Integrated chaos generators

In eg a ed Chaos Gene a o s MANUEL DELGADO-RESTITUTO, MEMBER, IEEE AND ANGEL RODRÍGUEZ-VÁZQUEZ, FELLOW, IEEE In i ed Pape Thispape su eys hedi e en designissues, omma hema ical model o silicon, in ol ed on he design o in eg a ed ci cui s o he gene a ion o chao ic beha io . Keywo ds—Analog CMOS, chaos, design me hodology, design sys em, nonlinea ci cui design. I. INTRODUCTION The design o elec onic ci cui s wi h cus omized con- ollable chao ic beha io has po en ial in e es in many applica ion scena ios such as ins umen a ion, analog signal p ocessing, and communica ion and anging sys ems. Rega ding ins umen a ion, chao ic ci cui s ep esen an e icien al e na i e o non epea able pseudo andom signal gene a ion. Such gene a o s a e use ul o he implemen a- ion o noise sou ces—bo h whi e and colo ed—which a e equen ly employed a speech p ocessing [1] and o es ing he dynamic beha io o elec onic sys ems [2], among many o he applica ions [3]. On he o he hand, chaos gene a o s can be used in analog signal p ocessing applica ions as a di he sou ce o imp o e he pe o mance o o he blocks. Fo ins ance, di he ing can be used o whi en he noise loo o modula o s, as well as o educe he (idle channel) spu ious ones, which a e in oduced du ing quan iza ion o di ec cu en (dc) inpu s (audible in oice-band appli- ca ions) [4], [5]. Also, di he ing can be used o imp o e he in eg al nonlinea i y o high-pe o mance Nyquis - a e analog- o-digi al con e e s [6]. In ano he applica ion, chaos gene a o s can be used, oge he wi h ce ain dynamic elemen ma ching mechanisms, o make digi al- o-analog e o s a e age o ze o o e mul iple sample ins ances [7]. Manusc ip ecei ed July 6, 2001; e ised Decembe 12, 2001. This wo k was suppo ed in pa by he C.I.C.Y.T, Spain, unde G an 1FD97-1611(TIC), in pa by he Spanish P.R.O.F.I.T. P ojec AFIN (FIT-070000-2001-843), and in pa by he EC P ojec INSPECT (ESPRIT 31103). The au ho s a e wi h he Ins i u e o Mic oelec onics o Se ille–Na ional Cen e o Mic oelec onics (IMSE-CNM), Se ille 41012, Spain (e-mail: [email p o ec ed]). Publishe I em Iden i ie S 0018-9219(02)05238-6. In anging sys ems, he nonpe iodici y o chao ic signals, as well as he apid deco ela ion o hei ime-shi ed sequences, make he use o chaos an in e es ing coding echnique o high esolu ion ada sys ems [8]. Finally, chao ic ci cui s play a p ominen ole in chaos-based digi al communica ion sys ems as hey supply he equi ed sample unc ions o which in o ma ion symbols a e mapped o [9]. In hese sys ems, chaos gene a o s, ins ead o con en ional equency syn hesize s, p o ide he communica ion ca - ie s, which a e modula ed by he digi al in o ma ion ha is ansmi ed. Inhe en o his chao ic modula ion, he digi al in o ma ion also expe imen s a bandwid h sp eading as a consequence o he wideband and noise-like spec al p op- e ies o chaos. This capabili y o simul aneous modula ion and sp eading, wi h an a p io i lowe sys em complexi y han adi ional sp ead spec um echniques, is dese ing a conside able esea ch in e es du ing he las yea s. In he a o emen ioned applica ions, chao ic ci cui s can be ealized by in e connec ing disc e e in eg a ed ci cui (IC) componen pa s on a p in ed ci cui boa d. Howe e , whene e sys em minia u iza ion and/o powe consump ion a e issues, chao ic ci cui s mus be ealized as monoli hic ICs, p e e ably in s anda d complemen a y me al–oxide–semiconduc o (CMOS) echnologies whe e hey can be embedded wi h o he digi al and analog ci cui y. The objec i e o his pape is, indeed, o su ey he di e en design echniques, bo h a sys em and ci cui le els, in ol ed in he monoli hic ealiza ion o chao ic ICs. Though he design o chao ic gene a o s can be a o ded om di e en pe spec i es as, o ins ance, by adjus ing he pa ame e s o well-known oscilla o s o phase-locked loop s uc u es[10], hispape ocusesonasys ema ics a e-space app oach, which lead o mo e gene al solu ions, based on he elec onic syn hesis o he sys em s a e equa ions. Following his app oach, Sec ion II e iews he ma hema ical models leading o chao ic beha io and iden i ies he basic building blocks equi ed o hei implemen a ions. They a e classi- ied in o linea (co e ed in Sec ions III and IV) and non- linea (desc ibed in Sec ion V) ope a o s. Finally, Sec ion VI 0018-9219/02$17.00 © 2002 IEEE PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 747 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. p esen s h ee chao ic IC p o o ypes which illus a e he ap- plica ion o he me hodological aspec s and ci cui concep s p e iously desc ibed. II. MATHEMATICAL MODELS FOR CHAOS GENERATION E e y ma hema ical model able o p oduce chao ic be- ha io has wo basic ing edien s: dynamics and nonlinea i y. Rega ding dynamics, models o chaos gene a ion can be classi ied in o disc e e- ime o con inuous- ime, depending on whe he he sys em e olu ion is desc ibed by nonlinea di e ence o di e en ial equa ions, espec i ely. Ano he possible classi ica ion is be ween au onomous o nonau- onomous sys ems, which depends on whe he he gene a o is able o no o sel -sus ain chao ic oscilla ions wi hou any ex e nal d i ing exci a ion. Because his las classi ica ion has a weak impac ega ding IC implemen a ion, we will ocus exclusi ely on he au onomous case. In he ollowing, we will sepa a ely e iew he basic ea- u es o disc e e- ime and con inuous- ime chaos gene a o s, iden i ying he basic ope a ions needed o hei syn hesis. Asal eadymen ioned,asys ema ics a e-spaceapp oachwill be used as he heo e ical amewo k o exp ess (and la e o implemen ) he di e en chao ic sys ems. A. Disc e e-Time Chaos Gene a o s Au onomous disc e e- ime sys ems (o disc e e maps, in sho )canbegene ally desc ibedby he ollowing h(delay) o de -dimensional ( -D) ini e-di e ence equa ion (FDE): (1) whe e symbolizes he disc e e- ime a iable, ep esen s he s a e ec o o he sys em a he h disc e e ime ins an , and is a -D ime-in a ian nonlinea ec o ield ha depends on he pa- ame e se . Fo he pu poses o signal gene a ion, we will assume ha sys em (1) is cha ac e ized by an in a ian se unde , such ha any ajec o y s a ing in emains con ined oi .Addi ionally, hemodelmayalsoincludea -D ou pu equa ion de ined in e ms o he mos ecen s a es o he sys em (2) whe e is he ou pu ec o o he disc e e map a he h ins an and is a unc ion, in gene al, non- linea and pa ame e ized by a ec o . Among he disc e e maps de ined by (1), i s -o de sys- ems ( ) play a majo ole as hey model mos o he elec onic chaos gene a o s p oposed so a . Thei s a e equa- ion may be w i en as (3) whe e , , , and is a nonlinea ime-in a ian ec o ield ( and ). Fig. 1 shows a block diag am o i s -o de disc e e-maps comp ising a linea sec ion, a nonlinea unc ion block connec ed in a (a) (b) Fig. 1. (a) Block diag am o a i s -o de FDE-based chaos gene a o . (b) Ope a ions encompassed in he 1 block [elemen in he inse o Fig. 1(a)]. Table 1 Sho Ca alog o Chao ic Disc e e Maps eedback loop [11] and an ou pu s age. The linea sec ion, included in he dashed box o Fig. 1(a), consis s o s a ic and dynamic elemen s. The s a ic elemen s ealize he ope a ions o summa ion and scaling (blocks labeled and ). On he o he hand, he dynamic elemen pe o ms sample-and-hold (S/H) and delay ope a ions, as shown in Fig. 1(b). Usually, such elemen is implemen ed by a single elec onic de ice which, he ea e , will be ep esen ed by he symbol in he inse o Fig. 1(a) and deno ed as delay elemen . The clock signal ixing he sampling pe iod o he delay elemen de e - mines he i e a ions o he eedback loop. Table1con ainsasho ca alogo i s -o de disc e emaps which ha e been implemen ed in elec onic o m, ei he by means o disc e e componen s o in eg a ed on silicon. Fo 748 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. Fig. 2. Block diag am o an ODE-based chaos gene a o . each en y, Table 1 shows he pa icula se ings o and ,acco ding o(3). Thede ini ionin e alo hemaps and hei pa ame e anges o achie e chao ic egime can be oundin he e e encesa ached o he i s columno Table1. An impo an conclusion ha can be d awn om Table 1 is ha chao ic