Digital Signal Processing Reference
In-Depth Information
representations of
x
(
n
),
s
(n)
and
v
(n), respectively, and W
(a)
=
A
(k,m)
equation (2) can be re-written as
Since
a
represents an instantaneous system,
A
(k,m)
=
A
(
k
) =
A
and is
constant for all frequency bands
k
and time
m.
In order to find an initial estimate of
A
the mutual information between
sensors is computed for each sub-band. The sub-band with the maximum
mutual information is chosen since it represents a band that exhibits the most
separation and hence, can best estimate initial
a
. For this estimation the
following equation is used:
where
k
is the chosen sub-band and
i
and
j
represent the signal received at the
and sensors. If is much larger than all the other at a particular
k
and
n
then equation (12) simplifies to:
This results in clusters of measurements that correspond to the arctangent
of the ratios of rows of Aj and Ai. Several methods could be used to find
these clusters. Some authors use peak picking of the histogram [7,11] and
others use potential function [4]. The peak picking of the histogram has the
disadvantage of difficulty in accurately picking the local maxima.
On the other hand, we use here a hierarchical clustering approach where
each observation is taken in succession and merged with nearest neighbor.
This is computationally less intensive than finding the two observations that
are closest together and then merging them. The result of hierarchical
clustering is used as an initial guess for the k-means clustering algorithm.
The means of the clusters that are obtained when the k-means clustering
algorithm converged are then used for the initial estimate of the mixing
matrix. The number of clusters is used as the estimate of the number of
sources to be separated. It was empirically observed that the combination of
