Design of IIR digital filter using Simulated Annealing

Transcription

1 Desgn of IIR dgtal flter usng Smulated Annealng Ranjt Sngh *, Sandeep K. Arya * Department of Electroncs and Comm. Engneerng, JMIT Radaur, INDIA Department of Electroncs and Comm. Engneerng, GJU Hsar, INDIA * Correspondng Author: e-mal: chauhan443@yahoo.co.n, Tel , Fax Abstract The Paper presents a smple computer-aded desgn approach for desgnng IIR dgtal flters. IIR flter s essentally a dgtal flter wth Recursve responses. There are many ways for the desgn of IIR Dgtal flters. Ths Paper Presents heurstc way for the desgnng IIR flters. A varety of Heurstc Technques are avalable for the desgnng of IIR Flters. Ths Paper s based only upon Smulated Annealng (SA) for fast convergence of result. SA s a well-known powerful global optmzaton algorthm, ntroduced n combnatoral optmzatons The Smulaton Program for Desgn of IIR dgtal flters usng MATLAB s presented n ths paper and results were found to be very encouragng. Keywords: Dgtal flters, IIR, SA, Optmzaton.. Introducton Over the past several decades the feld of Dgtal Sgnal Processng (DSP) has grown to mportant both theoretcally and technologcally. In DSP, there are two mportant types of systems. The frst type of systems performs sgnal flterng n tme doman and hence t s known as Dgtal Flters. The second type of systems provde sgnal representaton frequency doman and are known as Spectrum Analyzer. Dgtal flterng s one of the most powerful tools of DSP. Dgtal flters are capable of performance specfcatons that would, at best, be extremely dffcult, f not mpossble, to acheve wth an analog mplementaton. In addton, the characterstcs of a dgtal flter can be easly changed under software control. Dgtal flters are classfed ether as Fnte duraton unt pulse response (FIR) flters or Infnte duraton unt pulse response (IIR) flters, dependng on the form of unt pulse response of the system. In IIR system, the mpulse response sequence s of nfnte duraton.. Desgn of IIR Dgtal flter Dgtal flters are classfed as Recursve and Non- Recursve flters. The response of Recursve or IIR flters depend only upon Present and prevous nput of sgnal. IIR flters have the followng advantages:- They are Stable and can be realzed effcently n hardware. Because an IIR flter uses both a feed-forward polynomal (zeros as the roots) and a feedback polynomal (poles as the roots), t has a much sharper transton characterstc for a gven flter order. Lke analog flters wth poles, an IIR flter usually has nonlnear phase characterstcs. Sgnal feed back and IIR flters are recursve type flters. Analog flter to Dgtal flter converson s very easy. IIR flter s usually more effcent desgn n terms of computaton tme and memory requrement. The Dgtal IIR flters have varous stages for ther desgn. The flow chart of the Desgn of Dgtal flter s shown n Fg.. Fgure. Consder the IIR flter wth the nput-output relatonshp governed by y M ( k) + b y( k ) = a x( k ) START Performance Specfcaton Calculaton of Flter Coeffcents Realzaton Structurng Fnte Word length Effects Analyss & Soluton Hardware & Software Implementaton + Testng STOP = = L Proc. of the Internatonal Conference on Advanced Computng and Communcaton Technologes (ACCT ) Copyrght RG Educaton Socety ISBN:

2 Proc. of the Internatonal Conference on Advanced Computng and Communcaton Technologes (ACCT ) where x(k) and y(k) are the flter s nput and output, respectvely, and M ( L) s the flter order. The transfer functon of ths IIR flter can be wrtten n the followng general form: H A B = = = M L + a z = b z Dgtal Flters are desgned by usng the values of both the past outputs and the present nput, an operaton brought about by convoluton. If such flters subjected to an mpulse then ts output need not necessarly become zero. The nfnte mpulse response of such a flter mples the ablty of the flter to have an nfnte mpulse response. Ths ndcates that the system s prone to feedback and nstablty. IIR flters have Infnte Duraton Impulse Response, hence they can be matched to analog flters, all of whch generally have nfnte long mpulse response. Therefore the basc technque of IIR Flter desgn transforms well-known analog flters nto Dgtal Flters. Hence IIR flter desgns wll be treated as the characterstc of three wdely used analog flters, namely, Butterworth, Chebyshev (Type and Type) and Ellptc flters. Butterworth of maxmally flat flters have a monotonc ampltude frequency response whch s maxmally flat at zero frequency response and the ampltude frequency response decreases logarthmcally wth ncreasng frequency. The Butterworth flter has mnmal phase shft over the flter s band pass when companed to other conventonal flters. Chebyshev Flters are of two types.e Chebyshev I and Chebyshev II flters. Chebyshev I flters are all pole flters whch are equrpple n the pass band and are monotonc n the stop band.chebyshev II flters contan both poles and zeros exhbton a monotonc behavor n the pass band and equ-rpple n the stop band. Ellptc flters are characterzed by equ- rpples the both ther pass bands and stop band. They provde a realzaton wth the lowest order for a partcular set of condtons. 