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

A simple characterisation of the optimal stationary discrete Kalman filter is obtained in terms of the return-difference matrix for the associated feedback system. A ztransform spectral factorisation of the observation spectral-density matrix is developed from the discrete-time matrix Riccati equation, and is shown to generate directly the appropriate return-difference matrix. This leads to a proof of a necessary condition for optimality of discrete multivariable-feedback filter and regulator problems.