Towards global design of orthogonal filter banks and wavelets Conception orientée vers les bancs de fi ltres orthogonaux et à ondelettes

Transcription

1 5 Towards global desig of orthogoal filter baks ad wavelets Coceptio orietée vers les bacs de fi ltres orthogoaux et à odelettes Jie Ya ad Wu-Sheg Lu * This paper ivestigates several desig issues cocerig two-chael cojugate quadrature (CQ) filter baks ad orthogoal wavelets. It is well kow that optimal desigs of CQ filters ad wavelets i the least squares () or miimax sese are ocovex problems ad to date oly local solutios ca be claimed. By virtue of recet progress i global polyomial optimizatio ad a direct desig techique for CQ filters, we i this paper preset a desig strategy that may be viewed as our edeavors towards global solutios for CQ filters. Two desig scearios are cosidered, amely the least squares desigs with vaishig momet (VM) requiremet, ad equiripple (i.e. miimax) desigs with VM requiremet. Simulatio studies are preseted to verify our desig cocept for both ad miimax desigs of low-order CQ filters, ad to evaluate ad compare the proposed algorithms with existig desig algorithms for high-order CQ filters. Ce documet examie plusieurs publicatios de coceptio cocerat la quadrature cojuguée à deux caaux (Cojugate Quadrature - CQ) des baques de filtre et des odelettes orthogoales. Il est bie cou que les coceptios optimales de filtres de CQ et des odelettes au plus petit carré (Least Square - ) et où le ses du miimax sot des problèmes o covexes où jusqu à préset, seules des solutios locales peuvet être utilisées. Grâce au récet progrès das l optimisatio de polyômes globaux et d ue techique de coceptio directe pour les filtres CQ, ous présetos das ce documet otre stratégie de coceptio qui ue solutio globale pour les filtres CQ. Nous cosidéros deux scéarios de coceptio : le plus petit carré avec la coditio du momet de disparitio (Vaishig Momet - VM) et l equiripple (c est-à-dire le miimax) avec la coditio VM. Les études de simulatio sot illustrées pour valider otre coceptio du et pour la coceptio de miimax d ordre faible de CQ des filtres. Il est égalemet possible de comparer les algorithmes proposés avec des algorithmes de coceptio existats pour l ordre élevé de CQ. Keywords: orthogoal filter baks; wavelets; global optimizatio I It roductio The class of two-chael cojugate quadrature (CQ) filter baks, also kow as power-symmetric filter baks [4], is oe of the most wellkow buildig blocks for multirate systems ad wavelet-based codig systems, as it offers perfect recostructio (PR) ad other desirable properties. Despite the fact that may algorithms for the desig of CQ filters have bee proposed sice the 98s, see example refereces [], [2], []-[4] ad the work cited therei, to date oly locally optimal desigs ca be claimed. From a mathematical poit of view, this is primarily because the desig problems are iheretly ocovex, admittig may local solutios. I this regard, this paper is a attempt to develop feasible a strategy towards global desigs of CQ filters. The desig method proposed i this paper is made possible by virtue of recet progress i global polyomial optimizatio [6], [9] ad a direct desig techique for the CQ filters [] i cojuctio with our observatios o a commo patter shared amog globally optimal impulse resposes of low-order CQ filters ad a progressive desig procedure i terms of filter legth. Two desig scearios are cosidered: equiripple (i.e. miimax) desigs with vaishig momet (VM) requiremet ad least squares () desigs with VM requiremet. Cocerig the first, i digital filters the magitude of the largest amplitude-respose error is usually required to be as small as possible, thus miimax solutios are geerally preferred [5]. O the other had, i several applicatios especially telecommuicatios digital filters are required to have miimal stopbad eergy; hece solutios are of importace i these applicatios. Simulatio results for both ad miimax CQ filters are preseted to verify our desig cocept for * Jie Ya ad Wu-Sheg Lu are with the Departmet of Electrical ad Computer Egieerig, Uiversity of Victoria, Victoria, British Columbia V8W 