Ieee Transactions On Signal Processing, 1998, v. 46 n. 4, p

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1 Title M-Chael compactly supported biorthogoal cosie-modulated wavelet bases Author(s) Cha, SC; Luo, Y; Ho, KL Citatio Ieee Trasactios O Sigal Processig, 998, v , p. 4-5 Issued Date 998 URL Rights 998 IEEE. Persoal use of this material is permitted. However, permissio to reprit/republish this material for advertisig or promotioal purposes or for creatig ew collective works for resale or redistributio to servers or lists, or to reuse ay copyrighted compoet of this work i other works must be obtaied from the IEEE.

2 4 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 998 -Chael Compactly Supported Biorthogoal Cosie-Modulated Wavelet Bases S. C. Cha, Y. Luo, ad K. L. Ho Abstract I this correspodece, we geeralize the theory of compactly supported biorthogoal two-chael wavelet bases to M-chael. A sufficiet coditio for the M-chael perfect recostructio filter baks to costruct M-chael biorthogoal bases of compactly supported wavelets is derived. It is show that the costructio of biorthogoal M-chael wavelet bases is equivalet to the desig of a M-chael perfect recostructio filter bak with some added regularity coditios. A family of M-chael biorthogoal wavelet bases based o the cosiemodulated filter bak (CMFB) is proposed. It has the advatages of simple desig procedure, efficiet implemetatio, ad good filter quality. A ew method for imposig the regularity o the CMFB s is also itroduced, ad several desig examples are give. I. INTRODUCTION Wavelets are fuctios geerated from the dilatios ad traslatios of oe basic fuctio called the wavelet fuctio [], [3], [6]. The Haar system is cosidered the earliest example of such wavelet bases []. Grossma ad Morlet [] were the first to costruct a wavelet fuctio i the square itegrable real space H. Recetly, orthoormal wavelet bases have bee studied extesively both i the mathematical ad sigal processig commuities [] [8], [8]. I sigal aalysis, the wavelet trasform [3], which is a represetatio of a sigal i terms of a set of wavelet basis fuctios, allows the sigal to be aalyzed i differet resolutios or scales. The wavelet trasform makes a differet tradeoff i the time frequecy plae as compared with the short-time Fourier trasform. It has better time resolutio i high frequecy ad better frequecy resolutio i low frequecy. This property is very useful to detect discotiuity i ostatioary sigals, which usually have slowly varyig compoets with trasiet high-frequecy spikes. Multiresolutio approaches have bee popular i the computer visio commuity. Mallat defies a importat cocept of multiresolutio aalysis ad wavelet bases [4]. The theory of wavelets is also closely related to that of multirate perfect recostructio (PR) filter baks. Daubechies costructed compactly supported dyadic orthoormal wavelets based o iteratios of two-chael PR orthogoal filter baks with added regularity coditio []. Sice twochael orthogoal PR filter baks caot have otrivial liear-phase solutio, more geeral biorthogoal filter baks were studied [5], [6]. This approach has further bee exteded to M-chael orthoormal wavelet bases [7], [8]. It is show that ay square itegrable sigal ca be expaded i terms of dilatios ad traslatios of M 0 fuctios, 9 (j;k) i (x);i=;;;m0, which are called the M- chael wavelets. For a large class of sigals, M-chael wavelet decompositio gives a more compact represetatio tha the dyadic oe [7]. I additio, the discrete wavelet trasform provides a good approximatio to the KLT of several processig [7]. Like the dyadic Mauscript received February 5, 997; revised November 30, 997. This work is supported by the Hog Kog Research Grats Coucil ad the CRCG of the Uiversity of Hog Kog. The associate editor coordiatig the review of this paper ad approvig it for publicatio was Ali N. Akasu. The authors are with the Departmet of Electrical Egieerig, Uiversity of Hog Kog, Hog Kog ( sccha@eee.hku.hk; yluo@eee.hku.hk; klho@eee.hku.hk). Publisher Item Idetifier S X(98)05-4. case, it is possible to obtai M-chael wavelet bases from M- chael