

i1 ,i2 ,...,iL =1 i1 =i2 =···=iL

l



γ −1 il

−1 k=1 γ ik

l=1



 L 
 L cl     cl − ck   l=1  k=1 k=l

.

1 .

#4

" M −1 M cl cl π π 1− +tan−1 cot M 1+cl (M − 1) π 2 1 + cl M (9.485) The corresponding result for MRC is also obtained from (9.485), except that cl is now given by ×

cl = gPSK l

 l


−1 γ −1 ik

(9.486)

k=1

PDF and CDF of the Output SNR To obtain the PDF of the combiner output, it is convenient to express (9.476) as a partial-fraction expansion in the same manner as was done for the average SEP in the last section. Specifically MγGSC (s) =

L
i1 ,i2 ...,iL =1 i1 =i2 =...=iL

L l=1

γ −1 il min (l, Lc )



L
j =1

 −1 C j αj − s

(9.487)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

497

where j αj =

−1 k=1 γ ik

min (j , Lc )

L 

Cj =

,

αn − αj

−1

(9.488)

n=1 n=j

Then taking the inverse Laplace transform of (9.487), the PDF of γGSC is given by pγGSC (γGSC ) =

L

L


l=1

i1 ,i2 ...,iL =1 i1 =i2 =···=iL



γ −1 il min (l, Lc )

L


Cj e −αj γGSC

(9.489)

j =1

and the corresponding CDF obtained by integration of (9.489) is given by L


PγGSC (γGSC ) =

i1 ,i2 ,...,iL =1 i1 =i2 =···=iL

L l=1



γ −1 il min (l, Lc )

L
Cj  j =1

αj

1 − e −αj γGSC



(9.490)

Outage Probability Since the outage probability Pout is equal to the probability that the combiner output SNR falls below a threshold γth , then to compute this quantity for non-i.i.d. GSC with Rayleigh fading, we merely substitute γth for γGSC in (9.490). As an example, for the case of an exponentially decaying power delay profile, γ l = γ 1 exp (−δ (l − 1)) , l = 1, 2, . . . , L, the product factor and exponent in (9.490) become L

γ −1 il

l=1

min (l, Lc )

=

δ[L(L−1)]/2 γ −L 1 e c Lc !LL−L c

−j

,

αj =

γ1

j

k=1 e

δ(ik −1)

min (j , Lc )

(9.491)

As a numerical example, Fig. 9.73 illustrates the outage probability for GSC as computed from (9.490) together with (9.491) for L = 5, δ = 0.2 and various values of Lc . 9.11.2.3 I.I.D. Nakagami-m Case The “spacing” technique for ordered exponential RVs, which allows the necessary partitioning of the integrand for the MGF-based approach to be successfully applied, does not carry over to gamma-distributed variables, which are characteristic of the instantaneous SNR per path for Nakagami-m fading. Thus, an alternative approach is required to obtain analogous generic results for such channels. A partial solution to this problem was provided by the authors in a 1999 paper [179] presenting a performance analysis of two specific hybrid SC/MRC receivers, namely, SC/MRC-2/3 and SC/MRC-2/4. The final result for the average

498

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

10−1

Outage Probability

10−2

Lc = 1

10−3

Lc = 2

Lc = 3 Lc = 4

10−4

Lc = 5 10−5

10−6 0

5

10

15

Normalized Average SNR per Path [dB] Figure 9.73 Outage probability of GSC versus normalized average SNR of the first path γ 1 /γth over a Rayleigh fading channel with exponentially decaying power delay profile (L = 5 and δ = 0.2).

BER was shown to be expressible in terms of infinite series of hypergeometric functions suitable for numerical evaluation. However, the method used in that paper [179] does not allow for similar simplifications for the case of Lc > 2, and hence the analyses are not amenable to application to other SC/MRC receivers.

499

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

Furthermore, it is limited to binary coherent modulations and as such does not apply to the performance of SC/EGC or M-ary modulations such as M-PSK and M-QAM. In this section, applying the Dirichlet transformation [180] (a well-known technique found in classical textbooks on integral calculus [181, p. 492; 182, Ch. XXV] to simplify certain multiple integrals), we develop a well-structured procedure that allow us to obtain the average error rate for arbitrary L and Lc , and which is applicable for the performance analysis of not only SC/MRC with M-ary modulations but also SC/EGC receivers [183]. Specific results will be presented for a number of examples both numerically and as simple closed-form expressions. These results will be compared with the particular results corresponding to Rayleigh fading. As a byproduct of the general results, some interesting closed-form expressions will be presented in Appendix 9F for certain single and multiple definite integrals that heretofore appear not to have been reported in standard tabulations such as those in Refs. 36 and 53. Note that other approaches that solve for the performance of GSC in Nakagami-m environments are presented in Refs. 184 and 185.

GSC Input Joint Statistics Let βl = mγl / γ denote the normalized instantaneous SNR for the lth channel with PDF pβl (βl ) =

βlm−1 exp (−βl ) , (m)

βl ≥ 0

(9.492)

Consider β1:L ≥ β2:L ≥ · · · ≥ βL:L : the order statistics obtained by arranging the L L set {βl }L l=1 . Since the {αl }l=1 are assumed to be i.i.d. RVs, so are the {γl }l=1 and L L the {βl }l=1 , and the joint PDF of the {βl:L }l=1 is then given by [120, p. 185; 157, Eq. (9)] pβ1:L ,...,βL:L (β1:L , . . . , βL:L ) = L!

L

β1:L ≥ β2:L ≥ · · · ≥ βL:L ≥ 0

pβ (βl:L );

l=1

(9.493) which can be rewritten after substitution of (9.492) in (9.493) as L! pβ1:L ,...,βL:L (β1:L , . . . , βL:L ) = ( (m))L

L

m−1 βl:L

 exp −

l=1

L


 βl:L ;

l=1

β1:L ≥ β2:L ≥ · · · ≥ βL:L ≥ 0

(9.494)

MGF of GSC Output In this section, we first summarize the procedure proposed by Kabe [180] to obtain the MGF of any linear function of ordered gamma variates. We then show how this result can be used to get the MGF of the GSC output for an arbitrary Lc and L. In order to take advantage of the results of Kabe, let us first follow his notation and reorder the RVs from weakest to strongest by defining {xi:L }L i=1 as the reverse

ordered set corresponding to {βl }L l=1 , namely, xi:L = βL−i+1:L , i = 1, 2, . . . , L with

500

PERFORMANCE OF MULTICHANNEL RECEIVERS

the joint PDF obtained from (9.494) as L! px1:L ,...,xL:L (x1:L , . . . , xL:L ) = ( (m))L

L



m−1 xi:L

exp −

i=1

L


 xi:L ;

i=1

0 ≤ x1:L ≤ x2:L ≤ · · · ≤ xL:L The MGF of

L

 Ex1:L ,···,xL:L e

i=1 ui xi:L ,

s

L

i=1 ui

xi:L

(9.495)

where the {ui }L i=1 are constants, is given by 

 L   x3:L  x2:L
= ··· exp s ui xi:L 0 0 i=1 0   L-fold × px1:L ,...,xL:L (x1:L , . . . , xL:L ) d x1:L d x2:L · · · d xL:L m−1  x3:L  x2:L   ∞ L L! = ··· xi:L ( (m))L 0 0 0    i=1 L-fold  L 
× exp − (1 − ui s)xi:L d x1:L d x2:L · · · d xL:L 

∞

i=1

(9.496) which can be viewed as a multiple integral of the Dirichlet–Louiville type [182, Ch. XXV]. The difficulty in evaluating this L-fold integral is that the integration limits are functions of the integration variables themselves. This is where the Dirichlet transformation steps in to simplify the problem. In particular, using the following transformation [180, Eq. (2.2)] xi:L =

L

θl

(9.497)

l=i

so that 0 < θl < 1 for l = 1, 2, . . . , L − 1 and θL > 0, (9.496) can be rewritten as   1  1  ∞ L  L  L! lm−1 s i=1 ui xi:L = ··· θl Ex1:L ,...,xL:L e ( (m))L 0 0 0 l=1    (L−1)-fold   × exp −θL D(s; θ1 , θ2 , . . . , θL−1 ) dθL dθL−1 · · · dθ1 , (9.498) where each integral now has limits that are independent of the integration variables and the function D(s; θ1 , θ2 , . . . , θL−1 ) is defined by D(s; θ1 , θ2 , . . . , θL−1 ) = (1 − uL s) + θL−1 (1 − uL−1 s) + θL−1 θL−2 (1 − uL−2 s) + · · · + θL−1 θL−2 · · · θ1 (1 − u1 s).

(9.499)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

501

Note that D(s; θ1 , θ2 , . . . , θL−1 ) is independent of θL ; thus, the first integration in (9.498), specifically, the one on θL , is of the form 

∞ 0

  θLLm−1 exp −θL D (s; θ1 , θ2 , . . . , θL−1 ) dθL

(9.500)

which has the closed-form result [36, Eq. (3.381.4)] 

∞

  θLLm−1 exp −D (s; θ1 , θ2 , . . . , θL−1 )θL dθL =

(Lm) (D (s; θ1 , θ2 , . . . , θL−1 ))Lm 0 (9.501) Thus, applying (9.501) to (9.498) immediately reduces the L-fold integral to an L − 1-fold integral given by < L Ex1:L ,...,xL:L es i=1 ui L! (Lm) = ( (m))L



1

xi:L



= 1

···  0  0  (L−1)-fold

L−1

 θllm−1

(D (s; θ1 , θ2 ,. . . ,θL−1 ))−Lm dθL−1 · · · dθ1

l=1

(9.502) where each integral has finite limits that are independent of the integration variables. We now use Kabe’s procedure to derive the MGF of the total output SNR γGSC , which can be expressed in terms of the {xi:L }L i=1 as

γGSC =

Lc


γl:L =

l=1

c
γ
βl:L = ui xi:L m

L

L

l=1

i=1

(9.503)

where the weights {ui }L i=1 are defined by u1 = u2 = · · · = uL−Lc = 0 uL−Lc +1 = uL−Lc +2 = · · · uL =

γ m

(9.504)

Hence, using the result (9.502), the MGF of the total combined SNR can be written as L! (Lm) MγGSC (s) = ( (m))L



1



1

···  0  0  (L−1)-fold

L−1

 θllm−1

l=1

× (D (s; θ1 , θ2 , . . . , θL−1 ))−Lm dθL−1 · · · dθ1

(9.505)

502

PERFORMANCE OF MULTICHANNEL RECEIVERS

where D (s; θ1 , θ2 , . . . , θL−1 ) is found after the substitution of the weights of (9.504) in (9.499), giving D (s; θ1 , θ2 , . . . , θL−1 )

γ = 1 − s (1 + θL−1 + θL−1 θL−2 + · · · + θL−1 θL−2 · · · θL−Lc +2 θL−Lc +1 ) m + θL−1 θL−2 · · · θL−Lc +1 θL−Lc + · · · + θL−1 θL−2 · · · θ2 + θL−1 θL−2 · · · θ1 . (9.506) In view of the form of D (s; θ1 , θ2 , . . . , θL−1 ) in (9.506), the next integrations (i.e., the ones on θL−1 , θL−2 , . . .) can be computed in a recursive fashion [183]. Numerical Examples With the MGF of the SNR output in hand, we can compute the average SER for a wide variety of modulation schemes. As examples, Figs. 9.74–9.79 show the effects of Lc and L on the average error rate of BPSK, 8-PSK, and 16-QAM, for various values of the fading parameter m. The curves for m = 1 are in agreement with the Rayleigh fading results, as expected. Furthermore, results for BPSK with SC/MRC-2/3 and SC/MRC-2/4 match the results reported in Ref. 179. These numerical results confirm trends observed for Rayleigh fading in the sense that diminishing returns are obtained as the number of combined paths increases but a significant performance improvement can be gained by increasing the number of available diversity paths. In addition, Figs. 9.74, 9.76, and 9.78 (in which the number of available diversity paths L is fixed to 4 and the number of combined paths Lc is varied from 1 to 4) indicate that the more severe is the fading (i.e., the lower is the fading parameter m), the more diminishing are the returns obtained for an increasing number of combined paths. On the other hand, Figs. 9.75, 9.77, and 9.79 (in which the number of combined paths Lc is fixed to 2 and the number of available diversity paths L is varied from 2 to 4) show that the performance improvement gained by increasing the number of available diversity paths is more important for channels subject to a low amount of fading. 9.11.2.4 Partial-MGF Approach We have already observed that the evaluation of the MGF in (9.423) requires the solution of an Lc -fold nested integral. Specifically, for the i.i.d. case, using (9.419) for the joint PDF of the ordered SNRs, we obtain

 ∞   L−Lc   L Pγ γLc :L MγGSC (s) = L! pγ γLc :L e sγLc :L d γLc :L Lc 0  ∞  ∞   × pγ γLc −1:L e sγLc −1:L d γLc −1:L . . . pγ (γ1:L ) e sγ1:L d γ1:L γLc :L

γ2:L

(9.507)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

503

Average Bit Error Rate Pb(E )

(m = 0.5) 100 10−2 10−4

a

10−6

d 10−8 10−10

0

5

10

15

20

25

30

25

30

25

30

Average SNR per Bit per Path [dB]

Average Bit Error Rate Pb(E )

(m = 1) 100 10−2 10−4 10−6

a 10−8

d 10−10

0

5

10

15

20

Average SNR per Bit per Path [dB]

Average Bit Error Rate Pb(E )

(m = 2) 100 10−2 10−4 10−6

a

10−8 10−10

d 0

5

10

15

20

Average SNR per Bit per Path [dB] Figure 9.74 Average BER of BPSK versus the average SNR per bit path γ for L = 4 and (a) Lc = 1 (SC), (b) Lc = 2, (c) Lc = 3, and (d) Lc = 4 (MRC).

504

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Bit Error Rate Pb(E )

(m = 0.5) 100 10−2

a 10−4

b c

10−6 10−8 10−10

0

5

10

15

20

25

30

Average SNR per Bit per Path [dB]

Average Bit Error Rate Pb(E )

(m = 1) 100 10−2 10−4

a

10−6

b c

10−8 10−10

0

5

10

15

20

25

30

25

30

Average SNR per Bit per Path [dB]

Average Bit Error Rate Pb(E )

(m = 2) 100 10−2 10−4 10−6

a

10−8 10−10

c 0

5

10

b 15

20

Average SNR per Bit per Path [dB] Figure 9.75 Average BER of BPSK versus the average SNR per bit per path γ for Lc = 2 and (a) L = 2, (b) L = 3, and (c) L = 4.

505

Average Symbol Error Rate Ps(E )

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

(m = 0.5) 100 10−2 10−4

a d

10−6 10−8 10−10

0

5

10

15

20

25

30

Average Symbol Error Rate Ps(E )

Average SNR per Symbol per Path [dB] (m = 1) 100 10−2 10−4 10−6

a d

10−8 10−10

0

5

10

15

20

25

30

25

30

Average Symbol Error Rate Ps(E )

Average SNR per Symbol per Path [dB] (m = 2) 100 10−2 10−4 10−6

a 10−8 10−10

d 0

5

10

15

20

Average SNR per Symbol per Path [dB] Figure 9.76 Average SER of 8-PSK versus the average SNR per symbol per path γ for L = 4 and (a) Lc = 1 (SC), (b) Lc = 2, (c) Lc = 3, and (d) Lc = 4 (MRC).

Average Symbol Error Rate Ps(E )

506

PERFORMANCE OF MULTICHANNEL RECEIVERS

(m = 0.5) 100 10−2

a b

10−4

c

10−6 10−8 10−10

0

5

10

15

20

25

30

Average Symbol Error Rate Ps(E )

Average SNR per Symbol per Path [dB] (m = 1) 100 10−2 10−4

a

10−6

b c

10−8 10−10

0

5

10

15

20

25

30

Average Symbol Error Rate Ps(E )

Average SNR per Symbol per Path [dB] (m = 2) 100 10−2 10−4 10−6

a

10−8 10−10

b

c 0

5

10

15

20

25

30

Average SNR per Symbol per Path [dB] Figure 9.77 Average SER of 8-PSK versus the average SNR per symbol per path γ for Lc = 2 and (a) L = 2, (b) L = 3, and (c) L = 4.

507

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

Average Symbol Error Rate Ps(E )

(m = 0.5) 100 10−2

a

10−4

d 10−6 10−8

0

5

10 15 20 Average SNR per Symbol per Path [dB]

25

30

Average Symbol Error Rate Ps(E )

(m = 1) 100 10−2 10−4

a 10−6

d 10−8

0

5

10 15 20 Average SNR per Symbol per Path [dB]

25

30

25

30

Average Symbol Error Rate Ps(E )

(m = 2) 100 10−2 10−4

a

10−6

d 10−8

0

5

10 15 20 Average SNR per Symbol per Path [dB]

Figure 9.78 Average SER of 16-QAM versus the average SNR per symbol per path γ for L = 4 and (a) Lc = 1 (SC), (b) Lc = 2, (c) Lc = 3, and (d) Lc = 4 (MRC).

508

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Symbol Error Rate Ps(E )

(m = 0.5) 100 10−2

a b c

10−4 10−6 10−8

0

5

10 15 20 Average SNR per Symbol per Path [dB]

25

30

Average Symbol Error Rate Ps(E )

(m = 1) 100 10−2

a

10−4

b

c 10−6 10−8

0

5

10 15 20 Average SNR per Symbol per Path [dB]

25

30

Average Symbol Error Rate Ps(E )

(m = 2) 100 10−2 10−4

a 10−6 10−8

c

0

5

10 15 20 Average SNR per Symbol per Path [dB]

b

25

30

Figure 9.79 Average SER of 16-QAM versus the average SNR per symbol per path γ for Lc = 2 and (a) L = 2, (b) L = 3, and (c) L = 4.

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

509

To circumvent this computational bottleneck, Annamalai and Tellambura [186] defined the partial MGF of a RV Y as  My (s, x) =

∞

py (y) e sy d y

(9.508)

x

Note that 

∞

My (s, 0) = 

py (y) e sy d y = My (s) ,

0 ∞

My (0, x) =

py (y) d y = 1 − Py (x)

(9.509)

x

After some manipulation, they were able to show that (9.507) can be rewritten as a single integral involving the PDF, CDF, and partial MGF of γ , namely

 ∞   L−Lc   L pγ γLc :L e sγLc :L Pγ γLc :L MγGSC (s) = L Lc 0   L −1 × Mγ s, γLc :L c d γLc :L (9.510) At first glance it might appear that the only difference between (9.507) and (9.510) is that the (Lc − 1)-fold nested integral in the former is replaced by the partial MGF of the unordered fading SNR raised to the Lc − 1 power. However, we call the reader’s attention to the fact that the first factor on the right-hand side of (9.507) is L! whereas the same factor in (9.510) is merely L. This distinction comes about as a result of the manipulation of the (Lc − 1)-fold nested integral. The value of (9.510) as opposed to (9.507) lies in the ability of evaluating the partial MGFs for fading statistics of interest. For example, for the Rayleigh channel, it is straightforward to show that " # x 1 exp − (1 − sγ ) (9.511) Mγ (s, x) = 1 − sγ γ whereas for the Rician channel   . 2K (1 + K) x 1+K Q , 2 (1 + K − sγ ) Mγ (s, x) = 1 + K − sγ 1 + K − sγ γ

Ksγ × exp 1 + K − sγ

(9.512)

Similarly, for Nakagami-m fading,



sγ −m sγ mx 1 1− 1− m, Mγ (s, x) = (m) m γ m

(9.513)

510

PERFORMANCE OF MULTICHANNEL RECEIVERS

or for the case where m is integer, we obtain



# " sγ mx sγ −m Mγ (s, x) = 1 − 1− exp − m γ m "

# m−1
mx sγ m−1−k 1 1− × (m − k) γ m

(9.514)

k=0

The expression for the MGF of the combiner output as given in (9.510) can be written as a finite integral by applying the change of variables γLc :L = tan θ , resulting in MγGSC (s) = L

L Lc



π/2



1 + tan2 θ

 L−Lc Pγ (tan θ ) pγ (tan θ )

0



L −1 ×e Mγ (s, tan θ) c d θ

 π/2 L =L f (tan θ ) d θ Lc 0 s tan θ

(9.515)

Furthermore, letting θ = cos−1 y and applying a Gauss–Chebyshev quadrature approximation technique, we obtain 

 1 − y2 dy f MγGSC (s) = L  y 1 − y2 0  
2 n π L  1 − yk  ∼ f =L  Lc n yk

L Lc



1

(9.516)

k=1

where yk = cos [(2k − 1)π / (2n)] are the zeros of a Chebyshev polynomial in y of

order n. (Equivalently, f ( 1 − yk2 /yk ) = f (τk ), where τk = tan [(2k − 1) π / (2n)].) Finally, the extension of the partial-MGF method to the non-i.i.d. case has been considered by Ma and Pasupathy [187]. 9.11.2.5 I.I.D. Weibull Case Neither the MGF nor the partial-MGF approach leads to a tractable analytical solution for some fading scenarios, in particular, the case of the Weibull fading model discussed in Section 2.2.1.5. However, on the basis of Lieblein’s statistical results dealing with the moments and the joint moments of ordered Weibull samples [188], it can be shown that the average combined SNR at the GSC output over i.i.d. Weibull fading channels is given by [189] γ GSC

  Lc
L−l
(−1)i L−l L i =γ l (l + i)1+2/c l l=1

i=0

(9.517)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

511

where c is the Weibull fading parameter. It can also be shown that (9.517) reduces to (9.438) for the i.i.d. Rayleigh case by setting c = 2 in (9.517). The SNR variance at the GSC output over i.i.d. Weibull fading channels can also be shown to be given by [189]  L−l    Lc
L! 1 + 4c
(−1)i L−l i var[γGSC ] =    1+4/c 2 2 (L − l)!(l − 1)! (l + i) 1+ c l=1 i=0 γ2

+ 2γ 2

Lc Lc

j =1 i=2 i>j

−1 L−i i−j

L! (−1)k+n (L − i)!(i − j − 1)!(j − 1)! n=0 k=0

L−i i−j −1      2 1 + 2c B i−j +n−k 1 + 2c , 1 + 2c n k i+n ×   1+2/c 2 1 + 2c (i − j + n − k)(j + k)  L L−l L−l  2 c
L
(−1)i i 2 −γ l (l + i)1+2/c l l=1

(9.518)

i=0

where Bx (·, ·) is the xth-order incomplete beta function defined by Eq. (8.391) of Ref. 36. It can be shown that (9.518) reduces to (9.439) for the i.i.d. Rayleigh case by setting c = 2 in (9.518). Hence the AF at the GSC output over i.i.d. Weibull fading channels can be deduced as var[γGSC ] AFGSC =  2 γ GSC

(9.519)

which can be evaluated from (9.517) and (9.518). Using these exact results for the Weibull channels, approximate results for average combined SNR, SNR variance, and AF at the GSC output over i.i.d Nakagami, Rician, and Hoyt channels can be obtained by an AF-based mapping plotted in Figs. 2.2 and 2.3. As an example, Figs. 9.80 and 9.81 show the good match obtained for the normalized average combined SNR E[γGSC ]/ γ in heavier-than-Rayleigh fading (c = 1.75 or equivalently m = 0.76 and q = 0.53) and lighter-than-Rayleigh fading (c = 3 or equivalently m = 2.17 and K = 2.76) environments, respectively. In Figs. 9.80 and 9.81 the average combined SNR for Weibull fading is obtained from (9.517), whereas it is obtained via numerical integration for the Nakagami/Hoyt/Rice cases (with the appropriate mapping between the Nakagami/Hoyt/Rice and the Weibull fading parameters). As an additional numerical example, Fig. 9.82 quantifies the relatively significant diversity gain in the AF as L increases for a fixed Lc compared to the small decrease of the AF as Lc increases for a fixed L, especially as Lc approaches L. Note that in contrast to the normalized average combined SNR for which we had a good match between exact and approximate results for both the Rice and Nakagami models, Fig 9.82 shows that while the match is good for the Rician case, the AF values

512

PERFORMANCE OF MULTICHANNEL RECEIVERS

Normalized Average Combined SNR

Normalized Average Combined SNR per Symbol

5

4.5

4

3.5

3

2.5

Weibull (c = 1.75) Nakagami (m = 0.76) Hoyt (q = 0.53)

2

1.5

1

1

1.5

2

2.5

3

3.5

4

4.5

5

Number of Strongest Combined Resolvable Paths Lc Figure 9.80 Normalized average combined SNR γ GSC / γ versus the number of available paths L for various combined paths Lc .

are slightly different between the Weibull case (for c = 3) and the Nakagami case (for m = 2.17), especially for small Lc . 9.11.3 Generalized Selection Combining with Threshold Test per Branch (T-GSC) In this section, we discuss the behavior and performance of two conditional combining schemes in which each branch signal-to-noise ratio is tested against a threshold and applied to the combiner only if its value exceeds this threshold. The two schemes, respectively referred to as absolute threshold generalized selection combining (AT-GSC) and normalized threshold generalized selection combining (NT-GSC), differ from one another only in the manner in which the threshold is chosen (fixed for one scheme and variable for the other). However, when operating over a generalized fading channel, the two schemes have a markedly different behavior, as will be seen later on in the numerical results. As discussed in the previous section, conventional GSC first ranks the instantaneous branch output SNRs, then selects a subset of these corresponding to the ones with the largest value, and finally combines them in the fashion of MRC. Since,

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

513

Normalized Average Combined SNR per Symbol

5 Weibull (c = 3) Nakagami (m = 2.17) Rice (K = 2.76)

4.5

4

3.5

3

2.5

2

1.5

1

1

1.5

2 2.5 3 3.5 4 Number of Available Resolvable Paths L

4.5

5

Figure 9.81 Normalized average combined SNR γ GSC / γ versus the number of available paths L for various combined paths Lc .

in this form of GSC, the number of largest SNR branches is decided a priori, the scheme has a fixed processing complexity; however, it suffers from the fact that it potentially discards from combination many branches whose SNRs might be close in value to the ones selected or alternatively includes in the combination branches whose SNRs might be low. A receiver diversity combining scheme that alleviates the above-mentioned problem associated with conventional GSC was proposed [190] wherein the number of combined branches is allowed to be a variable whose value is determined in accordance with the strength of each of the branch SNRs. Specifically, the ratio of the instantaneous SNR of each branch to that of the best branch, namely, the maximum instantaneous SNR,24 is tested against a fixed normalized [values chosen in the interval (0,1)] threshold and applied to the combiner only if its value exceeds this threshold.25 When viewed this way, the scheme, which was given the acronym T-GSC by Sulyman and Kousa [190], resembles conventional GSC with 24 Sulyman

and Kousa [190] refer to each of these ratios as branch relative strength (BRS). the instantaneous SNR of each branch is compared to the product of the fixed normalized threshold and the instantaneous SNR of the strongest branch. Thus, the actual (unnormalized) threshold used to test each branch SNR (except for the strongest) is a RV.

25 Equivalently,

514

PERFORMANCE OF MULTICHANNEL RECEIVERS

Amount of Fading at the GSC Output Nakagami (m = 2.17) Rice (K = 2.76) Weibull (c = 3)

Amount of Fading at the GSC Output

L=2 0.25

0.2

L=3 0.15

L=4 L=5

0.1 1

1.5

2 2.5 3 3.5 4 Number of Strongest Combined Resolvable Paths Lc

4.5

5

Figure 9.82 Amount of fading at the GSC output versus the number of strongest combined resolvable paths Lc over Weibull channel with c = 3 (equivalent to m = 2.17 for Nakagami fading and K = 2.76 for Rician fading).

a variable (random) number of combined branches. The average BEP performance of a particular embodiment of this scheme in a Nakagami-m fading channel environment was determined by simulation in their paper [190] and compared with that of conventional GSC. An alternative threshold diversity combining scheme in which each branch SNR is, without normalization by the maximum, tested against a fixed predetermined threshold and applied to the combiner only if its value exceeds this threshold was considered in Ref. 191. The motivation for considering such a scheme was as follows. In many diversity combining scenarios, the number of available diversity paths can be quite high. This arises, for example, in the context of multipath diversity of ultrawideband spread-spectrum signals [192] or “collaborative” diversity of wireless sensor systems. In such systems, combining all these paths, although optimal from a performance standpoint, is often impractical from a total power (including processing and transmit power) consumption perspective. Hence, in order to avoid frequent charging of the mobile or portable units’ batteries and the corresponding downtimes, a scheme such as the above that limits the power consumption due to combining would be advantageous. For example, in the context of RAKE reception of ultrawideband spread-spectrum signals, this scheme would

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

515

resort to the processing power resources of the handsets to combine a particular path only if its quality is judged good enough to significantly improve the overall BEP performance. Similarly, this scheme would allow wireless sensors to reduce their average transmitted power by remaining in an idle mode as long the link quality between the transmitting and receiving nodes is below an acceptable SNR. The generic analysis approach that follows will assume independence among the fading channels; however, it does allow for the possibility of them being nonidentically distributed. For illustration purposes, we shall focus primarily on the identically distributed Rayleigh case. 9.11.3.1 Average Error Probability Performance In this section, we evaluate the average error probability performance of the ATGSC and NT-GSC schemes once again using the MGF-based approach. We begin with an evaluation of the MGF for the two different threshold cases. Moment Generating Function of the Combiner Output Instantaneous SNR AT-GSC As in previous sections, we consider a diversity receiver containing L branches with instantaneous SNR in the l th branch, γl , l = 1, 2, . . . , L. Define a fixed (independent of l ) branch threshold γth against which each branch SNR (γl ) will be tested. In particular, if γl equals or exceeds γth , it is included in the combiner output (assumed to be of the MRC type). Thus, for the purpose of analysis, we can define a set of RVs) 1 γl , γl ≥ γth (9.520) γl = 0, 0 ≤ γl < γth in which case the combiner output is given by γt =

L


γl

(9.521)

l=1

It is straightforward to show that the PDF of γl can be expressed in terms of the underlying PDF and CDF of γl by    γl = 0    Pγl (γth ) δ γl , 0,   0 < γl < γth pγl γl = (9.522) 

γl ≥ γth pγl γl , where δ (·) is the Dirac delta function. Note that γl is a combination of a discrete RV and a continuous RV distributed over the interval 0 ≤ γ1 ≤ ∞ and its PDF given by (9.522) is a valid one since  ∞  ∞  

  pγl γl d γl = Pγl (γth ) + pγl γl d γl

0 γth (9.523) = Pγl (γth ) + 1 − Pγl (γth ) = 1

516

PERFORMANCE OF MULTICHANNEL RECEIVERS

TABLE 9.6 Moment Generating Functions of Instantaneous SNR for lth Branch Contributing to Combiner Fading Channel

MGF, Mγ (s) " #  1 γth  + exp − 1 − sγ l 1 − sγ l γl



mγ ml , l th γl   ml

"

# sγ ml γth ml , 1− m l γ l 1 l  

ml sγ ml 1− m l l

γth 1- exp − γl

Rayleigh

1−

Nakagami-ml

1 − Q1

Rice

l





√

  Q1 

+

-   2Kl , 2 1 + Kl

2Kl sγ l  1−  1+Kl

γth γl



.   , 2 1 + Kl



+

γth γl

1 sγ l  1−  1+Kl



1−





exp −Kl 1 −

sγ l   1+Kl

  

1 sγ l  1−  1+Kl



From (9.522), the MGF of γl is given by  Mγl (s) =

∞

0

    pγl γl exp sγl d γl = Pγl (γth ) +



∞ γth

    pγl γl exp sγl d γl

(9.524) and thus under assumption of independent but not necessarily identically distributed branches, the MGF of γt is given by Mγt (s) =

L

Mγl (s) =

l=1

 L " Pγl (γth ) + l=1

∞ γth

    pγl γl exp sγl d γl

# (9.525)

or for the i.i.d branch case "  Mγt (s) = Pγ (γth ) +

∞

    pγ γ exp sγ d γ

#L (9.526)

γth

The per branch MGF, Mγl (s), in (9.524) can be evaluated in closed form for a number of different practical channel fading models. The results are summarized in Table 9.6. NT-GSC 26 For this scenario, we define a branch threshold γth that is a fixed fraction, ηth , of γmax = maxl γl , i.e., γth = ηth γmax with 0 ≤ ηth ≤ 1.27 Next, let γ1:L ≥ γ2:L ≥ · · · ≥ γL:L denote the ordered statistic corresponding to the set of 26

A performance analysis of this scheme was first considered by the authors in Ref. 191 and later extended by Annamalai et al. [193]. These analyses were later corrected by Zhang and Beaulieu [194], and as such the presentation given here follows the approach taken in Ref. 194. 27 Note that while η is fixed to a predetermined value, the actual threshold γ = η γ th th th max used to test each branch’s strength is a RV since γmax is itself a RV.

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

517

branch SNRs γ1 , γ2 , . . . , γL where, by definition, γmax corresponds to the ordered RV, γ1:L . Then, depending on the number of ordered RVs (starting with the largest) that equal or exceed the threshold, the combiner output can be expressed as a sum of these RVs. In particular, we obtain γt,Lc =

Lc


γi:L , Lc = 1, 2, . . . , L

(9.527)

i=1

where Lc is now an integer RV that represents the number of branches being combined in order of decreasing SNR starting with the one having the largest SNR. Equivalently, Lc is determined from the joint event γ1:L ≥ γ2:L ≥ . . . ≥ γLc :L ≥ ηth γ1:L ≥ γLc +1:L , γLc +2:L , . . . , γL:L and thus is a RV dependent on both the normalized threshold and the instantaneous branch SNRs. In this sense, NT-GSC with normalized per branch threshold testing can be viewed as a conventional GSC system whose number of branches being combined is random rather than fixed. By contrast, since the AT-GSC system combines the branches whose unordered SNRs equal or exceed the fixed threshold, it can be viewed as a conventional MRC system whose number of branches being combined is random rather than fixed.28 Since for NT-GSC the combiner outputs corresponding to each integer value of Lc represent disjoint events, then from the law of total probability, the average error probability is the sum of the average (over the fading) error probabilities for each of these events. Specifically, if P s (E; l) denotes the average SEP of the combiner output when l branches satisfying the joint event γ1:L ≥ γ2:L ≥ · · · ≥ γl:L ≥ ηth γ1:L ≥ γl+1:L , γl+2:L , . . . , γL:L are selected for combination, then the average SEP is given by P s (E) =

L


(9.528)

P s (E; l)

l=1

where, using the MGF-based approach, each P s (E; l) is determined from the MGF of γt for that particular value of l that, in accordance with the inequality on the ordered branch SNRs that must be satisfied, is given by  γl−1:L  γ1:L  ηth γ1:L  ∞ d γ1:L d γ2:L . . . d γl:L d γl+1:L Mγt,l (s) = 0

ηth γ1:L

 ×



γl+1:L

γL−1:L

d γl+2:L . . . 0

0

ηth γ1:L



exp s 0

l




γk:L

k=1

×pγ1:L ,γ2:L ,...,γL:L (γ1:L , γ2:L , . . . , γL:L ) d γL:L

(9.529)

where pγ1:L ,γ2:L ,...,γL:L (γ1:L , γ2:L , . . . , γL:L ) is the joint PDF of the ordered RVs. 28 Analogous to (9.527), at any instant the combiner output for the absolute threshold T-GSC system  c can alternatively be expressed as γt,Lc = L i=1 γ˜i , where γ˜i appears in the sum (and is equal to γi ) only if γi equals or exceeds γth and Lc is the number of such occurrences.

518

PERFORMANCE OF MULTICHANNEL RECEIVERS

Assuming the i.i.d. case,29 then, for the complete set of ordered statistics, γ1:L ≥ γ2:L ≥ · · · ≥ γL:L , the PDF is given by (9.311) with Lc = L, namely pγ1:L ,γ2:L ,...,γL:L (γ1:L , γ2:L , . . . , γL:L ) = L!

L

pγ (γl:L )

(9.530)

l=1

where it is clear that the γl:L values are not independent. Thus, from on the preceding description of the event Lc = l, we obtain  Mγt,l (s) = L! 



pγ (γ2:L ) e sγ2:L d γ2:L

ηth γ1:L



γ2:L

pγ (γ3:L ) e sγ3:L d γ3:L . . .

ηth γ1:L

γl−1:L



pγ (γl+2:L ) d γl+2:L 0

γl+2:L

×

(9.531)

γl+1:L

pγ (γl+1:L ) d γl+1:L 0

pγ (γl:L ) e sγl:L d γl:L

ηth γ1:L



ηth γ1:L

×

γ1:L

pγ (γ1:L ) e sγ1:L d γ1:L

0

× 

∞



γL−1:L

pγ (γl+3:L ) d γl+3:L . . . 0

pγ (γL:L ) d γL:L 0

which is a product of separable but nested integrals. By an application of the partial-MGF approach used to simplify the Lc -fold integral in the MGF of the combiner output SNR for GSC (see Section 9.11.2.4), the MGF in (9.531) can likewise be reduced to a single integral. In particular, using Eq. (3) from Ref. 194, we obtain  ∞ L−l   l−1 L Mγt,l (s) = l Mγ (s, ηth γ ) − Mγ (s, γ ) e sγ pγ (γ ) Pγ (ηth γ ) dγ l 0 (9.532) where Mγ (s, γ ) is the partial MGF of γ as defined in (9.501) and is evaluated in closed form for Rayleigh, Rician, and Nakagami-m fading in eqs. (9.504), (9.505), and (9.506), respectively. For example, for Rayleigh fading, whereupon γ is exponentially distributed, using (9.504) and (2.7) together with its corresponding CDF in (9.532), and evaluating the resulting integral, results in the closed-form expression [194, Eq. (17)] Mγt,l (s) = l × 29 Extension

L l


L−l
l−1 j =0 k=0

L−l j



l−1 k



(−1)j +k (9.533)

1 (1 − sγ )l [1 + (1 − ηth ) k + ηth (l − 1)] + ηth j (1 − sγ )l−1

to the non-i.i.d. case is somewhat more tedious and is treated in Ref. 194.

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

519

Using the identities



l−1
1 1 l−1 k = (−1) k 1+k l

(9.534)

k=0

1

L−l
0, L−l (−1)j = 1, j j =0

we obtain Mγt,l

! (s) !η

th =0

and

! Mγt,l (s) !ηth =1

=

l = 1, 2, . . . , L − 1 l=L

  0,

l = 1, 2, . . . , l − 1 1 = MγMRC (s) , l=L  (1 − sγ )L

 0,   

L−1   
L−1 L j =  j =0    1   , × (−1)j (1 − sγ ) + j

(9.535)

l = 2, 3, . . . , L = MγSC (s) l=1

(9.536) which verifies the results for the limiting cases of MRC and SC, the latter being obtained from (9.423) with Lc = 1. Average Error Probability Evaluation AT-GSC Having the MGF of the combiner output allows one to immediately evaluate the error probability of the receiver. We now illustrate the details of such a computation for the i.i.d. Rayleigh case using coherent BPSK or BFSK. For conventional MRC reception, the average BEP is given by (9.12), which for i.i.d. Rayleigh fading becomes  

k 

L−1 gγ
2k 1 1 MRC P b (E) = (9.537) 1− k 2 1 + gγ 4 (1 + gγ ) k=0

Expanding the MGF in (9.526) in a binomial series and substituting the result in (9.537) gives "

#L  ∞    g 1 π/2 Pγ (γth ) + P b (E) = pγ γ exp − 2 γ d γ dφ π 0 sin φ γth "

#l

 L ∞   L−l π/2 g 1
L  pγ γ exp − 2 γ d γ dφ = Pγ (γth ) l π sin φ 0 γth l=0

(9.538)

520

PERFORMANCE OF MULTICHANNEL RECEIVERS

Note that the average BEP computed from (9.538) includes the error event corresponding to no paths combined (i.e., Lc = 0). This event, which occurs with prob L ability Pr {Lc = 0} = Pγ (γth ) , and would result in an error probability equal to 12 , constitutes an outage and should be subtracted from (9.538). In addition, to  arrive at a true average BEP, one must renormalize the result by L l=1 Pr {Lc = l} =  L 1 − Pγ (γth ) . Thus, after some manipulation, the final result for the average BEP of AT-GSC is given by

"

# L
γth L−l L 1 − exp − P b (E) = L   l γ l=1 1 − 1 − exp − γγth 1

×

1 π



 π/2

0



1 1+

gγ sin2 φ

(9.539)



# l " gγ γth  dφ 1+ exp − γ sin2 φ

which unfortunately cannot be found in closed form. Nevertheless, the integral in (9.539) consists only of elementary functions and is routinely evaluated numerically. NT-GSC For NT-GSC, the event corresponding to zero reception cannot occur since the combiner will always have at least one contributing branch, namely, the one corresponding to γmax . Hence, in this case there is no need to modify the expression for the average BEP as was done following (9.538). Thus, from (9.528) and (9.533) we obtain for the Rayleigh fading case

 L
g 1 π/2 dφ P b (E) = Mγt:l − 2 π 0 sin φ l=1

=





L−l l−1

L
L

L − l l−1 l (−1)j +k j l k j =0 k=0

l=1

×

1 π

 0

π/2

" 1+

gγ sin2 φ

l

1

dφ

(9.540)

[1 + (1 − ηth ) k + ηth (l − 1)] l−1 #  + ηth j 1 + singγ2 φ

9.11.3.2 Outage Probability Performance In this section, we evaluate the outage probability performance of the T-GSC schemes. For AT-GSC, the expression for this performance will be given in terms of the PDF of the combiner output, while for NT-GSC, the outage probability will be obtained directly in terms of the MGF of this same output using the relation in (1.6). Thus, we begin this section with an evaluation of this PDF for the AT-GSC case.

521

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

AT-GSC To evaluate the outage probability of AT-GSC, we shall need the PDF of the instantaneous SNR of the combiner output. For the i.i.d. Rayleigh case, substituting (2.7) and its corresponding CDF in (9.526) and twice expanding in a binomial sum gives (also see Table 9.6)

L−l

l

L
1 γth L
l k Mγt (s) = (k + L − l) (−1) exp − l k γ γ l=0

k=0



×

L−l

1 1 γ

−s



γth exp (L − l) sγ γ

(9.541)

Recognizing the inverse Laplace transform

−1

L

1

1 s+a

n 4

 

1 γ n−1 exp (−aγt ) , (n − 1)! t =  δ (γt ) ,

γt ≥ 0, n ≥ 1

(9.542)

n=0

and also the property L−1 {exp (−bs) F (s)} = f (γt − b) u (γt − b)

(9.543)

where u (·) is the unit step function, then the inverse Laplace transform of (9.541), namely, the PDF of γt , becomes, after some manipulation

#

# " " L−1
L

1 γth l γth L 1− exp − pγt (γt ) = 1− exp − δ(γt )+ l γ γ (L − l − 1)! l=0 ×

L−l

  1 γt γt − γth (L − l) L−l−1 exp − u (γt − γth (L − l)) γ γ

(9.544) Note that the first term in (9.544) accounts for the event of no branches contributing to the combiner output. Recall that in the case of average BEP evaluation we removed this event from consideration. However, since in the next section we shall be dealing explicitly with outage probability, we wish to retain this term. Also note that the remaining portion of the PDF in (9.544) has jump discontinuities at the points γt = kγth , k = 1, 2, . . . , L that occur every time a new branch contributes to the combiner output (i.e., each time a branch SNR equals or exceeds the branch threshold γth ). Now we present the outage probability of the system based on the PDF of the output of the combiner found in the last section for the i.i.d. Rayleigh case. Letting T denote the system threshold against which the combiner output is compared, the

522

PERFORMANCE OF MULTICHANNEL RECEIVERS

outage probability becomes

# "  T γth L Pout = Pr {γt < T } = pγt (γt ) d γt = 1 − exp − γ 0





L−1 l
L
l 1 γth k exp − + (k + L − l) (−1) l k γ (L − l − 1)! l=0

 ×

"

T

γth (L−l)

k=0

γt − γth (L − l) γ

#L−l−1

# (γt − γth (L − l)) 1 d γt exp − γ γ "

(9.545) Letting m = L − l and x = [γt − γth (L − l)]/ γ , we can rewrite (9.545) as

# "  T γth L pγt (γt ) d γt = 1 − exp − Pout = Pr {γt < T } = γ 0





L L−m
L
L−m γth 1 k exp − + (−1) (k + m) m k γ (m − 1)! m=1 k=0  (T −mγth )/ γ x m−1 exp (−x) d x (9.546) × 0

Note that in (9.546) we only get a contribution to the integral for values of m such that mγth < T . Thus, the upper limit of the sum on m should be replaced with min {L, T /γth } where z denotes the integer part of z. The integral in (9.546) can be evaluated with the help of Eq. (2.321.2) in Ref. 36, leading, after some manipulation, to the final desired result  T pγt (γt ) d γt Pout = Pr {γt < T } = 0



# "

# min{L,T
/γth } L " γth L−m γth m 1 − exp − = exp − m γ γ m=0  

m−1

m−1−j
T − mγ 1 T − mγ th th  × 1 − exp − γ γ (m − 1 − j )! j =0

(9.547) Since the case γth = 0 corresponds to pure MRC, then, making this substitution in (9.547), we get Pout =

L−m L

L−m

L (−1)k m k m=1 k=0  

m−1 m−1−j
1 T T  × 1 − exp − γ (m − 1 − j )! γ j =0

(9.548)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

523

Note that the quantity in braces does not depend on the index k. Thus, making use of the second identity in (9.534), the only term in the sum on m that contributes is that corresponding to m = L, in which case (9.548) simplifies to

Pout

L−1

L−1

L−1−j T 1 T
T
1 T l = 1 − exp − = 1 − exp − γ γ l! γ (L − 1 − j )! γ j =0

l=0

(9.549) which is the outage probability of MRC with i.i.d. Rayleigh fading. Observe that if we choose T < γth , then min {L, T /γth } = 0, in which case Pout of (9.547) simplifies to

# " γth L Pout = 1 − exp − γ

(9.550)

Furthermore, if we choose T = γth , then min {L, T /γth } = 1 and only the m = 1 term in the sum on m in (9.547) survives. However, this term evaluates to zero, and thus Pout is again given by (9.550). NT-GSC According to the relationship in (1.6), the outage probability for this case can be found from  σ +j ∞ L
Mγt:l (−s) sT 1 e ds 2πj σ −j ∞ s l=1 /  0  L
Mγt:l − (σ + j ω) 2eσ T ∞ Re = cos ωT dω π σ + jω 0

Pout =

(9.551)

l=1

where, for the special case of i.i.d. Rayleigh paths, Mγt:l (s) is given by (9.533) and thus

Pout





L−l l−1  ∞ L 2eσ T
l−1 L

L − l j +k = l cos ωT (−1) j k l π 0 l=1 j =0 k=0 1 1 × Re (σ + j ω) (1 + (σ + j ω) γ )l−1 4 1 dω (9.552) × (1 + (σ + j ω) γ ) [1 + (1 − ηth ) k + ηth (l − 1)] + ηth j

Note that for a zero per branch threshold (i.e., ηth = 0), the fixed and normalized threshold GSC systems yield the same outage probability, namely, that corresponding to MRC as would be expected.

524

PERFORMANCE OF MULTICHANNEL RECEIVERS

9.11.3.3

Performance Comparisons

AT-GSC We want to compare the performance of AT-GSC with conventional GSC. To do this on the basis of processing complexity, we will want to know for AT-GSC the average number of branches that contribute to the combiner output, or equivalently, the average number of branches whose SNR equals or exceeds the threshold. Clearly, at each test against a threshold, the number of branches Lc that contribute to the combiner can range anywhere from 0 to L. The conditional probability that a given l out of L branches contribute to the combiner output is equal to the conditional probability that these l branches equal or exceed the l  L−l  . threshold (and the remaining L − l branches do not): 1 − Pγ (γth ) Pγ (γth ) Thus, for the i.i.d. case, the probability that any Lc = l out of L branches contribute to the combiner output is equal to

l  L−l L  Pr {Lc = l} = (9.553) 1 − Pγ (γth ) Pγ (γth ) l whereupon the average number of branches combined becomes L
Lc = l Pr {Lc = l} (9.554) l=0

which for the i.i.d. case becomes

L
 l  L−l  L  1 − Pγ (γth ) Pγ (γth ) Lc = l = L 1 − Pγ (γth ) l

(9.555)

l=0

Since the number of branches combined in AT-GSC is variable whereas in conventional GSC-Lc it is fixed, to allow a comparison between the two diversity schemes, one must define an appropriate basis for this comparison. Since the intention behind using either of these two suboptimum schemes instead of MRC is a reduction in complexity, then a fair basis of comparison is equivalent processing complexity, which is tantamount to choosing the AT-GSC threshold γth , such that the average number of branches combined Lc is equal to Lc . For example, to compare AT-GSC with SC, we set Lc of (9.555) equal to one. Thus, using (9.313) for Pγ (γ ), we have that

γth =1 (9.556) L exp − γ or, equivalently, γth = γ ln L Using the threshold of (9.557) in (9.539) gives



L
1 1 L−l L P b (E) = 1− L  l L 1 − 1 − L1 l=1   l  2 sin2 φ 1 π/2 − sin φ+gγ 2φ sin × dφ L π 0 sin2 φ + gγ

(9.557)

(9.558)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

525

which is to be compared with the average BEP of SC, which is given as SC P b (E)

=L

L−1
k=0

     (−1)k L−1 1 gγ k 1− 1+k 2 1 + k + gγ

(9.559)

In the more general GSC case, the threshold of (9.557) would become γth = γ ln

L Lc

(9.560)

with a corresponding average BEP P b (E) =

1

 1− 1− ×

1 π



π/2 0



L
Lc L−l L 1 − L l L Lc l=1

L

 

2

sin φ sin2 φ + gγ



L Lc

− sin2 φ+gγ sin2 φ

l

(9.561)

 dφ

This result is to be compared with the corresponding result for GSC-Lc for which the average BEP can be obtained from (9.438) combined with (5A.35) and (5A.36), namely GSC P b (E, Lc

 


L−l (−1)k L−l π L gγ k ; gγ , = l) = Il−1 2 1 + k/l l 1 + kl

(9.562)

k=0

combined with

m

.  1 c2 c2 1− ; c1 , c2 = 2 2 1 + c2 c2 − c1 

m−k  .

m−1 c2 c1
1 1 2k 1− − k 2 1 + c1 c2 − c1 [4 (1 + c1 )]k k=0 (9.563) Figure 9.83 is a plot of average BEP, namely, P b (E), as computed from (9.563) versus average SNR, namely, γ , for fixed L = 6 and values of Lc = Lc ranging from 1 to 6. Keeping in mind that the curve for Lc = 6 corresponds to MRC performance as given by (9.537), we observe that for sufficiently high SNR, the AT-GSC scheme, even with only Lc = 1, outperforms MRC. Although perhaps surprising at first, this result follows from the fact that, whereas the asymptotic (large-γ ) BEP behavior of MRC varies inversely as γ L , the asymptotic BEP behavior of AT-GSC can be shown to vary inversely as Lγ . Equivalently, on a logarithmic scale (such as that in Fig. 9.83), MRC has an inverse linear asymptotic performance (with Im

π

526

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

Lc = 1 Lc = 2 Lc = 3 Lc = 4 Lc = 5 Lc = 6

10−1

Average Bit Error Rate Pb(E )

10−2 10−3 10−4 10−5 10−6 10−7 10−8 −10

−8

−6

−4

−2 0 2 4 Average SNR per Bit per Path [dB]

6

8

10

Figure 9.83 Average BER of AT-GSC/BPSK versus average SNR with average number of paths combined as a parameter; L = 6.

slope proportional to L) whereas AT-GSC with Lc < L has an inverse exponential asymptotic performance. The apparent superior average BEP performance of AT-GSC over MRC at high SNRs is mitigated by a comparison of the outage probability of the two combining schemes. Specifically, the outage probability is computed by substituting the threshold of (9.560) into (9.547), which results in min{L,T /(γ ln L/Lc )}




Pout =

×

  

m=0

1−

L Lc

m

L m



Lc L

m

Lc L−m 1− L



m−1−j 

m−1
L T 1 T − m ln exp −  γ Lc (m − 1 − j )! γ j =0

For T → ∞, (9.564) becomes Pout



m L
Lc L−m Lc L 1− = =1 m L L m=0

(9.564)

(9.565)

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

527

100 10−1

Outage Probability

10−2 10−3 10−4

10

Lc = 1 Lc = 2 Lc = 3 Lc = 4 Lc = 5 Lc = 6

−5

10−6 10−7 10−8 −10

−8

−6

−4 −2 0 2 4 6 Normalized Average SNR per Bit per Path [dB]

8

10

Figure 9.84 Outage probability of AT-GSC versus average SNR with average number of paths combined as a parameter; L = 6.

whereas for T → 0, (9.564) becomes Pout

Lc L = 1− L

(9.566)

independent of γ . Thus, a plot of Pout versus normalized average SNR per branch (γ /T ) would range from Pout = 1 at γ /T = 0 to Pout = (1 − Lc /L)L at γ /T = ∞; thus, the AT-GSC scheme exhibits an irreducible outage probability in the limit of infinite average SNR. This behavior is clearly illustrated in Fig. 9.84, which plots outage probability Pout of (9.564) versus normalized SNR (γ /T ) for the same values of L and Lc as in Fig. 9.83. NT-GSC Analogous to the approach taken in the previous section, we compare the average BEP of NT-GSC with that of conventional GSC-Lc again on the basis of equivalent processing complexity; that is, we choose the NT-GSC normalized threshold ηth such that the average number of branches combined Lc is equal to Lc . Unlike AT-GSC, however, where the relation between the threshold and Lc is invertible [see (9.560)], for NT-GSC the solution for ηth as a function of Lc must

528

PERFORMANCE OF MULTICHANNEL RECEIVERS

be determined numerically. Specifically, for NT-GSC with i.i.d. fading, we obtain 



∞

Pr {Lc = l} = L!

γ1:L

pγ (γ1:L ) d γ1:L

pγ (γ2:L ) d γ2:L

0



ηth γ1:L



γ2:L

×

γl−1:L

pγ (γ3:L ) d γ3:L . . . 

ηth γ1:L



ηth γ1:L

×

pγ (γl:L ) d γl:L ηth γ1:L

pγ (γl+1:L ) d γl+1:L 

0

pγ (γl+2:L ) d γl+2:L 0

γl+2:L

×



γL−1:L

pγ (γl+3:L ) d γl+3:L . . . 0

(9.567)

γl+1:L

pγ (γL:L ) d γL:L 0

which for the Rayleigh case evaluates to Pr {Lc = l} = l

×

L l


L−l

l−1
m=0

k=0

l−1 m

L−l k





(−1)k

(−1)m

1 m + 1 + ηth (l − m − 1 + k)



(9.568)

Combining (9.568) with (9.554) and keeping L fixed provides the necessary solution for ηth as a function of Lc . Table 9.7 provides the results of this numerical solution for L = 6, specifically, the number of paths considered in Figs. 9.83 and 9.84. Using the values of ηth from this table in (9.11.3.1) gives the desired average BEP of NT-GSC. Similarly, the outage probability performance for NT-GSC can be determined from (9.552) combined with (9.533) using the values of ηth provided in Table 9.7 (see also Fig. 9.85). GSC/AT-GSC/NT-GSC Comparison Figures 9.86 and 9.87 compare the performance of GSC, AT-GSC, and NT-GSC for L = 6 and an average number of combined paths equal to 1 and 5, respectively. From Fig. 9.86, we can see that

TABLE 9.7 Optimum per Branch Thresholds for NT-GSC with Processing Complexity Equal to That of Conventional GSC with Lc Combined Paths out of L = 6 Available Paths Lc

ηth

1 2 3 4 5 6

1.0 0.5565 0.3337 0.187 0.081 0.0

529

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

100

Lc = 1 Lc = 2 Lc = 3 Lc = 4 Lc = 5 Lc = 6

10−1

Average Bit Error Rate Pb(E )

10−2 10−3 10−4 10−5 10−6 10−7 10−8 −10

−8

−6

−4

−2 0 2 4 Average SNR per Bit per Path [dB]

6

8

10

Figure 9.85 Average BER of NT-GSC/BPSK versus average SNR with average number of paths combined as a parameter; L = 6. 100 SC AT-SC NT-SC

Average Bit Error Rate Pb(E )

10−1

10−2

10−3

10−4

10−5

10−6 −10

−8

−6

−4

−2 0 2 4 Average SNR per Bit per Path [dB]

6

8

10

Figure 9.86 Comparison of the average BER of SC, AT-SC, and NT-SC for Lc = 1 and L = 6.

530

PERFORMANCE OF MULTICHANNEL RECEIVERS

100 GSC AT-GSC NT-GSC

Average Bit Error Rate Pb(E )

10−1

10−2

10−3

10−4

10−5

10−6 −10

−8

−6

−4

−2 0 2 4 Average SNR per Bit per Path [dB]

6

8

10

Figure 9.87 Comparison of the average BER of BPSK with GSC, AT-GSC, and NT-GSC for Lc = 5 and L = 6.

GSC (or, in this case, SC) and NT-SC have exactly the same performance, whereas we see from Fig. 9.87 that the same two schemes have a comparable performance. More generally, from these figures and other numerical experiments not included here, we noticed that conventional GSC slightly outperforms NT-GSC from both an average BEP and outage probability standpoints. This, of course, comes at the expense of a slightly higher complexity since GSC requires the ranking of all diversity branch strengths whereas NT-GSC just needs the knowledge of the branch relative strengths and therefore does not require full ranking. On another front, we see from Fig. 9.86 that AT-SC clearly outperforms SC, NT-SC, and actually even MRC (i.e., Lc = 6 out of L = 6). This might be surprising at first glance, but recall that the average BEP of AT-GSC is computed only for the fraction of time that one or more diversity paths are combined. So this superior average BEP performance of AT-GSC [which also holds at average SNR above 6 dB for Lc = 5 (see Fig. 9.87)] has to be contrasted by its poor (especially at high average SNR) outage probability performance as shown in Fig. 9.84. Hence the AT-GSC scheme can be viewed as a combining scheme that trades outage probability for a better average BEP performance. As such, this scheme is suitable for the transmission of non-delay-limited type of information since with this scheme, once the communication channel is in the ON state (i.e., no outage), the receiver will be able to operate not only at a lower

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

531

average BEP but also with a lower processing power or transmit power in the context of collaborative wireless sensor systems (recall that an AT-GSC with Lc = 1 and L = 6 outperforms MRC with Lc = 6 and L = 6 from an average BEP standpoint). 9.11.4

Generalized Switched Diversity (GSSC)

We now focus on the performance of a GSSC scheme [123]. This scheme involves first SSC followed by MRC or EGC, the operation of which is as follows: The incoming signal is received over an even number 2L of diversity branches that are grouped in pairs. Every pair of signals is fed to a switching unit that operates as per the rules of SSC. The output from the L switching units are connected to an MRC or EGC combiner. This scheme is motivated by the GSC scheme that was analyzed in the previous section and that inherits one of the main disadvantages of SC: the necessity of a “centralized,” continuous, and simultaneous monitoring of all the diversity branches. On the other hand, the GSSC scheme offers a “decentralized” simpler (although less efficient) solution, and can be viewed as a more practical implementation of GSC. In what follows, we evaluate the performance of the GSSC scheme and then compare it to GSC. A generalization of SEC is also available in Ref. 195. 9.11.4.1

GSSC Output Statistics

Joint PDF For simplicity let us assume that all the pairs of signals at the SSC unit inputs are i.i.d. Then the joint PDF at the MRC input is given by pγ1 ,γ2 ,...,γL (γ1 , γ2 , . . . , γL ) =

L

pγSSC (γl )

(9.569)

l=1

where pγSSC (γl ) denotes the SNR PDF at the output of the lth SSC unit and is given by 1 if γTl < γl Pγl (γl ) pγl (γl ) (9.570) pγSSC (γl ) = (1 + Pγl (γTl )) pγl (γl ) if γTl ≥ γl where Pγl (γl ) and pγl (γl ) are the CDF and PDF of at the SSC units individual branches, respectively, and are given in Table 9.5. Since the L MRC inputs are combined as per the rules of MRC, γGSSC = γ l=1 l , then, assuming independent fading across the SSC units, the MGF of the GSSC SNR output is just the product of the L MGFs of the SNRs at the L SSC outputs. In the particular case of Nakagami-m fading, using (9.280), the MGF of the GSSC output can be obtained as        γl ml γT ml

−ml L − m m , 1 − s , γ l l γ T sγ ml γl l 1 +  1− l MγGSSC (s) = ml (ml )

MGF L

l=1

(9.571)

532

PERFORMANCE OF MULTICHANNEL RECEIVERS

Similarly, for the Nakagami-n (Rician) fading case, using (9.281), the MGF of the GSSC output can be obtained as  −1  sγ l n2l sγ l exp 1− MγGSSC (s) = 1 + n2l 1 + n2l − sγ l l=1     √ 2(1 + n2l )γT  × 1 − Q1 nl 2, γl L



  +Q1 nl

*    + 2 2(1 + n2l ) + 1 + n l , ,2 − s γT  (9.572) γl 1 + n2l − sγ l

9.11.4.2 Average Probability of Error Using the MGF of the GSSC output SNR determined in the previous section, we can determine the average probability of error of several modulation schemes via the MGF-based approach. Differentiating the resulting expressions with respect to the L switching thresholds, it can be easily shown (because of the product form of the integrand) that the optimal thresholds of the individual switching units will yield the overall optimum performance. This means that GSSC performance can be optimized without much more computational complexity than conventional SSC. As an example, Fig. 9.88 compares the average BER performance of SC/MRC-2/4 with the performance of SSC/MRC-2/4 (using the optimum switching thresholds in all the switching units) for m = 0.5, m = 1, and m = 2. Note that SSC/MRC-2/4 suffers about a 1-dB penalty compared to SC/MRC-2/4 in the medium to high average SNR region. 9.11.5 Generalized Selection Combining Based on the Log-Likelihood Ratio Thus far, we have discussed several GSC schemes without regard to the optimality of their performance. In this section, we present, for binary signaling, a GSC scheme that selects Lc out of L branches based on the magnitude of the log-likelihood ratio (LLR). For slow-fading Rayleigh channels, it will be shown that such a scheme, herein referred to as LLR-based GSC [196], is the optimum GSC rule in that it minimizes average bit error probability.30 We shall also present a suboptimum GSC technique based on noncoherent envelope detection and motivated by an upper bound on the magnitude of the LLR. Such a technique, herein referred to as envelope-based GSC, leads to a simpler test than LLR-based GSC yet is still superior to conventional GSC with post-detection square-law combining. 30 Similar

LLR-based schemes for selection combining (i.e., Lc = 1) were first considered in Ref. 197.

533

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

(m = 0.5)

Average BER

100

MRC4 SC/MRC-2/4 SSC/MRC-2/4 MRC2

10−2 10−4 10−6 10−8

0

5

Average BER

20

25

(m = 1)

100

MRC-4 SC/MRC-2/4 SSC/MRC-2/4 MRC-2

10−2 10−4 10−6 10−8

0

5

10 15 Average SNR per bit per branch [dB]

20

25

(m = 2)

100

Average BER

10 15 Average SNR per bit per branch [dB]

MRC-4 SC/MRC-2/4 SSC/MRC-2/4 MRC-2

10−2 10−4 10−6 10−8

0

5

10 15 Average SNR per bit per branch [dB]

20

25

Figure 9.88 Comparison of the average BER of BPSK with (a) MRC-2, (b) SSC/MRC-2/4, (c) SC/MRC-2/4, and (d) MRC-4 over Nakagami-m fading channels.

9.11.5.1 Optimum (LLR-Based) GSC for Equiprobable BPSK Consider first the selection of two branches, say, i and j , out of the total available L branches. Then, following a development similar to that in Chapter 7 and furthermore letting hi = αi ej θi denote the complex channel gain, the LLR, i,j , for the transmitted signal x is given by i,j

3 2 √ ! √ 3 Pr x = Es !hi , yi , hj , yj 4 Es 2 ∗ 3= = ln 2 Re hi yi + h∗j yj √ ! ! N0 Pr x = − Es hi , yi , hj , yj

(9.573)

534

PERFORMANCE OF MULTICHANNEL RECEIVERS

where yi = hi x + ni is the equivalent lowpass received signal in the i th diversity branch and ni is the corresponding complex AWGN with variance N0 /2 per dimension. The maximum-likelihood (ML) decision rule based on observation of yi and yj and √ hi and hj is to decide in favor of √ perfect knowledge of the channel gains x = Es if i,j > 0 or in favor of x = − Es if i,j ≤ 0. To evaluate the error probability associated with this decision rule, we proceed as follows. Since from the law of total probability = < = <  !  ! Pr x = Es !hi , yi , hj , yj + Pr x = − Es !hi , yi , hj , yj = 1

(9.574)

then combining (9.573) and (9.574) gives 2 3 √ ! 1 Pr x = Es !hi , yi , hj , yj = 1 + e −i,j 3 2 √ ! 1 Pr x = − Es !hi , yi , hj , yj = 1 + e i,j

(9.575)

From the decision rule given, above, the conditional probability of error is given by = <  !   !  Pi,j E !hi , hj = Pr xˆ = Es , x = − Es !hi , yi , hj , yj = <  !  + Pr xˆ = − Es , x = Es !hi , yi , hj , yj

(9.576)

where xˆ denotes the detector’s decision on x. Since Pr {AB |C } ≤ P {A |C } or Pr {AB |C } ≤ P {B |C }, then using this in (9.576) gives the upper bound on conditional error probability = < = <  !  !  !  Pi,j E !hi , hj ≤ Pr x = − Es !hi , yi , hj , yj + Pr xˆ = − Es !hi , yi , hj , yj (9.577) √ Assuming that i,j > 0 or equivalently, xˆ = Es , then the first term is evaluated from (9.575), whereas the second term is equal to zero since it contradicts the assumption. Thus, we have the upper bound  !  Pi,j E !i,j ≤

1 1 + e i,j

(9.578)

Similarly, for the same assumption, to obtain a lower bound on the conditional error probability, we have that = <  !   !  Pi,j E !i,j = 1 − Pr xˆ = Es , x = Es !hi , yi , hj , yj = <  !  + Pr xˆ = − Es , x = − Es !hi , yi , hj , yj

COMBINING IN DIVERSITY-RICH ENVIRONMENTS

= <  ! ≥ 1 − Pr x = Es !hi , yi , hj , yj < =  ! − Pr xˆ = − Es !hi , yi , hj , yj =1−

535

(9.579)

1 1 = −i,j 1+e 1 + e i,j

where again the second term equates to zero since it contradicts the assumption. Had we assumed i,j ≤ 0, then upper and lower bounds would have been given by (9.578) and (9.579) with i,j replaced by −i,j . Finally, since the upper and lower bounds are identical, they must be equal to the true conditional error probability whereupon we obtain the desired result  !  Pi,j E !i,j =

1 1 + e |i,j |

(9.580)

Thus, we see that the sign of i,j is used to make the decision whereas the magnitude of i,j affects the reliability of the decision; thus, the larger |i,j | is, the smaller is the error probability. Since the conditional error probability is a monotonically decreasing function of |i,j |, we see that the optimum (in the sense of minimum error probability) GSC scheme is to choose that pair of branches that provides the largest |i,j |. For combining Lc branches, the optimum selection rule generalizes to selecting those branches, say, i1 , i2 , . . . iLc , that maximize the magnitude of i1 ,i2 ,...,iLc

/L 0 √ √ Lc c
3 2 4 Es 4 Es
∗ = Re hik yik = Re h∗ik yik N0 N0 k=1

(9.581)

k=1

resulting in a conditional error probability   ! P E !i1 ,i2 ,...,iLc =

1 1+e

|i1 ,i2 ,...,iL |

(9.582)

c

For Lc < L/2, an equivalent selection/data decision rule can be stated in terms of the ordered RVs Re{h∗1:L y1:L } ≥ Re{h∗2:L y2L } ≥ · · · ≥ Re{h∗Lc :L yLc :L }. Specifically, one can first make a single comparison between the magnitudes of the sum of the Lc smallest and the sum of the Lc largest Re{h∗i:L yi:L } terms [i.e.,  √  L √  c ∗ ∗ (4 Es /N0 )| L i=1 Re{hi:L yi:L }| vs. 4 Es /N0 | i=L−Lc +1 Re{hi:L yi:L }|] and then choose the larger of the two for making the data decision. For Lc ≥ L/2, it can be shown [196, App. A] that one can equivalently subdivide the group of branches into two arbitrarily selected subgroups of Lc and L − Lc branches, respectively, and make a single comparison between the magnitudes of the LLR for these two subgroups, again choosing the larger of the two for making the data decision. Interestingly enough, it is also shown [196, App. A] that, independent of Lc (provided it is greater than or equal to L/2), such an equivalent decision rule results in

536

PERFORMANCE OF MULTICHANNEL RECEIVERS

performance equivalent to L-branch MRC. Thus, we can conclude that there is no need to choose more than L/2 branches for the LLR-based GSC scheme, or equivalently, LLR-based GSC with L/2 branches is sufficient to achieve the conditional performance of L-branch MRC : √ L 3 4 Es
2 ∗ 1 (9.583) ;  = Re hi:L yi:L Pi,j (E |MRC ) = MRC | | 1 + e MRC N0 i=1

Finally, the average BER obtained by statistically averaging (9.582) over i1 ,i2 ,...,iLc serves as a lower bound on the average BER of any GSC scheme. 9.11.5.2 Envelope-Based GSC Once again we consider first the selection of two out of L branches. Since |Re{A + B}| ≤ |Re{A}| + |Re{B}| and Re{AB} ≤ |A||B|, then, from (9.573) we have √ ! ! ! 2 3! ! 2 3! !i,j ! ≤ 4 Es !Re h∗ yi ! + !Re h∗ yj ! i j N0 (9.584) √ ! ! ! ! 4 Es  ! ! ! ! |hi | |yi | + hj yj ≤ N0 Thus, generalizing to Lc out of L branches, we  propose ! to! !select ! those branches, Lc ! ! !yi !. Such a scheme h say, i1 , i2 , . . . iLc , that result in the maximum i k k k=1 is referred to as envelope-based GSC [196]. If we form the ordered RVs |h | |y1:L | ≥ |h2:L | |y2:L | ≥ · · · ≥ |hL:L | |yL:L |, then the maximum sum becomes 1:L Lc i=1 |hi:L | |yi:L | and hence the decision metric is given by EnvGSC =

√ Lc 3 2 4 Es
Re h∗i:L yi:L N0

(9.585)

i=1

which is a simpler test than that in (9.581). The corresponding expression for the conditional error probability is then P (E |EnvGSC ) =

1 1 + e |EnvGSC |

(9.586)

9.11.5.3 Optimum GSC for Noncoherently Detected Equiprobable Orthogonal BFSK   Consider first the case of selecting two out of L branches. Let yi = yi1, yi2 denote the pair of energy detector outputs corresponding to the i th branch, one of which (depending on the hypothesis) corresponds to signal plus noise and the other to noise only. Then, assuming i.i.d. fading (i.e., γ l = |hl |2 Es /N0 is independent of l ), the LLR for branches i and j is given by [196]



 1     γ γ 1 i,j = Yi + Yj (yi2 − yi1 ) + yj 2 − yj 1 = N0 1 + γ N0 1 + γ (9.587)

POST-DETECTION COMBINING

537

Hence, as before, the LLR-based selection rule is two choose that pair of branches that maximizes the magnitude of i,j in (9.587). Generalizing to selecthe optimum selection rule is based on the largest ! !tion of Lc out of L branches, !Yi + Yi + . . . + Yi !. For L-branch noncoherent EGC with post-detection squareLc 1 2  law combining, the decision is based on the sign of L l=1 Yl . Using a development similar to that in Ref. 196 App. A, and analogous to the comparison previous made between LLR-based GSC and MRC of coherent BPSK, it can be shown that for noncoherently detected orthogonal BFSK, LLR-based GSC with L/2 branches is sufficient to achieve the conditional performance of L-branch EGC with post-detection square-law combining. 9.12

POST-DETECTION COMBINING

As we mentioned earlier, among the simpler diversity combining schemes, the two most popular are dual-branch SC and SSC. These two combining schemes can be implemented not only in a pre-detection fashion (the most common fashion and the one we focused on thus far in this chapter) but also in a post-detection fashion. For instance, in the case of post-detection SC (post-SC), the receiver selects the branch with the largest signal plus noise (S + N) output (rather than the largest SNR as is the case with pre-detection SC) [198–200]. As with post-SC, post-detection SSC (post-SSC) does not need to monitor the instantaneous branch’s SNR, but rather obtains the information needed to implement the switching strategy from the same output quantity used for data detection (i.e., matched-filter output). In this section, we investigate the performance of post-SSC for noncoherent binary and then M-ary orthogonal FSK operating in the presence of slow flat fading modeled by Rayleigh and Rician distributions [201,202] and compare its performance to that of post-detection SC as well as traditional pre-detection SC (pre-SC) and SSC (pre-SSC). We limit ourselves here to dual-branch combining but extensions to the multibranch case are available in [203]. 9.12.1

System and Channel Models

9.12.1.1 Overall System Description We consider BFSK digital signaling over a slowly varying flat-fading channel. We assume that the two binary symbols of duration Tb are equiprobable, have the same energy Eb , and are transmitted as the carrier frequencies f1 and f2 , respectively. The transmitted signal is received by a dual-branch switched diversity system as shown in Fig. 9.89. 9.12.1.2 Channel Model We denote by {αl }2l=1 and {θl }2l=1 the random channel amplitudes and phases for antenna 1 and 2. We assume that the sets {αl }2l=1 and {θl }2l=1 are mutually independent and, by the slowly varying assumption, they remain constant over at least one symbol duration. We also assume that the separation between the two antennas is more than one-half a wavelength to guarantee fading uncorrelation (and therefore

538

Antenna 2

Antenna 1

Long-Term Average

Long-Term Average

Square- W22 Law Detector

Square- W21 Law Detector

BPF ( f1)

BPF ( f2)

Square- W12 Law Detector

BPF ( f2)

Square- W11 Law Detector

1

W2

sample at t = nTb

W1

2

Threshold wT

Threshold wT

−

+

−

+

−1

Data Bit +1 Decision

|W1(n)| (or |W2(n)|)

Absolute Value

Switch Driver

Comparator

Threshold Switch

Channel Switch

sample at t = nTb

Figure 9.89 Block diagram of noncoherent BFSK with post-detection SSC.

RF Front End

RF Front End

BPF ( f1)

POST-DETECTION COMBINING

539

full diversity gain) between the two received signals. While Section 9.12.4 considers non-line-of-sight scenarios for which the {αl }2l=1 are Rayleigh distributed, Section 9.12.5 considers more favorable radio propagation channels with more benign fading environments and for which the {αl }2l=1 are rather Rician distributed. Finally, the performance analysis is quite general and will cover the scenario where the average fading powers l = αl2 (l = 1, 2) are not necessarily equal. However, the important identically distributed special case where l = αl2 (l = 1, 2) are the same and equal to  will also be treated. 9.12.1.3 Receiver In Fig. 9.89, the switching is performed at discrete instants of time tn = nTb , where n is an integer. The switching strategy and mechanism illustrated in Fig. 9.89 will be explained in detail later on, but for the moment it suffices to assume that, at any given time, the switch is connecting one of the antennas to the receiver. Regardless of which antenna is connected to the receiver, after passing through the fading channel, the received signal is corrupted by an AWGN with two-sided power spectral density N0 /2. The AWGN is assumed to be statistically independent of the fading amplitudes {αl }2l=1 and phases {θl }2l=1 . The receiver employs noncoherent demodulation consisting of two bandpass filters, one tuned to frequency f1 and the other to frequency f2 , followed by square-law detection. Assuming that antenna 1 (2) is connected to the receiver, let W11 (W21 ) denote the output of the upper square-law detector (corresponding to frequency f1 ) and W12 (W22 ) denote the output of the lower square-law detector (corresponding to frequency f2 ). The receiver computes the decision variable W1 = W11 − W12 (W2 = W21 − W22 ) at time instants tn = nTb , and then selects the binary symbol corresponding to f1 if W1 ≥ 0 (W2 ≥ 0) or the binary symbol corresponding to f2 if W1 < 0 (W2 < 0). 9.12.2

Post-detection Switched Combining Operation

9.12.2.1 Switching Strategy and Mechanism Contrary to classical pre-detection SSC in which the switching is driven by the value of the instantaneous SNR of the connected antenna, the post-detection SSC scheme relies on the magnitude of the decision variables W1 or W2 (depending on which antenna is connected to the receiver at a given time) to assess the quality/state of the channel and to trigger when necessary a switch to the alternate antenna. More specifically, the switching mechanism operates in the following manner. Without loss of generality, assume that antenna 1 (2) is connected to the receiver. At time instants tn = nTb , the magnitude of W1 (W2 ) |W1 (n)| (|W2 (n)|) is compared to an absolute predetermined threshold wT1 (wT2 ). If |W1 (n)| (|W2 (n)|) is above wT1 (wT2 ), the receiver uses W1 (W2 ) to make a decision and remains connected to antenna 1 (2) for the next Tb seconds. On the other hand, if |W1 (n)| (|W2 (n)|) falls below wT1 (wT2 ), the receiver switches to antenna 2 (1), regardless of the magnitude |W2 (n)| (|W1 (n)|) of the output of the alternative

540

PERFORMANCE OF MULTICHANNEL RECEIVERS

(switch-to) antenna, and it uses W2 (W1 ) to make a decision. Mathematically speaking, and according to the switching mode of operation we just described, we can write the sequence of variables W (n) from which decisions on the data are being made as   W (n − 1) = W1 (n − 1) and |W1 (n)| ≥ wT1 or W (n) = W1 (n) if (9.588)  W (n − 1) = W2 (n − 1) and |W2 (n)| < wT2 and   W (n − 1) = W2 (n − 1) and |W2 (n)| ≥ wT2 or W (n) = W2 (n) if  W (n − 1) = W1 (n − 1) and |W1 (n)| < wT1

(9.589)

9.12.2.2 Switching Threshold Similar to the traditional pre-detection SSC, the setting of the predetermined threshold is an additional important system design issue for post-detection SSC. For instance, it should be high enough to prevent the antenna switching unit from almost being locked to one of the diversity branches resulting in a poor diversity gain. On the other hand, it should also be low enough to avoid the continuous switching between antennas, which results not only in a poor diversity gain but also in an undesirable increase in the rate of the switching transients on the transmitted data stream. In the subsequent sections, our analysis will show that there is indeed an optimal threshold (in the minimum average BER sense) for each branch. However, these optimal thresholds wT∗1 and wT∗2 depend on the local means (i.e., average fading power 1 and 2 ), and as such optimal switching threshold settings require (as illustrated in Fig. 9.89) an estimate of the short-term average fading powers 1 and 2 for antennas 1 and 2, respectively. 9.12.3

Average BER Analysis

Without loss of generality, suppose that the binary symbol corresponding to f1 is transmitted and that the switch is at a given time connected to antenna 1. Following the switching mode of operation given in (9.588) and (9.589) and the demodulation decision rules described at the end of section 9.12.1.3, an erroneous decision occurs if the magnitude of W1 exceeds the threshold wT1 but W1 < 0, or if the magnitude of W1 falls below the threshold wT1 (and therefore antenna 2 is connected) but W2 < 0. A similar observation can be made when the switch is at a given time connected to antenna 2. In mathematical terms and using the sequences W1 (n), W2 (n), and W (n) defined at the end of Section 9.12.2.1, we can write the probability of error under the equiprobable bits assumption as Pb (E) = Pr[W (n) = W1 (n) and W1 (n) < 0] + Pr[W (n) = W2 (n) and W2 (n) < 0]

(9.590)

POST-DETECTION COMBINING

541

which, in view of the aforementioned observations, can be rewritten as Pb (E) = Pr[|W1 (n)| ≥ wT1 and W1 (n) < 0] Pr[W (n − 1) = W1 (n − 1)] + Pr[|W1 (n)| < wT1 and W2 (n) < 0] Pr[W (n − 1) = W1 (n − 1)] (9.591) + Pr[|W2 (n)| ≥ wT2 and W2 (n) < 0] Pr[W (n − 1) = W2 (n − 1)] + Pr[|W2 (n)| < wT2 and W1 (n) < 0] Pr[W (n − 1) = W2 (n − 1)] Defining

p1= Pr[W (n − 1) = W1 (n − 1)]

(9.592)

and

p2= Pr[W (n − 1) = W2 (n − 1)]

(9.593)

as the probability (or equivalently the percentage of time) that antennas 1 and 2 are connected, respectively, we can write Pb (E) in (9.592) as Pb (E) = p1 (Pr[|W1 (n)| ≥ wT1 and W1 (n) < 0] + Pr[|W1 (n)| < wT1 and W2 (n) < 0]) +p2 (Pr[|W2 (n)| ≥ wT2 and W2 (n) < 0] + Pr[|W2 (n)| < wT2 and W1 (n) < 0])

(9.594)

which can be rewritten as Pb (E) = p1 (Pr[W1 (n) < −wT1 ] + Pr[−wT1 < W1 (n) < wT1 and W2 (n) < 0]) +p2 (Pr[W2 (n) < −wT2 ] + Pr[−wT2 < W2 (n) < wT2 and W1 (n) < 0])

(9.595)

Under the fading independence assumption between the two antennas, the decision variables W1 (n) and W2 (n) are also independent, and as such the probability of error Pb (E) in (9.595) can be written as   Pb (E) = p1 Pr[W1 (n) < −wT1 ] + Pr[−wT1 < W1 (n) < wT1 ] Pr[W2 (n) < 0]   +p2 Pr[W2 (n) < −wT2 ] + Pr[−wT2 < W2 (n) < wT2 ] Pr[W1 (n) < 0] (9.596) which can be expressed solely in terms of the CDFs FW1 (·) and FW2 (·) of the decision variables W1 and W2 , respectively, as     Pb (E) = p1 FW1 (−wT1 ) + FW1 (wT1 ) − FW1 (−wT1 ) FW2 (0)     +p2 FW2 (−wT2 ) + FW2 (wT2 ) − FW2 (−wT2 ) FW1 (0) (9.597)

542

PERFORMANCE OF MULTICHANNEL RECEIVERS

9.12.3.1 Identically Distributed Branches For i.i.d. fading on the two branches, FW1 (·) = FW2 (·)=FW (·), wT1 = wT2 =wT , and we can see by intuition that p1 = p2 = 12 . As a result, (9.597) reduces in this case to Pb (E) = FW (−wT ) + [FW (wT ) − FW (−wT )] FW (0)

(9.598)

9.12.3.2 Nonidentically Distributed Branches To find the average BER in this case, we need to evaluate just p1 and p2 . In view of the definitions of p1 in (9.592) and p2 in (9.593), as well as the post-detection SSC mode of operation (as described in Section 9.12.2.1), we can write p1 = Pr[(W (n − 2) = W1 (n − 2)

and |W1 (n − 2)| > wT1 )

or (W (n − 2) = W2 (n − 2) and |W2 (n − 1)| < wT2 )]

(9.599)

and p2 = Pr[(W (n − 2) = W2 (n − 2)

and |W2 (n − 2)| > wT2 )

or (W (n − 2) = W1 (n − 2) and |W1 (n − 2)| < wT1 )]

(9.600)

We then can write p1 and p2 in terms of the CDF of the magnitude of the decision variables W1 and W2 as p1 = p1 (1 − F|W1 | (wT1 )) + p2 F|W2 | (wT2 ) p2 = p2 (1 − F|W2 | (wT2 )) + p1 F|W1 | (wT1 )

(9.601)

Using the fact that the events W (n) = W1 (n) and W (n) = W2 (n) are mutually exclusive (i.e., p1 + p2 = 1), we can solve for p1 and p2 to get p1 =

F|W2 | (wT2 ) , F|W1 | (wT1 ) + F|W2 | (wT2 )

p2 =

F|W1 | (wT1 ) , F|W1 | (wT1 ) + F|W2 | (wT2 )

(9.602)

which can be written solely in terms of the CDF of the decision variables W1 and W2 as p1 =

FW2 (wT2 ) − FW2 (−wT2 ) FW1 (wT1 ) + FW2 (wT2 ) − FW1 (−wT1 ) − FW2 (−wT2 )

p2 =

FW1 (wT1 ) − FW1 (−wT1 ) FW1 (wT1 ) + FW2 (wT2 ) − FW1 (−wT1 ) − FW2 (−wT2 )

(9.603)

POST-DETECTION COMBINING

543

Combining (9.603) with (9.597) provides the general result for the average BER, which reduces when wT1 = wT2 = wT , FW1 (·) = FW2 (·) = FW (·), and p1 = p2 = 12 to the i.i.d. result given in (9.598). 9.12.4

Rayleigh Fading

In this subsection, we consider the average BER performance of noncoherent BFSK with post-detection SSC in a no-line-of-sight scenario and as such we assume that the fading affecting the two diversity antennas is Rayleigh distributed. Assuming that the switch is connected to antenna 1, then the decision variable W1 can be written as W1 = W11 − W12 = |2Eb α1 ej θ1 + N11 |2 − |N12 |2

(9.604)

where N11 and N12 are zero-mean complex Gaussian RVs with variance 4Eb N0 . We can rewrite W11 as R 2 I 2 | + |2Eb α1 sin θ1 + N11 | W11 = |2Eb α1 cos θ1 + N11

(9.605)

R I and N11 are zero-mean real Gaussian RVs with variances 2Eb N0 . where N11 Under the Rayleigh fading assumption, 2Eb α1 cos θ1 and 2Eb α1 sin θ1 are zeroR R I = 2Eb α1 cos θ1 + N11 and X11 = 2Eb α1 sin θ1 + mean Gaussian RVs. Hence X11 I N11 are zero-mean real Gaussian RVs with the same variance σ12 = 21 Eb + 2Eb N0 = 2Eb N0 (1 + γ 1 ), where γ 1 = 1 Eb /N0 is the average SNR of the first diversity branch. Therefore W11 is a central chi-square distribution with two degrees R I and N12 are zeroof freedom and parameter σ12 . On the other hand, since N12 2 mean real Gaussian RVs with the same variance σ2 = 2Eb N0 , W12 is also a central chi-square distribution with two degrees of freedom and parameter σ22 . Using the result Eq. (4.5) from Ref. 205 for the CDF of the difference of two independent central chi-square RVs with two degrees of freedom and with parameters σ12 = 2Eb N0 (1 + γ 1 ) and σ22 = 2Eb N0 , we obtain the CDF of the decision variable W1 as 

1 w   w≤0   2 + γ exp 4E N , b 0 1 (9.606) FW1 (w)

 γ 1 + w  1  , w≥0 exp −  1− 2 + γ1 4Eb N0 (1 + γ 1 )

Similarly, if the switch is connected to antenna 2, then going through the same steps, it can be shown that the CDF of the decision variable W2 is given by 

1 w   , w≤0 exp   2+γ 4Eb N0 2 FW2 (w) (9.607)

 1 + γ2 w   , w≥0 exp −  1− 2 + γ2 4Eb N0 (1 + γ 2 )

544

PERFORMANCE OF MULTICHANNEL RECEIVERS

Substituting (9.606) and (9.607) in (9.597) along with (9.603), we obtain the average BER for the non-i.i.d. case. 9.12.4.1 Identically Distributed Branches In the following subsections, we focus on the balanced branches case for which the fading is assumed to be i.i.d. on the two diversity paths. Average BER For the special i.i.d. Rayleigh branches cases, with γ 1 = γ 2 = γ and wT1 = wT2 = wT , we have FW1 (·) = FW2 (·) = FW (·), which when substituted in (9.598), leads to the average BER 1



wT wT 1+γ 1 exp − exp − +1− Pb (E) = 2+γ 4Eb N0 2+γ 4Eb N0 (1 + γ )

4 wT 1 (9.608) exp − − 2+γ 4Eb N0 Expressing this result in terms of the normalized threshold ηT = wT /(Eb2 ) and combining like terms, (9.608) can be written as 1 "



#4 1+γ ηT γ ηT γ 1 Pb (E) = 1+ exp − − exp − (9.609) 2+γ 2+γ 4 4(1 + γ ) Figure 9.90 plots the average BER of noncoherent BFSK with respect to the normalized switching threshold ηT for various values of the average SNR. All curves show a clear average BER minimum corresponding to the optimal switching threshold ηT∗ . In addition, for high values of average SNR, the curves decrease very sharply with respect to ηT before reaching their minimum. In contrast, these same curves tend to increase more slowly beyond the optimum ηT∗ and hence it is safer to set the optimal threshold for a slightly higher value then the optimal threshold rather than a slightly lower value. Note also that the choice of the switching threshold is more critical for high values of the average SNR per branch since the minimum average BER is more accentuated in these cases. Finally, it is also interesting to note that the optimal switching threshold ηT∗ is a decreasing function of the average SNR which implies that systems experiencing better channels will tend to switch more often to exploit the potentially better fading conditions on the alternative branch. Optimal Threshold For both extremes ηT = 0 and ηT = ∞, the average BER in (9.609) reduces to the BER of a single-branch (no-diversity) noncoherent BFSK receiver given by Pb− (E) =

1 2+γ

(9.610)

Since the average BER is a continuous function of the normalized threshold ηT , there is, as expected (and as illustrated in Fig. 9.90), an optimal value of ηT for

POST-DETECTION COMBINING

545

100

hT∗ = 2.374 g¯ = 5 dB Average Bit Error Rate Pb(E )

10−1 hT = 1.055

g¯ = 10 dB g¯ = 15 dB

hT∗ = 0.4547

10−2

g¯ = 20 dB

10−3 hT∗ = 0.1864

10−4

0

0.5

1

1.5 2 2.5 3 3.5 Normalized Switching Threshold hT

4

4.5

5

Figure 9.90 Average BER of noncoherent BFSK with post-detection SSC versus normalized switching threshold over i.i.d. Rayleigh channels.

which the average BER is minimal. This optimal value ηT∗ is a solution of the equation dPb (E) !! (9.611) !ηT =ηT∗ = 0 dηT Substituting (9.609) in (9.611), and solving for ηT∗ leads after some simplification to the desired solution ηT∗ =

4(1 + γ ) ln(1 + γ ) γ2

(9.612)

which is indeed a decreasing function of the average SNR. Minimum Average BER We now study the minimum average BER Pb∗ (E) for the i.i.d. Rayleigh case. This BER is obtained by substituting (9.612) into (9.609), which after some simplifications gives the following compact closed-form expression: # " γ 1 ∗ −1/ γ (9.613) 1− (1 + γ ) Pb (E) = 2+γ 2+γ

546

PERFORMANCE OF MULTICHANNEL RECEIVERS

1 0.9

f (x) g (x)

0.8

f (x) and g (x)

0.7 0.6 0.5 0.4 0.3 0.2 0.1 0

0

10

20

Figure 9.91

30

40

50 x

60

70

80

90

100

Plot of the f and g functions versus their arguments.

