Least-squares νth-order polynomial estimation of signals from observations affected by non-independent uncertainty

The least-squares νth-order polynomial filtering and fixed-point smoothing problems of uncertainly observed discrete-time signals are considered, when the variables describing the uncertainty in the observations are non-independent. By defining suitable augmented signal and observation vectors, the polynomial estimation problem of the signal is reduced to the linear estimation problem of the augmented signal. The proposed estimators do not require the knowledge of the state-space model generating the signal, but only the probability that the signal exists in the observations, the (2, 2) element of the conditional probability matrices of the sequence describing the uncertainty and the moments (up to the 2νth ones) of the signal and the observation noise.