# Simple Evaluation of the Nonlinearity Signature of  an ADC Using a Spectral Approach

Authors: E. J. Peralías; M. A. Jalón; A. Rueda
Journal: VLSI Design
 (2008)

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## Abstract

This work presents a new method to estimate the nonlinearity characteristics of analog-to-digital converters (ADCs). The method is based on a nonnecessarily polynomial continuous and differentiable mathematical model of the converter transfer function, and on the spectral processing of the converter output under a sinusoidal input excitation. The simulation and experiments performed on different ADC examples prove the feasibility of the proposed method, even when the ADC nonlinearity pattern has very strong discontinuities. When compared with the traditional code histogram method, it also shows its low cost and efficiency since a significant lower number of output samples can be used still giving very realistic INL signature values.


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## Body

## 1. Introduction

The parameters that characterize the transfer function of an ADC, such as the integral
nonlinearity (INL), are some of the most important specifications that must be
known to insure the correct operation of the ADC in a certain application. One
of the standardized methods to estimate these parameters is the code histogram test
[1, 2]. Its main drawback is the excessive cost of its
application, especially due to the large number of samples that must be
acquired (more than one million) and to the fact that this number increases, in
general, in an exponential way with the number of bits of the ADC. The
synchronization between the acquisition and the signal stimulation is also a nonelemental
task. On the contrary, this method can achieve a very precise measurement
independently of the shape of the transfer characteristic; it is a clear trade-off
between time and precision.These drawbacks make the histogram method
unfeasible for low-speed and high-resolution converters (>15 bits). For
these kinds of converters, the use of methods based on spectral processing can
be satisfactory acquiring only some tens of thousands of samples independently
of the ADC resolution [3–6]. Although this estimate leads to a partial
description of the static behavior of the ADC and, for example, it is not a
reliable method to calibrate the nonlinearity, such description may be enough
to show a malfunctioning and it may be very useful to set a test protocol [7–9].This paper presents a new and simple method for
ADC nonlinearity (INL) estimation using the spectral processing of its response
to a sine-wave excitation. The method does not require a concrete functional
form for the ADC transfer curve or for the INL, as [3, 4], nor to apply specific expansion series as in [6], although it is based on a continuous and
differentiable mathematical model of the converter transfer function. To reach
high precision in the INL estimation, the transfer function should have enough
smooth shape; however, we will show that it can be also applied in many cases
with strong discontinuities, giving INL signatures good enough to describe the
nonlinear behavior of the ADC, this validating its use for rapid production
test. The mathematical bases of this method are the standard definition of the
INL and the local variation of the ADC transfer function around each ideal
transition.The organization of this paper is as follows.
Section2 introduces the general hypotheses to apply the new
method and the mathematics to obtain the INL signature. The method adaptation
for its application through the spectral estimation is derived in Section 3. Section 4 shows some simulated and real examples of application
for the proposed method and Section 5 draws
the conclusions.


## 2. Modelling the Nonlinearity

The ADC basic model that we are considering supposes that the transfer curve: (1) is a smooth nonnecessarily polynomial functionZ(x),
continuous and differentiable with respect to the input, (2) it must be strictly increasing and therefore with a nonzero derivative. The description as a continuous function must be also understood in the way that the resolution of the converter is high enough to consider the quantization effect embedded in other noise contributions.Mathematically, the proposed input-output model
is(1)z=Z(x)+ε(x),∃∂xZ≠0,∀x∈[−VR,VR], where the error function ε(x) is of the same order than the quantization
error. Without any loss of generality, the input range will be considered
bipolar and centred at zero, [−VR,VR].Figure1 illustrates the nonlinearity
modeling. Both the ideal and a hypothetic real transfer function of an N-bit converter are shown. The curve Z(x)
is that obtained by fitting a low-density sampling of the real transfer curve. Considering
the previous conditions, a first-order Taylor’s
expansion of the mild transfer function Z(x) is going to be calculated around each
ideal transition, lk. The
set of points formed by the ideal transitions can be expressed as lk=q⋅k, k∈[−2N−1,2N−1−1],
where q=2VR/2N is the LSB of the N-bit ADC. Then, we have(2)Z(x)≈Z(lk)+∂xZ(lk)⋅(x−lk),∀x≈lk. Evaluating the expression at the corresponding
real transition, x=tk, where the output code value changes from k−1
to k,(3)k≈Z(lk)+∂xZ(lk)⋅qINLk, where it has been used that k≈Z(tk) and the standard INL definition INLk=(tk−lk)/q [1].Figure 1
Modelling the non-linearity: the first order Taylor’s expansion of the transfer functionZ(x) around an ideal transition lk. tk represents the real transition where the output code Z changes from k−1 to k.From (3), the following expression for the INL can be obtained:(4)INLk≈k−Z(lk)q∂xZ(lk). Notice that this expression can be evaluated only
if it is possible to obtain the derivative of the function Z(x).If the second derivative exists, an alternative
expression can be obtained using the second-order Taylor’s expansion. In any case, in this work
only the expression (4) will be used since the nonlinearity of the ADC is
considered very small: maxk{|INLk|}<2N−10(N>10).


