Design of Cosine Modulated Filter Bank Using Unconstrained Optimization Technique

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1 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coerncus Value (3): 6.4 Imac Facor (3): Desgn of Cosne Modulaed Fler Bank Usng Unconsraned Omzaon Technque Shaheen, Dnesh K. Dhaka RTU, Rajashan College of Engneerng for Women, Jaur, Rajashan, 36, Inda RTU, Rajashan College of Engneerng for Women, Jaur, Rajashan, 36, Inda Absrac: Ths aer rooses a unque mehod for desgnng of cosne modulaed-qmf bank usng unconsraned omzaon echnque. Objecve funcon of hs QMF bank s a quadrac equaon form by combnng he oal ass band error, mean square error of band ass and so band. We are usng he unconsraned omzaon mehod for mnmzng he objecve funcon. Resul shows ha he erformance s beer han he exsng mehod. Keywords: Prooye fler Cosne modulaon, hgh band wdh, NOI. Inroducon These day researchers are gvng a lo of aenon o he QMF bank because of s wde range of alcaon n he feld of dgal sgnal rocessng. Cosne modulaed QMF bank s one of he moran subclass of he Mul-rae fler banks. I s beng exensvely used because of fewer comlexy, ease n desgnng and low cos. Some alcaons of QMF banks are sub-band codng of seech and mage sgnals [ 6], seech and mage comresson [7,8], rans-mullexers [9 ], equalzaon of wreless communcaon channels [], source codng for audo and vdeo sgnals [3], desgn of wavele bases [4], sub-band acousc echo cancellaon [5], and dscree mul-one modulaon sysems [6]. The mos frequenly used fler bank among all he M-band fler banks s cosne-modulaed fler bank (CMFB) because he desgnng s easy and more realzable han ha of any oher fler banks [3, 6]. In CMFB all he flers are cosne modulaed verson of a low-ass rooye fler. So he desgn of whole fler bank reduces o he desgn of a sngle rooye fler. Imlemenaon of CMFB consss of one rooye and dscree cosne ransforms (DCT). Near erfec reconsrucon (NPR) fne mulse resonse (FIR) CMFB avods comuaon of large marx ses. There are wo yes of CMFB one s wh erfec reconsrucon [7] and he oher s seudo-qmf [8]. Unlke PR fler banks, n seudo-qmf alasng s canceled aroxmaely and he dsoron s aroxmaely a delay [3, 6] and he aroxmaon mroves wh he fler order. I could be also called a secal case of erfec reconsrucon QMF bank. I s found n many sgnals ha her energy s domnanly concenraed n a arcular regon of frequency. To save he bandwdh n such sgnals, sgnal can be comressed and decmaed. Bu smle way of comresson affecs he qualy of he sgnal. Fgure : Basc srucure of wo channel QMF Fgure : Frequency resonse of he fler So Quadraure Mrror Fler (QMF) comes as a soluon o hs roblem as saves he bandwdh and ncreases effcency whou comromsng he qualy of he sgnal. Fgure shows he ycal wo channel QMF bank and fgure shows he frequency resonse of hs fler. Ths fler sls he sgnal no wo sub-bands usng hgh-ass and low-ass flers H (Z) and H (Z) resecvely. These sgnals are decmaed, coded and ransmed on he ransmer sde. These sgnals are decoded, nerolaed and hen fnally assed hough he flers o recombne he sgnals. The reconsruced sgnal s no exac relca of he ransmed sgnal; suffers from hree yes of dsorons: amlude dsoron (AMD), hase dsoron (PHD) and alasng dsoron (ALD). Mos of he researchers sresses on he elmnaon or mnmzaon of hese dsorons o oban he erfec Volume 4 Issue 6, June 5 Paer ID: SUB Lcensed Under Creave Commons Arbuon CC BY

