Efficient design methods for multirate filter banks and their applications

MULTIRATE FILTER BANKS AND THEIR

APPLICATIONS

by

Hua Xu

M. Eng , S outheast U niversity, 1991

A D issertatio n S u b m itte d in P a rtia l Fulfillm ent of the R equirem ents for th e Degree of

D O C T O R O F P H IL O SO PH Y

in the D e p a rtm e n t of E lectrical and C om p u ter Engineering

W e accept this dissertatio n as conform ing to th e req u ired sta n d a rd

Dr. A. A ntoniou, C o-supervisor, D ept, of Elect, and C om p. Eng.

D r. W .-S . Lu) C o-supervisor, D ept, of Elect, and Com p. Eng.

Dr. P. A gatlipfflfs, D e p a rtm e n ta l M em ber, D ept, of E lect, an d C om p. Eng.

Dr.'TA-^’isher, O utsTdrrM cniber, D ept, of H ealth Inform ation Sen

Dr, M. 0 . A h m ad , E x te rn a l E xam iner (C oncordia U niversity) © I lu a X u, 1995

U niversity of V icto ria

All rig h ts reserved. T h e d issertation m ay not be reproduced in whole or in p a rt, by ph o to co p y in g or o th e r m eans, w ith o u t th e perm ission of th e au th o r,

Supervisors: Drs. A. Antoniou and W.-S. Lu.

Abstract

Over the last decade, multirate digit:.i.l signal processing techniques have found many

applications in speech and image compression, the digital audio industry, statistical

and adaptive signal processing, numerical solution of differential equations, and in

many other fields. Research activity in the area of multirate signal processing has

been growing quickly and is producing a large amount of new literature in the

design of multi rate digital filter banks, rrnltidimensional multirate systems, wavelet

representations, and their applications in audio and

im~ge

compression etc.

This tl:esis is dedicated to the design of multirate filter banks and their

ap-plications in audio and image compression, and it consists of three parts. Part I is

concentrated on the design of 1-D filter banks. Several methods are proposed for the

design of two-channel QMF banks with near-perfect reconstruction. The methods

have improved design efficiency and lead to QMF banks with satisfactory

perfor-mance. Low-delay QMF banks, which are highly desired in real-time applic:itions,

are also considered. A null-space projection method is then proposed to design

pc!rfcct reconstruction QMF banks with either linear-phase responses or low

recon-struction delays. On the design of multi-channel cosine-modulated QMF banks, an

efficient method is proposed which can be used in the design of both conventional

and low-delay QMF banks. Part II is devoted to studies on nonseparable 2-D

fil-ter banks where the design of four-channel hexagonal QMF banks and two-channel

diamond-shaped QMF banks are investigated, leading to several efficient design

methods. Part III describes applications of the 1-D and 2-D filter banks designed

by

the' design methods developed in Parts I and II of the thesis in audio and image

E xam iners:

Dr. A. A ntoniou, C o-supervisor, D ept, of Elect, and C om p. Eng.

---A---Dr. W .-S. Lu, C o-supervisor, D ept, of Elect, and Com p. Eng.

Dr. P. AgatMwdis, D e p a rtm e n ta l M em ber, D ept, of Elect, and Com]). Eng.

Dr. P. F ish er, O u tsid e M em ber, D ept, of H e a lth In form ation Sci.

A cknow ledgem ents

T h e a u th o r w ould like to th a n k his supervisors, Professors A. A ntoniou and W .-S. lai of th e D e p a rtm e n t of E lectrical and C om p u ter Engineering, for th eir encouragem ent, p a tie n ce a n d advice du rin g th e course of this research, and for th e ir help in the p re p a ra tio n of th is thesis.

F in a n cia l assistance provided by Professors A. A ntoniou and W .-S. Lu’s research g ran ts fro m M icronet, N C E P ro g ram , and NSERC is also gratefully acknowledged.

C o n ten ts

A b str a c t _ii D e d ic a tio n _iv A c k n o w le d g e m en ts V C o n te n ts _vi L ist o f T ab les _{x ii} L ist o f F ig u r es _{x v iii} L ist o f A b b r e v ia tio n s _{x ix} 1 In tr o d u c tio n ₁

1.1 Filter Banks and Design Considerations... *

1.1.1 Multirate Filter B a n k ...

1.1.2 Design P r o b le m s ...

1.2.1 Q uadrature-M 'irror F ilte r B a n k ...

1.2.2 F ilte r B anks w ith P erfect R e c o n s tr u c tio n ... 1.2,3 C osine-M odulated Q M F B a n k ... 1.2.4 Low-Delay F ilte r B anks ... 1.2.5 2-D F ilte r B a n k s ... 1.3 Scope o f th e Thesis

2 Im p ro v ed Ite r a tiv e M e th o d s for th e D esig n o f Q M F B an k s

2.1 I n t r o d u c t i o n ... 2.2 Design o f C onventional Q M F B a n k s ...

2.2.1 O b jectiv e F u n c t i o n ... 2.2.2 Im proved Ite ra tiv e A l g o r i t h m ... 2.2.3 Design E x a m p l e s ... 2.3 Design o f Low-Delay Q M F B a n k s ... 2.3.1 G eneral Tw o-C hannel Q M F B ank ... 2.3.2 Ite ra tiv e M e t h o d ... 2.3.3 E q u irip p le C om plex R econstruction E rror ... 2.3.4 Im proved Ite ra tiv e M e th o d ... 2.3.5 A C ase S t u d y ... ... 2.4 F u rth e r A nalysis on th e Ite ra tiv e M e th o d ... 2.4.1 T h e In itia l P o i n t ... 2.4.2 T h e Effect of S m oothing P a ra m e te r r ... 2.4.3 T h e Effect of W eighting a ... 2.5 C o n c lu s io n ... 6 7 7 8 9 11 14 14 17 17 18 22 24 21 27 32 37 43 50 51 51 53 54

3 T im e -D o m a in A p p roach es for th e D esig n o f Q M F B a n k s 55 3.1 I n tr o d u c ti o n ... 55 3.2 Design o f C onventional Q M F B a n k s ... 56 3.2.1 P ro b lem F o r m u la tio n ... 56 3.2.2 Ite ra tiv e A p p r o a c h ... -57 3.2.3 Design E x a m p l e s ... 60 3.3 Design of Low-Delay Q M F B a n k s ...62 3.3.1 P ro b lem F o r m u la tio n ... 62 3.3.2 Ite ra tiv e A p p r o a c h ...63 3.3.3 Design E x a m p l e s ... 67 3.4 C o n c lu s io n ...68

4 D e sig n o f Q M F B an k s w ith P e r fe c t R e c o n str u c tio n 70 4.1 I n t r o d u c t i o n ... 70

4.2 Design of L in ear-P hase Q M F B a n k s ...71

4.2.1 P ro b lem F o r m u la tio n ...71 4.2.2 N ull-Space P ro je c tio n A p p r o a c h ... 72 4.2.3 D esign E x a m p l e s ...75 4.3 Design o f Low-Delay Q M F B a n k s ... 77 4.3.1 P roblem F o r m u la tio n ..., ... 77 4.3.2 N ull-Space P ro je c tio n A p p r o a c h ...79 4.3.3 Design E x a m p l e s ...81 4.4 C o n c lu s io n ... 84

5 D e s ig n o f M u lti-C h a n n e l C o sin e -M o d u la ted Q M F B a n k s 86

5.1 I n tr o d u c ti o n ... 86

5.2 C onventional C osine-M odulated Q M F B a n k ... 88

5.2.1 R e v ie w ...88

5.2.2 New Ite rativ e M e t h o d ...f)l 5.3 Low -D elay C osine-M odulated Q M F B a n k ... 95

5.3.1 A G eneral C osine-M odulated Q M F B a n k ... 95

5.3.2 Design M e t h o d ...98

5.4 Design E xam ples ... 102

5.4.1 C onventional F ilte r B a n k s ... 103

5.4.2 F ilte r B anks w ith Low R econstruction Delays , ... 107

5.5 C o n c lu s io n ... JIG 6 Im p ro v ed D e sig n s o f 2 -D N o n sep a ra b le F ilte r B a n k s 112 6.1 I n t r o d u c t i o n ... 112

