Distortion and estimation of the autocorrelation function and spectrum of a randomly sampled signal

Abstract

The theory of sampled signals is an important topic in the study of communication and control systems. While a comprehensive theory is available to treat the equispaced sample case, much work remains to be done to completely understand the effects of random sampling.

This thesis provides two additions to the theory of random sampling. The first is a new theory for predicting the autocorrelation function (ACF) and spectrum of thesampling process. In this theory the random sampling process is described by its time-varying mean sampling rate and the complete set of inter-sample probability density functions. These results are then used to determine the autocorrelation function and spectrum of a signal after sampling. The theory predicts the ACF for both the stationary and nonstationary cases, and does not use any of the "small deviation from equispaced sampling" assumptions required by many random sampling theories. Thus, from the model developed here, several new results have been obtained for important sampling processes previously untreated in the literature. Notable among these is an analysis of Poisson sampling with dead time. It is expected that this theory will find application in the error analysis of equispaced sample systems, design of optimized random sampling schemes for specific applications, and generalization of the theory of "shot noise".

In the second part of this thesis, an estimator is developed for determining the ACF of a signal from its random samples. This method requires no a priori knowledgeof the sampling process and is shown to yield good estimates of the ACF even when the mean sampling rate is well below the Nyquist rate. Error sources in practical realizations of the estimator are explored and results for the variance of the estimate are determined. It is shown that this estimate approaches the maximum likelihood estimate at low mean sampling rates. Determination of a spectrum estimate from the ACF estimate is also discussed. This estimate has found application in determining the spectrum of the turbulent velocity fluctuations in the exhaust of a jet engine whenthey are measured by a laser velocimeter.