Lattice filter parameterization and modeling of nonstationary processes

Abstract

A general theory of constant-parameter modular lattice models for discrete-time nonstationary second-order processes is presented. A complete parametrization of such processes in terms of Schur and congruence coefficients is derived by developing a natural connection between the displacement structure of a covariance matrix and Schur's test for positive-definiteness of matrices. Schur coeffieients provide a simple solution to problems of covariance extension and rational spectral approximation for nonstationary covariances, and they coincide with the well-known reflection (or PARCOR) coefficients when the covariance is stationary. The congruence coefficients provide the time-varying gains of a tapped-delay-line realization of the whitening filter for the process. A constant-parameter realization of the same filter is derived by combining a lattice filter structure with a tapped delay line, both with time-invariant gains. This configuration also provides a recursive relation for the congruence coefficients (namely, a generalized Levinson-Szego recursion). The tapped-delay-line part of the realization can be eliminated by introducing the concept of admissibility. Admissibility also reduces the parametrization of a nonstationary process to Schur coefficients alone, in analogy to stationary processes, which are completely characterized by their PARCOR coefficients.