Analog Group Delay Equalizers Design Based on Evolutionary Algorithm

Transcription

1 RADIOENGINEERING, VOL. 5, NO., APRIL 6 Aalog Group Delay Equalzers Desg Based o Evolutoary Algorthm Přemysl ŽIŠKA, Mloš LAIPERT Dept. of Crcut Theory, Czech Techcal Uversty, Techcá, 66 7 Prague 6, Czech Republc pzsa@hppo.feld.cvut.cz, lapert@fel.cvut.cz Abstract. Ths paper deals wth a desg method of the aalog all-pass flter desgated for equalzato of the group delay frequecy respose of the aalog flter. Ths method s based o usage of evolutoary algorthm, the Dfferetal Evoluto algorthm partcular. We are able to desg such equalzers to be obtaed equal-rpple group delay frequecy respose the pass-bad of the low-pass flter. The procedure wors automatcally wthout a put estmato. The method s preseted o solvg practcal examples. Keywords Aalog all-pass flters, group delay frequecy respose, Dfferetal Evoluto algorthm.. Itroducto The geetc ad evolutoary algorthms were foud out as a powerful tool desg ad optmzato of electrcal crcuts ad systems. These algorthms are global mmzato algorthms whch smulate a evolutoal process the ature [5]. They store a umber of represetatos of solutos to a problem a so-called populato matrx. The geetc algorthms ca mmze ot oly a stadard fucto, but also hghly olear ad partly odfferetable fuctos wth may local mma. These algorthms are tme-cosumg, but, o the other had, computatoally robust ad lead to optmum results the soluto of complcated desg problems. Excellet results have bee obtaed e.g. a trasfer fucto approxmato problem soluto may cases of aalog or dgtal flters desg. For example, Marte ad Vodraš have dealt wth the approxmato of the aalog flter trasfer fucto wth cocurret requremets for magtude ad group delay frequecy respose the papers [], [3] ad maly the paper [4]. Stor proposed a method for the desg of a dgtal IIR flter wth cocurret requremets for magtude ad group delay frequecy respose usg the Dfferetal Evoluto (DE) algorthms. Here we propose a dfferet soluto of the smlar problem aalog doma. The magtude respose ad the group delay are optmzed dvdually cascade-coupled flter blocs. At ths pot, the optmzato ca be effectvely doe by adjustg parameters of each secto. The desg procedure of a aalog flter wth requremets for magtude ad group delay frequecy respose ca be dvded to two parts. The desg of the aalog flter by usg a stadard method s the frst well-establshed part ad the desg of the addtoal aalog all-pass flter s the secod part. It s appled coecto wth the ma flter for equalzato of the group delay frequecy respose of the flter. I geeral, a approxmato of the equalzer trasfer fucto represets a exget mathematcal problem. The preset desg methods are based o usage of classcal umercal methods, where a good estmato of the tal approxmato to esure covergece of the umercal method s eeded. These methods are mostly mathematcally complcated for programmg ad mplemetato. The dsadvatage of these classcal methods s also that they ca coverge to local optmum. Several such desg procedures have bee publshed papers [7], [8], [9]. Therefore, a ew very smple method of the equalzg all-pass flter desg based o evolutoary algorthm s preseted ths cotrbuto. The procedure s splt to two parts. At frst, we calculate the aalog all-pass flter trasfer fucto to obta the total group delay frequecy respose (of the low-pass flter wth appled all-pass flter) whch has uequal-rpple extremes the pass-bad. Subsequetly, the estmato of extremes s utlzed for the umercal solver to search correct shfted values of the allpass flter trasfer fucto complex poles to be obtaed equal-rpple form of the group delay the pass-bad. Thus, the procedure allows desgg flters wth equalrpple group delay frequecy respose the pass-bad. The Dfferetal Evoluto algorthm s used le umercal solver both parts. The method wll be demostrated o practcal examples. The whole desg procedure has bee programmed the MATLAB evromet.. The Dfferetal Evoluto Algorthm The Dfferetal Evoluto algorthm belogs to the evolutoary algorthms group. The algorthm was developed by K. Prce ad for the frst tme was preseted

