Abstract:

Spectral Factorization appeared first in N.Wiener's and A.N.Kolmogorov's papers as a mathematical method for solving certain estimation and prediction problems for stochastic processes. Namely Wiener in 1941, in an effort to make his own contribution to the war effort, was led to formulate the antiaircraft fire control problem as a linear prediction problem the solution of which required spectral factorization. Wiener also studied the problem of filtering signals out of noise and noted that similar approaches could be used for other problems in control theory and circuit theory. In fact, although Wiener's solution was not effective for the anti-aircraft problem, his emphasis on the statistical nature of communication problems and on seeking solutions that met specified optimization criteria influenced heavily the development of mathematical system theory. Spectral factorization has since found further applications in various branches of engineering and economics. Due to its importance, it is not surprising that numerous methods were developed for its solution.

Spectral factorization can be explicitly written, and is relatively easy to calculate, for scalar valued processes. However, even in the case of rational power spectrum, whenever the order of the polynomials involved is large, the factorization requires some non-trivial special methods. In addition to a few traditional methods of scalar spectral factorization named after Kolmogorov, Bauer, Levinson-Durbin, Wilson, etc. there exist also some relatively new methods as well.

A significantly more difficult task is to actually compute the spectral factor of a matrix-valued function, which arises in the case of vector-valued observations. The lack of efficient methods for such calculations is considered to be a major bottleneck that makes many theoretical developments in multidimensional signals and systems infeasible. Since Wiener's original efforts to create a computable method of multidimensional spectral factorization, dozens of papers addressed the development of appropriate algorithms. Nevertheless, the methods derived so far, even those that are algorithmic and very general, are far from being numerically sound computational procedures when the dimension of the factorized matrix is high. Specifically, it was assumed so far that the most computationally satisfactory way of computing the spectral factors is by solving a discrete-time algebraic Riccati equation, which is unfortunately a nonlinear equation with non-unique solution.

I will present a completely new approach to the multi-dimensional spectral factorization problem which was originated by my former adviser Prof. G.Janashia together with his students. Speaking about the advantages of the proposed method I will mention the following: 1) Methods of Complex Analysis are used for the solution of the problem; this turns out to be very effective since the problem itself is set in this branch of mathematics (other methods mostly use Functional Analysis or some System-theoretic approaches); 2) A dense class of matrix functions is identified for which an explicit spectral factorization is constructed; 3) Using flexible manipulations, the hard technical difficulties of the problem are completely absorbed by the mathematics leaving very few and simple procedures for computation. Namely, the multi-dimensional spectral factorization is reduced to the scalar factorization problem by solving in addition only one system of linear algebraic equations; 4) Although this system of linear equations might sometimes be of high dimension in order to achieve good accuracy, it is always positive definite and enjoys the so called displacement structure which further reduces the computational burden. In summary the longstanding mathematical problem, initiated by Wiener, of finding a computationally reliable matrix spectral factorization algorithm is solved.

During my talk I will describe the details of the proposed algorithm. A corresponding software implementation which proves the efficiency of the method will be demonstrated as well. I will mention also some connections of the established method of spectral factorization with the Corona Problem (solved by Carleson).