Correlation Between Static and Dynamic Parameters of A-to-D Converters: In the View of a Unique Test Procedure

Abstract

ADCs are fully characterized by both static and dynamic parameters. Testing methods usually combine a histogram-based approach with a spectral analysis to determine the complete set of ADCs parameters. In the view of a unique test procedure, this paper investigates the correlation between both kinds of parameters. Experimental results demonstrate that under appropriate test conditions, the dynamic parameters extracted from a classical FFT exhibit significant variations against ADC offset, gain and non-linearity errors, opening the way of a low-cost test strategy in the frequency domain.

Full text

Page 1

JOURNAL OF ELECTRONIC TESTING: Theory and Applications 20, 375–387, 2004
c ? 2004 Kluwer Academic Publishers. Manufactured in The United States.
Correlation Between Static and Dynamic Parameters of A-to-D Converters:
In the View of a Unique Test Procedure∗
F. AZA¨IS, S. BERNARD, Y. BERTRAND, M. COMTE AND M. RENOVELL
LIRMM—CNRS/University of Montpellier, 161, Rue Ada, 34392 Montpellier, Cedex 5, France
Received September 9, 2002; Revised May 12, 2003
Editors: F. Vargas and V. Champac
Abstract.
a histogram-based approach with a spectral analysis to determine the complete set of ADCs parameters. In the view
of a unique test procedure, this paper investigates the correlation between both kinds of parameters. Experimental
results demonstrate that under appropriate test conditions, the dynamic parameters extracted from a classical FFT
exhibit significant variations against ADC offset, gain and non-linearity errors, opening the way of a low-cost test
strategy in the frequency domain.
ADCs are fully characterized by both static and dynamic parameters. Testing methods usually combine
Keywords:
analog and mixed-signal testing, ADC test, spectral analysis
1.Introduction
At present, most part of the signal processing in ar-
eas like instrumentation, telecommunications, control
and consumer electronics is carried out at the digital
level.TheroleofAnalog-to-DigitalConverters(ADCs)
placed at the borders of the digital domain is getting a
particularrelevance,sincethesignaldegradationintro-
duced by these components cannot normally be recov-
ered by subsequent processing. The correct evaluation
ofthewholesystemperformancethereforerequiresthe
test of these mixed-signal devices. As the new genera-
tionsofADCsprovideincreasingspeedandresolution,
test requirements become more and more stringent, re-
sulting in ever more expensive test procedures. The
test cost frequently reaches half the price of the de-
vice itself. The markets of both stand-alone ADCs and
ADC macrocells to be embedded in complex ASICs
would benefit from the availability of low-cost test
methods.
∗This work has been selected from the 3rd IEEE Latin-American
Test Workshop (LATW2002).
The industrial testing of ADCs is still largely spec-
ification based. Parameters listed in the specification
of a converter can actually be divided in two groups,
one related to the transfer function and another that
expresses the deformation induced on the converted
signal. In the first group, one finds the so-called static
parameterssuchasoffset,gain,differentialandintegral
non-linearityerrors.Theseparametersareusuallyeval-
uated using a histogram-based approach [6, 7], which
is based on a statistical analysis of how many times
each code appears on the ADC output for a given in-
put signal. The main disadvantage of this technique re-
sidesinthehighnumberofsamplesrequiredtoachieve
statistically satisfactory results. The required number
of samples can become unacceptably large for high
resolution converters, especially in presence of noise.
Even present day Automated Test Equipment (ATE)
has problem with the efficient handling and processing
ofsuchlargesamplesets.ThesecondgroupofADCpa-
rameters is composed of the so-called dynamic param-
eters. Examples of such characteristics are the Signal-
to-Noise And Distortion ratio (SINAD), the Spurious-
Free Dynamic Range (SFDR) or the Total Harmonic

