This paper treats the least-squares linear filtering and smoothing problems of discrete-time signals from uncertain observations when the random interruptions in the observation process are modelled by a sequence of independent Bernoulli random variables. Using an innovation approach we obtain the filtering algorithm and a general expression for the smoother which leads to fixed-point, fixed-interval and fixed-lag smoothing recursive algorithms. The proposed algorithms do not require the knowledge of the state-space model generating the signal, but only the covariance information of the signal and the observation noise, as well as the probability that the signal exists in the observed values.

This paper presents recursive algorithms for the least mean-squared error linear filtering and fixed-interval smoothing estimators, from uncertain observations for the case of white and white plus coloured observation noises. The estimators are obtained by an innovation approach and do not use the state-space model, but only covariance information about the signal and the observation noises, as well as the probability that the signal exists in the observed values. Therefore the algorithms are applicable not only to signal processes that can be estimated by the conventional formulation using the state-space model but also to those for which a realization of the state-space model is not available. It is assumed that both the signal and the coloured noise autocovariance functions are expressed in a semi-degenerate kernel form. Since the semi-degenerate kernel is suitable for expressing autocovariance functions of non-stationary or stationary signal processes, the proposed estimators provide estimates of general signal processes.