Discrete Kalman filtering using a generalized companion form

For an th-order constant system with outputs, this short paper shows that the Kalman filter gain can be described by difference equations instead of the usual difference equations. This is the minimum number of difference equations required for the solution of the Kalman filter gain. For the special case of stationary processes, this short paper further shows that only combinations of the system noise covariance enter into determination of the Kalman filter gain. The results have application in filtering and in the quadratic regulator problems.