3P6, Caada. {jya, wslu}@ece.uvic.ca. The authors are grateful to the Natural Sciece ad Egieerig Research Coucil of Caada (NSERC) for supportig this work. H ( z) H ( z) Figure : A two-chael CQ fi lter bak. low-order CQ filters, ad to evaluate ad compare the performace of the proposed algorithms with existig desig algorithms for high-order CQ filters. This paper exteds i a sigificat way the work reported i [6]. II Notatio ad Backgroud II.A Two-chael orthogoal filter baks A two-chael causal FIR CQ filter bak cosists of a pair of aalysis filters H, H ad a pair of sythesis filters G ad G as show i Fig., where the four filters are related by [2] H( z) = z ( N ) H( z ) G( z) = H( z) G( z) = H( z) N where H( z)= h z is a lowpass FIR trasfer fuctio of = legth N with N eve. With (), the aliasig is elimiated, ad the PR is achieved if H( z ) satisfies G ( z) G ( z) () H ( z) H ( z ) + H ( z) H ( z )= 2 (2) Ca. J. Elect. Comput. Eg., Vol. 34, No. 4, Fall 29

2 6 CAN. J. ELECT. COMPUT. ENG., VOL. 34, NO. 4, FALL 29 Eq. (2) is equivalet to a set of N /2 equality costraits as N 2m h h+ 2m = δm for m=,,...,( N 2) / 2 = (3) where δ m is the Dirac sequece with δ = ad δ m = for ozero m. Eq. (3) is kow as the double shift orthogoality i the wavelet literature. I additio to the PR coditios, CQ filters may be required to meet other costraits such as possessig a certai umber of VMs for costructig wavelets []-[3]. It is kow that the umber of VMs of a CQ filter bak is equal to the umber of zeros of H( z ) at ω = π. Because a CQ filter has L vaishig momets if Thus a desig of CQ lowpass filter H( z ) havig L VMs ca be cast as miimize π H( ejω ) 2 dω ωa (5) subject to: costraits (3) ad (4) (6) where ω a is the ormalized stopbad edge of H( z ). I this paper, we also cosider the miimizatio of maximum istataeous power of lowpass filter H( z ) over its stopbad, subject to PR ad VM costraits. Thus the miimax desig ca be formulated as miimize max H( ejω ) ω ω π (4) (7) a subject to: costraits(3) ad (4) (8) II.B Polyomial optimizatio problems ) Polyomial Optimizatio Problems: A real-valued polyomial f ( x ) i -dimesioal space R ca be expressed as ( )= F where c( α ) R, x = [ x x2... x ], α = [ αα 2... α ] F Z+ the set of all vectors i R whose compoets are oegative itegers, ad x α = x α 2 x α 2... x α. The order (degree) of f ( x ) is defied as the largest α i. i A geeral polyomial optimizatio problem (POP) has the form ( ) (9) miimize f( x ) () subject to: fk ( x ) for k =,..., L () fk ( x ) = for k = L+,..., K (2) where fk ( x ) for k =,,..., K are real-valued polyomials. POPs iclude the followig problems as its special cases: liear programmig (LP), covex quadratic programmig (QP), semidefiite programmig (SDP) ad secod-order coe programmig (SOCP). More importatly, POPs stad for a substatially broader class that covers may ocovex optimizatio problems [6]. 2) Global Optimizatio of Problem: A recet breakthrough i the field is made by Lasserre [6] i which it is proved that whe the feasible regio i () (2) is compact (ot ecessarily covex), the global solutio of () (2) ca be approximated as closely as desired (ad ofte ca be obtaied exactly) by solvig a fiite sequece of SDP problems. A techical difficulty with the method of [6] is that the size of the SDP problems ivolved i a POP usually grows very quickly. This may cause umerical difficulties eve for POPs of moderate scales. More recetly, sparse SDP relaxatio [7] is proposed for global solutios of POPs of relatively larger scales with improved efficiecy. The method is supported by MATLAB toolbox SparsePOP2 [4] [5]. Aother MATLAB toolbox for POPs is GloptiPoly 3.4 [8], which is iteded to solve the geeralized problems of momets (GPM) that ca be viewed as a extesio of the classical problem of momets [9]. Because of the SDP relaxatio approach take by [6][7], to date the size of POP problems that GloptiPoly ad SparsePOP ca solve is quite limited. Nevertheless, the globally optimal solutios provided by these toolboxes for low-order CQ filter baks ad wavelets form oe of the key igrediets i the proposed desig techique. I additio, the availability of the global desigs