PR orthogoal filter baks with added regularity coditio. I this correspodece, we will derive a sufficiet coditio for the M-chael PR filter baks to costruct M-chael biorthogoal bases of compactly supported wavelets. It is foud that the lowpass filters i the PR filter bak have to satisfy similar regularity coditio as i the orthogoal case, ad the badpass ad highpass filters have to satisfy the admissibility coditio. The desig of M-chael wavelet bases is cosiderably more difficult tha the two-chael case due to the large umber of desig parameters i the M- chael PR filter bak ad the difficulty i meetig the regularity coditio exactly. I this correspodece, we propose to use cosiemodulated filter baks (CMFB s) [] [3], [6], [9] to costruct such a wavelet basis. The advatages of the CMFB are its low desig ad implemetatio complexities, good filter quality, ad ease i imposig the regularity coditio. Nguye ad Koilpillai [3] have also cosidered the desig of orthoormal cosie-modulated wavelets usig the orthogoal CMFB. The regularity coditios are imposed as additioal costraits i the optimizatio. Gopiath [4] classified the modulated filter baks accordig to the type of the discrete cosie or sie trasform with which they are associated ad geeralized the results of modulated wavelet tight frames. The outlie of the paper is as follows. I Sectio II, we shall briefly review the theory of the M-chael PR filter baks. Sectio III is devoted to a overview of the M-chael wavelet bases. I Sectio IV, the sufficiet coditio for the M-chael PR filter baks to costruct M-chael biorthogoal bases of compactly supported wavelets is derived. The theory ad desig of the CMFB is discussed i Sectio V. A ew method for imposig the regularity coditio o the CMFB ad several desig examples are give i Sectio VI. Fially, we summarize our results i the coclusio. II. THEORY OF THE M-CHANNEL PR FILTER BANKS Fig. shows the structure of a M-chael uiform filter bak with hi() ad gi() as the aalysis ad sythesis filters, respectively. A filter bak is said to be a PR filer bak if the iput ad output are equal except for a delay, [i.e., y() =x(0d)]. For perfect recostructio, hi() ad gi() have to satisfy certai PR coditios. These coditios ca either be expressed i the time domai or i the Z-trasform domai. For coveiece, we shall give the oe i the time domai. The filter bak has the PR property if it satisfies M0 hi(m + )gi(0m 0 ) = ( 0 ): (.) i=0 Aother equivalet coditio is give by hi()gj(0m` 0 )=(`)(i0j): (.) For otio coveiece, we shall itroduce the mirror image ~gi() of gi() as ~gi() =gi(0): (.3) Equatios (.) ad (.) ca the be rewritte as M0 hi(m + )~gi(m + ) = ( 0 ) (.4) i= X/98$ IEEE

3 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL (a) Fig.. (b) (a) M-chael uiform filter bak ad (b) Two-level M-chael tree-structured aalysis filter bak. ad h i ()~g j (M` + )=(`)(i0j): (.5) If the filter bak is orthogoal, h i() ad gi() will be time reverses of each other. The PR coditio is further simplified to ad M0 i=0 h i (M + )h i (M + ) = ( 0 ) (.6) h i ()h j (M` + )=(`)(i0j): (.7) III. OVERVIEW OF M-CHANNEL WAVELETS The theory of wavelets is closely related to that of multirate filter baks [6], [8]. It has bee show that discrete dyadic wavelets ca be obtaied from two-chael PR filter baks [], [5], [6] with added regularity coditio. Here, we shall cosider the geeral case of M- chael biorthogoal wavelet bases. There will be two dual bases, each geerated from a set of wavelet fuctios. First of all, we start with the discrete-time Fourier trasforms (scaled by M 0= )ofh0() ad ~g0() H0(!) =M 0= h0()e 0j! (3.) G0(!) =M 0= ~g0()e 0j! : (3.) By iteratig these discrete filters, it is possible to defie the Fourier trasform 8() ad 8() ~ of the scalig fuctio (x) ad its dual ~(x) by usig the ifiite products 8() =() 0= H0(M 0j ) (3.3) j= ad ~8() =() 0= G0(M 0j ): (3.4) j= These ifiite products ca oly coverge if H0(0) = G0(0) = : (3.5) Equatios (3.3) ad (3.4) will the coverge uiformly ad absolutely o compact sets to 8() ad ~ 8(), which are well-defied C fuctios. I additio, from the Paley Wieer theorem, it ca be show that the ifiite product is a etire fuctio of expoetial type, ad it is a Fourier trasform of a distributio with support i [N; N][5]. From (3.3) ad (3.4), we also have 8() =H0 M 8 M (3.6) ~8() =G0 M ~ 8 M : (3.7) Takig the iverse Fourier trasform leads to the well-kow twoscale differece equatios of (x) ad its dual (x) ~ as p (x) = M h0()(mx 0 ) (3.8) p ~(x) = M ~g0() (Mx ~ 0 ): (3.9) For orthoormal wavelets, the scalig fuctio (x) will be idetical to its dual. Equatios (3.8) ad (3.9) tell us that (x) ad (x) ~ ca

