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IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 2, APRIL 2001 461
Influence of Frequency Errors in the Variance of the
Cumulative Histogram
Francisco André Corrêa Alegria and António Manuel da Cruz Serra
Abstract—In this paper, the calculation of the variance in the
number of counts of the cumulative histogram used for the char-
acterization of analog-to-digital converters (ADCs) with the his-
togram method is presented. All cases of frequency error, number
of periods of the stimulus signal, and number of samples are con-
sidered, making this approach more general than the traditional
one, used by the IEEE 1057-1994 standard, where only a limited
frequency-error range is considered, leading to a value of 0.2 for
the variance. Furthermore, this value is an average over all cumu-
lative histogram bins, instead of a worst-case value, leading to an
underestimation of the variance for some of those bins.
The exact knowledge of this variance allows for a more efficient
test of ADCs and a more precise determination of the uncertainty
of the test result. This calculation was achieved by determining the
dependence of the number of counts on the sample phases, on the
transition voltage between codes, and on the stimulus signal phase.
Index Terms—ADC test, analog–digital conversion, frequency
error, histogram.
I. INTRODUCTION
T
with a known amplitude probability density function is used to
stimulate the converter. Several samples are acquired at a fre-
quency
and the cumulative histogram is computed. The cu-
mulative histogram for code
is the number of samples whose
digital conversion is equal to or lower than output code . The
converter transition levels and code bin widths are determined
by comparing the number of counts experimentally obtained
with the number expected from an ideal converter.
Usually a sinusoidal stimulus signal (with frequency
usedsinceitiseasilygeneratedwiththerequiredspectralpurity.
Toguaranteethatallcodeshaveanequalopportunityofbeing
stimulated, the number of samples must be acquired during an
integer number of periods of the input signal. Letting
the number of samples acquired and
periods, the stimulus and sampling frequencies must satisfy the
following relation:
HE histogram method is a tool widely used for the charac-
terizationofanalog-to-digitalconverters(ADCs).Asignal
) is
denote
the number of signal
(1)
Besides acquiring the samples during an integer number of
periods, it is also necessary for their phases to be evenly dis-
Manuscript received May 14, 2000; revised November 10, 2000.
The authors are with Telecommunications Institute and Department of Elec-
trical and Computer Engineering, Instituto Superior Técnico, Technical Univer-
sity of Lisbon, Lisbon 1049-001, Portugal.
Publisher Item Identifier S 0018-9456(01)02972-2.
tributed. To achieve this, the numbers
ally prime.
Therandomphasedifferencebetweenthesignalandthesam-
pling clock will make the number of counts in the cumula-
tive histogram a random variable. The results of the histogram
method will thus be a random process with a normal probability
densityfunction.Bycalculatingthevarianceofthisdistribution,
an uncertainty interval for the test result may be calculated.
This variance will also depend on the additive noise present
in the stimulus signal, in the converter itself [1], [2], and on the
samplingclockjitter[3].Inthiswork,welimitedthestudytothe
case where neither jitter nor additive noise are present, focusing
only in the random nature of the phase difference between the
sampling clock and the stimulus signal.
In practice, the referred frequencies do not verify (1) exactly,
causing the sample phases not to be uniformly distributed, as
is desirable. In this paper, we present a study of the influence
of errors in both frequencies on the variance of the cumulative
histogram.
andmust be mutu-
II. VARIANCE OF THE CUMULATIVE HISTOGRAM
Let us consider that the stimulus signal is sinusoidal with pe-
riod
and phase:
(2)
We considered, without loss of generality, that the signal has
1 V of amplitude and no offset voltage.
A. Sample Phases
Numbering thesamples from 0to
the first sample (
are defined by
Defining the phase as the relative position (from 0 to 1) of the
sampling instant ( ) in relation to the stimulus signal period
( ), each sample
will have a phase
and considering that
, the sampling instants
is the sampling interval.
) occurs at
where
given by
MantissaMantissa(3)
where the function Mantissa represents the fractional part of its
argument.
The sample phase is a periodic function of the variable
can be seen in Fig. 1 for samples 1 and 4.
