This paper discusses the least-squares linear filtering and smoothing (fixed-point and fixed-interval) problems of discrete-time signals from observations, perturbed by additive white noise, which can be randomly delayed by one sampling time. It is assumed that the Bernoulli random variables characterizing delay measurements are correlated in consecutive time instants. The marginal distribution of each of these variables, specified by the probability of a delay in the measurement, as well as their correlation function, are known. Using an innovation approach, the filtering, fixed-point and fixed-interval smoothing recursive algorithms are obtained without requiring the state-space model generating the signal; they use only the covariance functions of the signal and the noise, the delay probabilities and the correlation function of the Bernoulli variables. The algorithms are applied to a particular transmission model with stand-by sensors for the immediate replacement of a failed unit.

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.

This paper considers the least-squares linear estimation problem of signals from randomly delayed observations when the additive white noise is correlated with the signal. The delay values are treated as unknown variables, modelled by a binary white noise with values zero or one; these values indicate that the measurements arrive in time or they are delayed by one sampling time. A recursive one-stage prediction and filtering algorithm is obtained by an innovation approach and do not use the state-space model of the signal. It is assumed that both, the autocovariance functions of the signal and the crosscovariance function between the signal and the observation noise are expressed in a semi-degenerate kernel form; using this information and the delay probabilities, the estimators are recursively obtained.