Let f (x) = (1 + x)−1/x

(9.614)

We can see from Fig. 9.91 that for positive values of x, f (x) is bounded between e−1  0.36 and 1. This, combined with the fact that γ /(2 + γ ) ≤ 1, leads to Pb∗ (E) γ = g(γ ) = 1 − f (γ ) ≤ 1 2+γ Pb− (E)

(9.615)

Figure 9.91 plots the BER reduction factor g(x) as a function of x and shows that the post-detection SSC diversity gain improves as the average SNR per branch increases since the BER reduction factor decreases with increasing x. Comparison between Pre-detection and Post-detection SSC We now compare the minimum average BER of noncoherent BFSK with conventional predetection SSC and with post-detection SSC. For traditional pre-detection SSC, Abu-Dayya and Beaulieu showed that the average BER of noncoherent BFSK

POST-DETECTION COMBINING

547

over i.i.d. Rayleigh paths is given by [9, Eq. (15)] "



# γT 1 1 1 1 − exp − + exp −γT + Pb (E) = 2+γ γ 2 γ

(9.616)

where γT is the SNR switching threshold whose optimal value γT+ is given by [9, Eq. (16)]

γ γT+ = 2 ln 1 + 2

(9.617)

which, in contrast to the optimal threshold for post-detection SSC, is an increasing function of the average SNR. Substituting (9.617) in (9.616), and after some simplifications, the minimum average BER of noncoherent BFSK with pre-detection SSC Pb+ can be written in the compact form Pb+ (E)

1 = 2+γ



γ 1− 2+γ



γ 1+ 2

−2/ γ  (9.618)

which can be expressed in term of the f -function defined in (9.614) as Pb+ (E) =

# " γ 1 1− f (γ /2) 2+γ 2+γ

(9.619)

From Fig. 9.91, f (·) is an increasing   function of its argument. Therefore f (γ /2) ≤ γ γ f (γ ) ≤ 1 − 2+γ f (γ /2) , which in view of (9.615) and f (γ ), and thus 1 − 2+γ ∗ + (9.619) gives Pb (E) ≤ Pb (E) . This leads to the key conclusion that optimal postdetection SSC provides an improved BER performance over optimal pre-detection SSC in the whole average SNR range. Figure 9.92 illustrates this improved average BER performance of post-detection SSC compared to pre-detection SSC. This figure also shows that post-detection SSC comes closer to the performance of predetection SC [3, Sect. 5.5.2] but does not require the simultaneous monitoring of the instantaneous SNR at the output of the two diversity antennas. 9.12.4.2 Nonidentically Distributed Branches We now consider the effect of the branch average fading power unbalance on the average BER performance of noncoherent BFSK with post-detection SSC. In this case, the average BER in (9.597), along with (9.603), (9.606), and (9.607), has to be minimized with respect to ηT1 = wT1 /(1 Eb2 ) and ηT2 = wT2 /(2 Eb2 ), to reach the global optimal performance. As an example, Fig. 9.93 illustrates the existence of these optimum thresholds ηT∗1 and ηT∗2 for the case of γ 1 = 5 dB and γ 2 = 10 dB. Since the overall average BER can be viewed as a weighted sum (by p1 and p2 ) of the average BER on the individual antennas [as shown in (9.597)], rather than going through an exhaustive two-dimensional search, one simpler procedure

548

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

No Diversity Pre-Detection SSC Post-Detection SSC PreDetection SC

Average Bit Error Rate Pb(E )

10−1

10−2

10−3

10−4

10−5

10−6

0

5

10

15

20

25

30

Average SNR per Bit per Branch [dB] Figure 9.92 Comparison of the average BER of noncoherent BFSK with no diversity and with pre-detection and post-detection SSC over i.i.d. Rayleigh channels.

relies on minimizing these individual BERs. This “individual” optimization can be achieved by choosing the normalized thresholds as per (9.612): ηT∗l =

4(1 + γ l ) ln(1 + γ l ) (l = 1, 2). γ 2l

(9.620)

While this simple choice of thresholds does not result in the global minimum average BER, it yields nearly optimal results especially for a small unbalance between the diversity paths, as illustrated in Fig. 9.94. 9.12.5

Impact of the Severity of Fading

We now look at the impact of the severity of fading on post-SSC by analyzing its BER performance in a Rician environment [201]. Note that the analysis of this scheme over Nakagami-m fading in Ref. 201 has an error in characterizing the distribution of W11 and as such is not correct. This analysis has been revised and corrected by Haghani and Beaulieu [204].

549

POST-DETECTION COMBINING

0.2

Average Bit Error Rate Pb(E )

0.18 0.16 0.14 0.12 0.1 0.08 0.06 0.04 5

* = 3.397, hT2 * = 1.024 hT1 4 3 Norma 2 lized S 1 witchin g Thre shold h

0

T2

1

0

zed ormali

4 3 h 1 shold T e r h T hing

2

5

Switc

N

Figure 9.93 Average BER of noncoherent BFSK versus the normalized thresholds over unbalanced Rayleigh channels for γ 1 = 5 dB and γ 2 = 10 dB.

Average Bit Error Rate Pb(E )

100

Global Optimization Individual Optimization

10−1

g¯2 = g¯ 1 10−2

g¯2 = 2g¯ 1

10−3 g¯2 = 5g¯1

10−4

0

5

10

15

20

25

30

Average SNR per Bit per Branch [dB]

Figure 9.94 Comparison of the average BER of noncoherent BFSK over unbalanced Rayleigh channels with global and individual optimization.

550

PERFORMANCE OF MULTICHANNEL RECEIVERS

9.12.5.1 Average BER Under the Rician fading scenario and assuming that the switch is connected to antenna 1, then 2Eb α1 cos(θ1 ) and 2Eb α1 sin(θ1 ) are nonzero-mean Gaussian RVs. R R I I = 2Eb α1 cos θ1 + N11 and X11 = 2Eb α1 sin θ1 + N11 are nonzero-mean Hence X11 2 real Gaussian RVs with the same variance σ1 = X1 /(2(1 + K1 )), where K1 is the Rician factor of the first diversity path and   I 2   R 2 X1 = E X11 + X11         R 2 I 2 + E N11 = E (2α1 Eb cos θ1 )2 + (2α1 Eb sin θ1 )2 + E N11 = 4Eb2 1 + 2(2Eb N0 ) = 4Eb N0 (1 + γ 1 )

(9.621)

Therefore, under the Rician fading assumption, W11 as defined in (9.605) is a noncentral chi-square distribution with two degrees of freedom, a noncentrality parameter a 2 given by a2 =

K1 X1 4Eb N0 K1 (1 + γ 1 ) = 1 + K1 1 + K1

(9.622)

2Eb N0 (1 + γ 1 ) 1 + K1

(9.623)

and σ12 =

R I On the other hand, since N12 and N12 in (9.604) are still zero-mean real Gaussian 2 RVs with the same variance σ2 = 2Eb N0 , W12 is a central chi-square distribution with two degrees of freedom and parameter σ22 = 2Eb N0 . Using the result in [205, Eq. (4.33)] for the CDF of the difference between independent noncentral chisquare and central chi-square RVs, both with two degrees of freedom and with the above-mentioned a 2 , σ12 , and σ22 parameters, we obtain the CDF of the decision variable W1 in the Rician case as



 1 + K1 w K1 (1 + γ 1 )   exp − , w≤0 exp −   2 + γ 1 + K1 2 + γ 1 + K1 4Eb N0          √ w(1 + K ) 1 + K1  1  1−Q  2K1 , +   2Eb N0 (1 + γ 1 ) 2 + γ 1 + K1



FW1 (w) = w K1 (1 + γ 1 )    exp − × exp −   2 + γ 1 + K1 4Eb N0       

   1 + K1 w(2 + γ 1 + K1 )   , 2K1 , w ≥ 0.  ×Q 2 + γ 1 + K1 2(1 + γ 1 )Eb N0 (9.624)

POST-DETECTION COMBINING

551

Similarly, if the switch is connected to antenna 2, then, going through the same steps, it can be shown that the CDF of the decision variable W2 is given by



 1 + K2 w K2 (1 + γ 2 )   exp − , w≤0 exp −   2 + γ 2 + K2 2 + γ 2 + K2 4Eb N0          √ w(1 + K ) 1 + K2  2   1−Q 2K2 , +   2Eb N0 (1 + γ 2 ) 2 + γ 2 + K2



FW2 (w) = w K2 (1 + γ 2 )    exp − × exp −   2 + γ 2 + K2 4Eb N0       

   1 + K2 w(2 + γ 2 + K2 )   , 2K2 , w ≥ 0.  ×Q 2 + γ 2 + K2 2(1 + γ 2 )Eb N0 (9.625) Substituting (9.624) and (9.625) along with (9.603) in (9.597), we obtain the average BER for the non-i.i.d. Rician case. For the special, i.i.d. Rician branches cases with γ 1 = γ 2 = γ and K1 = K2 = K, FW1 (·) = FW2 (·) and the average BER reduces to   

/ √ K(1 + γ ) (1 + K)ηT γ 1+K exp − 2K, Pb (E) = 1−Q 2+K +γ 2+K +γ 2(1 + γ )



K(1 + γ ) ηT γ 1+K exp exp − + 2+K +γ 2+K +γ 4  

 (2 + γ + K)ηT γ 1+K , ×Q 2K 2+γ +K 2(1 + γ ) "

#

4 1+K K(1 + γ ) ηT γ + 1− exp − exp − (9.626) 2+K +γ 2+K +γ 4 As a double check, using Q(0, x) = e−x /2 , (9.626) reduces to the Rayleigh result in (9.609) for K = 0, as expected. For ηT = 0 (the receiver never switches antennas) and ηT = ∞ (the receiver always switches antennas), it is easy to check that the average BER in (9.626) reduces to the no-diversity result

1+K K(1 + γ ) Pb (E) = exp − (9.627) 2+K +γ 2+K +γ 2

Thus, there exists an optimum threshold that minimizes Pb (E) at each average SNR γ . This optimum threshold can be determined by differentiating Pb (E) of (9.626) with respect to ηT and equating the result to zero. Using the fact that

√  √ √ x+y d 1 Q( x, y) = − exp − I0 xy (9.628) dy 2 2

552

PERFORMANCE OF MULTICHANNEL RECEIVERS

then, after some manipulations, the following transcendental equation results: 

#  " ∗

ηT γ 1+K γ (1 + K) ∗ K(1 + K)γ ∗ I0 − exp − K + η ηT exp − 4 1+γ 4(1 + γ ) T 1+γ

1 ∗

η γ K(1 + γ ) 2+K +γ 1+K exp − − exp T − 2+γ +K 2+K +γ 1+γ 4   "

# K(1 + K) 2 + K + γ γ ηT∗ K(1 + K)γ ∗ × exp − + I0 ηT 2+K +γ 1+γ 4 1+γ   ∗

 ηT γ (2 + γ + K)ηT∗ γ 1+K 2K + exp Q , 4 2+γ +K 2(1 + γ ) ∗ 4 η γ =0 (9.629) + exp − T 4 whose solution is the optimum threshold ηT∗ . 9.12.5.2 Numerical Examples and Discussion Figure 9.95 plots the average BER of noncoherent BFSK with post-detection SSC versus normalized switching threshold over i.i.d. Rician channels for γ = 15 dB. This figure illustrates the existence of the optimum threshold for various Rician factors and shows that this optimum threshold is an increasing function of the Rician factor, which means that the switching rate should be reduced as the strength of the line of sight increases. Figure 9.96 compares the optimal average BERs of noncoherent BFSK with post-detection and pre-detection SSC [as per the average BER expression given in Ref. 10, Eq. (11)] along with the optimal threshold given by Ref. 10, Eq. (12)] over i.i.d. Rician channels. It is interesting to note that the gain of post-detection SSC over pre-detection SSC increases as the severity of fading decreases. 9.12.6 9.12.6.1

Extension to Orthogonal M-FSK System Model and Switching Operation

M-FSK System Model Consider the dual-branch diversity receiver illustrated in Fig. 9.97. For M-FSK the outputs of the M square-law detectors of the receiver at the nth uniform sampling instant t = nTs , where Ts is the symbol time, can be mathematically described by ! !2 Wi1 (n) = !2αi Es e j θi + Ni1 ! Wim (n) = |Nim |2 ,

m = 2, 3, . . . , M,

i = 1, 2

(9.630)

where i = 1, 2 denotes which of the two antennas the switch (receiver) is currently connected to. In (9.630), αi , θi are the fading amplitude and phase associated with

553

POST-DETECTION COMBINING

10−1

K=0

Average Bit Error Rate Pb(E )

h*T = 1.055

K=1 h*T = 1.091 10−2

K=5

h*T = 1.211

10−3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Normalized Switching Threshold hT

Figure 9.95 Average BER of noncoherent BFSK with post-detection SSC versus normalized switching threshold over i.i.d. Rician channels for γ = 10 dB. 100

Pre-Detection SSC Post-Detection SSC

Average Bit Error Rate Pb(E )

10−1

10−2

10−3

K=0 10−4

K=5 10−5

K = 10 10−6

0

5

10

15

20

25

30

Average SNR per Bit per Branch [dB]

Figure 9.96 Comparison of the average BER of noncoherent BFSK with pre-detection and post-detection SSC over i.i.d. Rician channels.

554

Antenna 2

Antenna 1

Figure 9.97

RF Front End

RF Front End

Square- W2M Law Detector

BPF ( f2)

BPF ( fM)

Threshold wT2

Threshold wT1

Choose Wmax2(n)=maxW2m(n) m

m

Data Symbol Decision

Switch Driver

Comparator

Threshold Switch

Channel Switch

Block diagram of noncoherent M-FSK with post-detection SSC.

Long-Term Average

Long-Term Average

Square- W 22 Law Detector

BPF ( f1)

Choose Wmax1(n)=maxW1m(n)

Square- W1M Law Detector sample at t = nTs Square- W21 Law Detector

BPF ( f2)

BPF ( fM)

Square- W 12 Law Detector

BPF ( f1)

sample at t = nTs

Square- W11 Law Detector

POST-DETECTION COMBINING

555

reception on antenna i, Es is the M-FSK symbol energy, and Nim , m = 2, 3, . . . , M, i = 1, 2 are complex, zero-mean Gaussian samples with variance 2N0 representing the AWGN on the M frequencies. Standard ML detection chooses the transmitted frequency corresponding to the largest of the Wim (n)s [i.e., Wmaxi (n) = maxm Wim (n)]. Switching Operation The switching between antennas is made from comparison of the maximum output of the square-law detectors to a fixed threshold wTi (i = 1, 2) at the end of the symbol time. In other words, similar to the binary case, the metric used now for data detection is identical to that used for switching and as such is equivalent to the metric used for antenna selection in post-SC in Ref. [200] model 1, and in Refs. [198] and [199]. The details are as follows. At each of the abovementioned uniform sampling instants, a decision is made whereby the channel switch in Fig. 9.97 either remains connected as it was (i.e., to the receiver output fed by the same antenna), or switches to the other position (i.e., to the receiver output fed by the other antenna). The decision as to whether a channel switch occurs depends on a comparison of the receiver output currently passing through the channel switch with a threshold. For example, if at time t = nTs we are connected to antenna 1, that is, if the receiver output passing through the channel switch is Wmax1 (n) = maxm W1m (n), then if Wmax1 > wT1 , the channel switch remains connected to this antenna (until at least the following sample time at t = (n + 1)Ts ) and the data decision is based on Wmax1 . On the other hand, if Wmax1 < wT1 , then a switch to antenna 2 occurs [the channel switch now remains in this position until at least the following sample time at t = (n + 1)Ts ] whereby Wmax2 (n) = maxm W2m (n) now appears at the switch output on which the data decision is based. 9.12.6.2

Average Probability of Error

Method of Analysis Assuming without loss in generality that the signal sent has frequency f1 corresponding to the first M-FSK frequency, then, a correct decision is made if Wmax1 (n) = W11 (n) and W11 (n) ≥ wT1 (remain connected to antenna 1), or Wmax1 (n) < wT1 (switch to antenna 2) and Wmax2 (n) = max W2m (n) = W21 (n). m Note that for the second condition (i.e., switch to antenna 2), it is not necessary that maxm W1m = W11 . In particular, the probability of a switch to antenna 2 (i.e., Pr{max W1m < wT1 }) is equivalent to the probability that all W1m values are m less than wT1 , which in mathematical terms is given by Pr{W11 < wT1 }[Pr{W12 < wT1 }]M−1 . Hence the probability of this correct symbol detection is Ps1 (C) = Pr{W11 (n) ≥ wT1 , W12 (n) < W11 (n), W13 (n) < W11 (n), . . . , W1M (n) < W11 (n)} + Pr{Wmax1 < wT1 , W22 (n) < W21 (n), W23 (n) < W21 (n), . . . , W2M (n) < W21 (n)}

(9.631)

A similar probability occurs if the receiver was connected to antenna 2 at time t = nT . Since W12 (n) , W13 (n) . . . , W1M (n) and W22 (n) , W23 (n) , . . . , W2M (n)

556

PERFORMANCE OF MULTICHANNEL RECEIVERS

are i.i.d. (corresponding to the noise-only branches), then under the fading independence assumption on the two diversity antennas, the above-mentioned probability can be mathematically expressed as 

∞

Ps1 (C) =

wT1



   M−1  M−1 pW11 (x) PW12 (x) d x+PW11 wT1 PW12 wT1 ∞

× 0

 M−1 pW21 (x) PW22 (x) dx

(9.632)

Similarly, assuming that the receiver is connected to antenna 2, we can write 

∞

Ps2 (C) =

wT2



   M−1  M−1 pW21 (x) PW22 (x) d x+PW21 wT2 PW22 wT2 ∞

× 0

 M−1 pW11 (x) PW12 (x) dx

(9.633)

Defining p1 and p2 as the probability (or equivalently the percentage of time) that antennas 1 and 2 are connected, respectively, we can write the average probability of correct symbol detection as Ps (C) = p1 Ps1 (C) + p2 Ps2 (C)

(9.634)

Following a procedure analogous to the one we presented for the binary case, the probabilities of being connected to antennas 1 and 2 can be obtained in terms of the CDF of the decision variables PW11 (·), PW12 (·), PW21 (·), and PW22 (·), as   P|Wmax | wTi  i pi =   P|Wmax | wTi + P|Wmaxi | wTi i     PWmax wTi − PWmax −wTi i i     =    , PWmax wTi − PWmax −wTi + PWmaxi wTi − PWmaxi −wTi i

i

i = 1, 2

(9.635)

where the overbar on the i subscript denotes the two’s complement of i, that is, if i = 1 then i = 2 and vice versa, and      M−1 , PWmaxi wTi = PWi1 wTi PWi2 wTi

i = 1, 2

(9.636)

Combining (9.635) and (9.634), we obtain the average SER as Ps (E) = 1 − Ps (C) = 1 − p1 Ps1 (C) − p2 Ps2 (C)

(9.637)

POST-DETECTION COMBINING

557

which for orthogonal M-FSK can be related to the average BER by Pb (E) =

M Ps (E) 2(M − 1)

(9.638)

For the special i.i.d. fading case, p1 = p2 = 12 , wT1 = wT2 = wT , and Ps1 (C) = Ps2 (C) = Ps (C) given by (9.632) with wT1 = wT . It is easy to check from (9.632) that for both extremes wT = 0 and wT = ∞, (9.632) reduces to the average probability of correct symbol detection Ps− (C) given by  ∞  ∞  M−1  M−1 − pW11 (x) PW12 (x) dx = pW21 (x) PW22 (x) dx Ps (C) = 0

0

(9.639) corresponding to a single-antenna (no-diversity) reception with noncoherent M-FSK detection. Hence, there is an optimum value of the switching threshold for which the average probability of correct symbol detection (or equivalently the average SER and BER) is minimum. This optimum threshold can be determined by differentiating Ps (C) [determined from (9.632)] with respect to wT and equating the result to zero, resulting after some manipulations in the transcendental equation Ps− (C) pW11 (wT ) PW12 (wT ) = (M − 1) pW12 (wT ) PW11 (wT ) 1 − Ps− (C)

(9.640)

Rayleigh Fading For this case, the fading amplitudes αi , i = 1, 2 are Rayleigh distributed and the fading phases θi , i = 1, 2 are uniformly distributed. After averaging over the fading, all the Wim (n) terms are central chi-square-distributed with two degrees of freedom with PDF and CDF     w 1 w exp − 2 , PWim (w) = 1 − exp − 2 , w≥0 pWim (w) = 2 2σim 2σim 2σim (9.641) having variances as   2 2 = σ2 = 2Es N0 , m = 2, 3, . . . , M, i = 1, 2 σim σi12 = 2Es N0 1 + γ i , (9.642)



where γ i = αi2 Es /N0 = i Es /N0 , i = 1, 2 is now the average symbol SNR corresponding to reception on the ith antenna. Substituting (9.641) together with (9.642) into (9.632) and (9.633), we obtain, after some manipulation and integral evaluation     

M−1
ηTi γ i 1 + m 1 + γ i (−1)m M −1   exp − Psi (C) = m 4 1 + γi 1 + m 1 + γi m=0    "

#M−1 ηT γ ηT γ + 1 − exp −  i i  1 − exp − i i 4 4 1 + γi M−1


(−1)m M −1   × , i = 1, 2 (9.643) m 1 + m 1 + γi m=0

558

PERFORMANCE OF MULTICHANNEL RECEIVERS



where we have introduced the threshold normalization ηTi = wTi /i Es2 . Substituting (9.643) in (9.637) and making use of (9.641) and (9.642) in (9.635) and (9.636), we obtain the average SER in closed form as

(−1)m M −1   m 1 + m 1 + γi i=1 m=0  /     ηTi γ i 1 + m 1 + γ i × pi exp − 4 1 + γi   0  "

# ηTi γ i M−1 ηTi γ i  1 − exp − +pi 1 − exp −  (9.644) 4 4 1 + γi

Ps (E) = 1 −

2 M−1



For the i.i.d. case where γ 1 = γ 2 = γ , ηT1 = ηT2 = ηT , and p1 = p2 = 1/2, (9.644) reduces to Ps (E) = 1 −

M−1
m=0

(−1)m 1 + m (1 + γ )



M −1 m





ηT γ 1 + m (1 + γ ) exp − 4 1+γ

# "

# ηT γ M−1 ηT γ 1 − exp − 4 (1 + γ ) 4

M−1
(−1)m M −1 × m 1 + m (1 + γ ) " − 1 − exp −

(9.645)

m=0

Using Eq. (1.47) of Ref. 206, the second summation in (9.645) can be expressed in terms of tabulated functions as  

(M − 1)! 1 + 1 M−1
1+γ (−1)m M −1   = m 1 1 + m (1 + γ ) 1+γ +M m=0

2+γ (9.646) ,M −1 = (M − 1) B 1+γ

where B (x, y) = (x) (y) / (x + y) is the beta function [36, Eq. (8.384.1)]. Hence the average SER can be written as Ps (E) = 1 −

M−1
m=0

(−1)m 1 + m (1 + γ )



M −1 m





ηT γ 1 + m (1 + γ ) exp − 4 1+γ



# ηT γ M−1 1 − exp − 4

2+γ ,M −1 × (M − 1) B 1+γ " − 1 − exp −

ηT γ 4 (1 + γ )

# "

(9.647)

POST-DETECTION COMBINING

559

Note that at ηT = 0 (the receiver never switches antennas) and ηT = ∞ (the receiver always switches antennas), the average SER of (9.647) is equal to M−1


(−1)m 1 + m (1 + γ ) m=0

2+γ ,M −1 = 1 − (M − 1) B 1+γ

Ps− (E) = 1 − Ps− (C) = 1 −



M −1 m



(9.648)

which is also the error probability for a single-antenna noncoherent M-ary orthogonal system in the presence of Rayleigh fading. The optimum threshold ηT∗ that minimizes Ps (E) given by (9.647) at each average SNR γ is obtained by using (9.641), (9.642), and (9.648) in (9.640), leading to the transcendental equation  ∗   

" ∗

# 1 − exp − ηT γ η γ 4 1 γ    exp T ηT∗ γ 1+γ 4 1+γ 1 − exp − 

 (M − 1)2 B 2+γ , M − 1 1+γ   = 1 − (M − 1) B 2+γ 1+γ , M − 1

4(1+γ )

(9.649)

which can also be obtained by direct differentiation of (9.647). Letting M = 2 in (9.647), the average BER for the binary case corresponding to BFSK becomes, after some simplification "



ηT γ 1 ηT γ 1 − exp − + (1 + γ ) exp − Pb (E) = 2+γ 4 (1 + γ ) 4

# ηT γ 2 + γ −γ exp − (9.650) 4 1+γ with the optimum threshold determined from the transcendental equation  ∗   

# 1 − exp − ηT γ " ∗ ηT γ 4 γ   = (1 + γ )2  (9.651) exp ∗γ  η 4 1+γ T 1 − exp − 4(1+γ )

Rician Fading For the Rician fading (with Rician factor Ki ) case, Wi1 , i = 1, 2 become noncentral chi-square distributed with PDF and CDF    √  w + ai2 1 ai w pWi1 (w) = exp − I0 , 2 2 2σi1 2σi1 σi12 (9.652)     ηTi γ i (1 + Ki )   2Ki , PWi1 (w) = 1 − Q , w ≥ 0, i = 1, 2 2 1 + γi

560

PERFORMANCE OF MULTICHANNEL RECEIVERS

with variance and noncentrality parameter σi12 =

 2Es N0  1 + γi , 1 + Ki

ai2 =

  Ki 4Es N0 1 + γ i 1 + Ki

(9.653)

whereas Wim , m = 2, 3, . . . , M, i = 1, 2 remain central chi-square distributed as before. Also, from (8.194) 

∞ 0

M−1
 M−1 pWi1 (x) PWi2 (x) dx = (−1)m





1 + Ki   1 + K i + m 1 + γi m=0   Ki m(1 + γ i )   × exp − (9.654) 1 + Ki + m 1 + γ i M −1 m

and furthermore  ∞  M−1 pWi1 (x) PWi2 (x) dx wTi

  1 + Ki Ki m(1 + γ i )  exp −    = (−1) 1 + Ki + m 1 + γ i 1 + Ki + m 1 + γ i m=0 *       + + 1+K η γ +m 1 + γ 1 + K T i i i i i  , i = 1, 2  ,    × Q,2Ki 1+Ki +m 1 + γ i 2 1 + γi

M−1


m

M −1 m



(9.655) which checks with (9.654) when wTi = 0. Substituting (9.654) and (9.655) in (9.632) and (9.633) and then making use of (9.637) gives, after some manipulation, the desired result for the average SER as Ps (E) = 1 −

2 M−1



(−1)m

i=1 m=0

M −1 m



  f Ki , m, γ i

  Ki m(1 + γ i )f Ki , m, γ i × exp − 1 + Ki /    ηTi γ i (1 + Ki )  1     2Ki f Ki , m, γ i , × pi Q 2 1 + γi f Ki , m, γ i *       " +

#M−1  + ηT γ i 1 + Ki  η γ T i   1 − exp − i +pi 1 − Q  2Ki , , i   4 2 1 + γi (9.656) where   f Ki , m, γ i =

1 + Ki ,  1 + Ki + m 1 + γ i

i = 1, 2

(9.657)

POST-DETECTION COMBINING

561

and p1 and p2 are still given by (9.635) but now using (9.652) to evaluate PWmaxi wTi in (9.636). For the i.i.d. case, γ 1 = γ 2 = γ and K1 = K2 = K, (9.656) reduces to

M−1
M −1 Ps (E) = 1 − f (K, m, γ ) (−1)m m m=0

K × exp − m(1 + γ )f (K, m, γ ) 1+K  /    1 ηT γ (1 + K) × Q 2Kf (K, m, γ ), 2 (1 + γ ) f (K, m, γ )    0  "

# √ ηT γ M−1 ηT γ (1 + K) 1 − exp − 2K, + 1−Q 2 (1 + γ ) 4 (9.658) Here again, since at ηT = 0 and ηT = ∞, the average SER is equal to

M−1
1+K M −1 − − m Ps (E) = 1 − Ps (C) = 1 − (−1) m 1 + K + m (1 + K + γ ) m=0

Km(1 + γ ) (9.659) × exp − 1 + K + m (1 + γ ) there exists an optimum threshold that is determined from (9.640) with Ps− (C) given by (9.12.5) and the remaining PDFs and CDFs determined from (9.641) with (9.642) and (9.652) with (9.653). For the binary case, letting M = 2 in (9.658), the average BER becomes after some simplification     √ ηT γ (1 + K) 2K, Ps (E) = 1 − Q 2 (1 + γ )

# "

#4 1 " K(1 + γ ) ηT γ 1+K 1 − exp − exp − × 1− 1− 2+K +γ 2+K +γ 4

K(1 + γ ) 1+K exp − + 2+K +γ 2+K +γ     ηT γ 2 + K + γ 1 + K  , (9.660) × Q  2K 2+K +γ 2 (1 + γ ) with optimum threshold determined from (9.640) using

K(1 + γ ) 1+K − − exp − Ps (E) = 1 − Ps (C) = 2+K +γ 2+K + γ together with (9.641) and (9.652).

(9.661)

562

PERFORMANCE OF MULTICHANNEL RECEIVERS

9.12.6.3 Numerical Examples In the following set of numerical examples, we study the behavior of the proposed post-SSC scheme and compare its performance to pre-SSC as well as post-SC and the alternative post-SSC analyzed for the binary case. To distinguish the two postSSC schemes, we refer to the one limited to binary FSK, as model 1-based post-SSC while we refer to the one presented in the M-ary case, which is applicable to not only binary but more generally to M-ary FSK, as model 2-based post-SSC.31 More specifically, in contrast to the detection decision used in model 2-based post-SSC, in model 1-based post-SSC, the detection decisions are based on the comparison of the difference of the two square-law detector outputs with zero. Figure 9.98 plots the average BER of noncoherent BFSK with model 2-based post-SSC versus the normalized switching threshold ηT for various values of the average SNR. All curves show a clear average BER minimum corresponding to the optimal switching threshold ηT∗ . In addition, similar to model 1-based post-SSC: (1) for high values of average SNR, the curves decrease very sharply with respect

100 hT* = 3.7029

Average Bit Error Rate Pb(E )

g¯ = 5 dB

10−1

hT* = 1.5944

g¯ = 10 dB g¯ = 15 dB

hT *= 0.6609

10−2

10−3

10−4

g¯ = 20 dB

η*T = 0.2610

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Normalized Switching Threshold hT Figure 9.98 Average BER of noncoherent BFSK with post-detection SSC versus normalized switching threshold over i.i.d. Rayleigh channels. 31 To avoid any confusion with the terminology used in [200], note that our model 1-based post-SSC corresponds to model 3 in the post-SC study of Ref. 200. On the other hand, our model 2-based post-SSC corresponds to model 1 in the post-SC study of Ref. 200.

563

POST-DETECTION COMBINING

100

Average Bit Error Rate Pb(E )

10−1

10−2

10−3

10−4 No Diversity Pre-Detection SSC Post-Detection SSC (Model 2) Post-Detection SSC (Model 1) Pre-Detection SC Post-Detection SC (Model 2) Post-Detection SC (Model 1)

10−5

10−6

0

5

10

15

20

25

30

Average SNR per Bit per Branch [dB]

Figure 9.99 Comparison of the average BER of noncoherent BFSK with no diversity, predetection SSC and SC, and post-detection SSC and SC (models 1 and 2).

to ηT before reaching their minimum while these same curves tend to increase more slowly beyond the optimum ηT∗ , and (2) the optimal switching threshold ηT∗ is a decreasing function of the average SNR, which implies that systems experiencing better channels will tend to switch more often to exploit the potentially better fading conditions on the alternative branch. Figure 9.99 compares the average BER of noncoherent BFSK with no diversity, pre-detection SSC and SC, and post-detection SSC and SC (models 1 and 2). Similar to the behavior observed in Ref. 200 for post-SC, model 1-based postSSC outperforms model 2-based post-SSC, but the latter is still offering better BER performance than pre-SSC. Similar to Fig. 3 in Ref. 199, which compared the performance of pre-SC and post-SC when used with M-ary FSK, Fig. 9.100 presents an analogous comparison for pre-SSC and post-SSC. Note that the SNR gain of post-SSC over pre-SSC slightly increases as M increases. It should be pointed out that the average BER of M-FSK with pre-SSC was computed following the procedure developed by Abu-Dayya and Beaulieu [9,10]. More specifically, the average SER can be shown to be given by

Ps (E) =

M−1
k=1

M −1 k k+1

(−1)k+1



k MγSSC − 1+k

(9.662)

564

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

Pre-Detection SSC Post-Detection SSC

Average Bit Error Rate Pb(E )

10−1 M=2 M=4

10−2

M=8

10−3 M = 16

M = 32

10−4 0

5

10

15

20

25

30

Average SNR per Bit per Branch [dB] Figure 9.100 Comparison of the average BER of noncoherent MFSK with pre- and postdetection SSC for various values of M and over Rayleigh fading channels.

where MγSSC (s) is as given in (9.279) and (9.281) for the Rayleigh and the Rician channels, respectively, and the optimal SNR switching threshold γT∗ is the solution of the transcendental equation M−1
k=1

 " #

(−1)k+1 M−1 k ∗ k Mγ − − e−kγT /(k+1) = 0 k+1 k+1

(9.663)

where Mγ (s) is given in Table 9.1. Finally, Fig. 9.101 plots the average BER of noncoherent BFSK with post-SSC versus normalized switching threshold over i.i.d. Rician channels for γ = 10 dB. This figure illustrates the existence of the optimum threshold for various Rician factors and shows that, similar to model 1-based post-SSC, this optimum threshold is an increasing function of the Rician factor which means that the switching rate should be reduced as the strength of the line of sight increases. Figure 9.102 compares the optimal average BERs of noncoherent BFSK with pre-SSC (as per the average BER expression given in Eq. (11) of Ref. 10 along with the optimal threshold given by Eq. (12) in that study [10]) and model 1- and 2-based post-SSC over

565

POST-DETECTION COMBINING

10−1 K=0

Average Bit Error Rate Pb(E )

hT* = 1.5864

K=1

hT* = 1.6654

10−2 K=5 hT* = 2.131

10−3

0

0.5

1

1.5

2

2.5

3

3.5

4

4.5

5

Normalized Switching Threshold hT

Figure 9.101 Average BER of noncoherent BFSK with post-detection SSC versus normalized switching threshold over i.i.d. Rician channels; Average SNR = 10 dB

Average Bit Error Rate Pb(E )

100

Pre-Detection SSC Post-Detection SSC (Model 2) Post-Detection SSC (Model 1)

10−1

10−2 K=0

K=5

10−3 K = 10

10−4

0

5

10

15

20

25

Average SNR per Bit per Branch [dB]

Figure 9.102 Comparison of the average BER of noncoherent BFSK with pre-detection and post-detection SSC (models 1 and 2) over i.i.d. Rician channels.

566

PERFORMANCE OF MULTICHANNEL RECEIVERS

i.i.d. Rician channels. It is interesting to note that the gain of model 1-based postSSC over model 2-based post-SSC increases as the severity of fading decreases.

9.13 PERFORMANCE OF DUAL-BRANCH DIVERSITY COMBINING SCHEMES OVER LOG-NORMAL CHANNELS So far, we have devoted a great deal of attention to studying the average probability of error and the outage probability performance of diversity combining systems in the presence of Rayleigh, Rice, and Nagakami-m fading. However, as mentioned in Chapter 2, the log-normal distribution is often found to be the most suitable PDF to fit empirical fading channel measurements [207,208,209,210]. The latter is particularly true for indoor radio propagation environments where terminals with low mobility have to rely on macroscopic diversity to overcome the shadowing from indoor obstacles and moving human bodies. Indeed, in such kinds of slowly varying channels, the small- and large-scale effects tend to get mixed and the log-normal statistics tend to dominate and to accurately describe the distribution of the channel path gain. However, unlike the Rayleigh, Rice, and Nagakami-m environments, it is not possible to obtain an exact and simple closed-form expression for the MGF of the combined output SNR once a log-normal variate is involved in the calculations. Thus, for example, average probability of error evaluation techniques based on the MGF-based approach we pursued so far will not provide exact results. In this section, we summarize the authors’ work in Ref. 211 showing that, in many cases, it is possible to evaluate the marginal and joint statistical moments of correlated and not necessarily identically distributed log-normal variates in closed form.32 As a result, performance measures that depend only on these moments such as average combined SNR and AF can likewise be obtained in closed form. Capitalizing on these results, we study and compare the average combined SNR and the AF for the three most popular dual-diversity combining systems, namely, MRC, SC, and SSC, allowing for the possibility of average power unbalance and correlation between the two branches. In each case, the combiner output moments are given first, from which one then obtains exact closed-form expressions for the average combined SNR and AF. Finally, we use the single finite-range integral representation of the two-dimensional Q-function discussed in Sections 4.1.2 and 4.1.3, to obtain simple single finite-range integral expressions for the outage probability of the output SNR corresponding to these dual-diversity receivers. 9.13.1

System and Channel Models

We consider a dual-branch diversity system over two correlated and not necessarily identically distributed log-normal fading channels. Under these conditions the 32 Ordinarily, the MGF of a distribution can be expressed in terms of the complete set of statistical moments, and thus one might anticipate that an MGF-based approach for evaluating average BER should be possible. Unfortunately, for the log-normal distribution and related distributions, the Taylor series that expresses the MGF exactly in terms of all its moments does not converge [212].

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

567

instantaneous SNRs γ1 and γ2 of the first and second branch, respectively, follow a log-normal PDF given by (2.43)   2  10 log10 γi − µi ξ exp − pγi (γi ) = √ , i = 1, 2 (9.664) 2σi2 2πσi γi where ξ = 10/ ln 10 = 4.3429, µi (in dB) is the mean of 10 log10 γi and σi (in dB) is the standard deviation of 10 log10 γi . The corresponding CDF is given by

µi − 10 log10 γi , i = 1, 2, γi ≥ 0 (9.665) Pγ1 (γi ) = Q σi where once again Q(·) is the standard one-dimensional Gaussian Q-function. In addition, the corresponding average SNR of the ith branch γ i = E[γi ], i = 1, 2, is given by (2.45) for k = 1 as   σi2 µi γ i = exp (9.666) + 2 ξ 2ξ while more generally the corresponding nth moment is given by (2.45)33   2 n3 n2 σi2 nµi + E γi = exp (9.667) ξ 2ξ 2 yielding an AF of (1.46) 

σi2 AF = exp ξ2

 −1

(9.668)

To model the correlation between the two diversity channels, we assume a joint log-normal PDF given by pγ1 ,γ2 (γ1 , γ2 ) =

ξ2  2π σ1 σ2 1 − ρ 2 γ1 γ2 /   2 2 10 log10 γ2 − µ2 10 log10 γ1 − µ1 1 × exp − + 1 − ρ2 2σ12 2σ22



0 10 log10 γ1 − µ1 10 log10 γ2 − µ2 , −2ρ √ √ 2σ1 2σ2 γ1 ≥ 0,

γ2 ≥ 0

(9.669)

where ρ is the correlation coefficient, which takes values between -1 and 1. When ρ = 0, the two diversity paths are uncorrelated. On the other hand, when positive 33 Note

that the result in (9.667) applies even when n is noninteger.

568

PERFORMANCE OF MULTICHANNEL RECEIVERS

correlation exists (i.e., 0 < ρ ≤ 1), if one diversity path is strong, it is likely that the other diversity path is also strong and vice versa. Finally, when negative correlation exists (i.e, −1 ≤ ρ < 0), if one diversity path is weak, it is likely that the other diversity path is strong and vice versa [213]. 9.13.2

Maximal-Ratio Combining

The combined SNR at the output of a dual branch MRC receiver is given by γMRC = γ1 + γ2

(9.670)

9.13.2.1 Moments of the Output SNR The nth moment of the output SNR is given by n ] = E[(γ1 + γ2 )n ]. E[γMRC

Using the binomial expansion, (9.671) can be written as  n


n n k n−k E[γMRC ] = E γ γ k 1 2 k=0 n


  n = E γ1k γ2n−k k

(9.671)

(9.672)

k=0

Using  the joint PDF of (9.669), it can be shown after some manipulations that E γ1k γ2n−k has the following simple closed-form expression:34

 k n−k  k 2 σ12 + (n − k)2 σ22 + 2ρk(n − k)σ1 σ2 kµ1 + (n − k)µ2 + = exp E γ1 γ2 ξ 2ξ 2 (9.673) Substituting (9.673) in (9.672) yields the moments at the MRC output as n E[γMRC ]

=

n


n

k

k 2 σ12 + (n − k)2 σ22 + 2ρk(n − k)σ1 σ2 kµ1 + (n − k)µ2 + × exp ξ 2ξ 2 k=0

(9.674) Average Output SNR The average combined SNR γ MRC at the MRC output can be obtained by setting n = 1 in (9.674), yielding



σ12 σ22 µ2 µ1 + 2 + exp + 2 = γ1 + γ2 γ MRC = exp (9.675) ξ 2ξ ξ 2ξ 34 Here

again, the relation in (9.673) does not require k and n to be integer. This would be important, for example, when computing the AF of coherent equal gain combining to be discussed in Section 9.13.2.3.

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

569

as expected from (9.670) and regardless of the fading correlation between the two diversity branches. In the case of identically distributed fading over the two branches, (9.675) reduces to

σ2 µ + 2 = 2γ γ MRC = 2 exp (9.676) ξ 2ξ Amount of Fading The second moment of the combined SNR at the MRC output can also be found by setting n = 2 in (9.674). Using this result, the AF at the MRC output can be written as 3 2 E (γ1 + γ2 )2 AFMRC =  2 − 1 γ1 + γ2 "

# "

# σ2 σ2 exp 2 µξ1 + ξ12 + exp 2 µξ2 + ξ22

σ12 +σ22 +2ρσ1 σ2 2 + 2 exp µ1 +µ + ξ 2ξ 2 (9.677) = "



#2 − 1 exp

µ1 ξ

+

σ12 2ξ 2

µ2 ξ

+ exp

+

σ22 2ξ 2

for the most general correlated and not necessarily identically distributed branches case. For the identically distributed case where µ1 = µ2 and σ1 = σ2 , (9.677) reduces to  2  2 exp σξ 2 + exp ρσ −2 ξ2 (9.678) AFMRC = 2 On the other hand, for the uncorrelated nonidentically distributed case, (9.677) reduces to

# "

# " # " σ12 σ22 σ12 +σ22 µ1 µ2 µ1 +µ2 + 2ξ 2 exp 2 ξ + ξ 2 + exp 2 ξ + ξ 2 + 2 exp ξ AFMRC = −1 "



#2 exp

µ1 ξ

+

σ12 2ξ 2

+ exp

µ2 ξ

+

σ22 2ξ 2

(9.679) Both (9.678) and (9.679) reduce to  AFMRC =

exp

σ2 ξ2

2



−1 (9.680)

for the i.i.d. case (ρ = 0, µ1 = µ2 , and σ1 = σ2 ). Hence in the i.i.d. case dualbranch MRC reduces by a factor of 2 the AF in a single log-normal faded channel (i.e., without diversity) as given in (9.668). More generally, it can be shown that, in the i.i.d case, L-branch MRC reduces by a factor of L the AF in a single lognormal faded channel. Figure 9.103 shows the amount of fading for MRC versus the

570

PERFORMANCE OF MULTICHANNEL RECEIVERS

Amount of Fading for MRC

102

101

r = 0.95 r = 0.75 r = 0.5 r = 0.25 r=0

100 4

5

6

7

8

9

10

Standard Deviation s [dB] Figure 9.103 Amount of fading for MRC versus the fading standard deviation for various values of the correlation coefficient.

fading standard deviation for various values of the correlation coefficient. For small values of the correlation coefficient, the impact of correlation is more accentuated for low values of the standard deviation. On the other hand, for larger values of the correlation coefficient, the increase in the amount of fading is equally important over the whole standard deviation range. 9.13.2.2 Outage Probability MRC The outage probability Pout of dual-branch MRC corresponds to the probability that γMRC falls below a predetermined threshold γth : MRC = P [γMRC = γ1 + γ2 ≤ γth ] Pout

(9.681)

Unfortunately, an exact closed-form expression for the distribution of the sum of correlated (or, for that matter, independent) log-normal RVs is not known [214,215]. One way to circumvent this difficulty is to rely on the extensions of the Wilkinson, Schwartz and Yeh, and Fenton (moment matching) approaches as developed in Ref. 216 and 217. Another method, which we present in what follows, relies on extending the “bounding” approach developed in [218] for the sum of independent log-normal RVs, to obtain upper and lower bounds (in the forms of single finite-range integrals) on the CDF of the sum of two correlated log-normal

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

571

RVs. More specifically, using the facts that γ1 + γ2 ≤ 2γmax and γ1 + γ2 ≥ 2γmin , MRC as given in (9.681) can where γmax = max(γ1 , γ2 ) and γmin = min(γ1 , γ2 ), Pout be bounded by γ  γ  th th MRC ≤ Pout (9.682) Pγmax ≤ Pγmin 2 2 where Pγmax (·) and Pγmin (·) are the CDF of γmax and γmin , respectively. On the basis of an alternative representation and symmetry relations of the two-dimensional Gaussian Q-function [presented in (6.42)], single finite-range integral expressions for Pγmax (·) and Pγmin (·) are presented in Sections 6.4.1 and 6.4.2, respectively. 9.13.2.3 Extension to Equal Gain Combining For dual diversity coherent EGC, the output SNR is given by √ √ 2 γEGC = 12 γ1 + γ2

(9.683)

√ √ Since the RVs γ1 and γ2 are also log-normal with logarithmic means µi /2 and logarithmic standard deviation σi /2, i = 1, 2, the moments in (9.672) and (9.673) can be used to obtain closed-form expressions for the average combined SNR and the AF at the EGC output. Similarly, following the procedure of Section 9.13.2.2, it can be easily shown that the outage probability of EGC can be bounded by .

.

γth γth EGC √ √ ≤ Pout ≤ P γmin (9.684) P γmax 2 2 √ √ where P√γmax (·) and P√γmin (·) are the CDF of γmax and γmin , respectively, and which can also be computed using the expressions presented in Sections 6.4.1 and 6.4.2, respectively, by replacing µi by µi /2 and σi by σi /2, i = 1, 2 in all these expressions. These substitutions lead to γ  γ  th th EGC ≤ Pout (9.685) Pγmax ≤ Pγmin 2 2 which (when compared with (9.684)) means that EGC and MRC offer a comparable diversity gain. This also suggests that this diversity gain is not significant, as we indeed will observe in some of our numerical results discussed at the end of Section 9.13.4.2. 9.13.3

Selection Combining

The SNR γSC at the output of a dual-branch SC receiver is given by γSC = max(γ1 , γ2 )

(9.686)

Hence the PDF of γSC can be expressed in terms of the joint PDF of γ1 and γ2 as  γSC  γSC pγ1 ,γ2 (γSC , γ2 ) dγ2 + pγ1 ,γ2 (γ1 , γSC ) dγ1 (9.687) pγSC (γSC ) = 0

0

572

PERFORMANCE OF MULTICHANNEL RECEIVERS

Substituting (9.669) in (9.687) and using the change of variables ui = 10 log10 (γi − √ µi )/( 2σi ), i =1, 2 in both integrals, we get after some manipulations and with the help of Eq. (7.4.32) in Ref. 53 (p. 303), namely 

   exp − ax 2 + bx + c 1 = 2

.

 .  2

 √ b − ac π 2 exp b , 2ax + 1 − 2Q a a a

a = 0 (9.688)

the PDF of the output SNR in terms of the Gaussian Q-function as

ξ (10 log10 (γSC ) − µ1 )2 pγSC (γSC ) = √ exp − 2σ12 2πσ1 γSC /   1 ρ × 1−Q − 10 log10 (γSC )  1 − ρ 2 σ2 1 − ρ 2 σ1 0 µ2 ρµ1 − + 1 − ρ 2 σ2 1 − ρ 2 σ1

(10 log10 (γSC ) − µ2 )2 ξ exp − +√ 2σ22 2πσ2 γSC  /  ρ 1 − 10 log10 (γSC ) × 1−Q  1 − ρ 2 σ1 1 − ρ 2 σ2 0 µ1 ρµ2 − + (9.689) 1 − ρ 2 σ1 1 − ρ 2 σ2 9.13.3.1 Moments of the Output SNR The nth moment of the output SNR is given by  n E[γSC ]=

∞ 0

n γSC pγSC (γSC ) dγSC

(9.690)

√ Substituting (9.689) in (9.690), and then letting x√= (10 log10 (γSC ) − µ1 )/( 2σ1 ) in the first integral and x = (10 log10 (γSC ) − µ2 )/( 2σ2 ) in the second integral, we obtain, after much simplification, the nth moment of γSC as n ] E[γSC

   √  1 σ2 1 − ρ 2 ∞ =√ exp (f (x)) Q − 2x d x ρσ2 π σ1 |1 − σ1 | −∞    √  1 σ1 1 − ρ 2 ∞ exp Q − 2x d x +√ (g(x)) 1 π σ2 |1 − ρσ σ2 | −∞

(9.691)

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

573

where 

2  2  2 σ 1 − ρ µ2 − µ1 2     f (x) = − x+√  2 2σ2 1 − ρ 2 σ1 1 − ρσ σ1 √  2 2σ2 1 − ρ 2 x + µ2 − µ1 ρσ σ1   +n 2 ξ 1 − ρσ σ1 

2  2  2 σ 1 − ρ − µ µ 1 1 2  x + √  g(x) = −   1 2σ1 1 − ρ 2 σ2 1 − ρσ σ2 √  1 2σ1 1 − ρ 2 x + µ1 − µ2 ρσ σ2   +n . 1 ξ 1 − ρσ σ2

(9.692)

By an extension of the definite integral found in Middleton’s work [219, p. 1072, Eq. (A.1.10a)], one can arrive at the following identity: 

∞ −∞

   √  exp −a 2 (x + µ)2 + bx + c Q ± 2x d x √



b2 π b − 2a 2 µ exp c + 2 − bµ Q ± √ √ = a 4a 2a 1 + a 2

(9.693)

Applying the identity in (9.693) to both integrals in (9.691) and simplifying gives the desired closed-form result, namely    nσ12 ρσ2 2

2 µ 1 − − µ − 2 1 2 n3 n σ1 ξ σ1  nµ1  E γSC = exp Q +  ξ 2ξ 2 2 2 σ + σ − 2ρσ σ + exp



nµ2 + ξ

1

µ − n2 σ22  1 Q  2ξ 2

1 2

2

µ2 −

nσ22 ξ

 1−

ρσ1 σ2

σ12 + σ22 − 2ρσ1 σ2

  

(9.694)

which holds for the most general case of correlated and not necessarily identically distributed fading channels. For correlated identically distributed fading over the diversity channels (i.e., µ1 = µ2 and σ1 = σ2 ), (9.694) reduces to the simpler result

2 n3 nµ n2 σ 2 nσ  = 2 exp √ 1 − ρ E γSC Q − + ξ 2ξ 2 2ξ

(9.695)

574

PERFORMANCE OF MULTICHANNEL RECEIVERS

On the other hand, for uncorrelated (i.e., ρ = 0) and not identically distributed fading over the diversity channels, (9.694) reduces to   nσ12

2 2 2 n3 n σ1 nµ1  µ2 − µ1 − ξ  Q + E γSC = exp  2 ξ 2ξ σ12 + σ22   nσ22 2

2 n σ2 nµ2  µ1 − µ2 − ξ  + exp (9.696) Q +  2 ξ 2ξ σ12 + σ22 Finally, for the i.i.d. case where ρ = 0, µ1 = µ2 = µ, and σ1 = σ2 = σ , both (9.695) and (9.696) simplify to the compact expression



2 n3 nσ nµ n2 σ 2 E γSC Q − + (9.697) = 2 exp √ ξ 2ξ 2 2ξ Note that for ρ = 1, since Q(0) = 12 , (9.695) further reduces to the result for the no-diversity case in (9.667), which mathematically confirms that the diversity advantage disappears with SC when the two branches are fully correlated. Average Output SNR The average SNR γ SC at the SC output can be obtained by setting n = 1 in the moments formulas of the previous section. For example, the uncorrelated case yields   σ12 2

σ µ1  µ2 − µ1 − ξ  + 12 Q  γ SC = exp  ξ 2ξ σ12 + σ22   σ22 2

σ µ2  µ1 − µ2 − ξ  + exp + 22 Q   ξ 2ξ σ12 + σ22     σ12 σ22 − µ − − µ − µ µ 2 1 1 2 ξ  ξ    = γ 1Q  (9.698)  + γ 2Q   2 2 2 2 σ1 + σ2 σ1 + σ2 whereas the correlated identically distributed fading case gives   .



σ  σ2 µ σ 1−ρ + 2 Q −√ γ SC = 2 exp 1 − ρ = 2γ Q − (9.699) ξ 2ξ ξ 2 2ξ Finally the i.i.d. case gives





µ σ σ σ2 γ SC = 2 exp + 2 Q −√ = 2γ Q − √ ξ 2ξ 2ξ 2ξ

(9.700)

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

575

Comparing (9.700) and (9.676), we see that for the i.i.d. case the average SNR penalty of dual diversity SC relative to MRC is simply given by γ SC / γ MRC =  √  Q −σ / 2ξ , which becomes diminishingly small for large σ . Amount of Fading The closed-form expressions for the moments can also be used to find very simple expressions for the AF, which, in the most general case, will be a function of µ1 , µ2 , σ1 , σ2 , and ρ. However, the most general result simplifies to more compact and simpler expressions for the two following special cases of interest. In particular, for correlated identically distributed branches, the AF simplifies to   √ 2 Q − 2 σ √1 − ρ ξ σ   −1 (9.701) AFSC = exp ξ 2 2 Q2 − √1 σ √1 − ρ 2ξ which further reduces in the i.i.d. case to AFSC = exp

2

σ ξ2

 √  Q − 2 σξ   −1 2 Q2 − √12 σξ

(9.702)

Figure 9.104 shows the amount of fading for SC versus the fading standard deviation for various values of the correlation coefficient. Note that the gradual increase in the AF as the correlation coefficient increases is about the same over the whole standard deviation range. 9.13.3.2 Outage Probability SC of dual-branch SC is given by The outage probability Pout SC Pout = P [γSC = max(γ1 , γ2 ) ≤ γth ] = Pγmax (γth )

(9.703)

which in view of Section 6.4.1 can be expressed in the form of single finiterange integrals. 9.13.4

Switched Combining

To avoid presenting lengthy equations, we will consider here only the correlated identically distributed case; however, similar expressions can be derived for the more general correlated and not identically distributed scenario [220]. Following the Abu-Dayya and Beaulieu derivation for the correlated dual-branch Nakagami case originally given in Ref. 9 (Sect. IV) and briefly presented in Section 9.9.1.3, the PDF of the SSC output SNR for the correlated log-normal case can be written as   γ T   pγ1 ,γ2 (γSSC , γ2 ) dγ2 , γSSC ≤ γT  0  γT (9.704) pγSSC (γSSC ) =   pγ1 ,γ2 (γSSC , γ2 ) dγ2 , γSSC > γT  pγ (γSSC ) + 0

576

PERFORMANCE OF MULTICHANNEL RECEIVERS

Amount of Fading for SC

102

101

r = 0.95 r = 0.75 r = 0.5 r = 0.25 r=0 100 4

5

6

7

8

9

10

Standard Deviation σ [dB]

Figure 9.104 Amount of fading for SC versus the fading standard deviation for various values of the correlation coefficient.

where γT is the switching threshold below which the receiver switches to the other diversity branch. Using (9.688) and some manipulations, the PDF in (9.704) can be written in the following closed-form    ρ   pγ (γSSC )Q  (10 log10 (γSSC ) − µ)    1 − ρ2σ       1   (10 log10 (γT ) − µ) , γSSC ≤ γT   − 1− ρ2σ  / pγSSC (γSSC ) =  ρ   (10 log10 (γSSC ) − µ) pγ (γSSC ) 1 + Q     1 − ρ2σ   0    1   γSSC > γT (10 log10 (γT ) − µ) ,   − 1 − ρ2σ (9.705) 9.13.4.1 Moments of the Output SNR The nth moment of the output SNR is given by  n ] E[γSSC

∞

= 0

n γSSC pγSSC (γSSC ) dγSSC

(9.706)

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

577

Substituting the SSC output PDF of (9.705) in (9.706) yields   ∞ ρ n n E[γSSC ] = (10 log10 (γSSC ) − µ) γSSC pγ (γSSC ) Q  0 1 − ρ2σ  1 − (10 log10 (γT ) − µ) d γSSC 1 − ρ2σ  ∞ n + γSSC pγ (γSSC ) d γSSC (9.707) γT

√ On the one hand, letting x = (10 log10 (γSSC ) − µ)/( 2σ) in the first integral of (9.707) and again using the identity in (9.693) allows us to obtain this integral in closed form. On the other hand, using the change of variables x = 10 log10 γ in the second integral of (9.707) allows us to obtain it in closed form with the help of (9.688). These manipulations result in the following simple closed-form expression for the moments of the SNR at the SSC output:

" 10 log10 (γT ) − µ nσ nµ n2 σ 2 n + − Q E[γSSC ] = exp ξ 2ξ 2 σ ξ

# nσρ 10 log10 (γT ) − µ +Q (9.708) − ξ σ Note again that similar to the SC diversity scheme, since Q(x) + Q(−x) = 1, for ρ = 1, (9.708) reverts to the result for the no-diversity case in (9.667), as expected. Average Output SNR The average SNR γ SSC at the SSC output can be obtained by setting n = 1 in the moments formulas of the previous section. More specifically, in the correlated case, we get



# " σρ 10 log10 (γT ) − µ 10 log10 (γT ) − µ σ +Q − − γ SSC = γ Q σ ξ ξ σ (9.709) which reduces to "



# 10 log10 (γT ) − µ σ −10 log10 (γT ) + µ γ SSC = γ Q − +Q (9.710) σ ξ σ for the i.i.d. case. It is interesting to mention at this point that the optimum switching threshold γT∗ for which the average output SNR is maximized can also be obtained in closed form. Indeed, differentiating (9.709) with respect to γT and setting the resulting expression to zero, it can be shown after some manipulations that the optimum threshold is given by

2

µ σ 2 (1 + ρ) ρσ ∗ = γ exp (9.711) + γT = exp ξ 2ξ 2 2ξ 2

578

PERFORMANCE OF MULTICHANNEL RECEIVERS

for the correlated case, which reduces to

σ2 µ γT∗ = exp + 2 =γ ξ 2ξ

(9.712)

for the uncorrelated case. Substituting (9.711) in (9.709) yields, after some simplifications, the optimum output average SNR as

σ 1−ρ (9.713) γ ∗SSC = 2γ Q − ξ 2 which reduces for the i.i.d. case to

σ γ ∗SSC = 2γ Q − 2ξ

(9.714)

Comparing (9.676), (9.699), and (9.713), we see that, √ because the Q-function is a decreasing function of its argument and because (1 − ρ)/2 ≥ (1 − ρ)/2 for 0 ≤ ρ ≤ 1, we have γ ∗SSC ≤ γ SC ≤ γ MRC

(9.715)

Thus, MRC outperforms SC which in turn outperforms optimal SSC, as expected. Figures 9.105–9.107 compare the average output SNR for MRC, SC, and SSC (with optimum switching threshold) versus the fading standard deviation for uncorrelated branches and correlated branches with ρ = 0.5 and ρ = 0.95, respectively. Contrary to the behavior of the average SNR at the MRC output, which is unaffected by correlation, the average SNR at the output of SC and SSC degrades gradually as the correlation coefficient increases. Amount of Fading Using the second and first moment formulas of the SNR at the SSC output obtained in (9.708), it can be shown that the AF is given in this case by       2 Q 10 log10 (γT )−µ − 2σ + Q 2σρ − 10 log10 (γT )−µ σ ξ ξ σ σ   AFSSC = exp     2  − 1 2 ξ 10 log10 (γT )−µ 10 log10 (γT )−µ σρ σ Q − ξ +Q ξ − σ σ (9.716) Unfortunately, minimization of the AF in (9.716) with respect to γT does not provide a closed-form result for the optimum switching threshold as was the case for the average SNR criterion. For this reason, the AF as given in (9.716) was minimized numerically. An illustrative example showing the dependence of the AF on γT is given in Fig. 9.108. Figure 9.109 shows the amount of fading for SSC (with optimal switching threshold) versus the fading standard deviation for various values of the correlation coefficient. The AF remains fairly unaffected by

579

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

Average Combined SNR [dB]

25

20

15

MRC SC SSC 10

4

5

6

7

8

9

10

Standard Deviation s [dB]

Figure 9.105 Comparison of the average output SNR for MRC, SC, and SSC (with optimum switching threshold) versus the fading standard deviation for uncorrelated branches.

Average Combined SNR [dB]

25

20

15

MRC SC SSC 10

4

5

6

7

8

9

10

Standard Deviation s [dB]

Figure 9.106 Comparison of the average output SNR for MRC, SC, and SSC (with optimum switching threshold) versus the fading standard deviation for correlated branches ρ = 0.5.

580

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Combined SNR [dB]

25

20

15

MRC SC SSC 10

4

5

6

7

8

9

10

Standard Deviation s [dB]

Figure 9.107 Comparison of the average output SNR for MRC, SC, and SSC (with optimum switching threshold) versus the fading standard deviation for highly correlated branches ρ = 0.95. 3

2.8 r = 0.95

Amount of Fading

2.6 r = 0.75 2.4 r = 0.5 2.2 r = 0.25 2 r= 0 1.8

0

2

4

6

8 gT [dB]

10

12

14

16

Figure 9.108 Dependence of the AF of SSC on the switching threshold for various values of the correlation coefficient; µ = 10 dB and σ = 5 dB.

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

581

Amount of Fading for SSC

102

101

r = 0.95 r = 0.75 r = 0.5 r = 0.25 r=0

100 4

5

6

7

8

9

10

Standard Deviation s [dB] Figure 9.109 Amount of fading for SSC (with optimum switching threshold) versus the fading standard deviation for various values of the correlation coefficient.

correlation for low values of the correlation coefficient. However, as soon as the correlation coefficient exceeds about 0.7, a noticeable increase in AF is observed. Figures 9.110 and 9.111 compare the AF for MRC, SC, and SSC (with optimum switching threshold) versus the fading standard deviation for uncorrelated branches and for various negative and positive values of the correlation coefficient, respectively. MRC clearly outperforms SC and SSC for negatively correlated or uncorrelated branches. However, as the correlation becomes positive and increases between the two branches, the diversity gain of MRC compared to SS and SSC reduces considerably and becomes negligible for ρ = 0.95. Comparing the various subplots, it is also interesting to note that the negative values of the correlation coefficient lead to the lowest values of the AF, which may be explained by the fact that, in such a correlation scenario, it is more likely that one branch is experiencing good channel conditions. 9.13.4.2 Outage Probability SSC is given by [9, Eq. (20)] The outage probability of dual-branch SSC Pout SSC = P [γSSC ≤ γth ] Pout

= P [γT ≤ γ1 ≤ γth ] + P [γ1 ≤ γth and γ2 ≤ γT ]

(9.717)

582

PERFORMANCE OF MULTICHANNEL RECEIVERS

Amount of Fading

101

MRC SC SSC

100

4

4.5

5

5.5

6

6.5

7

Standard Deviation s [dB] Figure 9.110 Comparison of the amount of fading of MRC, SC, and SSC (with optimum switching threshold) versus the fading standard deviation for uncorrelated branches ρ = 0.

which can be written in terms of one- and two dimensional Gaussian Q-functions as



µ − 10 log10 γT µ − 10 log10 γth −Q =Q σ σ

µ − 10 log10 γth µ − 10 log10 γT , ;ρ +Q σ σ

Pout

(9.718)

for γth ≥ γT and

Pout

µ − 10 log10 γth µ − 10 log10 γT , ;ρ =Q σ σ

(9.719)

for γth ≤ γT , which (in view of the alternative representations given in Section 6.4.1) can be expressed in the form of single finite-range integrals. Note that if we substitute γT for γth in (9.718) and (9.719), then the outage probability of SSC reduces to

Pout

µ − 10 log10 γth µ − 10 log10 γth , ;ρ =Q σ σ

(9.720)

DUAL-BRANCH DIVERSITY OVER LOG-NORMAL CHANNELS

(r = 0.5)

583

(r = 0.95)

Amount of Fading

101

Amount of Fading

101

SSC SC MRC 100

SSC SC MRC 100

4

5

6

7

4

Standard Deviation s [dB]

5

6

7

Standard Deviation s [dB]

(r = −0.5)

(r = −0.95)

Amount of Fading

101

Amount of Fading

101

SSC SC MRC 100

SSC SC MRC 100

4

5

6

Standard Deviation s [dB]

7

4

5

6

7

Standard Deviation s [dB]

Figure 9.111 Comparison of the amount of fading of MRC, SC, and SSC (with optimum switching threshold) versus the fading standard deviation for various values of the correlation coefficient.

which corresponds to the outage probability of SC, as given in (9.703) for identically distributed diversity branches. Since SC can be viewed as an optimal implementation of any switched diversity system, we can conclude that the optimal switching threshold in the minimum outage probability sense is given by γT∗ = γth .

584

PERFORMANCE OF MULTICHANNEL RECEIVERS

100

Outage Probability

10−1

10−2

10−3

10−4

10−5 −10

No Diversity SC/SSC(Optimal) MRC/EGC(UpperB) EGC(Simulation) MRC(Simulation) MRC/EGC(LowerB)

−5

0

5

10

15

20

25

30

Outage Threshold gth [dB]

Figure 9.112 Comparison of the outage probability for MRC, EGC, SC, and SSC (with optimum switching threshold) versus the outage threshold for highly correlated branches ρ = 0.9 (identically distributed branches with µ = 10 dB and σ = 5 dB).

As an example, Fig. 9.112 compares the outage probability of MRC, EGC, SC, and SSC (with optimum switching threshold) versus the outage threshold for highly correlated (ρ = 0.9) identically distributed branches (with µ = 10 dB and σ = 5 dB). On the basis of this figure and other numerical experiments and simulations, we notice that (1) as expected, MRC slightly outperforms EGC, which, in turn, outperforms SC and SSC (with optimum switching threshold); (2) EGC approaches the performance of MRC as the correlation increases; and (3) the upper and lower bounds on the outage probability of MRC and EGC get tighter as the correlation increases. In particular, while the lower bound of MRC remains relatively tight for various values of the correlation coefficient, the corresponding upper bound becomes loose (and, in particular, sometimes higher than the outage probability of SC or optimal SSC) for small or negative values of the correlation coefficient.

9.14

AVERAGE OUTAGE DURATION

So far, we have used the average combined SNR, the amount of fading (or equivalently the normalized standard deviation of the combined SNR), the outage probability, and the average probability of error, as performance measures of various diversity combining schemes. However, in certain communication system applications such as adaptive transmission schemes, these performance measures do

AVERAGE OUTAGE DURATION

585

not provide enough information for the overall system design and configuration. In that case, in addition to these measures, the frequency of outages [or equivalently the level crossing rate (LCR)] and the average outage duration (AOD) are important performance measures for the proper selection of the transmission symbol rate, interleaver depth, packet length, and/or time slot duration. As such, some studies have focused on the analysis of the AOD and the LCR of MRC [221–224], SC [223–226], SSC [227], and GSC [228] diversity combining techniques. In this section, as an example, we summarize the general method presented in Ref. 228 to obtain the LCR and AOD of GSC over independent diversity paths and present as special cases the LCR and AOD of MRC and SC. 9.14.1

System and Channel Models

9.14.1.1 Fading Channel Models We consider isotropic Rayleigh, Nakagami, and Rician types of fading. For all these fading models the time derivative of the signal amplitude process α˙ is always independent of the signal amplitude α and is normally distributed with zero mean but different variance depending on the type of fading. More specifically, it is well known that the PDF of α˙ [pα˙ (α)] ˙ is given by [2]

α˙ 2 1 exp − 2 ˙ = √ pα˙ (α) 2σ 2πσ

(9.721)

where for isotropic scattering σ 2 = x π 2 fm2 with fm the maximum Doppler frequency shift and x given by   , /m, x =  /(K + 1),

Rayleigh fading Nakagami fading Rician fading

(9.722)

9.14.1.2 GSC Mode of Operation Recall from Section 9.11.2 that GSC identifies the Lc paths with the strongest SNRs among the L available ones, then combines them as per the rules of MRC. Let α1 , α2 , . . ., αL denote the set of L non-i.i.d. random fading amplitudes associated with the L inputs of the GSC combiner. The corresponding average fading powers are 2 2 2 ≥ α2:L ≥ · · · αL:L ≥ 0 be the ordered denoted by l , l = 1, 2, . . . , L. Then let α1:L statistics obtained by arranging the instantaneous powers {αl2 }L l=1 in decreasing order. Thus, assuming that the noise powers of the input branches are the same, the total combined signal power αt2 can be written as αt2

=

Lc
l=1

2 αl:L

(9.723)

586

9.14.2

PERFORMANCE OF MULTICHANNEL RECEIVERS

Average Outage Duration and Average Level Crossing Rate

9.14.2.1 Problem Formulation An outage is declared whenever the instantaneous total combiner output amplitude αt falls below a predetermined threshold αth (i.e., αt < αth ). The AOD, T (αth ) (in seconds) is a measure used to determine the average length of time that the system remains in the outage status. Mathematically speaking, the AOD is given by [3] T (αth ) =

Pout N (αth )

(9.724)

where Pout = Pr[αt < αth ] is the outage probability and N (αth ) is the average LCR of the amplitude αt at level αth . The outage probability of GSC for various fading scenarios has already been studied, and closed-form expressions are available for some fading scenarios of interest in Section 9.11.2. Therefore, to compute the AOD in (9.724), we only need to calculate the LCR N (αth ). This LCR can be obtained from the well-known Rice formula provided, for example, in Ref. 3:  ∞ α˙ t pαt ,α˙ t (αth , α˙ t ) d α˙ t (9.725) N (αth ) = 0

where pαt ,α˙ t (αt , α˙ t ) is the joint PDF of αt and α˙ t . The method used to obtain pαt ,α˙ t (αt , α˙ t ) of GSC is basically the same for all fading scenarios and is presented in the text that follows. 9.14.2.2 General Formula for the Average LCR of GSC The joint PDF pαt ,α˙ t (αt , α˙ t ) can be expressed as  pαt ,α˙ t (αt , α˙ t ) = 

∞ ∞ 0

 ···

αLc :L



∞

α2:L

Lc -fold

pα˙ t |αt ,α1:L ,...,αLc :L (α˙ t |αt , α1:L , . . . , αLc :L )



×pαt ,α1:L ,...,αLc :L (αt , α1:L , . . . , αLc :L ) d α1:L · · · d αLc −1:L d αLc :L (9.726) where pαt ,α1:L ,...,αLc :L (αt , α1:L , . . . , αLc :L ) is the joint PDF of the combiner output envelope αt and the Lc highest envelopes α1:L , . . . , αLc :L and pα˙ t |αt ,α1:L ,...,αLc :L (α˙ t |αt , α1:L , . . . , αLc :L ) is the conditional PDF of α˙ t given αt , α1:L , . . . , αLc :L . From (9.723), the time derivative α˙ t of αt is given by Lc α˙ t =

l=1

αl:L α˙ l:L αt

(9.727)

We now present the derivation of the general formulas for the average LCR of GSC over non-i.i.d. diversity branches and then i.i.d. diversity branches.

587

AVERAGE OUTAGE DURATION

General Non-I.I.D. Case The PDF of the time derivative of the lth-order statistics pα˙ l:L (α˙ l:L ) can be obtained from L


pα˙ l:L (α˙ l:L ) =

pα˙ l:L |αl:L =αi (α˙ l:L |αl:L = αi )Pr[αl:L = αi ]

i=1 L


=

pα˙ i (α˙ l:L )Pr[αl:L = αi ]

(9.728)

i=1

where 

L


∞

Pr[αl:L = αi ] = 0

pαi (x)

l−1

[1 − Pαik (x)]

i1 ,i2 ,···,iL−1 =1 k=1 i1 =i2 =,···,=iL−1 =i

L−1 j =l

Pαij (x) d x

(9.729) Equation (9.727) indicates that conditioned on the αl:L , l = 1, . . . , Lc , and αt , α˙ t can be viewed as a linear combination of α˙ l:L . Therefore we can use either a Jacobian transformation or an MGF-based method to obtain the conditional PDF of α˙ t . Going back to the joint PDF pαt ,α˙ t (αt , α˙ t ) in (9.726), we can rewrite it as  ∞ ∞  ∞ pαt ,α˙ t (αt , α˙ t ) = ··· pα˙ t |αt ,α1:L ,...,αLc :L (α˙ t |αt , α1:L , . . . , αLc :L ) 

0

αLc :L

 Lc -fold

α2:L



×pαt |α1:L ,...,αLc :L (αt |α1:L , . . . , αLc :L ) ×pα1:L ,...,αLc :L (α1:L , . . . , αLc :L ) d α1:L . . . d αLc −1:L d αLc :L  ∞ ∞  ∞ = ··· pα˙ t |αt ,α1:L ,...,αLc :L (α˙ t |αt , α1:L , . . . , αLc :L ) 

0

αLc :L

 ×δ

 Lc -fold

αt2

−

Lc


α2:L

 

2 αl:L

l=1

×pα1:L ,...,αLc :L (α1:L , . . . , αLc :L ) d α1:L . . . d αLc −1:L d αLc :L (9.730) where δ(·) is the Dirac δ-function and pα1:L ,...,αLc :L (α1:L , . . . , αLc :L ) is known to be given by L


pα1:L ,...,αLc :L (α1:L , . . . , αLc :L ) =

pαi1 (α1:L ) . . . pαiL (αLc :L )

i1 ,i2 ,...,iL =1 i1 =i2 =,...,=iL

×

L j =Lc +1

Pαij (αLc :L )

c

(9.731)

588

PERFORMANCE OF MULTICHANNEL RECEIVERS

With the joint PDF pαt ,α˙ t (αt , α˙ t ) in hand, using (9.725), we have an analytical but complicated expression for the LCR for this non-i.i.d. case. We now show how these results reduce to relatively simple closed-form expressions for the i.i.d. case. Special I.I.D. Case Taking into account the i.i.d. assumption on the individual amplitudes α1 , . . . , αL , the PDF of the time derivative of the lth-order statistics pα˙ l:L (α˙ l:L ) can be obtained as pα˙ l:L (α˙ l:L ) =

L


pα˙ l:L |αl:L =αi (α˙ l:L |αl:L = αi )Pr[αl:L = αi ]

i=1

= pα˙ i (α˙ l:L )

L


Pr[αl:L = αi ] = pα˙ i (α˙ l:L )

(9.732)

i=1

where we made use of the i.i.d. property between the α˙ i s (hence their distributions are independent of their indexes) caused by the i.i.d. assumption of the αi s to get the second step of (9.732). Therefore the α˙ l:L s are i.i.d. Gaussian random variables with zero mean and variance 2 2 2 ]}L ˙ l2 ]}L σ 2 = {E[α˙ l:L l=1 = {E[α l=1 = x π fm

(9.733)

where x is defined in (9.722). On the basis of (9.727) and conditioned on αl:L , l = 1, . . . , Lc and αt , α˙ t can be viewed as a linear combination of i.i.d. Gaussian random variables, and as such is itself a conditional Gaussian random variable with zero mean and variance given by   σ2

c α˙ t |αt ,{αl:L }L l=1

= E|α ,{α t

Lc l:L }l=1

 

Lc ˙ l:L l=1 αl:L α

2 

αt2

 

(9.734)

Since the α˙ l:L s are independent, (9.734) can be rewritten as Lc σ2 Lc α˙ t |αt ,{αl:L }l=1

=

2 2 ˙ l:L ] l=1 αl:L E[α 2 αt

(9.735)

Substituting (9.723) in (9.735), we can rewrite (9.735) solely in terms of all αl:L s and α˙ l:L s, specifically  c 2 2 ] L E[α˙ l:L l=1 αl:L = Lc 2 l=1 αl:L

(9.736)

2 = E[α˙ l:L ] = x π 2 fm2 = σ 2

(9.737)

σ2 Lc α˙ t |αt ,{αl:L }l=1 which, in view of (9.733), results in σ2

L

c α˙ t |αt ,{αl:L }l=1

AVERAGE OUTAGE DURATION

589

Therefore, we have pα˙ t |αt ,α1:L ,...,αLc :L (α˙ t |αt , α1:L , . . . , αLc :L ) = {pα˙ l:L (α˙ t )}L ˙ t )}L ˙t) l=1 = {pα˙ l (α l=1 = pα˙ t (α (9.738) thus, α˙ t is independent of αt , α1:L , . . . , αLc :L and Gaussian distributed with zero mean and variance σ 2 = x π 2 fm2 . Going back to the joint PDF pαt ,α˙ t (αt , α˙ t ) in (9.726), we can now write  ∞ ∞  ∞ ··· pα˙ t (α˙ t ) pαt ,α˙ t (αt , α˙ t ) = 

0

αLc :L

 Lc -fold

α2:L



× pαt ,α1:L ,...,αLc :L (αt , α1:L , . . . , αLc :L ) d α1:L · · · d αLc −1:L d αLc :L = pα˙ t (α˙ t )pαt (αt )

(9.739)

Substituting (9.739) into (9.725) and noting that  ∞ σ α˙ t pα˙ t (α˙ t ) d α˙ t = √ 2π 0 we get the compact general formula for the LCR of GSC as σ N (αth ) = √ pαt (αth ) 2π where pαt (αt ) is the PDF of the GSC output envelope. 9.14.3

(9.740)

(9.741)

I.I.D. Rayleigh Fading

The general formula obtained in the last section can be applied to various fading channels by simply substituting in these generic expressions the PDFs and the outage probabilities of the combiner output amplitude of GSC with the corresponding fading scenarios. As an example, we use in the following section, (9.741) and (9.724) to obtain the LCR and AOD for GSC systems over i.i.d. Rayleigh fading channels. 9.14.3.1 Generic Expressions for GSC Using the PDF and CDF of the combiner output SNR of GSC over i.i.d. Rayleigh fading channels given in Section 9.11.2, we obtain the PDF and outage probability Pout of the GSC output envelope αt as   2 Lc −1 αt



L −αt2 / 2αt   e pαt (αt ) =   (Lc − 1)! Lc

Lc −1 L − Lc Lc l l l=1 

m  L
c −2 lαt2 1 −lαt2 /Lc  − × e − m! Lc 

+

L−L
c

(−1)Lc +l−1

m=0

(9.742)

590

PERFORMANCE OF MULTICHANNEL RECEIVERS

and Pout =

  

 



L
c −1 L 2 1 − e −αth / Lc   l=0  

2 αth 

l

l!

+

L−L
c

(−1)Lc +l−1

l=1





Lc −1 L − Lc Lc l l





m c −2  1 − e −(1+l/Lc )(αth2 /) L
l m 2 /
  −αth 1 − e − × −    Lc 1 + Ll c

m=0

k=0

k        k!    

2 αth 

(9.743) respectively. The LCR of GSC is obtained by substituting (9.742) in (9.741), yielding  Lc −1 2 αth .

L−L 
c L −α 2 /  2π  N (αth ) = e th  fm αth + (−1)Lc +l−1  (Lc − 1)! Lc  l=1



×



Lc −1 L
c −2 lα 2 Lc 1 L − Lc 2 e −lαth /Lc  − − th l l m! Lc  m=0

m 

    

(9.744) Then, substituting (9.743) and (9.744) in (9.724), we can get the AOD T (αth ) of GSC over i.i.d. Rayleigh fading channels. 9.14.3.2 Special Cases: SC and MRC The second-order statistics for GSC presented above yield in the limit the special cases corresponding to MRC and SC, as we discuss in what follows. Lc = 1 (SC) The LCR N (αth ) and AOD T (αth ) of the amplitude αt at level αth for the SC case can be obtained by setting Lc = 1 in (9.743) and (9.744) and then substituting the results into (9.724). In this case, it can be shown with the help of the binomial theorem and some extra manipulations that (9.744) reduces to .  L−1 2π 2 2 fm αth Le−αth / 1 − e −αth / (9.745) NSC (αth ) =  and the resulting AOD is given by 2

SC e αth / − 1 Pout TSC (αth ) = = NSC (αth ) L 2π  fm αth

(9.746)

AVERAGE OUTAGE DURATION

591

in agreement with the Dong and Beaulieu results given in Eqs. (18a) and (20a) of Ref. 225 (or equivalently the Yacoub et al. results [223, Eqs. (13) and (14)] for the Rayleigh fading condition (m = 1) and the Iskander and Mathiopoulos result [224, Eq. (18)]). Lc = L (MRC) Setting Lc = L makes (9.744) directly reduce to . NMRC (αth ) =

2π fm αth  (L − 1)!



2 αth 

L−1 e −αth / 2

(9.747)

and the resulting AOD is  (L − 1)! e TMRC (αth ) =

-

2 / αth

−

2π  fm αth



L−1

l   2 / αth

l=0

L−1 2 / αth

l!

(9.748)

which perfectly coincide with the Ko et al. results [221, Eqs. (15) and (17)] (or equivalently the Yacoub et al. results [223, Eqs. (38) and (39)] and the Iskander and Mathiopoulos result [224, Eq. (31)] for LCR). 9.14.4

Numerical Examples

The selected numerical examples focus on the i.i.d. Rayleigh fading scenario. First, Fig.√9.113 shows the AOD as a function of normalized outage threshold ρth = αth /  for various values of the number of available paths L and a fixed number of combined paths L √c = 2. Figure 9.114 plots the AOD versus normalized outage threshold ρth = αth /  for various values of the number of combined paths Lc and a fixed number of available paths L = 4. These two figures demonstrate that, for a fixed Lc , a relatively significant decrease in the AOD is obtained as the number of available diversity paths L increases, while, for a fixed number L of available diversity paths, diminishing diversity gain is obtained as the number of combined paths Lc increases. These observations are in agreement with those based on the previously studied performance criteria (i.e., average probability of error, average output SNR, or outage probability). In the context of ultrawideband communications over frequency-selective channels, spreading the transmitted signal as much as possible increases the number L of available multipaths at the receiver and as a result improves the diversity gain but on the other hand implies also a reduction in the average SNR per path for a fixed total “captured” average SNR γ t . This tradeoff and the optimum choice L∗ , which corresponds to an optimum spreading bandwidth, was studied in Ref. 229 for other performance criteria. We now study this optimum choice of L to minimize AOD and the frequency of outages (or equivalently LCR). Figures 9.115 and 9.116 plot the AOD and average LCR versus the number of available paths L for various values of the combined paths Lc and a fixed normalized total average captured

592

PERFORMANCE OF MULTICHANNEL RECEIVERS

Normalized Average AOD T(ath) × fm

101

No diversity (Analytical) No diversity (Simulation) GSC 2/2 (Analytical) GSC 2/2 (Simulation) GSC 2/3 (Analytical) GSC 2/3 (Simulation) GSC 2/4 (Analytical) GSC 2/4 (Simulation)

100

10−1

10−2 −10

−5

0

5

Normalized Outage Threshold rth = ath/√Ω in dB

√ Figure 9.113 Average outage duration versus normalized outage threshold ρth = αth /  for various values of the number of available paths L and a fixed number of combined paths Lc = 2.

Normalized Average AOD T(ath) × fm

101

GSC 1/4 GSC 2/4 GSC 3/4 GSC 4/4

100

10−1

10−2 −10

−5 0 Normalized Outage Threshold rth = ath/√Ω in dB

5

√ Figure 9.114 Average outage duration versus normalized outage threshold ρth = αth /  for various values of the number of combined paths Lc and a fixed number of available paths L = 4.

593

AVERAGE OUTAGE DURATION

102

Lc = 1, L*= 3 Lc = 2, L*= 5 Lc = 3, L*= 8

Normalized Average AOD T(ath) × fm

Lc = 4, L*= 11

Lc = 1

101

Lc = 2

100 *

L=3 Lc = 3

*

L=5

10−1

0

5

*

*

L=8

10

L = 11

15

20

Lc = 4

25

30

Number of Available Diversity Paths L

Figure 9.115 Average outage duration versus the number of available paths L for various values of the combined paths Lc and a fixed normalized total average captured SNR γ t /γth = 2 = 5dB. (L)/αth 101 L*= 2

Normalized Average LCR N(ath) /fm

100

Lc = 4

L*= 4 L*= 7 L*= 9

Lc = 3

10−1 Lc = 2

10−2

10−3

Lc = 1

10−4

10−5

Lc = 1, L*= 2 Lc = 2, L* = 4 Lc = 3, L* = 7 Lc = 4, L*= 9

0

5

10

15

20

25

30

35

40

45

50

Number of Available Diversity Paths L

Figure 9.116 Average level crossing rate versus the number of available paths L for various values of the combined paths Lc and a fixed normalized total average captured SNR γ t /γth = 2 = 5dB. (L)/αth

594

PERFORMANCE OF MULTICHANNEL RECEIVERS

2 SNR γ t /γth = L/αth = 5dB. It can be observed that a clear minimum for the AOD and LCR (or equivalently frequency of outages) is obtained, resulting in an optimum value of L. These optimum values are very close to the ones obtained for the outage probability criterion [229] for the same parameters. These figures also imply that the corresponding optimum spreading bandwidth increases with the number of combined paths Lc as is the case for the other performance criteria studied in Ref. 229.