beha io s can be ob ained om e y simple ma hema ical models. Indeed, a single s a e- a iable is e- qui ed o gene a e chaos, as occu s in he 1-D maps lis ed in he i s eigh ows o Table 1. The eby, simple monoli hic ealiza ions can be expec ed om he use o disc e e maps. In spi e o his s uc u al simplici y, he dynamic beha io o he sys em can be ex emely ich and complica ed. This will be illus a ed in Sec ion VI by means o he Be noulli map de ined in he i h ow o Table 1. B. Con inuous-Time Chaos Gene a o s As al eady men ioned, con inuous- ime chaos gene a o s a e hose ha can bedesc ibedbynonlinea di e en ialequa- ions.Among hem, we can u he dis inguishbe ween hose based on o dina y di e en ial equa ions (ODEs) and hose based on delay-di e en ial equa ions. The la e ha e been ecen lyp oposedas ane icien me hod o hegene a iono high ac al dimension chaos wi h no subs an ial inc ease on complexi y (a i s o de sys em is enough o p oduce chao ic beha io ) [24]. Ne e heless, hese sys ems a e s ill a om being well unde s ood and we will ocus on ODE-based sys- ems, o which a lo o esea ch has been done in he las decades. Au onomous con inuous- ime ODE-based chaos gene a- o s belong o he space o -D dynamical sys ems wi h nonlinea elemen s, de ined by he s a e equa ion (4) whe e is a diagonal ma ix de ining he ime-in eg a ion cons an s o he sys em, is he s a e ec o , , , , and is a nonlinea ec o ield (and ). Such sys ems can be mapped on o he analog compu e concep o Fig. 2. I con- sis s o a o wa d pa h con aining a linea ime-in a ian sub- sys em(includedin hedashedboxo Fig.2),a eedbackpa h including he nonlinea elemen s o , and an addi ional pa h o syn hesize he ou pu ec o .As Table 2 Ca alog o ODE-Based Au onomous Chao ic Oscilla o s can be seen, he only di e ence be ween he concep ual dia- g am in Fig. 2 and ha associa ed o i s -o de disc e e maps inFig.1(a)is heuseo in eg a o sins eado delayelemen s. This appa en ly mino change has, howe e , s ong implica- ions ega ding sys em design, as will be shown nex . Table 2 includes some exempla y ODE-based chao ic sys ems ound in he li e a u e. Condi ions on he di e en sys em pa ame e s o gua an ee chao ic beha io can DELGADO-RESTITUTO AND RODRÍGUEZ-VÁZQUEZ: INTEGRATED CHAOS GENERATORS 749 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. be ound in he e e ences a ached o he i s column o Table 2. The emaining columns indica e, espec i ely, ma ices , and he elemen s o he ec o ield in acco dance o he s a e ep esen a ion in (4) (in all cases, is a null ec o and is he iden i y ma ix). Ci cui demons a o s using o - he-shel disc e e elec onic de ices ha e been epo ed o all he examples in Table 2 and hose in ows 3, 7, 9, and 12 ha e been also implemen ed in monoli hic o m. Table 2 e eals a well-known ac : in au onomous ODE-based sys ems, h ee s a e a iables a e a leas e- qui ed o gene a e chaos i he nonlinea eedback pa h in Fig. 2 is memo yless. O he wise, i he ec o ield exhibi s hys e esis, as occu s in he las ow o Table 2, he jumps in he hys e e ic elemen s co espond o addi ional s a es [36]. This is clea ly in con as wi h disc e e maps o which single s a e- a iable sys ems a e enough o p oduce chao ic beha io . C. Gene al Conside a ions o he Design o Chao ic ICs In he p e ious wo sec ions,bo h he a chi ec u es and op- e a ions equi ed o he sys ema ic design o chaos gene a- o s using a s a e-space app oach ha e been iden i ied. One s ep ahead is o apply he app op ia e ans o ma ions on he ma hema ical models o make hem sui able o syn hesis in monoli hic o m. Such modi ica ions mus conside wo di e en aspec s ha a e ela ed, on he one hand, o he pa icula nonlinea ec o ield and,on heo he , o heo e alls a eequa ion o he dynamical sys em [de ined by (3) o disc e e maps o (4) o ODE-based gene a o s]. Fi s , le us conside he nonlinea ec o ield. The syn hesis o a bi a y nonlinea unc ions in IC o m can be achie ed by elying o sys ema ic ep esen a ion echniques whe e ope a o s a e closely ela ed o he nonlinea i ies a ailable a he design p imi i es (de ails a e gi en in Sec ion V). Ne e heless, o he sake o eliabili y and also o educe he ha dwa e complexi y o he design (and, hence, i s a ea and powe consump ion), nonlinea ec o ields should be made as “p imi i e-based” as possible in o de o educe he numbe o such elemen a y ope a o s. I is, he e o e, s ongly sugges ed o p ope ly al e he nonlinea ec o ield (i i de ia es oo much om a simple p imi i e-based ep esen a ion) while e aining he mos ele an ea u es o he a ge ed dynamic beha io . In pa icula , simpli ica ion s a egies based on piece- wise-linea (PWL) modeling a e specially appealing o IC ealiza ion because o he accu acy and simplici y o hei syn hesis—i is ul ima ely based on he con oled ansi ion be ween he ON and OFF s a es o ansis o s, as nonlinea p imi i e ope a o . An example o piecewise linea iza ion is gi en by he ODE-based sys ems in he ows 8 and 9 o Table 2, in which mul iplie s a e eplaced by simple PWL nonlinea i ies, namely, sign in e sion and absolu e alue ope a ions. Ano he ad an age o PWL modeling, in pa icula o high-accu acy IC implemen a ions, is ha he dynamical sys em becomes linea a each egion o he space pa i ion and, hence, well-de ined calib a ion [38], [39] and Fig.3. Annihila ion o chao icdynamics in he en map o B =2 . uning [40] mechanisms a e eadily applicable o p ecisely im each o he a ine cha ac e is ics. Ano he impo an issue o he choice o an IC-sui able nonlinea ec o ield is he obus ness o he sys em dy- namics [22], [42]. Because o he limi ed accu acy o analog ci cui implemen a ions, models o chaos gene a o s mus be obus enough so ha he una oidable echnological pa- ame e de ia ionsdono se e elydeg ade he p esc ibeddy- namic ea u es. A main consequence o his ac is ha some nonlinea i ies, which a e o en ound in heo e ical s udies, mus be p ecluded o elec onic chaos gene a ion, unless hey a e con enien ly ans o med. A ypical example is o - e ed by he en map, de ined in he ou h ow o Table 1. In o de o ob ain a uni o m dis ibu ion o he chao ic ime-se- ies, pa ame e is se o 2, as illus a ed in Fig. 3. In his con igu a ion, i o some ci cui impai men o noise con- ibu ion, he ajec o y jumps ou side he nominal in a ian se (shaded a ea in Fig. 3), he sys em e ol es a e a an- sien o he pa asi ic equilib ium poin , which a ises om he sa u a ion cha ac e is ics o he ci cui (long-dashed ec- angle in Fig. 3). As a esul , he chao ic beha io anishes and he nominal in a ian se collapses o he s able ixed poin .Toa oid hissi ua ion, hemap mus be ans o med so ha i exhibi s a basin o a ac ion la ge han i s nom- inal in a ian se , wi h a clea ance be ween hem de e mined by he maximum expec ed pe u ba ions in he ci cui im- plemen a ion. Di e en s a egies o achie e his goal can be ound in [15], [22], [41], [42]. Le us, now, conside he o e all s a e equa ion o he chao ic sys em. Fo simila easons o eliabili y and cos , i should be simpli ied be o e implemen a ion. This can be accomplished by, i s , de ining a amily o dynamical sys ems ha e ains almos all ea u es o he a ge ed model and, second, by iden i ying which elemen o such amily is he mos con enien om an IC pe spec i e. Essen ial o he i s s ep is he concep o linea conjugacy,1among dynamical sys ems [43], [44], as i gua an ees ha bo h he 1Two dynamics sys ems F ( 1 ) and H ( 1 ) a e said o be linea ly conjuga ed i he e exis s a nonsingula ma ix M such ha M  F = H  M (“  ” deno es composi ion). 750 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. o iginal model and he elemen s o i s linea ly conjuga ed amily exhibi he same quali a i e dynamics. In e es ingly enough, i has been shown ha o a wide class o dynamical sys ems, namely, hose which can be ep esen ed in Lu ’e o m,2linea conjugacy be ween sys ems wi h he same ec o ield is assu ed whene e he eigen alues o co esponding ma ices , and , a e iden ical [44]. This implies ha he amily o linea ly conjuga ed Lu ’e o ms buil upon a gi en nonlinea ec o ield can be exac ly de ined by less pa ame e s han hose nominally included in he ep esen a ions (3) and (4)— oge he wi h ec o . Hence, he e exis in ini ely many linea ly conjuga ed elemen s able o ep oduce he same quali a i e dynamics as he o iginal model, which allows one o es ablish a selec ion p ocedu e aimed o de e mine ha elemen mos sui able o IC ealiza ion.3Some ailo ing c i e ia o his selec ion p ocedu e a e [32]. 