IIR Band Pass and Band Stop Filter Design Employing Teaching-Learning based Optimization Technique

Transcription

1 Inernaonal Journal of Compuer Applcaons ( ) Volume 4 o.4, Ocober 4 IIR Band Pass and Band Sop Fler Desgn Employng Teacng-Learnng based Opmzaon Tecnque Damanpree Sng San Longowal Insue of Engneerng and Tecnology, Longowal, Inda J.S. Dllon San Longowal Insue of Engneerng and Tecnology, Longowal, Inda ABSTRACT In s paper newly developed eacng-learnng based opmzaon (TLBO) algorm s appled for desgnng band pass (BP) and band sop (BS) dgal IIR flers. TLBO s eursc algorm based on e socal penomenon of eacng-learnng process. Te effecveness of purposed algorm s valdaed by desgnng e BP and BS flers by approxmang e magnude response w L p -norm error creron, mnmzng pass band and sop band rpples along w guaraneed sably. Te resuls obaned employng TLBO are compared o ose obaned by e well known evoluonary algorms suc as erarccal genec algorm, ybrd aguc genec algorm and mmune algorm. Te resuls reveal a e purposed TLBO algorm gves beer opmal fler n erms of magnude response and rpples n pass band and sop band. Keywords IIR fler, eacng-learnng based opmzaon, magnude response, band pass, band sop, sably, L p -approxmaon error.. ITRODUCTIO Dgal flers are used n number of applcaon lke communcaon, radar sgnal processng, speec and mage processng, bomedcne, and sesmc exploraon. Te man purpose of usng dgal fler s o exrac e useful nformaon from e sgnal, and o remove e unwaned par of e sgnal. On e bass of mpulse response leng, dgal flers are classfed n o wo ypes: fne mpulse response (FIR) fler and nfne mpulse response (IIR) fler []. IIR fler gves beer performance w lesser number of coeffcens an FIR for e same desgn specfcaon. IIR flers are used wen sarp cu off and g rougpu are needed. However, ere are some dsadvanages of IIR dgal fler suc as []: ) nsably of IIR fler ) mul-modal error surface. Te sably problem can be easly andled by mposng sably consrans on e denomnaor coeffcens of IIR dgal fler. Due o non-lnear and mulmodal error surface of IIR dgal fler, e classcal graden based opmzaon ecnques canno fnd e global mnmum [3, 4]. In recen years, many researcer ave appled number of evoluonary algorms for e desgn of IIR dgal flers suc as: GA [5-8], mmune algorm (TIA) [9], abu searc [], parcle swarm opmzaon (PSO) [, ], Seeker opmzaon algorm (SOA) [3], wo-sage ensemble evoluonary algorm [4], Gravaon searc algorm [5] and many more. All e evoluonary and swarm nellgence based algorms are probablsc algorms and requre common conrollng parameers lke number of generaons, populaon sze, ele sze, ec. Besdes e common conrol parameers, eac algorm requres s own algorm-specfc conrol parameers. Te performance of e above menoned algorms s mmensely dependen upon proper unng of e algorm-specfc parameers. Te mproper unng of algorm-specfc parameers eer ncreases e compuaonal effor or yelds e local opmal soluon. To overcome e above dscussed drawbacks, Rao e al. [6, 7] proposed a eacng-learnng based opmzaon (TLBO) algorm based on e naural penomenon of eacnglearnng. Te mplemenaon of TLBO does no requre e deermnaon of any algorm specfc conrollng parameers wc makes e algorm robus and powerful. TLBO requres only common conrollng parameers lke populaon sze and number of generaons for s workng. Te nen of s paper s o use e effecveness of TLBO algorm for desgnng band pass (BP) and band sop (BS) IIR fler employng L p -norm error creron, and pass band rpples and sop band rpples of dgal IIR fler. Te desgned BP and BS IIR flers w TLBO algorm are compared w erarccal genec algorm (HGA) [7], ybrd aguc genec algorm (HTGA) [8] and aguc mmune algorm [TIA] [9] o fnd ou e comparave effecveness of e algorm and deermnaon of bes opmal IIR flers. Te paper s organzed as follows: In Secon, maemacal expresson of an a IIR fler and e obecve funcon are formulaed. In Secon 3, TLBO algorm s brefly dscussed for e IIR fler desgn problem. In Secon 4, compreensve and demonsrave ses of resuls and llusraons are gven o make a comparson of e sudy w exsng algorms. Fnally, Secon 