Page 2

376
Aza¨ ıs et al.
Distortion (THD). These parameters are usually evalu-
atedfromaFastFourierTransform(FFT)ofthedigital
samples acquired on the converter output when a pure
sine-wave is applied to its input [9]. Relatively small
sample sets are usually sufficient to get good estimates
oftheADCdynamicparameters.Completecharacteri-
zation of a converter can therefore be obtained by cou-
pling a histogram-based approach with a spectral anal-
ysis [4]. Finding a link between static and dynamic
parameters could enable one to deduce the complete
set of an ADC parameters from a unique test acqui-
sition and processing [2]. Spectral analysis would be
preferabletothehistogrammethodfortworeasons[8]:
firstly,itismorerepresentativeoftheconverterrunning
reality, and secondly it requires less samples, which
means a shorter processing time and reduced storage
resources, two major considerations in the perspective
of low-cost ADC testing. Hence, it is the objective of
this work to investigate whether the dynamic parame-
ters are sensitive to ADC static errors and under which
test conditions.
The paper is organized as follows. Section 2 defines
the dynamic parameters and Section 3 presents the ex-
perimental setup. Section 4 gives a preliminary study
on the sensitivity of the considered dynamic parame-
ters upon test conditions: the number of samples taken
intoaccountfortheFFT,thenumberofinputsignalpe-
riods during acquisition and the input signal amplitude
are successively treated. Then, the influence of static
parameters on dynamic performances is explored in
Section 5. Section 5.1 sets out the impact of an offset
error, while the consequences of a gain error are pre-
sented in Section 5.2. The study of non-linearity errors
is more complex than the one of other static errors.
Thus, Section 5.3 is divided in two parts: firstly a study
of influence of a single non-linearity on one code, sec-
ondly the influence of different shapes of global non-
linearity. Finally, Section 6 gives some concluding re-
marks.
2.Definitions
Spectral analysis of ADC is based on the exploitation
of the Fourier transform of the numerical samples ac-
quired on the converter output, when a pure sine-wave
isappliedtoitsinput.Whilethespectrumofapuresine-
wave consists of a unique line located at the frequency
of the signal, the spectrum of the corresponding quan-
tizedoutputthroughanidealconvertercomprisesnoise
components throughout the frequency range. These
parasitic elements are due to the quantization princi-
ple itself, that induces predictable quantization noise.
Inthecaseofarealconverteraffectedbynon-idealities,
additional noise and harmonics of the signal frequency
alsoappearinthespectrum.Therefore,theanalysisofa
real converter is conducted in comparison with the one
of a perfect ADC, in order to differentiate the intrinsic
quantization noise from the one resulting of the physi-
caldeviceimperfections.Spectraltestactuallyconsists
in processing the frequency components of the spec-
trum to determine the different test parameters listed
below.
The Signal-to-Noise Ratio (SNR) of an ideal ADC is
defined by the ratio of the input signal effective value
over the quantization noise effective value. A good es-
timation of the SNR value for a perfect n-bit converter
excited by a full scale signal is given by [5]:
SNRdB= 6.02 × n + 1.76(1)
As mentioned, real converters generate extra noise
and harmonics, which have to be considered in the
ADC performances. Consequently, the corresponding
parameter dedicated to real converters is the SINAD
(SIgnal-to-Noise And Distortion ratio), extracted from
the spectrum as follows [5]:
SINADdB= 20 × log10


S
??
f ?= finS2
i


(2)
where S is the input signal effective value, and Sithe
effective value of the ith harmonic component.
When the input signal peak-to-peak amplitude 2Ain
differs from the converter full scale FS (2Ain≤ FS),
a corrective term is applied to the SINAD to enable
comparisons and calculation of the effective number
of bits [5]:
SINADFS= SINAD2Ain+ 20 × log10
?FS
2Ain
?
(3)
The effective number of bits, neff, is a widely used per-
formance criteria, derived from the expression of the
SNR [5]:
neff=SINADFS− 1.76
6.02
(4)
This sets out the ADC effective resolution, that means
the equivalent number of bits a perfect ADC would

Page 3

Correlation Between Static and Dynamic Parameters of A-to-D Converters377
Fig. 1.Test setup.
have if all the noise observed on the spectrum was due
to quantization process.
To give an indication upon the harmonic distortion,
particularly needed in the audio field, the THD (Total
HarmonicDistortion)isdeterminedfromtheharmonic
compoentss Hklocated at multiple frequencies of the
input signal in the Nyquist band (from 0 to half the
sampling frequency) [5]:
THDdB= 20 × log10