for low-order filter baks helps demostrate the validity of our method for desigig globally optimal high-order filter baks ad wavelets. We shall illustrate these poits with details i the ext two sectios. III Least square desigs III.A Brief review of a direct method for local desigs A techique for direct desig of CQ filter baks ad wavelets is reported i []. The desig techique is simple ad gives local solutios of good quality. Sice it is oe of the igrediets of our desig method, below we sketch its mai steps. The desig formulatio i (5),(6) ad (7),(8) ca be expressed as miimize htqh (3) N 2m h h+ 2m δ m (4) = subject to: = N ( ) l h = (5) = where m=,,..., ( N 2) / 2, l =,,..., L, h = [ hh... hn ] T ad Q is a costat symmetric positive defiite Toeplitz matrix characterized by its first row [ π ωa si ωa... si( N ) ωa / ( N )]. Suppose we are i the k th iteratio to update the coefficiet vector from h ( k) to h( k+ ) = h( k) + d, ad we write (3) at h ( k + ) as ( k) ( k) ( k) h h+ 2m + h d + 2m (6) ( k) + dh + 2 m + dd + 2m = δm By assumig d is small i magitude ad eglectig the secod-order term i (6), we obtai ( k) ( k) ( k) ( k) ( k) h d+ 2m + dh+ 2m δm h h+ 2m um which ca be put i the form of C( k) d = u( k)

3 YAN / LU: TOWARDS GLOBAL DESIGN OF ORTHOGONAL FILTER BANKS AND WAVELETS 7.8 N = 6, L = 2.7 N = 8, L = 2 N =, L = N = 8, L = N = 8, L = N = 8, L = 2 N = 8, L = 3 N = 8, L = Figure 2: Patter of impulse resposes with differet filter legth N. The costraits (4) o VMs at h ( k + ) ca be expressed as Dd = v( k ) where v( k) = Dh ( k). The smalless of d ca be characterized as di β for i =, 2,..., N, which ca be put together as Ad b with T = N N β T problem ca be formulated as miimize dtqd+ 2dTq ( k) + κ (7) subject to: Ad b (8) C( k) u( k) d = ( k) (9) D v with q( k) = Qh ( k) ad κ a costat. Sice Q is positive defiite, (7) (9) is a covex quadratic programmig (QP) problem. III.B Global desig of low-order filter baks Evidetly, the problem i (3) (5) is a POP with N / 2+ L costraits, ad the maximum order of all the polyomials ivolved is two. For low-order filter baks, toolbox GloptiPoly 3.4 was foud to work well. For example, with N = 6, L =2 ad ωa =.56π, the software produces four globally optimal impulse resposes as h = h, flipud ( h ) ad flipud ( h ) where flipud ( h ) deotes a vector geerated by flippig vector h upside dow. We remark that the above four impulse resposes satisfy costraits (4) ad (5) ad yield the same miimum objective fuctio value as Figure 3: Patter of impulse resposes with various umber of VMs L. T h Qh = Also ote that h (as well as h ) possess miimum-phase as o zeros of their correspodig trasfer fuctios are outside the uit circle. For a miimum-phase filter, both the system fuctio ad its iverse are causal ad stable so that the filter ca be of practical use. Ufortuately, the software fails to work as log as the filter legth N is greater tha or equal to. O the other had, toolbox SparsePOP2 was foud to work for global desig of filter baks up to N =6. A techical problem with SparsePOP2 is that, ulike GloptiPoly 3.4 s ability to produce multiple global solutios, it requires settig a lower boud ad a upper boud for the impulse respose, ad oly oe global solutio that falls withi the bouds will be geerated. Our desig experieces suggest that the followig bouds work well: hd.5e h hd +.5e (2) where h d is the impulse respose of the legth N Daubechies filter [4] ad e is a N all-oe vector. III.C Potetially global desig of high-order filter baks I this sectio, we propose a method for potetially global desig of CQ filters with legth N that is too high for the above metioed software to hadle. ) Patter of impulse resposes of globally optimal filter baks: Although the curret versios of the software examied earlier are of limited use, it turs out that observatios made o the patter of the impulse resposes of low-order desigs do provide useful clues for tacklig the desig of high-order CQ filters. Our observatios are illustrated i Figs. 2 ad 3. Show i Fig. 2 are the impulse resposes of globally optimal miimum-phase lowpass CQ filters of legths N =6, 8 ad (all with L =2) obtaied usig