The Design of M-Channel Linear Phase Wavelet-Like Filter Banks

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1 The Degn of -Chnne Lner Phe Wveet-Le Fter Bn X.. Xe B. Peng X. J. G.. Sh nd A. Fernndez-Vzquez Schoo of Eectronc Engneerng Xdn Unverty X n Chn xmxe@m.xdn.edu.cn bpeng@eee.org m.xoun@gm.com gmh@xdn.edu.cnfernn@noep.mx Abtrct Th pper nvetgte -chnne ner-phe (LP) veet-e fter bn. The term veet-e utfed becue the fter bn of ner-perfectrecontructon (NPR). We frty derve ome condton for degnng -chnne LP NPR fter bn nd then propoe degn method of veete fter bn. The propoed veet-e fter bn obtned by mpong regurty contrnt on the LP NPR fter bn. By ung ome mpe gebrc trnformton e convert the orgn fter nto the veet-e fter thout nonner optmzton. Wth th method e not ony preerve the dered LP property but o obtn good NPR property the orgn NPR fter bn uch hgh topbnd ttenuton. Fny exmpe ho tht LP veete fter bn th even nd odd number of chnne re cpbe of chevng dfferent degree of regurty.. Introducton Wveet fter bn hve been ued extenvey n gn proceng. ny method hve been propoed for degnng veet fter bn [-9]. In ome ppcton uch mge proceng ner-phe (LP) property hghy dered. LP ytem o u to ue mpe ymmetrc extenon method to ccurtey hnde the boundre of fnte-ength gn. Hoever ony fe method cn be empoyed for degnng veet fter bn th LP property [4-9]. The method for degnng -chnne regur LP perfect recontructon (PR) fter bn ue the ttce tructure hch gurntee the PR LP nd regurty tructury [4-8]. Unfortuntey the number of chnne ncree th pproch become very compcted due to the ncreed effort on optmzng more prmeter. Addtony there hve ome mtton on the ength of fter or the number of chnne. Another degn crred out n tme domn [9]. The qudrtc cot functon defned PR contrnt mnmzed by ovng et of ner equton tertvey. Th method nvove the ccuton of the nvere mtrce. Unfortuntey the extence of thoe nvere mtrce cnnot be gurnteed. For the degn of LP veet fter bn the bove method hve to be empoyed. Whe n ome ppcton uch oy codng here m dtorton cn be toerted the PR property not necery nd the ner-perfect-recontructon (NPR) property uffcent. Therefore e rex the PR condton to get ome other good properte. In th pper e ntroduce mpe technque to degn -chnne LP veet-e fter bn. The term veet-e ued becue e rex the PR contrnt on the veet fter bn. Our degn bed on LP NPR fter bn hch re degned ung the derved condton nd the reutng fter re optm n the mnmx ene. The regurty contrnt re then mpoed ndvduy on ech fter by ung certn mpe gebrc trnformton. We get the veet-e fter bn thout optmzton procedure nvoved n the trnformton. oreover the LP nd NPR properte re t preerved. The degn exmpe ho tht th the propoed method LP veet-e fter bn th dfferent degree of regurty cn be obtned th hgh topbnd ttenuton. ot mportnty there no mtton on the number of chnne nd the ength of fter. The outne of th pper foo. Secton derve ome condton for the -chnne LP NPR fter bn. Secton preent the propoed method of the -chnne LP veet-e fter bn. Degn exmpe re gven n Secton 4. Concuon re drn n Secton 5.. LP NPR fter bn /8/$5. 8 IEEE 46 ICALIP8

