Artificial Evolution of Active Filters: A Case Study

Transcription

1 Artfcal Evoluton of Actve Flters: A Case Study Rcardo Salem Zebulum 1,2 Marco Aurélo Pacheco 2 Marley Vellasco Centre for Computatonal Neuroscence and Robotcs School of Bologcal Scences Unversty of Sussex, BN1 9QG Brghton, UK; emal: rcardoz@cogs.susx.ac.uk 2- Centro de Intelgênca Computaconal Aplcada (ICA) Departamento de Engenhara Elétrca PUC-Ro Brasl e-mal:salem@ele.puc-ro.br Abstract Ths artcle focuses on the applcaton of artfcal evoluton to the synthess of analog actve flters. The man objectve of ths research s the achevement of a new class of systems, wth advantageous features compared to conventonal ones, such as lower power consumpton, hgher speed and more robustness to nose. The partcular problem of desgnng the amplfer of an AM recever s examned n ths work. Genetc algorthms are employed as our evolutonary tool and two sets of experments are descrbed. The frst set has been carred out usng a sngle objectve, the desred frequency response of the crcut. In a second set of experments, three other objectves have been ncluded n the system. A new mult-objectve evaluaton methodology was conceved for ths second set of experments. A second approach for evolvng actve flters, usng programmable chps, s also dscussed n ths paper. 1 Introducton The evolutonary desgn of crcuts based on bpolar transstors s the man focus of ths work. Bpolar transstor technology s stll mportant for hgh speed applcatons n electroncs. Partcularly, our work addresses some mportant ssues of electronc crcuts evoluton, such as the multple-objectve nature of the task; the speed of the evolutonary tool to produce a desgn from scratch; and the mplementablty of the produced crcuts. The synthess of an amplfer for AM band s performed n ths work. Ths crcut works as a bandpass flter, amplfyng ncomng sgnals located n the AM frequency band. Partcularly, many practcal rado amplfers are stll mplemented through bpolar transstors technology [Sansen98]. Due to the fact that analog desgn s more complex for automaton than ts dgtal counterpart [Johns97], the use of search technques represents an nterestng alternatve. Recently, the evolutonary approach appled to analog desgn has been proposed by many authors, and promsng results have been acheved [Koza98][Layzell98][Lohn98][Stoca98. A Genetc Algorthm (GA) [Goldberg89][Holland75] wth nteger representaton s employed as the evolutonary tool n these experments. We present results for both sngle and multple objectves tasks. In the former, the desred frequency response of the amplfer s the only objectve taken nto account; n the later, power dsspaton, symmetrc excurson and nose are taken nto account as well. Addtonally, a comparson between the performance of genetc algorthms and hllclmbng [Blckle96] for ths problem s provded. We also compare the applcaton of extrnsc and ntrnsc methods [Zebulum98] for crcuts evoluton. Ths work s organsed n fve addtonal sectons. Secton 2 descrbes the behavour of the crcut to be synthessed. Secton 3 presents the man features of our genetc algorthm, ncludng representaton and ftness evaluaton functon. Secton 4 shows the results obtaned usng ths approach. Secton 5 provdes a dscusson on results obtaned usng a programmable analog chp durng the crcuts evaluaton step. Fnally, secton 6 concludes ths work. 2 Problem Descrpton We analyse the partcular problem of desgnng an amplfer for a rado recever tuned n the AM frequency band. Ths actve flter must amplfy ncomng sgnals nsde the frequency band rangng from.15 to 1.6MHz [Sansen98], whlst attenuatng sgnals outsde ths frequency band. Fgure 1 depcts a schematc of the desred system:

2 S Vout In La Ca R load Fgure 1 Basc schematc of the AM recever wth nductve source. In Fgure 1, L a represents the antenna nductance, and C a s a parastc capactance, assumng values of 3.5mH and 7pF respectvely [Sansen98]. The nput sgnal, I n, presents a magntude of 1.5µA (around - 117dB) [Sansen98]. The combnaton of L a and C a works as a trap to the ncomng sgnal I n. Ths trap has a resonant frequency at 1MHz. S represents the amplfer to be evolved, whch must be tuned to the AM frequency band defned above. Fnally, R load s the load output to be drven. Conventonal desgn of actve flters uses operatonal amplfers as crcut buldng blocks. In contrast, ths work does not enforce ths conventonal desgn prncple, and low-level buldng blocks, such as transstors, resstors and capactors, are utlsed. Although ths procedure ncreases the desgn complexty, novel crcuts are more lkely to be acheved. The dscusson secton of ths artcle presents an addtonal case study, where operatonal amplfers and swtched capactors are used as buldng blocks for the evolutonary system. 