5 concludes e paper.. FORULATIO OF IIR FILTER DESIG PROBLE IIR dgal flers are caracerzed by e followng dfference equaon []: y ( n) ( k) x( n k) () k y( n) bk x( n k) ak y( n k) () k k were (k) s e mpulse response of e fler, wc s eorecally nfne n duraon, b k and a k are e coeffcens of e fler, x (n) and y (n) are e dscree npu and oupu of e fler. Te ransfer funcon of IIR fler s defned as below: k ak z k H ( z) (3) k bk z k Were, are numeraor and denomnaor orders, respecvely. An mporan par of e IIR fler desgn process s o fnd suable values for e coeffcens b k and a k suc a some aspec of e fler caracerscs, suc as frequency response, beaves n a desred manner. An IIR dgal fler 38

2 Inernaonal Journal of Compuer Applcaons ( ) Volume 4 o.4, Ocober 4 can be expressed by e cascadng frs and second order secons [8] saed as: pu e H (, A u qu e (4) gve gve v ve ve were T x p, q,..., p, q, g, g,,,..., g, g,,, A [ ] and Vecor x denoes e fler coeffcens of dmenson S w S = and A s e gan. Te man goal of e desgn algorm of dgal IIR fler s o fnd a se of fler coeffcens o mnmze e magnude approxmaon error funcon n L p -norm [8, 9] and rpples n pass band and sop band.. Te magnude response s specfed a K equally spaced dscree frequency samples n pass-band and sopband. E ( denoes e absolue error L -norm of magnude response and E ( denoes e squared error L -norm of magnude response and are defned as gven below: K E ( H I ( ) H( (5) K ( H I ( ) H( E (6) Ideal magnude response H I ( ) of IIR fler s gven as:, for passband H I ( ) (7), for sopband Te rpple magnudes of pass-band and sop-band are o be mnmzed, wc are denoed by δ p ( and δ s ( respecvely. Rpple magnudes are defned as: H( mnh (, x p ( max ) and for passband p ( max H( (9) for sopband Aggregang all obecves and sably consrans, e mulcreron consraned opmzaon problem s saed as nmze O ( E( (a) nmze O ( E( nmze O3 ( p ( nmze O4 ( s ( Subec o: e sably consrans ( u,,..., ) (b) q u ( u,,..., ) (c) q u ( v,,..., ) (d) v v v ( v,,..., ) (e) v v ( v,,..., ) (f) (8) Desgn of IIR fler s a mul-obecve opmzaon problem (OOP) as several obecves are opmzed smulaneously as sown n Eq. (a). Te mulobecve consraned opmzaon problem for e desgn of dgal IIR fler s convered no a scalar consraned opmzaon problem by usng wegng meod as defned below: Te obecve funcon o be opmzed s defned as: 4 nmze f ( w O ( () a a a Subec o: Te sasfacon of sably consrans gven by equaon (b) o equaon (f). were w a s nonnegave real number called weg. In s paper wegs are aken same as gven by [9]. All e poles of desgned dgal IIR fler sould le nsde e un crcle for e fler o be sable. Terefore, e sably consrans by usng e Jury meod [9] ave been mposed on e denomnaor coeffcens as gven by equaon (b) o equaon (f). To sasfy e sably condons coeffcens ave been updaed w random varaon as gven n equaon (a) o equaon (c). Te varaon s gven as small so a e caracersc of populaon n TLBO sould no be canged. ;( qu ) q r q u ( ) u or ( qu ) qu ; Oerwse (a) v v v v ( r) v ( r) v ;( v ) ; Oerwse ;( v v ) or ( v v ) ; Oerwse (b) (c) 3. TLBO ALGORITH FOR THE DESIG OF IIR FILTER In s paper a recenly developed eursc opmzaon algorm namely Teacng-learnng based opmzaon (TLBO) proposed by Rao e al. [6, 7] s appled for e desgn of dgal IIR flers. Te useful arbue of TLBO s a requre few conrol parameers wc reman fxed rougou e opmzaon process and need mnmum unng. In e noble professon of Teacng, a eacer wo s e mos knowledgeable person n e class always movaes e sudens o acqure supreme knowledge by wc ey can mprove er academc performances. A eacer works ard o ncrease e average resul of e sudens / learners from nal level o s own level. However, n spe of e bes effor of e eacer, e sudens no only gan knowledge on e bass of e qualy of eacng delvered bu also based on er own qualy. Tere are oer means n wc learners can also gan knowledge by dscussng and sarng deas among oer learners n form of uorals and semnars. In e mplemenaon of TLBO for e desgn of dgal IIR fler e populaon s analogous o L number of learners n a class and eac learner s assgned S subecs (fler coeffcens). Te learner s represened as 39

