DIGITAL COMMUNICATIONS
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 1
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 1
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`DIGITAL COMMUNICATIONS
`Third Edition
`
`John G. Proakis, Ph.D., P.E.
`Department of Electrical and Computer Engineering
`Northeastern University
`
`McGraw-Hill, Inc.
`New York St. Louis San Francisco Auckland Bogota Caracas Lisbon
`London Madrid Mexico City Milan Montreal New Delhi
`San Juan Singapore Sydney Tokyo Toronto
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 2
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 2
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`This book was set in Times Roman.
`The editors were George T. Hoffman and John M. Morriss;
`the production supervisor was Leroy A. Young.
`The cover was designed by Tana Kamine.
`Project supervision was done by The Universities Press (Belfast) Ltd.
`R. R. Donnelley & Sons Company was printer and binder.
`
`DIGITAL COMMUNICATIONS
`
`Copyright © 1995, 1989, 1983 by McGraw-Hill, Inc. All rights reserved. Printed in the
`United States of America. Except as permitted under the United States Copyright Act
`of 1976, no part of this publication may be reproduced or distributed in any form or by
`any means, or stored in a data base or retrieval system, without the prior written
`permission of the publisher.
`
`This book is printed on recycled, acid-free paper containing 10%
`postconsumer waste.
`
`1 2 3 4 5 6 7 8 9 0 D O H D O H 9 0 9 8 7 6 5
`
`ISBN 0-07-051726-6
`
`Library of Congress Cataloging-in-Publication Data
`
`Proakis, John G.
`Digital communications / John G. Proakis.—3rd ed.
`p.
`cm.—(McGraw-Hill series in electrical and computer
`engineering. Communications and signal processing)
`Includes bibliographical references and index.
`ISBN 0-07-051726-6
`1. Digital communications.
`TK5103.7.P76
`1995
`621.382—dc20
`
`94-41620
`
`I. Title.
`
`II. Series.
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 3
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 3
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`CONTENTS
`
`Preface
`
`1 Introduction
`1-1 Elements of a Digital Communication System
`1-2 Communication Channels and Their Characteristics
`1-3 Mathematical Models for Communication Channels
`1-4 A Historical Perspective in the Development of Digital
`Communications
`1-5 Overview of the Book
`1-6 Bibliographical Notes and References
`
`2 Probability and Stochastic Processes
`2-1 Probability
`2-1-1 Random Variables, Probability Distributions,
`and Probability Densities
`2-1-2 Functions of Random Variables
`2-1-3 Statistical Averages of Random Variables
`2-1-4 Some Useful Probability Distributions
`2-1-5 Upper bounds on the Tail Probability
`2-1-6 Sums of Random Variables and the Central Limit
`Theorem
`2-2 Stochastic Processes
`2-2-1 Statistical Averages
`2-2-2 Power Density Spectrum
`2-2-3 Response of a Linear Time-Invariant System to a Random
`Input Signal
`2-2-4 Sampling Theorem for Band-Limited Stochastic Processes
`2-2-5 Discrete-Time Stochastic Signals and Systems
`2-2-6 Cyclostationary Processes
`2-3 Bibliographical Notes and References
`Problems
`
`xix
`
`l
`1
`3
`11
`
`13
`16
`16
`
`17
`17
`
`22
`28
`33
`37
`53
`
`58
`62
`64
`67
`
`68
`72
`74
`75
`77
`77
`
`xi
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 4
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 4