6.2 Design of H exagonal Q M F B a n k s ... 114

6 .2.1 T h e H exagonal Q M F B a n k . : ... 114

6.2.2 Im proved Design M e th o d ... 116

6.2.3 Ite ra tiv e M e t h o d ...122

6.3 D esign of D iam ond-S haped Q M F B a n k s ...127

6.3.1 F o rm u latio n of th e O b jective F u n c t i o n ...f 127 6.3.2 Ite ra tiv e A pproach ... 130

6.3.3 Im proved Ite ra tiv e A p p ro a c h ...133

7 F ilte r B a n k s for th e M P E G A u d io C o d e c 143

7.1 I n t r o d u c ti o n ... 143

7.2 Design of th e M P E G F ilter B a n k ... 145

7.3 Low -C elay F ilte r B ank for M P E G A udio C o d e c ... 146

7.3.1 Low-Delay F ilte r B a n k ... 140 7.3.2 Polyphase I m p le m e n ta tio n ... 148 7.3.3 C om parisons ... 153 7.4 C o n c lu s io n ...154 8 S u bb an d C o d in g o f Im a g es 157 8.1 I n t r o d u c ti o n ... 157

8.2 Q uin cu n x S ubb an d C oding of I m a g e s ...159

8 .2.1 F irs t S t a g e ... 161

8 .2.2 Second S t a g e ... 161

8.2.3 E x a m p l e s ... 164

8.3 Im age C om pression Using D iscrete V,avelet T ransform ...166

8.3.1 W avelet T r a n s f o r m ... 166

8.3.2 Im age Com pression U sing D W 'f ... 173

8.4 C o n c l u s i o n s ...177

9 C o n clu sio n s 1 8 0 9.1 S u m m a ry o f C o n t r ib u t io n s ...180

9.2 R eco m m en d atio n s for F u tu re R e s e a r c h ... 182

L ist o f Tables

2.1 C om parisons of th e pro p o sed m eth o d w ith th e m eth o d of Chen-1,00. 2.2 SN R for E xam ple 2.3... 2.3 R esults for E xam ples 2.4 an d 2.5... 2.4 R esults for E xam ples 2.6 and 2.7... 2.5 C om parisons of th e proposed m eth o d w ith th e m ethod by Nayebi. . 2.6 R esults for a ease s t u d y . ...

24 27 32 35 42 44

3.1 C om parisons of th e proposed m eth o d and th e m eth o d by Chon and

L ee... (j[

3.2 C om parisons betw een th e proposed m ethod and the m eth o d by Nayebi. 68

4.1 P erform ance evaluations o f filter banks for E xam ples 4 . 1 and 4.2, . . 75 4.2 O u tp u ts of th e filter banks of Exam ples 4.1 a n d 4.2 w ith a ram p in p u t. 77 4.3 P erform ance evaluations o f filter banks for E xam ples 4.3 and 4 .4 . . . 84 4.4 O u tp u ts of th e filter banks of E xam ples 4.3 and 4.4 w ith a ram p in p u t. 85

5.1 R esu lts for Exam ple? 5.1 a n d 5.2...105 5.2 R esults for E xam ples 5.3 and 5.4...107

6.1 T h e im pulse responses of E xam ples 6.1, 6.2 and 6.3... 120

6.2 P R E of E xam ples 6.1, 6.2 an d 6.3 and those o f Sim oncelli...120

6.3 T h e im pulse responses of E xam ples 6.4, 6.5 and 6.6... 125

6.4 R esults for E xam ples 6.4, 6.5 and 6.6... 125

6.5 T h e im pulse responses of Exam ples 6.7, 6.8 and 6.9... 134

6.6 R esults for E xam ples 6.7, 6.8 an d 6.9... 134

6.7 R esults for E xam ples 6.10, 6.11 a n d 6.12... 141

7.1 S N R of th e cu rren t M P E G filter b ank and th e filter b a n k designed. . 146

7.2 C om parisons of th e low-delay filter b an k and c u rren t M P E G filter b a n k ... 148

7.3 O p eratio n s in p o lyphase and direct im p le m e n ta tio n s ... 154

8.1 B it allocation and S N R values for th e exam ples...166

L ist o f F igu res

1.1 A g en eral m u ltira te filter b an k sy ste m ... 2

2.1 A two-bancl filter b a n k ... 17

2.2 (a) A m p litu d e responses o f th e filters in E xam ple 2. 1. (b) A m p litu d e

responses o f th e filters in E x am p le 2.2... 25 2.3 E x am p le 2.3 (a) A m p litu d e response of H0. (b) G roup delay of H0. . 27 2.4 E x am p le 2.4 (a) A m p litu d e responses of H0. (b) G roup delay of M(). . 31 2.5 E x am p le 2.5 (a) A m p litu d e response of H0. (b) G roup delay of l l 0. . 31 2.6 E x am p le 2.6 (a) A m p litu d e responses of th e analysis filters, (b) Mag­

n itu d e of th e com plex reco n stru ctio n e rro r 36 2.7 E x a m p le 2.7 (a) A m p litu d e responses of the analysis filters, (b) Mag­

n itu d e o f th e com plex reconstruction e rro r... 36

2.8 (a) A m p litu d e responses o f th e filters in E xam ple 2.8. (b) A m p litu d e

responses o f th e filters in E xam ple 2.9... 43

2.9 G ra p h ic display for fc* = 7 (a) A m plitude responses of analysis filters. (b) A m p litu d e responses o f filter banks, (c) G roup delay c h a ra c te r­ istic of Hq. (d) G roup delay characteristic of filter harm s... 45

‘2.10 G ra p h ic display for k<i = 11 (a) A m plitude responses of analysis fil­ ters. (b) A m p litu d e responses of filter banks, (c) G roup delay char­ a c te ristic of Ho- (d) G roup delay ch aracteristic of filter b a n k s... 46 2.1 L G ra p h ic display for ka = 15 (a) A m plitude responses of analysis fil­

ters. (b) A m p litu d e responses of filter banks, (c) G roup delay char­ a c te ris tic of Ho- (d) G roup delay ch aracteristic of filter b a n k s... 47 2.12 G ra p h ic display for kd = 23 (a) A m plitude responses of analysis fil­

ters. (b) A m p litu d e responses of filter banks, (c) G roup delay char­ a c te ristic o f H 0. (d) G roup delay ch aracteristic of filter b a n k s ...48 2.13 G rap h ic display for kd = 31 (a) A m plitude responses of analysis fil­

ters. (b) A m p litu d e responses of filter banks, (c) G roup delay char­ a c te ristic of Ho- (d) G roup delay ch aracteristic of filter b a n k s...49 2.14 T h e in itia l filter ( N = 32, kd = 15) o b tained b y th e least-squares

approach, (a) A m p litu d e response, (b) G roup d e l a y ...52 2.15 T h e in itia l filter ( N = 32, kd — 15) o b tained by th e shifting m eth o d .

(a) A m p litu d e response, (b) G roup d e la y ...52 2.16 S m o o th p a ra m e te r r versus ite ra tio n num bers, (a) In th e e x p erim en t

w hen N = 32. (b) In th e ex p erim ent when N — 24... 53 2.17 (a) a versus P R E . (b) a. versus S E A ... 54

3.1 E x a m p le 3.1 (a) A m p litu d e response of filter H0. (b) A m p litu d e re ­ sponse of filter b a n k ...61 3.2 E x am p le 3.2 (a) A m p litu d e response of filter H0. (b) A m p litu d e re­

sponse o f filte r b a n k ...62 3.3 E x am p le 3.3 (a) A m p litu d e responses of analysis lowpass filters, (b)

4.1 E x a m p le 4.1 (a) A m p litu d e responses of analysis lowpass and high- pass filters, (b) A m p litu d e response of th e o b ta in e d filter bank. . . . 76 4.2 E x a m p le 4.2 (a) A m p litu d e responses of analysis lowpass and high-

p ass filters, (b) A m p litu d e response of th e o b tain e d filter bank. . . . 76 4.3 E x a m p le 4.3 (a) A m p litu d e responses of th e analysis lowpass and

highpass filters, (b) A m p litu d e response of th e o b tain e d filter bank. (c) G roup delay of th e analysis lowpass filter (d o tte d line) and high- p ass filter (solid line), (d) G roup delay of th e filter b a n k ... 82 4.4 E x a m p le 4.4 (a) A m p litu d e responses of th e analysis lowpass and

hig h p ass filters, (b) A m p litu d e response of th e o b tain e d filter bank. (c) G ro u p delay o f th e analysis lowpass filter ( d o tte d line) a n d high- pass filter (solid line), (d) G roup delay of th e filter.b a n k ...83

5.1 M -channel filter b a n k ... 88 5.2 E x a m p le 5.1 (a) A m p litu d e response of th e p ro to ty p e filter, (b) A m ­

p litu d e responses of th e analysis filters... 104 5.3 E x a m p le 5.2 (a) A m p litu d e responses of th e analysis filters, (b) Am ­

p litu d e response of th e filter bank, (c) P lo t of th e aliasing error E(tu).