SparsePOP2 where the impulse resposes are plotted over ormalized iterval [, ] for better compariso. From the figure, it is clear that these impulse resposes are distictly differet from each other. Nevertheless, it is equally clear that they exhibit a similar patter: it starts with a short uphill to a peak, the goes dow to compoets of small values. I additio, viewig each impulse respose as a curve (fuctio), we see that the earest eighbor to a give curve associated with filter legth N is the curve associated with legth N + 2. Furthermore, for a fixed filter legth N, the impulse resposes of globally optimal CQ filters with various VMs are clustered ad exhibit a patter similar to that i Fig. 2. As a example, Fig. 3 shows the impulse resposes of lowpass CQ filters with N = 8 ad L =,, 2, 3, 4 obtaied usig either GloptiPoly 3.4 or SparsePOP2.

4 8 CAN. J. ELECT. COMPUT. ENG., VOL. 34, NO. 4, FALL Normalized frequecy Imagiary Part Real Part Figure 4: Magitude respose of H( z ) with N =96, L =3 ad ωa =.56 π. Figure 5: Zero-pole plot of H( z ) with N =96, L =3 ad ωa =.56 π. 2) A desig strategy: Both the ad miimax desigs of CQ filters as formulated i (5),(6) ad (7),(8) are ocovex problems that possess multiple local solutios, ad several (local) desig techiques for CQ filters are available i the literature [2], []-[4]. A recet additio to this rich field of research is a direct desig method [] that deals with problems (5),(6) ad (7),(8) by local covex approximatios i a sequetial maer, ad the method is show to produce satisfactory results. Brief reviews of the method for local ad local miimax desigs are sketched i III.A ad IV.A. Takig the above aalysis ito accout, the situatio facig the desiger may be summarized as follows: (i) global desigs of CQ filters are possible by usig the methods of [6][7], but oly for short filter legths; (ii) a commo patter exists amog the impulse resposes of globally optimal lowpass miimum-phase CQ filters of short legths ad, the optimal impulse respose of legth N + 2 falls withi a small viciity of the optimal impulse respose of legth N ; ad, (iii) a sequetial desig method that requires a reasoable iitial desig for producig a locally optimal desig is withi reach. A strategy for the desig of miimum-phase CQ filters of a log (eve) legth N is developed based o the above observatios, ad ca be described i steps as follows: a. Set a iitial workig filter legth N w, say, to 4, ad desig a globally optimal, miimum-phase, CQ filter of legth N w usig, for example, GloptiPoly 3.4. Deote the impulse respose obtaied by h w. b. Geerate a legth ( N w + 2 ) iterpolated versio of h w by, for example, liear iterpolatio. Deote the iterpolated vector by h wi. c. Apply the method of [] with h wi as its iitial poit (impulse respose) to desig a optimal CQ filter of legth N w + 2. Deote the impulse respose obtaied by h w. d. If N = N w + 2, output h w as the optimal desig ad termiate; otherwise, set Nw = N w + 2 ad repeat from Step b. Although o theoretical claim about the global optimality of the above desig methodology ca be made for large N, we speculate that the desigs obtaied by this approach are quite likely to be globally optimal. This is because i each roud of iteratio the iitial poit is sufficietly close to the global miimizer ad the algorithm i [] is kow to coverge to a earby miimizer. I the ext sectio, we provide experimetal evidece that supports our speculatio. III.D Desig examples ad performace evaluatio ) Performace of the proposed method for low-order desigs: The method just described was applied to desig lowpass CQ filters of legth N = 6, 8,..., 6. For all desigs, the ormalized stopbad edge was set to ωa =.56π ad the umber of VMs was set to L =. I each desig, toolbox GloptiPoly 3.4 was applied oly oce to (3) 5) to geerate a globally optimal miimumphase impulse respose with N =4 ad L =, deoted by h (4,). (4,) I the case of N =6, h was liearly iterpolated to legth 6 ad the used as the iitial poit to ru the desig algorithm (6,) i [], ad the impulse respose obtaied is deoted by ĥ. I the (6,) case of N =8, we first obtai impulse respose ĥ as above; (6,) the liearly