Hindawi Publishing Co po a ion
VLSI Design
Volume 2008, A icle ID 657207, 8pages
doi:10.1155/2008/657207
Resea ch A icle
Simple E alua ion o he Nonlinea i y Signa u e o
an ADC Using a Spec al App oach
E. J. Pe al´
ıas, M. A. Jal´
on, and A. Rueda
Cen o Nacional de Mic oelec ´
onica (CNM-CSIC), Ins i u o de Mic oelec ´
onica de Se illa (IMSE), Uni e si y o Se illa,
Edi icio CICA, A enue Reina Me cedes s/n, 41012 Se illa, Spain
Co espondenceshouldbeadd essed oE.J.Pe al
´
ıas, [email p o ec ed]
Recei ed 17 Oc obe 2007; Re ised 30 Ma ch 2008; Accep ed 16 May 2008
Recommended by Ma celo Lubaszewski
This wo k p esen s a new me hod o es ima e he nonlinea i y cha ac e is ics o analog- o-digi al con e e s (ADCs). The me hod
is based on a nonnecessa ily polynomial con inuous and diffe en iable ma hema ical model o he con e e ans e unc ion, and
on he spec al p ocessing o he con e e ou pu unde a sinusoidal inpu exci a ion. The simula ion and expe imen s pe o med
on diffe en ADC examples p o e he easibili y o he p oposed me hod, e en when he ADC nonlinea i y pa e n has e y s ong
discon inui ies. When compa ed wi h he adi ional code his og am me hod, i also shows i s low cos and efficiency since a
signi ican lowe numbe o ou pu samples can be used s ill gi ing e y ealis ic INL signa u e alues.
Copy igh © 2008 E. J. Pe al´
ıas e al. This is an open access a icle dis ibu ed unde he C ea i e Commons A ibu ion License,
which pe mi s un es ic ed use, dis ibu ion, and ep oduc ion in any medium, p o ided he o iginal wo k is p ope ly ci ed.
1. INTRODUCTION
The pa ame e s ha cha ac e ize he ans e unc ion o
an ADC, such as he in eg al nonlinea i y (INL), a e some
o he mos impo an speci ica ions ha mus be known
o insu e he co ec ope a ion o he ADC in a ce ain
applica ion. One o he s anda dized me hods o es ima e
hese pa ame e s is he code his og am es [1,2]. I s
main d awback is he excessi e cos o i s applica ion,
especially due o he la ge numbe o samples ha mus
be acqui ed (mo e han one million) and o he ac ha
his numbe inc eases, in gene al, in an exponen ial way
wi h he numbe o bi s o he ADC. The synch oniza ion
be ween he acquisi ion and he signal s imula ion is also a
nonelemen al ask. On he con a y, his me hod can achie e
a e y p ecise measu emen independen ly o he shape o
he ans e cha ac e is ic; i is a clea ade-offbe ween ime
and p ecision.
These d awbacks make he his og am me hod un easible
o low-speed and high- esolu ion con e e s (>15 bi s).
Fo hese kinds o con e e s, he use o me hods based
on spec al p ocessing can be sa is ac o y acqui ing only
some ens o housands o samples independen ly o he
ADC esolu ion [3–6]. Al hough his es ima e leads o a
pa ial desc ip ion o he s a ic beha io o he ADC and,
o example, i is no a eliable me hod o calib a e he
nonlinea i y, such desc ip ion may be enough o show a
mal unc ioning and i may be e y use ul o se a es p o ocol
[7–9].
This pape p esen s a new and simple me hod o ADC
nonlinea i y (INL) es ima ion using he spec al p ocessing
o i s esponse o a sine-wa e exci a ion. The me hod does
no equi e a conc e e unc ional o m o he ADC ans e
cu e o o he INL, as [3,4], no o apply speci ic expansion
se ies as in [6], al hough i is based on a con inuous and
diffe en iable ma hema ical model o he con e e ans e
unc ion. To each high p ecision in he INL es ima ion,
he ans e unc ion should ha e enough smoo h shape;
howe e , we will show ha i can be also applied in many
cases wi h s ong discon inui ies, gi ing INL signa u es good
enough o desc ibe he nonlinea beha io o he ADC, his
alida ing i s use o apid p oduc ion es . The ma hema ical
bases o his me hod a e he s anda d de ini ion o he INL
and he local a ia ion o he ADC ans e unc ion a ound
each ideal ansi ion.
