Detection of $F_0$s and Spectral Envelopes

In the previous process, the ML procedure allows to aqcuire local optimal solutions of $\mu_k$ without distinction of the true $F_0$s or the multiples of the true $F_0$s. Therefore, the true $F_0$s must somehow be discovered by replacing $\mu_k$ each by each to their multiples. Consider now that a degree of freedom is given to every $w_n^k$ and consequently allows to extract the spectral envelope, i.e., the relative amplitudes of the partials. If $\mu_k$ is lower than the true $F_0$, the model must be over-fit. From this point of view, the problem of obtaining the true $F_0$s and the spectral envelope can also be handled with the information criterion. The process shown below is done with all remaining tied-GMMs after the previous process.

1. Replace the representative means to $\mu_k\!\!+\!\log t$ where $t$ is an integer number whose initial value is 1. The number of Gaussians limited below the Nyquist log-frequency is denoted as $N_k^t$.
   
2. Estimate the ML model parameters by EM algorithm. Here we only update $w_n^k$ and should be updated to
   

   $$\bar{w}_n^k=\displaystyle\frac{1}{F}\int_{-\infty}^{\infty}P_{\theta}(n,k\vert x)dx.$$ (13)

   
   
   
   
3. Calculate AIC with equation (9). The number of free parameters here is $N_k^t$. If the AIC increases, the process should be interrupted and the $\mu_k\!\!+\!\log (n\!-\!1)$ is considered as the detected $F_0$, and if not, add $1$ to $t$ and return to step1.

図: Detected $F_0$ contour of a single speaker
図: Reference $F_0$ contour corresponding to Figure 3

表 1: Results for a single speaker

	Accuracy(%)
Speech file	Cepstrum	Proposed
`myisda01'	88.2	98.0
`myisda02'	88.4	99.0
`myisda03'	84.8	98.1
`myisda04'	85.1	92.4
`myisda05'	76.8	93.7
`fymsda01'	86.3	98.5
`fymsda02'	87.1	97.5
`fymsda03'	83.3	95.8
`fymsda04'	86.7	96.8
`fymsda05'	85.2	96.0