iterpolate ĥ to legth 8 ad use it as the iitial poit to ru the algorithm i [] to geerate impulse respose (8,) ĥ. The desigs for N =,..., 6 were carried out i a similar maer to produce impulse resposes h ˆ N. For compariso (,) ( N,) purposes, globally optimal impulse resposes h for N = 6, 8,..., 6 were obtaied by usig GloptiPoly 3.4 or Sparse- (,) POP2. It was foud that ˆ N ( N,) h ad h are practically idetical for all eve N from 6 to 6. We also remark that with the startig (4,) impulse respose h havig miimum-phase, the CQ filters so desiged all have miimum-phase, a desirable property for digital filters to be of practical use. 2) Performace of the proposed method for high-order desigs: Supported by the verificatio of our desig cocept as just illustrated, we ow proceed to apply the proposed method to desig high-order lowpass CQ filters with legth N up to 96. As a example, Fig. 4 shows the magitude respose of the CQ lowpass filter desiged by the proposed method with N = 96, L =3, ad ωa =.56π. The eergy of the filter over stopbad, i.e., the value of the objective fuctio hqh T i (3), was foud to be E 9 = For compariso, a CQ filter with the same desig specificatios, i.e., N = 96, L =3, ad ωa =.56π was desiged usig the algorithm of []. The iitial poit used i the desig was a liear-phase lowpass filter obtaied by the covetioal widow-based techique. The stopbad eergy of the CQ filter obtaied was foud to be E 9 =.394, which represets a % icrease compared with E. Like the low-order desigs, it was foud that all high-order CQ filters possess miimum-phase. As a example, Fig. 5 shows the zero-pole plot of the lowpass CQ filter of legth 96 obtaied by the proposed method. We observe that o zeros are outside the uit circle of the complex plae.

5 YAN / LU: TOWARDS GLOBAL DESIGN OF ORTHOGONAL FILTER BANKS AND WAVELETS 9 IV Miimax desigs IV.A Brief review of a direct method for local miimax desigs This sectio sketches the method for direct desig of miimax CQ filters. The techique is addressed i [] ad turs out to be of critical importace towards global miimax desigs. The problem i (7),(8) ca be formulated as N = 4, L = N = 6, L = miimize η (2) subject to: η2 H( e j ω ) 2 forω Ω (22) N 2m h h+ 2m = δm (23) = N ( ) l h =,where (24) =, ad is a set of K uiformly distributed frequecy grids over stopbad [ ωa, π ]. Defie c ( ω) = [ cos ω... cos( N ) ω] T ad s ( ω) = [ si ω... si( N ) ω] T, we ca write H ( ejω ) 2 i (22) as H( e j ω ) 2=[ h T c( ω)] 2 + [ h T s( ω)] 2 2 ( ) = c ω T ( ) 2 ω ( ω) T h T h s With a aalysis similar to that outlied i III.A, the problem i (2) (24) ca be coverted ito miimize η (25) subject to: T( ω)( h( k) + d) η forω Ω (26) Ad b (27) C( k) u( k) d = ( k) D v (28) which ca be solved as a secod-order coe programmig (SOCP) problem. IV.B Global miimax desig of low-order filter baks Obviously the objective fuctio as well as the costrait fuctios i (2) (24) are all polyomials. Therefore, (2) (24) is a POP with K + N /2+ L costraits ad the maximum order of all the polyomials ivolved is two. Toolbox GloptiPoly 3.4 was foud to work for CQ filter of order 4. With N =4, L =, ωa =.56π, ad Ω = { ωa, ωa +.25 π, ωa +.5 π,..., π} (which gives K =8), the toolbox was able to produce four globally optimal impulse resposes as (4,) h miimax = (4,) miimax (4,) h, flipud ( h miimax) ad flipud ( h miimax). The maximum istataeous eergy over stopbad for the above four impulse resposes was foud to be the same value η 2 = It was also (4,) (4,) observed that h miimax (as well as h miimax ) possess miimumphase. However, GloptiPoly 3.4 failed to work for N as small as 6 because of the relatively large umber of costraits i the miimax (4,) Figure 6: Patter of miimax impulse resposes with differet legth N. desig. O the other had, with the same bouds set for h as i (2) ad η 2, SparsePOP2 was able to produce global miimax desigs for N =4 ad 6. IV.C Potetially global miimax desig of high-order filter baks Like the observatios made o the desigs (see III.C.), globally optimal (i miimax sese) impulse resposes obtaied i IV.B appear to exhibit a patter similar to that i the case, as ca be see i Fig. 6. It is