Design of optimal IIR digital filter using Teaching-Learning based optimization technique

Transcription

1 Damanpree Singh, J. S. Dhillon Design of opimal IIR digial filer using Teahing-Learning based opimizaion ehnique DAMANPREET SINGH, J.S. DHILLON Deparmen of Compuer Siene & Engineering Deparmen of Elerial & Insrumenaion Engineering San Longowal Insiue of Engineering and Tehnology, Longowal, INDIA, Absra: - In his paper an Enhaned Teahing-Learning Based Opimizaion (ETLBO) algorihm is employed o design sable digial infinie impulse response (IIR) filer using L p -norm rierion. The original TLBO algorihm has been remodeled by merging he onep of opposiion-based learning and migraion for seleion of good andidaes and o mainain he diversiy, respeively. The muliobeive IIR digial filer design problem onsiders imizing he L p -norm approximaion and imizing he ripple magniude simulaneously while saisfying sabiliy onsrains on he oeffiiens of he filer. Weighed sum mehod and p-norm mehod are applied o solve he mulirierion opimizaion problem. Bes weigh paern is searhed using evoluionary searh mehod ha imizes he performane rieria simulaneously. The validiy of he mehod is demonsraed for he design of low pass (LP), high pass (HP), band pass (BP) and band sop (BS) IIR filers. The omparison of simulaion resuls wih oher exising mehods show ha he proposed ETLBO algorihm is superior in erms smaller L -norm, L -norm and smaller pass band and sop band ripples. Key-Words: - IIR filer, TLBO, magniude response, sabiliy, L p -approximaion. Inroduion Filers are mainly used for exoring informaive par of he signal and o remove undesirable omponen of he signal. Exoring of signal is required when a noise or some disurbane onaaes a signal []. Digial filers have araed he aenion of researhers due o large number of appliaion lie daa ommuniaion, video proessing, radar and opial ommuniaions, speeh proessing and many more. In erms impulse response, digial filers are widely aegorized as infinie impulse response (IIR) and finie impulse response (FIR) filers []. The seleion of digial filer for an appliaion is a edious as involving finding of opimum sruure in order o saisfy erain parameers of frequeny response, namely ripples in pass band, ransiion band widh and aenuaion in sop band. Digial IIR filers are preferred over FIR digial filers beause of higher ompuaional effiieny and aurae frequeny seleiviy. The wo problems wih he design of IIR digial filer are [3-4]: (i) endeny of he filer o beome unsable (ii) filer surfae is mulimodal in naure due o whih onvenional design opimizaion algorihms may su a loal ima. The sabiliy problem is handled by imposing sabiliy onsrains on he filer oeffiiens. Numerous evoluionary and meaheurisi opimizaion algorihms have been suessfully applied o handle he non-differeniable and mulimodal surfae of digial IIR filer. Some evoluionary opimizaion algorihms reenly applied for digial IIR filer are: genei algorihms [5-0], immune algorihm [], parile swarm opimizaion [-4], seeer-opimizaion-algorihm [5], predaor-prey opimizaion [6], heurisi searh mehod (HSM) [7], wo-sage ensemble evoluionary algorihm [8], graviaion searh algorihm [9] and many more. The main drawbas of he above algorihms are slow onvergene owards opimal soluion and he requiremen of algorihm speifi onrolling parameers in addiion o regular onrolling parameers lie size of populaion, number of ieraions, group size e. In order o overome he above drawbas, eahing-learning based opimizaion (TLBO) algorihm developed by Rao e al. [0-] has been applied o design he digial IIR filers. TLBO is a heurisi searh mehod inspired by he learning behavior of he sudens in a lass. There is no need o une any algorihm E-ISSN: Volume, 05