(d ) S p e c tru m s of th e in p u t signal and th e reco n stru ctio n e rro r 106 5.4 E x a m p le 5.3 (a) P ro to ty p e filter w ith artifacts, (b) A nalysis filters

w ith a rtifa c ts, (c) P ro to ty p e filter w ith reduced artifacts, (d) A nal­ ysis filters w ith reduced a rtifa c ts ... 103 5.5 E x a m p le 5.4 (a) P ro to ty p e filters, (b) G roup delay of th e p ro to ty p e

filte r (low -delay). (c) A nalysis filters (low-delay). (d) G roup delay of th e filter b a n k (low -delay)... I 'l l

6.2 Coefficients of a hexagonal filter w ith s y m m e tr y ... 117

6.3 S am pling a re a in form ing th e objective function, (a) To form Ey. (b) To form E2... 118

6.4 (a) A m p litu d e response of th e 4-ring filter by proposed m eth o d , (b) A m p litu d e response of th e 4-ring filter by Sim oncelli...121

6.5 A m p litu d e response of o b tain ed hexagonal filters, (a) 3-ring, (b) 4-ring, (c) 5-ring... 126

6.6 D iagram o f a 2-D nonseparable diam ond-shaped filter b a n k ... 127

6.7 (a) B an d C h aracteristics, (b) Q uincunx sam pling...128

6.8 T h e o rd er of th e a ( n i, n i ) in form ing vector y ... 129

6.9 S am pling areas (a) whole b an d , (b) sto p b a n d ... 130

6.10 E x am p le 6.8 (a) A m p litu d e response of H0. (b) A m p litu d e response o f H i... 133

6.11 R eordering of th e im p u lse responses of H0...136

6.12 S am pling a re a in th e sto p b a n d o f Ho... 137

6.13 E x am p le 6.10 (a) A m p litu d e response of H0. (b) A m p litu d e response o f H i... i 4 i 6.14 E x am p le 6.12 (a) A m p litu d e response of H0. (b) A m p litu d e response o f H i... 7.1 Sketch o f th e basic s tru c tu re of th e IS O /M P E C /A U D IO encoder. . .1 4 4 7.2 A m p litu d e responses of th e analysis filters in (a) th e c u rre n t M P E G filter b a n k , (b) th e filter b a n k d e s i g n e d ... 147

7.3 A m p litu d e responses of th e first th re e analysis filters in (a) th e c u rre n t M P E G filter bank, (b) th e filter b a n k designed...147

7.4 A m p litu d e responses of th e analysis filters in th e low-delay filter bank. 149 7.5 F rom to p to b o tto m : O riginal “H allelujah”, rec o n stru c te d from M PEG

filter b a n k , rec o n stru cted from th e low-delay filter b a n k ... 155 7.6 F rom to p to b o tto m : O riginal “Chinese G ong” , rec o n stru cted from

M P E G filter b a n k , reco n stru cted from the low-delay filter bank. . . . 156

8.1 S u b b a n d coding p ro ce d u re ... 158 8.2 (a) D iag ram of a quincunx sub b an d coding schem e, (b) S ubband

p a rtitio n ... 160 8.3 (a) D iam ond-shaped prefilter frequency response, (b) Q uincunx sam ­

pling g rid ... 162 8.4 (a) A m p litu d e response of filter Ho. (b) A m p litu d e response of filter

H i ...163 8.5 J P E G encoder d iag ra m ...163

8.6 A d ead -b an d unifo rm q u a n tiz e r... 165

8.7 D ecom position stages for “L enna” : F irst-stag e (a) Lowpass filtered im ag e (L). (b) H ighpass filtered im age (H ). Second-stage (c) Lowpass filte red im age (LL). (d) H ighpass filtered im age (L H )... 167

8.8 (a) O riginal “L enna” . (b) R eco n stru cted “Lenna” . (e) Difference

im a g e ... 168 8.9 D ecom position stage for “p lan e ” . F irst-stage (a) Lowpass filtered

im age (L). (b) H ighpass filtered im age (H). Second-stage (c) Lowpass filtered im age (LL). (d) H ighpass filtered im age ( L H ) , ... 169

8.10 (a) O riginal “p lan e” , (b) R eco n stru cted “plane” , (c) Difference im age. 170

8.12 (a) One stag e of 2-D separable su b b an d decom position, (b) S p e c tru m

p a rtitio n from one-stage subband decom position...175 8.13 Tw o-stage decom position of “Lenna” by D W T ... 176 8.14 (a) O riginal “Lenna” . (b) R eco n stru cted im age from J P E G , (c) R e­

c o n stru cted im age from D W T ... 178 8.15 (a) O riginal “b a b o o n ” , (b) R eco nstructed im age from J P E G , (c)

L ist o f A b b r e v ia tio n s

1-D one-dim ensional

2-D tw o-dim ensional

A P U ad d itio n s p e r u n it

B FG S B royden-F letcher-G oldfard-S hanno

bpp b its p e r pixel

D O T discrete cosine transform

D W T d iscrete w avelet tra n sfo rm F IR fin ite -d u ra tio n im pulse response H D T V high-definition television

H R in fm ite -d u ra tio n im pulse response ISO In te rn a tio n a l S ta n d a rd O rganization J P E G J o in t P ic tu re E x p e rt G roup

M F L O P S floating-point op eratio n s in m illions M P E G M oving P ic tu re E x p e rt G roup M PU m u ltip lic a tio n s p e r u n it N I n u m b er of ite ratio n s

P C R E p e a k com plex reco sn tru ctio n error P R E p e a k reco n stru ctio n error

Q M F q u a d ra tu re m irro r filter

S N R signal-to-noise ra tio

SN R s signal-to-noisc ra tio w ith a step input S N R , signal-to-noise ra tio w ith a random in p u t S T F T sh o rt-tim e Fourier transform

In tro d u ctio n

M ulti rate dig ital signal processing techniques find applications in speech and im age com pression, th e d ig ital audio in d u stry , s ta tistic a l and ad ap tiv e signal p ro ­ cessing, th e n um erical solution of differential equations, and in m an y o th e r fields. T h e discipline also fits n a tu ra lly w ith c e rtain special classes of tim e-frequency rep ­ resen tatio n s such as th e sh o rt-tim e Fourier tran sfo rm and th e wavelet tra n sfo rm , which are useful in analyzing th e tim e-varying n a tu re of signal sp ectra.

Over th e last decade, th e re has b een a trem endous grow th of a c tiv ity in th e a rea of m u ltira te signal processing, p erh ap s triggered by th e first book in th is field [1]. P a rtic u la rly im pressive .is th e a m o u n t of new lite ra tu re in m u ltira te d ig ita l filter banks, m u ltid im en sio n al m u ltira te system s, and w avelet representations.

In th e thesis several new m eth o d s w hich could be used in th e design of a wide range of one-dim ensional (1-D ) an d tw o-dim ensional (2-D) filter banks are in v esti­ g ated an d some of th e filter b anks designed are used in audio and im age com pression.

F ig u re L I : A general m u ltira te filter ban k system .

1.1

F ilter B anks and D esign Considerations

1.1.1

M u ltira te F ilter B ank

Fig. 1.1 shows th e block d iag ram of a general m u ltira te filter ban k system , in w hich th e sp e c tru m of th e in p u t signal, x ( n ) , is divided into frequency b ands using filters w ith im p u lse responses h m (n) in th e analysis filter bank. T hese channel signals are th e n m axim ally decimated in d iv id u ally a t a ra te of M : 1. w hich resu lts in th e aliased channel o u tp u ts

i M —l

K „ ( < ^ ) ( 1. 1)

M r= 0

For m o st ap p licatio n s th e filters in th e analysis filter bank have uniform passbarid. T herefore, th e set of d ec im ate d signals, {?/m(n)}, form a critically sam pled time-

frequency re p re se n ta tio n o f th e original signal, x{n).

In th e absence o f any processing, like quantization and coding, th e o u tp u t signal

x ( n ) is sim ply a rec o n stru c te d version of th e in p u t, x (n ). T he reconstruction is

accom plished by up-sampling th e channel signals to th eir original sam pling ra te , th en passing each signal th ro u g h a synthesis filter w ith im pulse responses <jmfn ),

an d sum m ing th e resu lts. T h e resu ltin g reconstruction eq u ation is given by

M —1 i M —1

= £

M

£

(

1

.