9.15 MULTIPLE-INPUT/MULTIPLE-OUTPUT (MIMO) ANTENNA DIVERSITY SYSTEMS So far in this chapter, we dealt mainly with receiver diversity systems such as the ones consisting of a single antenna at the transmitter and multiple antennas at the receiver. However, applications in more recent years have become increasingly sophisticated, thereby relying on the more general multiple-input/multipleoutput (MIMO) antenna diversity systems [230–233], which promise significant increases in system performance and capacity. In this context, Tse et al. [231] derived the joint MRC weights at both mobile unit and base station over fading channels, and the performance of the resulting MIMO MRC system was analyzed over Rayleigh and Rician fading in Refs. 232 and 233, respectively. In what follows, we first briefly describe MIMO MRC systems, then summarize the analytical approach developed in Ref. 233, which extends the Khatri distribution of the largest eigenvalue of central complex Wishart matrices [234] to the noncentral case, and finally apply this new result to the performance analysis of MIMO MRC systems over Rician fading channels. The resulting expressions are obtained in terms of generalized hypergeometric functions, generalized Marcum Q-functions, modified Bessel functions, or incomplete gamma functions. These expressions are also compared to the special cases dealing with the outage probability of MIMO MRC systems over Rayleigh fading channels [232] and traditional single-input/multiple-output (SIMO) MRC systems as covered earlier in this chapter. 9.15.1

System, Channel, and Signal Models

We consider a wireless link equipped with Lt antenna elements at the transmitter and Lr antenna elements at the receiver as shown in Fig. 9.117. The discrete equivalent Lr × 1 received vector r can be modeled as r = HD wt sD + n

(9.749)

where sD is the transmitted signal of the desired user and n is the AWGN vector with zero mean and covariance matrix N0 IR . Without loss of generality, we assume that sD has unit average power. In (9.749), wt represents the weight vector at the transmitter with wt 2 = Es (i.e, the power of the vector wt is restricted to be Es )

595

MULTIPLE-INPUT/MULTIPLE-OUTPUT (MIMO) ANTENNA DIVERSITY SYSTEMS

Wt,1

Branch 1 r1

Branch 1 X

SD

Wt,2

X

X

Wr,2*

Branch 2 r2

Branch 2

X

Wt,Lt

Wr,1*

+

X

Wr,Lr*

Branch Lr rLr

Branch Lt

Output

X

Figure 9.117 Block diagram of a wireless link equipped with Lt antenna elements at the transmitter and Lr antenna elements at the receiver.

and HD is the channel gain matrix of the desired user defined by    HD =  

hD,1,1 hD,2,1 .. .

hD,1,2 hD,2,2 .. .

··· ··· .. .

hD,Lr ,1

hD,Lr ,2

· · · hD,Lr ,Lt

hD,1,Lt hD,2,Lt .. .

    

(9.750) Lr ×Lt

where hD,i,j denotes the complex channel gain of the desired user from the j th transmitter antenna element to the ith receiver antenna element. In what follows, we also assume that both the transmitter and the receiver have perfect channel state information and as such know HD instantaneously. 9.15.2

Optimum Weight Vectors and Output SNR

Before analyzing the performance of the MIMO system of interest, we review the MIMO MRC combining scheme considered in Refs. 231,232,233. The optimum combining vector at the receiver (given the transmitting weight vector wt ) is well known to be given by wr = c HD wt

(9.751)

where c is a constant that does not affect the output SNR. The resulting conditional (on wt ) maximum SNR is given by γ =

1 H H w H HD wt N0 t D

(9.752)

According to the Rayleigh–Ritz theorem [235, Sect. 4.2.2], for any nonzero N × 1 complex vector x and a given N × N Hermitian matrix A, xH Ax ≤ x2 λmax ,

596

PERFORMANCE OF MULTICHANNEL RECEIVERS

where λmax is the largest eigenvalue of A. The equality holds if and only if x is along the direction of the eigenvector corresponding to λmax . On the basis of this fact, we may choose the transmitting weight vector as  wt = D Umax (9.753) where Umax (Umax  = 1) denotes the eigenvector corresponding to the largest eigenvalue of the quadratic form F = HH D HD

(9.754)

It is obvious that F is a Hermitian nonnegative definite matrix. Therefore the maximum output SNR is given by γ =

Es λmax N0

(9.755)

where λmax is the largest eigenvalue of the matrix HH D HD , or equivalently, the largest eigenvalue of HD HH . Note that λ > 0 with probability 1; therefore the max D maximum output SNR γ defined in (9.755) is always positive, as expected. Note that the signals are transmitted along the direction of the eigenvector corresponding to the largest eigenvalue of HH D HD , and as such the MIMO MRC scheme is also sometimes referred to as “beamforming.” 9.15.3 Distributions of the Largest Eigenvalue of Noncentral Complex Wishart Matrices In order to statistically characterize the SNR at the output of MIMO MRC diversity systems as given in (9.755), we present in this subsection some results presented in Ref. 233 dealing with the distribution of the largest eigenvalue of noncentral complex Wishart matrices. These results are presented here without derivations but detailed proofs for interested readers can be found in the Ref. 233 appendices. Let X be an m × n matrix whose columns are independent m-variate complex Gaussian vectors with covariance matrix  and let S =  −1 XXH , where the superscript H denotes the Hermitian conjugate transpose operator and (·)−1 the matrix inverse. When m ≤ n, S is said to follow a complex Wishart distribution. To characterize the distribution of the output SNR, we need the distribution of the largest eigenvalue of matrices of the type of S. Khatri [234] gave the CDF of the largest eigenvalue of S when the mean of X, E(X) = 0. Extension of Khatri’s result to the noncentral case [i.e., when E(X) = M] was derived in Ref. 233 and is briefly presented in what follows without proof. 9.15.3.1

CDF of S

Theorem 9.2 Let X be an m × n matrix whose columns are independent mvariate complex Gaussian vectors with covariance matrix  and let E(X) = M. Let

597

MULTIPLE-INPUT/MULTIPLE-OUTPUT (MIMO) ANTENNA DIVERSITY SYSTEMS

s = min(m, n), t = max(m, n), and 0 < φ1 < φ2 < · · · < φs be m nonzero eigenvalues of S =  −1 XXH . Then the CDF of the largest eigenvalue φs of S is given by e −tr() |(x)| |V| ( (t − s + 1))s

Pr(φs ≤ x) =

(9.756)

where tr(·) denotes the trace operator,  = diag(λ1 , . . . , λs ), where 0 < λ1 < λ2 < · · · < λs are s nonzero distinct eigenvalues of MH  −1 M, | · | denotes the determinant, s   s−j = (λi − λj ) |V| = det λi

(9.757)

i 1 and L2 > 1 are not necessarily equal, the L1 L2 two real constants A and B are such that AB < 0, and the {Xk }k=1 and {Yk }k=1 are all mutually independent complex Gaussian random variables (RVs) with means L1 L2 and {Y k }k=1 , respectively, and with variances {X k }k=1  2  µXX = 12 E Xk − Xk   2  (10.2) µY Y = 12 E Yk − Y k  which are independent of k.1 We are interested in the probability P given by  0 P = Pr {D < 0} = p (D) dD (10.3) −∞

which is the CDF of the variate D evaluated at zero. This problem is analogous to a reduced version of that treated by Proakis [24], (or equivalently Ref. 23, App. B and Appendix 9A of this book); the exception is that we now allow the two chisquare variates (sums of magnitude-squared complex Gaussian RVs) in (10.2) to have different degrees of freedom (L1 and L2 ). In Appendix 10A, we prove the following theorem. Theorem 10.1

The probability P can be expressed in closed form as   1 P = Q1 (a, b) − exp − a 2 + b2 I0 (ab) 2  L1 −1

1 2

I0 (ab) exp − 2 a + b2
L1 + L2 − 1 ηk + k [1 + η]L1 +L2 −1

+

−

 1

exp − 2

 L1 −1
a 2 + b2

[1 + η]L1 +L2 −1

L
2 −1 n=1

In (ab)

L2
−1−n k=0

k=0

In (ab)

L1
−1−n

n=1

L1 + L2 − 1 k

k=0

L1 + L2 − 1 k



 a n ηL1 +L2 −1−k b

n b ηk a (10.4)

where Q1 (·, ·) is the first-order Marcum

Q-function, In (·) is the nth-order modified L Bessel function [26, Sect. 8.43], = L!/(k!(L − k)!) and the parameters a, k b, and η are given by   2 L1   k=1 A X k a= (10.5) AµXX − BµY Y 1 This is the same assumption as that made in Ref. 23, App. B and Appendix 9A of this book. If the variances are not independent of k, then the approach taken by Proakis [23, App. B] and likewise that taken here cannot lead to a closed-form solution.

642

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS



L2

k=1 B

b=

−

η=−

B µY Y A µXX

 2 Y k 

AµXX − BµY Y

(10.6) (10.7)

Clearly (10.4) reduces to the Proakis result given in Eq. (9A.3) when L1 = L2 = L. This result in (10.4) can alternatively be expressed in terms of the generalized Marcum Q-function as was done in Appendix 9A for the original Proakis result (i.e., L1 = L2 = L). In particular, we obtain after much manipulation the following corollary. Corollary 10.1 The probability P can be written in terms of the generalized Marcum Q-function as  P = Q1 (a, b) − 1 −

+

1 (1 + η)L1 +L2 −1

L1 −1 L1 +L2 −1 k=0

(1 + η)

η k L1 +L2 −1

k



2

a + b2 exp − I0 (ab) 2

L

1
L1 + L2 − 1 ηL1 −l [Ql (a, b) − Q1 (a, b)] L1 − l l=2



L2
L1 + L2 − 1 L1 +l−1 [Ql (b, a) − Q1 (b, a)] − η L2 − l

(10.8)

l=2

where Ql (·, ·) is the lth-order (generalized) Marcum Q-function. As a check note that (10.8) reverts to [25, Eq. (60)] or equivalently Eq. (9A.7) of Appendix 9A for the case L1 = L2 = L. As was done in Appendix 9A, the result in (10.8) can be put in the form of a single finite-range integral using the alternative representation of the generalized Marcum Q-function discussed in Chapter 4. In particular, we get, after considerable manipulations, this second corollary. Corollary 10.2 The probability P can be written in the form of a single integral with finite limits and an integrand composed of simple (trigonometric and exponential) functions as P =

f (L1 , L2 ; ζ , η; θ) 2π (1 + η)L1 +L2 −1 −π 1 + 2ζ sin θ + ζ 2

2  b

 a 2 dθ , 0+ ≤ ζ = < 1 × exp − 1 + 2ζ sin θ + ζ 2 b ηL1



π



(10.9)

OUTAGE PROBABILITY IN INTERFERENCE-LIMITED SYSTEMS

643

where f (L1 , L2 ; ζ , η; θ) = f0 (L1 , L2 ; ζ , η; θ ) + f1 (L1 , L2 ; ζ , η; θ)

(10.10)

and2 

1 (1 + η)L1 +L2 −1
f0 (L1 , L2 ; ζ , η; θ ) = − + ηL1

L



l=1

+

L2
l=1

f1 (L1 , L2 ; ζ , η; θ ) =

L1 + L2 − 1 L2 − l

η−l



η

L1 + L2 − 1 L1 − l

l−1

ζ (ζ + sin θ) = 0

L1
L1 + L2 − 1 η−l ζ −l+1 L1 − l l=1      π  π  − ζ cos l θ + × cos (l − 1) θ + 2 2

L 2
L1 + L2 − 1 ηl−1 ζ l + L2 − l l=1      π  π  − ζ cos (l − 1) θ + × cos l θ + 2 2

(10.11)

As a check, note that (10.9) with (10.11) agrees with Eqs. (62) and (63) of Ref. 25 [or equivalently Eq. (9A.20)] when L1 = L2 = L. 10.1.2

Fading and System Models

10.1.2.1 Channel Fading Models We consider Rayleigh, Rice, and Nakagami types of fading. Let S denote the instantaneous faded signal power, and let  = ES [S] denote its short-term average. A key observation is that for the three fading models under consideration, S can be viewed as a chi-square variate since it can be written as S=

N


|Xn |2

(10.12)

n=1

where the {Xn }N n=1 are independent complex Gaussian RVs with common mean E[Xn ] = X and common variance µXX = 12 E{|Xn − Xn |2 }. In particular, for Rayleigh fading, S is a central chi-square variate with two degrees of freedom (N = 1), X = 0, and µXX = /2. For the Rician case, S is√ a noncentral chisquare variate with two degrees of freedom (N = 1), X = K/(K + 1), and 2 We

note that the term f0 (L1 , L2 ; ζ , η; θ) can be shown to be identically equal to zero. We include it here to allow comparison with the corresponding result in Appendix 9A when L1 = L2 = L.

644

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

µXX = /[2(K + 1)], with K ≥ 0 denoting the Rician factor. Finally, for the Nakagami case, S can be viewed as a central chi-square variate (N = m), X = 0, and µXX = /2m, where m is the fading parameter which in general can be any real number greater than or equal to 12 but that under this model is restricted to integer values. 10.1.2.2 Desired and Interference Signals Model We consider a cellular mobile radio system in which the desired and interfering signals may have different fading statistics. The desired signal is assumed to be received over L i.i.d. diversity (time, frequency, or space) paths and to be independent of the interfering signals. The corresponding L received desired instantaneous signal powers {SD,l }L l=1 are assumed to have the same average fading power D and be subject to slowly varying flat Rayleigh, Rician (with Rician factor KD ), or Nakagami (with fading parameter mD ) type of fading. The L replicas of the desired signal are combined, as per the rule of MRC, and the resulting total desired signal power SD is thus given by SD =

L


(10.13)

SD,l

l=1 N

I On the other hand, the NI active interfering signal powers {SI ,n }n=1 are assumed to be i.i.d. with the same short-term average fading power I and subject to slowly varying flat Rayleigh, Rician (with Rician factor KI ), or Nakagami (with fading parameter mI ) type of fading. We assume throughout our study that the co-channel interfering signals add up incoherently since this leads to a more realistic assessment of the co-channel interference in cellular systems [28]. Under this assumption the total instantaneous interference power SI is just the sum of the instantaneous powers of the NI active interferers and is therefore given by

SI =

NI


SI ,n

(10.14)

n=1

10.1.3

A Generic Formula for the Outage Probability

We consider interference-limited systems for which an outage is declared when the CIR λ = SD /SI falls below a predetermined protection ratio λth . Hence, the outage probability Pout is given by  SD (10.15) Pout = Pr λ = ≤ λth SI or equivalently

Pout = Pr [D = SD − λth SI ≤ 0]

(10.16)

In view of the key observation of Section 10.1.2.1 and under the diversity paths and interference signals i.i.d. assumptions discussed in Section 10.1.2.2, the

OUTAGE PROBABILITY IN INTERFERENCE-LIMITED SYSTEMS

645

N

I {SD,l }L l=1 are chi-square variates, and so are the {SI ,n }n=1 . Since the sum of i.i.d. chi-square variates is also a chi-square variate, we can conclude from (10.13) and (10.14) that SD and SI are chi-square variates, respectively. Hence, the RV D as given in (10.16) can be viewed as the difference of two chi-square variates (with not necessarily the same number of degrees of freedom) and the outage probability, as given in (10.16), is therefore in the form of the probability P in (10.3). To obtain this outage probability in closed form, we just need to set A = −λth and B = 1 in (10.1), whereas the other relevant parameters in (10.1) (i.e., L1 , L2 , Xk , Y k , µXX , and µY Y ) are set to different values according to the scenario of interest. The four scenarios under consideration in this chapter, namely, (1) both desired and interfering signals are subject to Nakagami fading (Nakagami/Nakagami), (2) both desired and interfering signals are subject to Rician fading (Rice/Rice), (3) desired signals are subject to Rician fading whereas interfering signals are subject to Nakagami fading (Rice/Nakagami), and (4) desired signals are subject to Nakagami fading whereas interfering signals are subject to Rician fading (Nakagami/Rice), are discussed in detail in the following subsections.

10.1.3.1 Nakagami/Nakagami Scenario To find the outage probability in the case where both the desired and interfering signals are subject to Nakagami fading, we just need to set L1 = mI NI , L2 = mD L, Xk = Y k = 0, µXX = I /2mI , and µY Y = D /2mD . Substituting these settings along with A = 1 and B = −λth in (10.5), (10.6), and (10.7), leads to a, b, and η in Table 10.1, which when substituted in (10.8) [or equivalently (10.4) or (10.9)] and using the fact that Ql (0, 0) = 1, (l = 1, 2, . . . , L), results in Pout = 

1+

k

NI −1 mI
mD L + mI NI − 1 D mI 1 mD L+mI NI −1 λth I mD k

D mI λth I mD

k=0

(10.17) For the no-diversity case (i.e., L = 1), (10.17) is equivalent to the Yao and Sheikh result [4, Eq. (13)] (as we show in Appendix 10B) and is the same form as the Abu-Dayya and Beaulieu formula [3, Eq. (7a)] for the further particular case of mD = mI = m. Furthermore, when the desired user is subject to Rayleigh fading (mD = 1), it can be shown using the binomial expansion, that (10.17) reduces to the compact Tellambura and Bhargava expression [9, Eq. (18)]

λth I −mI NI Pout = 1 − 1 + mI D

(10.18)

which further simplifies to the Sowerby and Williamson formula [1, Eq. (20)] given by

λth I −NI Pout = 1 − 1 + D when the interferers are also subject to Rayleigh fading (i.e., mI = 1).

(10.19)

646

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

TABLE 10.1 Parameters L1 , L2 , a, b, and η for the Various Scenarios under Consideration Desired/Interferers Rayleigh/Rayleigh Rayleigh/Rice

L1

L2

NI

a

b

η

0

0

D λth I

2KI NI  1 + (1 + KI ) λ D

0

(KI + 1)D λth I

0

mI D λth I

L    

NI

L

mI NI

L

0

NI

L

0

th I

Rayleigh/Nakagami Rice/Rayleigh

   

2LKD λth I 1 + (KD + 1) 

D (KD + 1)λth I

2LKD (KI + 1) λth I KI + 1 + (KD + 1) 

(KI + 1)D (KD + 1)λth I

D

Rice/Rice

NI

L

   

2NI (KD + 1)KI  KD + 1 + (KI + 1)  λD

   

I th

Rice/Nakagami

mI NI

L

0

Nakagami/Rayleigh

NI

mD L

0

Nakagami/Rice

NI

mD L

mI NI

mD L

Nakagami/Nakagami

   

D

   

2KI NI 

1 + (1 + KI ) m λD  D th I

0

2LKD

λ  (K +1) 1 + th mI D I D

mI D (1 + KD )λth I

0

D mD λth I

0

(KI + 1)D mD λth I

0

mI D mD λth I

10.1.3.2 Rice/Rice Scenario To find the outage probability in the case where both the desired and interfering signals are subject to Rice fading, we just need to set L1 = NI , L2 = L,  Xk = µXX =

 KI I , KI + 1

I , 2(KI + 1)

Yk = µY Y =

KD D , KD + 1 D 2(KD + 1)

Substituting these settings along with A = −λth and B = 1 in (10.5), (10.6), and (10.7), leads to a, b, and η in Table 10.1, which, when substituted in (10.8) [or equivalently (10.4) or (10.9)] yields the outage probability for the Rice/Rice scenario. For the no-diversity case (i.e., L = 1), it can be easily shown that the resulting expression is in agreement with the Tjhung et al. result [11, Eq. (13)],3 and it therefore reduces to all the special cases identified and discussed by Tjhung et al. [11, Sect. II-B] (in particular the Rice/Rayleigh case studied by Yao and Sheikh [2] and the Rayleigh/Rice case studied by Haug and Ucci [5]). 3 More precisely, the resulting expression is in agreement with the Tjhung et al. result [11, Eq. (13)] if a minor typo is corrected in Eq. (13) of Ref. 11 (the square root of the Bessel function argument should not extend over the denominator of the fraction).

OUTAGE PROBABILITY IN INTERFERENCE-LIMITED SYSTEMS

647

10.1.3.3 Rice/Nakagami Scenario To find the outage probability in the case where the desired user is subject to Rician fading whereas the interfering signals are subject √ to Nakagami fading, we just need to set L1 = mI NI , L2 = L, Xk = 0, Y k = (KD D )/(KD + 1), µXX = I /2mI , and µY Y = D /[2(KD + 1)]. Substituting these settings along with A = −λth and B = 1 in (10.5), (10.6), and (10.7), leads to a, b, and η in Table 10.1, which, when substituted in (10.8) [or equivalently (10.4) or (10.9)] and using the classical identities of the generalized Marcum Q-function given by Dillard [27, Eq. (9)] and Chapter 4 as Ql (0, w) = e −w

2 /2

l−1
(w2 /2)k

k!

k=0

Ql (u, 0) = 1,

,

l = 1, 2, . . . , L l = 1, 2, . . . , L

(10.20)

results in e −b /2 = (1 + η)mI NI +L−1 2

Pout

+

m


mI NI + L − 1 ηk k

I NI −1

k=0

m NI −k−1 2 n  mI NI + L − 1 k I
(b /2) η n! k

mI
NI −2 k=0

(10.21)

n=1

The outage probability of (10.21) can be shown to be in agreement with the Lin et al. expression [13, Eq. (14)] for the no-diversity case (i.e., L = 1) and with the Yao and Sheikh result [2, Eq. (13)] for the no-diversity case and Rayleigh interferers (i.e., mI = 1). 10.1.3.4 Nakagami/Rice Scenario To find the outage probability in the case where the desired user is subject to Nakagami fading whereas the interfering signals √ are subject to Rician fading, we just need to set L1 = NI , L2 = mD L, Xk = (KI I )/(KI + 1), Y k = 0, µXX = I /[2(KI + 1)], and µY Y = D /2mD . Substituting these settings along with A = −λth and B = 1 in (10.5), (10.6), and (10.7) leads to a, b, and η, which, when substituted in (10.8) [or equivalently (10.4) or (10.9)] and using the classical identities of the generalized Marcum Q-function given in (10.20), results in 

2 N
I −1 mD L + NI − 1 k e −a /2 mD L+NI −1 η − Pout = 1 − (1 + η) (1 + η)mD L+NI −1 k k=0 

mD L−2 m L−k−1
mD L + NI − 1 mD L+NI −k−1 D
(a 2 /2)n η + (10.22) n! k k=0

n=1

The outage probability of (10.22) can be shown to be in agreement with the Wang and Lea expression [29, Eq. (13)] for the no-diversity case (i.e., L = 1).

648

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

10.2 OUTAGE PROBABILITY WITH A MINIMUM DESIRED SIGNAL POWER CONSTRAINT As mentioned in the introduction of this chapter, in multiuser wireless communications systems that are not necessarily interference-limited, an outage occurs when either CIR or CNR is less than its respective predetermined threshold value. In what follows, we summarize the approach adopted in Ref. 33 to study these kinds of systems by considering two special fading combinations: Nakagami/Rician and Rician/Nakagami, for which closed-form expressions for outage probability are given in terms of the Marcum Q-function and modified Bessel functions. 10.2.1

Models and Problem Formulation

10.2.1.1 Fading and System Models We consider Nakagami and Rician types of fading. Let SD denote the instantaneous faded power of the desired signal, and let D = ESD [SD ] denote its short-term average. We assume that SD is subject to either Nakagami (with Nakagami parameter mD ) or Rician (with Rician factor KD ) type of fading depending on the scenario of interest. For Nakagami fading, the PDF of SD is given by pSD (SD ) =

mD D

mD

m −1 SD D m D SD exp − (mD ) D

(10.23)

while for Rician fading, the PDF becomes 

  KD (KD + 1) KD + 1 KD + 1 pSD (SD ) = exp −KD − SD I 0 2 SD D D D (10.24) where Ik (·) is the kth-order modified Bessel function. Note that both mD = 1 for Nakagami fading and KD = 0 for Rician fading lead to the Rayleigh case with the PDF given by pSD (SD ) = exp(−SD /D )/D . Similar to our treatment of interference-limited systems, we again assume for NI are analytical tractability that the power of the NI active interfering signals {Si }i=1 i.i.d. with the same short-term average fading power I and subject to Nakagami (with Nakagami parameter mI ) or Rician (with Rician factor KI ) type of fading. The PDF of the interfering signal pSi (Si ) is of the same form as in (10.23) and (10.24) for the desired signals, but with different parameters. Also, we again assume that the co-channel interfering signals add up incoherently since this leads to a more realistic assessment of the co-channel interference in cellular systems [28]. As such, under this assumption, the total instantaneous interference power SI is just the sum  I of the instantaneous powers of the NI active interferers: SI = N i=1 Si . 10.2.1.2 Outage Probability Definition In the systems that are both interference-limited and power limited, an outage is declared when either the CIR λ = SD /SI falls below a predetermined protection

OUTAGE PROBABILITY WITH A MINIMUM DESIRED SIGNAL POWER CONSTRAINT

649

ratio λth or the instantaneous power of the desired signal SD is less than a predetermined threshold Sth . Mathematically speaking, the outage probability Pout is thus given by Pout

 SD = Pr λ = ≤ λth or SD ≤ Sth SI

(10.25)

which can be written in terms of the PDF of SD and the PDF of the total instant interference power SI , pSI (SI ), as  Pout = 1 −

∞  SD /λth

Sth

10.2.2

pSI (SI ) d SI

0

pSD (SD ) d SD

(10.26)

Rice/I.I.D. Nakagami Scenario

In this section the simpler Rician/i.i.d. Rayleigh scenario is first investigated, and then the approach utilized is generalized to the more general Rician/i.i.d. Nakagami fading combination when the Nakagami fading parameter is restricted to integer values. 10.2.2.1 Rice/I.I.D. Rayleigh Scenario Let us assume that the received signal from the desired user is Rician distributed with parameters D and KD and that there are NI i.i.d. Rayleigh interferers in the system with average fading power I . General Case In this case, with the closure property of the gamma distribution, the total instantaneous interference power SI has a gamma PDF with the following form:

SI exp − pSI (SI ) = N I I I (NI − 1)! NI −1

SI

(10.27)

Applying the finite-sum representation of the gamma CDF and (10.24) in (10.26) gives, after switching the order of integration and summation, the outage probability as

Pout

 



 NI −1 2(KD + 1)Sth 1 KD + 1
= 1 − Q1 2KD , + D D n! n=0

n

  ∞ SD 1 1 + KD SD × exp −KD − + λth I λth I D Sth    KD (KD + 1)SD × I0 2 (10.28) d SD D

650

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

Using an appropriate change of variable in the integral (10.28), it can be shown, after some manipulations and with the help of Nuttall’s report [34], that the outage probability can be written in the following closed form

Pout = 1 − Q1

 

 2KD ,

2(KD + 1)Sth D



NI −1 An a2
Q2n+1,0 (a, b) + 2KD n!

(10.29)

n=0

where 



−1  2(KD + 1)λth I −n D An = 2 + exp −KD + KD 1 + D (KD + 1)λth I  

−1

D Sth D a = 2KD 1 + , b = 2 1 + KD + (KD + 1)λth I λth I D and Qm,n (., .) is the Nuttall Q-function, which was defined and discussed in Chapter 4. Since the computations of the Nuttall Q-functions involved in (10.29) are such that the sum of their indices is odd, all Nuttall functions Qm,n (a, b) can be expressed in terms of just the first-order Marcum Q-function and modified Bessel functions. Specifically, with the help of the following recursive relation [34] Qm,n (a, b) = (m + n − 1)Qm−2,n (a, b) + aQm−1,n+1 (a, b) + b

m−1

2

a + b2 exp − In (ab) 2

(10.30)

all the Nuttall Q-functions involved in (10.29) can be expressed just in terms of generalized Marcum Q-functions as shown in Fig. 10.1, which themselves are just the sum of first-order Marcum Q-functions and a finite number of weighted Bessel functions. It is not difficult to verify that the evaluation of the outage probability expression given in (10.29) involves computing NI + 1 generalized Marcum Q-functions and (NI − 1)NI /2 Bessel functions, both of which are, for example, available in MATLAB. Special Cases •

Single Rayleigh Interferer. If there is only one Rayleigh interferer (i.e., NI = 1), (10.29) reduces to

Pout

     a2 2(KD + 1)Sth a2 + Q1 (a, b) = 1 − Q1 2KD , exp −KD + D 2KD 2 (10.31)

OUTAGE PROBABILITY WITH A MINIMUM DESIRED SIGNAL POWER CONSTRAINT

651

n

Recursive relation of Nuttall functions, Eq. (10.30) Qm − 1, n + 1

Generalized Marcum Q-functions am−1Qm

Qm − 2, n

Qm, n

Nuttall functions in Eq. (10.46) Q2i + N, N − 1 i = 0, 1, ...

Nuttall functions in Eqs. (10.29) and (10.35) Q2i + 1,0 i = 0, 1, ... 0

m

Figure 10.1 Illustration of the calculation of the Nuttall Q-functions in (10.29), (10.35), and (10.46) using the generalized Marcum Q-functions.

in agreement with the Yao and Sheikh result [2, Eq. (4)]. For the interferencelimited case (i.e., Sth = 0), (10.31) further simplifies to 

−1

−1  D D exp −KD +KD 1 + Pout = 1+ (KD + 1)λth I (KD + 1)λth I (10.32) in agreement with the Yao and Sheikh result [2, Eq. (7)]. • Two Rayleigh Interferers. If there are two Rayleigh interferers, the outage probability is given by     2(KD + 1)Sth a2 2KD , [(A0 + (2 + a 2 )A1 )Q1 (a, b) + Pout = 1 − Q1 D 2KD

2 a + b2 (b2 I0 (ab) + abI1 (ab))] + A1 exp − (10.33) 2 where a, b, and An for n = 0, 1 are the same as in (10.29).

652

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

10.2.2.2 Extension to Rice/I.I.D. Nakagami Scenario Consider the case when the desired user is subject to Rician fading with parameters D and KD , whereas the NI i.i.d. interferers are subject to Nakagami fading with fading parameter mI and average fading power I . Relying on the MGF method or using the closure property of the gamma distribution, it can be shown that the PDF of SI is pSI (SI ) =

mI I

mI NI

SImI NI −1 m I SI exp − (mI NI ) I

(10.34)

This means that SI has a gamma distribution with parameters mI NI and I /mI . Following the same procedure as for the i.i.d. Rayleigh case, the outage probability can be found in the desired closed form

Pout

    2(KD + 1)Sth a2 = 1 − Q1 2KD , + D 2KD

mI
NI −1 n=0

An Q2n+1,0 (a, b) n!

(10.35) where An , a, and b are defined in the same way as in (10.29) except for using I /mI instead of I . It should be noted that (10.35) generalizes (10.21) [or equivalently Eq. (14) of Ref. 13] for an interference-limited environment and gives an alternative closed-form expression for the infinite sum representation given by Lin et al. [13, Eq. (13)]. 10.2.2.3 Numerical Examples Figure 10.2 shows the outage probability, as given in (10.29), as a function of the normalized “average” CIR D /(λth I ) for different Rician parameters KD with NI = 6. Note that the change in Rician parameter KD significantly affects the outage probability. However, it should also be noted that the Rician parameter affects the outage probability differently for different ranges of CIR. For low CIR, outage probability performance is slightly improved when KD is smaller, while for high CIR it degrades considerably when KD decreases. Figure 10.3 shows the outage probability, as given in (10.35), as a function of the normalized average CIR D /(λth I ) for different Rician parameters KD and Nakagami parameters mI with NI = 6. Note that in this case, increasing mI has little impact on outage performance, but the increase in KD improves the performance considerably, especially for the high-CNR range. As an additional numerical example, Fig. 10.4 shows the outage probability, as given in (10.35), for different Nakagami parameters mI with NI = 1. It is observed that for the small-number-of-interferers case, the effect of mI is larger, while still insignificant compared with that of KD , as shown in previous examples. Also note that as mI increases, the outage probability performance degrades for a lower CIR range but slightly improves for a higher CIR range before the minimum power requirement dominates.

653

OUTAGE PROBABILITY WITH A MINIMUM DESIRED SIGNAL POWER CONSTRAINT

100

KD = 0 KD = 1 KD = 5 KD = 10

Outage Probability, Pout

10−1

10−2

10−3

10−4

10−5

10−6

0

5

10

15

20

25

30

35

40

Normalized CIR, ΩD /lth ΩI in dB

Figure 10.2 Outage probability Pout as a function of the normalized CIR D /λth I for the Rician/i.i.d. Rayleigh scenario with NI = 6 and D /Sth = 20 dB.

100

mI = 1 mI = 2 mI = 4

−1

Outage Probability, Pout

10

KD = 1

10−2

10−3

KD = 5

10−4

10−5

0

5

10

15

20

25

30

35

40

Normalized CIR, ΩD /lthΩI in dB

Figure 10.3 Outage probability Pout as a function of the normalized CIR D /λth I for the Rician/i.i.d. Nakagami scenario with NI = 6 and D /Sth = 30 dB.

654

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

Outage Probability, Pout

100

mI = 1 mI = 2 mI = 4

10−1

10−2

0

5

10

15

20

25

30

Normalized CIR, ΩD /lthΩI in dB Figure 10.4 Outage probability Pout as a function of the normalized CIR D /λth I for the Rician/i.i.d. Nakagami scenario with NI = 1, KD = 1 and D /Sth = 20 dB.

10.2.3

Nakagami/I.I.D. Rice Scenario

In this section, the simpler Rayleigh/i.i.d. Rician scenario is studied first; then the more challenging Nakagami/i.i.d. Rician case is solved. 10.2.3.1 Rayleigh/I.I.D. Rice Scenario Let us assume that the received power of the desired signal is Rayleigh distributed with average D and that the NI interfering signal powers are i.i.d. Rician distributed with parameters I and KI . General Case Under the i.i.d. assumption, the PDF of the total instantaneous interference power SI can be obtained, using the closure property of the chi-square distribution or applying the MGF method, as 1 pSI (SI ) = ∗ I



SI KI NI ∗I

(NI −1)/2

  

KI NI SI SI exp −KI NI − ∗ INI −1 2 I ∗I (10.36)

OUTAGE PROBABILITY WITH A MINIMUM DESIRED SIGNAL POWER CONSTRAINT

655

where ∗I = I /(KI + 1). Substituting (10.36) and the Rayleigh PDF for the desired signal in (10.26) and using the definition of the generalized (mth-order) Marcum Q-function given in Chapter 4, it can be shown that the outage probability can be written as   

 ∞  1 2SD Sth SD + d SD QNI 2NI KI , exp − Pout = 1 − exp − D D Sth λth ∗I D (10.37) Using an appropriate change of variable in (10.37), and then applying the identity [35, Eq. (13)], leads after some manipulations to the final desired closed-form result for the outage probability as

Pout

  



 λth ∗I −NI 2Sth Sth = 1 − exp − 2NI KI , − 1+ 1 − QNI D λth ∗I D 

−1  D × exp −KI NI 1 + λth ∗I   



λth ∗I −1  S th D  1+ × QNI  2NI KI 1 + , 2 (10.38) D D λth ∗I

Special Cases •

Interference-Limited Case. For interference-limited systems (i.e., Sth = 0), simplifying (10.38) with Qm (a, 0) = 1, the outage probability reduces to 



NI

 D −1 D −1 Pout = 1 − 1 − 1 + exp −KI NI 1 + λth ∗I λth ∗I (10.39) which is the limiting expression that one obtains (through l’Hˆopital’s rule) from the Tjhung et al. formula [11, Eq. (13)] when setting KD = 0 in it. In the single-interferer case, where NI = 1, (10.39) reduces to  Pout = 1 −

D (1 + KI ) λ th I D 1 + (1 + KI ) λ th I



 exp −

KI D 1 + (1 + KI ) λ th I

 (10.40)

in agreement with the Haug and Ucci result [11, Eq. (15)]. • Rayleigh Interferers. When the interferer is assumed to be Rayleigh faded, we have KI = 0. By applying the property of the generalized Marcum Q-function

656

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

given in (4.73), it can be shown that (10.38) simplifies to



N


I −1 Sth D Sth n 1 Sth + exp − 1+ Pout = 1 − exp − D D λth I n! λth I



D λth I −NI Sth 1+ 1+ − exp − D λth I D



N
I −1 D n 1 Sth n 1+ × n! D λth I 

n=0

(10.41)

n=0

The closed-form result (10.41) is equivalent to the recursive solution offered by Sowerby and Williamson [1, Eqs. (12) and (14)]. For the single-interferer case, in which NI = 1, (10.41) further simplifies to

Pout

  

exp − Sth 1 + D D λth I Sth = 1 − exp − + D D 1 + λ th I

(10.42)

in perfect agreement with Eq. (3) of Ref. 1. 10.2.3.2 Extension to Nakagami/I.I.D. Rice Scenario Consider the case in which the desired user is subject to Nakagami fading with parameters D and mD , whereas the NI i.i.d. interferers are subject to Rician fading with the same fading parameter KI and the same average fading power I . Using a gamma PDF (corresponding to the Nakagami amplitude distribution) instead of the exponential PDF (corresponding to the Rayleigh amplitude distribution) in (10.37), the outage probability can be written as

Pout

mD Sth i mD Sth =1− exp − D D i=0   

 ∞ m −1  mD mD SD D 2(KI + 1)SD + QNI 2NI KI , λth I D (mD − 1)! Sth

m D SD d SD (10.43) × exp − D m
D −1

1 i!



With an appropriate change of variable and application of the following definitions a=



 2NI KI ,

b=

(KI + 1)D , mD λth I

 c=

2mD Sth D

(10.44)

OUTAGE PROBABILITY WITH A MINIMUM DESIRED SIGNAL POWER CONSTRAINT

657

the integral in (10.43) denoted by RmD can be shown through integrating by parts to satisfy the following recursive relationship:

RmD

2 c (c2 /2)mD −1 exp − QNI (a, bc) = RmD −1 + (mD − 1)! 2

NI +1 NI −1

1 b a2 [2(b2 + 1)]1−mD exp − 2 − √ (mD − 1)! a 2(b + 1) b2 + 1

 ab , b2 + 1 (10.45) × Q2(mD −1)+NI ,NI −1 √ b2 + 1

Therefore, the outage probability can be written, after some manipulation, in the following closed form

Pout = 1−

m
D −1 i=0

NI +1 NI −1 2 c 1 (c2 /2)i b [1 − QNI (a, bc)]− √ exp − i! 2 a b2 + 1

×exp −

m
D −1  ab [2(b2 + 1)]−i a2 2 ,c b + 1 Q2i+NI ,NI −1 √ 2(b2 + 1) i! b2 + 1 i=0

(10.46) where a, b and c are as given in (10.44). Note that (10.46) generalizes (10.22) [or equivalently Eq. (13) of Ref. 29] which are valid only for an interference-limited environment. It should be further noted that all the Nuttall Q-functions Qm,n (., .) in (10.46) satisfy m + n = 2i + 1 and as such they can be expressed solely in terms of generalized Marcum Q-functions and modified Bessel functions as discussed in Chapter 4. It should be observed that the general formula given in (10.46) is not applicable when a = 0, that is, when either KI = 0 or NI = 0. The case of KI = 0 corresponds to the Nakagami/i.i.d. Rayleigh combination, for which the outage probability can be easily obtained as a special case of the Yao and Sheikh result [4, Eq. (18)]. The case of NI = 0 leads to the noise-limited case, where the outage probability is well known to be given by Pout = 1 −

m
D −1 i=0

2 c (c2 /2)i exp − i! 2

(10.47)

10.2.3.3 Numerical Examples Figure 10.5 shows the outage probability, as given in (10.38), for different numbers of interferers NI . It is observed that as NI increases, the outage probability performance degrades significantly in the low CIR range but does not change much in the high CIR range since the minimum power requirement dominates in that range. Similar behavior was also observed for the Rician/i.i.d. Rayleigh case. Figure 10.6

658

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

Outage Probability, Pout

100

NI = 1 NI = 2 NI = 4 NI = 6

10−1

10−2 5

10

15

20

25

30

Normalized CIR, ΩD /λth ΩI in dB

Figure 10.5 Outage probability Pout as a function of the normalized CIR D /λth I for the Rayleigh/i.i.d. Rician scenario with KI = 5 and D /Sth = 20 dB.

Outage Probability, Pout

100

KI = 1 KI = 5 KI = 10

10−1

10−2 0

5

10

15

20

25

30

Normalized CIR, ΩD /lth ΩI in dB

Figure 10.6 Outage probability Pout as a function of the normalized CIR D /λth I for the Rician/i.i.d. Rayleigh scenario with NI = 1 and D /Sth = 20 dB.

OUTAGE PROBABILITY WITH DUAL-BRANCH SC AND SSC DIVERSITY

659

100

KI = 1 KI = 5

Outage Probability, Pout

10−2

KI = 10 mD = 2

10−4

10−6

10−8

mD = 4

10−10

10−12

0

5

10

15

20

25

30

35

40

Normalized CIR, ΩD /lth ΩI in dB

Figure 10.7 Outage probability Pout as a function of the normalized CIR D /λth I for the Nakagami/i.i.d. Rician scenario with NI = 6 and D /Sth = 30 dB.

shows the outage probability, as given in (10.38), as a function of the normalized average CIR D /λth I for different Rician parameters KI with NI = 1. Note that, unlike the Rice/i.i.d. Nakagami scenario, the change in KI has much smaller impact on the outage probability. However, it should be further noted that, similar to the Rician/i.i.d. Nakagami case, the Rician parameter affects outage probability differently for different ranges of CIR. In the low-CIR range, the outage probability performance is slightly improved when KI gets smaller. On the other hand, the outage probability degrades when KI decreases in the high-CIR range. Finally, Fig. 10.7 shows the outage probability, as given in (10.46), as a function of the normalized average CIR D /λth I for different Rician parameters KI and Nakagami parameters mD . Note that, as in the Rician/i.i.d. Nakagami case, increasing KI has little impact on outage performance, but increasing mD improves the performance considerably, especially in the high CIR range.

10.3 OUTAGE PROBABILITY WITH DUAL-BRANCH SC AND SSC DIVERSITY We saw in Chapter 9 that diversity combining is a well known method to mitigate the deleterious effect of channel fading. Diversity combining can also help reducing the impact of co-channel interference and therefore lower the system outage

660

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

probability [36]. However, because of the limited resources of the mobile terminals, it is seldom practical to deploy complicated diversity schemes (such as MRC, discussed in Section 10.1, multiple-input/multiple-output (MIMO) MRC addressed in Ref. 38, or even more sophisticated “optimum” schemes that will be covered in detail in Chapter 11), especially over the downlink, and hence the number of available diversity branches at the receiver is often limited to two. In this section, we highlight the main contributions of the work published in Ref. 37 and revisit two low-complexity dual-branch diversity schemes, namely, selection combining (SC) and switch-and-stay combining (SSC) (which were covered in Chapter 9 in the absence of co-channel interference) and investigate their outage probability in a single-interferer Rayleigh fading environment. This interference/fading scenario applies to sectorized or lightly loaded macrocellular networks for which the consideration of a minimum CNR constraint is important for an accurate system prediction. It also applies to the multiple-Rayleigh-interferer case under the (pessimistic) assumption that the interferers add up coherently [28]. Although more complicated interference and fading scenarios are also of interest, the closed-form results obtained in this work provide useful insight for the design of dual-diversity systems in the presence of co-channel interference. Recall that in SC, the diversity combiner always picks the best branch by continuously monitoring both branches whereas in SSC, only the selected branch needs to be continuously monitored and a switch occurs only when the selected branch becomes unacceptable. When a diversity combiner is subject to co-channel interference, the combining decision algorithm is not unique [36]. More specifically, the selection of branches for SC can be based on the total (desired plus interference) power, the desired signal power, or the CIR of the received signal at each branch output. Similarly, the switching between branches in SSC can also be triggered by different kinds of criteria. In general, the decision algorithm based on CIR provides the best performance for interference-limited systems, but is the most complex to implement. On the other hand, as shown in Ref. 6, the outage probability using the desired signal power and total power algorithm have nearly the same performance, but the algorithm based on total received power is easier to implement. Related work on the outage probability of SC and SSC diversity systems in the presence of co-channel interference has included, for example, the following. The Jakes textbook [36] presented outage probability results for the interferencelimited case (i.e., without a minimum CNR requirement) of SC in a single-interferer Rayleigh fading scenario using the desired signal power algorithm. On the basis of the total power algorithm, Schiff [39] presented the PDF of the output CIR with both SC and SSC in the single-interferer Rayleigh fading scenario. Sowerby and Williamson [40] considered the multiple-interferer case with SC over Rayleigh fading using both the CIR and the total power algorithms. Abu-Dayya and Beaulieu [6] generalized the analysis of the interference-limited case of SC and SSC to the Nakagami fading environment for all three algorithm mentioned above while still assuming branch independence. Finally, Okui considered dual-branch SC in correlated Nakagami-m fading using the desired signal power algorithm [41] and subsequently using the CIR algorithm [42].

OUTAGE PROBABILITY WITH DUAL-BRANCH SC AND SSC DIVERSITY

661

In what follows, we present closed-form expressions of the outage probability for dual-branch SC and SSC diversity schemes in a single-interferer Rayleigh fading environment and consider both minimum CIR and CNR constraints. As an example, the desired signal power algorithm is used for both diversity schemes for mathematical convenience. It should be noted that this algorithm provides the best performance among the three aforementioned algorithms over a wide range of CIR. This decision algorithm requires identification and separation of the desired and interferer signals, which can be achieved using different pilots for the different signals [36]. We limit ourselves here to the case of independent fading among the two diversity branches and perfect channel estimates. The effect of branch correlation as well as outdated or imperfect channel estimates (two usual impairments resulting from implementation constraints as discussed in Chapter 9) on these two diversity combining schemes in the presence of co-channel interference is addressed in Ref. 37. 10.3.1

Fading and System Models

Consider a dual-branch diversity system operating in a Rayleigh fading environi , i = 1, 2 denote the instantaneous faded power of the desired signal ment. Let SD received by the ith branch and D = ESD [SD ] be the common short-term average i for the two antennas. The PDFs of SD , i = 1, 2 are then identically given by pSi D (SD ) =

1 SD , exp − D D

SD ≥ 0

(10.48)

On the other hand, we assume that there is a single dominant interfering signal, independent of the desired signal, also subject to Rayleigh fading. The instantaneous faded powers of the interfering signals received by both branches are assumed to be identically distributed with the same short-term average I . Using the desired signal power algorithm, the combining decision is solely based on the power of the desired signal. Therefore, the diversity combiner will not affect the statistics of the interfering signal power, irrespective of whether the interfering signals of the two branches are correlated. Consequently, the PDF of the interfering signal power at the combiner output, denoted by SI , is of the same form as in (10.48) for the desired signals but with one different parameter (i.e., I instead of D ) and correspondingly the CDF of SI is given by  PSI (s) =

10.3.2

s 0

s , pSI (SI ) d SI = 1 − exp − I

s≥0

(10.49)

Outage Performance with Minimum Signal Power Constraint

In this section, the outage probability of the two diversity schemes is evaluated with the consideration of a minimum signal power constraint. Assuming

662

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

independent dual branches in a single-interferer Rayleigh fading environment, closed-form expressions of outage probability for both schemes are derived and some special cases are studied. For SSC, the optimal switching threshold minimizing the outage probability is also given in closed form. Finally, some numerical results of interest are presented and discussed. 10.3.2.1 Selection Combining For SC, the combiner continuously monitors both branches and always uses the branch with the largest desired signal power at its output. General Case In a Rayleigh fading environment, the PDF of the desired signal power at the SC combiner output SD is given by



2 SD SD exp − pSD (SD ) = 1 − exp − D D D

(10.50)

Substituting the CDF of the interferer signal power given in (10.49), evaluated at SD /λth , together with (10.50) in (10.26) gives, after some manipulations, the outage probability in the following closed form

 2 2  2 c 2 p 2 c2 (1 + p2 )c2 − + 2 exp − exp − Pout = 1 − exp − 2 p 2 1 + p2 2 (10.51) where   2Sth D and p = 1 + (10.52) c= D I λth Special Cases •

Interference-Limited Case. For interference-limited systems (i.e., Sth = 0), simplifying (10.51) with c = 0 yields Pout =

•

2 p2 (1 + p2 )

(10.53)

in agreement with Jakes’ result [36, Eq. (5.4-84)] for the dual-branch case. Noise-Limited Case. For noise-limited systems [i.e., λth = 0, or equivalently D /(λth I ) → ∞, and thus p2 → ∞], (10.51) reduces to  2 2 c Pout = 1 − exp − 2 consistent with Jakes’ result [36, Eq. (5.2-5)].

(10.54)

OUTAGE PROBABILITY WITH DUAL-BRANCH SC AND SSC DIVERSITY

663

10.3.2.2 Switch-and-Stay Combining For SSC, the desired signal power is monitored in a periodic fashion and the combiner switches from one branch to another at the instants the power of the desired signal crosses a preassigned switching threshold ST in the negative direction no matter what the power of the switch-to branch is. General Case In a Rayleigh fading environment, the PDF of the desired signal power at the SSC combiner output is given by



1 SD ST exp − 1 − exp − , SD ≤ ST (10.55) pSD (SD ) = D D D



ST SD 1 2 − exp − , SD > ST (10.56) exp − = D D D Substituting (10.49), evaluated at SD /λth , together with (10.55) in (10.26) yields a closed-form expression for the outage probability given by

 2

 c 1 d2 p 2 c2 exp − − 2 exp − Pout = 1 − 1 − exp − 2 2 p 2 2 2 2 d 1 p d − exp − + 2 exp − , Sth ≤ ST 2 p 2

 2

 c 1 d2 p 2 c2 exp − − 2 exp − , Sth > ST = 1 − 2 − exp − 2 2 p 2 where d =

√

(10.57) 2ST /D , and c and p are as defined in (10.52).

Optimum Switching Threshold It is not difficult to check that both ST = 0 and ST → ∞ cause (10.57) to reduce to

2 1 c p 2 c2 + 2 exp − (10.58) Pout = 1 − exp − 2 p 2 which is the outage performance of a single-branch (no-diversity) receiver as given by Sowerby and Williamson [1, Eq. (3)]. Since the outage probability is a continuous function of ST , there exists an optimal value that minimizes the outage probability. This optimal value ST∗ is a solution of the equation d Pout |S =S ∗ = 0 d ST T T

(10.59)

Substituting (10.57) in (10.59) and solving for ST∗ , the optimal switching threshold is obtained after some manipulations as ! ln[1 − exp(−c2 /2) + exp(−p2 c2 /2)/p2 ] ∗ (10.60) ST = max D , Sth 1 − p2

664

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

if ST∗ ∈ [Sth , ∞], while ST∗ = Sth if ST∗ ∈ [0, Sth ] because in this case, Pout is a monotonic decreasing function of ST . Combining the preceding two cases, we can conclude that (10.60) gives the optimal switching threshold in general. Special Cases •

Interference-Limited Case. For interference-limited systems (i.e., Sth = 0), simplifying (10.57) with c = 0 leads to Pout =





d2 p2 d 2 1 1 − exp − + exp − p2 2 2

(10.61)

From this new result, we can easily obtain the optimum threshold ST∗ by following the same procedure as before, which gives ST∗ =

•

ln p2 D p2 − 1

(10.62)

in agreement with the one obtained by simplifying (10.60) for c = 0. Noise-Limited Case. For noise-limited systems [i.e., λth = 0, or equivalently D /(λth I ) → ∞ and thus p2 → ∞], after simplification of (10.57), we get

 2  d2 c 1 − exp − , Pout = 1 − exp − 2 2 2

2 c + d2 c + exp − , = 1 − 2 exp − 2 2

Sth ≤ ST Sth > ST

(10.63)

which is consistent with the outage probability result for SSC obtained in Section 9.740 and the corresponding optimum switching threshold is Sth as pointed out in that section.

10.3.2.3 Numerical Examples Figure 10.8 shows some computer simulation results on the outage probability of SC in a single-interferer Rayleigh fading environment as a function of the normalized “average” CIR D /λth I for different decision algorithms and with a normalized CNR D /Sth = 10 dB. As shown in the figure, the CIR algorithm provides the best performance in the low-CIR range when the system is solely interference-limited, while the desired signal power algorithm and the total power algorithm outperform the CIR algorithm in the high-CIR range (i.e., when the system is noise-limited). Note that the desired signal power algorithm gives the best outage performance when the normalized average CIR D /λth I is sufficiently high (above 9 dB in this case). Figure 10.9 shows the outage probability of SC, as given in (10.51), as a function of the normalized average CIR D /λth I for different normalized CNR D /Sth . For D /Sth = 10 dB (see again Fig. 10.9),

665

OUTAGE PROBABILITY WITH DUAL-BRANCH SC AND SSC DIVERSITY

Outage Probability, Pout

100

Desired Signal Power Total Power CIR

10−1

10−2 0

5

10

15

20

25

30

35

40

Normalized CIR, ΩD /lth ΩI in dB

Figure 10.8 Monte Carlo simulations of the outage probability of SC in independent Rayleigh fading for the three different decision algorithms with D /Sth = 10 dB. 100 10−1

Outage Probability, Pout

10−2 10−3 10−4 10−5 ΩD /Sth = 0 dB ΩD /Sth = 10 dB

10−6

ΩD /Sth = 20 dB ΩD /Sth = 30 dB

10−7 10−8

ΩD /Sth = ∞ 0

5

10

15

20

25

30

35

40

Normalized CIR, ΩD /lth ΩI in dB

Figure 10.9 Outage probability Pout as a function of the normalized average CIR D /λth I for SC in independent Rayleigh fading.

666

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

100

Outage Probability, Pout

10−1

10−2

10−3

10−4 ΩD /Sth = 0 dB

10−5

ΩD /Sth = 10 dB ΩD /Sth = 20 dB ΩD /Sth = 30 dB

10−6

ΩD /Sth = ∞

10−7 0

5

10

15

20

25

30

35

40

Normalized CIR, ΩD /lth ΩI in dB

Figure 10.10 Outage probability Pout as a function of the normalized average CIR D /λth I for SC with optimal switching threshold S∗T /D in independent Rayleigh fading. 100 No diversity SSC with ST /ΩD = 0.5 SSC with ST /ΩD = 0.1

Outage Probability, Pout

SSC with optimal threshold SC

10−1

10−2 0

5

10

15

20

25

30

35

40

Normalized CIR, ΩD /lth ΩI in dB

Figure 10.11 Comparison of SC and SSC diversity schemes in independent Rayleigh fading with D /Sth = 10 dB.

OUTAGE RATE AND AOD OF MULTIUSER COMMUNICATION SYSTEMS

667

these results are in agreement with the simulation results for the desired signal power algorithm in Fig. 10.8, and as
D /Sth increases, the outage probability performance improves in the high-CIR range while few changes are observed in the lower-CIR range because the minimum CIR constraint dominates in that range. Similar behavior can also be observed for the outage probability of SSC as given in (10.57) and illustrated in Fig. 10.10. Figure 10.11 compares the outage performance of SC and SSC diversity techniques. As shown in the figure, both schemes provide considerable performance improvement over the no diversity case, especially in the high-CIR range. Finally, note that, as expected, even when using the optimal switching threshold for SSC, SC still clearly outperforms SSC especially in the low-CIR range. 10.4 OUTAGE RATE AND AVERAGE OUTAGE DURATION OF MULTIUSER COMMUNICATION SYSTEMS Outage rate and AOD [43–47] as well as the distribution of the outage duration [48] are also important performance measures of multiuser wireless communication systems. These performance criteria take into account the time correlation properties (i.e., the second-order statistics) of the fading channels and provide as such a dynamic representation of the system outage performance. Assuming that an outage is declared whenever either the output CIR λ falls below a pre-determined threshold λth or the received signal power of the desired user SD falls below another predetermined threshold Sth , the average outage duration T (λth , Sth ) is given by [46, Eq. 8] T (λth , Sth ) =

Pout N (λth , Sth )

(10.64)

where N (λth , Sth ) is the joint level crossing rate (LCR) (or equivalently the outage rate) at which the CIR process λ(t) crosses λth or the desired signal power process SD (t) crosses Sth and Pout = Pr[λ < λth or SD < Sth ] is the system outage probability when a minimum desired signal power constraint is present. Closed-form expressions for the outage probability Pout with a minimal signal power requirement have been presented for several fading scenario combinations of interest in Section 10.2. Therefore, only the LCR N (λth , Sth ) is needed to obtain the AOD. When both desired users and co-channel interferers are subject to Nakagamim fading, it has been shown [46] after considerable manipulations that this LCR with a minimum power requirement N (λth , Sth ) is given by
 (2m −1)/2 √ (NI mI , λnn Sn ) Sn D exp (−Sn ) 1 − N (λth , Sth ) = 2πfD (mD ) (NI mI ) √   2π  mD + NI mI − 12 , Sn (1 + λnn ) + (NI mI )(mD )   1/2 N m f2 λnnI I (10.65) × fD2 + I λnn (1 + λnn )mD +NI mI −(1/2)

668

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

where mD and mI are the Nakagami fading parameters of the desired and interfering users, respectively; D and I are the average signal powers of the desired and interfering users, respectively; Sn = Sth mD /D , λnn = (D mI )/(λth I mD ); and fD and fI are the maximum Doppler frequency shifts for the desired and interfering users, respectively. Special Cases •

Interference-Limited Case. The LCR N (λth ) for interference-limited systems can be obtained by setting Sth = 0 in (10.65), which results in √ N (λth ) =

 

1/2 2π mD + NI mI − 12 f2 fD2 + I (NI mI )(LmD ) λnn N m

×

λnnI I (1 + λnn )mD +NI mI −1/2

(10.66)

If fD = fI = fm and mD = mI = 1, (10.66) further reduces to the LCR of the interference-limited system in the Rayleigh/Rayleigh fading environment, which is given by NI  √   2πfm  NI + 12 1 λth I N (λth ) = λth I (NI ) D 1 + D •

(10.67)

in agreement with the original formula [43, Eq. (17)]. Power-Limited Case. Setting λth = 0 in (10.65), we obtain the LCR N (Sth ) of the signal power SD at level Sth (purely power-limited case), which is given by N (Sth ) =

m −1/2 √ Sn D exp (−Sn ) 2πfD (mD )

(10.68)

in agreement with the Yacoub et al. result [49, Eq. (17)]. Extensions of the general Nakagami/Nakagami outage rate formula given in (10.65) to other fading combinations and MRC diversity have been presented [46]. On the other hand, this outage rate formula is also derived for various lower-complexity SC algorithms in the presence of co-channel interference in Ref. [47]. As a first numerical example, Fig. 10.12 plots the AOD versus the normalized CIR threshold D /I λth when NI = 6, mD = 2, mI = 2, and for various values of the normalized signal power thresholds Sth /D . From this figure, we can see that the minimum desired signal power requirement causes a floor on the AOD, since for Sth > 0 increasing the normalized CIR threshold above a particular value does not reduce the AOD and the system becomes power-limited. This floor and

669

OUTAGE RATE AND AOD OF MULTIUSER COMMUNICATION SYSTEMS

101

Sth /ΩD = 0 Sth /ΩD = −20 dB Sth /ΩD = −10 dB Sth /ΩD = 0 dB

Average Outage Duration (AOD)

100

10−1

10−2

10−3

10−4

10−5

0

5

10

15

20

25

30

35

40

45

50

Normalized SIR Threshold ΩD /(λthΩI) in dB

Figure 10.12 AOD versus D /I λth for various values of Sth /D when L = 1, NI = 6, fD = fI = 40 Hz, mD = 2, and mI = 2. 105 104

ΩD /lthΩI = 0 dB ΩD /lthΩI = 20 dB ΩD /lthΩI = 40 dB ΩD /lthΩI = 160 dB

Average Outage Duration (AOD)

103 102 101 100 10−1 10−2 10−3 10−4 10−5 −40

−35

−30

−25

−20

−15

−10

−5

0

5

10

Normalized Signal Power Threshold Sth/ΩD in dB

Figure 10.13 AOD versus Sth /D for various values of D /I λth when L = 1, NI = 6, fD = fI = 40 Hz, mD = 2, and mI = 2.

670

OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

101

fD = 10 Hz, fl = 40 Hz fD = 20 Hz, fl = 40 Hz fD = 40 Hz, fl = 40 Hz fD = 80 Hz, fl = 40 Hz fD = 160 Hz, fl = 40 Hz

Average Outage Duration (AOD)

100

10−1

10−2

10−3

10−4

0

5

10

15

20

25

30

35

40

45

50

Normalized SIR Threshold ΩD /(lthΩI) in dB

Figure 10.14 AOD versus D /I λth for various values of the fD when L = 1, NI = 6, fI = 40 Hz, Sth /D = −20 dB, mD = 2, and mI = 2. 101

fD = 40 Hz, fl = 20 Hz fD = 40 Hz, fl = 40 Hz fD = 40 Hz, fl = 80 Hz fD = 40 Hz, fl = 160 Hz

Average Outage Duration (AOD)

100

10−1

10−2

10−3

10−4

0

5

10

15

20

25

30

35

40

45

50

Normalized SIR Threshold ΩD /(lth ΩI) in dB

Figure 10.15 AOD versus D /I λth for various values of fI when L = 1, NI = 6, fD = 40 Hz, Sth /D = −20 dB, mD = 2, and mI = 2.

REFERENCES

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the value of the normalized CIR threshold at which this floor occurs depend on Sth /D for fixed fading parameters. This can be seen from Fig. 10.13, which shows the AOD versus the normalized desired signal power threshold for various values of the normalized CIR threshold. On the other hand, Figs. 10.14 and 10.15 show the effects of the speeds of desired and interfering users on the AOD. Figure 10.14 plots the AOD versus the normalized CIR threshold D /I λth , NI = 6, fD = fI = 40 Hz, Sth/D =−20 dB, mD = 2, and mI = 2, and for various values of the desired user maximum Doppler frequencies. It is clear that the desired user Doppler frequency (or equivalently speed) has a significant impact on the system AOD; and from Fig. 10.15, we can also conclude that the interfering users speed has a negligible effect on the system AOD, especially in the medium to high normalized CIR threshold range.

REFERENCES 1. K. W. Sowerby and A. G. Williamson, “Outage probability calculations for multiple co-channel interferers in cellular mobile radio systems,” IEE Proc. (Pt. F), vol. 135, June 1988, pp. 208–215. 2. Y.-D. Yao and A. U. H. Sheikh, “Outage probability analysis for microcell mobile radio systems with co-channel interferers in Rician/Rayleigh fading environment,” IEE Electron. Lett., vol. 26, June 1990, pp. 864–866. 3. A. A. Abu-Dayya and N. C. Beaulieu, “Outage probabilities of cellular mobile radio systems with multiple Nakagami interferers,” IEEE Trans. Veh. Technol., vol. VT-40, November 1991, pp. 757–768. 4. Y.-D. Yao and A. U. H. Sheikh, “Investigation into co-channel interference in microcellular mobile radio systems,” IEEE Trans. Veh. Technol., vol. VT-41, May 1992, pp. 114–123. 5. J. R. Haug and D. R. Ucci, “Outage probability for microcellular radio systems in a Rayleigh/Rician fading environment,” Proc. IEEE Int. Conf. Communication (ICC’92), June 1992, pp. 312.4.1–312.4.5. 6. A. A. Abu-Dayya and N. C. Beaulieu, “Outage probabilities of diversity cellular systems with co-channel interference in Nakagami fading,” IEEE Trans. Veh. Technol., vol. VT41, November 1992, pp. 343–355. 7. T. T. Tjhung, C. C. Chai, and X. Dong, “Outage probability for a Rician signal in L Rician interferers,” IEE Electron. Lett., vol. 31, March 1995, pp. 532–533. 8. Q. T. Zhang, “Outage probability of cellular mobile radio in the presence of multiple Nakagami interferers with arbitrary fading parameters,” IEEE Trans. Veh. Technol., vol. VT-44, August 1995, pp. 661–667. 9. C. Tellambura and V. K. Bhargava, “Outage probability analysis for cellular mobile radio systems subject to Nakagami fading and shadowing,” Trans. IECE Japan, vol. E 78-B, October 1995, pp. 1416–1423. 10. Q. T. Zhang, “Outage probability in cellular mobile radio due to Nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol., vol. 45, May 1996, pp. 364–372.

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11. T. T. Tjhung, C. C. Chai, and X. Dong, “Outage probability for lognormal-shadowed Rician channels,” IEEE Trans. Veh. Technol., vol. VT-46, May 1997, pp. 400–407. 12. J. W. Stokes and J. A. Ritcey, “A general method for evaluating outage probabilities using Pade approximations,” Proc. IEEE Global Commun. Conf. (GLOBECOM’98), Sydney, Australia, November 1998, pp. 1485–1490. 13. J.-C. Lin, W.-C. Kao, Y. T. Su, and T.-H. Lee, “Outage and coverage considerations for microcellular mobile radio systems in a shadowed-Rician/shadowed-Nakagami environment,” IEEE Trans. Veh. Technol., vol. VT-48, January 1999, pp. 66–75. 14. C. Tellambura, “Co-Channel interference computation for arbitrary Nakagami fading,” IEEE Trans. Veh. Technol., vol. 48, March 1999, pp. 487–489. 15. C. Tellambura and A. Annamalai, “A unified numerical approach for computing the outage probability for mobile radio systems,” IEEE Commun. Lett., vol. 3, April 1999, pp. 97–99. 16. G. L. St¨uber, Principles of Mobile Communications. Norwell, MA: Kluwer Academic Publishers, 1996. 17. P. A. Bello, “Binary error probabilities over selectively fading channels containing specular components,” IEEE Trans. Commun. Technol., vol. COM-14, August 1966, pp. 400–406. 18. R. Price, “Some noncentral F-distributions expressed in closed form,” Biometrika, vol. 51, 1964, pp. 107–122. 19. J. Gil-Pelaez, “Notes on the inversion theorem,” Biometrika, vol. 38, 1951, pp. 481–482. 20. N. C. Beaulieu, “An infinite series for the computation of the complementary probability distribution function of a sum of independent random variables and its application to the sum of Rayleigh random variables,” IEEE Trans. Commun., vol. COM-26, September 1990, pp. 1463–1474. 21. V. A. Aalo and J. Zhang, “On the effect of co-channel interference on average error rates in Nakagami-fading channels,” IEEE Commun. Lett., vol. 3, May 1999, pp. 136–138. 22. M. K. Simon and M.-S. Alouini, “On the difference of two chi-square variates with application to outage probability computation,” IEEE Trans. Commun., vol. 49, no. 11, November 2001, pp. 1946-1954. 23. J. G. Proakis, Digital Communications, 3rd ed. New York, NY: McGraw-Hill, 1995. 24. J. G. Proakis, “On the probability of error for multichannel reception of binary signals,” IEEE Trans. Commun. Technol., vol. COM-16, February 1968, pp. 68–71. 25. M. K. Simon and M.-S. Alouini, “A unified approach to the probability of error for noncoherent and differentially coherent modulations over generalized fading channels,” IEEE Trans. Commun., vol. COM-46, December 1998, pp. 1625–1638. 26. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1994. 27. G. M. Dillard, “Recursive computation of the generalized Q-function,” IEEE Trans. Aerosp. Electron. Syst., vol. AES-9, July 1973, pp. 614–615. 28. R. Prasad and A. Kegel, “Improved assessment of interference limits in cellular radio performance,” IEEE Trans. Veh. Technol., vol. VT-40, May 1991, pp. 412–419. 29. L. C. Wang and C. T. Lea, “Co-channel interference analysis of shadowed Rician channels,” IEEE Commun. Lett., vol. 2, March 1998, pp. 67–69. 30. M. K. Simon, S. M. Hinedi, and W. C. Lindsey, Digital Communication Techniques—Signal Design and Detection. Englewood Cliffs, NJ: PTR Prentice Hall, 1995.

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31. J. C. Bic, D. Duponteil, and J. C. Imbeaux, Elements of Digital Communication. New York, NY: John Wiley, 1991. 32. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York, NY: Dover Publications, 1970. 33. H.-C. Yang and M.-S. Alouini, “Closed form formulas for the outage probability of multiuser wireless communication systems with a minimum signal power constraint,” IEEE Trans. Veh. Technol., vol. 51, no. 6, November 2002, pp. 1689–1698. 34. A. H. Nuttall, Some Integrals Involving the Q Function Research Report, Naval Underwater Systems Center, 1972. 35. A. H. Nuttall, “Some integrals involving the Qm function,” IEEE Trans. Inform. Theory, no. 1, June 1975, pp. 95-96. 36. W. C. Jakes, Microwave Mobile Communication, 2nd ed. Piscataway, NJ: IEEE Press, 1994. 37. H.-C. Yang and M.-S. Alouini, “Outage probability of dual-branch diversity systems in presence of co-channel interference,” IEEE Trans. Wireless Commun., vol. 2, no. 2, pp. 310–319, March 2003. 38. M. Kang and M.-S. Alouini, “A comparative study on the performance of MIMO MRC systems with and without co-channel interference,” Proc. IEEE Int. Conf. Communication (ICC’2003), Anchorage, AK, May 2003; full journal version to appear in a future issue of IEEE Trans. Commun. 39. L. Schiff, “Statistical suppression of interference with multiple Nakagami fading channels,” IEEE Trans. Veh. Technol., vol. VT-21, November 1972, pp. 121–128. 40. K. W. Sowerby and A. G. Williamson, “Selection diversity in multiple interferer mobile radio systems,” Electron. Lett., vol. 24, November 1988, pp. 1511–1513. 41. S. Okui, “Probability of co-channel interference for selection diversity reception in the Nakagami m-fading channel,” Proc. Inst. Elect. Eng., vol. 139, pt. 1, February 1992, pp. 91–94. 42. S. Okui, “Effects of CIR selection diversity with two correlated branches in the m-fading channel,” IEEE Trans. Commun., vol. COM-48, October 2000, pp. 1631–1633. 43. J.-P. M. G. Linnartz and R. Prasad, “Threshold crossing rate and average non-fade duration in a Rayleigh-fading channel with multiple interferers,” Archiv fur Elektronik und Ubertragungstechnik Electronics and Communication, vol. 43, November/December 1989, pp. 345–349. 44. A. Abrardo and D. Sennati, “Outage statistics in CDMA mobile radio systems,” IEEE Commun. Lett., vol. 4, no. 3, March 2000, pp. 83–85. 45. Z. Cao and Y.-D. Yao, “Definition and derivation of level crossing rate and average fade duration in an interference-limited environment,” Proc. IEEE Veh. Technol. Conf. (VTC’Fall 01), Atlantic City, NJ, October 2001, pp. 1608–1611. 46. L. Yang and M. -S. Alouini, “Average outage duration of multiuser wireless communication systems with a minimum signal power requirement,” Proc. IEEE 55th Veh. Technol. Conf. (VTC’Spring 02), Birmingham, AL, May 2002, pp. 1507–1511; full journal paper in IEEE Trans. Wireless Commun., vol. 3, no. 4, July 2004, pp. 1142–1153. 47. L. Yang and M.-S. Alouini, “Performance comparison of different selection combining algorithms in presence of co-channel interference,” Proc. IEEE Vehi. Technol. Conf. (VTC Fall’2003), Orlando, FL, October 2003; full journal paper to appear in a future issue of IEEE Trans. Veh. Technol.

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APPENDIX 10A. A PROBABILITY RELATED TO THE CDF OF THE DIFFERENCE OF TWO CHI-SQUARE VARIATES WITH DIFFERENT DEGREES OF FREEDOM Because Xk and Yk in (10.1) can be either zero or nonzero mean complex Gaussian RVs, |Xk |2 and |Yk |2 are in general noncentral chi-square-distributed. Making use of the CFs of nonzero-mean complex Gaussian RVs given in [30, App. 5A of Ref. 30] and the assumed independence of all RVs, the characteristic function of D is given by

L1

L2 1 1 ψD (j v) = 1 − 2j vAµXX 1 − 2j vBµY Y "  L   2 #  L2 1  2   k=1 A X k k=1 B Y k × exp j v + (10A.1) 1 − 2j vAµXX 1 − 2j vBµY Y Now let v1 = (2BµY Y )−1 v2 = − (2AµXX )−1

(10A.2)

and 

ξ1 =

L2


 2 B Y k 

k=1 

ξ2 =

L1


 2 A X k 

(10A.3)

k=1

in which case (10A.1) can be rewritten as

L2

!  −j v2 L1 j v1 ξ 1 j v2 ξ 2 j v1 ψD (j v) = exp j v − v + j v1 v − j v2 v + j v1 v − j v2     

L2

L1  v1 v2 j v (ξ1 + ξ2 ) − v 2 v1 ξ1 −v2 ξ2  −j v2 v1 v2 j v1 = exp   v + j v1 v − j v2 (v + j v1 ) (v − j v2 ) (10A.4)

APPENDIX 10A. CDF OF DIFFERENCE OF TWO CHI-SQUARES

675

Comparing (10A.4) with Proakis’ [23] Eq. (B-7), we see that we need to define   L1 L2

 2  2  v1 ξ 1 − v2 ξ 2     α1 = = −2AB µY Y Xk + µXX Yk v1 v2 k=1



α2 = ξ 2 + ξ 1 =

L1


 2 A X k  +

k=1

L2


k=1

 2 B Y k 

(10A.5)

k=1

which agrees with Proakis’ [23] Eqs. (B-6) and (B-8) combined, when L1 = L2 . Note that Proakis’ [23] Eq. (B-6) and thus our results here require that |C|2 − AB > 0 which for C = 0 implies that either A < 0 or B < 0 but not both (i.e., AB < 0). That’s why, in reality, (10.1) represents the difference of two chi-square variates. Using (10A.5), we can rewrite (10A.4) as

L2

L1 !  j v2 j v1 ξ 1 j v2 ξ 2 j v1 − exp j v − ψD (j v) = v + j v1 v − j v2 v + j v1 v − j v2 "

#

L2 −L1

L1 v1 v2 j vα2 − v 2 α1 j v1 v1 v2 = exp v + j v1 (v + j v1 ) (v − j v2 ) (v + j v1 ) (v − j v2 ) (10A.6) Comparing (10A.6) with [23, Eq. (B-7)] we see that the only difference is the addition of the factor [j v1 /(v + j v1 )]L2 −L1 and the exponent of the second factor is L1 instead of L. Despite the addition of this factor, we shall see that we can still use the same conformal mapping as in Proakis’ [23] Eq. (B-11) and get the claimed result in our (10.4) for P which is in the same form as Proakis’ [23] Eq. (B-21). From the definition of p in Proakis’ [23] Eq. (B-11), we find that 1+

j (v1 + v2 ) v2 p= v1 v + j v1

(10A.7)

Thus, the factor [j v1 /(v + j v1 )]L2 −L1 transforms to

L2 −L1  1 + v2 p L2 −L1 

v2 v1 v1 = 1+ p = v1 v1 + v2 1 + vv21 (10A.8) Adding this factor to what is already given in Proakis’ [23] Eqs. (B-12) and [23] (B-13) gives after some simplification of the factors   2 +α2 v1 −α2 v2 )  exp v1 v2 (−2α1(vv1 v+v 2 1 1 2) f (p) dp (10A.9) P = 2πj  (1 + v2 /v1 )L1 +L2 −1

j v1 v + j v1

L2 −L1

with now

L +L −1  1 + (v2 /v1 ) p 1 2 A2 (v2 /v1 ) A3 (v1 /v2 ) 1 exp p+ f (p) = pL1 (1 − p) v1 + v2 v1 + v2 p

(10A.10)

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OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

where A2 and A3 are still defined as per Eq.(B-10) of Ref. 23 A2 =

v12 v2 (α1 v1 + α2 ) v1 + v2

A3 =

v1 v22 (α1 v2 − α2 ). v1 + v2

(10A.11)

Equations (10A.9) and (10A.10) are now directly comparable to Proakis’ [23] Eqs. (B-12) and (B-13), respectively, with some slight modifications in the exponents. Fortunately, this does not hinder the remainder of the derivation. To evaluate the integral, we first make the simplifications of Proakis’ [23] Eq. (B-15), namely a 2  A3 (v1 /v2 ) = 2 v1 + v2 b2  A2 (v2 /v1 ) = 2 v1 + v2

(10A.12)

L +L −1 and then expand the function 1 + (v2 /v1 ) p 1 2 in the binomial series

L1 +L
2 −1 L1 + L2 − 1 v2 k v2 L1 +L2 −1 1+ p = pk k v1 v1

(10A.13)

k=0

Substituting (10A.12) and (10A.13) in (10A.10) gives

f (p) =

L1 +L
2 −1 k=0

L1 + L2 − 1 k



v2 v1

k

2

1a pk 1 2 exp + b p pL1 (1 − p) 2p 2 (10A.14)

and thus the integral becomes 1 2πj

 f (p) dp =

L1 +L
2 −1



k=0

×

1 2πj



L1 + L2 − 1 k pk



pL1 (1 − p)



exp

v2 v1

k

a2

(10A.15)

1 1 + b2 p dp 2p 2

which is to be compared with Proakis’ [23] Eq. (B-16). The integral on the right-hand side of (10A.15) is evaluated in Proakis’ [23] Eqs. (B-17)–(B-19). In particular, for our case, for 0 ≤ k ≤ L1 − 2

APPENDIX 10A. CDF OF DIFFERENCE OF TWO CHI-SQUARES

2

1 2 1a 1 + b exp p dp L −k (1 − p) 2p 2  p 1 L1
 −1−k n  b 1 2 = Q1 (a, b) exp In (ab) a + b2 + 2 a

1 2πj

677



(10A.16)

n=1

while for k = L1 − 1 2

  1a 1 1 1 2 exp + b2 p dp = Q1 (a, b) exp a + b2 2p 2 2  p (1 − p) (10A.17) and for L1 ≤ k ≤ L1 + L2 − 1 

1 2πj

1 2πj

 

2

  1 2 pk−L1 1a 1 2 2 + b p dp = Q1 (a, b) exp a +b exp 2p 2 2 (1 − p) −

k−L
1

 a n

n=0

b

(10A.18)

In (ab)

Using (10A.16)–(10A.18) in (10A.15) and collecting terms gives after considerable algebraic manipulation 1 2πj



!  1 2 a + b2 Q1 (a, b) − I0 (ab) 2

k L
1 −1 v2 L1 + L2 − 1 + I0 (ab) k v1

f (p) dp = (1 + v2 /v1 )L1 +L2 −1 exp 



k=0

+

L
1 −1

In (ab)

n=1

−

L
2 −1

L1
−1−n k=0

In (ab)

n=1

L2
−1−n k=0

L1 + L2 − 1 k L1 + L2 − 1 k

n k v2 b a v1

  L1 +L2 −1−k a n v 2

b

v1

(10A.19) which is to be compared with Proakis’ [23] Eq. (B-20). Finally, using (10A.19) in conjunction with (10A.9) and the identity  exp

v1 v2 (−2α1 v1 v2 + α2 v1 − α2 v2 ) (v1 + v2 )2



  1 = exp − a 2 + b2 2

(10A.20)

we get the final desired result for P given in (10.4). As a validity check, it can be easily seen that (10.4) reduces to the Proakis result [23, Eq. (B-21)] when L1 = L2 = L.

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OUTAGE PERFORMANCE OF MULTIUSER COMMUNICATION SYSTEMS

It is important to note that the result in (10.4) possesses symmetry in the following sense. Letting P = Pr{D < 0}, then from (10.1) and (10.3), it can be shown that    P = 1 − P  L1 →L2 , L2 →L1 (10A.21)  Xk →Yk , Yk →Xk A→−B, B→−A

Since the parameter exchanges in (10A.21) produce µXX → µY Y ,

µY Y → µXX

v1 → v2 ,

v2 → v1

α1 → α1 ,

α2 → −α2

a → b,

b→a

(10A.22)

making these replacements in (10.4) and using the well-known identity [31, Sect. A4.6.2, p. 577]

2 a + b2 I0 (ab) Q1 (a, b) + Q1 (b, a) = 1 + exp − 2

(10A.23)

results after much manipulation in the right hand sides of these equations satisfying (10A.21).

APPENDIX 10B. OUTAGE PROBABILITY IN THE NAKAGAMI/NAKAGAMI INTERFERENCE-LIMITED SCENARIO Consider a user receiving a desired signal over L i.i.d. Nakagami-mD diversity paths with a common short-term average fading power D . Assume that this user is subject to NI i.i.d interferers with Nakagami-mI fading and a common short-term average fading power I . Recall that Abu-Dayya and Beaulieu [3] as well as Yao and Sheikh [4] derived an expression for the outage probability corresponding to the nondiversity case (i.e., L = 1) of this scenario. Following that approach, it can be easily shown (with the help of Gradshteyn and Ryzhik, Eq. (3.194.1) of [26]) that the outage probability with respect to a threshold λth can be also obtained for the diversity case as Pout =

 (mD L + mI NI ) zmD L 2 F1 (mD L + mI NI , mD L; 1 + mD L; −z) (mD L)  (mD L)  (mI NI ) (10B.1) 

where z = λth I mD /D mI and 2 F1 (·, ·; ·; ·) is the Gaussian hypergeometric function [26, Sect. 9.1] and where, contrary to (10.17), the fading parameters mD and mI are not restricted to integer values in (10B.1). It should be noted that this outage probability expression (10B.1) corresponds to the average probability of error

679

APPENDIX 10B. OUTAGE PROBABILITY IN NAKAGAMI/NAKAGAMI

expressions derived by Aalo and Zhang [21, Sect. III] for the same scenario (i.e., Nakagami/Nakagami with MRC diversity). We now show that the result (10B.1) agrees with (10.17) when mD and mI are restricted to integer values. For mD and mI integers, we have  (mD L + mI NI ) (mD L + mI NI − 1)! =  (mD L + 1)  (mI NI ) (mD L)! (mI NI − 1)!

(10B.2)

From Eq. (15.3.3), of Ref. 32, 2 F1 (mD L

+ mI NI , mD L; 1 + mD L; −z) =

2 F1 (1

− mI NI , 1; 1 + mD L; −z) (1 + z)mD L+mI NI −1 (10B.3)

Next, from Eq. (15.4.1) of Ref. 32, we obtain

2 F1 (1

− mI NI , 1; 1 + mD L; −z) =

mI
NI −1 n=0

(1 − mI NI )n (1)n (−z)n n! (1 + mD L)n

(10B.4)



where (b)n =  (b + n) / (b) is the Pochhammer symbol [32, p. 256]. But (1)n = (1 + mD L)n = =

 (1 + n) = n!  (1)  (1 + mD L + n)  (1 + mD L) (mD L + n)! (mD L)!

(10B.5)

and (1 − mI NI )n = (1 − mI NI ) (2 − mI NI ) · · · (n − mI NI ) = (mI NI − 1) (mI NI − 1) · · · (mI NI − 1) (−1)n (mI NI − 1)! = (−1)n (mI NI − 1 − n)!

(10B.6)

Substituting (10B.5) and (10B.6) in (10B.4) gives 2 F1 (1

− mI NI , 1; 1 + mD L; −z)

= (mD L)! (mI NI − 1)!

mI
NI −1 n=0

1 zn (mI NI − 1 − n)! (mD L + n)!

(10B.7)

680

OUTAGE PERFORMANCEOF MULTIUSER COMMUNICATION SYSTEMS

Finally, using (10B.7) in (10B.3), multiplying by (10B.2), and making the substitution of index k = mI NI − 1 − n, gives mI NI −1 mD L+mI NI −1 Pout = =

k=0

zmD L+mI NI −1−k k mD L+mI NI −1

(1 + z) mI NI −1 mD L+mI NI −1  1 k k=0



1+

k z  1 mD L+mI NI −1 z

(10B.8)

which (in view of the substitution z = λth I mD /D mI ) agrees with (10.17).

11 OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION OVER FADING CHANNELS IN THE PRESENCE OF INTERFERENCE

Thus far in the book we have discussed the performance of digital communication systems perturbed only by a combination of AWGN and multipath fading. Under such noise-limited conditions, coherent diversity reception takes the form of a maximum-ratio combiner (MRC), which, as discussed in Chapter 9, is optimum from the standpoint of maximizing the signal-to-noise ratio (SNR) at the combiner output. In applications such as digital mobile radio, space diversity provided by an adaptive antenna array is an attractive means for providing such diversity [1]. In addition to combating multipath fading, space diversity can also be used in cellular radio systems to reduce the relative power of co-channel interferers (CCIs) that are present at each element of the array. When operating in this scenario, the appropriate diversity scheme to employ is one that combines the branch outputs in such a way as to maximize the signal-to-interference plus noise (SINR) ratio at the combiner output. Under such conditions, this scheme, which is referred to as optimum combining (OC), will achieve a larger output SINR than MRC and is thus highly desirable even when the number of interferers exceeds the number of antenna array elements. This improved SINR efficiency can manifest itself in the cellular mobile radio application as a reduction in the number of base stations and/or an increased channel capacity through greater frequency reuse. The maximization of output SINR using adaptive antenna arrays techniques has been studied extensively in the early literature [1–4] primarily in a pure AWGN Digital Communication over Fading Channels, Second Edition. By Marvin K. Simon and Mohamed-Slim Alouini ISBN 0-471-64953-8 Copyright  2005 John Wiley & Sons, Inc.

681

682

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

environment, specifically, in the absence of fading. The application of these principles to the slow-fading channel, which is typical of digital mobile radio applications, was first studied by Bogachev and Kiselev [5], who evaluated the bit error probability (BEP) performance of an optimum combiner for coherent binary orthogonal signals in the presence of a flat Rayleigh fading assumed to be independent between antenna elements and a single interferer with the same fading characteristics. Later, Winters [6] amplified on these analytical results and also provided computer (Monte Carlo) simulation results for the multiple interferer case. In both of these papers, comparisons were made with comparable systems employing MRC and significant performance improvement was demonstrated for the OC case. Subsequent to these early papers, many studies of OC followed which considered in detail such issues as: (1) the number of interferers relative to the number of antenna elements, (2) channel correlation due to nonideal space separation, (3) practical signal and interference models that allow for analytical results corresponding to the multiple interferer case, (4) simple upper bounds (as opposed to complicated exact expressions) on average BEP performance. In this chapter, we bring together the above analytical results under a single roof by applying the unifying framework of the moment generating function (MGF)based approach discussed in previous chapters of this book. We shall again see that such an approach not only simplifies the analytical expressions for average BEP associated with the various cases previously treated in the literature but also allows extension to a large variety of modulation schemes as well as slow fading channels other than those that are Rayleigh distributed. To illustrate the approach, we shall first treat the simple case of a single co-channel interferer impinging on an L-element antenna array. As we shall see there are, in principle, two approaches (one exact and one approximate) that can be taken to evaluate the average BEP for this scenario. What shall be important to observe is that a comparison of the two for the Rayleigh fading channel, assumed to be independent and identically distributed (i.i.d.) across the array, reveals that the simple (but approximate) approach yields performance results that are extremely close to those provided by the more complicated (but exact) approach [7]. From this observation which can also be readily justified by intuitive reasoning, we shall then draw the conclusion that for the remainder of the cases (i.e., other modulations and fading channel models, more than one interferer), it is sufficient to evaluate average BEP performance using the simpler approach. Indeed this is the assumption made in many of the above-cited references without, however, the mathematical and numerical justification offered here and originally presented in [7]. 11.1 11.1.1

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS Single Interferer; Independent, Identically Distributed Fading

Consider a communication receiver (typical of the base station of the reverse link (mobile-to-base) of a digital mobile radio system) which provides space diversity via an L-element antenna array. Assume that the antenna elements of the array are

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

683

placed sufficiently far apart so as to provide independent fading paths. Furthermore, assume that the fading is sufficiently slow as to allow coherent detection to be employed. Then the received signal vector r (t) at the outputs of the array elements may be expressed as1

r (t) = Pd cd sd (t) + PI cI sI (t) + n(t) (11.1) where sd (t) and sI (t) are the desired and interfering signals normalized such that Pd and PI represent their respective powers, cd , cI are the corresponding L channel propagation vectors with components cdl L l=1 , cI l l=1 , respectively, and n(t) is the AWGN vector each element of which has zero mean and variance σ 2 . Each vector is of dimension L. In the absence of fading, the elements of cd and cI are constant complex quantities each with unit magnitude and a phase determined by the relative distance of its associated antenna element from the reference antenna element (often taken as the center element of the array). In the presence of fading, the elements of cd and cI become complex random variables (RVs) with statistics dependent on the fading channel model assumed. For example, for Rayleigh fading, the elements of cd and cI would be i.i.d. complex Gaussian RVs with zero means and unit mean-square value.2 Finally, the desired signal, interference signal, and additive noise are assumed mutually independent as would be the case in a practical system. As in a conventional RAKE receiver, the components of r (t) are appropriately (complex) weighted and combined (summed) to form a decision statistic. The difference between the RAKE receiver for MRC and that for OC lies in the selection of the weight vector w. Specifically, for MRC the weights are selected to maximum the instantaneous SNR at the combiner output and thus w = cd /σ 2 . For OC the weights are selected to maximum the instantaneous SINR at the same location −1 cd , where Rni is the noise-plus-interference covariance matrix and thus w = Rni to be defined below. As such, the implementation of the RAKE receiver for OC requires complete knowledge of the channel corresponding to both the desired signal and the interferer. For this receiver, the maximum instantaneous SINR at the combiner output is given by [6] −1 γt = Pd cH d Rni cd

(11.2)

where the superscript H stands for the Hermitian (transpose complex conjugate) operation and the noise plus interference covariance matrix is given by 
 
H  2 = PI cI cH Rni = E PI cI sI (t) + n(t) PI cI sI (t) + n(t) I +σ I (11.3) 1

We assume for simplicity a baseband model corresponding to ideal coherent demodulation. assumption of equal unit mean-squared values for both the desired signal and interference propagation vector fading components results in no loss of generality and is made for consistency with the no-fading case.

