# Achievable ADC Performance by Postcorrection Utilizing Dynamic Modeling of the Integral Nonlinearity.

**ABSTRACT** There is a need for a universal dynamic model of analog-to-digital converters (ADC's) aimed for postcorrection. However, it is complicated to fully describe the properties of an ADC by a single model. An alternative is to split up the ADC model in different components, where each component has unique properties. In this paper, a model based on three components is used, and a performance analysis for each component is presented. Each component can be postcorrected individually and by the method that best suits the application. The purpose of postcorrection of an ADC is to improve the performance. Hence, for each component, expressions for the potential improvement have been developed. The measures of performance are total harmonic distortion (THD) and signal to noise and distortion (SINAD), and to some extent spurious-free dynamic range (SFDR).

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**ABSTRACT:**Integral nonlinearity (INL) for pipelined analog-digital converters (ADCs) operating at RF is measured and characterized. A parametric model for the INL of pipelined ADCs is proposed, and the corresponding least-squares problem is formulated and solved. The INL is modeled both with respect to the converter output code and the frequency stimuli, which is dynamic modeling. The INL model contains a static and a dynamic part. The former comprises two 1-D terms in ADC code that are a sequence of zero-centered linear segments and a polynomial term. The 2-D dynamic part consists of a set of polynomials whose parameters are dependent on the ADC input stimuli. The INL modeling methodology is applied to simulated and experimental data from a 12-bit commercial ADC running at 210 mega samples per second. It is demonstrated that the developed methodology is an efficient way to capture the INL of nowadays ADCs running at RF, and it is believed that the methodology is powerful for INL-based ADC postcorrection in wideband applications.IEEE Transactions on Instrumentation and Measurement 11/2010; · 1.36 Impact Factor - SourceAvailable from: Niclas Björsell[Show abstract] [Hide abstract]

**ABSTRACT:**An input-dependent integral nonlinearity (INL) model is developed for pipeline ADC post-correction. The INL model consists of a static and dynamic part. The INL model is subtracted from the ADC digital output for compensation. Static compensation is performed by adjacent sets of gains and offsets. Each set compensates a certain output code range. The frequency content of the INL dynamic component is used to design a set of filter blocks that performs ADC compensation in the time domain. The compensation scheme is applied to measured data of two 12-bit ADCs of the same type (Analog Devices AD9430). Significant performance improvements in terms of spurious free dynamic range (SFDR) are obtained.IEEE Transactions on Instrumentation and Measurement 01/2011; 60:3342-3350. · 1.36 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**The semiconductor industry tends to constantly increase the performances of developed systems with an ever-shorter time-to-market. In this context, the conventional strategy for mixed-signal component design, which is based only on analog design effort, will no longer be suitable. In this paper, a digital correction technique is presented for analog-to-digital converters (ADCs). The idea is to use a lookup table (LUT) for the online correction of integral nonlinearity (INL). The main challenge for this kind of technique is the cost in time and resources to estimate the actual INL of the ADC needed to load the LUT. In this paper, we propose to extract INL with a very rapid procedure based on spectral analysis. We validate our technique on a 12-bit folding-and-interpolating ADC and we demonstrate that the correction is efficient for a large range of application fields.IEEE Transactions on Instrumentation and Measurement 04/2011; · 1.36 Impact Factor

Page 1

Hindawi Publishing Corporation

EURASIP Journal on Advances in Signal Processing

Volume 2008, Article ID 497187, 10 pages

doi:10.1155/2008/497187

ResearchArticle

Achievable ADC Performance by Postcorrection Utilizing

Dynamic Modeling of the Integral Nonlinearity

Niclas Bj¨ orsell1and Peter H¨ andel2

1ITB/Electronics, University of G¨ avle, 801 76 G¨ avle, Sweden

2Signal Processing Lab., ACCESS Linnaeus Center, School of Electrical Engineering, Royal Institute of Technology,

100 44 Stockholm, Sweden

Correspondence should be addressed to Niclas Bj¨ orsell, niclas.bjorsell@hig.se

Received 1 May 2007; Revised 24 September 2007; Accepted 19 December 2007

Recommended by Boris Murmann

There is a need for a universal dynamic model of analog-to-digital converters (ADC’s) aimed for postcorrection. However, it is

complicated to fully describe the properties of an ADC by a single model. An alternative is to split up the ADC model in different

components, where each component has unique properties. In this paper, a model based on three components is used, and a

performance analysis for each component is presented. Each component can be postcorrected individually and by the method that

best suits the application. The purpose of postcorrection of an ADC is to improve the performance. Hence, for each component,

expressions for the potential improvement have been developed. The measures of performance are total harmonic distortion

(THD) and signal to noise and distortion (SINAD), and to some extent spurious-free dynamic range (SFDR).

Copyright © 2008 N. Bj¨ orsell and P. H¨ andel. This is an open access article distributed under the Creative Commons Attribution

License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly

cited.

