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
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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
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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
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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).
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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
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(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].
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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
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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
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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)].
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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
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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.
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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.
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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.
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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
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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
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