Constellation Shaping, Nonlinear Precoding, and Trellis Coding for Voiceband Telephone Channel Modems: with Emphasis on ITU-T Recommendation V.34

270 Pages·2002·22.874 MB·English

by  Steven A. Tretter (auth.)

CONSTELLATION SHAPING,
NONLINEAR PRECODING, AND
TRELLIS CODING FOR
VOICEBAND TELEPHONE
CHANNEL MODEMS
with Emphasis on ITU-T
Recommendation V.34
THE KLUWER INTERNATIONAL SERIES
IN ENGINEERING AND COMPUTER SCIENCE
CONSTELLATION SHAPING,
NONLINEAR PRECODING, AND
TRELLIS CODING FOR
VOICEBAND TELEPHONE
CHANNEL MODEMS
with Emphasis on ITU-T
Recommendation V.34
Steven A. Tretter
University ofM aryland, U.S.A.
SPRINGER SCIENCE+BUSINESS MEDIA, LLC
Library of Congress Cataloging-in-Publication Data
Tretter, Steven A.
Constellation shaping, nonlinear precoding, and trellis coding for voiceband telephone
Channel modems with emphasis on lTU-T recommendation V.34 / Steven A. Tretter.
p. cm-{The Kluwer international series in engineering and computer science; SECS 673)
lncludes bibliographical references and index.
ISBN 978-1-4613-5339-3 ISBN 978-1-4615-0989-9 (eBook)
DOI 10.1007/978-1-4615-0989-9
1. Modems. 2. Lattice theory. 3. Coding theory. 4. Trellis-coded modulation. 1. Title.
II. Series.
TK7887.8.M63 T74 2002
62 1.385' l-dc2 1
2002016134
Copyright © 2002 bYSpringer Science+Business Media New York
Originally published by Kluwer Academic Publishers in 2002
Softcover reprint of the hardcover 1s t edition 2002
AH rights reserved. No part ofthis work may be reproduced, stored in a retrieval
system, or transmitted in any form or by any means, electronic, mechanical,
photocopying, microfilming, recording, or otherwise, without the writtenrmission
from the Publisher, with the exception of any material supplied specifically for the
purpose ofbeing entered and executed on a computer system, for exclusive use by
the purchaser of the work.
Printed on acid-free paper.
Contents
Preface xi
1. BASICS OFLATIICETHEORY 1
1.1. DefinitionofaLattice 1
1.2. ExamplesofLattices 2
1.3. Sublattices. LatticePartitions, andCosets 7
1.4. Binary Lattices andCosetRepresentatives 11
1.5. FundamentalRegionsand Volumes, andVoronoi Regions 15
1.5.1 Formulafor theFundamentalVolume 17
1.5.2 LinearTransformationsandthe FundamentalVolume 18
1.5.3 FundamentalVolumeofaSublattice 18
1.6. PointSpacing,WeightDistributions, andThetaSeries 19
1.7. FundamentalCodingGain 22
2. PERFORMANCEMEASURESFORMULTIDIMENSIONAL
CONSTELLATIONS 25
2.1. Introduction 25
2.2. ConstellationFigureofMeritandSymbolErrorProbabilities 27
2.2.1 NormalizedBitRateand AveragePower 28
2.2.2 DefinitionofCFMandExamples 28
EXAMPLE2.1 One-DimensionalPAMConstellation 29
EXAMPLE2.2 M x M SquareGrid 31
EXAMPLE2.3 N-CubeGrid 33
2.2.3 AnApproximationtotheSymbolErrorProbabilityfor
Large SquareQAMConstellationsatHigh SNR 34
2.2.4 TheContinuousApproximation 35
2.3. Constituent2DConstellationsandConstellationExpansionRatio 36
vi CONTENTS
2.4. Peak-to-AveragePowerRatio 37
2.4.1 PARfor theM x M SquareGridandN-CubeGrid 37
2.4.2 PARfor aCircle 38
2.4.3 PARforthe N-Sphere 40
2.5. RepresentingCFM(C) in TermsofCoding Gain and Shaping
Gain 41
2.5.1 Why,c(A) isCalledtheFundamentalCodingGain 43
2.5.2 ShapingGain Properties andExamples 44