3. Smulated Annealng SA s a well-known powerful global optmzaton algorthm, ntroduced n combnatoral optmzatons.it s based on random moves, and has the ablty to overcome local mnma, found on the way toward a better mnmum, wth uphll moves. The synthess of a dgtal flter transfer functon requres that the values of the multpler coeffcents n the flter mplementaton be selected such that the transfer functon of the flter meets the prescrbed frequency response specfcaton. The flter Coeffcents are selected n such a way as to mnmze the dfference between the desred and the actual response. The desgn process seeks to mnmze ths "dfference" functon, known as the objectve or cost functon. Snce the number of flter coeffcents can be large, the number of confguratons that the flter system can assume becomes very large. An exhaustve search for a coeffcent set that mnmzes the objectve functon can only be attempted at great computatonal expense. To mnmze the computatonal effort, lmted systematc or random search methods are reled upon. Smulated annealng s a stochastc optmzaton technque that has been used for large-scale, dscrete space problems. The smulated annealng algorthm, whch s based on the theory of statstcal mechancs s, essentally an teratve random search procedure wth adaptve moves along the coordnate drectons. It permts uphll moves under the control of a probablstc crteron. It s the capacty to traverse "hlls" that makes smulated annealng an attractve algorthm for optmzng functons wth many varables and hence many local mnma. A close relatonshp has been shown to exst between statstcal mechancs and combnatoral mnmzaton. In partcular, the study shows that the process that occurs n nature n order for matter to acheve ts low temperature state can provde useful technques for use n mnmzaton. The study also shows that fndng the low temperature state alone n practce does not guarantee matter to be n one of ts lowest energy states. It s possble for the substance to be n a low temperature state and yet fal to mantan equlbrum at that temperature. A system n ths state s sad to be "metastable", that s t can easly get out of equlbrum. Experments that attempt to fnd the low temperature state of a substance begn by rasng the temperature of the substance n queston to a very hgh value that s the meltng pont of the substance. The temperature s then lowered n slow stages so that at each temperature value the substance s allowed suffcent tme to acheve thermal equlbrum. The process contnues untl the substance s "cold, that s untl no further changes n energy occur. The Metropols algorthm, whch has been wdely used n smulated annealng, was orgnally developed for use n the smulaton of a collecton of atoms n thermal equlbrum at a gven temperature. At each step of ths algorthm, an atom s gven a small random dsplacement and the resultng change n the energy of the system ( E) s computed. The method of smulated annealng conssts of three functonal relatonshps.. g(x): Probablty densty of state-space of D parameters x = x ; =, D.. h( E): Probablty for acceptance of new cost-functon gven the just prevous value. 3. T(k): schedule of annealng the temperature T n annealng-tme steps k,.e., of changng the volatlty or fluctuatons of one or both of the two prevous probablty denstes. The new dsplacement s accepted subject to the crteron: If E, the acceptance probablty s based on 336

3 Proc. of the Internatonal Conference on Advanced Computng and Communcaton Technologes (ACCT ) the chances of obtanng a new state wth energy Ek+ relatve to a prevous state wth energy h exp( E k / T ) E ) = exp( E / T ) + exp( E = + exp( Δ E / T ) = exp( Δ E / T ) + ( Δ k / T ) where E represents the energy dfference between the present and prevous values of the energes (consdered here as cost functons) approprate to the physcal problem,.e., E = Ek+ Ek. Ths essentally s the Boltzmann dstrbuton contrbutng to the statstcal mechancal partton functon of the system.unform random devates n the nterval (, l) are a convenent means of mplementng the random part h ( E). The accepted rearrangement s then used as the startng pont of the next step. By repeatng the step many tmes, the algorthm smulates the thermal moton of atoms n thermal contact wth a heat bath at temperature T. At very hgh values of temperature T, n the above expressons, vrtually any rearrangement wll be accepted. As the temperature falls to low values, rearrangements that ncrease the change n energy wll "re lkely be rejected. Ths strategy allows for controlled "uphll" excursons to be taken and hence to seek "hdden" arrangements wth lower energy levels. Usng the Metropols algorthm, smulated annealng has been used for dscrete optmzaton problems. Although the algorthm does not guarantee to fnd the global optmum, f the functon has many good near optmal solutons t wll fnd one. The smulated annealng desgn process conssts of frst "meltng" the system beng optmzed at a hgh temperature and then lowerng the temperature n slow stages untl the system s "cold", that s untl no further changes occur. As n statstcal mechancs, the smulaton must proceed long enough for the system to attan a steady state at each temperature. An annealng schedule s the sequence of temperatures and rearrangements of the coeffcent values attempted to reach steady state at each temperature. An mportant feature of smulated annealng s that the "gross" features of the functon appear at hgh values of temperature and the "fner" detal at lower values. 