The Engineer’s Error Coding Handbook

259 Pages·1996·6.922 MB·English

by  A. D. Houghton (auth.)

The Engineer's Error
Coding Handbook
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The Engineer's Error
Coding Handbook
A.D. Houghton
Department of Electronic and Electrical Engineering
The University of Sheffield
Sheffield
UK
CHAPMAN & HALL
London· Weinheim . New York· Tokyo· Melbourne· Madras
Published by Chapinan & Hall, 2-6 Boundary Row, London SE18HN, UK
Chapman & Hall, 2-6 Boundary Row, London SEI 8HN, UK
Chapman & Hall GmbH, Pappelallee 3,69469 Weinheim, Germany
Chapman & Hall USA, 115 Fifth Avenue, New York, NY 10003, USA
Chapman & Hall Japan, ITP-Japan, Kyowa Building, 3F, 2-2-1 Hirakawacho,
Chiyoda-ku, Tokyo 102, Japan
Chapman & Hall Australia, 102 Dodds Street, South Melbourne, Victoria 3205,
Australia
Chapman & Hall India, R. Seshadri, 32 Second Main Road, CIT East, Madras
600 035, India
First edition 1997
© 1997 A.D. Houghton
ISBN -13: 978-0-412-79070-6 e-ISBN -13 :978-1-4613-0447-0
DOl: 10.1007/978-1-4613-0447-0
Apart from any fair dealing for the purposes of research or private study, or
criticism or review, as permitted under the UK Copyright Designs and Patents
Act, 1988, this publication may not be reproduced, stored, or transmitted, in any
form or by any means, without the prior permission in writing of the publishers,
or in the case of reprographic reproduction only in accordance with the terms of
the licences issued by the Copyright Licensing Agency in the UK, or in
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Rights Organization outside the UK. Enquiries concerning reproduction outside
the terms stated here should be sent to the publishers at the London address
printed on this page.
The publisher makes no representation, express or implied, with regard to the
accuracy of the information contained in this book and cannot accept any legal
responsibility or liability for any errors or omissions that may be made.
A catalogue record for this book is available from the British Library
8
Printed on permanent acid-free text paper, manufactured in accordance with
ANSIINISO Z39.48-1992 and ANSIINISO Z39.48-1984 (Permanence of
Paper).
Contents
Preface xi
Glossary of terms xiii
PART ONE ERROR DETECTION
1 Introduction 3
1.1 Redundancy 3
1.2 Orthogonality 4
1.3 Data packaging 5
1.4 Summary 6
2 Parity checking 7
2.1 The horizontal and vertical parity check 7
2.2 Generating the parity bit 8
2.3 Discussion 11
3 Cyclic redundancy checking 12
3.1 Ground rules and terms 12
3.2 An overview of the CRC 13
3.3 The generator polynomial 14
3.4 Finite fields 15
3.5 Longhand calculation of the CRC 18
3.6 Performance 20
3.7 Hardware implementation 21
3.8 Discussion 24
PART TWO ERROR CORRECTION, AN INTRODUCTION 25
4 Hamming codes 27
4.1 Correcting a single bit 29
4.2 Extending the message size 31
4.3 Discussion 34
vi Contents
5 Error correction using the CRC 35
5.1 Locating single-bit errors 35
5.2 A hardware error locator 37
5.3 Multiple-bit errors 39
5.4 Expurgated codes 40
5.5 Soft decision decoding 41
5.6 Discussion 43
PART THREE TIME DOMAIN REED-SOLOMON CODING 45
6 Correcting errors in the time domain 47
6.1 Manipulating fInite fIeld elements 47
6.2 Introduction to the time domain 57
6.3 Calculating check symbols for one error 57
6.4 Example of a one-symbol error correction 59
6.5 Discussion 61
7 Correcting two symbols 62
7.1 Correcting by erasure 62
7.2 Finding check symbols for two errors 63
7.3 Example of a two-symbol error correction 69
7.4 Discussion 72
PART FOUR FREQUENCY
DOMAIN REED-SOLOMON CODING 75
8 Frequency domain coding 77
8.1 Calculating a Fourier transform 77
8.2 Using the frequency domain to encode a message 79
8.3 Correcting errors in the frequency domain 81
8.4 How does recursive extension work? 88
8.5 Correction limits 92
8.6 Summary of frequency domain coding 93
8.7 An example over GF(28) 95
8.8 Exceeding the error capacity of the code 100
8.9 Discussion 102
9 Mixed domain error coding 104
9.1 Advantages of time domain processing 104
Contents vii
9.2 Calculating the errors in the frequency domain 104
9.3 Finding the check symbols using matrix inversion 106
9.4 Finding the check symbols using the FFT 109
9.5 Discussion 112
PART FIVE PACKAGING THE DATA 113
10 Block-interleaving 115
10.1 Discussion 120
11 Multi-dimensional data structures 121
11.1 Multi-dimensional time domain coding 121
11.2 Two-dimensional frequency domain coding 125
11.3 Punctured codes 132
11.4 Discussion 133
12 Convolutional coding 135
12.1 Finding the correct path 138
12.2 Other convolutional coders 140
12.3 Discussion 141
12.4 Symbol-based convolutional coding 142
PART SIX PRACTICAL CONSIDERATIONS 153
13 Manipulating elements with hardware 155
13.1 Multiplying elements using XOR gates 155
13.2 A hardware time domain encoder/decoder 161
13.3 General hardware considerations 165
13.4 Bit error rates 168
14 Applications 172
14.1 Example application one 173
14.2 Example application two 176
14.3 Some real applications 179
14.4 Summary of ground covered 182
14.5 Bibliography 183
viii Contents
PART SEVEN EXERCISES 185
15 Questions 187
15.1 Parity 187
15.2 Cyclic redundancy checking 188
15.3 Manipulating finite field elements 189
15.4 Convolutional coding 192
16 Solutions 194
16.1 Parity 194
16.2 Cyclic redundancy checking 195
16.3 Manipulating finite field elements 201
16.4 Convolutional coding 210
Appendix A Some primitive polynomials 212
Appendix B Solutions to some key equations 217
Appendix C Frequency domain example 221
Appendix D Time domain error correction in three dimensions 242
Appendix E Correcting two-dimensional data in the frequency
domain 245
Appendix F Example of mixed domain error correction 249
Index 252
Dedicated to my parents
Hugh and Hazel Houghton