Wiener filtering - Signal Processing and Optimization for Transceiver Systems - page 796

y ( n )

x( n ) +

G ( z ) x ( n )

filter

w ( n )

Figure F.2 . Filtering of a signal contaminated by additive noise. Here x ( n )isan
M × 1 vector sequence and so is the output x ( n ) .

The filter G ( z ) which minimizes this quantity is called the Wiener filter for this
pair . We will see that this filter has a simple closed form expression

and can be expressed in terms of the various power spectra.

{ x ( n ) , w ( n ) }

♠ Theorem F.1. The Wiener filter. In the above setting, the optimal filter
G ( z ) which minimizes the mean square error E is given by

G ( z )= S xy ( z ) S − 1

yy ( z ) ,

(F . 12)

assuming [det S yy ( e jω )] is nonzero for all ω. The expressions

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

yy ( z )

(F . 13)

and

− S wy ( z )] S − 1 G ( z )=[ S yy ( z )

yy ( z )

(F . 14)

are equivalent forms of Eq. (F.12). ♦

Proof. The filter output is given by the matrix vector convolution
x ( n )=
m

g ( m ) y ( n − m ) , (F . 15)

where g ( m ) is the impulse response matrix, that is G ( z )= m g ( m ) z −m . Ap-
plying the orthogonality principle (Lemma F.1) we find that the best filter should
satisfy the condition

E [ e ( n ) y † ( n − k )] = 0 , for all k, (F . 16)

x ( n ) − x ( n ) . That is, where e ( n )=

E [ x ( n ) y † ( n x ( n ) y † ( n − k )] ,

− k )] = E [ for all k. (F . 17)

By joint wide sense stationarity, the above expectations are independent of n.
Thus the left-hand side is R xy ( k ) . Substituting from Eq. (F.15) the equation
simplifies to

R xy ( k )=
m

g ( m ) R yy ( k − m ) .

(F . 18)