Digital Signal Processing Reference
In-Depth Information
6.2.2.3 Nonlinear Decorrelation
Independence between signals can also be verified by means of the nonlinear
correlation, which can be seen as an extension of the concept of correlation
presented in Section 2.4.1. The nonlinear correlation between two random
variables is defined as
E
f
1
(
)
η
(
X
,
Y
)
=
X
)
f
2
(
Y
(6.30)
with
f
1
(
·
)
representing two arbitrary nonlinear functions.
If
X
and
Y
are independent, we have
and
f
2
(
·
)
E
f
1
(
)
=
E
f
1
(
)
E
f
2
(
)
X
)
f
2
(
Y
X
Y
(6.31)
The converse statement is only true if (6.31) holds for
all
continuous functions
f
1
(
·
)
that are zero outside a finite interval [230]. Nevertheless, it is
possible to employ the notion of nonlinear correlation to obtain very simple
BSS methods.
Let us consider that both
f
1
(
·
)
and
f
2
(
·
)
are smooth functions with deriva-
tives of all orders around the origin. In these conditions, (6.30) can be
expressed in terms of the Taylor expansion of these nonlinear functions
as [148]
and
f
2
(
·
)
∞
∞
E
f
1
(
)
=
f
(k)
1
f
(l
2
E
X
k
Y
l
X
)
f
2
(
Y
{
}
(6.32)
k
=
1
l
=
1
where
f
(i)
1
and
f
(i
2
denote the coefficients of the Taylor series. Hence, if
X
and
Y
are independent and either
E
X
k
Y
k
0, for all
k
, then the
nonlinear correlation is zero, and the variables are said to be nonlinearly
decorrelated.
Therefore, if we are trying to separate two signals
y
i
and
y
j
, the following
criterion could be employed:
min
E
f
1
y
i
f
2
y
j
{
}=
0or
E
{
}=
(6.33)
X
k
Y
k
It is important to note that the condition that either
E
{
}=
0or
E
{
}=
0
implicitly requires that
f
1
(
·
)
be an odd function, which means that
the corresponding Taylor series has only odd powers, otherwise the afore-
mentioned condition would imply that even moments like the variance are
zero.
It is worth pointing out that the nonlinear decorrelation may not be effec-
tive in all cases, since we cannot guarantee that an arbitrary pair of nonlinear
functions will lead to independent signals.
or
f
2
(
·
)
