Linear Genome Methodology for Analog Circuit Design

Transcription

1 Lnear Genome Methodology for Analog Crcut Desgn Shn Ando Inf. and Comm. Dept. School of Eng. Unv. of Tokyo Tokyo, Japan Abstract Ths paper descrbes a set of specalzed GA methods for the analog crcut desgn,.e., the component-lst chromosome, the mult-staged evoluton, and the sze reducng pressure. In the several experments, these methods have shown to acheve the robustness, the effcency n tme and the hardware consumpton. 1 INTRODUCTION The purpose of ths paper s to propose a set of GA methods n desgnng an analog crcut. Ths ncludes component-lst chromosome, mult-staged evoluton, and sze reducng pressure. These methods are tested and evaluated through a set of smulated experments. There have been a few representaton schemes proposed for analog electrc crcut. Ths ncludes a matrx representaton by Ktamura et al.[6] and a lnear crcutcreatng program by Lohn et al.[13]. There s also a major study of the crcut desgn usng Genetc Programmng, n whch Koza et al. has generated crcut synthess programs for varous useful crcuts [4]. The analog crcut synthess s a subject n whch the Evolvable Hardware s very useful. Hence the robustness of EHW system can compensate for the analog crcut s fraglty to extrnsc envronment. Murakawa et al. has developed an EHW chp for IF flter. In that study, the delcate tunng of component parameter resulted n sgnfcant ncrease n the yeld rate [8]. Based upon these prevous approaches, we propose the methods below for an analog crcut EHW system. Lst representaton for analog crcut Mult-staged evoluton Sze-reducng pressure We have conducted the followng experments to verfy the effcacy of these methods: Nose and error absorpton The detals of ths paper s to be presented at CEC2000 Htosh Iba Dept of Fronter Informatcs School of Fronter Scence. Unv. of Tokyo Tokyo, Japan Comparson to other representaton schemes Dvson of topology and parameter evoluton Crcut pressurzaton 2 GA ARCHITECHTURE Ths secton descrbes the detals of the GA system we have used for mplementng our methods and conductng the experments. 2.1 CIRCUIT REPRESENTATION SCHEME The crcuts are coded nto genes of analog components. A phenotype and the correspondng genotype are shown n Fg. 1. Fgure 1: Representaton Scheme 2.2 FITNESS DEFINITION Each ndvdual s evaluated on the devaton between the deal and actual response by frequency. The ftness functon s defned as below. 1 ftness = K ÁÂÃÂÄÂÃÅ ÆÂÄÂÇÂÄÈÉÅ ÂÃÂÂÄ Å ÂÇÂÂÄÅ ÆÂÃÂÇÂÄÉÅ K f F f R f 2 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 the roulette wheel selecton. We also used Evolutonary Strategy breedng of (µ+?)-es.

