Analog Circuit Design with Variable Length Chromosomes

Transcription

1 Analog Crcut Desgn wth Varable Length Chromosomes Shn Ando Unv. of Tokyo, Bunkyo-ku Hongo, Tokyo, Japan Htosh Iba Unv. of Tokyo, Bunkyo-ku Hongo, Tokyo, Japan Abstract- Ths paper proposes a system of evolvng analog crcut based on a varable length chromosomes. It should confrm the system s robustness, crcut scalablty, and effecency on tme and memory comsumpton. Methods featured here are chromosomes of component lst, multobjectve evoluton, and two-stage evoluton. Set of experments are shown n ths paper. Frst experment reconfrms the robustness suppled by the evolutonary method. Second compares several types of chromosome mplementaton. There are also experments to evaluate the two-stage and scalng method. 1 Introducton Ths paper descrbes the system of analog crcut evoluton usng varable length chromosomes. The feature of ths system wll be verfed through set of smulated crcut evoluton experment. There are reasons why evolutonal method was ntroduced to crcut desgn problems. One s to utlze the creatve ablty of evolutonary method to derve a new functon or topology. Another ablty of evolutonal method, n ths case GA, s adaptaton through adjustment of parameters. The second factor s very useful n the feld of analog crcut desgn because one of the man dsturbance n manufacturng analog crcut s the error of the component values. The values components such as resstors and capactors regularly dffer from expected or specfed values. They are also subject to the nfluence of envronmental effect such as temprature. The countermeasures for these varatons such as redundant mplementaton or lnear adaptve flters requre complcated method and much human experence. We expect to solve the problem on robustness presented above and add effcency to analog crcut desgn usng the Evolutonary Analog Crcut, whch we so call the system. It ncludes the lnear chromosomes mplementaton whch can also make possble the crcut scalablty and mult-stage evoluton. We also expect to mplement ths on Evolvable Hardware(here after EHW) to acqure much robustness. In EHW only the response of the crcut s evaluated and varance n value of the each componets are absorbed through repetve set of adjustment n topology and parameters and evaluaton of the resoponse n the genetc algorthm. In ths manner robustness aganst varance n components s acheved. There are many types of mpementaton proposed n evoluton of analog crcut. For example Genetc Programmng uses tree-type chromosomes and synthesze a crcut creatng program. In GA, there are matrx and lnear program mplementaton of analog crcut and succeeded n desgnng several passve flters. In the study of EHW, several Based upon these conventonal method, we used methods shown below to feature automatc crcut desgn and manufacture, topology and parameter evoluton, and adequate crcut scalng. Lst based chromosomes. Two stages of evoluton. Complexed ftness. Furthermore these experments wll evaluate the above method. Nose and error absorpton. Comparson wth GP. Sze reducton 2 Varance n Analog Crcut Component One of the largest defects of analog system s naccuracy. When analog crcut s mplemented as an ntegrated crcut, crcut component values nevtably dffer from desgned specfcaton. Ths s caused by error n component producng process or number of envronmental factors such as temprature. Ths s a qute large obstacle for desgnng and manufacturng precson analog devce. To realze a strct specfcaton for analog devce, complcated and emprcal desgnng process s requred.