2 P. ŽIŠKA, M. LAIPERT, ANALOG GROUP DELAY EQUALIZERS DESIGN BASED ON EVOLUTIONARY ALGORITHM 995. I the coferece Frst Iteratoal Cotest of Evolutoary Computato (sticeo) held Nagoya May 996, ths algorthm tured out to be the best evoluto type of algorthm for the real-valued fuctos solvg. The Dfferetal Evoluto (DE) algorthm s a parallel drect search method whch uses floatg-pot umber represetato to fd cotuous parameters. Ths techque utlzes NP D-dmesoal parameter vectors x, G,,,..., NP () as a populato for each geerato G. It meas, for each terato step of the mmzato process. NP does ot chage durg the evaluato. The tal populato s chose radomly f othg s ow about the system. Ths populato s geerated the MATLAB as NP D matrx, where NP s the umber of members of the populato ad D s the umber of uow varables the solved tas. Each matrx elemet s geerated as a radom umber wth uform probablty dstrbuto the form: x, j m+ r (max m) () where...np, j...d, ad the rage of the varables s (m..max). DE geerates ew parameter vectors by addg the weghted dfferece vector betwee two populato members to a thrd member. If the resultg vector yelds a lower objectve fucto value tha a predetermed populato member, the ewly geerated vector replaces the vector whch t was compared wth. The compared vector ca, but eed ot be part of the geerato process metoed above. Several varats of DE exst. But, we have foud out that the varat DE/best//b s the most effcet verso for the solved tas. Therefore, the scheme of the varat wll be descrbed greater detal. It wors as follows:. For each target vector x, G,,,..., NP, a tral vector v s geerated accordg to ( x x ) v xbest, G + F r, G r3, G (3) wth r, r 3 [, NP], teger ad mutually dfferet, ad F>. x best,g s the best vector, whch was obtaed as the best soluto the prevous algorthm rug. The tegers r ad r 3 are chose radomly from the terval [, NP] ad are dfferet from the rug dex. F s a real varable whch cotrols the amplfcato of dfferetal varato (x r, G - x r3, G ).. Each elemet of the chose target vector x, G,,,..., NP, ad each elemet of the tral vector v s selected (the frst elemet of both, the secod elemet of both ) ad for each couple a radom umber wth uform probablty dstrbuto the rage ( ) s geerated. Ths radom umber s compared wth the crossover costat CR. If the radom umber s greater tha CR, the the correspodg elemet of the vector x, G s serted to the ew vector u, G. Otherwse the correspodg elemet of the tral vector v s serted to the ew vector u, G. Thus, the ew vector u, G s obtaed. It troduces crossover operator evoluto theory. Ftess value of the vector u, G s compared wth ftess value of the target vector x, G. The vector wth lower ftess value s chose to the ew populato G+. Ths DE process s repeated utl a acceptable soluto s foud or a preselect umber of geeratos s performed. 3. Frst Part: Estmato of Group Delay Equalzer The put requremet s to desg a low-pass flter wth pre-assged cutoff frequecy, whch satsfes requremets for the magtude frequecy respose. Such flter ca be proposed usg some developmet system, for stace usg Sytfl Maple lbrary [6]. We wll deote a trasfer fucto of the desged flter the paper le H(s) by varable s defed as: s Σ + j Ω. (4) Thus, s s complex varable meag ormalzed frequecy. 3. A Aalog All-Pass Flter Defto Geerally, the group delay frequecy respose of the aalog flter ca be equalzed by a aalog all-pass flter cascade coecto wth the prevous ma flter. Geeral trasfer fucto of the eve order aalog all-pass flter ca be defed the form: A( s) ( s α + j β ) ( s α j β ) ( s + α + j β ) ( s + α j β ) where s the order of the all-pass flter Geeral trasfer fucto of the odd order aalog allpass flter ca be defed the form: A s) ( s + α ) ( s α + j β ) ( s α j β ) ( s + α ) ( s + α + j β ) ( s + α j β ) ( where the order of the all-pass flter s +. The total group delay of the cascade coecto of the low-pass flter ad the all-pass flter s defed as a sum of the group delays of each of them, so that: () s () () s () s s j Ω (5) (6) A H τ ( Ω) τ allpass ( Ω) + τ f ( Ω) Re + (7) A s H

3 RADIOENGINEERING, VOL. 5, NO., APRIL 6 3 where A(s) deotes the trasfer fucto of the all-pass flter ad A (s) deotes the dervatve A(s) by the varable s, H(s) deotes the trasfer fucto of the low-pass flter ad H (s deotes the dervatve H(s) by the varable s. 3. The Aalog All-Pass Flter Estmato Prcple Fudametal of the preseted procedure s the computato of the complex poles of the trasfer fucto of the all-pass secto wth the group delay frequecy respose whch equalzes the group delay frequecy respose of the low-pass flter. We fd the mmum of the τ usg the Dfferetal Evoluto algorthm. The mathematcal formulato of the objectve fucto s F ( x) max[ τ ( Ω) ] m[ τ ( Ω) ] + P (8) where vector x s composed of real ad magary parts of the complex poles of the all-pass flter trasfer fucto. P s the pealty fucto, whch ca be computed by N x f x < P. (9) otherwse N labels the umber of the uow searched varables ad x are elemets of the vector x. The pealty fucto P s cluded to the objectve fucto to guaratee stablty of the aalog all-pass flter. As t s ow, the real parts of the complex poles must be located the left part of the complex plae the varable s. Here we have used pealty fucto publshed [4]. The vector x opt, for whch the objectve fucto F(x) has mmum, s the wated soluto of the aalog all-pass flter estmato problem. 