2

)

m = 0 M r = 0

For ideal b a n d p a ss filters, it is sim ple to show th a t Eqn. (1.2) resu lts in an exact reco n stru ctio n of th e in p u t signal. For such system , th e aliasing term s, (i.e., te rm s involving r ^ 0), are zero w hich m odifies (1.2) to

M - l | X ( e T ) = X ( e ? u ) £ — f f m( e > ) G m(< > ) = X ( e * “) (1.3) m= 0 W if M - l |

E

= 1 (1 .4 ) m = 0 "

Since ideal filters a re n o t realizable, in p rac tic e th e channel signals are alw ays aliased. However, in a filter b a n k en v iro n m en t it is possible to choose p ro p er filters to cancel th e aliasing and achieve an a ly sis/sy n th e sis system s th a t achieve ex act reco n stru ctio n even th o u g h all th e in d iv id u al channel signals are aliased.

1.1.2

D esig n P rob lem s

T h e system in Fig. 1.1 can b e considered to be a hierarchical. A t th e lowest level th e re arc th e in d iv id u al filters w hich typically consist of lowpass, ban d p ass, and highpass filters. A t th e second level th e analysis filters are considered collectively as the analysis filte r b a n k an d th e synthesis filters are considered collectively as th e synthesis filte r b a n k . A t th e th ird level, th e analysis filter b ank and synthesis filter bank are viewed as a so-called analysis/synthesis system which is form ed by d irectly connecting th e analysis o u tp u ts to th e synthesis in p u ts w ith o u t any processing in betw een. T h e final level in th e h ierarchy is th e com plete system in w hich processing

or coding of th e analysis o u tp u ts is considered. In th e rest of th e section, design considerations a n d co n strain ts for each of th e levels are described.

A . F ilte r s

For th e filters in th e analysis filter bank, th e stopband, passb an d , and tran sitio n b a n d c h a ra c te ristic s of th e individual filters m ust be constrained to control the m a g n itu d e , ph ase, and aliasing d istortion of the channel signals. For th e filters in th e synthesis filte r b an k , in ad d itio n to th e reconstruction issues, exactly th e sam e p ro p e rtie s m u st b e addressed so as to control th e effects of th e processing d istortion in th e re c o n s tru c te d signal. If th e co m p u tatio n al com plexity is a critical issue, the ty p e of filters to b e used, like finite-duration im pulse response (FIR.) filter or infinite- d u ra tio n im p u lse response (H R) filter, th e p a ra m e te r sen sitivity o f th e filters, and im p le m e n ta tio n s tru c tu re m u st also b e considered during th e design.

B . F ilte r B a n k s

Since th e to ta l set of channel signals co n stitutes a tim e-frequency rep re sen ta tio n , i t is im p o r ta n t to ensure th a t th e union of frequency b ands of analysis filters or sy n thesis filters covers th e e n tire baseband. In addition, th e realization of th e filter b a n k needs to b e efficient.

C. A n a ly s is /S y n th e s is S y ste m

T h e goal of th e analy sis/sy n th esis system in th e absence of channel processing is to re c o n s tru c t th e in p u t signal a t th e o u tp u t. T herefore, th e distortion due to aliasing, an d th e system distortions in m ag n itu d e and phase responses m ust be m inim ized.

D . C o m p le te S y s te m

T h e co m p le te system explicitly includes th e analysis and synthesis filter banks, a n d th e processing on decim ated m u ltira te signals such as quan tizatio n and coding.

T h e final goal o f th e design is to m axim ize th e system perform ance in th e a c tu a l processing en vironm ent a n d reject th e disto rtio n in tro d u ced by th e processing.

In th is thesis, p rim a ry a tte n tio n will be given to th e first th re e design issues. C om plete system s w ith coding m echanism are used in C h a p te r 8 when applications of th e an aly sis/sy n th esis system s are addressed.

1.2

R eview of th e P revious Work

M ost o f th e work on m axim ally d ecim ated an aly sis/sy n th esis system s has been de­ veloped based on th e earlier work on th e frequency-dom ain processing of speech [2] an d on filter b a n k for T D M /F D M conversion [3]. A n early th eo re tic al fram e­ work for signal analysis an d reconstruction was fo rm u lated by P ortn o ff [4] using th e sh o rt-tim e Fourier tra n sfo rm (S T F T ). T his work provided insight w ith resp ect to th e narro w b an d -w id eb an d n a tu re of th e tim e-frequency rep re sen ta tio n , th e fil­ te r co n strain ts for reco n stru ctio n , a n d th e relationships betw een d iscrete-tim e and continuous-tim e analogues. However, th e question of how to design realizable m a x ­ im ally decim ated tim e-frequency system s able to ex actly re c o n stru c t th e in p u t was still unansw ered.

In 1976, C rochicre e t al. [5] in tro d u ced th e sub b an d coder as a new tech n iq u e for coding speech waveform s. In th is m fchod th e speech is sp lit in to four non-uniform frequency b ands each of which is m o d u la te d to th e baseband. T he signal in each channel is th en lowpass filtered and decim ated. T h e d ual process is applied for reco n stru ctio n . F ilte rs of 125 tap s are used to reject th e aliasing d isto rtio n .

1.2.1

Q uadrature-M irror F ilter B ank

•Almost a t th e sam e time-, Croisier e t a l .-[6] proposed a tw o-band filter ban k system now know n as q u a d ra tu re m irro r filter (Q M F) bank which consists of two analysis filters a n d tw o synthesis filters. C ertain frequency-dom ain relationships m u st be satisfied for th e filters to cancel th e aliasing. If linear-phase F IR filters are used, th e re will b e no phase d isto rtio n in th e o u tp u t signal. However, th e n o n -unity m ag n itu d e , w hich in tro d u ces m a g n itu d e distortion, still has to be m inim ized in th e design. A rem ark ab le fe a tu re of Q M F banks is th a t th ey can be im plem ented using th e efficient polyphase s tru c tu re [7] where m ultiplications and ad d itio n s can b e sh ared betw een th e lowpass an d highpass filtering. T h e tw o-channel Q M F bank system can b e cascaded in a tre e s tru c tu re to perform a variety of m u lti-b a n d sp e c tra l decom positions an d its use in an octave-band stru c tu re for th e subband coder has su b sta n tiv e ly im proved th e speech q u ality [8],

T h e p ro b le m of designing Q M F banks have been a su b ject for research since th e m id d le of 7G’s. J o h n sto n [9] designed a fam ily of Q M F banks by m inim izing a w eighted s to p b a n d -rip p le /sy ste m reco n stru ctio n error. T h e results have been widely used as a s ta n d a rd set of Q M F banks for m any years since th eir publication in 1980. L ater, B arnw ell [10] proposed a Q M F s tru c tu re using H R filters where e ith e r phase or m a g n itu d e d isto rtio n could be elim in ated . In [11] a tirne-dom ain algorithm was proposed for th e design o f Q M F banks which involves calculation of th e eigenvalue's and eigenvectors o f a m a trix in each step of an iterativ e procedure. Recently, Cher) a n d Lee 112] in tro d u c e d an ite rativ e algorithm th a t im proves design efficiency over th e design m eth o d s using conventional m inim ization.

1.2.2

F ilter B anks w ith P erfect R econ stru ction

In th e design m eth o d s m en tio n ed above, th e overall a m p litu d e d isto rtio n is m in ­ im ized b u t n o t com pletely elim in a te d , which leads to near-perfect rec o n stru c tio n Q M F banks. S m ith and B arnw ell [13] [14] , and M intzer [15] proposed m eth o d s for th e design of filter banks th a t yield perfect reconstruction of th e in p u t signals. T hese filters were te rm e d conjugate q u a d ra tu re filters or C Q F ’s [14]. In ad d itio n , S m ith an d Barnv/ell [16] [17] developed th e so-called A C -m atrix fo rm u latio n for an ­ alyzing m u lti-b a n d filter banks. In p a rtic u la r, necessary and sufficient conditions for N -band perfect rec o n stru c tio n filter banks were o btained.

B ased on certain polyphase rep resen tatio n s [7] of th e filter involved, V etterli [18] and by V aid y an ath an [19] proposed a variation of th e AC m a trix form ulation. V a id y a n ath a n re p o rte d a p a ra u n ita ry stru c tu re for th e polyphase m a trix which, alth o u g h was only sufficient to ensure perfect reco n stru ctio n , g reatly sim plified th e design pro ced u re. L ater, im provem ents to th e original schem e w ere re p o rte d [20] [21], R ecently, a Lagrange m u ltip lie r approach was proposed by Horng a n d W illson on th e design of tw o-channel perfect-reco n stru ctio n linear-phase F IR filter banks

[22].