2 3 System Archtecure In the EHW anlalog crcut evoluton, The topology and values of crcut components are adjusted through genetc operaton and GA wll evaluate each of the crcut and fnd the closest response wth avallable component. The system of Evolutonary Analog Crcut s composed as shown n Fg.1. Crcut Phenotype mplement Reconfgurable Hardware response Fg.1:Structure of Evolutonary Analog Crcut It conssts of reconfgurable hardware and GA smulator. Hardwares are resstors and capactors wth programmable value and topology. Crcut evoluton follow these steps. 1. Prepare ntal populaton 2. Adustment by genetc operaton 3. Implementaton of the phenotype 4. Response of the hardware contanng nose 5. Response s evaluated based on the objectve spec. 6. Inferor ndvduals are excluded from the group. 7. Return to 2. These are basc steps for genetc algorthms. Through repetton of adjustment and feedback precse specfcaton s acqured. 4 GA Smulator Specfcat on Envron ment evaluaton GA Smulator Populaton mutaton selecton crossover new ndvduals ndvduals/new Ths secton descrbes the detals of the GA smulator. 4.1 Chromosomes Implementaton There have been several chromosome representaon proposed for crcut structure. In the Genetc Programmng, tree-representaton s used and J.R.Koza evolved a crcut creatng program for seres of complcated crcuts. GP s sad to have advantage over GA n dervng new topology. Its defects are volumnous memory consumpton and convergence tme. Matrx representaton for crcut structure s proposed by Ktamura et al. and they succeeded n evolvng several analog flters. Its defects are that t requres prelmnary knowledge on objectve crcut sze and complexty. Lnear crcut creatng program mplementaton was used by J.D.Lohn n analog crcut synthess n [13]. Our chromosome mplementaton s lst representaton. Ths chromosome conssts from lst of components n crcut and s varable n length. The chromosomes s shown n Fg.2. N,,1, C,1,2,1nF R,,3,1k L,2,3,1mH C,,2,1pF Fg.2:Chromosome Implementaton The components are descrbed wth type, value, and locaton. In descrbng the locaton of the component we used MessyGA method whch s proposed by Goldberg to prevent GA from fallng nto local soluton. Zebulum also used ths representaton n synthess of actve flters. Each gene holds an allele for type, locaton and value parameter. Locaton s descrbed by par of ntegers whch represent the nodes n the crcut whch the component s connected to. Allele for components type are R, C, L, N,O whch represent resstors, capactors reactance, short crcut, open crcut. 4.2 Ftness Each ndvdual s evaluated based on the devaton between the deal and actual response by frequency. The ftness functon s defned as below. 1 ftness = K K f F f R f 2 eq. 1 Ths ftness s the mean of squared devaton between deal gan F f and obtaned gan R f at frequency f. The chromosomes wth lower ftness are selected to reproduce accordng to roullette wheel selecton. We also used Evolutonary Strategy(+)-ES. 4.3 Structure and Parameter Evoluton GA features strong global seach and quck convergence to quas-optmal soluton. On the other hand, stochastc search of GA can be neffcent from quasoptmal to optmal soluton. In the case of electrc crcut evoluton from scrach requres two dfferent tasks, whch are fndng the rough pattern of the flter and adjustng precsely to specfcaton. The frst task s acqurng suffcent topology or crcut structure and the second s fnely tunng the parameters. Though n our chromosome representaton, both structure and parameters of the components are