1.INTRODUCTION

The analog to digital converter (ADC) is a key component

in many applications, for example, radio base stations and

test and measurement instruments. In state-of-the-art de-

signed vector signal analyzers (VSAs), the ADC is the bot-

tle neck and an improvement in ADC performance directly

improves the VSA performance. Characterization and test-

ing of ADC’s are important in many different aspects. One

example is ADC postcorrection, where improvements in the

ADC characteristics are obtained by digital signal process-

ing methods, in particular the error occurrence is predicted

in order to compensate for error source effects. A survey of

error compensation methods is given in [1]. Postcorrection

is built on the model of the ADC. A survey of state-of-the-

art ADC modeling techniques and models may be found in

[2]. Normally, the ADC model is based on a characteriza-

tion performed in high-performance test setups, whereupon

an off-chip postcorrection algorithm is developed. In the lit-

erature, a majority of the proposed ADC characterization

methods describes the static properties of the converter. A

common solution for postcorrection based on a static model

is the use of lookup tables (LUTs); that is, the ADC output

is remapped using a table lookup, where the table entries

are such that some performance measure is improved, as for

example [3]. ADC postcorrection by table lookup methods

has shown to improve performance measures such as spuri-

ous free dynamic range (SFDR), total harmonic distortion

(THD), and signal-to-noise and distortion ratio (SINAD).

It has been shown that postcorrection based on LUTs that

do not take the dynamics of the ADC into account is band

limited (see, e.g., [4, 5]). Characterization and testing of the

dynamic effects of ADC’s are instrumental for the perfor-

mance of systems characterized by a wide bandwidth and

high-dynamic range, such as contemporary and future mo-

bile telephony systems requiring higher resolution and sam-

pling rates.

Error tables can be used to characterize and compensate

ADC’s characterized by dynamic effects with short memory.

Here, two types of tables that are normally considered are

phase plane [6] and state space [3, 7], respectively. In a phase

plane approach, the error is related to the amplitude and

slope of the input, while for the state space the error is re-

lated to the current and the previous sample amplitude. In

[8, 9], a further development of the state space method is

suggested, where a generalized approach is taken with full

Page 2

2EURASIP Journal on Advances in Signal Processing

ADC

Inverse

model

v(t)

k(n)

...

k(n −d)

? v(n)

Figure 1: Postcorrection by using an inverse model of the ADC.

flexibility between the dynamics (i.e., the number of delayed

samples) and the precision or number of bits of each sample.

Thus, the size of the multidimensional lookup table is kept at

a reasonable predetermined number. However, these meth-

ods are burdensome considering the time they take to train

the entries of the LUT, as well as the requirement on the size

of the memories. Accordingly, there is a need for dynamic

postcorrection that is easy to train and simple to implement.

A parametric model requires less memory size than an

LUT and does not need to be trained for every combination

of present and previous samples. A well-assigned model is

able to describe scenarios for which it is not trained, even

though one should strive to train the model with a stimulus

as realistic as possible for a given application. Different types

of parametric models have been suggested in the literature,

such as Volterra models and a variety of different box models

(e.g., Wiener and Hammerstein models). These models de-

scribe the nonlinear dynamic behavior well, but typical error

models of an ADC also include components that can not be

described by nonlinear dynamic models. That implies that

a postcorrection that compensate for multiple kinds of er-

ror behavior might be based on two or more models, where

each error behavior is compensated with a suitable postcor-

rection method. The purpose of this paper is to provide a

tool to evaluate to what extent a given parametric model can

improve the ADC performance, when the model is used for

postcorrection.

Postcorrection can be divided into two different meth-

ods. One method is to use an inverse ADC model and the

other is to add a correction term. When using LUT for post-

correction, the two methods are often denoted replacement

and correction, respectively. The inverse model corresponds

to replacement. The output code from the ADC is a table in-

dex. The code addresses a memory, where the memory value

of that address is an estimate of the analog input. The index

can also be compounded by one or more previous samples

(see Figure 1).

Figure 1 can also represent a correction method based

on inverse models. In other words, the method is based on

some mathematical system model and its inverse. Typically,

a model is characterized for the ADC under test. The model

gives an approximation of the input-output signal relation-

ship.Aninverse—possiblyapproximate—ofthemodeliscal-

culated, thereafter. The model inverse is used in sequence af-

tertheADC,henceoperatingontheoutputsamples,inorder

to reduce or even cancel the unwanted distortion.

Instead of replacing the output code from the ADC, one

can add a correction term. (see Figure 2). In postcorrection

using LUT, the output sample (possibly together with previ-

ous samples) addresses a correction term instead of an esti-

G

v(t)

ADC

? v(n)

?

INL[k,ω]

INL[k,ω]

Postcorrection

k[n]

Figure 2: Postcorrection of an ADC by adding a correction term.

The block G includes analog preprocessing, sample and hold, and

quantization.

0

1

2

3

4

5

6

7

Vmin

Output digital code k

0

Analog input V (Volt)

Vmax

Figure 3: The relationship between the analog input signal v and

the digital output code k from an ideal n = 3bits ADC (dashed

line) and a practical one (solid line).

mate of the input as in the replacement method. The cor-

rection term is added to the output code. In model-based

postcorrection, the postcorrection term is computed from

a mathematical model. The correction term is added to the

output code. In a static postcorrection, the correction term

corresponds to the ADC integral nonlinearity (INL).

2.BASIC PROPERTIES OF ADC NONLINEARITIES

The relationship between the analog input signal V [Volt]

and the digital output code k from an ideal ADC approxi-

mates the dotted staircase transfer curve shown in Figure 3.