## 3. Application of the Model Using a Spectral Approach

This section shows how to apply expression (4) in the case that a spectral measurement is used to
process the ADC response to a sinusoidal input.Let us assume that the input excitation is(5)x(t)=Acos(ωxt+φx)+B and that it covers all the input range without causing
the ADC saturation (A≈VR,B≈0). The
input frequency ωx is low enough to produce unimportant dynamic effects. The
phase-shift  φx and the offset B do not
need to be known a priori. The amplitude, A,
has to be known only when the gain of the ADC, g, is very different from the unity and/or it is wanted to estimate the gain error of the A/D conversion.For such input, the ADC output is a
superposition of harmonics of the excitation frequency,(6)Z(x(t))=C0+∑n≥1Cncos(ωnt+φn),ωn=nω1, where the frequency ωx and the phase-shift φx are identified with the frequency ω1 and the phase-shift φ1 of the main harmonic, respectively.
Rejecting, in a first approximation, the other harmonics (n>1), the input amplitude can also be related to the amplitude
of the 1st harmonic by means of the gain g of the ADC. Using a simple linear model Z(x)≈(g⋅x)/q+zos, where g is the gain and zos is the offset of the ADC [1], it can be obtained as follows:(7)C1=g⋅A/q,C0=(g⋅B)/q+zos,ω1=ωx,φ1=φx. A more suitable model,
but more costly, could still use the previous linear relationship but
considering as outputs the sine-wave signal z^(t)=CAcos(ωzt+φz)+CB that best fits to (6) in the sense of the least mean-squared error. In this
case, expressions (7) are still valid using the parameters CA, CB, ωz, and φz. In any case, whatever the model is, the
transfer function derivative in (4) can now be calculated in an indirect way:(8)∂xZ=∂tZ(x(t))∂tx(t)=∑n≥1ωnCnsin(ωnt+φn)ωxAsin(ωxt+φx)≈gq[1+∑n≥2CnC1nsin(nω1t+φn)sin(ω1t+φ1)], where the relations in (7) have been applied to the simplest model.Now, let us evaluate expressions (6) and (8) at the ideal transitions, Z(lk),∂xZ(lk).Ifτk are the time instants in which the sinusoidal input signal crosses
the ideal transitions of the ADC, x(τk)=lk (see Figure 2), it can be written as
(9)lk=Acos(ωxτk+φx)+B→ωxτk=−φx+Arccos(lk−BA). Applying relations (7),(10)δk=ω1τk≈−φ1+Arccos(g⋅k+zos−C0C1) that gives the phases used in (6) and (8)(11)ωnt+φn∣t=τk=nω1τk+φn=nδk+φn to obtain, respectively, Z(lk) and ∂xZ(lk). Finally, expression (4) for the INL becomes(12)INLk≈k−(C0+∑n≥1Cncos(nδk+φn))g⋅[1+∑n≥2(Cn/C1)(nsin(nδk+φn)/sin(δk+φ1))]. Notice that in (10) and (12) the only quantities to evaluate are the harmonic amplitudes {Cn},
the phase-shifts {φn},
and the gain g of the ADC. All of
them can be estimated using the spectral processing of the output. In general, zos in (10) can be considered null, which is equivalent to
consider that zos is cancelled out by the input offset. When g=1 and zos=0 are used in (10) and (12), the ADC intrinsic nonlinearity is being estimated,
that is, the nonlinearity signature without gain and offset effects. This is the
INL usually evaluated according to the standards [1, 2].Figure 2
Defining the input wave crossing pointsτk over the ADC ideal transitions lk. According to (10), these timing points allow work out the phase values of the output harmonics when the input wave crosses an ideal transition.The number of harmonics
that must be selected to apply (12) depends
on both the spectral discrimination that the total noise allows and the
reliability of the mathematical method used to estimate the harmonic parameters.
In all of our experiments, we have used a conservative criterion: the selected harmonics
are those with amplitudes at least 10 dB over the spectrum noise floor.
Mathematical methods to estimate spectra are basically based on discrete-time
Fourier transform (DTFT) [1, 2].In very good coherent
experiments (input frequency and sampling frequency are commensurable values),
and when the noise is small enough and well described by additive white model,
simple relationships can be used. Being acquiredL samples of the ADC output, {zi}i=1L,
with the sampling frequency fs and satisfying the input wave frequency fx=(M/L)fs with M and L relative prime integers (to
excite at least L ADC different
levels), the DTFT is obtained of the output register, {ζj}=FFT{zi}, selecting after that the most significant spectral lines respect to the noise
floor, {ζjn}n=1H. The main harmonic occurs at j1=M. The correct frequencies for
the other lines must be carefully identified because if high-order harmonics exist, ωn>ωs/2, their spectral lines are folded at the DTFT interval [0,fs/2]. With these considerations, expressions to
estimate the parameters involved in (12) are(13)ω1=2πfsM/L,ωn=nω1,C0=meani({zi}),Cn=(2/L)∥ζjn∥,φn=(−1)pnArg(ζjn),pn=k≥0{0,ωn/ωs∈[k,(2k+1)/2[,1,ωn/ωs∈[(2k+1)/2,(k+1)[,g=q⋅rms{zi}/rms{x(t)}, where rms{x(t)} is the root mean square of the input
wave and pn corrects the
inversion of estimated phase for the case that a spectral line is folded.Although it has been
suggested about (12) that the value of the gain g is not strictly necessary for the
intrinsic nonlinearity evaluation, in (13) it has been included an expression that uses the root
mean square for its estimation. This expression allows a simple way to evaluate g because it is easy to measure the
rms value of the input signal using a wattmeter. For a low-distortion ADC (such
as it has been considered for the application of the method), the difference of
the gain value estimated using this way is usually less than 0.5% respect to
the gain value estimated using a standardized method, such as that estimating g through the slope of the best-fitting line
to the transition set, or the slope of the straight line that joins the extreme
transitions on the ADC transfer function [1].IfR registers (R>1) of length L are acquired to average the noise, all of them must be consecutively taken to do not lose the phase information. This is the same as to trace a unique register with R⋅L output samples of L-sample periodicity. In this way, (13) is still used but with the averaged module and the
averaged phase spectrum values, (14)∥ζj∥¯=meanm=1,…,R{∥ζj(m)∥2},Arg(ζj)¯=meanm=1,…,R{Arg(ζj(m))},with{ζi(m)}m−1R=FFT{zi(m)}.If a coherent sampling is not possible,windowing to each register should be
applied to reduce the spectral leakage, {ζj(m)}m=1R=FFT{wizi(m)}, where {wi}i=1L is the convolution window. In general, the expressions in (13) have to be corrected since the estimates are biased
depending on the applied window. In [2, 10], there are some suggestions to select the most
appropriate window. For all of our noncoherent experiments, we have used a
4-term cosine-class window and an estimation method based on phase regression
presented in [11], which requires that R≥2.When high-order harmonics exist and they are
folded in the[0,fs/2]
band, a particular input frequency has to be selected in such a way that no
significant harmonic overlaps. Problems are minor if coherent sampling and no
windowing are applied. We have always selected test frequencies that lead
spectral lines to separate each other at least 10 bins, when 4-term cosine
window has been used.The DC component is evaluated by means of the
weighted mean of the samples, using as weight function the convolution window,(15)C0=meanm{∑iwizi(m)∑iwi}.


