Wiener filtering - Signal Processing and Optimization for Transceiver Systems - page 799

Theorem F.2. Power spectrum of error . Assume the noise w ( n ) is zero-
mean and uncorrelated to the signal x ( n ) , so that the Wiener filter is as in Eq.
(F.25). Then the error e ( n ) has the power spectrum
S ee ( e jω )=

♠

xx ( e jω )+ S − ww ( e jω ) − 1 ,

S − 1 (F . 26)

assuming that S ww ( e jω )and S xx ( e jω ) are invertible for all ω. Thus the error
spectrum is the inverse of the sum of inverses of the spectra of the signal and
the noise. ♦

Proof. Since x ( n )=[ g ∗ ( x + w )]( n ) , where g ( n ) is the impulse response of
G ( z )and ∗ denotes convolution, we see that

e ( n )=[ g ∗ x ]( n )+[ g ∗ w ]( n ) − x ( n ) . (F . 27)

The orthogonality principle dictates that e ( n ) be orthogonal to y ( m ) for all
n, m. Since x ( n ) is a linear combination of samples of y ( . ) , it then follows

that x ( n ) is orthogonal to e ( m ) for all n, m. Thus

x † ( n E [ e ( n ) − k )] = 0 ,

for all k , which implies that S e

x ( z )= 0 . Thus

S ee ( z )= S e , ( ( z )= S e

x ( z )

− S ex ( z )= − S ex ( z ) .

x − x )

From Eq. (F.27) we therefore obtain

S ee ( z )= − S ex ( z )= − G ( z ) S xx ( z ) − G ( z ) S wx ( z )+ S xx ( z ) .

In view of the assumption that S wx ( z )= 0 , it therefore follows that

S ee ( z )=( I − G ( z )) S xx ( z ) . (F . 28)

Substituting for G ( z ) from the Wiener filter expression (F.25) we have

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

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

Using this in Eq. (F.28) we have

S ee ( z )= S ww ( z )[ S xx ( z )+ S ww ( z )] − 1 S xx ( z ) ,

which indeed can be rewritten as Eq. (F.26).

The error covariance martrix is simply the integral of the power spectral matrix,
that is,

2 π

1
2 π

R ee (0) = E [ e ( n ) e † ( n )] , = S ee ( e jω ) dω,

0