The o ganiza ion o his pape is as ollows. Sec ion 2
in oduces he gene al hypo heses o apply he new me hod
and he ma hema ics o ob ain he INL signa u e. The
me hod adap a ion o i s applica ion h ough he spec al
es ima ion is de i ed in Sec ion 3.Sec ion 4 shows some
2VLSI Design
Low o de diffe en iable
ans e cu e Z(x) i ed o
a low densi y sampling
Ideal ans e cu e
Ac ual ans e cu e
unde a low densi y
sampling
A ideal ansi ion lk(poin A)
B
Z(x)≈Z(lk)+∂Z
∂x 


x=lk(x−lk)
ZZmax
Zmin
ADC
Nbi s
VR
Z
x
N
−VR
A
Z=k
q=1LSB lk=q.k k
x
Figu e 1: Modelling he non-linea i y: he i s o de Taylo ’s expansion o he ans e unc ion Z(x) a ound an ideal ansi ion lk. k
ep esen s he eal ansi ion whe e he ou pu code Zchanges om k−1 ok.
simula ed and eal examples o applica ion o he p oposed
me hod and Sec ion 5 d aws he conclusions.
2. MODELLING THE NONLINEARITY
The ADC basic model ha we a e conside ing supposes ha
he ans e cu e: (1) is a smoo h nonnecessa ily polynomial
unc ion Z(x), con inuous and diffe en iable wi h espec o
he inpu , (2) i mus be s ic ly inc easing and he e o e
wi h a nonze o de i a i e. The desc ip ion as a con inuous
unc ion mus be also unde s ood in he way ha he
esolu ion o he con e e is high enough o conside he
quan iza ion effec embedded in o he noise con ibu ions.
Ma hema ically, he p oposed inpu -ou pu model is
z=Z(x)+ε(x), ∃∂xZ/
=0, ∀x∈−VR,VR,(1)
whe e he e o unc ion ε(x) is o he same o de han
he quan iza ion e o . Wi hou any loss o gene ali y, he
inpu ange will be conside ed bipola and cen ed a ze o,
[−VR,VR].
Figu e 1 illus a es he nonlinea i y modeling. Bo h he
ideal and a hypo he ic eal ans e unc ion o an N-bi
con e e a e shown. The cu e Z(x) is ha ob ained by
i ing a low-densi y sampling o he eal ans e cu e.
Conside ing he p e ious condi ions, a i s -o de Taylo ’s
expansion o he mild ans e unc ion Z(x) is going o be
calcula ed a ound each ideal ansi ion, lk. The se o poin s
o med by he ideal ansi ions can be exp essed as lk=q·k,
k∈[−2N−1,2N−1−1], whe e q=2VR/2Nis he LSB o he
N-bi ADC. Then, we ha e
Z(x)≈Zlk+∂xZlk·x−lk,∀x≈lk.(2)
E alua ing he exp ession a he co esponding eal ansi-
ion, x= k, whe e he ou pu code alue changes om k−1
o k,
k≈Zlk+∂xZlk·qINLk,(3)
whe e i has been used ha k≈Z( k) and he s anda d INL
de ini ion INLk=( k−lk)/q [1].
F om (3), he ollowing exp ession o he INL can be
ob ained:
INLk≈k−Zlk
q∂
xZlk.(4)
No ice ha his exp ession can be e alua ed only i i is
possible o ob ain he de i a i e o he unc ion Z(x).
I he second de i a i e exis s, an al e na i e exp ession
can be ob ained using he second-o de Taylo ’s expansion.
In any case, in his wo k only he exp ession (4) will be used
since he nonlinea i y o he ADC is conside ed e y small:
maxk{|INLk|} <2N−10(N>10).
3. APPLICATION OF THE MODEL USING
A SPECTRAL APPROACH
This sec ion shows how o apply exp ession (4) in he case
ha a spec al measu emen is used o p ocess he ADC
esponse o a sinusoidal inpu .
