Signal Processing with Lapped Transforms

191 Pages·2016·44.16 MB·English

by  Malvar H.S.

Signal Processing with
Lapped Transforms
I' ,"
Signal Processing with
Lapped Transforms
Henrique S. IVlalvar
Universidade de Brasflia, Brazil
ARTECH HOUSE
Or
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SignAl PI'OC 88il1g with lupp d transform8 / IIOl11'iqu S. Malvar.
p. cm.
Includo8 bibliographical references and index.
ISJ3N0-89006-467-9
I. ignal processing-Digital techniques. 2. Signal processing
Mathematics. 3. Transformations (Mathematics) I. Title.
IL. TiLle: Lapped transforms. Ill. Series.
'1'J(5102.2M275 1991 91-35984
G2J.382'2-dc20 CIP
To my parents, Henrique and Gilsea
UdU~b Library Cataloguing in Publication Data
MlIlvar, Hcnrique S.
Si!:nal processing with lapped transforms.
I. 'Pitlc.
(;21.3822
ISBN 0 9006-467-9
(O1!)!)2 AHTECH HOUSE, INC.
(JAG untollStreet
NlIl'wood, MA02062
All rights reserved. Printed and bound in the United States ofAmerica. No
I)/'I't ofthis book may be reproduced or utilized in any form or by any means,
(llu't"onic or mechanical, including photocopying, recording, or by any
irlfOl'l1'1ution storage and retrieval system, withoul permission in writing
1"1'0111 the publisher.
Int I'national StandardBook Number: 0-8900G-~67-9
L.it rary ofCongress CatalogCard Numb 1': 91-35984
l098766~32J
Contents
Preface xiii
1 Introduction 1
1.1 Signal Models 2
1.1.1 Deterministic and Stochastic :LvIodels 2
1.1.2 Power Spectrum 4
1.1.3 Aut.orcgressive Models 7
1.1.4 Spectral Flatness 8
1.2 Block Transforms. 9
1.2.1 Basic Concepts 10
12.2 Discrete Fourier Transform 12
1.23 Discrete Hartley Transform 14
1.2.4 Karhuncn-Lotvc Tram.form 15
1.2.5 Discrete Cosine Transform. 19
1.2.6 Type-IV Dif:crete Cosine Trausfonn 20
1.2.7 Other Transforms 21
1.2.8 Two-Dimensional Transforms 22
1.3 De,-elopment of Lapped Transforms 22
2 Applications of Block TransfortllS 31
2.1 Signal Filtering 32
2.1.1 Efficient FIR Filtering 32
V11
,·/d (:ON'/'/IN"'S ('ON'/'/';N"'S ix
2.'1.' Mlllticlul.llnd Fill.l·rin,a: 38 ~.icJ Block Tra,m·:fnl'ln~V('l':"ll~ Filt.er Banks 127
..1.3 Adnpl,ivl' Fill,lTiuJ,!; 38 3.6 Applications. 130
2.' SW·Ct.1'11111 E~l.illlrt.l;inll . 44 3.61 Signal Filtering 130
,:1 ']'nln~f()rm Cntling 47 3.6.2 Adaptive Filtering 133
~."1 Otlll'!' Applira.t.itll1s 55 3.6.3 Spectrum Estimation. 134
2,rj Past. AIgoritlulls . 55 3.64 Signal Coding. 134
2.:i.] Discl'd,(' FOllricr Transform 56 3.6.5 Other Applications 135
2.~.2 Discn.~.t.c Hartley Transform 63 37 Summary 136
2.0.:1 Di~t:.l'('j.(' Cosine Transform. 67
4 Lapped Orthogonal Transforms 143
2..J.!1 Typ('-IV Discrete Cosine Transform 71
4.1 Theory of Lapped Transforms 144
2..J.G Complttational Complexities 74
4.1.1 Time-Domain Analysis. 146
,.6 llllllllary 75
4.1.2 Connection with Filter Banks 148
