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.