On the Well-Posedness of the Rational Covariance Extension Problem

Abstract

In this paper, we give a new proof of the solution of the rational covariance extension problem, an interpolation problem with historical roots in potential theory, and with recent application in speech synthesis, spectral estimation, stochastic systems theory, and systems identification. The heart of this problem is to parameterize, in useful systems theoretical terms, all rational, (strictly) positive real functions having a specified window of Laurent coefficients and a bounded degree. In the early 1980’s, Georgiou used degree theory to show, for any fixed “Laurent window”, that to each Schur polynomial there exists, in an intuitive systems-theoretic manner, a solution of the rational covariance extension problem. He also conjectured that this solution would be unique, so that the space of Schur polynomials would parameterize the solution set in a very useful form. In a recent paper, this problem was solved as a corollary to a theorem concerning the global geometry of rational, positive real functions. This corollary also asserts that the solutions are analytic functions of the Schur polynomials.

After giving an historical motivation and a survey of the rational covariance extension problem, we give a proof that the rational covariance extension problem is well-posed in the sense of Hadamard, i.e a proof of existence, uniqueness and continuity of solutions with respect to the problem data. While analytic dependence on the problem data is stronger than continuity, this proof is much more streamlined and also applies to a broader class of nonlinear problems. The paper concludes with a discussion of open problems.

This Research was supported in part by grants from AFOSR, TFR, the Göran Gustafson Foundation, and Southwestern Bell.