Channel Coding
Proefschrift
terverkrijging van degraad van doctor
aan de TechnischeUniversiteitDelft,
op gezagvan deRector Magni cus Prof.ir. K.F. Wakker,
inhet openbaar teverdedigen ten overstaan van een commissie,
doorhet College van Dekanen aangewezen,
op dinsdag 29maart 1994 te 16.00uur
door
Renatus Josephus van der Vleuten,
elektrotechnisch ingenieur,
Promotiecommissie:
RectorMagni cus
Prof.dr.ir. J. Biemond(promotor)
Dr.ir. J.H. Weber(toegevoegdpromotor)
Prof.dr. J.C.Arnbak
Prof.dr.ir. E.Backer
Prof.dr.ir. J.P.M.Schalkwijk
Prof.dr.ir. K.A. SchouhamerImmink
Prof.dr.ir. A.J.Vinck
CIP-DATA KONINKLIJKE BIBLIOTHEEK, DENHAAG
Vleuten,Renatus Josephusvander
Trellis-basedsourceandchannelcoding/ RenatusJosephus
vanderVleuten. -Delft: TechnischeUniversiteitDelft,
FaculteitderElektrotechniek. -Ill.
ThesisTechnischeUniversiteitDelft. -Withref. -With
summaryin Dutch.
ISBN90-5326-013-7
Subjectheadings: trellis-codedquantization/
trellis-coded modulation.
Copyright c
1994byRenatusJosephusvanderVleuten
All rights reserved. Nopart of this thesismay be reproduced or transmittedin
anyformorbyanymeans,electronic,mechanical,photocopying,anyinformation
storage and retrieval system, or otherwise, without written permission from the
Summary xi
I Trellis-Based Source Coding 1
1 Introduction 3
1.1 Multidimensional Quantization : ::: : : ::: :: :: 3
1.2 Codebook design :: :: : :: :: : :: : : : : : :: : : 4
1.3 Trellis Waveform Coding : : : :: : : : :: ::: : : :: 6
2 New Constructions of Trellis-Coded Quantizers 9
2.1 Introduction ::: : : ::: : : :: : :: : : :: : : : : : 9
2.2 Trellis Waveform Coding : : : :: : : : :: ::: : : :: 11
2.2.1 Codebook Design Methods : :: :: ::: :: :: 11
2.2.3 New Constructions : : :: : : : :: ::: : : :: 18
2.3 Trellis-Coded Quantization : : :: ::: :: ::: : : :: 19
2.4 Trellis-Coded Vector Quantization : : : : : ::: :: :: 21
2.5 Conclusions ::: : : ::: : : :: : :: : : :: : : : : : 22 3 Performance Evaluation 25 3.1 Introduction ::: : : ::: : : :: : :: : : :: : : : : : 25 3.2 Preliminaries : : : : : :: : : :: :: : :: : :: :: :: 26 3.2.1 Implementation Complexity :: :: ::: : : :: 26 3.2.2 Training Sequences :: :: :: : : : : : : :: : : 26 3.2.3 Con dence Intervals : :: ::: :: ::: : : :: 27 3.2.4 Codebook Optimization : ::: :: ::: : : :: 28 3.3 TWC and TCQ Experiments :: ::: :: ::: : : :: 30 3.4 TCVQExperiments: :: : :: :: :: : : : : : : :: : : 34 3.5 Gauss-Markov Sources : : :: : : : : : :: ::: : : :: 39 3.6 The M-Algorithm : : : :: : : :: : : : :: ::: : : :: 43 3.7 Discussion : ::: : : ::: : : :: : :: : : :: : : : : : 44
4 Rate Distortion Theory for Trellis Waveform Coding 49
4.1 Introduction ::: : : ::: : : :: : :: : : :: : : : : : 49
4.2 Rate Distortion Theory for Discrete Memoryless Sources 50
4.3 Discrete Alphabet Rate Distortion Theory :: : :: :: 52
4.4 Application toTrellis Waveform Coding : :: : :: :: 55
4.5 Conclusions ::: : : ::: : : :: : :: : : :: : : : : : 59
5 DCT Coding of Images Using TCQs 61
5.1 Introduction ::: : : ::: : : :: : :: : : :: : : : : : 61
5.2 The Discrete Cosine Transform : ::: :: ::: :: :: 62
5.3 Quantization : : : : : :: : : :: :: : :: : :: :: :: 63
5.4 Channel Error Protection :: :: :: : : : : : : :: : : 65
5.5 Image Coding Experiments: : :: ::: :: ::: : : :: 69
5.6 Conclusions ::: : : ::: : : :: : :: : : :: : : : : : 77
6 Discussion 79
II Trellis-Based Channel Coding 83
7 Introduction 85
7.1 Digital Communication :: :: :: : :: : : : : : :: : : 85
7.2 The Additive White Gaussian Noise Channel : : :: :: 87
7.3 Modulation for Bandwidth-Limited Channels : : :: :: 88
7.4 Trellis-Coded Modulation :: :: :: : : : : : : :: : : 90
8 TCM with Optimized Signal Constellations 95
8.1 Introduction ::: : : ::: : : :: : :: : : :: : : : : : 95
8.2 1-Dimensional Signal Constellations :: : : ::: :: :: 96
8.2.1 Optimization for R =1 :: ::: :: ::: : : :: 96
8.2.2 Extension toR>1 : : :: ::: :: ::: : : :: 99
8.3 2-Dimensional Signal Constellations :: : : ::: :: :: 106
8.3.1 PSK Constellations : : :: ::: :: ::: : : :: 106
8.3.2 QAM Constellations : : :: : :: : : : :: :: : 110
A Autocorrelation of the Fake Process 117
B Proof of White Spectrum for Construction B 119
C Proof of White Spectrum for Construction C 123
D Codebook Initializations for the TWCs and TCQs 125
Bibliography 127
Samenvatting 137
Acknowledgements 141
Curriculum Vitae 143
This thesis concerns the ecienttransmission of digital data,such as
digitized sounds orimages, froma sourceto itsdestination. To make
the best use of the limited capacity of the source-destination
chan-nel, a source coder is used to delete the less signi cant information.
To correct the occurring transmission errors, achannel coder is used.
Ecient techniques for source and channelcoding, based on trellises,
are respectively investigated in Part I and Part II of this thesis.
Part I: Trellis-Based Source Coding
There are two mainstreams in source coding: lossless source coding
and\lossy"sourcecoding,ordatareduction. Thelatterformofsource
coding is considered here.
The art of data reduction is commonly referred to as quantization.
The advances of digital technology have led toquantizers which
pro-cessseveralsourcesamplesatonceandareknownasvectorquantizers.
In general, the complexity of vector quantization increases
exponen-tially with increasing vector dimension, but there are some
reduced-complexityvariationsofvectorquantization, oneofwhichisknownas
trellis waveform coding. TWC (used to denote both trellis waveform
coding and a trellis waveform coder) can be improved by atechnique
PartI ofthisthesisreports on newlydesigned TCQs,whichare based
on a fake process approach. Using this approach, one tries to imitate
the original source by creating a fake process, which is generated by
feeding a random bit stream to the decoder. To evaluate the
perfor-mances of the new quantizers, experiments have been performed for
memorylessLaplacian, Gaussian, and uniform sources. For the
mem-oryless Gaussian and Laplacian sources,the proposed TCQs improve
upon all previously published results.
The discipline of information theory that treats quantization is
called rate distortion theory. Rate distortion theory for memoryless
continuous-amplitude sources with discrete representations is called
discrete-alphabet rate distortion theory. Computation of the
discrete-alphabetratedistortionfunctionnotonlyprovidestheasymptotically
achievable coding performance, but also the asymptotically optimal
representation symbols. Experiments have shown that TWCs using
those representation symbols can perform close to optimized TWCs.
This isalso thecase for TWCsusingthe maximum-entropyquantizer
representation symbols. At low complexities, the maximum-entropy
quantizer based TWCs outperform the rate distortion function based
TWCs.
The performance of the new TCQs has also been investigated for a
discrete cosine transform (DCT) image coding scheme. The
perfor-manceof the codingschemeusing TCQshas been compared withthe
performance whenusing Lloyd-Maxquantizers (LMQs). Itwasfound
that theperceivedimagequality isconsiderablyimprovedwhen using
TCQs compared to LMQs: the occurring edges are better preserved
and many fewer blocking e ects and much less background noise are
visible.
The main practical advantages of TCQ are that it does not need
en-tropy coding and thatit has an asymmetric complexity, concentrated
at the encoder. Further, a distinct practical advantage of TCQ over
Part II: Trellis-Based Channel Coding
A distinction can be made between power-limited and
bandwidth-limited channels; the latter are considered here. Important practical
examplesofsuchchannelsarethetelephonechannelandthemagnetic
recording channel.