2 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coerncus Value (3): 6.4 Imac Facor (3): reconsrucon (PR) or near erfec reconsrucon (NPR) sysem [4]. Alasng can be cancelled comleely by choosng he arorae synhess flers wh resec o he analyss fler hase dsoron can be elmnaed by usng he lnear hase FIR flers [5-6]. Amlude dsoron can be reduced usng comuer aded echnques [7]. So s clear from he equaon (5) now o ge he lnearhase ransfer funcon we need o have lnear-hase lowass lnear hase fler. One of he mos frequenly used fler bank among all he M-band fler banks s cosne-modulaed fler bank (CMFB) because he desgnng s easer and more realzable han ha of oher fler banks [-]. Mos of he researchers sresses on he elmnaon or mnmzaon of hese dsorons o oban he erfec reconsrucon (PR) or near erfec reconsrucon (NPR) sysem [8].. Basc Analyss of Fler Bank General equaon for he wo channels QMF s gven as Y( Z) H Z G Z H Z G Z X Z H Z G Z H Z G Z X Z The second erm n he above equaon reresens he alas comonen n he ouu. Alas comonen X(-Z) can be removed by choosng he synhess fler such ha o make he second erm zero such as G Z H Z and G Z H Z () () Afer removng he alas comonen from he equaon () overall ransfer funcon s gven by Z Z Y TZ H X Z G Z H Z G Z (3) Y(n) suffers from amlude dsoron (AMD)( f T(Z) s no consan for all value of ω,) and would suffer from hase dsoron (PHD)( f T(Z) doesn has lnear hase). So o elmnae AMD T(Z) should be all ass and o elmnae PHD T(Z) should be lnear hase. *When all he flers nvolved are lnear-hase flers ransfer funcon of he QMF T(Z) would behave lke lnear hase fler ha would remove he hase dsoron n he QMF. The relaon of analyss and synhess o each oher s gven as [9-] H Z H Z G Z H Z G Z H Z H Z,, (4) Usng equaon (3) and (4), we ge T Z H Z H Z (5) I s one of he effcen echnques whch rovde mnmum comuaonal cos for he desgn of fler banks. In CMFB all he flers are cosne modulaed verson of a low-ass rooye fler. In hs echnque all he analyss flers and synhess flers are smulaneously generaed by he low-ass rooye fler. So he desgn of whole fler bank reduces o he desgn of a sngle rooye fler [3-8]. Imlemenaon of CMFB consss of one rooye and dscree cosnes ransform (DCT). All of hese flers can be desgned usng eher he near erfec reconsrucon (NPR) or erfec reconsrucon (PR) [9]. Many romnen auhors has been suded he cosne modulaed fler banks wh PR and NPR condons and fnd ha NPR ye s more realzable, comuaonally effcen and more smle n comarson o he PR [4],[ -] 3. Formulaon of Desgn Problem Desgn of cosne modulaed-qmf banks sar wh he desgn of he low-ass rooye fler. All he oher flers are hen obaned by he cosne modulaon of hs lowass rooye fler []. We ake lnear-hase FIR fler for he basc low-ass rooye fler. Our am s o mnmze he objecve funcon of he rooye fler, whch s n our case s a quadrac equaon formed wh some real consans and oal ass band error, mean square error of band ass and so band of he fler. Objecve funcon v s gven by he equaon as L * E j * Es k E v * (6) L, j and k are real consans, and E, E s and E are he measure of mean-square ass-band error, mean-square so band error and oal error. E, E s and E can be exressed as E b '* q * b (7) q c c' d E s s he so band energy of low-ass rooye fler beween ω s o s gven by (8) E b * b ' (9) s * Volume 4 Issue 6, June 5 Paer ID: SUB Lcensed Under Creave Commons Arbuon CC BY

