Optimal Filtering of a Stationary Process with Respect to Measured Discrepancies

Abstract

It is known that an optimal filter in the linear-quadratic filtering and extrapolation problem for stationary processes with rational spectral densities has a rational transfer function. It is proved below that if only the errors of previous forecasts are measured rather than the values of one of the initial stationary processes, then the transfer function of the optimal filter need not be rational and it may contain interior singular functions in a circle. Standard methods of reducing of the filtering problem to minimization of a quadratic function on a linear manifold [1]-[3] turn out to be inapplicable. This is related to the essential nonlinearity of the set of all admissible transfer functions of the formative filters. The factorization of functions in Hardy spaces [4] is used for the solution. The problem is reduced to minimizing a nonquadratic function on a linear manifold, which is accomplished by