2 2.3 MULTI-STAGED EVOLUTION The Genetc Algorthms features the strong global search and quck convergence to a quas-optmal soluton. On the other hand, the stochastc search of GA can be neffcent from the quas-optmal to the optmal soluton. Evolvng an electrc crcut from the scratch requres two dfferent tasks,.e., fndng the rough layout of the crcut and adjustng precsely to the specfcaton. The frst task requres effcent topology search and the second requres fne-tunng of the parameters. Though both the structure and parameters of the components are confgurable n our component-lst representaton, t s neffcent to evolve them smultaneously. At the earler stage of the evoluton, the parameter adjustment has relatvely smaller affect on the crcut response and s less mportant compared to the topology alternaton. Meanwhle, at the fnal stage of the evoluton where a precse adjustment s requred, modfyng the topology changes the response so drastcally that t may degrade the search. Thus, we have dvded the evoluton nto two stages. At the frst stage, the man objectve s to acqure a proper topology and parameters wll be fxed to pre-settled values. At the second stage, the objectve s to realze a precse specfcaton usng the acqured layout as a fxed structure. At the frst stage or the structural stage, chromosomes shown n Fg.1 are used. At the second stage, or the parameter stage, arrays of s values are used as our chromosomes. The value of component s s adjusted accordng to eq.2. The s s are real numbers rangng from 1 to -1. Range of modfcaton s kept small for the applcablty n reconfgurable analog components of EHW. s Adjval = Val 10 Á Lmtng the varables at each stage also results n better ftness, faster convergence, and less memory consumpton. Secton 5.2 descrbes the expermental results usng ths method. 2.4 PRESSURIZING CIRCUIT SIZE One of the problems n Genetc Programmng and GA wth varable-length chromosome s the development of ntrons. At a certan pont n the evoluton, ntrons bloat up to huge amount and make the search awfully neffcent. Detals on the effect of ntrons are descrbed n [7]. In electrc crcuts, they appear as a set of components connected to the ground or a node. These ntrons are fatal to EHW applcaton because t results n consumng a large amount of hardware resources. There can be several measures to elmnate the ntrons. A method of mult-crtera evoluton s used for the dgtal crcut evoluton by Kalganova[5]. We have chosen to smply put a selectve pressure on the crcut sze. The ftness s adjusted as shown n eq.3, where E s the evaluaton of the response and P s the penalty for the crcut sze. P s defned as shown n eq.4, where N s the number of components n the crcut and represents the sze factor, and T s the modulus to control the ntensty of the pressure. ftness = E + P P = N T Á Snce ntrons have no effect on the crcut response, crcuts wth ntrons are subject to the elmnaton by the sze factor. Ths selectve pressure can be mpedtve to GA search when appled too excessvely or too early. Elmnatng ntrons too much causes crossover operaton to be semantcally destructve, and there are also dangers of abandonng dversty and deletng useful schema at the early stage of the evoluton. The ntensty of the pressure s controlled usng the modulus T, by properly settng the order of P and E n eq.3. At the early stage of evoluton, the term E should be predomnant. As the evoluton proceeds and the value of E decreases, the selectve pressure P should gan nfluence. Therefore unnecessary large crcuts are elmnated or modfed to the proper sze. A larger T value results n less accuracy because the large P overwhelms the small dfferences. Thus, the T has to be set accordng to the prorty of the crcut sze and requred accuracy. We have used an emprcal value for the followng experments descrbed n secton 7. 3 EXPERIMENTS The followng experments are conducted usng one or more of the methods descrbed above. 4 ROBUST DESIGN OF ANALOG CIRCUIT The desgn methods for varous 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 values vary from the one specfed n the desgnng process.

3 4.1 SPECIFICATION AND RESULTS The goal response s the band elmnatng response shown as a sold lne n Fg. 2. The central frequency of the stop band s 16kHz. The components used to compose ths crcut are shown n Table 3 Table 1: Ftness of Band Elmnaton Flter ÁÂÃÄÅÆ ÁÂÃÄÅÆÇÈÉÈÉÇ ÇÈÈ ÁÂ ÇÆÆÂÉÆ ÉÈÈ ÁÂ ÆÆÆÂÉÆ Â Á ÃÄÃÃÃÅÆÅÇÈÉÁ ÉÄÈÉÈÆÃÂÁ ÅÄÂÂÃÁ ÉÃ Á ÃÄÃÃÉÅÉÂÂÉÁ ÉÄÂÆÈÅÃÂÁ ÉÄÆÂÈÃÈÁ ÅÃ Á ÃÄÃÃÂÅÉÇÃÈÁ ÅÄÉÈÇÂÃÂÁ ÉÄÂÈÆÉÃÈÁ Fgure 2: Ideal and Actual Response of the Band Elmnaton Flter Á ÁÂÃÄÅ Fgure 3: Band Elmnaton Crcut Desgn The sold lne n Fg. 2 shows the response of a bandelmnator flter desgned as Fg. 3. However, when the crcut s manufactured from real components, because the components values dffer from the specfcaton, the response would not be dentcal to the sold lne. Actual analog components lke resstors and capactors could contan errors up to 20% of the specfed value. The dotted and broken lnes n Fg. 2 show the response when each component n crcut of Fg. 3 randomly contaned errors wthn 20%, 10%, and 5% of the desgned values, respectvely. The dfference caused by these errors s fatal n manufacturng precse analog devces. Therefore, we conducted a flter synthess experment under such a condton that components values are not exactly as specfed. Ths s to show how Evolutonary Analog Crcut can accommodate wth prelmnary errors. In EHW, the crcut s evaluated and modfed based on ts whole response, and not by the value of each component. Thus, the errors of the components are absorbed through the modfcaton of topology and parameter appled to the components as a whole. 5 COMPARISON WITH OTHER REPRESENTATION SCHEMES In ths secton, we show several flter syntheses usng lst chromosomes along wth other representaton schemes. To compare the results, we used the smlar objectve functon and GA parameters. 5.1 SPECIFICATION The experment descrbed here s based on "Synthess of an Asymmetrc Bandpass Flter" n Chap.31 of [4]. The objectve s to acqure an asymmetrc bandpass flter, whch s dffcult to desgn because of ts strngent and hghly asymmetrc specfcaton [4]. The deal and allowable characterstcs are defned as shown n Fg.4. The 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 101 frequences n the nterval between 10kHz and 200kHz n equal ncrements on a logarthm scale. The ftness s defned as n eq [ W ( d( f ) d( f )] F = 0 Weght W s calculated from the dfference between the response and the goal response at each observaton pont. The ftness s derved from the total product of the weght W and the dfference d. W n the pass-band s 10 f allowable, 100 f else. In the stop-band, W s set to 1 f allowable, 10 f not. Detaled descrpton s found n [4]. The GA parameters are shown n Table 2. Table 2: GA Parameters Populaton Generaton Crossover rate Mutaton rate Lst GP[4]