3 Evolutonary Algorthm A three operator evolutonary algorthm, ncludng selecton, crossover and mutaton operatons, s employed. The applcaton of an evolutonary algorthm to ths problem encompasses the choce of an effcent representaton and ftness evaluaton functon. Both are now descrbed. 3.1 Representaton An nteger representaton based on a lnear strng has been employed. Ths representaton has been used prevously by the authors n the synthess of operatonal amplfers [Zebulum98]. Fgure 2 depcts an example of ths knd of genotype-phenotype mappng for a common emtter amplfer. Power Supply Gene = [Connectng ponts, Component value,component nature] R = 1k Gene= [(1,2); 1nF, 2] 4 Gene1 = [(3,4); 1k; 1] 3 Out Component nature: = transstor 1 = resstor 2 = capactor ~ 1 2 In C=1nF Gene2 = [(3,2,); - ; ] External ponts:,1,3,4. Internal pont: 2 Fgure 2 Analog Crcut Representaton

3 The genotypes are made up of genes, each of whch encodes a partcular component. The gene determnes the nature, value and connectng ponts of the related component. Frst ntroduced n [Grbleby95] for the evoluton of passve flters topologes, ths representaton s here extended to nclude the assgnment of component values as well. The total number of connectng ponts s a parameter to be set n ths representaton. Ths parameter s crtcal to the effcency of the representaton: f too few connectng ponts are consdered, the number of possble topologes sampled by the evolutonary algorthm wll be lmted; conversely, f too many connectng ponts are consdered, a hgher number of unsmulatable topologes (wth floatng components) wll arse. Addtonally, each connectng pont may be classfed as nternal or external. Whle the former does not serve for any specal purpose, the latter s connected to one of the followng sgnals: power supply, ground, nput sgnal or probed output (Fgure 2). Due to the fact that a lnear strng s beng used, we wll refer to ths evolutonary system as a genetc algorthm [Goldberg89] throughout ths paper. 3.2 Evaluaton So far, most works concernng the evoluton of analog crcuts have used only one goal to be acheved by the evolved crcut. Nonetheless, the man challenge of applyng genetc algorthms or any other search technque to analog desgn s the multobjectve nature of ths task [Zebulum99]. Multobjectve optmsaton concerns the need to ntegrate vectoral performance measures wth the nherently scalar way n whch most optmsaton technques reward performance. Because genetc algorthms requre scalar ftness nformaton on whch to work, a scalarzaton of the objectve vectors s always necessary [Fonseca95]. After testng some standard methods for multpleobjectve optmsaton, the authors devsed a new one that s talored for ths class of problems. As t wll be observed, our method s based on artfcal neural networks learnng algorthms [Churchland92]. Gven a populaton of ndvduals, each one encodng an analog crcut n the way shown prevously, a measure of performance or ftness s assgned to each ndvdual n the followng way: Ftness = n = 1 w Fnorm (1) Accordng to the above equaton, the ftness s computed by a weghted sum, where w s a weght vector; n s the number of objectves; and F norm s the normalsed ftness vector. Ths vector s defned by: Fnorm = F F (2) F s the ndvdual s score wth respect to a partcular objectve, whereas the denomnator of the above expresson represents the average ftness, over all the ndvduals of the populaton, wth respect to the same objectve. Ths normalsaton s accomplshed to account for the fact that the objectves are measured n dfferent unts (decbels, Hertz, etc), and all of them must have the same nfluence n the ftness expresson. The man problem of ths approach s the one of settng the weghts values. It s desrable to use a strategy n whch the weghts are dynamcally updated accordng to the level of satsfacton of each objectve; and also to take nto account the user s specfcatons (desgn plan) for each partcular objectve. Based on these gudelnes, the followng weght updatng equaton has been adopted: w, t + 1 = α w, t + ( 1 α ) e, t (3) The above equaton uses an addtonal temporal ndex t, whch ponts to a partcular generaton of ndvduals. Hence, w,t+1 s the next value of the weght assocated to objectve. It s computed usng ts present value, w,t, and an error measure, e,t. Ths equaton s based on the Backpropagaton learnng algorthm for Artfcal Neural Networks (ANN) [Churchland92]. The term α used n equaton 3 can take real values from to 1, and wll balance the contrbuton of the error and of the current weght value n the updatng equaton. Through much expermentaton, we found that a value of α=.8 (Equaton 3) produced best results. Ths term s analogous to the momentum term used n the backpropagaton algorthm, whch s related to the stablty of the learnng process. The error e,t provdes a measure of the overall system performance for the partcular objectve, and t s computed by: e, t = F, t User User (4) Where User represents the user specfcaton for objectve. Therefore, the error s calculated by the

