Digital Signal Processing Reference
In-Depth Information
is derived from a modification of the constrained criterion in (4.33), which
leads to
maximize
c
4
y
)
(
n
(4.54)
c
2
y
)
2
(
n
which characterizes a normalized criterion. The cost function can be rewrit-
ten in terms of the cumulants of the transmitted signal as
c
4
y
)
l
|
4
(
n
c
4
(
s
(
n
))
g
(
l
)
|
=
(4.55)
c
2
y
)
2
l
|
2
2
2
c
2
(
s
(
n
))
(
n
g
(
l
)
|
f
4
(
g
)
where we define
l
|
4
g
(
l
)
|
f
4
(
g
)
=
(4.56)
l
|
2
2
g
(
l
)
|
for which we have
0
≤
f
4
(
g
)
≤
1
(4.57)
so that
f
4
(
1onlyif
g
presents a single non-null element, which corre-
sponds to the ZF solution. Moreover, it is possible to show that
g
)
=
f
4
(
g
)
=
f
4
(
α
g
)
(4.58)
for any non-null constant α.
Since the statistical characteristics of the transmitted signal are known,
the criterion in (4.54) is equivalent to the maximization of
f
4
(
)
,acrite-
rion whose origin can be traced back to the works of Wiggins [307] and
Donoho [101] about signal deconvolution.
In order to maximize
f
4
(
g
, the SEA explores a simple iteration employing
the
Hadamard exponent
of a vector. Let us consider, for the moment, that
g
is
real-valued. The
m
th Hadamard exponent of
g
is defined componentwise by
g
)
g
m
m
(
)
k
=
g
(
k
)
(4.59)
If
g
presents a unique dominant term in the
k
th position, the
m
th Hadamard
exponent
))
m
will converge to a ZF condition as
m
tends to infinity.
(
g
/
g
(
k
