It is an important problem in signal processing, system realization and system identification to find linear discrete-time systems which are consistent with given covariance parameters. This problem is formulated as a problem of finding discrete-time positive real functions which interpolate given covariance parameters. Various investigations have yielded several significant solutions to the problem, while there remains an important open problem concerning the McMillan degree. In this paper, we use more general input-output characteristics than covariance parameters and consider finding discrete-time positive real matrix functions which interpolate such characteristics. The input-output characteristics are given by the coefficients of the Taylor series at some complex points in the open unit disk. Thus our problem is a generalization of the interpolation problem of covariance parameters. We reduce the problem to a directional interpolation problem with a constraint and develop the solution by a state-space based new approach. The main results consist of the necessary and sufficient condition for the existence of the discrete-time positive real matrix function which interpolates the given characteristics and has a limited McMillan degree, and a parameterization of all such functions. These are a contribution to the open problem and a generalization of the previous result.

We present a minimal lattice realization of MIMO linear discrete-time systems which interpolate the desired Markov and covariance parameters. The minimal lattice realization is derived via a recursive construction algorithm based on the state space description and it parametrizes all the interpolants.