3.3 Practcal Example The method usablty to the all-pass flter desg wll be preseted o the soluto of the followg example. It s requred to equalze the group delay frequecy respose of the aalog low-pass flter, whch meet requremets to the magtude frequecy respose. The flter s proposed by applyg the Cauer approxmato. The trasfer fucto of the 6 th order aalog low-pass flter s defed: 6 6 H ( s) b s a s () where coeffcets are arraged Tab.. The group delay of the flter s show Fg.. The group delay error (the dfferece betwee the maxmum ad m mum of the group delay the pass-bad) s τ f 5.6 s the pass-bad ad we wat to equalze the group delay frequecy respose to receve a better group delay error. The optmzato tas was solved usg the DE algorthm tal settgs: NP5, CR.9, F.9, the rage of the uow varables (real ad magary parts of complex poles) was ( ). The group delay frequecy resposes were sampled at 4 equdstat pots the pass-bad. We have decded for llustrato of the descrbed method to use all-pass flters of the order 4, 5 ad 6 defed by approprate terms (5), (6). The algorthm foud the fal values show Tab.. The trasfer fuctos obtaed usg these values were smulated MATLAB. The resultat group delay frequecy resposes of the cascaded coectos of the flter ad the all-pass flters are show Fgs., 3 ad 4. 4 th order all-pass flter α β.4474 α β th order all-pass flter α α β α β th order all-pass flter α.9339 β α β α β Tab.. The foud parameters of the all-pass flters. Fg.. The group delay of the low-pass flter. Fg.. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 4. a 6 a a a a a a.97 b b 5 b b 3 b b b Tab.. Coeffcets of the low-pass flter. Fg. 3. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 5.

4 4 P. ŽIŠKA, M. LAIPERT, ANALOG GROUP DELAY EQUALIZERS DESIGN BASED ON EVOLUTIONARY ALGORITHM Fg. 4. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 6. costat value of the group delay, whch s approxmated Chebyshev sese, ε s the rpple of the group delay. Ω deotes the terval whch a approprate extreme wll be searched. N s the lower boud of the terval of Ω ad N s the upper boud of the terval of Ω. Mostly, the optmum values are N pots ad N 8 pots. Oly N, N, N + ad N +. We tae to accout samplg at N4 pots. The values of N ad N have to be re-couted for samplg by usg dfferet umber of sampled pots. 4. Secod Part: Fal Desg of Group Delay Equalzer I the prevous part, after low-pass flter group delay equalzato, we have obtaed lower value of group delay error all cases, but group delay frequecy resposes have uequal-rpple form the pass-bad. Now, we wll utlze the postos of the foud extremes Fgs., 3 ad 4 to search correctly shfted complex poles of the all-pass flter trasfer fuctos to acheve equal-rpple form of the group delay frequecy resposes. Let us have a pass-bad terval of the group delay frequecy resposes (low-pass flter + approprate all-pass flters) sampled at N equdstat pots. It s plotted for each appled all-pass flter the Fg. 5, 6 ad 7. I the fgures, the group delay frequecy resposes samplg at the N4 pots was used. Now, a so called error fucto for the sampled resposes wll be created: ( ) + ( ) ε max[ τ ( Ω )] ( ) + ( ) ε m[ τ ( Ω )] τ for eve e () τ for odd Fg. 5. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 4. Fg. 6. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 5. Ω ( p N, p + N ) () where,..,+, p deotes the posto (sample) of the approprate th extreme of the sampled respose, τ() s the Fg. 7. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 6. Importat: case of odd order all-pass flter desg, the parameter,..,+. Thus, N, N, N + ad N +. As metoed above, DE algorthm wll be used as a umercal solver. Therefore, a objectve fucto wll be defed. DE algorthm searches complex poles of the allpass flter trasfer fucto ad costats τ() ad ε to obta the mmum of the objectve fucto. Thus, the mmzato process of the objectve fucto leads to the desg of the all-pass flter. The mathematcal formulato of the objectve fucto for a eve order all-pass flter s defed: + F( x) P [ e ] +. (3) As metoed above, for odd all-pass flter desg, the objectve fucto s modfed so that,..,+. The vector x s composed of real ad magary parts of complex poles of the all-pass flter trasfer fucto ad the costats τ() ad ε. P s the pealty fucto, whch ca be computed by: M x f x < P. (4) otherwse M labels the umber of the uow searched varables ad x are elemets of the vector x. The pealty fucto P s cluded to the objectve fucto to guaratee stablty of the aalog all-pass flter. As t s ow, the real parts of the complex poles must be located the left part of the complex plae the varable s. The pealty fucto P also esures postve values of tal group delay τ() ad rpple ε. The vector x opt, for whch the objectve fucto F(x) has mmum, s the wated soluto of the aalog all-pass flter desg problem.