:1 Sigllal Processing in Subbands 81 4.2 The Lapped Orthogonal Transform 152
:1.1 tvl1l11.ira,l,(' Sign;)] Processing 82 4.2.1 Recursive LOT Optimization 152
31.1 DC'cimation and Interpolation 82 4.2.2 Quasiopt.imal LOTs 155
31.2 Cas(';'ule Connections. 86 4.3 Fast. Algorit.hms for t.he LOT 161
3.1.3 Polyphase Decompositions. 88 4.3.1 Structure of the Fast LOT. 161
:\.' Filt,c'1' Banks. 89 43.2 LOTs of Finit.e-Lengt.h Signals 166
3.2.1 St.ructures FB-I and FB-II . 89 4.4 Fast LOT for AI > 16 167
32.2 Signal ReconstructioIl 91 4.5 Coding Performances. 170
:3.2.3 Computat.ional Complexity 93 4.6 Summary 171
3.2.4 DFT Filt.er Banks 94
5 Modulated Lapped Transfonns 175
:3.3 Quadnl,tllre rvIirror Filters 100
5.1 The MLT 176
3.3.1 Two-channel QMF Banks 100
3.3.2 QMF Banks for AI > 2 106 5.2 Extended Lapped Transforms 180
52.1 Perfect. Reconstruction. 181
3.'1 PC'rfcet Reconstruction 109
5.2.2 Properties 185
3.4.1 Two-channel PR Filter Banks. 110
3.4.2 PR Filt.(·f B;mks for 111 > 2 119 5.3 Design Techniques 190
xi
193 7.2.1 Fin Filkril1~ 254
;'.4.1 Fast ELT rnr J\ = l (Mr:n 199 7.2.2 Adaptive Filtering 257
(;.4.2 F"ISt. ELT f,.,. l\" =2 202 7.3 Speech Coding 2Gl
205 7.4 Image Coding 2G5
,).4.4 Cnlllpnta/'.inllal Complexity 209 7.5 Other Applications 273
G.4..1': ELT~ (If Finitt~-Length Signals. 210 7.G Summary 274
G.5 'odinJ,'!; Pcrforlllnncc 212
Appendix A Tables of CQF Filter Coefficients 277
!).fj SllrnIHary 21(;
Appendix B Pragranls far Black Transfanns 281
n Ilicl':wchical Lapped Transfonlls 219
G.l NOlll1uiform Filter Banks 220 Appendix C Pragranls far Lapped TransfarIns 315
C.l.I Dirf'ct Form. 220
Appendix D Tables of ELT Butterfly Angles 345
G,].2 Tree Structures 222
224 Index 353
6.2.1 HLT with Octave Band Splitting 225
6.2.2 Generalized HLT . 227
1i.3 CUlllwdinJ):-; with \Vavelet Transforms 229
6.3.1 The HLT as a rvlultiresolut.ion Transform 230
G.3.2 Etluivalcnt Subband Filters 231
G.3.3 vVavclet, TrnIlsforms with the HLT 233
G.3.4 Regularity. 234
G.3.5 Examples of \iVavclet Transforms 235
6.3.G Computational Complexity 240
Cnding Performance 241
$11Illlnary 244
7 Applications of Lapped Transfornls 247
7.1 Sp(·d.rum Estimation. 247
7.2 Signal Filtrdng . 254
Preface
Digitalsignalprocessing(DSP) ha..<;; beenagrowingfield for mort>than threedecades.
Wit.h the availability of fast integrated circuits dedicated specifically to DSP ap
plications, we now live in a world where DSP is not just a hot research topic, but
part of our everyday life. If wc look at wha.t is attached t.o a standard telephone
line in our modern office, for example, wc sec modems) fa-x machines, and tapeless
i1nswering machines: all of which could not. exist without DSP, vVc could certainly
:-pcnd many pages describing examples of DSP applications..
