# Linear Minimum Mean-Square Error Filtering

Background :: Filtering

## Causal Wiener Filtering

The examples we have seen so far have produced
*
noncausal
*
filters, i.e., practically nonimplementable in many cases. We will
see what can be done now to make causal filters.

Let us take
, with
and
jointly and
individually W.S.S., and take
(filtering). Furthermore, assume
that the filter is time-invariant and
*
causal
*
. Then the
Wiener-Hopf equations can be written

The question is, how can this be solved? Because the limit does not proceed to , we can't use conventional transform techniques.

Here are some facts to help. Suppose
satisfies the
following condition:

(This is known as the Paley-Wiener condition.) Then it turns out that we can write

where

and is zero for negative times (that is, it is causal) and is zero for positive times (that is, it is anticausal). Moreover,

is also causal and

is also anticausal. The proof in general is rather difficult (we will skip it, but give some examples). This factorization is known as the

**spectral factorization**.