3 confgurable, we thought t was n effcent to evolve them smultaneously. In frst stage acqurng the proper topology, parameter adjustment wth relatvely smaller affect on crcut response would be less mportant to topology alternaton. And n fnal stages of evoluton where precse adjustment s requred alterng topology would be neffectve. We dvded the evoluton nto two stages. In frst stage, man objectve s to acqure proper topology or structure and parameters wll not be a varable. In the second stage, realzng precse specfcaton wll be the objectve and use the acqured topology as fxed structure. In the frst stage or the structural stage, chromosomes shown n Fg.2 s used. In second stage, or the parameter stage, array of s value s used as chromosome. The components are adjusted accordng to Eq.3. The s are real numbers from 1 to 1. Range of modfcaton s kept small for applcablty n reconfgurable analog components of EHW. Adjval 1 s = Val eq. 2 Lmtng the varables n each stage results n better ftness, faster convergence, and less memory consumpton. Secton 6.2 descrbe the experment on ths method. 4.4 Selectve Pressure on Crcut Sze One of the problems n Genetc Programmng and GA wth varable length chromosomes s development of ntrons. At certan pont n evoluton, ntrons bloat up to huge amount and makes the search awfully neffcent. Detals on effect of the ntrons s descrbed n [7]. In electrc crcuts, they appear as set of components connected to ground. These ntrons are crucal to EHW applcaton for t result n consumng large amount of hardware resources. There can be several measures to elmnate the ths ntrons, but we chose to smply put selectve pressure on crcut sze. The ftness s adjusted as shown n Eq.4, where E s the evaluaton of the response and P s the penalty for crcut sze. P s defned as n Eq.(5) where N s the number of components n the crcut and represent the sze factor, and T s the modulus to control the ntensty of the pressure. ftness E + P = eq. 3 P = N T eq. 4 Ths selectve pressure has ts defects when appled too excessvely or too early. Elmnatng too much ntrons s sad to make crossover operaton semantcally too destructve, and there are danger of abandonng dversty and elmnatng useful schema at early stage of evoluton. T can control the ntensty of the pressure by settng the order of P and E n Eq.4. At the early stage of evoluton, the term E should be predomnant. As the evoluton progress and value of E decrease, selectve pressure P should gan nfluence and unnecessary large crcuts are elmnated or modfed to proper sze. T has to be set accordng to prorty of crcut sze and requred accuracy, and fndng the proper T value s left for future work. We used emprcal values for the followng experments n secton 8. 5 Robustness Aganst Varance The desgn methods for many passve flters are well establshed. Yet, analog flters used n many devces are hard to manufacture. As we mentoned before, ths s because the components value vary from value specfed n desgnng process. Gan e+ Fg. 3:Ideal and Actual Response of the Band Elmnaton Flter vout speculaton 5% 1% 2% Snce ntrons have no affect on the crcut response, crcuts wth ntrons wll be subject to elmnaton by the sze factor.

4 Fg. 4:Band Elmnaton Crcut Desgn For example, sold lne n Fg. 3 shows the response of band elmnator flter desgned as Fg. 4. However, when the crcut s manufactured from real components, because the components values vary from specfcaton, response would not be dentcal to the sold lne. The analog components lke resstors and capactors could contan errors up to 2% of the specfed value, and dotted and broken lnes n Fg. 3 shows the response when each components n crcut of Fg. 4 randomly contaned errors wthn 2%,1%,5% of the desgned values. These dfference caused by the errors are fatal n manufactureng precse analog devces.we conducted a flter synthess experment under such condton where components values are not exactly as specfed, to show how Evolutonary Analog Crcut can accomodate wth such errors. 