Magnitude Approximation of IIR Digital Filter using Greedy Search Method

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1 Ranjt Kaur, Damanpreet Sngh Magntude Approxmaton of IIR Dgtal Flter usng Greedy Searh Method RANJIT KAUR, DAMANPREET SINGH Department of Eletrons & Communaton, Department of Computer Sene & Engnnerng Punjab Unversty, Patala Sant Longowal Insttute of Engneerng & Tehnology, Longowal INDIA Abstrat: - The paper presents a greedy searh method based on bnary suessve approxmaton-evolutonary searh () strategy to desgn stable nfnte mpulse response (IIR) dgtal flter usng L optmalty rteron. The stablty onstrants are well taen are of durng the desgn proedure. The flter desgned based on L -approxmaton error possesses flat pass-bands and stop-bands to that of the least square desgn. A omparson has been made wth other desgn tehnques, demonstratng that obtans better results for desgnng dgtal IIR flters than the exstng genet algorthm (GA) based methods. Key-Words: - Dgtal nfnte mpulse response (IIR) flters, Greedy algorthms, Stablty, Magntude response. Introduton The optmal desgn problem of dgtal nfnte mpulse response (IIR) flters has attrated muh attenton durng past deades. The IIR flter desgn problem has been taled usng varous optmzaton tehnques suh as p-error, weghted least square and rpple magntudes (toleranes) of both passband and stop-band []-[3]. Due to the suess of L p -norm mnmzaton, the desgns based on L p - normed mnmzaton has been suessfully appled n varous applatons suh as: fnte mpulse response (FIR) flter desgn [5]-[6] and IIR flter desgn [3]-[4]. For p=, ths generalzed L p rteron redues to the onventonal L approxmaton. The desgn of IIR flters s problemat as: ) the error surfae of IIR flters s usually nonlnear and multmodal, onventonal gradent-based desgn methods may easly get stu n the loal mnma of error surfae; ) IIR flters an be unstable and that s why the stablty onstrants must be nluded nto IIR flter desgn problems. Hene, t s neessary to use an effent optmzaton algorthm, robust to the loal mnma problem and possbly globally onvergent. Reently several optmzaton algorthms based on modern heursts optmzaton algorthms [7]- [0] have been proposed for the desgn of dgtal IIR flters suh as partle swarm optmzaton [3], smulated annealng [4], tabu searh [5], ant olony optmzaton [6], artfal mmune algorthm [7], hybrd taguh genet algorthm (HTGA) [8], mmune algorthm () [9], herarhal genet algorthm () [0] and many more. The ntent of ths paper s to apply a greedy searh method based on bnary suessve approxmaton evolutonary searh () to desgn stable IIR dgtal flter. The values of the flter oeffents are optmzed wth approah to aheve L -norm approxmaton error rteron n terms of magntude response. The redblty of.the proposed method has been demonstrated n [] to solve the eonomemsson load dspath problem by searhng the generaton pattern of ommtted unts. The problem formulaton and desgn methodology s detaled below. Problem Formulaton The transfer funton of IIR flter an be represented by asadng frst and seond order setons to avod the oeffent quantzaton problem whh auses nstablty. In asade realzaton oeffent range s lmted. The struture of asadng type dgtal IIR flter s [8]: H ( ω, x) = A where M = + a e + b e = N + + d e e + + d e e () E-ISSN: 4-66X 84 Volume 3, 04

