A New Spectrum Extension Method That Maximizes the Multistep Minimum Prediction Error-Generalization of the Maximum Entropy Concept

Given (n + 1) consecutive autocorrelations of a stationary discrete-time stochastic process, one interesting question is how to extend this finite sequence so that the power spectral density associated with the resulting infinite sequence of correlations is nonnegative everywhere. It is well known that when the Hermitian Toeplitz matrix generated from the given autocorrelations is positive definite, the problem has an infinite number of solutions and the particular solution that maximizes the entropy functional results in a stable all-pole model of order n. Since maximization of the entropy functional is equivalent to maximization of the minimum meansquare error associated with one-step predictors, in this paper the problem of obtaining admissible extensions that maximize the minimum meansquare error associated with A:-step (k < n) predictors, that are compatible with the given autocorrelations, is studied. It is shown here that the resulting spectrum corresponds to that of a stable ARM A (n, k − 1) process. The details of this true generalization of the maximum entropy extension are worked out here for a two-step predictor along with several other interesting conclusions.