Optimal filtering of discrete homogeneous fields

Abstract

The problem of linear optimal filtering of discrete homogeneous fields in the context of the Wiener—Kalman theory is studied in this chapter. The foundation of the theory of optimal filtering of stationary processes is the idea of the factorization of the spectral density of an observable homogeneous field which implies the representation of this density in the form of the product of the functions possessing special analytic properties determined by physical readability conditions for the optimal filter. In studies of the optimal filtering of homogeneous fields the difficulties associated with the factorization of multi-dimensional densities emerge. This raises the question of not only how the factorization can be performed but what meaning needs to be assigned to it as well. In particular, this problem is accounted for by the lack of an analog of the theorem being valid in the one-dimensional case of the rational spectral density representation in the form of the module squared of some rational function with special analytic properties. This peculiarity of the multi-dimensional case results from the fundamental concept, closely related to the causality properties of time evolving stochastic processes, of unidirectionality of elements of the ‘past—future’ type being unavailable. Should we seek to order a multi-dimensional ‘time’ set (e.g., a row scanning of stochastic texture) by artificle means it may be unnatural, the time series obtained may be either non-stationary or stationary but with a density other than rational.