Empirical Determination of Autocorrelation from Discrete-time Measurements†

Abstract

ABSTRACT Expressions are derived for the mean and variance of an estimate V(τ) of the autocorrelation function φ(τ) of a random signal, where the estimate is obtained from a set of samples of the signal, at discrete instants of time, taken from a record of finite length, the signal being assumed to be stationary and ergodic. The case of a Gaussian signal with zero mean value is discussed in detail, and for this case it is shown that a reasonable sampling frequency may be found by taking the samples close enough together to ensure that a smooth curve drawn through the sample points is a fairly accurate representation of the random signal. Expressions are also derived for the mean and variance of the estimate of autocorrelation obtained from samples which are not uniformly spaced. Such a sampling scheme may allow a better estimate to be obtained of a particular value of φ(τ), for a given amount of computational effort.