A practical guide to error-control coding using Matlab

293 Pages·2010·11.149 MB·English

by  Yuan Jiang

A Practical Guide to Error-Control
Coding Using MATLAB®
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The technical descriptions, procedures, and computer programs in this book
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author in a broad range of applications; however, they are provided as is,
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of the book titled A Practical Guide to Error-Control Coding Using MATLAB®
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claim all liability for direct, incidental, or consequent damages resulting from
use of the programs or procedures in this book or the associated software.
A Practical Guide to Error-Control
Coding Using MATLAB®
Yuan Jiang
artechhouse.com
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A catalog record for this book is available from the U.S. Library of Congress.
British Library Cataloguing in Publication Data
A catalogue record for this book is available from the British Library.
ISBN 13: 978-1-60807-088-6
Cover design by Patrick McCarthy
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10 9 8 7 6 5 4 3 2 1
Contents
Preface ix
1 Error Control in Digital Communications
and Storage 1
1.1 Error Control Coding at a Glance 1
1.1.1 Codes for Error Control 1
1.1.2 Important Concepts 5
1.2 Channel Capacity and Shannon’s Theorem 14
1.3 Considerations When Selecting
Coding Schemes 16
References 17
Selected Bibliography 17
2 Brief Introduction to Abstract Algebra 19
2.1 Elementary Algebraic Structures 19
2.1.1 Group 19
2.1.2 Field 22

i A Practical Guide to Error-Control Coding Using MATLAB®
2.2 Galois Field and Its Arithmetic 25
2.2.1 Galois Field 25
2.2.2 Arithmetic in GF (2m) 33
2.3 Implementation of GF (2m) Arithmetic 36
2.3.1 Arithmetic with Polynomial Representation 36
2.3.2 Arithmetic with Power Representation 39
2.3.3 A Special Case: Inversion 40
Problems 43
References 44
Selected Bibliography 44
3 Binary Block Codes 45
3.1 Linear Block Codes 45
3.1.1 Code Construction and Properties 45
3.1.2 Decoding Methods 51
3.1.3 Performance of Linear Block Codes 60
3.1.4 Encoder and Decoder Designs 65
3.1.5 Hamming Codes 66
3.2 Cyclic Codes 73
3.2.1 Basic Principles 74
3.2.2 Shift Register–Based Encoder and Decoder 81
3.2.3 Shortened Cyclic Codes and CRC 91
3.3 BCH Codes 95
3.3.1 Introduction 97
3.3.2 BCH Bound and Vandermonde Matrix 100
3.3.3 Decoding BCH Codes 101
Problems 110
References 111
Selected Bibliography 112
Contents ii
4 Reed-Solomon Codes 113
4.1 Introduction to RS Codes 113
4.1.1 Prelude: Nonbinary BCH Codes 113
4.1.2 Reed-Solomon Codes 117
4.2 Decoding of RS Codes 123
4.2.1 General Remarks 123
4.2.2 Determining the Error Location Polynomial 124
4.2.3 Frequency-Domain Decoding 135
4.2.4 Error and Erasure Decoding 140
4.3 RS Decoder: From Algorithm to
Architecture 143
4.3.1 Syndrome Computation Circuit 143
4.3.2 Architectures for Berlekamp-Massey
Algorithm 143
4.3.3 Circuit for Chien Search and Forney’s
Algorithm 149
4.4 Standardized RS Codes 149
Problems 150
References 151
5 Convolutional Codes 153
5.1 Fundamentals of Convolutional Codes 153
5.1.1 Code Generation and Representations 153
5.1.2 Additional Matters 161
5.2 Decoding of Convolutional Codes 165
5.2.1 Optimum Convolutional Decoding
and Viterbi Algorithm 166
5.2.2 Sequential Decoding 179
iii A Practical Guide to Error-Control Coding Using MATLAB®
5.3 Designing Viterbi Decoders 189
5.3.1 Typical Design Issues 189
5.3.2 Design for High Performance 197
5.4 Good Convolutional Codes 201
5.4.1 Catastrophic Error Propagation 202
5.4.2 Some Known Good Convolutional Codes 202
5.5 Punctured Convolutional Codes 202
Problems 210
References 210
Selected Bibliography 212
6 Modern Codes 213
6.1 Turbo Codes 213
6.1.1 Code Concatenation 213
6.1.2 Concatenating Codes in Parallel: Turbo Code 218
6.1.3 Iterative Decoding of Turbo Codes 228
6.1.4 Implementing MAP 250
6.2 Low-Density Parity-Check Codes 252
6.2.1 Codes with Sparse Parity-Check Matrix 254
6.2.2 Decoding and Encoding Algorithms 259
6.2.3 High-Level Architecture Design for
LDPC Decoders 270
Problems 272
References 274
Selected Bibliography 276
About the Author 277
Index 279
Preface
This book attempts to provide a comprehensible and practical introduction
to error control coding. The targeted readers are practicing engineers and
university students who have already set foot in this territory or plan to. To
achieve the goal, this book takes an approach that is somewhat different from
the approaches used by the many excellent textbooks currently available.
First, the book introduces MATLAB as a tool to facilitate the pre-
sentation of key concepts. The DVD that accompanies this book provides
more than 90 MATLAB programs with which readers can experiment. It is
the author’s hope that this fresh attempt does help readers in mastering the
subject.
Second, the book pays attention to the implementation of various de-
coding algorithms. Readers will find that a few practical issues have received
in-depth treatment in the book, such as implementation of Galois field arith-
metic, Viterbi decoder design, RS decoder design, and MAP architecture, to
name a few.
The organization of the book is standard. Readers may notice, however,
that many mathematical proofs and theorems have been omitted. This is
because this book emphasizes concepts and rationales. For those who wish
to explore further, a comprehensive list of references is given at the end of
each chapter. Note that the MATLAB functions marked with asterisks are
provided by the book not by the MATLAB software.
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