1.2.3

C o sin e-M o d u la ted Q M F B ank

F or uniform m u lti-b a n d system s it was shown th a t satisfactory perform ance of th e filter b a n k can b e achieved b y only canceling th e m a jo r aliasing com ponents in th e re c o n stru c te d signal [23] - [26]. T h e m a jo r benefit of th is approach, how ever, was th e im provem ent in c o m p u ta tio n a l efficiency. T h is is because each in d iv id u al filter

in th e b a n k is rep re sen te d as a cosine-rnodulation of th e baseb an d filter in th e form

By ex p loiting re d u n d a n t a rith m e tic presen t am ong th e bandpass filters, th ese p

ar-m o d u la te d Q M F banks) achieve th e ir enorar-m ous efficiency over tre e s tru c tu re d Q M Fs in m u lti-b a n d decom position. O ne of th e m o st im p o rta n t applications of cosine- m o d u la te d filter b anks is in th e M oving P ic tu re E x p e rt G roup (M P E G ) audio com ­ pression schem e [27].

Several m eth o d s for th e design of m ulti-channel cosine-m odulated Q M F b anks have b een proposed. In [26][28][29], near-perfect reconstruction filter banks wore de­ signed by m inim izing a w eighted o b jective function w hich is a fo u rth-order function o f th e coefficients of th e p ro to ty p e filter. In [30] a sp ectral factc ization approach for th e design of pseudo-Q M F banks w ith o u t using o p tim izatio n was proposed. In [31] a la ttic e s tru c tu re th a t leads to a p erfect reco n stru ctio n filter b a n k was presented. In [32] an e x te n d e d lap p e d tra n sfo rm is described w hich also achieves perfect re ­ co n stru ctio n filte r bank. Recently, in [33] a m eth o d for th e design of near-perfect rec o n stru c tio n Q M F banks was proposed. It appears t h a t this approach can achieve high s to p b a n d a tte n u a tio n and very low aliasing and a m p litu d e distortions.

1.2.4

L ow -D elay F ilter B anks

T h e re c o n stru c tio n delay is an im p o rta n t issue in real-tim e applications arid to d a te th e re are rela tiv e ly few resu lts available on th e design o f low-delay filter banks. For m an y p o p u lar filter banks such as tw o-channel Q M F banks and m u lti-channel cosine- m o d u la te d Q M F banks, th e reconstruction delay is fixed to b e N — 1 w here N is

(1.5)

cosine-th e filter lengcosine-th. W hen high o rder filters are needed in order to achieve satisfacto ry perform ance, or in a tre e s tru c tu re w here tw o-channel filter banks are cascaded, long re c o n stru c tio n delays are unavoidable, which is highly u n d esired for rea l-tim e applications.

In [34] [35], a tim e-d o m ain design m eth o d is stu d ie d for th e design of low -delay tw o-channel filter banks. Som e interestin g observations and useful com parisons were m ade a n d low-delay tre e -s tru c tu re system s were co n stru c te d using th e o b tain e d low- delay tw o-channel filter banks. For th e design of low-delay cosine-m odulated Q M F banks, a class of generalized cosine-m odulated filter banks has been proposed in [36] whose re c o n stru c tio n delay is n o t fixed b u t could b e chosen am ong several values. Some p relim in ary resu lts w ere also re p o rte d in [37] on low -delay cosine-m odulated filter b anks th ro u g h designing several filter banks w ith various-lengths b u t a fixed reco n stru ctio n delay.

1.2.5

2-D F ilter B anks

Since 1984, research on 2-D su b b a n d coding schem es has also b een intensive. V etterli [38] was th e first to p ropose tw o-band nonseparable Q M F b anks w ith qu in cu n x d ecim atio n la ttic e s, in w hich th e aliasing term s are com pletely cancelled. Using lin ear-p h ase filters w ith c e rtain restrictio n s on th e o rder of filters, it was shown th a t th e re c o n stru c te d signal coincides w ith th e in p u t signal u p to a c o n sta n t facto r a n d a sp a tia l shift. A separable 2-D Q M F ban k w here 1-D Q M Fs o p e ra te first on th e rows a n d th e n on th e colum ns of 2-D in p u ts or vice v ersa was also proposed in [38]. In 1986 W oods and O ’Neil [39] con stru cted a 2-D sub b an d im age coder using a separable 2-D Q M F ban k an d rep o rted th a t th e sub b an d coder has im proved th e signal-to-noise ra tio (SN R ) a n d t h a t th e subjective p erform ance com pared to th a t

o b ta in e d by using th e ad a p tiv e discrete-cosine tran sfo rm (D O T ). S m ith et al. [40] an d R a m s ta d [41] suggested several 2-D perfect reco n stru ctio n recursive filter banks based on sep arab le filters. In [42] b o th recursive and nonrecursive filter banks were investigated.

In a sep arab le 2-D Q M F b a n k th e frequency sp ectru m of th e in p u t signal is split in to c e n tra l low pass, h o rizontal highpass, vertical highpass, and diagonal highpass b ands in w hich th e diagonal highpass b a n d contains a m ix tu re of th e two o rien ta­ tions. To avoid th is pro b lem , a n o n separable four-band hexagonal Q M F ban k based on hex ag o n al sam p lin g was proposed by Simoncelli and Adelson [43] w here th e anal­ ysis a n d synthesis filters have a sim ilar s tru c tu re to th a t of 1-D Q M F banks. It has been show n t h a t aliasing in th e sy stem o u tp u t is cancelled and a m p litu d e distortion can b e m in im ized th ro u g h th e design. A nother im p o rta n t class of nonseparable 2- D filter b a n k s is th e class of tw o-band diam ond-shaped filter banks, in which th e frequency sp e ctru m is split in to diam ond-shaped lowpass a n d four-corner highpass b ands. T h is ty p e of filter banks has been applied for im age and video com pres­ sion in high-definition television (H D T V ) coding and th e resu lts o b tain e d ap p ear to b e sa tisfa c to ry [44] [45]. M ost existing m ethods for th e design of nonseparable 2-D diam o n d -sh ap ed filter banks are based on th e application of tran sfo rm atio n s to 1-D p ro to ty p e filters [45] - [47].

In th e a re a o f generalized m ultidim ensional ban k design, V iscito and A llebach [48] e x te n d e d th e th eo ry of p erfect reconstruction filter b anks to a rb itra ry down- sam pling la ttic e s in tro d u c ed in [49][50]. T h e schem e of [48] is based on th e concept of th e m u ltid im e n sio n al polyphase tran sfer m atrix . It was shown th a t if th e analysis p o lyphase m a trix is form ed by cascading co n stan t coefficient m atrices w ith diagonal shift m a tric e s b etw een th e m , its inverse can be readily evaluated. T h e synthesis

filters are th e n derived from th is m a trix inverse. In [51], a th eo ry of m u ltira te op er­ ations on a rb itra ry m ultidim ensional lattices and a n um erical o p tim isa tio n m e th o d for designing th e m ultidim ensional filter banks are proposed.

1.3

Scope o f th e Thesis

T his thesis consists of th re e p a rts. P a r t I, com prising C h ap ters 2 to 5, is c o n c en tra te d on several m eth o d s for th e design of 1-D filter banks; P a r t I I (C h a p te r 6) is devoted to studies on th e design of 2-D filter banks; P a r t III, com prising C h ap ters 7 an d 8, describes ap plications of th e 1-D and 2-D filter banks designed in P a rts I an d II of th e thesis in audio an d im age com pression.

T h e m a jo r concern of C h a p te r 2 is th e design of near-perfect tw o-channel Q M F banks from th e frequency dom ain. F irs t an im proved version of th e ite ra tiv e m e th o d , originally p ro posed in [12], is described. A sim ple and explicit fo rm u la fo r th e precise evaluation of in te g rals involved in th e objective function is derived. As will be show n in th e design exam ples, th is significantly reduces th e design com plexity. To achieve low re c o n stru c tio n delay in a Q M F b an k , a generalized tw o-channel Q M F b a n k is proposed, whose rec o n stru c tio n delay is ad ju stab le, leading to low-delay filter banks.

