Analytic theory of random fields estimation and filtering

Abstract

Let u(x)=s(x)+n(x), where u(x) is the observed random signal, s(x) is the useful signal, n(x) is noise, Open image in new window , where O is the domain of signal processing with a boundary Γ, and Open image in new window . Let A be some operator on s(x). The problem is to find a linear operator (estimate) Lu which estimates As optimally in the sense of least squares method, i.e. \(\varepsilon \equiv \overline {(Lu - As)^2 }\)= min. Here the bar denotes the mean value. If A=I (the identity operator) then the estimation problem is the filtering problem. The basic integral equation of the estimation theory is of the form Open image in new window , where h(y)=h(y,z) is the impulse reaction of the optimal filter (that is the optimal estimate is of the form Open image in new window is the covariance function of u, and \(f(x) = f(x,z) = \overline {u^ \star (x)s(z)}\). Here the asterisk denotes complex conjugation and it is assumed that \(\bar s = \bar u = \bar n = 0\), that is the mean values of the signals are zeros. The dependence on z in f and h is supressed since z enters as a parameter in the basic equation (0). The theory of equation (0) developed by the author is briefly surveyed in this paper and open problems are formulated.