4 dfference between the average value for objectve over all ndvduals, and the user specfcaton. The weghts values wll then reflect the state of the system at the partcular nstant t. All the weghts are ntalsed wth an equal arbtrary absolute value: f the correspondng objectve needs to be mnmsed, the weght must take a negatve ntal value, and a postve value f the correspondng objectve needs to be maxmsed. Concernng the actual mplementaton of measurements n the crcuts sampled by the GAs, there are two standards modes n whch ths procedure can be carred out, ntrnsc and extrnsc assessments [Zebulum98]. In the former, each ndvdual of the evolutonary algorthm s downloaded nto a programmable chp [Stoca98][Thompson98], whereas, n the latter, smulators accomplsh the performance measurements. Ths work concentrates on extrnsc evoluton, provdng, though, a dscusson on results acheved n ntrnsc evolutonary experments. 4 Results We present the results of two classes of experments, processng a sngle objectve and multple objectves respectvely. 4.1 Sngle Objectve Experments Ths frst class of experments took only the desred frequency response nto account. The ftness s gven by the followng equaton: Ftness = n = 1 w ( V out ( ) V ( )) (5) Where V out s the crcut output and V n s the ncomng sgnal at the nput of the crcut S, shown n Fgure 1. As these voltages are measured n decbels, (V out () V n ()) s the amplfer gan at a partcular frequency. n s the total number of output samples. The weghts w take postve values for frequency ponts nsde the AM band, and negatve values outsde ths band. These weghts are not related to the ones presented n secton 3.2, snce we are tacklng a sngle objectve optmsaton here. The weghts values have been set through expermentaton, assumng a value of +33 for frequency ponts between 15 khz and 2 MHz (passng band); -4 for frequency ponts above 2 MHz; and 1 for frequency ponts below 15 khz. A chromosome made up of 12 genes has been employed n ths set of experments. The components nature are chosen from four optons: npn transstor; pnp transstor; resstor; and capactor. Eght connectng ponts are avalable for the topology arrangement, four of them beng external ones (nput, n output, power supply and ground). A power supply of 3V, whch s typcal of rado batteres, has been used. The evolutonary algorthm can choose among eght dfferent values for resstors and capactors respectvely. The resstor values range from 75Ω to 5kΩ and the capactor values range from.1nf to 1µF. The authors strategy was to let a small number of dfferent component values avalable to the GA, and keep the desgn space n a manageable sze. As a consequence, an nteractve nvolvement wth an expert may be necessary to further mprove the evolved crcuts. The sze of the search space can be calculated from the above values. There are a around 4 x 8 3 dfferent genes n ths representaton 1. As each chromosome s consttuted by 12 genes, one can conclude that the sze of the search space s gven by (4 x 8 3 ) 12 possble solutons, whch s around In order to sample ths search space, we ran 1 executons of the GA, each one ncludng 4 ndvduals and 1 generatons. Each executon lasts around 3 mnutes n a Sun Ultra Enterprse 2 server wth one 3 MHz ultra sparc processor. The small sgnal analyss of the SPICE smulator has been used n ths set of the experments. It has been verfed that most executons produced crcuts that conformed well to the specfcaton. Fgure 3 depcts the schematc of one crcut acheved n ths set of experments; Fgure 4 shows ts frequency response. I n Vn L a C1 = 1n C a R2= 1k5 R3=1k5 R1=1k 3V C2=.1n Fgure 3 Schematc of an amplfer obtaned n the frst set of experments. From Fgure 3, t can be seen that the evolved soluton uses only 6 components (not consderng L a, C a, and R Load ); the other 6 components encoded n the chromosome were not effectvely contrbutng to the 1 Number of genes = (# dfferent components). (# connectng ponts) 2 (# dfferent component values) = 4 x 8 3 1k Out R Load