4 44 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 998 be writte as a liear combiatio of their cotracted (by M) ad shifted versios. Therefore, the space spaed by ad (j;k) (x)=m 0j= (M 0j x 0 k) ~ (x) =M 0j= ~ (M 0j x 0 k); Z (3.0) at a give resolutio j ca be viewed as a multiscale approximatio of a sigal f (x). To show that (x) ad (x) ~ ca be used to geerate a basis, we eed M 0 wavelet fuctios ad their duals to describe the remaiig details i the approximatio p i(x) = M h i ()(Mx 0 ) i =;;;M0 (3.) ad ~ i(x) = pm ~g i() (Mx ~ 0 ) i =;;;M0: (3.) The fuctios i(x) ad ~ i(x) ca oly be cadidates for geeratig Riesz bases of wavelets if they satisfy the admissibility coditio j9i()j jj d < ad j ~ 9i()j jj d < (3.3) where 9i() ad ~ 9i() deote the Fourier trasforms of i(x) ad ~ i(x), respectively. For i(x); ~ i(x) L (<); 9i(), ad ~ 9i() are cotiuous, ad (3.3) implies that ad () = 9i(0) = i(x) dx =0 () = ~ 9 i(0) = ~ i (x) dx =0: (3.4) Takig the Fourier trasform of (3.) ad (3.), we have 9i() = ~9i() = 0j(=M) p h i()e 8 M M (3.5) 0j(=M) p ~g i()e ~8 M M : (3.6) Sice 8(0)ad ~ 8(0) caot be zero [(3.3) (3.5)], this leads to the ecessary coditios o H i(z) ad Gi(z) ad p M h i() = Hi(0) = 0 p M ~g i() = Gi(0) = 0 i =;;;M0: (3.7) To show that the traslated ad dilated versios of i(x) ad ~ i(x) i (3.) ad (3.) geerate a Riesz basis, we eed to show the followig. C) The collectio (3.) ad (3.) costitutes a frame, i.e., 9 0 <AB <such that ad Akfk m0 ~Akfk m0 jhf; i= i= i ij Bkfk jhf; ~ i ij ~ Bkfk where ~A = A ; ~ B = B : (3.8) C) They are liearly idepedet. For dyadic orthoormal wavelets, the scalig fuctio (x) is idetical to its dual. Sice M is equal to two, there is oly oe wavelet fuctio (x). For the biorthogoal case, there are two scalig fuctios ad wavelet fuctios. The two coditios i C) ad C) ca be satisfied [5] if the associated multirate filter bak h i() ad gi() have the PR property ad the scalig fuctios that satisfy the decay coditios ad j8()j C( + jj) 0=0" j 8()j ~ C( + jj) 0=0" ; for some C ">0: (3.9) It had also bee proved i [5] that if H0(!) ad G0(!) satisfy the regularity coditio ad H0(z) = +z0 G0(z) = +z0 L Q(z) L Q 0 (z) (3.0) the the decay coditio is satisfied. If Q(z) (Q 0 (z)) is bouded above by a appropriate costat, the the regularity of the scalig fuctio (ad its dual) ca be estimated [5]. We shall come to this decay coditio for M-chael wavelets i Sectio IV. It is iterestig to ote that the admissibility coditio is automatically satisfied for M =. I this case, we have G0(z) =H(0z)ad G(z) =0H0(0z): (3.) Therefore, if H0(z) ad G0(z) satisfy the regularity coditio, i.e., have multiple zeros at! =, the H(z) ad G(z) will have the same umber of zeros at! = 0. For the M-chael orthoormal wavelets, (x) is idetical to its dual, but there are M 0 wavelet fuctios, i(x); i = ;;;M0[7], [8]. The ecessary coditios i (3.7) are also satisfied because for M-chael orthogoal filter bak, we have M0 jh i(!)j =: (3.) i=0 Sice H0(0) =, we must have H i(0) = 0; i=;;;m0. It had bee show i [8] that if H0(!) satisfies the K-regularity coditio +z0 ++z 0(M0) K H0(z) = Q(z) (3.3) M the the decay coditio will also be satisfied. IV. THEORY OF M-CHANNEL BIORTHOGONAL WAVELETS I this sectio, we geeralize the idea i [5] to the M-chael biorthogoal bases of compactly supported wavelets. We shall prove that if the scalig fuctios of a biorthogoal PR filter bak satisfy the decay coditios (3.9), the the collectio (3.) ad (3.) geerates a Riesz basis. I particular, we eed to establish (3.8), i.e., they geerate a frame, ad show that the collectio (3.) ad (3.) is liearly idepedet.

