# Development of Parameters towards Voice Bifurcations
## 1 Introduction
### Main Concept  
This paper focuses on the detection of mid-phonon bifurcations in pathological voice. The underlying motivation for such detection scheme is twofold. First, automatic segmentation of voice signal helps the selection of voice analysis methods and objective parameters. For example, the perturbation analysis is limited to Type-I signals [3]. Second, the detection outcomes (e.g., the number of mid-phonon bifurcations) can be used as objective measures, similar to MDVP's DSH measure but targeting broader Type-II characteristics. Frequent mid-phonon voice change can be perceived as poor voice quality, possibly worse than consistent Type-II or Type-III voice. As such, objective measures of bifurcations are potential strong contributors to objective voice assessment tools. To be able to detect frequent bifurcations as seen in Figure 1, a signal window length of shorter than 100 ms is desired. Two measures, named harmonic disturbance factor (HDF) and biphonia index (BI), are proposed to be used for this task. The biphonic and subharmonic spectral contents of Type-II voice are shown to have at least comparable strength to the second harmonic of such voice [29]. This observation leads to a family of voice parameters based on the frequencies of the spectral peaks. The basic premise here is to pick two spectral peaks: one, labeled \(f_{0}\), which corresponds to \(F_{0}\) of the voice, and another, labeled \(f_{1}\), which is indicative of the active voice type (Type I vs. Type II/III). The selection strategy of the second peak is critical, and this paper proposes a strategy for glottal source signals. Given these peaks, the ratio of these two frequencies is used to normalize the effect of the fundamental frequency; this ratio is defined as:  
\[\gamma\ =\frac{f_{1}}{f_{0}}. \tag{1}\]  
Table 1 summarizes the expected behaviors of the three quantities: \(f_{0}\), \(f_{1}\), and \(\gamma\). There is a clear segmentation in the expected values with three caveats. First, the measures of Type-III voice are expected to be random, and thus the estimates are inconsistent across successive observation windows, unlike the other voice types. Second, \(f_{1}\) for the Type-II voice with subharmonics could align with the proper pitch harmonic frequencies (\(nf_{0}\)), in which case it cannot be distinguished from Type I. Third, the strengths of the two spectral peaks of biphonic voice could be similar, thus their fundamental frequency, \(f_{0,1}\) and \(f_{0,2}\), assignments to \(f_{0}\) and \(f_{1}\) may switch back and forth across the phonation. With these caveats in mind, two short-term parameters on the frequency ratio and its expectations for each voice type are proposed as follows. **Harmonic Disturbance Factor (HDF):**_A short-term, narrowband voice measure to penalize if \(f_{1}\) is not a higher harmonic frequency than \(f_{0}\). Given the frequency ratio \(\gamma\) defined in (1), the_ HDF \(\in[0.0,\ 2.0]\smallsetminus\{1.0\}\) _is defined by:_  
\[\mathrm{HDF}\ =\ \left\{ \begin{array}{ll}|2-\gamma|&\text{if $\gamma\leq 2$,}\\ |\gamma\ \mathrm{mod}\ 1|&\text{if $\gamma>2$ and $(\gamma\ \mathrm{mod}\ 1)\leq 0.5$,}\\ |1-\gamma\ \mathrm{mod}\ 1|&\text{otherwise}\end{array} \right. \tag{2}\]  
_where (a mod b) is the modulo operator. The_ HDF _is expected to be zero if the voice is Type I and non-zero for Type-II voice._  
**Biphonia Index (BI):**_A short-term, narrowband voice measure to indicate the likelihood of vocal folds vibrating with two unrelated frequencies. Given the frequency ratio \(\gamma\)_ defined in (1), \(\mathrm{BI}\in[0,\ 1)\) is defined by:_  
\[\mathrm{BI}\ =\ \left\{ \begin{array}{ll}2\gamma-1&\text{if $0.5<\gamma\leq 1$,}\\ 2\gamma^{-1}-1&\text{if $1<\gamma\leq 2$,}\\ 0&\text{otherwise.}\end{array} \right. \tag{3}\]  
_The_ BI _of Type-I voice is expected to vanish, although_ BI = 0 _does not imply_ Type I_, unlike_ HDF_.