For the ideal ADC, the code transition levels Tk[Volt] within

the ADC range (Vmin, Vmax) [Volt] are given by

Tk= Q(k −1) +T1[Volt],(1)

where Q [Volt] is the ideal width of a code bin; in other

words, the full-scale range of the ADC is divided by the to-

tal number of codes (Vmax− Vmin)/2N, where N denotes the

number of bits. Further, T1is the ideal voltage correspond-

ing to first transition level, and T1is equal to Vmin+ Q or

Vmin+ Q/2 depending on the convention used: the “mid-

riser” convention or “mid-tread” convention, respectively

[10]. The code k spans k = 1,...,2N−1.

Due to imperfections in all practical ADC’s, the trans-

fercurveisnormallysomewhatdistorted,whichisillustrated

Page 3

N. Bj¨ orsell and P. H¨ andel3

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

INL [LSB]

0500 10001500

Transition level [LSB]

2000250030003500 4000

Figure4:ExemplarymeasuredINLfroma12bitcommercialADC.

by the solid line in Figure 3. The actual code transition level

T[k] [Volt] (i.e., the ideal and practical transition levels are

distinguishedbytheplacementoftheargumentk,viz.Tkand

T[k], resp.) is the voltage that results in a transition from

ADC output code k − 1 to k. The INL is described as the

difference between the ideal Tk in (1) and the actual T[k]

code transition levels of the ADC, after a correction has been

made for gain and offset errors [10, 11]. Given the ideal code

transition levels Tkin (1) and the measured levels T[k], the

correction is made by adjusting the gain G and offset Vosin

order to “minimize” the residual ε[k] (for k = 1,...,2N− 1)

[10],

ε[k] = Tk−G ·T[k] −VOS[Volt].

(2)

Equation(2) describesanoverdetermined setof2N−1equa-

tions with the two unknowns G and Vosthat are sought for.

AccordingtotheIEEEstandards[10],differentmethodsmay

be applied for determining the optimal (G, Vos)-pair such as

the “terminal-based” method [10] as used in this paper. The

INL as a percentage of the full scale (FS) range of the ADC is

given by the normalized residual in (2), that is,

INL[k] =100% ·ε[k]

2NQ

[% of FS].

(3)

TheINLisnormallyexpressedinleastsignificantbits(LSBs),

where a LSB is synonymous with one ideal code bin width Q

[Volt], that is, INL[k] = ε[k]/Q[LSB].

Differential nonlinearity (DNL) is the difference, after

correcting for the obtained static gain G, between a speci-

fied code bin width and the ideal code bin width Q, divided

by the ideal code bin width. The DNL is given as follows:

DNL[k] =W[k] −Q

Q

[LSB],(4)

where W[k] is the corrected width of code bin k, that is,

W[k] = G(T[k + 1] − T[k]). From (2), it follows that

ε[k + 1] − ε[k] = Q − W[k], and thus the relation between

INL[k] and DNL[k] is

DNL[k] = INL[k +1] −INL[k].

Theroot-mean-square(RMS)valueoftheDNLiscommonly

used and given by

(5)

DNLRMS=

?

1

2N−1

2N−1

?

k=1

DNL2[k]

?1/2

.

(6)

3.PARAMETRIC INL MODELING FOR

ADC POSTCORRECTION

In Figure 4, a typically measured INL from a commercial

ADC is plotted. As it is evident from the plot, the behav-

ior is a combination of a smooth wave or polynomial curve

and a prickly sawtooth wave. In the following analysis, the

INL will be broken up in two components; one representing

the smooth curve and one representing the prickly sawtooth

wave. The INL[k] is then described as

INL[k] =HCFINL?[k] +LCFINL[k],

where the first term is the contribution by the, so-called,

high-code frequency component and the second term by

the low-code frequency component, respectively. In [12], the

static INL model was expressed as a one dimensional image

in the code k domain consisting of the two components. The

smooth curve was entitled low-code frequency (LCF), to un-

derstand the meaning of low-frequency code, consider that

thecodeaxisrepresentatimeaxis.Accordingly,low-codefre-

quencymeansslowvariationoverthecodes,[13]component

denoted byLCFINL[k] and was represented by a polynomial

approximation:

(7)

LCFINL[k] = h0+h1k +h2k2+ ···hLkL,

where the hk’s are the polynomial coefficients and L is the

order of the polynomial. The parameters h0and h1are typ-

ically set to zero due to the fact that INL is calculated after

a correction has been made for gain and offset errors when

determining the INL [10]. The high code-frequency compo-

nentHCFINL?[k] is caused by a significant deviation between

the polynomial approximation (8) and the actual INL[k].

In [14], the high-code frequency component was further

divided into two parts:HCFINL[k] andNoiseINL[k], respec-

tively. The former term,HCFINL[k], depends on the physical

design of the component (designs such as pipeline, succes-

siveapproximation,oranyotherstructure)andismodeledas

piecewise linear [12, 15]. The latter component,NoiseINL[k],

isthepartof INL[k]thatcannotbedescribedbyanequation.

Thus, the INL[k] model in (7) is refined to

(8)

INL[k] =HCFINL[k] +LCFINL[k] +NoiseINL[k].

