Digital Signal Processing Reference
In-Depth Information
According to the above proposition, replacing the original observations with
x
(n)
t
(n)y(n)
reduces the mixture of
N
sources to a mixture of
(N
1
)
sources
only. For this result to hold,
y(n)
is required to contain a filtered version of a source
signal, which can be obtained by maximizing a MISO contrast as shown in the pre-
vious section. Combining these two ideas yields the following generic deflationary
source separation algorithm:
−
−
General algorithm for deflationary source separation
Set
x
(
1
)
(n)
•
=
x
(n)
.
•
For
p
1
)
, do:
1.
Extraction:
From the observations
x
(p)
(n)
, determine an estimate
y
p
(n)
of one source signal up to admissible ambiguities using a suitable MISO
contrast such as (
9.9
)-(
9.10
).
2.
Deflation:
Deflate the observations by removing the contribution of the
estimated source:
(a) Find a column vector filter
t
p
(n)
satisfying
=
1
,
2
,...,(N
−
E
x
(p)
(n)
t
(n)y
p
(n)
.
2
t
p
(n)
=
arg min
t
(n)
−
(b) Define the deflated observations
x
(p
+
1
)
(n)
as follows:
x
(p
+
1
)
(n)
x
(p)
(n)
t
p
(n)y
p
(n).
=
−
Estimate the last source as an arbitrary MISO filtered version of
x
(N)
(n)
.
•
In practice, one often deals with FIR filters and the above problem amounts to
the least squares solution of a linear system. Remark that in practical settings such
as noisy or short sample size scenarios, the sources can only be estimated with
some inaccuracies, and then the error term to be minimized in step (
9.11
) cannot be
perfectly canceled. As a result, estimation errors accumulate through successive de-
flation iterations [
14
]. This error propagation is probably the main drawback of the
deflation approach. In the remaining of the chapter, we turn our attention to prac-
tical algorithms for optimizing the kurtosis contrast in Step 1 of the above general
algorithm.
9.4 Optimization Methods
As seen in Sect.
9.3.1
, MISO source extraction can be accomplished by finding the
extraction filters maximizing the kurtosis contrast. This problem lacks closed-form
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