Digital Signal Processing Reference
In-Depth Information
In short, by applying the above described LEC, we can solve our
problem, i.e., the minimization of (4) iteratively by using the following two
steps:
1.
Find that min (This corresponds to minimizing
under the linear constraint:
2.
Use
from Step 1 and estimate
(This corresponds
to finding lines or directions of lines.)
From the above set of equations, it can be seen that to get a fairly good
initial estimate of
s
i.e. that is used in the step 1 to begin the iterative
process, a good initialization of
a
is needed. Note that even though splitting
of the minimization of Eq. (4) into two parts as described above is not
theoretically justified since
a
is not convex, however, we have found that
given a good initial estimate of
a
,
the “dual update” algorithm converges fast
and results in accurate final estimation of
a
and
s.
For the initial estimate of
a
an information theoretic based method was used which is described in detail
in [7].
Note that the “dual update” approach described above is not a single
maximum a posteriori estimation (MAP) of
a
.
However, it corresponds to
much more tractable joint MAP of
a
and
s
.
It is important to note that the initialization of
a
matrix should not be
confused with the classical approach of single MAP estimate of
a
and the
estimation of separated source signals by inverting the estimated
a
.
Instead,
the initialization is only for a good starting point for the iterative dual update
algorithm.
To summarize, the steps of our “dual update” algorithm are:
1.
Find that minimizes under the linear constraint
2.
Use from Step 1 to create a new estimate of the mixing matrix
Repeat Steps 1 and 2 until a convergence or stopping criterion is
met.
We start our “dual update” algorithm with an initial estimate of the
mixing matrix obtained using the technique based on mutual information and
angle thresholding technique described in [7].
3.
2.1.2
Generalization of the “dual update” algorithm for IM
By representing the Fourier (short time Fourier)
transform W
(x)
=
X
(k,m),
W
(s)
=
S
(k,m)
and W
(
v
) =
V
(k,m)
which are the time-frequency