C h a p te r 3 is concerned w ith th e design of Q M F banks from th e tim e dom ain. A new a p p ro ach is described in w hich th e p erfect reco n stru ctio n condition is form u­ la te d in th e tim e-d o m ain in stead of in th e frequency do m ain as in C h a p te r 2. A n ite rativ e m e th o d sim ilar to th a t in C h a p te r 2 is used to reduce th e design com plexity. For th e p u rp o se o f designing low-delay Q M F banks, a tim e-d o m ain p erfect recon­ stru c tio n co n d itio n is derived w hich can b e m inim ized efficiently by th e ite ra tiv e m eth o d .

C h a p te r 4 describes a null-space p ro jection m eth o d for th e design of tw o-channel perfect re c o n stru c tio n Q M F banks. In th e proposed m eth o d tw o filters, i.e., th e analysis low pass filter a n d th e synthesis lowpass filter, need to be designed. T h e analysis low pass filter is first designed by a conventional F IR filter design m eth o d . It is th e n followed by solving a constrained optim ization pro b lem using th e null-space p ro je c t m e th o d to o b tain th e synthesis lowpass filter. T h e proposed m eth o d is used in th e design of tw o-channel perfect reco nstruction Q M F banks w ith norm al and low re c o n stru c tio n delays.

C h a p te r 5 describes several m eth o d s for th e design of cosine-m odulated Q M F banks, w hich h av e been w idely used in m any applications due to th eir high design efficiency and im p lem e n ta tio n efficiency. In order to design conventional cosine- m o d u la te d Q M F banks, a n e w ite ra tiv e alg o rith m is proposed th a t greatly im proves th e design efficiency and leads to filter banks w ith high sto p b a n d a tte n u atio n and low aliasing an d a m p litu d e distortions. A general version of th e algorithm is then developed, w hich can be used to design low-delay cosine-m odulated Q M F banks. T his a lg o rith m is b ased on a w eighted ob jectiv e function th a t depends on th e erro r b etw een th e a c tu a l frequency response and th a t of a lincar-phasc ideal filter. A rti­ facts t h a t can occu r in th e am p litu d e responses of th e analysis and synthesis filters w hen designing low -delay filter banks can b e reduced significantly by sim ply adding one m ore erro r co m ponent to th e o b jectiv e function.

A lth o u g h sep arab le 2-D filter banks are easy to im plem ent, th e class of non­ separable 2-D filter banks a re believed to be m ore ad eq u ate for use in m any im age processing re la te d applications. In C h a p te r 6 several design m ethods for th e de­ sign of fo u r-channel hexagonal Q M F banks a n d tw o-channel diam ond-shaped Q M F banks a re proposed. It will b e d e m o n stra te d th a t th e hexagonal filter banks de­

signed using th e proposed m eth o d s are superior to th a t described in [43] in te rm s of design efficiency and system perform ance. T h e problem of designing diam o n d ­ shaped Q M F b anks is solved by using some ite rativ e m eth o d s. U nlike th e existing m eth o d s, no 1-D to 2-D tra n sfo rm atio n is required in our designs. T h e proposed m eth o d s show high design efficiency an d design flexibility.

In C h a p te r 7 a 32-band filter b a n k is designed using th e ite ra tiv e m e th o d p ro ­ posed in C hapter 5 and com parisons of th e filter banks designed w ith th e c u rre n t M P E C filter b a n k are m ade. In o rd er to reduce th e reco n stru ctio n del ;y of th e M P E G audio codec, a cosine-m odulated Q M F b a n k whose reco n stru ctio n delay is a b o u t h a lf of th a t in th e c u rre n t M P E G filter b a n k is designed a n d im p lem en ted using an efficient polyphase stru c tu re . Some sam ple sound signals are used to te s t th e designed filter bank.

In C h a p te r 8, sub b an d coding o f im ages is studied. A tw o-stage su b b an d coding system is c o n stru c te d using diam ond-shaped filter banks, th e D C T tra n sfo rm atio n , and HuffmfU coding. In ad d itio n , th e D aubechies w avelet tra n sfo rm is used in a separable 2-D m u lti-sta g e su b b an d coding system . C om parisons of th e w avelet tran sfo rm m eth o d w ith th e c u rre n t J o in t P ic tu re E x p e rt G roup (J P E G ) coding schem e are >.onducted using sam ple im ages.

Im p roved Itera tiv e M eth o d s for

th e D e sig n o f Q M F B anks

2.1

Introduction

T h e im p o rta n c e o f q u a d ra tu re -m irro r-filter (Q M F) banks in subbam l coding has been w idely recognized an d various analysis, design, and im plem entation issues p e r­ tain in g to th ese filters have b een intensively studied since th e m id 70’s. T h e q u a d ra ­ tu re m irro r s tru c tu re of Q M F banks leads to th e com plete cancellation of iriterband aliasing due to th e overlapping filter responses. If th e filters involved have sy m m e t­ rical im pulse responses of finite d u ratio n (F IR ), no phase distortion will occur ai,:l, therefore, th e design can be focused on selecting filter p a ra m ete rs so as to m inim ize th e sy ste m ’s a m p litu d e d isto rtio n .

Like th e design of conventional d igital filters, th e design of Q M F banks can be accom plished by using least-squares and m inim ax m ethods. In [52], th e reasoning of using th e least-squares criterio n for telecom m unication applications has been clar­ ified. O n th e o th e r h an d , in [12][53][54] it has been shown th a t m in im ax design can be accom plished if an a d e q u ate ly u p d a te d weighting function is included in a

least-squares o b jec tiv e fu n ction. In e ith e r th e non-w eighted or w eighted case, th e least-squares o b jec tiv e fu n ctio n is a fourth-order function of th e design p a ra m ete rs. Hence th e design is a ty p ic a l u n co nstrained, nonquadratic o p tim iz a tio n problem . Recently, an ite ra tiv e design m eth o d has been proposed by C hen an d Lee [12] in w hich th e least-squares o b jec tiv e function is m odified by assigning a set of a p p ro p ri­ a te values, which fo rm a v ector h , to a p a rt of th e design p a ra m e te rs in such a way as to o b tain a quadratic a n d globally convex objective fu n ctio n . U pon o b tain in g th e global m inim um p o in t, say f , th e value of h is u p d a te d accordingly and is th e n re-assigned to th e sam e p a rt of th e design p aram eters. T his pro ced u re is re p e a te d u n til f an d h becom e iden tical. Since in each ite ratio n f can be fo rm u la ted in closed form and only a few ite ra tio n s a re needed for convergence w hen a good in itia l h is used, th e alg o rith m is efficient a n d good perform ance is achieved in th e filter b an k . In th e m eth o d d escrib ed tw o in tegrals are involved in th e o b jectiv e fu n ctio n , w hich are ev aluated by d isc re tiza tio n . T his gives rise to two problem s. F irs t, th e solution o b tain ed ac tu a lly m inim izes th e discretized version of th e o b jectiv e fu n ctio n r a th e r th a n th e o b jec tiv e fu n ctio n itself. T his can degrade the p erform ance of th e Q M F bank designed. Second, in o rd er to reduce th e perform ance d e g ra d a tio n th e den sity o f sam ple points needs to b e high, w hich leads to increased co m p u ta tio n a l com ­ plexity. In this c h a p te r, we describe an im proved version of th e above alg o rith m in w hich th e integral d isc re tiza tio n is avoided by deriving a sim ple a n d ex plicit fo rm u la for th e precise e v alu atio n of th e tw o integrals. As a resu lt, filter b anks w ith b e tte r perform ance can b e designed w ith considerably reduced c o m p u ta tio n a l com plexity. Design exam ples a re d e m o n stra te d to show th e proposed alg o rith m an d com parisons are m ad e in term s of design efficiency.

reconstruc-tio n delay are h ighly desired. U nfortunately, in convenreconstruc-tional Q M F b anks th e filter len g th N is req u ired to b e fairly high in order to achieve satisfacto ry perform ance, C onsequently, t h e rec o n stru c tio n delay, which is (iV — 1) sam pling periods, can be­ com e to o long. A tim e-d o m ain approach to th e design of an aly sis/sy n th esis system s w ith low re c o n stru c tio n delays was proposed in [34]. In this ch a p te r, we propose a tw o-channel low -delay Q M F b ank, and th e iterativ e algorithm m entioned above is e x te n d e d to th e design of low-delay Q M F banks, first in a discretization version and th e n in an im p ro v ed version in w hich sim ple and explicit expressions arc derived for th e precise ev a lu a tio n of th e objective function. T his increases th e c o m p u tatio n al efficiency d u rin g th e design and im proves th e quality of th e filter banks designed. D esign ex am ples are illu stra te d and com parisons are m ad e w ith designs from [34] in term s of design efficiency, perform ance of th e resulting filter banks, im plem entation asp ects, etc. B y em ploying th e proposed m eth o d , a fam ily of tw o-channel low-delay Q M F b an k s w ere designed which could be used in different kinds of applications.

