ERROR CONTROL CODING ERROR CONTROL CODING From Theory to Practice Peter Sweeney University of Surrey, Guildford, UK JOHN WILEY & SONS, LTD Copyright © 2002 John Wiley & Sons, Ltd., Baffins Lane, Chichester, West Sussex PO19 1UD, England Phone (+44) 1243 779777 E-mail (for orders and customer service enquiries): cs- books@wiley.co.uk Visit our Home Page on www.wiley.co.uk or www.wiley.com All Rights Reserved 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 under the terms of the Copyright, Designs and Patents Act 1988 or under the terms of a licence issued by the Copyright Licensing Agency Ltd., 90 Tottenham Court Road, London W1P 0LP UK, without the permission in writing of the Publisher Requests to the Publisher should be addressed to the Permissions Department, John Wiley & Sons, Ltd., Baffins Lane, Chichester, West Sussex PO19 1UD, England, or emailed to permreq@wiley.co.uk, or faxed to (+44) 1243 770571 Other Wiley Editorial Offices John Wiley & Sons, Inc., 605 Third Avenue, New York, NY 10158–0012, USA Jossey-Bass, 989 Market Street, San Francisco, CA 94103–1741, USA Wiley-VCH Verlag GmbH, Pappelallee 3, D-69469 Weinheim, Germany John Wiley & Sons Australia, Ltd., 33 Park Road, Milton, Queensland 4064, Australia John Wiley & Sons (Asia) Pte Ltd., Clementi Loop #02–01, Jin Xing Distripark, Singapore 129809 John Wiley & Sons Canada, Ltd., 22 Worcester Road, Etobicoke, Ontario, Canada M9W 1L1 British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library ISBN 470 84356 X Typeset in 10/12pt Times by Kolam Information Services Pvt Ltd, Pondicherry, India Printed and bound in Great Britain by TJ International, Padstow, Cornwall This book is printed on acid-free paper responsibly manufactured from sustainable forestry in which at least two trees are planted for each one used for paper production The Principles of Coding in Digital Communications 1.1 1.2 1.3 1.4 1.5 1.6 1.7 Error Control Schemes Elements of Digital Communication Systems Source Encoding Error Control Coding Modulation The Channel Demodulation 1.7.1 Coherent demodulation 1.7.2 Differential demodulation 1.7.3 Soft-decision demodulation 1.8 Decoding 1.8.1 Encoding and decoding example 1.8.2 Soft-decision decoding 1.8.3 Alternative decoding approaches 1.9 Code Performance and Coding Gain 1.10 Information Theory Limits to Code Performance 1.11 Coding for Multilevel Modulations 1.12 Coding for Burst-Error Channels 1.13 Multistage Coding 1.14 Error Detection Based Methods 1.14.1 ARQ strategies 1.14.2 Error concealment 1.14.3 Error detection and correction capability of block codes 1.15 Selection of Coding Scheme 1.15.1 General considerations 1.15.2 Data structure 1.15.3 Information type 1.15.4 Data rate 1.15.5 Real time data processing 1.15.6 Power and bandwidth constraints 1 2 8 10 11 12 14 15 16 18 21 22 24 24 24 26 26 27 27 28 29 29 30 30 vi CONTENTS 1.15.7 Channel error mechanisms 1.15.8 Cost 1.16 Conclusion 1.17 Exercises 1.18 References 31 31 32 33 34 Convolutional Codes 35 2.1 2.2 2.3 2.4 2.5 2.6 2.7 2.8 2.9 35 35 36 37 38 39 39 41 42 42 42 45 45 51 53 55 56 57 62 64 64 66 2.10 2.11 2.12 2.13 2.14 2.15 2.16 2.17 2.18 2.19 Introduction General Properties of Convolutional Codes Generator Polynomials Terminology Encoder State Diagram Distance Structure of Convolutional Codes Evaluating Distance and Weight Structures Maximum Likelihood Decoding Viterbi Algorithm 2.9.1 General principles 2.9.2 Example of viterbi decoding 2.9.3 Issues arising Practical Implementation of Viterbi Decoding Performance of Convolutional Codes Good Convolutional Codes Punctured Convolutional Codes Applications of Convolutional Codes Codes for Multilevel Modulations Sequential Decoding Conclusion Exercises References Linear Block Codes 3.1 3.2 3.3 3.4 3.5 3.6 3.7 3.8 3.9 3.10 3.11 3.12 3.13 3.14 3.15 Introduction Mathematics of Binary Codes Parity Checks Systematic Codes Minimum Hamming Distance of a Linear Block Code How to Encode - Generator Matrix Encoding with the Parity Check Matrix Decoding with the Parity Check Matrix Decoding by Standard Array Codec Design for Linear Block Codes Modifications to Block Codes Dorsch Algorithm Decoding Conclusion Exercises References 67 67 67 68 69 70 70 71 73 75 76 78 81 83 83 85 CONTENTS Cyclic Codes 4.1 4.2 4.3 4.4 4.5 4.6 4.7 4.8 4.9 4.10 4.11 4.12 4.13 4.14 4.15 4.16 4.17 4.18 4.19 4.20 4.21 4.22 Introduction Definition of a Cyclic Code Example of a Cyclic Code Polynomial Representation Encoding by Convolution Establishing the Cyclic Property Deducing the Properties of a Cyclic Code Primitive Polynomials Systematic Encoding of Cyclic Codes Syndrome of a Cyclic Code Implementation of Encoding Decoding Decoder Operation Multiple-Error Correction Example of Multiple-Error Correction Shortened Cyclic Codes Expurgated Cyclic Codes BCH Codes