An alternative approach to the linear causal least-square filtering theory

We present in this paper an alternative approach to the discrete-and-continuous-time, linear, causal, least-square filtering of wide-sense stationary random processes. Various new and simplified solutions for the optimum filter and the minimum-mean-square error (MMSE) are given in a unified manner by using Hilbert-space techniques. Particular emphasis is placed on the closed-form solution obtained without explicit spectral factorization. In contrast to the Wiener theory, we impose the causality requirement on the linear minimum-phase filter-transfer function by using the conjugate Poisson-integral transformation before we perform the optimization operation. After optimization, we obtain a set of integral equations from which various symmetrical properties of the optimum transfer function and the MMSE appear in simple forms. When one of the spectral densities is arbitrary, while the other is rational and fixed, the filtering problem is reduced to the solution of a set of transcendental equations. In particular, the closed-form solutions for the optimum filter and the MMSE of an arbitrary signal disturbed by a fixed, additive, wise-sense Markov noise are given explicitly for discrete-and-continuous-time cases.