ON THE VARIANCE OF THE HISTOGRAM TEST OF ADCS

The histogram test is one of the most widely used methods for the characterization of analog-to-digital converters (ADCs), yielding estimates of the transition voltages and, from them, of the code bin widths, gain, offset and linearity errors. Since these estimates are computed from a finite number of samples that are affected by several non-idealities of the test setup, they behave as random variables whose uncertainty must be quantified in order to establish a confidence interval for the converter characteristics. This paper presents an overview of that uncertainty, with particular emphasis on the variance of the transition voltages obtained with the cumulative histogram.
The combined influence of the three dominant random effects — phase noise, input-equivalent additive noise and the random phase difference between the stimulus signal and the sampling clock — on the number of counts of the cumulative histogram is analysed within a unified framework, instead of treating each effect separately and adding the individual contributions as was done in earlier works. The variance of the number of counts is decomposed into the mean of the conditional variance and the variance of the conditional mean, and the dependence of each term on the standard deviations of the noise sources and on the transition voltage is examined. Building on this analysis, an asymptotically approximate but more accurate expression for the variance of the transition voltages is presented, whose maximum relative error is below 17%. It reproduces the saturation of the variance towards M/4 for large noise by means of a minimum function and accounts for the random phase difference through a maximum function, thereby avoiding the overestimation inherent in the expressions adopted in the de facto standard for ADC testing.
The effect of frequency errors, which cause the sample phases to depart from a uniform distribution, is also addressed, including the coherent-sampling condition, the role of the frequency ratio and of the Farey sequence, and the bound on the frequency error below which the variance of the number of counts remains lower than ¼. Theoretical predictions are validated both against results previously reported in the literature and against experimental measurements performed on real ADCs, showing good agreement in all cases.