2 Ranjt Kaur, Damanpreet Sngh x = [ a, b,..., am, bm,,, d, d,..., N, N, T d N, dn, A].The Vetor x denotes the flter oeffents of dmenson V wth V = M + 4N + and A s the gan of the flter. Dgtal flter desgn problem nvolves the determnaton of a set of flter oeffents whh meet performane spefatons suh as pass-band wdth and orrespondng gan, wdth of the stop-band and attenuaton, band edge frequenes, and tolerable pea rpple n the pass band and stop-band. The magntude response s spefed at K equally spaed dsrete frequeny ponts n pass-band and stop-band. The L p -norm approxmaton error for the magntude response s defned as [4]. p K / p e( x) = H d ( ω ) H ( ω, x) () = 0 In IIR flter desgn problem fxed grd approah s used [4]. For p=, the magntude response error denotes the L -norm error and s defned as gven below: K e( x) = H ( ω ) H ( ω, x) (3) = 0 d Desred magntude response H d ( ω ) of IIR flter s gven as:, for ω passband Hd ( ω ) = (4) 0, for ω stopband The desgn of ausal reursve flters requres the nluson of stablty onstrants. Therefore, the stablty onstrants gven by (5a) to (5e) whh are obtaned by usng the Jury method [] on the oeffents of the dgtal IIR flter n () are nluded n the optmzaton proess. Mathematally, IIR flter problem s formulated as below: Mnmze F = e ( x) (5) Subjet to: the stablty onstrants: + b 0 ( =,,..., M ) (5a) b 0 ( =,,..., M ) (5b) d 0 ( =,,..., N) (5) + d + d 0 ( =,,..., N) (5d) d + d 0 ( =,,..., N) (5e) The stablty onstrants gven by (5a) to (5e) have been fored to satsfy by updatng the oeffents wth random varaton. 3 Soluton Methodology A dret searh methodology s appled to examne tral solutons, sequentally. The proess of departng from a gven pont to the next mproved pont s alled a move. A move s termed a suess f objetve mproves; otherwse, t s a falure. The dret searh tehnque maes two types of move. Frst move s an exploratory move desgned to aqure nowledge onernng the behavor of the funton. Ths move s performed n the neghborhood of the urrent pont systematally to fnd the best pont around the urrent pont. Seond move s pattern move [3]. 3. Exploratory Move In exploratory move, the urrent pont s perturbed n postve and negatve dretons along eah varable one at a tme and the best pont s reorded. The urrent pont s hanged to the best pont at the end of eah varable perturbaton. If the pont found at the end of all varable perturbatons s dfferent than the orgnal pont, the exploratory move s a suess; otherwse the exploratory move s a falure. In any ase, the best pont s onsdered to be the outome of the exploratory move. 3. Evolutonary Searh Method Evolutonary method s proposed to searh the optmal value of flter oeffents. In ths method, V feasble solutons are generated for V number of flter oeffents. A (V) dmensonal hyperube of sde Δ s formed around the pont. x C represents flter oeffents from the urrent pont n the hyperspae. The better feasble soluton s obtaned from objetve funton of the problem. Another hyperube s formed around the better pont, to ontnue the teratve proess. All the orners of the hyperube represented n bnary (V) bts equvalent ode, generated around the urrent set of flter oeffents, are explored for the desred soluton smultaneously. Table I shows the pattern of flter oeffents for 3-flter oeffents where 3 bts ode s onsdered to represent the orners of the 3- dmensonal hyperube (Fg. ). Seral numbers of hyperube orners n demal are onverted nto ther bnary equvalent ode. The devaton from the urrent entre pont s obtaned by replang 0 s wth -Δ and s wth +Δ n ode assoated wth hyperube orners. As the number of flter oeffents nreases, the number of hyperube orners nreases exponentally. The proess of explorng the better soluton from all orners of the hyperube beomes tme onsumng, whh needs some effent searh tehnque that E-ISSN: 4-66X 85 Volume 3, 04

3 Ranjt Kaur, Damanpreet Sngh should explore all the orners of the hyperube wth mnmum number of funton evaluatons and omparsons. Table Flter Coeffent Vetor at Hyperube Corners Hyper ube Corners Possble ombnatons of 3-bts C C C 0 Dstane of hyperube orners from entre pont C C C x, x, x3 Pattern of flter oeffents at the hyperube orners x x x x x x + x x + x + x x + x x + x + x + x + Fg.. Three dmensonal hyperube representng orners n demal 3.3 Bnary Suessve Approxmaton (BSA) Strategy To redue the omputatonal burden, searh s performed on flter oeffents pattern explotng evolutonary optmzaton and BSA strategy to searh the optmal soluton. BSA strategy to searh the flter oeffents s elaborated n Fg., where the soluton proedure moves towards the optmal soluton by omparng two solutons at a tme represented by two orners of hyperube. Fg.. Bnary suessve approxmaton for 3-bts ode. The searh proess s started by ntalzng vetor deson varable x, gvng objetve F. To perform the BSA strategy by the teratve proess C s ntally seleted as follows: ; for ( = ) C = (6) 0 ;for ( =, 3,, V ) Two orners, wth referene to seleted orner, are generated for the omparson as follows: ; for + C = (7) C j ; for (j =,,..,, ( + ),..., V) E-ISSN: 4-66X 86 Volume 3, 04