4 5.2 RESULT The acqured response s shown n Fg. 4. The best response at the 400th generaton s shown by the broken lne labeled acqured. The dotted lne labeled as GP ndcates the response of the crcut obtaned n [4]. V goal (f) s 1V n the pass-band and 0V n the stop-band. Ftness was calculated from the total of 78 sample frequences,.e., 50 from the pass-band and 28 from stopband. We used a populaton of 500 ndvduals, and 200 generatons for each run as n [6]. Crossover rato, mutaton rato, and replacement rato are the same as shown n Table RESULT The broken lne n Fg. 5 shows the response of the best ndvdual at 200 th generaton. The devaton from the specfcaton remaned wthn W (=0.02V), 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. 6. Fgure 4: Acqured Asymmetrc Bandpass Flter Response The ftness of the best ndvduals was wth the acqured and wth the GP. Meanwhle, the dotted lne of the label Nelson shows the response of a human desgned prototype crcut. As can be seen n Fg. 4, the acqured response satsfes the allowable condton n every regon, and obtaned better response than the Nelson s heurstc method. In comparson wth GP, we were able to obtan very close response at the pass-band, and equally acceptable characterstc n the cut-off regon as well. 5.3 SPECIFICATION Next experment s conducted accordng to [6]. The objectve s to acqure an deal low-pass flter shown n Fg. 5. The pass-band ranges from 1Hz to 1300Hz and the stop-band s from 1300Hz to 100kHz, thus the cut-off frequency s 1300Hz. 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 0.02 n ths experment. For detals refer to [6]. F Ftness = W d W = 0 ( d( f ), f ) d( f ) ( f ) = V ( f ) V ( f ) ( d( f ), f ) goal 1 = 10 for for d out ( f) Wθ ( f ) > W d θ Fgure 5: Specfcaton and Acqured Response of Ideal Lowpass Flter ÁÂÃÄ Fgure 6: Acqured Lowpass Flter Crcut 6 MULTI-STAGE EVOLUTION The experment n ths secton shows the effect of dvdng the evoluton nto the structural and parameter stages.

