Toeplitz matrices for LTI systems, an illustration of their application to Wiener filters and estimators

The Wiener–Kolmogorov theory of filtering has been with us since the first half of the twentieth century. A later matrix-based approach which was more general was derived with the steady-state Kalman filter. This approach uses a novel method of representing causal and uncausal systems in the form of convolution matrices and leads to a Wiener solution which is much easier to calculate than either the Kalman or Wiener approaches. For coloured additive noise, it avoids the use of Diophantine equations. The key idea missing in previous work is the close link between polynomials and Toeplitz matrices which are lower triangular in form. There is already a reasonably sized literature in the mathematics field on such matrices and so the area is ripe for exploration. Although the method does not offer a different or better solution, it shows a completely new way of defining linear time-invariant (LTI) systems which is neither transfer-function nor state-space-based. This is achieved by exploiting the connection between polynomials and Toeplitz matrices. The application here is the Wiener filter but there could well be many more as this is a generic approach.