5 RADIOENGINEERING, VOL. 5, NO., APRIL Fal Results The optmzato tas was solved usg the tal settgs: NP5, CR.9, F.9, rage of the uow varables (real ad magary parts of complex poles, costats τ() ad ε) was ( ). The group delay frequecy resposes were sampled at 4 equdstat pots the pass-bad. The algorthm foud the fal values show Tab. 3. The trasfer fuctos obtaed usg these values were smulated MATLAB. The resultat group delay frequecy resposes of the cascaded coectos of the flter ad the all-pass flters are show Fgs. 8, 9 ad. 4 th order all-pass flter 5 th order all-pass flter 6 th order all-pass flter α β α β α α β α β α β α β α β τ 7.9s τ 6.58s τ 4.85s Tab. 3. The foud parameters of the fal all-pass flters. 5. Coclusos A ew ucovetoal method for the desg of the aalog group delay equalzers was preseted ths paper. The method allows desgg such all-pass flter to acheve a equal-rpple form of the total group delay frequecy respose the pass-bad. It s a very good property of the method developed for equalzato of aalog flters or aother delay les. The procedure cossts of two parts. At frst, the aalog all-pass flter trasfer fucto s calculated to be obtaed the total group delay frequecy respose (of the low-pass flter wth the appled all-pass flter) whch has uequal-rpple extremes the pass-bad. Thus, the respose s ot equrpple. Hece, the equalzato s performed the sese of achevable dfferece betwee maxma ad mma of the resultg respose. Therefore, we have to apply the secod step of the procedure as to acheve equrpple form of the resultg group delay. The procedure wors automatcally wthout a put estmato. Moreover, ths paper we appled oe of the most effcet types of evolutoary algorthms, the so called Dfferetal Evoluto algorthm. Due to the fact that the method s based o usage of evolutoary algorthm, the method s much more robust agast covergece to the local extremes tha classcal umercal methods. Acowledgemets The wor reported the paper was supported by the Czech Grat Agecy uder grat GAČR No. /3/H85 ad by the research program MSM Fg. 8. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 4. Fg. 9. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 5. Fg.. The group delay of cascaded coecto of the low-pass flter wth the appled all-pass flter of the order 6. Refereces [] DAVÍDEK, V., LAIPERT, M., VLČEK, M. Aalog ad Dgtal Flters (Aalogové a číslcové fltry Czech). ČVUT Praha,. [] MARTINEK, P., VONDRAŠ, J. New approach to flters ad group delay equalser trasfer fucto desg. I The 8th IEEE It. Cof. o Electrocs, Crcuts ad Systems ICECS, vol., p. 7. [3] MARTINEK, P., VONDRAŠ, J. Mult-crtero flter desg va Dfferetal Evoluto method for fucto mmzato. I st IEEE It. Cof. o Crcuts ad Systems for Commucatos ICCSC' Proceedgs. St. Petersburg (Russa): Sat-Petersburg State Techcal Uversty,, vol., p [4] STORN, R. Dfferetal Evoluto Desg of a IIR-Flter wth Requremets for Magtude ad Group Delay. Techcal Report TR-95-6, ICSI, May 995. [5] MICHALEWICZ, Z. Geetc Algorthms + Data Structures Evoluto Programs. Sprger-Verlag, Berl Hedelberg,.996. [6] HOSPODKA, J., BIČÁK, J. Sytfl - Sythess of electrc flters Maple. I MSW 4 [CD-ROM]. Waterloo, 4, vol.. [7] GREGORIAN, R., TEMES, G. Desg techques for dgtal ad aalog all-pass crcuts. IEEE Tras. o Crcuts ad Systems, 978, vol. 5, o., p [8] HENK, T. The geerato of arbtrary-phase polyomals by recurrece formulae. J. Crcut Theory Appl., 98, vol. 9, p [9] ZAPLATÍLEK, K., HÁJEK, K., DENK, M. Optmal all-pass etwor desg usg umercal optmzato loop. I Proc. of the th Electr. Devces Systems Cof. EDS 4, Bro, 4, p