Linear Kalman—Bucy filter with an autoregressive signal and noise

Abstract

In the Kalman—Bucy filter problem the observed process consists of a sum of a signal and of a noise. The filter begins simultaneously with the beginning of observation, and it is necessary to estimate a signal. As a rule, this problem is studied both for scalar and for vector Markovian processes. In this paper, the scalar linear problem is considered, but a signal and a noise are independent stationary autoregressive processes with orders larger than unit. The recurrent equations for the filter process, for its error, and for its condition correlations are obtained. These recurrent equations contain the previous estimates and some last observed data. The optimal definition of initial data is proposed. The algebraic equations for the limit values of the filter error and for the limit cross-correlations are delivered. The roots of these equations lead to a criterion of the filter process convergence. Some examples at that the filter process converges or not converges are given. The Monte- Carlo method is used for a control of theoretical formulas for the filter and its error.