where
r
xx
k Exnxn À k
8:2:11
is the autocorrelation function of x(n).
Example 8.5: Given an optimum filter illustrated in the following figure:
x
(n)
x
(n
-1)
w
1
z
-1
+
+
+
-
d
(n)
e
(n)
If Ex
2
n 1, Exnxn À 1 0:5, Ed
2
n 4, Ednxn À1, and
Ednxn À 1 1. Find x.
From (8.2.10), R
1
0:5
0:5
1
!
, and from (8.2.8), we have P À11
!
.
Therefore from (8.2.7), we obtain
x Ed
2
n À 2p
T
w w
T
Rw
4 À 2À1 1
1
w
1
!
1 w
1
1
0:5
0:5
1
!
1
w
1
!
w
2
1
À w
1
7:
The optimum filter w
o
minimizes the MSE cost function xn. Vector differentiation
of (8.2.7) gives w
o
as the solution to
Rw
o
p:
8:2:12
This system equation defines the optimum filter coefficients in terms of two correlation
functions ± the autocorrelation function of the filter input and the crosscorrelation
function between the filter input and the desired response. Equation (8.2.12) provides a
solution to the adaptive filtering problem in principle. However, in manyapplications,
the signal maybe non-stationary. This linear algebraic solution, w
o
R
À1
p, requires
continuous estimation of R and p, a considerable amount of computations. In addition,
when the dimension of the autocorrelation matrix is large, the calculation of R
À1
may
present a significant computational burden. Therefore a more useful algorithm is
obtained bydeveloping a recursive method for computing w
o
, which will be discussed
in the next section.
To obtain the minimum MSE, we substitute the optimum weight vector w
o
R
À1
p
for w(n) in (8.2.7), resulting in
x
min
Ed
2
n À p
T
w
o
:
8:2:13
362
ADAPTIVE FILTERING