Digital Signal Processing Reference
In-Depth Information
Given this channel model, the ZF condition is achieved using an FIR
equalizer, and no infinitely long equalizers were required. Thus, since
perfect equalization is attained for
w
, α
]
T
, it is expected that
they also represent ideal solutions of
J
CM
. This assumption, as proved by
the authors, is entirely correct. In addition, the authors consider possible
solutions given by
=±
[
1, 0,
...
,0,1]
T
w
eq
=±
[0,
...
(4.83)
where
E
2
|
x
(
n
)
|
E
4
=
R
2
(4.84)
|
x
(
n
)
|
This solution is not able to reduce ISI, and it is proven to be a minimum of
J
CM
with the aid of an analysis of the derivatives of the cost function.
Let us consider an example, in which the channel transfer function is
1
H
(
z
)
=
(4.85)
+
0.6
z
−
1
1
From the above reasoning, we are correctly led to expect that the points
[1, 0.6]
T
w
=±
(4.86)
be global minima of
J
CM
. However, there shall also exist solutions like (4.83),
solutions that are complete failures insofar as the equalization task is con-
cerned. As shown in
Figure 4.6
, the minima at
w
[1, 0.6]
T
are indeed the
best and lead to perfect equalization. On the other hand, as the work of Ding
et al. anticipates, there is a pair of “shallow minima” that have exactly the
form described in (4.83).
Finally, it is worth mentioning that the authors also show, with the aid of
simulations, that the bad minima continue to exist in the presence of noise
and that there are local minima even for an FIR channel model. These min-
ima, however, are not ineffective minima like the ones discussed above.
As a matter of fact, not all local minima are necessarily bad minima. It is
important to keep in mind that even the Wiener criterion possesses multi-
ple solutions depending on the perspective brought by distinct equalization
delays.
=±
