A modular current-mode high-precision winner-take-all circuit

557 A Modula Cu en -Mode High-P ecision Winne -Take- All Ci cui T. Se ano and B. Lina es-Ba anco Na ional Mic oelec onics Cen e (CNM), Ed. CICA, AV. Reina Me cedes s/n, 41012 SPAIN. Phone: 34-5-4239923 Fax: 34-5-4624506, E-mail: [email p o ec ed] Abs ac In his pape we p esen a Winne -Take-All (WTA) ci cui ealized using cu en -mode ci cui design echniques. The ope a ion o he WTA is based on cu en compa a o s and cu en mi o s. Speed and p ecision is de e mined p ima ily by he cha ac e is ics o he cu en mi o used. Using special cu en mi o s, esolu ions below 1% and se ling imes below loons we e simula ed. The ci cui emains ope a i e wi h easonable speed and p ecision o an inpu signal ange wide han wo decades. I equi es cu en inpu signals and p o ides ol age ou pu signals. I. In oduc ion In he las yea s he ield o cu en -mode signal p ocessing has been ecei ing conside able a en ion, and mo e ci cui designe s a e using cu en -mode ci cui design echniques o de elop compac and as A/D con e e s [I], il e s [2], neu al ne wo ks [31, and uzzy ope a o s [4]. In he a ea o ixu al ne wo ks and uzzy signal p ocessing a e y common building block is he Winne -Take-All (WTA) ci cui (also called Ma -ope a o ). Se e al WTA ci cui opologies ha e been epo ed in he li e a u e. Some o hem ope a e wi h ol age signals [5], and o he s ope a e wi h cu en inpu signals [6]. Howe e , all o hem ely on he good ma ching o he VT o an a ay o ansis o s. The numbe o ansis o s in his a ay is equal o he numbe o inpu s. This ac imposes ha all ansis o s o a WTA ha e o be inside he same chip (o , a leas , he same wa e ). I he e is a need o assemble a la ge neu al ne composed o se e al chips and wi h a WTA ci cui dis ibu ed among hese chips, he pe o mance in p ecision o he WTA may become d as ically deg aded. In his pape we p esen a cu en -mode WTA ci cui which equi es cu en inpu signals, and is he e o e sui able o a cu en signal p ocessing en i onmen . The ci cui can be ex ended be ween se e al chips, wi hou loss o p ecision o he o e all sys em pe o mance. In he p esen pape we will i s desc ibe he ma hema ical p inciple o ope a ion o he WTA. A e wa ds, we will s udy he s abili y o he equilib ium poin , ollowed by he e i ica ion o he global s abili y o he ci cui . Then we will conside se e al cu en mi o opologies o be used in he ci cui . Finally we will p esen Hspice simula ion esul s ha cha ac e ize he ope a ion o he cu en -mode WTA ci cui . 11. Desc ip ion o WTA Ope a ion Analy ically, he p oblem o inding he maximum alue among a se o inpu alues {xi} can be s a ed as he solu ion y o he ollowing nonlinea algeb aic equa ion N Y = CaiU(xi-Y) (1) i= 1 whe e ai a e posi i e pa ame e s ha need o mee some condi ions, and U(.) is he s ep unc ion 1 i x>o U(x) = E (0, 1) i x = 0 (2) i 0 i x<O Fo example, conside he special case o eq.( I) in which N=4, as shown in Fig. 1. In his case, he solu ion o eq.( 1) is gi en by he in e sec ion o he cu es , (y) = caiU (xi - y) and 2 (y) = y, which is poin A. A poin A we ha e (3) which is he maximum o he inpu se alues. Looking a Fig. 1 we can see ha i we impose he condi ion hen he cu es i(y) and i(y) will in e sec a he poin ha de ines he maximum o he se {xi}. In o de o implemen eq.(l) in an elec onic ci cui , some dynamics need o be added. P ac ical ci cui s a e desc ibed by ime-domain di e en ial equa ions, he e o e we need a di e en ial equa ion whose solu ion is eq.( 1). Conside , o example, he ollowing ime-domain i s o de di e en ial equa ion, (5) The s eady-s a e solu ion o his equa ion is p ecisely eq.( 1). Fo his solu ion o be s able, i is equi ed ha (6) y = x4 = max{xi> ai>xi Vi (4) N z$ = -y+ C a,u(xi-y) i=I - q <o d~ Solu ion This means ha he s abili y condi ion o eq.(5) is XI x2 x3 x4 Fig. 1: G aphical ep esen a ion o a pa icula case o eq.(l) 558 PMOS-mi o wi h mul iple ou pu s L >I1 Fig. 2: Ci cui diag am o comple e cu en -mode WTA - 1 - ai6(xi-y) <O (7) xi2y 6(.) being he de i a i e o he s ep unc ion. This condi ion is always ul illed. 