Since the channels cannot directly handle a bit stream, it has to be
converted, by a modulator, into a suitablewaveform. The traditional
forms of modulation for bandwidth-limited channels are pulse
ampli-tude modulation (PAM), phase shift keying (PSK), and quadrature
amplitude modulation (QAM). Bycombiningthe channel codingand
modulationfunctionsaperformance gainoverthetraditionaluncoded
modulation can be achieved. An e ective method for designing
com-bined coding and modulation codes is known as trellis-coded
modula-tion (TCM).TCM uses the symmetriesof binaryconvolutional codes
tomapthechannelsymbolsontothetrellis. Traditionally,thechannel
symbols are selected from the PAM, PSK, or QAM signal
constella-tions that are also used for uncoded modulation.
In Part II of this thesis, the implementation of a computer search
to jointly optimize the convolutional code, the mapping, and the
sig-nal constellation in a TCM scheme is discussed. As a result of the
search, TCM schemes that outperform the traditional TCM schemes
havebeen found. The performance gainsoverthe traditional schemes
are obtained without expanding the signal constellation and without
Introduction
1.1 Multidimensional Quantization
There are two main streams insource coding: lossless source coding,
or data compaction, and \lossy" source coding, or data reduction or
compression. The latter form of source coding is the topic of Part I
of this thesis. The art of data reduction is commonly referred to as
quantization.
In the early days of digital signal processing, the only task of the
quantizer was to perform an analog-to-digital conversion, scaling the
inputsamplesand rounding themtothenearestintegernumber. The
advances in digital technology have led to more complex
quantiz-ers. Although a high-resolution analog-to-digital conversion is still
the rst step of adigital signal processing system, it isoften followed
atsomepointinthesystembyasecondaryquantizationstep[Gers92].
Whereas the analog-to-digital converter is simply a means to obtain
a signal suitable for digital processing, the purpose of the secondary
the digital signal, at the cost of an increased distortion in its
repre-sentation of the original signal.
The amount of information necessary to describe a signal is called
the entropy and is expressed in bits per symbol (or sample). The
distortion isusually measured as the mean squared error (MSE). The
best quantizer is the one that results in the lowest distortion at a
certain xedentropy,or, alternatively,itis theone that resultsinthe
lowestentropyatacertain xeddistortion. Theoreticalboundstothe
performanceofanyquantizeraregivenbyratedistortiontheory(which
is discussedin Chapter 4).
A quantizerthat operateson asingle signalsampleatatime iscalled
a scalar quantizer; when it quantizes several samples at once, it is
known as a multidimensional or vector quantizer. VQ (which is here
used to denote both Vector Quantization and a Vector Quantizer)in
itsbasicformisastraightforwardextensionofthe1-dimensionalscalar
quantization. Speci cally,anN-dimensionalinputvectorxismapped
ontoanN-dimensionaloutputvectory,whereyistakenfroma nite
set, called the codebook.
Given a source vector and a codebook, the aim of the VQ is to
pro-duce the best representation vector, de ned as that vector from the
codebook which has the minimal distortion from the source vector.
Thus, for a given source vector, the VQ computes the distortion for
each codebook vector, selects the one having minimal distortion, and
transmitsthecorrespondingcodebookindex. Thisindexisusedatthe
decoder to select the corresponding vector from its codebook (which
is identical tothe codebook of the encoder).
1.2 Codebook design
ex-optimal quantizer minimizes theexpecteddistortion fora given
code-book dimension and cardinality. Basically, there are two approaches
towards the design of the codebook:
1. Stochastic codebook, the elements of which are chosen at
ran-dom from the set of input vectors. The input vectors can also
be approximated by vectors of random variables having
appro-priately adapted variances.
2. Optimized codebook,whichis obtained asthe result ofan
opti-mizationprocedurewhichadaptsthe codebookelementstobest
matcha training set of inputvectors.
The asymptotic performance (for large N) of the stochastic and
op-timizedcodebook VQsapproaches the rate distortion bound [Berg71,
Vite79].
The conditions forthe minimum-distortion quantizer were derivedby
Lloyd and Max [Lloy82 , Max60]. The rst (obvious) condition has
already been discussed: the quantizer should select the output
vec-tor that results in the minimal distortion. The second condition is
thateachoutputvectorbethe centroidofthoseinputvectorsthatare
mapped onto it. In practice, the statistics of the input vectors may
not beknown. In that case, the codebook can be designed using
rep-resentative training vectors. The VQ performance is then quanti ed
by itsperformance for vectors outside the training set.
A codebook design algorithm that has successfully been applied was
developed by Linde, Buzo, and Gray [Lind80]. Their extension of
Lloyd's algorithm is known as the LBG or K-means or generalized
Lloyd algorithm and has been listed in Figure 1.1. The drawbacks
of the algorithm are that it converges slowly and that it converges
quan-Step 1. Choosean initialcodebook.
Step 2. Encodethe training setusing the present codebook.
Step 3. Replace eachoutput vector by the centroid of those
in-put vectors that were mapped onto it (in Step 2). If
the distortion is suciently small then quit, else go to
Step 2.
Figure 1.1: The LBG codebook optimizationalgorithm.
timestodi erentinitialcodebooksandselectingthebestresult.
How-ever,thecomputationalburdenofthissearchproceduresoon becomes
prohibitive.
Finally, it should be remarkedthat quantizers designed for minimum
distortion, for a given codebook cardinality, in general are not
opti-mal in arate distortion sense, i.e. the same distortion, but ata lower
rate,may beobtainedbyanotherquantizer,havingalargercodebook
cardinality. Inthatcase,entropycoding(seee.g.[Gers92])ofthe
code-bookindexesis applied,i.e. variable-length codewordsare assignedto
the indexes. One of the approaches to nding such rate distortion
optimal quantizers is known as entropy-constrained vector
quantiza-tion (ECVQ) [Chou89, Gers92]. The design algorithm is similar to
theLBG algorithmandhasbeenlistedinFigure1.2. Amoredetailed
listing of this algorithm can befound in[Chou89] and [Gers92].
1.3 Trellis Waveform Coding
The complexity of VQ increases exponentially with increasing vector
dimensionN,buttherearesomevariationsofVQwhichtrytoreduce
Step 1. Choose an initial codebook and assign initial
code-words (variableor xedlength) tothe indexes.
Step 2. Encodethe training setusing the presentcodebook,
us-ing as the distortion measure the weighted sum of the
MSE between the input vector and a code vector and
the length of the codeword assigned to that codebook
vector.
Step 3. Replace eachoutput vector by the centroid of those
in-putvectorsthatweremappedontoit(inStep2). Assign
new variable-length codewords tothecodebook vectors,
basedontheirfrequencyofselectioninStep2. Ifthe
dis-tortion issuciently small then quit, else goto Step 2.
Figure 1.2: The ECVQ codebook optimization algorithm.
Oneofthesevariationsisknown astrellis waveformcoding. Intheory,
trelliswaveformcodersuseanin nite-dimensionalcodebook(N =1)
havingaspecialstructurethatallowsforreduced-complexity(relative
to general VQ) encoding and decoding procedures. In practice, the
highestoccurring dimension is nite sincethereis apractical limiton
the implementation complexity.
The design of trellis waveform coders is the topic of Chapter 2. The
performances of the new quantizers are then evaluated in Chapter 3
for sources with well-behaved distributions, in Chapter 4 inthe light
of the optimal performance theoretically achievable, as given by rate
distortion theory, and in Chapter 5 in an image coding application.
Chapter 6discusses trelliswaveform codingfrom atheoretical aswell
New Constructions of
Trellis-Coded Quantizers
2.1 Introduction
TWC(used todenotebothtrellis waveform codingand atrellis
wave-form coder) is a proven technique for source coding, with a long
his-tory[Vite79,Gers92]. Theencoderconsistsofacodebook anda nite
state machine, the state transitions of which specify the codebook
symbol which is used to represent the source symbol. All possible
statesequencesofthe nitestatemachinecanberepresentedaspaths
through a trellis. An exampleis depicted in Figure 2.1.
In Figure 2.1(a), the nodes of the trellis represent the state of the
machine and the branch values represent the representation symbol
of the quantizer. In this case, each source sample is represented by a
single bit (which completely speci es the path). For each branch, a
distortion functionisde ned, equal tothe squared di erencebetween
the source symbol and the representation symbol for that
@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ yy y yyy y y yyy 0.81 8.41 0.09 5.29 0.49 1.69 3.61 0.01 8.41 0.81 0.36 6.76 0.16 2.56 1.44 0.64 4.84 0.04 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ yy y yyy y y yyy 0.5 -1.5 0.5 -1.5 1.5 -0.5 0.5 -1.5 1.5 -0.5 0.5 -1.5 1.5 -0.5 0.5 -1.5 1.5 -0.5 (b) (a)
Figure 2.1: Trellis diagram for a 2-state 1-bit-per-sample trellis
waveformcoder: (a) theTWC, and(b)thedistortion trellisforthe
1.4 0.8 -1.4 1.1 -0.7
Table2.1: Example source symbols.
symbols listed in Table 2.1. For each path through the trellis, the
distortion is the sum of the distortions associated with the branches
ofthat path. Theencoder investigatesthe pathsthroughthe trellisin
orderto ndthepaththatminimizesthetotaldistortion. Theoptimal
algorithm forthis search isthe Viterbi algorithm [Vite71,Forn73 ]. In
the rathertrivial example,the bestpath|therepresentation symbols
of whichhavebeenindicated inbold typeinFigure 2.1(a)|results in
a total distortion of 1:71.