2.5.3 Ultimate ShapingGainand2DDistribution 47
2.6. CodingandShapingFactorsoftheConstellationExpansionRatio 49
2.7. Factorsofthe Peak-to-AveragePowerRatio 53
2.8. OptimumTradeoffsofShapingGain withCER and PAR 55
s
3. PRINCIPLES OFCONVOLUTIONALANDTRELLIS CODES 61
3.1. TheHuffmanD-Transform 61
3.1.1 Two-SidedTransformofaDelayedSequence 62
3.1.2 One-SidedTransformofaDelayedSequence 63
3.1.3 D-TransformofaConvolution 64
3.2. TransferFunctions and Realizations 64
3.2.1 Type 1DirectFormRealization 65
3.2.2 Type2DirectFormRealization 66
3.3. DescriptionofaConvolutionalCodebyits GeneratorMatrix 67
3.4. SystematicFormofaConvolutionalCode 69
3.5. TheParityCheckMatrix andSyndromes 71
3.6. InverseCheckMatrixorInverseSyndromeFormer 73
3.7. TheCodeTrellis 75
3.8. WeightDistributions andErrorCorrectionProperties 76
3.9. Trellis CodedModulation(TCM) 79
3.10. BriefReview ofthe ViterbiDecodingAlgorithm 85
3.11. TheFundamentalCodingGainofaTrellis Code 89
4. TRELLIS SHAPING 91
4.1. Trellis ShapingBasedonLatticePartitions 92
4.1.1 TheTrellis ShapingEncoder 92
4.1.2 TheReceiver 97
4.1.3 SelectionofaSpecificConstellation 97
4.2. Trellis ShapingonRegions 106
CONTENTS vii
4.2.1 EssentialPropertiesofTrellisShapingBasedonLattice
Partitions 107
4.2.2 TheTrellis ShapingEncoderfor ShapingonRegions 109
4.2.3 TheReceiverfor ShapingonRegions 112
4.2.4 Peak-to-AverageRatioConsiderations 112
4.2.5 CERaandPAR2Constraintswiththe4-StateUngerboeck
ShapingCode 112
5. NONLINEARPRECODINGMETHODSTO
REMOVEINTERSYMBOLINTERFERENCE 117
5.1. TomlinsonlHarashimaPrecoding 118
5.2. LTFlMotorola/GDCPrecoding 122
5.3. PrecodingandNoiseWhitening 128
5.3.1 TheFirst-OrderLinearPredictor 131
6. TRELLIS PRECODING 133
6.1. TrellisPrecodingBasedon ShapingonRegions 133
6.1.1 TheTransmitter 134
6.1.2 TheReceiver 139
6.1.3 An ExampleofaTrellis PrecodingSystem 139
6.2. Trellis PrecodingBasedonLatticePartitions andLinearCodes 143
6.3. ExperimentalPerformanceResults 144
7. MAPPING DATATOCHANNELSYMBOL
FRAMES BYAMODULUS ENCODER 147
7.1. TheAT&TFractional BitRateModulusConverter 148
7.2. TheV.90ModulusEncoder 152
8. CONSTELLATIONSHAPINGBYSHELLMAPPING 157
8.1. General SystemDescription 158
8.2. RingWeights andtheNumberofFramesofEachWeight 161
8.3. LexicographicalOrdering ofRing Frames 162
8.4. TheDecodingAlgorithm 166
8.5. TheEncodingAlgorithm 171
Appendix8.A. Justificationfor the MotorolaWeightFunction 178
Appendix8.B. ShellMappingProgram 180
9. THEFOURDIMENSIONALCONSTELLATIONUSEDBY
ITU-TV.34 MODEMS 187
9.1. The2D Constellationandits Partitioning 187
viii CONTENTS
9.1.1 Generatingthe2DConstellationby90DegreeRotations
of4Z2 + (1,1) 187
9.1.2 Partitioningthe 2D Constellationinto 8Subsets 189
9.1.3 A Method for Determining the Binary Subset Label
from theCoordinatesofa2DPoint 192
9.2. Framing 193
9.3. The4D Constellation 195
9.3.1 MappingFrames andInitial4DPointSelection 195
9.3.2 Mappingthe Initial4DPointIntotheFinal4DPoint 197
9.3.3 90° RotationalInvarianceofthe 4DConstellation 198
9.3.4 Partitioningofthe4DConstellation 200
9.3.5 Slicing4DPointsto PartitionChainBinary Variables 203
10. THE COMBINED PRECODING AND TRELLIS CODING
SCHEMEFORY.34 205
10.1. TheNonlinearPrecoder 205
