Return-difference matrix properties for optimal stationary Kalman-Bucy filter

A strikingly simple characterisation of the optimal stationary Kalman-Bucy filter is obtained in terms of the return-difference matrix for the associated feedback system. The spectral factorisation of the observation spectral-density matrix is shown to generate directly the appropriate return-difference matrix. This leads to a physical interpretation of the mechanism by which signal and noise are separated, which could form the basis of an approach to filter design.