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`Xii
`
`CONTENTS
`
`3 Source Coding
`3-1 Mathematical Models for Information
`3-2 A Logarithmic Measure of Information
`3-2-1 Average Mutual Information and Entropy
`3-2-2
`Information Measures for Continuous Random Variables
`3-3 Coding for Discrete Sources
`3-3-1 Coding for Discrete Memoryless Sources
`3-3-2 Discrete Stationary Sources
`3-3-3 The Lempel-Ziv Algorithm
`3-4 Coding for Analog Sources—Optimum Quantization
`3-4-1 Rate-Distortion Function
`3-4-2 Scalar Quantization
`3-4-3 Vector Quantization
`3-5 Coding Techniques for Analog Sources
`3-5-1 Temporal Waveform Coding
`3-5-2 Spectral Waveform Coding
`3-5-3 Model-Based Source Coding
`3-6 Bibliographical Notes and References
`Problems
`
`4 Characterization of Communication Signals
`a n d Systems
`4-1 Representation of Bandpass Signals and Systems
`4-1-1 Representation of Bandpass Signals
`4-1-2 Representation of Linear Bandpass Systems
`4-1-3 Response of a Bandpass System to a Bandpass Signal
`4-1-4 Representation of Bandpass Stationary Stochastic
`Processes
`4-2 Signal Space Representation
`4-2-1 Vector Space Concepts
`4-2-2 Signal Space Concepts
`4-2-3 Orthogonal Expansions of Signals
`4-3 Representation of Digitally Modulated Signals
`4-3-1 Memoryless Modulation Methods
`4-3-2 Linear Modulation with Memory
`4-3-3 Nonlinear Modulation Methods with Memory
`4-4 Spectral Characteristics of Digitally Modulated Signals
`4-4-1 Power Spectra of Linearly Modulated Signals
`4-4-2 Power Spectra of CPFSK and CPM Signals
`4-4-3 Power Spectra of Modulated Signals with Memory
`4-5 Bibliographical Notes and References
`Problems
`
`5 Optimum Receivers for the Additive White
`Gaussian Noise Channel
`5-1 Optimum Receiver for Signals Corrupted by AWGN
`5-1-1 Correlation Demodulator
`5-1-2 Matched-Filter Demodulator
`
`82
`82
`84
`87
`91
`93
`94
`103
`106
`108
`108
`113
`118
`125
`125
`136
`138
`144
`144
`
`152
`152
`153
`157
`157
`
`159
`163
`163
`165
`165
`173
`174
`186
`190
`203
`204
`209
`220
`223
`224
`
`233
`233
`234
`238
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 5
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 5
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`82
`g2
`84
`g7
`91
`93
`94
`103
`106
`108
`108
`H3
`H8
`125
`125
`136
`138
`144
`144
`
`1 S?
`
`152
`153
`'57
`157
`
`159
`163
`1^3
`165
`165
`173
`174
`186
`190
`203
`204
`209
`220
`223
`224
`
`233
`233
`234
`238
`
`CONTENTS
`
`xiii
`
`5-1-3 The Optimum Detector
`5-1-4 The Maximum-Likelihood Sequence Detector
`5-1-5 A Symbol-by-Symbol MAP Detector for Signals
`with Memory
`5-2 Performance of the Optimum Receiver for Memoryless
`Modulation
`5-2-1 Probability of Error for Binary Modulation
`5-2-2 Probability of Error for M-ary Orthogonal Signals
`5-2-3 Probability of Error for M-ary Biorthogonal Signals
`5-2-4 Probability of Error for Simplex Signals
`5-2-5 Probability of Error for M-ary Binary-Coded Signals
`5-2-6 Probability of Error for M-ary PAM
`5-2-7 Probability of Error for M-ary PSK
`5-2-8 Differential PSK (DPSK) and its Performance
`5-2-9 Probability of Error for QAM
`5-2-10 Comparison of Digital Modulation Methods
`5-3 Optimum Receiver for CPM Signals
`5-3-1 Optimum Demodulation and Detection of CPM
`5-3-2 Performance of CPM Signals
`5-3-3 Symbol-by-Symbol Detection of CPM Signals
`5-4 Optimum Receiver for Signals with Random Phase in AWGN
`Channel
`.