From e x p e rim e n ts, it is found th a t th e ite rativ e m eth o d can achieve filter banks o f good p e rfo rm a n c e w ith high design efficiency. To investigate th e m eth o d fu rth e r, som e analysis h a s b een done concerning such aspects as th e choice of initial points an d convergence speed.

x ( n )

1____

H , ( z )

F igure 2.1: A tw o-band filter bank.

2.2

D esign of Conventional QM F Banks

2.2.1 O b jective Function

C onsider th e tw o-band filter ban k d ep icted in Fig, 2.1. T h e in p u t-o u tp u t rela tio n is given by

X { z ) = ^[H0( z ) G 0(z) + H l ( z ) G l ( z ) ] X ( z )

+ \ [ H o ( - z ) G 0(z) + H ^ G t i z W - z ) (2.1)

w here th e second te rm on th e rig h t-h a n d side represents aliasing due to th e dec­ im atio n o p e ra tio n . By assum ing th a t H i { z ) — Ho( —z), C o ( z ) = H 0(z), and G\ (z) = —Hq(—z), w hich, in general, are referred to as quadrature-mirror rela tio n ­

ships, we o b tain a Q M F b an k in w hich th e aliasing te rm is com pletely elim in ated , an d (2.1) becom es

X ( z ) = i I H f c ) - H l ( - z ) \ X ( z ) (2.2) If F I R filters w ith sy m m etrical im pulse responses are used in th e Q M F b a n k , which is referred to as th e conventional Q M F l a n k , th e n th e frequency response of filter Ho can be expressed as

where

M h(uj) = 2 h Tc(w) (2.8b)

h ~ [ h 0 hi • • • hff/2-i ]T (2.8c.)

c(u>) = [ cos ( N — l)w /2 • • • cos or/2 f (2.3d) and N is th e le n g th of th e filter, w hich is assum ed to be even. From (2.2) and (2.8), th e p e rfe c t rec o n stru c tio n condition of th e Q M F bank assum es th e form

T(w ) = M l ( u ) + M l i w + tt) = 1 (2.4) and th e sy stem delay is (iV — 1) sam pling periods. In order to design a filter bank satisfying (2.4), an ob jectiv e function is defined as

E — E i -j- otE2 (2.5a) w here E i = J * [ T { u ) - l]2du> (2.5b) and E 2 = r Ml{ u) dto (2.5c) Jws

T h e p a ra m e te r a is a positive w eight th a t can be used to control th e sto p b a n d a tte n u a tio n for Ho and u s is th e sto p b a n d edge.

2.2.2

Im proved Iterative A lgorithm

T h e b a sic id ea in th e ite ra tiv e algorithm [12] is th a t instead of d ire c tly m inim izing th e o b jec tiv e fu n ctio n in (2.5a), which is a fourth-order function of p a ra m e te r vector h , one m inim izes th e m odified ob jectiv e function

w here E[ = A r ' H - 1]2^ (2.6b) Jo E'2 = f R f ] ( u ) d u (2.6c) Juja T '(w ) = M h( w ) M / ( u ) + M h(oj + tt ) M j ( u + i t) (2.6d)

Mj(ui) — 2 f 2 c(o>) (2.6e)

f = [ / o / i ••• /AT/2-i]r (2.6f)

un d er th e assu m p tio n th a t coefficient vector h is fixed w ith resp ect to th e m in im iza­

tion. O bviously, as a function of f , th e m odified objective function E ' is quadratic and globally convex and, therefore, its m inim um can be o b tain e d easily. H aving o b tain e d th e m in im u m p o in t of E' , say f, vector h is u p d a te d using a lin ear com bi­

n a tio n o f f an d h as

h := (1 - r)h + r f (2.7)

w here r is a sm oothing p a ra m e te r betw een 0 and 1. W ith an ap p ro p riate choice of r , w hich is found to be in a vicinity of 0.5, f will quickly converge to h and an

o p tim a l design p a ra m e te r v ector results. In th e approach in [12], d iscretizatio n is used to ev alu a te E[ and E L An a lte rn a tiv e approach is to derive a closed-form and ex p licit fo rm u latio n of E ' as a q u a d ra tic function of f w ith o u t discretizing E[ and E'r As will be show n, th is form ulation in conjunction w ith th e ite ra tiv e p rocedure

described above leads to im proved com p u tatio n efficiency a n d p erform ance in th e designed filter b an k .

From (2.6b) an d (2.6d), E[ can be w ritten as

w here

n<

u = Jo

[M ^(w) c (w) + A^ ( w+ 7r)c (w+7r)][A/,l(w)c(a;)+iV4(w+7r)c(w+7r)]'-/\/w (2.9)

w ith c (uj) given by (2.3d) and JV /'2 -l AT/2-1 M l ( u ) = A Y h Y h n h m C OS n = 0 m=0 N - 1 \ 1 n — w 2 ) . cos \ ( N ~ l \ , ( r o ---(2.10)

ive analysis shows th a t th e (i, j ) t h e n try of m a trix U in (2.9) is given by

N / 2 - I N / 2 - 1 J V / 2 - 1 J V / 2 - 1 z - 8 = * Y Y h"hm | Y [1 + H / ' ] W \ , 1 < i, j < N / 2 (2 .1 1) n = 0 m = 0 U = i J where S(k,) = 1 , ki = 0 0 , otherw ise k i = (3 + 7 -f £ -f rp k i = (3 -f 7 -f £ — t? ^3 = /? + 7 — £ - f

T), h\

=

0

+ 7 - £ —

r)

h = /? — 7 -f £ + t], k$ = (3 — 7 + £ — ?| k 7 = /3 - ' y - £ - \ - r } , ks = {3 - 7 - £ - r/ (3 = n N - 1 2 y V - l 7 = m , A ^ + l 5 = — 5--- * + 1 . v = _ ---j k[ = rn — j + 1, /4 = m + j — N

K = K , K = K

y —

~y y

—

~y

'•'5 — 2 ) 6 ~ "'1 ij _ 7„' L' _ 7/ f i / J — f l < ) j » l / g — H / j Now from (2.6c), E '2 = 4 fr f c(u>)cT (u;) f/w L/o/j = 4 f U , f (2 . 12)

w here th e (?!, f ) t h e n try of U s is given by „ TV — l

l ~--- I 'J3 COS j _ i --- — | w. 1 T V -1 duj

i = j

(2.13) | (tt — ws) /2 — sin[(2i — IV - l)w s]/( 4 i — 2 N — 2) ,

{ cjn[(i - j ) u s] / (2j - 2i) - sin[(z + j - N - l ) u a]/(2i + 2j - 2 N - 2) , i ^ j E q uation (2.13) was also u sed in [11] in a tim e-dom ain fo rm u latio n of th e design problem . N ote th a t m a trix U s is independent of h and can b e pre-calcu lated as long as th e filter le n g th N an d sto p b a n d edge ivs are specified.

T h e o b jec tiv e function B ' can now be expressed as

E ' = 4 fr Q f - 87thTf + 7T (2.14)

where

Q = U + cvUs (2.15)

From (2,9) an d (2.12) it follows th a t m a trix Q is positive definite and, therefore, w ith a fixed h th e global m inim um of E ' is given by

O n th e basis of th e preceding analysis, an ite rativ e algorithm can now be con­ s tru c te d as follows:

A lg o r ith m 2.1

S tep 1 Use a conventional m eth o d (e.g., th e window m eth o d ) to design a linear

phase, low pass, F IR filter of length N w ith sto p b a n d edge w.,, and use th e coefficient vector of th e filter o b tain e d as th e in itia l h.

S tep 2 Use (2.11) to co m pute m a trix U .

S tep 3 F orm m a trix Q using (2.15), and com pute f in (2.16).

S tep 4 If || h — f || < e, where

t

is a prescribed tolerance, o u tp u t f as the design

re su lt an d stop. O therw ise, u p d a te h using (2.7) w ith a r close to 0.5

and re p e a t from Step 2.

A rem ade on th e evaluation of m a trix U is ap p ro p riate a t this point. From (2.9)

it is clear th a t U is sym m etric, and so only N ( N + 2)/ 8 entries of U need to be cal­

culated. For each e n try of U , ^ f _ l [1 + (—1 ) ] A(A'/) is a sim ple com bination of eight delta functions, each of which is very sparse and easy to determ in e. C onsequently,

designs using A lg o rith m 2.1 can b e accom plished very quickly.