Le us assume ha he inpu exci a ion is
x( )=Acos ωx +ϕx+B(5)
and ha i co e s all he inpu ange wi hou causing he
ADC sa u a ion (A≈VR,B≈0). The inpu equency ωx
is low enough o p oduce unimpo an dynamic effec s. The
phase-shi ϕxand he offse Bdo no need o be known a
p io i. The ampli ude, A, has o be known only when he
gain o he ADC, g,is e ydiffe en om he uni y and/o i
is wan ed o es ima e he gain e o o he A/D con e sion.
Fo such inpu , he ADC ou pu is a supe posi ion o
ha monics o he exci a ion equency,
Zx( )=C0+
n≥1
Cncosωn +ϕn,ωn=nω1,(6)
E. J. Pe al´
ıas e al. 3
whe e he equency ωxand he phase-shi ϕxa e iden i ied
wi h he equency ω1and he phase-shi ϕ1o he main
ha monic, espec i ely. Rejec ing, in a i s app oxima ion,
he o he ha monics (n>1), he inpu ampli ude can also
be ela ed o he ampli ude o he 1s ha monic by means o
he gain go he ADC. Using a simple linea model Z(x)≈
(g·x)/q +zos,whe egis he gain and zos is he offse o he
ADC [1], i can be ob ained as ollows:
C1=g·A/q,
C0=(g·B)/q +zos,
ω1=ωx,
ϕ1=ϕx.
(7)
A mo e sui able model, bu mo e cos ly, could s ill use he
p e ious linea ela ionship bu conside ing as ou pu s he
sine-wa e signal z( )=CAcos(ωz +ϕz)+CB ha bes i s o
(6) in he sense o he leas mean-squa ed e o . In his case,
exp essions (7) a e s ill alid using he pa ame e s CA,CB,
ωz,andϕz. In any case, wha e e he model is, he ans e
unc ion de i a i e in (4) can now be calcula ed in an indi ec
way:
∂xZ=∂ Zx( )
∂ x( )=n≥1ωnCnsinωn +ϕn
ωxAsinωx +ϕx
≈g
q1+
n≥2
Cn
C1
nsinnω1 +ϕn
sinω1 +ϕ1,
(8)
whe e he ela ions in (7) ha e been applied o he simples
model.
Now, le us e alua e exp essions (6)and(8) a he ideal
ansi ions, Z(lk), ∂xZ(lk).
I τka e he ime ins an s in which he sinusoidal inpu
signal c osses he ideal ansi ions o he ADC, x(τk)=lk(see
Figu e 2), i can be w i en as
lk=Acos ωxτk+ϕx+B
−→ ωxτk=−ϕx+A ccos
lk−B
A.(9)
Applying ela ions (7),
δk=ω1τk≈−ϕ1+A ccos
g·k+zos −C0
C1(10)
ha gi es he phases used in (6)and(8)
ωn +ϕn| =τk=nω1τk+ϕn=nδk+ϕn(11)
o ob ain, espec i ely, Z(lk)and∂xZ(lk). Finally, exp ession
(4) o he INL becomes
INLk≈k−C0+n≥1Cncos nδk+ϕn
g·1+n≥2Cn/C1nsinnδk+ϕn/sinδk+ϕ1.
(12)
No ice ha in (10)and(12) he only quan i ies o e alua e
a e he ha monic ampli udes {Cn}, he phase-shi s {ϕn},
(Ideal ans e cu e)
Z
k
j
k+1
k
k−1
j
−VRljlk+VR
x
τk
τj
x( )=Acos(ωx +ϕx)
(Sinusoidal inpu wa e)
Figu e 2: De ining he inpu wa e c ossing poin s τko e he ADC
ideal ansi ions lk. Acco ding o (10), hese iming poin s allow
wo k ou he phase alues o he ou pu ha monics when he inpu
wa e c osses an ideal ansi ion.
and he gain go he ADC. All o hem can be es ima ed using
he spec al p ocessing o he ou pu . In gene al, zos in (10)
can be conside ed null, which is equi alen o conside ha
zos is cancelled ou by he inpu offse . When g=1andzos =
0a eusedin(10)and(12), he ADC in insic nonlinea i y is
being es ima ed, ha is, he nonlinea i y signa u e wi hou
gain and offse effec s. This is he INL usually e alua ed
acco ding o he s anda ds [1,2].
