A complete parameterization of all positive rational extensions of a covariance sequence

In this paper we formalize the observation that filtering and interpolation induce complementary, or “dual,” decompositions of the space of positive real rational functions of degree less than or equal to n. From this basic result about the geometry of the space of positive real functions, we are able to deduce two complementary sets of conclusions about positive rational extensions of a given partial covariance sequence. On the one hand, by viewing a certain fast filtering algorithm as a nonlinear dynamical system defined on this space, we are able to develop estimates on the asymptotic behavior of the Schur parameters (1918) of positive rational extensions. On the other hand we are also able to provide a characterization of all positive rational extensions of a given partial covariance sequence. Indeed, motivated by its application to signal processing, speech processing, and stochastic realization theory, this characterization is in terms of a complete parameterization using familiar objects from systems theory and proves a conjecture made by Georgiou (1983, 1987). Our basic result, however, also enables us to analyze the robustness of this parameterization with respect to variations in the problem data. The methodology employed is a combination of complex analysis, geometry, linear systems, and nonlinear dynamics