1) Low Complexi y: Because sys em pa ame e s mus be mapped in o physical de ices, hose ec o ields wi h a minimum numbe o di e en nonze o en ies in , and — hey a e e e ed o as canonical elemen s—a e a p io i he bes sui ed in e ms o a ea and powe consump ion. In pa icula , hose con igu a ions wi h p opo ional o a uni a y ec o a e p e e ed because he ec o ield exhibi s a single nonlinea block. 2) Op imum Dynamic Range: The dynamic ange o a chaos gene a o is maximized as long as all i s s a e a iables a e able o swing up o a maximum ole able le el imposed by he powe supply o he ci cui [1], [47]. The p ocedu e by which his maximiza ion can be achie ed is scaling and basically consis s on ap- plying a con enien simila i y ans o ma ion on he s a e ec o . I is wo h poin ing ou ha scaling does no a ec he sys em a chi ec u e (null en ies o ma ices and emain unal e ed a e scaling), bu he canonical p ope y o he o iginal sys em may be los , i.e., sys em pa ame e s, ini ially wi h iden- ical magni ude, u n o be di e en a e scaling, hus leading o an inc ease on he sys em complexi y. 3) Reduced Misma ch: Ra io accu acy (o ma ching) o simila componen s is enhanced as long as ci cui elemen s a e buil by eplica ing a gi en uni a y de ice [48]. Thus, i sys em pa ame e s a e ela ed by in ege a ios, he IC imp o es in accu acy and, a he layou le el, in modula i y and in eg a ion densi y. This imp o emen , howe e , educes as he sp ead o sys em pa ame e alues inc eases [48]. Thus, he uni a y elemen s eplica ion app oach mus be accompanied, in some cases, by echniques aiming 2Dynamical sys ems in Lu ’e o m a e sys ems de ined by (3) and (4) in which he ec o ield ( 1 ) , assumed memo yless, depends on w x , whe e w 2< . Fo ou pu poses, i will u he assumed ha Lu ’e o ms a e obse able in he classical sense o con ol heo y [45]. 3I is wo h no ing ha mul idimensional PWL ep esen a ion wi h pa - allel bounda y planes [46] can be also exp essed in Lu ’e o m and, hence, hey a e also sui able o sys em le el op imiza ion—an addi ional ad an- age on he use o PWL models o chaos gene a ion. o educe he sp ead o sys em pa ame e s [1]. Once again, his can be achie edby using a p ope simila i y ans o ma ion on he s a e a iables. A inal (and c i ical) sys em-le el conside a ion ha mus be add essed on he design o chaos gene a o s is o e alua e he ole ance o he dynamic beha io agains pa ame e de- ia ions. Such de ia ions a e due o he ac ha physical ci cui componen s (e.g., capaci o s, ope a ional ampli ie s, compa a o s, e c.) de ia e om nominal alues o design in en because o a a ie y o nonideali ies which can be g ouped in o h ee main ca ego ies, namely, noise, s a ic, and dynamic [39]. Noise ca ego y basically comp ises he e - o s due o he mal noise gene a ed by solid-s a e de ices. On he o he hand, misma ch o ideally iden ical de ices, which esul s om uncon olled echnological pa ame e s in he ab ica ion p ocess, and dc- ela ed e o s such as o se , signal-independen cha ge injec ion, and ini e dc gain o ac- i e componen s can be g ouped as s a ic nonideali ies. Sa u- a ion cha ac e is ics ha esul om he upwa d limi ed dy- namic ange o he ci cui elemen s can be also seen as an s a ic nonideali y. Finally, dynamic e o s sums all equency dependen nonideali ies such as signal-dependen cha ge in- jec ion, limi ed dynamic accu acy in compa a o s, limi ed slew- a e, and limi ed gain-bandwid h p oduc in ampli ie s. In o de o ie he deg ada ion o he chao ic dynamics o he abo e nonideali ies, each o he e o sou ces mus be con enien ly modeled and inco po a ed in he nominal ep- esen a ions (3) o (4) [22], [32]. Then, a wo s -case anal- ysis, oge he wi h exhaus i e simula ions o he sys em in- cluding all nonideal e ec s, mus be made o de e mine he speci ica ions o he di e en building blocks o he a chi- ec u e. This b idges he sys em and ci cui le els in he de- sign ou e o he chaos gene a o . O cou se, he e may be cases in which he calcula ed block equi emen s a e beyond he limi a ions imposed by he echnological p ocess. This occu s ei he when he speci ica ions o he chaos gene a o (usually gi en in e ms o ou pu s a is ics) a e oo es ic i e o when he sys em a chi ec u e shows a la ge sensi i i y o some pa ame e a ia ions, making i imp ac ical o silicon implemen a ion. In his las case, i he ma hema ical model belongs o a amily o linea ly conjuga ed sys ems, a new el- emen ha is less sensi i e o pa ame e inaccu acies mus be ound. In gene al, he e is no a simple way o link chao ic sys em pe u ba ions and de ia ions on s a is ic pe o mance o he han by long- un simula ions. Only o PWL chao ic models, whe e he sys em beha es linea ly a each egion, a classical sensi i i y analysis [49] on he eigen alues pa - e n—which de e mines he quali a i e dynamics o he gen- e a o —wi h espec o he ci cui componen s can be use ul o es ima e how a he dynamic beha io de ia es om he nominal one [32]. D. Concluding Rema ks In his sec ion, we ha e explo ed di e en al e na i es o chaos gene a o s, gi en selec ion c i e ia o high-le el op i- miza ion, and iden i ied he basicope a ions in ol ed in hei implemen a ion. Such ope a ions can be classi ied be ween DELGADO-RESTITUTO AND RODRÍGUEZ-VÁZQUEZ: INTEGRATED CHAOS GENERATORS 751 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. (a) (b) (c) (d) Fig. 4. Basic concep s o he con inuous- ime dynamics. (a) Open-loop in eg a o . (b) Mille in eg a o . (c) Pa asi ics o in eg a ed capaci o s. (d) Fi s -o de equency-domain model o a ansconduc o . linea and nonlinea and wi hin he i s g oup, be ween dy- namic (con inuous- ime in eg a o s and delay elemen s) and s a ic (signal weigh ing and summa ion) ope a o s. In he ollowing sec ions, we will p esen some gene al ideas and concep s o he IC ealiza ion o hese ope a- ions, paying special a en ion o he nonideali ies which a ec hem, as hey ul ima ely de e mine he accu acy and ope a ion speed o he chaos gene a o . III. LINEAR OPERATORS FOR CONTINUOUS-TIME GENERATORS A. In eg a o s Because monoli hic induc o s a e only easible a e y high equencies,4capaci o s a e he basic dynamic p im- i i es o ODE-based chao ic ICs. S a e a iables a e, hence, ol ages and he dynamic upda ing o hese ol - ages is ealized by d i ing he s a e capaci o s h ough cu en s. Fig. 4 shows wo al e na i e implemen a ions o his dynamic upda ing: he open-loop [see Fig. 4(a)] and he Mille [see Fig. 4(b)] s uc u es. In bo h cases, he exci a ion is ob ained o con enience as he esul o a linea ol age- o-cu en ans o ma ion—using a anscon- duc o — om an in e media e ol age , i.e., . Ideally, bo h ci cui s ob ain (5) which co esponds o he beha io o an in eg a o wi h nom- inal ime cons an ( s ands o he h s a e a iable o he sys em). The di e ences be ween hese al e na i e ealiza ions a ise when pa asi ics a e accoun ed. In he o egoing anal- ysis, conside ed pa asi ics a e he ollowing. 4In e es ingly enough, some (in eg a able) classical oscilla o s based on passi e esonan ci cui s, such as he Colpi s oscilla o [28], can exhibi chao ic beha io upon p ope pa ame e se ing, hus gi ing he possibili y o gene a ing chao ic signals in he gigahe z ange. Table 3 Time-Cons an E o and App oxima ed Poles o he Open-Loop and Mille S uc u es 1) Those associa ed wi h he capaci o [see Fig. 4(c)], consis ingo woaddi ionalcapaci o s(bo omand op pla es). 2) The i s -o de small-signal pa asi ics o he anscon- duc o , namely: ou pu esis ance , ou pu capaci ance , and equency-dependen anscon- duc ance [see Fig. 4(d)]. 