2.2.3

D esig n E xam p les

•Two F I R Q M F b anks have been designed by using A lgorithm 2,1. T h e p a ra m ete rs used in th e two designs, which will be referred to as E xam ples 2.1 and 2,2, are

N = 32, a = 1, iuB = 0.67T, r = 0.7, e = lO- 3 , and N = 80, a = 1, u)„ ~ 0.557T,

r = 0.7, e = 10~3, respectively. T h e in itia l h was [ 1 0 • • • 0 ~ }T . For com parison purposes th e m e th o d of Chen and Lee [12] was applied to design two F JR Q M F banks w ith th e sam e design p a ra m ete rs and initial h . T he com parisons were m ade in te rm s of

• N u m b er of floating-point operations in m illions (M F L O PS )

• th e m in im u m sto p b a n d a tte n u a tio n

UJS <U/<7T

• th e p eak -to -p eak passbancl ripple

A a = ^ < , [ - 201o g 1 0 |f f o (e ,'u,) l ]

j \ u / \ w p

A ’ = o S S , [201o« 1o | a ( ei " ) |] - o m m _ [201og l o\ H 0 ( e >

v.'here cup is th e p assb an d edge,

P e a k rec o n stru c tio n error

P R E = m ax

W

• Signal-to-noise ra tio

energy of th e signal

energy of th e reco n stru ctio n noise S x \ n )

S N R = 101og10

10 °S J0 2

w here k,i is th e reco n stru ctio n delay.

B o th th e proposed m eth o d and th e m eth o d of [12] were p ro g ram m ed using M AT- LAB (version 4.1) and run on a Sun SPA RC statio n . T h e n u m b er of frequency sam pling p o iu ts was set to 8 N w hen im plem enting th e m e th o d of [12], w here N is th e filter length. T h e results are sum m arized in Table 2.1 w here SNR* and SNRr denote th e S N R w ith a step in p u t a n d a ran d o m in p u t, respectively. T h e am pli­ tu d e responses of filter H 0 designed by th e proposed m eth o d and th e m eth o d o f [12] w ith two sets of design p a ra m e te rs specified above are d ep icted in Fig. 2.2 (a) and

T able 2.1: C om parisons of th e proposed m ethod w ith th e m eth o d of Chen-Lee.

E xam ple 2.1 E xam ple 2,2

P roposed Chen-Lee P roposed Chen- Lee

M FL O PS 0.62 2.55 8.11 36.83 A a (dB ) 35.20 36.SI 44,69 45.19 A p (dB) 0.0124 0.0139 0.0042 0.0045 P R E (dB ) 0.0148 0.0154 0.0091 0.0093 S N R 5 (dB ) 84.1 83.6 83.9 S2.7 SN Rr (dB) 69.1 68.4 76.5 74.8

(b), respectively. As can be observed from th e design results, th e proposed m eth o d shows consisten t im provem ent over th e m eth o d of [12] especially in term s of design efficiency w here th e co m p u tatio n com plexity can b e reduced by as m uch as 75%.

2.3

D esign of Low -Delay QMF Banks

2.3.1 G eneral T w o-C h an n el Q M F B ank

In Sec. 2.2.1, th e im pulse responses of th e filters in a Q M F filter bank are assum ed to be sy m m e tric a l so th a t th e y are of lin ear phase w ith group delay ( N —1 )/2 , whore N is th e le n g th o f th e filter. T his gives a filter bank th a t has linear phase response and th e re c o n stru c tio n delay is fixed a t ( N — 1) sam pling period. U nlike th e conventional Q M F b a n k , h e re we consider a Q M F b a n k which im poses no sy m m e try co n straints on th e im pulse responses of th e filters. From [55] we know th a t op tim i'/atio n m ethods can be used to design F IR filters w ith group delay less than ( N — l ) / 2 . Suppose th a t we have designed a lowpass F IR filter H0 w ith group delay k d j 2 < ( N ~ I ) /2 w here

♦10 M e t h o d o f C h e n - L e e : d o t t e d lim M e t h o d o f C h e n - L e e : d o t t e d lil ■-40 P r o p o s e d m e t h o d : s o l i d l i n e P r o p o s e d m e t h o d : s o l i d li n e

A

-70 •100 •80 •120 0.1 0.3 0.4 0 5 N o r m a l i z e d f r e q u e n c y 1 0.2 0.6 0.7 0.1 0.2 0 4 0.5 0.6 0.7 0.8 N o r m a l i z e d f r e q u e n c y 0.3 (a) (b)

F igure 2.2: (a) A m p litu d e responses of th e filters in E xam ple 2.1. (b) A m p litu d e responses of th e filters in E x am p le 2.2.

kd is assum ed to b e an o d d n u m ber, th e n its frequency response can b e expressed

as

H 0(ejw) = \H0(ejw)\e~jujkd/ 2

So w ith th e sam e q u a d ra tu re -m irro r relationship am ong analysis and synthesis filters assum ed as in a conventional Q M F b an k , th e aliasing term in th e o u tp u t is cancelled arid th e freq u en cy response of th e filter bank is given as

= \H0( e ^ ) \ e - ^ k“ + \H0( e ^ w^ ) \ e ~ ^ w^ k^

= [ |tf 0V w)| + |//o (e J(w+,r))|] e~iwkd (2.17) T herefore, if

l«oV ")l +

= 1

for all 0 < u> < 7T, th e p erfect reco n stru ctio n will be achieved while th e rec o n stru c tio n delay is kd < N — 1, w hich is less th a n th a t in a conventional Q M F b an k , a n d it

can b e m ad e s mal l if k d is small. T he Q M F bank o b tain e d can be referred to as a tw o-channel low -delay Q M F ban k and its perfect reconstruction condition in th e z do m ain can b e expressed as

H l ( z ) - H & ~ z ) =

B ased on th e above analysis, a design m eth o d can b e developed in w hich a w eighted o b jec tiv e fu n ctio n is form ed as

E = Y J wiE i (2.18) i = l w ith E i = E l ^ o V " ) “ H 20 ( c J ^ ) - e~iujkd|2 W E2 = Y 1 \ M ^ ) - H ^ ) \ 2 a•

w here to,- for i - 1,2 are weights and H i ( e ju>) is th e frequency response of an ideal low pass filter w ith group delay k d / 2.

T h e m in im iz atio n of th e ob jectiv e function in th e above design procedures can b e achieved b y o p tim iz a tio n . VVe have em ployed a quasi-N ew ton o p tim izatio n algo­ r ith m b a se d on th e B royden-F letcher-G oldfard-S hanno (B FG S) u p d a tin g form ula and th e in ex a c t line search described in [55] to perform th e o p tim izatio n . T h e n u m b er o f variables is N . A fo rtran program has been w ritten to im plem ent th e alg o rith m .

A filter b a n k was designed w ith filter length N = 32 and system delay k d = 9, w hich is referred as E xam ple 2.3. F irst th e coefficients of filter H0 are d eterm ined by m in im izin g th e o b jectiv e function given in Eqn. (2.18) and th en th re e o th er analysis an d synthesis filters a re o b tain through th e q u a d ra tu re -m irro r relation as

T able 2.2: S N R for E xam ple 2.3. kd SN R s (dB) SN R- (dB) 9 78.87 75.99 | - , 0 a 3 2 '20 O •30 •50 0.4 0.6

N orm alized Frequency

0.2 0.8 •10 •15 4 ^ 3 *.» _Q. .25_" 5 .30 •35 •40 -45 0 A 0.6

N orm alized Frequency

0.8

0.2

(a) (b)

F igure 2.3: E x am p le 2.3 (a) A m p litu d e response of H0. (b) G roup delay of Ho.

in a conventional Q M F bank. T h e a m p litu d e response and group delay c h a ra c te ristic of filter Ho are show n in Fig. 2.3 (a) and (b), respectively. It is observed t h a t in th e passb an d th e g ro u p delay c h a ra c te ristic is flat. T he S N R values for th e o b tain e d filter b a n k a re listed in T able 2.2.

2.3.2

Iterative M eth od

In Sec. 2.3.1, a tw o-channcl Q M F b a n k is considered whose rec o n stru c tio n delay could b e lower th a n t h a t in a conventional Q M F bank. Since th e o b jectiv e function involved in th e design is usually highly nonlinear, its m in im izatio n w ith sta n d a rd