5 6.1 SPECIFICATION The target response s an deal hgh-pass flter depcted as a sold lne n Fg. 7. The cut-off frequency s at 30kHz, and 14 observaton ponts were taken at an nterval of a geometrc rato rangng from 100kHz to 1MHz. In the structure evoluton phase, the settled values were used as shown n Fg. 7. GA parameters are shown n Table 3. Table 3: Crcut Components Specfcaton Element type Values Resstances Condensers Cols 10k,1M 1nF,1pF 100 H,10mH 6.2 RESULT Fg. 8 shows the ftness of the best ndvdual n each generaton. Ths ftness s averaged over 3 runs. The broken lne labeled sngle step denotes one-stage evoluton, n whch the topology and the parameters were smultaneously evolved. And dotted lne labeled 2step ndcates that of the mult-stage evoluton. The arrow shows where the parameter evoluton started. The responses acqured by two methods are shown n Fg. 7. Fg. 8:Ftness by Generaton for Hghpass Flter Evoluton The response of the sngle-step evoluton s gven as the broken lne labeled sngle step, whereas that of the twostage evoluton s provded by the dotted lne labled as 2 step. The acheved ftness was for the multstage and for the one-stage. It s perceved from Fg. 8 that whle the smultaneous evoluton converged after 100 generatons, the mult-stage evoluton resumed the search by enterng the parameter evoluton. 7 SELECTIVE PRESSURE ON THE CIRCUIT SIZE 7.1 SPECIFICATION We have smulated a crcut evoluton usng the selectve pressure descrbed n secton 2.4. The objectve response s the bandpass flter shown n Fg. 9. Fgure 7: Specfcaton and Acqured Response of Hghpass Flter Fg. 9:Objectve Bandpass Flter Response Ftness defnton was adjusted as n eq.3, and T modulous was set to be We have conducted 5 runs wth a populaton of 500 and 200 genereratons. Other parameters followed that of Table 2. Only the topology was modfed n the course of evoluton as the crcut sze was fxed n the parameter evoluton. 7.2 RESULT The responses of the best ndvduals at the 40th and 150th generatons for a typcal tral are shown n. The phenotypes are shown n Fg.13 and 14. The ftness value at the fnal generaton was e-11. The ftness and crcut sze wth generatons are shown n Fg. 11 and 12. It can be seen from Fg. 10 that by the 40th generaton, the response fullflled the specfcaton. At generaton 40, whle the nfluence of the pressure was nconsderable, electrcal ntrons were exstent as shown n Fg. 13.

6 However, as the evoluton proceeded, those portons were removed as seen n Fg. 14. Fg. 11 and Fg. 12 show that the adapton at the earler stage of the evoluton was done by aqurng the proper crcut and at the later stage, by gettng rd of the unnecessary components. Fgure 13: Best Indvdual at Generaton 40 Fg. 10:Response of the Best Indvduals at Generatons 40 and 150 Fgure 14: Best Indvdual at Generaton 150 Fg. 11: Ftness by generaton Fgure 12: Crcut Sze by Generaton 8 CONCLUSIONS In the experments we have shown, each of the proposed methods has respectvely mproved the effcency n the crcut desgn. These methods are expected to work ndependently, but we plan to mplement a system ntegratng all of the methods n the future. The sze reducng pressure has seemed to be effectve n generatng not only a smaller crcut but also a ftter crcut.e., crcuts wth more accurate response. We plan to conduct an experment to verfy that pont. Snce the sze parameter s a very restrcted factor n many exstng re-confgurable hardware, the sze-reducng objectve has not been studed as much. But as more elastc hardware develop; we beleve that the method should become a major subject for EHW. We have to note that the lst-component genome s a very general crcut representaton, and s not drectly applcable to varous types of the exstng EHW hardware. However, proper restrctve settngs could easly make ths apt for many types of crcut generatng systems. The proposed mult-stage evoluton was a specalzed method for the analog crcut desgn. We expect t to be effectve n many evolutonary crcut generaton. In addton to the former experment, we are plannng pursue

7 the mean ftness ncrease (or decrease) after crossovers and mutatons n each evolutonary stage. The purpose s to compare the effcency of genetc operatons n snglestaged evoluton and the structural and parameter evoluton. 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 Dr.Tetsuya Hguch, who has gve us helpful advce and suggestons. References [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 Frequency 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 [10] 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 Ishzuka, Evolvable Analog Crcut usng Varable Length Chromosomes, 56 th IPSJ Natonal Conference, n Japanese, 1999 [12] Ival Nelson, A C-T flter compler-from specfcaton to layout, Analog Integrated Crcuts and Sgnal Processng, 7(1):21-33, 1995 [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.205 [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 Hardware, Sprnger-Verlag, Berln, 1998, pp [15] Stuart J. Flockton, Kevn Sheehan, A System for Intrnsc Evoluton of Lnear and Non-lnear Flters, The Frst NASA/DOD Workshop on Evolvable Hardware, 1999 [16] Tatana Kalganova, Julan F. Mller and Terence C. Fogarty, Some Aspects of an Evolvable Hardware Approach for Multple-Valued Combnatonal Crcut Desgn, Proc. of the Second Int l Conf. on Evolvable Systems: From Bology to Hardware, Sprnger-Verlag, Berln, 1998