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Filtering and fixed-point smoothing from an innovation approach in systems with uncertainty

The least mean-squared error linear estimation problem of signals in systems with uncertain observations has been widely studied, since many practical situations (for example, in Communication Theory) are modelled by this kind of systems. The systems with uncertain observations are characterized by the fact that the observation equation contains, besides additive noise, a multiplicative noise component, defined by a sequence of Bernoulli random variables, which determines the presence of the signal in the observations. Under the hypothesis that the state-space model is known, Nahi [6] and, subsequently, Monzingo [5] obtained the solution of this problem assuming independence of the variables that characterize the uncertainty and, also, that the additive noises of the signal and the observation are uncorrelated. Later on, these results were generalized by Hermoso and Linares ([1], [2]) to the case in which the additive noises are correlated. In the papers mentioned above, the algorithms are obtained by starting from the general expression of the estimators as a linear function of the observations. In contrast, the technique based on innovations, consisting of converting the observation process into the innovation process, and expressing the estimators as a linear function of the innovations, allows to make a simpler derivation of the estimation algorithms. This technique is justified by two