??
kH2
S
k


(5)
In practice, only the first few harmonics of the output
signal are actually treated as distortion, while the re-
maining distortion is treated as noise. In this paper, we
will differentiate the Total Harmonic Distortion com-
puted with all harmonics in the Nyquist band from the
THD1–5computed only with the first five harmonics.
The SFDR, Spurious Free Dynamic Range, is the
ratio of the greatest spectral component, which may
appear anywhere in the spectrum and not necessarily
in an harmonic bin, over the signal [5]:
SFDRdB= 20 × log10
?max(Si)
S
?
(6)
The floor level of noise represents the average level
of noise excluding the fundamental and the harmonics
considered in the THD1−5. Many definitions can be
found in the literature for this parameter. In this paper,
we adopt the one that is the closest to the noise mean
power:

Bm= 20 × log10

??
f ?= finS2
i−?5
k=1H2
k
N
2− 7

(7)
3.Experimental Setup
The typical test setup for ADC dynamic testing on a
classical ATE is illustratedFig. 1. The waveform syn-
thesizer generates a sine-wave signal with an input fre-
quency fin, an amplitude Ainand an offset Vo. This
stimulusisappliedontheconverterinputandtheresult-
ing samples on the converter output are acquired in the
capture memory at the rate of the sampling frequency
fs. These samples are then transferred to the CPU for
further processing: dynamic parameters are extracted
from a FFT on the digital sample set. Note that co-
herent sampling is usually used to guarantee that each
sample carries unique and independent information.
Coherence consists of an integer number N of sam-
ples acquired at frequency fsthat are equally spaced
overanintegernumber M ofidenticalsignalperiodsat
frequency fin, with N and M relatively prime. When
coherentsamplingisusedinthefrequencyanalysisdo-
main,thestimulusfundamentalcomponentandeachof
itsharmonicsfallpreciselyonsinglefrequenciesofthe
spectrum.Thisfactallowsaneasierlocationandamore
precise power measurement, which then leads to better
results for the parameter measurements. Implementing
coherence on ATE requires synchronization between
the synthesizer, the ADC under test and the capture
memory.
This experimental setup has been implemented on
the HP VEE software, a measurement programming
environment that can be used for both simulation and
physical test. For simulation, we consider a classical
model of ADC presenting a stair shaped transfer func-
tion.Incaseofanidealn-bitADC,thetransferfunction
manifests(2n−1)equallyspacedtransitionlevelsover
the full scale (FS) range of the converter. The width
of a step is a quantum or Least Significant Bit (LSB),

Page 4

378
Aza¨ ıs et al.
given by:
q = 1
LSB =FS
2n
(8)
In case of a real converter, the transfer function is
affected by some non-idealities characterized by the
static parameters. An offset error can be simply mod-
eled by adding (or subtracting) the same quantity to
all transition levels, resulting in an horizontal shift of
the ideal transfer function. A gain error is modeled by
multiplying all transition levels by the same factor, re-
sultinginacompressionordilationoftheidealtransfer
function. Non-linearity errors are modeled by individ-
ual variations of the transition levels, resulting in a de-
viation of the actual transfer function from the ideal
one.
4. Dynamic Parameters vs. Test Conditions
In order to determine whether static errors have an
influence on the dynamic test parameters measured
through spectral analysis, it is important to evaluate
the sensitivity of these dynamic parameters to the test
conditions. Investigations have been performed vary-
ing the number of samples N considered to perform
the FFT, the number of periods M of the input sine-
wave during acquisition and the input signal ampli-
tude Ain. Results presented in this section are based
on a 6-bit ADC but remain valid for higher resolution
ADCs.
4.1.Influence of the Number of Samples
Thenumberofsamplestakenintoaccountfortheanal-
ysisisanimportantfactortoconsiderforlow-costtest-
ing, since large records necessitate more acquisition
and computation time and larger storage memory. For
illustration, Fig. 2 givesthe measured SINAD and Bm
for several input signals presenting a small phase shift.
RegardingtheSINADparameter,itclearlyappearsthat
the measured value does not depend on the number of
samplesprovideditishighenough.Similarresultshave
been observed for all dynamic parameters, except Bm.
Hence, we will use 1024 samples for each FFT com-
putation afterwards in order to leave out of account
the uncertainty on the synchronization. Regarding the
mean level noise Bm, the measured value decreases
as the number of samples gets higher but appears not
really sensitive to an input signal phase shift. Addi-
Fig. 2.Dynamic parameters vs. number of samples.
tionalexperimentshavepointedoutthatthisparameter
does not depend on other factors than the number of
samples used for the FFT computation. Therefore, in
further sections, we will not present the study of its
invariability versus the setup conditions.
4.2.Influence of the Number of Input Periods
Spectral analysis is generally performed at ADC max-
imal sampling frequency and nominal input signal fre-
quency. The number of periods is adjusted to collect
the required sample set with respect to the coherence
sampling condition since it is widely accepted that the
number of periods of the input signal has no impact on
the measured parameter values. In practice, we found
out that this number of periods only alters the THD
computed using all the harmonics in the Nyquist band
(seeFig. 3). Indeed, when the number of periods in-
creases, extra bins appear between the harmonic bins.
As the number of bins is a constant equal to N/2,
the number of harmonics considered in the Total Har-
monic Distortion value decreases. In contrast, a stable
value is measured whatever the number of periods if
the TDH is evaluated using only a limited number of
harmonics.