3 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coerncus Value (3): 6.4 Imac Facor (3): b b J (6) s c * c' d () E s he square error of he rooye fler gven as E b '* d H r N /...cos / T () c cos () d=c evaluaed a ω = π/, H r s he amlude funcon and We omze he gven objecve funcon usng Marquard omzaon mehod o ge he mnmum value. Cosne Modulaed Fler Bank Afer desgnng he low-ass fler all flers of analyss and synhess secons are obaned by cosne modulaon of he rooye fler usng he followng relaons. real ak* n magak * n hn H ( k, n) * * n and n s gven as (3) n cos( *(* k )*(* n ) /(4* nbands)) (4) n sn( *(* k )*(* n ) /(4* nbands)) (5) Equaon (4) and (5) reresens he analyss and synhess flers of he fler bank resecvely. And n bands s he number of bands n fler banks., n vares from o N- and k vares from o M-. There are wo man advanages of cosne modulaed fler banks are, frs one s he cos of whole analyss flers reduces o he cos of one rooye fler lus he cos of modulang and second one s we need no omze so many arameers. The number of arameers need o be omze becomes fewer. 4. Desgn Algorhm Here we are omzng he objecve funcon usng Marquard omzaon mehod. If b s he mnmum value a he h se hen b + a + h se can be calculaed by usng Marquard omzaon mehod as follows J H a I, (7) s he graden of he objecve funcon, H s he Hessan marx, a s a h sage eraon consan and I s N/ x N/ deny marx s a consan and H D (8) ' D d *d ( d cos N / ) and / N / a Dh (9) () Se nal values of α, e, a >, b <, and N. () Se nal desgn vecor h h h h... h N / T such ha energy s un remans whn a secfed olerance,.e., k u h e (3) Se he eraon number =, and b = h. (4) Comue ф usng Eq. (9) a he desgn vecor b. () (5) Comue he Hessan marx H usng Eq. (8) and he marx J usng Eq. (7). (6) Comue he new or mroved aroxmaon b b J (7) Comue he value as gven n () u, a he on b +, and f value s no sasfed hen choose he omum on as b, so he rocedure and go o se (). (8) If he condon a (9) s sasfed a he on b +, comue he objecve funcon ф + a he on b +. If, choose he omum on as b, so he rocedure and go o se (). If <ф, se and b = b +. (9) Se he new eraon number as = +, a = a b and go o se (3). () Comue h = (/)b and so he rocedure. Volume 4 Issue 6, June 5 Paer ID: SUB Lcensed Under Creave Commons Arbuon CC BY

4 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coerncus Value (3): 6.4 Imac Facor (3): Resuls and Dscusson Fgure 3: Flow char for he omzaon rocess A wo channel Quadraure mrror fler wh cosne modulaon s resened here wh roosed algorhm usng MATLAB. Volume 4 Issue 6, June 5 Paer ID: SUB Lcensed Under Creave Commons Arbuon CC BY

5 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coerncus Value (3): 6.4 Imac Facor (3): Fgure 4(a): Gan of he analyss fler Fgure 4 (d): Amlude dsoron funcon grah Fgure 4(a) and 4(b), show he gan and reconsrucon error grah of for he analyss fler resecvely. Whle fgure 4(c) and 4(d) gves he aenuaon and dsoron grah for he same resecvely. Table gves erformance summary of he roosed fler a he dfferen fler orders. Table : Performance Parameers of he Fler Fler order Bands CPU me NOI A s Fgure 4(b): Reconsrucon error grah Smulaon resul show ha he roosed mehod gves he beer resul n erms of comuaon me, number of eraon and so-band edge aenuaon. Fgure 4(c): Aenuaon grah of analyss fler Effecveness of he roosed algorhm s shown n erms of shar cu off, broader bandwdh, less NOI and less CPU me requred. Grahs here shown s a N=3. References Volume 4 Issue 6, June 5 [] R. E. Crochere, Sub-band codng, The Bell Sysem Techncal Journal, vol. 6, no. 7, , 98. [] R. Bregovc and T. Saramak, A