`5-4-1 Optimum Receiver for Binary Signals
`5-4-2 Optimum Receiver for M-ary Orthogonal Signals
`5-4-3 Probability of Error for Envelope Detection of M-ary
`Orthogonal Signals
`5-4-4 Probability of Error for Envelope Detection of Correlated
`Binary Signals
`5_5 Regenerative Repeaters and Link Budget Analysis
`5-5-1 Regenerative Repeaters
`5-5-2 Communication Link Budget Analysis
`5-6 Bibliographical Notes and References
`Problems
`
`6 Carrier and Symbol Synchronization
`6-1 Signal Parameter Estimation
`6-1-1 The Likelihood Function
`6-1-2 Carrier Recovery and Symbol Synchronization
`in Signal Demodulation
`6-2 Carrier Phase Estimation
`6-2-1 Maximum-Likelihood Carrier Phase Estimation
`6-2-2 The Phase-Locked Loop
`6-2-3 Effect of Additive Noise on the Phase Estimate
`6-2-4 Decision-Directed Loops
`6-2-5 Non-Decision-Directed Loops
`6-3 Symbol Timing Estimation
`6-3-1 Maximum-Likelihood Timing Estimation
`6-3-2 Non-Decision-Directed Timing Estimation
`
`244
`249
`
`254
`
`257
`257
`260
`264
`266
`266
`267
`269
`274
`278
`282
`284
`285
`290
`296
`
`301
`302
`308
`
`308
`
`312
`313
`314
`316
`319
`320
`
`333
`333
`335
`
`336
`337
`339
`341
`343
`347
`350
`358
`359
`361
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 6
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 6
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`CONTENTS
`
`6-4 Joint Estimation of Carrier Phase and Symbol Timing
`6-5 Performance Characteristics of ML Estimators
`6-6 Bibliographical Notes and References
`Problems
`
`Channel Capacity and Coding
`7-1 Channel Models and Channel Capacity
`7-1-1 Channel Models
`7-1-2 Channel Capacity
`7-1-3 Achieving Channel Capacity with Orthogonal Signals
`7-1-4 Channel Reliability Functions
`7-2 Random Selection of Codes
`7-2-1 Random Coding Based on M-ary Binary-Coded Signals
`7-2-2 Random Coding Based on M-ary Multiamplitude Signals
`7-2-3 Comparison of R$ with the Capacity of the AWGN
`Channel
`7-3 Communication System Design Based on the Cutoff Rate
`7-4 Bibliographical Notes and References
`Problems
`
`Block and Convolutional Channel Codes
`8-1 Linear Block Codes
`8-1-1 The Generator Matrix and the Parity Check Matrix
`8-1-2 Some Specific Linear Block Codes
`8-1-3 Cyclic Codes
`8-1-4 Optimum Soft-Decision Decoding of Linear Block Codes
`8-1-5 Hard-Decision Decoding
`8-1-6 Comparison of Performance between Hard-Decision and
`Soft-Decision Decoding
`8-1-7 Bounds on Minimum Distance of Linear Block Codes
`8-1-8 Nonbinary Block Codes and Concatenated Block Codes
`8-1-9
`Interleaving of Coded Data for Channels with Burst
`Errors
`8-2 Convolutional Codes
`8-2-1 The Transfer Function of a Convolutional Code
`8-2-2 Optimum Decoding of Convolutional Codes—
`The Viterbi Algorithm
`8-2-3 Probability of Error for Soft-Decision Decoding
`8-2-4 Probability of Error for Hard-Decision Decoding
`8-2-5 Distance Properties of Binary Convolutional Codes
`8-2-6 Nonbinary Dual-/c Codes and Concatenated Codes
`8-2-7 Other Decoding Algorithms for Convolutional Codes
`8-2-8 Practical Considerations in the Application of
`Convolutional Codes
`8-3 Coded Modulation for Bandwidth-Constrained Channels
`8-4 Bibliographical Notes and References
`Problems
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 7
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 7
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`365
`367
`370
`371
`
`374
`375
`375
`380
`387
`389
`390
`390
`397
`
`399
`400
`406
`406
`
`413
`413
`417
`421
`423
`436
`445
`
`456
`461
`464
`
`468
`470
`477
`
`|
`
`|
`|
`
`i
`
`I
`:
`I
`
`483
`I
`486
`b
`489
`492 |
`492
`I
`I
`500
`
`506
`511
`526
`528
`
`I
`I
`I
`I
`
`CONTENTS
`
`XV
`
`9 Signal Design for Band-Limited Channels
`9-1 Characterization of Band-Limited Channels
`9-2 Signal Design for Band-Limited Channels
`9-2-1 Design of Band-Limited Signals for No Intersymbol
`Interference—The Nyquist Criterion
`9-2-2 Design of Band-Limited Signals with Controlled ISI—
`Partial-Response Signals