therefore atural to follow the strategy described i III.C.2 for miimax desig of high-order filter baks. IV. D Desig examples ad performace evaluatio ) Performace of proposed method for a low-order desig: The desig strategy i III.C.2 for potetially global miimax desig was applied to desig miimax lowpass CQ filters. We set ωa =.56π, L =, ad Ω cotais K = frequecy grids. GloptiPoly 3.4 was applied to (2) (24) to geerate a globally optimal miimumphase impulse respose with N =4 ad L =, deoted by h miimax. (4,) (4,) Impulse respose h miimax was the liearly iterpolated to legth 6 ad used as the iitial poit to ru the miimax algorithm of []. (6,) The impulse respose obtaied is deoted by ĥ miimax. For compariso, SparsePOP2 was applied to problem (2) (24) to geerate (6,) globally optimal impulse respose h miimax. The two impulse resposes, ĥ miimax ad h miimax were foud to be practically ide- (6,) (6,) tical, givig support to our desig cocept for the miimax desigs. (4,) We also ote that with the startig impulse respose h miimax havig miimum-phase, the CQ filter ĥ miimax obtaied possesses (6,) miimum-phase as well. 2) Performace of proposed method for high-order desigs: (6,) Followig the desig approach outlied above, ĥ miimax was iterpolated to legth 8 ad used as the iitial poit for the miimax algorithm of [] to geerate ĥ miimax. This process was repeated ad (8,) high-order lowpass miimax CQ filters with legth N up to 96 were desiged. The filters obtaied ted to be equiripple. As a example, Fig. 7 shows the magitude respose of the miimax lowpass CQ filter desiged by the proposed method with N = 96, L =3, ωa =.56π ad Ω cotaiig K = uiformly distributed frequecy grids. The maximum istataeous eergy η 2 over stopbad was foud to be E 9 miimax = For compariso, a miimax CQ filter with the same specificatios was desiged usig the miimax algorithm of []. The iitial poit used i the desig was a liear-phase lowpass filter obtaied by the covetioal widow-based techique. The maximum istataeous eergy over stopbad of the CQ filter was foud to be E 9 miimax = 7.265, which represets a 4% icrease compared with E. As i the low-order desigs, miimax

6 2 CAN. J. ELECT. COMPUT. ENG., VOL. 34, NO. 4, FALL 29 Table Coefficiets of H ( z ) of [4] ad from global desig ( ) of [4] ( ) from global desig Table 2 Filter performace compariso we also foud that all high-order miimax CQ filters produced by the proposed method possess miimum-phase. As a example, Fig. 8 depicts the zero-pole plot of the miimax lowpass CQ filter of legth 96 desiged from the proposed method 3) Comparisos with other existig methods: I this sectio, we compare the global miimax desig of filter baks with two existig methods that are well established i the literature. a) Compariso with a half-bad fi lter based method i [4]: A simple ad effective method for two-chael CQ filter baks is described i Sec of [4]. The method first desigs a zero-phase lowpass FIR half-bad filter W( z ) of order 2( N ) by, for example, the Parks-McClella algorithm. Oe the defies Y( z) = W( z) + δ with δ the peak stopbad ripple of W( z ) to esure Ye ( jω ) for all ω. The followig desig specificatios were chose for the compariso: legth N =2, stopbad edge ωa =.6π ad umber of VMs L = (example of [4]). The resultig coefficiets of the CQ filter [4] ad optimized lowpass CQ filter (algorithm proposed i IV, umber of frequecy grids K set to 3) are listed i Table, colums ad 2 respectively. The maximum istataeous eergy η 2 i stopbad ad the largest magitude error amog all N /2 equatios i (23) for the two desigs are give i Table 2. The magitude resposes of the two filters are depicted i Fig. 9. It is apparet that the proposed desig techique is able to produce CQ filters with reduced istataeous stopbad eergy ad more accurate satisfactio of the PR coditios. Imagiary Part Real Part Figure 8: Zero-pole plot of miimax H( z ) with N =96, L =3 ad ωa =.56π Normalized frequecy Figure 7: Magitude respose of miimax H( z ) with N =96, L =3 ad ωa =.56π Normalized frequecy Figure 9: Magitude resposes of miimax H( z ) with N =2 of the global desig (solid lie) versus that of H( z ) i [4] (dashed lie).

