Wiener filter design using polynomial equations

A simplified way of deriving 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 tool for obtaining these equations, for a given problem structure, is described. It is based on the evaluation of orthogonality in the frequency domain, by means of canceling stable poles with zeros. Comparisons are made to previously known derivation methodologies 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 multistage filtering problems. To illustrate, two examples are discussed: a filtering and a generalized deconvolution problem. A new solvability condition for linear polynomial equation appearing in scalar problems is also presented