`9-2-3 Data Detection for Controlled ISI
`9-2-4 Signal Design for Channels with Distortion
`9-3 Probability of Error in Detection of PAM
`9-3-1 Probability of Error for Detection of PAM with Zero ISI
`9-3-2 Probability of Error for Detection of Partial-Response
`Signals
`9-3-3 Probability of Error for Optimum Signals in Channel
`with Distortion
`9-4 Modulation Codes for Spectrum Shaping
`9-5 Bibliographical Notes and References
`Problems
`
`_
`_
`10 Communication through Band-Limited Linear
`Filter Channels
`10-1 Optimum Receiver for Channels with ISI and AWGN
`10-1-1 Optimum Maximum-Likelihood Receiver
`10-1-2 A Discrete-Time Model for a Channel with ISI
`10-1-3 The Viterbi Algorithm for the Discrete-Time White
`Noise Filter Model
`10-1-4 Performance of MLSE for Channels with ISI
`10-2 Linear Equalization
`10-2-1 Peak Distortion Criterion
`10-2-2 Mean Square Error (MSE) Criterion
`10-2-3 Performance Characteristics of the MSE Equalizer
`10-2-4 Fractionally Spaced Equalizer
`10-3 Decision-Feedback Equalization
`10-3-1 Coefficient Optimization
`10-3-2 Performance Characteristics of DFE
`10-3-3 Predictive Decision-Feedback Equalizer
`10-4 Bibliographical Notes and References
`Problems
`
`11 Adaptive Equalization
`11-1 Adaptive Linear Equalizer
`11-1-1 The Zero-Forcing Algorithm
`11-1-2 The LMS algorithm
`11-1-3 Convergence Properties of the LMS Algorithm
`11-1-4 Excess MSE Due to Noisy Gradient Estimates
`11-1-5 Baseband and Passband Linear Equalizers
`11-2 Adaptive Decision-Feedback Equalizer
`11-2-1 Adaptive Equalization of Trellis-Coded Signals
`
`534
`534
`540
`
`542
`
`548
`551
`557
`561
`561
`
`562
`
`565
`566
`576
`576
`
`583
`584
`584
`586
`
`589
`593
`601
`602
`607
`612
`617
`621
`621
`622
`626
`628
`628
`
`636
`636
`637
`639
`642
`644
`648
`649
`650
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 8
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 8
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`HL
`
`xvi
`
`11-3 An Adaptive Channel Estimator for ML Sequence Detection
`11-4 Recursive Least-Squares Algorithms for Adaptive Equalization
`11-4-1 Recursive Least-Squares (Kalman) Algorithm
`11-4-2 Linear Prediction and the Lattice Filter
`11-5 Self-Recovering (Blind) Equalization
`11-5-1 Blind Equalization Based on Maximum-Likelihood
`Criterion
`11-5-2 Stochastic Gradient Algorithms
`11-5-3 Blind Equalization Algorithms Based on Second-
`and Higher-Order Signal Statistics
`11-6 Bibliographical Notes and References
`Problems
`
`12 Multichannel and Multicarrier Systems
`12-1 Multichannel Digital Communication in AWGN Channels
`12-1-1 Binary Signals
`12-1-2 M-ary Orthogonal Signals
`12-2 Multicarrier Communications
`12-2-1 Capacity of a Non-Ideal Linear Filter Channel
`12-2-2 An FFT-Based Multicarrier System
`12-3 Bibiliographical Notes and References
`Problems
`
`13 Spread Spectrum Signals for Digital Communications
`13-1 Model of Spread Spectrum Digital Communication System
`13-2 Direct Sequence Spread Spectrum Signals
`13-2-1 Error Rate Performance of the Decoder
`13-2-2 Some Applications of DS Spread Spectrum Signals
`13-2-3 Effect of Pulsed Interference on DS Spread Spectrum
`Systems
`13-2-4 Generation of PN Sequences
`13-3 Frequency-Hoppped Spread Spectrum Signals
`13-3-1 Performance of FH Spread Spectrum Signals in AWGN
`Channel
`13-3-2 Performance of FH Spread Spectrum Signals in Partial-
`Band Interference
`13-3-3 A CDMA System Based on FH Spread Spectrum Signals
`13-4 Other Types of Spread Spectrum Signals
`13-5 Synchronization of Spread Spectrum Signals
`13-6 Bibliographical Notes and References
`Problems
`
`14 Digital Communication through Fading
`Multipath Channels
`14-1 Characterization of Fading Multipath Channels
`14-1-1 Channel Correlation Functions and Power Spectra
`14-1-2 Statistical Models for Fading Channels
`
`652
`654
`656
`660
`664
`
`664
`668
`
`673
`675
`676
`
`680
`680
`682
`684
`686
`687
`689
`692
`693
`
`695
`697
`698
`702
`712
`
`717
`724
`729
`
`732
`
`734
`741