2 In th ecton e gve ome condton for the degn of LP NPR fter bn. x( n) h ( n) h ( n) h n f ( n) f f ( n) n xˆ( n) Fg.. -chnne unform fter bn Fg. ho the gener tructure of -chnne mxmy decmted unform fter bn th the ny nd ynthe fter beng H ( z ) nd F ( z ). The nput-output retonhp cn be expreed ˆ ( X z = T z X z + A z X zw ) () = π here W = e. The over ytem trnfer functon T( z) = H( z) F( z) () = nd the trnfer functon re = A ( z) = H ( zw ) F ( z). () The recontructed gn xn ˆ dffer from the nput gn xn due to three reon: mptude dtorton phe dtorton nd ng error. For the NPR LP fter bn Te ( ω ) pproxmtey houd be contnt nd A ( z ) pproxmtey be zero. To emnte phe dtorton e hve the foong reton beteen ny nd ynthe fter f( n) = h( N n) (4) N or F( z) = z H( z ) here N the order of ech fter. Subttutng (4) nto () gve N z T( z) = H( z ) H( z) (5) = hch ho tht T( z ) of LP. Wth regrd to mptude dtorton e ume tht the topbnd ttenuton uffcenty hgh nd the pbnd ft enough. Thu the probem of reducng the mptude dtorton cn be formuted the foong contrnt H( e ) + H+ ( e ) = (6) for ( + ) π ε < ω < ( + ) π + ε here the vue of ε te the hf of the trnton bnddth. Th requrement cn be redy performed by ung the Pr-cCen gorthm th hch the dered mgntude repone epecy n the trnton bnd cn be pproxmtey pecfed for ech fter. The t ue cnceton. Under the umpton tht the ny nd ynthe fter tfy the tme-revere property (4) nd ny nd ynthe fter hve the me ength e obtn the necery nd uffcent condton for pproxmte cnceton foo: ) The ny fter h ( n ) = tfy n ternte ymmetry property tht h ( n ) re ymmetrc nd nt-ymmetrc terntvey; ) The mgntude repone of ech overpped term H( z) H( zw ) need to hve the me mount tht the trnton bnd of fter houd hve the me hpe. The deted proof gven n Appendx. Accordng to the bove condton expreed n (4) nd (6) for emntng phe nd mptude dtorton e the necery nd uffcent condton for pproxmte cnceton e empoy the Pr- ccen gorthm to degn -chnne unform LP NPR fter bn. The reutng ytem h good ytem performnce nd ech fter optm n the mnmx ene.. Degn of LP veet-e fter bn In th ecton e decrbe mpe technque to mpoe regurty on the LP NPR fter bn degned n Secton to obtn LP veet-e fter bn... Formuton for cng- nd veet-e fter In n -chnne veet fter bn the cng fter denoted H z h zero of order K t the frequency pont ω = π =.e. H z the K-regur op fter. Therefore t cn be decompoed H z = C z Q z (7) here the comb fter Cz ( ) Cz = + z + + z / nd Q ( z) tfe Q ( e π ). The correpondng veet fter denoted H ( z ) = K 47

3 hve K zero t the dc frequency. So e the cng fter e decompoe H ( z ) nto to component H ( z) = D( z) R ( z) = (8) here Dz = ( z ) K nd R ( z ) tfe R (). In our propoed veet-e fter bn thoe fter H ( z) = re generted by trnformng the fter H ( z ) = of the LP NPR fter bn. Thu n the veet-e fter bn the cng nd veet fter re renmed cng-e nd veet-e fter. We denote them H ( z ) nd H z = repectvey. The deted trnformton procedure be decrbed n next ubecton... Degn of the fter Scng-e fter Our go to obtn H ( z ) of order N from H ( z ) th LP nd