3) The small-signal pa asi ics associa ed o he op-amp. Ob iously, hese a e dependen on he op-amp a chi- ec u e. He e, we assume ha he op-amp is in e nally compensa ed, has low ou pu impedance (negligible o analysis pu poses), and can be modeled as [48] (6) Fi s o all, no e ha in he s uc u es o Fig. 4(a) and (b), he capaci o e minal labeled is connec ed o a low- impedance poin (a poin whe e he ol age changes only sligh ly o la ge cu en anges). In Fig. 4(a), he e minal is di ec ly connec ed o an al e na ing cu en (ac) g ound, while in Fig. 4(b), he low-impedance ea u e is achie ed by heop-amp ou pu node.Consequen ly, he wos uc u esa e insensi i e o , i.e., he pa asi ic has i ually no in luence on he ci cui beha io .5Le us now sepa a ely analyze he ci cui s o Fig. 4(a) and (b). In he s uc u e o Fig. 4(a), he pa asi ic capaci o s and a e connec ed in pa allel wi h he nominal capaci o . This makes he in eg a o ime cons an o de ia e om i s nominal alue as , whe e he ime-con- s an e o is gi en in Table 3. In addi ion, he pa asi ic esis ances connec ed o he node p oduce losses in he in eg a ion and, hence, he dynamic beha io de ia es om he nominal one ep esen ed by . The ac ual ans e unc ion is (7) whe e is he low- equency pole c ea ed by he pa allel connec ion o and (see Table 3) and 5This is no exac ly ue as his capaci o may in luence he ansien e- sponse o he op amp, especially when he op amp has a single-s age a chi- ec u e [48]. 752 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. Fig. 5. Equi alen ci cui o he analysis o he Mille con igu a ion. ep esen s he ansconduc o equency esponse.6 The ans e unc ion models he so-called dynamic e o o he in eg a o . This e o is negligible only o hose equencies, whe e . Neglec ing a his poin he in luenceo he ansconduc o , hese equenciesa ede ined by . Conside now he Mille con igu a iono Fig.4(b).To i s o de , he subci cui o med by he op amp and he capaci o can be ep esen ed by he equi alen ci cui a he igh o node in Fig. 5—ob ained by applying he Mille heo em. Analysis o his ci cui ob ains (8) which con ains wo poles a and , espec i ely, and displays ime cons an e o s in he passband . Assuming ha and a e la gely sepa a ed and ha ,oneob ains hepole exp essionsshowninTable3.Wi hin hepassband equency ange, whe e he ci cui ope a es as an in eg a o , (8) can be app oxima ed by (9) hus leading o he exp ession o gi en in Table 3. I shows ha he ime cons an e o is in e sely dependen on he op-amp dc gain and, hence, e y small. Compa ing he Mille and he open-loop con igu a ions, he ollowing conclusions can be d awn. 1) In he Mille con igu a ion, he ime cons an e o is a enua ed by . Hence, he Mille in eg a o exhibi s supe io pe o mance ega ding he in luence o he pa asi ic capaci ances. I is a consequence o he ac ha , in he passband, he op amp exhibi s e y small inpu esis ance gi en by , which domi- na es o e o he impedances connec ed o his node.In he limi , as , his esis ance becomes null and he op-amp inpu becomes a i ual g ound. 2) The low- equency co ne o he passband, gi en by , is much smalle o he Mille han o he open loop. In he la e , he ou pu conduc ance man- i es s as such in he exp ession o , while, o he 6To i s -o de analysis, he equency dependence o ansconduc ances can be modeled by using a single pole T ( s )  (1 + s=! ) . This model can be alid o equencies up o ens o megahe z. Fo mo e de ailed models, see [50]. Mille con igu a ion, i mani es s a enua ed by . This is ano he posi i e consequence o eedback. 3) The high- equency co ne is smalle o he Mille con igu a ion—a nega i e consequence o eedback. In he open-loop con igu a ion, he high- e- quency beha io is limi ed by he dynamic esponse o he ansconduc o , while in he Mille one, i is also limi ed by . Assuming ha he op amp and he ansconduc o a e op imized, i is likely ha he la e exhibi s a e- quency ange wide han , hus, in e ing poo e equency esponse o he Mille con igu a ion han o he open-loop one. Summa izing, hep e iousanalysisshows ha heopen-loop con igu a ion is p e e able o high- equency applica ions, hough i may equi e p edis o ion o compensa e o he ime cons an e o s. On he con a y, he Mille con igu a- ion is mo e app op ia e o low and medium equencies, e- qui ing no p edis o ion. No e, howe e , ha he deg ada ion o he equency esponse in he Mille s uc u e is mainly a consequence o he model used o he op amp. High- e- quency ad an ages o he open-loop s uc u e a e no so e - iden i cus om op amps wi hou in e nal compensa ion a e used [51]. In addi ion, equency esponse o Mille s uc- u e may pe haps be enhanced by ac i e compensa ion ech- niques [49] o p ope ly shape he in eg a o high- equency esponse and, hus, combine he ea u es o accu acy, small losses, and la ge equency bandwid h in o a single s uc u e. Ano he compa ison be ween he wo con igu a ions con- ce ns hei sui abili y o ICimplemen a ion.Speci ically, he ac ha ac g ounded capaci o s (i.e., hose ha ha e one o hei e minals ied oei he heposi i eo henega i epowe supply) a e be e sui ed han loa ing capaci o s. B. Signal Summa ion The ci cui s o Fig. 4(a) and (b) can be ex ended o pe - o m summing in eg a ion by ou ing all he ol age- o-cu - en ans o ma ion ou pu s (each associa ed wi h a summing e m) o node and le ing Ki cho cu en law (KCL) o wo k. In his way, he basic s uc u e o implemen (4), con- cep ually shown in Fig. 6(a), is de ined.No e ha e e y sum- ming e m has an ou pu conduc ance and an ou pu capac- i ance. Hence, a node , he equi alen conduc ance and capaci ance a e gi en, espec i ely, by (10) whe e and a e mean alues o he indi idual conduc ances and capaci ances, espec i ely, and is he numbe o exci a ions [acco ding o (4) ]. A e subs i u ing by and by in he exp essions o Table 3, we no ice ha inc eases p opo - ionally wi h o he open-loop con igu a ion. The same enla gemen is obse ed in he Mille in eg a o . Howe e , he whole e o o his con igu a ion is s ill a enua ed DELGADO-RESTITUTO AND RODRÍGUEZ-VÁZQUEZ: INTEGRATED CHAOS GENERATORS 753 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. Fig. 6. (a) Ob aining he s a e a iable upda ing cu en as he summa ion o M cu en componen s. (b) Using a second-gene a ion cu en con eyo o isola e he summing node om he s a e a iable node. (c) Concep o he ealiza ion o cu en con eyo s. by . As a coun e pa , he equency beha io o he open-loop in eg a o emains i ually unchanged, while he alue o o he Mille con igu a ion dec eases in e sely p opo ional o . A s a egy o a enua e he e o s caused by he summa- ion o signals is o isola e he node whe e he cu en s a e agg ega ed om ha whe e he esul ing cu en is applied o he s a e capaci o . This is ep esen ed in Fig. 6(b) o he open-loop con igu a ion, al hough i can be used wi h he Mille con igu a ion as well. The “glue” componen is a cu - en con eyo [52].Ac ually, hecu en con eyo inFig.6(b) is o he so-called second gene a ion, whose ideal beha io is desc ibed by (11) On he one hand, i c ea es a i ual g ound be ween he e minals and . On he o he , i ealizes a cu en ol- lowe ope a ion be ween he e minals and . Depending on he pola i y o he cu en ans e be ween he and e minals, he con eyo can be posi i e (CCII+) o nega i e (CCII-), which co espond espec i ely o heplus and minus signs in (11). In p ac ice, he inpu e minals o he cu en con eyo can be ealized by a anging wo MOS ansis o s in eedback con igu a ion a ound an op amp, as depic ed in he concep ual ci cui o Fig. 6(c). Then, he nega i e and he posi i e componen s o he inpu cu en can be oo ed o he ou pu node by using cu en mi o s [52]. Ob iously, he cu en con eyo p oduces new e o s ha mus be aken in o conside a ion o p ope design. Fi s -o de analysis o hese e o s can be ound in [53]. (a) (b) (c) Fig. 7. S uc u es o ol age- o-cu en con e sion in he case o (a) low ou pu esis ance, (b) loa ing sel -coduc o , and (c) g ounded sel -conduc o . C. Basic S a egies o Vol age- o-Cu en T ans o ma ion—Signal Weigh ing Along his sec ion, ol age- o-cu en ans o ma ion has been modeled h ough a ansconduc o , i.e., a com- ponen whose ou pu esis ance—modeled h ough in Fig. 4(d)—is la ge by cons uc ion. Also, he anscon- duc o inpu esis ance has been implici ly assumed in ini e and, consequen ly, loading e o s a he ansconduc o d i ing node ha e been dis ega ded. Howe e , in p ac ice, ol age- o-cu