Thenumbe o ha monics ha mus beselec ed oapply
(12) depends on bo h he spec al disc imina ion ha he
o al noise allows and he eliabili y o he ma hema ical
me hod used o es ima e he ha monic pa ame e s. In all o
ou expe imen s, we ha e used a conse a i e c i e ion: he
selec ed ha monics a e hose wi h ampli udes a leas 10 dB
o e he spec um noise loo . Ma hema ical me hods o
es ima e spec a a e basically based on disc e e- ime Fou ie
ans o m (DTFT) [1,2].
In e y good cohe en expe imen s (inpu equency
and sampling equency a e commensu able alues), and
when he noise is small enough and well desc ibed by
addi i e whi e model, simple ela ionships can be used.
Being acqui ed Lsamples o he ADC ou pu , {zi}L
i=1,wi h
he sampling equency sand sa is ying he inpu wa e
equency x=(M/L) swi h Mand L ela i e p ime in ege s
( o exci e a leas LADC diffe en le els), he DTFT is
ob ained o he ou pu egis e , {ζj}=FFT{zi}, selec ing
a e ha he mos signi ican spec al lines espec o he
noise loo , {ζjn}H
n=1. The main ha monic occu s a j1=
M. The co ec equencies o he o he lines mus be
4VLSI Design
ca e ully iden i ied because i high-o de ha monics exis ,
ωn>ω
s/2, hei spec al lines a e olded a he DTFT in e al
[0, s/2]. Wi h hese conside a ions, exp essions o es ima e
he pa ame e s in ol ed in (12)a e
ω1=2π
sM/L,
ωn=nω1,
C0=mean
izi,
Cn=(2/L)
ζjn
,
ϕn=(−1)pnA gζjn),
pn=
k≥0⎧
⎨
⎩
0, ωn/ωs∈k,(2k+1
/2,
1, ωn/ωs∈(2k+1)/2, (k+1)
,
g=q· mszi/ msx( ),
(13)
whe e ms{x( )}is he oo mean squa e o he inpu wa e
and pnco ec s he in e sion o es ima ed phase o he case
ha a spec al line is olded.
Al hough i has been sugges ed abou (12) ha he
alue o he gain gis no s ic ly necessa y o he in insic
nonlinea i y e alua ion, in (13) i has been included an
exp ession ha uses he oo mean squa e o i s es ima ion.
This exp ession allows a simple way o e alua e gbecause i
is easy o measu e he ms alue o he inpu signal using
awa me e .Fo alow-dis o ionADC(suchasi hasbeen
conside ed o he applica ion o he me hod), he diffe ence
o he gain alue es ima ed using his way is usually less han
0.5% espec o he gain alue es ima ed using a s anda dized
me hod, such as ha es ima ing g h ough he slope o he
bes - i ing line o he ansi ion se , o he slope o he
s aigh line ha joins he ex eme ansi ions on he ADC
ans e unc ion [1].
I R egis e s (R>1) o leng h La e acqui ed o a e age
he noise, all o hem mus be consecu i ely aken o do
no lose he phase in o ma ion. This is he same as o ace
a unique egis e wi h R·Lou pu samples o L-sample
pe iodici y. In his way, (13) is s ill used bu wi h he a e aged
module and he a e aged phase spec um alues,
ζj=meanm=1,...,Rζ(m)
j
2,
A gζj=meanm=1,...,RA gζ(m)
j,
wi h ζ(m)
jR
m=1=FFTz(m)
i.
(14)
I a cohe en sampling is no possible, windowing o each
egis e should be applied o educe he spec al leakage,
{ζ(m)
j}R
m=1=FFT{wiz(m)
i},whe e{wi}L
i=1is he con olu ion
window. In gene al, he exp essions in (13)ha e obe
co ec ed since he es ima es a e biased depending on he
applied window. In [2,10], he e a e some sugges ions
o selec he mos app op ia e window. Fo all o ou
noncohe en expe imen s, we ha e used a 4- e m cosine-
class window and an es ima ion me hod based on phase
eg ession p esen ed in [11], which equi es ha R≥2.