`743
`744
`752
`753
`
`758
`759
`762
`767
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 9
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 9
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`652
`654
`656
`660
`664
`
`664
`668
`
`673
`675
`676
`
`680
`680
`682
`684
`686
`687
`689
`692
`693
`
`695
`697
`698
`702
`712
`
`717
`724
`729
`
`732
`
`734
`741
`743
`744
`752
`753
`
`758
`759
`762
`767
`
`CONTENTS XVli
`
`14-2 The Effect of Characteristics on the Choice
`of a Channel Model
`14-3 Frequency-Nonselective, Slowly Fading Channel
`14-4 Diversity Techniques for Fading Multipath Channels
`14-4-1 Binary Signals
`14-4-2 Multiphase Signals
`14-4-3 M-ary Orthogonal Signals
`14-5 Digital Signaling over a Frequency-Selective, Slowly Fading
`Channel
`14-5-1 A Tapped-Delay-Line Channel Model
`14-5-2 The RAKE Demodulator
`14-5-3 Performance of RAKE Receiver
`14-6 Coded Waveforms for Fading Channels
`14-6-1 Probability of Error for Soft-Decision Decoding of Linear
`Binary Block Codes
`14-6-2 Probability of Error for Hard-Decision Decoding of
`Linear Binary Block Codes
`14-6-3 Upper Bounds on the Performance of Convolutional
`Codes for a Raleigh Fading Channel
`14-6-4 Use of Constant-Weight Codes and Concatenated Codes
`for a Fading Channel
`14-6-5 System Design Based on the Cutoff Rate
`14-6-6 Trellis-Coded Modulation
`14-7 Bibliographical Notes and References
`Problems
`
`15 Multiuser Communications
`15-1
`Introduction to Multiple Access Techniques
`15-2 Capacity of Multiple Access Methods
`15-3 Code-Division Multiple Access
`15-3-1 CDMA Signal and Channel Models
`15-3-2 The Optimum Receiver
`15-3-3 Suboptimum Detectors
`15-3-4 Performance Characteristics of Detectors
`15-4 Random Access Methods
`15-4-1 ALOHA System and Protocols
`15-4-2 Carrier Sense Systems and Protocols
`15-5 Bibliographical Notes and References
`Problems
`
`Appendix A The Levinson-Durbin Algorithm
`
`Appendix B Error Probability for Multichannel
`Binary Signals
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 10
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 10
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`XV111
`
`CONTENTS
`
`Appendix C Error Probabilities for Adaptive Reception
`of M-phase Signals
`C-l Mathematical Model for an M-phase Signaling
`Communications System
`C-2 Characteristic Function and Probability Density
`Function of the Phase 6
`C-3 Error Probabilities for Slowly Rayleigh Fading
`Channels
`C-4 Error Probabilities for Time-Invariant and Ricean
`Fading Channels
`
`Appendix D Square-Root Factorization
`References and Bibliography
`Index
`
`887
`
`887
`
`889
`
`891
`
`893
`
`897
`
`899
`
`917
`
`Petitioner Sirius XM Radio Inc. - Ex. 1008, p. 11
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 11
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`486
`
`DIGITAL COMMUNICATIONS
`
`the same node at t — S. It has been found experimentally (computer simula­
`tion) that a delay 8^5K results in a negligible degradation in the performance
`relative to the optimum Viterbi algorithm.
`
`8-2-3 Probability of Error for Soft-Decision Decoding
`
`The topic of this subsection is the error rate performance of the Viterbi
`algorithm on an additive white gaussian noise channel with soft-decision
`
`decoding.
`In deriving the probability of error for convolutional codes, the linearity
`
`property for this class of codes is employed to simplify the derivation. That is,
`we assume that the all-zero sequence is transmitted and we determine the
`probability of error in deciding in favor of another sequence. The coded binary
`the convolutional code, denoted as {cjm>
`digits for
`the y'th branch of
`m = 1, 2,..., n} and defined in Section 8-2-2, are assumed to be transmitted by
`binary PSK (or four-phase PSK) and detected coherently at the demodulator.
`The output of the demodulator, which is the input to the Viterbi decoder, is
`the sequence {rjm, m = 1,2,..., n\ j = 1, 2,...} where rjm is defined in (8-2-9).
