Fundamentals of Convolutional Coding

443 Pages·1999·4.174 MB·English

by  Rolf Johannesson, Kamil Sh. Zigangirov

Fundamentals
of Convolutional
Coding
ROLF JOHANNESSON
KAMIL Sh. ZIGANGIROV
John B. Anderson, Series Editor
IEEE Press
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Piscataway, NJ 08855-1331
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Roger F. Hoyt, Editor in Chief
J. B. Anderson A. H. Haddad M. Padgett
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Kenneth Moore, Director of IEEE Press
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Marilyn G. Catis, Assistant Editor
IEEE Communications Society, Sponsor
COMM-S Liaison to IEEE Press, Salah Aidarous
Information Theory Society, Sponsor
IT-S Liaison to IEEE Press, Sanjeev Kulkarni
Vehicular Technology Society, Sponsor
VT-S Liaison to IEEE Press, J. R. Cruz
Cover photo: After Gottfried Ungerboeck, Courtesy of IBM,
Zurich Laboratory, Switzerland
Cover design: William T. Donnelly, WT Design
Technical Reviewers
Daniel J. Costello, Jr., University of Notre Dame
John Proakis, Northeastern University
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FUNDAMENTALS OF
CONVOLUTIONAL CODING
Rolf Johannesson
Lund University, Sweden
Kamil Sh. Zigangirov
Lund University, Sweden
IEEE Communications Society, Sponsor
IEEE Information Theory Society, Sponsor
IEEE Vehicular Technology Society, Sponsor
IEEE
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Printed in the United States of America
10 9 8 7 6 5 4 3 2 1
ISBN 0-7803-3483-3
IEEE Order Number: PC5739
Library of Congress Cataloging-in-Publication Data
Johannesson, Rolf, 1946-
Fundamentals of convolutional coding / Rolf Johannesson, Kamil Sh.
Zigangirov.
p. cm. -- (IEEE Press series on digital and mobile
communication)
"IEEE Communications Society, sponsor. IEEE Information Theory
Society, sponsor. IEEE Vehicular Technology Society, sponsor."
Includes bibliographical references and index.
ISBN 0-7803-3483-3
1. Coding theory. 2. Convolutions (Mathematics). 3. Error-
correcting codes (Information theory) I. Zigangirov, K. Sh.
II. IEEE Communications Society. III. IEEE Information Theory
Society. IV. Vehicular Technology Society. V. Title. VI. Series.
TK5102.92.J64 1998
003' .54--dc21 98-36706
CIP
To
Regina, Katrin, Peter, and Hanna
and
Ira, Dima, and Valja

Contents
PREFACE xi
ACKNOWLEDGMENTS xiii
CHAPTER 1 INTRODUCTION 1
1.1 Why Error Control? 1
1.2 Block Codes-A Primer 6
1.3 A First Encounter with Convolutional Codes 16
1.4 Block Codes versus Convolutional Codes 21
1.5 Capacity Limits and Potential Coding Gain Revisited 23
1.6 Comments 25
Problems 26
CHAPTER 2 CONVOLUTIONAL ENCODERS-
STRUCTURAL PROPERTIES 31
2.1 Convolutional Codes and Their Encoders 31
2.2 The Smith Form of Polynomial Convolutional
Generator Matrices 38
2.3 Encoder Inverses 45
2.4 Equivalent and Basic Encoding Matrices 52
2.5 Minimal-Basic Encoding Matrices 55
2.6 Minimal Encoding Matrices and Minimal Encoders 61
2.7 Canonical Encoding Matrices* 73
2.8 Minimality via the Invariant-Factor Theorem* 87
vii
viii Contents
2.9 Syndrome Formers and Dual Encoders 91
2.10 Systematic Convolutional Encoders 96
2.11 Comments 103
Problems 103
CHAPTER 3 DISTANCE PROPERTIES
OF CONVOLUTIONAL CODES 109
3.1 Distance Measures-A First Encounter 109
3.2 Active Distances 117
3.3 Properties of Convolutional Codes
via the Active Distances 123
3.4 Lower Bound on the Distance Profile 128
3.5 Upper Bounds on the Free Distance 132
3.6 Time-Varying Convolutional Codes 136
3.7 Lower Bound on the Free Distance 139
3.8 Lower Bounds on the Active Distances* 143
3.9 Distances of Cascaded Concatenated Codes* 149
3.10 Path Enumerators 153
3.11 Comments 158
Problems 159
CHAPTER 4 VITERBI DECODING 163
4.1 The Viterbi Algorithm Revisited 163
4.2 Error Bounds for Time-Invariant Convolutional Codes 168
4.3 Tighter Error Bounds for Time-Invariant
Convolutional Codes 181
4.4 Upper Bounds on the Output Error Burst Lengths 186
4.5 Error Bounds for Periodically Time-Varying
Convolutional Codes 195
4.6 Lower Error Bounds for Convolutional Codes 203
4.7 Error Bounds for Time-Varying Convolutional Codes 211
4.8 Error Bounds for Finite Back-Search Limits 220
4.9 Tailbiting Trellises 223
4.10 Quantization of Channel Outputs 230
4.11 Comments 233
Problems 234
CHAPTER 5 LIST DECODING 239
5.1 List Decoding Algorithms 239
5.2 List Decoding-Performance 242
5.3 The List Minimum Weight 247
5.4 Upper Bounds on the Probability of Correct Path Loss 255
Contents Ix
5.5 Lower Bound on the Probability of Correct Path Loss 261
5.6 Correct Path Loss for Time-Invariant
Convolutional Codes 264
5.7 Comments 266
Problems 267
CHAPTER 6 SEQUENTIAL DECODING 269
6.1 The Fano Metric 269
6.2 The Stack Algorithm 274
6.3 The Fano Algorithm 276
6.4 The Creeper Algorithm* 278
6.5 Simulations 288
6.6 Computational Analysis of the Stack Algorithm 289
6.7 Error Probability Analysis of the Stack Algorithm 296
6.8 Analysis of the Fano Algorithm 305
6.9 Analysis of Creeper* 310
6.10 Comments 313
Problems 313
CHAPTER 7 ITERATIVE DECODING 317
7.1 Iterative Decoding-A Primer 317
7.2 The Two-Way Algorithm for APP Decoding 321
7.3 The Two-Way Algorithm for Tailbiting Trellises 330
7.4 The One-Way Algorithm for APP Decoding 334
7.5 Low-Density Parity-Check Convolutional Codes 337
7.6 Comments 344
Problems 344
CHAPTER 8 CONVOLUTIONAL CODES
WITH GOOD DISTANCE PROPERTIES 347
8.1 Computing the Distance Spectrum 347
8.2 Some Classes of Rate R = 1/2 Convolutional Codes 351
8.3 Low-Rate Convolutional Codes 357
8.4 High-Rate Convolutional Codes 360
8.5 Tailbiting Trellis Encoders 362
8.6 Comments 370
CHAPTER 9 MODULATION CODES 371
9.1 Bandlimited Channels and QAM 373
9.2 Coding Fundamentals 378
9.3 Lattice-Type Trellis Codes 384
x Contents
9.4 Geometrically Uniform Trellis Codes 384
9.5 Decoding of Modulation Codes 387
9.6 Comments 388
Problems 389
APPENDIX A MINIMAL ENCODERS 393
APPENDIX B WALD'S IDENTITY 399
BIBLIOGRAPHY 407
INDEX 419
ABOUT THE AUTHORS 427