Digital Signal Processing Reference
In-Depth Information
T
—
pair is close to one of the best Wiener solutions, which are
[−
0.271, 0.586
]
T
(for
d
for
d
=
2—, the best of the three, and
[
0.406, 0.120
]
=
1), and that
the worst Wiener solution (for
d
=
0) has no counterpart in the CM cost
function.
This situation is also observed in other scenarios, and some works in the
literature indeed studied the relationship between the minima of these crite-
ria and their cost values [264, 312, 313]. An interesting idea that arises from
these works is to focus on CM minima close to Wiener solutions associated
with certain delays.
In fact, the application of the CM criterion to problems in which an FIR fil-
ter is used to equalize an FIR channel often leads to an analytical framework
in which CM minima are more or less closely related to a set of good Wiener
solutions. If we recall that perfect inversion solutions should be present in
both criteria, it is expected that Wiener solutions associated with a small
MSE should have counterparts in the CM cost function. On the other hand,
solutions obtained from equalization delays that lead to a large MSE might
not even be present therein, a fact that is also observed in the previous
example.
The analysis in a general case is rather complex, and certain aspects
thereof can still be considered to be open. However, if we only consider the
special case of binary signals with i.i.d. samples, it is possible to obtain a sim-
ple relationship between the CM and Wiener criteria. Under this assumption,
the CM cost function can be expressed as
E
y
d
2
s
n
d
2
y
s
n
J
CM
(
w
)
=
(
n
)
−
−
(
n
)
+
−
(4.100)
where, in this case,
E
s
4
n
d
=
1.
It is useful to recall the important Cauchy-Schwarz inequality [78], which
guarantees that
−
E
a
2
E
b
2
E
ab
2
≤
(4.101)
This result, together with (4.100), leads to
E
y
d
4
E
y
d
4
s
n
s
n
J
CM
(
w
)
≤
(
n
)
−
−
(
n
)
+
−
≤
J
F
(
w
)
J
F
(
−
w
)
(4.102)
As
J
CM
≥
0, we may write
J
F
J
CM
≤
J
F
(
−
(
w
)
w
)
(4.103)
