Statistical Signal Processing in Engineering

12Optimal Filtering

Estimation for stochastic processes is carried out by linear filtering, and when stationary and Gaussian, the filtering can be designed to be optimal in the MMSE sense. In addition, if random processes are non‐Gaussian, linear filtering is so convenient that LMMSE is a routinely employed tool.

12.1 Wiener Filter

MMSE estimators, as derived in Chapter 11, are constrained by a finite number of observations N. When removing the constraint of limited length, an MMSE estimator can be arbitrarily long in WSS Gaussian processes. This is the optimal Wiener filter that is based on the auto and cross‐correlation sequences as detailed below.

Let be the impulse response of a linear filter (estimator) that is used on a zero‐mean WSS process x[n] to estimate the sample θ[n] of another WSS process

The estimation error

depends on the impulse response a[n] to be evaluated. The z‐transform of the estimation error is

and the estimator is defined in terms of . Since the process ...