4 Ranjt Kaur, Damanpreet Sngh 0 ; for C = C j ;for (j =,,...,(- ), (8) ( + ),...,V) In referene to these two orners, vetors of flter oeffents are generated as depted n Table I. Mathematally, t s represented n generalzed form: xm = x + m ; ( m =, ) (9) ( =,,, V) where + f Cm = m = f Cm = 0 (0) ( m =, ) ( =,,, V ) Intal value of nrement to flter oeffents s deded by max mn = ( x x ) δ () Compute x m usng (9) and then objetve funtons at x and ( x ) x are evaluated as follows: Fm = f m ( m =, ) () Mnmum of the two ponts s seleted to be ompared wth the rest of the orners, generated subsequently by (7) and (8). { F, F } F = mn ; (3) The seleted orner for the generaton of next two orners s = C f F < F C C < f F F ( =,,, V ) (4) Table Comparson of Number of Funton Evaluatons Number of Number of Number of omparsons flter orners of by BSA oeffents hyperube ( V ) method (V) ( V)) , ,9 6 7,3, ,398,046,5,04 84 The proess s repeated tll all the orners of hyperube are explored by BSA method and overall mnmum s seleted to fnd the new enter pont for the next teraton. So, ths proedure ends when last element of C vetor ontans or the last branh of the bnary suessve approxmaton tree s reahed whh ensures that all the orners are explored. In ths method the number of omparsons s redued by a large amount. Ths s elaborated n Table for dfferent number of flter oeffents. 3.4 Pattern Move The pattern move s desgned to utlze the nformaton aqured n the exploratory move, and aomplsh the mnmzaton of the funton by movng n the dreton of the establshed "pattern". A new pont s found by jumpng from the urrent best pont x along a dreton onnetng the prevous best pont x and s exeuted as gven below. + x = x + η x x ( =,,,. (5) ( ) V) 4 Desgn Examples and Comparsons For desgnng dgtal IIR flter 00 equally spaed ponts are set wthn the frequeny doman [0,π] and for the purpose of omparson, the lowest order of the dgtal IIR flter s set exatly the same as that gven n [0] for the LP, HP, BP, and BS flters. Therefore, n ths paper, the order of the dgtal IIR flter s a fxed number not a varable n the optmzaton proess. The objetve of desgnng the dgtal IIR flters s to mnmze the objetve funton gven by (5) wth the stablty onstrants stated by (5a) to (5e) under the presrbed desgn ondtons gven n Table 3. The examples of [9] and [0] are onsdered to test and ompare the performane of the proposed approah. From the evaluated results wth the proposed method presented n Table 4 and depted n Fg. 3, t an be observed that, for the LP, HP, BP, and BS flters, the proposed approah gves the smaller L -norm approxmaton errors and the better magntude performanes n both pass-band and stop-band than the genet algorthm based method gven n [9] and [0]. The pole zero dagrams for LP, HP, BP and BS flters are presented n Fg. 4. It an be observed that the desgned flters follow the stablty onstrants mposed n the desgn proedure as all the poles le nsde the unt rle. The poles magntude and angles n radan are gven by (0.8590, ± 0.675), (0.670, 0) for LP, (0.8597, ±.498), (0.659, 3.46) for HP, (0.879, ±.890), (0.876, ±.9470), (0.7, ±.5683) for BP and (0.786, E-ISSN: 4-66X 87 Volume 3, 04

