A NOVEL NETWORK METHOD DESIGNING MULTIRATE FILTER BANKS AND WAVELETS

Transcription

1 A NOVEL NEWORK MEHOD DESIGNING MULIRAE FILER BANKS AND WAVELES Yng an Deparmen of Elecronc Engneerng and Informaon Scence Unversy of Scence and echnology of Chna Hefe 37, P. R. Chna E-mal: ABSRAC A new unfed mehod for desgnng boh paraunary cosne-modulaed FIR fler bans and cosne-modulaed waveles s proposed n hs paper. hs problem has been formulaed as a quadrac-consraned leas-squares (QCLS) mnmzaon problem n whch all consran marces are symmerc and posve defne. Furhermore, a specfc analog neural newor whose energy funcon s chosen as he combned cos of he QCLS mnmzaon problem s bul for our desgn problem n real me. I s que easy and effcency o oban he analyss and synhess flers wh hgh sopband aenuaon and cosne modulaed waveles wh compac suppor by hs mehod. A number of smulaons show he effecveness of hs mehod and he correcness of he heorecal analyss n hs paper. Keyword: neural newor, cosne-modulaed FIR fler ban, quadrac-consraned leas-square, waveles. INRODUCION Recenly, he cosne-modulaed fler ban (CMFB) wh perfec reconsrucon (PR) characer n he feld of mulrae sgnal processng has emerged as an aracve choce of fler ban (FB) wh respec o mplemenaon cos and desgn savng. I s shown [,5] ha he M polyphase componens of he prooype fler can be grouped no M power-complemenary pars, where each par s mplemened as a wo-channel lossless lace fler ban. he lace coeffcens are opmzed o mnmze he sopband aenuaon of he prooype fler, bu hs s a hghly nonlnear opmzaon problem wh respec o lace coeffcens. Consequenly, s dffcul o oban he PR CMFB wh hgh sopband aenuaon. Recenly, several auhors [4,5,9] formulae he desgn problem as a quadrac-consraned leas squares (QCLS) problem and oban he CMFB wh hgh sopband aenuaon. In hs paper, we conver he QCLS problem n [4,5] no a very smple forma whose consran marces are of symmerc and posve defne properes, and hen recas hem no an analog neural newor framewor. Afer ha, we can no only oban cosne-modulaed fler bans wh hgh sopband aenuaon bu also realze real-me fler ban desgn n erms of modern analog VLSI echnology. I s shown [] ha fler bans sasfyng regulary condons can be used o generae orhonormal bases of compacly suppored waveles. hese regulary condons can also be ranslaed no some addonal consrans mposed on he frequency response of he lowpass fler of he CMFB so as o desgn cosnemodulaed compacly suppored waveles. hs paper s organzed as follows. In secon II, we presen he quadrac-consran (QC) formulaon of he desgn problem of mulrae cosne-modulaed fler bans and waveles. In secon III, we buld a specfc analog neural newor o solve he opmzaon desgn n real me. wo examples are gven n secon VI. Fnally, we conclude hs paper wh some remars.. QUADRAIC-CONSRAIN FORMULAIONS.. QC Formulaon for Cosnemodulaed Fler Ban he mpulse responses of he analyss and synhess flers of he cosne-modulaed fler ban, h ( n) and f ( n), are cosne-modulaed versons of a prooype fler denoed by () n, whch can be gven by p π ( M 4) π N h n = p n cos + n + (a)

2 f ( n) = h ( N n) (a) where N denoes he fler lengh, M denoes he number of channels of he fler ban, n N and M. Here, he lenghs of flers, whose z-ransform of mpulse response can be expressed as H ( z) and F ( z), respecvely, are he same and are assumed o be mulples of M, ha s, N = mm. As shown n [,4], when he prooype fler has lnear-phase, he PR condons for he CMFB wh even M can be wren as ~ ~ G() z G() z + GM+ () z GM+ () z = for even M (a) M m = M G( z) = p + M z (b) where, G (z) are he polyphase componens of he prooype fler, p( n) denoes he mpulse response of prooype fler. Hereafer we only consder he even M case snce he odd M case s very smlar. I s well nown ha a smple fler desgn mehod s drecly opmzng he mpulse response p ( n) of fler bans under he condon of mnmzng objecve funcon subjec o requred consrans (e.g., PR condons). For hs purpose, he PR condons n () should be ransformed and wren as a se of QC n erms of p ( n ) p C p (3) p, r = [ ] where = p (), p(),, p( mm ) C, and, r s a famly of symmerc and posve defne marces. Noe ha goes from o M whereas r s n he range of o m-, here herefore are mm consrans n Eq.