New Derivation of the Filter and Fixed-Interval Smoother with Correlated Uncertain Observations

A least-squares linear fixed-interval smoothing algorithm is derived to estimate signals from uncertain observations perturbed by additive white noise. It is assumed that the Bernoulli variables describing the uncertainty are only correlated at consecutive time instants. The marginal distribution of each of these variables, specified by the probability that the signal exists at each observation, as well as their correlation function, are known. The algorithm is obtained without requiring the state-space model generating the signal, but just the covariance functions of the signal and the additive noise in the observation equation.