Uppsala universitet

Derivation and Design of Wiener Filters Using Polynomial Equations

A Ahlén and M Sternad

In C T Leondes, ed: Control and Dynamic Systems,
Vol 64: Stochastic Techniques in Digital Signal Processing Systems
pp 353-418, Academic Press, New York, NY, 1994.

In this chapter, a polynomial approach to filter design is presented. Our goal is to demonstrate its utility in the area of signal processing and communications. By studying specific model structures, considerable engineering insight can be gained.

Minimization of mean-square error criteria by linear filters is considered. We focus on the optimization of realizable discrete-time IIR-filters, to be used for prediction, filtering or smoothing of signals. Stochastic models of possibly complex-valued signals are assumed known.

The basis for our discussion is a general linear filtering problem, outlined in Section 2. In Section 3, it is discussed how the classical Wiener and inner-outer factorization approaches relate to the polynomial methods, based on variational arguments and completing the squares. The purpose of this discussion is not only to compare advantages and drawbacks, but also to emphasize similarities, and to link and increase understanding of the different viewpoints. To understand how they relate to one another, design equations for a simple scalar filtering problem are derived using each approach.

The polynomial approach, based on variational arguments, is then used to study a collection of signal processing and communications problems in Sections 4-6. We discuss deconvolution (Section 4), numerical differentiation and state estimation (Section 5) and decision feedback equalization (Section 6). The selected special problems have features of general interest: multisignal estimation (Section 4), discrete time design based on a continuous time problem formulation (Section 5), and the approximation of a problem involving a static nonlinearity by a linear-quadratic problem (Section 6). A summarizing discussion of characteristics and suitability of the polynomial approach is found in Section 7.

1. Introduction
2. A set of filtering problems
3. Derivation methods
4. Multisignal deconvolution
5. Differentiation and state estimation
6. Decision feedback equalization
7. Concluding discussion

Appendix A: Scalar polynomial Diophantine equations
Appendix B: Unstable models
Appendix C: Polynomial matrix Diophantine equations
Appendix D: Matlab algorithms for filter design.

Related publications:
Paper in IEEE Trans. AC 1995, on a probabilistic approach to multivariable robust filtering and open-loop control.
Paper in IEEE Trans. SP 1991 on Wiener filter design using polynomial equations.
Paper in IEEE Trans. ASSP 1989 on design of scalar deconvolution estimators.
Paper in IEEE Trans. SP 1991 on the differentiating filters of Chapter 5.
Paper in IEEE Trans. IT 1990 on the decision feedback equalizer of Chapter 6.
Later book chapter (Springer 1996), which includes also robust design.

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