en ans o ma ion is some imes ealized using ci cui s whose inpu and/o ou pu esis ances a e no la ge by cons uc ion— o ins ance, MOS ansis o s ope a ing in he ohmic egion unde s ong in e sion [54]. Fo ans o ma ion ci cui s ha ing low inpu esis ance, he only way o a enua e loading e o s is d i ing he inpu node wi h low ou pu esis ance. On he o he hand, o hose ha ing low ou pu esis ance, he loading p oblems can be a enua ed by eso ing o one o he s uc u es o Fig. 7. In Fig. 7(a), he ou pu node is clamped a a ixed alue , hus annulling spu ious cu en con ibu ions o due o node ol age luc ua ions. On he o he hand, Fig. 7(b) and (c) is app op ia e whene e he ol age- o-cu en ans o - ma ion is ealized by exploi ing he sel -conduc ance o ei he an ac i e, i.e., composed o MOS an- sis o s, o a passi e esis o . O he impo an issues on he design o ol age- o-cu en ans o ma ion ci cui s a e b ie ly e iewed in he ollowing. P og ammabili y: I basically e e s o he possibili y o scaling ansconduc ances h ough elec ical con ol 754 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. a iables. Fo ins ance, he ansconduc ance o a MOS ansis o in he sa u a ion egime unde s ong in e sion depends on he la ge-signal ansconduc ance ac o and on he ga e ol age o e d i e (see Fig. 12 in Sec ion V o de ails). Two possible con olling scena ios, hence, a ise: 1) aking ad an age o he dependence o on ansis o geome y and o he MOS ansis o ope a ion as an analog swi ch o ealize digi ally con oled alues; 2) aking ad an age o he dependence on biasing con- di ions o ealize analog-con oled ansconduc ance alues. I is wo h poin ing ou ha p og ammabili y is he basic mechanism o signal weigh ing and, hence, o he imple- men a ion o he coe icien s o , and in (4). Linea i y: Ano he impo an issue ega ding signal weigh ingis ogua an eelinea i yo heo e allinpu –ou pu cha ac e is ics. Because p imi i e componen s a e essen- ially nonlinea (see Fig. 12), linea i y on ansconduc ances mus be achie ed by p ope ly combining di e en elemen s. Many di e en s a egies ha e been p oposed o nonlinea cancella ion as, o ins ance, by using di e en ial con ig- u a ions, by applying eedback, h ough in e se unc ion echniques, e c. Some o hese s a egies a e e iewed in [52]. Scaling Fac o Accu acy: Scaling ac o accu acy has wo aces: absolu e accu acy and a io accu acy. The o me e e s o exac ness in absolu e alues o ansconduc ances and has ole ances o a ound 30%. Absolu e accu acy is impo an in cases whe e iming is ele an . In hese cases, a uning mechanism mus be inco po a ed o he ci cui o educe he ole ances o abou 1%–2% [40], [49]. On he o he hand, a io accu acy, which was al eady con- side ed as a selec ion c i e ia in Sec ion II-C, can be made qui e good—up o 0.1%—depending on he de ice a eas, shapes, and dis ances [48]. IV. LINEAR OPERATORS FOR DISCRETE-TIME GENERATORS The implemen a ion o delay elemen s o disc e e maps always elies on he use o capaci o s o s o ing and e- ie ing in o ma ion in he o m o ol ages, swi ches o cha ging and/o discha ging capaci o s in esponse o a con- ol signal, and ac i e de ices o de ining he condi ions o cha ge ans e . Main di e ence among analog sampled-da a echniques come om he physical a iables which is ul i- ma ely used o con ey he in o ma ion. Such a iables can be in he o m o ol ages [swi ched-capaci o (SC) ech- nique [11]), cu en s [swi ched-cu en (SI) echnique [16]], iming cha ac e is ics o a pulse ain (pulsewid h [13] o pulse-posi ion modula ion echniques [17]), o phase angles (phase-locking echnique [14]), among o he possibili ies. In his pape , we will ocus on SC and SI echniques. The SC echnique equi es op amps, as ac i e de ices, and linea capaci o s, as holding elemen s [1], [56], [57]. High-quali y capaci o s (high linea i y, educed ol age, and empe a u e dependence, and good ma ching p ope ies) a e a ailable in echnologies ha o e pa allel-pla e s uc u es sepa a ed by hin oxide [58], [59]. I such s uc u es a e no a ailable, as in pu e digi al CMOS echnologies, capaci o s a ecommonlyimplemen ed byexploi ing he hin-oxidega e capaci ance o MOS ansis o s [60]. MOS-based capaci o s usually exhibi la ge capaci ance pe uni a ea and be e ma ching han pa allel-pla e s uc u es, bu su e om sig- ni ican nonlinea i ies and pa asi ic capaci ances, and mus be con enien ly biased o gua an ee a low- esis i i y con- duc ing laye unde he ga e. As a esul , SC ci cui s buil on digi al echnologies ha e ine i ably poo e pe o mance han hose implemen ed on analog-o ien ed p ocesses. An al e na i esampled-da a app oach ha a oids he need o highly linea capaci o s is he SI echnique [61]. In his case, capaci o s a e simply o med by he inpu pa asi ics o ansconduc o s, hus, ende ing he app oach specially appealing o s anda d digi al p ocesses. Un o una ely, his no able simpli ica ion is a he expense o pe o - mance deg ada ions. Ne e heless, in applica ions equi ing mode a e accu acy, he complexi y and a ea consump ion o SI ci cui s is gene ally lowe han ha o SC ci cui s pe o ming he same unc ion, which makes SI echnique a allback al e na i e when low-cos ab ica ion is manda o y. A. Swi ched-Capaci o Linea Ope a o s Conside he basic S/H s uc u e o Fig. 8(a) [1]. Analog swi ches a e con oled by a clock wi h wo nono e lapping phases, as shown in Fig. 8(c). Swi ches labeled ( espec- i ely, ) u n ON in synch oniza ion wi h he i s ( espec- i ely, second) clock phase.7The ci cui ope a es as ollows. In he acquisi ion phase, swi ches labeled a e ON and he opampiscon igu edasauni y-gainampli ie .Assuming ha he op amp is ideal, he inpu ol age is sampled by capac- i o . In he holding phase, swi ch labeled is ON and he bo om pla e o he samplingcapaci o isconnec ed o he op-amp ou pu . Since he op pla e o emains connec ed o he in e ing inpu o he op amp, heou pu ol agedu ing he holding phase keeps he p e iously sampled inpu . Al o- ge he , he ope a ion o he S/H ci cui can be desc ibed by he ollowing ecu si e equa ion: (12) hus p o iding uni y-gain hal -cycle delay o he inpu ol age du ing he holding phase and null ou pu du ing acquisi ion. Full-cycle delay elemen s, as equi ed by (3), can be ealized by simply cascading wo hal -delay s ages wi h al e na ing S/H clock phases. Taking ad an age o he holding ope a ion, SC echniques allowsimple ealiza ionso heagg ega ionandscaling unc- ions. Conside , o ins ance, he SC ci cui o Fig. 8(b) and assume he op amp is ideal. Du ing phase , ol ages a e sampled by capaci o s , while capaci o is discha ged as a esul o he i ual g ound a he inpu e minals o he op amp. Du ing he nex phase, 7By con en ion, any a bi a y signal s ( 1 ) obse ed a he end o he i s ( espec i ely, second) clock phase will be deno ed as s ( k +1 = 2) [ espec- i ely, s ( k ) ] k =0 ; 1 ; ... ,whe e T is he clock signalpe iod [see Fig.8(c)]. DELGADO-RESTITUTO AND RODRÍGUEZ-VÁZQUEZ: INTEGRATED CHAOS GENERATORS 755 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. Fig. 15. SC schema ics o Be noulli map. (a) (b) Fig. 16. (a) Onse o pa asi ic s able poin s in he Be noulli map due o imp ope se ing o A . (b) S a egy o a oid locked s a es. Op-amp and ela ed capaci o s pe o m he weigh ed summa ion in (37) and in oduce a hal -cycle delay. Op-amp is used o implemen he emaining hal delay s age and comple e he concep o Fig. 1(a). Pa ame e s in he map a e con oled by he capaci o s , , and and he dc ol age as ollows: (38) The nonlinea i y is ealized ia a phase- e e se swi ch a - angemen con olled by a dynamic compa a o . Depending on he alue o , his a angemen makes o be ei he added o sub ac ed a he inpu o he op-amp , he eby yielding he sign ope a o in (37). The compa a o consis s o an inpu o se canceled ampli ie , ollowed by a egene - a i e sense ampli ie and a NOR-based la ch [57]. Ope a ion o he ci cui in Fig. 15 is desc ibed by (37) whene e op amps wo k in hei linea egion. I any o he ampli ie s en e s in sa u a ion, he ci cui no longe imple- men s (37) and locks a pa asi ic s able poin s close o he powe ails. This undesi able si ua ion can be a oided by p ope ly se ing pa ame e . To illus