## 4. Application Examples

### 4.1. Simulated Experiments

This section shows the simulation results obtained applying the introduced method in
two different models for the ADC. The first converter, ADC1, has a
very regular transfer curve and a smooth INL. For the second one, ADC2,
the transfer curve is nonmonotonic and its INL has very strong discontinuities.


#### 4.1.1. Nonspectral Approach (DC Sweep) on Low-Speed, High-Resolution ADC1 Model

Before applying (12), let us show the immediate approximation of (4). The ADC1 has been excited using a DC
signal whose value is changing inside the input range. For each value of the
input, x¯i, it is obtained an output register to calculate the corresponding average code z¯i. Using the set {(x¯i,z¯i)} it can be built the Mth order
best-fitting polynomial model, ZM(x), which allows to apply (4) directly.ADC1 is a high-level model of a 14-bit
ΣΔ converter, with reference voltages 0.0 V and
5.0 V but with a practical input range [0.5 V,4.5 V]. The model reproduces
the nonlinear, noisy, and frequency behavior. The sampling frequency is fs=100 kHz. The noise due to
both the input and the ADC, referred to the input, is approximately 2LSBs rms
white noise. A DC sweep with about 4000 points has been carried out in the
range [0.7 V, 4.3 V]. For each DC value, it has been taken a register of 50
points, being the obtained code in the range Ik=[−5792,5801]. Using this dataset, the
M=32nd-order best-fitting
polynomial has been calculated by means of the first class Chebyshev base. The
choice of the order is deduced from the harmonic significant number found in
the experiment that will be described in next subsection. For each value k of Ik, it has been evaluated ZM(lk), ∂xZM(lk), and INLk in (4).On the other hand, it has been applied the
standard sinusoidal histogram method [1, 2] to determine a good estimate
of the INL (real INL from now
on) in order to establish a reference for comparison purpose. The obtained
results are depicted in Figure 3.(a) INL estimates of the example ADC1. In thick (black) line the one obtained by (4), in thin (blue) line the real INL evaluated by the Histogram method. (b) Differences between both estimates in (a).
(a)(b)Figure3(a) shows the superposition of both
estimates. The thick line is the curve calculated by (4). To show the reliability achieved, in Figure 3(b)
it is depicted the difference between both curves, which is affected by a
systematic 1/2 LSB error. This
is because, for convenience, (2) and (3)  have been
obtained using the transitions and not the code centres. This is not very
important since, in general, the offset value of the ADC is not used in the
expression (12) and so it must be corrected by means of the
elimination of its mean value. In any way, it can be noticed the high accuracy
achieved by the method. This is justifiable since the INL curve of this ADC
follows a very smooth behavior.


