# Development of Parameters towards Voice Bifurcations
## 2 Materials and Methods
### Tonal Frequency Selection Algorithm for Glottal Source Signals
#### 2.1.2 Main Algorithm  
The PSD of the detrended samples \(\mathcal{X}\) is estimated with a windowing function with minimal sidelobe leakage. The use of such a windowing function is critical to avoid erroneously picking sidelobe peaks as real spectral peaks. Additionally, sufficient zero-padding is applied to obtain a denser representation of the underlying discrete-time Fourier transform. Let the estimated PSD \(P_{xx}(f)\) as a function of frequency \(f\) in Hz.  
All spectral peaks in \(P_{xx}(f)\) are identified with a simple peak-picking technique followed by quadratic peak interpolation. Let \(\mathcal{F}_{\text{all}}\) be the set of the frequencies of all the  
\begin{table}
\begin{tabular}{l c c c} \hline \hline
**Type-II Voice** & \(\Upsilon\) & **HDF** & **BI** \\ \hline Period-2 subharmonic * & \(1/2,3/2\) & \(3/2,1/2\) & \(0,1/3\) \\ Period-3 subharmonic * & \(2/3,4/3\) & \(4/3,2/3\) & \(1/3,1/2\) \\
2:3 biphonia ** & \(2/3,3/2\) & \(4/3,1/2\) & \(1/3\) *** \\
3:5 biphonia ** & \(3/5,5/3\) & \(7/5,1/3\) & \(1/5\) *** \\
7:9 biphonia ** & \(7/9,9/7\) & \(11/9,5/9\) & \(5/9\) *** \\ \hline \hline \end{tabular}  
* 2 possible values: \(f_{0}=F_{0};f_{1}\) to be one of the nearest subharmonic frequencies to \(F_{0}\) over and under. ** 2 possible values: 2 ways to assign \(F_{0}\)’s of 2 harmonic series to \(f_{0}\) and \(f_{1}\). *** Both \(\Upsilon\) values yield in the same BI value.  
\end{table}
Table 2: Expected HDF and BI values for different Type-II voicings.  
peaks. Note that \(\mathcal{F}_{\text{all}}\) includes all the local peaks, possibly including those of the noise and window sidelobe peaks. The first peak frequency is chosen at the global maxima:  
\[f_{0}=\underset{f\in\ \mathcal{F}_{\text{all}}}{\text{argmax}}P_{xx}(f)\quad P_{0}=P _{xx}(f_{0}) \tag{4}\]  
Because \(\mathcal{X}\) has been detrended, \(f_{0}>0\) is guaranteed, thus \(\gamma\) is finite, as per (1).  
Denoting the remaining peaks by \(\mathcal{F}_{1}=\mathcal{F}_{\text{all}}\smallsetminus\{f_{0}\}\), the next strongest peak is detected as a point of reference:  
\[f_{\text{ref}}=\underset{f\in\ \mathcal{F}_{1}}{\text{argmax}}P_{xx}(f),\quad P _{\text{ref}}=P_{xx}\Big{(}f_{ref}\Big{)} \tag{5}\]  
This reference peak is used to form a threshold to define a set of candidates for the second peak \(f_{1}\):  
\[\mathcal{F}_{\text{can}}=\{f:f\in\ \mathcal{F}_{1},\ P_{xx}(f)>\rho P_{\text{ref} },f>f_{g}\}, \tag{6}\]  
where \(\rho\) is a relative threshold parameter and \(f_{g}\) is a minimum frequency parameter to avoid selecting a spurious near-dc peak. Finally, the chosen candidate in \(\mathcal{F}_{\text{can}}\) is the one such that its HDF value is closest to 1.0:  
\[f_{1}=\underset{f\in\ \mathcal{F}_{\text{can}}}{\text{argmin}}|1-\text{ HDF}(f)|. \tag{7}\]  
Here, \(\text{HDF}(f)\) represents the HDF value computed from a given arbitrary \(f\) by \(\gamma=f/f_{0}\) and (2). This selection criterion favors the biphonic case (in which the other fundamental frequency is closer to \(f_{0}\)) followed by subharmonic frequencies, then by the harmonic frequencies. For a Type-I voice, \(\mathcal{F}_{\text{can}}\) is expected to contain only the harmonic frequencies, while a Type-II voice may produce \(\mathcal{F}_{\text{can}}\) with both non-harmonic and harmonic frequencies. The selection strategy in (7) guarantees to pick non-harmonic frequencies if present.