A static model of an ADC and in particular the correspond-

ing INL[k] is in general not sufficient to accurately describe

an ADC in a wideband application. Hence, the dynamic be-

havior also needs to be included in the model, which can be

(9)

Page 4

4EURASIP Journal on Advances in Signal Processing

done by adding amplitude information from either previous

sample amplitudes or estimates of the input slope, which is

state-space and phase plane modeling, respectively. The dy-

namic behavior of the INL can alternatively be described as a

frequencydependency,thatis,differentsinewaveteststimuli

result in different INL data. One may note that frequency se-

lective LUTs for ADC postcorrection was considered in [16].

In order to stress the dependency of some of the components

in (9) on the stimuli frequency ω, (9) is rewritten as

INL[k,ω] =HCFINL[k]+LCFINL[k,ω]+NoiseINL[k,ω],

(10)

where ω denotes the normalized frequency variable,

ω =2π f

fs

, (11)

where f is the actual frequency in Hertz and fsis the sam-

pling frequency.

The main purpose of the model (10) is to use it for

ADC postcorrection. The structure of the components of the

model may, to some extent, be affected by the aim to find a

dynamicmodelthatiseasytotrainandsimpletoimplement.

Even though the behavior models are black-box mod-

els, the arguments for having a staticHCFINL[k] can be jus-

tified based on some knowledge of the ADC design. The

hardware structure of an ADC is consisting of two sections.

First is an analog signal processing section with an ampli-

fier and sample-and-hold circuits followed by a section per-

forming the quantization. The high-code frequency compo-

nentHCFINL[k] is mainly representing the imperfections in

the quantizer, which are (at least in a first approximation)

static and thus depend on the code k only and not on the

test frequency. One favorable feature with considering the

high-code frequency component to be static is that the size

of the lookup table will be minimized. The low-code fre-

quency componentLCFINL[k,ω] is a two-dimensional para-

metric model describing a dynamic behavior. Due to the pa-

rameterization, the postcompensation can be implemented

by numerical calculations.

The componentLCFINL[k,ω] is described as a nonlin-

ear dynamic model and can be described by different model

structures. In [17], the input-output relation of the ADC was

explored based on measured Volterra kernels [18]. In partic-

ular, in that paper it was argued for employing a constrained

nonlinear Volterra model known as the parallel Hammer-

stein model [19]. Based on the promising results in [17], the

parallel Hammerstein model is used in this work to analyze

the nonlinear dynamic parts of the integral nonlinearity, as

well. The basic Hammerstein model is given by a static non-

linearity, followed by a linear filter. The difference between

the ordinary Hammerstein model and the parallel structure

is that the contributions of different orders l are now fil-

tered by different filters defined by their frequency functions

Hl[ω], respectively. Starting with the polynomial nonlinear-

ity i (8), a frequency dependency is incorporated by letting

the polynomial coefficient be frequency dependent, that is,

LCFINL[k,ω] = h0(ω)+h1(ω)k +h2(ω)k2+ ··· +hL(ω)kL.

(12)

The above dynamic nonlinearity can be described in

terms of a parallel Hammerstein model with the lth single-

input multiple-output given by kl, and the zero phase linear

filterswithfrequencyfunctionhl(ω).Insummary,theparal-

lel Hammerstein structure with a polynomial nonlinearity is

a natural generalization of the static polynomial model (8).

Although in this paper constraints are imposed both on the

static nonlinearity as well as the phase characteristics of the

bank of linear filters, the obtained dynamic model will be

used throughout this paper in order to analyze and compen-

sate for the nonlinear dynamic parts of ADC integral nonlin-

earity.

TheLCFINL[k,ω] is a continuous function by construc-

tion and thus models employing Volterra Kernels or a box-

model for the transfer function are appropriate. Moreover,

the noncontinuous behavior is modeled by the remaining

terms, that is,HCFINL[k] andNoiseINL[k,ω], respectively.

The complete block scheme over the employed dynamic INL

model is given in Figure 5. The high-code frequency compo-

nentHCFINL[k] depends on the code k only, and not on the

testfrequency. Further,HCFINL[k]isasin[15]assumedtobe

piecewise linear in the code k; in other words, it is described

by the first-order polynomial α0+α1k within a limited set of

neighboring codevalues kp−1 ≤ k < kp;denoted asthe code

interval Kp. TheHCFINL[k] is thus modeled such that

HCFINL[k] = α0[p]+α1[p]?k − kp−1

where p refers to the ordered code interval

?,(13)

Kp:?k | kp−1≤ k < kp

?, (14)

where p = 1,...,P. The initial value of k0is given by k0= 1,

and the upper end point is, by definition, kP= 2N. Typically,

the number of intervals P is small compared with the total

number of codes, P ? 2N− 1 (see, e.g., the INL curve given

in Figure 3).

Two different postcorrection methods based on the theo-

riesgiveninSection 3werepresentedin[14,20],respectively.

In both papers, the different componentsLCFINL[k,ω] and

HCFINL[k] were postcorrected separately and the dynamics

were concentrated to the low-code frequency component.

4.IDENTIFICATION OF INL MODEL PARAMETERS

FROM MEASUREMENTS

In [15], the dynamic characterization of an ADC, when us-

ing a plurality of different test frequencies in the measure-

ment setup, was considered. In particular, the different test

frequencies are denoted by the integer m, that is, a one-to-

one map to the employed set of test frequencies f1··· fMin

Hertz.Thisorderingoftestfrequenciesattachesthenotation,

that is, the normalized frequency ω is below replaced by the

integer m corresponding to the actual test frequency fm(in

Hertz).