5.1 Specfcaton The goal response s the band elmnatng response shown as sold lne n Fg. 3. The central frequency of the stop band s 16kHz. The components used to compose ths crcut are shown n Table 2. But each components value are not exact and contan errors upto certan maxma for each of the experment. We set the maxmum errors to 5, 1, and 2% and conducted 5 runs for each case respectvely. Result s shown n Table 1. Nos e Sample crcut 2 th generaton 4 th generaton e e e e e e-7 Table 1:Ftness of Band Elmnaton Flter In EHW, crcut smodfed accordng to the whole response of the crcut, and not by the value of each component. Thus errors n each component wll be absorbed through topology and parameter modfcaton of the components as a whole. 6 Comparson wth Other Representaton In ths secton we show several flter synthess usng lst chromosomes along wth other representatons. To compare the result, we used smlar objectve functon and GA parameters. 6.1 Specfcaton The experment descrbed here under s based on Synthess of an Asymmetrc Bandpass Flter n Chap.31 of [4]. The objectve s to acqure an asymmetrc bandpass flter descrbed n [4] and [12] as dffcult to desgn because ts specfcatons are both strngent and hghly asymmetrc. The deal and allowable characterstcs are defned as shown n Fg.3. Sold lne labeled deal ndcates the bounds of deal characterstcs and the broken lne labeled allowable ndcates the allowable range. The crcut behavor s observed at 11 frequences n the nterval between 1kHz and 2kHz n equal ncrements on a logarthm scale. The ftness s defned as n Eq.6. [ W ( d( f )) d ( f )] F eq. 5 = 1 Weght W s calculated from the dfference between the response and the goal response wth each observaton pont, and the total product of the weght W and the dfference d becomes the ftness. Weght n the pass-band s 1 f allowable, 1 f else. In the stop-band, weght s 1f allowable, 1 f not. Detaled descrpton s found n [4]. The parameters of the GA are shown n Table 2. Populato n Generato n Crossove r rate Mutaton rate Lst based GP [4] Table 2:GA Parameters 6.2 Result The acqured crcut response s shown n Fg. 5. Gan deal allowable nelsen GP acqured Fg. 5:Acqured Asymmetrc Bandpass Flter Response The best response of the 4 th generaton s shown as the broken lne labeled acqured. The dotted lne labeled GP ndcates the response of the crcut obtaned n [4]. The ftness of the best ndvduals was wth the acqured and 224. wth the GP. Meanwhle, dotted lne of the label Nelson shows the response of human desgned prototype

5 crcut. The acqured response satsfes the allowable condton n the every regon, and obtaned better response than the Neloson s heurstc method. In comparson wth GP, we were able to obtan very close response at the pass-band, and equally acceptable characterstc at cut-off regon as well. 6.3 Specfcaton Next experment s conducted based on [6]. The objectve s to acqure an deal low-pass flter shown n Fg. 6. The pass-band s from 1Hz to 13Hz and stop-band s from 13Hz to 1kHz, thus cut-off frequency s at 13Hz. The ftness s defned as gven n eq.6. d(f ) s the dfference between the goal gan V goal (f ) and the actual gan V out (f) at F+1 sample frequences defned as eq.7. The weghted functon W s defned by eq.8. The value of W s set to.2 n ths experment. For detals refer to [6]. F Ftness = W d = ( d ( f ), f ) d( f ) ( f ) V ( f ) V ( f ) goal = eq. 7 out ( f ) Wθ ( f ) > W eq. 6 1 for d W( d( f ), f ) = eq. 8 1 for d θ Vgoal(f) s 1V n the pass-band and V n stop-band. Ftness was calculated from total of 78 sample frequences, 5 from the pass-band and 28 from stop-band. We used the populaton of 5 ndvduals, and 2 generatons for each run as n [6]. Crossover rato, mutaton rato, and replacement rato are the same as n Table Result Fg. 6 shows the response of the best ndvdual of 2 th generaton. The devaton from specfed band remaned wthn W (=.2V), and ts ftness was whle the ftness of the best ndvdual obtaned n [6] was The phenotype of the best ndvdual s shown n Fg specfcaton response vout Fg. 7:Aqured Lowpass Flter Crcut 7 2stage Evoluton The experment n ths secton dvde the evoluton nto structural and parameter stage. 