111. Cu en -Mode Ci cui Implemen a ion The ci cui implemen a ion ha ollows implemen s he ma hema ical model o eq.(l) wi h he coe icien s {ai} de ined as The ci cui ha implemen s each o he e ms aiU(xi-y) o eq.(l) can be seen in Fig. 2. The equi alences be ween his ci cui pa ame e s and he pa ame e s in eq.(l) a e as ollow: (9) In i s s eady s a e, he WTA ci cui (see Fig. 2) sol es he ollowing equa ion, ai = xi Vi (8) x.=I. a.=Ii 11 y = I, aiU(xi-y) =Iloi N I, = c IOi wi h I,i = IiU(Ii-Io) (10) i= 1 Fo he cu en compa a o a simple in e e can be used, o a high pe o mance compa a o [7]. The p ecision o he WTA ci cui o Fig. 2 is limi ed by he cu en mi o s misma ch e o s. Cu en eplica ion h ough cu en mi o s is subjec o e o s p oduced by he misma ch be ween ansis o s inside he same cu en mi o s. The e o e, i i is needed o ex end he WTA among se e al chips, he p ecision can be made insensi i e o in e -chip elec ical pa ame e s sp ead by keeping all he ansis o s o he same cu en mi o inside he same chip. This can be achie ed by using he assembling echnique depic ed in Fig. 3, which equi es he addi ion o only one ex a cu en mi o . By his app oach, he p ecision o he o e all WTA ci cui is insensi i e o elec ical pa ame e s sp ead be ween di e en chips. IV. S abili y Conside a ions In his Sec ion we will i s s udy he condi ions o make s able he unique s eady-s a e equilib ium poin , ollowed by a global s abili y s udy o he comple e sys em. The s abili y o he equilib ium poin is based on local conside a ions and is he e o e pe o med h ough a small signal linea ci cui analysis. The global s abili y, howe e , needs nonlinea la ge signal conside a ions, and is based on Liapuno 's s abili y heo y [8]. A. Local S abili y o he Equilib ium Poin I he ope a ion o he WTA ci cui o Fig. 2 is s able, he e should be a unique s eady s a e in which all N-mi o cells, excep one, a e OFF (Zoi=O), and Z0=Zim. In his case, he ac i e pa o he ci cui is gi en by he diag am shown in Fig. 4(a), whe e ansis o MI is in i s ON s a e and d i ing I II 1; I I I All II I I II YYI I 1 Fig. 3: Modula in e -chip connec ion o cu en -mode WTA (PMOS-mi o I Fig. 4: S eady s a e ep esen a ion o WTA cu en -mode ci cui . (a) Ci cui diag am, (b) small signal equi alen ci cui he comple e I,, cu en . Fig. 4(b) ep esen s he co esponding small signal equi alen ci cui , whe e he main delay elemen s ha e been included. Elemen s C, and G, ep esen he inpu impedance o he cu en compa a o , A i s ol age gain, and O, accoun s o i s in e nal delay. Conduc ances go, ep esen he ou pu impedance o he NMOS cu en mi o , and o he PMOS cu en mi o gip is i s inpu impedance, g, i s ou pu impedance, and O , de ines i s delay. Fo ansis o MI, only he s a ic pa ame e s g,, and gdsn a e conside ed. I s delay has been included in pa ame e OA. Analysis o his small signal equi alen ci cui yields he ollowing second o de cha ac e is ics equa ion, which is s able i (12) The e o e, s able s eady-s a e WTA ope a ion equi es as cu en mi o s and compa a o s, bu domina ing inpu compa a o dynamics and small compa a o ol age gain. B. Global S abili y o he Sys em No e howe e , ha now he e a e N dominan dynamic elemen s in he comple e WTA du ing he ansien egimes: capaci o s C, a he inpu o each cu en compa a o . The e o e, he dynamics o he sys em is no a i s o de one as was supposed in eq.(5), bu i is o o de N. Ne e heless, Cc(gmn+gon)'A 1 1 >-+- gmngon OA "p 559 I b) (d) Fig. 5: Cu en mi o &ologies: (a) simple, (b) cascode, (c) (;;kgula ed cascode [lo], (d) ac i e-inpu&], (e) ac i e-inpu egula ed cascode 1121 s abili y analysis can be pe o med h ough he use o Liupuno S s abili y heo ems [8]. The dynamics o he WTA, assuming capaci o s C, a e he dominan dynamic elemen s, a e desc ibed by he ollowing se o N i s o de nonlinea di e en ial equa ions, N CCGxi = - G, ( xi - M) - Ii + Ii U ( M - xj) (1 3) j= 1 whe e M is he ol age ip poin a he compa a o s inpu , and he cell cu en l<,i is desc ibed now by, No e ha , since IOi = IiU(VM- xi) (14) (15) he e s ill exis s he unique s eady-s a e solu ion o eq.