Digital Signal Processing Reference
In-Depth Information
where
J
H
(
y
)
corresponds to the Jacobian of
H
(
·
)
, which is given by
det
cos
(
s
2
)
sin
(
s
2
)
|
det
J
H
(
y
)
|=
s
1
sin
(
s
2
)
−
s
1
cos
(
s
2
)
cos
2
sin
2
=|−
s
1
(
(
s
2
)
+
(
s
2
))
|=|
s
1
|
(6.106)
Substituting (6.103) and (6.106) in (6.105), and noticing that
s
1
=
y
1
+
y
2
,
we obtain
π
exp
2
π
exp
2
π
exp
−
y
1
+
y
2
y
1
2
y
2
2
−
−
2
p
y
1
y
2
(
y
1
,
y
2
)
=
=
2
(6.107)
Thus, we notice that
y
is composed of independent components—since its pdf
is the product of functions of the pdf of
y
1
and
y
2
—even though
y
represents a
mixture of
s
1
and
s
2
.
The above example shows that it is possible to obtain nonlinear mixing
mappings that preserve independence. This result was first observed by Dar-
mois back in 1951, in a nonlinear factor analysis context [281]. In the context
of BSS, though, this problem was studied in [150].
In a certain sense, we could argue that the main difficulty is due to the
great flexibility of nonlinear mappings. Since there is no reference signal to
guide the adaptation of the separating system, we saw that it is possible to
recover independent signals without separating the sources. Thus, in order
to avoid these undesirable solutions, the existing algorithms for the non-
linear case try to restrict the degree of flexibility of these mappings. In this
context, a possible approach to solve the nonlinear mixing problem is to con-
sider nonlinear mappings to which all theoretical backgrounds developed
for the linear case are still valid. In such case, even though the developed
solutions are restricted to a smaller number of practical scenarios, several
existing tools remain applicable.
A first attempt in this direction was made in [6, 310], in which only
mild nonlinearities—provided by a multilayer perceptron—were consid-
ered. Nonetheless, even under this assumption, it is possible to obtain
undesirable solutions [200], indicating that other constraints must be taken
into account.
6.6.2 Post-Nonlinear Mixtures
Taleb and Jutten introduced the PNL model in [280], which is depicted in
The PNL model is particularly useful when the sensors present some sort
of nonlinear distortion. Mathematically, the observed signals
x
are given by
