Wiener filtering - Signal Processing and Optimization for Transceiver Systems - page 798

and

T = − R wy (0)

R − 1 R yy (0)

yy (0) .

(F . 21 c )

From this we can verify the following:

(a) T = I if and only if R wy (0) = 0 ,thatis,the noise w ( n )andthe
noisy signal y ( n ) are orthogonal for each n. This is weaker than the
requirement for G ( z )= I in remark 1 above.
(b) When the noise w ( n ) and the signal x ( n ) are orthogonal for each n,
that is, R xw (0) = 0 ,then

T = R xx (0)[ R yy (0)] − 1 = R xx (0)[ R xx (0) + R ww (0)] − 1 . (F . 22)

This is the counterpart of Eq. (F.19) for the memoryless case.

A very important practical case arises when the Wiener filter is constrained to

be an FIR matrix of the form G ( z )= n =0 g ( n ) z −n . In this case it is possible
to develop a time-domain formulation of the optimal filtering problem and solve
for the optimal impulse response matrices g ( n ) directly.

F.3 Wiener filter for zero-mean uncorrelated noise

We now consider the special case of the Wiener filtering problem where the
additive noise w ( n ) has zero mean, that is,

E [ w ( n )] = 0 , (F . 23)

and is uncorrelated to the signal, that is,

E [ x ( n ) w † ( m )] = 0 , (F . 24)

for all m, n. This is a common assumption made in the analysis of communication
systems. By definition, the condition (F.24) says that the random processes are
orthogonal, but the term uncorrelated is equally appropriate because of the zero-
mean property (F.23). Since Eq. (F.24) implies that S xw ( z )= 0 , the Wiener
filter reduces to Eq. (F.19), that is,

G ( z )= S xx ( z )[ S xx ( z )+ S ww ( z )] − 1 .

(F . 25)

Since this is a popular special case arising in communication applications, we
shall derive an expression for the mean square value of the error

e ( n )= x ( n ) − x ( n ) .

In the following discussions, z is an abbreviation for e jω and G ( z )istranspose
conjugation on the unit circle. (Recall from Sec. 1.6 that G ( z )=[ G (1 /z ∗ )] † .)