An influence of a window function and a varying pitch within the short
time single analysis frame inevitably cause widening of the spectral harmonics which
makes it difficult to extract the precise value of s and to separate close partials.
First we assume that each widened partial is a probability distribution
of frequencies, approximated by a Gaussian
Therefore, a single harmonic structure can then be modeled by a tied
Gassian mixture model (tied-GMM), in which their means have
only 1 degree of freedom.
In log-frequency scale, means of tied-GMM are denoted here as
ideally corresponds to the of th sound and denotes
the index of partials.
We then introduce a model of multiple harmonic structures
which is a mixture
of tied-GMMs whose model parameter
is denoted as