Given the trellis structure, the question is how to design the trellis
codebook, i.e.howtochoosethe branchrepresentationsymbols. This
is the topic of Section 2.2. Section 2.3 and Section 2.4, respectively,
treattwospecialcasesofTWC,namelytrellis-codedquantizationand
trellis-coded vector quantization. Section 2.5 concludes the chapter.
2.2 Trellis Waveform Coding
2.2.1 Codebook Design Methods
Traditionally, therehave been twomethods fordesigning trellis
code-books (they are the TWC equivalents of the vector-quantizer design
methods of Section 1.2). The rst, based on the asymptotic
optimal-ity proof [Vite79], stochastically populates the trellis with randomly
chosen samplesfromthe sourcedistribution. Althoughingeneralthis
method is very complex, Pearlman et al. have shown that it can be
considerably simpli ed at the cost of a relatively small increase in
symbolsateachstep), whichare consideredinthis thesis,can achieve
performances close tothose of time-varying TWCs.
The second codebook design method optimizes a given initial
code-book;analgorithmisdescribedin[Stew82 ]. Thealgorithmisbasedon
the LBG algorithm fordesigning a VQ [Lind80](see also Section1.2)
and hasthe samedrawbacks: itconvergestoalocal optimum andthe
convergencecan be very slow,dependingon the initialcodebook, the
number of representation symbols, and the required accuracy. Thus,
ndinga good codebook usingthis algorithmis atrial-and-error
pro-cedure.
Although both methods have been successfully applied, their
disad-vantageisthattheyareessentiallynon-constructive. The rstmethod
justpicksarandomcode; thesecondmethodpicksarandomcodeand
tries to improve it. A rst constructive design method was given by
MarcellinandFischer[Marc90a ],whomaptherepresentationsymbols
deterministicallyontothetrellisaccordingtoaconvolutionalcode
(in-terestingly, itwasobservedalready in [Free88] that optimized
uncon-strained trellis codes tend to have a great deal of regularity, but the
link to convolutional codes was not made). The performance of the
TWCs of [Marc90a ] in general is good and in some cases superior to
all previous results from the quantization literature, which was our
reason for investigating new constructionsof TWCs.
2.2.2 The Fake Process Approach
The new TWC constructions are based on the fake process approach
of [Lind78]. Using this approach, one tries to imitate the original
source by a \fake process," which is generated by a random walk
through atime-invariant trellis. The sequence of representation
sym-bolsproducedinthiswayshouldhavethesamestatisticsasthesource.
In particular,asshown in[Lind78], asanecessary(but not sucient)
spec-@ @ @ @ @ @ @ @ @ {{ {{ A B C D
Figure 2.2: Trellis diagram of a general 2-stateTWC.
memory,abetterperformanceisobtainedbyincorporatingtheTWCs
intoapredictivecoding scheme(suchasdescribedin[Ayan86])which
decorrelates(whitens)thesourcesamples(seealsoSection3.5). Thus,
since it is assumed that the sequence of source sampleshas a at (or
white) spectrum, the representation sequence should also have this
property. While for randomly populated, time-varying trellises this
requirement is ful lled by de nition, for deterministically populated,
time-invarianttrellisesitisnot. Therefore,inorderto ndouthowto
generate white representationsequences, astudyis madeof the
spec-trum,i.e.theautocorrelation,ofsequencesgeneratedbytime-invariant
trellises with uncorrelatedinputs.
As an example, consider the 2-statetrellis shown inFigure 2.2. The
spectrum of agenerated sequence fx
k g is at if R( )=E[x k x k+ ]=0; (2.1)
for j j>0. Writing out (2.1) for =1 gives
R(1) = AP(A)fAP(AjA)+BP(BjA)g
+BP(B)fCP(CjB)+DP(DjB)g
Since independentinputs are assumed(2.2) simpli es to R(1) = AP(A) P(A)+P(B) fAP(A)+BP(B)g+ BP(B) P(C)+P(D) fCP(C)+DP(D)g+ CP(C) P(A)+P(B) fAP(A)+BP(B)g+ DP(D) P(C)+P(D) fCP(C)+DP(D)g =
fAP(A)+CP(C)gfAP(A)+BP(B)g
P(A)+P(B)
+
fBP(B)+DP(D)gfCP(C)+DP(D)g
P(C)+P(D)
: (2.3)
Finally, substituting a = AP(A);b = BP(B);c = CP(C);and d =
DP(D)into(2.3) results in R(1)= a+c b+d ! (a+b)=fP(A)+P(B)g (c+d)=fP(C)+P(D)g ! : (2.4)
Thus R(1) can be written as the inner (dot) product of two vectors.
To ensurethat R(1)0 itsuces that one of the vectorsin (2.4) be
the zero vector. So
fb = a^d= cg_fc= a^d= bg: (2.5)
Interpreting the two solutions in (2.5) shows that the rst solution
impliesthatthebranchesemanatingfromastateshouldhaveopposite
values and the second solution implies that the branches entering a
state shouldhaveoppositevalues,provided thatthe twobranches are
selected(bytheencoder)withequalprobability. Thereadercaneasily
verify that (2.5) also guarantees that R( ) is zero for > 1. The
2-stateexamplecanbeextendedasfollowsfortrellisescorrespondingto
q-ary shift registers.
Consider a TWC having q states S l ,1 l q , with q branches n
P P P P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ S S S S S S S S S S Sw w w w w w w w w w w w w w w w P P P P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ S S S S S S S S S S Sw w w w w w w w w w w w w w w w Y 1 Y 2 Y 2 Y 1 Y 3 Y 4 Y 4 Y 3 Y 5 Y 6 Y 6 Y 5 Y 7 Y 8 Y 8 Y 7 Y 1 Y 2 Y 3 Y 4 Y 5 Y 6 Y 7 Y 8 Y 2 Y 1 Y 4 Y 3 Y 6 Y 5 Y 8 Y 7 W 1 W 9 W 2 W 10 W 3 W 11 W 4 W 12 W 5 W 13 W 6 W 14 W 7 W 15 W 8 W 16 W 1 W 2 W 3 W 4 W 5 W 6 W 7 W 8 W 9 W 10 W 11 W 12 W 13 W 14 W 15 W 16 S 1 S 1 S 2 S 2 S 3 S 3 S 4 S 4 S 5 S 5 S 6 S 6 S 7 S 7 S 8 S 8 (a) (b)
Figure 2.3: Trellis diagram of an 8-state trellis waveform coder
for q =2: (a) the states are numbered S
l
, 1 l 8; the branches
have representation symbols W
k
, 1 k 16, and (b) example of the
symmetryof the underlying convolutionalcode.
branchfromstateS
dl=qe +rq 1,0r q 1,tostateS l isassignedthe representation symbol W l+rq
,where dte denotes the smallestinteger
not less than t. The rate, R, equals n b/sample. As an example, in
Figure 2.3(a) an 8-state TWC with branch values W
k
and states S
l
is shown for q = 2. As derived in Appendix A, assuming all trellis
branchesareselectedwithequal probability(itisshowninSection3.7
that this is a good approximation) the autocorrelation of the fake
process generated by a random walk through the trellis and denoted
byR( ), can be writtenas R( )= q (+ +1) q +1 X i=1 2 4 0 @ q X j=1 W i+(j 1)q +1 1 A 0 @ q X j=1 W (i 1)q +j 1 A 3 5
for 1 +1. For obtaining R( ) =0, according to (2.6) there
are two trivial solutions (the equivalents of (2.5)):
q X j=1 W i+(j 1)q +1 =0, and (2.7) q X j=1 W (i 1)q +j =0; (2.8) for 1 +1and1 i q +1 .For = 1, (2.7) and
(2.8), respectively, state that the sum of the values of the branches
entering or leavingeach state should be zero, in order for R( )to be
zero. Based on this observation, in [Vleu91 ], for q = 2, TWCs were
constructed and their performances evaluated. It was found,
how-ever,thatTWCsbasedon convolutionalcodes|inparticularthose of
rate1=|haveabetterperformance. Theyuseaone-to-onemapping
from the convolutional-code symbols to the representation symbols.
The generalization toq =2 n
, developedhere, assumesthe underlying
rate 1= q-ary convolutional code has a symmetry of its q
di erent
branchsymbolsY
m
as speci ed by the following set of equations:
W
1+(q
(2r(mmod2) r+(m 1)modq)+q((m 1)divq)+r)modq
+1 =Y
m
; (2.9)
for 1m q
,0r q 1. ForTWC, the branch values Y
m
rep-resent real numbers, of course. Examples of the symmetry are shown
in Figure 2.3(b) and Figure 2.4, for q = 2 and q = 4, respectively.