10.1.1 ThePrecoderInputandOutput 207
10.1.2 ThePredictionFilterOutput 207
10.1.3 TheModuloBox 207
10.1.4 Why the Precoderis theInverseofH(z) 208
10.2. TheTrellis Encoders 209
10.3. Viterbi Decodingof4DTrellis Codes 212
10.4. MoreDetails onthe Wei 16-StateCode 213
10.4.1 GeneratorandCheckMatrices 213
10.4.2 Invarianceto90DegreeRotations 214
10.4.3 TheFundamentalCodingGain 215
10.4.4 TheOriginalWei 16-StateConvolutionalEncoder 215
10.5. Using the ModuloEncoderto Makey(n) aTrellis Sequence 216
10.6. SuperframeSynchronization 219
10.6.1 Compensatingfor SuperframeBitInversions 222
10.7. ReceiverOperation 223
11.FASTEQUALIZERADJUSTMENTBYUSINGAPERIODIC
TRAININGSEQUENCE 227
11.1. TheV.34 PeriodicTraining Sequence 227
11.1.1 The Periodic Autocorrelation Function and CAZAC
Sequences 227
11.1.2 ConstructingaCAZACSequenceofLengthMK2from
oneofLength M 229
11.1.3 TheY.34CAZAC Sequence 233
CONTENTS ix
11.2. TheOptimal FractionallySpacedEqualizer 233
11.2.1 DerivationoftheOptimumLinearEqualizer 237
11.2.2 MSEfortheOptimumLinearEqualizer 243
11.3. FindingtheInitial EqualizerTapsbyUsingthe FFf 244
11.3.1 TheComplexCross-CoupledandRealPhase-Splitting
Equalizers 246
11.3.2 ComputingEqualizerCoefficientsby Using the FFf 249
References 255
Index
261
Preface
Thisbookisessentiallyahistoryofthedevelopments insignalconstellation
design, nonlinear pre-equalization, and trellis coding made by ITV-T Study
GroupXVIIduring itsmeetings tocreatearecommendation forastate-of-the
art voiceband telephone channel modem. It started deliberations in the fall
of 1991 and by June of 1994 finalized the recommendation which was then
given the V.34label. Thebookisheavily basedonnotesIprepared for aseries
of lectures given to engineers at Pencil Datability Networks (now the Signal
Processing Group of Nortel Networks, Germantown, MD) during the ITV
T V.34 committee deliberations to keep them abreast of the latest technical
proposals.
During the deliberations, the study group was called the V.fast committee
and was jokingly called the V.1ast committee by members believing that no
further improvements in voiceband modems would be made. The initial V.34
recommendation allowed data rates from 2400 to 28,800 bits per second in
increments of2400 bps. It was later amended to extended the maximum rate
to 33,600 bps. More recently, the Y.90 and Y.92 recommendations were ap
proved and allow rates up to 56,000 bps in the downstream direction, that is,
from the server to the client modem, by taking advantage ofthe downstream
digital networkand PCMcodec in the local office. TheY.90recommendation
specifies using V.34 modulation in the upstream direction while V.92 allows
PCM encoding in both the downstream and upstream directions in addition to
V.34 modulation upstream.
The period during which the V.34 recommendation was being formulated
wasaveryexcitingtime. Newtechniques werebeingproposedanddiscovered
continuallyduringthedeliberations. Digitalsignalprocessor(DSP)technology
wasrapidly improving in termsofspeed, smallersize, moreinternal RAM and
ROM,andsignificantlyreducedcost. TheDSPtechnological advancesallowed
the committee to consider techniques that were significantly more complex