The structure of state covariances and its relation to the power spectrum of the input

We study the relationship between power spectra of stationary stochastic inputs to a linear filter and the corresponding state covariances, and identify the structure of positive-semidefinite matrices that qualify as state covariances of the filter. This structure is best revealed by a rank condition pertaining to the solvability of a linear equation involving the state covariance and the system matrices. We then characterize all input power spectra consistent with any specific state covariance. The parametrization of input spectra is achieved through a relation to solutions of an analytic interpolation problem which is analogous, but not equivalent, to a matricial Nehari problem.

I. Introduction

Consider a finite-dimensional linear filter which is driven by a multivariable stationary stochastic process, and suppose that nothing else is known about the input process. We are interested in the following two basic questions. First, is there anything we can say about the structure of the covariance of the state vector which is independent of the specific input? In particular, how can tell whether a given positive–semidefinite matrix qualifies as the state covariance of the filter for a suitable input process? Second, assuming knowledge of the state-covariance, what are all admissible power spectra for the input which are consistent with the particular state-covariance matrix?