5 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL A. The Collectio Geerates a Frame Suppose that (x) ad (x) ~ satisfy the decay coditio (3.9); it ca be show that A) jhf ; i ij Ckf k j;k ad jhf ; ~ i ij C 0 kf k (4.) A) M0 f (x) = hf; i= M0 = hf; i= ~ i i i i i ~ i : (4.) Due to space limitatios, the proof is omitted. Iterested readers are referred to [5]. The proof is similar to the two-chael filter bak situatio [5]. The upper boud i (3.8) is immediate from (4.). For the lower boud, we have kf k = C sup jhf; gij kgk= sup kgk= sup kgk= M0 i= M0 i= M0 i= jhf; M0 i= jhf; jhf; i ij i ij jh ~ i ;gij = jh ~ i ;gij = [By (4.)] (By Schwartz the iequality) i ij = : [By (4.)] B. The Collectio Is Liearly Idepedet We first show that B) i ad ~ i ;j;k Zare liearly idepedet if ad oly if h i ; ~ (j ;k ) i i=(j0j 0 )(k0k 0 )(i0i 0 ): (4.3) This is the biorthogoal relatio for Riesz bases. B) A sufficiet coditio for (4.3) to hold is h (0;`) ; ~ (0;k) i=(k0`): (4.4) That is, (x 0 `) ad (x ~ 0 k) are biorthogoal. I Appedix A, we shall show that (4.4) is satisfied if (x) ad ~(x) satisfy the decay coditio (3.9). Proof: B) If (4.3) is satisfied, the ay f i the closed liear spa of the i with (i; ) 6= (i 0 ;j 0 ;k 0 )satisfies hf; ~(j ;k ) i i = (j ;k ) 0. It follows that i is ot i this closed liear spa. From (4.), we have (j ;k ) i = M0 i= h (j ;k ) i ; ~ i i i (j ;k ) ad hece, [ 0 h i ; ~(j ;k ) (j ;k ) i i] i = M0 (j ;k ) h i= i ; ~ i i i. If the i; 6=i; j ;k i are liearly idepedet, the this implies h i ; B) First, we shall show that (j ;k ) i i=(j0j 0 )(k0k 0 )(i0i 0 ): h (0;k) ; ~ (0;`) i= k` iff h ; ~ (j; `) i = k`: (4.5) Equatio (4.5) follows from the defiitio of h (0;k) ; ~ (0;`) i h (0;k) ; ~ (0;`) i= (x0k) ~ (x0`)dx ad the chagig of variable x = M 0j u. Usig the relatioship betwee (x), [ ~ (x)], ad (x), [ ~ (x)], we have (x) = i (x) = h 0() (j0;mk+) (x) (4.6) h i () (j0;mk+) (x): (4.7) We shall verify that (4.3) holds for j = j 0. From (4.6) ad (4.7), we have h i ; ~ (j; `) i i = h i()~gi (m)h (j0;mk+) ; ~ (j0;m`+m) i ; m = h i()~gi [M(k 0 `) +]=(k0`)(i0i 0 ) [from (.5) ad (4.5)]. (4.8) Similarly, from (4.6) ad (4.7) h i ; ~ (j; `) i = h i ()g 0 (m)h (j0;mk+) ; ~ (j0;m`+m) i ; m = h i()~g0[m(k 0 `) +]=(k0`)(i)=0: [from (.5) ad (4.5)]. (4.9) Sice, for j < j 0 ; ~ (j ;k ) i ca be writte as a liear combiatio of ~ (j; `), it follows that h i ; ~ (j ;k ) i i=0; if j<j 0 : (4.0) Similarly, sice for j>j 0 ; i ca be writte as a liear combiatio of the (j; `), it follows that h This proves B). i ; ~ (j ;k ) i i=0; if j>j 0 : (4.) C. K-Regularity I this sectio, we shall show that if the lowpass aalysis ad sythesis filters satisfy some regularity coditio (K-regularity [7], [8]), the the scalig fuctios ad wavelets will satisfy the decay coditios i (3.9). H 0(!) is said to be K-regular if it ca be factored as H 0 (z) = +z0 +z 0 ++z 0(M0) M K Q(z) (4.)