The low- and high-code frequency components are pa-

rameterized and a least-square method was derived for

the estimation of the parameter values from the obtained

Page 5

N. Bj¨ orsell and P. H¨ andel5

k[n]

α0[1]+α1[1](k −k0)

α0[2]+α1[2](k −k1)

...

α0[p] +α1[p](k −kp−1)

N[·]

k2[n]

...

kL[n]

NoiseINL

h2(ω)

hL(ω)

...

+

+

+

LCFINL

INL

HCFINL?

HCFINL

k∈kp

Figure 5: A block scheme over the complete INL model. The nonlinear block N[·] is a polynomial.

measurements. A closed-form solution to the estimation

problem was derived. The method in [15] is reviewed be-

low. The high-code frequency component is assumed to be

piecewise linear in the code k, that is, described by (13) and

(14). Accordingly, there are P sets of polynomial coefficients

{α0[p],α1[p]}, which are gathered in the parameter vector η

of size 2P. That is,

η =

⎛

⎜

⎜

⎜

⎜

⎜

⎝

⎜

α0[1]

α1[1]

...

α0[P]

α1[P]

⎞

⎟

⎟

⎟

⎟

⎟

⎠

⎟

.

(15)

The parameter vector η in (15) describes the local gain and

offset in INL for the different code intervals Kp. As pre-

viously mentioned, the high-code frequency component is,

at least approximately, independent of the input test fre-

quency. The low-code frequency component models the re-

maining dynamic behavior of the INL. The latter component

LCFINL[k,ω] is modeled by a polynomial of order L as in

(12). Consider the signal model

LCFINL[k,m] = f[k]Tθ[m],(16)

where T denotes transpose. In (16), the vector θ[m] =

?

possibly dependent on the test frequency as indicated by the

?

One may note that each entry in the parameter vector de-

scribes the gain of the corresponding zero-phase filter in the

parallel Hammerstein model. In order to estimate the pa-

rameters in the dynamic model of the integral nonlinear-

ity, experiments are performed where in each experiment a

sinewave Histogram test is used in order to pick up a set of

INL data. Collecting all experimental data corresponding to

the unique test frequency fm[Hertz] in an vector ym, a least

squares fit is employed in order to fit the model parameters,

h2[m] ··· hL[m]

?Tis the parameter vector (i.e., θ[m] is

k2··· kL?T

argument m) and f[k] =

is the regressor.

so that ym is as close as possible (in least squares) to our

model, that is,

ym=

⎛

⎜

⎜

⎝

INL[1,m]

...

INL?2N− 1,m?

⎞

⎟

⎟

⎠,

m = 1,...,M.

(17)

In (17), ymis the vector with experimental data and the

right-hand side in our model (10), where the frequency vari-

able used in (10) has been replaced by its corresponding in-

teger value m.

We introduce (where p spans p = 1,...,P)

gp=

⎛

⎜

⎜

⎝

1

...

1 kp− kp−1−1

0

...

⎞

⎟

⎟

⎠

(18)

with the convention that k0 = 1 and kP = 2N. Further, we

introduce the Vandermonde matrix f of size 2N− 1 × L + 1

given by

f =

⎛

⎜

⎜

⎝

10

...

11

···

1L

...

?2B−1?0 ?2B− 1?1

···

?2B−1?L

⎞

⎟

⎟

⎠.

(19)

Then we may put the model that describes the experimental

integral nonlinearity on a vector form as

ym= gη +kθ[m] +em,(20)

where η is given by (15). Here, g is introduced as the block

diagonal matrix with the gp’s defined by (18) on its main di-

agonal, that is, g = blockdiag(g1,...,gp).

For M sets of test frequencies, we have to augment

the model in (20) by incorporating the multiple data sets

Page 6

6 EURASIP Journal on Advances in Signal Processing

{y1,...yM} and expanding the parameter vector with the m-

dependent components, that is,

⎛

⎜

? ?? ?

⎜

⎝

y1

...

yM

⎞

⎟

⎟

⎠

y

=

⎛

⎜

?

⎜

⎝

g f

...

g

...

f

⎞

⎟

⎟

⎠

???

[GF]

⎛

⎜

?

⎜

⎝

⎜

⎜

η

θ[1]

...

θ[M]

??

θ

⎞

⎟

⎟

⎠

⎟

⎟

?

⎡

⎣η

⎤

⎦

+

⎛

⎜

? ?? ?

⎜

⎝

e1

...

eM

⎞

⎟

⎟

⎠

e

.

(21)

The least-squares (LS) solution of (21) is by a block ma-

trix notation given by

?? η

In [15], it is shown that the above LS solution can be sepa-

rated into one solution for η and one for each of the M fre-

quency independent parameter vectors θ(m). The complex-

ityoftheLSsolutionissignificantlyreducedbyexploitingthe

sparesstructureoftheinvolvedmatrices,leadingtoM+1sets

of solutions, that is,

?

?θ[m] =?fTf?−1fT?ym−g? η?,

In (23), y is defined as the average (over all test frequencies)

of all INL data, that is,

?θ

?