7.1 Specfcaton Target response s an deal hgh-pass flter depcted as a sold lne n Fg. 8. Cut-off frequency s 3kHz, and 14 ponts were taken at an nterval of geometrc rato rangng from 1kHz to 1MHz as the observaton ponts. Wth the structure evoluton phase, the settled values were used as shown n Table 3. GA parameters are as shown n Table 2. Element types Values Resstances 1k,1M 7.2 Result Condensers Cols.6.5 1nF,1pF 1H,1mH Table 3:Crcut Components Specfcaton specfcaton sngle step 2 step Gan.6 gan Fg. 6:Specfcaton and Acqured Response of Ideal Lowpass Flter e+ frequency Fg. 8:Specfcaton and Acqured Response of Hgh-pass Flter

6 Fg. 9 shows the ftness of the best ndvdual by generaton. Ths ftness s average of 3 runs. The broken lne labeled sngle step denotes one-stage evoluton where topology and parameters are smultaneously evolved. And dotted lne labeled 2step ndcates that of 2 stage evoluton. An arrow s shown where the parameter evoluton starts. The response acqured by each evoluton are shown n Fg. 8. The response of sngle-step evoluton s gven n broken lne labeled sngle step whereas the two-stage evoluton s provded by the dotted lne labled 2 step. The ftness s for two-stage and for one stage. It s perceved from Fg. 9 whle smultaneous evoluton converges after 2th generaton, two-stage evoluton resumes the search by enterng the parameter evoluton..1 sngle evoluton 2step evoluton Fg. 1:Objectve Bandpass Flter Response Ftness defnton s adusted as n Eq.4, and T modulous s set to 1-6. We conducted 5 runs wth 5 populaton and 2 genereratons. Other parameters follows that of Table 2. Only the topology was modfed n the course of evoluton as crcut sze wll be fxed n parameter evoluton. 8.2 Result Responses of the best ndvdual at the 4th and 15th generatons of a typcal tral are shown n Fg. 11. The actual crcut phenotype of each ndvdual s shown n Fg. 14 and Fg. 15. Ftness of the fnal generaton of ths run was e-11. The ftness and crcut sze by generaton s shown n Fg. 12 and Fg th 4th.25.1 Gan generaton Fg. 9:Ftness by Generaton n Hghpass Flter Evoluton 8 Selectve Pressure on Crcut Sze e+ Fg. 11:Response of the Best Indvduals from Each Generaton 8.1 Specfcaton We smulated a crcut evoluton usng the selectve pressure referred to n secton 4.4. Objectve response s the bandpass flter shown n.1.1 ftness.35.3 specfcaton Gan.1.25 Gan e Fg. 12:Ftness by generaton e+

7 35 crcut sze 9 Dscusson Fg. 13:Crcut Sze by Generaton It can be obtaned that by the 4th generaton, response fullflled the specfcaton. At 4th generaton, whle the nfluence of the pressure s nconsderable, electrcal ntrons can be seen n Fg. 14, but as can be seen n Fg. 15, those porton are deleted as the evoluton progress. Fg. 12 and Fg. 13 shows that adapton n earler stage of evoluton done by aqurng the proper crcut and n the later stage, t s done by gettng rd of the unnecessary components. Fg. 14:Best Indvdual of Generaton 4 Fg. 15:Bes t Indvdual of Generaton Robustness Aganst Varaton In experment shown n secton5, 9.2 Comparson wth Other Representaton In experments shown n secton6.1, lst mplementaton was able to acqure an crcut wth almost equvalent ftness. Generally, GP has the advantage n fndng topology and structure. But, n fndng crcut structure for the farly dffcult flter shown n secton 6.1, lst representaton was able to acheve equvalent ftness. Meanwhle, usng the GA, amount of calculaton as n populatons and generatons can be kept small and memory consumpton stays low because of the dfference n chromosomes mplementaton. From ths experment, t can be sad that the GA and lst representaton has the adequecy n crcut desgn stage Evoluton In two stages of evoluton as we proposed, frst stage, or the structural evoluton, cause dynamc change n response and ftness, whle n later stage, the parameter evoluton, response wll be altered by smaller degrees to adapt to strngent specfcaton wth hgh accuracy. Consderng the smultaneous evoluton of topology and component value, n earler stage when ftness mprove rapdly n prmary convergence, effect of parameter beng modfable s so small. In the later stage, topology modfcaton affect too much and newly created crcuts do not survve. It also overwhelms the parameter modfcaton and makes t mpossble to make lttle changes. Ths 2stage evoluton also contrbutes to lessen the memory consumpton by lmtng the varables n each stage. 