6 46 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 998 (a) (b) Fig.. (c) Four-chael orthogoal CMFB wavelet bases. Filter legth is 40. (a) Prototype filter i frequecy domai. (b) I time domai. (c) Aalysis filter bak. where z = e j!, ad Q(z) is a trigoometric polyomial. The precise relatioship betwee K-regularity of the scalig filters [h 0() ad g 0 ()] ad the smoothess of the scalig fuctios ad wavelets is ukow eve i the two-chael case. Usig the techiques i [5], it is possible to estimate the regularity of the scalig fuctio. I fact, we ca prove the followig [5]. If for some value k 0 B k =maxjq()q(m)q(m k0 )j =k <M L0= (4.3) the 8() satisfies the decay coditio j ~ 8()j C( + jj) 0(=)0" ; with " = L 0 0 log B k log M > 0: (4.4) V. THEORY OF THE COSINE-MODULATED FILTER BANK I this sectio, we shall itroduce the theory of the CMFB ad its desig procedure. The desig procedure will be used i Sectio VI to costruct M-chael wavelet bases. More details of CMFB theory ca be foud i [] [3], [6], ad [9]. I the CMFB, the aalysis filter bak f k () ad sythesis filter bak g k () are obtaied by modulatig the prototype filters h() f k () =h()c k; ; g k () =h()c k; k =0;;;M0 =0;;;N0 (5.) where M is the umber of chaels, ad N is the legth of the filters. The cosie modulatio that we use is c k; = cos (k +) M 0 N 0 +(0) k 4 : (5.) The proof will be give i Appedix B. I summary, to costruct M-chael biorthogoal bases of compactly supported wavelets, the lowpass filters of the associated M- chael PR biorthogoal filter baks should satisfy the K-regularity coditio, ad the highpass ad badpass filters should satisfy the admissibility coditio. This modulatio is also closely related to the exteded lapped trasform (ELT) proposed i []. It is oted that if h() is a liearphase filter, amely, h() =h(n00), the f k () ad g k () will be time reverses of each other, ad we obtai the orthogoal CMFB. O the other had, if h() is ot a liear-phase filter, the we obtai a biorthogoal CMFB.

7 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL (a) (b) (c) (d) Fig. 3. Scalig ad wavelet fuctios of four-chael orthogoal CMFB wavelet bases. (a) Scalig fuctio. (b) Wavelet fuctio-. (c) Wavelet fuctio-. (d) Wavelet fuctio-3. Lettig H(z) = M0 z 0k G k=0 k (z M ) be the type-i polyphase decompositio [9] of the prototype filter, it ca be show [6], [9] that the PR coditios are give by G k (z)g M0k0(z)+G M+k (z)g M0k0(z)=z 0 : (5.3) For orthogoal CMFB, the PR coditios are further simplified to ~G k (z)g k (z)+ G ~ M+k (z)g M+k (z)= (5.4) where G ~ k (z) = z m0 G M00k(z). It ca be see that the PR coditios i (5.3) ad (5.4) deped oly o the prototype filter h(). There will be M= PR coditios whe M is eve ad bm=c whe M is odd. Therefore, the desig ad implemetatio complexities ca be greatly reduced. I the orthogoal case, the umber of free parameters is further reduced by half due to the liear-phase property of the prototype filter. Sice the filter baks are frequecy-shifted versios of the prototype filter, the objective fuctio i the optimizatio is 8= jh(e i! )j d! (5.5)! where! s is the stopbad cutoff frequecy whose value should be chose betwee =M ad =M. It is also possible to replace the itegral i (5.5) by a summatio. It has the advatage of beig able to put differet weightigs to differet parts of the stopbad ad provides more cotrol over the stopbad atteuatio. The desig problem is formulated as the costraied optimizatio mi jh(e j! )j d! (5.6) h! subjected to the PR coditios i (5.3) or (5.4) for orthogoal ad biorthogoal CMFB. The desig procedure is similar to the oe that we had itroduced i [6] ad the optimizatio is performed usig the NCONF (DNCONF) subroutie of the IMSL library. I the case of biorthogoal CMFB, the liear-phase requiremet of the prototype filter is relaxed. Therefore, we have more freedom i choosig its coefficiets. I particular, biorthogoal CMFB ca be used to realize filter baks with differet delays. VI. THE DESIGN OF M-CHANNEL CMFB WAVELET BASES I this sectio, we are goig to costruct a family of M-chael wavelet bases called cosie-modulated wavelet bases usig the CMFB. Accordig to the coditios derived i Sectio IV, the lowpass aalysis (sythesis) filter should satisfy the K-regularity coditio F 0 (z) =C[ + z0 + +z 0(m0) ] K B(z) (6.)