NPR properte beng preerved. Next e utrte ho to get the cng-e fter H ( z ) from the orgn op fter H ( z ). Conder Qz = H ( z) Cz from (7). By repcng H ( z ) th H ( z ) e hve Q ( z ) Q( z) = H( z) C( z). (9) Athough Qz n LP FIR fter Q ( z ) genery n LP IIR fter. We rerte Q ( z ) n frequency domn N Q( e ) = e QR ( e ) () here N = N Nc ( N c the order of the comb fter Cz ) nd QR ( e ω ) the mptude repone of ( Q ) e ω ). Snce Q ( z ) h poe on the unt crce ( e π = ).e. the correpondng functon ( Q ) e ω h nfnte t the frequency pont ω = π e need to remove thoe nfnte to get the op fter H ( z ). Furthermore our cng-e fter H ( z ) houd be LP. We thu te the foong operton mng the bove to requrement tfed N ( ( ω Δ ) ( ω+δ e QR e QR e + )) ω = ω Q ( e ) = N Q( e ) = e QR ( e ) othere ω [ π] Δ > =. () Here e et the nfnte t ω = ω to be the verge vue of t neghborhood n ω ±Δ n order to eep ( Q ) e ω t op fter nd thu the mpue repone decy fter. Addtony the reutng ( Q ) e ω t n LP fter hch cn be een from t frequency expreon n (). The reuted ( Q ) e ω tbe nd the bg coeffcent of t mpue repone q ( n ) re octed t n = N th N beng the ymmetrc center. No e te the ( N +) effectve coeffcent of q n to obtn the LP FIR fter qn. Th operton done by ppyng rectngur ndo n of ength ( N +) to the equence q n tht qn = nq ( n). () Fny from the reutng ymmetrc FIR fter qn e obtn the dered K-regur LP op fter (ee (7)) n tme domn foo h ( n) = c( n) q( n). () Tht h n obtned by convovng cn nd qn. In the bove trnformton the frequency repone of H ( z ) pproxmtey preerved n the reutng H ( z ). The fter ength the me nd the LP property mntned. Thu the NPR property of the veet-e fter bn cn be preerved. Wveet-e fter For the veet-e fter H ( z ) = e need mpoe zero t the dc frequency. Th cn be obtned by foong the mr procedure e the cng-e fter. So e not expn t n det nymore. ) Dvde the orgn fter H ( z ) by D( z ) n (9) R z = H z D z (4) nd R z cn be rertten n frequency domn N R( e ) = e RR ( e ) (5) here N = N Nd ( N d the order of D( z ) ). ) Set the nfnte of ( R ) e ω to pec vue N ( ( Δ ) ( +Δ e RR e RR e + )) ω = R ( e ) = N R( e ) = e RR ( e ) ω ω [ π] Δ > =. (6) ) Appy rectngur ndo ( n ) of ength N +) to the equence r n : ( 48

4 r( n) = ( n) r ( n) =. (7) 4) Convove r ( n ) th dn: h( n) = r( n) d( n). (8) Thu the dered LP veet-e fter h ( n ) re obtned n the tme domn. Wth the ny fter modfed nto the regur one e chooe the ynthe fter the tme revered veron of the ny fter n (4) to get the LP veet-e fter bn. hon n Fg. (d). Fny convove Qe ( ω ) th Cz e get the cng-e fter H ( z ). The veet-e fter re obtned by foong the procedure. Fg. (e) pot the mgntude repone of ny fter of 4-chnne -regur LP veet-e fter bn. The -regur one hon n Fg. (f). Tbe ho the pecfcton of th exmpe. The cng- nd veet-e functon We cn terte the fter of the veet-e fter bn to get the functon φ ( t) nd ψ ( t ) =. In frequency domn they re expreed ω ˆ( φω ) = H = (9) ˆ ω ω ψ( ω) = H H = = () here H () =. The functon φ ( t ) nd ψ ( t) = re ced cng-e nd veet-e functon repectvey. 