Digital Signal Processing Reference
In-Depth Information
The problem of nonlinear BSS has received a great deal of attention from
the 1990s on. Important efforts were established within a neural computation
framework, like Burel's work [50] and proposals based on self-organizing
maps [221]. A key reference in the context of post-nonlinear mixtures is Taleb
and Jutten's work [280], and the study of this class of mixtures is still a
prolific research subject [3,281].
6.1 The Problem of Blind Source Separation
Figure 6.1 depicts the general problem of BSS, in which a set of informa-
tion signals is submitted to the mixing and eventually distorting effect of a
MIMO system. The resulting signals are captured by a set of sensors. The
purpose of source separation techniques is to recover the original signals
from the sensor outputs, by means of an appropriate separating system. Sim-
ilarly to the equalization case, we talk about BSS when the mixing system is
unknown and the desired signals are not available for any kind of training
procedure.
The problem may be formulated as follows: let us consider a set of
N
signals, denoted sources, whose samples form a source vector
s
(
n
)
, and a set
of
M
signals, the observations, organized in a vector
x
. The observations
represent, in general, a mixture of the different source signals and can be
expressed as
(
n
)
x
(
n
)
=
F
(
s
(
n
)
,
...
,
s
(
n
−
L
)
,
n
(
n
)
,
n
)
(6.1)
where
F
is the mixing mapping.
Clearly, if
F
(
·
)
is known a priori, the sources can be estimated by obtain-
ing the optimal inverse mapping (assuming that it exists and the noise is
negligible). Moreover, if it is possible to rely on a training sequence, it should
(
·
)
Source 1
s
1
(
n
)
Source 2
s
2
(
n
)
Estimate 1
y
1
(
n
)
Sensor 1
x
1
(
n
)
Sensor 2
x
2
(
n
)
Estimate 2
y
2
(
n
)
Mixing
system
Separating
system
Source
N
s
N
(
n
)
Sensor
M
x
M
(
n
)
Estimate
M
y
M
(
n
)
FIGURE 6.1
The general source separation problem.