8 48 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 998 (a) (b) (c) (d) Fig. 4. Four-chael biorthogoal CMFB wavelet bases. Filter legth is 40. (a) Prototype filter i frequecy domai. (b) I time domai. (c) Aalysis filter bak. (d) Sythesis filter bak. where B(z) is a polyomial i z, ad C is a costat. Moreover, all the highpass or badpass filters should have at least oe zero at! =0. Due to the low desig ad implemetatio complexities of the CMFB, it has bee used to desig compactly supported orthoormal cosie-modulated wavelet bases [3]. Our approach is to decompose the prototype filter H(z) ito two parts: H(z) =Q(z)P(z) (6.) ad determie the polyomial P (z) such that after modulatio F 0 (z) will have the required zeros at!` =(`)=M; ` =;;;M0. From (5.), we ote that f 0 () is derived from h() usig the cosie modulatio where f 0() =h()c 0; =h()cos =h() cos M + ' M 0N0 + 4 (M 0 N +) ' = ; =0;;;N0: (6.3) 4M Therefore, the frequecy respose of F 0 (e j! ), is give by F 0(e j! )=e j' H[e j(!0=m) ]+e 0j' H[e j(!+=m) ]: (6.4) It meas that H(e j! ) is shifted alog the frequecy axis by =M ad 0(=M. If!` are zeros of F 0 (e j! ), the the right-had side of (6.4) should also be zero. This will be the case if H(e j! ) have zeros at!` 6 (=M). Therefore, for the M-chael CMFB to have the K-vaishig momet, H(e j! ) should have zeros of order K at! m =(4m6)=M, m =;;;M0. Hece the polyomial P (z) is give by P (z) = M0 m= [z 0 0 e j(4m+=m) ] [z 0 0 e j(4(m0m)0=m) ] K : (6.5) By multiplyig P (z) with Q(z) that cotais the free parameters, the prototype filter H(z) will always satisfy the regularity coditio of a M-chael wavelet. This desig procedure ca be used i both orthogoal ad biorthogoal cases because the regularity coditio is idetical i both cases. It is iterestig to ote that due to the frequecy shift property of CMFB, all the highpass ad badpass filters will have the same umber of zeros at! =0ad satisfy (3.7) automatically. A umber of orthogoal ad biorthogoal cosie-modulated wavelet bases with differet values of M ad K were desiged. Fig. (a) (c) shows a example of the prototype filter ad aalysis filter bak of a orthogoal wavelet bases. The legth of the filters is N =40with

9 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL (a) (b) (c) (d) Fig. 5. Scalig ad wavelet fuctios of four-chael biorthogoal CMFB wavelet bases. (a) Scalig fuctio. (b) Dual scalig fuctio. (c) Wavelet fuctio-. (d) Wavelet fuctio-. M = 4ad K =. Fig. 3 shows the correspodig scalig ad wavelet fuctios. Figs. 4 ad 5 show aother example of a fourchael biorthogoal CMFB ad its correspodig wavelet bases. It ca be see from Figs. (a) ad 4(a) that the prototype filters have the desired zeros at 3=8; 5=8; 7=8 ad their cojugates. Due to the regularity coditio, the degree of freedom i Q(z) is reduced (for the same filter legth), ad the cutoff frequecy of the prototype filter is elarged. As a result, the overlap betwee adjacet aalysis/sythesis filters is also icreased. The scalig ad wavelet fuctios i Figs. 3 ad 5 are obtaied from iteratig the correspodig two-level tree-structured filter bak [Fig. (b)]. It should be oted