5 crcut s behavour. The most nterestng aspect of ths crcut s ts parsmony, whch stems from the fact that only one objectve had to be fulflled by the GA. It s also nterestng to note that ths amplfer s confgured n a conventonal common-emtter topology. The crcut connected n ts collector, R2, R3 and C2, works as a low-pass flter, wth cut-off frequency around 5 MHz; the capactor C1 attenuates low frequency sgnals. The resstor R1 sets the DC operatng pont of the amplfer. Even though temperature varatons were not taken nto account n the ftness evaluaton functon, ths basng confguraton s advantageous to compensate effects of temperature changes [Laker94]. (exponental selecton wth parameter c equal to.9 [Blckle96]). The graph of Fgure 5 shows the average ftness obtaned n both experments. Although the GA outperformed hllclmbng, the latter performed surprsngly well for ths problem. Ftness 12, 8, 4, -4, -8, -12, 5-16, -5-2, Generatons Gan(dB) , 1,1,1,,1E+7 1E+8 1E+9 Frequency (Hz) Fgure 4 Frequency response of the amplfer shown n Fgure 3. The graph of Fgure 4 shows the amplfer s gan n the frequency doman. Focusng on the AM frequency band, the mnmum gan acheved by the crcut s 28 db, at 1.5MHz; and the maxmum gan s 34dB, at 4kHz. Conventonal crcuts present an average mnmum gan of 16dB; and an average maxmum gan of 37dB [Sansen98]. Further desgn mprovements can be accomplshed by optmsng the resstors and capactors values. For nstance, a smple nspecton of the evolved crcut shows that reducng the value of C1 s a way to enhance the transfer functon, by shftng the lower half of the crcut passng band from 2kHz to 5 khz. Fnally, we performed a comparson between the performance of our GA and the hllclmbng search technque [Blckle96]. A total of 2 GA executons were performed, each one processng 4 ndvduals along 1 generatons. The same number of ndvduals has been processed n the hllclmbng method,.e., 2 executons processng one ndvdual along 4 generatons. We note that the GA was optmsed n terms of mutaton rate (around 1 mutaton per genotype) and of selecton pressure Fgure 5 Average ftness along the generatons for two experments: GA (full lne) and Hllclmbng (traces). In the case of hllclmbng, the ftness values were taken wthn an nterval of 4 generatons, n order to match the number of 1 ftness ponts. 4.2 Mult-Objectves Experments Although hllclmbng and GAs produced comparable performances n the sngle-objectve experments, the genetc algorthm s better talored for the multple-objectve applcaton. Ths s due to the fact that, as t s descrbed n secton 3.2, our mult-objectve ftness evaluaton method requres the computaton of an average over a set of ndvduals (equaton (4)). Snce hllclmbng focuses on only one ndvdual at a tme, t can not be appled n the context of ths technque. Four objectves have been consdered: the frequency response ftness, computed n the way shown n equaton (5); the mnmsaton of the power dsspaton; the maxmsaton of the Maxmal Symmetrc Excurson (MSE); and the mnmsaton of the ntegrated output nose. The power dsspaton and the ntegrated output nose are drectly measured by the smulator; the MSE s maxmsed by keepng the DC value of the output voltage, V dc (out), at half of the power supply value (1.5V); ths s accomplshed by mnmsng the quantty V dc (out) 1.5. These four objectves are aggregated n the way shown n equaton (1), and the weghts are updated through the expresson presented n equaton (3). The desgn plan (user s specfcatons) used n ths experment was set to: ftness of the frequency response (equaton (5)) equal to 1; power consumpton equal to.1 mw; value of V dc (out)

6 1.5 equal to.5v, correspondng to a 1.V excurson n the output; and ntegrated output nose equal to 12dB. We ran 1 GA s executons, each one ncludng 4 ndvduals and 1 generatons. The overall experment lasted around 4 hours n one 3 MHz ultra sparc processor. The SMASH [SMASH93] smulator was used n ths set of the experments. The graphs of Fgure 6 dsplay the average values taken by the four objectves durng ths experment. It can be seen that the GA tres to optmse the frequency response, MSE, and dsspaton, keepng control of the output nose smultaneously. Fgure 7 shows the schematc of the best crcut acheved n ths second set of experments. 8, 1.4 7, 1.2 Ftness of Frequecy Response 6, 5, 4, 3, 2, 1, Vdc MSE , Generatons Generatons (A) (B) Dsspaton (Watts) Output Voltage Nose (db) Generatons Generatons (C) (D) Fgure 6 Average values of the objectves along the evolutonary process: frequency response(a); MSE (B); Dsspaton (C); Nose (D).

