Kalman filtering revisited

Infinite-horizon Kalman filtering is re-examined and generalized to include a class of nonstationary and nonergodic disturbances. This revision is achieved by defining a generalized infinite-horizon filtering problem using a flexible functional analytic signal description. It is shown that the solution to the generalized filtering problem is equivalent to the solution of the corresponding standard filtering problem with noise covariance matrices having in the generalized problem a different, more general, meaning than in the standard problem. In the generalized problem these covariance matrices are majorizing matrices that have a precise meaning even in the nonstationary and nonergodic signal case. This result justifies in a nice way the wide practice of interpreting the noise covariance matrices in Kalman filtering as design variables