Geometric Aspects of the Covariance Partial Realization Problem

In this paper, both Euclidean geometric and non-Euclidean geometric descriptions for the regions of the solution set to the covariance partial realization problem are discussed. In particular, it is shown that all the solution functions are located inside a Euclidean disk - the Weyl disk, and a non-Euclidean disk - the Apollonian disk. The radius of each disk depends on the norm of the parameterizing Schur functions. Moreover, it is pointed out that the well known maximum entropy solution is exactly the hyperbolic center of the Apollonian disk; while, the associated power spectral density function is the geometric mean center of the real region where the Weyl disk is projected to along the imaginary axis. Sensitivity analysis of the Weyl disk radius and Apollonian disk center due to the perturbation of the given finite covariance data is presented.