2 Damanpree Singh, J. S. Dhillon speifi onrolling parameers in order o implemen TLBO, hus maing i more robus. The inen of his paper is o inrodue enhanemen in original TLBO o improve is exploraion and exploiaion apabiliies, by iniializing wih good andidaes and mainaining he diversiy. The onep of opposiion-based learning is employed for iniializaion and evoluion of populaion. Furher migraion has been applied o mainain he diversiy and searh spae exploraion, and avoid premaure onvergene. The unique ombinaion of broad exploraion and furher exploiaion yields a powerful opion o solve mulimodal opimizaion problems ha designs IIR filers. The mulirierion opimizaion problem of digial IIR filer design is onvered ino salar onsrained opimizaion problem using weighing p-norm mehod. The weighing ehnique is used o generae non-inferior soluions, whih allow explii rade-off beween onfliing obeive levels. Evoluionary searh ehnique is employed o searh for he weighage paern. The paper analyzes he performane of Enhaned TLBO (ETLBO) algorihm for designing digial IIR filers using L p - approximaion rierion and he obained resuls are ompared wih hierarhial genei algorihm (HGA) [8], hybrid aguhi genei algorihm (HTGA) [0], aguhi immune algorihm (TIA) [] and heurisi searh mehod (HSM) [7] for he low-pass (LP), high-pass (HP), band-pass (BP), and band-sop (BS) filers for validaion. The paper is sruured as follows. The digial IIR filer design problem is formulaed in Seion. Seion 3 elaboraes he implemenaion of ETLBO algorihm for he digial IIR filer design. The performane of ETLBO is evaluaed and ompared wih he design obained by various researhers in Seion 4. Finally, Seion 5 onludes he ouomes of he wor. Problem Formulaion To design a digial IIR filer a se of opimum filer oeffiiens are searhed in order o mee various obeives summarized below: Minimize he absolue L -norm of magniude response. Minimize he squared L -norm of magniude response. Minimize he magniude pass band ripples. Minimize he magniude sop band ripples. The IIR digial filer is denoed by he following ransfer funion: M a z 0 H ( z) () M b z 0 M w al e H(, x) A w l b le () N w w m e me w w m dme dme where Veor X=[a, b,..., a M, b M,,, d, d,..., N, N, d N, d N, A] T of dimension S, wih S=M+4N+ denoes he filer oeffiiens. A represens he gain of he filer. The main goal of he design algorihm of digial IIR filer is o searh for filer oeffiiens a and b suh ha he magniude response in erms of L p -norm [0-] and ripples in pass band and sop band are imized. The magniude response is speified a K evenly disribued disree poins of frequeny in pass band and sop band. Absolue magniude response is represened by E (x) and squared magniude response is denoed by E (x): E E K ( x) HI ( i) H( i, x) i0 (3) H ( ) H(, x K ( x) I i i ) i0 (4) Ideal magniude response H I (ω i ) of IIR digial filer is saed as:, fori passband H I ( i ) (5) 0, fori sopband The pass band and sop band ripples o be imized are denoed by δ p (x) and δ s (x) respeively: H (, x) H (, x) p ( x) i i for i passband and i i (6) s ( x) H ( i, x) i (7) for i sopband Aumulaing all above menioned rieria's, he mulirierion onsrained opimizaion problem is formulaed as below: E-ISSN: Volume, 05

