Realization of Power Spectra from Partial Covariance Sequences

In this paper, we address the problem of realization of a spectral density function from incomplete information about the underlying stochastic process. The standing assumption is the availability of an (incomplete) partial sequence of covariance samples of the process. We study the set of rational extensions of this finite sequence to an infinite covariance function that agrees with the available samples. The classical theory of orthogonal polynomials (with respect to the unit circle) and the theory of moments have been utilized extensively in a variety of engineering problems, including the one we are dealing with. These have been known to provide a unifying framework for a variety of current spectral estimation techniques (maximum entropy method, Pisarenko's Harmonie decomposition, etc.). In this work, we consider and study the set of all covariance realizations of dimension lower than or equal to the length of the partial sequence (and equal to the dimension of the maximum entropy realization). The ME Solution is a point in this set. Other points correspond to “pole-zero” models. A general formula is obtained for recursively updated “pole-zero” models of dimension increasing with the data record. Information about the “zeros” is obtained from the asymptotic behavior of the “partial autocorrelation coefficients.” Our approach combines techniques from the theory of orthogonal polynomials on the unit circle, the theory of moments, and also techniques from degree theory/topology. Our objective is to develop a theory that will provide a frame for constructing recursively “pole-zero” realizations of increasing dimension.