5 Ranjt Kaur, Damanpreet Sngh ±0.9509), (0.703, ±.76) for BS flters. The frst number n the parentheses s the magntude of pole and the seond number s the angle n radans. The stablty of flter s not nfluened by the zeros lyng outsde the unt rle. The desgned IIR flter models obtaned by the proposed approah are gven below. Table 3 Presrbed Desgn Condtons on LP, HP, BP and BS Flters. Flter type Pass-band Stop-band Maxmum Value of H ( ω, x) Low-Pass Hgh-Pass Band-Pass Band-Stop 0 ω 0. π 0.3π ω π 0.8π ω π 0 ω 0. 7π 0 ω 0. 5π 0.4π ω 0. 6π 0.75 ω π 0 ω 0. 5π 0.75 ω π 0.4π ω 0. 6π ( z )( z z ) H LP ( z) = ( z )( z z ) (6) ( z.74660)( z z ) H HP ( z) = ( z )( z z ) (7) ( z z 0.849)( z z.0377)( z z ) H = BP ( z) (8) ( z z )( z z )( z z ) ( z z )( z z ) H BS ( z) = ( z z )( z z ) (9) LP Flter Approah Approah[9] Approah[0] HP Flter Approah Approah[9] Approah[0] Table 4 Desgn Results for LP, HP, BP and BS Dgtal IIR Flter L -norm error Pass-band performane (Rpple magntude) Stop-band performane (Rpple magntude) H(e ). 07 (0.0973) H ( e ) (0.577) H(e ). 000 (0.086) H ( e ) (0.638) H ( e ). 009 (0.39) H ( e ) (0.80) H ( e ).03 (0.068) H ( e ) 0.50 (0.50) H ( e ). 000 (0.077) H ( e ) (0.44) H ( e ). 003 (0.0779) H ( e ) (0.89) E-ISSN: 4-66X 88 Volume 3, 04

6 Ranjt Kaur, Damanpreet Sngh BP Flter Approah Approah[9] Approah[0] BS Flter Approah Approah[9] Approah[0] H ( e ).005 (0.049) H ( e ) (0.066) H ( e ). 000 (0.039) H ( e ) (0.0679) H ( e ). 000 (0.044) H ( e ) (0.77) H ( e ).007 (0.076) H ( e ) 0.36 (0.36) H ( e ). 000 (0.074) H ( e ) (0.78) H ( e ) ) H ( e ) (0.76) Fg. 3. Magntude response of LP, HP, BP and BS flter respetvely Fg. 4. Pole-zero plots for LP, HP, BP and BS flter respetvely 5 Conluson In ths paper, a novel searh methodology has been appled for the desgn of stable dgtal IIR flters based on L -norm approxmaton error. On the bass of results obtaned for the desgn of dgtal IIR flter, t s onluded that the proposed method s better method as ompared to the exstng GA based methods and t satsfes presrbed ampltude spefatons onsstently. The man advantage of the method s that there s no need to ompute the dervatve of the funtons and soluton s obtaned wth relatve small nown number of omparson and funton evaluatons. In order to overome the lmtaton of the nteratve method, t s proposed to searh the optmal pattern wth the help of BSA. The method s equally applable to solve mult-objetve optmzaton problems and falls n the ategory of nteratve soluton proedure. Referenes: [] W.-S. Lu, S. Pe, and C. Tseng, A weghted least-squares method for the desgn of stable - D and -D IIR dgtal flters, IEEE Trans. Sgnal Proess., vol. 46, no., pp. 0, Jan [] C. Tseng and S. Lee, Mnmax desgn of stable IIR dgtal flter wth presrbed magntude and phase responses, IEEE Trans. Cruts Syst. I, Reg. Papers, vol. 49, no. 4, pp , Apr. 00. [3] C. Tseng, Desgn of stable IIR dgtal flter based on least p-power error rteron, IEEE Trans. Cruts Syst. I, vol. 5, no. 9, pp , Sep [4] A. G Dezy, Synthess of reursve flters usng the mnmum p-error rteron, IEEE Trans. Audo Eletroaoust., vol. 0, no.4, pp , 97. E-ISSN: 4-66X 89 Volume 3, 04

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