Ach-ances in DSP have been so many that. specialized arCilS within it are thcm
~elves becoming new fields. Among them, wc may cite spe<:'ch processing, image_
processing. adaptive filtering, and ffiltltirat.esignal processing. In all of thef:c areas,
fast transform~are frequently used, because it. is often more efficient to process a
:-;ignal in the t.ransform domain.
The purpose of this book is to present to the reader a. complete reference on
t.he theory and applications of a new family of transforms, called lopped l'fal1S
/0,.,n8 (LTs). These trnnsforms can be used in many applications, such a!' filtering,
coding, spectral estimation, and any OtlH.:'fS where a tradition,,} block tnlllsform is
employed,suchas t.he discrete Fourier transform (DFT) or the discret.ecosine trans
form (DCT). In manycases. LTswill lead toa bet.ter complexityversusperformance
trade-off than other transfonus. Until now, the theory of lapped transforms and
many oftheir applications have appeared in theses, journal articles, and conference
proceedings. This i:- the first. book in which all ofthe known result.s are put together
in an organized form.
INe believe that. thi:- book is a u!'ieful referencefor designengineers, graduate stu
dents, and researchers involveo with DSP ;:\pplications that make use of fast traIlS·
forms. Many signal processing systern:o; employingfast. transforms arc presented, as
well as evaluations of implementations of those syst.cms. Thus, the reader with a
practical application in mind will be able to put LTs to work to/his or her benefit
X1Jl
XI'
illllllt'.liul.·ly. P(lrlhllt. plll'pn:-;I', \V,'!WV" inl'1lld,·d ill t.hl~ ;q)lll'nrlin'~ lislillj,(s of "0111 :I II~v·1 of ddllil previollsly ullavailable ill t.llf' lit.erat,l1rc. The chaptet· ends with a
pul('1" 1'1'0,ll;l'lllI1S wil,il fi'''1. alp;ol'itlllllS for lap\)!'d lransful'Ills, a.... \Vdl ;1.... program::; for jIlI'c)[l,tical dit'('llssion of t.he .oding performance of the MLT and ELT.
Il'lIdilillnal Illcwk transforms hasl'd (lU OpLill1ii'pd alp;oritllllls.
The hicrarrhic"llapped transform (HLT) is discussed in Chapter 6. HLTs ore
As is IIH' (';tS(' wit.h allY u,~'" lopi,', tlwrC' ,lr(' many inh'n~stingthC'oretical issues lI!"cful for multiresolutionsignal analysis and coding because HLTs are in filet filter
illVnlvill,l!; LTs, for l~xilmplC', t.he rcliltionships that exist among LTs, multirate filter hanks with subbands of unequal widths, and impulse responses ofdifferent lengths.
hlllLk~, aliiI disndl' wilw·ld t,rallsforl1l:-:i. Throughout t.he book. there are sections 'Tn"'C' structures for the HLT are discussed, as well as the connections with the
<l1'vlIII'd '·lltin·l)' t,o thcs£' a.nd ot.her t.llC'oretical aspects. The reader is only assumed discrete wavelet transform.
In !litVI' it solid backgrouud in the bnsic theory of discrete-time signal processing, Applications of lapped transforms are discussed in Cha.pter 7. Examples of
IllcllHlillj.!; Ill!' flllldanH'ntals offnndom signal representat.ion and int.roductory linear
the use of LTs in signal filtering a.rc considered, with emphasis on adaptive filters
nl}\,·1 U btccUy, this book will be eyen more useful to those already familiar
fnl, lIdllll and variable filters, The use of LTs in signal coding is also discussed, with many
with t,ll'~ irllJlll'lfwutation of signal processing systems t.hat. employ fa~t transf()rm~
exarnples ofthe results obtained with speech f\.nd image coders based on LTs. From
01' flll.·1' hnl1ks. these results, it becomes clear th..t oneofthe main advantagesoflapped transforms
'l11lptf'l'l st.a.rt.s witha briefre\'iewofsignal modelsand the basicdefinitions and over traditional block transforms is the strong reduction in the discontinuit,if's in
l'IIIIII'l'til'sof t,mditional block trilIl~formsand lapped transforms. A brief history of the reconstructed signal at the block boundaries, the so-called blod..in.g cJJ(cJs.