a e his poin , Fig. 16 shows he open-loop ans e cha ac e is ics o he map, in- cluding op-amp ol age sa u a ions, o wo di e en alues o and hesame alueo . InFig.16(a),pa - asi ic s able poin s and appea a he in e sec ions o he ans e unc ion cha ac e is ics wi h he bisec ing line. This makes he ci cui o e ol e, a e a ansien , o ei he o , des oying any chao ic beha io . On he o he hand, o Fig. 16(b) nospu ious equilib ia appea andchao ic wa e o ms a e obus ly gene a ed. Necessa y condi ions o gua an ee his las si ua ion a e (39) Fig. 17. Mic opho og aph o he SC Be noulli map p o o ype. Fig. 18. Measu ed open-loop ans e cha ac e is ic o he SC Be noulli map o (a) di e en alues o A and (b) di e en alues o B and B . Measu ed spec a o di e en B , B se ings o (c) B = B =61 = 32 and (d) B =47 = 32 , B =39 = 32 . whe e deno es heopamp’sposi i e(nega i e) sa u a ion le el. In e es ingly enough, he condi ion gi es ise o he c ea ion o a clea ance be ween he in a ian se o he sys em and i s basin o a ac ion, which gua an ees ha , unde small pe u ba ions, ajec o ies a e always ein- jec ed in o he in a ian se . O he s a egies o achie e his goal can be ound in [15], [22], [41], and [42]. Fig. 17 shows he mic opho og aph o a p og ammable p o o ype o he ci cui in Fig. 15 [15]. In his p o o ype, he slopes o he cha ac e is ic— o and o —can be sepa a ely con oled by means o wo bina y weigh ed capaci o s wi h six con ol bi s each. Also, an addi ional con ol bi can be used o selec i ely open o close he eedback loop. Fig. 18(a) shows a amily o cu es o di e en alues o ol age and slopes and ixed a 61/32. On he o he hand, Fig. 18(b) shows a se o ans e cha ac e is ics ob ained o di e en alues o and wi h chosen so ha V. Measu emen s in closed loop we e also made o all possible combina ions o and alues in- side he chao ic egime. Fig. 18(c) and (d) show he spec a ob ained o wo o hese combina ions using a clock e- quency o kHz. Fla spec a we e ob ained o he cases . This is illus a ed in Fig. 18(c), ob- ained o . The spec um is la up o 75 kHz (35% o he clock equency) wi h a maximum de i- a ion o 1 dB, which ende s he ci cui well sui ed o whi e 762 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. Fig. 19. P og ammable cu en -mode scaled delay block. (a) (b) Fig. 20. Cu en -mode ealiza ion o he Be noulli map nonlinea i y. (a) Ci cui schema ic. (b) Implemen ed cha ac e is ic. noise gene a ion. On he o he hand, o , gene a ed noise becomes colo ed, as shown in Fig. 18(d). Now, le us conside implemen a ion o (37) in cu - en -mode domain [16]. The scaled delay ope a ion is ealized as a cascade o wo ack-and-hold SI s ages wi h complemen a y phase clocks. As indica ed in Fig. 19, he igh mos ansconduc o has a pa allel digi ally p og ammable s uc u e con oled by a digi al wo d o 4 bi s . This makes pa ame e B bina y-p o- g ammable om 1.0 o 2.0 a s eps o 0.25—in p ac ice, he scaling ac o o he ansconduc o con olled by bi is made sligh ly less han o make pa ame e lowe han 2.0 and, hence, a oiding di e gen o bi s. Fig. 20(a) shows a concep ual schema ic o he ealiza ion o he PWL cha ac e is ics o Fig. 20(b). I s ope a ion elies on he cu en ec i ie o Fig. 14(a). Posi i e inpu cu en s a e ou ed o node while, simul aneously, he ol age e ol es o he high logic s a e, u ning ON and OFF. Thus, a cu en (ob ained by KCL) is di ec ed o he ou pu node h ough he ansis o — he igh -hand piece o Fig. 20(b) is implemen ed in his manne . Simila ly, nega i e inpu cu en s u n ON and a cu en , ob ained by KCL a node , is deli e ed o he ou pu node. Fig. 21 shows a mic opho og aph o he SI Be noulli map p o o ype [16]. I includes some ex a ci cui y o enable es ing he ou pu cu en and o open o close he eedback loop. Fig. 22(a) shows he measu ed PWL cu en ans e cha - ac e is ics ob ained om he p o o ype. De ia ion om he ideal cha ac e is ic o inpu cu en s be ween 20 A o 20 A is less han 0.2%. Fig. 22(b) shows a de ail o he globalcha ac e is ics, inwhich heinpu cu en swings om 21 pA o 21 pA. I is in ended o illus a e he esolu ion Fig. 21. Mic opho og aph o he SI Be noulli map p o o ype. Fig. 22. (a) Measu ed cha ac e is ic o he nonlinea block. (b) De ail o he disc imina ion unc ion. (c) Measu ed cu en wa e o m. (d) Powe densi y spec um. achie ed in he cu en disc imina ion which, as al eady an- icipa ed in Sec ion V-B, amoun s o a ew picoampe es. Fig. 22(c) and (d) illus a es he closed loop ope a ion o he p o o ype o a clock equency o 500 kHz. Fig. 22(c) shows he measu ed cu en wa e o m a he ou pu o he delay block o (ac ually, a sligh ly lowe alue as men ioned be o e), while Fig. 22(d) shows i s associa ed powe densi y spec um. The wa e o m o Fig. 22(c) shows ha appa en ly coinciden alues o esul in qui e di e en alues a e ew i e a ions, he eby con i ming he expec ed unp edic ably ea u e. Rega ding Fig. 22(d), de ailed measu emen s shows a e y la spec um om dc up o abou 30% o he clock equency (de ia ion in his ange was o less han 1 dB). I is illus a i e o compa e pe o mance o his ci cui o ha o he SC ci cui in Fig. 15. A ea occupa ion o he SI p o o ype is abou one o de o magni ude smalle han o he SC p o o ype. Also, o hal he powe consump ion, he speedo heSIp o o ypeisabou h ee imesg ea e han ha ob ained om he SC p o o ype. This con i ms he sui abili y o he SI echnique o mode a e sys em equi emen s. DELGADO-RESTITUTO AND RODRÍGUEZ-VÁZQUEZ: INTEGRATED CHAOS GENERATORS 763 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. Fig. 23. Op imized Gm-C ealiza ion o he Chua’s oscilla o . B. Chua’s Oscilla o Fig. 23 shows he simpli ied schema ic o an in eg a ed p o o ype o he Chua’s oscilla o [32]. I implemen s, in- deed, a modi ied e sion o such oscilla o , ob ained om he op imiza ion p ocedu e desc ibed in Sec ion II-C. The esul ing model is sligh ly di e en o ha shown in Table II and de ined by ma ices (40) whe e . The non- linea unc ion is s ill gi en by . In Fig. 23, all he in eg a ing capaci o s a e assumed iden- ical and he linea ansconduc o s ha e been implemen ed by building a uni a y block wi h gain and connec ing in pa allel as many o such uni s as indica ed by he alues o , , and . On he o he hand, he nonlinea ansconduc o has been designed so ha i s ou pu cu en also includes he linea e m associa ed o he i s en y o , i.e., (41) Fig. 24 shows he ci cui used o he PWL unc ion con- sis ing o a on -end ansconduc o and a nonlinea ci cui ha ope a es in cu en -mode domain based on he high-ac- cu a e ec i ica ion mechanism desc ibed in Sec ion V-B. Two u he ci cui le el aspec s ha e been conside ed in he design o he schema ic o Fig. 23. One is he addi ion o dummy de ices so ha all he in eg a ion nodes exhibi he same capaci ance by cons uc ion. Acco dingly, he global ime cons an o he ci cui is gi en by , whe e is he o al capaci ance a he s a e a iable nodes. Since pa asi ics a e nonlinea and depend on he ope a ing poin o he ci cui , mo e han 80% o he o al capaci ance is con ibu ed by he nominal in eg a ing capaci ance . Fig. 24. Implemen a ion o he PWL ansconduc o . Fig. 25. Chip mic opho og aph. Fig. 26. Rou e o chaos in a silicon p o o ype o he Chua’s oscilla o . Limi cycle o (a) pe iod 1, (b) pe iod 2, and (c) pe iod 4. (d) Bi h o Rössle -like a ac o . (e) Rössle -like a ac o . ( ) Bi h o double-sc oll a ac o . (g) Double-sc oll a ac o . (h) Pe iodic window. (i) Double-sc oll a ac o close o sa u a ion. 764 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. A second aspec is he in oduc ion o a uning mechanism [49] (no shown in Fig. 23) o educe he absolu e ole ance o he ci cui ime cons an below 2% [32]. Fig. 25 shows a mic opho og aph o he chao ic oscilla o , which includes he onchip uning scheme, and o he auxil- ia y ci cui y o biasing and measu emen pu poses. Powe dissipa ion is less han 1.8 mW o a symme ical biasing o 2.5 V. The ab ica ed p o o ype is able o ep oduce he whole bi u ca ion sequence leading o he chao ic a ac o s o he oscilla o ,as shownin Fig.26[32].Thedi e en phase po ai s (p ojec ions on he plane ) has been ob ained by p og essi ely inc easing pa ame e , while keeping he o he sys em pa ame e s ixed. As can be seen, he pic u e book e eals a pe iod-doubling ou e o chaos, including pe- iodic windows, as well as Rössle -like and double-sc oll a ac o s. VII. SUMMARY Th ough p ope design echniques encompassing consid- e a ions bo h a sys em and ci cui le els, i is possible o de- sign compac and obus chao ic ICs in CMOS echnologies. This pa es he way o he in eg a ion in silicon o many o he applica ions al eady de ised o nonlinea dynamics. REFERENCES [1] R. G ego ian and G. C. Temes, Analog MOS In eg a ed Ci cui s o Signal P ocessing. New Yo k: Wiley, 1986. [2] J. Shoukens, “Su ey o exci a ion signals o FFT based signal an- alyze s,” IEEE T ans. Ins um. Meas., ol. 37, pp. 342–352, Sep . 