(3). In addon, s evden ha here s a normalzed condon ncluded n (3). Because of lmed space, he dealed dervaon process of Eq.(3) s omed here. Ineresng readers can refer o he relevan maerals and references [5-9]. Snce he sysem s mos approxmae perfec reconsrucon n hs case and accordng o he PR heory of mulrae fler ban, s suffcen o only mnmze he sopband energy of he prooype fler under he quadrac-consrans of Eq.(3) n even M case. he cos of sopband energy of he prooype fler, expressed by π jω φ = ( π / M ) + ε P( e ) dω, can be furher smplfed, n vecor and marx noaon, as φ = p Qp (4) where Q s a real, symmerc, and posve-defne marx whose enry ough o be specfed by M, N and ε whch denoes he bandwdh of ranson band of he prooype fler... QC Formulaon for Orhonormal Cosne-modulaed Wavele I s evden ha fler bans ha sasfy regulary condons can be used o generae orhonormal bases of compacly suppored waveles accordng o leraure. Here, we wan o oban a cosne modulaed PR fler ban whose lowpass fler H ( z) s maxmally regular. I has been shown [] ha an Mh-band orhonormal wavele can be obaned from an M-channel PR fler ban by usng an nfne ree srucure. Moreover, he Mh-band wavele would have L vanshng momens f and only f he funcon H ( e j ω ) has zeros of order L a frequences ωl = lπ / M, l M and H = M. hrough a lo of cumbersome formula dervaons, he former condon of above can be furher expressed n erms of he fler s coeffcen vecor p as follows d dω H jω ( e ) ω l = p W p = Noe ha W l whose dagonal elemens are real s H Herman marx because Wl = Wl. hus, Eq.(5) can be furher smplfed as p l (5) R e Wl p = (6) where funcon Re () aes he real par of s augmen. On he oher hand, he laer of he above condons, H = M, could be also convered no he form of p W p = (7) where W s a symmerc and posve defne marx. By combnng (6) wh (7) ogeher, we can oban a group of equvalen regulary condons n p whch are as follows p R p = p R lp = (8) where R = W and R l = γre( Wl ) + W. he choce of he scalar γ mus be done o guaranee ha R l s a se of absolue dagonally domnan marces. herefore, R l are posve defne marces accordng o classcal marx heory. I s evden ha Eq.(8) ncludes LM + addonal consraned condons whch ensure ha

3 desgned cosne modulaed wavele has L vanshng momens and hen he propery of compac suppor. In summary, by gaherng he regulary condons wh he PR condons n Eq.(3), we are able o desgn he requred cosne modulaed PR wavele wh specfed vanshng momens..3. Uned QC Formulaon for PR CMFB and Waveles From he dscusson of wo subsecons above, we have nown ha boh he PR condons of cosnemodulaed fler bans and regulary condons mposed on s lowpass analyss fler of prooype fler can be convered no a se of same formaon of quadracconsrans whose consran marces are symmerc and posve defne as shown n (3) and (8). hus, he desgn problem of boh CMFB and s correspondng waveles can be unfed as a unfcaon mehod ha can be expressed as a nd of QCLS mnmzaon problem subjec o a se of quadrac-consrans, ha s () 3 Eq. hop = mn p Qp s.. (9) p Eq.(8) By solvng (9) for mpulse response of prooype fler and consderng Eq.(), we can oban PR cosnemodulaed waveles wh gven vanshng momens. Obvously, hs mehod s able o mpose arbrary order of vanshng momens on he desgned cosne-modulaed waveles accordng o he acual requremens. 