=

??GT

FT

??

G F

??−1?GT

FT

?

y.

(22)

? η =

gTπ⊥g

?−1gTπ⊥y,(23)

m = 1,...,M.

(24)

y =1

M

M

?

m=1

ym.

(25)

Further, π⊥is given by

π⊥= I −f?fTf?−1fT, (26)

where I denotes the unity matrix of proper size.

We note that (23) is a linear combination of the averaged

INL data (25). Regarding (24), one may note that this is the

least-squares solution corresponding to the detrended data

ym−g? η.

5.PERFORMANCE ANALYSIS

The purpose of postcorrection is to improve the perfor-

mance for the system, where the ADC’s are used. Commonly

used figures of merits are THD, SFDR, and SINAD. For a

model-based postcorrection based on the model structure

described in Section 4, and given that all model parame-

ters are estimated, questions of relevance include the follow-

ing. What are the achievable performance improvements for

these figures of merits? How advanced must the postcorrec-

tion method be to meet the demands for the system?

InSection 5.1,wewillformulatetherelationshipbetween

agivenINLmodelandtheADCperformance,hence,thepo-

tential of improvement of an optimal postcorrection. Below,

the influence of the three terms in (10) on the ADC per-

formance measures are investigated separately. By gradually

correcting for each INL term, the performance will improve

step bystep. We consider the case whena multiterm dynamic

modeloftheINL(suchas(10)above)isavailable,andisused

for postcorrection as illustrated in Figure 2.

The termNoiseINL[k,ω] in (10) is throughout the paper

modeled as noise, so thatNoiseINL[k,ω] = e[k,ω] is assumed

to be zero mean independently identically distributed noise

both in the code k and in test frequency ω. The general rela-

tion (10) is accordingly reduced to

INL[k,ω] =LCFINL[k,ω]+HCFINL[k] +e[k,ω].

In the following sections, we will describe the different

components in more detail and how they affect the perfor-

mance of the ADC. The conditions for the analysis are that

the ADC performance is limited by the effects of the inte-

gral nonlinearity. In the following analysis, an expression for

the THD, SINAD (and to some extent also the SFDR) as

functions of the INL model parameters is developed. Con-

sequently, the derived expressions constitute an upper limit

on the performance improvements by postcorrection elim-

inating the effects of INL. That is, under the condition that

theestimatedINLusedforpostcorrectioncorrespondstothe

actual INL of the ADC.

(27)

5.1. EffectsoftheINLlow-codefrequencycomponent

onADCperformance

TheLCFINL[k,ω] is a weakly nonlinear dynamic transfer

function (12). Since its transfer function is a polynomial,

LCFINL[k,ω] will mainly affect the harmonic distortion such

as THD, and also SFDR since the 2nd and 3rd harmonics are

usually the limiting spurs in SFDR. To get a measure on how

the ADC performance depends on this component, an eval-

uation in the frequency domain will be done.

For simplicity, but without loss of generality, the analysis

is performed in continuous time. Let the time domain input

v[t] to a static nonlinear system with output x[t] be a unit

amplitude cosine with zero initial phase

v[t] = cosω0t =ejω0t+e−jω0t

2

,(28)

where t is the absolute time, and j is the imaginary unit j =

√−1. The harmonic distortion on the output x[t] of a static

nonlinearity driven by input v[t] is given by a combination

of (28) and (12), that is,

x[t] = h0+h1ejω0t+e−jω0t

2

+h2

?ejω0t+e−jω0t

?L

2

?2

+ ···+hL

?ejω0t+e−jω0t

2

.

(29)

Note that, we here include h0 and h1 since the output

from the ADC is a sum of a linear quantization of the input

v[t] (where h0and h1are included) and the nonlinear error

model given by INL[k,ω] (i.e., described by h2(ω)···hL(ω)

according to (12)). The figures of merits used in the perfor-

mance analysis below are defined for the output from the

ADC.

Page 7

N. Bj¨ orsell and P. H¨ andel7

The even exponents produce harmonic distortion prod-

ucts at the ADC level and on even multiples of the funda-

mental frequency. Odd exponents will result in distortion

at odd overtones. For example, the third-order component

reads

?ejω0t+e−jω0t

2

?ej3ω0t+e−j3ω0t+3?ejω0t+e−jω0t?

?1

In a more general case of order n, one can find the coeffi-

cients from the binomial theorem,

h3

?3

= h3

8

?

= h3

4cos3ω0t +3

4cosω0t

?

.

(30)

(c +d)l=

l?

i=0

?

l

i

?

cl−idi.

(31)

For example, the output vl[t] will have the following set

of frequencies:

⎧

2l−1

i=1

×cos?(l −i+1)ω0t?

1

2l−1

2

2

⎛

⎝

vl[t] =

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

1

l?

?1 −(−1)i

2

?⎛

⎜

⎝

l

i −1

2

⎞

⎠

⎟

l odd,

⎛

⎜

⎝1

⎛

⎝

⎜

l

l

⎞

⎠+

⎞

⎟

l−2

?

i=0

?1+(−1)i

2

?

×

⎜

l

i

2

⎟

⎠cos?(l −i)ω0t?

⎞

⎠

⎟

l even.

