Explicit solutions to the singular discrete finite-time linear estimation problem

The problem of optimal filtering of linear discrete-time processes is considered. It is assumed that the driving and measurement noise signals are stationary white, and that the number of measurements is finite. Explicit expressions for the filtering error covariance and the corresponding Kalman gain matrices are obtained for the limiting case where the measurement noise intensity tends uniformly to zero. Similar expressions are derived for the limiting singular smoothing problem with finite number of measurements.