2 The

684

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

where I is the L × L identity matrix. In the absence of interference, (11.3) becomes a purely diagonal matrix and thus (11.2) simplifies, as it should, to the instantaneous SNR of the combiner output for MRC, namely γt =

L L  Pd H Pd  2 c c = α = γl d dl σ2 d σ2 l=1

(11.4)

l=1



2 where αdl is the fading amplitude of the l th element of cd and γl = αdl Pd /σ 2 is its corresponding instantaneous SNR. Since, as we have seen in Chapter 9, the form of (11.4) is highly desirable from the standpoint of evaluating average BEP since for the i.i.d. fading assumption it allows the MGF of γt to be expressed in product form, we shall find it expedient for the OC case to diagonalize the covariance matrix of (11.3) with the hope of again applying the same MGF-based approach. The diagonalization of Rni is accom plished by applying a unitary matrix transformation U to it; thus,  = U H Rni U is a diagonal matrix with elements λ1 , λ2 , . . . , λL corresponding to the eigenvalues3 of Rni . The rows of U are the corresponding complex eigenvectors which from the properties of a unitary matrix are orthonormal. From the definition of , the −1  = U −1 U H . Thus, substituting this result inverse of Rni is easily found to be Rni in (11.2) gives 

−1 H H −1 γt = Pd cH d U  U cd = Pd s  s = Pd

L  |sl |2 l=1

λl

(11.5)

where s = U H cd is the transformed desired signal propagation vector with components sl , l = 1, 2, . . . , L. It is clear from (11.5) that, conditioned on the set of eigenvalues λl , l = 1, 2, . . . , L, the MGF of γt will be expressible in product form if the transformed instantaneous signal powers |sl |2 , l = 1, 2, . . . , L are mutually independent. To see how this condition can be satisfied, we proceed as follows. If the elements of cd are modeled as complex Gaussian RVs (as for Rayleigh or Rician fading) or sums of complex Gaussian RVs (as for Nakagami-m fading), then a linear operation (e.g., multiplication by U H ) on cd results in a vector (i.e., s) whose components are again complex Gaussian RVs. Furthermore, from the orthonormal property of the rows of U , the mean-square value of the components of s are all equal to unity (as is the case for cd ). In addition, these components are mutually uncorrelated, and since they are Gaussian, they are mutually independent. Thus, we conclude that the transformed desired signal propagation vector s has statistics identical to those of its untransformed version cd , and therefore for analytical pur2 in (11.5). Doing so allows us to rewrite poses we can replace |sl |2 by |cdl |2 = αdl (11.5) as γt =

L L  σ2 Pd  σ 2 2 α = γl σ2 λl dl λl l=1

(11.6)

l=1

3 In general these eigenvalues are RVs, although, as we shall see shortly, for the single-interferer case, only one of them, say, λ1 , is random.

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

685

and hence the conditional (on the eigenvalues) MGF of γt , Mγt |λ1 ,λ2 ,...,λL (s), is given by the product Mγt |λ1 ,λ2 ,...,λL (s) =

L 

Mγ

l=1

σ2 s λl

(11.7)

where Mγ (s) is the MGF of any of the γl s with mean value γ d = Pd /σ 2 , which also represents the average SNR of the desired signal per antenna. Before proceeding with the evaluation of average BEP, we first specify the eigenvalues for the single-interferer case. It has been shown in several places in the literature [5–8] that the eigenvalues of the covariance matrix of (11.3) are given by  L L      PI |cI n |2 + σ 2 = PI αI2n + σ 2 , l=1 λl = (11.8) n=1 n=1    2 σ , l = 2, 3, . . . , L that is, L − 1 of them are constant and one of them, λ1 , is a RV. Making this substitution in (11.7) gives Mγt |λ1

 L−1 Mγ (s) = Mγ (s)



σ2 s λ1

(11.9)

To exactly evaluate the average BEP of coherent BPSK using an OC receiver, we must average the conditional (on the fading) BEP over the fading distribution of the combiner output statistic. In particular  ∞ 
 Pb (E) = Q 2γt pγt (γt ) d γt 0 (11.10)  ∞  ∞ 
 = Q 2γt pγt (γt |λ1 ) d γt pλ1 (λ1 ) d λ1 σ2

0

where Q (x) is as before the Gaussian Q-function and pγt (γt |λ1 ) is the probability density function (PDF) of the combiner output SINR conditioned on the single random eigenvalue λ1 [with PDF pλ1 (λ1 )] and is ordinarily found by first evaluating the conditional MGF, namely, Mγt |λ1 (s), and then taking its inverse Laplace transform. Although direct evaluation of (11.9) may be possible, it typically involves complicated analysis that includes first determining the PDF pγt (γt |λ1 ) in closed form and then successively performing the remaining integrations over the Gaussian Q-function and the eigenvalue probability distribution. Quite often the closed-form expressions obtained at any stage in the process are given in terms of functions not readily available in standard software packages such as Mathematica and are in a form that provides little insight into their dependence on such system parameters as Pd , PI , and σ 2 .

686

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

11.1.1.1 Rayleigh Fading—Exact Evaluation of Average Bit Error Probability To illustrate the above-mentioned point, Shah et al. [7] evaluate (11.10) for the Rayleigh channel. First, the conditional MGF of γt is from (11.9) found to be Mγt |λ1 (s) = 

1  L−1  1 − s Pλ1d 1 − sγ d

(11.11)

which has the inverse Laplace transform [9, p. 410] λ1 γtL−1 exp pγt (γt |λ1 ) =



− λP1dγt







L − 1; L;

1 F1



σ 2  (L) γ d

L

 λ1 −σ 2 γt



Pd

,

(11.12)

γt ≥ 0, λ1 ≥ σ 2 , L ≥ 1 where 1 F1 (·; ·; ·) is Kummer’s confluent hypergeometric function [10, p. 1085, Eq. (9.210)]. Performing the first integration (on γt ) in (11.10) gives 

∞

Pb (E |λ1 ) =

Q


 2γt pγt (γt |λ1 ) d γt

0

  L+1/2   Pd λ1 −σ 2 1 1 3 λ L + P  L + F , 1, L − 1; , L; , 1 2 d 2 2 2 λ1 +Pd λ1 +Pd 1 = −  L √ 2 L+1/2 2 π σ  (L) γ d (λ1 + Pd ) (11.13) where F2 (·, ·, ·; ·, ·; ·, ·) is Appell’s hypergeometric function of two variables [10, p. 1080, Eq. (9.180.2)]. Since each αI2n in (11.8) has a chi-square distribution, then λ1 has the PDF pλ1 (λ1 ) =



  λ1 − σ 2 1 2 L−1 λ − σ exp − , 1 PI  (L) PIL

λ1 ≥ σ 2

(11.14)

Finally, the average BEP is obtained by averaging (11.13) over the PDF in (11.14) in accordance with (11.10), namely   L+(1/2)  ∞  L−1 1 λ1 1  L + 12 Pd Pb (E) = − √ λ1 − σ 2  L L L+(1/2) 2 2 2  P (L) (λ1 + Pd ) σ πσ  (L) γ d I



1 3 Pd λ1 − σ 2 λ1 − σ 2 F2 L + , 1, L − 1; , L; d λ1 × exp − , PI 2 2 λ1 + P d λ1 + Pd (11.15)

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

687

which cannot be obtained in closed form except for the special case of L = 1, which has the result   

2 + σ Pd Pd + σ 2 P 1 d  Q 2 Pb (E) = − π exp 2 PI PI PI (11.16) 



 1 γ 1 + γd 1 + γd Q = − π d exp 2 2 γI γI γI 

where γ I = PI /σ 2 is the average interference-to-noise ratio per antenna. We recall from Chapter 9 that, using the alternative form of the Gaussian Q-function, allows us to express the BEP directly in terms of the MGF of γt . Specifically, conditioned on λ1 , we have Pb (E |λ1 ) =

1 π



π/2 0

Mγt |λ1

−



1 sin2 θ

dθ

(11.17)

Then, the average BEP is given by 1 Pb (E) = π



∞  π/2

−

1



d θpλ1 (λ1 ) d λ1 sin2 θ 

L−1

  1 σ2 1 1 ∞ π/2 Mγ − 2 = Mγ − d θpλ1 (λ1 ) d λ1 π σ2 0 λ1 sin2 θ sin θ (11.18) which for the Rayleigh channel becomes   L−1    2 θ sin sin2 θ 1 ∞ π/2   d θpλ1 (λ1 ) d λ1 Pb (E) = 2 π σ2 0 sin2 θ + γ d sin2 θ + σ γ d σ2

0

Mγt |λ1

λ1

(11.19) Performing the integral on λ1 , we first obtain, after much manipulation



 

 1 1 1 L π/2 γd γd  1 − L, 1 + exp 1 + γI sin2 θ γ I sin2 θ γ I 0





γd γd 1 +L 1+ dθ (11.20)  −L, 1 + 2 2 sin θ sin θ γ I

1 Pb (E) = π



where  (a, x) is the complementary incomplete gamma function [10, p. 950, Eq. (8.353.3)]. If one wants to simplify the notation a bit (which will be convenient when extending the results to other modulations), one defines

γd 1  = f (θ ) 1+ (11.21) 2 γ sin θ I

688

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

in which case (11.20) simplifies to4 Pb (E) =

1 π



1 γI

L

π/2

  exp {f (θ )} (1 − L, f (θ )) +Lγ I f (θ ) (−L, f (θ )) d θ

0

(11.22) Clearly, (11.19) or (11.22) together with (11.21) is considerably simpler in form than (11.15) and, as we shall see momentarily, allows us to immediately draw conclusions for certain special cases.   For the special case of no interferer, PI = 0 λ1 = σ 2 , we immediately see from (11.19) that 

π/2

Pb (E) =



L

sin2 θ sin2 θ + γ d

0

dθ

(11.23)

which corresponds to the performance of coherent PSK with MRC and an L-element array. For the special case of a single infinite power interferer, PI = ∞ (λ1 = ∞), Eq. (11.19) simplifies to 

π/2

Pb (E) =



sin2 θ sin2 θ + γ d

0

L−1 dθ

(11.24)

Thus, on the basis of exact expressions for BEP, we observe that for the infinite power interferer, the array uses up one entire order of diversity in its attempt to cancel it. This same conclusion was reached by Shah et al. [7] on the basis of “upper” bounds on conditional BEP and by Winters [6] on the basis of approximate expressions for average BEP (obtained by replacing λ1 by its mean value λ1 = σ 2 + LPI ). We shall ourselves pursue the legitimacy of this approximation momentarily. 4 Using

an analogous approach based on an alternative form of the complementary error function (equivalently the Gaussian Q-function) in Appendix 4A, Eq. (4A.1), Aalo and Zhang [11] were able to arrive at a closed-form expression for average BEP, namely   k 

 L−2 1 γ d  2k 1    Pb (E) = 1 − k 2 γd + 1 4 γd + 1 k=0

 L−1 2 (L) −Lγ I

  

γd + 1 π γ γ + 1 d d   erfc  exp Lγ I Lγ I Lγ I





 −

1

L−2 γ d  (2k)! − γd + 1 k! k=0

which checks numerically with (11.22).

−Lγ I   4 γd + 1

k  

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

689

Before proceeding, we wish to point out that the conclusion reached above from a comparison of the no interferer and infinite power interferer could have been reached earlier from the context of a generalized slow-fading channel. This can easily be seen by evaluating (11.9) for the cases of PI = 0 λ1 = σ 2 and  L PI = ∞ (λ1 = ∞), resulting in Mγt |λ1 =σ 2 (s) = Mγ s, γ d and Mγt |λ1 =∞ (s) =  L−1 Mγ s, γ d , respectively, which from (11.17) establishes the desired conclusion. 11.1.1.2 Rayleigh Fading—Approximate Evaluation of Average Bit Error Probability To simplify the analysis, several authors [5–8] have made the assumption of replacing λ1 by its mean value λ1 = σ 2 + LPI in the conditional MGF of γt (and likewise the same assumption in the conditional PDF of γt ) and then computing the average BEP from Pb (E) ∼ =



∞

Q 0

 
  2γt pγt (γt |λ1 ) λ1 =λ1 d γt

(11.25)

Clearly this avoids evaluating the PDF of the eigenvalue λ1 and integration over this RV in (11.10) and also makes the expression for average BEP independent of the probability statistics of the interferer; that is, it is necessary only to know its average power. When this substitution is made in the conditional PDF of (11.12), the following closed-form expression results for Rayleigh fading [6, Eq. (25)]:5    ! 1 + Lγ I γd γd 1 Lγ I Pb (E) =  + − L−1 − 1 + Lγ I 1 + γd 1 + Lγ I 1 + Lγ I + γ d 2 −Lγ I  "  #$ L−2 k   k γd (2i − 1)!! −Lγ I − 1− 1+ (11.26)  i 1 + γd i! 2 + 2γ d 

k=1

i=1

This result should also agree with that obtained by substituting λ1 = σ 2 + LPI for λ1 in (11.19): 1 Pb (E) = π

5





π/2 0



sin2 θ sin2 θ +

γd 1+Lγ I

 

sin2 θ sin2 θ + γ d

L−1 dθ

(11.27)

The result in (11.26) was obtained from the expression in Ref. 5 for the BEP of coherent binary orthogonal signaling by replacing γ d by 2γ d reflecting the 3 dB difference between orthogonal and antipodal signaling in Gaussian noise. Also, although not explicitly stated in either reference, the result in (11.26) is valid only for L ≥ 2. The lack of validity for L = 1 (i.e., no diversity) can easily be seen by observing that for γ I = 0 (i.e., no interference), (11.26) would yield Pb (E) = 0, which is an incorrect result.

690

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

The result in (11.27) can be also be evaluated in closed form using (5A.56):  " # !



L−2 1 γ d  2k 1 1 L−1−k Pb (E) = 1−   k 1 − − k 2 1 + γd Lγ I 4 1 + γd k=0 

$ 1 L−1 γd − − (11.28) 1 + Lγ I + γ d Lγ I The equivalence between (11.28) and (11.26) can be established mathematically (after much tedious manipulation). Furthermore, (11.28) is valid for L = 1 and is thus the more general result. Figures 11.1–11.3 illustrate the evaluation of Pb (E) from (11.20) (the exact result) and from (11.28) (the approximate result) as a function of average total SNR Lγ d for two- and four-element arrays and for the special case of equal average desired signal and interference powers (or equivalently γ d = γ I .) We observe that the difference between the curves using (11.20) and (11.28) is small. There is a simple, intuitive explanation for why the average BEPs are as close as they appear. For either L = 2 or L = 4, the array has a sufficient number of degrees of freedom to suppress the single interferer regardless of whether or not it is degraded by fading. Thus, the performance is affected predominantly by the interferer’s average power (which is all that is needed in the approximate evaluation case) rather than its complete statistical description (which is what is needed in the exact evaluation case). As we shall soon see, obtaining exact results in useful form for more complex fading channels is difficult. Thus, in view of the observation above, we shall resort to using the approximate approach for evaluating average BEP in these cases

10−1

Average BEP

L=2

10−2

L=4

10−3

BEP (exact) BEP (approximate) 10−4

0

2

4

6

8

10

12

14

16

Average total SNR (dB)

Figure 11.1 Average BEP of the optimum combiner in Rayleigh fading with a single fading co-channel interferer; γ d = γ I .

691

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

Average BEP

10−1

L=2

BEP (exact) BEP (approximate) 10−2

6

7

8

9

10

11

12

Average total SNR (dB)

Figure 11.2 Average BEP performance (magnified scale); γ d = γ I , L = 2 (courtesy of Shah et al. [7]).

Average BEP

10−2

L=4

BEP (exact) BEP (approximate) 10−3

6

7

8

9

10

11

Average total SNR (dB)

Figure 11.3 Average BEP performance (magnified scale); γ d = γ I , L = 4.

12

692

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

according to the same intuitive argument made for the simpler Rayleigh channel. To illustrate the difficulty of the exact evaluation method and the relative simplicity of the approximate approach, we present next the average BEP results for the Rician and Nakagami-m fading channels. Before doing this, however, we first present results for the Rayleigh channel corresponding to coherent M-PSK and QAM. 11.1.1.3 Extension to Other Modulations From the treatment of coherent M-PSK in the presence of fading discussed in Chapter 8, the conditional SEP can be written as

 gPSK 1 (M−1)π/M dθ (11.29) Mγt |λ1 − 2 Ps (E |λ1 ) = π 0 sin θ where gPSK = sin2 π /M. Thus, following the same procedure as for binary PSK, then, after averaging over the PDF of λ1 , we obtain [analogous to (11.22)]   1 1 L π/2 exp {fPSK (θ )}  (1 − L, fPSK (θ )) Ps (E) = π γI (11.30) 0  + Lγ I fPSK (θ )  (−L, fPSK (θ )) d θ where now

γ s gPSK 1  = fPSK (θ ) 1+ γI sin2 θ

(11.31)

and γ s denotes the average desired signal SNR per symbol per antenna. For QAM with M = 2k signal points, the conditional SER is given by



 π/2 4 gQAM 1 dθ Mγt |λ1 − 2 Ps (E |λ1 ) = 1− √ π sin θ M 0 (11.32)



gQAM 4 1 2 π/4 − 1− √ Mγt |λ1 − 2 dθ π sin θ M 0 

where gQAM = 3/ [2 (M − 1)]. Thus, analogous to (11.22), the average SER is given by

L  π/2 % & 1 1 4 1− √ Ps (E) = exp fQAM (θ ) π γI M 0      ×  1 − L, fQAM (θ ) +Lγ I fQAM (θ )  −L, fQAM (θ ) d θ (11.33)

 % & 1 2 1 L π/4 4 1− √ exp fQAM (θ ) − π γI M 0      ×  1 − L, fQAM (θ ) +Lγ I fQAM (θ )  −L, fQAM (θ ) d θ

693

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

where now

γ s gQAM 1  = fQAM (θ ) 1+ γI sin2 θ

(11.34)

and again γ s denotes the average desired signal symbol SNR per antenna. 11.1.1.4 Rician Fading—Evaluation of Average Bit Error Probability Analogous to (11.11), the conditional MGF is now  1  (s) =   L−1  1 Pd 1 1 − s 1+K 1 − s γ d λ1 1+K 

Mγt |λ1

!

" #$ Pd K γd λ1 × exp s + (L − 1) 1 1 Pd 1+K 1 − s 1+K γd 1 − s 1+K λ1 (11.35) Also, the largest eigenvalue has a noncentral chi-square PDF given by



(L−1)/2

1 + K (L−1)/2 λ1 − σ 2 1+K pλ1 (λ1 ) = PI LK PI



 λ1 − σ 2 × exp − LK + (1 + K) PI   

2 λ − σ 1 , λ1 ≥ σ 2 × IL−1 2 LK (1 + K) PI

(11.36)

  For L = 1 and y = λ1 − σ 2 /PI , Eq. (11.36) reduces to     
py (y) = (1 + K) exp − K + (1 + K) y I0 2 K (1 + K) y ,

y≥0

(11.37) which is the standard PDF for the square of a Rician RV. Also, for K = 0, using the asymptotic (small argument) form of IL−1 (x) in (11.36), namely IL−1 (x) ∼ =

1  x L−1  (L) 2

(11.38)

the PDF of (11.36) can be shown to reduce to that in (11.14), as it should.

694

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

The conditional BEP for coherent PSK is from (11.17) and (11.35), given by  1 1 π/2 Pb (E |λ1 ) =   L−1 π 0 γd Pd 1 1 1 + 1+K 1 + 1+K sin2 θ λ1 sin2 θ    Pd  1 K λ 1   (11.39) × exp − Pd  1 + K sin2 θ 1 + 1 2 1+K λ1 sin θ #$ γd + (L − 1) dθ γd 1 1 + 1+K 2 sin θ To exactly evaluate the average BEP, we must average (11.39) over the PDF in (11.36). After some algebraic manipulation, we get the following result:  L−1

 1 + K (L−1)/2 1 π/2 sin2 θ Pb (E) = exp (−LK) (1 + K) 1 LK π 0 sin2 θ + 1+K γd " #  γd (L − 1) K × exp − 2 1 1+K sin θ + 1+K γd    γd  ∞ sin2 θ K 1+γ I y   × exp − γd 1 + K sin2 θ + 1 γ d 0 sin2 θ + 1 1+K 1+γ I y

1+K 1+γ I y


 × y (L−1)/2 exp {− (1 + K) y} IL−1 2 LK (1 + K) y d y d θ

(11.40)

Unfortunately, the integral on y cannot be obtained in closed form. Thus, we now resort to the approximate approach corresponding to substituting λ1 = LPI + σ 2 for λ1 in the MGF of (11.35). When this is done, the approximate average BEP from (11.39) becomes   L−1  π/2 2 2 θ θ sin sin 1   Pb (E) = γd 1 1 π 0 sin2 θ + 1+K γd sin2 θ + 1+K 1+Lγ I    γd  K  (L − 1) γ d  1+Lγ I × exp − + dθ γd 1  1 + K sin2 θ + 1 sin2 θ + 1+K γd  1+K 1+Lγ I

For K = 0, (11.41) reduces to   sin2 θ 1 π/2  Pb (E) = γd π 0 sin2 θ +

1+Lγ I

which is identical to (11.27).

(11.41)  

sin2 θ sin2 θ + γ d

L−1 dθ

(11.42)

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

695

100

Average BEP

10−1

K=0 K=2 K=4

10−2

K = 10 10−3

10−4

10−5

0

2

4

6

8

10

12

14

16

Average Total SNR, Lgd (dB)

Figure 11.4 Average BEP of the optimum combiner in Rician fading with a single fading co-channel interferer; γ d = γ I , L = 2.

Figures 11.4 and 11.5 illustrate the approximate evaluation of Pb (E) from (11.41) as a function of average total SNR Lγ d with Rician factor K as a parameter for two- and four-element arrays again assuming γ d = γ I . 11.1.1.5 Nakagami-m Fading—Evaluation of Average Bit Error Probability For this fading channel, we can model each of the interference vector components as a sum of m i.i.d. complex Gaussian RVs, each with zero mean and variance 1/m. In this way, each component of the interference vector (e.g., cI n ) still has unity mean-square value. Then, after transformation by the unitary matrix, the new vector s, which corresponds to a weighted sum of the components of cI , will still have i.i.d. complex Gaussian components and its properties will thus be preserved * 2 as in the Rayleigh and Rician cases. Also, the RV L l=1 |cI l | is now central chisquare-distributed with 2mL*degrees of freedom, each of which has variance 1/2m. 2 2 Thus, the PDF of λ1 = PI L l=1 |cI l | + σ is now given analogous to (11.14) by

  λ1 − σ 2 mmL 2 mL−1 , λ1 ≥ σ 2 λ pλ1 (λ1 ) = − σ exp −m 1 PI  (mL) (PI )mL (11.43)

696

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

100

10−1

K=0 K=2

10−2

K=4

Average BEP

10−3

10−4

10−5

K = 10

10−6

10−7

10−8 10−9 0

2

4

6

8

10

12

14

16

Average Total SNR, Lgd (dB)

Figure 11.5 Average BEP of the optimum combiner in Rician fading with a single fading co-channel interferer; γ d = γ I , L = 4.

Similarly, analogous to (11.11), the conditional MGF of the combiner output SNR is now 1 Mγt |λ1 (s) =  m(L−1)   γ Pd 1 − s md 1 − s mλ 1 Using (11.44), the conditional BEP of (11.17) is given by  1 1 1 π/2 Pb (E |λ1 ) = m d θ  m(L−1)  π 0 γd d 1 + mλ Psin 1 + m sin2 θ 2θ 1

(11.44)

(11.45)

  For no interferer λ1 = σ 2 , (11.45) becomes Pb (E) =

1 π



π/2 0

 1+

1 γd m sin2 θ

mL d θ

(11.46)

697

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

which is equivalent to MRC combining with mL orders of diversity and channels, each with 1/mth of the power. For an infinite power interferer (λ1 = ∞), we get 1 Pb (E) = π



π/2

 1+

0

1 γd m sin2 θ

m(L−1) d θ

(11.47)

Thus, analogous to the Rayleigh and Rician cases, the diversity is reduced by m orders of magnitude in attempting to cancel the interferer where each channel now has only 1/mth of the power. If we attempt to get the exact expression for average BEP by averaging (11.45) over the PDF in (11.43), we obtain, after some simplification 1 Pb (E) = π  (mN) 

∞

× 0



m γI



mN 

π/2 0

m

1+z 1+

γd m sin2 θ

 1+

+z

1 γd m sin2 θ

L−1

zmL−1 exp −



(11.48)

m z dz dθ γI

Unfortunately, the integral on z cannot be obtained in closed form, so once again we must resort to the approximate approach. Substituting λ1 = LPI + σ 2 for λ1 in the MGF of (11.44) and then applying (11.17) results in the approximate average BEP:  m  m(L−1)  2 θ sin2 θ 1 π/2  sin  dθ (11.49) Pb (E) = γ d /m π 0 sin2 θ + γ d /m sin2 θ + 1+Lγ I

Analogous to Figs. 11.4 and 11.5, Figs. 11.6 and 11.7 illustrate the approximate evaluation of Pb (E) from (11.49) as a function of Lγ d with Nakagami-m factor m as a parameter for two- and four-element arrays again assuming γ d = γ I . 11.1.2 Multiple Equal Power Interferers; Independent, Identically Distributed Fading Consider now a scenario where more than a single co-channel interferer exists at each antenna element. On the one hand, to increase network capacity, the cellular system might be operating in an interference-limited environment, in which case the number of co-channel interferers NI could typically exceed the number of antenna elements L. Under such conditions, which is typical of CDMA, the number of degrees of freedom provided by the array is insufficient to allow a partitioning of the observation space into distinct noise and interferer subspaces; that is, the optimum combiner processes both the noise and interference together as a single entity and cannot cancel all the interfering signals. On the other hand, in certain other practical cellular mobile applications, for instance, in a TDMA system such as Global System for Mobile (communication) (GSM), most of the interference

698

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

100

m=1

10−1

Average BEP

m=2

10−2

m=4 10−3

10−4

10−5 0

2

4

6

8

10

12

14

16

Average Total SNR, Lgd (dB) Figure 11.6 Average BEP of the optimum combiner in Nakagami-m fading with a single fading co-channel interferer; γ d = γ I , L = 2.

can be due to only a limited number of dominant interferers. Here the array has sufficient degrees of freedom to partition the observation space into distinct noise and interferer subspaces and combat them as separate entities (much like the case of the single interferer discussed in the previous section). As we shall soon see, the treatment of this scenario is a natural extension of the single-interferer case and therefore will be the first one discussed. Exact and approximate evaluation of average BER for optimum combining receivers in the presence of multiple narrowband interferers has been considered by several authors. Specifically, Cui et al. [12,13]6 considered optimum combining receivers for DPSK modulation with NI = 2 and Rayleigh fading. Their results also yield an upper bound on the performance of coherent BPSK and QAM based on the approach taken in Ref. 14. Optimum combining for coherent BPSK systems with arbitrary NI < L and Rayleigh fading was considered by Winters and 6 Also considered in these papers is the average BEP performance of MRC receivers in the presence of an arbitrary number of interferers.

699

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

100

10−1 m=1 10−2

m=2

Average BEP

10−3

10−4

10−5

m=4

10−6

10−7

10−8

0

2

4

6

8

10

12

14

16

Average Total SNR, Lgd (dB) Figure 11.7 Average BEP of the optimum combiner in Nakagami-m fading with a single fading co-channel interferer; γ d = γ I , L = 4.

Salz [15] and Winters et al. [16]. However, their results were presented only in the form of an upper bound on average BEP, which unfortunately is not accurate for low BEP and a large number of antenna elements. Most recently, Aalo and Zhang [11] obtained exact average BEP results (not in closed form, however) for optimum combining coherent BPSK systems with NI = 2 and again Rayleigh fading. The approach taken there was the same as that used to derive closed-form results for the single interferer case as mentioned previously in footnote 4 of this chapter. A more generic approach to the problem of spatial combining in the presence of multiple equal power narrowband interferers was considered by Haimovich and Shah [17] and Shah and Haimovich [18,19],7 who assumed that the number of interferers is sufficiently large as to justify an interference-limited environment; thus, AWGN was ignored. The results of this work will be presented in Section 11.1.2.2. For the case where the number of interferers is less than the number of array 7 Shah and Haimovich [19] compare EGC and MRC with OC and in addition allow for either Rayleigh, Rician, or no fading on the desired signal when the fading on the interference is Rayleigh distributed.

700

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

elements, Villier [8] assumed that the CCI is dominated by the strongest interferers NI 8 and that the approximate approach for evaluating error probability, replacing the random eigenvalues by their mean value, is once again valid. (We shall say more about this assumption momentarily.) As we shall see, this approach allows for consideration of generalized fading channels as well as a variety of modulation types. Before considering this approach and the performance results derived from it, we first present the exact average BEP results for the two interferer, coherent BPSK, Rayleigh channel case as given by Aalo and Zhang [11] starting with a generalization of the system model to the multiple-interferer scenario. 11.1.2.1 Number of Interferers Less than Number of Array Elements Analogous to (11.1), the received signal vector r (t) at the outputs of the array elements may be expressed as r (t) =




Pd cd sd (t) +

NI 


PI cIn sIn (t) + n(t)

(11.50)

n=1

where sIn (t) and cIn are now, respectively, the signal and propagation vectors associated with the nth interfering signal. Assuming that the NI interference signals are mutually independent, then, analogous to (11.3), the noise-plus-interference covariance matrix Rni is given by   N H  NI I
   
Rni = E PI cIn sIn (t) + n(t) PI cIn sIn (t) + n(t)   n=1 n=1 (11.51) NI  2 c In c H = PI In + σ I n=1

Since the maximum instantaneous SINR at the combiner output is still given by −1 is now the inverse of (11.51), then once again the the relation in (11.2), where Rni eigenvalue decomposition of Rni yields the result in (11.5), where now there exist L − NI nonrandom eigenvalues λNI +1 , λNI +2 , . . . , λL , each with value σ 2 and NI random eigenvalues λ1 , λ2 , . . . , λNI . Although we still obtain a product form for the MGF of the combiner output SINR, namely Mγ λ t

1 ,λ2 ,...,λNI

2

NI  L−NI  σ Mγ s (s) = Mγ (s) λn

(11.52)

n=1

the difficulty now lies in the determination of the NI random eigenvalues that are related in a complex manner to the total interference power received by the array. For the case where the interference sources are represented by mutually orthogonal 8 Any

other interfering signal is assumed to have a power level significantly lower than that of the NI strongest ones and as such is included in the additive noise term n(t).

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

701

propagation vectors, the random eigenvalues λ1 , λ2 , . . . , λNI become equal in form to that of (11.8) with PDFs given by [see (11.14)] pλn (λn ) =

 L−1 1 λn − σ 2 L  (L) PI

λn − σ 2 , × exp − PI

(11.53) λn ≥ σ ,

n = 1, 2, . . . , NI

2

When the number of interferers is restricted to two and the fading is Rayleigh distributed, the two random eigenvalues can be found exactly. Using the results given in Refs. 12 and 13 and later in Ref. 11, we have   + γ1 + γ2 1 λ1 = σ 2 1 + + (γ1 − γ2 )2 + 4ρ12 γ1 γ2 2 2 (11.54)   + γ1 + γ2 1 2 2 λ2 = σ 1 + − (γ1 − γ2 ) + 4ρ12 γ1 γ2 2 2 *L 2  * 2 2 where γn = L l=1 αIn l PI /σ = γI l=1 αIn l , n = 1, 2 is the instantaneous SNR for the nth interferer with αIn l the Rayleigh fading amplitude on its l th branch and has the PDF [analogous to (11.14)]

 1 γn L−1  , γn ≥ 0, n = 1, 2  L γn exp − γI pγn (γn ) = (11.55)   (L) γ I 0, otherwise Also, in (11.54), ρ12 is the normalized correlation between the two interference propagation vectors cI1 and cI2 and is a beta-distributed RV with PDF given by ! pρ12 (ρ12 ) =

(L − 1) (1 − ρ12 )L−2 , 0 ≤ ρ12 ≤ 1 0, otherwise

(11.56)

Note that for uncorrelated vectors (i.e., ρ12 = 0), (11.54) simplifies to * interference 2 2 α + σ , n = 1, 2, which, as alluded to above, has λn = σ 2 (1 + γn ) = PI L l=1 In l the form of (11.8). To evaluate the average BEP for coherent BPSK, we proceed analogous to (11.17) and (11.18). Specifically, from (11.52) the conditional MGF for Rayleigh fading is now Mγt |λ1 ,λ2 (s) =  = 

1 − sγ d 1 − sγ d

L−2  L−2 

1 1 − s Pλ1d



1 2

1 − sγ d σλ1

1 − s Pλ2d



 (11.57) 2

1 − sγ d σλ2



702

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

Then, the average BEP can be written as 1 Pb (E) = π 1 = π

∞  ∞  π/2



σ2



σ2

π/2



Mγt |λ1 ,λ2

0

−

d θpλ1 (λ1 ) pλ2 (λ2 ) d λ1 d λ2 ∞ 1

dθ

sin2 θ + γ d

0

sin2 θ  ∞

L−2

sin2 θ



1

0

0

0

 1+

γd σ2 sin2 θ λ1

1 

× pρ12 (ρ12 ) d ρ12 pγ1 (γ1 ) pγ2 (γ2 ) d γ1 d γ2

1+

γd σ2 sin2 θ λ2



(11.58)

where λ1 , λ2 are, in turn, expressed in terms of the RVs γ1 , γ2 , and ρ12 as in (11.54). The statistical average over ρ12 [inner integral of the triple integral in (11.58)] can be evaluated in closed form. Using results from Refs. 12 and 13, it can be shown that this average can be put in the form 

1 0

 1+

γd σ2 sin2 θ λ1

1 

1+

γd σ2 sin2 θ λ2

 pρ12 (ρ12 ) d ρ12

(L − 1) (1 + γ1 + γ2 )   =  γ γ 1 + sin2d θ 1 + sin2d θ + γ1 + γ2



1 0

(1 + au) L−2 u du (1 + bu)

(11.59)

where we have made the change of variables u = 1 − ρ12 and 

a=

γ1 γ2 , 1 + γ1 + γ2



b= 

1+

γd sin2 θ

γ1 γ2   γ 1 + sin2d θ + γ1 + γ2

(11.60)

Evaluating the integral in (11.59) as in Eq. (49) of Ref. 13, then after some manipulation, we arrive at the closed-form result (see also Ref. 11) 

1 0

 1+

γd σ2 sin2 θ λ1

1 

= 1 + (L − 1) ×

"L−3  k=0



1+

γd σ2 sin2 θ λ2

b−a ab

 pρ12 (ρ12 ) d ρ12



1 1 + ln (b + 1) k (L − 2 − k) (−b) (−b)L−2

(11.61) #

Substituting (11.61) into (11.58) leaves a triple integral for evaluating Pb (E), which, according to Aalo and Zhang [11], “can be easily evaluated numerically.” Motivated by the desire to obtain a simple expression for assessing the average BEP of the OC in the presence of multiple interferers with NI < L, Villier

703

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

[8] proposes using an approximate approach analogous to that for the singleinterference case wherein each random eigenvalue is replaced by its average over the fading distribution, and all of them are made equal in accordance with what would be obtained from λ1 of (11.8) (i.e., λi = LPI + σ 2 , i = 1, 2, . . . , NI ). The validity of this approximate approach is justified for the case where the interference power level is high (relative to the desired signal power) and the number of antenna elements is considerably greater than the number of dominant interferers, which mathematically to λi  Pd and L  NI . In such instances,  translates  ,NI σ2 the product n=1 Mγ s; λn γ d in (11.52) tends to become insignificant com L−NI  which then dominates pared to the remainder of the product Mγ s; γ d the MGF. A further justification corresponds to the scenario where the interferers are indeed orthogonal, in which case, on the basis of our previous discussion, the mean values of the random eigenvalues would become equal and precisely given by λi = LPI + σ 2 , i = 1, 2, . . . , NI . Proceeding under the assumption of the approximate approach described above, then, analogous to (11.11), the MGF for the Rayleigh fading channel becomes Mγt |λ1 (s) = 

1 − sγ d

1 L−NI 

1−

s Pλd 1

NI = 

1 NI L−NI  γd 1 − s 1+Lγ 1 − sγ d I

(11.62) and likewise, analogous to (11.27), the average BEP of coherent BPSK with OC is given by 1 Pb (E) = π





π/2 0



sin2 θ sin2 θ +

γd 1+Lγ I

NI  

sin2 θ sin2 θ + γ d

L−NI dθ

(11.63)

Note the similarity in the form of (11.63) (NI -independent interferers with Rayleigh fading) with that in (11.49) (one interferer in Nakagami-m fading). Furthermore, Villier [8] gives a closed-form result for the NI -independent interferer, Rayleigh fading case, which, from the similarity of the integrals mentioned above, would therefore imply that the single interferer in Nakagami-m fading performance [see (11.49)] would have a similar closed-form result (this is left as an exercise for the reader). For the former, we have from Eq. (20) of Ref. 8 that N −1 "NI −1



 1 + Lγ I I −Lγ I k γd B I Pb (E) =  k k L−1 1 + Lγ I 1 + Lγ I 2 −Lγ I k=0 

I −1   L−N  k    −Lγ I Ck Ik γ d − 1 + Lγ I

k=0

#

(11.64)

704

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

where the coefficients Bk and Ck are given by9 

Ak



Bk = L−1 ,

Ck =

k

N I −1

k  n L−1  An , n

n=0



Ak = (−1)NI −1+k

NI −1

NI 

k

(NI − 1)!

(11.65) (L − n)

n=1 n=k+1

and 1 Ik (c) = k! 



∞

x k erfc

√

 cx e −x d x

(11.66)

0

which has the series form Ik (c) = 1 −

# " k  c (2n − 1)!! 1+ 1+c n!2n (1 + c)n

(11.67)

n=1

where the double factorial notation denotes the product of only odd integers from 1 to 2k − 1. An alternative closed-form expression for the average BEP can be obtained from the results in Ref. 17 using the approach in Eqs. (11.12) and (11.13) but with λ1 replaced by λ1 . In particular, the inverse Laplace transform of (11.62) is [17, Eq. (71)]       λ1 −σ 2 γt λ1 γt L−1  Ni γt exp − Pd 1 F1 L − NI ; L; Pd    λ 1  pγt γt λ1 = ,  L σ2  (L) γ d γt ≥ 0, NI < L

(11.68)

which clearly reduces to (11.12) with λ1 replaced by λ1 when NI = 1. Integrating the Gaussian Q-function over the PDF in (11.64) as in (11.13) gives [17, 9 Note

that by convention

,NI

k n

= 0 for n > k. Also, for NI = 1, by convention the prod-

uct n=1,n=k+1 (L − n) = 1 and the only non-zero-valued coefficients are A0 = B0 = 1 and Ck = 1, k = 0, 1, . . . , L − 2. For NI > 1, the coefficients Ak , Bk , and Ck clearly depend on both NI and L.

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

705

Eq. (83)])10  Pb (E) =

∞

Q




    2γt pγt γt λ1 d γt

0

=

1 − 2



λ1 σ2 

NI

  L+(1/2)   L + 12 Pλd 1  L √ π (L) γ d

 Pd λ1 − σ 2 3 1 1 × F2 L + , , L − NI ; , L; − , 2 2 2 λ1 λ1   L+(1/2)   γd 1 NI  L + 2 1+Lγ I 1  = − 1 + Lγ I  L √ 2 π (L) γ d

1 1 γd 3 Lγ I × F2 L + , , L − NI ; , L; − , 2 2 2 1 + Lγ I 1 + Lγ I

(11.69)

where again F2 (·, ·, ·; ·, ·; ·, ·) is Appell’s hypergeometric function of two variables [10, p. 1080, Eq. (9.180.2)]. Using the functional relation between hypergeometric functions of two variables [10, p. 1083, Eq. (9.183.2)], the BEP of (11.69) can be rewritten as  NI    L + 12 (Pd )L+(1/2) λ1 1 Pb (E) = −  L  L+(1/2) √ 2 σ2 π (L) γ d λ1 + P d   Pd λ1 − σ 2 3 1 × F2 L + , 1, L − NI ; , L; , 2 2 λ1 + Pd λ1 + Pd L+(1/2)   γd 1  L +   2 1+γ d +Lγ I 1 N = − 1 + Lγ I I  L √ 2 π  (L) γ d

1 γd 3 Lγ I × F2 L + , 1, L − NI ; , L; , 2 2 1 + γ d + Lγ I 1 + γ d + Lγ I (11.70) which reduces to (11.13) with λ1 replaced by λ1 when NI = 1. Figures 11.8 and 11.9 illustrate Pb (E) as computed from (11.64) [or equivalently from (11.70)] versus average total SNR, Lγ d , for four- and eight-element arrays with multiple interferers and equal desired signal and interference powers. 10 Equation (83) of Haimovich and Shah [17] has a typographical error. A factor of ρ (N −r) should appear in the denominator.

706

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

gd = gI −1

10

NI = 3

NI = 2 10−2 Pb(E )

NI = 1

10−3

10−4 0

2

4

6

8

10

12

14

16

Average Total SNR, Lgd (dB) Figure 11.8 Average BEP versus average total SNR for optimum combining with Rayleigh fading and multiple interferers; L = 4.

Also included for comparison are the corresponding results for the single-interferer case, which for the four-element array are obtained from Fig. 11.3. The numerical results clearly show the increased performance penalty produced by the additional CCI. 11.1.2.2 Number of Interferers Equal to or Greater than Number of Array Elements The scenario where the number of interfering signals is no less than the number of array elements is treated in Ref. 18 for the case of Rayleigh fading on both the desired signal and the CCI, in Refs. 19 and 20 for the case of Rician fading on the desired signal and Rayleigh fading on the CCI, and in Ref. 40 for the case of Rician fading on the desired signal and Nakagami-m fading on the CCI. In particular, assuming that the system is interference-limited (hence, that thermal

707

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

gd = gI NI = 7

10−1

NI = 5

Pb(E)

10−2

NI = 3 10−3 NI = 1 10−4

10−5

10−6

10−7 0

2

4

6

8

10

12

14

16

Average Total SNR, Lgd (dB) Figure 11.9 Average BEP versus average total SNR for optimum combining with Rayleigh fading and multiple interferers; L = 8.

noise can be neglected), for the former case it was shown [18] that11 pγt (γt ) =

 (NI + 1)  (L)  (NI + 1 − L)



Pd PI

NI +1−L 

Pd PI

γtL−1 NI +1 , + γt

γt ≥ 0

(11.71) where γt now denotes the signal-to-interference ratio (SIR) at the combiner output. This PDF is an example of one that has a finite number of finite moments with all remaining higher-order moments being infinite. We will show this explicitly after 11 Shah and Haimovich [18] also note that the PDF in (11.71) does not depend on the form of the covariance matrix of the interference propagation vectors. Thus, the performance of the optimum combiner obtained from using this PDF is the same regardless of whether or not the fading at the L receiver elements is independent. It is to be emphasized, however, that this statement is true only for the equal power interferer and NI ≥ L case as is being considered here.

708

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

evaluating the MGF. For the latter case, it was shown [19,20,40] that  (NI + 1) pγt (γt ) =  (L)  (NI + 1 − L)  × 1 F1 NI + 1; L;





Pd PI

γtL−1 −LK NI +1 e + γt (11.72)



LKγt Pd PI

NI +1−L

Pd PI

+ γt

γt ≥ 0

,

Using Kummer’s transformation [10. Eq. (19.212.1)], it was shown [20] that (11.72) could be simplified to pγt (γt ) = (NI + 1 − L) ×

×

Pd PI

Pd PI

NI +1−L

"  Pd # γtL−1 PI NI +1 exp −LK Pd PI + γt + γt

NI +1−L n=0



NI  L+n−1

n!



LKγt Pd PI

+ γt

n ,

γt ≥ 0

(11.73)

Clearly, for K = 0, that is, for a Rayleigh desired user, only the n = 0 term in the summation survives, in which case (11.73) reduces to (11.71), as it should. To compute the MGF associated with (11.71), we make use of a result in Eq. (12) of Ref. 21 (p. 234) for the generalized Stieltjes integral 

∞ 0

ρ−λ λ−ρ xλ −ax d x =  (λ + 1) a 2 −1 y 2 e 1/2ay Wk,m (ay) , ρe (x + y)

λ−ρ+1 −λ − ρ ,m = k= 2 2

(11.74)

where Wk,m (z) is the Whittaker function defined by Gradshteyn and Ryzhik [10, Eq. (9.222)].12 Then, letting x = γt , a = s, λ = L − 1, ρ = NI + 1, y = Pd /PI , we 12 Note

that the condition on the integral definition of Wk,m (z) in Eq. (9.222) of Ref. 10, namely, m − k > − 21 , is satisfied since here

m−k =

L − NI − 1 1 (−L − NI ) − =L− 2 2 2

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

709

obtain after simplification  Mγt (−s) =

∞

pγt (γt ) e −sγt d γt

0

=

 (NI + 1)  (L)  (NI + 1 − L)



Pd PI

NI +1−L 

∞



0

Pd PI

γtL−1 −sγ NI +1 e t d γt + γt





Pd Pd NI −L/2  (NI + 1) 1 Pd W −L−NI , L−NI −1 s s s = exp  (NI + 1 − L) PI 2 PI PI 2 2 (11.75) To obtain the moments of γt , we must evaluate the derivatives of the MGF in (11.75) at s = 0. To obtain these, we first rewrite (11.75) using the integral definition of the Whittaker function [10, Eq. (9.222)] to give Mγt (−s) =

 (NI + 1)  (NI + 1 − L)  (L)



∞

0



Pd t L−1 exp −s t dt PI (1 + t)NI +1

(11.76)

Then % & dn E γtn = (−1)n n Mγt (−s) |s=0 ds n  ∞ Pd t n+L−1  (NI + 1) = dt  (NI + 1 − L)  (L) PI (1 + t)NI +1 0

(11.77)

The integral in (11.77) can be evaluated using Eq. (3.241.4) of Ref. 10, which is restricted to n < NI − L. When this done, the following results: % &  (L + n)  (NI + 1 − L − n) E γtn =  (NI + 1 − L)  (L)



Pd PI

n

n Pd L (L + 1) · · · (L + n − 1) = , (NI − L) (NI − L − 1) · · · (NI + 1 − L − n) PI

The first moment corresponding to n = 1 is simply given by E {γt } =

L (NI − L)



Pd PI

n < NI − L (11.78)

(11.79)

which agrees with Eq. (14) of Ref. 18 and varies linearly with L when the number of interferers is large compared to the number of antenna elements. For n ≥ NI − L, the integral in (11.77) diverges and thus, as stated above, only a finite number (NI − L − 1) of moments are finite. Assuming that the number of interferers is large and that the total interference can be modeled as a Gaussian RV [18], the conditional error probability for coherent

710

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

detection is still described by a Gaussian Q-function and as before, the average BEP of OC can be computed entirely from knowledge of the MGF; for instance, for BPSK, we have, analogous to (11.17) Pb (E) =

1 π



π/2 0

Mγt −

1 sin2 θ

dθ

(11.80)

Alternatively, a closed-form expression for the average BEP corresponding to coherent BPSK is obtained [18] by direct integration of the conditional BEP over the PDF in (11.71) with the result "

  Pd NI +1−L  L−NI − 12  (L+1) 1 1 Pb (E) = + √ 2 2 π (L)  (NI +1 − L) PI  (L − NI − 1)

3 Pd × 2 F2 NI + 1, NI + 1 − L; NI − L + , NI − L + 2; 2 PI 1/2



Pd 1 1 −2  NI − L +  L+ PI 2 2

 1 1 1 3 Pd × 2 F2 L + , ; L − N I + , ; (11.81) 2 2 2 2 PI p

q

. /0 1 . /0 1 where p Fq (·, · · · , ·; · · · , · · · , · · ·; · · ·) is the generalized hypergeometric series defined in Ref. 10, Eq. (9.14.1). Again the simplicity of (11.80) combined with (11.65) compared with (11.81) is to be observed. 11.1.3 Comparison with Results for MRC in the Presence of Interference As mentioned previously, using MRC in the presence of interference results in suboptimum performance in that it produces a smaller SINR at the combiner output than does OC. As such, it is of interest to evaluate the performance of MRC with interference and then compare the amount by which it suffers relative to that obtained with OC. On the basis of the approximate approach (replacing the eigenvalues by their statistical means), the analytical results describing the performance of MRC in the presence of interference can be directly obtained from the results for the performance of MRC in the absence of interference (see Chapter 9) by replacing the average SNR with the average SINR. For example, for Rayleigh fading, analogous to (9.6), the appropriate expression for average BEP would be 1 Pb (E) = L 2

 1−

 γ 1+γ

 L L−1 k

  1 γ L−1+k (11.82) 1+ k 2k 1+γ k=0

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

711

where now 

γ =

γd 1 + NI γ I

(11.83)

The result in (11.82) agrees with that obtained by Winters [6] for the singleinterferer case and that obtained by Villier [8] for the multiple-interferer case. We further note that (11.82) is also obtained as the limit of (11.26) (for L ≥ 2) or (11.28) (for L ≥ 1) when γ I approaches zero, in which case γ of (11.83) becomes equal to γ d . For the case were the system is interference-limited and the interference is assumed to be modeled by a Gaussian distribution (Shah and Haimovich [19] justify this assumption using histograms obtained from computer simulations), the exact approach discussed in Section 11.1.2.2 is applied to derive results for MRC performance in the presence of CCI. Specifically, for the Rayleigh/Rayleigh fading case, the PDF of the SIR γt is given by13

γtL−1  (NI + L) Pd NI pγt (γt ) = γt ≥ 0 (11.84)  NI +L ,  (L)  (NI ) PI Pd + γ t PI whereas for the Rice/Rayleigh case the simplified version of the result, analogous to (11.73), becomes [20] "  Pd # NI Pd γtL−1 PI pγt (γt ) = NI  NI +L exp −LK Pd PI Pd PI + γt PI + γt

(11.85) NI + L − 1   n NI  NI − n LKγt , γt ≥ 0 × P d n! + γt n=0

PI

which again reduces to (11.84) when K = 0. Comparing (11.85) with (11.73) or (11.84) with (11.71), we observe that they are of similar form; namely, the OC results can be obtained from the MRC results by replacing NI by NI − L + 1. Thus, the approach taken to compute the MGF in (11.75) for the Rayleigh/Rayleigh case can also be used here. Specifically, using (11.74) with now x = γt , a = s, λ = L − 1, ρ = NI + L, y = Pd /PI , we obtain, after simplification





Pd Pd (NI −1)/2 1 Pd  (NI + L) Mγt (−s) = W −2L−NI +1 , −NI s s s exp  (NI ) PI 2 PI PI 2 2 (11.86) Hence, the average BEP can be computed from (11.80) with Mγt (−s) as in (11.86). 13 Note that the PDF for MRC in (11.84) is not restricted to L ≥ NI , as was the case for (11.71) corresponding to OC.

712

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

Analogous to (11.81), a closed-form expression for Pb (E) was found by Shah and Haimovich [19] by direct integration of the conditional BEP over the PDF in (11.71) with the result 1 Pb (E) = √ 2 π (NI )  (L) × 2 F2

"

Pd PI

NI



1 2

 − NI  (NI + L)  (−NI )

1 Pd NI + L, NI ; NI + , NI + 1; 2 PI





Pd −2 PI

1/2

1  NI − 2





√ 1 1 3 1 3 Pd + π  (NI )  (L) − NI , ; × L + 2 F2 L + , ; 2 2 2 2 2 PI (11.87) Borrowing on numerical results presented in Ref. 19, Fig. 11.10 is a plot of average BEP versus SIR per channel Pd /PI , as computed from (11.81) for OC and (11.87) for MRC for two values of the number of antennas, namely, L = 3 and L = 6, and also NI = 18 interfering sources. It is observed that OC can provide improved performance over MRC even when the number of interfering sources is much larger than the number of antennas. For example, for Pb (E) = 10−3 and L = 6, OC requires 1 dB less SIR than MRC. Figure 11.11 explicitly shows the improvement of OC over MRC as a function of the average BEP for the same numbers of antennas as in Fig. 11.10 (clearly for a single antenna, the two combining schemes produce identical performance). By “improvement” in Fig. 11.11 is meant the reduction in required SIR per channel to obtain a given average BEP using OC as compared with MRC. We observe from this figure that, for a fixed number of antenna elements, the improvement is quite insensitive to the value of average BEP over a range of four decades. On the other hand, for a fixed average BEP, a noticeable improvement is obtained as the number of antenna elements increases. The extension of these results to the case of Nakagami-m fading has been considered by Aalo and Zhang [22]. In particular, letting md and mI respectively denote the Nakagami-m parameters on the desired signal and interference, then, from Ref. 22, the result for the average BEP of coherent BPSK with MRC in our notation is given by √  π 1  (md L + mI NI )  md L + 12 Pb (E) = − 2  (md L)  (mI NI ) cos (π mI NI ) " mI NI Pd 1 1 ×     2mI NI PI  md L + 12  mI NI + 12

1 Pd × 2 F2 mI NI , md L + mI NI ; mI NI + 1, mI NI + ; 2 PI

(11.88)

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

713

100 10−1

Average probability of bit error

L=3 10−2

L=6

10−3 10−4

10−5

10−6

10−7

Optimum Combining Maximal Ratio Comb. 0

2

4

6

8

10

12

14

16

18

20

Pd /PI (dB)

Figure 11.10 Average BEP versus channel SIR for Rayleigh fading; NI = 18 (courtesy of Shah and Haimovich [19]).

1 Pd 1/2   PI  (md L + mI NI )  32 − mI NI

 1 1 3 3 Pd , md L + ; , − mI NI ; × 2 F2 2 2 2 2 PI

−

An identically equivalent result can be obtained from (11.87) by replacing L with md L and NI with mI NI . Using this same replacement in (11.81) gives the analogous result for OC: "

Pd mI NI +1−md L 1 1 Pb (E) = + √ 2 2 π (md L)  (mI NI + 1 − md L) PI    md L − mI NI − 12  (L + 1) ×  (md L − mI NI − 1) × 2 F2 mI NI + 1, mI NI + 1 − md L; mI NI (11.89)

1/2

Pd 3 Pd 1 −2 −md L + , mI NI − md L + 2;  mI NI − md L + 2 PI PI 2

#

1 1 1 1 3 Pd ×  md L + 2 F2 m d L + , ; m d L − m I N I + , ; 2 2 2 2 2 PI

714

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

1.6

L=6

1.4

Improvement (dB)

1.2

1

0.8

0.6

L=3

0.4

0.2

L=1 0

10−4

10−3

10−2

10−1

Average probability of bit error Figure 11.11 Improvement of OC over MRC in Rayleigh fading; NI = 18 (courtesy of Shah and Haimovich [19]).

More recently, Aalo and Zhang [23] extended the result in (11.88) to the case where the additive Gaussian noise cannot be neglected. Specifically, for integer values of md the following result is obtained " √



md L−1 k 1−µ  1 2k  µ k  k  (mI NI + 1) Pb (E) = 1− m N k i 2 µ I I 4  (mI NI ) µi k=0 i=0

 1 mI × mI NI + 1, mI NI + 1 − k + ; (11.90) 2 µγ I where 

µ=

md γ d + md

(11.91)

and  (a, b; x) is the confluent hypergeometric function of the second kind, which is related to the confluent hypergeometric function of the first kind 1 F1 (a; b; x) in Eq. (9.210.2) of Ref. 10.

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

715

11.1.4 Multiple Arbitrary Power Interferers; Independent, Identically Distributed Fading In Section 11.1.2, we restricted ourselves to the case where the interferers were assumed to have equal power. The extension to the case of arbitrary power levels for the interferers has been considered in Ref. 24. For M-PSK modulation, the average SEP is obtained in the form of a single integral with finite limits and bounded integrand, whereas for BPSK, the average BEP is obtained in closed form. In what follows, the results are presented in summary form. Define the power of the nth interferer by PI n . Then, the reliability function R (γt ) of the output SINR, defined as the complement of the CDF of γt , can be written for OC as [25]  m−1 I L−N γt 1  R (γt ) = 1 − P (γt ) =    (m) γ d m=1

+

*L−m

L 

i 1 i=0 Ci γt  NI   (m) , m=max(L−NI +1,1) γt + PPIdn





m−1 

γt γt   exp −  γd γd

n=1

(11.92) where Ci is a coefficient determined from the expansion n=1 [γt + (Pd /PI n )] = *NI i i=0 Ci γt , which by mathematical induction can be proved to be given by ,NI

Ci =



NI  Pd n i

n1 +n2 +...+nNI =NI −i,ni ∈{0,1} i=1

PI i

(11.93)

11.1.4.1 Average SEP of M-PSK For OC with M-PSK the average SEP is obtained by averaging (9.14) over the PDF of γt which is obtained from (11.92) as pγt (γt ) = −dR (γt ) dγt . Using the method of integration by parts and performing some straightforward manipulations and change of variables, the final result is given by Ps (E) =

π  π/2     π M − 1 sin M − √ tan2 φ R tan2 φ exp − sin2 M M π 0   π sec2 φ d φ × erfc − tan φ cos M

(11.94)

It should be noted that even though the argument of R (·) could become infinite, the range of R (·) and thus the integrand are finite. For the special case of BPSK (i.e., M = 2), the result in (11.94) can indeed be obtained in closed form [24]; however, the resulting expression involves, at the

716

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

very least, quadruple sums and, in the opinion of the authors of this book, is no more computationally convenient than (11.94), which already simplifies to  π/2     1 1 R tan2 φ exp − tan2 φ sec2 φ d φ (11.95) Pb (E) = − √ 2 π 0 11.1.4.2 Numerical Results Figure 11.12 is a plot of average SEP versus average desired signal SNR γ d , as determined from (11.94) for L = 8 antenna elements and NI = 6 interferers and several different values of modulation order M. The SIRs (Pd /PIn ), for the interferers are chosen equal to 10, 10, 2, 2, 0, and 0 dB, respectively. Figure 11.13 is an alternative presentation of these results whereby the average SEP is plotted versus the number of antenna elements L for NI = 32 interferers and a fixed average SNR γ d = 10 dB. The SIRs for 16 of the interferers were chosen equal to 0 dB while the SIRs for the remaining 16 interferers were chosen equal to 2 dB. Also included in Figs. 11.12 and 11.13 are simulation results, which agree exactly with those obtained from the theory. Figure 11.14 illustrates the average SEP versus SIR (assumed here to be equal for all interferers) for L = 4 antenna elements, γ d = 10 dB, and various numbers of interferers. Both the desired signal and all interferers are QPSK modulations. The results reveal that although the interferers are in fact not Gaussian distributed 100

Average SEP

10−1

10−2

10−3

16-PSK (Simulation) 16-PSK (Analysis) 8-PSK (Simulation) 8-PSK (Analysis) QPSK (Simulation) QPSK (Analysis) BPSK (Simulation) BPSK (Analysis)

10−4

10−5

0

1

2

3

4

5

6

7

8

9

10

Average SNR per symbol (dB) Figure 11.12 Average SEP versus average symbol SNR for L = 8 antenna elements and NI = 6 interferers (courtesy of Lao and Haimovich [24]).

717

PERFORMANCE OF DIVERSITY COMBINING RECEIVERS

Average SEP

10−1

10−2

10−3

16-PSK (Simulation) 16-PSK (Analysis) 8-PSK (Simulation) 8-PSK (Analysis) QPSK (Simulation) QPSK (Analysis) BPSK (Simulation) BPSK (Analysis)

10−4

10−5 5

10

15

20

25

30

L (Number of antenna elements)

Figure 11.13 Average SEP versus the number of antenna elements L for NI = 32 interferers and average symbol SNR L = 10 dB (courtesy of Lao and Haimovich [24]).

100

NI = 4 (Analysis) NI = 4 (Simulation) NI = 3 (Analysis) NI = 3 (Simulation) NI = 2 (Analysis) NI = 2 (Simulation) NI = 1 (Analysis) NI = 1 (Simulation)

Average SEP

10−1

10−2

10−3

10−4 −10

−5

0

5

10 SIR (dB)

15

20

25

30

Figure 11.14 Average SEP versus SIR for QPSK modulation; L = 4 antenna elements and average symbol SNR = 10 dB (courtesy of Lao and Haimovich [24]).

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OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

random signals, the assumption of such made in the analysis provides results that are still very close to those obtained by simulation regardless of the number of interferers and SIR levels. A similar conclusion was drawn by Lao and Haimovich [26]. Before moving on to a different aspect of the subject, we mention in passing some other contributions that are pertinent to the material described in Sections 11.1.2 and 11.1.4. By expressing the OC decision statistic as a sum of quadratic forms of Gaussian RVs, Lao and Haimovich [26] obtain exact closedform expressions for the BEP for OC of M-PSK and BPSK in the presence of AWGN and multiple, equal power interferers. As considered above, the aggregate interference plus noise is assumed to be Gaussian, and both the desired signal and interference are assumed to be subjected to Rayleigh fading. Previous to this disclosure, the same problem was analytically solved in the form of approximate expressions [8] or exact expressions with integral forms [27,28]. 11.1.5 Multiple-Symbol Differential Detection in the Presence of Interference In Section 9.4.3 we studied the behavior of multiple-symbol differential detection (MSDD) of M-PSK with MRC in the presence of slow fading and AWGN. Here we extend the results given there to include the presence of a single interfering source. The covariance matrix of the interference plus noise Rni is assumed known and constant over the observation interval. The MSDD decision metric is derived according to the principle of MLSE. Following this we present a closed-form result for the pairwise error probability (PEP) that can then be used to compare performance with that of OC of M-PSK with differential encoding. 11.1.5.1 Decision Metric Using the formulation as in Section 11.1.1, it is shown in [29] that the optimum decision metric (to be maximized) for Rayleigh fading and an observation interval of Ns symbols is, analogous to (9.155), given by (for simplicity, we assume a uniform additive noise profile) η=

L  l=1

 2 N s −2   * s −i−2 γ dl   −j N β k m=0 i−m       xn−Ns +1,l + x e n−i,l  λl /σ 2 λl /σ 2 /Ns + γ dl  i=0

(11.96) where xn−i = U H rn−i with U the eigenvector matrix associated with the noiseplus-interference covariance matrix and λl , l = 1, 2, . . . , L the corresponding set of eigenvalues. Thus, for OC, the decision metric becomes a weighted sum of terms that are squared magnitudes of correlations of whitened observations with the hypothesized transmitted symbols. 11.1.5.2 Average BEP Consider now the special case of a uniform power profile for the channels associated with the desired signal, namely, γ dl = γ d , l = 1, 2, . . . , L profile,% and &further normalize the interference channel gain vector such that E cH I cI =

OPTIMUM COMBINING WITH MULTIPLE TRANSMIT AND RECEIVE ANTENNAS

719

* 2 E{ L l=1 |cI l | } = L. Then, following a discussion similar to that given in Sections 9.4.3.2 and 9.4.3.3, the average BEP is approximately upper-bounded by  (9.165) with P 12 (E) |δ|=|δmax | (for Ns > 2) given by [29]  P 12 (E) |δ|=|δmax | =

where

µ2L−1 2 (µ1 − µ2 ) (µ3 − µ2 )L−1 (µ4 − µ2 )L−1 L−1 L−2   (L − 1 + k) µ4 1 − (−1)L µ3  (L − 1)  (k + 1) k=0 " µL−1−k µ1 1 × 1− µ1 − µ2 (µ1 − µ4 )L−1−k #  L−1+k µ µL−1−k µ2 3 2 + µ1 − µ2 (µ2 − µ4 )L−1−k (µ3 − µ4 )L−1+k

 +     ζ P ± ζ 2 Pd2 + 4 Ns Pd + λ1 ζ λ1 d      ,   2λ1 Ns Pd + λ1 µi = +     ± ζ 2 Pd2 + 4 Ns Pd + λ2 ζ λ2 ζ P  d      ,  2λ2 Ns Pd + λ2

(11.97)

i = 1, 2 (11.98) i = 3, 4



with ζ = Ns2 − |δ|2max and in accordance with the assumptions applied above to (11.8), λ1 = LPI + σ 2 ,

λ2 = λ2 = σ 2

(11.99)

For small SIR (i.e., Pd /PI 1) and large SNR (i.e., Pd /σ 2  1), it was shown [29] that the ratio of the average BEP for OC to the approximate upper bound of MSDD (for Ns > 2) is given by

Pb (E) |OC 1 L−1 = 1− (11.100) Pb (E) |MSDD Ns Thus, it follows that when the observation interval becomes large, the performance of OC of M-PSK with differential encoding approaches that of MSDD with MRC. Figure 11.15 plots the average BEP of DPSK versus SNR for an SIR Pd /PI = −6 dB and several values of observation interval (in symbols) Ns . Also shown are simulations results for these same cases that can be seen to agree quite well with the analytical results. We can also observe, as mentioned above, that for Ns = 40 the results are indeed quite close to those for OC with differential encoding. Figure 11.16 plots the same average BEP versus SIR for an SNR Pd /σ 2 = 10 dB and two values of Ns . Here we see that over a wide range of SIRs, for Ns = 40, the results are again quite close to those for OC with differential encoding.

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OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

Average Bit Error Probability Pb(E)

100

MSDD: Ns = 2 (Analysis) MSDD: Ns = 2 (Simulation) MSDD: Ns = 7 (Analysis) MSDD: Ns = 7 (Simulation) MSDD: Ns = 40 (Analysis) MSDD: Ns = 40 (Simulation) OC: (Simulation)

10−1

10−2

10−3

10−4

0

1

2

3

4

5

6

7

8

9

Average SNR per bit (dB)

Figure 11.15 Average BEP versus SIR for DPSK modulation; L = 4 antenna elements and SIR = −6 dB (courtesy of Lao and Haimovich [29]).

Average Bit Error Probability Pb(E)

10−3

MSDD: Ns = 2 MSDD: Ns = 40 OC

10−4

10−5 −50

−40

−30

−20

−10

0

10

20

30

40

50

SIR (dB)

Figure 11.16 Average BEP versus SIR for DPSK modulation; L = 4 antenna elements and average bit SNR = 10 dB (courtesy of Lao and Haimovich [29]).

OPTIMUM COMBINING WITH MULTIPLE TRANSMIT AND RECEIVE ANTENNAS

721

11.2 OPTIMUM COMBINING WITH MULTIPLE TRANSMIT AND RECEIVE ANTENNAS Whereas initial work on optimum combining in the presence of co-channel interference dealt mainly with a single antenna at the transmitter and multiple antennas at the receiver, applications in more recent years have become increasingly sophisticated thereby relying on the more general multiple-input/multiple-output (MIMO) antenna systems [30–32], which promise significant increases in system performance and capacity. In this context, Wong et al. [30,32] derived the joint optimal antenna weights at both mobile unit and base station over interference-limited fading channels and evaluated the average BER performance of the resulting optimized MIMO system by Monte Carlo simulations. In this section, we summarize work [33,34] on an interesting connection between results in the statistical literature dealing with the distribution of the largest eigenvalue of certain quadratic forms in complex Gaussian vectors [35] and the performance analysis of the optimized MIMO system proposed by Wong et al. [30,32]. We adopt the following assumptions: (1) both the desired and interfering signals are subject to Rayleigh fading, (2) the interferers have equal average power, and (3) the effect of thermal noise is neglected, which is reasonable for interference-limited systems in which the number of antenna elements Lr at the receiver is less than or equal to the number of interferers NI . Extensions of the work presented here to Rician fading for the desired user, unequal power interferers, and the presence of interferers is available in Refs. 34 and 36. 11.2.1

System, Channel, and Signals Models

We consider a TDMA system equipped with Lt antenna elements at the transmitter and Lr antenna elements at the receiver as shown in Fig. 11.17. It is assumed that there exist a total of NI co-channel interferers from neighboring cells and that the system is interference-limited and therefore the effect of thermal noise can be neglected. The discrete equivalent Lr × 1 received vector at the receiver can be modeled as r = sD HD wt +

NI 


I sI ,n HI ,n

(11.101)

n=1

where sD is the transmitted signal of the desired user and sI ,n is the transmitted signal of the nth interferer. Without loss of generality, we assume that sD and sI ,n have unit average power. In (11.101), wt represents the weight vector at the transmitter with wt 2 = D (i.e, the power of the vector wt is restricted to be D ) and I denotes the short-term average power of the NI co-channel interferers.14 In we have normalized sD and sI ,n to unit average power, then the average signal power PD = D and the average power of the interferers PI = I . 14 Since

722

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

wt,1

Branch 1 r1

Branch 1 X

SD

wt,2 X

X

X

wr,2 *

Branch 2 r2

Branch 2

wt,Lt

wr,1 *

+

X

wr,Lr *

Branch Lr rLr

Branch Lt

Output

X

Figure 11.17 Block diagram of a TDMA system equipped with Lt antenna elements at the transmitter and Lr antenna elements at the receiver.

(11.101), HD is the channel gain matrix for the desire user defined by    HD =  

hD,1,1 hD,2,1 .. .

hD,1,2 hD,2,2 .. .

··· ··· .. .

hD,Lr ,1

hD,Lr ,2

· · · hD,Lr ,Lt



hD,1,Lt hD,2,Lt .. .

   

(11.102) Lr ×Lt

where hD,i,j denotes the complex channel gain for the desired user from the j th transmitter  antenna element to the T ith receiver antenna element. On the other hand, HI ,n = hI ,1,n , hI ,2,n , . . . , hI ,Lr ,n, is the complex channel vector of the nth interferer at the receiver. Hence the channel vectors of the NI co-channel interferers can be written in matrix form as    HI =  

hI ,1,1 hI ,2,1 .. .

hI ,1,2 hI ,2,2 .. .

··· ··· .. .

hI ,Lr ,1

hI ,Lr ,2

· · · hI ,Lr ,NI

hI ,1,NI hI ,2,NI .. .

    

(11.103)

Lr ×NI

Finally, (11.101) can be rewritten as r = sD HD wt + where sI = (sI ,1 , sI ,2 , . . . , sI ,NI )T .


I HI sI

(11.104)

OPTIMUM COMBINING WITH MULTIPLE TRANSMIT AND RECEIVE ANTENNAS

11.2.2

723

Optimum Weight Vectors and Output SIR

The optimum combining vector at the receiver (given the transmitting weight vector wt ) is well known to be given by wr = αR−1 HD wt

(11.105)

where α is a constant that does not affect the output SIR and R = HI HH I . The resulting conditional (on wt ) maximum SIR is given by γt =

1 H H −1 w H R HD wt I t D

(11.106)

According to the Rayleigh–Ritz theorem [37, Sect. 4.2.2], for any nonzero N × 1 complex vector x and a given N × N Hermitian matrix A, 0 < xH Ax ≤ x 2 λmax , where λmax is the largest eigenvalue of A and · denotes the norm. The equality holds if and only if x is along the direction of the eigenvector corresponding to λmax . On the basis of this fact, we may choose the transmitting weight vector as
wt = D Umax (11.107) where Umax ( Umax = 1) denotes the eigenvector corresponding to the largest eigenvalue of the quadratic form: −1 F = HH D R HD   H −1 = HH HD D HI HI

(11.108)

It is obvious from its definition that R is a Hermitian matrix and is positive-definite with probability 1 when NI ≥ R (hence R is invertible with probability 1). Thus, F is a Hermitian matrix. Therefore the maximum output SIR is given by γt =

D λmax I

(11.109)

where λmax is the largest eigenvalue of the matrix F defined in (11.108). 11.2.3

PDF of Output SIR and Outage Probability

When the desired user and co-channel interferers are all subject to Rayleigh fading, HD :Lr × Lt and HI :Lr × NI are independent matrices whose columns are i.i.d. complex multivariate normal vectors with zero-mean and covariance matrix . Hence the matrix F in (11.108) is exactly in the form of the matrix F2 in (11A.4) which we are characterizing statistically in Theorem 11.2 and Corollary 11.1 in Appendix 11A.

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OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

11.2.3.1 PDF of Output SIR If we let s = min(Lt , Lr ), t = max(Lt , Lr ), and r = min(NI , NI + Lt − Lr ), then by Corollary 11.1, the PDF of the output SIR γt in (11.109) when NI ≥ Lr is given by pγt (γt ) =

s 

(r + t − i + 1) (r − i + 1)(s − i + 1)(t − i + 1) i=1 

   I I γt (I γt /D )t−s   × β  r+t−2s+2 D (1 + I γt /D ) D + I γt 



I γt −1 γs u(γt ), (11.110) × tr β D + I γt

where 

I γt D + I γt

1  

2   I γt I γt  γs = D + I γt  D + I γt  .. ..  . . 

s−1

s  I γt I γt D + I γt D + I γt

s−1 I γt D + I γt

s I γt D + I γt .. .

2s−2 I γt D + I γt

··· ··· ..

.

···

            s×s

(11.111) and β[I γt /(D + I γt )] is an s × s matrix defined in the same way as in Theorem 11.2. As a double check of (11.110), Fig. 11.18 is a comparison between the theoretical PDF of the output SIR in (11.110) and Monte Carlo simulations when Lt = 5, Lr = 4, NI = 6, and D /I = 5. It is evident that there is a very good match between the analytical and simulation results. 11.2.3.2 Outage Probability The outage probability is an important statistical measure to assess the quality of service provided by the system. It is defined as the probability of failing to achieve a specified SIR value γth sufficient for satisfactory reception. Therefore, the outage probability is simply the CDF of the output SIR evaluated at γth . By Theorem 11.2, the outage probability when NI ≥ Lr is given by Pout = Pr{γt ≤ γth } =

s 

(r + t − i + 1) (r − i + 1)(s − i + 1)(t − i + 1) i=1 

   I γth   × β D + I γth 

(11.112)

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OPTIMUM COMBINING WITH MULTIPLE TRANSMIT AND RECEIVE ANTENNAS

0.03 Theoretical PDF of m Monte-Carlo Simulations

Probability Density Function of m

0.025

0.02

0.015

0.01

0.005

0 0

10

20

30

40

50

60

70

80

90

100

Output SIR m

Figure 11.18 Comparison between the theoretical PDF of the output SIR and Monte Carlo simulations when Lt = 5, Lr = 4, NI = 6, and D /I = 5.

11.2.3.3 Special Case When Lt = 1 Now we shall show that for the special case when Lt = 1 and 1 ≤ Lr ≤ NI (the traditional receiver diversity), the results for the PDF of the output SIR and for the outage probability reduce to those given by Shah and Haimovich [18]. Since Lr ≥ Lt = 1 , then s = 1, t = Lr , and r = NI + 1 − Lr . Note that now in (11.110), both

I γt = β I γt (Lr , NI − Lr + 1) β D +I γt D + I γt and γs = 1 reduce to scalars. Therefore 





  I γt I γt  tr β −1 β γs = 1  D + I γt  D + I γt and, after simplification, the PDF of SIR pγt (γt ) becomes (NI + 1) pγt (γt ) = (NI − Lr + 1)(Lr ) ×



D I

NI −Lr +1

γtLr −1 NI +1 u(γt ), 1 ≤ Lr ≤ NI D + γ t I

(11.113)

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OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

which, substituting PD for D and PI for I , is in perfect agreement with the PDF result in (11.71). On the other hand, the outage probability Pout in (11.112) now reduces for Lt = 1 to Pout =

(NI + 1) (NI − Lr + 1)(Lr ) ×β

I γth D +I γth

(Lr , NI − Lr + 1)

(11.114)

Applying the identity Eq. (8.391) in Ref. 10 relating the incomplete beta function to the Gaussian hypergeometric function 2 F1 (·, ·; ·; ·) [10, Sect. 9.1] in (11.114), Pout can be rewritten as  Lr I γth D +I γth (NI + 1) Pout = (NI − Lr + 1)(Lr ) Lr

I γth (11.115) × 2 F1 Lr , Lr − NI ; Lr + 1; D + I γth Using a Gaussian hypergeometric function identity [10, Eq. (9.131.1)] in (11.115), Pout can be finally written as

I γth Lr (NI + 1) Pout = (NI − Lr + 1)(Lr + 1) D

I γth × 2 F1 Lr , NI + 1; Lr + 1; − (11.116) D which again is in perfect agreement with the outage probability result in Eq. (16) of Ref. 18. 11.2.4

Key Observations

11.2.4.1 Distribution of Antenna Elements An interesting observation can be made from the definitions of s, t, and r, that is, if Lt ≥ Lr , a system with Lt transmitter antenna elements, Lr receiver antenna elements, and NI co-channel interferers is equivalent to a system with Lr transmitter antenna elements, Lt receiver antenna elements, and NI − (Lt − Lr ) cochannel interferers. Generally, this implies that a system with a single transmitter antenna element and multiple receiver antenna elements (SIMO) will always outperform the one with multiple transmitter antenna elements and a single receiver antenna element (MISO) for fixed numbers of total antenna elements and cochannel interferers. 11.2.4.2 Effects of Correlation between Receiver Antenna Pairs Another interesting observation is that when the desired user and co-channel interferers have equal nonsingular correlation matrix , then the structure of the matrix has no effect on the output SIR, that is, the correlation between receiver antennas does not affect the system performance.

OPTIMUM COMBINING WITH MULTIPLE TRANSMIT AND RECEIVE ANTENNAS

11.2.5

727

Numerical Examples

Figure 11.19 shows the effect of the number of transmitter antenna elements Lt on the PDF of the output SIR when Lr = 2, NI = 6, and D /I = 5. It indicates that increasing the number of transmitter antenna elements will increase the chance of taking on a larger output SIR and therefore improve the performance of the system. Figure 11.20 plots the outage probability Pout versus normalized threshold D /I γth in decibels with the number of transmitter antenna elements Lt as a parameter when Lr = 3 and NI = 6, which again confirms that a lower outage probability, namely, that better performance, can be achieved by increasing the number of transmitter antenna elements. Figure 11.20 also verifies the correctness of the analytical result on the outage probability by Monte Carlo simulations. Figure 11.21 plots the outage probability Pout versus normalized threshold D /I γth in decibels with the number of co-channel interferers NI as a parameter when Lt = 4 and Lr = 3. It shows that a larger number of co-channel interferers degrades the system performance by increasing the outage probability. The effect of different distribution schemes of a fixed number of antenna elements between the transmitter and the receiver on the outage probability Pout is shown in Fig. 11.22. Note that this figure confirms the observation made in Section 11.2.4.1, for example, the scheme with Lt = 5, Lr = 1 (or Lt = 4, Lr = 2) is always worse

0.35 Lt = 1, Lr = 2, NI = 6 Lt = 2, Lr = 2, NI = 6 Lt = 3, Lr = 2, NI = 6 Lt = 4, Lr = 2, NI = 6

Probability Density Function of gt

0.3

0.25

0.2

0.15

0.1

0.05

0

0

2

4

6

8

10

12

14

16

18

20

Output SIR gt Figure 11.19 Effect of the number of transmitter antenna elements Lt on the PDF of the output SIR when Lr = 2, NI = 6, and D /I = 5.

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OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

100 Lt = 1, Lr = 3, NI = 6 Lt = 2, Lr = 3, NI = 6 Lt = 3, Lr = 3, NI = 6 Lt = 4, Lr = 3, NI = 6 Monte Carlo, Lt = 4, Lr = 3, NI = 6 Monte Carlo, Lt = 2, Lr = 3, NI = 6

Outage Probability Pout

10−1

10−2

10−3

10−4

10−5

10−6

0

2

4

6

8

10

12

14

16

18

20

Normalized Threshold ΩD /(ΩI gth) in dB

Figure 11.20 Outage probability Pout versus normalized threshold D /I γth in decibels with the number of transmitter antenna elements Lt as a parameter with Monte Carlo simulations when Lr = 3 and NI = 6. 100

Lt Lt Lt Lt

Outage Probability Pout

10−1

= 4, Lr = 3, NI = 12 = 4, Lr = 3, NI = 10 = 4, Lr = 3, NI = 8 = 4, Lr = 3, NI = 6

10−2

10−3

10−4

10−5

10−6

0

1

2

3

4

5

6

7

8

9

10

Normalized Threshold ΩD /(ΩIgth) in dB

Figure 11.21 Outage probability Pout versus normalized threshold D /I γth in decibels with the number of co-channel interferers NI as a parameter when Lt = 4 and Lr = 3.

REFERENCES

100

Lt = 5, Lr = 1, NI = 12 Lt = 1, Lr = 5, NI = 12 Lt = 4, Lr = 2, NI = 12 Lt = 2, Lr = 4, NI = 12 Lt = 3, Lr = 3, NI = 12

10−1

Outage Probability Pout

729

10−2

10−3

10−4

10−5

10−6

0

2

4

6

8

10

12

14

16

18

20

Normalized Threshold ΩD /(ΩIgth) in dB

Figure 11.22 Outage probability Pout versus normalized threshold D /I γth in decibels with different distribution schemes of the number of antenna elements between the transmitter and the receiver when Lt + Lr = 6 and NI = 12.

than the one with Lt = 1, Lr = 5 (or Lt = 2, Lr = 4) for a fixed number of cochannel interferers. Another important observation from Fig. 11.22 is that it seems preferable to distribute the number of antenna elements evenly between the transmitter and the receiver (i.e., Lt = 3 and Lr = 3) for minimum outage probability given a total number of transmitter and receiver antenna elements. It is interesting to note that this is similar to the conclusion obtained via Monte Carlo simulations for the minimum average BER criterion [32, Fig. 3].

REFERENCES 1. W. C. Jakes, Microwave Mobile Communications. New York, NY: John Wiley, 1974. 2. B. Widrow, P. E. Mantey, L. J. Griffiths, and B. B. Goode, “Adaptive antenna systems,” Proc. IEEE, vol. 55, December 1967, p. 2143. 3. S. R. Applebaum, “Adaptive antenna systems,” IEEE Trans. Antennas Propag., vol. AP24, September 1976, p. 585. 4. R. A. Monzingo and T. W. Miller, Introduction to Adaptive Arrays. New York, NY: John Wiley, 1980. 5. V. M. Bogachev and I. G. Kiselev, “Optimum combining of signals in space-diversity reception,” Telecommun. Radio Eng., vol. 34/35, no. 10, October 1980, pp. 83–85.

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6. J. H. Winters, “Optimum combining in digital mobile radio with co-channel interference,” IEEE Trans. Veh. Technol., vol. VT-33, August 1984, pp. 144–155. 7. A. Shah, A. M. Haimovich, M. K. Simon, and M.-S. Alouini, “Exact bit error probability for optimum combining with a Rayleigh fading co-channel interferer,” IEEE Trans. Commun., vol. 48, no. 6, June 2000, pp. 908–912. 8. E. Villier, “Performance analysis of optimum combining with multiple interferers in flat Rayleigh fading,” IEEE Trans. Commun., vol 47, no. 10, October 1999, pp. 1503–1510. 9. J. V. DiFranco and W. L. Rubin, Radar Detection. Englewood Cliffs, NJ: Prentice-Hall, 1968. 10. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1994. 11. V. A. Aalo and J. Zhang, “Performance of antenna array systems with optimum combining in a Rayleigh fading environment,” IEEE Commun. Lett., vol. 4, no. 4, April 2000, pp. 125–127. 12. J. Cui, D. D. Falconer, and A. U. H. Sheikh, “Analysis of BER for optimum combining with two co-channel interferers and maximal ratio combining with arbitrary number of interferers,” Proc. PIMRC ‘96, October 1996, pp. 53–57. 13. J. Cui, D. D. Falconer, and A. U. H. Sheikh, “Performance evaluation of optimum combining and maximal ratio combining in the presence of co-channel interference and channel correlation for wireless communication systems,” Mobile Networks Appl., vol. 2, 1997, pp. 315–324. 14. G. J. Foschini and J. Salz, “Digital communications over fading radio channels,” Bell Sys. Tech. J., vol. 62, February 1983, pp. 429–456. 15. J. H. Winters and J. Salz, “Upper bounds on the bit error rate of optimum combining in wireless systems,” Proc. Veh. Technol. Conf., 1994, pp. 942–946; see also IEEE Trans. Commun., vol. 46, no. 12, December 1998, pp. 1619–1624. 16. J. H. Winters, J. Salz, and R. Gitlin, “The impact of antenna diversity on the capacity of wireless communication systems,” IEEE Trans. Commun., vol. 42, no. 2/3/4, February/March/April 1994, pp. 1740–1751. 17. A. M. Haimovich and A. Shah, “The performance of space-time processing for suppressing narrowband interference in CDMA communications,” Wireless Personal Commun., vol. 7, August 1998, pp. 233–255. 18. A. Shah and A. M. Haimovich, “Performance analysis of optimum combining in wireless communications with Rayleigh fading and co-channel interference,” IEEE Trans. Commun., vol. 46, no. 4, April 1998, pp. 473–479. 19. A. Shah and A. M. Haimovich, “Performance analysis of maximal ratio combining and comparison with optimum combining for mobile radio communications with co-channel interference,” IEEE Trans. Veh. Technol., vol. 49, no. 7, July 2000, pp. 1454–1463. 20. C. Chayawan and V. A. Aalo, “On the outage probability of optimum combining and maximal ratio combining schemes in an interference-limited Rice fading channel,” IEEE Trans. Commun., vol. 50, no. 4, April 2002, pp. 532–535. 21. A. Erdelyi et al., Table of Integral Transforms, vol. 2, New York, NY: McGraw-Hill, 1954. 22. V. A. Aalo and J. Zhang, “On the effect of co-channel interference on average error rates in Nakagami-fading channels,” IEEE Commun. Lett., vol. 3, no. 5, May 1999, pp. 136–138.

REFERENCES

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23. V. A. Aalo and J. Zhang, “On the effect of co-channel interference on average error rates in Nakagami-fading channels,” IEEE Trans. Veh. Technol., vol. 50, no. 2, March 2001, pp. 497–503. 24. D. Lao and A. Haimovich, “Exact average symbol error probability of optimum combining with arbitrary interference power,” IEEE Commun. Lett. vol. 8, no. 4, April 2004, pp. 226–228. 25. H. Gao, P. J. Smith, and M. V. Clark, “Theoretical reliability of MMSE linear diversity combining in Rayleigh-fading additive interference channels,” IEEE Trans. Commun., vol. 46, no. 5, May 1998, pp. 666–672. 26. D. Lao and A. Haimovich, “Exact closed-form performance analysis of optimum combining with multiple co-channel interferers and Rayleigh fading,” IEEE Trans. Commun., vol. 51, no. 6, June 2003, pp. 995–1003. 27. M. Chiani, M. Z. Win, A. Zanella, and J. H. Winters, “Exact symbol error probability for optimum combining in the presence of multiple co-channel interferers and thermal noise,” Proc. Global Telecommunications Conf., vol. 2, 2001, pp. 1182–1186. 28. D. Lao and A. Haimovich, “New error probability expressions for optimum combining with MPSK modulation,” Proc. IEEE Veh. Technol. Conf. (VTC2002-Fall ), vol. 4, September 2002, pp. 2293–2297. 29. D. Lao and A. Haimovich, “Multiple-symbol differential detection with interference suppression,” IEEE Trans. Commun., vol. 51, no. 2, February 2003, pp. 208–217. 30. K.-K. Wong, K. B. Letaief, and R. D. Murch, “Investigating the performance of smart antenna system at the mobile and base stations in the down and uplinks,” Proc. IEEE VTC’98, Ottawa, Ontario, Canada, May 1998, pp. 880–884. 31. I.-T. Lu and J.-S. Choi, “Space-time processing for broadband multi-channel communication systems using smart antennas at both transmitter and receiver,” Proc. IEEE Int. Conf. Communication (ICC’99), Vancouver, British Columbia, Canada, June 1999, pp. 1–5. 32. K.-K. Wong, R. S. K. Cheng, K. B. Letaief, and R. D. Murch, “Adaptive antennas at the mobile and base stations in an OFDM/TDMA system,” IEEE Trans. Commun., vol. 49, January 2001, pp. 195–206. 33. M. Kang and M.-S. Alouini, “Performance analysis of MIMO systems with co-channel interference over Rayleigh fading channels,” Proc. IEEE Int. Conf. Communications (ICC’2002), New York, NY, April 2002, pp. 391–395; full journal version to appear in a future issue of IEEE Trans. Wireless Commun. 34. M. Kang and M.-S. Alouini, “Quadratic forms in complex Gaussian matrices and performance analysis of MIMO systems with co-channel interference,” Proc. IEEE Int. Symp. Information Theory (ISIT’2002), Lausanne, Switzerland, June 2002; full journal version in IEEE Trans. Wireless Commun., vol. 3, no. 2, March 2004, pp. 418–431. 35. C. G. Khatri, “Distribution of the largest or the smallest characteristic root under null hyperthesis concerning complex multivariate normal populations,” Annals. Math. Stat., vol. 35, December 1964, pp. 1807–1810. 36. M. Kang, L. Yang, and M.-S. Alouini, “Performance analysis of MIMO systems in presence of co-channel interference and additive Gaussian noise,” 37th Annual Conf. Information Sciences and Systems (CISS’03), Johns Hopkins Univ., Baltimore, MD, March 2003. 37. R. A. Horn and C. R. Johnson, Matrix Analysis. New York, NY: Cambridge University Press, 1985.

