Time-Variant and Time-Invariant Lattice Filters for Nonstationary Processes.

摘要

The structure of second-order processes is exposed by specification of whitening filters and modeling filters, or equivalently by Cholesky decompositions of the covariance matrix and its inverse. We shall show that these filters can be obtained as a cascade of lattice sections, each specified by a single so-called reflection coefficient parameter. For stationary processes, the reflection coefficient will be time-invariant. For nonstationary processes we can use the displacement rank concept either to find a simple time-update formula for the reflection coefficients or to replace them by a time-invariant vector reflection coefficient of size governed by the displacement rank of processes. These results are obtained in a quite direct way by using a geometric (Hilbert-space) formulation of the problem, combined with old results of Yule (1907) on update formulas for partial correlation coefficients and of Schur (1917) and Szego (1939) on the classical moment problem. (Author)