4. Proposed algorthm The proposed smulated annealng algorthm s derved from that used for functons of contnuous varables and adapted to the optmzaton of functons of dscrete varables. Fgure s a pseudo code lstng of the man steps of the proposed smulated annealng algorthm. The ntalzaton phase reads the desred frequency magntude specfcatons of the flter, computes the ntal value for the control parameter T, calculates the correspondng ntal cost functon, and sets up the teraton lmts. The man loop, whch performs the annealng, repeatedly performs a cycle of random moves along coordnate drectons n turn, calculates the objectve functon, and apples the acceptance crteron untl equlbrum s consdered establshed at that temperature. If, wthn the man loop, the optmzaton process satsfes the desred specfcatons the process s able to termnate. Annealng Algorthm () Intalze: passband and stopband tolerance regons etc; Obtan startng temperature; Generate random ntal coeffcent set; Compute:current magntude response error; olderror = current error; optmum_error = current error: set: upper_lmt = lmt based on sze of problem; lower_lmt = value based on sze of problem; temp_dec = ; for (=; <= number of step adjustments; ++) nsucc = ; coordnate = ; for (k=l ; k<=upper_lmt; k++) perform random move along specfed coordnate drecton; calculate objectve functon(new error); f(acceptance crteron && stable) update error(olderror = newerror); record total accepted ponts; record ponts n coordnate dr; f( newerror = opterror) record optmum pont; else restore last pont; next coordnate; f(lower_lmt s reached) break; compute statstcal data for temperature determnaton; update step: reset ponts n coordnate dr. to ; decrement the temperature; ncrement temp_dec; f(temp_dec = = Nt) start from optmum pont; f(stoppng crtera s satsfed) stop smulated annealng; compute fnal frequency response; wrte fnal coeffcent set; Fgure : Proposed Smulated Annealng Algorthm. The determnaton of steady state n the smulated annealng algorthm s referred to as the "nner-loop" crteron. Most smulated annealng algorthms do not detect the steady state condton explctly. Rather a large constant number of teratons s appled at each temperature after whch the system s assumed to have attaned steady state. Applyng a fxed teraton count n ths manner has the effect of slowng 337

4 Proc. of the Internatonal Conference on Advanced Computng and Communcaton Technologes (ACCT ) down the rate of convergence. Ths s due to the fact that at hgh temperature values most of the functon evaluatons do not contrbute drectly to the optmzaton process. In the proposed algorthm an attempt has been made to speed up the rate of convergence of the optmzaton process by alterng the manner n whch the annealng parameters are : (a) Startng Temperature: The method used n the proposed algorthm for calculatng the startng temperature s based on the requrement for numercal stablty when computng the exponental functon n the Metropols crteron. The proposed algorthm starts by randomly perturbng as much of the coeffcent search space as possble whle computng the change n the objectve functon at each pont. The maxmum change n the objectve functon s recorded and the startng temperature s made ten tmes ths maxmum change. (b) Temperature decrement scheme: Usng a fxed temperature decrement factor slows the rate of convergence. In addton, t s possble to mss the hgh ponts of the surface beng explored whch could lead the routne nto fndng and gettng stuck n a local mnmum. The result s an exponental decrease wth the temperature fallng quckly ntally and hence spendng less tme evaluatng functons that do not contrbute to the fnal convergence of the algorthm. To avod a drastc temperature decrease, the temperature rato s set to have a lower bound, say.5. (c) Inner-Loop Crteron: Most smulated annealng algorthms do not mplement an explct equlbrum detecton scheme. The problem wth not dong so s that t s easy to get caught n local mnma of hgh energy. An easy way around ths s to have a large teraton count at each temperature whch yelds many functon evaluatons. (d) Termnaton Crteron: Smple termnaton tests nvolve determnng when the cost has not changed sgnfcantly over several temperature