4. Exmpe Exmpe I 4-chnne LP veet-e fter bn We degn 4-chnne LP veet-e fter bn th regurty of degree K = nd. The order of fter N = 79. We frt degn n LP NPR fter bn ccordng to the decrpton n Secton. The fter re contructed by empoyng the Pr-cCen gorthm th FIRP functon n tb hch n optm degn n the mnmx ene nd cn pecfy the mgntude repone of the fter. The mgntude repone of the ny fter re potted n Fg. (). The mptude dtorton E pp the ng error E nd the topbnd ttenuton A re hon n Tbe th K =. Bed on the degned LP NPR fter bn e ue the trnformton n Secton. to get the veet-e fter bn. Fg. (b)-(d) utrte the degn procedure for -regur op ny fter H ( z ). The orgn op fter H ( z ) frty dvded by comb fter hch h zero t frequency pont π / = reutng n n LP untbe fter Q ( z ) hch h poe on the unte crce. It frequency repone hon n Fg. (b). Ue Eq. () to remove the poe nd thu get the LP tbe fter Q ( e ω ) potted n Fg. (c). Appy rectngur ndo to Q ( e ω ) nd e get Qe ( ω ) Fg.. gntude repone of () 4-chnne LP NPR fter bn (b) Q( z ) (c) Q( z ) (d) Qz (e) 4- chnne LP -regur veet-e fter bn (f) - regur one. By tertng the correpondng to-eve treetructured -regur veet-e fter bn e get the cng- nd veet-e functon φ( t) nd ψ ( t ) =. They re potted n Fg.. A cn be een the functon re ymmetrc or nt-ymmetrc. 49

5 Fg.. The cng- nd veet-e functon from veet-e fter bn th -regurty. Tbe. Compron beteen NPR nd veet-e LP fter bn n term of A E pp E ( = 4 N = 79 ). Degree of A S (db) E pp E Regurty K = K = K = Exmpe II 5-chnne LP veet-e fter bn bn th even nd odd number of chnne ho tht the propoed method cn cheve dfferent degree of regurty nd hgh topbnd ttenuton. ot mportnty nce the ength of fter nd the number of chnne cn be choen freey th method more gener thn the extng method. We gve the LP veet-e fter bn th = 5 K = nd. The order of fter 8. Fg. 4()-(c) gve the mgntude repone of the orgn LP NPR fter bn the - nd -regur LP veete fter bn. By tertng to-eve tree tructured -regur veet-e fter bn e get the cngnd veet-e functon hon n Fg. 4(d)-(h). The pecfcton re hon n Tbe. Tbe. Compron beteen NPR nd A E veet-e LP fter bn n term of pp E nd ( = 5 N = 8 ) Degree of A S (db) E pp E Regurty K = K = K = A cn be een from Tbe nd the orgn fter bn the - nd -regur veet-e fter bn hve mot the me E pp nd E. A the degree of regurty ncree the topbnd ttenuton A h certn degrdton. Hoever the degned veete fter bn t hve hgh topbnd ttenuton. 5. Concuon In th pper e preent ne method for degnng -chnne LP veet-e fter bn. The gener de n th degn technque to mpoe the regurty condton on degned NPR LP fter bn hch obtned ccordng to the derved condton. We te ome gebrc trnformton on the orgn fter mng t rezng the formuton tht chrcterze the veet-e fter. In th trnformton there no optmzton procedure he the LP nd NPR properte cn t be preerved. Degn exmpe of the NPR LP veet-e fter Fg. 4. gntude repone of ny fter of 5-chnne () NPR LP fter bn (b) -regur nd (c) -regur veet-e fter bn. (d)-(h) The cng- nd veet-e functon from veet-e fter bn th regurty. 6. Appendx Snce the LP fter h ( n ) ymmetrc or ntymmetrc t cn be expreed h n = h( N n) = (A.) here defned gn functon by h ( n) ymmetrc = =. h ( n) nt-ymmetrc Combnng (4) nd (A.) e obtn f ( n ) = ( ) h ( n). Then from () e get the functon A z = H( z) H( zw) =. = 44