1111' dl'VI'\{)plllf'llt of lapped transforms is also presented. In Chapter 2, some of the
The appendices presentvaluable information for the reader interestedin putting
Ilppli(·ld.iQ!lS of block transforms are discussed, with emphasis on the DFT. DHT,
t.he ideas in this book to work immediately in his (or her) application t.hat requires
llllt! till' 0 ,T. The current state-of-art of fast. algorithms for these transforms is
a. block transform or a filter bank. "Vhen the desired number of bands is two,
1',·vi'·Wt·t!, aud the best known procedures for their computation are presented.
a good alternative for the implementation of perfect-reconstruction filter banks is
'i'lli' !'iISi('s of multirate signal proce!'sing are discussed in Chapter 3, with the
t.he conjugate '1uadraturc filters (CQFs). Thus. wc have included" table of CQF
pilipmll' n[st.lldying maximally-decimated filter bi1nks, which arc essential building coefficients in Appendix A. Computer programs for fast computation of~oll1eofthe
ld.wk:'l SlIbh;llld coding systems. Special attention is gi\'en t.o quaclratw'c mirror
(If most conllTIonly l1~ed blocktransforms1.\representcdin Appendix B. In Appendix C
11It1'1' (QMF) b'lllks a.nd perfect. reconst.ruction (PR) filter banks, including conju thereareseycraltablesofbut.terflyanglesfor theMLT andin Appendix Dcomputer
I
,LVll,t· '1lladratllrf' filtf'rs (CQF). The chapter discll:-isf:'S tile fundamental idea that programs for fast computa.tion of LTs are presented. The progra.ms are all written
11'1111111'0"01 (,lIding i::; in fact a special case of ~llbband coding, and also discllssrs in the (~C"..language for increased port<lbility.
litl.'lty till' i,pplications of subband signal processing.
III '1lilpttT 4. the theory and propertie!i of the lapped orthogoni'l} t.ri'lnsform
(Ij()T) ILn' ~t.lIdiedin dctnil. The theoretical aspeds leading to tlH' PR proPf'rty of
Illpp,·d ll'an!oiforms are discussed. within the context that. a lapped t.ransform is a
lull11lid ~'xlf'nsion of CL regul"r block transform. This exten!'ion is directed towards
I'llllill,l!. till' jransforminto a filter bank with impro\'edfrequency resolution. Design
1"j'hlliqlll'Salld fast algorithmsfor the LOT are presented. The coding perform<lnce
or 1111' LOT, which is better than that of the DCT. is also diHussecl.
Thl' llIodulat.ed lapped trill1sfonn (i\IILT) family of LTs is studied in Chapter 5.
t0A'·t111'1' wit.h it.s gCllf'ralizf'd vrrsioll, t.he l'xh.'udt'd lapped transform (ELT). A dp
!l,il,'d dis<"ut'silln of t.he df'sign tcchniques for t.lll'p;:rnf>ration ofoptimized :\ILTsand
I'~l.../TH is prl'sf'utcd. F'lst algorit.hms for 1.111' MLT and ELT are also present.ed at
l'I{/~1"I\ '/~
Acknowledgements
T~ere are many people who had a strong intiuencc on tIll' ma.terial presellted ill
thIs.book, ar~d to whom I would like to thank. Professor Da,vid H. Staelin, my
thesIs supervIsor at M.LT. imd a good friend, has alwa.ys been vcry supportive and
encouraging, giving me the right advice on everything. He was the originatorofthe Chapter 1
term lapp,d transform. The research team that developed the basic ideas behind
the lapped orthogonal transform at M.LT., back in 1984, also inclnded Philippe
Cassereau, Brian Hinman, and Jeff Bernstein. 11any encouraging discussions on
the theory and applications oflapped transforms and related topics were held with Introduction