Sincethe underlyingconvolutionalcodedoesnotneed tobeexplicitly
speci ed (contrary to [Marc90a ], where Ungerboeck's codes [Unge82]
were assumed), no actual convolutional code is required for the
con-struction. In fact, there are many convolutional codes that t (2.9).
For q = 2, for example, the convolutional codes used in [Zeha90 ] for
Quasi-Orthogonal and Super-Orthogonal codes of degree 1 t (2.9);
their generatorpolynomials are: g
1 (x)=1+x , andg j (x)= x j 1 , for
Y 16 Y 15 Y 14 Y 13 Y 15 Y 16 Y 13 Y 14 Y 14 Y 13 Y 16 Y 15 Y 13 Y 14 Y 15 Y 16 Y 12 Y 11 Y 10 Y 9 Y 11 Y 12 Y 9 Y 10 Y 10 Y 9 Y 12 Y 11 Y 9 Y 10 Y 11 Y 12 Y 8 Y 7 Y 6 Y 5 Y 7 Y 8 Y 5 Y 6 Y 6 Y 5 Y 8 Y 7 Y 5 Y 6 Y 7 Y 8 Y 4 Y 3 Y 2 Y 1 Y 3 Y 4 Y 1 Y 2 Y 2 Y 1 Y 4 Y 3 Y 1 Y 2 Y 3 Y 4 Y 16 Y 15 Y 14 Y 13 Y 12 Y 11 Y 10 Y 9 Y 8 Y 7 Y 6 Y 5 Y 4 Y 3 Y 2 Y 1 Y 15 Y 16 Y 13 Y 14 Y 11 Y 12 Y 9 Y 10 Y 7 Y 8 Y 5 Y 6 Y 3 Y 4 Y 1 Y 2 Y 14 Y 13 Y 16 Y 15 Y 10 Y 9 Y 12 Y 11 Y 6 Y 5 Y 8 Y 7 Y 2 Y 1 Y 4 Y 3 Y 13 Y 14 Y 15 Y 16 Y 9 Y 10 Y 11 Y 12 Y 5 Y 6 Y 7 Y 8 Y 1 Y 2 Y 3 Y 4
Figure 2.4: Example of the symmetry of the underlying
2.2.3 New Constructions
Thecorrespondence between therepresentationsymbolsandthe
sym-bols of the underlying convolutional code is not uniquely speci ed.
Therefore,threeconstructionsareconsidered: constructionA,a
\triv-ial" construction (based on (2.7)) (because of the structure of the
underlying convolutional code, as given by (2.9), (2.7) and (2.8) are
equivalent), and constructions B and C, two \non-trivial"
construc-tions. Theyall result ina whitespectrum, forequal branch
probabil-ities.
For construction A, in addition to the symmetry speci ed by (2.9),
the representation symbolshave the following relation:
Y 2k = Y 2k 1 ; (2.10) for 1 k q
=2. By combining (2.10) and (2.9), it follows
imme-diately from (2.8) or (2.7) that the construction results in a white
spectrum. An example of the construction is shown inFigure 2.5(a).
In the example, Y 1 = A, Y 2 = A, Y 3
= B, etc. Interestingly, the
constructionissimilartothatoftheSuper-Orthogonalcodesofdegree
1, asde ned in[Zeha90 ], which are designed fortrellis-coded
modula-tion (TCM) (TCM is describedin Part II of this thesis).
For construction B, in addition to the symmetry speci ed by (2.9),
the representation symbolshave the following relation, for > 1:
Y k 1+q =2+q 2((k 1)modq) = Y k ; (2.11) for 1 k q
=2. The proof that the construction results in a white
spectrum is givenin Appendix B. An example of the construction is
showninFigure2.5(b). Now,Y
1 =A,Y 2 =B,Y 3 =C,etc.
Construc-tion B is the construction that was initially developedin[Vleu91 ].
Finally, for construction C, in addition to the symmetry speci ed by
(2.9),therepresentationsymbolshavethefollowingrelation,for>1:
P P P P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ S S S S S S S S S S Sw w w w w w w w w w w w w w w w P P P P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ S S S S S S S S S S Sw w w w w w w w w w w w w w w w P P P P P P P P P P P P P P P P Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q Q @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ S S S S S S S S S S Sw w w w w w w w w w w w w w w w A B B A C D D C B A A B D C C D A A A A B B B B C C C C D D D D A B B A A B B A C D D C C D D C (a) (b) (c)
Figure 2.5: Examplesof the proposed constructions for q=2, =3:
(a) constructionA, (b) construction B, and(c) constructionC.
for 1 k q
=2. The proof that the construction results in a white
spectrum is givenin Appendix C. An example of the construction is
shown in Figure 2.5(c). In this case, Y
1 =A, Y 2 =B, Y 3 = A,etc. 2.3 Trellis-Coded Quantization
Inspired by Ungerboeck's trellis-coded modulation (TCM) technique
knownincommunicationtheory[Unge82 ,Unge87a ,Unge87b ](seealso
Part II of this thesis), Marcellin and Fischer [Marc90a] recognized
that TWC can be improved by a technique which they call
trellis-coded quantization (TCQ). It is similar to TWC, but, instead of a
singlecodebook element,the nitestatemachineinthis casespeci es
aset ofcodebook elements. Theencoder nowinvestigatesall allowed
@ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ yy y yyy y y yyy 0.25 0.75 0.25 0.75 1.25 1.75 -1.75 -1.25 -0.75 -0.25 0.25 0.75 1.25 1.75 0.25 -1.25 -0.75 -0.25 @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ @ y y yyy D 1 D 1 D 1 D 1 D 1 D 2 D 2 D 2 D 2 D 3 D 3 D 3 D 3 (b) (a)
Figure 2.6: Trellis diagram of a 2-state 2-bit-per-sample
trellis-coded quantizerfor q=2: (a) thegenerictrellis,and(b)the trellis
for the source sequenceshown inTable 2.1 (on p.11).
An example of TCQ is given in Figure 2.6. In Figure 2.6(a), the
trellis codebook consists of four sets: D
0 = f 1:75;0:25g, D 1 = f 1:25;0:75g, D 2 = f 0:75;1:25g, and D 3 = f 0:25;1:75g.For
each branch the set member closest to the source value is selected;
Figure 2.6(b) shows the trellis for the source symbols listed in T
a-ble2.1 (onp.11). The trellisisthensearchedforthebestpath,which
has been indicated in bold type in Figure 2.6(b); the total distortion
equals 0.87. The ratefor this exampleistwobit persample sinceone
bit isused tospecify whichsetmemberhas been selectedand one bit
is used tospecify which branch has been chosen.
The TWC constructionsof Section2.2.3 are easily extendedto TCQ.
Consider again the trellis havingq states S l ,1l q ,with q =2 n
branchesenteringand leavingeachstate. Now,the branchfrom state
S 1,0rq 1,tostateS is assignedtheset W
quantizing at R b/sample (R =n;n+1;:::), each set contains 2 R n
representation symbols.
Y
m
now denotes the set fy
m;1 ;y m;2 ;:::;y m;2 R ng and Y m is
used to denote the set f y
m;1 ; y m;2 ;:::; y m;2 R ng. As an
ex-ample, in Figure 2.5 replace A by fa
1 ;a 2 ;:::;a 2 R ng, A by f a 1 ; a 2 ;:::; a 2 R ng, etc.
Constructions A, B, and C again give a white spectrum, assuming
thatall setmembersare usedwith thesame probability; theproofs of
Appendix B and Appendix Care easily extendedto TCQ.
2.4 Trellis-Coded Vector Quantization
In[Fisc91 ],trellis-codedvectorquantization(TCVQ)wasinvestigated.
WhileforTCQthebranchsetscontainscalars,forTCVQtheycontain
vectors. Thus, TCQ can be seen as 1-dimensional TCVQ.
The TCQ constructions of Section 2.3 can be extended to TCVQ as
follows. Consideragainthetrellishavingq statesS l ,1l q ,with q =2 n
branches entering and leaving each state. Again, the branch
from state S dl=qe+rq 1,0 r q 1, to state S l is assigned the set W l+rq
. Now, for quantizing at R b/sample using N-dimensional
representation vectors,each setcontains 2 NR n
vectors.
Y
m
denotesthesetofN-dimensionalvectorsfy
m;1 ;y m;2 ;:::;y m;2 NR ng and Y m
isusedtodenotethesetf y
m;1 ; y m;2 ;:::; y m;2 NR ng.As
an example,in Figure 2.5 replace A by
8 > > > > < > > > > : (a 1;1 a 1;2 ::: a 1;N ) (a 2;1 a 2;2 ::: a 2;N ) . . . . . . . . . . . . (a NR n a NR n ::: a NR n ) 9 > > > > = > > > > ; ;
A by 8 > > > > < > > > > : ( a 1;1 a 1;2 ::: a 1;N ) ( a 2;1 a 2;2 ::: a 2;N ) . . . . . . . . . . . . ( a 2 NR n ;1 a 2 NR n ;2 ::: a 2 NR n ;N ) 9 > > > > = > > > > ; ; etc.