that K-regularity is oly a sufficiet coditio to satisfy the decay coditio i (3.9). Therefore, wavelets with higher K-regularity are ot ecessarily smoother tha other wavelets with lower K-regularity if the filter legths of both systems are the same. Hece, there is a tradeoff betwee the umber of free parameters i the polyomial Q(z) ad the order of zeros that are imposed. VII. CONCLUSION I this correspodece, a theory of M-chael biorthogoal bases of compactly supported wavelets costructed from M-chael PR filter baks is preseted. It is foud that the lowpass filters i the PR filter bak have to satisfy a similar regularity coditio as i the orthogoal case, ad the badpass ad highpass filters have to satisfy the admissibility coditio. The cosie-modulated filter bak (CMFB) is used to costruct a family of such wavelet bases that has the advatages of low desig ad implemetatio complexities, good filter quality, ad ease i imposig the regularity ad admissibility coditio. A ew method for imposig the regularity o the CMFB is also itroduced, ad several desig examples are give. APPENDIX A Here, we shall show that if both H 0 () ad G 0 () satisfy the coditio (3.9), the (x 0 `) ad ~ (x 0 `) are biorthogoal. The proof is similar to the two-chael case i [5]. Proof: First, we defie the sequece U() = p ~U () = p It ca be proved that j= j= H 0(M 0j ) [0; ](M 0 ) ~H 0 (M 0j ) [0; ](M 0 ): u (x)~u (x0 `)dx = `0 :

10 50 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL 998 (e) (f) (g) (h) Fig. 5. (Cotiued.) Scalig ad wavelet fuctios of four-chael biorthogoal CMFB wavelet bases. (e) Wavelet fuctio-3. (f) Dual wavelet fuctio-. (g) Dual wavelet fuctio-. (h) Dual wavelet fuctio-3. To prove (x) ~ (x 0 `) dx = `0, it suffices to prove L 0 lim! U () =8()ad L 0 lim! ~ U () = ~ 8(). Now O the other had ju ()j = p L si(=) M si(m 0 =) jq(m 0`)j [0; ](M 0 ): `= j si j jfor jj =: Hece, jsi(m 0 =)j 0 [0;](M 0 ) M jj 0, which implies si(=) M si(m 0 =) [0;](M 0 ) si(=) C( + jj) 0 : = Writig = k 0 + q with 0 q<k, `= Q(M 0`) sup jq()j sup jq()j q j#(m k` )j ` =0 q C( + jj) log B = log M : Puttig it all together, we have ju ()j C 0 ( + jj) 0L+log B = log M where C 0 is idepedet of. Sice U () coverges poitwise to 8(), the Lebesgue domiated theorem implies that U () teds to 8() i L (<). The L -covergece of ~ U () is proved aalogously. Therefore, (x 0 `) ad ~ (x 0 `0) are biorthogoal. APPENDIX B Here, we shall prove that 8() satisfies the decay coditio (4.7) if H 0(!) has K-regularity. We will cosider the case jj ad jj separately jj : Cosider j8()j = H 0 (M 0j ) ( + CM 0j jj) j= j= j= exp(c 0 M 0j jj) = exp C 0 jj M 0 which is uiformly bouded for jj. Sice exp[cjj=m 0 ] is mootoic icreasig i the rage [0, ] ad j8()j C( +

