An initial-value solution of the least-squares estimation problem with degenerate covariance

Abstract

In the case of degenerate covariance an initial-value solution of the optimal estimate of a signal in the presence of white Gaussian noise can be evaluated with the aid of a Cauchy system for a matrix Riccati differential equation, whose nonlinear form is convenient for computer-aided determination. In the present paper, with the aid of invariant imbedding, we show how to get an initial-value solution ofthe impulse response function in the case of degenerate covariance. Furthermore, it is also shown that a real-time solution of the linear least-squares filtering problem is given by the Cauchy system for the newly introduced function, which is a time integral of the stochastic process weighted by the auxiliary function.