It should be noted that, in general, constructions A, B, and C no
longer guarantee a white spectrumfor TCVQ. A white spectrumcan
be guaranteed, however,by forcingthe representation vectors tohave
a certain structure. This was done for the case of q =2, N =2, and
R =1=2, for the Laplacian source,in [Vleu92 ], but the performances
obtained for this case are lower than the performances obtained for
the constructions proposed in this thesis, which use unconstrained
representation vectors. As argued in [Eyub93], this observation is
true ingeneral: althoughstructured quantizerscan beasymptotically
optimal forlarge dimensions,forsmall dimensionstheyare inferiorto
unconstrained quantizers. Experimentsperformedwiththe optimized
TCVQsshowthattheydogenerate awhitespectrum(astheyshould,
sincegeneratingawhitespectrumisanecessaryconditionforthefake
process, as wasshown in[Lind78]).
2.5 Conclusions
Threedi erentconstructionsofTWCs, TCQs,andTCVQshavebeen
proposed. They are based on a fake process approach. By enforcing
certainsymmetryproperties,ithas beenguaranteedfortheTWCand
TCQ constructions that a random walk through the trellis results in
anuncorrelatedsignal,irrespectiveoftheactualtrelliscodebook. This
cannot be guaranteed for the TCVQconstructions.
construc-the trellis are based on underlying convolutional codes, the
Performance Evaluation
3.1 Introduction
In Chapter 2, three new constructions of TWCs, TCQs, and TCVQs
havebeen proposed. In this chapter,the quantizers will be optimized
forspeci csourcesandtheirperformanceswillbedetermined. Soasto
be able tocompare the performances with results from the
quantiza-tionliterature,memorylessLaplacian,Gaussian, anduniformsources,
as wellas an AR(1) Gauss-Markov source,are used.
Section 3.2 discusses some issues that have to be resolved before
ac-tual experiments can be performed. Section 3.3 then evaluates the
TWCsand TCQsandthe sameisdoneinSection3.4for theTCVQs.
Section 3.5 discusses the application of TWCs and TCQs to
Gauss-Markov sources. The performance of the M-algorithm, a well-known
reduced-state search algorithm, isdetermined in Section3.6 and
Sec-tion 3.7 discusses why the newly constructed TCQs outperform the
previous TCQ construction of [Marc90a]. Section 3.8, nally,
3.2 Preliminaries
3.2.1 Implementation Complexity
In order to make a fair comparison of the various quantizers, they
shouldbecomparedatthesamerateand complexity. Thecomplexity
de nition criticallydependson the kindof implementationone has in
mind. In particular, two extreme cases can be distinguished, viz., a
low-speed serial (or software) implementation and a high-speed
par-allel implementation. For the serial implementation, the complexity
de nition of [Marc90a ], i.e. the number of multiplications, additions,
and comparisons, is suitable, but for a parallel implementation the
complexity de nitionof [Forn73 ,Vite79,Unge87b ], i.e.the numberof
state transitions inthe trellis, ismore appropriate. The latter
de ni-tionisusedhere. Thus,thecomplexityequalstheproductofthe
num-berofstates,the numberofbranches (sets)perstate,andthe number
of (1- or multi-dimensional) vectorsperset: q q2 NR n =2 n+NR . 3.2.2 Training Sequences
To determine the performances of the proposed quantizers,
ex-periments have been performed for samples from memoryless
uni-form, Gaussian, and Laplacian sources, the probability-density
func-tions (PDFs) of which respectivelyare:
f(x)= ( 1 2 p 3 if jxj< p 3 0 otherwise ; (3.1) f(x)= 1 p 2 e x 2 2 2 ,and (3.2) f(x)= 1 p e jxj p 2 ; (3.3)
where 2
is the variance. In the experiments, the Gaussian and
Laplacian sources have 2
= 1, while the uniform source has 2
=
4=3. The gure of merit is the signal-to-noise ratio (SNR),
de- ned as 10log
10
(S=D) dB, where S is the source variance and D
is the quantization error variance (the distortion). The C-language
routines used for generating the random samples use the ran3
function from [Pres88], which implements a routine suggested by
Knuth [Knut81]; ran3 returns a uniform random deviate between 0
and 1. The routine for generating samples from the Gaussian
distri-bution implements the direct method suggested in [Abra65] and the
routine for generating samples from the Laplacian distribution is the
inverse of the integralof (3.3).
Fortheexperiments,atrainingsetofN100000 independentrandom
vectors(N 2
100000i.i.d. samples)isused. Thereasonforthisisthat
100 000 samples, as used in [Marc90a ], turned out not to be enough
for TCVQ, in several experiments. Therefore, as a rule of thumb,
N 2
100 000 samples are used and the nal performance ismeasured
on an i.i.d. sequence not in the training set. It should be remarked
thatfor100000 i.i.d.samples(aswerealsousedin[Marc90a]),forthe
TWCandTCQexperiments,theperformancesobtainedforsequences
notinthetrainingsetarethesameasforsequencesinsidethetraining
set.
3.2.3 Con dence Intervals
To enable the computation of the signi cance, or reliability, of
the computed SNR values, the samples are divided into 100
se-quences (each consisting of N 1000 random vectors). To compute
the con dence intervals, for each of the T = 100 experiments both
the source variance S
i
and noise variance D
i
are considered to be
random variables. The con dence interval is overestimated in this
way, since S
i
in reality is known exactly. The total source and
noise variances are computed as S = 1 P T i=1 S i and D = 1 P T i=1 D i .
Since each experiment involves N 1000 vectors, it is valid to
as-sume that S
i
and D
i
are normally distributed. Thus for S and D
the 100% con dence intervalsare (S z
S = p T;S+z S = p T) and (D z D = p T;D+z D = p T), where 2 S = 1 T 1 P T i=1 (S i S) 2 , 2 D = 1 T 1 P T i=1 (D i D) 2 ,and z
is chosensuch that
Z z z f T 1 (y)dy= ; (3.4) wheref T 1
(y)isthePDFofStudent'st-distributionwithT 1degrees
of freedom [LG89 ]. The probability of both S and D being inside
their respective con dence intervals is and the resulting 2
100% con dence intervalfor S=D is:
(S=D)2 S z S = p T D+z D = p T ; S+z S = p T D z D = p T ! : (3.5) For 2 =0:95, z
=2:27, as can be obtained by solving (3.4), either
numerically (used here)or by table lookup ( 0:975).
3.2.4 Codebook Optimization
To optimize the codebook, 100 iterations were performed using an
algorithm based on that described in [Stew82], but adapted to
main-tain the structures prescribedby the respective constructionsand
ex-tended to TCQ and TCVQ. Although convergence is reached in less
than 100 optimization steps for small trellises atlow rates, large
trel-lises at higher rates require about 100 steps, in our experience. The
optimization algorithm is listed inFigure 3.1. In [Stew82 ], in Step 2,
eachrepresentationsymbolofgeneration k+1isthecentroidof those
elements of the training sequence that were encoded by the
corre-spondingrepresentationsymbolofgenerationk. Fortheconstructions
presented in this thesis, the same sets of representation symbols Y (k)
m
and Y (k)
m
eachoccuratqbranchesofthetrellis. Therefore,inStep2,
noweachrepresentationsymbolofY (k+1)
m
Step 0. Initialization. Given are a training sequence and the initial codebook, C (0) . Set k=0. Step 1. Using C (k)
, the codebook for generation k, encode the
training sequence.
Step 2. Findtheoptimal codebook,C (k+1)
,forgenerationk+1.
Step 3. If k<99, then replace k by k+1 and goto Step 1.
Step 4. Halt with C (100)
as the nal codebook.
Figure 3.1: Codebook optimizationalgorithm.
occurrences of the corresponding representation symbols of Y (k)
m and
the negatives of those elements of the training sequence that were
encoded by any of the q occurrences of the corresponding
representa-tion symbolsof Y (k)
m
. Representationsymbols ontowhichno source
symbols are mapped are updated to zero (the average source value).
Althoughtheconvergenceofthealgorithmisslow,thecodebooksthat
are nally obtained inthe experiments give good results.
For TWC and TCQ, the initial trellis codebooks are chosen
deter-ministically using uniformly spaced levels from the interval ( 2;2).
Contrary to a random initialization, this choice of initial codebooks
guarantees a certain minimal distance both inside each set and
be-tween the sets of the branches entering and leaving each state. The
same initial codebooks are used for all sources. The speci c
initial-izations for constructions A, B, and C can be found inAppendix D.
3.3 TWC and TCQ Experiments
For TWC and TCQ at R =1, R =2, R = 3, and R = 4, the SNR
results are listed in Table 3.1, Table 3.2, and Table 3.3, for
quanti-zation of the Laplacian, Gaussian, and uniform sources, respectively.