7 Out In La Rn=1k Ca Q11 R6=1k5 Q2 Q1 C7 Rload=1k C7=1n C1 = 5n 1n C9=1n R=1k 3 V Fgure 7 Best amplfer acheved n the second set of experments. In the above crcut, transstors Q11 and Q2 work n the lnear regon and transstor Q1 works n the reverse regon. Q2 s performng the sgnal amplfcaton and delverng t to the par Q1 and Q11, whch s settng the DC output to the desred value. As shown n the schematc, ths desgn can be mproved by a smple nspecton: the value of C 1 may be decreased from 5nF to 1nF; and the nput mpedance of the amplfer may be ncreased by nsertng R n. These changes mprove the passng band boundares wthn the AM frequency regon, by shftng t to the rght. The graphs of Fgure 8 show the frequency response of the crcut, wth and wthout these changes Gan (db) -12 Gan (db) , 1, 1,1,, 1E+7 1E+8 1E+9 Frequency (Hz) , 1, 1,1,, 1E+7 1E+8 1E+9 Frequency (Hz) (A) (B) Fgure 8 Gan of the crcut shown n Fgure 7: wthout changes (A); mproved desgn (B).

8 We can draw a comparson between the performance of the crcut obtaned n the frst set of experments and the mproved verson of the crcut shown n Fgure 7. Both of them dsplay a gan close to 3dB at the frequency of 5kHz. The desgn of Fgure 7 presents an ntegrated nose n the output of -94 db, aganst -83dB observed n the crcut of Fgure 3. A reference value for nose n bpolar amplfers can be taken from [Johns97], where the output nose for a common emtter amplfer, wth DC operatng pont optmsed for nose attenuaton, assumed a value around 8dB at 3K (the same temperature used n our experment). The power consumpton of the above crcut s 5. mw, aganst 3.9mW of the crcut depcted n Fgure 3. Ths dfference s due to the fact that the amplfyng transstor of the above crcut, Q2, s dranng more current from the power supply than the sngle transstor of the frst crcut. Fnally, the MSE of the crcut n Fgure 7 s around 1.5V (DC output value s 1.5V), aganst.9v for the crcut shown n Fgure 3. Therefore, we can conclude that the second crcut s better n terms of output excurson and nose. An mportant ssue concernng crcuts evoluton through smulaton s ther mplementablty. It has been verfed, n a prevous work [Zebulum98], that smulators mght bas transstors n overvoltage and overcurrent condtons, whch could not be reproduced n practce. The mnmsaton of the crcut dsspaton along the evolutonary process, accomplshed n ths experment, was the means whereby these condtons could be avoded. 5 Dscusson We present, n ths fnal secton, related results acheved through ntrnsc evoluton. We have used the feld programmable analog array MPAA2, from Motorola [Motorola97]. Ths chp s archtecture conssts of an array of operatonal amplfers, connected through swtched capactors. The GA controls the crcut connectvty, capactors values and other programmable features encoded n a lnear btstrng. Further detals can be found n [Zebulum98]. In a partcular experment, we focused on the evoluton of a bquad low-pass flter [Johns96], wth cutoff frequency close to 1kHz. The GA was allowed to manpulate a chp s regon ncludng two operatonal amplfers, whch s the standard sze of conventonal bquad flters [Motorola97]. The ftness evaluaton functon was computed by comparng the transent response of the sampled crcuts wth the one of a target flter. Ths transent response