Philippe Delsarte

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.

Weights of linear codes and strongly regular normed spaces

Starting from a theorem on the distance matrix of a projective linear code, one introduces an axiomatic definition of a strongly regular normed space. it is then shown that every such normed space admits a representation by means of a projective code. As a particular case, this yields a one-to-one correspondence between two-weight projective codes over prime fields and some strongly regular graphs.

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 geometric approach to a class of cyclic codes

Abstract From an extension of the classical concept of Euclidean geometry (see for instance [6]), one defines a new class of cyclic codes called here “geometric codes” (primitive and non-primitive), generalizing those introduced by several authors [7,8,10,11,14]. The codes are defined as generated over a prime field by a set of vectors having remarkable geometric properties. These properties are used to decode the dual codes in a step-by-step manner using majority logic (see [2]).

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.

Onr-partition designs in hamming spaces

The concept of the combinatorial matrix of an unrestricted code and the notion of anr-partition design admitted by a code are introduced and discussed in detail. The theory includes a characterization of completely regular codes, and a combinatorial interpretation of the fact that the distinct rows of the distance distribution matrix of a code are linearly independent. In general, it is possible to compute the distance distribution matrix of any code admitting a given partition design by solving a well-defined system of linear equations; this is an efficient technique provided the number of classes in the partition is relatively small.

Spectral enumerators for certain additive-error-correcting codes over integer alphabets

The paper contains a study of certain block codes, with integer coordinate symbols, devised for the correction of some types of additive errors. The codes are defined by a check equation over a finite Abelian group; thus they appear as an extension of the Varshamov and Constantin—Rao codes. The main results of the paper are concerned with the error-correcting capability, including a decoding method for generalized Varshamov codes, and principally with enumeration problems. A general formula for spectral enumerators is obtained and applied to the Varshamov and Constantin—Rao codes.

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.

On the role of the partial trigonometric moment problem in AR speech modelling

The partial trigonometric moment problem is shown to provide a unifying framework for several speech modelling techniques, such as the classical LPC antoregressive model, the line spectral pairs and composite sinusoidal waves models, and the Toeplitz eigenvector model for formant extraction, From a mathematical viewpoint, this moment problem can be identified to an extension problem in the class of impedance functions or equivalently in the class of nonnegative definite Toeplitz matrices.