#### 4.1.2. Spectral Approach (Sine Input) on Low-Speed, High-Resolution ADC1 Model

For the same ADC1 converter, it has
been simulated a set of 35 experiments that use a sinusoidal input and
calculate the INL from (12). To estimate the parameters from the spectrum, it has
been used the procedure described in [11]. The amplitude of the sinusoidal input is
approximately −3 dBFS, the offset is about 100LSBs, and the frequency is near fs/83. The phase has been
evenly spread inside the range [−π,π] all over the
experiments. The equivalent noise at the input is approximately 1LSB rms white
noise. In each experiment, R=4 consecutive
registers of L=4096 samples are used.
A set of 500 samples have been eliminated at the beginning of the four registers
to reduce the settling errors. A typical spectrum is shown in Figure 4(a).
As the background noise is about −112 dBFS, the spectral lines selected are
those that have the amplitude higher than −102 dBFS. The typical selection
detects about 15 harmonics with orders up to 30th. Figure 4(b) shows a
comparison between the real INL (thick line) and the estimated (thin line) using the histogram method but with only
the above four acquired registers. It is obvious that the number of acquired
samples is not still enough to sketch the INL shape. On the other hand, the results
obtained in one of the experiments using the spectral estimation and (12) are shown in Figures 5(a)
and 5(b). As it can be
noticed, our method gives good enough results with the 4·4096 acquired
samples. Both INL  curves have been depicted with the offset corrected, since
the offset of the ADC1 has been supposed null zos=0.
To show how robust the estimate from (12) is, Figure 5(c) depicts the differences between the
INL of each experiment and the real INL.(a) Typical averaged magnitude spectrum obtained from the ADC1 output. It is used for the selection of the spectral lines taken as harmonics. (b) Comparison between two ADC1 INL Histogram estimations: in thin (blue) line the one estimated with very few samples, (the four registers acquired to apply the new method). In thick (black) line the real INL.
(a)(b)(a) Overlapped ADC1 INL curves: in thick (black) line the one obtained by spectral estimation and (12). In thin (blue) line the real INL.
(b) Difference between the two above INL signatures.
(c) Differences obtained for all of the experiments on the ADC1.
(a)(b)(c)


#### 4.1.3. Spectral Approach (Sine Input) on Medium-Speed, High-Resolution Model ADC2

The ADC2 is a high-level model of a 16-bit
pipeline converter that operates with reference voltages of −2.0 V
and 2.0 V. The sampling frequency is fs=5.0 MHz.
The noise due to both the input and the ADC, referred to the input, is
approximately 1LSB rms white noise. The amplitude of the input signal is
approximately −0.2 dBFS, the offset is about 10LSBs and the frequency is near fs/222.The phase has been evenly spread allover the
experiments inside the range[−π,π]. In these experiments,
it has been taken two consecutive registers with 32768 samples each. In this example,
the background noise appears at about −130 dBFS, so the selected spectral lines
are those that are harmonics with amplitudes higher than −120 dBFS.The so irregular and discontinuous structure of
the INL of this ADC leads to a typical selection about 150 harmonics with
orders up to 600th. Figure6 shows the typical results obtained. Figure 6(a)
shows a comparison between the real INL (thick line) and the one estimated (thin line) using the histogram method with
the above two acquired registers. The number of acquired samples is not still
enough to sketch the INL shape. However and in spite of such a discontinuous
structure of the nonlinearity of the ADC2, the estimate from (12) absolutely follows the real INL evaluated using the standardized histogram method
(Figure 6(b)). The difference between both estimates has been depicted in Figure 6(c).
Biggest differences occur at the higher transitions, where the smoothing effect
due to the limited number of harmonics that has been selected is more evident. Even
if the gain is not evaluated, the intrinsic nonlinearity of the ADC (the one where
the gain and the offset are corrected) still can be extracted. Figure 7(a) shows the results
obtained for the ADC2 making g=1 in (10). Although the difference is noticeable, if the gain
and offset are eliminated from both curves (subtracting their best-fitting
lines), a good approximation can be achieved as Figure 7(b) shows.(a) Comparison between two ADC2 INL Histogram estimations: in thin (blue) line the one estimated with very few samples (the two registers acquired to apply the new method). In thick (black) line the real INL, (no gain effect has been included).
(b) Overlapped ADC2 INL estimations: the estimated using the spectral approach and (12), and the real INL (gain effects are included).
(c) Difference between the above INL signatures in (b).
(a)(b)(c)(a) Thick (black) line: estimated ADC2 INL using (12) and taking the gain equal to 1. In thin (blue) line the real INL where gain effects are included.
(b) Overlapping of the two above estimations after the gain and offset effects have been corrected.
(a)(b)


### 4.2. Real Experiments

This section shows the results obtained applying the introduced method to a real ADC.
This converter has a transfer curve with very strong discontinuities.


