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. Zele,
D.
J.
Alls o , and
G.
Liang, “CMOS
Con inuous-Time Cu en -Mode Fil e s o High-F equency
Applica ions,”
IEEE
J.
Solid-s a e Ci cui s,
ol. 28, Ma ch
[3]
A.
Rod iguez-Vizquez,
S.
Espejo,
R.
Dominguez-Cas o,
J.
L.
Hue as, and E. Sinchez-Sinencio, “Cu en -Mode Techniques
o he Implemen a ion
o
Con inuous- and Disc e e-Time
Cellula Neu al Ne wo ks,”
IEEE T ans.
on
Ci cui s and
Sys ems,
ol. CAS-40, Ma ch 1993, pp. 132-146.
[4] T. Ke ne , C. Hei e, and
K.
Schumache , “Analog CMOS
Realiza ion o Fuzzy Logic Membe ship Func ions,”
IEEE
J.
Solid-s a e Ci cui s,
ol. 28, July 1993, pp. 857-861.
[5]
J.
Choi and
B.
Sheu, “A High-P ecision VLSI Winne -Take-All
Ci cui o Sel -o ganizing Neu al Ne wo ks,”
IEEE
J.
Solid-S a e Ci cui s,
ol.
28,
May 1993, pp. 576-584.
[6]
J.
Lazza o, R. Ryckebush, M. A. Mahowald, and C. Mead,
“Winne -Take-All Ne wo ks o O(n) Complexi y,” in
Ad ances
in Neu al In o ma ion P ocessing Sys ems,
ol.
1, D.
S.
Tou e zky (Ed.), Los Al os, CA: Mo gan Kau mann, 1989, pp.
[7] R. Dominguez-Cas o, A. Rod iguez-Vizquez, F. Medei o and
J.
L.
Hue as, “High Resolu ion CMOS Cu en Compa a o s,”
P oc. o he 1992 Eu opean Solid-s a e Ci cui s Con e ence
[8]
N.
Mino sky,
Theo y
o
Nonlinea Con ol Sys ems,
Mac
G aw-Hill Inc., 1969.
[9] M.
J.
M. Pelg om, A. C.
J.
Duinmaije , and A. P.
G.
Welbe s,
“Ma ching P ope ies
o
MOS T ansis o s,”
lEEE
J.
Solid-S a e
Ci cui s,
ol. 24, Oc obe 1989, pp. 1433-1440.
[
1
O]E. Sackinge and W. Guggenbuhl, “A High-Swing,
High-Impedance MOS Cascode Ci cui ,”
IEEE
J.
Solid-S a e
Ci cui s,
ol.
25,
Feb ua y 1990, pp. 289-298.
[
1
1]E Medei o, A. Rod iguez-V izquez, R. Dominguez-Cas o, and
J.
L. Hue as, “Simula ion Based Au oma ed Analog Design
using S a is ical Op imiza ion,”
P oc.
o
he 1992 Eu opean
Solid-S a e Ci cui s Con e ence
(ESSCIRC’92), pp. 259-262.
[
12lT. Se ano and B. Lina es-Ba anco, “The Ac i e-Inpu
Regula ed-Cascode Cu en Mi o ,”
IEEE T ans.
on
Ci cui s
and Sys ems
( o appea
as
an Exp ess Le e ).
1993, pp. 323-329.
703-7
1
I.
(ESSCIRC’92), pp. 242-245.