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 HE histogram method is a tool widely used for the charac- terization of analog-to-digital converters (ADCs). A signal 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 ) is used since it is easily generated with the required spectral purity. To guarantee that all codes have an equal opportunity of being stimulated, the number of samples must be acquired during an integer number of periods of the input signal. Letting denote the number of samples acquired and the number of signal periods, the stimulus and sampling frequencies must satisfy the following relation: (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 and must be mutu- ally prime. The random phase difference between the signal and the sam- 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 density function. By calculating the variance of this distribution, 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 sampling clock jitter [3]. In this work, we limited the study to the 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. 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 the samples from 0 to and considering that the first sample ( ) occurs at , the sampling instants are defined by where is the sampling interval. 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 given by Mantissa Mantissa (3) where the function Mantissa represents the fractional part of its argument. The sample phase is a periodic function of the variable as can be seen in Fig. 1 for samples 1 and 4. As can be seen in Fig. 1, for some values of the phase of sample 1 is greater than the phase of sample 4 and for other 0018–9456/01$10.00 © 2001 IEEE