3 X x, x,... xs and f(x ) represen e fness funcon for learner. Te funconng of TLBO based upon wo man pases namely Teacer pase and Learner pase. Teacer Pase A learner avng e mnmum value of fness funcon value calculaed usng Eq. (9) s desgnaed as eacer x for curren eraon. A eacer res o ncrease / mprove e mean score of all e learners n eac of e subec alloed owards s own mean score. So, e mean fness of e class s ncreased by e eacer accordng o s / er own capably. Te mean (τ ) for S subecs alloed o e sudens s evaluaed and a randomly weged dfferenal vecor (Dff ) from curren mean and varous desred mean vecors [] s calculaed as sown below: L x, (,,..., S ) (3) L Dff ( x ( TF ) R() (,,..., S ) (4) were τ s mean of subec for all learners of a class; x s e score of e eacer n subec; TF s e eacng facor; R s a unform generaed random number beween (,). Te eacng facor ( TF ) s one of e val aspec a faclaes e convergence of TLBO. Te value of T f decdes abou e volume of effec a eacer as on e oupu of a learner. In s paper e value of TF s eurscally seleced as or as sown below: TF ROUD(. R()) (5) Te weged dfferenal vecor (Dff ) generaed usng Eq. (4) s added o curren score of learners n dfferen subecs o generae new learners: xnew x Dff (,,..., S) (6) Te newly generaed learner w a beer fness value replaces e exsng learner n e class. Learner Pase Te concep of uorals and semnars of class room eacng sysem s followed n e learner pase as e knowledge acqured by e learners / sudens n eacer pase s furer dssemnaed among learners emselves roug dscussons, sarng of noes and presenaons. Two arge learners namely and m are seleced randomly suc a m. Te (z z )(z H BP (z) (z z )(z (z z ) (z. 6339z ) Inernaonal Journal of Compuer Applcaons ( ) Volume 4 o.4, Ocober 4 resulan new learners afer sarng / excange of know-ow are generaed as follows: x R() ( x xm ); f ( X ) f ( X ) m xnew (7) x R() ( xm x ); Oerwse were (,,..., S) Afer one successful compleon of Teacer and Learner pase, algorm s made o updae eacer value before e sar of nex eraon. In s TLBO algorm maxmum number of eraons s cosen as e soppng creron. If e soppng creron s no sasfed, e above procedure s repeaed w ncremened value 4. DESIG EXAPLES AD COPARISOS For e purpose of comparson desgn condons aken for desgnng BP and BS flers are same as [7] and are gven n Table. Te frequency range from o s dvded n o frequency samplng pons. In e proposed TLBO approac e combnaon of four crera, L -norm approxmaon error, L -norm approxmaon error, rpple magnudes of pass-band and rpple magnude of sop-band are consdered for desgnng IIR fler. Tese four crera are conrary o eac oer n mos suaons. Te fler desgner needs o adus e wegs of crera o desgn e fler dependng on e fler specfcaons. For e purpose of comparson e wegs w, w, w 3 and w 4 are se same as n [9] for BP and BS flers respecvely. Te performance of fler desgned w TLBO algorm are presened and compared w e resuls obaned by [7], [8] and [9] n Table and Table 3 for BP and BS flers respecvely. Te obaned magnude response of BP and BS flers desgned w TLBO and [7], [8] and [9] are presened n Fgure and Fgure for e purpose of comparson. Te bes opmzed numeraor coeffcens and denomnaor coeffcens obaned by e TLBO approac for BP and BS flers are gven by Eq. (8) and Eq. (9) respecvely. Te obaned resuls reveal a e purposed approac gves e beer performance n erms of magnude response and rpples n pass band and sop band z ) z ) (z z 583. )(z z 65. ) H BS (z) (9) (z. 8579z. 5478)(z z ) (8) Table. Prescrbed desgn condons for BP and BS flers Fler ype Pass band Sop band Order BP and.75 6 BS. 5 and