1988. [3] M. Gup a, “Applica ions o elec ical noise,” P oc. IEEE, ol. 63, pp. 996–1010, July 1975. [4] S. R. No swo hy, “Quan iza ion e o s and di he ing in 61 modula o s,” in Del a–Sigma Da a Con e e s—Theo y, Design, and Simula ion, S. R. No swo hy, R. Sch eie , and G. C. Temes, Eds. Pisca away, NJ: IEEE P ess, 1997, ch. 3. [5] S. Hein, “Exploi ing chaos o supp ess spu ious ones in gene al double-loop 61 modula o s,” IEEE T ans. Ci cui s Sys . II, ol. 40, pp. 651–659, Oc . 1993. [6] R. Jewe , K. Poul on, K.-C. Hsieh, and J. Doe nbe g, “A 12b 128MSample/s ADC wi h 0.05LSB DNL,” in ISSCC Diges o Technical Pape s, San F ancisco, CA, Feb. 1997, pp. 138–139. [7] L. R. Ca ley, “A noise-shaping code opology o 15+ bi con- e e s,” IEEE J. Solid-S a e Ci cui s, ol. 24, pp. 267–273, Ap . 1989. [8] W. T. Walke , “Rada sys em u ilizing chao ic coding,” U.S. Pa en 5321409, June 1994. [9] G. Kolumban, M. P. Kennedy, and L. O. Chua, “The ole o synch o- niza ion in digi al communica ions using chaos—Pa II: Chao ic modula ionand chao ic synch oniza ion,” IEEET ans. Ci cui sSys . I, ol. 45, pp. 1129–1140, No . 1998. [10] M. J. Ogo zalek, “O e iew o elec onic chaos gene a ion, con ol, and applica ions,” P oc. SPIE, ol. 2612, pp. 2–13, Oc . 1995. [11] A. Rod íguez-Vázquez, J. L. Hue as, A. Rueda, B. Pé ez-Ve ú, and L. O. Chua, “Chaos om swi ched-capaci o ci cui s: Disc e e maps,” P oc. IEEE, ol. 75, pp. 1090–1106, Aug. 1987. [12] G. C. McGonigal and M. I. Elmas y, “Gene a ion o noise by elec- onic i e a ion o he logis ic map,” IEEE T ans. Ci cui s Sys ., ol. 34, pp. 981–983, Aug. 1987. [13] T. Mo ie, S. Sakabayashi, M. Naga a, and A. Iwa a, “CMOS ci cui s gene a ing a bi a y chaos by using pulsewid h modula ion echniques,” IEEE T ans. Ci cui s Sys . I, ol. 47, pp. 1652–1657, No . 2000. [14] G. M. Be ns ein and M. A.Liebe man, “Secu e andom numbe gen- e a ion using chao ic ci cui s,” IEEE T ans. Ci cui s Sys ., ol. 37, pp. 1157–1164, Sep . 1990. [15] A. Rod íguez-Vázquez, M. Delgado-Res i u o, S. Espejo, and J. L. Hue as, “Swi ched-capaci o b oadband noise gene a o o CMOS VLSI,” Elec on. Le ., ol. 27, no. 21, pp. 1913–1915, Oc . 1991. [16] M. Delgado-Res i u o, F. Medei o, and A. Rod íguez-Vázquez, “Nonlinea swi ched-cu en CMOS IC o andom signal gene a- ion,” Elec on. Le ., ol. 29, no. 25, pp. 2190–2191, Dec. 1993. [17] P. Chiang, W. Dally, and E. Lee, “Monoli hic chao ic communica- ions sys em,” in P oc. IEEE In . Symp. Ci cui s and Sys ems, ol. III, May 2001, pp. 325–328. [18] C. S. Pe i and J. A. Connelly, “A noise-based IC andom numbe gene a o o applica ions in c yp og aphy,” IEEE T ans. Ci cui s Sys . I, ol. 47, pp. 615–621, May 2000. [19] M. Delgado-Res i u o and A. Rod íguez-Vázquez, “Swi ched-cu - en chao ic neu ons,” Elec on. Le ., ol. 30, no. 5, pp. 429–430, Ma . 1994. [20] H. Ho io and K. Suyama, “Expe imen al e i ica ion o signal ansmission using synch onized SC chao ic neu al ne wo ks,” IEEE T ans. Ci cui s Sys . I, ol. 42, pp. 393–395, July 1995. [21] G. Cauwenbe ghs, “Del a–Sigma cellula au oma a o analog VLSI andom ec o gene a ion,”IEEE T ans. Ci cui s Sys . II, ol.46,pp. 240–250, Ma . 1999. [22] M. Delgado-Res i u o and A. Rod íguez-Vázquez, “Mixed-signal map-con igu able in eg a ed chaos gene a o o chao ic commu- nica ions,” IEEE T ans. Ci cui s Sys . I, ol. 48, pp. 1462–1474 , Dec. 2001. [23] M. Delgado-Res i u o, A. Rod íguez-Vázquez, S. Espejo, and J. L. Hue as, “A chao ic swi ched-capaci o ci cui o 1 = noise gene - a ion,” IEEE T ans. Ci cui s Sys . I, ol. 39, pp. 325–328, Ap . 1992. [24] L. La ge , V. S. Udal so , J. P. Goedgebue , and W. T. Rhodes, “Chao ic dynamics o oscilla o s based on ci cui s wi h VCO and nonlinea delayed eedback,” Elec on. Le ., ol. 36, no. 3, pp. 199–200, Feb. 2000. [25] A. S. Elwakil and M. P. Kennedy, “Cons uc ion o classes o ci - cui -independen chao icoscilla o susingpassi e-onlynonlinea de- ices,”IEEE T ans.Ci cui sSys . I, ol.48,pp. 289–307, Ma . 2001. [26] A. Tamase icius, “A ep oducible analogue ci cui o chao ic syn- ch oniza ion,” Elec on. Le ., ol. 33, no. 13, pp. 1105–1106, Jun. 1997. [27] A. S. Elwakil, K. N. Salama, and M. P. Kennedy, “A sys em o chaos gene a ion and i s implemen a ion in monoli hic o m,” in P oc. IEEE In . Symp. Ci cui s and Sys ems, ol. V, May 2000, pp. 217–220. [28] M. P. Kennedy, “Chaos in he Colpi s oscilla o ,” IEEE T ans. Ci - cui s Sys . I, ol. 41, pp. 771–774, No . 1994. [29] A. S. Elwakil and M. P. Kennedy, “Cu en con eyo chaos gene a- o s,” IEEE T ans. Ci cui s Sys . I, ol. 46, pp. 393–398, Ma . 1999. [30] L. O. Chua and G. N. Lin, “Canonical ealiza ion o Chua’s ci cui amily,” IEEE T ans. Ci cui s Sys ., ol. 37, pp. 885–902, July 1990. [31] A. Rod íguez-Vázquez and M. Delgado-Res i u o, “CMOS design o chao ic oscilla o s using s a e a iables: A monoli hic Chua’s ci - cui ,” IEEE T ans. Ci cui s Sys . II, ol. 40, pp. 596–613, Oc . 1993. [32] M. Delgado-Res i u o and A. Rod íguez-Vázquez, “Design consid- e a ions o in eg a ed con inuous- ime chao ic oscilla o s,” IEEE T ans. Ci cui s Sys . I, ol. 45, pp. 481–495, Ap . 1998. [33] J. M. C uz and L. O. Chua, “An IC chip o Chua’s ci cui ,” IEEE T ans. Ci cui s Sys . II, ol. 40, pp. 614–625, Oc . 1993. [34] O. A. González, G. Han, J. Pineda de Gy ez, and E. Sánchez- Sinencio, “Lo enz-based chao ic c yp osys em: A monoli hic im- plemen a ion,” IEEE T ans. Ci cui s Sys . I, ol. 47, pp. 1243–1247, Aug. 2000. [35] A. Tamase icius, A. Cenys, G. Mykolai is, A. Namajunas, and E. Lindbe g, “Hype chao ic oscilla o wi h gy a o s,” Elec on. Le ., ol. 33, no. 7, pp. 542–543, Ma . 1997. [36] S. Nakagawa andT. Sai o,“An RC OTA hys e esis chaos gene a o ,” IEEE T ans. Ci cui s Sys . I, ol. 43, pp. 1019–1021, Dec. 1996. [37] J. E. Va ien os and E. Sánchez-Sinencio, “A 4-D chao ic oscilla o based on a di e en ial hys e esis compa a o ,” IEEE T ans. Ci cui s Sys . I, ol. 45, pp. 3–10, Jan. 1998. [38] M.Gus a sson,J.J. Wilkne ,and N.N. Tan,CMOS Da aCon e e s o Communica ions. No well, MA: Kluwe Academic, 2000. DELGADO-RESTITUTO AND RODRÍGUEZ-VÁZQUEZ: INTEGRATED CHAOS GENERATORS 765 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. [39] J. Goes, J. C. Vi al, and J. E. F anca, “Sys ema ic design o op- imiza ion o high-speed sel -calib a ed pipelined A/D con e e s,” IEEE T ans. Ci cui s Sys . II, ol. 45, pp. 1513–1526, Dec. 1998. [40] Y. P. Tsi idis and J. O. Voo man, Eds., In eg a ed Con inuous-Time Fil e s. Pisca away, NJ: IEEE P ess, 1993. [41] G. Mazzini, G. Se i, and R. Ro a i, “Chao ic complex sp eading sequences o asynch onous DS-CDMA—Pa I: Sys em modeling and esul s,” IEEE T ans. Ci cui s Sys . I, ol. 44, pp. 937–947, Oc . 1997. [42] S. Callega i, R. Ro a i, and G. Se i, “Robus ness o chaos in analog implemen a ions,” in Chao ic Elec onic in Telecommunica ions,M. P. Kennedy, R. Ro a i, and G. Se i, Eds. Boca Ra on, FL: CRC P ess, 2000, ch. 12. [43] S. Wiggins, In oduc ion o Applied Nonlinea Dynamical Sys ems and Chaos. Be lin, Ge many: Sp inge -Ve lag, 1990. [44] C. W. Wu and L. O. Chua, “On linea opological conjugacy o Lu ’e sys ems,” IEEE T ans. Ci cui s Sys . I, ol. 43, pp. 158–161, Feb. 1996. [45] M. Vidyasaga , Nonlinea Sys ems Analysis, 2nd ed. Englewood Cli s, NJ: P en ice-Hall, 1993. [46] S. M. Kang and L. O. Chua, “A global ep esen a ion o mul idi- mensional piecewise-linea unc ions wi h linea pa i ions,” IEEE T ans. Ci cui s Sys ., ol. 5, pp. 938–940, No . 