Yves V. Genin

Orthogonal polynomial matrices on the unit circle

This paper proposes a natural matrix extension of the classical theory of orthogonal polynomials on the unit circle introduced by Szego. As a result, orthogonal polynomial matrices appear to be a unifying concept in various mathematical aspects of circuit and system theory.

A generalization of the Levinson algorithm for Hermitian Toeplitz matrices with any rank profile

The paper describes a recursive algorithm for solving Hermitian Toeplitz systems of linear equations, without any restriction on the ranks of their nested Toeplitz subsystems. Such a general algorithm is needed, e.g., to obtain the eigenfilters for signal processing applications, or to compute the inverse of a nondefinite Toeplitz matrix. The regular portion of the algorithm is made of the classical Levinson recursion. The singular portion requires solving some well-defined systems of linear equations with gradient structure. The dimension of each of these sytems equals the amplitude of the corresponding singularity.

A simple proof of Rudin's multivariable stability theorem

It is shown how Rudins multivariable stability theorem can be proved by using simple one-variable arguments, exclusively. In particular, no use is made of multivariable homotopy, in contrast with the original proof. The efficacy of the approach presented here is further illustrated by deriving a new stability test as well as elementary and independent proofs for the classical criteria.

Pseudo-lossless functions with application to the problem of locating the zeros of a polynomial

This paper contains an investigation of the class of pseudolossless rational functions F(p) = N(p)/D(p) , which are characterized by the property \Re F(p) = 0 for \Re p = 0 . The index of such a function, counting the zeros of the polynomial N(p)+ D(p) in the right half-plane \Re p > 0 , enjoys some very useful decomposition properties. It is shown how an appropriate index theory of pseudo-lossless functions provides a framework in which the most classical results concerning the problem of locating the zeros of a polynomial can be unified, simplified, and generalized.

Half-plane Toeplitz systems

A detailed study is made of positive definite half-plane Toeplitz systems and of the corresponding least-squares inverse approximaion problems. The general question is to minimize a given functional over the space of two-variable functions with a prescribed half-plane support. For some particular supports, such as those considered by Marzetta, the minimizing functions enjoy remarkable recurrence, stability, and convergence properties. Simple derivations of these properties are given and various new results are obtained. As an application, it is shown how the half-plane spectral factor of a given magnitude function can be inversely approximated by stable pseudepolynomials.

Multichannel singular predictor polynomials

The concept of singular predictor polynomials relative to a positive definite block-Toeplitz matrix is considered. These predictors are defined in terms of the classical multichannel predictors in a fairly natural manner. It is shown that the singular predictors satisfy a simple three-term recurrence relation, which gives rise to an efficient Levinson-type algorithm for computing the classical predictors of a given length. An associated Schur-type method can be used to determine the reflection coefficients of the prediction filters. The simplifications resulting from some symmetries in the data are examined, with special emphasis on the centrohermitian structure met in two-variable prediction problems. Finally, the Caratheodory-Fejer interpolation problem for matrix-valued functions is shown to be solvable with the help of a function-theoretic version of the Schur-type algorithm, and a duality relation is exhibited in the special case of lossless functions. >

A simple approach to spectral factorization

It is shown that the problem of spectral factorization of rational matrices can be approached by simple arguments, relying only on elementary complex analysis and standard matrix algebra.

A polynomial approach to the generalized Levinson algorithm based on the Toeplitz distance

A polynomial approach to the generalized Levinson algorithm based on the Toeplitz distance concept is given. It turns out that most properties of the standard Levinson algorithm admit natural generalizations, including the three-term recurrence relations, the Christoffel-Darboux formula, and the reflection coefficients (Schur-Szego parameters) obtainable from the data via an extension of the Schur algorithm. The theory of \sum -lossless transfer functions is shown to play the same illuminating role in the problem as the theory of Szego orthogonal polynomials in the standard Levinson algorithm.

Planar least squares inverse polynomials: Part I-Algebraic properties

A general analysis of the planar least squares inverse polynomials is presented. The algebraic properties of these polynomials are discussed in detail and applied to the stability problem related with the Shanks et al. conjecture. The theory of the orthogonal polynomial matrices on the unit circle is shown to provide interesting new results concerning planar least squares inverse polynomials.

Stability of linear predictors and numerical range of a linear operator (Corresp.)

The zeros of a predictor polynomial are shown to belong to the numerical range of a linear operator associated with the particular prediction problem considered. Application of this result to the autocorrelation and postwindowed cases shows that the predictor polynomials enjoy a well-defined stability margin which depends in particular on the length of the data sequence. The generalization of these results to the multichannel case is also discussed.