Optimal sequential smoothing with uncertain observations

Abstract

Bayes optimal recursive algorithms that do not require growing memory are obtained for the problems of fixed-interval, fixed-point and fixed-lag smoothing with uncertain observations. It is assumed that the signal sequence to be estimated is Markov and that the observations may contain the noise alone or the signal corrupted by noise (not necessarily additive). The uncertainty in the observations is governed by a Markov sequence, and the observation noise is an independent sequence. Under these assumptions, recursive algorithms are devised for the a posteriori density ƒ(Xk YN ), for the three types of smoothing problem. The algorithms also yield a detection scheme of the sequential likelihood ratio test type, as to the presence or absence of the signal at each observation. The Bayes fixed-interval smoothing algorithm is applied to a Gauss-Markov example. The simulation results for this example indicate that the MSE performance of the Bayes smoother is significantly better than that of the linear smoother.