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38. C. G. Khatri, “Classical statistical analysis based on a certain multivariate complex Gaussian distribution,” Annals Math. Stat., vol. 36, February 1965, pp. 98–114. 39. M. A. Golberg, “The derivative of a determinant,” Am. Math. Monthly, vol. 79, December 1972, pp. 1124–1126. 40. M. Kang, M.-S. Alouini, and L. Yang, “Outage probability and spectrum efficiency for cellular mobile systems with smart antennas,” IEEE Trans. Commun., vol. 51, no. 12, December 2002, pp. 1871–1877.

APPENDIX 11A. DISTRIBUTIONS OF THE LARGEST EIGENVALUE OF CERTAIN QUADRATIC FORMS IN COMPLEX GAUSSIAN VECTORS 11A.1

General Result

The distributions concerning the eigenvalues of certain Hermitian matrices derived from complex Gaussian vectors have long been studied by statisticians in the context of classical multivariate analysis [e.g., 35,38]. In this chapter, we are specially interested in the distribution of the largest eigenvalue of the q × q matrix  −1 XH YYH X, where X: p × q and Y: p × n are independent matrices whose columns are independent identically distributed (i.i.d.) p-variate complex normal vectors with zero mean and covariance matrix , and where (·)H denotes the Hermitian conjugate transpose operator. First, we shall cite some background material from the statistical literature in the form of a theorem due to Khatri. We then deduce the statistics of the largest eigenvalue for our communication application of interest. Theorem 11.1: (From Khatri [35]) Let X: p × q and Y: p × n , q ≥ p and n ≥ p, be independent matrices whose columns are i.i.d. p-variate complex normal vectors with zero mean and covariance matrix , and 0 < w1 ≤ w2 ≤ · · · ≤ wp ≤ 1 be the p nonzero eigenvalues of the q × q matrix  −1 X F1 = XH YYH + XXH

(11A.1)

then the cumulative distribution function (CDF) of the largest eigenvalue wp when n ≥ p and q ≥ p is given by [35] Pr{wp ≤ x} = c |β(x)|

(11A.2)

where c=

p  i=1

(n + q − i + 1) (n − i + 1)(p − i + 1)(q − i + 1)

(11A.3)

| · | denotes the determinant operator, (·) is the gamma function [10, Sect. 8.31], and β(x) is a p × p matrix function of the scalar x ∈ (0, 1) with entries {β(x)}i,j =

APPENDIX 11A. DISTRIBUTIONS OF THE LARGEST EIGENVALUE

733

βx (q − p + i + j − 1, n − p + 1), where βx (·, ·) is the incomplete beta function [10, Eq. (8.391)] (For further details, see Ref. 35). 11A.2

Special Case

In this chapter, we are interested in the distribution of the largest eigenvalue of the quadratic form  −1 X F2 = XH YYH

(11A.4)

Consider the following facts: 1. It can be shown that the relation between the eigenvalues of F1 , wi , i = 1, · · · p in (11A.1) and those of F2 in (11A.4), say, φi , i = 1, . . . , p, is given by [38] wi =

φi 1 + φi

(11A.5)

2. When q ≤ p and n ≥ p, the joint PDF of the eigenvalues of F1 can be obtained by making the following substitution in the joint PDF of the eigenvalues of F1 when q ≥ p and n ≥ p [38]: (p, q, n) −→ (q, p, n + q − p) On the basis of these two facts, we can obtain the CDF of the largest eigenvalue of F2 for (1) q ≥ p, n ≥ p and (2) q ≤ p, n ≥ p. The result is summarized in the following theorem. Theorem 11.2 Let X: p × q and Y: p × n be independent matrices whose columns are i.i.d. p-variate complex normal vectors with zero mean and covariance matrix . Let s = min(p, q), t = max(p, q), and r = min(n, n + q − p); then the  −1 CDF of the largest eigenvalue of the matrix F2 = XH YYH X is given by 

   x  Pr(φs ≤ x) = cs β (11A.6) 1+x  where cs =

s  i=1

(r + t − i + 1) (r − i + 1)(s − i + 1)(t − i + 1)

(11A.7)

and β[x/(1 + x)] is an s × s matrix function of the scalar x/(1 + x), x ∈ (0, ∞), with entries 

 x β = βx/(1+x) (t − s + i + j − 1, r − s + 1) 1 + x i,j

734

OPTIMUM COMBINING—A DIVERSITY TECHNIQUE FOR COMMUNICATION

The PDF of the largest eigenvalue of F2 can be found by taking the derivative of the CDF with respect to x. A compact form of this PDF may be obtained by using the following classical formula for the derivative of determinants [39, Eq. (9)]:

d d detA(t) = detA(t)tr A−1 (t) A(t) , (11A.8) dt dt where tr(·) is the trace operator. After some manipulations, the final result can be summarized in the following corollary: Corollary 11.1 The PDF of the largest eigenvalue of F2 as defined in Theorem 11.2 is given by 





  φs φs −1   tr β s pφs (φs ) = cs β 1 + φs  1 + φs ×

φst−s u(φs ) (1 + φs )r+t−2s+2

(11A.9)

where s, t, and r are as defined in Theorem 11.2,  1     φs   s =  1 + φs  ..   . 

s−1  φs 1 + φs





φs 1 + φs φs 1 + φs .. . φs 1 + φs

and u(·) is the unit step function.

s−1 φs ··· 1 + φs

s φs ··· 1 + φs .. .. . .

2s−2 φs ··· 1 + φs

2

s

            

(11A.10)

s×s

12 DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA) In its generic form, direct-sequence code-division multiple access (DS-CDMA) is a spread-spectrum (SS) technique for simultaneously transmitting a number of signals representing information messages from a multitude of users over a channel employing a common carrier.1 The method by which the various users share the channel is the assignment of a unique pseudonoise (PN)-type code to each user (which accompanies the transmission of the information) with orthogonal-like properties that allows the composite received signal to be separated into its individual user components, each of which can then be demodulated and detected. The deployment of the code (assumed to be represented by a binary waveform with PN properties2 ) at the transmitter (spreading process), that is, the superposition onto the binary information waveform, is accomplished by a simple multiplication [which is equivalent to modulo-2 addition of their (0,1) representations]; hence the term direct-sequence modulation. Similarly, the removal of the code at the receiver (despreading process) is also accomplished by the identical multiplication operation. For our purposes in this chapter, we shall assume that the receiver is perfectly capable of regenerating the transmitted codes corresponding to each of the users’ transmissions, and as such we shall ignore all synchronization issues dealing with the acquisition and tracking of these codes at the receiver. A complete discussion of techniques for accomplishing these functions and their impact on system performance can be found in Ref. [1] (Part 4, Chaps. 1 and 2). 1 Later on we shall address a multiple-carrier version of this modulation that has become of interest in recent years. 2 Depending on the particular application, the codes may or may not be purely orthogonal; however, in either event they are chosen to have large autocorrelation and small crosscorrelation. A detailed discussion of the design of codes for (SS) applications such as CDMA is beyond the scope of this book but can be found in Ref. [1] (Part 1, Chap. 5).

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

735

736

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

The DS-CDMA technique has its roots in the literature dealing with military network applications (for a complete historical perspective on early SS systems employing such modulations, see Ref. [1], Part 1, Chap. 2), where their usage was primarily to combat intentional jamming introduced by an enemy. As such, the communication channel was typically modeled as additive white Gaussian noise (AWGN) combined with jamming of one sort or another (see Ref. [1], Part 1, Chaps. 3 and 4 and Part 2, Chap. 1). More recently, however, DS-CDMA has secured a strong foothold in the commercial market primarily because of its adoption as the IS-95 standard that governs digital cellular telephony here in the United States and elsewhere. It is this and other related wireless communication applications that motivate the results presented in this chapter since here the primary channel of interest is the fading channel along with the possibility of narrowband interference. In fact, it is the inclusion of this possibility that has stimulated researchers to investigate a multiple-carrier form of DS-CDMA since, as we shall see later on in the chapter, this particular form offers significant advantage over the traditional single-carrier version in combating such interference. In view of the discussion above, it therefore seems natural to divide the chapter into two main sections, namely, single-carrier DS-CDMA and multiple-carrier DS-CDMA. Each of these sections will focus on the average bit error rate (BER) performance of the corresponding DS-CDMA system when transmitted over a generalized fading channel such as those modeled in previous chapters of this book. As before, the emphasis will be on using the alternative forms of the classic functions developed in Chapter 4 to simplify the resulting expressions. Although we shall restrict ourselves to binary DS modulation (spreading waveforms), we shall allow for the possibility of other than a rectangular pulse shape to represent the PN code chip waveform. 12.1

SINGLE-CARRIER DS-CDMA SYSTEMS

A large number of papers deal with the performance of DS-CDMA systems over frequency-selective fading channels [2–13]. In this section, we apply the MGFbased approach presented in Chapter 9 to derive the average BER performance of binary DS-CDMA systems operating over these channels [14]. The results presented in this section are applicable to systems that employ RAKE reception with coherent maximal-ratio combining (MRC). Extensions to other types of detection/diversity combining techniques is straightforward (in view of the results presented in Chapter 9) and are therefore left as an exercise for the reader. Also although specifically developed for receiver type of diversity, the analysis presented in this section can be used with some modifications to assess the performance of transmit diversity CDMA systems [15,16]. 12.1.1

System and Channel Models

12.1.1.1 Transmitted Signal We consider a binary DS-CDMA system with Ku independent users simultaneously sharing a channel, each transmitting with power P at a common carrier

SINGLE-CARRIER DS-CDMA SYSTEMS

737

frequency fc = ωc /2π , using a data rate Rb = 1/Tb and a chip rate Rc = 1/Tc . The kth user (k = 1, 2, . . . , Ku ) is assigned a unique code sequence {ak,j } of chip elements (+1, −1), so that its code waveform is given by +∞


ak (t) =

ak,j PTc (t − j Tc )

(12.1)

j =−∞

where the function PT (·) denotes the chip pulse of duration T . In the single-carrier case we assume that PT (·) is a unit rectangular pulse, whereas in the multicarrier case we will consider Nyquist pulses. The code sequence {ak,j } is assumed to be periodic with period equal to the processing gain PG = Tb /Tc . The data signal waveform bk (t) given by +∞


bk (t) =

bk,j PTb (t − j Tb )

(12.2)

j =−∞

is binary phase-shift-keyed (BPSK) onto the carrier at fc , which is then spread by that user’s code sequence and transmitted over the channel. The resulting kth user’s transmitted signal sk (t) is thus given by √ sk (t) = 2P ak (t) bk (t) cos(ωc t) (12.3) The composite transmitted signal s(t) at the input of the channel can then be expressed as s(t) =

K √


2P ak (t) bk (t) cos(ωc t)

(12.4)

k=1

12.1.1.2 Channel Model DS-CDMA systems involve a spreading process that results in a transmitted signal whose bandwidth is much wider than the channel coherence bandwidth, and therefore undergoes frequency-selective fading. Following our discussion in Chapter 2, this type of fading is typically modeled by a linear filter, which for the kth user is characterized by a complex-valued lowpass equivalent impulse response [17–19] hk (t) =

Lp


αk,l e−j θk,l δ(t − τk,l )

(12.5)

l=1 L

p , where δ(.) is the Dirac delta function, l is the propagation path index, and {αk,l }l=1 Lp Lp {θk,l }l=1 , and {τk,l }l=1 are the random path amplitudes, phases, and delays, respecLp Lp Lp tively. We assume that the sets {αk,l }l=1 , {θk,l }l=1 , and {τk,l }l=1 are mutually independent. In (12.5) Lp is the number of resolvable paths (the first path is the reference path whose delay τ1 = 0) and is related to the ratio of the maximum delay spread τmax to the chip time duration Tc .

738

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

L

L

p p We assume slow fading, so that Lp is constant over time, and {αk,l }l=1 , {θk,l }l=1 , Lp and {τk,l }l=1 are all constant over a symbol interval. If the different paths of a given impulse response are generated by different scatterers, they tend to exhibit negligible correlations [20,21]. In this case it is reasonable to assume that the Lp are statistically independent random variables (RVs). We will make this {αk,l }l=1 assumption in our analysis, but we remind the reader that this assumption can be relaxed for Nakagami-m fading channels in view of the results presented in Section 9.7.4. We denote the fading amplitude of the kth user lth resolved path by 2 is assumed to be independent of αk,l , which is a RV whose mean-square value αk,l k and is denoted by l . After passing through the fading channel, the signal is perturbed by additive white Gaussian noise (AWGN) with a one-sided power spectral density denoted by Lp . N0 . The AWGN is assumed to be independent of the fading amplitudes {αk,l }l=1 2 Eb )/ Hence, the instantaneous SNR per bit of the lth channel is given by γk,l = (αk,l N0 , where Eb is the energy per bit, and the average SNR per bit of the lth channel is given by γ l = (l Eb )/N0 .

12.1.1.3 Receiver With multipath propagation, it follows from (12.5) and (12.4) that the received signal r(t), whose signal component is the time convolution of s(t) and h(t), may be written as r(t) =

√

2P

Lp Ku



αk,l ak (t − τk,l ) bk (t − τk,l ) cos(ωc (t − τk,l ) + θk,l ) + n(t)

k=1 l=1

(12.6) where n(t) is the receiver AWGN random process. We consider an L-branch (finger) MRC RAKE receiver, as shown in Fig. 9.2. The optimal value for L is Lp , but L may be chosen less than Lp because of receiver complexity constraints. Let us consider the kth user receiver. Each of the L paths to be combined is first coherently demodulated through multiplication by the unmodulated carrier cos[ωc (t − τk,l ) + θk,l ] and then lowpass-filtered to remove the second harmonics of the carrier. All these operations assume that the receiver is correctly time- and phase-synchronized at every branch (i.e., perfect carrier recovL ery, and perfect phase {θk,l }L l=1 and time delay {τk,l }l=1 estimates). Using MRC (see Section 9.2) and assuming perfect knowledge of the fading amplitude on each finger, the L lowpass filter outputs {rm,l }L l=1 are individually weighted by their respective fading amplitudes, and then combined by a linear combiner yielding the decision variable rk =

L
l=1

αk,l rk,l ;

k = 1, 2, . . . , Ku

(12.7)

SINGLE-CARRIER DS-CDMA SYSTEMS

12.1.2

739

Performance Analysis

Without loss of generality, let us consider the kth user’s performance. The decision variable rk may be written as the sum of a desired signal component and three interference/noise components [10] rk = ±

 L


 2 αk,l

L
 Eb + αk,l (IS + IM + N )

l=1

(12.8)

l=1

where IS is the self-interference component induced by the autocorrelation function of the kth user’s spreading code, IM is the multiple-access interference (MAI) component induced by the other Ku − 1 users on the desired user, and N is a zeromean AWGN component with variance σN2 = N0 /2. Eng and Milstein showed that IS can be considered to be a zero-mean Gaussian RV with variance [10, Eq. (9)] σS2 =

T − 1 1 Eb 2 PG

(12.9)

Lp where T = l=1 l /1 can be interpreted as the normalized (to the first path) total average fading power. Similarly, under the standard Gaussian approximation (large numbers of users) [3,5,6,10], IM can be modeled as a zero-mean Gaussian RV with variance [10, Eq. (6)] 2 σM =

2(Ku − 1)T 1 Eb 6 PG

(12.10)

Under these assumptions rk may be considered to be a conditional Gaussian RV L (conditioned on {αk,l }L l=1 ) with a conditional mean E[rk /{αk,l }l=1 ] and a conditional L variance var[rk /{αk,l }l=1 ] given by E [rk |{αk,l }L l=1 ] = ±

 L


 2 αk,l



Eb

(12.11)

l=1

var [rk |{αk,l }L l=1 ] =

L


 2  2 2 σN + σS2 + σM αk,l

(12.12)

l=1

Assuming that the data bits +1 or −1 are equiprobable, the kth user conditional SNR, SNR({αk,l }L l=1 ), is given as 2  (E[rk |{αk,l }L  l=1 ]) SNR {αk,l }L l=1 = 2var[rk |{αk,l }L l=1 ]   L
Eb 2 αk,l = Ne l=1

(12.13)

740

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

where Ne /2 is the equivalent two-sided interference plus noise power spectral density defined as Ne  2 2 = σN + σS2 + σM 2  (2Ku + 1)T − 3 N0 1 Eb + = 6 PG 2

(12.14)

with γ 1 = (1 Eb )/N0 the average received SNR per bit corresponding to the first path. 12.1.2.1 General Case In view of (12.13) and (12.14), and since we are assuming BPSK modulation, the average BER performance expression obtained in (9.11) applies here for DSCDMA by replacing N0 with Ne , or equivalently, replacing the average SNR per bit of the lth path γ l = l Eb /N0 by γ l,e

l Eb = = γl Ne



 1+

−1 (2Ku + 1) T − 3 γ1 3 PG

(12.15)

12.1.2.2 Application to Nakagami-m Fading Channels The performance of coherent DS-CDMA systems over Nakagami-m frequencyselective fading channels with MRC [10,12] and postdetection EGC [9,13,22] RAKE reception has received considerable attention in the more recent literature. In particular, Eng and Milstein [10] have provided a BER performance analysis of coherent DS-CDMA systems in a Nakagami-m fading environment with an equally spaced exponentially decaying power delay profile (l = 1 eδ(l−1) , l = 1, 2, . . . , Lp ), where δ is the power decay factor. Their analysis relies on a classical PDF-based approach and uses a Nakagami approximation [23, Eq. (80)] to the PDF of the sum of squares of independent non–identically distributed Nakagami-m RVs, which leads to a closed-form approximation to the BER in terms of the Gauss hypergeometric series, 2 F1 (·, ·; ·; ·). The approximation is accurate for small values of the power decay factors δ but looses its accuracy as δ increases. More recently Efthymoglou et al. [12] applied the Gil-Pelaez lemma [12, Eq. (26)] to obtain exact BER performance in the Nakagami-m fading environment with arbitrary fading parameters along the different resolvable paths. Applying the approach described above to the problem treated in Refs. 10 and 12 and using the MGF corresponding to Nakagami-m fading given in Table 2.1 yields, after some manipulations, the following expression for the average BER 1 Pb (E) = π

L  π/2

0

l=1

1+

γ e e−(l−1)δ ml sin2 φ

−ml dφ

(12.16)

where γ e = 1 Eb /Ne can be viewed as the equivalent signal-to-noise-plusinterference ratio per bit corresponding to the first path, and which may be expressed

MULTICARRIER DS-CDMA SYSTEMS

741

using (12.15) as  γe =

1 (2Ku + 1)T − 3 + 3PG γ1

−1 (12.17)

Lp −(l−1)δ In this case, T reduces to T = l=1 e = (1 − e−Lp δ )/(1 − e−δ ). The exact and simple form of the average BER in (12.16) has advantage over previous equivalent forms that either are an approximation [10, Eq. (16a)] or are expressed in terms of an integral with an infinite upper limit and with a much more complicated integrand [12, Eq. (33)]. Many such DS-CDMA performance analysis problems (for other types of channel models [24] or other types of detection/diversity combining [9,13,22]) may be solved using the approach described above.

12.2

MULTICARRIER DS-CDMA SYSTEMS

Multicarrier code-division multiple-access (MC-CDMA) systems have received widespread interest due to their potential for high-speed transmission and their effectiveness in mitigating the effects of multipath fading and in rejecting narrowband interference. Different classes of these systems have been presented and studied by various researchers [25]. Here we focus on one particular scheme proposed by Kondo and Milstein [26]. In the absence of partial-band interference (PBI) this scheme was shown to achieve performance equivalent to RAKE reception of single carrier DS-CDMA systems operating over frequency-selective Rayleigh channels with a flat multipath intensity profile. However, the MC-CDMA system has the advantage of not requiring a continuous spectrum since the various subbands over which the modulated signal is transmitted can be chosen in various parts of the “unused” spectrum. In addition, the proposed MC-CDMA scheme was shown to outperform single carrier systems in the presence of PBI and can therefore be favorably used in an overlay CDMA/FDMA or CDMA/TDMA situation without the need for sophisticated adaptive notch filtering to excise the interference. Several studies have been based on the work of Kondo and Milstein. In particular, Rowitch and Milstein examined the performance of coded versions of the original scheme [27]. The effect of fading correlation between the various subbands was studied in Refs. 28 and 29. In this section, we present generic expressions for the average BER performance of MC-CDMA systems operating over generalized fading channels with and without PBI [30]. The results are again applicable to systems that employ coherent detection with MRC but can be easily extended, in view of the results in Chapter 9, to other detection and combining technique combinations. Aside from providing equivalent forms for known expressions corresponding to the performance of MC-CDMA over Rayleigh fading channels [26], utilization of the MGF-based approach for this specific application yields a solution for many scenarios that would be otherwise difficult to analyze in simple form. More specifically, this approach is particularly useful for

742

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

the performance of MC-CDMA systems operating in a microcellular environment where the fading tends to follow a Rice or Nakagami-m type of distribution. Indeed, as we shall see in the following sections, because of the PBI, the variance of the subband correlator outputs may be different, and hence computing the PDF of the conditional SNR, which can be done for the Rayleigh fading case, becomes a difficult task for other types of fading. In fact, even in the absence of PBI, the variance of the subband correlator outputs may be different because of the variation in the average fading power across the band. This is particularly true when the various subbands are not contiguous (i.e., subcarriers are far apart) because of the dependence of the path loss on the carrier frequency [31, Sect. 2.5]. 12.2.1

System and Channel Models

12.2.1.1 Transmitter Consider a BPSK multicarrier coherent DS-CDMA system with Ku independent users each transmitting with power P . The users are simultaneously sharing an available bandwidth BW = (1 + α)/Tc , where Tc is the chip duration of a corresponding single-carrier wideband DS-CDMA system, and α (0 ≤ α ≤ 1) is the rolloff factor of the chip wave shaping Nyquist filter. The available spectrum BW is divided into (not necessarily contiguous) Mf equal bandwidth subbands each of width BWMf approximately equal to the coherence bandwidth of the channel (see Fig. 12.1). Each subband is assigned a carrier [at frequency fl (l= 1, 2, · · ·, Mf )] that is DS-CDMA-modulated with the same user information at bit rate 1/Tb and

fc

f

BW = Mf BWMf

f1

f2

.....

fMf

f

BWMf Figure 12.1 Comparison between single- and multicarrier DS-CDMA system power spectral densities.

MULTICARRIER DS-CDMA SYSTEMS

743

Chip Pulse Shaping Filter Data Sequence

2 cos(2pf1t + qk,1)

{dh(k)}

{cn(k)} PN Code Sequence

Transmitted Multicarrier DS Signal

H( f )

Impulse Modulator ∞

∑ dh(k)cn(k)d(t − MfTc),

n = −∞

PG′ h = n/(Tb /MfTc)

∞

2cos(2pfMf t + qk,Mf )

∑

dh(k)cn(k)h(t − MfTc), n = −∞ Identical DS Signal is Modulated Onto Each of the Mf Carriers

Figure 12.2

Transmitter of multicarrier DS-CDMA system for the kth user.

chip rate 1/(Mf Tc ) (see Fig. 12.2). Each user is effectively assigned a specific periodic code sequence of chip elements (+1, −1) and of processing gain per subband PG = PG/Mf . We assume deterministic subband PN codes with ideal autocorrelation function. The use of bandlimited (Nyquist-shaped) spreading waveforms with wave shaping filter transfer function denoted by H (f ) guarantees that the DS waveforms do not overlap. 12.2.1.2 Channel Following the system design and modeling assumptions of Kondo and Milstein [26], the number of subbands as well as their bandwidths are chosen so that the separate subbands fade slowly and nonselectively. Under these assumptions,   exp(j θk,l ), the channel transfer function of the lth subband for the kth user is αk,l M

M

f f   where the {αk,l }l=1 are the fading amplitude RVs and {θk,l }l=1 are independent, uniformly distributed RVs over [0, 2π ]. The average fading power of the lth subband  2 ) and is assumed to be independent of k. is denoted by l = (αk,l

12.2.1.3 Receiver The receiver consists of a bank of Mf matched filters followed by MRC (see Fig. 12.3). Each of the received modulated subband carriers is first passed through a bandpass chip matched filter H ∗ (f ), then coherently demodulated, sampled, despread, and summed. All these operations assume that the receiver is correctly phase- and time-synchronized at every branch (i.e., perfect carrier recovery and bit synchronization). We denote by X(f ) = H (f )H ∗ (f ) = |H (f )|2 the overall frequency response of the chip wave shaping Nyquist filter and assume that X(f ) is

744

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

sample at t = nMfTc

H *( f − f1)

2 cos(2pf1t + qk,1)

. . . .

r(t)

1 PG′-1 rk,1 w ∑ (•) k,1 PG′n = 0

LPF

+ H *( f + f1)

{cn(k)} correlation over one bit interval

removes second harmonic of carrier

sample at t = nMfTc

H *( f − fM ) + H ( f + fM )

f

LPF

f

2 cos(2pfM t + qk,M ) f

f

r

rk,M 1 PG′-1 ∑ (•) PG′n = 0

f

*

. . . .

wk,Mf

{cn(k)}

Figure 12.3 Receiver of multicarrier DS-CDMA system for kth user (ideal coherent demodulation).

a root raised-cosine frequency response given by  1 W   , 0 ≤ |f | < (1 − α)   W 2  

    1 1 2π |f | W W X(f ) = 1−sin , −π (1 − α) ≤ |f | ≤ (1 + α)  2W 2α W 2 2     W   0, (1 + α) ≤ |f |. 2 (12.18) with W = 1/Tc = 1/(Mf Tc ) for multicarrier, and W = 1/T for single carrier. In c √ addition, we normalize the chip correlators by 1/ PG (Kondo and Milstein [26] do not normalize). Finally, the Mf test statistics rk,l are individually weighted by the coefficients wk,l and then combined as per the rules of MRC to form the decision  Mf wk,l rk,l . variable rk = l=1 12.2.1.4 Notations For clarity we show in this section the equivalence between multicarrier and singlecarrier system parameters. To distinguish between them, we add a prime to the multicarrier system parameters: •

Subband • Subband • Subband • Subband

chip time Tc = Mf Tc processing gain PG = Tb /(Mf Tc ) = PG/Mf chip energy Ec = Ec [(P /Mf )Mf Tc = P Tc ] bit energy Eb = Eb /Mf = PG Ec = PG Ec

MULTICARRIER DS-CDMA SYSTEMS

12.2.2

745

Performance Analysis

Without loss of generality, let us consider the kth user’s performance. Two cases are of interest. In the first case the received signal is affected by fading, MAI, and AWGN. The second case corresponds to the scenario where the received signal is affected not only by the fading, MAI, and AWGN but also by Gaussian PBI. 12.2.2.1

Conditional SNR

Case 1: No Partial-Band Interference The decision variable of the kth user may be written as the sum of a desired signal component and two interference/noise components [26] rk =

Mf


Mf
 wk,l rk,l = ± wk,l αk,l

l=1

l=1



Mf

Eb
+ wk,l (IMl + N ), Mf

(12.19)

l=1

where Eb = P Tb is the energy per bit, IMl is the MAI term induced by the other Ku − 1 users in the lth subband, and N is the zero-mean AWGN component with variance σN2 = N0 /2. Under the standard Gaussian approximation (valid for large number of users), IMl can be modeled as a zero mean Gaussian RV with variance 2 σM given by (assuming Nyquist chip pulses) l 2 σM = l

(Ku − 1)Ec l 2Tc



∞ −∞

X2 (f ) d f =

(Ku − 1)Eb l  α 1 − 2Mf PG 4

(12.20)

Assuming that the MAI and AWGN are independent, we define an equivalent additive interference/noise with two-sided power spectral density (Ku − 1)Eb l  Nel α  N0 2 = σN2 + σM + 1 − = l 2 2Mf PG 4 2 

  (Ku − 1) α N0 1+ 1− γ =  2 Mf PG 4 l

(12.21)

where γl = (l Eb /N0 ) represents the average SNR/bit of the lth subband. Under  Mf }l=1 ) with these assumptions rk is a conditionally Gaussian RV (conditioned on {αk,l M

M

f f   conditional mean E[rk |{αk,l }l=1 ] and conditional variance var(rk |{αk,l }l=1 ) given by

  Mf
Eb  Mf   }l=1 ] = ±  wk,l αk,l E[rk |{αk,l Mf l=1

 Mf var[rk |{αk,l }l=1 ]

Mf
 2 Nel . wk,l = 2 l=1

(12.22)

746

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

Assuming that there is an equal probability that the data bits are + or −1, we get  Mf }l=1 ), as the kth user conditional SNR, SNR({αk,l   2  Mf   E r |{α } k k,l l=1  Mf   SNR {αk,l }l=1 =  Mf 2var rk |{αk,l }l=1 M 2 f  l=1 wk,l αk,l Eb = Mf Mf wk,l 2 Ne l=1

(12.23)

l

The maximum conditional SNR is obtained when the various frequency diversity subbands are weighted as per the rule of MRC by the coefficients     Mf  E rk,l |{αk,l }l=1 2αk,l Eb  =  wk,l = (12.24) M f  Nel Mf var rk,l |{αk,l }l=1 Substituting (12.24) in (12.23), we obtain the kth user maximum conditional SNR,  Mf }l=1 ), at the MRC output as SNRmax ({αk,l  Mf SNRmax ({αk,l }l=1 )

Mf    2 Eb
αk,l = Mf Nel l=1

=

Mf E b
  2 βl αk,l N0 Mf

(12.25)

l=1

with 

α   −1 (Ku − 1)  1 − βl = 1 + γ Mf PG 4 l

(12.26)

Kondo and Milstein [26] compared the performance of their proposed multicarrier system with a wideband single-carrier coherent CDMA system with L-finger RAKE reception [10]. For the special case of a single user (Ku = 1), the MAI term vanishes and the maximum conditional SNR of the multicarrier system, as given by (12.25), reduces to M

f  }l=1 )= SNRmax ({αk,l

Mf E b
  2 αk,l N0 Mf

(12.27)

l=1

On the other hand, it is well known that the maximum conditional SNR, L SNR(k) max ({αk,l }l=1 ), of a single-carrier system with L-finger RAKE reception and in

MULTICARRIER DS-CDMA SYSTEMS

747

which the self-interference is negligible is given by [10] SNRmax ({αk,l }L l=1 ) =

L E b
 2 αk,l N0

(12.28)

l=1

Thus, for exact equivalence between the single-carrier system and the multicarrier system, what we need is L = Mf ,

 αk,l αk,l =  Mf

(12.29)

For the multiuser case the maximum conditional SNR of the multicarrier system is given by (12.25), whereas the maximum conditional SNR of the single-carrier system is given by

−1 L α (Ku − 1)T  E b
  2 L 1− γ (12.30) αk,l 1+ SNRmax ({αk,l }l=1 ) = N0 PG 4 1 l=1

Hence, in the multiuser case even if L = Lp and even if the conditions in (12.29) are met, comparing (12.25) and (12.30), we see that we do not have in general equivalence between the single- and multicarrier systems. However, equivalence can be achieved in the particular case of a uniform average fading power delay profile (l = , l = 1, 2, . . . , Lp ) for single-carrier systems and a uniform average fading power across the frequency band for multicarrier systems (l =  , l = 1, 2, . . . , Mf ). Indeed, in this special case if L = Lp = Mf and if the single-user conditions (12.29) are met, then we have T = L and  =  /Mf , and thus T 1 = L = L

 =  Mf

(12.31)

which is the condition necessary for equivalence between (12.25) and (12.30). Case 2: Partial-Band Interference Consider now the presence of a PBI jammer modeled as a bandlimited white Gaussian noise with bandwidth WJ = BWMf and power spectral density SnJ (f ) such as   ηJ , f − WJ ≤ |f | ≤ f + WJ J J SnJ (f ) = (12.32) 2 2 2  0, elsewhere where fJ denotes the jammer carrier frequency. The decision variable of the kth user may now be written as the sum of a desired signal component and three interference/noise components [26]  Mf Mf Mf

Eb
 wk,l rk,l = ± wk,l αk,l + wk,l (N + IMl + IJl ) (12.33) rk = Mf l=1

l=1

l=1

748

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

where IJl is the Gaussian PBI present in the lth subband with variance  1 ∞ 2  NJl = σJl = SnJ (f − fl ) + SnJ (f + fl ) X(f ) d f 2 2 −∞

(12.34)

Assuming that the PBI is independent of the MAI and AWGN, we can define a new equivalent additive interference/noise with two-sided power spectral density Nel 2 = σN2 + σM + σJ2l l 2 

Ku − 1  α   NJl N0 1+ 1− = γ + 2 Mf PG 4 l N0

(12.35)

Note that even if the average fading power is uniform across the subbands, some of the Nel s terms will still depend on l because of the presence of the PBI NJl in these subbands. Since the various frequency diversity subbands are weighted as per the rule of MRC by the weights (12.24), the kth user maximum conditional  Mf }l=1 ), is still given by (12.25), where βl is now given by SNR, SNRmax ({αk,l  α   NJl −1 Ku − 1  βl = 1 + 1− γ + Mf PG 4 l N0

(12.36)

Because of the assumption that the bandwidth of the PBI is equal to the bandwidth of one subband, two subcases have to be considered. First, if the jammer carrier fJ coincides with one of the system subcarriers, say, fν (ν = 1, 2, . . . , Mf ), then the PBI completely overlaps the νth subband and JSRν γ ν NJν , = N0 PG (1 + α) NJl = 0, l = ν N0

ν = 1, 2, . . . , or Mf (12.37)

where JSRν = (ηJ WJ )/(ν Eb /Tb ) represents the interference (jamming) to average signal power ratio in the νth subband. Now if the jammer carrier fJ is between two of the system subcarriers, say, fν < fJ < fν+1 , then the PBI partially overlaps the νth and (ν + 1)th subbands and it can be shown by substituting (12.18) in (12.34) that

  (1 + α) |f | α π (1 + α) |f | JSRν γ ν NJν 1− + sin = N0 PG (1 + α) 4 WJ /2 2π 2 α WJ /2

  NJν+1 JSRν+1 γ ν+1 (1 + α) |f | α π (1 + α) |f | − sin = N0 PG (1 + α) 4 WJ /2 2π 2 α WJ /2 NJl = 0, N0

l = ν, ν + 1

where f = fJ − fν and |f | ≤ αWJ /(α + 1).

(12.38)

749

MULTICARRIER DS-CDMA SYSTEMS

For single-carrier systems affected by Gaussian PBI, the maximum conditional SNR becomes SNRmax ({αk,l }L l=1 ) =

 L (Ku − 1)T  α E b
 2 NJ −1 1+ 1 − γ1 + αk,l N0 PG 4 N0 l=1

(12.39) where NJ is as in (12.34) so that NJ JSR T γ 1 = N0 PG

(12.40)

 where JSR = ηJ WJ /[( L l=1 l )Eb /Tb ] represents the interference to total average signal power ratio. Note that even in the special case of uniform power delay profile and uniform power distribution across the band, in the presence of PBI, we do not have equivalence between single- and multicarrier systems. In fact, if we employ the same equivalence conditions as for the no-interference case, then the multicarrier system has an SNR advantage in mitigating the PBI since, in general, some of the NJl s will be equal to zero. This will be confirmed in Section 12.2.3 on numerical examples. 12.2.2.2 Average BER Mf ) Gaussian RV with a Since the output of the MRC is a conditional (on the {αk,l }l=1  Mf conditional SNR given by (12.25), the kth user’s conditional BER, Pb (E|{αk,l }l=1 ), is given by    Mf  2Eb
  2    Mf Pb (E|{αk,l }l=1 ) = Q  βl αk,l  N0 Mf

(12.41)

l=1

where Q(·) denotes the Gaussian Q-function. The goal is to evaluate the system performance in terms of the users’ average BER, which requires averaging of the Mf conditional BERs as given by (12.41) over the random fading amplitudes {αl }l=1 . Recall that the classical PDF-based approach for solving this problem is to first   Mf  find the PDF of γt = l=1 βl (αk,l )/N0 and then average (12.41) over that PDF. This is, in fact, the approach used by Kondo and Milstein since for Rayleigh fading this PDF can be either found in closed form or evaluated by residue calculations. However, because γt is a weighted sum of RVs, finding its PDF for other fading conditions of interest (such as Rician or Nakagami-m with or without a uniform average fading power across the subbands) is a difficult task. To circumvent this difficulty, we apply the alternative MGF-based approach. Independent Fading across the Subbands Following the procedure in Section 9.2.3, we partition the conditional BER (12.41) in a separable product

750

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

form, thereby obtaining the unconditional BER by independently averaging over the fading of each subband resulting in an average BER expression given by 1 Pb (E) = π



Mf π/2

0

 Mγl −

l=1



βl Mf sin2 φ

dφ

(12.42)

where Mγl (s) denotes the MGF of the lth subband SNR/bit as given in Table 2.1. Correlated Nakagami-m Fading As discussed in Refs. 28 and 29, fading correlation among the various subband fading amplitudes induces a certain performance degradation. Under these conditions, using a procedure similar to the one adopted in Section 9.7.4.1, the average BER performance of MC-CDMA systems can be expressed as Pb (E) =

1 π



π/2

0

 Mγt −

1 sin2 φ

dφ

(12.43)

where Mγt (s) is the MGF of the combined output SNR with arbitrary correlated Nakagami-m faded subbands and that can be found in closed form based on Eq. (2.3) in Ref. 32 as  Mγt (s) = Eγ1 ,γ2 ,...,γM

f

=



 Mf
β l exp s γ   Mf l l=1

Mf  −m

sβl −m    1− det Cij M ×M f f mMf

(12.44)

l=1

where & Cij =

1,  √ ρij 1 −

mMf sβj

−1

i=j , otherwise

(12.45)

with ρij the fading power correlation coefficient between subbands i and j . 12.2.3

Numerical Examples

We present in this section some numerical examples illustrating the effect of the severity of fading on the performance of MC-CDMA systems operating over a Nakagami-m fading channel with uniform average fading power profile across the band. Using the same system parameters as the ones in Kondo and Milstein [26] [i.e., Mf = 4, α = 0.5, Ku = 50, PG = 128, WJ = BWMf , and fJ = fl (l = 1, 2, 3, 4)], we plot the BER performance of both systems in terms of γ  = Mf γ . Figures 12.4–12.6 compare the performance of an MC-CDMA system

MULTICARRIER DS-CDMA SYSTEMS

751

100 Single Carrier Multicarrier

Average Bit Error Rate (BER)

d 10−1

c d

10−2

c b

10−3

0

5

10

15

20

b a

25

30

Average Signal–to–Noise–Ratio (SNR) [dB]

Figure 12.4 BER comparison between SC-CDMA and MC-CDMA systems over Nakagamim fading channels [m = 0.5; (a) JSR = −∞ dB, (b) JSR = 10 dB, (c) JSR = 20 dB, (d) JSR = 30 dB]. 100 Single Carrier Multicarrier

Average Bit Error Rate (BER)

d 10−1

d 10−2

c c 10−3

10−4 0

a

5

10

15

20

25

b

b

30

Average Signal-to-Noise-Ratio (SNR) [dB]

Figure 12.5 BER comparison between SC-CDMA and MC-CDMA systems over Nakagami-m fading channels [m = 1; (a) JSR = −∞ dB, (b) JSR = 10 dB, (c) JSR = 20 dB, (d) JSR = 30 dB].

752

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

100 Single Carrier Multicarrier d

Average Bit Error Rate (BER)

10−1

10−2

c

10−3

d c b b a

10−4

10−5 0

5

10

15

20

25

30

Average Signal-to-Noise-Ratio (SNR) [dB] Figure 12.6 BER comparison between SC-CDMA and MC-CDMA systems over Nakagamim fading channels [m = 2; (a) JSR = −∞ dB, (b) JSR = 10 dB, (c) JSR = 20 dB, (d) JSR = 30 dB].

with its corresponding SC-CDMA system (with a flat power delay profile) for m = 0.5, m = 1, and m = 2, respectively. The results for the Rayleigh case (m = 1) check with the results published in [26]. Note first that both MC-CDMA and SCCDMA systems are more sensitive to the JSR for channels with a lower amount of fading since we observe a higher dynamic range in the BER performance for higher values of m. Furthermore, regardless of the severity of fading, the performance of SC-CDMA and MC-CDMA are almost the same for negligible JSR but MC-CDMA outperforms SC-CDMA for high values of JSR. However, the relative difference between the MC-CDMA and SC-CDMA systems tends to increase as the amount of fading decreases (i.e., higher m), which means that MC-CDMA is an even better choice in a microcellular environment. Figures 12.7, 12.8, and 12.9 compare the performance of aligned MC-CDMA systems with the misaligned ones for m = 0.5, m = 1, and m = 2, respectively. Aligned systems correspond to the case of fJ = fl (l = 1, 2, 3, 4) whereas misaligned systems correspond to the case fJ = (fl + fl+1 )/2 (l = 1, 2, 3). For all values of m, the system performance is better when only one subband is affected by the interference. Furthermore, the relative difference between the two systems slightly increases for channels with a lower amount of fading.

MULTICARRIER DS-CDMA SYSTEMS

753

100

Average Bit Error Rate (BER)

Aligned Misaligned 10−1

d c 10−2

10−3

c

0

5

10

15

d

a

b

20

25

30

Average Signal-to-Noise-Ratio (SNR) [dB]

Figure 12.7 Average BER of MC-CDMA systems over Nakagami-m fading channels [m = 0.5; (a) JSR = −∞ dB, (b) JSR = 10 dB, (c) JSR = 20 dB, (d) JSR = 30 dB]. 100

Average Bit Error Rate (BER)

Aligned Misaligned 10−1

d

10−2

c

10−3

10−4

0

5

10

15

c

d

a

b

20

25

30

Average Signal-to-Noise-Ratio (SNR) [dB]

Figure 12.8 Average BER of MC-CDMA systems over Nakagami-m fading channels [m = 1; (a) JSR = −∞ dB, (b) JSR = 10 dB, (c) JSR = 20 dB, (d) JSR = 30 dB].

754

DIRECT-SEQUENCE CODE-DIVISION MULTIPLE ACCESS (DS-CDMA)

100 Aligned Misaligned

Average Bit Error Rate (BER)

10−1

10−2

d c 10−3

c a

d b

10−4

10−5

0

5

10

15

20

25

30

Average Signal-to-Noise-Ratio (SNR) [dB]

Figure 12.9 Average BER of MC-CDMA systems over Nakagami-m fading channels [m = 2; (a) JSR = −∞ dB, (b) JSR = 10 dB, (c) JSR = 20 dB, (d) JSR = 30 dB].

REFERENCES 1. M. K. Simon, J. K. Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications Handbook, 2nd ed. New York, NY: McGraw-Hill, 1994; originally published in three parts as Spread Spectrum Communications, Computer Science Press, 1984. 2. G. Turin, “The effects of multipath and fading on the performance of direct sequence CDMA systems,” IEEE J. Select. Areas Commun., vol. SAC-2, April 1984, pp. 597–603. 3. E. A. Geraniotis and M. B. Pursley, “Performance of coherent direct-sequence spreadspectrum communications over specular multipath fading channels,” IEEE Trans. Commun., vol. COM-33, June 1985, pp. 502–508. 4. H. Xiang, “Binary code-division multiple-access systems operating in multipath fading, noisy channels,” IEEE Trans. Commun., vol. COM-33, August 1985, pp. 775–784. 5. E. A. Geraniotis, “Direct-sequence spread-spectrum communications multiple-access communications over nonselective and frequency-selective Rician fading channels,” IEEE Trans. Commun., vol. COM-34, August 1986, pp. 756–764. 6. J. S. Lehnert and M. B. Pursley, “Error probabilities for binary direct-sequence spreadspectrum communications with random signature sequences,” IEEE Trans. Commun., vol. COM-35, February 1987, pp. 87–98.

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22. V. Aalo, O. Ugweje, and R. Sudhakar, “Performance analysis of a DS/CDMA system with noncoherent M-ary orthogonal modulation in Nakagami fading,” IEEE Trans. Veh. Technol., vol. VT-47, February 1998, pp. 20–29. 23. M. Nakagami, “The m-distribution—a general formula of intensity distribution of rapid fading,” in Statistical Methods in Radio Wave Propagation. Oxford, U.K.: Pergamon Press, 1960, pp. 3–36. 24. J. R. Foerster and L. B. Milstein, “Analysis of hybrid, coherent FDMA/CDMA systems in Ricean multipath fading,” IEEE Trans. Commun., vol. COM-45, January 1997, pp. 15–18. 25. K. Fazel and G. P. Fettweis, Multi-Carrier Spread-Spectrum. Boston, MA: Kluwer Academic Publishers, 1997. 26. S. Kondo and L. B. Milstein, “Performance of multicarrier DS CDMA systems,” IEEE Trans. Commun., vol. COM-44, February 1996, pp. 238–246. 27. D. N. Rowitch and L. B. Milstein, “Coded multicarrier DS-CDMA in the presence of partial band interference,” Proc. IEEE Mil. Commun. Conf. (MILCOM’96), McLean, VA, October 1996, pp. 204–209. 28. R. E. Ziemer and N. Nadgauda, “Effect of correlation between subcarriers of a MCM/DSSS communication system,” Proc. IEEE Veh. Technol. Conf. (VTC’96), Atlanta, GA, April 1996, pp. 146–150. 29. W. Xu and L. B. Milstein, “Performance of multicarrier DS CDMA systems in the presence of correlated fading,” Proc. IEEE Veh. Technol. Conf. (VTC’97), Phoenix, AZ, May 1997, pp. 2050–2054. 30. M. K. Simon and M.-S. Alouini, “BER performance of multicarrier DS-CDMA systems over generalized fading channels,” Proc. Communication Theory Mini-Confe. in Conjunction with IEEE Int. Conf. Communication (ICC’99), Vancouver, British Columbia, Canada, June 1999, pp. 72–77. 31. G. L. St¨uber, Principles of Mobile Communications. Norwell, MA: Kluwer Academic Publishers, 1996. 32. A. S. Krishnamoorthy and M. Parthasarathy, “A multivariate gamma-type distribution,” Annals Math Stat., vol. 22, 1951, pp. 549–557.

PART 5 CODED COMMUNICATION SYSTEMS

13 CODED COMMUNICATION OVER FADING CHANNELS Thus far we have considered the performance of uncoded digital communication systems over fading channels. As such it has been necessary to model the fading channel in only a single-symbol interval, Ts , since for uncoded transmission, decisions are made on a symbol-by-symbol basis. When error correction coding is applied to the transmitted modulation and decisions are based on an observation of the received signal much longer than Ts , it becomes necessary to consider the variation of the fading channel from symbol interval to symbol interval. For the case of slow fading, the fading parameters (e.g., amplitude, phase) are typically treated as being constant over many symbol intervals, thereby introducing unintentional memory into the channel and an associated degradation of performance. A common method for breaking up these fading bursts without disturbing the intentional memory introduced by the coding is to employ interleaving at the transmitter and deinterleaving at the receiver. The combination of interleaving and deinterleaving acts in such a way as to produce fades that are independent from symbol to symbol whereupon the fading channel once again becomes memoryless. In studying the performance of coded communications over memoryless channels (with or without fading), the results are given as upper bounds on the average bit error probability (BEP). In principle, there are three different approaches to arriving at these bounds, all of which employ obtaining the so-called pairwise error probability, or the probability of choosing one symbol sequence over another for a given pair of possible transmitted symbol sequences, followed by a weighted summation over all pairwise events. The first approach, which typically gives the weakest but simplest to evaluate result, upper-bounds the pairwise error probability by a Chernoff bound and then further upper-bounds the summation over all pairwise events by a union bound; thus, it treats the pairwise events as if they were independent where in reality they are correlated. Bounds on average error probability and Digital Communication over Fading Channels, Second Edition. By Marvin K. Simon and Mohamed-Slim Alouini ISBN 0-471-64953-8 Copyright  2005 John Wiley & Sons, Inc.

759

760

CODED COMMUNICATION OVER FADING CHANNELS

average BEP arrived at in this fashion are referred to as union–Chernoff bounds and are typically the most common form found in the literature, for example, for trellis-coded M-PSK modulation over Rayleigh and Rician fading channels, see Refs. 1 and 2 (later documented in tutorial fashion in Refs. 3–5). Furthermore, evaluation of the union bound portion of the overall upper bound is conveniently accomplished using the transfer function bound method originally proposed by Viterbi [6] and later documented in tutorial form by Viterbi and Omura [7, Ch. 4]. The second approach exactly evaluates the pairwise error probability but considers only a finite number of pairwise events, specifically, those that are dominant, in place of the true union bound that accounts for all pairwise events. Examples of this approach can be found in Refs. 8 and 9. In the limiting case, only the single dominant error event corresponding to the minimum distance between the correct and incorrect sequences is considered, which results in the simplest of this form of upper bound.1 The third approach also exactly evaluates the pairwise error probability but accounts for all pairwise events by using the transfer function bound to evaluate the true union bound. Clearly, of the three approaches this form will result in the tightest upper bound; however, it is, in general, more complex to evaluate analytically. Significant contributions using this approach can be found in the literature [10–13] focusing on trellis-coded modulation (TCM) transmitted over fading channels. The degree to which these contributions differ from each other depends on the nature of the detection schemes, namely, coherent versus differentially coherent, the statistics of the fading channel, such as Rayleigh, Rician, and the amount of knowledge concerning the state of the channel, specifically, the availability of channel state information (CSI). For example, the approach taken by Tellambura [10] is not readily applicable to differential detection, and the approach taken by Kim and others [11,12] is easily computed only for Rayleigh and Rician channels where the difference of the decision metrics can be modeled as a quadratic form in complex Gaussian random variables (RVs). On the other hand, the method taken in Ref. 13 has the advantage that it can be extended to include a larger class of fading channels other than just Rayleigh and Rician at the same time, allowing for both coherent and differentially coherent detection of a variety of different modulation schemes of the form discussed earlier in this textbook. Unfortunately, however, as we shall demonstrate, this method is not practical when other than perfect CSI knowledge is available. In this chapter, we will focus on the results obtained from the third approach since these provide the tightest upper bounds on the true performance. The first emphasis will be placed on evaluating the pairwise error probability with and without CSI, following which we shall discuss how the results of these evaluations can be used via the transfer bound approach to evaluate average BEP of coded modulation transmitted over the fading channel. For the method in Ref. 13, we shall soon show that the use of interleaving/deinterleaving to create a memoryless fading 1 Upper bounds found by this approach are not true upper bounds but rather are approximate upper bounds because of the limited number of error events considered.

COHERENT DETECTION

761

channel results in decision statistics akin to those obtained when diversity combining is employed to enhance system performance for the uncoded communication case (see Chapter 9). Because of this analogy, we shall therefore find it possible to apply the unified approach to coded communication over the memoryless fading channel in much the same way as it was used to simplify the evaluation of performance for multiple reception of uncoded communications. In particular, by using the alternative representations of the classic functions given in Chapter 4, we shall be able to exactly evaluate the pairwise error probability in the form of a product of integrals each with finite limits and an integrand composed of tabulated functions. In the situations where the method in Ref. 13 is not practical, we shall turn to the method in Ref. 12, which requires evaluation of an integral with doubly infinite limits whose integrand is a product of characteristic functions. Although in principle, results can once again be obtained for a variety of fading channel models and modulation/coding types, to allow comparison with previously obtained results obtained by, say, the first approach, we shall specifically focus on the combination of M-PSK modulation with trellis coding. We begin by considering the case of ideal coherent detection.

13.1 13.1.1

COHERENT DETECTION System Model

Consider the block diagram of a trellis-coded M-PSK system illustrated in Fig. 13.1. Random binary i.i.d. information bits are inputted to a rate nc /(nc + 1) trellis encoder whose output symbols are then block-interleaved to break up fading bursts as per the discussion above.2 Groups of nc + 1 interleaved code symbols are mapped (in accordance with the set partitioning method of Ungerboeck [14]3 ) into M-PSK signals chosen from a set of M = 2nc +1 members. Thus, for example, a rate- 12 encoder (nc = 1) would be combined with a QPSK (M = 4) modulation. The inphase (I) and quadrature (Q) components of the mapped signal point are then modulated onto quadrature carriers (with or without pulse shaping) for transmission over the fading channel. The usual additive white Gaussian noise (AWGN) is added at the input to the receiver, which first demodulates the I and Q signal components, soft-quantizes the results of these demodulations, and then passes them through a block deinterleaver to recreate the codewords temporarily scrambled by the interleaver at the transmitter. The soft-quantized, deinterleaved code symbols are then passed to the trellis decoder, which implements a Viterbi algorithm with a metric depending on whether or not channel state information is available. For our purposes here, we shall assume that such information is either perfectly known or 2 As was done in Ref. 2, we shall assume for the purpose of analysis an infinite interleaving depth, thereby resulting in an ideal memoryless channel. In practice, the depth of interleaving would be finite and chosen in relation to the maximum duration of fade anticipated. 3 A detailed discussion of trellis-coded modulation (TCM) and its application is beyond the scope of this textbook. The reader is referred to Ref. 3 for a thorough treatment of this subject.

762

CODED COMMUNICATION OVER FADING CHANNELS

RF Carrier Input Information Bits

Trellis Encoder

Block Interleaver

Signal Set Mapping

Digital Pulse Shaper and Quadrature Modulator

Fading Channel

Output Bits

Buffer

Trellis Decoder

Viterbi Algorithm

Metric Computer

Block Deinterleaver

Q-Bit Quantizer

Quadrature Demodulator

RF Carrier

Figure 13.1 Block diagram of a trellis-coded M-PSK system.

x Encoder

Memoryless Coding Channel p(y|x, a), p(a)

y Decoder

a Figure 13.2 Simple analysis block diagram.

totally unknown without regard to the manner in which this information is obtained (e.g., via pilot tone techniques). Finally, the tentative soft decisions from the Viterbi decoder are stored in a buffer whose size is typically a design parameter (dependent on the nature of the encoded information, e.g., speech) but for simplicity of analysis will be assumed to be infinite. A mathematical model for this system can be derived from the simple analysis block diagram illustrated in Fig. 13.2. The block labeled encoder includes both the binary input/binary output trellis encoder together with mapping onto the M-PSK signal set. Hence, the output of the encoder is a succession of coded M-PSK symbols, which, for a sequence of length N , is denoted by x = (x1 , x2 , . . . , xN )

(13.1)

where the kth element of x, namely, xk , represents the normalized4 transmitted M-PSK symbol (in complex form) at time k and, in general, is a nonlinear function assume that the M-PSK symbols are normalized such that |xn | = 1, specifically, the signals lie on the perimeter of the unit circle. The actual transmitted M-PSK symbol in the nth interval is given by √ 2Es xn , where Es = nc Eb is the symbol energy with Eb denoting the information bit energy. 4 We

COHERENT DETECTION

763

of the state of the encoder sk and the information symbol uk , representing the i.i.d. information bits at its input [i.e., xk = f (sk , uk )]. The transition from state to state is defined by a similar nonlinear relation, namely, sk+1 = g (sk , uk ). Corresponding to the transmission of x, the channel outputs the sequence y = (y1 , y2 , . . . , yN )

(13.2)

where the kth element of y, namely, yk , representing the channel output at time k is given by
yk = αk 2Es xk + nk

(13.3)

Here αk is the fading amplitude for the kth transmission and nk is a zero-mean   complex Gaussian RV with variance σ 2 = N0 per dimension (i.e., E |nk |2 = 2N0 ). An observation of y reveals that the maximum-likelihood (ML) receiver chooses as the transmitted information bit sequence the one that maximizes the a posteriori probability p (u |y ) or equivalently (since the information bit sequences are equiprobable) the likelihood p (y |u ). Receivers that make decisions in this fashion are referred to as maximum-likelihood sequence estimators (MLSEs)5 and are practically implemented by the Viterbi algorithm [15]. 13.1.2

Evaluation of Pairwise Error Probability

The first step in evaluating the average error probability is to compute the pairwise error probability associated with the transmitted M-PSK symbol sequences, namely,  the probability of choosing the sequence xˆ = xˆ1 , xˆ2 , . . . , xˆN when in fact x = (x1 , x2 , . . . , xN ) was transmitted given that these are the only two possible choices. We refer to this occurrence as an error event of length N. The assumption of infinite interleaving/deinterleaving, namely, an ideal memoryless channel, allows one to express the channel probabilities as p (y |x, α ) =

N 

p (yn |xn , αn )

(13.4)

n=1

and p (α) =

N 

p (αn )

(13.5)

n=1 5

Strictly speaking, the term maximum-likelihood sequence estimator is reserved for sequences whose length N approaches infinity. However, it has become common practice to use this terminology even when the sequence length is finite since practical implementations of the ML decision rule such as the Viterbi algorithm begin making decisions prior to an infinite observation time interval.

764

CODED COMMUNICATION OVER FADING CHANNELS

Since conditioned on αn and xn , yn of (13.3) is a Gaussian RV with PDF    √ yn − αn 2Es xn 2 1 exp − p (yn |xn , αn ) = 2π σ 2 2σ 2

(13.6)

then, for the case of perfectly known channel state information, substituting (13.6) in (13.4) and taking the natural logarithm of the result gives the decision metric (ignoring unnecessary scale factors) 

m (y, x;α) =

N

m (yn , xn ; αn ) = −

n=1

N  2


  yn − αn 2Es xn 

(13.7)

n=1

In the absence of channel state information, the optimum decision metric would be obtained by averaging (13.4) over (13.5) to obtain p (y |x ) and then again taking the natural logarithm. This procedure would yield a decision metric whose form depends on the actual fading PDF assumed (see Chapter 7). To simplify matters, we propose (as was done in Ref. 1) a suboptimum decision metric that treats the channel as if it were purely Gaussian, namely 

m (y, x) =

N

N  2


  m (yn , xn ) = − yn − 2Es xn 

n=1

n=1

(13.8)

and that therefore is not a function of the fading sequence α = (α1 , α2 , . . . , αN ). We now proceed to evaluate the pairwise error probability for these two extreme cases of channel state information. 13.1.2.1 Known Channel State Information   Since for the decision metric of (13.7), the sequence  xˆ = xˆ1 , xˆ2 , . . . , xˆN would be chosen over x = (x1 , x2 , . . . , xN ) whenever m y, xˆ ;α ≥ m (y, x;α), then the pairwise error probability (i.e., the probability of an error event of length N ) for this case is given by  N N

 

(13.9) P (x → xˆ |α) = Pr m yn , xˆn ; αn ≥ m (yn , xn ; αn ) |x n=1

n=1

where the conditioning on x in the right-hand side indicates the fact that the components of the observation y are to be computed assuming that x was transmitted. Substituting (13.7) into (13.9) and recalling that both x and xˆ have components with unit squared magnitude, we obtain  N N



   ∗ ∗ P (x → xˆ |α) = Pr αn Re yn xˆn ≥ αn Re yn xn |x  = Pr

n=1

n∈η

n=1

  ∗  αn Re yn xˆn − xn ≥ 0 |x



(13.10)

COHERENT DETECTION

765

where η is the set of all n for which xˆn = xn . On the basis of (13.3), the decision variable to be compared with the zero threshold in (13.10) is Gaussian 2   ∗  √

and variance N0 n∈η αn2 xˆn − xn  . with mean 2Es n∈η αn2 Re xn xˆn − xn   2 Thus, since for constant envelope signal sets such as M-PSK where |x|2 = xˆ  , it is straightforward to show that       x − xˆ 2 = 2 Re x x − xˆ ∗

(13.11)

then the pairwise error probability immediately evaluates to  P (x → xˆ |α) = Q

2 Es 2  αn xˆn − xn  2N0 n∈η

(13.12)

which has a form analogous to that obtained for the probability of error of uncoded BPSK transmitted over a multichannel with maximal-ratio combining (MRC) employed at the receiver 2 (see Chapter 9). In fact, for convolutionally encoded BPSK where xˆn − xn  = 4 for all n ∈ η, (13.12) would simplify to  P (x → xˆ |α) = Q

2Es 2 α N0 n∈η n

(13.13)

and hence the number of diversity channels for the uncoded application can be seen to be directly analogous to the number of symbols that are in error in the coded case. In Ref. 1, the conditional (on the fading) pairwise error P (x → xˆ |α) was upper (Chernoff)-bounded to allow averaging over the fading statistics (a feat not possible using the classic definition of the Gaussian Q-function), thereby obtaining a closedform upper bound on P (x → xˆ ). In particular, it was shown there that P (x → xˆ ) ≤ D

 α d 2 x,xˆ

(13.14)

where the overbar denotes statistical averaging over the random vector α, 2    2 αn xˆn − xn  d 2 x, xˆ =

(13.15)

n∈η

and where   Es  = D exp − 4N0

(13.16)

766

CODED COMMUNICATION OVER FADING CHANNELS

is the so-called Bhattacharyya parameter [7]. Since the fading amplitudes are independent, it was then possible to write (13.14) as the product 

P (x → xˆ ) ≤

D αn |xˆn −xn | 2

α 2 n

(13.17)

n∈η

which is in a form that lends itself to the application of the transfer function bound approach for computing an upper bound on average BEP. The individual terms of the product in (13.17) were evaluated in Ref. 1 for Rayleigh and Rician fading amplitude statistics. Since it was demonstrated in Chapter 4 that the alternative form of the Gaussian Q-function has the analytic advantages of the Chernoff bound without the disadvantage of being a bound, it is reasonable to apply that approach here to the conditional pairwise error probability in much the same manner as it was used in Chapter 9 to unify and simplify the analysis of multichannel reception of BPSK with MRC. In particular, using the alternative form of Q(x) given in (4.2), we can express (13.13) as 1 P (x → xˆ |α ) = π 1 π

=



π/2

 exp −

0



Es



4N0 sin2 θ

n∈η

  2 αn2 xˆn − xn  d θ



 d 2 x,xˆ

π/2

[D (θ )]

dθ

(13.18)

0

where analogous to (13.15)   D (θ ) = exp −



Es

(13.19)

4N0 sin2 θ

Hence, the unconditional pairwise error probability is given by 1 P (x → xˆ ) = π 1 = π



α

π/2

d 2 (x,ˆx)

[D (θ )] 

dθ =

0 π/2 0



D (θ )

α 2 n

αn2 |xˆn −xn |

1 π



π/2

α

[D(θ )]d 2 (x,ˆx) d θ 0

(13.20)

dθ

n∈η

which is to be compared with the upper bound in (13.17). The statistical average required by each term in the product of (13.20) can be written as D (θ )

αn2 |xˆn −xn |

α 2 n

 = 0

∞

 exp −

αn2 Es 4N0 sin2 θ

   xˆn − xn 2 pa (αn ) d αn n

(13.21)

COHERENT DETECTION

767

Since  for a rate nc /(nc + 1) trellis encoder nc information bits produces one M = 2nc +1 -ary symbol, then, in terms of the bit energy-to-noise ratio (Eb /N0 ), (13.21) can be rewritten as    ∞ α 2 2 n nc α 2 Eb  αn2 |xˆn −xn |  xˆn − xn pa (α) d α D (θ ) = exp − (13.22) 4N0 sin2 θ 0 where we dropped the n subscript on α since the fading variables are all identically  distributed. Alternatively, in terms of the instantaneous SNR per bit γ = α 2 Eb /N0 , (13.22) becomes   2  ∞ α 2 n nc xˆn − xn  αn2 |xˆn −xn | pγ (γ ) d γ D (θ ) = exp −γ (13.23) 4 sin2 θ 0 Integrals of the form in (13.23) were considered in Chapter 5 for a wide variety of fading channel types. For example, for Rayleigh fading, using (5.5), we would obtain6 α 2 n

D (θ )αn |xˆn −xn | 2

=

1+

1 = 2 nc |xˆn −xn | γ 4 sin2 θ

sin2 θ 2 n |xˆ −x | γ sin2 θ + c n 4 n

(13.24)



where γ = α 2 Eb /N0 is the average SNR per bit. For Rician fading, using (5.11) we would obtain   2 nc |xˆn −xn | γ α n K 4 sin2 θ 2 1+K 2   (13.25) D (θ )αn |xˆn −xn | = exp − 2 2  nc |xˆn −xn | γ nc |xˆn −xn | γ 1 + K + 4 sin2 θ 1 + K + 4 sin2 θ which clearly reduces to (13.24) for K = 0. Finally, for Nakagami-m fading, using (5.15), the analogous expression to (13.24) and (13.25) becomes  m α 2 2 n θ 1 sin 2  D (θ )αn |xˆn −xn | = (13.26) m =  2 2 nc |xˆn −xn | γ nc |xˆn −xn | γ 2 sin θ + 1+ 4m 2 4m sin θ

Note that had one chosen to use the upper bound on pairwise error probability (as was done in Ref. 1) of (13.17) rather than the exact result of (13.20), the terms that would be required for the  product in the former equation would be obtained simply by setting θ = π /2 sin2 θ = 1 in (13.24), (13.25), and (13.26), respectively as discussed in Chapter 4: is D αn |xˆn −xn | 2

α 2 n

α 2 n

= D (θ )αn |xˆn −xn | 2

 θ =π/2

(13.27)

 2 the special case of BPSK modulation wherein xn − xˆn  = 4, a result for the pairwise error probability similar to (13.20) together with (13.24) was obtained by Hall and Wilson [16]. 6 For

768

CODED COMMUNICATION OVER FADING CHANNELS

13.1.2.2 Unknown Channel State Information When channel state information is not available, then the expression for the pairwise error probability analogous to (13.9) becomes P (x → xˆ |α) = Pr

 N

N  

m yn , xˆn ≥ m (yn , xn ) |x

n=1

(13.28)

n=1

Substituting (13.8) into (13.28) and recalling that both x and xˆ have components with unit squared magnitude, we obtain  P (x → xˆ |α) = Pr







Re yn xˆn − xn

∗ 

≥ 0 |x

(13.29)

n∈η

where η is the set of all n for which xˆn = xn . On the basis of (13.3), the decision variable to be compared with the zero threshold in (13.29) is now Gaussian 2 with   ∗  √



and variance N0 n∈η xˆn − xn  . Once mean n∈η αn 2Es Re xn xˆn − xn again using the relation in (13.11), which is valid for constant envelope signal sets such as M-PSK, the pairwise error probability immediately evaluates to      2 2     Es  n∈η αn xˆ n − xn   P (x → xˆ |α) = Q  2

  2N0   n∈η xˆ n − xn   2      E

 xˆn − xn 2  s   = Q  αn    2    2N0 n∈η   k∈η xˆ k − xk

(13.30)

The pairwise error probability in (13.30) has a form analogous to that obtained for the probability of error of uncoded BPSK transmitted over a multichannel with equal gain combining (EGC) employed at the receiver (see Chapter 9). In fact, for 2 convolutionally encoded BPSK where xˆn − xn  = 4 for all n ∈ η, (13.30) would simplify to   2    2Es 1

P (x → xˆ |α) = Q  αn  N0 Lη n∈η

(13.31)

where Lη is the number of elements in the set η or equivalently the Hamming distance between the correct and incorrect sequences. Hence, the number of diversity channels for the uncoded application can again be seen to be directly analogous to the number of symbols that are in error in the coded case.

COHERENT DETECTION

769

To obtain the unconditional pairwise error probability, one must average (13.31) over the i.i.d. fading statistics of the αn s. In particular, defining   xˆn − xn 2 

= αT = αn   αn dn2 2   n∈η n∈η k∈η xˆ k − xk 



(13.32)

then the Lη -fold integral obtained by averaging (13.31) over the fading amplitudes can be collapsed to a single integral: 



∞

P (x → xˆ ) =

Q 0

Es 2 α 2N0 T

pαT (αT ) d αT

(13.33)

Applying the alternative representation of the Gaussian Q-function to (13.33) gives

1 P (x → xˆ ) = π



π/2 0



∞

 exp −

0

Es 4N0 sin2 θ

 αT2

pαT (αT ) d αT d θ

(13.34)

Direct evaluation of (13.34) requires finding the PDF of αT defined in (13.32), which, in general, is difficult even when the αn s are i.i.d. Instead, we follow the procedure given in Chapter 9 by first representing pαT (αT ) in terms of its characteristic function. Thus   Lη  π/2  ∞    1  ψαl j vdl2  P (x → xˆ ) = 2π 2 0 −∞ l=1

  J (v,θ) # $% &    ∞  Es   2 × α exp − − j vα d α  dv dθ T T T 2  0  4N0 sin θ

(13.35)

Despite the apparent similarity between (13.35) and Eq. (9.61), the difficulty here lies in the fact that the weight dl2 of (13.32) that appears in the argument of the lth characteristic function does not depend only on the squared Euclidean distance for the lth branch of the error sequence but because of its normalization also depends on the total squared Euclidean distance of the entire error sequence. Even in the case of BPSK modulation, each of these weights would be normalized by the length Lη of the error sequence. Because of this, it will not be possible to obtain an integral form for P (x → xˆ ) where the integrand is a product of terms, each of which depends only on the squared Euclidean distance associated with that term. Thus, we abandon this method for the case of unknown CSI and instead turn to the inverse Laplace transform method first introduced

770

CODED COMMUNICATION OVER FADING CHANNELS

for problems of this type by Cavers and Ho [9]7 with additional generalizations reported in Ref. [12], and additional methods for evaluation later explored by Biglieri et al. [17,18]. 8 Consider z with PDF p (z) and moment ( ∞a RV ( z generating function (MGF) sz Mz (s) = −∞ e pz (z) dz. Then the CDF P (z) = −∞ p (y) dy is related to Mz (s) by (see Chapter 9) 1 P (z) = 2πj



σ +j ∞ σ −j ∞

Mz (−s) sz e ds s

(13.36)

where σ is chosen such that the vertical line of integration lies in the region of convergence (ROC) of the bilateral Laplace transform. Since P (a) = Pr {z ≤ a}, then Pr {z ≤ 0} = P (0) =

1 2πj



σ +j ∞ σ −j ∞

Mz (−s) ds s

(13.37)

Methods for evaluating integrals of the type in (13.37) are discussed in detail in Appendix 9B. Hence, we shall draw on these results only when needed to perform numerical evaluations. When applied to the difference decision metric ) *  N 

1 z= N n=1 2 m (yn , xn ) − m yn , xˆ n |x = n=1 zn , (13.37) results in the evaluation of the pairwise error probability. Since, as we have already discussed, the interleaving/deinterleaving operation makes the zn terms independent, then denoting the MGF of zn by Mzn (s), we obtain from (13.37) that

P (x → xˆ ) =

1 2πj



σ +j ∞ σ −j ∞

N 1 Mzn (−s) d s s

(13.38)

n=1

   √    ∗  Using (13.3), the RV zn = Re yn xn − xˆn ∗ |x = Re αn 2Es xn xn − xˆn +  ∗   nn xn − xˆn |x is conditionally (on the fading) Gaussian with mean µzn |αn =  2  √  ∗   Re αn 2Es xn xn − xˆn and variance σz2n |αn = N0 xˆn − xn  . Thus, using 7 Cavers and Ho [9] refer to this method as the characteristic function method. As we shall see momentarily, in keeping with the distinction made in Chapter 5 between the moment generating function and the characteristic function of a RV in terms of their relations to the Laplace and Fourier transforms, respectively, their method [9] is more appropriately called a moment generating function method. Also, in their definition of characteristic function, Cavers and Ho [9] do not reverse the sign of the exponent in the Laplace transform and as such are not consistent with the traditional definition of this function. 8 The definition of moment generating function used here is the generalization of that introduced in Chapter 5 to the case where the underlying RV takes on both positive and negative values. As such the MGF is now the bilateral (as opposed to unilateral) Laplace transform of the PDF with the sign of the exponent reversed.

COHERENT DETECTION

771

(13.11), the unconditional RV zn has PDF 

∞

pzn (zn ) = 

pzn (zn |αn ) pan (αn ) d αn

0 ∞

1  2 2π N0 xˆn − xn     2  √ zn − 12 αn 2Es xˆn − xn   pan (αn ) d αn (13.39) × exp −  2 2N0 xˆn − xn 

=



0

Since for a Gaussian RV Y with mean µY and variance σY2 the MGF is given by MY (s) = e sµY +σY s

2 2 /2

(13.40)

then applying (13.40) to the conditional Gaussian RV zn |αn , we have from (13.39) that the MGF of zn is  Mzn (s) = =

∞

−∞  ∞

pzn (zn ) e szn d zn
 2  2 e (1/2) sαn 2Es xˆn − xn  + (1/2)N0 xˆn − xn  s 2 pan (αn ) d αn

0

+
, 2 α n = e (1/2) sαn 2Es + N0 s 2 xˆn − xn 

(13.41)

or in terms of the MGF of the fading random variable αn = α  √   xˆn − xn 2   2E s 2 Mzn (s) = e (1/2) N0 s 2 xˆn − xn  Mα s 2

(13.42)

Finally, substituting (13.42) into (13.38) gives the pairwise error probability in the desired product form P (x → xˆ ) =

1 2πj



σ +j ∞

σ −j ∞

 2 
  , 1 + (1/2) e N0 s 2 xˆn − xn  Mα −s 2Es xˆn − xn 2 /2 d s s n∈η

(13.43) where the nth term of the product depends only on the squared Euclidean distance for the nth branch of the sequence and not on the distance properties of the entire sequence as in (13.35). Also, in (13.43) it has again become possible to replace the product over all branches by the product over only thosefor which xˆn = xn (i.e.,  2  √ n ∈ η) since, for the terms where xˆn = xn , the MGF Mα s 2Es xˆn − xn  /2 is equal to unity.

772

13.1.3

CODED COMMUNICATION OVER FADING CHANNELS

Transfer Function Bound on Average Bit Error Probability

To compute the true upper (union) bound (TUB) on the average BEP, one sums the pairwise error probability over all error events (sequence pairs) corresponding to a given transmitted sequence, weighting each term by the number of information bit errors associated with that event, then statistically averages this sum over all possible transmitted sequences finally dividing by the number of input bits per transmission. In mathematical   terms, if P (x) denotes the probability that the sequence x is transmitted, n x, xˆ the number of information bit errors committed by choosing xˆ instead of x, and nc the number of information bits per transmission, then the average BEP has the TUB Pb (E) ≤

    1

P (x) n x, xˆ P x → xˆ nc x

(13.44)

x=xˆ

An efficient method for computing this weighted sum was originally proposed by Viterbi [6] for convolutional codes transmitted over the AWGN channel and is referred to as the transfer function  bound approach. To apply this approach one must be able to write P x → xˆ in a product form where the nth term of the product is associated with the nth branch of the particular path through the state diagram defined by the error event. Also, evaluation of this nth term must depend only on the distance between the nth branch of the correct and incorrect sequences and not on the distance properties of the entire sequence. Alternatively, one can write P x → xˆ in the form of an integral whose integrand is a product of terms satisfying the above-mentioned condition. In this case, it is possible to evaluate the TUB on average BER by first applying the transfer function bound approach to the integrand (conditioned on the integration variable) and then performing the required integration. For trellis codes, we have seen that the appropriate distance measure is Euclidean   distance and thus for the AWGN channel P x → xˆ would take the form [see (13.20), omitting the averaging on the fading]9  2 1 π/2  P (x → xˆ ) = D (θ )|xˆn −xn | d θ (13.45) π 0 n∈η   where D (θ ) is still  defined as in (13.19). To incorporate n x, xˆ into the product, we define n xn , xˆn as the number of bit errors in the nth interval of the error event,     in which case n x, xˆ = N n=1 n xn , xˆ n . Then, defining an indicator variable I , we can rewrite the second summation in (13.44) as 

    1 π/2 ∂  2 n x, xˆ P x → xˆ = D (θ )|xn −xˆn | I n(xn ,xˆn ) |I =1 d θ π 0 ∂I n x=xˆ

x=xˆ

9 From hereon in, for simplicity of notation, we shall denote the product over the N branches of an error event by n with the understanding that it need only be taken over those branches for which an error occurs.

COHERENT DETECTION

1 = π 1 = π 

773





π/2

0



π/2



 2 ∂  D (θ )|xn −xˆn | I n(xn ,xˆn ) |I =1  d θ ∂I n .

0

x=xˆ

/ ∂ T (D (θ ) , I ) |I =1 d θ ∂I

(13.46)

where T (D, I ) is the transfer function associated with the error state diagram [3] of a particular TCM scheme and in general depends on the transmitted sequence x. The form of T (D, I ) is typically a ratio of polynomials in D and I , as will become clear when we consider some examples. Finally, combining (13.44) and (13.46) gives the TUB on average BER for the AWGN channel: . /  1 π/2 ∂ 1

Pb (E) ≤ P (x) T (D (θ ) , I ) |I =1 d θ nc x π 0 ∂I

(13.47)

For a large class of trellis codes, a symmetry property exists such that for the purpose of evaluation of the TUB, the correct sequence x can always be chosen as the all-zeros sequence, thus avoiding the necessity of averaging over all possible transmitted code sequences in (13.44). Codes of this type are referred to as uniform error probability (UEP) codes and will be the only ones considered in this chapter, although the generic methods to be discussed also, in principle, apply when the UEP criterion is not valid. Thus, for UEP TCM schemes, (13.44) simplifies to 1 Pb (E) ≤ π

π/2 .

 0

/ 1 ∂ T (D (θ ) , I ) |I =1 d θ nc ∂I

(13.48)

Had we applied the Chernoff bound to the pairwise error probability rather than obtain its exact form, then the equivalent result to (13.47) would become Pb (E) ≤

 ∂ 1

P (x) T (D, I ) I =1,D=exp(−Es /4N0 ) nc x ∂I

(13.49)

or for UEP TCM schemes Pb (E) ≤

 1 ∂ T (D, I ) I =1,D=exp(−Es /4N0 ) nc ∂I

(13.50)

both of which are looser than the TUB. To find the TUB on the average BEP for TCM transmitted over the fading channel, we simply substitute in (13.44) the expressions found in Section 13.1.2.1 or 13.1.2.2 for the pairwise error probability corresponding to the cases of perfectly known CSI or unknown CSI, respectively. We now present the specific results for the two cases of CSI knowledge.

774

CODED COMMUNICATION OVER FADING CHANNELS

13.1.3.1 Known Channel State Information Comparing the integrand of (13.20) with (13.45), we observe the analogy between α 2 n 2 2 D (θ )αn |xˆn −xn | of the former and D (θ )|xn −xˆn | of the latter. Thus, on the basis of the discussion above, the average BEP for trellis coded M-PSK transmitted over the slow fading channel has a TUB analogous to (13.48): 10 Pb (E) ≤

1 π

 0

π/2

  1 ∂  T D (θ ), I I =1,D(θ )=e−Es /4N0 sin2 θ d θ nc ∂I

(13.51)

The averages over the fading required in (13.51) have been evaluated in closed form in Section 13.1.2.2; for example, see (13.24) for Rayleigh fading, (13.25) for Rician fading, and (13.26) for Nakagami-m fading. 13.1.3.2 Unknown Channel State Information For this case, the appropriate product form of the pairwise error probability integrand is (13.43). Thus, by analogy with (13.48), the TUB on the average BEP is given by 1 Pb (E) ≤ 2πj



σ +j ∞ σ −j ∞

. / 1 1 ∂ T (s, I ) |I =1 d s s nc ∂I

(13.52)

where T (s, I ) is the transfer function computed from the state transition diagram for 2 the AWGN channel with the label D (θ )|xn −xˆn | on each branch between transitions replaced by 
  2    2  2  D s; xˆn − xn  = e (1/2) N0 s 2 xˆn − xn  Mα −s 2Es xˆn − xn  /2 13.1.4

(13.53)

An Alternative Formulation of the Transfer Function Bound

A variation of the previous approach to computing the transfer function was introduced by Divsalar [19] and is referred to as the pair-state method. It is particularly useful for non-UEP codes since it circumvents the averaging over the transmitted code sequences by incorporating it in the transfer function itself. In this method,   a pair-state transition diagram is constructed wherein each pair-state Sk = sk , sˆk corresponds to a pair of states sk and sˆk in the trellis diagram. One is said to be in a correct pair-state when sk = sˆk and  an incorrect pair-state when sk = sˆk . A transition between pair-states Sk = sk , sˆk and Sk+1 = sk+1 , sˆk+1 in the transition diagram corresponds to a pair of transitions in the trellis diagram (i.e., sk to sˆk and sk+1 to sˆk+1 ). Since associated with each transition in the pair is an M-PSK symbol and a corresponding information symbol (a sequence of nc information bits), the transition between two pair-states in the transition diagram is characterized by the 10 The

implication of the simple notation D (θ) is that the label D (θ)|xn −xˆn | on each branch between 2

transitions be replaced by D (θ)αn |xn −xˆn | 2

α 2 n

.

COHERENT DETECTION

775

squared δ 2 between the corresponding M-PSK symbols and the Hamming distance dH between the corresponding information bit sequences. From the discussion above, in the absence of fading (i.e., the AWGN channel), each branch between pair-states in the transition diagram has a gain G of the form G=

1 2 I dH D δ n c 2

(13.54)

where, as before, I is an indicator variable and D is the Bhattacharyya parameter defined in (13.19). The summation in (13.54) accounts for the possibility of parallel paths between states of the trellis diagram. Since the pair-state method accounts for all possible transmitted symbol sequences and their probability, then the union–Chernoff bound would be given by (13.50) (which was formally restricted to UEP codes) where the transfer function T (D, I ) is now computed from the pair-state transition diagram on the basis of the gains of (13.54). Extending this approach to exact evaluation of the pairwise error probability, if instead of (13.54) we were to label each branch with a gain G (θ ) =

1 2 I dH D (θ )δ n c 2

(13.55)

with D (θ ) as in (13.19), then the TUB would be given by (13.48), where again the transfer function T (D (θ ) , I ) is now computed from the pair-state transition diagram on the basis of the gains of (13.55). In the presence of fading, the appropriate substitutions for D and D (θ ) as discussed in Section 13.1.3, would result in the upper bounds on average BEP. Since our interest is in the TUB, we shall consider only the case where the pair-state gains are as in (13.55) for the AWGN and their equivalences for the fading channel. 13.1.5

An Example

Consider the case of rate- 12 -coded QPSK using a two-state trellis. The signal constellation and appropriate set partitioning [14] are illustrated in Fig. 13.3 and the corresponding trellis diagram and pair-state transition diagram are shown in Figs. 13.4 and 13.5, respectively. The dashed branch in the trellis diagram corresponds to a transition resulting from a “0” information bit, whereas the solid branch corresponds to a transition resulting from a “1” information bit. The branches are labeled with the M-PSK output symbol that is transmitted because the abovementioned information bit is input to the encoder. The branches of the pair-state transition diagram are labeled with the gains computed from (13.56). For example, for the transition from the pair-state “0,0” to the pair-state “0,1” in Fig. 13.5 or equivalently the pair of transitions from state “0” to state “0” and state “0” to state “1” in Fig. 13.4 the corresponding output M-PSK symbols are “0” and “2.” Since the signal constellation is normalized to unit radius circle, then, from Fig. 13.3, the squared Euclidean distance between symbols “0” and “2” is δ 2 = 4. Similarly, the transition from state “0” to state “0” is the result of transmitting a single “0” information bit, whereas the transition from state “0” to state “1” is the

776

CODED COMMUNICATION OVER FADING CHANNELS

1

2

0

0

1

3

1

2

0

3 0

1

0

1

1

0

2

3 Figure 13.3 Set partitioning of QPSK signal constellation.

0

0

2

1

3

0 1

0

0 1

1

Figure 13.4 Trellis diagram and QPSK signal assignment.

result of transmitting a single “1” information bit. Thus, the Hamming distance between these two information bits is dH = 1. Since there are no parallel branches in the trellis diagram and nc = 1, then, in accordance with (13.55), the gain associated with the transition from pair-state “0,0” to pair-state “0,1” is a = 12 I D 4 (θ ) (see Fig. 12.5). The gains b = 12 I D 2 (θ ) and c = 12 D 2 (θ ) follow from similar considerations.

777

COHERENT DETECTION

ξe

ξc ξa

b 0,0 1/2

a

c

0,1

a

I

F

b c

1/ 2

1,1

1

c b

a

0,0

1,0

a

ξI = 1

1,1

1 ξF

b ξb ξd

ξf

Figure 13.5 Pair-state transition diagram.

Defining the states of the pair-state diagram by ξ with the input state having value unity, the transfer function is obtained by solving the following set of state equations: T (D (θ ) , I ) = ξe + ξf ξe = ξf = c (ξc + ξd ) , ξa = ξb =

ξc = ξd = b (ξc + ξd ) + a (ξa + ξb )

1 2

(13.56)

resulting in T (D (θ ) , I ) =

4ac I D 6 (θ ) = 1 − 2b 1 − I D 2 (θ )

(13.57)

Thus, for the AWGN channel, the TUB on average BEP would be given by 1 Pb (E) ≤ π =

1 π



π/2 0

 0

π/2

.

/  d 1 π/2 D 6 (θ ) T (D (θ ) , I ) |I =1 d θ =  2 d θ dI π 0 1 − D 2 (θ ) 1 0 exp − 2N3Esinb 2 θ 0 (13.58)  0 12 d θ b 1 − exp − 2N Esin 2θ 0

778

CODED COMMUNICATION OVER FADING CHANNELS

For the fading channel with known channel state information, the transfer function would become 

α



T D (θ ), I =

I D 2α 2(θ ) D 4α 2(θ ) 1 − I D 2α 2(θ )

α

α

(13.59)

from which we would obtain the TUB 1 Pb (E) ≤ π



π/2 0

α

α

D 2α 2(θ ) D 4α 2(θ ) dθ  α 2 1 − D 2α 2(θ )

(13.60)

where the statistical averages are obtained from (13.24), (13.25), and (13.26) for Rayleigh, Rician, and Nakagami-m channels, respectively. For example, for Rayleigh fading, using (13.24) gives the simple TUB     sin2 θ + γ2 1 π/2 2 2 4 dθ (13.61) sin θ Pb (E) ≤ π 0 γ sin2 θ + γ As a check on the results, the union–Chernoff bound would be obtained by replacing the integrand in (13.61) by its value at θ = π /2, resulting in  γ 2 1+ 2 Pb (E) ≤ 2 (13.62) 1+γ γ which agrees11 with [1, Eq. (49)]. For the case of no channel state information, the branch gains of Fig. 13.5 would become   

I I 2 2 a = e 2N0 s Mα −2s 2Es , b = e N0 s Mα −s 2Es 2 2 (13.63) 
 1 N0 s 2 Mα −s 2Es c= e 2 and thus from (13.57), the transfer function is given by   √   2N s 2   √ 2 e 0 Mα −2s 2Es I e N0 s Mα −s 2Es T (s, I ) =   √  1 − I e N0 s 2 Mα −s 2Es   √    √ 2 I e 3N0 s Mα −s 2Es Mα −2s 2Es =   √  1 − I e N0 s 2 Mα −s 2Es

(13.64)

11 The result in (13.63) is actually one-half of the result in Eq. (49) of Ref. 1 since the bound on the alternative representation of the Gaussian Q-function obtained by replacing the integrand by its value θ = π /2 (see Chapter 4) is one-half of the result obtained from the conventional Chernoff bound.

