Model of Harmonic Structures

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 $F_0$s and to separate close partials. First we assume that each widened partial is a probability distribution of frequencies, approximated by a Gaussian distribution model. 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 $\mu$$_k\!\!=\!\{\mu_k,\cdots,$ $\mu_k\!+\!\log n,\cdots,\mu_k\!\!+\!\log N_k\}$ where $\mu_k$ ideally corresponds to the $\log F_0$ of $k$th sound and $n$ denotes the index of partials. We then introduce a model of multiple harmonic structures $P_{\theta}(x)$ which is a mixture of $K$ tied-GMMs whose model parameter $\theta$ is denoted as

$$\{\theta\}=\{ \mbox{\boldmath$\mu$} _k, \mbox{\boldmath$w$} _k,\sigma~\vert~k\!=\!1,\cdots,K\},$$ (1)

where $w$$_k=\{w_1^k,\cdots,w_n^k,\cdots,w_{N_k}^k\}$ and $\sigma$ indicate the weights and variances (which are briefly assumed here as a constant) of the respective Gaussian distributions.