6 (A.) Under the umpton tht the topbnd ttenuton hgh enough t cn be found tht A ( z ) = re cued by four overpped term or to pr: { ( ) ( H ) zw H z H zw H z } nd { H ( + ) ( H ) ( + ) z H zw H z } =. We cn ee tht f the trnton bnd of the dcent ny fter re ymmetrc bout the nterecton the mgntude repone of the overpped term hve the me mount. Further e notce tht f the phe of thoe term re the me then by choong propery the gnfcnt ng cn n prncpe be cnceed competey. In ht foo e gve the deted ny of the phe property. Before proceedng e me n umpton tht the fter hve the me ength. To verfy the phe property of H ( z) ( zw ) e expre the LP fter H ( z ) th z H = e ω ( ) ( ) N + e H ( e ) H ( e ) for ymmetry ; + H ( e ) = N + e H ( e ) H ( e ) for nt-ymmetry (A.) here + H ( e ) ω π H ( e ) = π < ω < (A.4) ω π H ( e ) = H ( e ) π < ω <. For mpcty e ony conder one perod of π < ω π. For ymmetrc h ( n ) H( z) H( zw ) th z = e ω π nd W = e h the foong form ( ω π ) ( ω π ) N H ( e ) H ( e ) = e ( ) ( ) ( H e ω + H e ω π + H e ω H e ω + π + H e H e + H e H e + + ( ω π) ( ω π). (A.5) Snce e ume tht the topbnd ttenuton + ( ) gnfcnty hgh the term H ( e ω + ) H ( e ω π ) ( ω π ) nd H ( e ) H ( e ) n (A.5) cn be negected. Therefore (A.5) become ( ω π ) ( ω π ) N H ( e ) H ( e ) = e + ( ) ( ) ( H e ω H e ω π H e ω + H e ω + π ). (A.6) ) Wth nt-ymmetrc h ( n ) the excty me reut (A.6) cn be derved. Thu e concude tht the phe of H( z) H( zw ) = re the me regrde of the ymmetrc or nt-ymmetrc property of h ( n ). 7. Acnoedgment Th pper upported by NSF of Chn ( ) Progrm for Ne Century Exceent Tent n Unverty (NCET-7-656) PCSIRT (IRT645) 8. Reference [] R. A. Gopnth nd C. S. Burru On cone-moduted veet orthonorm be IEEE Trn. Imge Proce. vo. 4 no. pp Feb [] T. Q. Nguyen nd R. D. Kop The theory nd degn of rbtrry ength cone-moduted fter bn nd veet tfyng perfect recontructon IEEE Trn. Sgn Proce. vo. 44 pp r [] S. C. Chn Y. Luo nd K. L. Ho -chnne compcty upported borthogon cone-moduted veet be IEEE Trn. Sgn Proce. vo. 46 no. 4 pp. 4-5 Apr [4] A. K. Somn P. P. Vdynthn nd T. Q. Nguyen Lner phe pruntry fter bn: theory fctorzton nd degn IEEE Trn. Sgn Proce. vo. 4 no. pp Dec. 99. [5] S. Orntr T. D. Trn P. N. Heer nd T. Q. Nguyen Lttce tructure for regur pruntry ner-phe fterbn nd -bnd orthogon ymmetrc veet IEEE Trn. Sgn Proce. vo. 5 no. pp Nov.. [6] S. Orntr T. D. Trc nd T. Q. Nguyen A c of regur borthogon ner-phe fter bn: theory tructure nd ppcton n mge codng IEEE Trn. Sgn Proce. vo. 49 no. pp. -5 Dec.. [7] Y. J. Chen S. Orntr nd K. Amrtung Dydcbed fctorzton for regur pruntry fter bn nd -bnd orthogon veet th tructur vnhng moment IEEE Trn. Sgn Proce. vo. 5 no. pp. 9-7 Jn. 5. [8] Y. J. Chen S. Orntr nd K. Amrtung Theory nd fctorzton for c of tructury regur borthogon fter bn IEEE Trn. Sgn Proce. vo. 54 no. pp Feb

7 [9]. Iehr nd T. Q. Nguyen Tme-domn degn of ner-phe PR fter bn IEEE ICASSP vo. pp