As can be seen in Fig. 1, for some values of
sample 1 is greater than the phase of sample 4 and for other
as
the phase of
0018–9456/01$10.00 © 2001 IEEE
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462IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 2, APRIL 2001
Fig. 1. Representation of ???? for ? = 1 and ? = 4.
values of
samples,
the opposite is true. The phases of two different
and, are equal when
Mantissa
Mantissa(4)
where
is an integer. Solving forwe have
(5)
We can thus conclude that the ordering of the sample phases
changes every time that
is a rational number with a denomi-
nator between 1 and
( and
0 to
). The increasing sequence of all rational numbers
with denominator equal or lower than
Series of order
[4]. For instance, in the case of
we have the following Farey series:
are integers ranging from
is called the Farey
= 5,
(6)
As will be seen later, the values of the elements of the Farey
Series are the values of
where the variance has a local max-
imum. The Farey Series does not have an analytical description.
For the calculus of the variance of the number of counts in
the cumulative histogram, we will need to order the samples
with increasing values of phase . The resulting sequence will
be noted as
.
B. Phase Interval
Consider an example where
frequency error:
signal, the sample values (dots), and the sample phases
symbols) are represented for
to the cumulative histogram bin are the ones inside interval
Since in this example (1) is not satisfied, the distribution of the
sample phases is not uniform.
When thestimulussignalphase( )changes, thesamplesthat
belong to a cumulative histogram bin are not the same. The case
where
rad is shown in Fig. 3, where we can count 13
samples (in Fig. 2 we could count only ten samples).
Eachbinhasadifferentvoltagetransitionlevel(
fore a different phase interval
. The variable , represented in Figs. 2 and 3, is
the difference between the start of the phase interval
smallest value of thephase of all samples. (This is equal to 0be-
cause we chose the time reference so that the sampling instant
of the first sample was 0.)
= 19,= 2and thereis a 14%
. In Fig. 2, the stimulus
(plus
rad. Samples belonging
.
)andthere-
whose length is given by
and the
Fig.2.
with sample values ??? and phases (+) in the case of ? ? ???? rad.
Representationofthe stimulussignal andtransitionvoltage? together
Fig.3.
with sample values ??? and phases (+) in the case of ? ? ???? rad.
Representationofthe stimulussignal andtransitionvoltage? together
The value of
and on the stimulus signal phase. Traditionally, that phase is not
controlled and will vary randomly from test to test.
In order to determine the variance of the number of counts
in a bin of the cumulative histogram, we need to know its value
for different phase differences between the sampling clock and
the stimulus signal
.
Considering asarandomvariable,thenumberofcountsina
bin of the cumulative histogram will also be a random variable,
function of
and of the deterministic variable . The variable
is the end of the phase interval
Observing Figs. 2 and 3 we can construct the function
by counting the number of samples that have a phase in the
interval
whose endpoints are
greater than the value of
the phase interval has two segments,
one from
toand another from
(Fig. 3). The contour representation of
Fig. 4.
Considering that
will depend on the voltage transition level
.
and . When the value ofis
to
can be seen in
(7)
we can construct the contour representation of
from that of
(Fig. 4).
To determine the variance of the random variable
(Fig. 5)
(8)
we consider that the random variable
between 0 and 1:
is uniformly distributed
or
(9)
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CORREA ALEGRIA AND DA CRUZ SERRA: INFLUENCE OF FREQUENCY ERRORS IN THE VARIANCE OF THE CUMULATIVE HISTOGRAM463
Fig. 4.Contour representation of ??????.
and thus the variance of the random variable is
(10)
Theaveragevalueof
When
samples are counted as belonging to that interval independently
ofthevalue of .Consequently, wehave
situation, when
all samples are counted as belonging to
the phase interval and we have
forsomeisgivenby.
the phase interval has a length of 0 and thus no
.Intheopposite
.
C. Absence of Frequency Error
We considered it appropriate to establish a bridge between
the calculations presented here and results obtained in [2] and
[3] for the case of an absence of frequency error.
When there is no frequency error, all the sample phases are
equally spaced and the contour representation of
comestheonerepresentedinFig.6.Delimitedbyverticaldotted
lines, one can see
different intervals of
. Numbering the intervals from 1 to
their endpoints by
and
From the contour representation of Fig. 6, we can reach the
representation of
, for a generic value
It can be seen that the area under the line
of one rectangle (
) that always has the same area, whatever
the value of
whose area depends on the value of
be-
with a width of
, we can express
.