7 YAN / LU: TOWARDS GLOBAL DESIGN OF ORTHOGONAL FILTER BANKS AND WAVELETS 2 Table 3 Filter legth N = 8 Table 4 Filter legth N = 6 Table 5 Filter legth N =32 b) Compariso with the method of Smith-Barwell: The techique is developed i [2] for tree-structured aalysis/recostructio systems ad has sice bee a popular bechmark for performace evaluatio ad compariso as it also provides accurate umerical results of CQ filters of high quality. For compariso, we applied the algorithm proposed i IV to desig miimax lowpass CQ filters of legth 8, 6 ad 32 to meet the requiremet of 4 db miimum stopbad atteuatio; miimax desigs of CQ filters with the same filter legths ad stopbad atteuatio are also reported i [2]. The umbers of frequecy grids for N =8, 6 ad 32 were set to K = 25, 28 ad 5 respectively, ad the umber of vaishig momets was set to zero i all three desigs. The desig results i terms of maximum istataeous stopbad eergy η 2 ad largest equatio error i magitude are show i Tables 3, 4 ad 5 for the CQ filters of legth N =8, 6 ad 32, respectively. It is observed that i all three istaces the proposed method produces desigs with improved performace. IV.E Remarks o complexity of the proposed algorithms As made clear i III.C ad IV.C, both the ad miimax algorithms are progressive i terms of filter order. As such, more computatios relative to covetioal desig algorithms are required especially for high order CQ filters because the umber of iteratios is early N /2. O the other had, we ote that because of the way the iitial poit is costructed, each iteratio is doe with high efficiecy ad, as a result, the total CPU time for desigig a high-order CQ filter remais fairly reasoable. For example, it took 22 secods o a PC laptop with a.66 GHz dual core processor to desig a CQ filter of legth N =96. The time take to desig a miimax CQ filter of legth N =96 is relatively loger but still bearable (4857 secods). Based o the evaluatios of a large umber of desigs we carried out to date, the complexity of both the ad miimax desigs was foud to have a approximately liear growth versus filter legth. This meas for example that a miimax desig of a CQ filter of legth 2N will take twice as log as that for a miimax CQ filter of legth N. Fially, we stress that desigig filters ad filter baks is typically a off-lie (o real-time) task ad to get a desig doe very quickly is of secodary importace whe compared to the performace the CQ filter ca achieve. V Coclusio A ew method for the desig of two-chael orthogoal filter baks ad wavelets has bee proposed. Attemptig to develop a methodology for global desig of CQ filters, the proposed method is built o some recet progress i global polyomial optimizatio ad a direct desig techique for CQ filters, i cojuctio with several critical observatios o the globally optimal impulse resposes ad a progressive desig procedure i terms of filter legth. Several desig examples have bee preseted to verify the desig cocept ad demostrate the performace of the proposed algorithms i compariso with several existig methods. Refereces [] W.-S. Lu ad T. Hiamoto, Direct Desig of Orthogoal Filter Baks ad Wa velets, ISCAS 29, pp. 8-84, Taipei, Taiwa, May 24-27, 29. [2] M.J.T. Smith ad T.P. Barwell III, Exact recostructio techiques for tree-structured subbad coders, IEEE Tras. Sigal Processig, vol. 34, pp , Jue 986. [3] W.