COHERENT DETECTION

with the corresponding TUB  √   3 √  σ +j ∞ 2 3N0 s 2 Mα −s 2Es Mα −2s 2Es 1 e 1 ds Pb (E) ≤ )  √ *2 2πj σ −j ∞ s 1 − e N0 s 2 Mα −s 2Es

779

(13.65)

For even the simplest of fading channels (e.g., Rayleigh), the MGF is not available in a simple form involving elementary functions that lend themselves to integration. Nevertheless, as we shall see momentarily, for the Rayleigh fading channel it is still relatively straightforward to obtain results using the method of Gauss–quadrature (in particular, Gauss–Hermite) integration. The procedure is as follows. The MGF of a Rayleigh RV with mean-square value α 2 = is given by √ /    2 . √ 1 1 s2 π s − + s (13.66a) , ; − F Mα (s) = exp 1 1 4 2 2 4 2 where 1 F1 (·, ·; ·) is the Kummer confluent hypergeometric function [20, p. 504, Eq. (13.2)] or equivalently using the relation between 1 F1 (·, ·; ·) and the complementary error function erfc (·):  √ √  2  s π √ s s exp erfc − (13.66b) Mα (s) = 1 + 2 4 2 √ Renormalizing the complex integration variable in (13.65) as ξ = s N0 γ , the TUB can be written as 3  σ +j ∞ 2 3ξ 2 /γ 1 e Mα (−ξ ) Mα (−2ξ ) 1 Pb (E) ≤ dξ (13.67) ) *2 2πj σ −j ∞ ξ 1 − e ξ 2 /γ Mα (−ξ ) √ where now σ = σ N0 γ and 4    2 1 1 ξ2 π ξ  A (ξ ) ; A (ξ ) = 1 F1 − , ; − +ξ (13.68a) Mα (ξ ) = exp 2 2 2 2 2 or equivalently 4 Mα

(ξ ) = 1 +

 2   ξ ξ π ξ exp erfc − √ 2 2 2

(13.68b)

In order to evaluate the bound on average BEP in (13.67), one has the option of using either of two Gauss–quadrature methods. First, substituting (13.68a) into (13.67) and simplifying gives   Pb (E) ≤

1 2πj



σ +j ∞

σ −j ∞





ξ 2 γ γ+3 1 A (−ξ ) A (−2ξ )   e  2  d ξ   γ +2  ξ ξ 2 2γ 1−e A (−ξ )

(13.69)

780

CODED COMMUNICATION OVER FADING CHANNELS

To evaluate the integration along the vertical line in (13.69), we recognize that along this line the complex integration variable can be expressed as ξ = σ + j ω, where σ is fixed and ω varies from −∞ to ∞. Thus, making this change of variables in (13.69) gives     2 γ +3      ∞  e (σ +j ω) γ A −σ − j ω A −2σ − 2j ω  1 1   Pb (E) ≤  dω  2    2 γ +2  2π −∞ σ + j ω  σ +j ω ( ) 2γ 1−e A (−σ − j ω) (13.70) which is of the form12    ∞  ∞ 1 x 2 2 dx Pb (E) ≤ e −aω f (ω) d ω = √ e −x f √ a −∞ a −∞

(13.71)

and thus can be evaluated by Gauss–Hermite integration [20, p. 890, Sect. 25.4.46], namely 

∞

−∞



exp −x

2



 f

x √ a



dx ∼ =

Np

 Hxn f

n=1

xn √ a

 (13.72)

where {xn } are the Np zeros of the Np -order Hermite polynomial HNp (x) and Hxn are corresponding weight factors defined by 

Hxn = 

√ 2Np −1 Np ! π 2 ) *2 Np HNp −1 (xn )

(13.73)

The zeros and the weight factors are both tabulated in Ref. 20 (p. 924, Table 25.10) for various polynomial orders. Typically, Np = 20 is sufficient for excellent accuracy. The second approach to evaluating an upper bound on average BEP is to substitute (13.68b) into (13.67) and use the Gauss–Chebyshev method of Appendix 9B. In particular 3  σ +j ∞ 2 3ξ 2 /γ  σ +j ∞ 1 e Mα (−ξ ) Mα (−2ξ ) 1 1  1 = f (ξ ) d ξ d ξ ) *2 2 /γ 2πj σ −j ∞ ξ 2πj ξ ξ σ −j ∞ 1−e M (−ξ ) α

1 ∼ = n

n/2

k=1

)

     * Re f σ + j σ τk + j Im f σ + j σ τk ,

τk = tan

(2k − 1) π 2n (13.74)

12 Note

that f (ω) is a complex function of ω and thus it might appear that the upper bound is also 2 complex. However, the imaginary part of f (ω) will be an odd function of ω and thus since e−aω is an even function of ω, the imaginary part of the integral will evaluate to zero.

DIFFERENTIALLY COHERENT DETECTION

781

where the choices of n and σ are discussed in Appendix 9B.13 This approach is perhaps the simpler of the two in that it does not involve computation of the zeros and weight factors of the Hermite polynomials. The MGF of a Nakagami-m RV with mean-square value α 2 = is given by   2 2 1 1 2 − m, ; −s s 2 2 Mα (s) = exp 1 F1 4m 4m    3 √  m + 12 3 s2 +s √ (13.75) 1 F1 1 − m, ; − 2 4m m (m) √ Again renormalizing the integration variable in (13.65) as ξ = s N0 γ , the TUB can be written as in (13.67), where now  2 ξ Mα (ξ ) = exp A (ξ ; m) (13.76) 2m with 4

    ξ2 3 2  m + 12 +ξ A (ξ ; m) = 1 F1 1 F1 1 − m, ; − m  (m) 2 2m (13.77) [Note that for m = 1 (i.e., Rayleigh fading), A (ξ ; 1) reduces to A (ξ ) of (13.68a), as it should.] It should now be obvious that the TUB on average BEP is given by (13.70) with the appropriate substitution of A (ξ ; 1) for A (ξ ), which again can be evaluated using the Gauss–Hermite integration method. 

13.2 13.2.1

1 ξ2 1 − m; ; − 2 2 2m



DIFFERENTIALLY COHERENT DETECTION System Model

Consider the block diagram of a trellis-coded M-DPSK system14 illustrated in Fig. 13.6. The only difference between this block diagram and that of Fig. 13.1 is the inclusion of a differential (phase) encoder prior to the modulator and the replacement of the coherent demodulator by a differential (phase) detector. Thus, if xk still denotes the kth trellis-coded M-PSK symbol corresponding to the information symbol uk , then the actual M-PSK symbol transmitted over the channel is15 vk = vk−1 xk

(13.78)

13 Specifically, the function f (ξ ) in (13.74) should be numerically minimized and the resulting value of ξ > 0 at which this minimum occurs is then equated to σ . Note that the minimization must be performed for each γ , and thus the value of σ used to evaluate (13.74) is a function of γ . 14 By “M-DPSK,” we refer in this chapter to the conventional (two-symbol observation) form of differentially detected M-PSK. 15 Note that the transmitted M-PSK symbols are still normalized such that |v | = 1. k

782

CODED COMMUNICATION OVER FADING CHANNELS

RF Carrier Input Information Bits

Differential Encoder Trellis Encoder

Block Interleaver

Digital Pulse Shaper and Quadrature Modulator

Signal Set Mapping Delay Ts

Fading Channel

Differential Detector

Output Bits Trellis Decoder

Block Deinterleaver

Q-Bit Quantizer

Quadrature Demodulator Delay Ts RF Carrier

Figure 13.6

Block diagram of a trellis-coded M-DPSK system.

Analogous to (13.3), the fading channel output at time k is
wk = αk 2Es vk + nk

(13.79)

where the noise sample nk has the same properties as for the coherent detection case and the output of the differential detector is ∗ yk = wk wk−1

(13.80)

If we again assume that the fading is independent from symbol to symbol, then we can represent the combination of the differential encoding/detection operations and the fading channel as a memoryless coding channel whose input is the information M-PSK symbol xk and whose output is the decision variable yk . As such, Fig. 13.2 also represents a simple block diagram of the trellis-coded M-DPSK system that is suitable for performance analysis the primary difference being that, conditioned on the fading, the memoryless coding channel is no longer AWGN. This is easily seen by combining Eqs. (13.78)–(13.80), which yields  ∗ 

∗ = αk 2Es vk−1 xk + nk αk−1 2Es vk−1 + nk−1 yk = wk wk−1 = αk αk−1 2Es xk + noise (non-Gaussian) terms

(13.81)

The optimum decision metric still depends on the availability or lack thereof of CSI. Such metrics for multichannel reception of differentially detected M-PSK were considered in Chapter 7 and also in Ref. 2. For the case of perfect CSI, the branch decision metric is complicated (involving the zeroth-order Bessel function), and thus theoretical analysis of the average BEP is difficult, if not impossible. Even for the case of no CSI, depending on the statistics of the fading amplitude,

DIFFERENTIALLY COHERENT DETECTION

783

the optimum branch metric can also be quite complicated. [The one case that has a simple solution is the branch metric for Rayleigh fading with no CSI, which happens to be the same as that for the pure AWGN channel (see Chapter 7).] In view of this difficulty, we will follow the approach taken in Ref. 2 and assume the simpler Gaussian metric for both the known and unknown CSI cases. This approach was also taken by Johnston and Jones [21] in analyzing the performance of a blockcoded M-DPSK system over a Rayleigh fading channel. Thus, analogous to (13.7) and (13.8) the decision metrics for the differentially detected M-PSK case become 

m (y, x;α) = −

N

|yn − αn αn−1 2Es xn |2

(13.82)

n=1

and 

m (y, x) = −

N

|yn − 2Es xn |2

(13.83)

n=1

corresponding, respectively, to perfectly known and unknown CSI. Since, as mentioned above, the Gaussian decision metric is (at least for Rayleigh fading) optimum for unknown CSI, we shall focus our attention first on this case since the suboptimality of the metric for known CSI will tend to reduce the performance improvement. 13.2.2

Performance Evaluation

In this section we evaluate the TUB on the average BEP starting first with evaluation of the pairwise error probability. 13.2.2.1 Unknown Channel State Information As for the coherent detection case, we must first find the pairwise error probability in a form suitable for application of the transfer function bound approach. The first approach to try is find the conditional pairwise error probability and then average this result over the fading PDF. Substituting (13.81) into (13.83), the conditional pairwise error probability of (13.28) becomes 

 ∗   ∗ ≥ 0 |x Re wn wn−1 xˆn − xn P (x → xˆ |α) = Pr  = Pr

n∈η



 ∗   ∗ xn − xˆn < 0 |x Re wn wn−1



(13.84)

n∈η

Since wn and wn−1 are conditionally (on the fading) complex Gaussian, the probability required in (13.84) is an extension of the problem considered in Appendix 8A to a sequence of weighted RVs. In particular, because of the assumption of ideal

784

CODED COMMUNICATION OVER FADING CHANNELS

interleaving/deinterleaving, the decision variable in (13.84) is a sum of independent complex conjugate products of Gaussian RVs, which would appear to be a special case of the quadratic form considered by Proakis [22, Appendix B]. However,  ∗ because each Gaussian product RV in the sum is weighted by xˆn − xn , a constant that, in general, depends on the summation index n, the approach taken by Proakis [22], which would lead to a conditional pairwise error probability expressed in terms of the Marcum Q-function, does not apply here. Nevertheless, because of the independence of the terms in the sum, we can still apply the method used in Section 13.1.2.2, in particular, Eq. (13.38), where we must now   ∗   ∗ find the MGF of zn = Re wn wn−1 xn − xˆn |x . The solution is presented in Appendix 13A, which for the case of slow fading (i.e., αn−1 = αn = α) gives the following result   α 2 Es 1 N0 Mzn (s) =  2 exp   1 − sN0 xn − xˆn 

  ) * α xn − xˆn 2 2s 2 N 2 + sN0  0   2   1 − sN0 xn − xˆn 

(13.85) where the overbar denotes statistical averaging with respect to the fading RV α.  Defining, as before, the instantaneous SNR per bit γ = α 2 Eb /N0 and recalling that Es = nc Eb , then (13.85) can be rewritten as   2 ) * γ nc xn − xˆn  2s 2 N02 + sN0 1 Mzn (s) =  2 exp γ   2   1 − sN0 xn − xˆn  1 − sN0 xn − xˆn  (13.86) Statistical averages of the type required in (13.86) are in fact Laplace transforms of the PDF pγ (γ ) (with exponent reversed) and have been evaluated in Chapter 5 for a wide variety of fading channels. Finally, making the change of complex variables ξ = sN0 in (13.86) and then substituting the result in (13.38) gives the pairwise error probability with an integrand in the desired product form, namely P (x → xˆ ) =

1 2πj



σ +j ∞

σ −j ∞

1 M (−ξ )d ξ ξ n∈η zn

(13.87)

where

Mz n



 2   γ nc xn − xˆn  2ξ 2 + ξ (ξ ) =  2 exp γ  2 1 − ξ 2 xn − xˆn  1 − ξ 2 xn − xˆn  1

(13.88)

For the fast-fading case where αn−1 = αn but because of the interleaving and deinterleaving the products αn−1 αn are still independent of each other, then (13.87)

785

DIFFERENTIALLY COHERENT DETECTION

is still valid with now Mz n (ξ ) =

1−

ξ2

1   xn − xˆn 2 

 2 ) * γn−1 ,γn √ nc xn − xˆn  (γn−1 + γn ) ξ 2 + γn−1 γn ξ × exp (13.89)  2 1 − ξ 2 xn − xˆn  Note that evaluation of (13.89) now requires second-order fading statistics. 13.2.2.2 Known Channel State Information Comparing the decision metrics of (13.82) with (13.83), then, following the same approach as for the unknown CSI case, we immediately see that to evaluate the pairwise error probability as given by (13.38), we now need to obtain the MGF   ∗   ∗ of the decision variable zn = Re αn αn−1 wn wn−1 xn − xˆn |x or, for the slow  ∗   ∗ fading assumption, zn = Re αn2 wn wn−1 xn − xˆn |x . The approach taken in Appendix 13A is still appropriate and, with suitable redefinition of the constant C in Eq. (13A.7), we obtain the following result:  exp Mzn (s) =

  αn 2 Es αn4 N xn −xˆn | 2αn2 s 2 N02 +sN0 | 0 2  1− sN0 αn2 |xn −xˆn |

  2  1 − sN0 αn2 xn − xˆn 

(13.90)

Analytical evaluation of the statistical averages of (13.90) in closed form is not possible even for the simplest of fading channels such as the Rayleigh. Similar evaluation problems were noted in Ref. 2 in connection with trying to Chernoff-bound this very same pairwise error probability. Furthermore, Divsalar and Simon [2] reported that computer simulation results corresponding to several examples revealed that, for the Gaussian metric under consideration, little performance was gained from having knowledge of CSI available at the receiver. Thus, because of the difficulty associated with the analysis, we shall abandon pursuit of the known CSI case as was done there. 13.2.3

An Example

Consider the same example as in Section 13.1.5, namely, rate- 12 -coded QPSK using the two-state trellis illustrated in Fig. 13.4. Here we are interested in computing the TUB on average BER when differential detection is employed at the receiver and no CSI is available. We now develop the specific result for the case of slow Rayleigh fading. Results for other fading channel models follow directly from the Laplace transforms given in Chapter 5.

786

CODED COMMUNICATION OVER FADING CHANNELS

Using the Laplace transform of Eq. (5.5) to evaluate the statistical average over the fading, the MGF of (13.88) becomes Mz n (ξ ) = =



    −1 xn − xˆn 2 2ξ 2 + ξ 1−γ  2 1 − ξ 2 xn − xˆn 

1

 2 1 − ξ 2 xn − xˆn 

(13.91)

1

 2 ) * 1 − xn − xˆn  (2γ + 1) ξ 2 + γ ξ

2 Replacing D (θ )|xn −xˆn | with Mz n (−ξ ), the branch gains of Fig. 13.5 are now given by

 I I  a = Mz n (−ξ ) |x −xˆ |2 =4 = n n 2 2  I I  b = Mz n (−ξ ) |x −xˆ |2 =2 = n n 2 2  1 1  c = Mz n (−ξ ) |x −xˆ |2 =2 = n n 2 2

2

1 ) * 1 − 4 (2γ + 1) ξ 2 − γ ξ

2 2

1 ) * 1 − 2 (2γ + 1) ξ 2 − γ ξ 1 ) * 1 − 2 (2γ + 1) ξ 2 − γ ξ

3 3 (13.92) 3

Thus, from (13.57), the transfer function is I 4ac ) ** ) ) * * = ) 2 1 − 2b 1 − 4 (2γ + 1) ξ − γ ξ 1 − 2 (2γ + 1) ξ 2 − γ ξ − I (13.93) which from the TUB analogous to (13.52), namely T (ξ , I ) =

Pb (E) ≤

1 2πj



σ +j ∞

σ −j ∞

1 ∂ T (ξ , I ) |I =1 d ξ ξ ∂I

(13.94)

results in 3 * ) 1 − 2 (2γ + 1) ξ 2 − γ ξ ) ) ** ) *2 d ξ σ −j ∞ 4 1 − 4 (2γ + 1) ξ 2 − γ ξ (2γ + 1) ξ 2 − γ ξ (13.95) For the case of fast Rayleigh fading, the MGF of (13.88) can be evaluated using a result in Ref. 23 that gives 1 Pb (E) ≤ 2πj



Mz n (ξ ) =

σ +j ∞

1 ξ

2

1  2 )    *   2γ + 1 + 1 − ρ 2 γ 2 ξ 2 + γ ρξ 1 − xn − xˆn

(13.96)

NUMERICAL RESULTS—COMPARISON BETWEEN THE TRUE UPPER BOUNDS

787

where ρ is the correlation between the underlying complex Gaussian fading variables whose amplitudes are αn−1 and αn . Note that for ρ = 1 (13.96) reduces to the result for slow fading given in (13.91), as it should. Analogous to (13.93) the transfer function is now T (ξ , I ) =

I

** ) )  1−4 2γ +1+(1−ρ 2 )γ 2 ξ 2 −γ ρξ ) ) * *  2 2 2 × 1−2 2γ +1+(1−ρ )γ ξ −γ ρξ −I

(13.97)

which can then be used to obtain the TUB from (13.94):   )  *  2 2  σ +j ∞ 2 1  1 − 2 2γ + 1 + 1 − ρ γ ξ − γ ρξ  1 ) ) **  Pb (E) ≤   dξ 4 1−4 2γ +1+(1−ρ 2 )γ 2 ξ 2 −γ ρξ 2πj σ −j ∞ ξ ) *  2 × 2γ +1+(1−ρ 2 )γ 2 ξ 2 −γ ρξ (13.98)

13.3 NUMERICAL RESULTS—COMPARISON BETWEEN THE TRUE UPPER BOUNDS AND UNION–CHERNOFF BOUNDS In this section we compare the various TUBs derived thus far with the corresponding union–Chernoff bounds obtained previously [1, 2, 23]. For the purpose of illustration, we shall present numerical results for the rate- 12 -coded QPSK with two-state trellis example and only for the Rayleigh channel. For coherent detection with known CSI, the TUB and union–Chernoff bounds are given in (13.61) and (13.62). Figure 13.7 is a plot of these two upper bounds. We observe a uniform superiority of about 1.5 dB for the TUB relative to the union–Chernoff bound. For coherent detection with unknown CSI, the TUB is obtained by evaluating either (13.70) or (13.74). The union–Chernoff bound is obtained from [1, Eqs. (58), (60)] and is given by Pb (E) ≤ min  λ≥0

ξ1 ξ2 D −24λ

2

1 − ξ2 D −8λ2

2

(13.99)

where D = e −γ /4 

    √ √ ξ1 = 1 − 4 πλγ exp (2λγ )2 Q 2 2λγ    √ √ 2λγ ξ2 = 1 − 2 πλγ exp (λγ )2 Q

(13.100)

Superimposed on Fig. 13.8 is the TUB and union–Chernoff bounds for the no CSI case as given above.

788

CODED COMMUNICATION OVER FADING CHANNELS

1E+00

1E−01

Upper Bounds on Average BEP

Union-Chernoff Bound 1E−02 True Upper Bound

1E−03

1E−04

1E−05

1E−06 0

5

10

15 γ , dB

20

25

30

Figure 13.7 Upper bounds on the average bit error probability for rate- 12 trellis-coded QPSK with two-state trellis-coherent detection with known channel state information, Rayleigh fading.

For differential detection with unknown CSI and slow fading, the TUB is obtained from (13.95), which can be evaluated using the Gauss–Chebyshev quadrature technique described in Appendix 9B.16 In this regard, it is convenient to first write the transfer function T (ξ , I ) of (13.93) in its infinite series form, namely   T (ξ , I ) = 4ac 1 + 2b + (2b)2 + · · · =)

I ** ) ) ** ) 2 1 − 4 (2γ + 1) ξ − γ ξ 1 − 2 (2γ + 1) ξ 2 − γ ξ 

 ×

)

1 − 2 (2γ +

1) ξ 2



1+I   (13.101) * I2 − γξ + + · · · 2 2 [1−2[(2γ +1)ξ −γ ξ ]]

the particular integrand in (13.95), the best value of σ (denoted by c in (9B.15)) to guarantee quick convergence can be easily evaluated as σ = 2(2γγ+1) .

16 For

789

NUMERICAL RESULTS—COMPARISON BETWEEN THE TRUE UPPER BOUNDS

Bit Error Probability (No Channel State Information)

100

Union-Chernoff Bound True Union Bound

Bit Error Probability

10−1

10−2

10−3

10−4

10−5

10−6

0

5

10 15 20 Average SNR per Bit [dB]

25

30

Figure 13.8 TUB and union–Chernoff bounds on average BEP performance of coherent detection with unknown CSI in slow Rayleigh fading.

from which the derivative required in (13.94) has the corresponding infinite series form 1 ∂ ** ) ) ** ) T (ξ , I ) |I =1 = ) 2 ∂I 1 − 4 (2γ + 1) ξ − γ ξ 1 − 2 (2γ + 1) ξ 2 − γ ξ 2 +) ) ** ) ) **2 1 − 4 (2γ + 1) ξ 2 − γ ξ 1 − 2 (2γ + 1) ξ 2 − γ ξ 3 +) ** ) ) **3 +· · · ) 1−4 (2γ + 1) ξ 2 − γ ξ 1−2 (2γ + 1) ξ 2 − γ ξ (13.102) Substituting (13.102) in (13.94) and applying the Gauss–Chebyshev technique of Appendix 9B term by term until additional terms produce a negligible change in the result is the most efficient method of evaluating (13.95). The union–Chernoff bound on average BEP for differential detection with unknown CSI and slow fading is given by    γ2 2  4 1 + 2γ + 2 (2γ + 1) ∂  = T (ξ , I )  Pb (E) ≤ γ I =1,ξ = 2(2γ +1) ∂I γ 4 (1 + γ )2

(13.103)

790

CODED COMMUNICATION OVER FADING CHANNELS

Bit Error Probability over Slow Rayleigh Fading Channels

100

Union-Chernoff Bound True Union Bound

Bit Error Probability

10−1

10−2

10−3

10−4

10−5

10−6 10

12

14

16

18 20 22 Average SNR per Bit [dB]

24

26

28

30

Figure 13.9 TUB and union–Chernoff bounds on average BEP performance of differential detection with unknown CSI in slow Rayleigh fading.

which agrees with Ref. 2, Eq. (40) after being specialized to the case of symmetric QPSK. Figure 13.9 is an illustration of the TUB and the union–Chernoff bound. Finally, for differential detection with unknown CSI and fast fading, the TUB is obtained from (13.98), which again is conveniently evaluated using the Gauss–Chebyshev quadrature technique.17 Again, analogous to (13.102), an infinite series representation of the derivative of the transfer function in (13.97) is particularly helpful in efficiently carrying out the evaluation. The corresponding union–Chernoff bound is given by   1 + 2ζ ∂  T (ξ , I )  , = 2 Pb (E) ≤ γρ   I =1,ξ = 2 2γ +1+(1−ρ 2 )γ 2 ∂I 4ζ (1 + 4ζ ) 

ζ=

ρ2γ 2   * 4 2γ + 1 + 1 − ρ 2 γ 2 )

(13.104)

which agrees with Eqs. (34)–(35) of Ref. 23 and furthermore reduces to (13.103) when ρ = 1 (slow fading). Note that in the limit of large γ and ρ = 1, the bound 17 Here,

the best value of σ to guarantee quick convergence becomes σ =

γ ρ  .  2 2γ +1+ 1−ρ 2 γ 2

NUMERICAL RESULTS—COMPARISON BETWEEN THE TRUE UPPER BOUNDS

791

100

Bit Error Probability

10−1 10−2 10−3 10−4 10−5 10−6 10

12

14

16

18 20 22 Average SNR per Bit [dB]

24

26

28

30

12

14

16

18 20 22 Average SNR per Bit [dB]

24

26

28

30

100

Bit Error Probability

10−1 10−2 10−3 10−4 10−5 10−6 10

Figure 13.10 TUB (solid line) and union–Chernoff (dashed line) bounds on average BEP performance of differential detection with unknown CSI in fast Rayleigh fading; land–mobile channel: (a) fb Tb = 0.05; (b) fb Tb = 0.1.

on BEP as given by (13.104) approaches a finite value given by 2    2 1 − ζ2 2 − ζ2 Pb (E) ≤ ζ4

(13.105)

which represents an “error floor”; thus, regardless of how large we make the average SNR, the upper bound predicts a nonzero BEP.

792

CODED COMMUNICATION OVER FADING CHANNELS

In order to obtain numerical results for this case, we must apply an appropriate correlation model for the fading process. Mason [24] has tabulated the autocorrelation function (or equivalently, the power spectral density) for various types of fast fading processes of interest. These results were given in Table 2.1, where fd denotes the Doppler spread and for convenience the variance of the fading process has been normalized to unity. Figure 13.10 is an illustration of the union Chernoff bound and the TUB as computed from (13.104) and (13.98), respectively, for the land mobile channel with fd Ts as a parameter. The value of fd Ts = 0 corresponds to the case of slow fading as per the results in Fig. 13.9.

REFERENCES 1. D. Divsalar and M. K. Simon, “Trellis coded modulation for 4800–9600 bits/s transmission over a fading mobile satellite channel,” IEEE J. Select. Areas Commun., vol. 5, no. 2, February 1987, pp. 162–175. 2. D. Divsalar and M. K. Simon, “The performance of trellis coded multilevel DPSK on a fading mobile satellite channel,” IEEE Trans. Veh. Technol., vol. 37, no. 2, May 1988, pp. 78–91; see also ICC ’87 Conf. Rec., June 7–10, 1987, Seattle, WA, pp. 21.2.1–21.2.7. 3. E. Biglieri, D. Divsalar, P. J. McLane, and M. K. Simon, Introduction to Trellis Coded Modulation with Applications. New York, NY: Macmillan, 1990; currently distributed by Prentice-Hall, Englewood Cliffs, NJ. 4. K. Y. Chan and A. Bateman, “The performance of reference based M-ary PSK with trellis coded modulation in Rayleigh fading,” IEEE Trans. Veh. Technol., vol. 41, no. 5, May 1992, pp. 190–198. 5. S. B. Slimane and T. Le-Ngoc, “Tight bounds on the error probability of coded modulation schemes in Rayleigh fading channels,” IEEE Trans. Veh. Technol., vol. 44, no. 2, February 1995, pp. 121–130. 6. A. J. Viterbi, “Convolutional codes and their performance in communication systems,” IEEE Trans. Commun. Technol., vol. COM-19, no. 5, October 1971, pp. 751–772. 7. A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding. New York, NY: McGraw-Hill, 1979. 8. R. G. McKay, P. J. McLane, and E. Biglieri, “Error bounds for trellis coded MPSK on a fading mobile satellite channel,” IEEE Trans. Commun., vol. 39, no. 12, December 1991, pp. 1750–1761. 9. J. K. Cavers and P. Ho, “Analysis of the error performance of trellis coded modulations in Rayleigh fading channels,” IEEE Trans. Commun., vol. 40, no. 1, January 1992, pp. 74–80. 10. C. Tellambura, “Evaluation of the exact union bound for trellis coded modulations over fading channels,” IEEE Trans. Commun., vol. 44, no. 12, December 1996, pp. 1693–1699. 11. J.-H. Kim, P. Ho, and J. K. Cavers, “A new analytical tool for evaluating the bit-errorrate of trellis coded modulation in Rician fading channel,” Proc. VTC ’97, Phoenix, AZ, May 1997, vol. 3, pp. 2012–2016.

APPENDIX 13A. EVALUATION OF A MOMENT GENERATING FUNCTION

793

12. J. K. Cavers, J.-H. Kim, and P. Ho, “Exact calculation of the union bound on performance 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. 13. M. K. Simon and D. Divsalar, “Some new twists to problems involving the Gaussian probability integral,” IEEE Trans. Commun., vol. 46, no. 2, February 1998, pp. 200–210. 14. G. Ungerboeck, “Channel coding with multilevel/phase signals,” IEEE Trans. Inform. Theory, vol. IT-28, no. 1, January 1982, pp. 55–67. 15. A. J. Viterbi, “Error bounds for convolutional codes and an asymptotically optimum decoding algorithm,” IEEE Trans. Inform. Theory, vol. IT-13, no. 2, April 1967, pp. 260–269. 16. E. K. Hall and S. G. Wilson, “Design and analysis of turbo codes on Rayleigh fading channels,” IEEE J. Select. Areas Commun., vol. 16, no. 2, February 1998, pp. 160–174. 17. E. Biglieri, C. Caire, G. Taricco, and J. Ventura-Traveset, “Simple method for evaluating error probabilities,” IEE Electron. Lett., vol. 32, February 1996, pp. 191–192. 18. E. Biglieri, C. Caire, G. Taricco, and J. Ventura-Traveset, “Computing error probabilities over fading channels: A unified approach,” Eur. Trans. Telecommun., vol. 9, no. 1, February 1998, pp. 15–25. 19. D. Divsalar, Performance of Mismatched Receivers on Bandlimited Channels, PhD dissertation, Univ. California, Los Angeles, 1978. 20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th ed. New York, NY: Dover Press, 1972. 21. D. A. Johnston and S. K. Jones, “Spectrally efficient communication via fading channels using coded multilevel DPSK,” IEEE Trans. Commun., vol. 29, no. 3, March 1981, pp. 276–284. 22. J. Proakis, Digital Communications, 3rd ed. New York, NY: McGraw-Hill, 1995. 23. D. Divsalar and M. K. Simon, “Performance of trellis coded MDPSK on fast fading channels,” ICC ’89 Conf. Rec., Boston, MA, June 11–14, 1989, pp. 9.1.1–9.1.7. 24. L. J. Mason, “Error probability evaluation of systems employing differential detection in a Rician fast fading environment and Gaussian noise,” IEEE Trans. Commun., vol. 35, no. 1, January 1987, pp. 39–46.

APPENDIX 13A. EVALUATION OF A MOMENT GENERATING FUNCTION ASSOCIATED WITH DIFFERENTIAL DETECTION OF M-PSK SEQUENCES The decision variable associated with the pairwise error probability of differential detection of M-PSK sequences transmitted over a fading channel with no channel state information available to the receiver is a sum of RVs of the form18  ∗   ∗ xn − xˆn zn = Re wn wn−1

(13A.1)

18 For simplicity of notation, we shall omit the conditioning on x with the understanding that the RVs wn−1 and wn are evaluated assuming that x is the transmitted vector.

794

CODED COMMUNICATION OVER FADING CHANNELS

where wn and wn−1 are ∗ conditionally (on the fading) complex Gaussian random variables and xn − xˆn is a complex constant. Of interest in this appendix is the evaluation of the moment generating function (MGF) of zn . Recalling (13.78) and (13.79), we can express (13A.1) as  ∗  

∗ zn = αn 2Es vn−1 xn + nn αn−1 2Es vn−1 + nn−1 (13A.2) xn − xˆn Proakis [22, App. B] considers a quadratic form of complex Gaussian RVs Xn , Yn , namely dn = A |Xn |2 + B |Yn |2 + CXn Yn∗ + C ∗ Xn∗ Yn

(13A.3)

in which A, B, and C are constant weights. Our interest is in the case where A = B = 0 and Xn , Yn are uncorrelated whereupon (13A.3) becomes   dn = 2 Re C ∗ Xn∗ Yn (13A.4) whose MGF is as given by Proakis [22, App. B, Eq. (B-5)] which for this special case becomes 2  3 v 2 ξ1n s 2 + ξ2n s v2 exp Mdn (s) = (13A.5) −s 2 + v 2 −s 2 + v 2 with 2 1 2 1 1 0 1 0 E Xn − Xn  , µyy = E Yn − Y n  2 2  1 v= 4µxx µyy |C|2  2   2 ξ1n = 2 |C|2 Xn  µyy + Y n  µxx , 1 0 ∗ ∗ ∗ ξ2n = CXn Y n + C ∗ Xn Y n = 2 Re C ∗ X n Y n

µxx =

(13A.6)

Comparing (13A.1) with (13A.4), we draw the equivalences Xn = wn−1 , Yn = wn , C =

xn − xˆn 2

(13A.7)

Recognizing the normalizations |vn−1 | = |vn | = |xn | = 1, the parameters in (13A.6) then become

    Xn  = αn−1 2Es , Y n  = αn 2Es  1  E |nn−1 |2 = N0 , 2  1  µyy = E |nn |2 = N0 , 2

µxx =

APPENDIX 13A. EVALUATION OF A MOMENT GENERATING FUNCTION

1    N0 xn − xˆn    2   + αn2 Es xn − xˆn 2 , ξ1n = N0 αn−1  2 ∗    ξ2n = 2αn−1 αn Es Re xn xn − xˆn = αn−1 αn Es xn − xˆn 

795

v=

(13A.8)

where the last equality in ξ2n is obtained by using (13.11). Substituting (13A.8) in (13A.5) gives the conditional MGF of zn as Mzn (s) |αn−1 , αn =

1  2 1 − sN0 xn − xˆn     )  *  Es xn − xˆn 2 α 2 + α 2 s 2 N 2 + αn−1 αn sN0  n n−1 0 N0 × exp   2     1 − sN0 xn − xˆn  

(13A.9) Since the unconditional PDF of zn , namely, pzn (zn ), is obtained by averaging the conditional PDF pzn |αn (zn |αn ) over the PDF of αn , then the unconditional MGF is also the average of the conditional MGF over the PDF of αn [e.g., see (13.41) of the main text for the case of a Gaussian zn ]. Finally, then, the unconditional MGF of zn is, from (13A.9) Mzn (s) =

1   2  1 − sN0 xn − xˆn     )  * αn−1 ,αn  Es xn − xˆn 2 α 2 + α 2 s 2 N 2 + αn−1 αn sN0  n n−1 0 N0 × exp   2    1 − sN0 xn − xˆn  

(13A.10)

14 MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING Thus far in the book, diversity, which acts to reduce the deleterious effects of multipath fading, has been discussed only in terms of its application in the receiver portion of the overall communication system. From an error probability standpoint, we have seen that receive diversity has the effect of steepening the rate of descent of the curve of SEP or BEP versus average SNR while at the same time leaving the nature of the variation of these error probabilities with average SNR unchanged. In this sense, the term diversity gain or diversity order refers to the increase in the slope of the error probability versus average SNR curve. Equivalently, in the limit of large average SNR, whereupon the error probability versus average SNR in many cases behaves approximately as P (E) ≈ γ −Gd , the exponent Gd represents the diversity gain. Furthermore, on the basis of such an asymptotic behavior, the difference in average SNR at fixed error probability (i.e., the horizontal separation of the error probability performance curves plotted on a log-log scale) between a system with no diversity (i.e., Gd = 1) and one with diversity order Gd > 1 continues to increase with increasing SNR and, in the limit of infinite SNR, becomes unbounded. Separate and apart from the improvement in system performance attributed to receive diversity is the possibility of additional improvement brought about by the introduction of error correction coding at the transmitter and its associated decoding at the receiver such as that discussed in Chapter 13. In many instances, coding affects the error probability versus SNR curve in such a way that asymptotically, in the limit of large SNR, the uncoded and coded error probability performance curves become parallel and as such the difference in SNR (horizontal separation) Digital Communication over Fading Channels, Second Edition. By Marvin K. Simon and Mohamed-Slim Alouini ISBN 0-471-64953-8 Copyright  2005 John Wiley & Sons, Inc.

797

798

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

between these curves approaches a fixed finite amount. This asymptotic SNR separation is what is commonly referred to as coding gain. Mathematically speaking, if asymptotically the error probability behaves as P (E) ≈ (Gc γ )−Gd , then in the limit of large SNR, the coding gain Gc represents the horizontal shift in the error probability performance relative to the benchmark curve P (E) ≈ γ −Gd . In addition to a horizontal shift in the error probability versus SNR curve, coding can also steepen the rate of descent of this curve and as such can contribute to diversity. Thus, a system that incorporates both error correction coding and multiple receive antennas is capable of providing both diversity and coding gain, and its asymptotic (large average SNR) error probability performance would again behave as P (E) ≈ (Gc γ )−Gd , where Gd now reflects the diversity contributions from both the coding and the antenna multiplicity. This notion of describing diversity and coding gain in terms of the asymptotic behavior of the error probability performance curve is nicely described and analytically quantified in Ref. 1. In this chapter, we first introduce the notion of transmit diversity, which shifts the complexity associated with the implementation of a diversity receiver to the transmitter with the hope of accomplishing the same purpose, specifically, providing diversity gain, with as little compromise as possible in average SNR performance.1 In the simplest of terms, we transmit the same signal information (over a multitude of symbol intervals and not necessarily in the same sequence) simultaneously from a multitude of antennas and appropriately combine the faded versions of these transmissions received in the multitude of symbol intervals in such a way as to provide a diversity gain.2 The most obvious application of such a concept is in the forward link of a mobile communication system where the complexity associated with the diversity implementation has now been moved from the mobile unit to the base station. Although it is possible to achieve the same diversity gain using transmit instead of receive diversity, the former incurs an SNR loss (assuming a fixed total transmit power for both schemes). Thus, the next logical step in improving system performance when transmit diversity is employed would be to apply error correction coding/decoding (but now in more than one dimension) so as to provide coding gain for recovering this loss in SNR while maintaining the diversity gain achieved. Indeed, the combination of transmit diversity in the form of multiple antennas coupled with error correction coding is what is generically referred to as space-time coding (STC),3 whose ultimate goal is to design codes that achieve both the maximum attainable diversity and rate independent of the channel statistics and modulation while at the same time achieving a good coding gain performance as well as maintaining a simple maximum-likelihood (ML) decoding scheme. One example of such a scheme, referred to as space-time trellis coding (STTC), is the combination of a trellis-coded modulation (TCM) with transmit (and possibly 1 It

is also possible to combine receive diversity with transmit diversity to increase the diversity gain still further. Such systems that employ multiple transmit antennas and multiple receive antennas are generically referred to as multiple-input/multiple-output (MIMO) systems. 2 In the simplest of schemes, the multitude of transmitters and the multitude of symbol intervals constitute the same numeric value. 3 It should be noted that the term space-time coding is not meant to imply that both diversity and coding gains are always achieved.

A HISTORICAL PERSPECTIVE

799

receive) diversity and, in a sense, is analogous to the notion of multiple trelliscoded modulation (MTCM) [2], where the multiplicity (number of symbols per trellis branch) is now associated with the space dimension, namely, the number of transmit antennas. While STTCs perform extremely well in the sense of being able to provide full diversity, full rate, and coding gain, they are suffer from the fact that they require a relatively high decoding complexity. By contrast, the combination of multiple transmit antennas with block codes, referred to as space-time block coding (STBC), is able to provide full diversity and full rate and requires considerably less decoding complexity but at a cost of reduced performance (coding gain). Thus, in keeping with the ultimate goal mentioned above, it should not be surprising that researchers have investigated schemes that stem from a combination of STTC and STBC, a number of which will be discussed later on in the chapter.

14.1

A HISTORICAL PERSPECTIVE

It is interesting to note that while the origins of receive diversity for combating multipath date back to the classic work of Brennan [3], the notion of transmit diversity is, relatively speaking, quite recent. Perhaps the earliest traces of such an idea appear in the work of Wittneben [4]4 and Seshadri and Winters [6], both reported in 1993, who suggested various delay transmit antenna diversity techniques for application in base stations.5 (Wittneben also refers to his particular scheme as modulation diversity since the same modulation method is used at all antennas; the distinction among them is different modulation parameters.) These schemes focused on attempts to achieve the same diversity gain as that obtained with receive diversity, sacrificing, however, the effective gain in receive SNR by a factor equal to the number of antennas that is achieved by the latter. It wasn’t until a number of years later that researchers thought of combining transmit diversity with error correction coding (and possibly also receive diversity) [8–10]. Since that time, a large number of papers extending these principles and exploring their application in modem design have appeared in the literature, examples of which for block and trellis coding can be found in Refs. 11–22. To a large extent, the papers that deal with space-time coding emphasize the design criteria and construction of the two-dimensional codes and their decoding algorithms rather than the methods by which the error probability of the overall system is evaluated and the associated accuracy of these methods. With regard to the latter, for the most part, the standard analysis techniques previously applied to time coding alone have been extended to allow evaluation of pairwise error probability (either exactly or by an upper bound) and upper bounds on (or approximations to) average bit error probability for space-time codes. In keeping with the theme of this book, our primary goal in this chapter will be to once again demonstrate that 4 To be chronologically precise, the scheme that was analyzed in Ref. 4 was first reported by the same author in Ref. 5 for a SIMULCAST application. 5 Winters [7] also subsequently expanded on and provided an analytical basis for the Monte Carlo simulation results originally proposed in 1993 [6].

800

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

the MGF-based approach [23] that has been so successful in allowing evaluation of the more conventional single-transmitter communication systems is equally as useful in the case of multichannel transmission, in particular, the MIMO scenario. Indeed, the approach will largely parallel that in Chapter 13, and as such we shall focus primarily on the trellis-coded application. As we shall see in the technical presentation that follows, whether one achieves transmit diversity gain and/or coding gain with multiple antennas and space-time coding depends on the model assumed for the rate of variation of the channel, specifically, fast versus slow (often called quasi-static) fading. In the case of the former, the fading is assumed to vary independently from symbol to symbol, whereas for the latter the fading is assumed to be constant over the duration of a block of symbols but varies from block to block. It is furthermore important that we consider both cases in our discussions since the differences between the two not only influence the system performance but also play a significant role in the criteria used to design the optimum codes. However, in keeping with the spirit of this textbook, we shall not discuss the means by which these designs are obtained but rather the tools used to analyze the performance of communication systems that employ the codes themselves. We begin our discussion by describing the basic idea of transmit diversity and its relation to the more traditional concept of receive diversity. To keep matters simple, we shall make certain idealistic assumptions that allow for a quick comparison between the two diversity concepts.

14.2

TRANSMIT VERSUS RECEIVE DIVERSITY—BASIC CONCEPTS

The basic idea of transmit diversity is to create at the input of a single receive antenna a multichannel signal that resembles in form the signal that would be received by a multitude of antennas in a system having a single transmit antenna communicating over a multipath channel. More specifically, a given information signal is redundantly transmitted from a multitude of transmit antennas over a set of flat-fading channels to create at the receiver a signal having the appearance of one transmitted from a single antenna over a multipath fading channel such as that discussed in Chapter 9. The manner in which the various transmitted components are separated at the receiver to allow for transmit diversity is analogous to the way in which the various multipath components are resolved in a system employing receive diversity.6 Furthermore, the method used for creating a set of transmitted signal replicas that allows for separation at the receiver is what distinguishes one transmit diversity scheme from another. A simple way of achieving this separation is to assign the transmitted signal replicas different CDMA codes (or different delays of the same code) in much the same way as that discussed in Chapter 11 for a multiuser communication system. 6 Recall that in a receive diversity system, it is the application of a pseudonoise direct-sequence spreadspectrum modulation to the transmitted signal that allows resolution of the various multipath components in the receiver.

TRANSMIT VERSUS RECEIVE DIVERSITY—BASIC CONCEPTS

801

The difference here is that instead of trying to separate the various communications from different users, we are trying to separate the various transmitted components from the same user to create a set of independent diversity channels. The basic transmit diversity system is illustrated in Fig. 14.1. In each transmission interval, an information symbol is spread by a set of Lt different CDMA codewords (assumed here to be orthogonal), producing a set of signals that characterize the outputs of the Lt transmit antennas in that interval.7 Each of these signals is transmitted over a flat-fading channel, the collection of which for the purpose of our discussion will be assumed to be i.i.d. (i.e., 1 = 2 = · · · = Lt = ) in Fig. 14.1. Since the total available transmitted power Pt is a finite resource, then, for the i.i.d. channel assumption, a good strategy for power allocation to the Lt transmitted signals would be to share the total power equally among them; thus, each transmitted signal would have power Pi = Pt /Lt , i = 1, 2, . . . , Lt . The optimization of transmitted power allocation for the non-i.i.d. (both correlated channels and nonidentical channel average powers) case has also been considered [24,25] but is not germane to our simple conceptual discussion here. The received signal is despread through an appropriate set of filters matched to the spreading codes, thereby separating the various transmitted replicas components. We note that whereas in a receive diversity system the AWGN components that exist in each of the receiver branches are independent of one another because they are associated with different antennas, in the transmit diversity system, the noises at the matched-filter outputs are also mutually independent because of the projection of the single noise source associated with the single receive antenna on the multitude of orthogonal despreading codes. Next, assuming, for simplicity, perfect knowledge of the statistics of the channels, the outputs of the despreading matched filters are weighted by the complex conjugate of the corresponding channel gain between the appropriate transmit antenna and the receive antenna and combined in the fashion of MRC. The output of the MRC is then used to make an ML decision on the transmitted symbol in the same manner as that discussed in Chapter 9. In view of the description above, it should be apparent that from an ideal mathematical standpoint, the received signal can be modeled as a set of Lt independent slowly varying flat-fading components analogous to that discussed Section 9.1.2, and thus one might anticipate that the analysis techniques presented there for receive diversity systems would also apply to the transmit diversity case. In fact, a straightforward application of the MGF-based approach to the transmit diversity system discussed above would reveal a performance having a diversity gain equal to that of an MRC receive diversity system with Lt receive antennas (a single transmitting antenna assumed) but an average SNR reduced by a factor of Lt . For example, a plot of average BER versus average SNR in decibels for the transmit diversity system 7 Note that, depending on the number of transmit antennas, that is, the number of orthogonal codewords required, the narrowband information signal may or may or not become wideband. Thus, for a small number of transmit antennas, the spreading sequences can be short, and as such the channel created may not necessarily become frequency-selective. This is in contrast to an alternative transmit diversity scheme suggested in Ref. 6, where the intent is to create a frequency-selective fading channel, thereby necessitating the use of MLSE or alternatively a RAKE receiver.

802

Information Symbols

Spreading Code Lt

Spreading Code 2

Spreading Code 1

n(t)

Fading AWGN Channel

aLt(a2Lt = ΩLt)

a2(a22 = Ω2)

a1(a21 = Ω1)

Spreading Code Lt

Spreading Code 2

Spreading Code 1

Figure 14.1 Mathematical model of a basic transmit diversity scheme.

Transmitter

√PLt

√P2

√P1

Receiver

*

aLt(a2Lt = ΩLt)

*

a2(a22 = Ω2)

*

a1(a21 = Ω1)

to ML Detector

ALAMOUTI’S DIVERSITY TECHNIQUE

803

would be parallel to that of the equivalent receive diversity system but translated to the right by 10 log10 Lt . In an effort to recover this inherent SNR loss, researchers then combined transmit diversity with error correction coding in an integrated fashion, which is what has generically been referred to as spacetime coding. Before launching into the more generic problem of evaluating the error probability performance for arbitrary space-time codes transmitted over generalized fading channels, we start our discussion with a specific transmit diversity scheme proposed by Alamouti [26] that has the advantage of being not only simple to implement (no spreading/despreading codes are required) but again simple to comprehend in terms of its comparison with the optimum receive diversity (MRC) technique. Indeed, Alamouti’s technique laid the foundation for many of the contributions to the subject that followed in the literature, some of which can be viewed as direct extensions of his work. The motivation for using Alamouti’s technique and its later generalizations, namely, the so-called orthogonal designs (to be discussed in the following section), as opposed to the CDMA-based transmitter diversity technique that was just described, is that the latter loses rate—the rate of transmission is 1/Lt symbols per channel use—while the orthogonal designs can achieve exactly the same performance at a rate of 1 symbol per channel use for real designs, or at least a rate of 12 symbol per channel use for complex designs [14].

14.3 ALAMOUTI’S DIVERSITY TECHNIQUE—A SIMPLE TRANSMIT DIVERSITY SCHEME USING TWO TRANSMIT ANTENNAS Consider first the simple communication system illustrated in Fig. 14.2 that employs two transmit antennas and a single receive antenna. For each group of b bits the transmitter generates a signal from a complex signal constellation with 2b elements, namely, s1 , s2 , . . . , s2b . Let xn = [x1(n) , x2(n) ]T denote the column

(n) c11 (n)

(n) n(n) 1 , n2

(n)

x1 , − (x2 )*

(n) c12

y1(n), y2(n)

Channel Estimator (n)

Combiner

(n) (n) x˜ 1(n), x˜2(n) Maximum- xˆ1 , xˆ2

(n) (n) c11 , c12

Likelihood Detector

(n) (n) c11 , c12

(n)

x2 , (x1 )* Figure 14.2

The two-branch Alamouti transmit diversity scheme with one receive antenna.

804

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

vector corresponding to the pair of complex symbols generated for a particular group of 2b input bits, each chosen from the abovementioned constellation, to be transmitted in the nth transmission interval,8 nTs ≤ t ≤ (n + 2) Ts , from the two transmit antennas where the T superscript denotes the transpose operation. The communication system is such that for each group of, say, b bits pair of complex symbols x1(n) , x2(n) are chosen from a signal constellation For example, xn could consist of a pair of independently chosen M -PSK symbols or alternatively a pair of such symbols chosen from a four-dimensional 2 × M-PSK constellation as might be the case if the input data were first trellis-encoded [17]. In Alamouti’s scheme [26] this joint information is sent twice per transmission interval (hence the information symbol rate remains unchanged) each time with a different assignment to the two transmit antennas. Thus, an “Alamouti symbol” consists of two information symbols, and in our notation, a transmission interval corresponds to the duration of an Alamouti symbol. Specifically, antenna 1 and antenna 2 first transmit x1(n) and x2(n) , respectively, in the interval nTs ≤ t ≤ (n + 1) Ts ; then they transmit (−x2(n) )∗ and (x1(n) )∗ in the interval (n + 1) Ts ≤ t ≤ (n + 2) Ts —a simple 2 × 2 space-time block code. Thus, for a block of N successive information vectors x1 , x2 , . . . , xN , the transmitted sequences from each antenna would be a block of 2N complex symbols as follows. Antenna 1 would transmit x1(1) , (−x2(1) )∗ , x1(2) , (−x2(2) )∗ , .., x1(N) , (−x2(N) )∗ , whereas antenna 2 would transmit x2(1) , (x1(1) )∗ , x2(2) , (x1(2) )∗ , .., x2(N) , (x1(N) )∗ . The channel during the nth transmission interval is described by the Lr × Lt (n) matrix Cn = [cj(n) i ] where cj i denotes the complex channel gain (fading) between the i th transmit antenna and the j th receive antenna and the cj(n) i terms are assumed to be independent of one another. For example, for a Rayleigh fading channel, cj(n) i would be a complex Gaussian random variable (RV) with average fading power 2  = |cj(n) i | assumed to be independent of n. For the case currently under consid(n) (n) c12 ]. Note that the preceding notation eration, Cn is given by the row vector [c11 implies that the channel gains are assumed to be constant over a transmission interval that corresponds to two symbol intervals. However, depending on the fast/slow-fading assumption, the channel matrix either varies as a function of n or is constant with respect to n over a block of transmissions. Denoting by xn1 = [x1(n) , x2(n) ]T , xn2 = [−x2(n) , x1(n) ]∗T the column vectors representing the two pairs of symbols successively transmitted over the channel in the nth transmission interval, then the corresponding pair of successive signal samples at the receiver [matched-filter outputs at times t = (n + 1) Ts and t = (n + 2) Ts ] is given by (n) (n) (n) (n) (n) y1(n) = Cn xn1 + n(n) 1 = c11 x1 + c12 x2 + n1
∗
∗ (n) (n) (n) x1(n) + n(n) y2(n) = Cn xn2 + n(n) + c12 2 = c11 −x2 2

(14.1)

8 We use the term “transmission interval” here to refer to the time intervals needed to transmit information symbols a pair at a time.

ALAMOUTI’S DIVERSITY TECHNIQUE

805

(n) where n(n) 1 and n2 constitute a pair of i.i.d. complex zero-mean Gaussian noise 2 2 samples each with variance σ 2 per dimension (i.e., E{|n(n) i | } = 2σ , i = 1, 2) and are assumed to be independent of the channel gains. Assuming perfect knowledge of the complex channel gains, that is, perfect channel state information (CSI), the receiver uses the sequence y1(n) , y2(n) to construct


∗
∗ (n) (n) y2(n) y1(n) + c12 x˜1(n) = c11
∗
∗ (n) (n) y2(n) x˜2(n) = c12 y1(n) − c11

(14.2)

to be used in forming the ML metric for making a decision on the information signal vector xn = [x1(n) , x2(n) ]T corresponding to the nth transmission interval.9 Substituting (14.1) into (14.2) gives     
∗
∗  (n) 2  (n) 2 (n) (n) (n) (n) (n) + x˜1(n) = c11 c12  x1 + c11 n1 + c12 n2  x˜2(n)

    
∗
∗  (n) 2  (n) 2 (n) (n) (n) (n) (n) = c11  + c12  x2 + c12 n1 − c11 n2

(14.3)

Note that even though x˜1(n) and x˜2(n) both depend on the same pair of noise com ponents n(n) , n(n) , the effective Gaussian noise RVs N˜ (n) = (c(n) )∗ n(n) + c(n) (n(n) )∗ 1

2

1

11

1

12

2

 (n) ∗ (n) (n) (n) ∗ and N˜ 2(n) = (c12 ) n1 − c11 (n2 ) are uncorrelated (and therefore independent):


∗  
∗

∗  (n) (n) (n) (n) = c11 c12 σ 2 − c12 c11 σ2 = 0 E N˜ 1(n) N˜ 2(n) In view of this independence, one can view the two equations in (14.3) as representing the input–output relationship of two parallel, independent channels, or, two realizations of the same single-input/single-output channel (since the two parallel channels have the same SNR). This means that existing one-dimensional (1D) codes (e.g., 1D TCM) could easily be combined with, in general, orthogonal designs, by serial-to-parallel (one to Lt ) converting the 1D encoder output and then feeding it to the orthogonal design. Applying the observations in (14.1), the ML metric is given by 2


  (n) (n) (n) (n)  m y(n) , x(n) = y1(n) − c11 x1 + c12 x2  
∗
∗ 2
  (n) (n) −x2(n) + c12 x1(n) + y2(n) − c11  9 Note

(14.4)

that the combining scheme as described by (14.2) is different from that of MRC. Nevertheless, as we shall see shortly, the decision variables that result are precisely the same as those for MRC.

806

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

which, using (14.2), can be written as 2  2
∗  
∗  

     m y(n) , x(n) = y1(n)  + y2(n)  − 2Re x˜1(n) x1(n) − 2Re x˜2(n) x2(n)              (n) 2  (n) 2  (n) 2  (n) 2  (n) 2  (n) 2 + c11  + c12  x1  + c11  + c12  x2  (14.5) or equivalently    2  

(n) (n)   (n) 2  (n) 2  (n) 2  (n) 2   (n) 2  (n) 2  m y ,x = y1  + y2  − x˜1  − x˜2  + c11  + c12  − 1 x1(n)   2  2  2  2      (n)   (n)   (n) 2  (n) (n)  + x˜1(n) − x1(n)  + c11  + c12  − 1 x2  + x˜2 − x2  (14.6) x1(n)

x2(n)

Since a decision on and based on (14.6) is independent of and x˜2(n) , the metric simplifies to  2  2   2  2
 (n)     (n)    (n) (n) = c11  + c12  − 1 x1(n)  + x˜1(n) − x1(n)  m x˜ , x

y1(n) , y2(n) , x˜1(n) ,

   2  2      (n) 2  (n) 2   + c11  + c12  − 1 x2(n)  + x˜2(n) − x2(n) 

(14.7)

Finally, assuming that the transmitted information signals x1(n) and x2(n) are independently chosen, a ML decision can be made separately on each of them using the metric10   2    2  2
 (n)   (n)   (n) 2  (n) (n)  m x˜i(n) , xi(n) = c11 i = 1, 2  + c12  − 1 xi  + x˜i − xi  , xi(n)

xˆi(n)

(14.8) if and only if

= with the corresponding decision rule, as follows. Choose     2      (n) 2  (n) 2  (n) 2  (n) (n)  c11  + c12  − 1 xˆi  + x˜i − xˆi      2     (n) 2  (n) 2  (n) 2  (n) (n)  ≤ c11 + − 1 for all xi(n) = xˆi(n)  c12  xi  + x˜i − xi 

(14.9)

For equal energy signal constellations such as M -PSK, |xi(n) |2 is constant and thus (14.9) reduces to the minimum squared Euclidean distance rule, as follows. Choose xi(n) = xˆi(n) if and only if 2  2   (n)  (n) (n)  (n)  x˜i − xˆi  ≤ x˜i − xi 

for all xi(n) = xˆi(n)

(14.10)

10 A generalization of the metric in (14.8) will also be used later on in the chapter when discussing the TCM case.

ALAMOUTI’S DIVERSITY TECHNIQUE

807

Note that in (14.10) knowledge of the channel does not appear to be explicitly needed to implement the decision rule. However, the reader is reminded that knowledge of the channel is needed to build the decision variables x˜1(n) and x˜2(n) from the received observables y1(n) and y2(n) in accordance with (14.2). Furthermore, applying the ML principle directly to the observables, it can formally be shown [27] that the [in the sense of maximizing the joint likelihood function  optimal (n) (n) p(y1(n) , y2(n) x1(n) , x2(n) , c11 , c12 ) ] receiver is indeed one that bases its decision on (n) (n) x˜1 and x˜2 of (14.2). Thus, for the perfectly known channel, the Alamouti scheme is optimal. Note that this optimality statement is independent of the statistics of the channel since the proof of optimality is based on the conditional (on the channel gains) likelihood function. The performance of the Alamouti scheme, however, will be heavily dependent on the statistics [e.g., probability density functions (PDFs), relative correlation] of the channel gains. To compare the performance of the Alamouti transmit diversity scheme with an equivalent two-branch MRC scheme, we note that the combined signals in (14.3) are identical in form (except for phase rotations of the noise components, which have no effect on the effective SNR) to what would be obtained at the output of the MRC combiner for the same pair of successively transmitted symbols. Thus, from the standpoint of diversity gain, the two schemes both achieve a diversity of order 2. However, under the assumption of a fixed total amount of power available at the transmitter, then, for the transmit diversity scheme, in each symbol interval the power is split equally between the two antennas; specifically, x1(n) and x2(n) each contain half the symbol energy and the same is true for (−x2(n) )∗ and (x1(n) )∗ . By contrast, in the MRC scheme, since only one transmit antenna is employed, then only one symbol is transmitted in each symbol interval and the total energy is allocated to it. Thus, from the standpoint of SNR, the power efficiency of the Alamouti transmit diversity scheme suffers a 3-dB loss with respect to that of the MRC receive diversity scheme. Stated another way, a curve of average BEP versus bit energy-to-noise ratio in decibels for the Alamouti transmit diversity scheme would be parallel to that of the equivalent curve for MRC and shifted 3 dB to the right. This analogy with the performance of MRC would hold for all channel scenarios, including the possibility of different statistical PDFs for the two channels gains as well as correlation between them. In order to achieve higher orders of diversity, Alamouti [26] generalized his simple transmit diversity scheme to allow for more than one receive antenna (still, however, using two transmit antennas). Specifically, he showed that for two transmit antennas and Lr receive antennas, it is possible to achieve a diversity of order 2Lr , which would then be equivalent (from a diversity gain standpoint) to an MRC system with 2Lr receive antennas. For the purpose of illustration, we present the details for the special case of two transmit antennas and two receive antennas as was done in Ref. 26. The appropriate communication system is illustrated in Fig. 14.3. The encoding of the pair of information symbols at the transmitter is done the same way as previously described, specifically, in the nth transmission

808

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

(n) n(n) 1 , n2 (n) c11

(n)

y1(n), y2(n) Channel Estimator

(n)

x1 , − (x2 )*

( n)

(n) c21

(n) c12

Combiner

(n) n(n) 3 , n4

Likelihood Detector

(n) (n) c21 , c22

(n) c22 (n)

( n)

(n) (n) c11 ,c12 c11 , c12 ˜x1(n), x˜2(n) Maximum- xˆ (n), xˆ (n) 1 2

(n) (n) c21 , c22

Channel Estimator

(n)

x2 , (x1 )*

y3(n), y4(n)

Figure 14.3 The two-branch Alamouti transmit diversity scheme with two receive antennas.

interval xn1 = [x1(n) , x2(n) ]T is transmitted followed by xn2 = [−x2(n) , x1(n) ]∗T . Since the channel gain matrix is now described by Cn =

(n) c11

(n) c12

(n) c21

(n) c22

(14.11)

then, analogous to (14.1), the four received samples are now

y1(n)

= Cn xn1 +

y2(n)

y3(n) y4(n)



= Cn xn2 +

n(n) 3 n(n) 4

 =

n(n) 2

n(n) 1

(n) (n) (n) (n) c11 x1 + c12 x2 (n) (n) (n) (n) c21 x1 + c22 x2

 +



n(n) 1

n(n) 2


∗
∗  (n) (n) (n) −x2(n) + c12 x1(n) c11 n3   =
∗
∗  + (n) (n) n(n) −x2(n) + c22 x1(n) c21 4 

(14.12) (n) (n) (n) where again n(n) , n , n , n are i.i.d. complex zero-mean Gaussian noise sam1 2 3 4 ples, each with variance σ 2 per dimension. Assuming as before perfect knowledge of the channel gains, the receiver uses the sequence y1(n) , y2(n) , y3(n) , y4(n) to construct the two decision variables needed for ML detection as
∗
∗
∗
∗ (n) (n) (n) (n) x˜1(n) = c11 y3(n) + c22 y4(n) y1(n) + c21 y2(n) + c12
∗
∗
∗
∗ (n) (n) (n) (n) y3(n) − c21 y4(n) x˜2(n) = c12 y1(n) + c22 y2(n) − c11

(14.13)

809

GENERALIZATION OF ALAMOUTI’S DIVERSITY TECHNIQUE

Substituting (14.12) into (14.13) gives x˜1(n)

          (n) 2  (n) 2  (n) 2  (n) 2 (n) = c11  + c12  + c21  + c22  x1

x˜2(n)


∗
∗
∗
∗ (n) (n) (n) (n) (n) n(n) + c11 n(n) n(n) + c22 1 + c21 2 + c12 n3 4           (n) 2  (n) 2  (n) 2  (n) 2 (n) = c11  + c12  + c21  + c22  x2

(14.14)


∗
∗
∗
∗ (n) (n) (n) (n) (n) n(n) + c12 n(n) n(n) − c21 1 + c22 2 − c11 n3 4 Again it is straightforward to show that the effective Gaussian noise RVs   (n) ∗ (n) (n) ∗ (n) (n) (n) ∗ (n) (n) ∗ (n) ∗ (n) N˜ 1(n) = (c11 ) n1 + (c21 ) n2 + c12 (n3 ) + c22 (n4 ) and N˜ 2(n) = (c12 ) n1 + (n) ∗ (n) (n) (n) ∗ (n) (n) ∗ (c22 ) n2 − c11 (n3 ) − c21 (n4 ) are uncorrelated and thus independent. Analogous to (14.9), then, the decisions on x1(n) and x2(n) can be made separately, now using the following rule. Choose xi(n) = xˆi(n) if and only if            2  (n) 2  (n) 2  (n) 2  (n) 2  (n) 2  (n) (n)  + + + − 1 c11  c12  c21  c22  xˆi  + x˜i − xˆi             2  (n) 2  (n) 2  (n) 2  (n) 2  (n) 2  (n) (n)  ≤ c11 + + + − 1 c c c  12   21   22  xi  + x˜i − xi  

(14.15)

for all xi(n) = xˆi(n) or for equal energy signal constellations, choose xi(n) = xˆi(n) if and only if  2  2  (n)  (n) (n)  (n)  x˜i − xˆi  ≤ x˜i − xi 

for all xi(n) = xˆi(n)

(14.16)

which is identical to (14.10). Finally, comparing the Alamouti transmit diversity scheme with two transmit and two receive antennas to an equivalent MRC scheme with one transmit and four receive antennas, we readily observe that both achieve a diversity order of 4; however, using the same reasoning as discussed previously, for a fixed total transmitter power, the former would again suffer a 3-dB loss relative to the latter.

14.4 GENERALIZATION OF ALAMOUTI’S DIVERSITY TECHNIQUE TO ORTHOGONAL SPACE-TIME BLOCK CODE DESIGNS Using the theory of orthogonal designs [28] studied by Hurwitz and Radon back in the 1920s, Alamouti’s scheme can be generalized to a number of transmit antennas greater than two [14]. Specifically, consider an Lt × Lt matrix X with real elements chosen from ±x1 , ±x2 , . . . , ±xLt such that the rows or columns are orthogonal

810

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

with each other. Such orthogonal matrices, referred to as Hurwitz–Radon matrices, were shown [28] to exist only for values of Lt = 2, 4, 8. Examples of such matrices corresponding to Lt = 2 and Lt = 4, respectively, are as follows: 

xa xb

−xb xa

xa  xb X4 =   xc xd

−xb xa −xd xc

X2 = 

 (14.17) −xc xd xa −xb

 −xd −xc   xb  xa

(14.18)

In (14.17), the subscripts a, b can be either permutation of the indices “1, 2,” whereas in (14.18), the subscripts a, b, c, d can be any permutation of the indices “1, 2, 3, 4.” In terms of the space-time coding problem, we associate a column of X with the Lt symbols transmitted from the Lt antennas in a given symbol interval (time slot). Thus, for example, letting a = 1, b = 3, c = 4, d = 2 in (14.18), simultaneous transmission of the group of symbols x1 , x3 , x4 , x2 would be followed by transmission of −x3 , x1 , −x2 , x4 , −x4 , x2 , x1 , −x3 , and −x2 , −x4 , x3 , x1 . Since over a total of Lt time slots we are transmitting information from Lt symbols, the information symbol rate is 1, or equivalently looking at X as a block code, the code rate is 1. Using X to describe the symbol transmissions in the nth transmission interval from Lt antennas to a single receiver with a channel characterized in this same interval by the channel matrix Cn = [cj(n) i ], it is proved [14] that such block code designs achieve a diversity of order Lt and have a simple ML decoding algorithm that is based only on linear processing. As pointed out above, the limitation of requiring that the elements of X be real, for example, M -AM or BPSK signal constellations in the application to space-time coding, is that such matrix designs can be achieved only for Lt = 2, 4, 8. In terms of the space-time coding application, this limitation is synonymous with limiting the number of transmit antennas, or equivalently the diversity order, to 2, 4, and 8. A similar limitation occurs for the design of X with elements chosen from complex signal constellation, such as QAM or M -PSK. Here orthogonal designs exist only for 2 × 2 matrices, specifically, STBCs with two transmit antennas (i.e., the Alamouti scheme).11 To allow arbitrary diversity orders to be achieved, Tarokh and his collaborators introduced the notion of generalized complex orthogonal designs [14] wherein the elements of matrix X, now of dimension Lt × Lp (i.e., where X is nonsquare), are allowed to be complex and chosen from 0, ±x1 , ±x1∗ , ±x2 , ±x2∗ , . . . , ±xk , ±xk∗ in such a way as to provide orthogonal  rows and columns. Furthermore, whereas t 2 for the real orthogonal designs XXT = ( L i=1 xi )I where I is the Lt × Lt identity 11 It is important to point out that the above limitations on the possible sizes of square orthogonal matrices assume arbitrary signal constellations. For specific signal constellations, such as QPSK, it is possible to design square orthogonal STBCs with more than two transmit antennas that achieve both full diversity and full rate. An example of such designs can be found in Ref. 29.

GENERALIZATION OF ALAMOUTI’S DIVERSITY TECHNIQUE

811

matrix, for the  generalized complex orthogonal designs the analogous property is XX∗T = C( ki=1 |xi |2 )I with C an integer constant. Applying such designs to the space-time coding problem, where, as before, the number of columns Lp of X corresponds to the number of symbols per transmission interval (code block) and the number of rows Lt of X corresponds to the number of transmit antennas, then, since there are now k symbols transmitted in a given transmission interval, the code rate is k/Lp . From the description above, it becomes immediately clear that the Alamouti scheme is a special case of a space-time complex orthogonal block code corresponding to Lp = Lt = k = 2. Specifically, if we define the complex orthogonal matrix X by  X=

x1 x2

−x2∗ x1∗

 (14.19)

then, if we associate the first and second columns of X with the transmitted symbol column vectors xn1 = [x1(n) , x2(n) ]T and xn2 = [−x2(n) , x1(n) ]∗T defined in Eq. (14.1), we immediately see the association. As an example of a complex orthogonal design that generates a space-time block code with diversity order 3 and can be applied, for example, to a system with 3 transmit and 1 receive antenna, consider the matrix X given by 

xa  X =  xb xc

−xb xa −xd

−xc xd xa

−xd −xc xb

xa∗ xb∗ xc∗

−xb∗ xa∗ −xd∗

−xc∗ xd∗ xa∗

 −xd∗  −xc∗  xb∗

(14.20)

where again the subscripts a, b, c, d can be any permutation of the indices 1, 2, 3, 4. The block code generated from the code construction matrix in (14.20) has a block length of eight symbols during which only four different complex symbols (and their complex conjugates) are transmitted. Thus, the effective code rate is 48 = 12 , resulting in a reduction of data throughput by a factor of 12 as compared to the square orthogonal designs. Also note from examination of each row of X, that a given transmitter sends each symbol and its complex conjugate one and only one  time and therefore XX∗T = 2( 4i=1 |xi |2 )I (i.e., C = 2). As was true for the square real orthogonal designs, space-time block codes generated from generalized complex orthogonal matrices have a simple ML decoding algorithm that is based only on linear processing. Furthermore, it has been shown that for any number of rows in X (i.e., any number of transmit antennas), there exists a construction that produces a code design that achieves full diversity but at a code rate of only 12 . For three and four transmit antennas, codes can be designed that achieve full diversity at three quarters of the full transmission rate. Also, while in principle any diversity order is now achievable, the higher the diversity order, the larger the SNR loss when compared with the equivalent (same diversity order) single transmit antenna–multiple receive antenna system. Specifically, a space-time

812

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

block-coded system with Lt transmit antennas and Lr receive antennas would suffer a 10 log10 Lt Lr dB SNR penalty when compared with the equivalent order Lt Lr receive diversity system.

14.5 ALAMOUTI’S DIVERSITY TECHNIQUE COMBINED WITH MULTIDIMENSIONAL TRELLIS-CODED MODULATION As discussed in the previous section, while space-time block codes achieve the maximum diversity gain with simple decoding algorithms (as has been demonstrated here for the Alamouti scheme), unfortunately they are not accompanied by coding gain. Perhaps the simplest method of achieving both diversity and coding gain is to combine multidimensional TCM with Alamouti’s scheme. One such method was investigated in Ref. 17, where a four-dimensional trellis-coded (2 × M)-PSK modulation was employed. Since this combination of trellis coding with the basic two-transmit antenna Alamouti scheme is conceptually the simplest form of spacetime TCM, we discuss its behavior here as a prelude to the presentation that follows in the next section dealing with the more general type of space-time TCM having an arbitrary number of transmit antennas. A block diagram of the system is illustrated in Fig. 14.4. In each transmission interval b bits enter the trellis encoder and k = 2m are generated, which are used to select a pair of M -PSK (M = 2m ) symbols from a 2 × M-PSK four-dimensional constellation. The pair of symbols is transmitted from two antennas using the Alamouti encoding scheme described in the previous section. The channel matrix Cn is (n) now 2 × Lr in size, and the received symbols, y1(n) , y2(n) , . . . , y2L , corresponding r to the nth transmission interval defined by (n) (n) (n) (n) yl(n) = cl1 x1 + cl2 x2 + n(n) l
∗
∗ (n) (n) (n) (n) yl+Lr = cl1 −x2 + cl2 x1(n) + n(n) l+Lr ,

l = 1, 2, . . . , Lr

(14.21)

are used to construct the decision variables x˜1(n) =

Lr


(n) cl1

∗

yl(n) +

l=1

Lr 


∗ (n) (n) yl+L cl2 r

l=1

L   Lr 
r  2  2  ∗    (n)   (n)  (n) (n) (n) cl1 = n(n) x1(n) + cl1  + cl2  l + cl2 nl+Lr l=1

l=1

L   r  2  2    (n)   (n)  = x1(n) + N˜ 1(n) cl1  + cl2  l=1

x˜2(n)

=

Lr
 l=1

(n) cl2

∗

yl(n)

−

Lr  l=1


∗ (n) (n) yl+L cl1 r

813

2 × M-PSK Trellis Encoder

2x2 Alamouti Block Encoder

(n)

(n) x2 ,

(n) (x1 )*

(n) c12

r

cL(n),2

(n) c22

r

cL(n),1

r

r

yL(n)− 1, yL(n)

(n) n(n) 3 , n4

y3(n), y4(n)

y1(n), y2(n)

Alamouti Receiver (n) (n) Maximum(Channel x˜ 1 , x˜2 Likelihood Estimators, Sequence Combiner) Detector

Figure 14.4 The two-branch Alamouti transmit diversity scheme combined with multidimensional trellis-coded modulation.

x2(n)

x1(n)

(n)

x1 , − (x2 )*

(n) c21

(n) c11

(n) n(n) 1 , n2

Trellis Decoder

814

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

L   Lr 
r  2  2  ∗    (n)   (n)  (n) (n) (n) cl2 = n(n) − c n x2(n) + cl1  + cl2  l l1 l+Lr l=1

l=1

L   r  2  2    (n)   (n)  = x2(n) + N˜ 2(n) cl1  + cl2 

(14.22)

l=1

which is the generalization of (14.13) to Lr antennas. Since the transmitted symbols are trellis-coded, the receiver must now perform maximum-likelihood sequence detection in space and time, which for an observation block of N transmission intervals uses the metric

 L   N  2 r  2  2   2  2
           (n) (n) (n) (n) (n) ˜ X = m X, c  + c  − 1 x  + x˜ − x  l1

n=1 i=1

l2

i

i

i

l=1

(14.23) or for equal energy constellations such as 2 × M-PSK N  2  2
   (n) (n)  ˜ X = m X, x˜i − xi 

(14.24)

n=1 i=1

˜ is defined accordingly. where X = [x1 , x2 , . . . , xN ] and X 14.5.1 Evaluation of Pairwise Error Probability Performance on Fast Rician Fading Channels The first step in evaluating the average BEP of the system in Fig. 14.4 is to compute the pairwise error probability (PEP), the probability of choosing the space-time ˆ when in fact the sequence X was transmitted. Upper (Chernoff) bounds sequence X on this probability were computed in Ref. 17. Here we evaluate exact expressions for the PEP. Conditioned on a realization of the channel over the entire block of length N, this probability is given by 

  ˆ ˜ X > m X, ˜ X ˆ |C P (X → X|C) = Pr m X, (14.25) 

  ˜ X − m X, ˜ X ˆ > 0|C = Pr m X, Assuming the equal energy 2 × M-PSK case, then substituting (14.22) into (14.24) and simplifying gives L  

2 N r  2  2    2         (n) (n) (n) (n) ˆ P (X → X|C) = Pr Z ≥ cl1  + cl2  xi − xˆi  

n=1

l=1

i=1

(14.26)

815

ALAMOUTI’S DIVERSITY TECHNIQUE COMBINED

where Z is conditionally (on the channel fading) a zero-mean Gaussian RV given by N

∗ ∗ 
 Z= 2Re xˆ1(n) − x1(n) N˜ 1(n) + xˆ2(n) − x2(n) N˜ 2(n)

(14.27)

n=1

Recognizing that the variance of Z can be computed as L 

2 N r  2  2    2        (n) (n) (n) (n)  σZ2 = 4σ 2 cl1  + cl2  xi − xˆi  n=1

l=1

(14.28)

i=1

then the conditional PEP can be expressed in terms of a Gaussian Q-function by   ! ˆ P (X → X|C) = Q  

x1(n)

    2 Lr  (n) 2  (n) 2 2  (n)  (n)  n=1 i=1 xi − xˆ i   l=1 cl1  + cl2    4σ 2 

N

x2(n)

Assuming that and are normalized M -PSK symbols such that E{|x2(n) |2 } = 1, then σ 2 = N0 /2Es and (14.29) becomes

(14.29) =

E{|x1(n) |2 }



 L 

2 N r  2  2    2   E       s (n) (n) (n) (n) ˜ P (X → X|C) = Q ! cl1  + cl2  xi − xˆi   2N0 n=1

l=1

i=1

(14.30) Under the assumption of fast Rican fading, the squared channel gain magnitudes (n) 2 (n) 2 |cl1 | and |cl2 | are i.i.d. noncentral chi-square RVs with parameter K and meansquare value . Because of the resemblance of the right-hand side of (14.30) with that for an equivalent MRC system having 2Lr antennas, the MGF-based approach can be used to obtain the unconditional PEP. Specifically, using the MGF for the SNR in a Rician channel as given in (2.17), and evaluating it at s = −1/2 sin 2 θ , we immediately obtain %

 N π/2 &

2Lr

(1 + K) sin θ   2     γ (n) (n) 2 0 n=1 (1 + K) sin2 θ + s i=1 xi − xˆ i  4    2   γ  2Lr K 4s 2i=1 xi(n) − xˆi(n)    × exp −  2  d θ γ s 2  (n) (n)  2 (1 + K) sin θ + 4 i=1 xi − xˆ i 

ˆ = 1 P (X → X) π

 

2

816

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

   N 2 π/2  & 1 (1 + K) sin θ   =   2   π 0    γ (n) (n) 2 2 n=1 (1 + K) sin θ + s x − xˆ  %

4

i=1

i

i

2lr  2  (n) (n)    i=1 xi − xˆ i    × exp − dθ  2     γ  (1 + K) sin2 θ + 4s 2i=1 xi(n) − xˆi(n)   

γ K 4s

2

(14.31) where, as always, γ s = Es /N0 denotes the average symbol SNR. For the Rayleigh fading case (K = 0), (14.31) reduces to  2Lr % π/2 & N 2 sin θ   ˆ = 1 P (X → X)   2  d θ π 0 γ s 2  (n) (n)  2 n=1 sin θ + i=1 xi − xˆ i  4 (14.32)   2Lr % N sin2 θ 1 π/2 &   =    2   d θ π 0 γ s 2  (n) (n)  2 n=1 sin θ + i=1 xi − xˆ i  4 Denoting the squared Euclidean distance between the transmitted and chosen 2 × M-PSK symbols in the nth symbol interval by 2  2       dn2 = x1(n) − xˆ1(n)  + x2(n) − xˆ2(n)  (14.33) then the product on n in (14.28) and (14.29) can be replaced by a product on η, where η is the set of all n for which dn2 = 0; that is, the cardinality of η is the ˆ multidimensional symbolwise Hamming distance dH between X and X. Comparing the PEP of an analogous single transmit/single receive antenna trellis-coded M -PSK system, which for a fast Rician fading channel is given by the combination of (13.20) and (13.25), with the PEP of the trellis-coded 2 × M-PSK Alamouti scheme as given in (14.31), we observe that the latter implies a coding gain dependent on the multidimensional distance properties of the trellis code as well as the addition of a diversity gain equal to 2Lr . The total diversity gain of the trellis-coded M -PSK Alamouti scheme can be determined by explicitly examining the asymptotic (large SNR) behavior of (14.31). For example, for the Rayleigh channel, we first obtain the Chernoff bound on PEP by setting θ = π /2 in the integrand of (14.32) and then letting γ s become large resulting in . 2 1/d H γ ]−2Lr dH . Thus, we conclude that, for the fast fading ˆ ≈ C[( P (X → X) s η dn ) channel, in so far as the PEP is concerned, the combination of space-time transmit diversity offered by the Alamouti code, receive diversity offered by MRC, and trellis coding results in a multiplicative diversity of order 2Lr dH . It is also interesting to note that, for large SNR, the asymptotic coding gain behaves inversely as the product of the branch squared Euclidean distances [see (14.33)], as is typical of the performance of trellis codes over fading channels [30].

817

ALAMOUTI’S DIVERSITY TECHNIQUE COMBINED

Finally, we note that the Chernoff bounds on PEP obtained in [16] can be readily obtained from (14.31) and (14.32) by upper-bounding the integrands in these equations by their value at θ = π /2. This follows immediately from the fact that upper-bounding the integrand of the alternative representation of the Gaussian Q-function given in (4.2) by its value at θ = π /2 results in the Chernoff bound on this function. 14.5.2 Evaluation of Pairwise Error Probability Performance on Slow Rician Fading Channels For the slow-fading case, the channel gains are assumed to be constant over the duration of the N -symbol block length, specifically, a data frame. Notationwise, (n) 2 (n) 2 we replace |cl1 | by |cl1 |2 and |cl2 | by |cl2 |2 . As such, the conditional PEP of (14.26) becomes  ˆ P (X → X|C) = Pr Z ≥

L r 



N 2  2    (n)   (n) |cl1 |2 + |cl2 |2 (14.34) xi − xˆi 

l=1

n=1 i=1

where Z is still defined as in (14.27) with, however, the effective noise components now given by N˜ 1(n) =

Lr  

(n) + c n (cl1 )∗ n(n) l2 l l+Lr ,

l=1

N˜ 2(n) =

Lr   (n) − c n (cl2 )∗ n(n) l1 l l+Lr l=1

(14.35) Again the conditional PEP of (14.34) can be expressed in terms of a Gaussian Q-function analogous to (14.30) by 



N 2 L r 2     (n)

 E  s (n) ˆ |cl1 |2 + |cl2 |2 P (X → X|C) = Q ! xi − xˆi   2N0 l=1

n=1 i=1

(14.36) which after averaging over the Rician slow-fading statistics gives

ˆ = P (X → X)

%



2Lr

(1 + K) sin θ   2      γ (n) (n) N 2 0 (1 + K) sin2 θ + 4s n=1 i=1 xi − xˆi    2 2  (n) γ  (n)  2Lr K 4s N − x ˆ x  n=1 i=1 i i   × exp −  2  d θ   γs (n)  N 2  (n) 2 (1 + K) sin θ + 4 n=1 i=1 xi − xˆ i  1 π

π/2

 

2

(14.37)

818

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

or for the Rayleigh channel ˆ = 1 P (X → X) π



%

π/2 0

 

2Lr 2

sin2 θ +

γs 4

  2   (n) (n)  i=1 xi − xˆ i 

sin θ N 2 n=1

dθ

(14.38)

Since 

dE2 =

N  2  2   (n) (n)  xi − xˆi 

(14.39)

n=1 i=1

represents the multidimensional squared Euclidean distance associated with the entire error event path, then we see that, on slow-fading channels, the PEP depends on this entire path distance rather than the individual branch distances and furthermore the trellis code does not contribute to the diversity. At high average SNR, the performance is dominated by the error event path producing the small2 , and thus to est value of dE2 , namely, the squared free Euclidean distance, dfree obtain the maximum coding gain, one should design the code to achieve the 2 . This is the identical criterion used for designing optimum largest value of dfree trellis codes on the AWGN channel. It should also be noted that the importance of Euclidean distance as a code design criterion comes about because the Alamouti scheme is an orthogonal block code design. Without orthogonality, other metrics become important.

14.6

SPACE-TIME TRELLIS-CODED MODULATION

The next step in the evolution of the theory presented in this chapter is to consider the more general space-time trellis coding scenario where the number of transmit antennas Lt and the number of receive antennas Lr are both arbitrary. Also, whereas in the previous section the trellis code was designed to act in combination with the 2 × Lr block code of the Alamouti transmit diversity scheme, here the trellis code by itself attempts to accomplish both the transmit diversity and coding gain functions. In what follows we use the approach taken in Ref. 31 to evaluate the PEP and average BEP of such schemes, which closely follows the method presented in the previous section but in a somewhat more general context. Since now each information symbol available at each antenna is transmitted only once, that is, the transmission interval is now equal to the symbol interval and of duration Ts , the received samples (matched-filter outputs) are given by yj(n) =

Lt  i=1

(n) (n) cj(n) i xi + nj ,

j = 1, 2, . . . , Lr ,

n = 1, 2, . . . , N

(14.40)

SPACE-TIME TRELLIS-CODED MODULATION

819

2 where n(n) j is a sample of a zero-mean, variance 2σ = N0 /Es complex Gaussian process representing the additive thermal noise. As before, these noise samples are i.i.d. and independent of the channel gains. Assuming, as in the previous section, that the receiver has perfect knowledge of the channel state, then the appropriate ML metric to be minimized by the choice of X is

 2 Lt Lr  N     (n)  (n) (n)  cj i xi  m (Y, X) = yj −   n=1 j =1

(14.41)

i=1

and the corresponding conditional PEP is given by12 

 N  /

/ E /Cn xn − xˆ n /2  ˆ |{Cn } ) = Q ! s P (X → X 2N0

(14.42)

n=1

To compute the average PEP, we again apply the MGF-based technique, resulting in ˆ = 1 P (X → X) π

%

π/2 % ∞

 exp −



2 sin2 θ   % 1 1 π/2 dθ M − = π 0 2 sin2 θ 0

 p () d d θ

0

(14.43)

where M (s) is the MGF of =

N

/ Es  / /Cn xn − xˆ n /2 2N0

(14.44)

n=1

and we define the squared norm of a matrix by sum of the magnitude squared  the N 2 r of all its elements, specifically, ALr ×N 2 = L i=1 |ali | or equivalently the l=1 ∗ T trace of the matrix ALr ×N (ALr ×N ) . Note that to the extent that the MGF can be evaluated in closed form for a given fading channel type, the relation in (14.43) is exact and can be evaluated as a single integral with finite limits. For the slow-fading model, we would have Cn = C independent of n, in which case (14.44) becomes
/2 Es / / ˆ / (14.45) = /C X − X / 2N0 12 A similar analysis is performed in Ref. 32 using, however, the inverse Laplace transform approach for evaluation of the PEP.