3. NEURAL NEWORK FOR DESIGNING FB AND WAVELE In recen years, due o he poneerng wor of Hopfeld and Kennedy e. al [], neural newors mplemened by analog VLSI echnology have been receved exensve sudes and appled o a varey of problems where processng me s crcal [,3,7,8,9]. Rgh here, we would le o use hs mehod o deal wh he desgn problem n hand. 3.. Energy Funcon Accordng o Eq.(9), we can buld he cos of he consraned opmzaon problem and map as he energy funcon of an analog neural newor l (, β) = Q + β J X X X p X = () where β s a posve consan, p ( X) funcon. Noe ha qualfed penaly funcon p ( X) s a penaly should be sasfed such condons: (a) s a connuous and dfferenable funcon of unnown column vecor X whch s corresponded o he vecor p n (9), (b) p( X), X. (c) p ( X ) = ff X A X =. where A s referred o as a consran marx. An ordnary p X p X = X A X. choce of s gven by 3.. Neural Newor s Dynamcs Accordng o he energy funcon n (), he neural newor dynamcs should be such ha he me dervave of J s negave. hus, we can defne he moon equaon for he h neuron as du() J = d v() ( a) v() = f ( u() ) =,, I. ( b) where u () and v () are he npu and oupu of he h neuron a nsan, ndvdually, f ( u ) s a monooncally ncreasng acvaon funcon, and I s he number of neurons n he newor. he me dervave of energy funcon of he neural newor can be gven by I dj J du() = ' f ( u () ) d = v () d () ' where f ( u) s he dervave of f ( u ) wh respec o s augmen u. I can be easly deduced from Eqs.() and () ha he neural newor wh dynamcs gven by () has sable saonary pons a he local mnma of J Performance Analyss In hs subsecon, we wll analyze he Lyapunov sably, global convergence propery and QCLS opmzaon compuaon ably of he proposed neural newor above. For space savng, Only necessary and mporan heorems whou proofs are gven o mae our problem be complee here. Ineresng readers, please refer o he references [5,8,9] For convenen expressons of followngs, we need o defne he feasble soluon space of (), whch consss of all he equlbrum pons of J, as Θ= X QX= ν X AX, where: ν X = β( X AX) =, s he soluon of heorem : Suppose V V ( V )

4 =, hen ( ) () sasfyng V V J V, V s a monooncally decreasng funcon of me. heorem : X s a sable global mnmzer of J n () f and only f X sasfes X = max X A A, X Θ =,, l, where X M sands for weghed norm of X wh respec o marx M. In summary, accordng o heorems of above, we can conclude ha he neural newor descrbed by () has only one unque global sable mnmzer whle he oher equlbrum pons are unsable mnmzers. hus, he desgn of paraunary CMFB and waveles can be n deed carred ou by our neural newor mehod wh random nalzaon. he oupu of he neural newor s jus he mpulse response coeffcens of he prooype fler of he desgned fler ban and wavele when s seled down. Furhermore, he whole desgn process may be accomplshed n magnude order of crcu me consan when s mplemened by advanced analog VLSI echnology. For convenen comparson, we also plo an 8-channel PR CMFB desgned by convenonal wo-channel lossless lace (C_LA) parameerzed mehod n Fg.(a) n dashed lne. I can be seen ha he sopband aenuaon by our NN desgn mehod s abou 3dB lower han hose by he convenonal C_LA mehod. he consran errors n hs case are very small and can be consdered o be neglgble for many praccal purposes even hough penaly facors are large fne consans. 4.. Desgn Example II For M=8 and m=3, we desgn he compacly suppored orhornormal cosne-modulaed waveles wh wo vanshng momens. he opmzed scalng funcon and seven compac suppor wavele funcons by he proposed neural newor mehod are shown n Fgure. 