Digital Signal Processing Reference
In-Depth Information
N
H
y
i
−
log
det
W
mi
W
(6.26)
i
=
1
6.2.2.2 A Criterion Based on Higher-Order Statistics
Another possible criterion to implement ICA explores the joint cumulants
of the involved signals [54,74]. The main idea is to explore the fact that the
joint cumulant
c
,
X
k
is
always null, for any order (see Section 2.3.3). For instance, considering only
two random variables
X
and
Y
, in order to ensure that they are independent,
the following should hold:
(
X
1
,
...
,
X
k
)
of a set of independent variables
X
1
,
...
X
s
,
Y
q
c
(
X
,
...
,
X
s terms
,
Y
,
...
,
Y
q terms
)
=
c
(
)
=
0
(6.27)
for any
s
,
q
.
Even though independence between signals, in general, is related to all
joint cumulants, signal separation can be achieved using only fourth-order
cumulants [74]. The restriction, however, is that there be no more than one
source with null kurtosis. Notice that this restriction includes the Gaussian
distribution as a special, and perhaps the most representative, case. There-
fore, for a wide range of applications, BSS can be performed solely based on
the information brought by the fourth-order joint cumulants.
Let
c
=
1,
...
,
∞
denote the fourth-order joint cumulant between signals
y
i
,
y
j
,
y
k
,and
y
l
. Then, according to the previous discussion, the separat-
ing structure should be chosen such that
c
(
y
i
,
y
j
,
y
k
,
y
l
)
(
y
i
,
y
j
,
y
k
,
y
l
)
be minimal for any
combination of indices
i
,
j
,
k
,and
l
, except for
i
=
j
=
k
=
l
, the case in which
c
. Therefore, a possible optimization criterion to reach
this condition is given by
(
y
i
,
y
j
,
y
k
,
y
l
)
=
c
4
(
y
i
)
mi
W
c
y
i
,
y
j
,
y
k
,
y
l
)
2
(
(6.28)
where
denotes all possible combinations of
i
,
j
,
k
,and
l
, except for
i
l
.
If we consider that the data has been prewhitened, so that we should look
for an orthogonal separating matrix, it is possible to show that the criterion
in (6.28) is equivalent to
=
j
=
k
=
c
y
i
,
y
i
,
y
i
,
y
i
)
2
max
W
(
(6.29)
i
=
1,
...
,
N
where
N
is the number of signals to be recovered.
