Schur and Levinson algorithms for nonstationary processes

It is known that a covariance matrix of a stationary discrete-time process can be uniquely characterized by a set of partial correlation coefficients (or matrices, for a vector process) which can be efficiently computed by the Levinson, or better by the Schur algorithm, each requiring O(N²) operations for an N×N covariance matrix. In this paper we point out that the same is true for any covariance matrix, stationary or not, but the corresponding nonstationary Levinson- and Schur-type algorithms require O(N³) operations which is the same amount as required by any direct method of matrix inversion. However, by introducing a classification of processes in terms of their closeness to stationarity we can obtain natural extensions of the stationary algorithms that now require only O(αN²) operations, where α is our measure of closeness to stationarity. As in the stationary case, both algorithms can be implemented by a cascade of time-invariant ladder (or lattice) sections. Some implications for the definition of a power spectral density of α-stationary processes are also noted.