-S. Lu, Digital Filter Desig: Global Solutios via Polyomial Optimizatio, APCCAS 26, pp , Sigapore, Dec 26. [4] H. Waki, S. Kim, M. Kojima, ad M. Muramatsu, SparsePOP: A sparse semidefiite programmig relaxatio of polyomial optimizatio problems, Research Report, Tokyo Istitute of Techology, March 25. [5] H. Waki, S. Kim, M. Kojima, M. Muramatsu, ad H. Sugimoto, User maual for SparsePOP: A sparse semidefiite programmig relaxatio of polyomial optimizatio problems, Research Report, Tokyo Istitute of Techology, March 25. [6] J. B. Lasserre, Global optimizatio with polyomials ad the problem of momets, SIAM J. Optimizatio, vol., o. 3, pp. 796 C87, 2. [7] H. Waki, S. Kim, M. Kojima, ad M. Muramatsu, Sum of squares ad semidefiite programmig relaxatios for polyomial optimizatio problem with structured sparsity, SIAM J. Optimizatio 7 (26) [8] D. Herio, J. B. Lasserre, ad J. Lofberg, GloptiPoly 3: momets, optimizatio ad semidefiite programmig, LAAS-CNRS Report, Toulouse, Frace Sep 28. [9] J. B. Lasserre, A semidefiite programmig approach to the geeralized problem of momets, Mathematical Programmig, 2:65-92, 28. [] O. Rioul ad P. Duhamel, A Remez exchage algorithm for orthoormal wavelets, IEEE Tras. Circuits Syst., II, vol. 4, pp , Aug [] W. Lawto ad C.A. Michelli, Desig of cojugate quadrature filters havig specified zeros, Proc. ICASSP, vol. 3, pp , 997. [2] D.B.H. Tay, Zero-piig the Berstei polyomial: A simple desig techique for orthogoal wavelets, IEEE Sigal Processig Letters, vol. 2, o. 2, pp , Dec. 25. [3] D.B.H. Tay, Least squares desig of orthogoal wavelets via the zeropiig techique, Proc. ICASSP, vol. 3, pp , 26. [4] P.P. Vaidyaatha, Multirate Systems ad Filter Baks, Pretice-Hall, 993. [5] A. Atoiou, Digital Sigal Processig: Sigals, Systems, ad Filters, McGraw- Hill, 25. [6] J. Ya ad W.-S. Lu, Towards global desig of orthogoal filter baks ad wavelets, 29 IEEE Pacific Rim Cof. o Commuicatios, Computers, ad Sigal Processig, pp , Victoria, Caada, Aug , 29. Jie Ya received the B.Eg. degree i electrical egieerig from Southeast Uiversity, Najig, Chia, i 28. He is curretly a graduate studet pursuig his M.A.Sc degree i electrical ad computer egieerig at the Uiversity of Victoria ad a studet member of IEEE. His research iterests iclude digital sigal processig, desig of digital filters ad filter baks, ad applicatio of optimizatio methods. Wu-Sheg Lu received the B.Sc. degree i Mathematics from Fuda Uiversity, Shaghai, Chia, i 964, ad the M.S. degree i Electrical Egieerig ad the Ph.D. degree i Cotrol Sciece from the Uiversity of Miesota, Mieapolis, USA, i 983 ad 984, respectively. He was a post-doctoral fellow at Uiversity of Victoria, Victoria, B.C., Caada, i 985 ad a visitig assistat professor with Uiversity of Miesota i 986. Sice 987, he has bee with the Uiversity of Victoria where he is a professor. His research iterests are i digital sigal processig ad applicatio of optimizatio methods. He is the co-author with A. Atoiou of Two-Dimesioal Digital Filters (Marcel Dekker, 992) ad Practical Optimizatio Algorithms ad Egieerig Applicatios (Spriger, 27). Dr. Lu is a Fellow of Egieerig Istitute of Caada, ad a Fellow of the IEEE..