820

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

14.6.1 Evaluation of Pairwise Error Probability Performance on Fast Rician Fading Channels To evaluate the average BEP, we again make use of the assumed i.i.d. properties of the channel, whereupon (14.43) evaluates to

ˆ = P (X → X)

=

1 π

1 π

%

N π/2 & 0

%

 exp −

n=1

Es 4N0 sin2 θ

/

/ /Cn xn − xˆ n /2

Cn dθ



(n) 2 cj i L t  
 Es   (n) (n) exp − cj i xi − xˆi(n)   d θ 2   4N sin θ 0 n=1 j =1 i=1

Lr N & π/2 & 0

(14.46) ξ

where the notation ( · ) denotes statistical expectation with respect to the random quantity (single variable or vector of variables) ξ . Since for the Rician   t (n) (n) (n) channel Zj(n) = L i=1 cj i (xi − xˆ i ) is a complex Gaussian RV with variance  (n) (n) L t |xi − xˆi |2 , which is independent of j , then, defining the normalσn2 =  i=1 

ized RV ζj(n) = Zj(n) /σn , and noting that the ζj(n) s are i.i.d., (14.46) can be rewritten in terms of this new notation as 

  Lr ζ (n)   2  (n)   t  (n) % N  γs L  2  i=1 xi − xˆ i   1 π/2 &   (n)   dθ   ˆ ζ exp − P (X → X) =    2 π 0 4 sin θ   n=1



(14.47) where, without loss in generality, we have dropped the j subscript on ζj(n) . Finally, averaging over the noncentral chi-square-distributed random variable |ζ (n) |2 , equivalently making use of the Rician MGF in (2.17), gives the desired result

ˆ = P (X → X)

1 π

%

π/2 0

    N & 2 (1 + K) sin θ     2      γ (n) (n) L t n=1 (1 + K) sin2 θ + s x − xˆ  4

i=1

i

i

Lr 2 Lt  (n) (n)    − x ˆ x  i i=1 i   × exp − dθ    2  γ  t  (n) (n)   − x ˆ x (1 + K) sin2 θ + 4s L   i i=1 i 

K

γs 4

(14.48)

SPACE-TIME TRELLIS-CODED MODULATION

821

which for the Rayleigh channel simplifies to

ˆ = 1 P (X → X) π

π/2 0

 N &    

Lr



%

n=1

2

sin2 θ +

γs 4

sin θ  2   Lt  (n)  (n) i=1 xi − xˆ i 

dθ

(14.49)

Comparing (14.48) and (14.49) with (14.31) and (14.32), respectively, it is readily apparent that the factor of 2 in diversity contributed by the Alamouti transmit diversity scheme is absent in the latter and the number of transmit antennas acts only to produce a coding gain dependent on the multidimensional distance properties of the space-time trellis code. 14.6.2 Evaluation of Pairwise Error Probability Performance on Slow Rician Fading Channels As required by (14.43), we need to evaluate the MGF of , which for the slowˆ 2 in the form fading case is defined in (14.45). We first rewrite C(X − X) Lr ∗ T ∗ T ∗ ˆ ˆ l=1 cl (X − X)(X − X ) (cl ) , where cl denotes the l th row of the constant channel gain matrix C. Then, for the Rician channel where cl is a complex Gaussian vector, with mean cl , using a result from Turin [33], it can be shown that −Lr γs 2 (1 + K)

 ˆ − ILt ×Lt − s X − X

  

ˆ X∗ − X ˆ∗ M (s) = det ILt ×Lt − s X − X

T

   0 1+K × exp −Lr c ILt ×Lt   −1
 

∗ γ T s ˆ T × X∗ − X c∗ 2 (1 + K)

(14.50)

Note that we have dropped the l subscript on cl since, because of the i.i.d. assumption, all rows of the channel gain matrix have identical means. Furthermore, since the mean of any element of this matrix, µ = E{cij }, has magnitude mean-square value |µ|2 = K/ (1 + K), then (14.50) can be rewritten as   

ˆ X∗ − X ˆ∗ M (s) = det ILt ×Lt − s X − X 





T

γs 2 (1 + K)

−Lr


 ˆ × exp −Lr K × tr ILt ×Lt − ILt ×Lt − s X − X


ˆ∗ × X −X ∗



T

γs 2 (1 + K)

−1  (14.51)

822

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

where tr [A] denotes the trace of the matrix A. For the Rayleigh channel, (14.51) simplifies to   

ˆ X∗ − X ˆ∗ M (s) = det ILt ×Lt − s X − X

T

γs 2

−Lr (14.52)

∗−X ˆ ˆ ∗ )T is a real Lt × Lt matrix whose ij th element is the Note that (X − X)(X correlation of the i th and the complex conjugate of the j th rows of the symbol ˆ Thus, if the rows of X − X ˆ are orthogonal (as can be the case error matrix X − X. ∗−X ˆ ˆ ∗ ) T will be an Lt × Lt for certain error paths in the trellis), then (X − X)(X ∗ ˆ ˆ ∗ ) T {γ s /[2 (1 + K)]} will diagonal matrix and the matrix ILt ×Lt − s(X − X)(X −X likewise be diagonal with elements

1−

N  2  sγ s  (n) (n)  xi − xˆi  , 2 (1 + K)

i = 1, 2, . . . , Lt

n=1

For this case, then, we have 




det ILt ×Lt =

Lt &

ˆ −s X−X






i=1

sγ s 1− 2 (1 + K)

ˆ∗ X −X



γs 2 (1 + K)

N 2   (n) (n)  xi − xˆi  ∗



T

(14.53)

n=1

and

−1  

∗ T γs ∗ ˆ ˆ tr ILt ×Lt − ILt ×Lt − s X − X X − X 2 (1 + K)  2    sγ (n) (n) N s Lt − 2(1+K)  n=1 xi − xˆ i  =  2 sγ s N  (n) (n)  i=1 1 − − x ˆ x n=1 i i  2(1+K)

(14.54)

Substituting (14.53) and (14.54) into (14.51) gives the desired result

M (s) =

Lt &

  1−

i=1

N  2  sγ s  (n) (n)  xi − xˆi  2 (1 + K)

−1

n=1

2 Lr N  (n) (n)    − x ˆ x n=1  i i   × exp −K   2      sγ (n) (n) N   s 1 − 2(1+K) n=1 xi − xˆ i    

sγ

s − 2(1+K)

(14.55)

823

SPACE-TIME TRELLIS-CODED MODULATION

which for the Rayleigh channel reduces to L

 N t 2 −Lr & sγ s   (n) (n)  M (s) = 1− xi − xˆi  2 i=1

=

Lt &

n=1

1−

i=1

sγ s 2

N  

2  (n) (n)  xi − xˆi 

(14.56)

−Lr

n=1

Finally, from (14.43), the PEP for the slow Rician and Rayleigh fading channels become    % π/2  Lt  2 & (1 + K) sin θ   ˆ = 1 P (X → X)   2   π 0    γ (n) (n) N  i=1 (1 + K) sin2 θ + s x − xˆ  4

n=1

i

i

lr 2 N  (n) (n)    n=1 xi − xˆ i    × exp − dθ  2   γ s N  (n) (n)  2  θ + − x ˆ + K) sin (1 n=1 xi i  4 

γ K 4s

(14.57) and ˆ = 1 P (X → X) π

%

π/2 0

  Lt &    i=1

Lr 2

sin2 θ +

γs 4

sin θ  2   d θ N  (n)  (n) n=1 xi − xˆ i 

(14.58)

Recalling the form of (14.48) and (14.49) for the fast-fading case, we see that insofar as the evaluation of the PEP is concerned, the role of the number of transmitters and the length of the error event path swap places for the two different fading models; in other words, for the slow-fading model, the number of transmitters once again contributes a factor of Lt to the total diversity. We hasten to remind the reader that this statement is valid only for those error event paths where the symˆ has orthogonal rows. As we shall show in the example that bol error matrix X − X follows, this condition is one that can be satisfied and allows evaluation of the PEP in closed form. ˆ does not have orthogonal Even in the case where the symbol error matrix X − X rows, it is still possible to obtain a product form for the MGF of . Specifically, the matrix whose determinant is required in (14.51) and (14.52) can be diagonalized, in which case we obtain −Lr   

γs ˆ X∗ − X ˆ∗ T det ILt ×Lt − s X − X 2 (1 + K) (14.59)   r −Lr & γs 1+ λi = 2 (1 + K) i=1

824

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

where r ≤ Lt and λi are, respectively, the rank and i th nonzero eigenvalue of ∗−X ˆ ˆ ∗ ) T . Thus, for example, using (14.59) in (14.52) and substituting (X − X)(X the corresponding result obtained for the MGF in (14.43), the PEP for the slow Rayleigh fading channel becomes [analogous to (14.58)] ˆ = 1 P (X → X) π

%

π/2 0

r  & i=1

sin2 θ sin2 θ +

γs 4 λi

 Lr dθ

(14.60)

which can be evaluated in closed form using (5A.75). Before proceeding, it is worth pointing out that exact expressions as well as upper bounds on PEP for space-time trellis codes have also been obtained by other researchers using a variety of techniques. Specifically, in Ref. 10 the PEP is upper-bounded with the traditional Chernoff bound in the same manner as done in Ref. 34 for time coding, for instance, TCM alone. More recently, Aktas and Fitz [35] considered the distance spectrum of space-time trellis codes and along with that an upper bound on the PEP in combination with the union bound to compute average BEP. In Ref. 36, a method for exactly evaluating PEP using an eigenanalysis based on Turin’s results [33] for the characteristic function associated with a quadratic form of a complex Gaussian vector is developed. This method also led to an upper bound on PEP that is asymptotically tight at high SNR and tighter than the Chernoff bound. Finally, in Ref. 37, the characteristic function technique previously used in the performance of TCM [38] is extended to the space-time problem to allow for exact evaluation of PEP by residue methods. The advantage of the MGF-based method presented in this paper is that (1) in certain cases it can allow for evaluation of the PEP (and therefore approximation to the average BEP) in a simple closed form, and (2) it can allow for direct evaluation of the transfer function upper bound on average BEP for the fast fading case. The latter will be demonstrated in a later section of the chapter. 14.6.3

An Example

At this point in our discussion, it is instructive to present an example of the evaluation of the PEP for space-time codes partially with the intent of validating the row orthogonality of the symbol error matrix assumed in the slow-fading case. For simplicity, we shall consider the Rayleigh fading case. Consider the four-state QPSK space-time code discussed by Tarokh et al. [9,10] (also in Ref. 37) with Lt = 2 transmit antennas and illustrated here in Fig. 14.5. The labeling ii /kk along each branch of the trellis refers to the pair of input bits (ii ) and the corresponding pair of output symbols (kk ) that result from the transition between the states at the beginning and end of the branch. The output symbol notation k refers to the integer multiple of π /2 that characterizes the phase in the complex representation of the complex QPSK symbol [i.e., ej (kπ/2) ]. The QPSK symbols are the elements ˆ matrices associated with the trellis. of the X and X Assuming that the correct path is the all-zeros sequence (zero phase for all symbols), then for the shortest error event path of length N = 2 illustrated by

SPACE-TIME TRELLIS-CODED MODULATION

00/00 01/01 10/02 11/03

00/00

825

00/00

2 /0 10

00 /2 0

00/10 01/11 10/12 11/13

00/20 01/21 10/22 11/23

00/30 01/31 10/32 11/33

1 QPSK Signal Point Constellation

2

0

3

Figure 14.5 Trellis diagram and signal point constellation for four-state QPSK space-time code.

shading in Fig. 14.5, we have   1 1 X= , 1 1

ˆ = X



1 −1 −1 1



with corresponding symbol error matrix and complex conjugate product     

∗ T 0 2 4 0 ∗ ˆ ˆ ˆ X−X= , X−X X −X = 2 0 0 4

(14.61)

(14.62)

ˆ are orthogonal (as previously stated) and so are Note that the rows of X − X ∗ T ∗ ˆ ˆ the rows of (X − X)(X − X ) . For the fast-fading case, substituting (14.61) in (14.49) gives ˆ = 1 P (X → X) π

% 0

π/2

 2Lr  −2Lr % sin2 θ γs 1 π/2 1+ dθ = dθ π 0 sin2 θ sin2 θ + γ s (14.63)

826

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

1.0E−1

PEP

1.0E−2

1.0E−3

1.0E−4

1.0E−5 0

2

4

6

8

10

12

14

16

18

20

Avg. Symbol SNR (dB) Figure 14.6 PEP performance of four-state space-time code over i.i.d. Rayleigh fading channel with two transmit antennas and one receive antenna—Length 2 error event.

The integral in (14.63) can be found in closed form using (5A.4a). In particular  1 k   2L r −1     1 γ 1 2k s ˆ =

 P (X → X) 1− k  2 1 + γs 4 1 + γs k=0

(14.64)

The PEP of (14.64) is plotted in Fig. 14.6 for the case of Lr = 1 and is in exact agreement with Fig. 2 of Ref. 37 obtained by other means as mentioned previously. For the slow fading case, the result in (14.56) gives the MGF −2Lr

M (s) = 1 − 2sγ s

(14.65)

Finally, substituting (14.65) in (14.43) results in the identical PEP as for the fastfading case given by (14.63) or equivalently the closed form of (14.64). The fact that the two results are identical is somewhat of a coincidence because of the particular choice of the system parameters characterizing the example chosen for illustration: N = Lt = 2. The important point to note, however, is that for the fastfading case, the diversity factor of 2 in the exponent of the integrand in (14.63) comes from the length of the error event path N, whereas the same factor for the slow-fading case comes from the number of transmitters Lt .

SPACE-TIME TRELLIS-CODED MODULATION

14.6.4

827

Approximate Evaluation of Average Bit Error Probability

In this section, we use the PEPs derived earlier to evaluate in closed form an approximation to the average BEP by accounting only for error events of lengths less than or equal to H. In the next section, we shall demonstrate the accuracy of this approximation by comparing it to the true upper bound obtained from the transfer function of the code that accounts for error events of all lengths. For the purpose of illustration, we shall concentrate our attention on the example discussed in Section 14.6.3. 14.6.4.1 Fast-Fading Channel Model Assuming transmission of the all-zeros sequence,13 then, for the 4-state code in Fig. 14.5, there are three error event paths of length 2 and nine error event paths of length 3. Of the three error events of length 2, two of them are of type I and one is of type II, with corresponding PEPs: 2Lr % π/2  2 sin θ 1 ˆ I= P (X → X) dθ π 0 sin2 θ + γ s /2 (14.66) 2Lr % π/2  2 θ 1 sin ˆ II = P (X → X) dθ π 0 sin2 θ + γ s The two paths of type I contribute a total of three bit errors, whereas the one path of type II contributes one bit error. Thus, if we were to choose to approximate the average BEP by considering only error event paths of minimum length (i.e., H = 2), we would have the closed-form result  1 k   2L r −1     1 γ /2 3 2k s

 Pb (E) ∼ 1− = k  4 1 + γ s /2 4 1 + γ s /2 k=0 (14.67)  1 k   2L r −1     1 γs 1 2k 

+ 1− k  4 1 + γs 4 1 + γs k=0

Similarly, the PEPs of the nine possible error event paths of length 3 can be grouped into three different types. In particular, defining 2Lr   Lr % sin2 θ sin2 θ 1 π/2 ˆ P (X → X)III = dθ π 0 sin2 θ + γ s /2 sin2 θ + γ s 2Lr  Lr % π/2  2 2 θ θ 1 sin sin ˆ IV = P (X → X) dθ π 0 sin2 θ + γ s sin2 θ + 2γ s 13 We note that although space-time codes may, in general, be nonuniform, that is, that the average BEP can depend on the transmitted sequence, we consider here only the case where the transmitted path is the all-zeros one as was done, for example, in Ref. 37.

828

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

ˆ V= 1 P (X → X) π

%

π/2



sin2 θ

Lr 

sin2 θ + γ s /2 Lr  sin2 θ × dθ sin2 θ + 3γ s /2 0

sin2 θ

Lr

sin2 θ + γ s (14.68)

then, noting that there are four length 3 event paths of type III that contribute a total of 12 bit errors, one length 3 error event path of type IV that contributes 2 bit errors, and four length 3 error event paths of type V that contribute a total of 10 bit errors, the average BEP corresponding to H = 3 is approximated by  ˆ I + P (X → X) ˆ II + 12P (X → X) ˆ III Pb (E) ∼ = 12 3P (X → X) (14.69) ˆ IV + 10P (X → X) ˆ V +2P (X → X) where the first two terms account for the length 2 error events as before [see (14.67)] whereas the last three terms account for the length 3 error events as described above. ˆ I and P (X → X) ˆ II , the PEPs P (X → X) ˆ III , P (X → As was the case for P (X → X) ˆ V can be evaluated in closed form using the results in Appendix ˆ IV and P (X → X) X) 5A. The approximate average BEP performances for Lr = 1 and Lr = 2 are plotted in Fig. 14.7 (also see Figs. 3 and 4 of Ref. 37 corresponding to the curves labeled H = 3.) Also shown in Fig. 14.7 for comparison are analogous results obtained from (14.67) when only error event paths of length 2 are accounted for.

1.0E+0 Length 2 and 3 Error Events

1.0E−1

Length 2 Error Events

Lr = 1

Transfer Function Bound

Average BEP

1.0E−2 Lr = 2 1.0E−3

1.0E−4

1.0E−5

1.0E−6 4

6

8

10

12

14

16

18

Avg. Symbol SNR (dB)

Figure 14.7 Average BEP performance of four-state space-time code over i.i.d. Rayleigh fading channel with two transmit antennas; fast fading.

SPACE-TIME TRELLIS-CODED MODULATION

829

14.6.4.2 Slow-Fading Channel Model ˆ for the three length 2 paths, one finds Evaluating the symbol error matrix X − X ∗−X ˆ ˆ ∗ )T that each matrix has orthogonal rows, and thus the products (X − X)(X are all diagonal. Hence, it is straightforward to show that the PEPs accounting only for length 2 paths are of the two types given in (14.66) and the average BEP is once again given by (14.67). Extending now the analysis to the case where length 3 error events are considˆ ered, it can be shown that for none of the nine events of such length does the X − X ∗ T ∗ ˆ ˆ matrix have orthogonal rows. However, the determinants of the (X − X)(X − X ) matrices can still be easily evaluated and fall into three different types. After some effort it can be shown, interestingly enough, that the form of the average BEP that accounts for both length 2 and length 3 error events is once again given by (14.69), ˆ IV , and P (X → X) ˆ V are now given by ˆ III , P (X → X) where, however, P (X → X)  2Lr % π/2 2 sin θ   ˆ III = 1 P (X → X)  2

 dθ  2 π 0 sin2 θ + γ s − γ 2s /4  2Lr % π/2 2 sin θ   ˆ IV = 1 P (X → X) (14.70)  2

 dθ 2 π 0 2 2 sin θ + 2γ s − γ s  2Lr % π/2 2 sin θ   ˆ V= 1 P (X → X)  2

 dθ  2 π 0 sin2 θ + 3γ s /2 − γ 2s /2 Unfortunately, the form of the integrals in (14.70) does not have a readily available closed form, and thus one must evaluate these numerically. Illustrated in Fig. 14.8 are the approximated average BEP performances as computed from (14.69) together with (14.70) for Lr = 1 and Lr = 2. Also shown for comparison are the analogous results obtained from (14.67) where only error event paths of length 2 are accounted for. Comparing Figs. 14.7 and 14.8, we observe that the convergence to the true upper bound (that takes into account error event paths of all lengths) on BEP is slower for the slow-fading case than for the fast-fading case. Equivalently, to obtain an upper bound that asymptotically approaches the true BEP performance, one must consider larger values of H for the slow-fading case than for the fastfading case. To demonstrate this point, the 27 error event paths of length 4 were evaluated for the slow-fading case. Of the 27 possible PEPs associated with these paths, it was found that there are only 10 different ones, all of which are of the form  2Lr % π/2 2 sin θ   ˆ = 1 (14.71) P (X → X)  2

 dθ  2 π 0 2 2 sin θ + aγ s − bγ s

830

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

1.0E+0 Length 2 and 3 Error Events

1.0E−1

Length 2 Error Events

Average BEP

1.0E−2

Lr = 1

1.0E−3

1.0E−4

Lr = 2

1.0E−5

1.0E−6 4

6

8

10

12

14

16

18

Avg. Symbol SNR (dB)

Figure 14.8 Average BEP performance of four-state space-time code over i.i.d. Rayleigh fading channel with two transmit antennas; slow fading.

with appropriately chosen values of a and b. Table 14.1 summarizes the values of a and b for these 10 different PEPs along with the combined number of bit errors associated with the paths that produce each of them. Using the results of this table combined with the previous results obtained for length 2 and 3 error event paths, the approximate average BEP for H = 4 was computed, and the results are superimposed on Fig. 14.8. We observe that even with H = 4, the upper bound still has not reached its asymptotic behavior.

TABLE 14.1 PEP Parameters and Combined Number of Bit Errors for Error Events of Length 4. a

b

3/2 2 2 5/2 2 3/2 2 3/2 5/2 3

1 5/4 1 5/2 2 0 1/4 1/2 1 4

Combined No. of Bit Errors 9 16 8 14 8 9 16 18 7 3

SPACE-TIME TRELLIS-CODED MODULATION

831

14.6.5 Evaluation of the Transfer Function Upper Bound on Average Bit Error Probability In this section, we use the previously computed PEPs in combination with the transfer function bound method discussed in Section 13.1.3 to upper-bound the average BEP. Again, for simplicity, we consider only the Rayleigh case. 14.6.5.1 Fast-Fading Channel Model Recall that, for the AWGN channel, one computes the transfer function of the code T (D, I ), and then obtains the upper bound on BEP from Eq. (13.48). For the space-time code in the presence of fast fading, matters become a bit more complicated since one must now keep track of the D and I labeling on a per 2 branch basis. In particular, the label D (θ )|xn −xˆn | on each branch in  t of the trellis 2 )]Lr the AWGN case is now replaced by [sin2 θ /(sin2 θ + (γ s /4) L |x − x ˆ | i,n i=1 i,n in accordance with (14.49). This branch label can be evaluated in only a finite  t 2 , which number of ways corresponding to the possible values for L |x − x ˆ | i,n i=1 i,n depend on the modulation assumed. For example, for the case at hand where Lt = 2 Lt 2 and the modulation is QPSK, i=1 |xi,n − xˆi,n | can assume values of only 2, 4, 6, and 8 (see the QPSK constellation in Fig. 14.5). Thus, for this case, we define the branch labels  Dk (θ ) =

Lr

sin2 θ sin2 θ + k

γs 4

,

k = 2, 4, 6, 8

(14.72)

and use these on the trellis of Fig. 14.5 as appropriate (see Fig. 14.9), where we have also included the I labeling whose exponent reflects the number of bit errors per branch. [For completeness of notation, we also define D0 (θ ) = 1 corresponding to xn = xˆ n .] With this labeling, one computes the transfer function, which will now be a ratio of polynomials in D2 (θ ) , D4 (θ ) , D6 (θ ) , D8 (θ ), and I, from which the result analogous to (13.48) for the AWGN channel becomes Pb (E) ≤

1 π

% 0

π/2

1 ∂ T ({Dk (θ )} , I ) |I =1 d θ nc ∂I

(14.73)

Here the notation {Dk (θ )} , I implies that the transfer function is computed with the above-mentioned replacement labeling. For the four-state code trellis in Fig. 14.5, the transfer function bound can be evaluated (with some effort) as

 

I 3+ I 2 D22 (θ ) + I D42 (θ ) − I 2 + I 3 4 × D22 (θ ) D8 (θ ) + D43 (θ ) − 2D2 (θ ) D4 (θ ) D6 (θ )  

2 T ({Dk (θ )} , I ) = 2 3 1− 3 I + I D4 (θ ) −2I D84 (θ ) + I + I × D4 (θ ) D8 (θ ) − D6 (θ ) (14.74)

832

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

D0(q),I 0 D2(q),I 1 D4(q),I 1 D2(q),I 2

D2(q),I 0 D4(q),I 1 D6(q),I 1 D4(q),I 2

D4(q),I 0 D6(q),I 1 D8(q),I 1 D6(q),I 2

D2(q),I 0 D4(q),I 1 D6(q),I 1 D4(q),I 2 Figure 14.9 Trellis diagram for four-state space-time code for QPSK with relabeled branches.

from which 3D22 (θ ) + D42 (θ ) − 6D22 (θ ) D8 (θ ) − 4D43 (θ ) + 10D2 (θ ) D4 (θ ) D6 (θ )

∂ T ({Dk (θ )} , I ) |I =1 = 3 42 ∂I 1 − 2D4 (θ ) − D8 (θ ) + 2D4 (θ ) D8 (θ ) − 2D62 (θ ) 4D44 (θ ) − 8D2 (θ ) D42 (θ ) D6 (θ ) + 3D22 (θ ) D82 (θ ) − 6D2 (θ ) D4 (θ ) D6 (θ ) D8 (θ ) +3 42 1 − 2D4 (θ ) − D8 (θ ) + 2D4 (θ ) D8 (θ ) − 2D62 (θ ) 4D22 (θ ) D62 (θ ) + 3D42 (θ ) D62 (θ ) +3 42 1 − 2D4 (θ ) − D8 (θ ) + 2D4 (θ ) D8 (θ ) − 2D62 (θ ) (14.75) As a simple check on our previous result based on H = 2, we observe that for large SNR, the first two terms in the numerator of (14.75) dominate: ∂ T ({Dk (θ )} , I ) |I =1 ∼ = 3D22 (θ ) + D42 (θ ) ∂I

(14.76)

Thus, letting nc = 2, we get the approximation 1 Pb (E) ∼ = π 1 = π

%

π/2

0

% 0

π/2

4 13 2 3D2 (θ ) + D42 (θ ) d θ 2   2Lr  2Lr  sin2 θ sin2 θ 1  dθ + 3 2 sin2 θ + γ s /2 sin2 θ + γ s

(14.77)

which, when evaluated using (5A.4a), gives the result in (14.67). Similarly, for H = 3, expanding the denominator polynomial of (14.75), taking its reciprocal,

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

833

and then keeping the leading terms in the numerator up to order 3, we would get ∂ T ({Dk (θ )} , I ) |I =1 ∼ = 3D22 (θ ) + D42 (θ ) + 12D22 (θ ) D4 (θ ) ∂I

(14.78)

+ 2D42 (θ ) D8 (θ ) + 10D2 (θ ) D4 (θ ) D6 (θ ) which when substituted in (13.48) gives the identical result to (14.69) combined with (14.68). The transfer function upper bound on average BEP computed from the above is superimposed on the results of Fig. 14.7. We observe that at average symbol SNRs greater than about 10 dB, the three-term approximation of average BEP as given in (14.69) is sufficient to achieve the transfer function upper bound. Finally, we note from a comparison of (14.49) and (14.32) that, for the fastfading case, the transfer function bound approach applies equally well for upperbounding the average BEP of the multidimensional trellis-coded Alamouti diversity scheme discussed in Section 14.5. 14.6.5.2 Slow-Fading Channel Model For the fast-fading channel model, we saw that the integrand of each PEP (i.e., the MGF) corresponding to an error event path of length N was composed of a product of N factors of the type Dk (θ ) defined in (14.72). Because of this product form over the trellis branches that constitute the error event path for each PEP, the transfer function bound approach was directly applicable to evaluating an upper bound on average BEP. Unfortunately, for the slow-fading channel, such a product form (over the number of trellis branches in the error event paths) is not immediately evident from the form of the MGF in (14.52), and thus the transfer function bound approach is not directly applicable for evaluating average BEP in this manner. To obtain a tight upper bound on the average BEP, one would resort to a continuation of the method described in Section 14.6.4 for larger values of H, which is best performed by a computer search method. Finally, we note from the form of (14.38), that, for the slow-fading case, the transfer function bound approach is also not applicable for upper-bounding the average BEP of the multidimensional trellis-coded Alamouti diversity scheme discussed in Section 14.5.

14.7 OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE-TIME TRELLIS CODES Thus far we have seen that STBCs are capable of providing full diversity and a very simple decoding scheme; however, despite their name, their main objective is not to provide additional coding gain. By contrast, STTCs, which also can provide full diversity as well as coding gain, come at a cost of higher decoding complexity. In Section 14.5 we demonstrated the first example of marrying these two alternative code designs so as to achieve additional coding gain beyond that achievable by the STTC alone. In this section, we consider two other types of concatenated spacetime trellis/block codes that further exploit the advantages of combining the two

834

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

forms of STC. From a performance evaluation standpoint, the importance aspect of these forms of concatenated STTC is that they both lead to a block diagonal form ∗−X ˆ ˆ ∗ )T , thereby allowing one to readily apply the same analysis of (X − X)(X approaches that were previously used in evaluating the STTC schemes alone. 14.7.1

Super-Orthogonal Space-Time Trellis Codes

Two other examples of the concatenation of an outer trellis code with an inner space-time block code can be found in Refs. 18 and 22; however, the shortcoming of these schemes is that they do not provide the highest rate. In an effort to overcome this shortcoming, the notion of a super-orthogonal space-time trellis code (SOSTTC) was introduced [39] that first described a parameterized class of STBCs based on the Alamouti 2 × 2 code and then assigned constituent members of this class to the states of an MTCM based on the set partitioning construction.14 The results presented in Ref. 39 focus primarily on the design of these new codes according to the rank and coding gain distance principles originally introduced in Ref. 10 for STCs. In keeping with the main theme of our book and, in particular, this chapter, our interest here is in evaluation of the performance of these codes as characterized by their PEP and approximate average BEP for both the fast-fading and slow (quasi-static)-fading channels [46]. 14.7.1.1 The Parameterized Class of Space-Time Block Codes and System Model We start by reconsidering the concatenated scheme of a trellis code and the Alamouti block code discussed in Section 14.5. Since each of the two symbols x1 , x2 in the Alamouti code is selected from a 2D constellation of size M = 2m , then, for the specific Alamouti structure in (14.19), a total of M 2 = 22m possible orthogonal signal matrices can be generated by this code. Viewing each of these matrices as a 4D signal point (strictly speaking, it is not 4D), then the task of the outer trellis coder is to select one of these 4D signal points to be transmitted in response to the 2b input bits. The limitation of such a construction is that it does not result in the use of all the possible 4D signal constellations; that is, the Alamouti code by itself does not create all possible orthogonal 2 × 2 signal matrices for a given constellation. For example, all additional 2 × 2 orthogonal codes whose elements are composed of only x1 , x2 and their negatives and complex conjugates, namely {X} =

x1 x2

x2∗ −x1∗

x1 x2

x2∗ −x1∗

,

,

−x1 x2 −x1 −x2

x1 x2∗ , −x2 x1∗



x1 −x2∗ x2∗ , −x1∗ x2 −x1∗ x2∗ x1∗



(14.79)

14 Earlier conference versions of Ref. 39 appear in Refs. 40 and 41. One specific example of the general SOSTTCs was independently presented in Ref. 42. Finally, similar code designs were independently introduced in Refs. 43–45.

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

835

would have behavior similar to that of the Alamouti code in (14.19) and as a group would allow for the generation of these additional 2 × 2 orthogonal signal matrices. The union of all the codes in (14.79) together with (14.19) is referred to as a “super-orthogonal code set” [39]. Whether all the constituents of (14.79) are needed to generate all the possible 4D signal points depends on the size of the constellation for the symbols x1 , x2 . For example, if x1 , x2 are chosen from a BPSK constellation (i.e., M = 2), then there are eight possible 2 × 2 orthogonal signal matrices that are realizations of the super-orthogonal code set and are given by  {X} =

       −1 1 −1 −1 1 1 , , , −1 −1 1 −1 −1 1         −1 1 1 −1 1 1 −1 −1 , , , 1 1 −1 −1 1 −1 −1 1 1 −1 1 1

(14.80)

The first four of these can be realized from the Alamouti code in (14.19), whereas the second four can be realized using only the second constituent code in (14.79). Thus, for BPSK it is sufficient to define the super-orthogonal code set as being composed of {X} =

x1 x2

−x2∗ x1∗

,

x2∗

−x1

(14.81)

x1∗

x2

Another way of representing the code set in (14.81) is X=

x1 e j θ x2

−x2∗ e j θ x1∗

(14.82)

where θ = 0 provides the first member of the code set in (14.81) and θ = π provides the second. With this as an example, a class of SOSTTCs was described in Ref. 39 that uses an outer trellis code and an inner block code based on (14.82) wherein the symbols transmitted from antenna 1 are phase-shifted by θ radians relative to the Alamouti design. Specifically, antenna 1 and antenna 2 first trans(n) mit x1(n) ej θ and x2(n) , respectively, in the interval nTs ≤ t ≤ (n + 1) Ts ; then they (n) transmit (−x2(n) )∗ ej θ and (x1(n) )∗ in the interval (n + 1) Ts ≤ t ≤ (n + 2) Ts , and θ (n) is allowed to vary from transmission interval to transmission interval. (n) Corresponding to the transmission of xn1 = [x1(n) ej θ , x2(n) ]T followed by xn2 = (n) [(−x2(n) )∗ ej θ , (x1(n) )∗ ]T , the corresponding set of successive signal samples at the receivers (matched-filter outputs at times t = (n + 1) Ts and t = (n + 2) Ts ) is given by (n)

(n) (n) j θ (n) (n) x1 e + cl2 x2 + n(n) yl(n) = cl1 l
∗ (n)
∗ (n) (n) (n) (n) jθ yl+Lr = cl1 −x2 e + cl2 x1(n) + n(n) l+Lr ,

l = 1, 2, . . . , Lr

(14.83)

836

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

where, as before, n(n) are i.i.d. complex zero-mean Gaussian noise samples, each i with variance σ 2 per dimension (i.e., E{|ni(n) |2 } = 2σ 2 , i = 1, 2) and are assumed to be independent of the channel gains. Assuming perfect knowledge of the complex channel gains, that is, perfect chan(n) to nel state information (CSI), the l th receiver uses the pair of samples yl(n) , yl+L r construct
∗
∗ (n) (n) (n) (n) yl+L e −j θ yl(n) + cl2 x˜l(n) = cl1 r (14.84)

∗ ∗ (n) (n) (n) (n) j θ (n) (n) y x˜l+L = c y − c e , l = 1, 2, . . . , L r l l+Lr l2 l1 r to be used in forming the ML metric for making a decision on the information signal vector xn = [x1(n) , x2(n) ]T corresponding to the nth transmission interval. Substituting (14.83) into (14.84) gives     
∗
∗ (n)  (n) 2  (n) 2 (n) (n) (n) (n) n(n) e −j θ n(n) x˜l = cl1  + cl2  x1 + cl1 l + cl2 l+Lr     
∗
∗  (n) 2  (n) 2 (n) (n) (n) (n) j θ (n) n(n) x˜l+L = + n(n) ,  c cl2  x2 + cl2 l − cl1 e l+Lr l1 r l = 1, 2, . . . , Lr

(14.85)

(n) ∗ −j θ Note again that the effective Gaussian noise RVs N˜ l(n) = (cl1 ) e n(n) l +  (n) (n) (n) ∗ (n) (n) (n) (n) (n) ∗ j θ ∗ cl2 (nl+Lr ) and N˜ l+Lr = (cl2 ) nl − cl1 e (nl+Lr ) are uncorrelated (and therefore independent). 

14.7.1.2

(n)

Evaluation of the Pairwise Error Probability

Fast Rician Fading Channels According to an observation of N blocks (2N symbols), each described by (14.83), the ML metric corresponding to the correct path is given by Lr  N  2
 (n)  (n) (n) (n) (n) (n)  m (Y, X) = yl − cl1 x1 e j θ + cl2 x2  n=1 l=1 (14.86) 
∗ (n)
∗ 2 
  (n) (n) (n) −x2(n) e j θ + cl2 x1(n) + yl+L − cl1  r For the incorrect path, the corresponding metric is given by (14.86) with xi(n) , i = 1, 2 and θ (n) replaced by xˆi(n) , i = 1, 2, and θˆ (n) , respectively. Conditioned on a realization of the channel over the entire N -block observation, the PEP is given by 
  ˆ ˆ |C P (X → X|C) = Pr m (Y, X) > m Y, X (14.87) 
  ˆ > 0|C = Pr m (Y, X) − m Y, X

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

837

ˆ into (14.87) Substituting (14.86) and the corresponding expression for m(Y, X) gives, after much simplification 

 Lr 



 E T T s ˆ cl

∗ P (X → X|C) = Q ! c∗l  2N0

(14.88)

l=1

(1) (1) (2) (2) (N) (N) where cl = [cl1 cl2 cl1 cl2 · · · cl1 cl2 ] and
is a 2N × 2N block diagonal  ˆ n , n = 1, 2, . . . , N along its diagonal = matrix with 2 × 2 elements
n Xn − X ˆ where Xn = [xn1 xn2 ] and Xn = [ˆxn1 xˆ n2 ] are the correct and incorrect signal matrices in the nth block interval. Note that because of the block diagonal form of
,

T the product matrix

∗ is also block diagonal. One method15 of evaluating the average of (14.88) over the channel statistics is to employ the Craig form of the Gaussian Q-function in (4.2), which results in



∗ T ∗ T  Lr cl Es l=1 cl

exp − dθ 2 4N0 sin θ 0 

T ∗ T  % Lr cl Es cl

∗ 1 π/2 & exp − = dθ 2 π 0 4N0 sin θ

1 ˆ P (X → X|C) = π

%

π/2

(14.89)

l=1

Averaging now over the channel, because of the independence of the channel gain vectors cl associated with the Lr receivers, the unconditional PEP can be expressed in the familiar MGF product form, namely ˆ = P (X → X)

1 π

%

Lr π/2 & 0

l=1

 Mξl −

1 sin2 θ

 dθ

(14.90)

where 

ξl =

T ∗ T Es cl

∗ cl 4N0

(14.91)

is a quadratic form of complex variables with MGF Mξl (s). To evaluate the MGF, we again make use of the result of Turin [33] for the characteristic function of a quadratic form that results in  

∗ T


∗ T −1 ∗ T γs Es I − s c



c exp s 4N l l 4(1+K) 0  Mξl (s) = (14.92) 

T γs det I − s 4(1+K)

∗ 15 A

similar analysis is performed in Ref. 32 using an inverse Laplace transform approach together with a Gauss–Chebyshev numerical quadrature rule to approximately evaluate the PEP.

838

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

which for the Rayleigh case reduces to   

T −1 γ Mξl (s) = det I − s s

∗ 4

(14.93)

Furthermore, if we assume that the channel gains have identical statistics, then

Lr .Lr and hence l=1 Mξl (s) = Mξl (s)

ˆ = 1 P (X → X) π

%



T γ exp − Lr (1+K) cl 4(1+K)s sin2 θ

∗  


T −1 ∗ T γ × I + 4(1+K)s sin2 θ

∗ cl

π/2


 det I +

0

γs
4(1+K) sin2 θ




∗

T Lr

dθ

(14.94)

for the Rician channel and ˆ = 1 P (X → X) π

π/2 

%

 det I +

0



γs 4 sin2 θ





 ∗ T

−Lr dθ

(14.95)

for the Rayleigh channel. Evaluation of the determinant in (14.94) and (14.95) is simplified by the block

T diagonal form of

∗ and can be expressed as  det I +

γs 4 (1 + K) sin θ 2

T

∗

 =

N
&

(n) (n) 1 + a11 + a22 + det An

 (14.96)

n=1

where An =

γs 4 (1 + K) sin2 θ


n



T
∗n =



(n) a11

(n) a12

(n) a21

(n) a22

(14.97)

with (n) a11 (n) a12

(n) a22

 2  2  γs  (n) j θ (n)  (n) −j θ (n) (n) j θˆ (n)  (n) −j θˆ (n)  = − xˆ1 e − xˆ2 e x1 e  + x2 e  4 (1 + K) sin2 θ 
∗ 
γs (n) ˆ (n) x2(n) − xˆ2(n) = x1(n) e j θ − xˆ1(n) e j θ 2 4 (1 + K) sin θ 

∗
(n) ˆ (n) ∗ (n) x1(n) − xˆ1(n) = a21 − x2(n) e −j θ − xˆ2(n) e −j θ  2  2  γs  (n)  (n) (n)  (n)  = − x ˆ + − x ˆ x  x 1 1 2 2  4 (1 + K) sin2 θ (14.98)

839

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

Finally, then, the PEP for the Rayleigh channel would be given by ˆ = 1 P (X → X) π =

1 π

%

π/2



0

%

N
&

1+

(n) a11

+

(n) a22

+ det An

  −Lr dθ

n=1 π/2



0

 N 
  −Lr 
  &  (n) 2 (n) (n) 1 + a11 1 + a22 − a12  dθ

(14.99)

n=1

As a check for consistency, we note that for the orthogonal STTC case we would have (n) (n) = a22 = a11

γs 2

4 sin θ

2  2   (n) (n)  x1 − xˆ1  = i=1

γs 4 sin2 θ

dn2

(14.100)

(n) (n) a12 = a21 =0

and thus (14.99) simplifies to the previously derived result in (14.32). Similar results can be obtained for the Rician channel, except that now one needs to first simplify the matrix  

∗ T

∗ T −1 γs γs I +



4 (1 + K) sin2 θ 4 (1 + K) sin2 θ in order to evaluate the argument of the exponential in the numerator of the integrand of (14.94). Once again, because of the block diagonal nature of the matrix

T

∗ , it can be shown that γs



G=



 ∗ T



4 (1 + K) sin2 θ  . G1 0 .  0 G2 0 .  0 0 . . =   . . . GN−1 0 0 . 0

 I+ 0 0 . 0 GN

4 (1 + K) sin2 θ 

Gn = An (I + An )

−1

 ∗ T

−1





    

where Gn is the 2 × 2 Hermitian matrix 



γs

An + (det An ) I = = det [I + An ]

(14.101)



(n) g11

(n) g12

(n) g21

(n) g22

(14.102)

with An as defined in (14.97). The elements of Gn are obtained as follows:    (n) 2 (n) (n) (n) a11 + a11 a22 − a12  (n) =
g11  
   (n) 2 (n) (n) 1 + a11 1 + a22 − a12 

840

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

(n) g12 =


(n) g21 =


(n) g22

(n) 1 + a11

(n) 1 + a11







(n) a12

    (n) 2 (n) 1 + a22 − a12  (n) a21

    (n) 2 (n) 1 + a22 − a12 

(14.103)

   (n) 2 (n) (n) (n) a22 + a11 a22 − a12  =
 >1 
   (n) 2 (n) (n) 1 + a11 1 + a22 − a12 

Finally, we obtain N

T K   (n) (n) (n) (n) g11 + g12 cl G c∗l =  + g21 + g22 1+K

(14.104)

n=1

which, when substituted in (14.94) together with (14.96), yields  Lr 
     (n) (n) (n) (n) N   % π/2  exp −K  g + g + g + g n=1 11 12 21 22 ˆ = 1 
 P (X → X) dθ   
  .N  π 0   (n) 2 (n) (n)    1 + a 1 + a − a12  n=1 11 22 (14.105) Again as a check, for the orthogonal STTC case, the matrix coefficients in (14.7.1.1) reduce to (n) (n) = g22 = g11 (n) g12

=

(n) g21

(n) a11 (n) 1 + a11

=

γs 2 4 dn 2

(1 + K) sin θ +

γs 2 4 dn

(14.106)

=0

Also 

  2
 (n) 2 (n) (n) (n) 1 + a11 1 + a22 − a12 =  = 1 + a11



(1 + K) sin2 θ +

γs 2 4 dn

2

(1 + K) sin2 θ

(14.107) Substituting (14.106) and (14.107) into (14.105) gives a result identical to (14.31), as it should. Slow Rician Fading Channels For the slow-fading case, the conditional PEP of (14.88) still applies where the channel gain vector corresponding to the l th receive antenna now becomes cl = [cl1 cl2 cl1 cl2 · · · cl1 cl2 ]. Also, averaging over the channel statistics still produces an unconditional PEP with an integrand in product form as in (14.90). Thus, the difference in the results between the slow- and

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

841

fast-fading cases strictly lies in the evaluation of the Gaussian quadratic form in (14.91).

T In view of the block diagonal form of

∗ and the form of cl above, it is immediately apparent that the quadratic form of (14.91) can, for the slow fading case, be simplified to  N  

∗ T ∗ T Es c˜ l
n
n c˜ l ξl = 4N0

(14.108)

n=1

where c˜ l = [cl1 cl2 ] is the vector of channel gains corresponding to the l th receive antenna in any single block (two-symbol) interval. Hence, replacing cl by c˜ l and

∗ T

T 

∗ by N n=1
n
n and then applying Turin’s result, it can be shown that the PEP now becomes      ˆ =1 P (X → X) π

%

(n) N n=1 a11 +

exp −Lr K

π/2




0

1+

1+

N

(n) n=1 a11

(n) (n) N N N n=1 a22 +2 det n=1 An +2 Re n=1 a12  (n) N (n) N n=1 a11 + n=1 a22 +det n=1 An

N

+

N

(n) n=1 a22

+ det



N n=1 An

Lr

dθ

(14.109) or for the Rayleigh case ˆ = 1 P (X → X) π

%

π/2 0

 1+

N 

(n) a11

+

n=1

N 

(n) a22

+ det

N 

n=1

−Lr An

d θ (14.110)

n=1

An Example As an example,16 consider the rate r = 1 BPSK 2-state code (see Fig. 7 of Ref. 39) whose trellis diagram is illustrated in Fig. 14.10, where two sets each containing two pairs of BPSK symbols are assigned to each state; specifically, there is a pair of parallel paths between each pair of states. The labeling notation

0 (θ = 0)

00/00 01/11 01

10/

10

11/

10/

01

1 (θ = π)

00/00

11/

10

01/11

Figure 14.10 Trellis diagram for rate r = 1, two-state BPSK super-orthogonal STTC. 16 Additional

examples can be found in Ref. 44.

842

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

ii /kk is the same as that in Fig. 14.5. For the parallel paths (i.e., N = 1), we have that dn2 = 22 + 22 = 8. Thus, for the fast-fading Rayleigh channel, the PEP associated with the parallel paths is from (14.32), given by 2Lr % π/2  2 θ 1 sin ˆ = dθ (14.111) P (X → X) π 0 sin2 θ + 2γ s which can be evaluated in closed form using (5A.4a) as  1

k   2L r −1     1 2γ 1 2k s ˆ =

 P (X → X) 1− k  2 1 + 2γ s 4 1 + 2γ s k=0

(14.112)



and for Lr = 1 varies asymptotically (large γ s ) as 3/ 64γ 2s . For an error event path of length N = 2 with respect to the all-zeros path as the correct one, we would have x1(1) = x2(1) = 1, x1(2) = x2(2) = 1, xˆ1(1) = 1, xˆ2(1) = −1, xˆ1(2) = −1, xˆ 2(2) = 1 and θ (1) = θˆ (1) = 0, θ (2) = 0, θˆ (2) = π . Evaluating the matrix elements in (14.98) gives (1) (2) = a11 = a11

γs 2

sin θ

(1) (2) = 0, a12 =− a12

,

γs 2

sin θ

(1) (2) = a22 = a22

,

γs sin2 θ (14.113)

Thus, from (14.99), we have  2  2  2 −Lr % γs γs γs 1 π/2 ˆ 1+ 1+ P (X → X) = − dθ π 0 sin2 θ sin2 θ sin2 θ  2Lr  Lr % 1 π/2 sin2 θ sin2 θ = dθ (14.114) π 0 sin2 θ + γ s sin2 θ + 2γ s The PEP of (14.114) can also be evaluated in closed form using (5A.58) with the result

L −1 r r −1  k+1  

 2L

 1 2L −1 r ˆ =2 P (X → X) (14.115) Bk Ik 2γ s − C k Ik γ s 2 k=0

k=0



where 

Bk = 



Ak , 3Lr − 1 k 

Ak = (−1)Lr −1+k



Ck =

 Lr − 1 k (Lr − 1)!

L r −1



n=0

k n



3Lr − 1 n

 An , (14.116)

Lr & n=1 n=k+1

(3Lr − n)

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

843

and 5 Ik (c) = 1 −

k  c (2n − 1)!! 1+ 1+c n!2n (1 + c)n

(14.117)

n=1

For a single receive antenna (Lr = 1), (14.115) simplifies to 1 1   2γ γ 1 1 s s ˆ =  + 3+

P (X → X) 1−4 2 1 + 2γ s 1 + γs 2 1 + γs

(14.118)



which varies asymptotically (large γ s ) as 5/ 64γ 3s . For the slow-fading Rayleigh channel, the PEP for the parallel paths is still given by (14.111). For the error event path of length 2, we now have that γs

2 

4 sin2 θ

n=1



T
n
∗n

=

(1) (2) a11 + a11

(1) (2) a12 + a12

(1) (2) a21 + a21

(1) (2) a22 + a22

=

γs



4 sin2 θ

8 −4 −4 8



(14.119) and thus det I +

2 

γs 2

4 sin θ

T
n
∗n

n=1



1+

2γ s

−

γs



sin2 θ   2γ s  − 2 1+ sin θ sin2 θ    2 γs γs =1+4 +3 sin2 θ sin2 θ  = det  

=

sin2 θ γs

(14.120)

sin4 θ + 4γ s sin2 θ + 3γ 2s sin4 θ

Finally, then, the PEP is given by Lr sin4 θ dθ sin4 θ + 4γ s sin2 θ + 3γ 2s 0  Lr  Lr % 1 π/2 sin2 θ sin2 θ = dθ π 0 sin2 θ + 3γ s sin2 θ + γ s

ˆ = 1 P (X → X) π

%

π/2



(14.121)

which once again can be evaluated in closed form using Eq. (5A.58) as ˆ = P (X → X)

   Lr L r −1



 3 1 k 2 Bk Ik 3γ s − k+1 Ck Ik γ s 4 3 k=0

(14.122)

844

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

where Bk and Ck are as given in (14.116) with 3Lr replaced by 2Lr . For a single receive antenna, (14.122) simplifies to 1 1

3γ γ 1 1 3 s s ˆ = P (X → X) + 1− 2 2 1 + 3γ s 2 1 + γs

(14.123)



which varies asymptotically (large γ s ) as 1/ 16γ 2s . This is to be compared with the corresponding special case of (14.118) for the fast-fading channel. 14.7.1.3 Extension of the Results to Super-Orthogonal Codes with More than Two Transmit Antennas Consider the case of a full-rate square real orthogonal design with four transmit antennas. For this case, the expansion of the orthogonal code matrix to allow the design of super-orthogonal codes corresponds to the introduction of separate phase shifts θ1 , θ2 , θ3 on the symbols transmitted from antennas 1, 2, and 3, respectively. As such, the super-orthogonal STTC has block code constituents of the form 

x1 e j θ1

 x e j θ2 2 X=  x e j θ3 3 x4

−x2 e j θ1

−x3 e j θ1

x1 e j θ2 −x4 e j θ3 x3

x4 e j θ2 x1 e j θ3 −x2

−x4 e j θ1



−x3 e j θ2   x2 e j θ3  x1

(14.124)

assigned to each branch of the trellis. Following the same maximum-likelihood analysis approach as for the two-antenna case, it is straightforward to show that for the fast-fading case, the PEP is once again given by (14.88), where the channel gain vector corresponding to the l th receive antenna is now given by cl = (2) (N) (n) (n) (n) (n) (n) [c(1) l cl · · · cl ] with cl = [cl1 cl2 cl3 cl4 ] and
is a 4N × 4N block diagonal matrix with elements  ˆn Dn = Xn − X  (n) (n) (n) (n) (n) (n) (n) (n)  (n) j θ (n) j θˆ (n) j θ (n) j θˆ (n) j θ (n) j θˆ (n) j θ (n) j θˆ −x2 e 1 + xˆ2 e 1 −x3 e 1 + xˆ3 e 1 −x4 e 1 + xˆ4 e 1   x1 e 1 − xˆ1 e 1    (n) (n) (n) (n) (n) (n) (n) (n)  (n) j θ2 (n) j θˆ2 (n) j θ2 (n) j θˆ2 (n) j θ2 (n) j θˆ2   x (n) e j θ2 − xˆ (n) e j θˆ2 x e − x ˆ e x e − x ˆ e −x e + x ˆ e  2  2 1 1 4 4 3 3 =   (n) (n) (n) (n) (n) (n) (n)   (n) j θ (n)  j θˆ jθ j θˆ jθ j θˆ jθ j θˆ (n) (n) (n) (n) (n) (n) (n)  x3 e 3 − xˆ3 e 3 −x4 e 3 + xˆ4 e 3 x1 e 3 − xˆ1 e 3 x2 e 3 − xˆ2 e 3    (n) (n) (n) (n) (n) (n) (n) (n) x4 − xˆ4 x3 − xˆ3 −x2 + xˆ2 x1 − xˆ1

(14.125) All the results that follow from this point on would parallel those of the two-antenna case with the understanding that all 2 × 2 matrix operations are now replaced by 4 × 4 matrix operations. For example, An of (14.97), whose elements are key to the resulting expressions for average PEP, would now be a 4 × 4 matrix obtained by multiplying
n of (14.125) by its transpose conjugate. Similar substitutions would take place for the slow-fading case.

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

845

14.7.1.4 Approximate Evaluation of Average Bit Error Probability Following the procedure taken in Section 14.6.4, we use the PEPs derived previously to evaluate in closed form an approximation to the average BEP, Pb (E), accounting only for error events of lengths N less than or equal to H. In the next section, we shall demonstrate the accuracy of this approximation by comparing it to the true upper bound obtained from the transfer function of the code, which accounts for error events of all lengths. For the purpose of illustration, we shall concentrate our attention on the example discussed in Section 14.7.1.2. Assuming transmission of the all-zeros sequence, then, for the two-state code in Fig. 14.10, there is a single error event path of length 1 and four error event paths of length 2. The single error event path of length 1 has a PEP of type I, whereas the four error event paths of length 2 all have PEPs of type II. The path of type I contributes one bit error, whereas the four paths of type II contribute a total of 12 bit errors. Thus, if we were to choose to approximate the average BEP by considering only error event paths corresponding to H = 2, then  ˆ I + 12P (X → X) ˆ II (14.126) Pb (E) ∼ = 12 P (X → X) ˆ I and P (X → X) ˆ II are given where for the fast-fading Rayleigh channel P (X → X) by the closed-form expressions in (14.112) and (14.118), respectively. If we now include paths of length 3, then there are a total of eight of these, all of which have a PEP of type III, which for the fast-fading Rayleigh channel is given by ˆ III P (X → X)

1 = π

%

π/2



sin2 θ

2Lr 

sin2 θ + γ s

0

2Lr

sin2 θ

dθ

sin2 θ + 2γ s

(14.127)

which again can be evaluated in closed form using (5A.58). Since the eight paths of length 3 contribute a total of 28 bit errors, the average BEP corresponding to H = 3 is approximated by  ˆ I + 12P (X → X) ˆ II + 28P (X → X) ˆ III (14.128) Pb (E) ∼ = 12 P (X → X) We conclude this section by noting that similar approximations can be used to evaluate the average BEP for the slow-fading case. For example, for H = 2, the ˆ I is still given average BEP is again approximated by (14.126), where P (X → X) ˆ II is now given by the closed-form expression obtained by (14.112) but P (X → X) from (14.122) [e.g., (14.123) for Lr = 1]. Similarly, for H = 3, the average BEP is ˆ II , still approximated by (14.128) where, in addition to using (14.122) for P (X → X) ˆ III is now given by it can be shown that P (X → X) ˆ III = 1 P (X → X) π

%

π/2 0



sin4 θ sin4 θ + 6γ s sin2 θ + 7γ 2s

Lr dθ

(14.129)

846

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

14.7.1.5 Evaluation of the Transfer Function Upper Bound on the Average Bit Error Probability To demonstrate the degree of accuracy of the approximate evaluations of the previous section, we proceed as in Section 14.6.5 to use the exact PEPs to compute a true upper bound on the average BEP for the fast-fading case. Whereas for the 2 STTCs the appropriate replacement for the label D (θ )|xn −xˆn | on each branch of the  γ (n) (n) 2 −Lr t trellis in the AWGN case was [1 + 4 sins2 θ L , from the analogy i=1 |xi − xˆ i | ] between the product form of the integrand in (14.49) and (14.99), we see that for the (n) (n) (n) 2 −Lr SOSTTCs the appropriate replacement is [(1 + a11 )(1 + a22 ) − |a12 | ] . Apply(n) (n) (n) ing the definitions of a11 , a22 , and a12 in (14.98) (with K = 0 for the Rayleigh case) to label the branches of the trellis in Fig. 14.10 as per the previous statement, then, after a bit of effort, one arrives at the pair-state transition diagram illustrated in Fig. 14.11 whose transfer function is computed as  T (D (θ ) , I ) = 2

c (1 − 2b) + 2ad 1 − 2b

 (14.130)

where a=

1 2



 I + I 2 D1 (θ ) ,

c = 12 I D2 (θ ) ,

b=

(1 + I ) D3 (θ )

 d = 12 I + I 2 D3 (θ ) 1 2

(14.131)

c b a 0,0

d 0,1

c 1/2

0,0

b

a

1

d

F

I a c

1/2

d

b a

1,1

1

d 1,0

1,1

b c Figure 14.11

Pair-state transition diagram for trellis diagram of Fig. 14.10.

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

847

with  D1 (θ ) =

Lr

2

sin θ

D3 (θ ) =

sin2 θ + 2γ s

D2 (θ ) =

,

sin2 θ + γ s 



2Lr

sin2 θ

sin2 θ

2Lr

sin2 θ + 2γ s

(14.132)

6 = D2 (θ )

Substituting (14.131) in (14.130) and simplifying gives I D 2 (θ ) [1 − D3 (θ )] − I 2 D2 (θ ) D3 (θ ) + I 2 + 2I 3 + I 4 D1 (θ ) D3 (θ ) T (D (θ ) , I ) = 1 − (1 + I ) D3 (θ )

(14.133)

with derivative ∂ T (D (θ ) , I ) |I =1 = ∂I

D2 (θ ) − 4D2 (θ ) D3 (θ ) + 12D1 (θ ) D3 (θ ) + 4D2 (θ ) D32 (θ ) − 20D1 (θ ) D32 (θ ) [1 − 2D3 (θ )]2

(14.134)

Since for the example under consideration nc = 2, then the desired true upper bound on average BEP becomes Pb (E) ≤

=

1 π

%

1 2π

π/2 0

%

1 ∂ T (D (θ ) , I ) |I =1 d θ nc ∂I

π/2

D2 (θ ) − 4D2 (θ ) D3 (θ ) + 12D1 (θ ) D3 (θ ) + 4D2 (θ ) D32 (θ ) − 20D1 (θ ) D32 (θ ) [1 − 2D3 (θ )]2

0

(14.135) dθ

To check the validity of (14.135) in relation to the three-term approximate expression in (14.128), we expand 1/ [1 − 2D3 (θ )]2 as 1 + 4D3 (θ ) + 12D32 (θ ) + · · ·, which, on substitution in (14.135), gives 1 Pb (E) ≤ 2π

%

π/2 0

3 4 D2 (θ ) + 12D1 (θ ) D3 (θ ) + 28D1 (θ ) D32 (θ ) + · · · d θ (14.136)

Recognizing that ˆ I= P (X → X)

1 π

ˆ II = 1 P (X → X) π ˆ III = 1 P (X → X) π

%

π/2

D2 (θ ) d θ 0

%

π/2

D1 (θ ) D3 (θ ) d θ 0

%

0

π/2

D1 (θ ) D32 (θ ) d θ

(14.137)

848

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

then the leading three terms of (14.136) result in the three-term approximation of (14.128) as would be expected. 14.7.1.6 Numerical Results Consider the rate r = 1, two-state code with BPSK modulation and a single receive antenna whose trellis diagram is illustrated in Fig. 14.10. Figure 14.12 shows the PEP for fast- and slow-fading Rayleigh channels. The PEP for the error event path of length 1 (N = 1) is the same for both fast- and slow-fading channels and is given in (14.112). The PEP is also plotted for the error event path of length 2 (N = 2) using (14.118) and (14.123). As expected, the average PEP is smaller for a fast-fading channel. The error event path of length 2 is the worst case for the slow-fading channel, while the worst case for the fast-fading channel is the error event path of length 1. Furthermore, as mentioned earlier, the asymptotic (largesymbol SNR) behavior of the PEP for the slow-fading case varies inversely as γ 2s , whereas that for the fast-fading case varies inversely as γ 3s , which clearly shows the dependence of the diversity performance on the nature of the channel fading variation. Figures 14.13 and 14.14 show the average BEP of the code for fastand slow-fading channels, respectively. In these figures, the approximate average BEP performances are plotted accounting for error event lengths up to H using

10−1

N = 2 fast fading N = 2 slow fading N=1

10−2

PEP

10−3

10−4

10−5

10−6

0

2

4

6

8

10

12

14

16

18

20

Average symbol SNR (dB) Figure 14.12 Pairwise error probability performance of rate r = 1, two-state, BPSK superorthogonal STTC over slow- and fast-fading Rayleigh channels; one receive antenna; r = 1 bit/sec/Hz.

849

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

100

Len. 1 error Len. 2 error Len. 3 error

BEP (fast fading)

10−1

10−2

10−3

10−4

10−5

10−6

0

2

4

6

8

10

12

14

16

18

20

Average symbol SNR (dB)

Figure 14.13 Average bit error probability performance of rate r = 1, two-state, BPSK superorthogonal STTC over fast-fading Rayleigh channels; one receive antenna; r = 1 bit/sec/Hz.

10−2

Len. 1 error Len. 2 error Len. 3 error Simulated BEP

BEP (slow fading)

10−3

10−4

10−5

10−6

5

10

15

20

Average symbol SNR (dB)

Figure 14.14 Average bit error probability performance of rate r = 1, two-state, BPSK superorthogonal STTC over slow-fading Rayleigh channels; one receive antenna; r = 1 bit/sec/Hz.

850

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

ˆ for H = 1, (14.126) for H = 2, and (14.128) for H = 3. Pb (E) ∼ = 12 P (X → X) We observe that while considering error events lengths up to H = 2 is sufficient for calculating the average BEP for fast-fading channels, error events with longer lengths are needed to estimate the average BEP over slow-fading channels, as can be seen by a comparison with simulation results provided for the true BEP. This slower convergence of the PEP to the average BEP as a function of the length of paths considered for slow fading relative to fast fading is consistent with a similar observation made previously in Section 14.6.4 for orthogonal STTCs. 14.7.2

Super-Quasi-Orthogonal Space-Time Trellis Codes

In Section 14.4, we discussed the theory of complex orthogonal designs and the associated orthogonal designed STBCs that, for an arbitrary number of transmit antennas, provided full diversity and achieved the highest possible transmission rate. Keeping in mind that whereas full diversity is the right design choice for high SNRs, full transmission rate is more important at low SNRs, researchers then switched their train of thought and began to focus on the latter by somewhat relaxing the full orthogonality requirement on the code design. Along these lines, the notion of a quasi-orthogonal STBC (QOSTBC) was introduced [47] that partitioned the code matrix into subspaces (groups of vectors) that were orthogonal with each other but not necessarily orthogonal within the groups. In deviating from the fully orthogonal design, a sacrifice was made in simplicity of decoding whereby the partitioning of the decoding metric now allowed only for group-by-group symbol decoding. Although the initial QOSTBC designs achieved full rate at half the maximum possible diversity [47,48] by allowing for different phase rotations of the signal constellations used to design the code, further research led to designs that achieved both full rate and full diversity, still, however, requiring symbol decoding on a group basis [49–51]. The next natural step in the evolution of efficient STCs was to couple the notions of super-orthogonality and quasi-orthogonality, thereby forming superquasi-orthogonal space-time trellis codes (SQOSTTCs). The design of such codes was introduced in Ref. 52, and an analysis of their coding gain distance as well as simulation results for their error probability performance was presented there. In this section, we further extend the MGF-based approach, previously applied in this chapter to STTCs, STBCs, and SOSTTCs to evaluate the exact PEP of SQOSTTCs [53]. The analytical results obtained will be applied to some simple examples to demonstrate their use in obtaining system performance over a broad range of SNRs as well as allowing for straightforward comparisons with other STC designs. 14.7.2.1 Signal Model Consider the Alamouti code described by (14.19), which we now denote by X12 , where the subscript “12” is used to represent the indeterminates x1 and x2 in the transmission matrix. As has now been noted many times, such a code is an orthogonal block code that achieves both full diversity and full rate, and each of the two symbols can be decoded independently. Suppose now that we create a square

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

851

block code with a 4 × 4 transmission matrix for the nth block as follows:  Xn =

X12 X34

−X∗34 X∗12

x1(n)

  (n)  x2  =  x (n)  3  x4(n)

∗
− x2(n)
∗ x1(n) ∗
− x4(n)
∗ x3(n)

x3(n) x4(n) x1(n) x2(n)

∗
− x4(n)
∗ x3(n) ∗
− x2(n)
∗ x1(n)

        

(14.138)

As before, the i th row corresponds to the complex constellation symbols (e.g., M -PSK) transmitted from the i th antenna in the nth block interval of duration 4Ts . We know from the discussion in Section 14.4 that such a code cannot be orthogonal since the largest square orthogonal code with complex elements that can be constructed is 2 × 2. Furthermore, while this code is full rate, the minimum rank of Xn is only equal to two and thus the code achieves only half the maximum attainable diversity. To improve the code design dictated by (14.138) to one that has both full rate and full diversity, one chooses x3(n) and x4(n) from a rotated M -PSK constellation [with respect to that used for choosing x1(n) and x2(n) ]. Equivalently, one can include the constellation phase rotation directly in the matrix representation of (14.138) by multiplying x1(n) and x2(n) by ej φ1 and multiplying x3(n) and x4(n) by ej φ2 , where now all x1 , x2 , x3 , x4 come from the same M -PSK constellation. In what follows, we shall adopt this representation. To now integrate this QOSTBC with the notion of a SOSTTC, one further augments Xn by multiplying its i th row by ej θi , where θi is a multiple of 2π /M that preserves all the characteristics of the code without expanding the size of the constellation. Although such an augmentation would provide four additional degrees of freedom in designing the code, it is shown [52] that only two of the four phase rotations are needed to achieve the desired characteristics. In view of the above, the appropriate signal transmission matrix for an SQOSTTC becomes

∗

∗  e j θ1 e j φ1 x1(n) −e j θ1 e j φ1 x2(n) e j θ1 e j φ2 x3(n) −e j θ1 e j φ2 x4(n) 
∗

∗    j θ
j φ (n)  j θ2 e j φ1 x (n) j θ2 e j φ2 x (n) j θ2 e j φ2 x (n)   e 2 e 1 x2 e e e 1 4 3   ∗ ∗ 

Xn =    e j φ2 x (n) j φ2 (n) j φ1 (n) j φ1 (n) − e x4 e x1 − e x2   3 
∗
∗  (n) (n) (n) (n) j φ j φ j φ j φ e 2 x3 e 1 x1 e 2 x4 e 1 x2 

(14.139) Note that the subspace corresponding to rows 1 and 3 and the subspace corresponding to rows 2 and 4 are orthogonal to each other. However, the rows within each of these subspaces are not orthogonal. The class of SQOSTTCs designed in accordance with (14.139) will always be full rate and full diversity and the symbols can be decoded in pairs.

852

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

14.7.2.2 Evaluation of Pairwise Error Probability Without going into great detail, suffice it say that the identical analysis techniques used for evaluating the PEP for SOSTTCs in Section 14.7.1.2 apply equally well here for the SQOSTTCs. Thus, the presentation will be brief and merely highlight the differences where appropriate. For simplicity, we shall present only the Rayleigh fading case. Fast Rayleigh Fading Channels The PEP for this case is given by (14.95), which

T again, because of the block diagonal structure of

∗ , can be expressed as ˆ = 1 P (X → X) π

%

N π/2 &

0

(det [I + An ])−Lr d θ

(14.140)

ˆn
n = Xn − X

(14.141)

n=1

where now  An = aij(n) =

γs 2

4 sin θ

T
n
∗n ,



is a 4 × 4 matrix whose properties for the case of interest here depend on the specific parameters assigned to the quasi-orthogonal signal transmission matrix in (14.139). Slow Rayleigh Fading Channels Taking note of the difference between (14.91) and (14.108), for the slow-fading case the PEP of (14.95) becomes ˆ = 1 P (X → X) π =

1 π

%

π/2





det I +

0

%

π/2





det I +

0

γs

N 

4 sin2 θ

n=1

N 

T
n
∗n

−Lr dθ (14.142)

−Lr An

dθ

n=1

Note that for error events of length N = 1, i.e., parallel paths in the trellis, we see from a comparison of (14.140) and (14.142) that the PEPs are identical. Before going to specific examples, we investigate a bit further the evaluation of the determinants in (14.140) and (14.142) for pairs of trellis branches that originate from the same state, such as parallel paths. For this case we would have θ1 = θˆ1 , θ2 = θˆ2 , φ1 = φˆ 1 , φ2 = φˆ 2 , and 

dn2

 0

T 
n
∗n =  ∗  ξ1,n 0

0



0

ξ1,n

dn2

0

0

dn2

ξ2,n    0 

∗ ξ2,n

0

dn2

(14.143)

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

853

where dn2 =

4  2   (n) (n)  xi − xˆi  i=1



∗  ξl,n = e j θl 2Re e j (φ1 −φ2 ) x1(n) − xˆ1(n) x3(n) − xˆ3(n) 
∗ 
, l = 1, 2 + x2(n) − xˆ2(n) x4(n) − xˆ4(n)

(14.144)

Then, from the definition of An in (14.141), it is straightforward to show that 

2  2  2 γs γs 2   1+ ξ1,n dn − det [I + An ] = 4 sin2 θ 4 sin2 θ (14.145) 

2  2  2 γs γs 2   1+ ξ2,n × dn − 4 sin2 θ 4 sin2 θ

14.7.2.3 Examples As an example of the application of the results presented above, consider a rate r = 1 BPSK two-state code (see Fig. 2a of Ref. 52) whose trellis diagram is illustrated in Fig. 14.15. The states are labeled with the four parameters φ1 , φ2 , θ1, , θ2 S0

0 (f1 = 0,f2 = p/2,q1 = 0,q2 = 0)

S1

S1

1 (f1 = p/2,f2 = 0,q1 = 0,q2 = 0) S0 0000  1111 1100  0011 S0 =  0101 1010  1001 0110 

0001 1110  1101  0010 S1 =  0100 1011  1000 0111 

The output symbol indices 0, 1 in S0 and S1 correspond to the output symbols 1, −1, respectively.

Figure 14.15 Trellis diagram for rate r = 1, two-state BPSK super-quasi-orthogonal STTC.

854

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

that define the code as per the transmission matrix in (14.139). The branches are labeled with the four output symbols (as represented by indices) that result from the transition between the states at the beginning and at the end of the branch. Consider first the parallel paths (i.e., N = 1) between the “0” states. Then, from (14.144), for any incorrect parallel path, we have that 
∗  
ξ1,n = ξ2,n = 2Re −j x1(n) − xˆ1(n) x3(n) − xˆ3(n) 
∗ 
+ x2(n) − xˆ2(n) x4(n) − xˆ4(n)

(14.146)

Clearly, since for BPSK the xi(n) s and xˆi(n) s are all real, then ξ1,n = ξ2,n = 0. Furthermore, the smallest value of dn2 corresponds to an incorrect path for which two of the four symbols differ from those of the correct path, in which case dn2 = 8. Thus, the worst-case PEP of the parallel paths for either fast or slow fading is given by ˆ = 1 P (X → X) π

%

π/2



4Lr

sin2 θ sin2 θ + 2γ s

0

dθ

(14.147)

which clearly reveals the full diversity of 4Lr and can be evaluated in closed form from (5A.4a) as  1

k   4L r −1     2γ 1 1 2k s ˆ = 

P (X → X) (14.148) 1− k  2 1 + 2γ s 4 1 + 2γ s k=0

An identical result would be obtained for the parallel paths between the “1” states. Consider next an error event path of length N = 2 and, for simplicity of the discussion, assume that the correct path is the all-zeros path between the “0” states, i.e., x1(n) = x2(n) = x3(n) = x4(n) = 1, n = 1, 2. For the first branch of this error event, the correct and incorrect paths both leave from the “0” state and thus (14.144) applies. Hence, once again ξ1,1 = ξ2,1 = 0 and now the smallest value of d12 corresponds to an incorrect path for which only one of the four symbols differs from that of the correct path, in which case d12 = 4 and a PEP given by (14.147) or (14.148) with 2γ s replaced by γ s . For the second branch of this error event, the correct path leaves from the “0” state and the incorrect path leaves from the “1” state. Thus, for this case, we would have θ1 = θˆ1 = 0, θ2 = θˆ2 = 0, φ1 = 0, φˆ 1 = π /2, φ2 = π /2, φˆ 2 = 0, and hence analogous to (14.143) 

D22  0

T
2
∗2 =   2 0

0 D22 0 2

2 0 D22 0

 0 2   0  D22

(14.149)

855

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

where  2  2  2  2         D22 = x1(2) − j xˆ1(2)  + x2(2) − j xˆ2(2)  + j x3(2) − xˆ3(2)  + j x4(2) − xˆ4(2)   2  2  2  2         = 1 − j xˆ1(2)  + 1 − j xˆ2(2)  + 1 + j xˆ3(2)  + 1 + j xˆ4(2)  
∗

∗  x1(2) − j xˆ1(2) j x3(2) − xˆ3(2) + x2(2) − j xˆ2(2) j x4(2) − xˆ4(2) 


   
(14.150) = 2Re −j 1 − j xˆ1(2) 1 − j xˆ3(2) + 1 − j xˆ2(2) 1 − j xˆ4(2)

2 = 2Re




Here, regardless of how many of the four symbols that differ between the correct and incorrect paths, D22 = 8. Also, from Fig. 14.15 we find that for all eight of the parallel incorrect branches, |2 | = 4. Thus, analogous to (14.145), we obtain  det [I + A2 ] =

1+



2

γs

γs

2

2 |2 |2

D22 − 4 sin2 θ 4 sin2 θ  2  2 2 2γ s γs 1+ = − sin2 θ sin2 θ 2  2  sin2 θ + 3γ s sin2 θ + γ s = sin2 θ sin2 θ

(14.151)

Thus, for the fast-fading channel, the worst-case PEP corresponding to the error event of length 2 is determined from (14.140) as ˆ = 1 P (X → X) π

%

π/2 0



6Lr 

sin2 θ sin2 θ + γ s

2Lr

sin2 θ sin2 θ + 3γ s

dθ

(14.152)

which can be evaluated in closed form using (5A.58) with the result ˆ = P (X → X)



729 256

Lr 2L r −1 

k



2 Bk Ik 3γ s

k=0



6Lr −1  k

 2 1  − C k Ik γ s 3 3

k=0

(14.153) 

where 

Bk = 

Ak , 8Lr − 1 k



Ck =

2L r −1  n=0



k n



8Lr − 1 n

 An ,

856

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

 2Lr − 1 k (2Lr − 1)!

 

Ak = (−1)2Lr −1+k

2Lr &

(8Lr − n)

(14.154)

n=1 n=k+1

and Ik (c) is as given by (14.117). For the slow-fading case, we have for the smallest value of d12 = 4: det I +

2 

 1+

An =

n=1

 1+

=



γs 4 sin2 θ 3γ s

+

d12

2

 −

sin2 θ

D22

  2

 −

γs

2

2

4 sin2 θ

|2 |

2

2 2

γs sin2 θ

(14.155) Thus, the worst-case PEP corresponding to the error event of length 2 is ˆ = 1 P (X → X) π

%

π/2



2Lr

sin4 θ

dθ sin4 θ + 6γ s sin2 θ + 8γ 2s  2Lr  2Lr % 1 π/2 sin2 θ sin2 θ = dθ π 0 sin2 θ + 4γ s sin2 θ + 2γ s 0

(14.156)

which can again be evaluated in closed form using (5A.58) as ˆ = 22Lr −1 P (X → X)

2L −1 r  k=0

r −1  k+1 

 2L

 1 Bk Ik 4γ s − Ck Ik 2γ s 2

(14.157)

k=0

with now  

Bk = 



Ak , 4Lr − 1 k 

Ak = (−1)2Lr −1+k



Ck =

2L r −1 

 2Lr − 1 k (2Lr − 1)!



n=0

k n



4Lr − 1 n

 An , (14.158)

2Lr &

(4Lr − n)

n=1 n=k+1

The previous example corresponded to a full-rate code. We now consider an example of a non-full-rate SQOSTTC that still achieves full diversity but trades the reduction in rate for an improvement in coding gain as reflected by the asymptotic

OTHER COMBINATIONS OF SPACE-TIME BLOCK CODES AND SPACE TIME TRELLIS CODES

857

(large SNR) behavior of the PEP performance curve. The specific example to be considered is a system with again four transmit antennas and the class of half-rate (r = 1) QPSK codes for which x3 = x1 , x4 = x2 , and φ2 − φ1 = π /4. For pairs of trellis branches that originate from the same state, for example, parallel paths, we have from (14.144) that dn2

2  2   (n) (n)  =2 xi − xˆi  i=1

  2  2      (n) (n) j π/4 ξl,n  = 2Re e xi − xˆi 

(14.159)

i=1

=

2  2 √  1  (n) (n)  2 xi − xˆi  = √ dn2 , 2 i=1

l = 1, 2

Substituting (14.159) into (14.145) and using the result in (14.140), we see that for code designs whose parallel paths dominate, the PEP for either fast or slow fading is given by  2Lr % π/2 2 sin θ 1 ˆ =   
P (X → X) γ s dn2 1 2 π 0 √ sin θ + 1 + 2

 ×

2

4

(14.160)

2Lr

sin θ  
γ d2 sin θ + 1 − √12 s4 n

dθ

2

For a full-rate r = 2 QPSK two-state code such as that in Fig. 2b of Ref. 52, the worst-case PEP for either fast or slow fading is given by [53]  2Lr % π/2 2 sin θ 1 ˆ =   
P (X → X) π 0 sin2 θ + 1 + √1 γ  ×

2

sin θ
sin2 θ + 1 −

2

s



dθ

(14.161)

2Lr √1 2

 γs

Comparing (14.160) with (14.161), we readily see that a half-rate two-state QPSK code will achieve an SNR improvement over its full-rate counterpart by an amount equal to dn2 /4. 14.7.2.4 Numerical Results In this section, we provide numerical results for the previous examples. First, we consider the rate r = 1, two-state code with a single receive antenna. Figure 14.16 shows the PEP for fast- and slow-fading Rayleigh channels. The PEP for the error

858

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

10−2

Len. 1 error Len. 2 (slow fading) Len. 2 (fast fading)

10−3

10−4

PEP

10−5

10−6

10−7

10−8

10−9 0

2

4

6

8

10

12

14

16

Average symbol SNR (dB)

Figure 14.16 Pairwise error probability performance of rate r = 1, two-state code with BPSK modulation; slow and fast fading, length 1 and 2 error event paths; r = 1 bit/sec/Hz.

event path of length 1, N = 1, is the same for both fast- and slow- fading channels and is given in Eq. (14.148). The PEP is also plotted for the error event path of length 2, N = 2, using Eqs. (14.153) and (14.157). The error event path of length 1 (parallel paths) is the worst case for both slow- and fast-fading channels. Next we consider the PEPs of the full-rate r = 2, two-state code and the nonfull-rate r = 1 code with a single receive antenna in Fig. 14.17. Both codes use a QPSK constellation although the non-full-rate code provides a lower rate and decoding complexity compared to that of the full-rate code. The average PEP is plotted for the error event paths of length 1 (parallel paths), which is the same for fast- and slow-fading Rayleigh channels. Equations (14.160) and (14.161) are used in Fig. 14.17, where dn2 = 8 for the former case. As can be seen from the figure, and is evident from (14.160) and (14.161), the non-full-rate code provides a 3 dB SNR improvement by reducing the rate. It also provides a lower decoding complexity. Note that a non-full-rate r=1.75 code with similar performance can also be designed [52].

14.8

DISCLAIMER

From a time perspective, although the subject of space-time coding could still be considered in its infancy, from a contribution standpoint one would have to

REFERENCES

10−1

859

r = 2 bits/sec/Hz, full-rate r = 1 bits/sec/Hz, non full-rate

10−2

PEP (parallel)

10−3

10−4

10−5

10−6

10−7

0

2

4

6

8

10

12

14

16

Average symbol SNR (dB) Figure 14.17 Pairwise error probability performance of full-rate r = 2 and half-rate r = 1, 2 state codes with QPSK modulation; slow and fast fading, length 1 error event paths.

conclude that it has already reached a high level of accomplishment. Indeed, in the space allocated to a single chapter such as this one, it is not possible to cover the myriad of advancements to the basic idea that have been put forth since its inception a number of years back. In fact, the advancement of the subject has by now become so extensive that such coverage would require a textbook of its own. Indeed, a number of such textbooks now exist in the marketplace. At the risk of omitting particular contributors, the authors conclude the chapter without mentioning any other related and significant contributions.

REFERENCES 1. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun., vol. 51, no. 8, August 2003, pp. 1389–1398. 2. D. Divsalar and M.K. Simon, “Multiple trellis-coded modulation,” IEEE Trans. Commun., vol. 36, no. 4, April 1988, pp. 410–419. 3. D. Brennan, “Linear diversity combining techniques,” Proc. IRE, vol. 47, June 1959, pp. 1075–1102.

860

MULTICHANNEL TRANSMISSION—TRANSMIT DIVERSITY AND SPACE-TIME CODING

4. A. Wittneben, “A new bandwidth efficient transmit antenna modulation diversity scheme for linear digital modulation,” IEEE Int. Conf. Communication (ICC’93 ), May 1993, pp. 1630–1634. 5. A. Wittneben, “Basestation modulation diversity for digital SIMULCAST,” IEEE Veh. Technol. Conf. (VTC’91 ), St. Louis, MO, 1991, pp. 848–853. 6. N. Seshadri and J. H. Winters, “Two signaling schemes for improving the error performance of FDD transmission using transmitter antenna diversity,” IEEE Veh. Technol. Conf. (VTC’93 ), Secaucus, NJ, May 1993, pp. 508–511. 7. J. H. Winters, “The diversity gain of transmit diversity in wireless systems with Rayleigh fading,” IEEE Int. Conf. Communication (ICC’94 ), June 1994, pp. 1121–1125; see also IEEE Trans. Veh. Technol., vol. 47, no. 1, February 1998, pp. 119–123 for an expanded version of the conference paper. 8. J.-C. Guey, M. P. Fitz, M. R. Bell, and W.-Y. Kuo, “Signal design for transmitter diversity wireless communication systems over Rayleigh fading channels,” IEEE Veh. Technol. Conf. (VTC’96 ), Atlanta, GA, 1996, pp. 136–140; see also IEEE Trans. Commun., vol. 47, no. 4, April 1999, pp. 527–537. 9. V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Int. Conf. Communication (ICC’97 ), Montreal, Canada, June 1997, pp. 299–303. 10. V. Tarokh, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criterion and code construction,” IEEE Trans. Inform. Theory, vol. 44, no. 2, March 1998, pp. 744–765. 11. V. Tarokh, A. F. Naguib, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Mismatch analysis,” IEEE Int. Conf. Communication (ICC’97 ), Montreal, Canada, June 1997, pp. 309–313. 12. V. Tarokh, A. F. Naguib, N. Seshadri, and A. R. Calderbank, “Space-time codes for high data rate wireless communication: Performance criteria in the presence of channel estimation errors, mobility, and multiple paths,” IEEE Trans. Commun., vol. 47, no. 2, February 1999, pp. 199–207. 13. V. Tarokh, A. F. Naguib, N. Seshadri, and A. R. Calderbank, “Combined array processing and space-time coding,” IEEE Trans. Inform. Theory, vol. 45, no. 4, May 1999, pp. 1121–1128. 14. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inform. Theory, vol. 45, no. 5, July 1999, pp. 1456–1467. 15. A. F. Naguib, V. Tarokh, N. Seshadri, and A. R. Calderbank, “A space-time coding modem for high-data-rate wireless communications,” IEEE J. Select. Areas Commun., vol. 16, no. 8, October 1998, pp. 1459–1478. 16. V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: Performance results,” IEEE J. Select. Areas Commun., vol. 17, no. 3, March 1999, pp. 451–460. 17. J. Yuan, W. Firmanto, and B. Vucetic, “Trellis coded 2 x MPSK modulation with transmit diversity,” IEEE Int. J. Commun. and Networks, vol. 3, no. 3, September 2001, pp. 273–279. 18. S. M. Alamouti, V. Tarokh, and P. Poon, “Trellis-coded modulation and transmit diversity: Design criteria and performance evaluation,” Universal Personal Commun., vol. 2, 1998, pp. 917–920.

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19. S. Sandhu, R. Heath, and A. Paulraj, “Space-time block codes versus space-time trellis codes,” IEEE Int. Conf. Communication (ICC’01 ), Helsinki, Finland, June 2001, vol. 4, pp. 1132–1136. 20. M. J. Borran, M. Memarzadeh, B. Aazhang, “Design of coded modulation schemes for orthogonal transmit diversity,” IEEE Int. Symp. Information Theory (ISIT’01 ), 2001, p. 339. 21. A. Yongacoglu and M. Siala, “Space-time codes for fading channels,” IEEE Veh. Technol. Conf. (VTC’99 ), Houston, TX, April 1999, pp. 2495–2499. 22. S. Siwamogsatham and M. P. Fitz, “Robust space-time codes for correlated Rayleigh fading channels,” IEEE Trans. Signal Proc., vol. 50, no. 10, October 2002, pp. 2408–2416. 23. M. K. Simon and M.-S. Alouini, “A unified approach to the performance analysis of digital communications over generalized fading channels,” IEEE Proc., vol. 86, September 1998, pp. 1860–1877. 24. J. K. Cavers, “Optimized use of diversity modes in transmitter diversity systems,” IEEE Veh. Technol. Conf. (VTC’99 ), Houston, TX, April 1999, pp. 1768–1773. 25. Y. C. Ko and M.-S. Alouini, “Estimation of Nakagami fading channel parameters with application to optimized transmitter diversity systems,” IEEE Int. Conf. Communication (ICC’01 ), Helsinki, Finland, June 2001. 26. S. M. Alamouti, “A simple transmitter diversity scheme for wireless communications,” IEEE J. Select. Areas Commun., vol. 1, October 1998, pp. 1451–1458. 27. J. D. Brown, Performance of Optimal Dual Transmit Antenna Diversity System for Unknown Correlated Rayleigh Fading Channels, master of science (engineering) thesis, Queen’s Univ., Kingston, Ontario, Canada, April 2002. 28. J. Radon, “Lineare scharen orthogonaler matrizen,” Abhandlungen aus dem Mathematischen Seminar der Hamburgishen Universitat, vol. I, 1922, pp. 1–14. 29. L. A. Dalton and C. N. Georghiades, “A full rate, full diversity four antenna orthogonal space-time block code,” (in press). 30. E. Biglieri, D. Divsalar, P. J. McLane, and M. K. Simon, Introduction to Trellis Coded Modulation with Applications. New York, NY: Macmillan, 1990; subsequently distributed by Prentice-Hall, Englewood Cliffs, NJ. 31. M. K. Simon, “Evaluation of average bit error probability for space-time coding based on a simpler exact evaluation of pairwise error probability,” Int. J. Commun. and Networks, vol. 3, no. 3, September 2001, pp. 257–264. 32. G. Taricco and E. Biglieri, “Exact pairwise error probability of space-time codes,” IEEE Trans. Inform. Theory, vol. 48, no. 2, February 2002, pp. 510–513. 33. G. L. Turin, “The characteristic function of Hermetian quadratic forms in complex normal random variables,” Biometrika, June 1960, pp. 199–201. 34. D. Divsalar and M. K. Simon, “Trellis coded modulation for 4800-9600 bits/s transmission over a fading mobile satellite channel,” IEEE J. Select. Areas Commun., vol. 5, no. 2, February 1987, pp. 162–175. 35. D. K. Aktas and M. P. Fitz, “Computing the distance spectrum of space-time trellis codes,” Wireless Communication and Networking Conf. (WCNC ), 2000, vol. 1, pp. 51–55. 36. M. P. Fitz, J. Grimm, and S. Siwamogsatham, “A new view of performance analysis techniques in correlated Rayleigh fading,” Wireless Communication and Networking Conf. (WCNC ), 1999, vol. 1, pp. 139–144.

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37. M. Uysal and C. N. Georghiades, “Error performance analysis of space-time codes over Rayleigh fading channels,” Int. J. Commun. and Networks, vol. 2, no. 4, December 2000, pp. 351–355. 38. J. K. Cavers and P. Ho, “Analysis of the error performance of trellis coded modulations in Rayleigh fading channels,” IEEE Trans. Commun., vol. 40, no. 1, January 1992, pp. 74–83. 39. H. Jafarkhani and N. Seshadri, “Super-orthogonal space-time trellis codes,” IEEE Trans. Inform. Theory , vol. 49, no. 4, April 2003, pp. 937–950. 40. N. Seshadri and H. Jafarkhani, “Super-orthogonal space-time trellis codes,” IEEE Int. Conf. Communication (ICC’2002 ), New York, NY, April 2002. 41. H. Jafarkhani and N. Seshadri, “Optimal set-partitioning for super-orthogonal space-time trellis codes,” IEEE Int. Symp. Inform. Theory (ISIT ), June 2002. 42. M. Ionescu, K. K. Mukkavilli, Z. Yan, and J. Lilleberg, “Improved 8- and 16-state space-time codes for 4PSK with two transmit antennas,” IEEE Commun. Lett., vol. 5, no. 7, July 2001, pp. 301–303. 43. S. Siwamogsatham and M. P. Fitz, “Improved high-rate space-time TCM via orthogonality and set partitioning,” IEEE Wireless Communication and Networking Conf. (WCNC ), March 2002. 44. S. Siwamogsatham and M. P. Fitz, “Improved high-rate space-time TCM via concatenation of expanded orthogonal block code and M-TCM,” IEEE Int. Conf. Communication (ICC’2002 ), New York, NY, April 2002. 45. S. Siwamogsatham and M. P. Fitz, “Improved high-rate space-time codes via expanded STCB-MTCM constructions,” IEEE Int. Symp. Information Theory (ISIT ), June 2002. 46. M. K. Simon and H. Jafarkhani, “Performance evaluation of super-orthogonal spacetime trellis codes using a moment generating function-based approach,” special issue on signal processing for multiple-input, multiple-output (MIMO) wireless communication systems of IEEE Trans. Signal Proc. vol. 51, no. 11, November 2003, pp. 2739–2751; also see GLOBECOM 2003, San Francisco, CA. 47. H. Jafarkhani, “A quasi-orthogonal space-time block code,” IEEE Trans. Commun., vol. 49, no. 1, January 2001, pp. 1–4. 48. O. Tirkkonen, A. Boariu, and A. Hottinen, “Minimal non-orthogonality rate 1 space time block code for 3+ Tx antennas,” IEEE 6th Int. Symp. Spread-Spectrum Technology and Applications (ISSSTA 2000 ), September 2000, pp. 429-432. 49. O. Tirkkhonen, “Optimizing space-time block codes by constellation rotations,” Finnish Wireless Communications Workshop (FWWC ), October 2001. 50. W. Su and X. Xia, “Signal constellations for quasi-orthogonal space-time block codes with full diversity,” IEEE Trans. Inform. Theory (in press). 51. N. Sharma and C. B Papadias, “Improved quasi-orthogonal codes through constellation rotation,” IEEE Trans. Commun. (in press). 52. H. Jafarkhani and N. Hassanpour, “Super-quasi-orthogonal space-time trellis codes,” available at http://www.ece.uci.edu/∼hamidj. 53. M. K. Simon and H. Jafarkhani, “Error probability performance evaluation of superquasi-orthogonal space-time codes based on a moment-generating function approach,” to be presented at GLOBECOM 2004, Dallas, TX.

