Kalman like filtering and smoothing for reciprocal sequences

The MMSE filtering problem of reciprocal Gaussian sequences in additive white Gaussian noise is solved in a recursive and causal form. The solution, based on the innovations method, is expressed in terms of a set of recursive equations formally similar to those of the well-known Kalman filter; it gives as by-product the solution of the MMSE smoothing problems (fixed-point, fixed-interval, fixed-lag). The performance of the proposed estimators is also given by recursive expressions