Digital Signal Processing Reference
In-Depth Information
is valid (however, the results obtained for complex signals are quite similar).
Under this assumption we get to
R
2
1
1
J
CM
()
=
−
(4.95)
3
+
J
SW
()
This expression reveals that the CM cost function can be understood as a
sort of deformation of the SW cost function. The rest of the analysis amounts
essentially to a study of the nature of this deformation. First, let us study the
deformation with respect to monotonicity. This can be done with the aid of
the following expression:
R
2
dJ
CM
()
dJ
SW
=
(4.96)
()
]
2
[3
+
J
SW
()
Since this expression is always greater than zero, the deformation is mono-
tonic. Furthermore, the expression reveals that all stationary points in
of
the two cost function coincide, i.e., that
1
=
1
=
dJ
CM
()
d
dJ
SW
()
d
0
⇔
0
(4.97)
Finally, it is possible to show that
J
SW
(
)
−
J
SW
(
)
1
2
R
2
J
CM
(
)
−
J
CM
(
)
=
(4.98)
1
2
+
J
SW
(
+
J
SW
(
[3
)
][3
)
]
1
2
which means that the deformation is order-preserving, i.e., that
J
CM
(
1
) >
J
CM
(
2
)
⇔
J
SW
(
1
) >
J
SW
(
2
)
(4.99)
As (4.97) shows, all stationary points of the two cost functions coincide. In
addition to that, (4.96) and (4.99) reveal that the classification of each point is
identical, which renders the equivalence even stronger. However, we should
notice that this equivalence is restricted to the stationary points: the criteria
in their entirety are not identical, which means that there exist performance
aspects that may be different for both methods and their correspondent
algorithms.
4.7.2 Some Remarks Concerning the Relationship between the Constant
Modulus/Shalvi-Weinstein and the Wiener Criteria
The relationships between the DD and Wiener criteria and between the CM
and SW criteria are established within similar statistical frameworks: in the