9.4 Pressurng Crcut Sze One emprcal part n ths system s decdng the T modulus n eq.4. In secton 8, we have succeeded n settng the relatvty of penalty and response evaluaton. Fg. 12 and Fg. 13 shows that n earler stage of evoluton the ftness s mproved by acqurng better response, and n later stage, sze factor manly contrbute to the ftness mprovement. But general way to calculate the T s remaned to be studed. 1 Concluson In ths study, we proposed methods shown below to be used n the mplementaton of Evolutonary Analog Crcut. - Component lst representaton of the crcut. - 2stage evoluton - Selectve pressure on crcut sze

8 These methods were appled to experments shown n ths paper, and result show that these methods are effectve for analog EHW. The equpment of the GA system wth reconfgurable hardware s to be promoted as a prospect for the future. Acknowledgments We would lke to thank Professor Shnzo Ktamura for provdng us wth precous and nformatve materals for the study. We also wsh to show our grattude to the EHW researchng group of ETL ncludng Mr.Tetsuya Hguch, who has gve us helpful advce and suggestons. Bblography [1] D. Keymeulen, H.Sakanash, M.Murakawa, I. Kajtan, E.Takahash, K.Toda, M. Salam, N.Kajhara, and N. Otsu, Real-World Applcatons of Analog and Dgtal Evolvable Hardware, IEEE Transactons on Evolutonary Computaton, Vol.3, No.3, 1999 [2] Goldberg, D.E. and Deb, K. and Karpupta, H. and Hark, G. Rapd, Accurate Optmzaton of Dffcult Problems usng Fast Messy Genetc Algorthms, Proc. 5 th Int. Jont Conf. On Genetc Algorthms(ICGA93), 1993 [3] J.R.Koza and F.H.Bennett III and D. Andre and M.A. Keane and F.Dunlap, Automated Synthess of Analog Electrcal Crcut by Means of Genetc Programmng, IEEE Transactons on Evolutonary Computaton, Vol.1, No.2, 1997 [4] John R. Koza and Forrest H. Bennett III and Davd Andre and Martn A. Keane, Genetc Programmng III, Morgan Kaufmann Publshers Inc., 1999 [5] M. Iwata, I. Kajtan, H. Yamada, H. Iba, and T. Hguch, A Pattern Recognton System usng Evolvable Hardware, Parallel Problem Solvng from Nature - PPSN IV, Lecture Notes n Computer Scence 1141, pp , Sprnger-Verlag, [6] Masaya Koyabu, Hajme Murao, Shnzo Ktamura, Automatc Desgn of Electrcal Crcut by Genetc Algorthm, SICE 24 th Sympodum of Intellgent Systems, Mar. 1997, n Japanese [7] Melane Mtchell, An ntroducton to genetc algorthms, 1996 [8] Murakawa, Masahro and Yoshzawa, Shuj and Adach, Tosho and Suzuk, Shro and Takasuka, Kaoru and Hguch, Tetsuya, Analogue EHW Chp for Intermedate Flter, Proceedngs of the Second Internatonal Conference on Evolvable Systems, 1998 [9] R. Zebulum, M. Pacheco and M. Vellasco, Analog Crcut Evoluton n Intrnsc and Extrnsc Mode, Proceedngs of Second Internatonal Conference of Evolvable Systems, vol.1478 p ,1998 [1] R. Zebulum, M. Pacheco and M. Vellasco, Artfcal Evoluton of Actve Flters: A Case Study, The Frst NASA/DOD Workshop on Evolvable Hardware, 1999 [11] Shn Ando, Htosh Iba, Mtsuru Ishzu ka, Evolvable Analog Crcut usng Varable Length Chromosomes, 56 th IPSJ Natonal Conference, n Japanese, 1999 [12] Ival Nelson, Analog Integrated Crcuts and Sgnal Processng, [13] J.D.Lohn, S.P.Colombano, A Crcut Representaton Technque for Auomated Crcut Desgn, IEEE Trans. on Evolutonary Computaton, Sept. 1999, Vol.3 Num.3 p.25 [14] J.D.Lohn, S.P.Colombano, Automated Analog Crcut Synthess usng a Lnear Representaton, Proc. of the Second Int l Conf. on Evolvable Systems: From Bology to Hardwa re, Sprnger-Verlag, Berln, 1998, pp