When high-o de ha monics exis and hey a e olded
in he [0, s/2] band, a pa icula inpu equency has o be
selec ed in such a way ha no signi ican ha monic o e laps.
P oblems a e mino i cohe en sampling and no windowing
a eapplied.Weha ealwaysselec ed es equencies ha lead
spec al lines o sepa a e each o he a leas 10 bins, when 4-
e m cosine window has been used.
The DC componen is e alua ed by means o he
weigh ed mean o he samples, using as weigh unc ion he
con olu ion window,
C0=mean
miwiz(m)
i
iwi.(15)
4. APPLICATION EXAMPLES
4.1. Simula ed expe imen s
This sec ion shows he simula ion esul s ob ained applying
he in oduced me hod in wo diffe en models o he
ADC. The i s con e e , ADC1, has a e y egula ans e
cu e and a smoo h INL. Fo he second one, ADC2, he
ans e cu e is nonmono onic and i s INL has e y s ong
discon inui ies.
4.1.1. Nonspec al app oach (DC sweep) on low-speed,
high- esolu ion ADC1model
Be o e applying (12), le us show he immedia e app ox-
ima ion o (4). The ADC1has been exci ed using a DC
signal whose alue is changing inside he inpu ange. Fo
each alue o he inpu , xi, i is ob ained an ou pu egis e
o calcula e he co esponding a e age code zi. Using he
se {(xi,zi)}i can be buil he M h o de bes - i ing
polynomial model, ZM(x), which allows o apply (4) di ec ly.
ADC1is a high-le el model o a 14-bi ΣΔ con e e , wi h
e e ence ol ages 0.0 V and 5.0 V bu wi h a p ac ical inpu
ange [0.5 V,4.5 V]. The model ep oduces he nonlinea ,
noisy, and equency beha io . The sampling equency is
s=100 kHz. The noise due o bo h he inpu and he
ADC, e e ed o he inpu , is app oxima ely 2LSBs ms whi e
noise. A DC sweep wi h abou 4000 poin s has been ca ied
ou in he ange [0.7 V, 4.3 V]. Fo each DC alue, i has been
aken a egis e o 50 poin s, being he ob ained code in he
ange Ik =[−5792,5801]. Using his da ase , he M=32nd-
o de bes - i ing polynomial has been calcula ed by means
o he i s class Chebyshe base. The choice o he o de
is deduced om he ha monic signi ican numbe ound in
he expe imen ha will be desc ibed in nex subsec ion. Fo
each alue ko Ik, i has been e alua ed ZM(lk), ∂xZM(lk),
and INLkin (4).
On he o he hand, i has been applied he s anda d
sinusoidal his og am me hod [1,2] ode e mineagood
es ima e o he INL ( eal INL om now on) in o de o
es ablish a e e ence o compa ison pu pose. The ob ained
esul s a e depic ed in Figu e 3.
Figu e 3(a) shows he supe posi ion o bo h es ima es.
The hick line is he cu e calcula ed by (4). To show he
eliabili y achie ed, in Figu e 3(b) i is depic ed he diffe ence
E. J. Pe al´
ıas e al. 5
Es ima ed INL: DC inpu
−11
−9
−7
−5
−3
−1
LSB (@14bi s)
00.511.522.533.544.55
Inpu ange ( ol s)
(a)
Diffe ence be ween he abo e cu es
0
0.2
0.4
0.6
0.8
1
LSB (@14bi s)
00.511.522.533.544.55
Inpu ange ( ol s)
(b)
Figu e 3: (a) INL es ima es o he example ADC1. In hick (black)
line he one ob ained by (4), in hin (blue) line he eal INL
e alua ed by he His og am me hod. (b) Diffe ences be ween bo h
es ima es in (a).
be ween bo h cu es, which is affec ed by a sys ema ic 1/2
LSBe o .Thisisbecause, o con enience,(2)and(3)ha e
been ob ained using he ansi ions and no he code cen es.
This is no e y impo an since, in gene al, he offse alue o
he ADC is no used in he exp ession (12) and so i mus be
co ec ed by means o he elimina ion o i s mean alue. In
any way, i can be no iced he high accu acy achie ed by he
me hod. This is jus i iable since he INL cu e o his ADC
ollows a e y smoo h beha io .