3 Damanpree Singh, J. S. Dhillon Minimize F ( x) E ( x) Minimize F ( x) E Minimize F ( x) ( x) 3 p ( x) (8a) Minimize F4 ( x) s ( x) Sube o: he sabiliy onsrains 0 ( l,,..., M) (8b) b l 0 ( l,,..., M) (8) b l 0 ( m,,..., N) (8d) d m m m N dm dm 0 ( m,,..., N d d 0 ( m,,..., ) (8e) ) (8f) IIR digial filer design as is a muli-obeive opimizaion problem (MOOP) beause several obeives are opimized simulaneously as shown in Eq. (8a). The muliobeive onsrained opimizaion as for he design of IIR digial filer is onvered ino a salar onsrained opimizaion problem by using weighing mehod as defined below: L Minimize f ( x) F ( x) (9) Sube o: The saisfaion of sabiliy onsrains saed in Eqs. (8b) o (8f). where F (x) is he h obeive funion and α is nonnegaive real number alled weigh assigned o h obeive funion. This approah yields meaningful resuls when solved many imes for differen values of α (=,,...,L). The p-norm weighing paerns are eiher presumed on he basis of deision maer s inuiion or simulaed wih suiable sep size variaion. Weighed sum ehnique auses problem when he lower boundary of funion spae is no onvex [], beause no every non-inferior soluion will have a supporing hyperplane. In his paper weigh Paern searh based on evoluionary searh mehod is applied o searh he normalized weighs, α (=,,...,L) assigned o pariipaing obeives. The digial IIR filer design requires he saisfaion of sabiliy onsrains. The sabiliy onsrains o be imposed on he oeffiiens of IIR digial filer as saed in Eqs. (8b) o (8f) are obained by employing Jury mehod [3]. The value of filer oeffiiens are updaed wih a random variaion as given below in order o saisfy he sabiliy onsrains The are is aen ha he amoun of variaion is small enough so ha i should no hange he haraerisi of he populaion. b l ( r) ; ( b l ) 0 b l or ( b l ) 0 (0a) b Oherwise l ; d m( r) ; ( dm) 0 d m (0b) dm ; Oherwise d m ( r) ; ( dm dm) 0 d m or ( dm dm) 0 (0) d Oherwise m ; where r is a uniform random number whose value varies beween [0, ]. 3 IIR Filer Design Using ETLBO ETLBO based on he noble onep of eahinglearning [0-] is a reenly developed populaion based opimizaion ehnique. The unique feaure of ETLBO is ha i requires o une few onrol parameers. The growh of every soiey o a grea exen is influened and dependen upon he qualiy of eahers in he soiey. ETLBO effiienly explores he nowledge base of a eaher o inrease he now-how of learners / sudens. A eaher pus his bes effor in order o inrease he mean sore of all learners in eah alloed sube owards is own mean sore. So, he mean finess of he lass is inreased by he eaher aording o his / her own apabiliy. The learners also furher improve heir nowledge base by ineraing and sharing informaion wih eah oher. In he implemenaion of ETLBO for he design of digial IIR filer NL he number of learners in a lass represen he populaion and eah learner has been assigned S subes (filer oeffiiens). The i h learner is represened as X x x,... x and i i, i f(x i ) represen he finess funion for i h learner. X X.... XS f ( X) X X.... X S f ( X ) lass.. X i... f ( X i ) X NL X NL.... X NLS f ( X NL) Iniializaion of Class The mars for all he subes of learners in a lass are iniialized wih he help of random searh. Global searh is applied o explore he saring poin and hen he saring poin is perurbed in loal searh spae o reord he bes saring poin. The searh proess is sared by iniializing he variable i x using Eq. (): is E-ISSN: Volume, 05

4 Damanpree Singh, J. S. Dhillon x i x R()( x x ) () ( i,,..., NL;,,..., S ) where R is a uniform random generaed number beween (0,). S is number of subes alloed o eah learner. NL number of learners in a lass. is he ieraion ouner. and are he imum and imum values x x of h deision variable (filer oeffiien) of veor X. Opposiion-based Learning The heory of opposiion-based learning [4] is applied o enhane he onvergene rae of ETLBO. The noion behind opposiion-based learning is o sele beer urren andidae soluion by omparing he urren populaion and is opposie populaion. The opposiion-based learning is applied using Eq. () o