#### 4.2.1. Spectral Approach (Sine Input) on High-Speed, Low-Voltage Experimental Prototype ADC

The ADC under test is a
fully differential 12-bit pipeline converter prototype in a 120 nm CMOS
technology with reference voltages −1 V and 1 V. The sinusoidal input has been nonbuffered
AC coupled to the ADC and generated using the Agilent N8241A AWG, with amplitude
of −0.1 dBFS and a frequency of 500 kHz approximately. In this case, as a good
coherent experiment has been done, only a register of 4090 samples has been
acquired using a 20 MHz sampling master clock (neither averaging nor windowing has been applied).
Figure8(a) shows a typical nonwindowed magnitude spectrum, with the floor
noise at about −100 dBFS. The harmonics selected were those with the amplitude
higher than −90 dBFS, and the typical selection takes about 45 harmonics with
orders up to 150th.(a) Typical magnitude spectrum obtained from the prototype Pipeline ADC output.
(b) Comparison between two INL estimations: In thin (blue) line, thereal INL. In thick (black) line the INL estimation from the spectral approach and (12) using the spectrum in (a) (gain and offset effects have been corrected).
(a)(b)Figure8(b)
compares the INL pattern obtained using the spectral approach (from (12)) with the real INL estimated using the standard histogram method. Notice that the INL evaluated
from our proposed method describes the shape of the real INL good enough, even in hard discontinuities. Best
results are obtained if more registers have been acquired, since noise
averaging improves both uncertainty and the number of selected harmonics.


## 4.1. Simulated Experiments

This section shows the simulation results obtained applying the introduced method in
two different models for the ADC. The first converter, ADC1, has a
very regular transfer curve and a smooth INL. For the second one, ADC2,
the transfer curve is nonmonotonic and its INL has very strong discontinuities.


### 4.1.1. Nonspectral Approach (DC Sweep) on Low-Speed, High-Resolution ADC1 Model

Before applying (12), let us show the immediate approximation of (4). The ADC1 has been excited using a DC
signal whose value is changing inside the input range. For each value of the
input, x¯i, it is obtained an output register to calculate the corresponding average code z¯i. Using the set {(x¯i,z¯i)} it can be built the Mth order
best-fitting polynomial model, ZM(x), which allows to apply (4) directly.ADC1 is a high-level model of a 14-bit
ΣΔ converter, with reference voltages 0.0 V and
5.0 V but with a practical input range [0.5 V,4.5 V]. The model reproduces
the nonlinear, noisy, and frequency behavior. The sampling frequency is fs=100 kHz. The noise due to
both the input and the ADC, referred to the input, is approximately 2LSBs rms
white noise. A DC sweep with about 4000 points has been carried out in the
range [0.7 V, 4.3 V]. For each DC value, it has been taken a register of 50
points, being the obtained code in the range Ik=[−5792,5801]. Using this dataset, the
M=32nd-order best-fitting
polynomial has been calculated by means of the first class Chebyshev base. The
choice of the order is deduced from the harmonic significant number found in
the experiment that will be described in next subsection. For each value k of Ik, it has been evaluated ZM(lk), ∂xZM(lk), and INLk in (4).On the other hand, it has been applied the
standard sinusoidal histogram method [1, 2] to determine a good estimate
of the INL (real INL from now
on) in order to establish a reference for comparison purpose. The obtained
results are depicted in Figure 3.(a) INL estimates of the example ADC1. In thick (black) line the one obtained by (4), in thin (blue) line the real INL evaluated by the Histogram method. (b) Differences between both estimates in (a).
(a)(b)Figure3(a) shows the superposition of both
estimates. The thick line is the curve calculated by (4). To show the reliability achieved, in Figure 3(b)
it is depicted the difference between both curves, which is affected by a
systematic 1/2 LSB error. This
is because, for convenience, (2) and (3)  have been
obtained using the transitions and not the code centres. This is not very
important since, in general, the offset value of the ADC is not used in the
expression (12) and so it must be corrected by means of the
elimination of its mean value. In any way, it can be noticed the high accuracy
achieved by the method. This is justifiable since the INL curve of this ADC
follows a very smooth behavior.