4.1.2. Spec al app oach (sine inpu ) on low-speed,
high- esolu ion ADC1model
Fo he same ADC1con e e , i has been simula ed a se
o 35 expe imen s ha use a sinusoidal inpu and calcula e
he INL om (12). To es ima e he pa ame e s om he
spec um, i has been used he p ocedu e desc ibed in [11].
The ampli ude o he sinusoidal inpu is app oxima ely
−3 dBFS, he offse is abou 100LSBs, and he equency is
nea s/83. The phase has been e enly sp ead inside he ange
[−π,π] all o e he expe imen s. The equi alen noise a
he inpu is app oxima ely 1LSB ms whi e noise. In each
expe imen , R=4 consecu i e egis e s o L=4096 samples
a e used. A se o 500 samples ha e been elimina ed a he
beginning o he ou egis e s o educe he se ling e o s. A
ypical spec um is shown in Figu e 4(a). As he backg ound
noise is abou −112 dBFS, he spec al lines selec ed a e hose
ha ha e he ampli ude highe han −102 dBFS. The ypical
selec ion de ec s abou 15 ha monics wi h o de s up o
30 h. Figu e 4(b) shows a compa ison be ween he eal INL
1
2
3
4
5 6
7
9 11 12 13 14 20 22 24
29 32
15
−120
−100
−80
−60
−40
−20
0
ADC1ou pu a e aged magni ude spec um:
R×L=4×4096 samples
(dBFS)
01020304050
F equency (kHz)
−102 dBFS
(a)
Es ima ed INL: his og am es compa ison
−10
−6
−2
2
6
10
LSB (@14 bi s)
00.511.522.533.544.55
Inpu ange ( ol s)
His og am es wi h he
4×4096 acqui ed samples
Real INL
(b)
Figu e 4: (a) Typical a e aged magni ude spec um ob ained om
he ADC1ou pu . I is used o he selec ion o he spec al lines
aken as ha monics. (b) Compa ison be ween wo ADC1INL
His og am es ima ions: in hin (blue) line he one es ima ed wi h
e y ew samples, ( he ou egis e s acqui ed o apply he new
me hod). In hick (black) line he eal INL.
( hick line) and he es ima ed ( hin line) using he his og am
me hod bu wi h only he abo e ou acqui ed egis e s. I
is ob ious ha he numbe o acqui ed samples is no s ill
enough o ske ch he INL shape. On he o he hand, he
esul s ob ained in one o he expe imen s using he spec al
es ima ion and (12) a e shown in Figu es 5(a) and 5(b).As
i can be no iced, ou me hod gi es good enough esul s
wi h he 4·4096 acqui ed samples. Bo h INL cu es ha e
been depic ed wi h he offse co ec ed, since he offse o he
ADC1has been supposed null zos =0. To show how obus
he es ima e om (12)is,Figu e 5(c) depic s he diffe ences
be ween he INL o each expe imen and he eal INL.
4.1.3. Spec al app oach (sine inpu ) on medium-speed,
high- esolu ion model ADC2
The ADC2is a high-le el model o a 16-bi pipeline con e e
ha ope a es wi h e e ence ol ages o −2.0 V and 2.0 V.
The sampling equency is s=5.0 MHz. The noise due
o bo h he inpu and he ADC, e e ed o he inpu , is
app oxima ely 1LSB ms whi e noise. The ampli ude o he

6VLSI Design
Es ima ed INL: sine-wa e inpu
−5
−3
−1
1
3
5
LSB (@14 bi s)
00.511.522.533.544.55
Inpu ange ( ol s)
(a)
Diffe ence be ween he abo e cu es
−0.5
−0.3
−0.1
0.1
0.3
0.5
LSB (@14 bi s)
00.511.522.533.544.55
Inpu ange ( ol s)
(b)
Diffe ences ob ained o 35 diffe en expe imen s
−0.5
−0.3
−0.1
0.1
0.3
0.5
LSB (@14 bi s)
00.511.522.533.544.55
Inpu ange ( ol s)
(c)
Figu e 5: (a) O e lapped ADC1INL cu es: in hick (black) line
he one ob ained by spec al es ima ion and (12). In hin (blue) line
he eal INL.(b)Diffe ence be ween he wo abo e INL signa u es.
