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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.

and must 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 variableis

(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.