reord he alernaive saring poin and saring poin x i is furher explored using: xi NL, x R()( x x ) () ( i,,..., NL;,,..., S ) Ou of NL learners, bes NL learners onsiue a lass o iniiae he proess. For he global searh, bes learner is seleed ou of lass of learners. Furher he opposiion-based learning is also employed for generaing new learners afer he ompleion of learner phase using Eq. (3): U L xi NL, x x xi (3) ( i,,..., NL;,,..., S ) where x x U L xi ; ( i,,..., NL) (,,..., S) x ; ( i,,..., NL) (,,..., S) i Teaher Phase The bes learner is seleed among all he learners in a lass based upon he finess funion value alulaed using Eq. (9) and a as eaher x for urren ieraion. The mean ( ) for S subes alloed o he sudens is evaluaed and a randomly weighed differenial veor (Diff ) from urren mean and various desired mean veors [5] is alulaed as shown below: NL xi, (,,..., S ) (4) NL Diff where i R( ) ( x ( T ) (,,..., S ) (5) f is mean of h sube for all learners of a lass; x is he sore of he eaher in h sube; T f is he eahing faor; R is a uniform generaed random number beween {0,}. The eahing faor ( T f ) is one of he vial aspe ha failiaes he onvergene of ETLBO. In his paper he value of T f is heurisially seleed as or as shown below: ROUND(.0 R()) (6) T f The weighed differenial veor (Diff ) generaed using Eq. (5) is added o urren sore of learners in differen subes o generae new learners: xnew x Diff (,,..., S ) (7) i i The newly generaed learner wih a beer finess value replaes he exising learner in he lass. Learner Phase The nowledge aquired by he learners in eaher phase is furher disseaed among learners hemselves hrough sharing of noes, disussions and presenaions. The seond phase of ETLBO emulaes his sharing of nowledge by learners among hemselves. Two arge learners namely i and m are seleed randomly suh ha i m. The resulan new learners afer sharing / exhange of now-how are generaed as follows: x i R() ( xi xm ); f ( X i) f ( X m) xnewi (8) xi R() ( xm xi ); Oherwise where (,,..., S ) Migraion The derease in he abiliy of exploraion of searh spae by learners may lead o premaure onvergene. In order o inrease he diversiy of he learners random individuals are inrodued ino eah generaion from he global searh spae. In order o inrease he exploraion of he searh spae, i is randomly seleed 0.3NL learners o sar migraion operaion. The h sube sore of i h learner is randomly regeneraed as: ( G x ) G R() ( x G ) ; x i ( x x ) (9) G R() ( x G ) ; Oherwise where =,,...,S, G is he global bes mars, R and β are uniform random number. A he end of eah ieraion if he funion value obained by he bes learner is beer han he funion value of global bes han value of eaher is updaed for nex ieraion. E-ISSN: Volume, 05

5 Damanpree Singh, J. S. Dhillon Evoluionary Weigh Paern Searh The opimal weigh paern is obained by perforg evoluionary searh. One weigh assigned o an obeive is onsidered dependen weigh required o mee he equaliy onsrain required o ensure normalized weigh paern and searh is performed on res of weighs. So, in his mehod, L- feasible soluions are generaed for (L- ) weighs assigned o pariipaing obeives exep weigh assigned o dependen obeive. A (L-) dimensional hyperube of side Δ i is formed C around he poin. wi represens weigh paern ha is assigned o obeives from he urren poin in he hyperspae. The beer feasible soluion is obained from obeive funion of he filer design performane index. Anoher hyperube is formed around he beer poin, o oninue he ieraive proess. All he orners of he hyperube represened in binary (L-) bis equivalen ode, generaed around he urren se of assigned weigh paern of unis, are explored for he desired soluion, simulaneously. Table