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The deflation method
Marc Castella
We give a brief description of the deflation method implemented in the
toolbox. Assume that the mixing model is the following one:
where:
-
is the observation vector
-
is the source vector
-
is the impulse response of the mixing filter with
inputs and outputs.
The contrast functions implemented in the toolbox allow one to extract
one source from the mixture. After one source has been restored or if
a filtered version of one source has been extracted, one can subtract
its contribution in the observation signals. In so doing, the problem
of separating sources from the mixture simplifies to the problem
of separating sources. The so-called ``deflation'' method in
source separation is based on this idea. More precisely, the algorithm
(which has been implemented in Deflation.m) is the following
one:
- initialization: ; extraction of the first source
from the observations
.
- for
,
- subtract the contribution of the source extracted at the
-th stage; this yields a signal
with
components but which corresponds to a reduced mixture of
sources. (This is done in the program
SubtractSource.m).
- extraction of a -th source
from the modified
observations
.
At stage 2a, in order to subtract from the observations
the contribution of the -th source, we compute:
|
(1) |
where
is the impulse response of a filter
with entry and outputs. Since the sources are mutually
independent, this impulse response is obtained as the minizer of the
following quantity:
|
(2) |
In practice, the filters have finite impulse response and the above
problem amounts to the least square solution of a linear system.
Note that deflation methods often lead to an accumulation of errors.
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Marc Castella
2006-10-16