(Fig. 7).
is composed
and smaller rectangles
:
(11)
The total area is given by
(12)
and the variance (10) can thus be expressed as
(13)
Fig. 5.Contour representation of ??????.
Fig. 6.
phases.
Contour representation of ?????? in the case of equally spaced sample
Fig. 7.Representation of ? ???.
Equation (13) represents an inverted parabola for each of the
different intervals of. These parabolas can be observed in
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464IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 50, NO. 2, APRIL 2001
Fig. 8.Representation of ? for different values of ? and ? ?? ? ??.
Fig. 8, in vertical planes of constant , with
0.6, and 0.8.
equal to 0.2, 0.4,
D. Results in the presence of frequency error
In Fig. 8, a three-dimensional representation of
sented when
varies from 0.2 to 0.8 and when
to 1. Analyzing Fig. 8, we find that when
with a denominator of 5 (
with 5, we have a minimum in the value of the variance ( : 0.2,
0.4, 0.6, and 0.8). Also, when the value of
ments of the Farey series of order 4 (
maximum of the variance ( : 0.25, 0.33, 0.5, 0.66, and 0.75).
The value of the variance of the counts in the cumulative his-
togram is used to determine the total number of samples that
mustbeacquiredtoguaranteethattheresultsoftheintegralnon-
linearity (the difference between the actual and the ideal transi-
tion voltages), obtained by the histogram test, have an uncer-
tainty smaller than some chosen value. The expression tradi-
tionally used for this determination (14) is the one developed
by Blair [2] and later adopted by IEEE Standard 1057-1994 [5].
is pre-
varies from 0
is a rational number
) and a numerator mutually prime
is one of the ele-
) we have a local
(14)
where
has
in the standard.
is the number of records to be acquired. Each record
samples. The other symbols used in (14) are as defined
The value 0.2 present in (14) was computed by averaging the
variance over all values of
for a certain value of frequency
error. We believe, in accordance with [3], that the worst-case
valueshouldbeusedinsteadoftheaveragevalue.Weproposein
this paper that (14) should be used to estimate the total number
of samples required, but the value 0.2 should be replaced by the
maximum value of variance, obtained from Fig. 8, for the range
of values of
determined by the stimulus signal and sampling
clock frequency errors.
III. CONCLUSION
The expressions commonly used in the determination of the
number of samples required for the characterization of ADCs
with the histogram method are the ones presented in [2] and
usedbyIEEEStandard1057-1994[5].Theworkpresentedhere
demonstrates that these expressions can be easily expanded
when the frequency error is greater than the limit established
in [2]. It suffices to substitute the value of the variance in the
counts in the cumulative histogram calculated there (0.2) with
the value obtained by the method presented here.
This enables a more realistic estimation of the number of
samples required, making the ADC test faster. A better com-
putation of the variance of the number of counts in the cumu-
lative histogram also allows for a more exact computation of
the uncertainty interval of the results obtained by the histogram
method, namely the value of the transition levels. Those values
are usually obtained from the cumulative histogram by using
.
Unfortunately, there is not an analytical expression for the
calculation of the variance. However, since a computer is usu-
ally used to process the results of the test, this extra numerical
calculation is not a problem.
REFERENCES
[1] J. Doernberg, H.-S. Lee, and D. A. Hodges, “Full-speed testing of A/D
converters,” IEEE J. Solid-State Circuits, vol. 19, pp. 820–827, Dec.
1984.
[2] J. Blair, “Histogram measurement of ADC nonlinearities,” IEEE Trans.
Instrum. Meas., vol. 43, pp. 373–383, June 1994.
[3] G. Chiorboli and C. Morandi, “About the number of records to be
acquired for histogram testing of A/D converters using synchronous
sinewave and clock generators,” in Proc. 4th Workshop ADC Modeling
and Testing, Bordeaux, France, Sept. 1999, pp. 182–186.
[4] R. Graham, D. Knuth, and O. Potashnik, Concrete Mathematics, 2nd
ed.Reading, MA: Addison-Wesley, 1994.
[5] IEEE Standard for Digitizing Waveform Recorders, IEEE Std. 1057-
1994, Dec. 1994.