shows he weigh paern for 4-obeives where 3 bis ode is onsidered o represen he orners of he 3- dimensional hyperube (Figure ) beause one weigh is aen as dependen/sla weigh. Serial numbers of hyperube orners in deimal are onvered ino heir binary equivalen ode. The deviaion from he urren enre poin is obained by replaing 0 s wih -Δ and s wih +Δ in ode assoiaed wih hyperube orners. As he number obeives are inreased, he number of hyperube orners inreases exponenially. The proess of exploring he beer soluion from all orners of he hyperube beomes ime onsug, whih needs some effiien searh ehnique ha should explore all he orners of he hyperube wih imum number of funion evaluaions and omparisons. The weighs are generaed as given below: L wi wi i ( i,,..., L; i ;,,..., ) (0) where Δ i is he disane of he orners of he hyperube from he poin around whih he hyperube is generaed. w w i where wi {0,} The weigh of h obeive is alulaed as: w L i w i ( i ) () () L w { w (,,..., L)} (,..., ) (3) The weighs are obained as w L (,,..., L;,..., ; w 0) (4) w The normalized weighs are generaed as desribed above and he bes funion value is designaed as global bes. The above proedure is repeaed wih inremened value unil he value of reahes he imum value of ieraions speified. 4 Resuls and Comparisons In his seion LP and HP IIR digial filer design examples of HGA [8], HTGA [0] TIA [] and HSM [7] are onsidered o invesigae he performane of filer designed wih ETLBO algorihm. The design speifiaion in erms of pass band and sop band u-off frequenies are onsidered as given in Table. The inen is o design he IIR digial filer by imizing he obeive funion as given in Eq. (9) while meeing he sabiliy onsrains given by Eqs. (8b) o (8f). For he design of IIR digial filer 00 evenly disribued poins are hosen in he frequeny span [0,]. ETLBO onsiders L -norm, L -norm, pass band ripples and sop band ripples for designing IIR digial filer. In mos of he ases he above menioned rieria's are onfliing in naure. Depending upon he speifiaion of he filer he weighage o be given o eah rieria has o be deided by he designer. Weighs are adused using evoluionary searh mehod. In he purposed heurisi approah larger value of weighs w 3 and w 4 are hosen o obain small ripple magniude of boh pass-band and sop-band. The weighs w, w, w 3 and w 4 are se o be,, 6.6, and.4, respeively, for he LP, HP, BP and BS filer. The resuls obained by employing ETLBO are given and ompared wih HGA [8], HTGA [0], TIA [] and HSM [7] in Tables 3-6. The magniude response diagrams of LP, HP BP and BS digial IIR filers designed wih he proposed ETLBO are presened in Figure. The opimized value of numeraor and denoaor oeffiiens of LP, HP, BP and BS filers, obained by employing ETLBO are given by Eq. (5), Eq. (6), Eq. (7) and Eq. (8) respeively. E-ISSN: Volume, 05

6 Damanpree Singh, J. S. Dhillon Hyper ube Corners Table : Generaion of weigh paern a hyperube orners for 4 - obeives Possible ombinaions of 3- Disane of bis hyperube orners Paern of weigh generaion veor a he from enre poin hyperube orners C w C C 0, w, w w w w w w w w w w w w w w w w w w w w w w w w w 3 (0) 7 () (00) ( x,x,x 6 (0) 3 ) (00) 5 (0) 0 (000) 4 (00) Figure : Three dimensional hyperube represening filer oeffiiens ( z )( z z ) H LP ( z) (5) ( z )( z.40477z ) ( z )( z z.00036) H HP ( z) (6) ( z )( z z ) (z H BP (z) (z (z (z z ) (z z ) (z z ) z ) z ) z ) (z z ) (z z ) H BS (z) (z z ) (z z ) (7) (8) E-ISSN: Volume, 05