Dr. A. Brian<;on, C. Clapp, M. Cruvinel Dr G de Jagcr Prof P S R D'InI.Z,
. , .• (~ I •.••
R. Duarte, Dr. P. Duhame!, Dr. S. Erics~on, Dr. D. Le Gall, Dr. F. A. O. N<lscirnento.
A. Popat, Prof. K. R. Rao. R. Saraiva Jr.. Dr. J. Shapiro, Prof. M. J. T. Smith,
Dr. B. Szabo, Dr. A. Tabatabai, Prof. P. P. Vaidyanathan, Prof. M. Vett.erli. Prof.
A. S. Willsky, and Dr. G. Womell.
This is Cl book on signal processing with lappcd transforms (LTs). At first, this
In part,icular.l wouldlike toexpressmysinccre thanks to Ricardo L. de Quciroz might seem to be an obscuresubject because LTs arc relatiyely new. This is not S0,
f~rhis ~lanysuggestions on themanuscript, and for carrying out allofthe computer however, and LTs are becoming more attractive for a wide variety of applications.
simulatIOn!'ofthe applications oflapped trrlnsforms to image processing. I am also This is mainly because LTs are a, special family of filter bank~ t.hat can be ea.sily
thankful to Edual'do M. Rubino, for writing a family of TEXt device drivers that designed and implement.ed. even for Cl large number of ~ubbands. Throughout. this
allowed this book to be entirely typeset by the author. Eduardo also wrote t.he book, we shall see thnt LTs can lead to better syst.cm performance than other
soft.w~re that produced thehalf-tonc imf\gesofChapter 7. Theencouragement. that more usual transfonns, like the discrete Fourier transform (DFT) or discrete cosine
I rccel\'ed from Mark vValsh. Pamela Ah!, Kim Field, and Denllis Ricci of Artech transform(DCT), in applicat.ionssuchasimage coding,speech coding. and adaptiye
House helped keep my peace of mind as I was writing this book. It was certainly filtering. In any application where a block transform or a filt.er bank is employed. a
a pleasure working with them. lapped transformcan"hobeused, since block transfol'm~i:Uld lappedtran:o:fol'mscan
always be viewed a~ special cases offilter banks. As wc will see in later chapters, in
The financial support from the Brazilian Government., through t.he Constlho
many cases LTs \vil1lead to n bet.lcrsignal represent.at.ion or reduced computational
Nacionnl de. Dcsen,vo!tlimenlo C'icuUfico e TecnolOgico - CNPq, is gratefully acknowl
ed,c;ed. CNPq supported mo~t of my research on lapped transforms s'ince 1987 complcxity, or both) when compi'lred to the most commonly used block transforms
or filter hi'lnks.
I.hrough grants nos. 404.963·87,404.519-88,300.159_90, and 600.047-90. '
Finally..a special note ofgratitude goes to my wife Regina Helena. mydaughter Before we start discussing LTs, it is important, t,hat we re\'iew the basics of
Ana Beatnz, and my son Henrique. Each hour spent writing thi~ book was an dis<:rete-time signal represent.at.ion, sot.hat wemakcdear what wemeitn by a signal.
bOilI' taken away from thcm; a.nd there \vere many, many such hours. Viithout t.heir This is what we shall do in Section 1.1, where we will folIo"\\' a statistical approach
patience and understnnding. t.his book could not have been written. .
towi"!rrl signal modeling. Looking at. signals as sample functions of st.ochastic pro
cesses helps to predict quite accurately average:-ystem performance. ,",Ve must also
review the ba.sic concepts behind h'adit.ional block transforms to support, our later
discussion ofLTs, and this is the goal ofSection 1.2. A briefintrodndion to lapped
ITE/X is a trademark of the American M<ll.hematical Society.
transforms. including Cl re....icw of their history, is presented in Section 1.3.