Cyclic Codes for Burst-Error Correction Conclusion Exercises References vii 87 87 87 88 88 89 90 91 92 93 94 94 96 100 100 101 103 104 106 107 110 110 112 Finite Field Arithmetic 113 5.1 5.2 5.3 5.4 5.5 5.6 113 113 114 115 117 119 119 120 120 120 120 121 121 123 124 125 127 129 131 132 134 5.7 5.8 5.9 5.10 5.11 5.12 5.13 5.14 5.15 Introduction Definition of a Finite Field Prime Size Finite Field GF(p) Extensions to the Binary Field - Finite Field GF(2m) Polynomial Representation of Finite Field Elements Properties of Polynomials and Finite Field Elements 5.6.1 Roots of a polynomial 5.6.2 Minimum polynomial 5.6.3 Order of an element 5.6.4 Finite field elements as roots of a polynomial 5.6.5 Roots of an irreducible polynomial 5.6.6 Factorization of a polynomial Fourier Transform over a Finite Field Alternative Visualization of Finite Field Fourier Transform Roots and Spectral Components Fast Fourier Transforms Hardware Multipliers using Polynomial Basis Hardware Multiplication using Dual Basis Hardware Multiplication using Normal Basis Software Implementation of Finite Field Arithmetic Conclusion viii CONTENTS 5.16 Exercises 5.17 References BCH Codes 6.1 6.2 6.3 6.4 6.5 6.6 6.7 6.8 6.9 6.10 6.11 6.12 6.13 6.14 6.15 Reed Solomon Codes 7.1 7.2 7.3 7.4 7.5 7.6 7.7 7.8 7.9 7.10 7.11 7.12 7.13 7.14 7.15 7.16 Introduction Specifying Cyclic Codes by Roots Definition of a BCH Code Construction of Binary BCH Codes Roots and Parity Check Matrices Algebraic Decoding BCH Decoding and the BCH Bound Decoding in the Frequency Domain Decoding Examples for Binary BCH Codes Polynomial Form of the Key Equation Euclid's Method Berlekamp—Massey Algorithm Conclusion Exercises References Introduction Generator Polynomial for a Reed Solomon Code Time Domain Encoding for Reed Solomon Codes Decoding Reed Solomon Codes Reed Solomon Decoding Example Frequency Domain Encoded Reed Solomon Codes Further Examples of Reed Solomon Decoding Erasure Decoding Example of Erasure Decoding of Reed Solomon Codes Generalized Minimum Distance Decoding Welch—Berlekamp Algorithm Singly Extended Reed Solomon Codes Doubly Extended Reed Solomon Codes Conclusion Exercises References Performance Calculations for Block Codes 8.1 8.2 8.3 8.4 8.5 8.6 8.7 Introduction Hamming Bound Plotkin Bound Griesmer Bound Singleton Bound Gilbert—Varsharmov Bound Error Detection 135 136 137 137 137 138 138 140 143 144 146 147 149 149 151 152 153 154 CONTENTS 8.8 8.9 8.10 8.11 8.12 8.13 8.14 8.15 8.16 8.17 Random-Error Detection Performance of Block Codes Weight Distributions Worst Case Undetected Error Rate Burst-Error Detection Examples of Error Detection Codes Output Error Rates using Block Codes Detected Uncorrectable Errors Application Example - Optical Communications Conclusion Exercises Multistage Coding 9.1 9.2 9.3 9.4 9.5 9.6 9.7 9.8 9.9 9.10 Introduction Serial Concatenation Serial Concatenation using Inner Block Code 9.3.1 Maximal length codes 9.3.2 Orthogonal codes 9.3.3 Reed Muller codes 9.3.4 High rate codes with soft-decision decoding Serial Concatenation using Inner Convolutional Code Product codes Generalized Array Codes Applications of Multistage Coding Conclusion Exercises References 10 Iterative Decoding 10.1 10.2 10.3 10.4 10.5 10.6 10.7 10.8 10.9 10.10 10.11 10.12 10.13 10.14 10.15 10.16 Index Introduction The BCJR Algorithm BCJR Product Code Example Use of Extrinsic Information Recursive Systematic Convolutional Codes MAP Decoding of RSC Codes Interleaving and Trellis Termination The Soft-Output Viterbi Algorithm Gallager Codes Serial Concatenation with Iterative Decoding Performance and Complexity Issues Application to Mobile Communications Turbo Trellis-Coded Modulation Conclusion Exercises References ix 181 182 184 185 185 186 188 190 192 192 195 195 195 196 196 197 198 198 199 200 203 206 208 208 209 211 211 211 212 214 215 217 220 222 225 231 232 233 233 235 235 236 239 This page intentionally left blank 228 ERROR CONTROL CODING Figure 10.23 Example Tanner graph The decoding algorithm operates alternately over rows and columns to find the most likely code vector c that satisfies the condition cHT = Let V(i) denote the set of wr bits that participate in check i Let C(j) denote the set of wc checks that check bity The probabilities that bity' is or 1, given the parity checks other than check i, are written P? and P\j The values are initialized to the a priori probabilities />? and/;? of bit values and for each bity (i.e the initial values are the same for all values of /) The probabilities that check / is satisfied by a value or in bit j given the current values of the other bits in V(i), are denoted Q^ and Q]J The horizontal (row) step is: Define APij =P0ij- P1ij For each i compute A(?,y as the product of APij for all j' = j SetQ0ij= 1/2(1 + A0;,), Q}j = 1/2(1 - A