Generalization of maximum entropy spectrum extension method

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 correlations is positive-definite the problem has an infinite number of solutions and the particular solution that maximizes entropy results in a stable all-pole model of order n. Since maximization of entropy is equivalent to maximization of the minimum mean square error associated with one-step predictors, the problem of obtaining admissible extensions that maximize the minimum mean square error associated with k-step (k ≤ n) predictors, which are compatible with the given correlations, is studied. It is shown that the resulting spectrum corresponds to that of a stable ARMA (n, k-1) process. The details of this particular extension method are worked out for a two-step predictor.