7 Damanpree Singh, J. S. Dhillon Table : Presribed design ondiions on LP, HP, BP and BS filers Filer ype Pass-band Sop-band Maximum Value of H(, x) Low-Pass High-Pass Band-Pass Band-Sop Table 3: Design resuls for LP filer employing imizaion of E X ) E ( X ) ( X ) ( ) Mehod Order L -norm L - norm ETLBO HSM [7] TIA [] HTGA [0] HGA. [8] Pass-band performane (Ripple magniude) 0.97 H(e ω ).006 (0.0943) H(e ω ).0 (0.087) 0.90 H(e ω ).000 (0.0988) H(e ω ).000 (0.0996) H(e ω ).009 (0.39) ( p s X Sop-band performane (Ripple magniude) H(e ω ) 0.7 (0.7) H(e ω ) 0.50 (0.38) H(e ω ) 0.43 (0.43) H(e ω ) 0.47 (0.47) H(e ω ) 0.80 (0.80) Table 4: Design resuls for HP filer employing imizaion of E X ) E ( X ) ( X ) ( ) Mehod Order L -norm L -norm ETLBO HSM [7] TIA [] HTGA [0] HGA. [8] Mehod ( p s X Pass-band performane (Ripple magniude) H(e ω ).004 (0.0497) H(e ω ).008 (0.0504) H(e ω ).000 (0.0533) H(e ω ).000 (0.0540) 0.94 H(e ω ).003 (0.0779) Sop-band performane (Ripple magniude) H(e ω ) 0.57 (0.57) H(e ω ) (0.477) H(e ω ) (0.457) H(e ω ) (0.457) H(e ω ) 0.89 (0.89) Table 5: Design resuls for BP filer employing imizaion of E X ) E ( X ) ( X ) ( ) Order L -norm L -norm ETLBO HSM [7] TIA [] HTGA [0] HGA. [8] Pass-band performane (Ripple magniude) H(e ω ).005 (0.07) H(e ω ).004 (0.047) H(e ω ).000 (0.094) H(e ω ).000 (0.034) H(e ω ).000 (0.044) ( p s X Sop-band performane (Ripple magniude) H(e ω ) (0.055) H(e ω ) (0.067) H(e ω ) (0.0658) H(e ω ) 0.07 (0.07) H(e ω ) 0.77 (0.77) E-ISSN: Volume, 05

8 Damanpree Singh, J. S. Dhillon Table 6: Design resuls for BS filer employing imizaion of E X ) E ( X ) ( X ) ( ) ( p s X Mehod Order L -norm L -norm ETLBO HSM [7] TIA [] HTGA [0] HGA. [8] Pass-band performane (Ripple magniude) H(e ω ).006 (0.049) H(e ω ).008 (0.0434) H(e ω ).000 (0.0440) H(e ω ).000 (0.0437) H(e ω ).000 (0.080) Sop-band performane (Ripple magniude) H(e ω ) 0.07 (0.07) H(e ω ) (0.060) H(e ω ) 0.64 (0.64) H(e ω ) 0.03 (0.03) H(e ω ) 0.76 (0.76) Figure : Magniude response of LP, HP, BP and BS IIR filer using ETLBO approah employing E X ) E ( X ) ( X ) ( ) rierion. ( p s X Figure 3: Pole-Zero of LP, HP, BP and BS IIR filer using ETLBO approah employing E X ) E ( X ) ( X ) ( ) rierion ( p s X E-ISSN: Volume, 05

9 Damanpree Singh, J. S. Dhillon The sruinizing of he resuls presened in Tables 3-6 reveal ha ETLBO obains smaller L -norm approximaion s, he smaller L norm approximaion s, and beer magniude performanes in boh pass-band and sop-band han HGA [8], HTGA [0], TIA [] and HSM [7]. The designed LP, HP, BP and BS IIR digial filer wih ETLBO are esed for sabiliy by drawing pole-zero diagrams shown in Figure 3. I an be observed from Figure 3 ha he designed filers follow he sabiliy onsrains imposed in he design proedure as all he poles lie inside he uni irle. The sabiliy of filer is no influened by he zeros lying ouside he uni irle. 5 Conlusion In his paper a heurisi algorihm ETLBO is suessfully applied o design digial IIR filer and gives subsanial improvemen in erms of resuls and onvergene. The performane of he original TLBO is enhaned wih he inroduion of he onep of opposiion-based learning and migraion for saring wih good populaion of learners and mainain he diversiy of he learners, respeively. ETLBO is very feasible o design he digial IIR filers, pariularly when he ompliaed onsrains, he design requiremens, and he muliple rieria are all involved. The designed opimal filers mee he sabiliy rierion, gives beer performane in erms of L p -approximaion for magniude response and ripples in pass band and sop band in omparison o exising mehods. The advanage of applied ETLBO algorihm is ha i do no requires o une any algorihm-speifi parameers. Referenes: [] J. G. Proais and D. G. Manolais, Digial Signal Proessing: Priniples, Algorihms, and Appliaions. New Delhi: Pearson Eduaion, In., 007. [] A. V. Oppenheim, e al., Disree-Time Signal Proessing. NJ, Englewood Cliffs: Prenie Hall, 999. [3] J. H. Li and F. L. Yin, Genei opimizaion algorihm for designing IIR digial filers, Journal of China Insiue of Communiaions China, Vol. 7, 996, pp. 7. [4] W.