Linear generalized Kalman-Bucy filter

Abstract

The linear generalized Kalman Bucy filter problem is studied. An observed process is a sum of a useful signal and a noise. A signal and a noise are independent stationary auto-regressive processes with orders exceeding 1. The filter estimates a signal by using an observed process. Two algorithms of filter are considered, recurrent and direct. In frames of the recurrent algorithm to find the next in turn estimation of a signal the current observation and some last previous filter estimations are used. The direct algorithm uses all previous observations directly. For the both algorithms the errors of estimation are found. The advantages and locks of both algorithms are discussed. Calculations at the recurrent algorithm does depend on time of observation. The direct algorithm is reduced to a linear algebraic system, order of that increases with a time. On the other side, the direct algorithm converges with growth of time in all cases, and the recurrent algorithm sometime may not converge. Numerical examples are given.