Digital Signal Processing Reference
In-Depth Information
Modification of initial estimate of A for a complex case:
Next, in the case
of convolutive mixture
A
(
k
) is complex. So the initial estimate of the mixing
matrix described in section 2.1 needs to be modified. It is modified as
follows. The ratio in polar coordinates of the
and
rows for the
column of A is:
Using an argument similar to Equation (12) that
is larger than all
other sources results in:
where is the phase difference and represent the angle operator. This
shows that the estimation of
A
(
k
) requires two components - the ratio of
magnitudes of
A
(
k
) elements to obtain
and the difference in phase between
the elements to obtain
The remaining procedure is same as before as described in section 2.1 in
that the clustering approach is used to determine the initial estimate of
A
(
k
).
Since is between 0 and and is between 0 and is appropriately
weighted in the clustering so that the same amount of weight will be placed
on the components as the components. A value of that is slightly larger
than 0 should be considered closed to a value that is slightly less than If
the phase difference is close to 0 or then the clustering algorithm could
see two clusters. In order to avoid this possibility the histogram of is
computed and the values are shifted so that the discontinuity will occur at a
point that would not divide a cluster.
An example scatter plot of versus is shown in Figure 18-1. From this
figure it can be seen that three clusters corresponding to three sources are
formed without much overlap whose mean values are pretty close to the true
values (circled x). Unfortunately, due to the ambiguity in the scale factor the
actual phase values of the mixing matrix cannot be determined. However, the
use of the phase difference greatly improves the robustness of the
separation for convolutive mixtures and complex IM.
Lastly, the “dual update” algorithm for underdetermined IM chooses a
particular frequency sub-band by using mutual information measure to