11 IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 46, NO. 4, APRIL jj) 0(=) 0 " is mootoic decreasig i the same rage, we ca always choose a proper value of C such that the decay coditio is satisfied. Therefore, we eed oly to cosider the case jj. jj: Sice we have `= +e 0i + e 0j + +e 0(M0)j M = 0 e 0jM M 0 e 0j = e0j(m0)= si(m=) M si = M = 0jM(M 0 e ) 0 e0j(m ) `= = e 0j= Hece, we have 0j(M0)M = e si[m 0(`0) =] M si M 0`= `= si[m 0(`0) =] M si M 0`= 8() =() 0= H 0(M 0j ) j= =exp(0jl=) si(=) (=) = si(=) e0j= : = L #(M 0k`) with #() =Q(=M)Q(=M ) Q(=M k ). Sice jj >, there exists `0 0 so that M k` jjm k(` +). By the same argumet as i the case jj `=` + #[M 0k`] = `=0 #[M 0ki M 0k(` +) ] i=0 [6] M. Vetterli ad C. Herley, Wavelets ad filter baks: Theory ad desig, IEEE Tras. Sigal Processig, vol. 40, pp. 07 3, Sept. 99. [7] H. Zou ad A. H. Tewfik, Discrete orthogoal M-bad wavelet decompositios, i Proc. IEEE ICASSP, 99, vol. IV, pp [8] P. Steffe, P. N. Heller, R. A. Gopiath, ad C. S. Burrus, Theory of regular M-bad wavelet bases, IEEE Tras. Sigal Processig, vol. 4, pp , Dec [9] P. P. Vaidyaatha, Multirate Systems ad Filter Baks. Eglewood Cliffs, NJ: Pretice-Hall, 99. [0] M. J. T. Smith ad T. P. Barwell, III, Exact recostructio techiques for tree-structured sub-bad coders, IEEE Tras. Acoust., Speech, Sigal Processig, vol. ASSP-34, pp , Jue 986. [] R. D. Koilpillai ad P. P. Vaidyaatha, Cosie-modulated FIR filter baks satisfyig perfect recostructio, IEEE Tras. Sigal Processig, vol. 40, pp , Apr. 99. [] H. S. Malvar, Exteded lapped trasforms: Properties, applicatios, ad fast algorithms, IEEE Tras. Sigal Processig, vol. 40, pp , Nov. 99. [3] T. Q. Nguye ad R. D. Koilpillai, The theory ad desig of arbitrarylegth cosie-modulated filter baks ad wavelets, satisfyig perfect recostructio, IEEE Tras. Sigal Processig, vol. 44, pp , Mar [4] R. A. Gopiath, Modulated filter baks ad wavelets A uified theory, i Proc. ICASSP, 996, p [5] S. C. Cha, Theory of M-chael biorthogoal compactly supported wavelets bases, It. Rep., Uiv. Hog Kog, Dec [6] Y. Luo, S. C. Cha, ad K. L. Ho, Theory ad desig of arbitrarylegth biorthogoal cosie-modulated filter baks, IEEE ISCAS, 997, p. 49. [7] A. H. Tewfik ad M. Kim, Fast positive defiite liear system solvers, IEEE Tras. Sigal Processig, vol. 4, pp , Mar [8] Special Issue o Wavelets, Proc. IEEE, vol. 84, Apr [9] T. Q. Nguye ad P. N. Heller, Biorthogoal cosie-modulated filter bak, Proc. IEEE ICASSP, 996, vol. III, pp is bouded idepedetly of sice jm 0k(` +) j. O the other had `=` + #(M 0k`) k(` +) Bk k+log jj= log M Bk C 0 ( + jj) log B = log M : Fially, we have the desired result j8()j C( + jj) 0L+log B = log M. ACKNOWLEDGMENT The authors would like to thak Dr. K. M. Hog with the Departmet of Electrical ad Electroic Egieerig, Uiversity of Hog Kog, for useful discussios. REFERENCES [] A. Grossma ad J. Morlet, Decompositio of Hardy fuctios ito square itegrable wavelets of costat shape, Soc. Id. Appl. Math. J. Math., vol. 5, pp , 984. [] I. Daubechies, Orthoormal bases of compactly supported wavelets, Commu. Pure Appl. Math., vol. XLI, pp , 988. [3], The wavelet trasform, time- frequecy localizatio, ad sigal aalysis, IEEE Tras. Iform. Theory, vol. 36, pp , Sept [4] S. G. Mallat, Multiresolutio approximatios ad wavelet orthoormal bases of L (<), Tras. Amer. Math. Soc.,, vol. 35, pp , 989. [5] A. Cohe, I. Daubechies, ad J. Feauveau, Biorthogoal bases of compactly supported wavelets, Commu. Pure Appl. Math., vol. 45, pp , 99.