The need for estimating the auto- or crosscorrelation functions of nonstationary random processes frequently arises in communication and self-adaptive systems. In most situations only one sample function can he observed over finite time. It is the purpose of this work to establish a firm theoretical basis for such a measurement of time-varying correlation functions, and the emphasis here is on suitable estimation procedures rather than specific measurement techniques. Second order stochastic processes are used as a mathematical model. The minimum mean square error between a weighted time average and the true (ensemble average) correlation function is investigated. This procedure leads to an optimum weighting function which can be obtained numerically under the Gaussian assumption. The results of such an analysis justify the much simpler finite integration-time average as an estimate. By employing a bilinear approximation in time t and delay t to the true correlation function, the mean value and variance of the simple finite-time average are found. A minimum upper bound on the mean square error is used as a criterion for an optimum observation time of such an estimate. Explicit results, however, require again the Gaussian assumption. These approaches are based mainly on a strict error analysis. A more direct approach, that leads to approximants which are random variables with unknown properties, is also outlined. The restrictions and difficulties are discussed. Examples to support the proposed estimation procedures are presented and explicit results appear mostly in graphical form.
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