(c) Diffe ences ob ained o all o he expe imen s on he ADC1.
inpu signal is app oxima ely −0.2 dBFS, he offse is abou
10LSBs and he equency is nea s/222.
The phase has been e enly sp ead allo e he expe imen s
inside he ange [−π,π]. In hese expe imen s, i has been
aken wo consecu i e egis e s wi h 32768 samples each.
In his example, he backg ound noise appea s a abou
−130 dBFS, so he selec ed spec al lines a e hose ha a e
ha monics wi h ampli udes highe han −120 dBFS.
The so i egula and discon inuous s uc u e o he
INL o his ADC leads o a ypical selec ion abou 150
ha monics wi h o de s up o 600 h. Figu e 6 shows he
ypical esul s ob ained. Figu e 6(a) shows a compa ison
be ween he eal INL ( hick line) and he one es ima ed
( hin line) using he his og am me hod wi h he abo e wo
acqui ed egis e s. The numbe o acqui ed samples is no
s ill enough o ske ch he INL shape. Howe e and in spi e
o such a discon inuous s uc u e o he nonlinea i y o he
ADC2, he es ima e om (12) absolu ely ollows he eal
Es ima ed INL: his og am es compa ison
−30
−20
−10
0
10
20
30
LSB (@16 bi s)
−2−1.5−1−0.50 0.511.52
Inpu ange ( ol s)
His og am es wi h he2×32768
acqui ed samples
Real INL
(a)
Es ima ed INL: sine-wa e inpu
−20
−10
0
10
20
LSB (@16 bi s)
−2−1.5−1−0.50 0.511.52
−3
−1
1
3−0.3−0.2−0.10 0.10.20.3
Inpu ange ( ol s)
(b)
Diffe ence be ween he abo e cu es
−3
−2
−1
0
1
2
3
LSB (@16 bi s)
−2−1.5−1−0.50 0.511.52
Inpu ange ( ol s)
(c)
Figu e 6: (a) Compa ison be ween wo ADC2INL His og am
es ima ions: in hin (blue) line he one es ima ed wi h e y ew
samples ( he wo egis e s acqui ed o apply he new me hod). In
hick (black) line he eal INL, (no gain effec has been included).
(b) O e lapped ADC2INL es ima ions: he es ima ed using he
spec al app oach and (12), and he eal INL (gain effec s a e
included). (c) Diffe ence be ween he abo e INL signa u es in (b).
INL e alua ed using he s anda dized his og am me hod
(Figu e 6(b)). The diffe ence be ween bo h es ima es has
been depic ed in Figu e 6(c). Bigges diffe ences occu a
he highe ansi ions, whe e he smoo hing effec due o
he limi ed numbe o ha monics ha has been selec ed is
mo e e iden . E en i he gain is no e alua ed, he in insic
nonlinea i y o he ADC ( he one whe e he gain and he
offse a e co ec ed) s ill can be ex ac ed. Figu e 7(a) shows
he esul s ob ained o he ADC2making g=1in(10).
Al hough he diffe ence is no iceable, i he gain and offse
a e elimina ed om bo h cu es (sub ac ing hei bes -
E. J. Pe al´
ıas e al. 7
Es ima ed INL wi hou gain co ec ion (gain=1)
−20
−10
0
10
20
LSB (@16 bi s)
−2−1.5−1−0.50 0.511.52
Inpu ange ( ol s)
(a)
Es ima ed INL wi hou offse and gain effec s
−5
−3
−1
1
3
5
LSB (@16 bi s)
−2−1.5−1−0.50 0.511.52
Inpu ange ( ol s)
(b)
Figu e 7: (a) Thick (black) line: es ima ed ADC2INL using (12)
and aking he gain equal o 1. In hin (blue) line he eal INL
whe e gain effec s a e included. (b) O e lapping o he wo abo e
es ima ions a e he gain and offse effec s ha e been co ec ed.
i ing lines), a good app oxima ion can be achie ed as
Figu e 7(b) shows.