Forall SNRvalueslisted,the95%con denceintervalcorrespondstoa
tolerance of no more than 0.003 dB(this result di ers from the
toler-ances givenin[Marc90a ]which rangefrom0.02to0.15 dB;apossible
explanation is that in [Marc90a] it is incorrectly assumed that the
source variance is the same for each of the 100 parts of the training
sequence). For R=1, n equals 1, for R =2,n equals 1 or 2,and for
R =3 andR =4, nequals 1,2,or3(for \pure"TWC,R =n). Note
that the numbers ofstates in the experiments have been restricted to
be powersof q, soas to have an underlying q-aryconvolutional code.
The constructions are easily extended to di erent numbers of states,
however.
TWCsandTCQsatthesamerate,havingthesame numberofstates,
havethesame complexity,accordingtothede nition ofSection3.2.1.
When comparing the SNR results listed in Table 3.1, Table 3.2, and
Table 3.3atthe samecomplexities, itcanbeobservedthat,generally,
construction C gives the best performance (except for the Laplacian
source at R = 1), although the di erences with the other
construc-tions are small. It can also be observed that, generally, the
perfor-mances decrease as the number of (di erent) representation symbols
is decreased (i.e. as q is increased). The TCQs thus outperform the
TWCs, but the di erencesdecreaseasthe complexity (or the number
of representation symbols) increases.
In [Vleu93a], it was shown that at the same number of states, i.e.
at the same complexity, the proposed construction-B TCQs
outper-formthe TCQsof [Marc90a ], forthe Laplacianand Gaussian sources.
For the uniform source the performances of the proposed TCQs
R q States Compl. Symb. A B C 1 2 4 8 4 3.98 4.35 4.33 2 8 16 8 4.31 4.82 4.83 2 16 32 16 4.76 5.16 5.10 2 32 64 32 5.13 5.39 5.35 2 64 128 64 5.51 5.54 5.54 2 128 256 128 5.65 5.69 5.68 2 256 512 256 5.81 5.85 5.79 2 2 16 64 32 10.62 10.63 10.68 2 64 256 128 11.20 11.24 11.27 2 256 1024 512 11.58 11.59 11.65 4 16 64 16 10.29 10.28 10.38 4 64 256 64 11.15 11.15 11.20 4 256 1024 256 11.55 11.56 11.67 3 2 64 512 256 17.11 17.18 17.16 4 64 512 128 17.11 17.11 17.12 8 64 512 64 16.75 16.84 16.86 4 2 64 1024 512 23.00 22.92 22.97 4 64 1024 256 22.97 22.98 22.97 8 64 1024 128 22.69 22.73 22.78
Table 3.1: Experimental SNRs (in dB), complexities, and number
of (different) representation symbols for constructions A, B, and
C,forTWC/TCQ ofthe Laplaciansource atR=1,R=2,R=3,and
R q States Compl. Symb. A B C 1 2 4 8 4 4.78 5.02 5.05 2 8 16 8 4.97 5.16 5.19 2 16 32 16 5.20 5.30 5.30 2 32 64 32 5.31 5.39 5.39 2 64 128 64 5.43 5.49 5.49 2 128 256 128 5.49 5.56 5.56 2 256 512 256 5.56 5.63 5.61 2 2 16 64 32 10.88 11.00 11.05 2 64 256 128 11.22 11.29 11.28 2 256 1024 512 11.44 11.45 11.48 4 16 64 16 10.81 10.89 10.97 4 64 256 64 11.21 11.30 11.18 4 256 1024 256 11.41 11.38 11.48 3 2 64 512 256 17.21 17.24 17.24 4 64 512 128 17.18 17.19 17.23 8 64 512 64 17.02 17.08 17.10 4 2 64 1024 512 23.14 23.16 23.16 4 64 1024 256 23.12 23.14 23.15 8 64 1024 128 23.01 23.04 23.03
Table 3.2: Experimental SNRs (in dB), complexities, and number
of (different) representation symbols for constructions A, B, and
C, for TWC/TCQ of the Gaussiansource atR=1, R=2, R=3, and
R q States Compl. Symb. A B C 1 2 4 8 4 6.14 6.22 6.25 2 8 16 8 6.20 6.30 6.32 2 16 32 16 6.27 6.37 6.37 2 32 64 32 6.33 6.42 6.43 2 64 128 64 6.39 6.48 6.47 2 128 256 128 6.46 6.51 6.52 2 256 512 256 6.50 6.55 6.56 2 2 16 64 32 12.64 12.76 12.78 2 64 256 128 12.79 12.86 12.90 2 256 1024 512 12.88 12.96 12.98 4 16 64 16 12.61 12.71 12.77 4 64 256 64 12.77 12.84 12.90 4 256 1024 256 12.87 12.91 12.98 3 2 64 512 256 19.01 19.12 19.13 4 64 512 128 19.02 19.09 19.13 8 64 512 64 19.00 19.04 19.08 4 2 64 1024 512 25.14 25.27 25.27 4 64 1024 256 25.16 25.23 25.25 8 64 1024 128 25.19 25.17 25.21
Table 3.3: Experimental SNRs (in dB), complexities, and number
of (different) representation symbols for constructions A, B, and
C, for TWC/TCQ of the uniform source at R=1, R=2, R =3, and
R 1 2 3 States VW MF VW MF VW MF 8 4.83 4.47 10.18 9.56 15.87 15.00 16 5.16 4.92 10.68 10.47 16.48 16.20 32 5.39 5.13 10.98 10.73 16.90 16.43 64 5.54 5.35 11.27 10.98 17.18 16.79 128 5.69 5.49 11.44 11.16 17.43 16.84 256 5.85 5.54 11.67 11.22 17.57 16.96 512 5.95 | 11.81 | | | LM 3.01 3.01 7.54 7.54 12.64 12.64 RD 6.62 6.62 12.66 12.66 18.68 18.68
Table 3.4: SNRs (in dB) of the proposed TCQs (VW) compared
with the TCQs of [Marc90a] (MF), the Lloyd-Max quantizer
perfor-mance (LM), and the rate distortion bound (RD), for the Laplacian
source at R=1, R=2,and R=3.
of the proposed TCQs (the best results listed in Table 3.1 and T
a-ble 3.2) with the best results obtained in [Marc90a ], the Lloyd-Max
quantizer ([Lloy82 , Max60]) performance, and the rate distortion
bound [Berg71 ]. For the cases not listed in Table 3.1 and Table 3.2,
the results have been obtained for construction B.
For the Laplacian and Gaussian sources, the proposed TCQs in fact
improve upon all previous results found in the literature (as listed
in [Marc90a ]), asshown inTable 3.6.
3.4 TCVQ Experiments
For TCVQ, the initial trellis codebooks are chosen randomly using
i.i.d. samplesfromthedistributiontobequantized,bothbecausegood
(al-R 1 2 3 States VW MF VW MF VW MF 8 5.19 5.19 10.83 10.70 16.64 16.33 16 5.30 5.27 11.05 10.78 16.90 16.40 32 5.39 5.34 11.14 10.85 17.11 16.47 64 5.49 5.43 11.30 10.94 17.24 16.56 128 5.56 5.52 11.41 10.99 17.39 16.61 256 5.63 5.56 11.48 11.04 17.43 16.64 512 5.68 | 11.57 | | | LM 4.40 4.40 9.30 9.30 14.62 14.62 RD 6.02 6.02 12.04 12.04 18.06 18.06
Table 3.5: SNRs (in dB) of the proposed TCQs (VW) compared
with the TCQs of [Marc90a] (MF), the Lloyd-Max quantizer
perfor-mance (LM), and the rate distortion bound (RD), for the Gaussian
source at R=1, R=2,and R=3.
Source R States VW LIT
Lapl. 1 512 5.95 5.76 2 512 11.81 11.45 3 256 17.57 17.20 Gauss. 1 512 5.68 5.56 2 512 11.57 11.04 3 256 17.43 16.78
Table3.6: SNRs(in dB) of the proposedconstruction-B TCQs (VW)
compared with the performances found in the literature (LIT,as
listedin[Marc90a]),for theLaplacianandGaussiansourcesatR=1,
N q Compl. Symb. Laplacian Gaussian uniform 2 2 256 128 5.65 5.46 6.47 4 256 64 5.69 5.53 6.53 3 2 1024 256 5.65 5.46 6.46 4 1024 128 5.79 5.55 6.52 8 1024 64 5.78 5.56 6.54 4 2 2048 512 5.69 5.46 6.46 4 2048 256 5.85 5.55 6.52 8 2048 128 5.84 5.56 6.54
Table 3.7: Experimental SNRs (in dB), complexities, and number
of (different) representation symbols for 64-state construction-C
TCVQof the Laplacian, Gaussian,anduniformsourcesatR=1,for
severalvalues of N and q.
approximately white spectrum. Table 3.7 lists the performances of
several64-stateconstruction-C TCVQs atR=1;the 95% con dence
intervalscorrespondtoatoleranceofnomorethan0.003dB.Itcanbe
observed that, contraryto the results givenin Section3.3 for N =1,
for the Gaussian and uniform sources, the performances increase as
q is increased, eventhoughthe number of representationsymbols
de-creases with q. For the Laplacian source, q = 8 achieves virtually
the same performance as q = 4, using half as many representation
symbols. Further, for the Gaussian and uniform sources, it can be
observedfromTable 3.7 thatincreasing the numberof representation
symbols, or their dimension, beyond a certain value does not result
in a higher performance; the same performance can be obtained at a
lower complexity,by using lower-dimensional representation symbols.