( 10). Unde hese ci cums ances, he ollowing scala unc ion', d ,d Io - Ii = G, ( xi - M) + C - ( xi - W) U (VXi - Vw) (16) +GCE I ~'(B)~.S+C(~~-~I,)U(V~~-VM) 1 i lo is a Liapuno unc ion o he N h o de WTA sys em. Compu ing he ime de i a i e o his unc ion yields, N .k = U' ( xi - VM) xpi PI= ~~IJ(l-u( xJ- M))~-GC( xl-~M) =I -I[ whe e pi is, by eq.(13), equal o C, ,,, hus N (18) Since U(.) is mono onically inc easing (U' (x) 0, Vx), i hen esul s ha ou WTA ci cui will make dec ease in ime he Liapuno unc ion E. Since E is bounded om below he ci cui desc ibed by eq.( 13) will ha e a s eady s a e ( xl = 0) which is a minimum o he Liapuno unc ion E. By Liapuno 's heo ems [8] his means ha eq.( 13) desc ibes a s able sys em ha con e ges asymp o ically o i s unique solu ion gi en by (see eq.(3)), .2 .k = -c, U ( Vx1 - VM) x, 2 0 I= 1 Io = max { Ii} (19) I. In o de o apply Liapuno 's heo ems le us app oxima e he s ep unc ion U(.(.) by !he ollowing con inuous and di e en iable unc ion wi h EzO bu close o ze o, U (z) = ___ 1 1 + %. V. Elec ion o Cu en Mi o s The p ecision o he o e all WTA cu en mode ci cui is de e mined by he elec ion o he kind o cu en mi o s used. Since a cu en compa a o has i ually no o se [7], he cu en e o a he inpu o each cu en compa a o is de e mined by mi o misma ches. The e o a he posi i e cu en I, a ailable a he inpu o each cu en compa a o esul s om one p-mi o e lec ion, p eceded by an n-mi o e lec ion o he winning cell, (20) while he e o o he nega i e cu en li a he cu en compa a o s inpu s esul s om a single n-mi o e lec ion, The o al cu en e o a he inpu o each cu en compa a o is he e o e gi en by, (22) Howe e , he e o in oduced by a cu en mi o is no only he andom misma ch con ibu ion, as conside ed by eqs.(20)-(22), bu also i s sys ema ic e o con ibu ion, which esul s om di e en d ain- o-sou ce ol ages a he e lec ing ansis o s, poo impedance coupling, and inhe en nonlinea MOS ansis o ope a ion. The andom misma ch e o con ibu ion o a cu en mi o is almos opology independen and is de e mined p ima ily by he a ea o he e lec ing ansis o s [9]. Howe e , he sys ema ic e o con ibu ed by a cu en mi o is highly opology dependen . Conside , o example, he i e cu en mi o opologies o Fig. 5. I hey all ha e he same e lec ing ansis o s (MI and M2) sizes (W=lOOpm and L=20pm), VD= 1.5 and hey a e loaded by hei complemen a y PMOS cu en mi o s wi h equi alen inpu impedance, hei o al cu en e o de ined as is shown in Fig. 6. This e o is exp essed in bi s, and is a unc ion o he inpu cu en linl which in his case a ies om IO@ o 8OOM. Fo he ac i e-inpu egula ed cascode cu en mi o , wo cu es a e gi en: one wi h A=lOO and he o he wi h A=1000, whe e A is he ol age gain o he di e en ial ol age ampli ie s. Also shown in Fig. 6 is he andom misma ch e o con ibu ion (0, in eq.(23)) which is app oxima ely he same o all he opologies. Acco ding o ou p ecision equi emen s, Fig. 6 can help us o selec he mos app op ia e cu en mi o opology o ou applica ion. In ou case we would like o main ain a good accu acy o e a e y wide signal ange. The e o e, we will choose he ac i e-inpu egula ed cascode cu en mi o . VI. Simula ion Resul s o2 (I,) = 0; + 0; 02(li) = 02 N (21) O;, al = 02 (I,) + 02 (Ioi) = 20; + 0; "To al = A$ys ema ic + (23) A cu en -mode WTA p o o ype has been designed o he digi al 1.5pm single-poly CMOS p ocess p o ided by ES2 560 5.0 1 10.0 9.0 - 8.0 ) - ._ s ._ 0 E a 7.0 6.0 . ,. k ’ ’ e-05 1 e-04 Ii absolu e ela i e Table I1 l0pA 50pA l00pA 500pA I.0mA 2.0mA 272nA 640nA 977nA 3.21pA 5.93pA 11.6pA 2.72% 1.28% 0.98% 0.64% 0.58% 0.55% VII. Conclusions We ha e p esen ed a cu en -mode WTA ci cui ha can be ex ended by in e -chip modula connec ions wi hou loss o p ecision in i s ope a ion. S abili y o he ci cui has been s udied heo e ically and e i ied h ough Hspice simula ions. A design has been pe o med and cha ac e ized h ough ex ensi e simula ions. P ecision alues below I % ha e been ob ained. VIII. Re e ences [I] D. G. Nai n and C. A. Salama, “Cu en -Mode Algo i hmic Analog- o-Digi al Con e e s,” IEEE J. Solid-s a e Ci cui s, ol. 25, Augus 1990, pp. 997-1004. [2] S. S. Lee, R. H. 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