New recursive estimators from correlated interrupted observations using covariance information

Abstract

Least-squares linear and quadratic filtering and fixed-point smoothing algorithms are derived to estimate a signal from uncertain observations perturbed by an additive white noise. The random variables describing the uncertainty are correlated only at consecutive time instants, and this correlation, as well as the probability that the signal exists in each observation, is known. Recursive algorithms are obtained without requiring the state-space model generating the signal, but just some moments of both the signal and the additive noise in the observation equation. For the linear estimation algorithms, only the second-order moments are required, and the autocovariance function of the signal must be expressed in a semi-degenerate kernel form. The quadratic estimation algorithms use, in addition, the moments up to the fourth one, and they require the autocovariance and cross-covariance functions of the signal and their second-order powers in a semi-degenerate kernel form. This form for expressing autocovariance functions is not very restrictive, since it covers many general stochastic processes, including stationary and non-stationary processes.