Digital Signal Processing Reference
In-Depth Information
where
E
y
d
4
s
n
J
F
(
w
)
=
(
n
)
−
−
(4.104)
which corresponds to the cost function associated with the least mean-fourth
algorithm (a discussion about the least mean-fourth supervised approach
can be found in [301]).
Equation 4.103 provides an upper bound for the CM cost function
that is composed of a product of cost functions based on the fourth-order
moment of the error signal. In order to better understand this relationship, in
Figure 4.8 we show the upper bound and the original cost function. It is pos-
sible to observe that the upper bound changes for different values of
d
.This
may be understood as being indicative of the fact that the CM cost function is
associated with different equalization delays, as mentioned earlier. An over-
all upper bound composed of the surfaces associated with all equalization
delays is also illustrated Figure 4.8d.
Concluding, in simple terms, we believe it is valid to state that the analy-
sis of the CM criterion from the standpoint of supervised approaches should
2
1
-1
2
1
-1
-1
-1
-0.5
-0.5
-0.5
-0.5
0
0
0
0
w
1
w
0
w
0
w
1
0.5
0.5
0.5
0.5
(a)
(b)
1
1
1
1
2
1.
1
0.
0
-1
2
1
-1
-1
-1
-0.5
-0.5
-0.5
-0.5
0
0
0
0
w
1
w
1
0.5
0.5
w
0
w
0
0.5
0.5
(c)
(d)
1
1
1
1
FIGURE 4.8
Derived upper bound for the channel
H
1.5
z
−
1
and different equalization delays:
(
z
)
=
1
+
(a)
d
=
0, (b)
d
=
1, (c)
d
=
2, and (d) overall upper bound.