`The Viterbi soft-decision decoder forms the branch metrics defined by
`(8-2-14) and from these computes the path metrics
`
`CM(i) = 2 (0 . = 2
`
`2 r j m(2c$- 1)
`
`(8-2-16)
`
`7 = 1
`
`where i denotes any one of the competing paths at each node and B is the
`number of branches (information symbols) in a path. For example, the all-zero
`path, denoted as i = 0, has a path metric
`
`cm<°> = 2 2 (•
`7 = 1 m = 1
`
`-V¥c + n j m)(-l)
`
`= Vf cBn + 2 2
`
`-2-17)
`
`length, we
`Since the convolutional code does not necessarily have a fixed
`derive its performance from the probability of error for sequences that merge
`with the all-zero sequence for the first
`time at a given node in the trellis. In
`particular, we define the first-event
`error probability as the probability that
`another path that merges with the all-zero path at node B has a metric that
`exceeds the metric of the all-zero path for the first time. Suppose the incorrect
`path, call it i = 1, that merges with the all-zero path differs from the all-zero
`path in cl bits, i.e., there are d Is in the path i = 1 and the rest are Os. The
`probability of error in the pairwise comparison of the metrics CM<0) and CM
`
`is
`
`P2(d) = P(CM W
`
`CM m) = P(CM W - CM m 5= 0)
`
`P2(d) =
`
`22 2
`L
`/=1 m = 1
`
`(1) _ ^(0)-
`„(c
`jm
`
`)5=0
`
`(8-2-18)
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 12
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`CHAPTER 8: BLOCK AND CONVOLUTIONAL CHANNEL CODES
`
`487
`
`Since the coded bits in the two paths are identical except in the d positions,
`(8-2-18) can be written in the simpler form
`
`\ / = i
`
`/
`
`(8-2-19)
`
`where the index / runs over the set of cl bits in which the two paths differ and
`the set {/•/} represents the input to the decoder for these d bits.
`The {/-/} are independent and identically distributed gaussian random
`variables with mean —V% and variance %N0. Consequently the probability of
`error in the pairwise comparison of these two paths that differ in d bits is
`
`P2(d) = Q { ^ ~ d )
`
`= Q(\/2y hR cd)
`
`(8-2-20)
`
`where yb = %,/N 0 is the received SNR per bit and R c is the code rate.
`Although we have derived the first-event error probability for a path of
`distance d from the all-zero path, there are many possible paths with different
`distances that merge with the all-zero path at a given node B. In fact, the
`transfer function T(D) provides a complete description of all the possible
`paths that merge with the all-zero path at node B and their distances. Thus we
`can sum the error probability in (8-2-20) over all possible path distances. Upon
`performing this summation, we obtain an upper bound on the first-event error
`probability in the form
`•
`
`GO
`Pc55 S a dP 2(d)
`lt=ll free
`CO
`
`f"
`
`« 2 a dQ(S2y bR cd)
`d=dfTCC
`
`(8-2-21)
`
`where a d denotes the number of paths of distance d from the all-zero path that
`merge with the all-zero path for the first time.
`
`There are two reasons why (8-2-21) is an upper bound on the first-event
`error probability. One is that the events that result in the error probabilities
`{Pz(d)} are not disjoint. This can be seen from observation of the trellis.
`Second, by summing over all possible d
`dCree, we have implicitly assumed that
`the convolutional code has infinite length. If the code is truncated periodically
`after B nodes, the upper bound in (8-2-21) can be improved by summing the
`error events for dfree =£ (/ =s B. This refinement has some merit in determining
`the performance of short convolutional codes, but the effect on performance is
`negligible when B is large.
`
`The upper bound in (8-2-21) can be expressed in a slightly different form if
`the Q function is upper-bounded by an exponential. That is,
`
`Q(S2y hR cd) ^ e ~ ^ " = D"\ D = e- n*
`
`(8-2-22)
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 13
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`488 DIGITAL COMMUNICATIONS
`
`If we use (8-2-22) in (8-2-21), the upper bound on the first-event
`
`error
`
`probability can be expressed as
`
`Pe<T{D)\ D = e-y b K c
`
`(8-2-23)
`
`Although the first-event
`
`error probability provides a measure of the
`
`performance of a convolutional code, a more useful measure of performance is
`the bit error probability. This probability can be upper-bounded by the
`procedure used in bounding the first-event
`error probability. Specifically, we
`know that when an incorrect path is selected, the information bits in which the
`selected path differs from the correct path will be decoded incorrectly. We also
`know that the exponents in the factor N contained in the transfer function
`T(D, N) indicate the number of information bit errors (number of Is) in
`selecting an incorrect path that merges with the all-zero path at some node B.