4 Inernaonal Journal of Compuer Applcaons ( ) Volume 4 o.4, Ocober 4 Table. Desgn resuls for BP fler eod L -norm error L -norm error TLBO Approac TIA Approac [9].69.9 HTGA Approac [8] TIA Approac [7] Pass-band performance (Rpple magnude).9886 H(e ω ).5 (.7).986 H(e ω ). (.94).976 H(e ω ). (.34) 956 H(e ω ). (.44) Table 3. Desgn resuls for BS fler Sop-band performance (Rpple magnude) H(e ω ).55 (.55) H(e ω ).658 (.658) H(e ω ).7 (.7) H(e ω ).77 (.77) eod L -norm error L -norm error TLBO Approac TIA Approac [9] HTGA Approac [8] TIA Approac [7] Pass-band performance (Rpple magnude).969 H(e ω ).8 (.47).956 H(e ω ). (.44).9563 H(e ω ). (.437) 9 H(e ω ). (.8) Sop-band performance (Rpple magnude) H(e ω ) () H(e ω ).7 (.7) H(e ω ).3 (.3) H(e ω ).76 (.76). TLBO eod. TIA Approac HTGA Approac. eod of Tang e al Fg : responses of BP fler usng TLBO approac and e meod gven n [9], [8] and [7]. TLBO Approac. TIA Approac HTGA Approac. eod of Tang e.al Fg : responses of BS fler usng TLBO approac and e meod gven n [9], [8] and [7] 4

5 Inernaonal Journal of Compuer Applcaons ( ) Volume 4 o.4, Ocober 4 5. COCLUSIO A eursc opmzaon algorm namely TLBO s successfully appled o desgn BP and BS dgal IIR fler. Te desgned opmal flers obaned by employng TLBO mee e sably creron and gves beer performances n erms of L p -approxmaon error for magnude response and rpples n pass band and sop band n comparson o GA based meods. Te man caracerscs of e TLBO algorm over oer GA meods are s smplfed numercal srucure and s ndependence on a number of parameers o defne e algorm s performance. TLBO s a powerful searc and applcable opmzaon meod for e problem of dgal fler desgn problems. 6. REFERECES [] J. G. Proaks and D. G. anolaks, Dgal Sgnal Processng: Prncples, Algorms, and Applcaons. ew Del: Pearson Educaon, Inc., 7. [] A. V. Oppenem, e al., Dscree-Tme Sgnal Processng. J, Englewood Clffs: Prence Hall, 999. [3] A. Anonou, Dgal Sgnal Processng: Sgnals, Sysems and Flers: cgraw Hll, 5. [4] W.-S. Lu and A. Anonou, "Desgn of dgal flers and fler banks by opmzaon: a sae of e ar revew," presened a e Proceedng of European Sgnal Processng Conference, Fnland,. [5] S. P. Harrs and E. C. Ifeacor, "Auomac desgn of frequency samplng flers by ybrd genec algorm ecnques," IEEE Transacons on Sgnal Processng, vol. 46, pp , 998. [6] J. H. L and F. L. Yn, "Genec opmzaon algorm for desgnng IIR dgal flers," Journal of Cna Insue of Communcaons Cna, vol. 7, pp. 7, 996. [7] K. S. Tang, e al., "Desgn and opmzaon of IIR fler srucure usng erarccal genec algorms," IEEE Transacons on Indusral Elecroncs, vol. 45, pp , 998. [8] J.-T. Tsa, e al., "Opmal desgn of dgal IIR flers by usng ybrd aguc genec algorm," IEEE Transacons on Indusral Elecroncs, vol. 53, pp , 6. [9] J.-T. Tsa and J.-H. Cou, "Opmal desgn of dgal IIR flers by usng an mproved mmune algorm," IEEE Transacons on Sgnal Processng, vol. 54 pp , 6. [] A. Kalnl and D. Karaboga, " A new meod for adapve IIR fler desgn based on Tabu searc algorm," Inernaonal Journal of Elecroncs and Communcaons (AEÜ), vol. 59, pp. 7, 5. [] S. Cen and B. L. Luk, "Dgal IIR fler desgn usng parcle swarm opmsaon," Inernaonal Journal of odellng, Idenfcaon and Conrol, vol. 9, pp ,. [] P. Upadyay, e al., "Crazness based parcle swarm opmzaon algorm for IIR sysem denfcaon problem," AEU - Inernaonal Journal of Elecroncs and Communcaons, vol. 68, pp , 4. [3] C. Da, e al., "Seeker opmzaon algorm for dgal IIR fler desgn," IEEE Transacons on Indusral Elecroncs, vol. 57, pp. 7-78,. [4] B. L, e al., "Fxed-pon dgal IIR fler desgn usng wo-sage ensemble evoluonary algorm," Appled Sof Compung vol. 3, pp , 3. [5] S. K. Saa, e al., "Gravaon searc algorm: Applcaon o e opmal IIR fler desgn," Journal of Kng Saud Unversy-Engneerng Scences vol. 6, pp. 69-8, 4. [6] R. V. Rao, e al., "Teacng-learnng-based opmzaon: a novel meod for consraned mecancal desgn opmzaon problems," Compuer-Aded Desgn, vol. 43, pp ,. [7] R. V. Rao, e al., "Teacng-learnng-based opmzaon: a novel opmzaon meod for connuous non-lnear large scale problems," Informaon Scences, vol. 83, pp. 5,. [8] G. Vanuysel, e al., "Effcen ybrd opmzaon of fxed-pon cascaded IIR fler coeffcens," n IEEE Inernaonal Conference on Insrumenaon and easuremen Tecnology, Ancorage, AK,, pp [9] I. Jury, Teory and Applcaon of e Z-Transform eod ew York: Wley, 964. []. Sng, e al., "Opmal coordnaon of dreconal over-curren relays usng Teacng Learnng-Based Opmzaon (TLBO) algorm," Inernaonal Journal of Elecrcal Power and Energy Sysems vol. 5, pp. 33-4, 3. IJCA T : 4