Linear Generalized Kalman–Bucy Filters

Abstract

The linear generalized Kalman–Bucy filter problem has been studied in this work. A sum of a useful signal and a noise is an observed process. A signal and a noise are independent stationary auto-regressive processes with orders exceeding one. The filter estimates a signal, using an observed process. Two algorithms of filter are considered: a recurrent one and a direct one. Within the recurrent one, to find the next estimate of a signal, we use the current observation and several previous filter estimates. The direct algorithm uses all previous observations directly. The errors of estimates are found for both algorithms. The advantages and disadvantages of both algorithms are discussed in this paper. Calculations at the recurrent algorithm depend on the observation time. The direct algorithm is reduced to a linear algebraic system such that its order increases in time. On the other hand, the direct algorithm always converges in time, while this is not guaranteed for the recurrent algorithm. Numerical examples are given here.