To further investigate the in uence of the representation symbol
di-mensionontheTCVQperformance,experimentshavebeenperformed
for construction C, for several rates and dimensions, at a constant
complexity. Table 3.8 lists the SNRs obtained for the experiments
with a complexity of 256 at R =1=2, R =1, R = 2, and R =3; the
Source
R N q States Symb. Laplacian Gaussian Uniform
1/2 2 2 128 128 2.96 2.72 3.09 4 4 64 64 3.00 2.74 3.11 8 4 16 64 2.97 2.64 2.99 1 1 2 128 128 5.68 5.56 6.52 2 4 64 64 5.70 5.53 6.50 4 4 16 64 5.58 5.41 6.43 2 1 2 64 128 11.27 11.28 12.90 2 4 16 64 10.78 11.03 12.78 3 1 2 32 128 16.85 17.06 19.04 2 2 4 128 15.68 16.26 18.65
Table 3.8: Experimental SNRs (in dB) and number of (different)
representationsymbolsfor construction-C TCVQof the Laplacian,
Gaussian,and uniform sources at R=1=2, R=1, R=2, andR=3,at
a complexityof 256.
and complexity, increasing N while not simultaneously increasing q
decreases the performance, whereas simultaneously increasing N and
q can increase the performance. In Table 3.8, those performance
in-creases occur in particular in those cases where no parallel branches
are used in the trellis. In Table 3.7 as well, increasing q in general
increases the performance. The explanation for the observation that
increasingqdoesnotalways increasetheperformance(aswasalso
ob-servedinSection3.3)couldbetheassociatedreductionofthe number
of representation symbols.
TheobservationthatsimultaneouslyincreasingN andq,ataconstant
rate and complexity (whilenot using parallel branches), increasesthe
performance agreeswiththetheoreticalboundon thedistortion given
for thiscase inequation (7.4.37)of[Vite79], which, inthe notation of
this thesis, is:
DD(R n )+ d 0 q (Rn Rn(D))=C0 (1 q (R n R n (D)) 2 =2R n C 0 ) 2 e N(+1)Rn(Rn Rn(D))=C 0 ;
Complex. TCQ:MF TCVQ:FMW TCVQ:VW TCQ:VW
32 4.92 5.05 5.15 5.16
64 5.13 5.22 5.34 5.39
Table 3.9: SNRs (in dB), at the same complexity, for the TCQs
of [Marc90a](TCQ:MF),the TCVQs of [Fisc91](TCVQ:FMW), the
pro-posed q =2 construction-B TCVQs (TCVQ:VW), and the proposed
q=2 construction-B TCQs (TCQ:VW), for the Laplacian source, at
R=1. For theTCVQs, N=2.
whereR
n
istherate,expressedinnatspersymbol,D(R
n
)andR
n (D)
are the performance bounds given by rate distortion theory (see also
Chapter 4)and d
0
andC
0
areconstants. Since,ataconstantrateand
complexity,N(+1)isconstant,the exponentofthe bounddoesnot
depend on q. The fraction, however,does decreasewith q, explaining
the performance increaseas q increases.
In [Fisc91], two experiments were presented for a memoryless
Lapla-ciansource,atR =1. Table3.9showsacomparison,atthesame
com-plexities, of the performances of the TCQs of [Marc90a ], the TCVQs
of [Fisc91 ], the proposed TCVQs (q = 2, construction B), and the
proposed TCQs (q = 2, construction B). The proposed TCVQs
out-perform those of [Fisc91 ], but the proposed TCQs are still superior.
In [Wang92], di erent TCVQs and more results were presented. The
SNRs presented in [Wang92] were computed inside the training
se-quenceof1000000samplesofamemorylessGaussiansource. To
com-paretheperformancesoftheproposedTCVQswiththoseof[Wang92],
experiments were performed with the proposed TCVQs, also
us-ing 1 000 000 samples, for several cases selected from the tables
in [Wang92]. The performances were measured both inside and
out-side the training set. Table 3.10, in which the proposed TCVQs are
compared with those of [Wang92], clearly shows that, in the case of
VW WM
R N q Inside Outside Inside
0.5 2 2 2.62 2.62 2.63
1 4 4 5.42 5.41 5.33
2 4 4 11.20 11.09 11.22
3 2 4 16.90 16.89 16.62
Table 3.10: SNRs (in dB), inside and outside the training set, for
the proposed16-state construction-C TCVQs (VW)and the 16-state
TCVQsof[Wang92](WM),fortheGaussiansource,forseveralrates,
R,anddimensions, N.
3.5 Gauss-Markov Sources
Although the experiments discussed in Section 3.3 have shown that
for memoryless sources the TCQs have performances equal or
supe-rior to the TWCs, this could change for Gauss-Markov sources, since
TCQs are not optimal in this case [Marc90a]. Therefore, additional
experiments have been performed for Gauss-Markov sources, which
are de ned by x k = L X j=1 a j x k j +w k ; (3.7) where x k
is the source output, L is the order (memory length) of
the source, a
j
is a real coecient, and w
k
is a sample of a
memory-less Gaussian source as de nedby (3.2). An algorithm for predictive
TWC (theextension ofwhichtoTCQ istrivial) of Gauss-Markov (or
generalautoregressive)sourcesisgivenin[Ayan86 ] and[Gers92].
Ba-sically,foreachtrellisstate,apredictionofthecurrentinputsampleis
made, based on the previous representation symbolsof the best path
entering the state:
^ x k;l = L X j=1 ^ a j ^ x k j;l ; (3.8) where l,1 l q
, indicates the dependency on the state. The
Step 0. Initialization. Given are a training sequence, the
ini-tial codebook, C (0)
, and the initial prediction
coe-cients ^a (0) j ,1j L. Set k =0and t =0. Step 1. Using C (k+10t)
, the codebook for generation k+10t,
and a^ (t)
j
,encode the training sequence.
Step 2. Find the updated codebook, C
(k+1+10t)
, for generation
k+1+10t.
Step 3. If k<9, then replacek byk+1 and go toStep 1.
Step 4. Find the updated prediction coecients, ^a (t+1)
j .
Step 5. If t< 9, then replace t by t+1, set k = 0, and go to
Step 1.
Step 6. Halt with C (100)
as the nal codebook and ^a (10)
j
as the
nal prediction coecients.
Figure 3.2: Predictive codebookoptimization algorithm.
codebook and the prediction coecients (especially for lowrates, the
correlationofthe representationsymbolswillbedi erentfromthat of
the input samples).
TheoptimizationalgorithmusedhereislistedinFigure3.2. Itisbased
onthealgorithmgivenin[Ayan86],butisdi erentintwoaspects. The
rst is Step 4. The predictor update equationsused in [Ayan86 ] are:
L X j=1 ^ a j K X k=1 ^ x k j ^ x k i = K X k=1 (x k y k )^x k i ; (3.9)
for 1i L, where K is the length of the training sequence (in our
case, K = 100 000) and y
k
is the codebook element that is used to
represent the residualx
k ^ x
k
. Forlowrates, (3.9) overestimates the
that ofthe representationsequence; theestimate^a
j
divergesfromthe
realvaluea
j
. Agoodestimatefora
j isobtainedbyreplacing(x k y k ) in (3.9) by x^ k ,resulting in: L X j=1 ^ a j K X k=1 ^ x k j ^ x k i = K X k=1 ^ x k ^ x k i ; (3.10) for 1iL.
The second di erence is that in our algorithm the prediction
coe-cients are updated once every 10 codebook updates, giving a better
estimate of these coecients (in our experiments) than the method
of [Ayan86] where the prediction coecients are updated after every
codebook update. In total, 10 updates are performed for the
pre-diction coecients, bringing the total number of codebook updates
to 100.
Experiments have been performed, for construction C, for an AR(1)
source (L = 1) having a
1
=0:9. The resulting SNRs are listed in
Table 3.11, together with the results obtained in [Marc90a ], the
dif-ferentialpulsecodemodulation(DPCM,seeforexample[Jaya84 ])
per-formance,andthe ratedistortion bound. Forour SNRs,the95%
con- dence intervalscorrespond to atolerance ofno more than 0.003 dB.
Comparing the performances of the TWCs and the TCQs in T
a-ble 3.11, they can be seen to be the same. A comparison with T
a-ble 3.2 shows that the di erences between the performances of the
TWCs and the TCQs are smaller for the AR(1) source than for the
Gaussian source,butforR =2andR=3the TWCsarenot superior
to the TCQs. For example, as can be seen from Table 3.2, the
16-state construction-B TCQfor R=2has an SNRof 11.00dB andthe
correspondingTWChasan SNRof10.89dB|a di erenceof0.11 dB.
