A Linear Method for the Design of Optimal Nonlinear Filters for Non-Gaussian Processes

Abstract

Attention is devoted in the present paper to obtaining the optimal mean square estimate X^(t) of X(t) given the observations {Y(s): t0≤ s≤ t }, where X and Y are jointly defined second-order non-Gaussian (i.e., not necessarily Gaussian) real-valued stochastic processes on an interval I: t0≤ t ≤ tf of the real line with known statistics. The underlying filter is represented by a Volterra functional series belonging to a generalized Fock space Fρ. Generalized Fock spaces are reproducing kernel Hilbert spaces of nonlinear functionals introduced by de Figueiredo and Dwyer (1980). In such a setting. a new linearequation in the space Fρ, which the optimal nonlinear filter must satisfy. is derived. This equation reduces to the Wiener-Hopf equation in the linear Gaussian case. Finally. a new class of nonlinear filters is presented which is optimal for a wide class of non-Gaussian stochastic processes. and is simple to implement.