10.3.1 Concept of Minimum Mean-Square-Error Estimation

Wiener filtering is a classical method of optimal estimation of the original signal x( t) or { x _n} based on the measured (observed) signal y( t), { y _n} in the presence of stochastic noise ?( t), { ? _x}. The noise component precludes the precise restoration of the original and therefore we have to be satisfied with only an approximate estimation or { } which would be optimal in a suitably defined sense.

The task is, in its most general form, formulated using the concept that both the original signal and noise, and consequently even the observed signal as derived via the distortion model, are realisations of stochastic processes. Then we can interpret these signals as members of function families or { }, or { }, or { } and the equations, which we shall deal with in the following paragraphs, therefore as members of families of equations where every member applies to a particular combination of realisations.

The criterion of optimality is based on the notion of the error signal or { }; even this signal is, of course, stochastic and all its possible realisations form the family or { }. The Wiener restoration considers as the optimum restoration such an approach that, when applied to all possible combinations of the mentioned process realisations, leads to such estimates that their ensemble mean...