# Publication [1.1] of Tomás Oliveira e Silva

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## Reference

**Tomás António Mendes Oliveira e Silva**, "Sobre os Filtros de Kautz e sua Utilização na Aproximação de Sistemas Lineares Invariantes no Tempo," Ph.D thesis, Universidade de Aveiro, Jul. 1994. [In Portuguese.]

## Abstract

Let *H* be the transfer function of a strictly stable, causal, and linear system, and let *h* be its impulse response.
If one does not assume a specific form for *H*, or, what is the same, a specific model for the system, the question arises as how to approximate that transfer function by another with a known form.

A related problem, studied intensively in the last decades, consists in assuming that *H* has a known form, usually a rational fraction.
In that case the problem becomes one of estimation (of the parameters of *H*) and not one of approximation.

One way to solve the approximation problem consists in representing *H* by an orthonormal and complete set of functions and in using only the first terms of this representation to build the approximation of *H*.
In this thesis we propose and study the application of the Kautz functions and sequences to this problem.

It is considered not only the problem of finding approximations of *H* when this function is known, that is, when *h* is known, but also the more general problem of approximating *H* when the input and output signals of the system are known.
In this last case the approximation is performed by the so-called Kautz filters.
In both cases the quality of the approximation is measured by its squared error, weighted by the power spectrum of the input signal in the second case.

The study of the approximation problem in the general case is preceded by the study of some particular cases.

The first of these cases is that of the Laguerre filters, corresponding to the usage of the Laguerre functions and sequences in the approximation of *h*.
This case is studied with considerable detail, being presented some original results that are of great practical interest.

The observation that the Laguerre series expansions are equivalent to Laurent series expansions lead us to suggest the combination of two or more of these expansions to improve the speed of convergence (to zero) of the squared error of the approximation. The orthonormalization of the functions or sequences involved in this kind of approximations lead us directly to the Kautz functions and sequences, which in turn lead us to the Kautz filters.

After the study of another particular case, corresponding to the combination of two Laguerre filters with complex conjugate poles, the general case is studied, for which we have found a very elegant mathematical formulation.

## BibTeX entry

@PhdThesis { Oliveira.e.Silva-1994-1-FKASL, author = {Oliveira e Silva, Tom{\'a}s Ant{\'o}nio Mendes}, title = {Sobre os Filtros de {K}autz e sua Utiliza{\c c}{\~a}o na Aproxima{\c c}{\~a}o de Sistemas Lineares Invariantes no Tempo}, school = {Universidade de Aveiro}, year = {1994}, address = {Aveiro, Portugal}, month = Jul, comment = {In Portuguese.} }