(32)

The components vl[t] are the input to the linear parts of the

parallelHammersteinmodel,whereeachbranchvl[t]willbe

filtered by the corresponding zero-phase linear filter. The re-

sulting output of the parallel Hammerstein model y[t] reads

y[t] = A0+A1cos?ω0t?

+A2cos?2ω0t?+ ···+ALcos?Lω0t?,

where the real-valued A-coefficients are obtained from fil-

tering the components vl[t] given in (32) by the individual

zero-phase filters in the parallel Hammerstein model

?

1

24

?ω0

?

0

23

A3=1

22

0

24

(33)

A0=1

22

2

?

h2(0)+1

?

1

?

?+1

?

1

?

1

4

2

?

h4(0)+ ··· ,

?

2

(34)

A1= h1

?+1

22

3

?

h3

?ω0

24

5

?

h5

?ω0

?+ ··· ,

(35)

A2=1

21

2

?

?

h2

?2ω0

?3ω0

?+1

?+1

4

?

?

h4

?2ω0

?3ω0

?+ ··· ,

?+ ··· .

(36)

?

3

h3

5

h5

(37)

For an arbitrary order q, one has

⎧

i=q

Aq=

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

L?

?1+(−1)q

2

?

1

2i−1

⎛

⎝

⎛

⎝

⎜

i

i − q

2

⎞

⎠hi

⎞

⎠hi

⎟

?qω0

?

q even,

L?

i=q

?1 − (−1)q

2

?

1

2i−1

⎜

i

i − q

2

⎟

?qω0

?

q odd.

(38)

Forapuresinusoidalinputsignal,theamplitudeA1isthe

fundamentaltoneandallAq,whereq/ =1,aredistortionprod-

ucts. Thus, the figure of merit THD (and eventually SFDR)

can be expressed as a function of the amplitudes, as outlined

below.

5.1.1.Totalharmonicdistortion(THD)

For a pure sinewave input of a specified amplitude and fre-

quency, the THD is the root sum of squares (RSS) of all the

harmonic distortion components including their aliases in

the spectral output of the ADC. Normally, THD is estimated

by the RSS of the second through the tenth harmonics, in-

clusive. THD is often expressed as a dB ratio with respect to

the root mean square (RMS) amplitude of the output com-

ponent at the input frequency.

The total harmonic energy (THE) for the specific subset

of harmonics is defined by [10]

?

whereRisthelengthofthedatarecord,Y[ωn]isthecomplex

value of the spectral component at frequency ωn. Further, ωn

is the nth harmonic frequency of the discrete Fourier trans-

form(DFT)oftheADCoutputdatarecord,M isthenumber

of samples in the data record, and n is the set of harmonics

over which the sum is taken. The absolute value of Y[ωn] is

the amplitude An.

EmployingtheaboveexpressionfortheLCFINL,theresult

reads

THE =1

R2

n

??Y?ωn

???2,(39)

LCFTHE =1

R2

L?

l=2

??Y?ωl

???2=1

R2

L?

l=2

A2

l,(40)

where L is the order of the polynomial in (12).

The total harmonic distortion is given by the ratio

THD =

with respect to the RMS amplitude of the fundamental com-

ponent of the output. The maximum achievable gain in the

postcorrection of the ADC is accordingly given by the total

harmonic distortion from the low-code frequency compo-

nent,

√THE/A1. The THD is often expressed as a dB ratio

LCFTHD = 20log10

?√LCFTHE

A1

?

.

(41)

5.1.2.Spurious-freedynamicrange(SFDR)

Spurious-free dynamic range (SFDR) is the frequency do-

main difference in dB between the input signal level and the

Page 8

8EURASIP Journal on Advances in Signal Processing

level of the largest spurious or harmonic component for a

large, pure sinewave signal input. Including harmonics re-

flectscommon usageof the termSFDR.Normallythe second

or third harmonic in (36) or (37) will be the limiting spuri-

ous. Under condition that postcorrection can eliminate these

harmonics, the SFDR will be limited by a nonharmonic or

high-order harmonic distortion. An exact values for achiev-

able performance it thus not possible to find.

5.1.3.Signaltonoiseanddistortionratio(SINAD)

Thesignal-to-noise-anddistortionratio(SINAD)istheratio

of the signal to the total noise. Unless otherwise specified, it

is assumed to be the ratio of RMS signal to RMS noise, in-

cluding harmonic distortion, for sinewave input signals [10].

To study the effects ofLCFINL, SINAD is evaluated in the fre-

quency domain.

Both the signal and the total noise can be determined

from the DFT of data records. Let Eavm[ω] equal the resid-

ual spectrum of Yavmafter the bins at ωm= 0 (DC) and test

frequencies, ω0, have all been set to zero (excised from the

spectrum). Then, the RMS noise is found from the sum of all

the remaining Fourier components,

rms noise =1

R

?M−1

m=0

?

??Eavm

?ωm

???2

?1/2

.

(42)

The contribution to the noise fromLCFINL[k] isLCFTHE.

The noise will thus be reduced to

rms noise =1

R

?M−1

m=0

?

???Eavm

?ωm

???2−LCFTHE

??1/2

.

(43)

To conclude this section, the achievable effect from elim-

inatingthelow-codefrequencycomponentbyusingpostcor-

rection is for THD given by (41) and for SINAD by (43). In

indication of potential, SFDR improvement is given by (38),

if the limiting spurious is an overtone.