### 4.1.2. Spectral Approach (Sine Input) on Low-Speed, High-Resolution ADC1 Model

For the same ADC1 converter, it has
been simulated a set of 35 experiments that use a sinusoidal input and
calculate the INL from (12). To estimate the parameters from the spectrum, it has
been used the procedure described in [11]. The amplitude of the sinusoidal input is
approximately −3 dBFS, the offset is about 100LSBs, and the frequency is near fs/83. The phase has been
evenly spread inside the range [−π,π] all over the
experiments. The equivalent noise at the input is approximately 1LSB rms white
noise. In each experiment, R=4 consecutive
registers of L=4096 samples are used.
A set of 500 samples have been eliminated at the beginning of the four registers
to reduce the settling errors. A typical spectrum is shown in Figure 4(a).
As the background noise is about −112 dBFS, the spectral lines selected are
those that have the amplitude higher than −102 dBFS. The typical selection
detects about 15 harmonics with orders up to 30th. Figure 4(b) shows a
comparison between the real INL (thick line) and the estimated (thin line) using the histogram method but with only
the above four acquired registers. It is obvious that the number of acquired
samples is not still enough to sketch the INL shape. On the other hand, the results
obtained in one of the experiments using the spectral estimation and (12) are shown in Figures 5(a)
and 5(b). As it can be
noticed, our method gives good enough results with the 4·4096 acquired
samples. Both INL  curves have been depicted with the offset corrected, since
the offset of the ADC1 has been supposed null zos=0.
To show how robust the estimate from (12) is, Figure 5(c) depicts the differences between the
INL of each experiment and the real INL.(a) Typical averaged magnitude spectrum obtained from the ADC1 output. It is used for the selection of the spectral lines taken as harmonics. (b) Comparison between two ADC1 INL Histogram estimations: in thin (blue) line the one estimated with very few samples, (the four registers acquired to apply the new method). In thick (black) line the real INL.
(a)(b)(a) Overlapped ADC1 INL curves: in thick (black) line the one obtained by spectral estimation and (12). In thin (blue) line the real INL.
(b) Difference between the two above INL signatures.
(c) Differences obtained for all of the experiments on the ADC1.
(a)(b)(c)


### 4.1.3. Spectral Approach (Sine Input) on Medium-Speed, High-Resolution Model ADC2

The ADC2 is a high-level model of a 16-bit
pipeline converter that operates with reference voltages of −2.0 V
and 2.0 V. The sampling frequency is fs=5.0 MHz.
The noise due to both the input and the ADC, referred to the input, is
approximately 1LSB rms white noise. The amplitude of the input signal is
approximately −0.2 dBFS, the offset is about 10LSBs and the frequency is near fs/222.The phase has been evenly spread allover the
experiments inside the range[−π,π]. In these experiments,
it has been taken two consecutive registers with 32768 samples each. In this example,
the background noise appears at about −130 dBFS, so the selected spectral lines
are those that are harmonics with amplitudes higher than −120 dBFS.The so irregular and discontinuous structure of
the INL of this ADC leads to a typical selection about 150 harmonics with
orders up to 600th. Figure6 shows the typical results obtained. Figure 6(a)
shows a comparison between the real INL (thick line) and the one estimated (thin line) using the histogram method with
the above two acquired registers. The number of acquired samples is not still
enough to sketch the INL shape. However and in spite of such a discontinuous
structure of the nonlinearity of the ADC2, the estimate from (12) absolutely follows the real INL evaluated using the standardized histogram method
(Figure 6(b)). The difference between both estimates has been depicted in Figure 6(c).
Biggest differences occur at the higher transitions, where the smoothing effect
due to the limited number of harmonics that has been selected is more evident. Even
if the gain is not evaluated, the intrinsic nonlinearity of the ADC (the one where
the gain and the offset are corrected) still can be extracted. Figure 7(a) shows the results
obtained for the ADC2 making g=1 in (10). Although the difference is noticeable, if the gain
and offset are eliminated from both curves (subtracting their best-fitting
lines), a good approximation can be achieved as Figure 7(b) shows.(a) Comparison between two ADC2 INL Histogram estimations: in thin (blue) line the one estimated with very few samples (the two registers acquired to apply the new method). In thick (black) line the real INL, (no gain effect has been included).
(b) Overlapped ADC2 INL estimations: the estimated using the spectral approach and (12), and the real INL (gain effects are included).
(c) Difference between the above INL signatures in (b).
(a)(b)(c)(a) Thick (black) line: estimated ADC2 INL using (12) and taking the gain equal to 1. In thin (blue) line the real INL where gain effects are included.
(b) Overlapping of the two above estimations after the gain and offset effects have been corrected.
(a)(b)


