A mean-weighted square-error criterion for optimum filtering of nonstationary random processes

A procedure for use in the design of a physically realizable fime-invariant linear system for optimum filtering of a nonstationary random process in the presence of nonstationary random noise is presented in this paper. First, a new criterion for system performance is defined. On the basis of this criterion, an integral equation for the optimum physically realizable weighting function is derived, and it is shown that in some cases an exact solution to this equation can be obtained through the use of double Fourier transforms. Then the use of a technique to obtain an approximation to the solution to the integral equation is discussed. This theoretical background is followed by an illustrative example in which the method is used to design the optimum physically realizable linear time-invariant filter for a Brownian-motion signal contaminated by Markovian noise. It is shown here that if the designer is constrained by the requirement that the system be a digital filter with finite memory, then an exact solution can be found. Application of the method in cases where the random processes are stationary is discussed next, and the suggested approach is illustrated in an example.