Digital Signal Processing Reference
In-Depth Information
TABLE 3.1
Comparison between Wiener Solutions for
Different Noise Levels
Noise Variance
Wiener Solution
EQM
T
No noise
[
0.978,
−
0.337
]
0.0216
T
0.001
[
0.977,
−
0.337
]
0.0227
T
0.01
[
0.968,
−
0.331
]
0.0322
T
0.1
[
0.883,
−
0.280
]
0.1174
3
3
2
2
1
1
0
0
-1
-1
-2
-2
-3
-3
0
200
400
600
800
1000
0
200
400
600
800
1000
(a)
n
(b)
n
FIGURE 3.7
(a) Channel and (b) equalizer outputs in the presence of noise.
It is clear from Table 3.1 that the addition of noise modifies the performance
of the optimal equalizer. In fact, the minimization of the MSE leads the equalizer
to attempt to solve two distinct tasks: to cancel the IIS and to mitigate the noxious
effects of the noise. Since the number of parameters is fixed, this “double task”
becomes more difficult as the additive noise is more significant, which is reflected
by an increase in the residual MSE. This is also illustrated in Figure 3.7, in which
the channel and equalizer outputs are presented for the case in which
σ
ν
=
0.1. In
such case, recovery is not perfect, which confirms that two optimal solutions (those
corresponding to Figures 3.6 and 3.7) can present distinct performance behaviors
according to the conditions under which the filtering task is accomplished.
Example 3.2 (System Identification)
According to Figure 3.2, let us suppose that the unknown system to be identified
is characterized by the following response:
x
(
n
)
=
h
0
s
(
n
)
+
h
1
s
(
n
−
1
)
+
h
2
s
(
n
−
2
)
+
ν
(
n
)
(3.33)
where
ν
(
n
)
is the AWGN
s
(
n
)
is composed by i.i.d. samples with unit variance