`If we multiply the pairwise error probability P2(d) by the number of incorrectly
`decoded information bits for the incorrect path at the node where they merge,
`we obtain the bit error rate for that path. The average bit error probability is
`upper-bounded by multiplying each pairwise error probability P2(d) by the
`corresponding number of incorrectly decoded information bits, for each
`possible incorrect path that merges with the correct path at the Bth node, and
`
`summing over all d.
`The appropriate multiplication factors corresponding to the number of
`information bit errors for each incorrectly selected path may be obtained by
`differentiating T(D, N) with respect to N. In general, T(D,N) can be
`
`expressed as
`
`CO
`
`T(D,N)= 2 a dD dN f { d )
`t d=dlrcc
`
`(8-2-24)
`
`where f(d) denotes the exponent of N as a function of d. Taking the derivative
`of T(D, N) with respect to N and setting N = 1, we obtain
`
`"'V
`
`n=\
`
`d=d{rcc
`
`= J P„D"
`d=dfrcc
`
`(8-2-25)
`
`where pd =
`
`a df(d). Thus the bit error probability for k = 1 is upper-bounded
`
`by
`
`CO
`
`P b< 2 PcW)
`d=d free
`
`oo
`
`< 2 /3(,Q(V2^M)
`d=dltcc
`
`(8-2-26)
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 14
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`t-event error
`
`(8-2-23)
`
`asure of the
`erformance is
`nded by the
`jecifically, we
`
`s in which the
`sctly. We also
`isfer function
`ber of Is) in
`some node B.
`of incorrectly
`•e they merge,
`
`probability is
`P2(d) by the
`)its, for each
`Bth node, and
`
`le number of
`ie obtained by
`D, N) can be
`
`(8-2-24)
`
`the derivative
`
`(8-2-25)
`
`upper-bounded
`
`(8-2-26)
`
`CHAPTER 8: BLOCK AND CONVOLUTIONAL CHANNEL CODES
`
`489
`
`If the Q function is upper-bounded by an exponential as indicated in (8-2-22)
`then (8-2-26) can be expressed in the simple form
`
`P„< 2 P,D
`d—cl[KC
`
`0=e-«,«c
`
`dT(D, N)
`< -
`
`dN
`
`/V=l,D=e-w>«c
`
`(8-2-27)
`
`If k>l, the equivalent bit error probability is obtained by dividing (8-2-26)
`and (8-2-27) by k.
`
`The expressions for the probability of error given above are based on the
`assumption that the code bits are transmitted by binary coherent PSK. The
`results also hold for four-phase coherent PSK, since this modulation/
`
`demodulation technique is equivalent to two independent (phase-quadrature)
`binary PSK systems. Other modulation and demodulation techniques, such as
`coherent and noncoherent binary FSK, can be accommodated by recomputing
`the pairwise error probability P2(d). That is, a change in the modulation and
`demodulation technique used to transmit the coded information sequence
`affects only the computation of P2{d). Otherwise, the derivation for Ph remains
`the same.
`Although the above derivation of the error probability for Viterbi decoding
`of a convolutional code applies to binary convolutional codes, it is relatively
`
`easy to generalize it to nonbinary convolutional codes in which each nonbinary
`symbol is mapped into a distinct waveform. In particular, the coefficients {/3r/}
`in the expansion of the derivative of T(D, N), given in (8-2-25), represent the
`number of symbol errors in two paths separated in distance (measured in terms
`of symbols) by d symbols. Again, we denote the probability of error in a
`pairwise comparison of two paths thafare separated in distance by d as P2{d).
`Then the symbol error probability, for a /c-bit symbol, is upper-bounded by
`
`CO
`
`Pm 2 PdPiid)
`
`d—d(rcc
`
`The symbol error probability can be converted into an equivalent bit error
`probability. For example, if 2k orthogonal waveforms are used to transmit the
`&-bit symbols, the equivalent bit error probability is PM multiplied by a factor
`2^-1/(2* _
`as shown in Chapter 5.
`
`8-2-4
`
`Probability of Error for Hard-Decision Decoding
`
`We now consider the performance achieved by the Viterbi decoding algorithm
`
`on a binary symmetric channel. For hard-decision decoding of the convolu­
`tional code, the metrics in the Viterbi algorithm are the Hamming distances
`between the received sequence and the
`surviving sequences at each
`node of the trellis.