## 4.1.1. Nonspectral Approach (DC Sweep) on Low-Speed, High-Resolution ADC1 Model

Before applying (12), let us show the immediate approximation of (4). The ADC1 has been excited using a DC
signal whose value is changing inside the input range. For each value of the
input, x¯i, it is obtained an output register to calculate the corresponding average code z¯i. Using the set {(x¯i,z¯i)} it can be built the Mth order
best-fitting polynomial model, ZM(x), which allows to apply (4) directly.ADC1 is a high-level model of a 14-bit
ΣΔ converter, with reference voltages 0.0 V and
5.0 V but with a practical input range [0.5 V,4.5 V]. The model reproduces
the nonlinear, noisy, and frequency behavior. The sampling frequency is fs=100 kHz. The noise due to
both the input and the ADC, referred to the input, is approximately 2LSBs rms
white noise. A DC sweep with about 4000 points has been carried out in the
range [0.7 V, 4.3 V]. For each DC value, it has been taken a register of 50
points, being the obtained code in the range Ik=[−5792,5801]. Using this dataset, the
M=32nd-order best-fitting
polynomial has been calculated by means of the first class Chebyshev base. The
choice of the order is deduced from the harmonic significant number found in
the experiment that will be described in next subsection. For each value k of Ik, it has been evaluated ZM(lk), ∂xZM(lk), and INLk in (4).On the other hand, it has been applied the
standard sinusoidal histogram method [1, 2] to determine a good estimate
of the INL (real INL from now
on) in order to establish a reference for comparison purpose. The obtained
results are depicted in Figure 3.(a) INL estimates of the example ADC1. In thick (black) line the one obtained by (4), in thin (blue) line the real INL evaluated by the Histogram method. (b) Differences between both estimates in (a).
(a)(b)Figure3(a) shows the superposition of both
estimates. The thick line is the curve calculated by (4). To show the reliability achieved, in Figure 3(b)
it is depicted the difference between both curves, which is affected by a
systematic 1/2 LSB error. This
is because, for convenience, (2) and (3)  have been
obtained using the transitions and not the code centres. This is not very
important since, in general, the offset value of the ADC is not used in the
expression (12) and so it must be corrected by means of the
elimination of its mean value. In any way, it can be noticed the high accuracy
achieved by the method. This is justifiable since the INL curve of this ADC
follows a very smooth behavior.


## 4.1.2. Spectral Approach (Sine Input) on Low-Speed, High-Resolution ADC1 Model

For the same ADC1 converter, it has
been simulated a set of 35 experiments that use a sinusoidal input and
calculate the INL from (12). To estimate the parameters from the spectrum, it has
been used the procedure described in [11]. The amplitude of the sinusoidal input is
approximately −3 dBFS, the offset is about 100LSBs, and the frequency is near fs/83. The phase has been
evenly spread inside the range [−π,π] all over the
experiments. The equivalent noise at the input is approximately 1LSB rms white
noise. In each experiment, R=4 consecutive
registers of L=4096 samples are used.
A set of 500 samples have been eliminated at the beginning of the four registers
to reduce the settling errors. A typical spectrum is shown in Figure 4(a).
As the background noise is about −112 dBFS, the spectral lines selected are
those that have the amplitude higher than −102 dBFS. The typical selection
detects about 15 harmonics with orders up to 30th. Figure 4(b) shows a
comparison between the real INL (thick line) and the estimated (thin line) using the histogram method but with only
the above four acquired registers. It is obvious that the number of acquired
samples is not still enough to sketch the INL shape. On the other hand, the results
obtained in one of the experiments using the spectral estimation and (12) are shown in Figures 5(a)
and 5(b). As it can be
noticed, our method gives good enough results with the 4·4096 acquired
samples. Both INL  curves have been depicted with the offset corrected, since
the offset of the ADC1 has been supposed null zos=0.
To show how robust the estimate from (12) is, Figure 5(c) depicts the differences between the
INL of each experiment and the real INL.(a) Typical averaged magnitude spectrum obtained from the ADC1 output. It is used for the selection of the spectral lines taken as harmonics. (b) Comparison between two ADC1 INL Histogram estimations: in thin (blue) line the one estimated with very few samples, (the four registers acquired to apply the new method). In thick (black) line the real INL.
(a)(b)(a) Overlapped ADC1 INL curves: in thick (black) line the one obtained by spectral estimation and (12). In thin (blue) line the real INL.
(b) Difference between the two above INL signatures.
(c) Differences obtained for all of the experiments on the ADC1.
(a)(b)(c)


## 4.1.3. Spectral Approach (Sine Input) on Medium-Speed, High-Resolution Model ADC2

The ADC2 is a high-level model of a 16-bit
pipeline converter that operates with reference voltages of −2.0 V
and 2.0 V. The sampling frequency is fs=5.0 MHz.
The noise due to both the input and the ADC, referred to the input, is
approximately 1LSB rms white noise. The amplitude of the input signal is
approximately −0.2 dBFS, the offset is about 10LSBs and the frequency is near fs/222.The phase has been evenly spread allover the
experiments inside the range[−π,π]. In these experiments,
it has been taken two consecutive registers with 32768 samples each. In this example,
the background noise appears at about −130 dBFS, so the selected spectral lines
are those that are harmonics with amplitudes higher than −120 dBFS.The so irregular and discontinuous structure of
the INL of this ADC leads to a typical selection about 150 harmonics with
orders up to 600th. Figure6 shows the typical results obtained. Figure 6(a)
shows a comparison between the real INL (thick line) and the one estimated (thin line) using the histogram method with
the above two acquired registers. The number of acquired samples is not still
enough to sketch the INL shape. However and in spite of such a discontinuous
structure of the nonlinearity of the ADC2, the estimate from (12) absolutely follows the real INL evaluated using the standardized histogram method
(Figure 6(b)). The difference between both estimates has been depicted in Figure 6(c).
Biggest differences occur at the higher transitions, where the smoothing effect
due to the limited number of harmonics that has been selected is more evident. Even
if the gain is not evaluated, the intrinsic nonlinearity of the ADC (the one where
the gain and the offset are corrected) still can be extracted. Figure 7(a) shows the results
obtained for the ADC2 making g=1 in (10). Although the difference is noticeable, if the gain
and offset are eliminated from both curves (subtracting their best-fitting
lines), a good approximation can be achieved as Figure 7(b) shows.(a) Comparison between two ADC2 INL Histogram estimations: in thin (blue) line the one estimated with very few samples (the two registers acquired to apply the new method). In thick (black) line the real INL, (no gain effect has been included).
(b) Overlapped ADC2 INL estimations: the estimated using the spectral approach and (12), and the real INL (gain effects are included).
(c) Difference between the above INL signatures in (b).
(a)(b)(c)(a) Thick (black) line: estimated ADC2 INL using (12) and taking the gain equal to 1. In thin (blue) line the real INL where gain effects are included.
(b) Overlapping of the two above estimations after the gain and offset effects have been corrected.
(a)(b)


