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As mentioned above, knowledge of the distributions on and
can be used to create distributions on the values
from equations 9 through 11. The
distributions on need not be Gaussian to use the technique
described here. Assuming this simplifies the situation, however,
and informal experiments have shown that we may fairly model the
amplitude of the STFT coefficients this way by considering their
distribution as the positive values of a zero-mean Gaussian.
Specifically, it may be shown that doing so leads to
where represents the variance for source at a
given frequency. (We again recall that these distributions, and
their related and values are source dependent,
and different for each frequency. Clearly, will be
larger for frequencies corresponding to the active range of a
given voice or instrument.)
We may calculate then, the probability that a set of data (in
the form of values given by equation 4) was
generated by the presence of sources and via:
where
and
refers to the variance in the
distributions of expressions 14 through 16.
To achieve our goal of determining the two most likely sources at
a given point in time-frequency space, we first determine
for the point's values using
equation 17 and considering every possible
combination. Then we substitute in our result to
equation 13 which allows us to take into account prior
probabilities. By allowing or to be ``NULL'' and
assigning a value corresponding to the noise floor as the variance
of
, we effectively can include the one-source
combinations used in the DUET system as well.
Next: A Pathological Musical Example
Up: BAYESIAN TWO SOURCE MODELING
Previous: Bayesian Framework
Aaron S. Master
2003-10-30