4.2. Real expe imen s
This sec ion shows he esul s ob ained applying he in o-
duced me hod o a eal ADC. This con e e has a ans e
cu e wi h e y s ong discon inui ies.
4.2.1. Spec al app oach (sine inpu ) on high-speed,
low- ol age expe imen al p o o ype ADC
TheADCunde es isa ullydiffe en ial 12-bi pipeline
con e e p o o ype in a 120 nm CMOS echnology wi h
e e ence ol ages −1 V and 1 V. The sinusoidal inpu has
been nonbuffe ed AC coupled o he ADC and gene -
a ed using he Agilen N8241A AWG, wi h ampli ude o
−0.1 dBFS and a equency o 500 kHz app oxima ely. In
his case, as a good cohe en expe imen has been done,
only a egis e o 4090 samples has been acqui ed using
a 20 MHz sampling mas e clock (nei he a e aging no
windowing has been applied). Figu e 8(a) shows a ypical
nonwindowed magni ude spec um, wi h he loo noise
a abou −100 dBFS. The ha monics selec ed we e hose
wi h he ampli ude highe han −90 dBFS, and he ypical
selec ion akes abou 45 ha monics wi h o de s up o 150 h.
Figu e 8(b) compa es he INL pa e n ob ained using he
spec al app oach ( om (12)) wi h he eal INL es ima ed
1
3
5 6 7 8 9
13 14
15
16 20
21
22
23
25
27
28
30
31
33
35
36
37
38
41 42
45 47
48 50 51 52 53 60
74
76 100
131
147
2
P o o ype pipelined ADC: magni ude spec um
R×L=1×4090 samples
−120
−100
−80
−60
−40
−20
0
(dBFS)
01.22.53.756.27.58.710
F equency (MHz)
−90dBFS
(a)
P o o ype pipelined ADC: es ima ed INL signa u es by
spec al and his og am me hods
Real INL Spec al es ima ion
−4
−3
−2
−1
0
1
2
3
4
LSB (@12 bi s)
−1−0.75 −0.5−0.25 0 0.25 0.50.75 1
Inpu ange ( ol s)
(b)
Figu e 8: (a) Typical magni ude spec um ob ained om he
p o o ype Pipeline ADC ou pu . (b) Compa ison be ween wo INL
es ima ions: In hin (blue) line, he eal INL. In hick (black) line
he INL es ima ion om he spec al app oach and (12) using he
spec um in (a) (gain and offse effec s ha e been co ec ed).
using he s anda d his og am me hod. No ice ha he INL
e alua ed om ou p oposed me hod desc ibes he shape
o he eal INL good enough, e en in ha d discon inui ies.
Bes esul s a e ob ained i mo e egis e s ha e been acqui ed,
since noise a e aging imp o es bo h unce ain y and he
numbe o selec ed ha monics.
5. CONCLUSIONS
In his pape , a new me hod o he INL es ima ion o
ADCs has been p esen ed which is based on a con inuous
model o he ADC ans e unc ion. The me hod uses
a spec al p ocessing o he ADC ou pu o es ima e i s
ha monic ampli udes and phase-shi s om which he
INL signa u e is de i ed. Diffe en ADC examples wi h
e y diffe en nonlinea i y pa e n ha e been pe o med o
alida e he p oposed me hod. The ob ained esul s ha e
been compa ed wi h hose ob ained om he adi ional
his og am me hod and ha e p o en no only he easibili y
o he new me hod, e en when he ADC nonlinea i y has
e y s ong discon inui ies, bu also i s low cos and efficiency
since a signi ican lowe numbe o ou pu samples can be
used s ill gi ing e y ealis ic INL signa u e alues.
8VLSI Design
ACKNOWLEDGMENTS
The au ho s would like o hank Jes´
us Ruiz and D . Manuel
Delgado, bo h om Mic oelec onic Ins i u e o Se ille, o
allow he applica ion o ou me hod o hei p o o ype ADC
exposed in Sec ion 4.2. This wo k is in pa suppo ed by he
Spanish P ojec TEC2007-68072 and he Andalusian P ojec
EXC/2005/TIC-927.
REFERENCES
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[9] V. Ke z´
e ho,S.Be na d,P.Cau e ,andJ.M.Janik,“A i s s ep
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