IEEE TRANSACTIONS ON INSTRUMENTATION AND MEASUREMENT, VOL. 52, NO. 1, FEBRUARY 2003 69 Variance of the Cumulative Histogram of ADCs Due to Frequency Errors Francisco André Corrêa Alegria and António Manuel da Cruz Serra Abstract—The variance in the number of counts of the cumu- lative histogram, used for the characterization of analog-to-dig- ital converters (ADCs) with the Histogram Method, is calculated without any restrictions regarding the magnitude of the frequency errors on the stimulus and sampling signals, number of periods of the stimulus signal, and number of samples. The formulation adopted allows a graphical interpretation of the problem that helps future developments still needed in this particular subject. 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. Numerical simulation and experimental results that validate the theory are shown. Index Terms—Analog-to-digital conversion, analog-to-digital converter (ADC) test, frequency error, histogram method. I. INTRODUCTION T HE HISTOGRAM method [1] is a tool widely used for the characterization 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 frequency and the cumulative histogram is computed. The cumulative 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 deter- mined by comparing the number of counts experimentally ob- tained with the number expected from an ideal converter. 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. Denoting by the number of samples acquired and by the number of signal periods, the aforementioned frequencies must satisfy the fol- lowing relation [3]: (1) Besides acquiring the samples during an integer number of pe- riods, it is also necessary for their phases to be evenly dis- tributed. To achieve that, the numbers and must be mu- tually prime [3]. Manuscript received May 29, 2001; revised November 6, 2002. This work was supported by the Portuguese national research project entitled “New measurement methods in Analog to Digital Converters testing,” Reference POCTI/ESE/32698/1999. The authors are with the Telecommunications Institute and Department of Electrical and Computer Engineering, Instituto Superior Técnico, Technical University of Lisbon, Lisbon, Portugal, and also with the Laboratorio de Medidas Electricas, Instituto Superior Tecnico, Technical University of Lisbon, Lisbon, Portugal (e-mail: falegria@lx.it.pt; acserra@ist.utl.pt). Digital Object Identifier 10.1109/TIM.2003.809083 The initial random phase of the signal in each record of ac- quired samples will make the number of counts in the cumula- tive histogram a random variable. The results of the histogram method will thus be random with a probability density function that can be considered normal. By computing the variance of that distribution, an uncertainty interval for the test results may be calculated. In practice, the referred frequencies do not verify (1) exactly, causing the sample phases to not be uniformly distributed, as it is desirable. This contributes to an increase in the variance of the results. Due to the nonlinear relationship between the number of counts of the cumulative histogram and the transition voltages and sample phases, it is difficult to obtain an analytical expres- sion for the variance of the number of counts. What has been done in the past is to determine an expression for the error of the frequency relation that guarantees a certain maximum value for the variance of the number of counts of the cumulative histogram [6]. What needs to be done yet is to determine a sim- ilar expression for the variance of the number of counts of the histogram. The analytical approach taken in [6] is not easily extrapo- lated for other situations, so, in this work, we present a different formulation accompanied by a graphical interpretation that will help, in the future, to determine a limit for the frequency ratio that guarantees a maximum for the variance of the number of counts of the histogram. In [8]–[10], we have presented previous developments of this work; however, here, a new formulation is used that better il- lustrates the ideas we wish to transmit, allowing the reader an easier understanding of our work. II. PRELIMINARIES The value of the variance of the counts in the cumulative his- togram is used to determine the total number of samples that must be acquired to guarantee that the results of the INL, ob- tained by the histogram test, have an uncertainty smaller than a given chosen value. The expression traditionally used for this determination is the one developed by Jerome Blair [2] and later adopted by the IEEE 1057-1994 standard [3] (2) where is the maximum allowed uncertainty in LSB, is the confidence level, in the number of bits, is the amount of overdrive, * is the input-equivalent noise standard deviation, is the full-scale voltage, and is the number of records to be acquired. Each record has samples. The value 0.2 present 0018-9456/03$17.00 © 2003 IEEE