5.2.Effectsofthehigh-codefrequencycomponenton

ADCperformance

Since both THD and SFDR are evaluated in the frequency

domain, these components can be described by a transfer

function with an amplitude dependent piecewise constant

amplification. How this will affect the harmonic distortion

can be found from its DFT, where the error fromHCFINL[k]

will result in a piecewise constant amplification:

⎧

α2sin(ωt)+O2

...

αPsin(ωt)+Op

yHCF(t) =

⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎩

α1sin(ωt)+O1

sin(ωt) < k1,

k1≤ sin(ωt) < k2,

...

kP−1≤ sin(ωt),

(44)

where O is an offset and yHCF(t) is an virtual, internal signal

in the INL model (Figure 5), namely, theHCFINL output for

a single tone stimulus.

The harmonic distortion is highly dependant on the

structure ofHCFINL[k] given in (13). Thus, one needs to

solve and the DFT for (44) for the specific ADC used to get

information about the effects on the THD and SFDR. How-

ever, one property ofHCFINL[k] worth mentioning is that

the amplitudes from the harmonics fromHCFINL[k] are not

necessarily declining with the order of the harmonic. For a

smooth nonlinearity (as in the low-code frequency compo-

nent) the amplitude of harmonics decreases with higher or-

der, but this not necessarily the case forHCFINL[k].

Since we are not able to find a general expression for the

effect fromHCFINL[k] in the frequency domain, we will eval-

uate SINAD based on the results from [21], where the max-

imum achievable SINAD for a N-bit ADC is for a floating-

point postcorrection given by

SINAD =

22N

1+3 ·DNL2

RMS

, (45)

where DNLRMSwas defined in (6). In [22], (45) was further

developed to include a correction algorithm with a fixed-

point resolution for correction values.

Assume that we have eliminated the errors due to the

low-code frequency component. That will give us from (27)

the remaining INL[k] to beHCFINL?[k],

HCFINL?[k] =HCFINL[k] +e[k].

The differential nonlinearity DNL[k] is found from (5), that

is,

(46)

DNL[k]

=HCFINL?[k +1] −HCFINL?[k]

= e[k +1] − e[k]

⎧

⎪⎪⎪⎩

+

⎪⎪⎪⎨

α1[p]

α0[p] −α0[p −1] −α1[p −1]?k −1 −kp−2

kp−1< k < kp

?

(47)

k = kp−1.

Since {e[k]} are random variables independent and identi-

cally distributed, DNL2

RMSyields

DNL2

RMS

=

1

2N− 2

2N−2

?

k=1

DNL2[k]

=

1

2N− 2

2N−2

?

k=1

ErrorDNL2[k]

+

1

2N−2

P?

p=1

??α1[p]2?kp−kp−1

??

+?α0[p]−α0[p −1]−α1[p −1]?kp−1−1 −kp−2

??2?

,

(48)

Page 9

N. Bj¨ orsell and P. H¨ andel9

whereErrorDNL[k] is the DNL[k] related to e[k]. The poten-

tial for improvement fromHCFINL[k] is thus,

HCFDNL2

RMS

=

1

2N−2

+?α0[p] −α0[p −1] −α1[p −1]?kp−1−1 −kp−2

P?

p=1

??α1[p]2?kp−kp−1

??

??2?

(49)

.

The improvement in terms of SINAD is given by inserting

(49) into (45).

5.3.ModelErrorseffectonADCperformance

Thethirdcomponentin(27)isconsideredasazeromeanin-

dependently and identically distributed noise. Thus no har-

monic distortion can be expected; only nonharmonic distor-

tion is present. The effect on THD will thus be zero, and this

componentaffectsSINADonly.ThepotentialforSINADim-

provement is [21]

SINAD =

?2N?2

?

1+3 ·ErrorDNL2

RMS

?,(50)

wheretheroot-mean-squarevalueofDNLwasintroducedin

(6).

In conclusion, the performance measures THD and

SINAD (and to some extent SFDR) have been developed at

a function of the model parameters. The objective has been

to provide a tool for the user to get an estimate of what per-

formance improvement can be achieved from an ideal post-

correction.

6. CONCLUSIONS

In this paper, a three-component model of an ADC aimed

for postcorrection is presented. Each component has its own

properties; (i) the low-code frequency component captures

the dynamic component and it is modeled by a polyno-

mial followed by linear filters, that is, a parallel Hammerstein

model, (ii) the high-code frequency component is static and

piecewise linear in the code k, and (iii) the model error is

assumed to be zero mean and independent identically dis-

tributed noise.

The purpose is to provide a tool to evaluate to max-

imum achievable performance improvement for a model-

based postcorrection. For each of the three components, per-

formance analyses in THD and SINAD are presented. The

improvement potential for SINAD as well as the effect on

THD from the low-code frequency component is given by

thecoefficient in themodel of theADC, whilethe effectfrom

high-code frequency component on THD requires a DFT

analysis.

ACKNOWLEDGMENT

ThisworkwassupportedbyEricssonAB,FreescaleSemicon-

ductorNordicAB,InfineonTechnologiesNordicAB,Knowl-

edge Foundation, NOTE AB, Racomna AB, Rohde&Schwarz

AB, and Syntronic AB.

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