Table 3.11 shows that for the AR(1) source, the 16-state TCQ and
TWC have the same performance, i.e. 17.95 dB. A comparison with
Table 3.5 shows that the performance di erences between the
pro-posed TCQs and those of [Marc90a ] are much smaller for the AR(1)
R q States VW MF DPCM RD 1 2 4 11.34 11.19 10.00 13.23 2 8 11.63 11.60 2 16 11.82 11.89 2 32 12.08 12.13 2 64 12.19 12.22 2 128 12.31 12.41 2 256 12.38 12.49 2 2 16 17.95 17.95 16.07 19.25 2 64 18.35 18.24 2 256 18.59 18.41 4 16 17.95 | 4 64 18.34 | 4 256 18.61 | 3 2 64 24.32 23.90 21.69 25.27 4 64 24.32 | 8 64 24.27 |
Table3.11: Experimental SNRs (in dB) for predictive TWC/TCQ of
the AR(1) source for construction C (VW), at R =1, R =2, and
R =3, compared with the predictive TCQs of [Marc90a] (MF), the
the proposed TCQsare superior tothoseof[Marc90a ]formemoryless
sources,the reasonfor therelatively lowerperformance forthe AR(1)
source could be the optimization algorithm that was used, which is
di erentfromthat of [Marc90a]. In [Marc90a ], it isassumedthat the
predictionresidualsareGaussianand,consequently,ascaledversionof
the optimal codebookfor the Gaussian sourceis used toquantize the
residuals. Nevertheless,our codebook should converge to acodebook
adaptedtotheresiduals'PDFandagainourTCQsshouldoutperform
those of [Marc90a ]. The reasonthis isnot the case is that(as our
ex-periments show) the codebook update, Step 2 in Figure 3.2 (which
is the same as Step 2 in Figure 3.1), can actually decrease the
per-formance. In our experiments, this occurred more often as the rate
decreased.
Another explanation for the lower performance is the fact that,
due to the quantization errors, the prediction error signal is
non-white[Jaya84 ],con ictingwith theassumptionofmemorylesssources
in the design of the proposed TCQs.
3.6 The M-Algorithm
The main disadvantage of the Viterbi algorithm is the fact that its
complexity increases exponentially with the constraint length, .To
circumvent this problem, a (suboptimal) reduced-state search
algo-rithmcan beapplied. Onesuchalgorithm isthe M-algorithm [Jeli71 ,
Jaya84 , Gers92]. An ecient implementation of this algorithm has
been found [Vinc88a ,Vinc88b, Vinc90].
ContrarytotheViterbialgorithm, whichfollowsq
pathsthroughthe
trellis(i.e.one perstate),theM-algorithmonlyfollowsM q
paths
through the trellis. At each step, the M paths are extended to qM
paths and of these paths only the M best ones (i.e. the ones having
Source
R Laplacian Gaussian Uniform
1 6 5.52 5.47 6.48 7 5.59 5.52 6.52 8 5.61 5.49 6.52 2 6 11.26 11.28 12.88 7 11.28 11.32 12.91 8 11.27 11.32 12.92 1/2 6 2.88 2.71 3.05 7 2.91 2.74 3.07 8 2.93 2.74 3.10
Table 3.12: Experimental SNRs (in dB) using the M-algorithm with
M=64forq=2construction-BTWCs,TCQs,andTCVQs(N =2), at
R=1, R=2, andR =1=2,respectively, for the Laplacian, Gaussian,
anduniformsources.
Forq =2,variousconstruction-B TWCs,TCQs,andTCVQs(N =2)
at R =1, R = 2, and R =1=2, respectively, have been designed for
the Laplacian, Gaussian, and uniform sources using the M-algorithm
with M =2 k
, k =0;1;:::;9. Training sequences of 800 000 i.i.d.
samples were used and the performances were measured outside the
training set. Itwasfound that the quantizer performance mainly is a
function of M only. Increasing the number of states, 2
, results only
inaninsigni cantperformanceimprovement. Toillustrate thetypical
behaviour, Table 3.12 lists the obtained performances for M = 64,
as a function of .For = 6 the Viterbi algorithm performance is
obtained.
3.7 Discussion
TheobservationthattheproposedTCQshaveperformancesequal(for
The di erences between the proposed TCQs and those of [Marc90a ]
are thattheyarebasedon di erentconvolutional codesand thatthey
useadi erentnumberof(di erent)representationsymbols. Wedonot
knowwhethertheTCQconstructionof[Marc90a]generallyguarantees
a white spectrum.
The di erent convolutional codes probably do not account for the
performance di erences: the di erent constructions, A, B, and C,
presented in this thesis have about the same performances. Also,
in[Marc90a ], asearchwasperformedto nd convolutional codes with
better performances thanUngerboeck's codes, but littleimprovement
was obtained.
The di erence inthe numberof di erent representationsymbols
pro-vides a better explanation for the performance gain. As shown
in [Eyub93], the gain of a TCQ over a uniform scalar quantizer can
be separated (asymptotically, at high rates) into two components:
the granular gain and the boundary gain. The granular gain arises
fromamore ecientlocalspacecovering. Forthe mean-squarederror
distortion measure, used in this thesis, the granular gain is at most
0.255b/sample. The boundarygainarisesfromamoreecientglobal
space covering,i.e. it iscausedby the ability of the TCQto adaptits
representation symboldensity to the source density. Whereas for the
uniformsourcethereisnoboundarygain,fornon-uniform sourcesthe
boundary gain can bemuchlarger than the granular gain.
In [Marc90a ], forthe Gaussian andLaplacian sources,respectively,at
most4and8di erentsetsofrepresentationsymbolsareused,whereas
theproposedconstructionsuseq
di erentsetsofrepresentation
sym-bols fora q
-state TCQ.Since the proposed TCQsuse moredi erent
representation symbols, they are better able to adapt to the source
density. The conjecture that the gain of the proposed TCQs over
those of [Marc90a ] is attributable to the boundary gain is supported
by the observation that, for the uniform source, the proposed TCQs
do not provide a gain over those of [Marc90a ]. It is also supported
States Src R 8 16 32 64 128 256 512 LM RD Lap 1 0.94 0.96 0.96 0.97 0.98 0.98 0.98 1.00 1.00 2 1.87 1.91 1.92 1.95 1.95 1.96 1.95 1.72 2.00 3 2.84 2.88 2.91 2.92 2.93 2.92 | 2.57 3.00 Gau 1 0.99 0.99 0.99 1.00 0.99 0.99 0.99 1.00 1.00 2 1.96 1.97 1.98 1.97 1.98 1.98 1.97 1.91 2.00 3 2.94 2.95 2.96 2.96 2.96 2.94 | 2.82 3.00
Table 3.13: Entropies of the proposed construction-B TCQs
com-pared with the Lloyd-Max quantizer (LM) and rate distortion
the-ory (RD) values, for the Laplacian and Gaussian sources at R =1,
R=2,and R=3.
representationsymbolsare selectedwith equalprobability. Table3.13
lists the entropiesof the construction-B TCQs,for theLaplacian and
Gaussian sources,as a function of the rate and the number of states.
The entropies increase with the number of states and it can be seen
that it is a good approximation to assume that all branches are
se-lected with equal probability, for the proposed TCQs.
The better the representation-symbol density of the TCQ matches
the source density, i.e. the higher the boundary gain, the more all
representation symbols will be used with equal probability. Thus,
the entropy indicates how wellthe TCQ exploits the boundary gain.
In [Marc90a ] the entropies were not determined. We conjecturethat
they are lower than the entropies for the proposed TCQs.
3.8 Conclusions
In the experimentsfor the memorylessGaussian, Laplacian, and
uni-form sources, at the same rate and complexity, the proposed TCQs
com-as q is increased, even though the number of representation symbols
decreases with q. The bestTCVQ performance isobtained by
simul-taneously increasing N and q, ata constant rate and complexity.
For the memoryless Gaussian and Laplacian sources, the proposed
TCQs at1, 2, and 3b/sample improve upon all previously published
results (as listed in [Marc90a]). For the uniform source, the
perfor-mance equals that of [Marc90a ]. We conjecturethat the gains of the
proposed TCQs over those of [Marc90a] are attributable to a higher
Rate Distortion Theory for
Trellis Waveform Coding
4.1 Introduction
The discipline ofinformation theory that treatsquantization is called
ratedistortiontheory. Itsmain ndingisthatthereisafunctionR(D),
the rate distortion function,whichspeci es the e ectiverate R ofthe
sourcewhenitsoutputsmustbereproducedwithanaveragedistortion
of no more than D. The foundations of rate distortion theory were
laid by Shannon,in1948 [Shan48a,Shan48b] and1959 [Shan59]. The
book on the same subject, written by Berger [Berg71] in 1971, has
become a classic.
As an extension to the traditional rate distortion theory for
continuous-amplitude sources with a continuous representation and
discrete-amplitude sourceswith discrete representations,Pearlman et
al. developed a rate distortion theory for memoryless
continuous-amplitude sources with discrete representations, which they call