-S. Lu and A. Anoniou, "Design of digial filers and filer bans by opimizaion: a sae of he ar review," presened a he Proeeding of European Signal Proessing Conferene, Finland, 000. [5] R. Kaur, e al., Design of Opimal L Sable IIR Digial filer Using Real Coded Genei Algorihm, Inernaional Journal of Compuer Siene, Vol. 39, 0, pp [6] R. Kaur, e al., Digial IIR Filer Design using Real Coded Genei Algorihm, Inernaional Journal of Informaion Tehnology and Compuer Siene, Vol. 5, 03, pp [7] N. E. Masorais, e al., Design of wodimensional reursive filers using genei algorihms, IEEE Transaions on Ciruis and Sysems I: Fundamenal Theory and Appliaions, Vol. 50, 003, pp [8] K. S. Tang, e al., Design and opimizaion of IIR filer sruure using hierarhial genei algorihms, IEEE Transaions on Indusrial Eleronis, Vol. 45, 998, pp [9] C.-W. Tsai, e al., Sruure-speified IIR filer and onrol design using real sruured genei algorihm, Applied Sof Compuing, Vol. 9, 00, pp [0] J.-T. Tsai, e al., Opimal design of digial IIR filers by using hybrid aguhi genei algorihm, IEEE Transaions on Indusrial Eleronis, Vol. 53, 006, pp [] J.-T. Tsai and J.-H. Chou, Opimal design of digial IIR filers by using an improved immune algorihm, IEEE Transaions on Signal Proessing, Vol. 54, 006, pp [] S. Chen and B. L. Lu, Digial IIR filer design using parile swarm opimisaion, Inernaional Journal of Modelling, Idenifiaion and Conrol, Vol. 9, 00, pp [3] P. Upadhyay, e al., Craziness based parile swarm opimizaion algorihm for IIR sysem idenifiaion problem, AEU - Inernaional Journal of Eleronis and Communiaions, Vol. 68, 04, pp [4] S. Mandal, e al., Novel Parile Swarm Opimizaion for Low Pass FIR Filer Design, WSEAS Transaions on Signal Proessing, Vol. 8, 0, pp. -0. [5] C. Dai, e al., Seeer opimizaion algorihm for digial IIR filer design, IEEE Transaions on Indusrial Eleronis, Vol. 57, 00, pp [6] B. Singh, e al., Predaor Prey Opimizaion Mehod for he Design of IIR Filer, WSEAS Transaions on Signal Proessing, Vol. 9, 03, pp [7] R. Kaur, e al., Heurisi Searh Mehod for he Design of IIR filer, WSEAS Transaions E-ISSN: Volume, 05

10 Damanpree Singh, J. S. Dhillon on Signal Proessing, Vol. 8, 0, pp [8] B. Li, e al., Fixed-poin digial IIR filer design using wo-sage ensemble evoluionary algorihm, Applied Sof Compuing, Vol. 3, 03, pp [9] S. K. Saha, e al., Graviaion searh algorihm: Appliaion o he opimal IIR filer design, Journal of King Saud Universiy-Engineering Sienes Vol. 6, 04, pp [0] R. V. Rao, e al., Teahing-learning-based opimizaion: a novel mehod for onsrained mehanial design opimizaion problems, Compuer-Aided Design, Vol. 43, 0, pp [] R. V. Rao, e al., Teahing-learning-based opimizaion: a novel opimizaion mehod for oninuous non-linear large sale problems, Informaion Sienes, Vol. 83, 0, pp. 5. [] M. R. Lighner and S. W. Direor, Muliple rierion opimizaion for he design of eleroni iruis, IEEE Transaions on Ciruis and Sysems, Vol. CAS-8, 98, pp [3] I. Jury, Theory and Appliaion of he Z- Transform Mehod. New Yor: Wiley, 964. [4] H. Tizhoosh, "Opposiion-based learning : a new sheme for mahine inelligene," presened a he Proeedings of he Inernaional Conferene on Compuaional Inelligene for Modelling Conrol & Auomaion, Ausria, 005. [5] M. Singh, e al., Opimal oordinaion of direional over-urren relays using Teahing Learning-Based Opimizaion (TLBO) algorihm, Inernaional Journal of Elerial Power and Energy Sysems Vol. 50, 03, pp E-ISSN: Volume, 05