4. SIMULAIONS 4.. Desgn Example I In order o verfy our analyses and dervaons n precedng secons, we desgn an 8-channel PR CMFB as he frs example by usng he proposed neural newor when M=8 and m=3. hus, fler lengh s N=48. When we choose he penaly facor o be and nalze newor randomly, he desgnng newor can sar o evolve and sele down n magnude order of crcu me consan (abou hundreds of nanoseconds). he magnude responses of he desgned prooype fler P () z and correspondng analyss fler ban H () z are shown n Fgure. ( a) () c φ () b ψ () ( d) ψ () ψ 3 () () e ψ 4 () ( f ) ψ 5 () ( g) ψ 6 () () h ψ 7 () (a) (b) Fgure Magnude responses of he prooype fler (a) and analyss flers (b) of our desgned 8-channel CMFB Fgure he desgned scalng funcon φ( ) and seven wavele funcons { ψ, 7} by our proposed neural newors

5 5. CONCLUSION hs paper proposed a new unfed mehod for desgnng boh paraunary cosne-modulaed FIR fler bans and cosne-modulaed compacly suppored waveles. hs problem s equvalen o an QCLS mnmzaon problem n whch all consran marces are symmerc and posve defne. For he propose of effcenly solvng he QCLS opmzaon problem, we bul an analog neural newor and analyzed s performance n deal. By usng he neural newor o solve he correspondng mnmzaon problem, we easly and flexbly obaned cosne modulaed fler bans wh larger sopband aenuaon, and cosne modulaed compacly suppored waveles. A las par of hs paper, wo compuer smulaons were gven o suppor our heory and mehod. ACKNOWLEDGEMENS hs projec was suppored by Chna Posdocoral Scence Foundaon and parly by Naonal Naural Scence Foundaon of Chna. REFERENCES [] Kennedy, M. P., Chua, L. O., Neural newors for nonlnear programmng, IEEE rans. On Crcu and Sysems, vol.35, no.5, pp , 988. [] Kolplla, R. D., Vadyanahan, P. P., "Cosnemodulaed FIR Fler Bans Sasfyng Perfec Reconsrucon," IEEE rans. Sgnal Processng, vol. SP-4, pp , Aprl 99. [3] Maa, C.Y., Shanbla, M., "Lnear and Quadrac Programmng Neural Newor Analyss," IEEE rans. Neural Newor, vol.3, no.6, pp , Nov. 99. [4] Nguyen,. Q., Kolplla, R. D., "he heory and Desgn of Arbrary-lengh Cosne-modulaed Fler Bans and Waveles, Sasfyng Perfec Reconsrucon," IEEE rans. Sgnal Processng., vol.sp-44, no.3, pp , March 996. [5] an, Y., Gao, X., He, Z., "Desgn of Perfec Reconsrucon Cosne-modulaed QMF Bans," Proceedngs of IEEE ISCAS'97, pp , June 997. [6] an, Y., He, Z., Lnear Phase Paraunary Cosne modulaed Fler Ban Desgn Formula Proceedngs of 3rd IEEE In. Conf. on Elecroncs, Crcu & Sysems (ICECS'96), Rodos, Greece, Ocober 996. [7] an, Y., He, Z., "Neural Newor Approaches for he Exracon of he Egensrucure," Neural Newors for Sgnal Processng VI -- Proceedngs of he 996 IEEE Worshop, pp.3-3, Japan, Sep. 996, [8] an, Y., Lu, Z., "On Marx Egendecomposon by Neural Newors," Neural Newors World, (Inernaonal Journal on Neural and Mass-Parallel Compung and Informaon Sysems), vol. 8, no. 3, pp , June 998. [9] an Y., Gao X., He Z., "Neural newor desgn approach of cosne-modulaed FIR fler ban and compacly suppored waveles wh almos PR propery," Sgnal Processng, vol.69, no., pp.9-48, Ocober 998. [] Zou, H., ewf, A.H., "Dscree Orhogonal M- band Waveles Decomposons," Proc. of ICASSP'9, May 99. Bography Yng an was born n Yngshan couny, Schuan, Chna, n 964. He receved hs B.S., M.S., and Ph.D. degrees n 985, 988, and 997, respecvely. Durng he was a research scens and lecurer. He has publshed abou 6 academc papers and four boos. Now he s an assocae professor and posdocoral research fellow n Unversy of Scence and echnology of Chna (USC), Hefe, P. R. Chna. Hs curren research neress nclude neural newor heory and s applcaons, nellgen compuaonal scence, mulrae sgnal processng, wavele ransform, mage processng, paern recognon, nellgen sysems ec.