was taken by applyng a 1 khz square wave to the crcut s nput. The graphs of Fgure 9 compare the tme and frequency doman responses, respectvely, of the evolved crcut and the target flter. Ths experment processed an order of 1 3 ndvduals over many GA executons. The transent analyss s more attractve than the frequency analyss for ntrnsc evoluton, because the evaluaton step s less tme consumng, not requrng the computaton of the Fast Fourer Transform (FFT). Another approach to smplfy the ftness evaluaton functon s to consder only frequency ponts nsde the passng band. We ran another experment focusng only on the crcut s gan at a partcular frequency. A sne wave wth ampltude equal to 1Vpp and 2kHz frequency was appled to the crcut s nput. The ftness evaluaton functon used as target output sgnal a 2Vp-p and 2kHz sne wave. The GA manpulated the connectons of only one operatonal amplfer. The frequency response of the obtaned crcut s plotted n Fgure 1. It can be seen that the crcut behaves as a band pass flter, wth maxmum gan of 2.2, at the frequency of 2kHz. In ths case, the amplfer gan s lmted by feedback connectons. Even though the ftness evaluaton functon does not enforce partcular values of cutoff frequences, we observed that most of the evolved solutons were narrow band amplfers.

9 Output (Volts) Gan Tme (1/9kHz) 5, 1, 15, 2, 25, 3, 35, 4, 45, 5, Frequency(Hz) (A) (B) Fgure 9 Intrnsc evoluton - Comparson between the evolved crcut (lne) and target flter (ponts) response: (A) tme doman; (B) frequency doman Gan , 1, 15, 2, 25, 3, 35, 4, 45, 5, Frequency (Hertz) Fgure 1 - Gan of the amplfer obtaned n the experment usng the programmable analog chp. Each GA executon of the experments reported above lasted around 4 hours. Most of the executon tme s domnated by the chp downloadng tme. The genotypes are around 5 bts long, resultng n a very large genome space (~1 15 ), compared to the extrnsc experments. One of the most mportant aspects n ntrnsc evoluton s the fact that the evolved crcuts wll always work n realty. However, the use of smulators, n the context of extrnsc evoluton, provdes a more straghtforward way to accomplsh multple performance measures, such as frequency response, dsspaton and output nose. 6 Conclusons Ths work nvestgated the applcaton of genetc algorthms to the synthess of an actve flter for AM band. A new method to aggregate multple specfcatons nto the ftness functon was ntroduced. Four objectves have been addressed n our experments: frequency response; power consumpton; symmetrc excurson; and output nose. The search space sampled n these experments has been lmted, by restrctng the number of possble component values. Conversely, we have followed a partcular strategy for desgn mprovement, consstng of nspecton and small changes n the evolved crcut. The authors are currently studyng problems of mpedance couplng between the trap (L a and C a ) and the evolved amplfer, by ncludng ths as another objectve of the evolutonary process. The results are encouragng when compared to conventonal crcuts [Sansen98]. In another set of experments, a programmable analog chp was appled to the ntrnsc evoluton of flters. The authors have been nvestgatng dfferent approaches for ftness evaluaton; currently, the use of the FFT s beng studed.