1978. [47] G.G oenewold,“Thedesign o highdynamic angecon inuous- ime in eg a ablebandpass il e s,”IEEET ans.Ci cui sSys ., ol.38,pp. 838–852, Aug. 1991. [48] K. R. Lake and W. M. C. Sansen, Design o Analog In eg a ed Ci - cui s and Sys ems. NY: McG aw-Hill, 1994. [49] R. Schaumann, M. S. Ghausi, and K. R. Lake , Design o Analog Fil e s. Englewood Cli s, NJ: P en ice-Hall, 1990. [50] A. Rod íguez-Vázquez, B. Lina es-Ba anco, J. L. Hue as, and E. Sánchez-Sinencio, “On he design o ol age-con olled sinusoidal oscilla o s using OTAs,” IEEE T ans. Ci cui s Sys ., ol. 37, pp. 198–211, Feb. 1990. [51] K. D. Pe e son, A. Nedungadi, and R. L. Geige , “Ampli ie design conside a ions o high equency monoli hic il e s,” in P oc. Eu . Con . Ci cui Theo y and Design, Sep . 1987, pp. 321–326. [52] C. Toumazou, F. J. Lidgey, and D. G. Haigh, Eds., Analogue IC Design: The Cu en -Mode App oach. London, U.K.: Pe e Pe e- g inus, 1990. [53] A. Fab e, O. Saaid, and H. Ba helemy, “On he equency lim- i a ions o he ci cui s based on he second gene a ion cu en con eyo ,” An. In eg . Ci cui s Signal P ocessing, ol. 7, no. 2, pp. 113–129, Ma . 1995. [54] M. Banu and Y. P. Tsi idis, “Fully in eg a ed ac i e RC il e s in MOS echnology,” IEEE J. Solid-S a e Ci cui s, ol. SSC-18, pp. 644–651, Dec. 1983. [55] C. C. Enz, F. K ummenache , and E. A. Vi oz, “An analy ical MOS ansis o model alid in ALL egions o ope a ion and dedica ed o low- ol age and low-cu en applica ions,” An. In eg . Ci cui s Signal P ocessing, ol. 8, pp. 83–114, July 1995. [56] R. Unbehauen and A. Cichocki, MOS Swi ched-Capaci o and Con- inuous-Time In eg a ed Ci cui s and Sys ems. Heidelbe g, Ge - many: Sp inge -Ve lag, 1989. [57] C. C. Enz and G. C. Temes, “Ci cui echniques o educing he e ec s o op-amp impe ec ions: Au oze oing, co ela ed double sampling, and choppe s abiliza ion,” P oc. IEEE, ol. 84, pp. 1584–1614, No . 1996. [58] D. J. Alls o and W. C. Black, “Technological conside a ions o monoli hic MOS swi ched-capaci o il e ing sys ems,” P oc. IEEE, ol. 71, pp. 967–986, Aug. 1983. [59] J. Shyu, G. C. Temes,and F. K ummenache , “Random e o s e ec s in ma ched MOS capaci o s and cu en sou ces,” IEEE J. Solid- S a e Ci cui s, ol. SSC-19, pp. 948–955, Dec. 1984. [60] A. T. Beh , M. C. Schneide , S. N. Filho, and C. G. Mon o o, “Ha - monic dis o ion caused by capaci o s implemen ed wi h MOSFET ga es,” IEEE J. Solid-S a e Ci cui s, ol. SSC-27, pp. 1470–1475, Oc . 1992. [61] C. Toumazou, J. B. Hughes, and N. C. Ba e sby, Eds., Swi ched-Cu en s: An Analogue Technique o Digi al Tech- nology. London, U.K.: Pe e Pe eg inus, 1993. [62] F. Medei o, B. Pé ez-Ve dú, and A. Rod íguez-Vázquez, Top-Down Design o High-Pe o mance Sigma–Del a Modula o s. No well, MA: Kluwe Academic, 1999. [63] W. M. C. Sansen, H. Qiu ing, and K. A. I. Halonen, “T ansien anal- ysis o cha ge ans e in SC il e s—Gain e o and dis o ion,” IEEE J. Solid-S a e Ci cui s, ol. 22, pp. 268–276, Ap . 1987. [64] G. Fische and G. S. Moschy z, “On he equency limi a ions o SC il e s,” IEEE J. Solid-S a e Ci cui s, ol. 19, pp. 510–518, Aug. 1984. [65] A. Ma ques, V. Peluso, M. S eyae , and W. Sansen, “A 15-b esolu- ion 2-MHz Nyquis a e 61 ADC in a 1-  m CMOS echnology,” IEEE J. Solid-S a e Ci cui s, ol. 33, pp. 1065–1075, July 1998. [66] D. Cline, “Noise, speed, and powe ade-o s in pipelined analog o digi al con e e s,” Ph.D. disse a ion, Uni . Cali o nia,, Be keley, CA, No . 1995. [67] N. Tan, Swi ched-Cu en Design and Implemen a ion o O e sam- pling A/D Con e e s. No well, MA: Kluwe Academic, 1997. [68] J. H. Shieh, M. Pa il, and B. J. Shen, “Measu emen and analysis o cha ge injec ion in MOS analog swi ches,” IEEE J. Solid-S a e Ci cui s, ol. 22, pp. 277–281, Ap . 1987. [69] G. Wegmann, E. A. Vi oz, and F. Rahali, “Cha ge injec ion in analog MOS swi ches,” IEEE J. Solid-S a e Ci cui s, ol. 22, pp. 1091–1097, Dec. 1987. [70] M. Delgado-Res i u o, A. Rod íguez-Vázquez, and F. Vidal, “Non- linea syn hesis using in eg a ed ci cui s,” in Encyclopedia o Elec- ical and Elec onics Enginee ing, J. G. Webs e , Ed. Chiches e , U.K.: Wiley, 1999. [71] D. H. Sheingold, Nonlinea Ci cui s Handbook. No wood, MA: Analog De ices Inc., 1976. [72] E. See inck, Analysis and Syn hesis o T anslinea In eg a ed Ci - cui s. Ams e dam, The Ne he lands: Else ie , 1988. [73] B. Gilbe , “Cu en -mode ci cui s om a anslinea iewpoin : A u o ial,” in Analogue IC Design: The Cu en -Mode App oach,C. Toumazou, F. J. Lidgey, and D. G. Haigh, Eds. London, U.K.: Pe e Pe eg inus, 1990, ch. 2. [74] K. Bul and H. Wallinga, “A class o analog CMOS ci cui s based on hesqua e-lawcha ac e is icso an MOS ansis o in sa u a ion,” IEEE J. Solid-S a e Ci cui s, ol. 22, pp. 357–365, June 1987. [75] R. Wiege ink, Analysis and Syn hesis o MOS T anslinea Ci - cui s. No well, MA: Kluwe Academic, 1993. [76] N. I. Khachab and M. Ismail, “A nonlinea CMOS analog cell o VLSI signal and in o ma ion p ocessing,” IEEE J. Solid-S a e Ci - cui s, ol. 26, pp. 1689–1699, No . 1991. [77] B.J. Hos icka, W. B ockhe de,U.Klein, and R.Schwee ,“Designo nonlinea analog swi ched-capaci o ci cui s using building blocks,” IEEE T ans. Ci cui s Sys ., ol. CAS-31, pp. 354–368, Ap . 1984. [78] A. Rod íguez-Vázquez, R. Domínguez-Cas o, F. Medei o, and M. Delgado-Res i u o, “High- esolu ion CMOS cu en compa a o s: Design and applica ions o cu en -mode unc ion gene a ion,” An. In eg . Ci cui s Signal P ocessing, ol. 7, pp. 149–166, Ma . 1995. Manuel Delgado-Res i u o (Membe , IEEE) e- cei ed heDoc o enCienciasFísicasdeg ee om he Uni e si y o Se ille, Se ille, Spain in 1996. In 1990, he joined he Resea ch S a o he Depa men o Analog and Mixed-Signal In eg a ed Ci cui Design o he Ins i u e o Mic oelec onics, Uni e si y o Se ille, Se ille, Spain. Since 1998, he has been a Tenu ed Scien is o he Spanish Council o Resea ch (CSIC). He cu en esea ch in e es s include he design o analog and mixed-signal VLSI ci cui s o nonlinea signal p ocessing, including ision chips, neu o uzzy con olle s, and chao ic ci cui s o communica ions, he design and modeling o in eg a ed ci cui s o wi eless communica ion, and he design o eusabili y o analog and mixed-signal ci cui blocks. 766 PROCEEDINGS OF THE IEEE, VOL. 90, NO. 5, MAY 2002 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply. Angel Rod íguez-Vázquez (Fellow, IEEE) was bo n in Se ille, Spain. He is a P o esso o Elec onics wi h he De- pa men o Elec onics and Elec omagne ism, Uni e si y o Se ille, Se ille, Spain. He is also a Membe o he Resea ch S a o he Ins i u e o Mic oelec onics o Se ille–Na ional Cen e o Mic oelec onics (IMSE-CNM), Se ille, Spain, whe e he heads a esea ch g oup on analog and mixed-signal VLSI. His cu en esea ch in e es s include he design o analog in e aces o mixed-signal VLSI ci cui s, CMOS image s and ision chips, neu o uzzy con olle s, symbolic analysis o analog in eg a ed ci cui s, and op imiza ion o analog in eg a ed ci cui s. D . Rod íguez-Vázquez ecei ed he Young Scien is Awa d o he Se ille Academy o Science in 1992, he IEEE Ci cui s and Sys ems Socie y Guillemin–Caue Awa d in 1995 and he Bes Pape Awa d o he Eu opean Con e ence on Ci cui Theo y and Design in 1995. He was an Associa e Edi o o he IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART I: FUNDAMENTAL THEORY AND APPLICATIONS om 1993 o 1995, a Gues Edi o o he IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART I: FUNDAMENTAL THEORY AND APPLICATIONS Special Issue on Low-Vol age and Low-Powe Analog and Mixed-Signal Ci cui s and Sys ems in 1995, a Gues Edi o o he IEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS—PART II: ANALOG AND DIGITAL SIGNAL PROCESSING Special Issue on Ad ances in Nonlinea Elec onic Ci cui s in 1999, and a Chai o he IEEE Ci cui s and Sys ems Analog Signal P ocessing Commi ee in 1996. DELGADO-RESTITUTO AND RODRÍGUEZ-VÁZQUEZ: INTEGRATED CHAOS GENERATORS 767 Au ho ized licensed use limi ed o: Uni e sidad de Se illa. Downloaded on Ap il 13,2020 a 14:49:35 UTC om IEEE Xplo e. Res ic ions apply.