`
`As in our treatment of soft-decision decoding, we begin by determining the
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 15
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`Iakagami-ni
`blem 14-15)
`
`(14-3-14)
`
`methods is
`a nonfading
`
`mi-rn fading
`with in as
`We observe
`1, which is
`hand, when
`
`nribed above,
`methods for
`
`can be found
`statistics, the
`
`m .2.5
`
`4
`
`30
`
`(d13)
`
`CHAPTER 14: DIGITAL COMMUNICATION THROUGH FADING MULTIPATH CHANNELS
`
`777
`
`reader may refer to the papers by Esposito (1967), Miyagaki et al. (1978),
`Charash (1979), Al-Hussaini et al. (1985), and Beaulieu et al. (1991).
`
`14-4 DIVERSITY TECHNIQUES FOR FADING
`MULTIPATH CHANNELS
`
`Diversity techniques are based on the notion that errors occur in reception
`when the channel attenuation is large, i.e., when the channel is in a deep fade.
`If we can supply to the receiver several replicas of the same information signal
`transmitted over independently fading channels, the probability that all the
`signal components will fade simultaneously is reduced considerably. That is, if
`p is the probability that any one signal will fade below some critical value then
`p L is the probability that all L independently fading replicas of the same signal
`will fade below the critical value. There are several ways in which we can
`provide the receiver with L independently fading replicas of the same
`information-bearing signal.
`One method is to employ frequency diversity. That is, the same information-
`bearing signal is transmitted on L carriers, where the separation between
`successive carriers equals or exceeds the coherence bandwidth (AD, of the
`channel.
`A second method for achieving L independently fading versions of the same
`information-bearing signal is to transmit the signal in L different time slots,
`where the separation between successive time slots .equals or exceeds the
`coherence time (/.1t),.. of the channel. This method is called time diversity.
`Note that the fading channel fits the model of a bursty error channel.
`Furthermore, we may view the transmission of the same information either at
`different frequencies or in difference time slots (or both) as a simple form of
`repetition coding. The separation of the diversity transmissions in time by (At),
`or in frequency by (4), is basically a form of block-interleaving the bits in the
`repetition code in an attempt to break up the error bursts and, thus, to obtain
`independent errors. Later in the chapter, we shall demonstrate that, in general,
`repetition coding is wasteful of bandwidth when compared with nontrivial
`coding.
`Another commonly used method for achieving diversity employs multiple
`antennas. For example, we may employ a single transmitting antenna and
`multiple receiving antennas. The fatter must be spaced sufficiently far apart
`that the multipath components in the signal have significantly different
`propagation delays at the antennas. Usually a separation of at least 10
`wavelengths is required between two antennas in order to obtain signals that
`fade independently.
`A more sophisticated method for obtaining diversity is based on the use of a
`signal having a bandwidth much greater than the coherence bandwidth (AA of
`the channel. Such a signal with bandwidth W will resolve the multipath
`components and, thus, provide the receiver with several independently fading
`signal paths. The time resolution is 11W. Consequently, with a multipath
`
`Petitioner Sirius XM Radio Inc. – Exhibit 1031, p. 16
`Sirius XM v. Fraunhofer – IPR2018-00690
`U.S. Patent No. 6,314,289
`
`

`

`778
`
`DIGITAL COMMUNICATIONS
`
`spread of Ti„ s, there are T,„W resolvable signal components. Since 7;„
`1/(g)„ the number of resolvable signal components may also be expressed a
`vv/(6,f),. Thus, the use of a wideband signal may be viewed as just anothe
`method for obtaining frequency diversity of order L vv/(tine. The optirnui
`receiver for processing the wideband signal will be derived in Section 14-5. It i'
`called a RAKE correlatar or a RAKE matched filter and was invented by Pric
`and Green (1958).
`There are other diversity techniques that have received some consideratio
`in practice, such as angle-of-arrival diversity and polarization diversity
`However, these have not been as widely used as those described above.
`
`14-4-1 Binary Signals
`
`We shall now determine the error rate performance for a binary digit
`communications system with diversity. We begin by describing the mathemat
`cal model for the communications system with diversity. First of all, we assum
`that there are L diversity channels, carrying the same information-bearin
`signal. Each channel is assumed to be frequency-nonselective and slowly fadi
`with Rayleigh-distributed envelope statistics. The fading processes among t
`L diversity channels are assumed to be mutually statistically independent. T
`signal in each channel is corrupted by an additive zero-mean white gaussi
`noise process. The noise processes in the L channels are assumed to b
`mutually statistically independent, with identical autocorrela