10 Acknowledges The authors wsh to thank CAPES, brazlan federal agency, and Motorola for the support; and Dr. Adran Thompson and Mr. Paul Layzell, from the Unversty of Sussex, for the mportant suggestons. References: [Blckle96] - Blckle, T., Theory of Evolutonary Algorthms and Applcaton to System Synthess, Doctoral Thess, Swss Federal Insttute of Technology, Zurque, [Churchland92] - Churchland, P. S., Sejnowsk, T. J., The Computatonal Bran, MIT Press, [Fonseca95] - Fonseca, C. M, Flemng P. J., An Overvew of Evolutonary Algorthms n Multobjectve Optmzaton, Evolutonary Computaton, MIT Press, Vol. 3, No.1, pp.1-16, [Goldberg89] - Goldberg, D., Genetc Algorthms n Search, Optmzaton and Machne Learnng, Addson-Wesley Publshng Company, Inc., Readng, Massachusetts, [Grmbleby95] - Grmbleby, J. B., Automatc Analogue Network Synthess Usng Genetc Algorthms, Proceedngs of the Frst IEE/IEEE Internatonal Conference on Genetc Algorthms n Engneerng Systems (GALESIAS - 95), pp.53-58, UK, [Holland75] - Holland, J. H., Adaptaton n Natural and Artfcal Systems: An Introductory Analyss wth Applcatons to Bology, Control, and Artfcal Intellgence. Ann Arbor, MI: Unversty of Mchgan Press Second edton. Cambrdge, MA: The MIT Press [Johns97] - Johns, D. A., Martn, K., Analog Integrated Crcut Desgn, John Wley & Sons, Inc., [Koza98] - Koza, John R., Bennett III, Forrest H, Andre, Davd, Keane, Martn A, and Dunlap, Frank, Automated synthess of analog electrcal crcuts by means of genetc programmng, IEEE Transactons on Evolutonary Computaton. 1(2). pp , [Laker94] - Laker, K. R., Sansen, W., Desgn of Analog Integrated Crcuts and Systems, Mc. Graw- Hll Inc., 1994 [Layzell98] Layzell, P., A New Research Tool for Intrnsc Hardware Evoluton, n Proceedngs of the Second Internatonal Conference on Evolvable Systems: From Bology to Hardware (ICES98), Lausanne, Swtzerland, September, 23-26, M.Spper, D.Mange and A. Pérez-Urbe (edtors), vol. 1478, pp , LNCS, Sprnger-Verlag, [Lohn98] Lohn, J. D., Colombano, S. P., Automated Analog Crcut Synthess Usng a Lnear Representaton, n Proceedngs of the Second Internatonal Conference on Evolvable Systems: From Bology to Hardware (ICES98), Lausanne, Swtzerland, September, 23-26, M.Spper, D.Mange and A. Pérez-Urbe (edtors), vol. 1478, pp , LNCS, Sprnger-Verlag, [Motorola97] - Motorola Semconductor Techncal Data, Advance Informaton Feld Programmable Analog Array 2-Cell Verson MPAA2, Motorola, Inc., [Sansen98] - Sansen, W., Low-Nose Wde-Band Amplfers n Bpolar and CMOS Technologes, n notes on the course: Low-nose Analog CMOS &BCMOS Desgn, promoted by the Imperal College, February, 25-27, London, 1998 [SMASH93] - SMASH User and Reference Manual, Dolphn Integraton, France, [Stoca98] - Stoca, A., Fukunaga, A., Hayworth, K., Salazar-Lazaro, C., Evolvable Hardware for Space Applcatons, n Proceedngs of the Second Internatonal Conference on Evolvable Systems: From Bology to Hardware (ICES98), Lausanne, Swtzerland, September, 23-26, M.Spper, D.Mange and A. Pérez-Urbe (edtors), vol. 1478, pp , LNCS, Sprnger-Verlag, [Thompson98] Thompson, A., On the Automatc Desgn of Robust Electroncs Through Artfcal Evoluton, n Proceedngs of the Second Internatonal Conference on Evolvable Systems: From Bology to Hardware (ICES98), Lausanne, Swtzerland, September, 23-26, M.Spper, D.Mange and A. Pérez-Urbe (edtors), vol. 1478, pp , LNCS, Sprnger-Verlag, [Zebulum98] - Zebulum, R.S., Pacheco, M.A., Vellasco, M., Analog Crcuts Evoluton n Extrnsc and Intrnsc Modes, n Proceedngs of the Second Internatonal Conference on Evolvable Systems: From Bology to Hardware (ICES98), Lausanne, Swtzerland, September, 23-26, M.Spper, D.Mange and A. Pérez-Urbe (edtors), vol. 1478, pp , LNCS, Sprnger-Verlag, [Zebulum99] - Zebulum, R.S, Pacheco, M.A., Vellasco, M., A Novel Mult-Objectve Optmsaton Methodology Appled to the Synthess of CMOS Operatonal Amplfers, n revew by the Journal of Sold-State Devces and Crcuts, ISSN , edted by Wlhelmus Van Noje.