## 4.2. Real Experiments

This section shows the results obtained applying the introduced method to a real ADC.
This converter has a transfer curve with very strong discontinuities.


### 4.2.1. Spectral Approach (Sine Input) on High-Speed, Low-Voltage Experimental Prototype ADC

The ADC under test is a
fully differential 12-bit pipeline converter prototype in a 120 nm CMOS
technology with reference voltages −1 V and 1 V. The sinusoidal input has been nonbuffered
AC coupled to the ADC and generated using the Agilent N8241A AWG, with amplitude
of −0.1 dBFS and a frequency of 500 kHz approximately. In this case, as a good
coherent experiment has been done, only a register of 4090 samples has been
acquired using a 20 MHz sampling master clock (neither averaging nor windowing has been applied).
Figure8(a) shows a typical nonwindowed magnitude spectrum, with the floor
noise at about −100 dBFS. The harmonics selected were those with the amplitude
higher than −90 dBFS, and the typical selection takes about 45 harmonics with
orders up to 150th.(a) Typical magnitude spectrum obtained from the prototype Pipeline ADC output.
(b) Comparison between two INL estimations: In thin (blue) line, thereal INL. In thick (black) line the INL estimation from the spectral approach and (12) using the spectrum in (a) (gain and offset effects have been corrected).
(a)(b)Figure8(b)
compares the INL pattern obtained using the spectral approach (from (12)) with the real INL estimated using the standard histogram method. Notice that the INL evaluated
from our proposed method describes the shape of the real INL good enough, even in hard discontinuities. Best
results are obtained if more registers have been acquired, since noise
averaging improves both uncertainty and the number of selected harmonics.


## 4.2.1. Spectral Approach (Sine Input) on High-Speed, Low-Voltage Experimental Prototype ADC

The ADC under test is a
fully differential 12-bit pipeline converter prototype in a 120 nm CMOS
technology with reference voltages −1 V and 1 V. The sinusoidal input has been nonbuffered
AC coupled to the ADC and generated using the Agilent N8241A AWG, with amplitude
of −0.1 dBFS and a frequency of 500 kHz approximately. In this case, as a good
coherent experiment has been done, only a register of 4090 samples has been
acquired using a 20 MHz sampling master clock (neither averaging nor windowing has been applied).
Figure8(a) shows a typical nonwindowed magnitude spectrum, with the floor
noise at about −100 dBFS. The harmonics selected were those with the amplitude
higher than −90 dBFS, and the typical selection takes about 45 harmonics with
orders up to 150th.(a) Typical magnitude spectrum obtained from the prototype Pipeline ADC output.
(b) Comparison between two INL estimations: In thin (blue) line, thereal INL. In thick (black) line the INL estimation from the spectral approach and (12) using the spectrum in (a) (gain and offset effects have been corrected).
(a)(b)Figure8(b)
compares the INL pattern obtained using the spectral approach (from (12)) with the real INL estimated using the standard histogram method. Notice that the INL evaluated
from our proposed method describes the shape of the real INL good enough, even in hard discontinuities. Best
results are obtained if more registers have been acquired, since noise
averaging improves both uncertainty and the number of selected harmonics.


## 5. Conclusions

In this paper, a new method for the INL estimation of ADCs has been presented
which is based on a continuous model of the ADC transfer function. The method
uses a spectral processing of the ADC output to estimate its harmonic amplitudes
and phase-shifts from which the INL signature is derived. Different ADC examples
with very different nonlinearity pattern have been performed to validate the
proposed method. The obtained results have been compared with those obtained
from the traditional histogram method and have proven not only the feasibility of
the new method, even when the ADC nonlinearity has very strong discontinuities,
but also its low cost and efficiency since a significant lower number of output
samples can be used still giving very realistic INL signature values.



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*Source: 657207-2008-07-22.xml*