general-urose omzaon aroach for desgnng wo-channel fr flerbanks, IEEE Transacons on Sgnal Processng, vol. 5, no. 7, , 3 [3] P. P. Vadyanahan, Mulrae Sysems and Fler Banks, Prence Hall, Englewood Clffs, NJ, USA, 993. [4] R. Bregovc and T. Saramak, Two-channel FIR flerbanks-a uoral revew and new resuls, n Proceedngs of he nd Inernaonal Worksho on Transforms Fler banks, vol. 4, , Brandenburg, Germany, 999. [5] Jorg Klewer, On he Relaonsh beween Pseudo- QMF Desgns and Perfec-Reconsrucon Soluons for Modulaed Fler Banks, IEEE. 5 (9) (4) [6] J. H. Nussbaumer and M. Veerl, Pseudo quadraure mrror flers, n Proceedng Conference DSP Florence 8,. 8- (98). [7] T. A. Ramsad, Cosne modulaed analyss-synhess fler banks wh crcal samlng and erfec Paer ID: SUB Lcensed Under Creave Commons Arbuon CC BY

6 Inernaonal Journal of Scence and Research (IJSR) ISSN (Onlne): Index Coerncus Value (3): 6.4 Imac Facor (3): reconsrucon, n Proc. IEEE In. Con$ Acous., Seech Sgnal Processng, Torono, Canada, May 99, [8] R. V. Cox, The desgn of unformly and nonunformly saced seudo QMF, IEEE Trans. Acous., Seech Sgnal Processng, vol. ASSP-34,. 9-96, Oc. 986 [9] Rohweler, J. H., Polyhase quadraure flers A new subband codng echnque. In Proc. of Inernaonal Conference on ASSP 83, Boson, Vol. 4, 983, [] P. L. Chu, Quadraure mrror fler desgn for an arbrary number of equal bandwdh channels, IEEE ransacon on acousc, seech, sgnal rocessng, vol. ASSP 33, no.,. 3-8, 985. [] J. Klewer, On he relaonsh beween seudo-qmf desgn and erfec-econsrucon soluons for modulaed fler banks, IEEE, , 999. [] T. Q. Nguyen, Near erfec reconsrucon seudo QMF, IEEE ransacon on sgnal rocessng, vol. 4, no.,.65-76, 994. [3] C. D. Creusere, and S. K. Mra, A smle mehod for desgnng hgh qualy rooye flers for M-band seudo QMF banks, IEEE Transacons on Sgnal Processng, vol. 43,. 5-7, 995. [4] R.K. Son, A. Jan and R. Saxena, Desgn of NPR- Tye Cosne Modulaed Flerbank Usng Combnaonal Wndow Funcons, IJCNSS., 3, [5] T.Q. Nguyen, Near-erfec-reconsrucon seudo- QMF banks, IEEE, X, 994. [6] A. Kumar, G.K. Sngh and R.S. Anand, An mroved mehod for desgnng rooye fler for M-band seudo QMF banks, ICCET, 9. [7] J.Klewer and A. Merns, Desgn of araunary oversamled cosne modulaed fler banks, IEEE Trans. Acous., Seech Sgnal Processng, 73-76, 997. [8] Y-P. Ln and P. P. Vadyanahan, Lnear hase cosne modulaed maxmally decmaed fler banks wh erfec reconsrucon, IEEE Transacon on sgnal rocessng, X, 995. [9] R.D.Kollla and P.P. Vadyanahan, A secral facorzaon aroach o seudo-qmf desgn, IEEE,. [] J.Prncen, The desgn of nonunform modulaed flerbanks, IEEE, X, 995. [] F.Cruz-Roldan, P.Amo-Loez, P.Marn-Marn and F.Loez-Ferreras, Alernang analyss and synhess flers: A new seudo-qmf bank, Academc ress, ,. [] Anamka Jan and A. P. Goel, Desgn of M-channel seudo near erfec reconsrucon QMF bank for mage comresson, sgnal and mage rocessng: An nernaonal Journal, vol. 3, No. 4,. Auhor Profle Shaheen has receved B.ech n Elecroncs and communcaon engneerng from MU, Gaya. She s a fnal year suden of M.ech n dgal communcaon, Rajashan college of Engneerng for Women, RTU, Koa, Inda. Volume 4 Issue 6, June 5 Dnesh K. Dhaka has receved M.ech n Dgal comm. Sysem from SBCET, Jaur. He has eachng exerence of more han 6 years and has ublshed more han aers on nernaonal journal. He s asssan rofessor and M.ech cordnaor n Rajashan college of Engneerng for Women, RTU, Koa, Inda. Paer ID: SUB Lcensed Under Creave Commons Arbuon CC BY