reductons. An adaptve step scheme, used for the contnuous varable case, s employed n ths algorthm. Ideally, then, at termnaton the step sze must be the smallest nteger value possble, that s unty. In the new algorthm developed here, termnaton s effected when the average cost has not changed sgnfcantly over four temperature reductons or when all the component values of the step vector are at the smallest desgnated value. The end temperature, when approprately computed, can also be used as a stoppng crteron n conjuncton wth any of the above condtons. "nner-loop'' crteron for detectng equlbrum at a fxed temperature. (e) Functon Evaluatons: In order to speed the annealng process, trgonometrc functons are evaluated by the use of a "look-up" table over the dscrete frequency range to x radans. Ths scheme s made possble by the dscrete nature of the problem. In evaluatng the cost functon, only the contrbuton of the ncremental value of the altered coeffcent s evaluated resultng n further speed mprovement. 5. Smulaton and Results Ths secton presents the smulaton framework for the Desgn of IIR flter usng Smulated Annealng. The MATLAB smulaton s carred for certan specfcatons such as Rpb=.43, Rsb=.6, Wpb=7854, Wsb=9635 and Wss=48 for IIR flter. The Plots of Frequency Response and Pole-Zero of tradtonal and proposed scheme are shown below n Fgure 3 to Fgure 6. Magntude (db) -5 Butterworth IIR Lowpass Dgtal flter Response Normalzed Frequency Phase (Radans) Imagnary Part Magntude (db) Phase (Radans) Normalzed Frequency Fgure 3 Pole-Zero Plot, Order of Butterworth LP Flter s Real Part Fgure 4 Butterworth IIR Lowpass Dgtal flter Response Usng SA Normalzed Frequency Normalzed Frequency Fgure 5 338

5 Proc. of the Internatonal Conference on Advanced Computng and Communcaton Technologes (ACCT ) Imagnary Part x 4 Pole-Zero Plot usng SA, Order of Butterworth LP Flter s Real Part x 4 Fgure 6 6. Concluson A smulated annealng algorthm has been used to obtan fnte precson IIR flter coeffcents to approxmate a desred magntude frequency response. Both tradtonal and Proposed IIR dgtal flters have been desgned to meet a specfed magntude response. The Plots of Frequency Response, Impulse Responses and Pole-Zero of Butterworth low pass flter s ntroduced. The Pole-Zero plots dscusses the stablty of IIR dgtal flters. Further, the other Evolutonary algorthm can be dscussed for the desgn of IIR flters. Nomenclature IIR Infnte Impulse Respons SA Smulated Aneealng E Energy T Temperature [7] Devleeschouwer E., Grenez F., 99 An effcent procedure for the desgn of a large class of analog and dgtal flters, IEEE Trans. Crcuts Systm.-II, vol. 39, no., pp [8] Chauhan R.S., Kamboj A.K., 8, MATLAB based Desgn of Dgtal FIR flter usng wndow technque, IEEE Conf. on Art. Int. Systems (AIS), pp. 8-, [9] Pham D.T., Karaboga D.,, Intellgent Optmzaton Technques: genetc algorthms, tabu search, smulated annealng and neural networks, Advanced manufacturng seres. Sprnger, London. []Rao S.S, Ramasubrahmanyan A., 996, Desgn of dscrete co-effcent FIR flters by smulated evoluton, IEEE Sgnal Processng Letters, vol.3, no.5, pp []ProakS J. G., Manolaks D. G., 7, Dgtal Sgnal Processng: Prncples,Algorthms,andApplcatons, 4 th Edton, Pearson Educaton, Inc, New Delh. Bographcal notes Ranjt Sngh receved B. Tech. and M. Tech from Natonal Insttute of Technology, Jalandhar n 999 and 7, respectvely. He s a Assstant Professor n the Department of Electroncs and Communcaton Engneerng, JMIT Radaur, Inda. Presently he s dong Ph.D. from GJU Hsar Hs research nterests nclude Sgnal and Systems, Control Engg., dgtal sgnal processng and evolutonary Computaton. Sandeep Kumar Arya s a Charman n the Department of Electroncs and Communcaton Engneerng, Guru Jambheshwar Unversty of Scence and Technology, Hsar Inda. He has more than 5 years of experence n teachng and research. Hs current area of research ncludes Optcal communcaton, Sgnal and Systems, Control Engg. and Optmzaton Technques. He has publshed number of papers n referred nternatonal journals. He has also presented number of research artcles n natonal and nternatonal conferences. References [] Chua L.O., Ln P.M., 975 Computer-Aded Analyss of Electronc Crcuts: Algorthm and Computatonal Technques, Prentce-Hall, INC. [] Krkpatrck S., Gelatt C.D., Vecch M. P., 983, Optmzaton by smulated annealng, Jr. Scence, vol., pp [3] Dethorn E.J., Munson D.C., 986, Fnte word length FIR dgtal flter desgn usng smulated annealng IEEE Proc. Int. Symp. on Crcuts and Systems, vol. no.5, pp [4] Catthoor F., Man H.D., Vandewalle J., 988, Smulated-annealng based optmzaton of coeffcent and data word-lengths n dgtal flters, Int. J. Crcut Theory Appl., vol. 6, pp [5] Benvenuto N., Marches M., 989, Dgtal flters desgn by smulated, IEEE Trans. Crcuts Syst, vol 36, pp [6] Kacelenga R.V., Graumann P.V., and Turner L.E., 99, Desgn of flters usng smulated annealng, IEEE Proc. Int. Symp. on Crcuts and Systems (New Orleans, LA), pp