15 CAPACITY OF FADING CHANNELS The Shannon capacity of a channel defines its theoretical upper bound for the maximum rate of data transmission at an arbitrarily small BER, without any delay or complexity constraints. Therefore, the Shannon capacity represents an optimistic bound for practical communication schemes, and also serves as a benchmark against which to compare the spectral efficiency of all practical adaptive transmission schemes [1]. Goldsmith and Varaiya derived in [2] the capacity of a single-user flat-fading channel with channel measurement information at the transmitter and receiver for various adaptive transmission policies. In this chapter, we first apply the general theory that they developed in [2] and present closed-form expressions for the capacity of Rayleigh fading channels under different adaptive transmission schemes with and without diversity combining [3]. In particular, we consider three adaptation policies: optimum simultaneous power and rate adaptation, constant power with optimum rate adaptation, and channel inversion with fixed rate. We then investigate the impact of maximal-ratio diversity (MRC) on the Shannon capacity in conjunction with each of these adaptive transmission schemes. Note that an analytical evaluation of the capacity in a Rayleigh fading environment with the constant power policy has been carried out [4–6]. Extension of the capacity results presented in this chapter to multiple-input/multiple-output (MIMO) systems is briefly discussed at the end of this chapter but more details on this topic are available in the literature [7–13].

15.1

CHANNEL AND SYSTEM MODEL

A block diagram of the adaptive transmission system under consideration in this first part of this chapter is shown in Fig. 15.1. We assume that the channel changes at a rate much slower than the data rate, so that the channel remains constant over

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

863

864

CAPACITY OF FADING CHANNELS

Feedback Path Slowly Varying Rayleigh Channel Input Data

Pilot

Channel

Modulator

Data

Transmitter

Channel Estimator Carrier Recovery

AGC

Commun. Mode Selector Demodulator

Output Data

Receiver

Figure 15.1

Adaptive transmission system block diagram.

hundreds of symbols. We call this a slowly varying channel. We assume a Rayleigh fading channel so that the probability distribution function (PDF) of SNR, γ , is given by the exponential distribution in (2.7), namely pγ (γ ) =

e−γ / γ , γ

γ ≥0

(15.1)

where γ is the average received SNR. We consider MRC diversity combining of the received signal. As mentioned in Chapter 9, MRC diversity combining requires that the individual signals from each branch be weighted by their signal voltage-to-noise power ratios, then summed coherently. In the following MRC analyses, thorough knowledge of the branch amplitudes and phases is assumed. Since MRC with perfect combining is the optimum receive diversity scheme, it provides the maximum capacity of singleinput/multiple-output (SIMO) systems. Let γ k denote the average SNR on the kth branch. For independent branch signals and equal average branch SNRs γk = γ

for all

k ∈ {1, 2, 3, . . . , L}

(15.2)

the PDF of the received SNR at the output of a perfect L-branch MRC combiner is a chi-square distribution with 2L degrees of freedom [14, (5.2-14), p. 319] given by pγMRC (γ ) =

γ L−1 e−γ / γ , (L − 1)! γ L

γ ≥0

(15.3)

We assume throughout our analyses that the variation in the combiner output SNR γ is tracked perfectly by the receiver. We also assume that the variation in γ is sent back to the transmitter via an error-free feedback path. The time delay in this feedback path is assumed to be negligible compared to the rate of the channel variation. All these assumptions, which are reasonable for high-speed data transmission over a slowly fading channel, allow the transmitter to adapt its power and/or rate relative to the actual channel state.

OPTIMUM SIMULTANEOUS POWER AND RATE ADAPTATION

15.2

865

OPTIMUM SIMULTANEOUS POWER AND RATE ADAPTATION

Given an average transmit power constraint, the channel capacity of a fading channel with received SNR distribution pγ (γ ) and optimum power and rate adaptation [< C >OPRA (in bits per second)] is given by Goldsmith and Varaiya [2] as  
∞ γ < C >OPRA = B log2 (15.4) pγ (γ ) dγ γo γo where B (in Hertz) is the channel bandwidth and γo is the optimum cutoff SNR level below which data transmission is suspended. This optimum cutoff must satisfy the equation 
+∞  1 1 pγ (γ ) dγ = 1 − (15.5) γo γ γo To achieve the capacity (15.4), the channel fade level must be tracked at both receiver and transmitter, and the transmitter has to adapt its power and rate accordingly, allocating high power levels and rates for good channel conditions (γ large), and lower power levels and rates for unfavorable channel conditions (γ small). Since no data are sent when γ < γo , the optimum policy suffers a probability of outage Pout , equal to the probability of no transmission, given by
γo
+∞ Pout = P [γ ≤ γo ] = pγ (γ ) dγ = 1 − pγ (γ ) dγ (15.6) 0

15.2.1

γo

No Diversity

Substituting (15.1) in (15.5), we find that γo must satisfy     γo γo − E1 =γ E0 γ γ where En (x) is the exponential integral of order n defined by
∞ En (x) = t −n e− xt dt, x≥0

(15.7)

(15.8)

1

In particular, E0 (x) = e−x /x, so (15.7) reduces to   γo e−γo /γ − E1 =γ γo /γ γ

(15.9)

Let x = γo /γ and define f (x) =

e−x − E1 (x) − γ x

(15.10)

866

CAPACITY OF FADING CHANNELS

Note that df (x)/dx = −e−x /x 2 < 0 for all x ≥ 0. Moreover, from (15.10), limx→0+ f (x) = +∞ and limx→+∞ f (x) = − γ < 0. Thus we conclude that there is a unique xo for which f (xo ) = 0 or, equivalently, there is a unique γo that satisfies (15.9). An asymptotic expansion of (15.9) shows that as γ → +∞, γo → 1. Our numerical results show that γo increases as γ increases, so γo always lies in the interval [0,1]. Substituting (15.1) in (15.4), and defining the integral Jn (µ) as


∞

Jn (µ) =

t n−1 ln t e−µt dt;

µ > 0; n = 1, 2, . . .

(15.11)

1

we can rewrite the channel capacity < C >OPRA as < C >OPRA = B log2 (e)

  γo γo J1 γ γ

(15.12)

The evaluation of J1 (µ) is derived in Appendix 15A and given in (15A.14). Using that result, we obtain the capacity per unit bandwidth < C >OPRA /B [in bits per second per Hertz (bps/Hz)] as   γo < C >OPRA = log2 (e) E1 B γ

(15.13)

Using (15.9) in (15.13), the optimum capacity per unit bandwidth reduces to the simple expression < C >OPRA = log2 (e) B



e−γo / γ −γ γo / γ

 (15.14)

Using (15.1) in the probability of outage equation (15.6) yields Pout = 1 − e−γo /γ 15.2.2

(15.15)

Maximal-Ratio Combining

Inserting the SNR distribution (15.3) in (15.5), we see that with MRC combining γo must satisfy    L, γγo γo γ

  γo = (L − 1)! γ −  L − 1, γ

(15.16)

where (·, ·) is the complementary incomplete gamma function defined in (15A.8). Let x = γo / γ and define fMRC (x) =

(L, x) − (L − 1, x) − (L − 1)! γ x

(15.17)

OPTIMUM RATE ADAPTATION WITH CONSTANT TRANSMIT POWER

867

Note that [dfMRC (x)]/dx = [−(L, x)]/x 2 < 0 for all x ≥ 0 and L ≥ 2. Since limx→0+ fMRC (x) = +∞ and limx→+∞ fMRC (x) = −(L − 1)! γ < 0, we conclude that there is a unique xo such that fMRC (xo ) = 0 or, equivalently, there is a unique γo that satisfies (15.16). An asymptotic expansion on (15.16) shows that γo → 1 as γ → +∞, so γo ∈ [0, 1]. Comparing this with the results in Section 15.2.1, we see that as the average received CNR grows to infinity, the optimum cutoff value is the same for both diversity methods. Substituting (15.3) in (15.4) we obtain the channel capacity with MRC diversity, < C >MRC OPRA (in bps), in terms of the integral JL (.) as < C >MRC OPRA =

B log2 (e) (L − 1)!



γo γ



L JL

γo γ

 (15.18)

The evaluation of Jn (µ) is derived in Appendix 15A and given in (15A.13). Using this result, we obtain the capacity per unit bandwidth < C >MRC OPRA /B (in bps/Hz) as    L−1   Pk (γo / γ ) < C >MRC γo OPRA + = log2 (e) E1 B γ k

(15.19)

k=1

where Pk (.) denotes the Poisson distribution defined in (15A.10). The corresponding MRC is obtained by substituting (15.3) into (15.6), and using probability of outage Pout Eq. (8.381.3) of Ref. 15 (p. 364) and our Eq. (15A.9):   γo MRC Pout = 1 − PL (15.20) γ

15.3 OPTIMUM RATE ADAPTATION WITH CONSTANT TRANSMIT POWER With optimum rate adaptation to channel fading and a constant transmit power, the channel capacity < C >ORA (in bps) becomes [2]
∞ log2 (1 + γ ) pγ (γ ) dγ (15.21) < C >ORA = B 0

which was initially introduced by Lee [4] as the average channel capacity of a flatfading channel, since it is obtained by averaging the capacity of an AWGN channel CAWGN = B log2 (1 + γ )

(15.22)

over the distribution of the received SNR γ . In fact, (15.21) represents the capacity of the fading channel without transmitter feedback (i.e., with the channel fade level known at the receiver only) [5,16,17], under some mild regularity conditions on the fading process. In the following analysis, we first obtain the channel capacity

868

CAPACITY OF FADING CHANNELS

without diversity (correcting some minor errors in Ref. 4), and then derive analytical expressions as well as simple accurate asymptotic approximations of the capacity improvement with MRC diversity. 15.3.1

No Diversity

Substituting (15.1) into (15.21), the channel capacity < C >ORA of a Rayleigh fading channel is obtained as
∞ e−γ / γ dγ (15.23) < C >ORA = B log2 (1 + γ ) γ 0 Defining the integral In (µ) as
∞ In (µ) = t n−1 ln(1 + t) e−µt dt;

µ > 0; n = 1, 2, . . .

(15.24)

I1 (1/ γ ) γ

(15.25)

0

we can rewrite < C >ORA as < C >ORA = B log2 (e)

Using the result of (15B.9) from Appendix 15B, we can write < C >ORA /B (in bps/Hz) as   1 < C >ORA (15.26) = log2 (e) e1/ γ E1 B γ Note that the exponential integral function of first order, E1 (.), is related to the exponential integral function, Ei (.), used in [4] by E1 (x) = −Ei (−x). Using the first series expansion for E1 (.) [4, Eq. (2.32)] and substituting it into (15.26) yields   +∞ k  (−1/ γ ) < C >ORA = log2 (e) e1/ γ −E + ln γ − (15.27) B k · k! k=1

where E is the Euler constant (E = 0.577215665). Therefore, for γ  1, we get the following approximation for (15.26):   1 < C >ORA  log2 (e) e1/ γ −E + ln γ + (15.28) B γ Using the second series expansion for E1 (.) given by Lee [4, Eq. (2.33)] and substituting it in (15.26) yields < C >ORA = B log2 (e)

n  k=1

(−1)(k+1) (k − 1)! γ k + Rn

(15.29)

CHANNEL INVERSION WITH FIXED RATE

869

where Rn is a remainder term. Taking the limit as the channel bandwidth approaches infinity yields lim < C >ORA = log2 (e)

B→+∞

No

(15.30)

where < S > [in watts (W)] is the average carrier power and N0 (in W/Hz) is the noise power density per unit bandwidth. 15.3.2

Maximal-Ratio Combining

Substituting (15.3) into (15.21), we obtain the channel capacity < C >MRC ORA (in bps) with MRC in terms of the integral IL (.) as < C >MRC ORA = B log2 (e)

IL (1/ γ ) (L − 1)! γ L

(15.31)

Using (15B.7) from Appendix 15B, we can rewrite < C >MRC ORA /B (in bps/Hz) as L−1  ( − k, 1/ γ ) < C >MRC ORA 1/ γ = log2 (e) e B γk k=0

(15.32)

Note that by using Eq. (8.359.1) of Ref. 15 (p. 951), the capacity with the MRC diversity scheme (15.32) with a single branch (L = 1) reduces to (15.26), as expected. Note also that by using two additional equations from Gradshteyn and Ryzhik [15, Eqs. (8.352.3) and (8.359.1), pp. 950, 951], one may express (15.32) in terms of the Poisson distribution as [6, Eq. (2.33)]       L−1  Pk (1/ γ )PL−k (−1/ γ ) < C >MRC 1 1 ORA E1 + = 1 log2 (e) PL − (15.33) B γ γ k k=1

Moreover, using the first series expansion for E1 (.) [4, Eq. (2.32)] in (15.33), we obtain an asymptotic approximation (γ  1) for < C >MRC ORA /B as      M−1  Pk (1/ γ )PM−k (−1/ γ ) < C >MRC 1 1 ORA  log2 (e) PM − −E + ln γ + + B γ γ k k=1

(15.34)

15.4

CHANNEL INVERSION WITH FIXED RATE

The channel capacity when the transmitter adapts its power to maintain a constant SNR at the receiver (i.e., inverts the channel fading) was also investigated [2]. This technique uses fixed-rate modulation and a fixed code design, since the channel after channel inversion appears as a time-invariant AWGN channel. As a result, channel inversion with fixed rate is the least complex technique to implement, assuming that

870

CAPACITY OF FADING CHANNELS

good channel estimates are available at both transmitter and receiver. The channel capacity with this technique [< C >CIFR (in bps)] is derived from the capacity of an AWGN channel and is given in Ref. 2 as  < C >CIFR = B log2

1 1+ ∞ 0 pγ (γ )/γ dγ

 (15.35)

Channel inversion with fixed rate suffers a large capacity penalty relative to the other techniques, since a large amount of the transmitted power is required to compensate for the deep channel fades. A better approach is to use a modified inversion policy that inverts the channel fading only above a fixed cutoff fade depth γo . The capacity with this truncated channel inversion and fixed-rate policy [< C >TIFR (in bps)] was derived in Ref. 2 to be  < C >TIFR = B log2

1 1+ ∞ γo pγ (γ )/γ dγ

 (1 − Pout )

(15.36)

where Pout is as given by (15.6). The cutoff level γo can be selected to achieve a specified outage probability or, alternatively, to maximize (15.36). The choice of γo is examined in more detail in the following sections. 15.4.1

No Diversity

By substituting the SNR distribution (15.1) in (15.35), we find that the capacity of a Rayleigh fading channel with total channel inversion, < C >CIFR , is zero. However, with truncated channel inversion the capacity per unit bandwidth < C >TIFR /B (in bps/Hz) can be expressed in terms of γ and γo as   γ < C >TIFR e−γo / γ = log2 1 + B E1 (γo / γ )

(15.37)

As mentioned in Ref. 2, this capacity is maximized for an optimum cutoff CNR γo∗ that increases as a function of γ . Recall that we proved the existence of a unique optimum cutoff CNR for the optimum adaptation policy in Section 15.2.1. However, for optimum adaptation the optimum cutoff SNR γo was always bounded between [0,1] (i.e., smaller than 0 dB), whereas for this policy γo∗ it can be bigger than 0 dB. This means that, for a fixed γ , truncated channel inversion has both a smaller capacity (see Fig. 15.4) and a higher probability of outage (see Fig. 15.5) than the optimum policy of Section 15.2. 15.4.2

Maximal-Ratio Combining

We obtain the capacity per unit bandwidth for total channel inversion with MRC diversity combining, < C >MRC CIFR , by substituting (15.3) into (15.35) and using

NUMERICAL EXAMPLES

871

Eq. (3.381.4) of Ref. 15 (p. 364): < C >MRC CIFR = log2 (1 + (L − 1) γ ) B

(15.38)

Note that the capacity of this policy for a Rayleigh fading channel with an Lbranch perfect MRC combiner (15.38) is the same as the capacity of a set of L − 1 parallel, independent AWGN channels. Truncated channel inversion improves the capacity (15.38) at the expense of outMRC . The capacity of truncated channel inversion with MRC comage probability Pout MRC bining, < C >TIFR , is obtained by inserting (15.3) in (15.36) and using Eq. (3.381.3) of Ref. 15 (p. 364):   (L − 1)! γ (L, γo / γ ) < C >MRC TIFR = log2 1 + (15.39) B (L − 1, γo / γ ) (L − 1)! Using property (15A.9) of the complementary incomplete gamma function, we can rewrite (15.39) as the simple expression     < C >MRC (L − 1) γ γo TIFR PL , L≥2 (15.40) = log2 1 + B PL−1 (γo / γ ) γ It can be shown that, for a fixed γ , the maximizing cutoff CNR γo∗ increases when MRC diversity is used. In addition, as we will show in Section 15.2, the capacity improvement provided by truncated channel inversion (γo = γo∗ ) compared to total channel inversion (γo = 0) is relatively small, and this little improvement comes at the expense of a higher probability of outage (see Fig. 15.7). This suggests that, as long as diversity is used, total channel inversion is a better alternative than truncated channel inversion. 15.5

NUMERICAL EXAMPLES

In Fig. 15.2, channel capacity without diversity, < C >ORA , given by (15.26), as well as its asymptotic approximation (15.28), are plotted against γ . This figure also displays the capacity per unit bandwidth of an AWGN channel, CAWGN given in (15.22). With these results we find, for example, that • •

For γ = 10 dB, < C >ORA  2.91 B, whereas CAWGN  3.46 B. For γ = 25 dB, < C >ORA  7.50 B, whereas CAWGN  8.31 B.

Therefore, the channel capacity of a Rayleigh fading channel is reduced by 15.9% for γ = 10 dB and by 9.75% for γ = 25 dB in. Note in Fig. 15.2 that the asymptotic approximation (15.28) closely matches the exact average capacity (15.26) when γ > 5 dB. Figure 15.3 shows plots of < C >MRC ORA /B (15.32) as well as its asymptotic approximation (15.34) as functions of the average SNR per branch γ for various

872

CAPACITY OF FADING CHANNELS

Capacity per Unit Bandwidth /B [Bits/Sec/Hz]

10 Rayleigh Channel: Exact Rayleigh Channel: Approximation AWGN Channel

9 8 7 6 5 4 3 2 1 0

0

5

10 15 20 Average Received SNR [dB]

25

30

Figure 15.2 Average channel capacity per unit bandwidth for a Rayleigh fading and an AWGN channel versus average SNR γ (with no diversity).

Capacity per Unit Bandwidth /B [Bits/Sec/Hz]

10 Rayleigh Channel: Exact Rayleigh Channel: Approximation AWGN Channel

9

c

8

b a

7 6 5 4 3 2 1 0

0

5

10 15 20 Average Received SNR [dB] per Branch

25

30

Figure 15.3 Average channel capacity per unit bandwidth for a Rayleigh fading channel with maximal-ratio combining diversity versus average SNR per branch γ [(a) L = 1, (b) L = 2, (c) L = 4].

873

NUMERICAL EXAMPLES

values of L. Note the large diversity gain obtained by two-branch combining; the capacity with two branches in fading exceeds that of a single-branch AWGN channel. Figure 15.3 also displays the capacity per unit bandwidth of an array of M independent AWGN channels with optimum combining (MRC): MRC CAWGN = log2 (1 + Lγ ) B

(15.41)

Note that the capacity of an array of L independent Rayleigh channels with MRC combining approaches the capacity of an array of L independent AWGN channels as L tends to infinity. Note also the diminishing capacity returns that are obtained as the number of branches increases. Recall from Chapter 9 that this diminishingreturns characteristic is also exhibited when the performance evaluation is based on average probability of error and outage probability. Finally, note again that the asymptotic approximation (15.34) closely matches the exact average capacity (15.32) when γ ≥ 5 dB. Figure 15.4 shows the calculated channel capacity per unit bandwidth as a function of γ for the different adaptation policies without diversity combining. From this figure we see that the optimum power and rate adaptation (15.14) yields a small increase in capacity over just rate adaptation (15.26), and this small increase in capacity diminishes as γ increases. The corresponding outage probability (15.15)

Capacity per Unit Bandwidth /B [Bits/Sec/Hz]

10 Optimal Power and Rate Optimal Rate and Constant Power Total Channel Inversion Truncated Channel Inversion

9 8 7 6 5 4 3 2 1 0

0

5

10 15 20 Average Received SNR [dB]

25

30

Figure 15.4 Channel capacity per unit bandwidth for a Rayleigh fading channel versus average SNR γ for different adaptation policies when L = 1.

874

CAPACITY OF FADING CHANNELS

100

Probability of Outage Pout

Optimal Adaptation Truncated Channel Inversion

10−1

10−2

10−3

0

Figure 15.5

5

10 15 20 Average Received SNR [dB]

25

30

Outage probability of the optimum adaptation and truncated channel inversion.

for the optimum adaptation and truncated channel inversion (with optimum cutoff γo∗ ) are shown in Fig. 15.5. Figure 15.6 shows the channel capacity per unit bandwidth as a function of γ for the different policies with MRC diversity. As the number of combining branches increases, the capacity difference between optimum power and rate adaptation versus optimum rate adaptation alone becomes negligible for all values of γ . For any L, fixed-rate transmission with total channel inversion suffers the largest capacity penalty relative to the other policies. However, as L increases, the fading is progressively reduced, and this penalty diminishes remarkably. Thus, as L increases, all capacities of the various policies converge to the capacity of an array of L independent AWGN channels (15.41). However, it is not possible in practice to completely eliminate the effects of fading through space diversity since the number of diversity branches is limited. This is especially true for the downlink (base station to mobile), since mobile receivers are generally constrained in size and power. Since channel inversion is the least complex technique, there is a tradeoff of complexity and capacity for the various adaptation methods and diversity combining techniques. The diversity gain for all policies is quite important, especially for total channel inversion. For example, in Fig. 15.6 we see that the capacity with total channel inversion and two-branch MRC exceeds that of a single-branch system with optimum adaptation. Note that this figure also illustrates the typical

875

NUMERICAL EXAMPLES

10

Capacity per Unit Bandwidth /B [Bits/Sec/Hz]

9 c 8 7 b 6 5 a 4 3 2

Optimal Power and Rate Optimal Rate and Constant Power Total Channel Inversion Truncated Channel Inversion

1 0

0

5

10 15 20 Average Received SNR [dB] per Branch

25

30

Figure 15.6 Channel capacity per unit bandwidth for a Rayleigh fading channel versus average SNR γ for different adaptation policies with MRC diversity: (a) L = 1, (b) L = 2, (c) L = 4.

100 Optimal Adaptation Truncated Channel Inversion

Probability of Outage Pout

10−1

10−2

10−3

10−4

10−5

10−6

10−7

0

5

10 15 20 Average Received SNR [dB] per Branch

25

30

Figure 15.7 Outage probability of the optimum adaptation and truncated channel inversion with MRC diversity (L = 2).

876

CAPACITY OF FADING CHANNELS

diminishing returns obtained as the number of branches increases. In addition, for L = 1 total channel inversion suffers a large capacity penalty relative to truncated channel inversion. However, as the number of combining branches L increases, the effect of fading is progressively reduced, and this penalty diminishes remarkably. In particular, as L increases, we see that truncated channel inversion yields a small increase in capacity over total channel inversion, and this small increase in capacity diminishes as the average SNR γ and/or the number of combined branches L increase. The corresponding outage probability (15.20) for the optimum adaptation and truncated channel inversion (with optimum cutoff γo∗ ) policies with MRC are shown in Fig. 15.7.

15.6

CAPACITY OF MIMO FADING CHANNELS

Thus far in this chapter, we have considered the capacity of SISO and SIMO fading channels. In this final section of the chapter, we extend the results to the case of multiple transmit antennas, in particular, the MIMO channel. In keeping with the spirit of Chapter 14, we make the following assumptions: (1) the transmitters have no knowledge of the channel but the receivers track (i.e., have perfect knowledge of) the channel, (2) the total transmit power is constrained and divided equally among the transmit antennas regardless of their number, and (3) the channel between each transmit antenna and each receive antenna is statistically modeled as quasistatic i.i.d. flat wherein the fading is fixed over a burst (frame) but can randomly change from burst to burst. Under these assumptions it has been shown [7] that for a MIMO system with Lt transmit antennas, Lr receiver antennas, and a channel gain matrix C = [cj i ], the conditional channel capacity (in bps/Hz) is given by   C 1 E s ∗ T = log2 det ILr ×Lr + C C B L t N0

(15.42)

where ILr ×Lr is the identity matrix of order Lr and cij denotes the normalized channel gain between the j th transmit and the i th receive antennas. For the transmit diversity-only (MISO) case with a single receive antenna and Lt transmit antennas, (15.42) simplifies to   Lt C 1 Es  2 |c1l | = log2 1 + B L t N0

(15.43)

l=1

On the other hand, for the optimum (MRC) SIMO system with a single transmit antenna and Lr receive antennas, we have previously observed that the conditional channel capacity is given by   Lr C Es  |cl1 |2 = log2 1 + B N0 l=1

(15.44)

REFERENCES

877

Thus, from a comparison of (15.43) and (15.44) we observe that because of the constraint of a fixed total transmitter power uniformly distributed among the Lt transmit antennas in the MISO case, for Lt = Lr = L the MISO system suffers an SNR loss of a factor of Lt relative to the optimum SIMO system in order to achieve the same conditional capacity. An analogous loss was identified in Chapter 14 in the context of achieving equivalent diversity between the two systems. In an i.i.d. Rayleigh environment, Telatar showed that the average capacity is given by [8] 1

= B ln(2)




∞

ln(1 + ρy)e −y

0

s−1  k=0

k! y t−s t−s 2 L (y) dy, (k + t − s)! k

(15.45)

where t = max(Lt , Lr ), s = min(Lt , Lr ), ρ = Es /(N0 Lt ) is the normalized transmitting SNR per branch, and Lan (·) is the generalized Laguerre polynomial. Using the functional relation between the generalized Laguerre polynomial and the confluent hypergeometric function given by Gradshteyn and Ryzhik [15, Eq. (9.972.1)], noting that the resulting confluent hypergeometric function terminates as a finite sum because its first parameter is a negative integer, and integrating term by term, it can be shown after some additional manipulations that (15.45) can be written in terms of complementary incomplete gamma functions as [13] 1  (t − s + k)! e 1/ρ

= B ln(2) [(t − s)!]2 k! k=0  l  2k   (−k)m (−k)l−m (t − s + l)! × (t − s + 1)m m!(t − s + 1)l−m (l − m)! s−1

l=0

×

m=0

t−s+l+1  q=1

  1 q−(t−s+l+1) ρ  q − (t − s + l + 1), , ρ

(15.46)

where (a)n = a(a + 1) · · · (a + n − 1) with (a)0 defined to be 1.

REFERENCES 1. A. J. Goldsmith and S.-G. Chua, “Adaptive coded modulation for fading channels,” IEEE Trans. Commun., vol. COM-46, May 1998, pp. 595–602. 2. A. Goldsmith and P. Varaiya, “Capacity of fading channels with channel side information,” IEEE Trans. Inform. Theory, vol. IT-43, November 1997, pp. 1896–1992. 3. M.-S. Alouini and A. Goldsmith, “Capacity of Rayleigh fading channels under different adaptive transmission and diversity techniques,” IEEE Trans. Veh. Technol., vol. VT-48, July 1999, pp. 1165–1181. 4. W. C. Y. Lee, “Estimate of channel capacity in Rayleigh fading environment,” IEEE Trans. Veh. Technol., vol. VT-39, August 1990, pp. 187–190.

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CAPACITY OF FADING CHANNELS

5. L. Ozarow, S. Shamai, and A. Wyner, “Information theoretic considerations for cellular mobile radio,” IEEE Trans. Veh. Technol., vol. VT-43, May 1994, pp. 359–378. 6. C. G. G¨unther, “Comment on “Estimate of channel capacity in Rayleigh fading environment,” IEEE Trans. Veh. Technol., vol. VT-45, May 1996, pp. 401–403. 7. G. J. Foschini and M. J. Gans, “On limits of wireless communication in a fading environment when using multiple antennas,” Wireless Personal Commun., vol. 6, March 1998, pp. 311–335. 8. I. E. Telatar, “Capacity of multi-antenna Gaussian channels,” Eur. Trans. Telecommun., vol. 10, November–December 1999, pp. 585–596. 9. P. J. Smith and M. Shafi, “On a Gaussian approximation to the capacity of wireless MIMO systems,” Proc. Int. Conf. Communication (ICC’2002), New York, NY, April 2002. 10. M. Chiani, “Evaluating the capacity distribution of MIMO Rayleigh channels,” Proc. IEEE Int. Symps. Advanced Wireless Communication (ISWC’2002), Victoria, British Columbia, Canada, September 2002, pp. 3–4. 11. S. Jayaweera and V. Poor, “On the capacity of multi-antenna systems in the presence of Rician fading,” Proc. IEEE Veh. Technol. Conf. (VTC’2002 Fall), Vancouver, Canada, September 2002. 12. Z. Wang and G. B. Giannakis, “Outage mutual information of space-time MIMO channels,” Proc. 40th Annual Allerton Conf. Communication, Control, and Computing (Allerton’2002), Monticello, IL, October 2002. 13. M. Kang and M.-S. Alouini, “On the capacity of MIMO Rician channels,” Proc. 40th Annual Allerton Conf. Communication, Control, and Computing (Allerton’2002), Monticello, IL, October 2002. 14. W. C. Jakes, Microwave Mobile Communication, 2nd ed. Piscataway, NJ: IEEE Press, 1994. 15. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, 5th ed. San Diego, CA: Academic Press, 1994. 16. T. Ericson, “A Gaussian channel with slow fading,” IEEE Trans. Inform. Theory, vol. IT-16, May 1970, pp. 353–355. 17. R. J. McEliece and W. E. Stark, “Channels with block interference,” IEEE Trans. Inform. Theory, vol. IT-30, January 1984, pp. 44–53.

APPENDIX 15A. EVALUATION OF Jn (µ) The integral Jn (µ) defined in (15.11) is evaluated using partial integration:
1

∞


u dv = lim (uv) − lim (uv) − t→∞

t→ 1

+∞

v du

(15A.1)

1

First, let u = ln t

(15A.2)

APPENDIX 15A. EVALUATION OF Jn (µ)

879

Thus du =

dt t

(15A.3)

Then, let dv = t n−1 e−µt dt

(15A.4)

Performing n − 1 successive integration by parts yields [15, Eq. (2.321.2), p. 112] v = −e−µt

n  (n − 1)! t n−k (n − k)! µk

(15A.5)

k=1

Substituting (15A.2), (15A.5), and (15A.3) in (15A.1), we see that the first two terms go to zero. Hence Jn (µ) =


∞ n  (n − 1)! t n−k−1 e−µt dt µk (n − k)! 1

(15A.6)

k=1

The integral in (15A.6) can be written in a closed form with the help of Eq. (3.381.3) in Ref. 15 (p. 364), giving Jn (µ) =

n−1 (n − 1)!  (k, µ) µn k!

(15A.7)

k=0

where (·, ·) is the complementary incomplete gamma function (or Prym’s function, as it is sometimes called) defined by [15, Eq. (8.350.2), p. 949]
(α, x) =

∞

t α−1 e−t dt

(15A.8)

x

For k positive integers k≥2

(k, µ) = (k − 1)! Pk (µ);

(15A.9)

where Pk (·) denotes the Poisson distribution defined as Pk (µ) = e

−µ

k−1 j  µ j =0

j!

(15A.10)

For k = 0 [15, Eq. (8.359.1) p. 951], we obtain (0, µ) = E1 (µ)

(15A.11)

880

CAPACITY OF FADING CHANNELS

where E1 (·) is the exponential integral of a first-order function defined as
+∞ −xt e E1 (x) = dt (15A.12) t 1 Thus, for n positive integers, (15A.7) can be written as   n−1  Pk (µ) (n − 1)! Jn (µ) = E1 (µ) + µn k

(15A.13)

k=1

In particular, when n = 1, (15A.13) reduces to J1 (µ) =

E1 (µ) µ

(15A.14)

EVALUATION OF In (µ)

APPENDIX 15B.

The integral In (µ) defined in (15.24) is also evaluated using partial integration:
∞
∞ u dv = lim (uv) − lim (uv) − v du (15B.1) 0

t→+∞

t→0

0

First, let u = ln(1 + t)

(15B.2)

Thus du =

dt 1+t

(15B.3)

Then, let dv = t n−1 e−µt dt

(15B.4)

Performing n − 1 successive integration by parts yields [15, Eq. (2.321.2), p. 112] v = −e

−µt

n  (n − 1)! t n−k (n − k)! µk

(15B.5)

k=1

Substituting (15B.2), (15B.5), and (15B.3) in (15B.1), we see that the first two terms go to zero. Hence
∞ n−k −µt n  (n − 1)! t e dt In (µ) = k µ (n − k)! 0 1+t k=1

(15B.6)

APPENDIX 15B. EVALUATION OF In (µ)

881

The integral in (15B.6) can be written in a closed form with the help of Eq. (3.383.10) of Ref. 15 (p. 366), giving In (µ) = (n − 1)! eµ

n  ( − n + k, µ) µk

(15B.7)

k=1

where (·, ·) is the complementary incomplete gamma function defined in (15A.8). Note that when n = 1, (15B.7) reduces to I1 (µ) = eµ

(0, µ) µ

(15B.8)

I1 (µ) = eµ

E1 (µ) µ

(15B.9)

which can be written as

where E1 (·) is the exponential integral of first order defined by (15A.12).

INDEX Absolute threshold generalized selection combining (AT-GSC) average error probability evaluation, 519–520 performance, 515–516 defined, 512 outage probability performance, 521–523 performance comparisons, with NT-GSC, 524–531 Additive white Gaussian noise (AWGN) average error probability performance, 142 characteristics of, generally, 19, 25, 38, 46, 55, 61, 62, 65, 67, 70, 76 coded communication, 761, 772–773, 775, 777, 783 fading channel capacity, 869–874 Gaussian Q-function, 84 multicarrier DS-CDMA systems, 745, 748 multichannel receivers, 312, 317, 331 multichannel transmission, 801–802, 818 optimum combining, 683, 699 optimum receivers, 189–190, 195, 199, 201, 211, 215, 218 performance, detection methods binary signaling, generic results, 251 differentially coherent, 245–251 ideal coherent, 224–237 noncoherent, 242 nonideal coherent, 237–241 partially coherent, 242–245 single carrier DS-CDMA systems, 738 single-channel receivers, 223–251 Alamouti’s diversity technique, using two transmit antennas

characteristics of, 803–809 combined with multidimensional trellis-coded modulation, 812–818 generalization to orthogonal space-time block code designs, 809–812 Amount of fading implications of, 12–13 multichannel receivers, dual-branch diversity combining, 567, 578–581 Amplitudes, fading channel known known delays, unknown phases, 198–199 known phases and delays, 191–195 unknown known delays and unknown phases, 199–219 known phases and delays, 195–198 unknown phases and delays, 219–222 Antennas Alamouti’s diversity technique, see Alamouti’s diversity technique diversity system, see Multiple-input/multiple-output (MIMO) antenna diversity systems multiuser communication systems, multiple transmit and receive correlation between receiver antenna pairs, 726 distribution of antenna elements, 726 numerical examples, 727–729 optimum weight vectors and output SIR, 723 outage probability, 723–725 PDF of output SIR, 723–724

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

883

884

INDEX

Antennas (Continued) special cases, 725–726 system, channel, and signal models, 721–722 Antipodal signaling, 52 Arbitrary two-dimensional signal constellation error probability integral, 142–144 Automatic gain control (AGC), 314 Average fading power, multichannel receivers, 316 Average outage duration (AOD) multichannel receivers average level crossing rate and, 586–589 I.I.D. Rayleigh fading, 589–591 numerical examples, 591–594 overview, 584–585 system and channel models, 585 multiuser communication system, 667–671 Beckmann distribution, 28–30 Bessel function, 94, 96, 100–103, 114–115, 239, 346, 619, 650, 657 Beta function, 138, 140 Bhattacharyya parameter, 766 Binary DPSK (BDPSK), multichannel receivers multiple-symbol differential detection (MSDD), 370 outage probability, 411–413 selection combining, 415 switch-and-examine combining, 445 switched diversity, 464–465 Binary frequency-shift-keying (BFSK) average error probability performance, 137 defined, 58 multichannel receivers coherent, 413–415 coherent equal gain combining, 334, 339 noncoherent, BDPSK and, 411–413 noncoherent equal gain combining, 343, 365 post-detection combining, 564–565 optimum combining, 694 optimum receivers, detection of, 203–204, 207, 210 single-channel reception characteristics of, 243 fading channels, 267, 270, 273, 279–280, 283 Binary phase-shift-keying (BPSK) characteristics of, 51, 68 multichannel receivers characteristics of, 546–549, 552 correlation models, 401 differentially encoded, 235 selection combining, 415–417, 485–487

switch-and-examine combining, 441 switched diversity, 428, 469 multichannel transmission, 853 optimum combining, 698, 701, 710, 715 selection combining, 533–537 Binomial expansion, 10 Bit error probability (BEP), average alternative closed-form expressions, 135–137 coded communication differentially coherent detection, 783 over fading channels, 759–760 transfer function bound, 772–774 TUB, comparison with union-Chernoff bounds, 788–791 conditional, 319–320 for decision statistic, quadratic form of complex Gaussian random variables, 619–625 integrals involving arbitrary two-dimensional signal constellation error probability integral, 142–144 Gaussian-Q function, 123–131, 144–145 incomplete gamma function, 137–141 Marcum-Q function, 131–135 M-PSK error probability integral, 141–142, 145–148 Marcum Q-function, 100 multichannel receivers bounds on, 352–353 characteristics of, generally, 317, 320, 619–624 noncoherent equal gain combining, 334, 338–339 scan-and-wait combining, 448, 452, 454–455 selective combining, 519–520, 525–528, 530 multichannel transmission characteristics of, 801–803, 845 fast-fading channel model, evaluation of, 827–828 slow-fading channel model, evaluation of, 829–830 optimum combining diversity combining receivers, 685 multiple arbitrary power interferers and, 718 multiple-symbol differentiation detection (MSDD), 718–720 Nakagami-m fading, 695–697 number of interferers and, 701–703, 705, 710, 712 Rayleigh fading, 682, 686–692

INDEX

optimum receivers and, 203–204, 207, 210–212 as performance criterion, 6–12 single-channel receivers implications of, 223–229, 232–234, 236, 238, 241, 247–248, 251 fading channels, 273–281 space-time trellis-coded modulation approximate evaluation of, 827–830 transfer function upper bound evaluation, 831–833 Bit error rate (BER), average coded communication, 785 multicarrier DS-CDMA systems, 749–750 multichannel receivers correlation models, 400 multiple-symbol differential detection with diversity combining, 368–371 noncoherent equal gain combining, 351 outdated/imperfect channel estimates, 463–464 post-detection combining, 540–553, 557, 561–565 selection combining, 480–487, 498 switch-and-examine combining, 441, 443, 445–446 switch-and-stay combining, 435–436, 438–439 switched diversity, 463–465, 469 noncoherent equal gain combining, 347–352 Bivariate Rayleigh CDF, 172–174 Bivariate Rayleigh PDF, 170–172 Carrier reconstruction loop, 64 Carrier synchronization architectures, 65, 273–274 loop, 64 Carrier-to-interference ratio (CIR), outage probability, 639, 648, 657–661, 664–667 Carrier-to-noise ratio (CNR), 639, 648, 660–661, 664 Carrier tracking loop, 53 CDMA code/codewords, 801 Cellular mobile radio system, 644 Channel state information (CSI), 191, 193, 760, 764–771, 773–774, 781, 785 Characteristics function (CHF), 11n Chernoff bound, 85–86, 97–98, 231, 249, 759, 765–766, 773 Chi-square distribution, 550 Chi-square variates in outage probability, multiuser communication systems, 640–645 Classical functions, alternative representations Gaussian Q-function, 84–93, 120–122

885

Marcum Q-function, 93–113, 118 Nuttall Q-function, 113–117 overview of, 83, 117–118 Closed-form expressions, 135–137, 141, 143, 147 Co-channel interferers (CCI), 681, 700, 706, 711, 721–722 Coded communication systems coherent detection example of, 775–781 pairwise error probability, evaluation of, 763–771 system model, 761–763 transfer function bound alternative formulation of, 774–775 on average bit error probability (BEP), 772–774 differentially coherent detection example of, 785–787 performance evaluation, 783–785 system model, 781–783 true upper bounds (TUBs), comparison with Union-Chernoff bounds, 787–792 Code-division multiple access (CDMA), 311, 697 Coding gain, 798 Coherent detection differentially, 71–78, 117 ideal, 45–58 noncoherent, 66–68, 117 nonideal, 62–66 optimum receivers, 191–195, 217–219, 221–222 partially, 68–71 Coherent equal gain combining, multichannel receivers average output SNR, 332–333 characteristics of, generally, 331 exact error rate analysis approximate, 340–341 asymptotic, 342 binary signals, 333–339 M-PSK signals, extension to, 339–340 outage probability combining, 380–382 performance, 380–381 receiver structure, 331 Combined (time-shared) shadowed/unshadowed fading, 37 Communication, types of coded, see Coded communication differentially coherent detection conventional detection (two-symbol observation), 73–74

886

INDEX

Communication, types of (Continued) M-ary differential phase-shift-keying (M-DPSK), 71–73, 117–118 multiple-symbol detection, 76–78 ideal coherent detection M-ary frequency-shift-keying (M-FSK), 56–58 M-ary phase-shift-keying (M-PSK), 50–54, 75 minimum-shift-keying (MSK), 58–62 multiple amplitude modulation (M-AM), 47–49 multiple amplitude-shift-keying (M-ASK), 47–48 offset QSPK (OQPSK), 55–56 overview of, 45–47 π /4-QSPK, 54–55 quadrature amplitude modulation (QAM), 48–50 quadrature amplitude-shift-keying (QASK), 48–50 staggered QPSK (SQPSK), 55–56 noncoherent detection, 66–68, 117 nonideal coherent detection, 62–66 partially coherent detection, conventional detection multiple-symbol observation, 69–71 one-symbol observation, 68–69 π /4-differential QPSK (π /4-DQPSK), 78 Complementary CDF (CCDF), 450–451 Composite gamma/log-normal distribution, 33–34 fading channel, 21 Composite log-normal shadowing/Nakagami-m fading channel characteristics of, generally, 140–141 Gaussian-Q function and, 128–131 Marcum-Q function and, 134–135 Composite microscopic-macroscopic diversity, 315 Composite multipath/shadowing characteristics of, generally, 33 composite gamma/log-normal distribution, 33–34 K distribution, 34–35 Rician shadowed distribution model, 36 Suzuki distribution, 34 Computer software packages, 171, 173, 685 Conditional BEP, 6–7, 9 Continuous-phase-frequency-shift-keying (SPFSK), 58–59 Correlation models, channel dual branches with nonidentical fading, 390–392

dual correlation model, 396 exponential correlation model, 396–397 identically distributed branches with constant correlation, 392–394 with exponential correlation, 394 intraclass correlation/constant correlation model, 396 nonidentically distributed branches with arbitrary correlation, 395–398 numerical examples, 399–404 overview, 389–390 tridiagonal correlation model, 397 Correlative fading, PDF and CDF bivariate Rayleigh, 170–147 for maximum of two Nakagami-m random variables, 177–180 for maximum of two Rayleigh random variables, 175–177 two log-normal random variables maximum of, 180–183 minimum of, 183–185 Costas loop, 64, 238–239, 273 Cumulative distribution function (CDF) bilateral RVs, 9–10 bivariate Rayleigh, 172–174 inverse Laplace transform, numerical techniques, 625–627 Marcum Q-function and, 100 multichannel receivers average duration outage, 589 dual-branch diversity combining, 567, 570 multiple-input/multiple-output (MIMO) antenna diversity system, 597–598 post-detection combining, 541–543, 550–551, 556, 559 selection combining, 497, 518, 521 switch-and-examine combining, 440, 444 switch-and-stay combining, 433–434 switched diversity, 420–421, 424 optimum combining, 724 outage probability implications of, 5–6 multiuser communication systems, 640–643, 649, 661–662, 674–678 Weibull distribution, 26 Decay, power delay profile (PDP), 38 Decision matrices, diversity combining, 367–368 Definite integrals, 634–635 Degradation, 274 Degrees of freedom, 550, 640–643, 674–678 Deinterleaving, 759, 784 Delays, fading channel known known amplitudes and phases, 191–195

INDEX

known amplitude and unknown phases, 198–199 known phases and unknown amplitudes, 195–198 unknown amplitudes and phases, 199–219 unknown, amplitudes and phases, 219–222 Delay statistic, multichannel receivers, 448–450 Demodulation, 51–52, 57, 64–65, 76 Desired forms, 8 Differential decoding, 53–54 Differential detection/detectors, 64, 73 Differential encoding, 53–55, 144 Differentially coherent detection AGWN performance M-ary differential phase-shift-keying (M-DPSK), 245–250 π /4-differential QPSK (π /4-DQPSK), 250–251 conventional detection (two-symbol observation), 73–74 in fading communications characteristics of, 284–285 M-ary differential phase-shift-keying (M-DPSK), fast fading, 290–293 M-ary differential phase-shift-keying (M-DPSK), slow fading, 285–290 M-ary differential phase-shift-keying (M-DPSK), 71–73, 117–118 multiple-symbol detection, 76–78 optimum receivers characteristics of, 211–214, 217–218 Nakagami-m fading, 217, 218–219 Rayleigh fading, 214–216, 218 single-channel receivers, M-ary differential phase-shift-keying (M-DPSK) conventional detection, two-symbol observation), 245–249 with multiple-symbol detection, 249–250 Differentially coherent receivers, 199 Differentially encoded BPSK, single-channel receivers and, 235, 240 Differential phase encoding, 53, 215 Differential phase-shift-keying (DPSK) multichannel receivers, 343 optimum combining, 698 Differential quadrature phase-shift-keying (DQPSK), multichannel reception, 343, 370–371 Differentiation, 4 Digital portable communications, 315 Dirac delta function, 515 Direct-sequence code-division multiple access (DS-CDMA) channel models

887

multicarrier, 743 single, 737–738 numerical examples, 750–754 overview of, 735–736, 741–742, 744 performance multicarrier, 745–750 single, 739–741 receiver multicarrier, 743–744 single, 738 transmitter single, 736–737 multicarrier, 742–743 Diversity combining complexity-performance tradeoffs, 316 concept, 312 mathematical modeling, 312–313 multichannel receivers multiple-symbol differential detection (MSDD), 367–375 optimum, 375–379 techniques, survey of, 313–316 Diversity gain, 797 Diversity order, 797 Diversity-rich environments, combining in generalized selection combining based on log-likelihood ratio, 532–537 characteristics of, 469–512 with threshold test per branch (T-GSC), 512–531 generalized switched diversity (GSSC), 531–532 two-dimension diversity schemes, 466–469 Diversity selection combining, 158 Diversity systems, 12, 32 Dual-branch diversity combining schemes, over log-normal channels characteristics of, 566 maximal-ratio combining, 568–571 selection combining, 571–575 switched combining, 575–584 system and channel models, 566–568snr) Dual-branch post-detection combining identically distributed, 542, 544–547 nonidentically distributed, 542–543, 547–548 Dual-branch selective combining outage probability, multiuser communication system fading and systems models, 661 numerical examples, 664–667 outage performance with minimum signal power constraint, 661–663 Dual-diversity combining receiver, 169

888

INDEX

Eigenvalue, noncentral complex wishart matrices, 596–604 800/900 MHz frequency, 25 Envelope fluctuations, in fading channels, 17–18 Equal gain combining (EGC) coded communication, evaluation of, 768 defined, 11 multichannel receivers dual-branch diversity combining, 571 noncoherent, see Noncoherent equal gain combining switched diversity, 467–468 noncoherent, see Noncoherent equal gain combining optimum receivers, detection of, 203 Euler summation-based technique, 625–626 Fading channel capacity channel and system model, 863–864 channel inversion with fixed rate characteristics of, 869–870 no diversity, 870 maximal-ratio combining, 870–871 I n (µ), evaluation of, 880–881 MIMO fading channels, 876–877 numerical examples, 871–876 optimum rate adaptation with constant transmit power characteristics of, 867–868 no diversity, 868–869 maximal-ratio combining, 869 optimum simultaneous power and rate adaptation characteristics of, 865 no diversity, 865–866 maximal-ratio combining, 866–867 T n (µ), evaluation of, 878–880 Fading channels, see specific types of fading channels capacity of, see Fading channel capacity characteristics of, generally envelope fluctuations, 17–18 fast fading, 18 frequency-flat fading, 18–19 frequency-selective fading, 18–19 phase fluctuations, 17–18 slow fading, 18 multichannel receivers, noncoherent equal gain combining, 356–364. See Correlation models, channel optimum receivers for known amplitudes and delays, unknown phases, 198–199 known amplitudes, phases, and delays, 191–195

known delays, unknown amplitudes and phases, 199–219 known phases and delays, unknown amplitudes, 195–198 overview of, 189–191 unknown amplitudes, phases, and delays, 219–222 outage probability, multiuser communication systems, 643–644 performance detection differentially coherent, 284–294 ideal coherent, 252–267 imperfect channel estimation, 294–301 noncoherent detection, 281–282 nonideal coherent, 267–281 partially coherent, 282–284 SNR and, 21 Fading distribution, multichannel receivers, 316 Fast fading, 18, 784 First propagation, 38–39 Fixed-rate transmission, 869–871, 874 Flat-fading channels implications of, 381 modeling of characteristics of, 19–20 combined (time-shared) shadowed/ unshadowed fading, 37 composite multipath/shadowing, 33–37 log-normal shadowing, 32–33 multipath fading, 20–32 Fourier transform, 10, 337 Frequency-flat fading, 18–19 Frequency mismatch, 62 Frequency-selective fading, characteristics of, 18–19 Gauss-Chebyshev quadrature-based technique, 510, 626–627 Gauss-Chebyshev quadrature rules, 10 Gaussian-Hermite quadrature integrals, 123, 128–129, 134, 317, 338, 639 Gaussian noise, 118 Gaussian probability integral, 84 Gaussian Q-function characteristics of, generally, 7, 179 coded communication, 765 higher powers, alternative representations of, 90–93 integrals involving composite log-normal shadowing/ Nakagami-m fading channel, 128–131 higher-order integer powers, 144–145 log-normal shadowing channel, 128 Nakagami-m fading channel, 126–127, 145

INDEX

Nakagami-n (Rice) fading channel, 126 Nakagami-q (Hoyt) fading channel, 125 overview of, 123–124 Rayleigh fading channel, 125, 144–145 multichannel receivers dual-branch diversity combining, 567 implications of, 320, 334–335 switched diversity, 429 symbol error rate (SER), 324 one-dimensional case, 84–86, 88–90 optimum combining, 704 optimum receivers, 196 proofs of alternative forms, 120–122 single reception, 225–226, 229–232, 236, 239–240 symbol error rate (SER), 330 two-dimensional case, 86–90 Generalized selection combining (GSC), multichannel receivers based on log-likelihood ratio (LLR) envelope-based, 536 optimum for equiprobable BPSK, 533–536 optimum for noncoherently detected equiprobable orthogonal BPSK, 536–537 performance analysis characteristics of, 470–473 I.I.D. Nakagami-m case, 497–502 I.I.D. Rayleigh case, 472–492 I.I.D. Weibull case, 510–512 non-I.I.D. Rayleigh case, 492–497 partial-MGF approach, 502–510 with threshold test per branch (T-GSC), 512–531 Generalized switched diversity (GSSC), 531–532 Generic closed-form formula, 640 Global optimization, 465 Global System for Mobile (GSM), 697 Gradshteyn-Ryzhik equation, 325 Gray code bit mapping, 225, 227, 232–233, 246 Hamming distance, 775–776 Harmonics, 57 High-speed data transmission, 864 Hoyt distribution, see Nakagami-q (Hoyt) distribution Hoyt fading channel characteristics of, 21–23, 28–29 Gaussian-Q function and, 125 incomplete gamma functions and, 139 Marcum-Q function and, 133 multipath fading, 22–23 Hybrid diversity schemes, multichannel receivers generalized techniques, 315 multidimensional techniques, 315–316

889

Ideal coherent detection in communication systems M-ary frequency-shift-keying (M-FSK), 56–58 M-ary phase-shift-keying (M-PSK), 50–54 minimum-shift-keying (MSK), 58–62 multiple amplitude modulation (M-AM), 47–49 multiple amplitude-shift-keying (M-ASK), 47–48 offset QSPK (OQPSK), 55–56 overview of, 45–47 π /4-QSPK, 54–55 quadrature amplitude modulation (QAM), 48–50 quadrature amplitude-shift-keying (QASK), 48–50 staggered QPSK (SQPSK), 55–56 fading channels differentially encoded M-ary phase-shift-keying (M-PSK), 258–261 M-ary frequency-shift-keying (M-FSK), 262–267 M-ary phase-shift-keying (M-PSK), 256–258 minimum-shift-keying (MSK), 267 multiple amplitude-shift-keying (M-ASK), 253–254 multiple amplitude modulation, (M-AM), 253–254 offset QPSK (OQPSK), 262 π /4-QPSK, 258–261 quadrature amplitude modulation (QAM), 254–256 quadrature amplitude-shift-keying (QASK), 254–256 staggered QPSK (SQPSK), 262 single channel receivers, performance over AWGN channel M-ary frequency-shift-keying (M-FSK), 236–237 M-ary phase-shift-keying (M-PSK), 228–235 minimum-shift-keying (MSK), 237 multiple amplitude modulation (M-AM), 224–225 multiple amplitude-shift-keying (M-ASK), 224–225 offset QPSK (OQPSK), 235–236 π /4-QPSK, 234–235 quadrature amplitude modulation (QAM), 225–234

890

INDEX

Ideal coherent detection, single channel receivers (Continued) quadrature amplitude-shift-keying (QASK), 225–234 staggered QPSK (SQPSK), 235–236 Imperfect channel estimation, single-channel receivers characteristics of, generally, 294–295 Rayleigh fading, signal model and symbol error probability evaluation, 295–297 special cases M-PSK, 297–300 M-QAM, 300–301 Incomplete gamma functions, integrals involving characteristics of, 137–137 composite log-normal shadowing/Nakagami-m fading channel, 140–141 log-normal shadowing channel, 140 Nakagami-m fading channel, 140 Nakagami-n (Rice) fading channel, 139 Nakagami-q (Hoyt) fading channel, 139 Rayleigh fading channel, 138 Interference, error probability performance, 137 Interference-limited systems, outage probability average outage duration (AOD), 667–671 channel fading models, 643–644 desired and interference signals models, 644 dual-branch SC and SSC diversity fading and systems models, 661 outage performance with minimum signal power constraint, 661–667 overview, 659–661 generic formula for, 644–46 with minimum desired signal power constraint models and problem formulation, 648–649 Nakagami/I.I.D. Rice scenario, 654–659 Rice/I.I.D. Nakagami scenario, 649–654 outage rate, 667–671 related to CDF of the difference of two chi-square variates with different degrees of freedom, 640–643, 674–678 Interleaving, 759–780, 784 Irreducible error probability, 274, 293 K distribution, 34–35 Land-mobile satellite systems, 33, 37 Laplace transform characteristics of, generally, 6, 124–126, 129, 132–134, 138–139, 143–144 coded communication, 769–770, 785–786 correlation models, 394 inverse, implications of, 361 multichannel receivers, selection combining, 475, 497, 521

optimum combining, 704 Largest eigenvalue of certain quadratic forms in complex Gaussian vectors, distributions of, 732–734 Level crossing rate (LCR) average outage duration multichannel receivers, 585–589, 591–594 multiuser communication system, 667–668 defined, 14 Line-of-sight (LOS) path, Rayleigh fading, 20 Log-likelihood ratio (LLR), selection combining, 532–537 Log-normal random variables, PDF and CDF maximum of, 180–183 minimum of, 183–185 Log-normal shadowing characteristics of, 21, 32–33 Gaussian-Q function and, 128 incomplete gamma functions and, 140 Marcum-Q function and, 133–134 Loop SNR, 64, 273–274 MacLaurin series expansion, 273, 276, 279 Macrodiversity, 316 M-ary communication system, 10–11 M-ary differential phase-shift-keying (M-DPSK) coded communication, 782–783 communications, 71–73, 117 optimum receivers, detection of, 211 single-channel receivers characteristics of, 229 conventional detection, two-symbol observation, 245–249 fast fading, 290–293 multiple-symbol detection, 249–250 slow fading, 285–290 M-ary frequency-shift-keyed (M-FSK) characteristics of, generally, 56–58, 68 multichannel receivers noncoherent equal gain combining, 353–367 optimum diversity combining, 378–379 orthogonal extension, multichannel receivers post-detection combining, 552–566 single-channel receivers, 236–237, 242–243, 262–267 M-ary phase-shift-keyed (M-PSK) signal characteristics of, generally, 7, 50–54, 66, 68–69, 71, 73, 75, 77–78 coded communication characteristics of, 762, 765, 774–775, 781–782 differential detection of sequences, association with MGF, 793–795 differentially encoded, single-channel receivers, 258–261

INDEX

error probability performance, see M-PSK average error probability performance multichannel receivers coherent equal gain combining, 339–340 selection combining, 481, 489 symbol error rate (SER), 322–323, 326, 330, 339–340 multichannel transmission, 806, 810, 812, 815 optimum combining, 715–718 optimum receivers, detection of, 213–214, 217 single-channel receivers characteristics of, generally, 227–234 differentially encoded, 234–235 fading channels, 256–258, 297–300 partially coherent detection, 244 Maple V, 171 Marcum Q-function bounds on, 97–100, 105–113, 118 characteristics of, 172–174, 179 for decision statistic, quadratic form of complex Gaussian random variables, 619–621, 623–624 defined, 7 first-order, 93–100, 105–108, 114, 116 fourth order, 106–108 generalized (mth order), 100–113 integrals involving composite log-normal shadowing/ Nakagami-m fading channel, 134–135 log-normal shadowing channel, 133–134 Nakagami-m fading channel, 133 Nakagami-n (Rice) fading channel, 133 Nakagami-q (Hoyt) fading channel, 133 overview of, 131–132 Rayleigh fading channel, 132 multichannel receivers, switched diversity, 423, 432 noncoherent equal gain combining, 346 outage probability, multiuser communication systems, 641–642, 650, 655 second order, 106–108 single-channel receivers, 242 Mathematica, 171, 173, 685 MATLAB, 171, 650 Maximal-ratio combining (MRC) characteristics of, generally, 5 coded communication, 765 fading channel capacity, 864, 866–867, 869–871 multichannel receivers average outage duration (AOD), 590 correlation models, 398–399 dual-branch diversity combining, 568–571 MGF-based approach, 320–326

891

multiple-input/multiple-output (MIMO) antenna diversity system, 595–596 outage probability performance, 379–380, 386, 388, 426–427 outdated/imperfect channel estimates, 457–458, 464 scan-and-wait combining, 453–454 selection combining, 469–472, 477, 481, 485, 487, 499, 519, 523–526 switch-and-stay combining, 438 switched diversity, 424, 430–432, 464, 466–468 multichannel transmission, 801, 803, 807, 809, 816 optimum receivers, 193, 710–714 outage probability, multiuser communication systems, 640, 644 overview of, 316–317 PDF-based approach, 319 receiver structure, 317–318 selection combining, 410 SER expressions, bounds and asymptotic, 326–330 Maximum a posteriori (MAP), 64 Maximum-likelihood (ML) decision rule, 191, 534 Maximum-likelihood (ML) receiver, 763 Maximum-likelihood sequence estimation (MLSE), 367, 763 Minimum mean-squared error (MMSE), 294, 298–299 Minimum-shift-keying (MSK) characteristics of, 58–62 single-channel reception, 237, 267, 273 Mobile communication system, 798 Modulation diversity, 799 Moment generating function (MGF) average bit error rate probability, 12 average error probability performance, 143 coded communication characteristics of, 770–771, 779, 781, 784, 786 differential detection of M-PSK sequences, association evaluation, 793–795 composite gamma/log-normal, 21 log-normal shadowing, 21, 32 multichannel receivers correlation models, 392–396, 398 noncoherent equal gain combining, 356–359 selection combining, 405–406, 473–474, 476, 480, 483, 493–494, 499–510, 516–517, 531–532 switch-and-examine combining, 440–441, 445

892

INDEX

Moment generating function (MGF), multichannel receivers (Continued) switch-and-stay combining, 434–435, 437 switched diversity, 422 multichannel transmission, 816, 823–824, 833, 837 Nakagami-m fading, 21, 25 Nakagami-n (Rice) fading, 21 Nakagami-q (Hoyt) fading, 21–22 optimum combining, 682, 684–687, 689, 697, 703, 708 outage probability, 5–6 multiuser communication systems, 652, 654 Rayleigh fading, 21–22 signal-to-noise ratio, 4–5 Weibull distribution, 27 unconditional, 9 Moment generating function (MGF)-based approach, multichannel receivers, 320–326 Monte Carlo simulations, 665–666, 721, 728–729 M-PSK average error probability performance characteristics of, 141–142, 151–152 integer powers of characteristics of, 145–146 Rayleigh fading channel, 146–148 characteristics of, generally, 141 integer powers of overview, 145–146 Rayleigh fading channel, 146–148 Nakagami-m fading channel, 142 Rayleigh fading channel, 142 M-QAM multichannel receivers average SER, 324–326, 330 selection combining, 481 single-channel reception, 300–301 Multicarrier code-division systems (MC-CDMA) characteristics of, 741–742 numerical examples, 750–754 performance analysis average BER, 749–750 conditional SNR, 745–749 system and channel models channel, 743 notation, 744 receiver, 743–744 transmitter, 742–743 Multichannel receivers, performance of average outage duration and average level crossing rate, 586–589 I.I.D. Rayleigh fading, 589–591 numerical examples, 591–594

overview, 584–585 system and channel models, 585 bit error probability, 619–624 coherent equal gain combining approximate error rate analysis, 340–341 asymptotic error rate analysis, 342 average output SNR, 332–333 characteristics of, generally, 331 exact error rate analysis, 333–340 receiver structure, 331 definite integrals, 634–635 diversity combining complexity of, 316 concept, 312 mathematical modeling, 312–313 techniques, survey of, 313–316 diversity-rich environments, combining in generalized selection combining, 469–512 generalized selection combining based on log-likelihood ratio, 532–537 generalized selection combining with threshold test per branch (T-GSC), 512–531 generalized switched diversity (GSSC), 531–532 two-dimension diversity schemes, 466–469 dual-branch diversity combining schemes over log-normal channels characteristics of, 566 maximal-ratio combining, 568–571 selection combining, 571–575 switched combining, 575–584 system and channel models, 566–568 fading correlation, impact of identically distributed branches with constant correlation, 392–394 identically distributed branches with exponential correlation, 394 nonidentically distributed branches with arbitrary correlation, 395–398 numerical examples, 399–404 overview, 389–390 two correlated branches with nonidentical fading, 390–392 hybrid combining techniques, see Hybrid diversity schemes, multichannel receivers maximal-ratio combining (MRC) MGF-based approach, 320–326 overview of, 316–317 PDF-based approach, 319 receiver structure, 317–318 SER expressions, bounds and asymptotic, 326–330

INDEX

multiple-input/multiple-output (MIMO) antenna diversity systems characteristics of, 594 largest eigenvalue of noncentral complex Wishart matrice, distribution of, 596–604 optimum weight vectors and output SNR, 595–596 system, channel, and signal models, 594–595 noncoherent and differentially coherent equal gain combining DPSK, DQPSK, and BFSK, 343–353 M-ary orthogonal FSK, 353–367 multiple-symbol differential detection (MSDD) with diversity combining, 367–375 numerical techniques, inversion of Laplace transform of CDFs Euler summation-based technique, 625–626 Gauss-Chebyshev quadrature-based technique, 626–627 optimum diversity combining, noncoherent FSK characteristics of, 375–377 comparison with noncoherent equal gain combining receiver, 377–378 extension to M-ary orthogonal FSK case, 378–379 outage probability characteristics of, 379 coherent EGC, 380–381 MRC and noncoherent EGC, 379–380 numerical examples, 381–388 outdated or imperfect channel estimates characteristics of, 456–457 maximal-ratio combining, 457–458 noncoherent EGC over Rician fast fading, 458–460 numerical results, 464–466 selection combining, 461–462 switched diversity, 462–464 power correlation coefficient of correlated Rician random variables relation with correlation coefficient of their underlying complex Gaussian random variables, 627–631 pure techniques, see Equal gain combining (EGC); Maximal-ratio combining (MRC); Selection combining; Switched combining selection combining average output SNR, 406–409, 475–476, 494 average probability of error, 411–417, 480–487

893

average SER, 494–496 CDF of output SNR, 497 characteristics of, 404–405 GSC input joint PDF, 472–473, 492–493 GSC input joint statistics, 499 MGF of GSC output, 499–502 MGF of output SNR, 405–406, 473–474, 493–494 outage probability, 409–411, 476–480, 497 PDF of output SNR, 474–475, 496–497 SNR penalty, 487–492 switched diversity characteristics of, 417–418 dual-branch switch-and-stay combining, 419–439 multibranch switch-and-examine combining, 439–456 theorem proofs, 473–474, 631–634 Multipath diversity, 316 Multipath fading Beckmann distribution, 28–30 defined, 20 Nakagami-m model, 24–25 Nakagami-n (Rice) model, 23–24 Nakagami-q (Hoyt) model, 22–23 Rayleigh model, 20–22 spherically-invariant random process model (SIRP), 30–32 Weibull distribution, 25–28 Multipath intensity profile (MIP), 38 Multiple amplitude modulation, (M-AM) characteristics of, 47–49 multichannel receivers, average SER, 323–324 single-channel receivers characteristics of, generally, 224–225 fading channels, 253–254 Multiple amplitude-shift-keying (M-ASK) characteristics of, 47–48 single-channel receivers characteristics of, generally, 224–225 fading channels, 253–254 Multiple-input/multiple-output (MIMO) systems antenna diversity systems, multichannel receivers characteristics of, 594 largest eigenvalue of noncentral complex Wishart matrice, distribution of, 596–604 optimum weight vectors and output SNR, 595–596 system, channel, and signal models, 594–595 outage probability, multiuser communication systems, 660 transmission, 798

894

INDEX

Multiple-input/single-output (MISO) system, 601, 603, 726, 876 Multiple interferers, optimum combining, 697–718 Multiple symbol detection, see Differentially coherent detection Multiple-symbol differential detection (MSDD) multichannel receivers asymptotic behavior, 371 average bit error rate performance, 368–371 decision metrics, 367–368 numerical results, 372–375 optimum combining average BEP, 718–720 decision metric, 718 Multiple trellis-coded modulation (MTCM), 799 Multiuser communication systems direct-sequence code-division multiple access (DS-CDMA), 735–756 optimum combining, 681–734 outage performance, 639–680 Nakagami-m fading channel characteristics of, generally, 21, 24–25 composite gamma/log-normal distribution, 33 correlative PDF and CDF, 177–180 evaluation of definite integrals associated with bounds, upper and lower, 165–167 exact closed-form results, 149–165 Gaussian-Q function and, 126–127, 145 incomplete gamma functions and, 140 Marcum-Q function and, 133 M-PSK error probability integrals, 142, 146 multicarrier systems, DC-CDMA, 750 multichannel receivers characteristics of, generally, 323 coded communication, 778, 781 coherent equal gain combining, 340 correlation models, 392, 397–398, 403 multiple-symbol differential detection (MSDD), 372–375 noncoherent equal gain combining, 353 outdated/imperfect channel estimates, 461 post-detection combining, 548 selection combining, 497–502, 509–512, 518 switch-and-stay combining, 435–436 switched diversity, 420, 422–423, 426, 429, 465, 469 optimum combining, 692, 695–697, 703, 712 optimum receivers characteristics of, 196–198 detection by, 206–211, 217–219, 221 outage probability, multiuser communication systems

characteristics of, 640, 643–644, 648 Nakagami/I.I.D. Rice scenario, 654, 656–657 Nakagami/Nakagami scenario, 645–646, 678–680 Nakagami/Rice scenario, 647 outage rate and average outage duration, multiuser communication systems, 667–671 single carrier systems, DS-CDMA, 740–741 single-channel receivers, 275, 279, 282, 289 Nakagami-n (Rice) fading, see Rician fading channel Nakagami-q (Hoyt) fading, see Hoyt fading channel Narrowband land-mobile satellite modeling, 33 Noisy reference loss, 273–281 Noncentrality parameter, 94 Noncoherent detection optimum receivers characteristics of, 199–200, 219 Nakagami-m fading, 206–211, 221 Rayleigh fading, 201–206, 220–221 single-channel receivers AWGN channel performance, 242 fading channels, 281–282 Noncoherent equal gain combining multichannel receivers DPSK, DQPSK, and BFSK, 343–353 M-ary orthogonal FSK, 353–367 multiple-symbol differential detection with diversity combining, 367–375 outage probability performance, 379–380 optimum diversity, comparison with, 377–378 Noncoherent frequency-shift-keying (FSK), optimum diversity combining characteristics of, 375–377 comparison with noncoherent equal gain combining receiver, 377–378 M-ary orthogonal FSK, extension to, 378–379 Nonideal coherent detection, single-channel receivers AWGN channel performance, 237–241, 267 fading channels, 267–281 Nonlinear phase estimation techniques, 65 Normalized threshold generalized selection combining (NT-GSC) average error probability evaluation, 520 performance, 516–519 defined, 512 outage probability performance, 523 performance comparisons, with AT-GSC, 527–531

INDEX

Numerical techniques, inversion of Laplace transform of CDFs Euler summation-based technique, 625–626 Gauss-Chebyshev quadrature-based technique, 626–627 Nuttall Q-function, 113–117, 650, 657 Offset QPSK (OQPSK) characteristics of, 55–56, 59–60, 63 single-channel receivers AWGN channel performance, 235–236, 241 fading channels, 262, 267, 273 Optical scintillation, 35 Optimum combining (OC) characteristics of, 681–682 defined, 681 distributions of largest eigenvalue, quadratic forms in complex Gaussian vectors, 732–734 diversity combining receivers comparison with results for MRC in presence of interference, 710–714 multiple arbitrary interferers, 715–718 multiple equal power interferers, 697–710 multiple-symbol differential detection in presence of interference, 718–720 single interferer, 682–697 with multiple transmit and receive antennas correlation between receiver antenna pairs, 726 distribution of antenna elements, 726 numerical examples, 727–729 optimum weight vectors and output SIR, 723 outage probability, 723–725 PDF of output SIR, 723–724 special cases, 725–726 system, channel, and signal models, 721–722 Optimum diversity combining, noncoherent FSK characteristics of, 375–377 comparison with noncoherent equal gain combining receiver, 377–378 extension to M-ary orthogonal FSK case, 378–379 Optimum reception, performance evaluation fading channels, optimum receivers for, 189–222 multichannel receivers, 311–635 single-channel receivers, 223–309 Orthogonal signaling, 58 Orthogonal systems, noncoherent, 10 Outage duration average, multichannels receivers

895

and average level crossing rate, 586–589 I.I.D. Rayleigh fading, 589–591 numerical examples, 591–594 overview, 584–585 system and channel models, 585 as performance criterion, 13–14 Outage performance, multiuser communication systems average outage duration, 667–671 with dual-branch SC and SSC diversity, 659–667 interference-limited systems, 640–647, 678–680 with minimum desired signal power constraint, 648–659 outage rate, 667–671 overview of, 639 Outage probability diversity combining, 170 multichannel receivers characteristics of, 379 coherent EGC, 380–381 dual-branch diversity combining, 570–571 MRC and noncoherent EGC, 379–380 multiple-input/multiple-output (MIMO) antenna diversity system, 599–600, 604 numerical examples, 381–388 selection combining, 409–411, 476–480, 497 multiuser communication systems, optimum combining with multiple transmit and receive antennas, 723–725 as performance criterion, 5–6 Outdated or imperfect channel estimates, multichannel receivers characteristics of, 456–457 maximal-ratio combining, 457–458 noncoherent EGC over Rician fast fading, 458–460 numerical results, 464–466 selection combining, 461–462 switched diversity, 462–464 Pair-state method, 774 Pair-state transition diagram, 774–775 Pairwise error probability (PEP) coded communication, evaluation of known channel state information, 764–767 overview of, 759, 763–764 unknown channel state information, 768–772 defined, 244 multichannel transmission, Rician fading evaluation

896

INDEX

Pairwise error probability (PEP), multichannel transmission (Continued) fast, 814–817, 820–821 slow, 821–824 Parseval’s theorem, 11 Partial-band interference (PBI), multicarrier systems, DS-CDMA, 747–749 Partially coherent detection conventional detection multiple-symbol observation, 69–71, 244–245 one-symbol observation, 68–69, 242–244 single-channel receivers, AWGN performance, 242–245 Paths estimation, scan-and-wait combining, 450–451 Phase fluctuations, in fading channels, 17–18 Phase-locked loop (PLL) noisy reference loss, 273 nonideal coherent detection and, 64 single reception, 238 Phases, fading channel known known amplitudes and delays, 191–195 known delays and unknown amplitudes, 195–198 unknown known amplitudes and delays, 198–199 known delays and unknown amplitudes, 199–219 unknown amplitudes and delays, 219–222 Phase-shift-keying (PSK), 7 π /4-differential QPSK (π /4-DQPSK) defined, 78 single-channel receivers, detection of, 250–251 π /4-QPSK, single-channel receivers, 54–55, 234–235, 258–261 Pilot-symbol-assisted modulation (PSAM), 294, 298 Pilot tone-aided detection, 64 Post-detection combining, multichannel receivers average BER analysis, 540–543 channel model, 537, 539 operation switching strategy, 539–540 switching threshold, 540 orthogonal M-FSK, extension to, 552–566 Rayleigh fading, 543–548 receiver, 539 severity of fading, impact of, 548–552 system description, 537–538 Power correlation coefficient of correlated Rician random variables relation with correlation

coefficient of their underlying complex Gaussian random variables, 627–631 Power decay factor, 39 Power delay profile (PDP), 38, 201, 403, 469 Precoded MSK, 60, 62–63 Probability density function (PDF) bit error rate probability, 10 bivariate Rayleigh, 170–172 coded communication, 764, 770, 783–784 composite gamma/log-normal, 21 fading channel capacity, 864 Gaussian-Q function, 125 log-normal shadowing, 21 Marcum Q-function and, 93 multichannel receivers average duration outage, 586–589 characteristics of, 319 coherent equal gain combining, 336–337 correlation models, 390–393, 395 dual-branch diversity combining, 567, 571, 575, 577 multiple-input/multiple-output (MIMO) antenna diversity system, 598–602 post-detection combining, 559 selection combining, 472–475, 477, 492–493, 496–497, 499–500, 502, 516, 518, 520–521, 531 switch-and-examine combining, 440, 444 switch-and-stay combining, 434, 437 switched diversity, 421–422, 424, 467 multichannel transmission, 807 multiuser communication systems multiple transmit and receive antennas, 723–724 optimum combining, 685–686, 689, 694–695, 697, 704, 707, 710, 715 outage probability, 640, 648, 654–656, 661, 723–724 Nakagami-m fading, 21 Nakagami-n (Rice) fading, 21, 23–24 Nakagami-q (Hoyt) fading, 21 optimum combining, 723–724 optimum receivers, 192, 195–196, 198–199, 201, 219–221 outage probability, 5, 640, 648, 654–656, 661, 723–724 Rayleigh fading, 21 signal-to-noise ratio and, 5 Weibull distribution, 25–26 Quadrature amplitude modulation (QAM) average error probability performance, 130, 142, 151, 153 characteristics of, 48–52

INDEX

defined, 7 single-channel receivers characteristics of, 225–234 fading channel, 254–256, 267, 273, 279–280 Quadrature amplitude-shift-keying (QASK) characteristics of, 48–50 single-channel receivers, 225–234, 254–256 Quadriphase-shift-keying (QPSK), see Quadrature amplitude modulation (QAM) Radar clutter, 35 RAKE receivers multicarrier DS-CDMA systems, 746 multichannel, 315, 321, 381, 384, 468–469, 471, 514 optimum combining, 683 single carrier DS-CDMA systems, 736, 738 Rayleigh fading channel capacity of, 871–876 CDF, 175–177 characteristics of, 20–22, 28 coded communication, 760, 778–779, 781, 786, 789 evaluation of definite integrals associated with bounds, upper and lower, 165–167 exact closed-form results, 149–165 Gaussian-Q function and, 125, 144–145 incomplete gamma functions and, 138 Marcum-Q function and, 132 M-PSK error probability integral, 142, 145–148 multicarrier DS-CDMA systems, 749 multichannel receivers average outage duration, 589–591 coherent equal gain combining, 332 multiple-input/multiple-output (MIMO) antenna diversity system, 600, 603 noncoherent equal gain combining, 353, 360–364 outage probability, 410, 426, 428 post-detection combining, 543–548, 557–559 scan-and-wait combining, 453–454 selection combining, 407–408, 472–497, 418–520 switch-and-examine combining, 443–444 switched diversity, 420–422, 424, 426, 467 symbol error rate (SER), 323, 325–326 multichannel transmission, 823, 838 noisy reference loss evaluation, 273–281 optimum combining average bit error probability, 682, 686–692, 698–699, 703 number of interferers, 706, 711

897

optimum receivers characteristics of, 195–196 detection by, 201–206, 214–216, 218, 220–221 outage probability, multiuser communication systems characteristics of, 643–644, 660 multiple Rayleigh interferers, 651 Rayleigh/I.I.D. Rice scenario, 654–656 selective combining, 664–667 single Rayleigh interferer, 650–651 PDF, 175–177 single-channel receivers characteristics of, 277, 279–281 signal model and symbol error probability evaluation for, 295–297 Rayleigh/log-normal PDF, 34–35 Receive diversity, multichannel transmission, 800–803 Received vector, 229, 232 Rician fading channel characteristics of, 21, 23–24, 28–29, 37 coded communication, 760, 767, 778 Gaussian-Q function and, 126 incomplete gamma functions and, 139 Marcum-Q function and, 133 multicarrier DS-CDMA systems, 749 multichannel receivers characteristics of, generally, 323 multiple-input/multiple-output (MIMO) antenna diversity system, 601 noncoherent equal gain combining, 363–364 outdated/imperfect channel estimates, 458–460 post-detection combining, 550, 552, 559–561, 564 selection combining, 509, 511, 518 switched diversity, 421–422, 426, 429 multichannel transmission fast, 814–817, 820–821 slow, 817–818, 821–822 multipath fading, 23–24 optimum combining, 692–693, 695, 706, 711 outage probability, multiuser communication systems characteristics of, 643–644, 648 Rice/I.I.D. Rayleigh scenario, 657–659 Rice/Nakagami scenario, 647, 649, 652 Rice/Rayleigh scenario, 649–650 Rice/Rice scenario, 646 pairwise error probability performance fast, 820–821 slow, 821–824, 840–841

898

INDEX

Rician fading channel (Continued) shadowed distribution model, 36 single-channel receivers, 276–278

Scan-and-wait combining (SWC), multichannel receivers defined, 440 numerical examples, 452–456 operating assumptions, 446–447 performance of, 447–456 system and channel models, 446 Scattering, 35 Selection combining (SC) characteristics of, 169–170 instantaneous, 178 multichannel receivers average output SNR, 406–409, 475–476, 494, 574–575, 577–581 average probability of error, 411–417, 480–487 average SER, 494–496 CDF of output SNR, 497 characteristics of, 404–405 GSC input joint PDF, 472–473, 492–493 GSC input joint statistics, 499 MGF of GSC output, 499–502 MGF of output SNR, 405–406, 473–474, 493–494 outage probability, 409–411, 476–480, 497, 575, 581–584 outdated/imperfect channel estimates, 461–462 partial-MGF approach, 502–510 PDF of output SNR, 474–475, 496–497 SNR penalty, 487–492 Self-adaptive receivers, 191 Set partitioning method, 761 Severity of fading, multichannel receivers, 316 Shadowing, multichannel receivers, 316 Signal vector, 229 Signal-to-interference plus noise ratio (SINR), optimum receivers, 681–682, 685, 710 Signal-to-interference ratio (SIR), optimum combining, 712, 716, 719–720, 723–724, 727 Signal-to-noise ratio (SNR) average, as performance criterion, 4–5 coded communication, 767, 784, 791 conditional, in multicarrier DS-CDMA, 745–749 equivalent loop, 238–239 fading channel capacity, 864, 866–867, 870, 875, 877

instantaneous, 123–126, 128, 131–133, 175, 178, 233, 274, 313, 358, 469–470, 499, 515–519, 521, 567 large-symbol, 225, 227 multichannel receivers average bit error rate, 321 average output, 406–409 coherent equal gain combining, 331–333, 380–381 correlation models, 395, 401–402 dual-branch diversity combining, 566–569, 571–572, 574–581 MGF of output, 405–406 multiple-input/multiple-output (MIMO) antenna diversity system, 595–596, 599–603 multiple-symbol differential detection (MSDD), 373, 375 noncoherent equal gain combining, 347, 353, 367 optimum receivers and, 193, 196–197, 203, 210 outage probability, 424–426 outdated/imperfect channel estimates, 462–463, 465 post-detection combining, 544, 546–547, 559, 562–564 receiver structure and, 319 scan-and-wait combining, 447–448, 452–456 selection combining, 404, 406–409, 415–417, 471–476, 487–494, 496–498, 513–518, 521, 525–527, 529–533 switch-and-examine combining, 441, 443 switch-and-stay combining, 419–420, 423–424, 435, 437–438, 443–446 switched combining, 314–315 switched diversity, 423–424, 464–465, 467–468 symbol error rate (SER), 327–330 multichannel transmission, 801, 805, 816 optimum combining, 683–684, 695–696, 705 single carrier DS-CDMA system, 739 single-channel receivers, 225, 227–228, 230–231, 238, 273–275 Single carrier systems, DS-CDMA performance analysis characteristics of, 739–740 general case, 740 Nakagami-m fading channels, 740–741 system and channel models channel model, 737–738 receiver, 738 transmitter, 736–737

INDEX

Single-input/multiple-output (SIMO) MRC systems, 594, 601, 603, 726, 864, 876–877 Single-input/single-output (SISO) systems, 876 Single interferers, optimum combining, 682–697 Slow fading, characteristics of, 18, 145, 788 Space-time block coding (STBC), defined, 799 Space-time coding defined, 798 disclaimer, 858–859 Space-time trellis code (STTCs) designs, multichannel transmission orthogonal, 809–812 super-orthogonal space-time trellis codes average bit error probability (BEP), approximate evaluation of, 845 characteristics of, 834 extension to, with more than two transmit antennas, 844 numerical results, 848–850 pairwise error probability evaluation, 836–844 parameterized class and system model, 834–836 transfer function upper bound, evaluation on average bit error probability, 846–848 super-quasi-orthogonal space-time trellis codes characteristics of, 850 examples of, 853–857 numerical results, 857–858 pairwise error probability, evaluation of, 852–853 signal model, 850–851 Space-time trellis-coded modulation average bit error probability, approximate evaluation of fast-fading channel model, 827–828 slow-fading channel model, 829–830 characteristics of, 818–819 defined, 798 example of, 824–826 multichannel transmission, system design, see Space-time trellis-coded (STTC) designs, multichannel transmission pairwise error probability performance on Rician fading channels, evaluation of fast, 820–821 slow, 821–824 transfer function upper bound evaluation on average BEP fast-fading channel model, 831–833 slow-fading channel model, 833 Spacing technique, 497

899

Spherically-invariant random process model (SIRP), 30–32 Spherically-invariant RV (SIRV), 30–32 Spread-spectrum (SS) technique, 735 Square-law detection, 66, 68 Staggered QPSK (SQPSK), see Offset QSPK Stein’s unified analysis of error probability performance, 304–309 Super-orthogonal space-time trellis codes average bit error probability (BEP), approximate evaluation of, 845 characteristics of, 834 extension to, with more than two transmit antennas, 844 numerical results, 848–850 pairwise error probability evaluation fast Rician fading channels, 836–840 slow Rician fading channels, 840–844 parameterized class and system model, 834–836 transfer function upper bound, evaluation on BEP, 846–848 Super-quasi-orthogonal space-time trellis codes characteristics of, 850 examples of, 853–857 numerical results, 857–858 pairwise error probability, evaluation of, 852–853 signal model, 850–851 Suzuki distribution, 34 Switch-and-examine combining (SEC) defined, 418 multichannel receivers characteristics of, 439–440 multibranch, 440–446 scan-and-wait combining, 446–456 Switch-and-stay combining (SSC) characteristics of, 315 multichannel receivers branch correlation, effect of, 436–439 branch unbalance, effect of, 433–436 outdated/imperfect channel estimates, 462–463 performance over independently identically distributed branches, 419–432 outage probability, multiuser communication systems fading and systems models, 661 numerical examples, 664–667 outage performance with minimum signal power constraint, 663–664 Switched diversity branch correlation, effect of, 436–439 branch unbalance, effect of, 433–436

900

INDEX

Switched diversity (Continued) characteristics of, 417–418 dual-branch switch-and-stay combining (SSC) branch correlation, effect of, 436–439 branch unbalance, effect of, 433–436 performance over independently identically distributed branches, 419–432 multibranch switch-and-examine combining characteristics of, 439–440 classical multibranch SEC, 440–443 multibranch SEC with post-selection, 443–446 scan-and-wait combining, 446–456 multichannel receivers, outdated/imperfect channel estimates, 462–464 Symbol error outage (SEO), 328 Symbol error probability (SEP) defined, 7 multichannel receivers, selection combining, 482–483 optimum combining, 715–718 single-channel receivers and, 223–226, 229, 234, 246, 248–250, 295–297 Symbol error rate (SER) multichannel receivers asymptotic, 328–330 bounds, 326–328 coherent equal gain combining, 341–342 correlation models, 399–400 maximal-ratio combining (MRC), 317 M-PSK signals, 322–323 post-detection combining, 557–560 selection combining, 481–482, 489–491, 494–496 square M-QAM signals, 324–326 optimum combining receivers, 692 System performance measures amount of fading (AF), 12–13 average outage duration (AOD), 13–14 bit error probability (BER), 6–12 outage probability, 5–6 signal-to-noise ratio (SNR), average, 4–5

average error probability evaluation, 519–520 average error probability performance, 516–519 characteristics of, 512–515 outage probability performance, 520–523 performance comparisons, 524–531 Tikhonov distribution, 64, 73, 240 Transfer function upper bound, evaluation on average BEP fast-fading channel model, 831–833 slow-fading channel model, 833 Transmit diversity Alamouti’s diversity technique, using two transmit antennas characteristics of, 803–809 combined with multidimensional trellis-coded modulation, 812–818 generalization to orthogonal space-time block code designs, 809–812 historical perspective, 799–800 overview of, 797–799 receive diversity vs., basic concepts of, 800–803 Trellis-coded modulation (TCM) characteristics of, 760, 773, 775, 787, 798 multidimensional, 812–818 space-time, see Space-time trellis coded modulation Tropospheric propagation, 35 True upper bound (TUB), see Coded communication, coherent detection Two-dimension diversity schemes, 466–469

TDMA system, optimum combining, 697, 721–722 Threshold test per branch, generalized selection combining with (T-GSC), multichannel receivers

Weibull distribution, 25–28 Weibull fading channels, 510–512 Wise-Gallagher representation, 32 Wishart matrices, noncentral complex, 596–604

UHF (ultrahigh frequency), 36 Unified MGF-based approach, 8 Uniform error probability (UEP) codes, 773 Urban environment, 38 Viterbi algorithm, 761, 763