Page 5

Correlation Between Static and Dynamic Parameters of A-to-D Converters379
Fig. 3.Dynamic parameters vs. number of periods.
4.3. Influence of the Input Signal Amplitude
In a classical testing environment, it is extremely dif-
ficult to precisely guarantee the value of the generated
input signal amplitude. Nevertheless, the values of all
the dynamic parameters except Bmdepend on this sig-
nal amplitude. In order to evaluate the potential error
on the extracted parameters, experiments have been
conducted to determine the maximal variation of the
dynamic parameters for a large range of input signal
amplitudes.
Spectral analysis is usually performed using a sine-
wavewithapeak-to-peakamplitudeslightlylowerthan
the full scale of the ADC. Fig. 4presents the evolution
oftheADCparametersfordecreasingamplitudesfrom
fullscale.TheSINADandSFDRaresensitivetoampli-
tudevariationsbutinatolerablerange.Atotalvariation
of 0.6 dB is observed for the SINADFS, which corre-
spondstoarelativeaccuracyof1.6%.Inthesameway,
a total variation of 2.8 dB is observed for the SFDR,
which corresponds to a relative accuracy of 4.6%. In
contrast, the TDH1–5presents an extreme sensitivity
with a total variation higher than 30 dB. A deviation
in the input signal peak-to-peak amplitude less than
0.1 quantization step (0.1 LSB) can actually result in a
variation of 20 dB or more in the measured harmonic
distortion,correspondingtoarelativeaccuracyof50%.
Amplitude deviations of this size are quite common
in a production environment, where deviations around
1% may occur, yielding a deviation of more than one
and a half quantization step in a 6-bit converter [3].
The present-day solution to this sensitivity problem
is noise dithering [1]. But this technique has several
drawbacks, in particular the relatively high number of
samples required to achieve a significant reduction of
thesensibilityoftheTHDandtheinfluenceofthenoise
dither on the SINAD parameter.
In contrast to conventional dynamic test, one can
apply a signal with a peak-to-peak amplitude slightly
higherthattheADCfullscale.Fig.5presentstheevolu-
tion of the ADC parameters for increasing amplitudes
fromfullscale.Thetotalvariationobservedinthiscase
appears quite important but much more predictable.
For instance considering an input signal with a peak-
to-peak amplitude 2 LSB higher than FS, a deviation
of ±0.1 LSB in the amplitude results in a variation of
about 0.4 dB in the SINAD, 0.7 dB in the SFDR and
0.8 dB in the TDH1–5, corresponding to relative accu-
racy of 1.3%, 2% and 2.2% respectively. The relative
accuracyforagivenamplitudedeviationaroundanom-
inalpointisthereforegreatlyimprovedbytheuseofan
input signal with an amplitude higher than full scale,
even if measured values do not directly represent the
ADC performances.
5. Dynamic Parameters vs. Static Errors
In order to evaluate the influence of static errors
on the dynamic parameters, experiments have been
conducted introducing various errors in the converter
model, in particular offset, gain and non-linearity er-
rors. Note that the previous section has stated out that
the SINAD, SFDR and THD1–5parameters are not sen-
sitivetothenumberofsamples(providedthatitishigh
enough) neither to the number of input periods. Hence,