Uppsala universitet

Wiener Filter Design Using Polynomial Equations

Anders Ahlén and Mikael Sternad

IEEE Transactions on Signal Processing, vol 39, pp 2387-2399, November 1991. © 1991 IEEE.

Also: Internal Report UPTEC 90057R, Dept. of Technology, Uppsala University.

Paper In Pdf.

The classical concept of orthogonality is here utilized in a novel way, within the polynomial equations approach to linear filtering problems. As a result, the process of deriving estimator design equations, for a given problem and model structure in polynomial form, is simplified significantly.

A simplified way of deriving of realizable and explicit Wiener filters is presented. Discrete time problems are discussed, in a polynomial equation framework. Optimal filters, predictors and smoothers are calculated by means of spectral factorizations and linear polynomial equations.

A new tool for obtaining these equations, for a given problem structure, is described. It is based on evaluation of orthogonality in the frequency domain, by means of cancelling stable poles with zeros. Comparisons are made to previously known derivation methodology such as ``completing the squares'' for the polynomial systems approach and the classical Wiener solution. The simplicity of the proposed derivation method is particularly evident in multisignal filtering problems. To illustrate, two examples are discussed: a filtering and a generalized deconvolution problem. A new solvability condition for linear polynomial equations appearing in scalar problems is also presented.

Matlab m-files
for polynomial equation design of Wiener estimators for scalar signals:
fnfilt.m Function for estimator design.
abstar.m Used by fnfilt.m.
addcent.m Used by fnfilt.m.
spefac2.m Spectral factorization, by polynomial roots, used by fnfilt.m
polysolve.m Solution of Diophantine equation. used by fnfilt.m
sylv.m Form sylvester matrix. used by polysolve.m

Related publications:
Book chapter (Academic Press 1994), with additional aspects on the methodology.
Paper in IEEE Trans. AC 1993, where the method is applied on LQG control problems.
Paper in IEEE Trans AC 1995, where method is used for deriving robust filters.

| Research on polynomial methods | Main entry in list of publications |
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