Iterative Error Correction: Turbo, Low-Density Parity-Check and Repeat-Accumulate Codes

356 Pages·2009·2.73 MB·English

by  Sarah J. Johnson

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IterativeErrorCorrection
Turbo,Low-DensityParity-CheckandRepeat–AccumulateCodes
Iterative error correction codes have found widespread application in cellular
communications,digitalbroadcasting,deepspacecommunications,andwireless
LANs.Thisself-containedtreatmentofiterativeerrorcorrectionpresentsallthe
keyideasneededtounderstand,design,implement,andanalyzethesepowerful
codes.
Turbo, low-density parity-check, and repeat–accumulate codes are given
equal, detailed coverage, with precise presentations of encoding and decod-
ingprocedures.Workedexamplesareintegratedintothetexttoilluminateeach
newideaandpseudo-codeisincludedforimportantalgorithmstofacilitatethe
reader’s development of the techniques described. For each subject, the treat-
ment begins with the simplest case before generalizing. There is also coverage
ofadvancedtopicssuchasdensity-evolutionandEXITchartsforthosereaders
interested in gaining a deeper understanding of the field. This text is ideal for
graduate students in electrical engineering and computer science departments,
aswellaspractitionersinthecommunicationsindustry.
SarahJ.JohnsonisaResearchFellowintheSchoolofElectricalEngineeringand
Computer Science at the University of Newcastle, Australia. She is a member
oftheIEEEInformationTheoryandCommunicationsSocieties.

Iterative Error Correction
Turbo, Low-Density Parity-Check and
Repeat–Accumulate Codes
SARAH J. JOHNSON
UniversityofNewcastle,NewSouthWales
CAMBRIDGE UNIVERSITY PRESS
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Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521871488
© Cambridge University Press 2010
This publication is in copyright. Subject to statutory exception and to the
provision of relevant collective licensing agreements, no reproduction of any part
may take place without the written permission of Cambridge University Press.
First published in print format 2009
ISBN-13 978-0-511-69129-4 eBook (NetLibrary)
ISBN-13 978-0-521-87148-8 Hardback
Cambridge University Press has no responsibility for the persistence or accuracy
of urls for external or third-party internet websites referred to in this publication,
and does not guarantee that any content on such websites is, or will remain,
accurate or appropriate.
ToRoandClara

Contents
Preface pagexi
Notation xv
Commonlyusedabbreviations xix
1 Channels,codesandcapacity 1
1.1 Binaryinputmemorylesschannels 2
1.2 Entropy,mutualinformationandcapacity 8
1.2.1 Ameasureofinformation 8
1.2.2 Entropy 10
1.2.3 Capacity 13
1.3 Codesanddecoding 16
1.3.1 Decoding 18
1.3.2 Performancemeasures 24
1.4 Bibliographicnotes 29
1.5 Exercises 30
2 Low-densityparity-checkcodes 34
2.1 Introduction 34
2.2 Errorcorrectionusingparitychecks 34
2.2.1 Low-densityparity-checkcodes 37
2.3 Encoding 40
2.3.1 (Almost)linear-timeencodingforLDPCcodes 44
2.3.2 Repeat–accumulatecodes 48
2.4 Decoding 48
2.4.1 Messagepassingonthebinaryerasurechannel 50
2.4.2 Bit-flippingdecoding 54
2.4.3 Sum–productdecoding 61
2.5 Bibliographicnotes 71
2.6 Exercises 72
viii Contents
3 Low-densityparity-checkcodes:propertiesandconstructions 75
3.1 Introduction 75
3.2 LDPCproperties 75
3.2.1 Choosingthedegreedistribution 76
3.2.2 Girthandexpansion 82
3.2.3 Codewordandpseudo-codewordweights 84
3.2.4 Cyclicandquasi-cycliccodes 89
3.2.5 Implementationandcomplexity 95
3.3 LDPCconstructions 97
3.3.1 Pseudo-randomconstructions 97
3.3.2 StructuredLDPCcodes 105
3.4 LDPCdesignrules 114
3.5 Bibliographicnotes 117
3.6 Exercises 118
4 Convolutionalcodes 121
4.1 Introduction 121
4.2 Convolutionalencoders 121
4.2.1 Memoryorderandimpulseresponse 125
4.2.2 Recursiveconvolutionalencoders 127
4.2.3 Puncturing 131
4.2.4 Convolutionalcodetrellises 132
4.2.5 Minimumfreedistance 135
4.3 Decodingconvolutionalcodes 136
4.3.1 Maximumaposteriori(BCJR)decoding 136
4.3.2 LogMAPdecoding 148
4.3.3 Maximumlikelihood(Viterbi)decoding 156
4.4 Bibliographicnotes 160
4.5 Exercises 161
5 Turbocodes 165
5.1 Introduction 165
5.2 Turboencoders 165
5.3 Iterativeturbodecoding 169
5.4 Turbocodedesign 176
5.4.1 Interleavinggain 176
5.4.2 Choosingthecomponentcodes 178
5.4.3 Interleaverdesign 182
5.4.4 Designrules 188
5.5 Factorgraphsandimplementation 190
5.5.1 Implementationandcomplexity 191
5.5.2 Stoppingcriteria 194