Digital Signal Processing Reference
In-Depth Information
1.5
1.5
1
1
0.5
0.5
0
0
-0.5
-0.5
-1
-1
-1.5
-1.5
0
200
400
600
800
1000
0
200
400
600
800
1000
(a)
n
(b)
n
FIGURE 3.6
(a) Channel and (b) equalizer outputs in the absence of noise.
As mentioned above, the optimal MSE is not necessarily null. In fact, this
means that some kind of performance analysis of the Wiener solution must always
be carried out. In this example, we may inquire whether the Wiener equalizer
is efficient enough to mitigate IIS and provide correct symbol recovery. In order
to answer this question, let us first calculate the MSE associated with the Wiener
solution:
−
10
0.978
−
)
=
σ
d
−
p
T
w
w
=
J
MSE
(
w
1
=
0.0216
(3.29)
0.337
In view of the magnitude of the involved signals, this residual MSE indicates
that the equalization task has been carried out efficiently. Such conclusion is
confirmed by Figure 3.6, which shows the channel and equalizer outputs. The
latter is indeed concentrated around the corrected symbols
1.
Let us now consider what happens if the channel also introduces additive
noise, assumed to be Gaussian and white (which leads to the classical acronym
AWGN—additive white Gaussian noise) and, moreover, independent of the
transmitted signal. Thus, the received signal will be
+
1and
−
x
(
n
)
=
s
(
n
)
+
0.4
s
(
n
−
1
)
+
ν
(
n
)
(3.30)
stands for the AWGN, with variance
σ
ν
. From (3.30), the correlation
matrix becomes
where
ν
(
n
)
1.16
+
σ
ν
0.4
R
=
(3.31)
+
σ
ν
0.4
1.16
while the correlation vector remains as shown in (3.25). The Wiener solution is
now given by
1.16
−1
1
0
+
σ
ν
0.4
R
−1
p
w
w
=
=
(3.32)
σ
ν
0.4
1.16
+
In order to evaluate the noise effect, we consider three different variance
levels:
σ
1
=
0.1,
σ
2
=
0.01,
σ
3
=
0.001.
Table 3.1
presents the Wiener solutions
for the different levels of noise, as well as the residual MSE's.
