Discrete-Time Linear Filtering in Arbitrary Noise

The Kalman filter is a recursive Best Linear Unbiased Estimator (BLUE) for a linear dynamic system with uncorrelated white process and measurement noises. It has been extended to the case where the noises are Markov and/or crosscorrelated for the same time instant. This paper presents optimal batch and semi-recursive filters and a suboptimal recursive filter for a linear discrete-time system with arbitrarily colored (not necessarily Markov) noises that are arbitrarily cross-correlated and correlated with the initial state of the system. They are generalizations of the Kalman filter for the case of arbitrary additive noise of known first two moments. Numerical examples are provided. They demonstrate the superiority in terms of performance and efficiency of the proposed recursive filter.