# Development of Parameters towards Voice Bifurcations
## 4 Discussion
### Sensitivity of HDF and the Proposed Frequency Selection Algorithm  
The effectiveness of HDF and BI depends on the algorithm to select \(f_{0}\) and \(f_{1}\). The proposed periodogram-based algorithm has been shown to provide consistently accurate estimates to HDF and BI in the presented four cases. The configuration used (Table 1) has been carefully chosen for the four cases presented in this paper, and the results in Figures 4 and 5 illustrate the considerations needed for the parameter selection process.  
Figure 4 shows how the frequency selection process is affected by the relative threshold parameter \(\rho\), which defines the \(f_{2}\) candidate selection threshold relative to the second strongest peak in the spectrum. In general, more aggressive (lower) \(\rho\) yields more candidate peaks, thus there is a higher likelihood of selecting non-harmonic peaks. With the base configuration (\(\rho=1/4\)), the algorithm misses the presence of a weak-but-consistent period-doubling subharmonic present at the beginning, viz. \(t\in(0,0.1)\). These subharmonic peaks are -25 dB below the \(f_{0}\) peaks, as shown in Figure 6. By lowering the threshold (\(\rho=1/8\) and \(1/16\)), these subharmonic peaks can be detected. However, lowering also causes the detection of non-harmonic peaks during the following segment at \(t\in(0.1,\ 0.3)\), over which visual inspection of the spectrogram suggests Type-I voicing. This leaves us a question: How small of a subharmonic peak is considered perceptible?The selection threshold \(\rho\) for \(f_{1}\) ought to be drawn based on the audibility of the signal component, which the spectral peaks represent. [PERSON] and [PERSON] [PERSON] and [PERSON] (1988) suggest that the power of non-harmonic peaks of the Type-I signal must be at least an order of magnitude (or -10 dB) lower than that of the first harmonic. Under this guideline, the observed subharmonics in Figure 6 are weak enough to be ignored (-25 dB under the first harmonic). However, it may be too conservative as a -10 dB threshold leaves out the second harmonic, which is -20 dB below the first harmonic and may have perceptual relevance. [PERSON] and [PERSON] [PERSON] and [PERSON] (1988) found that perception of roughness is consistently reported (albeit at a five-out-of-ten rating) above 20% amplitude modulation. The amplitude modulation effect that was tested in their study would introduce period-doubling subharmonics, but how the 20% figure translates to the relative power of subharmonic peaks was not reported. Future experimental study is necessary to identify the audible amount of subharmonics to solidify the peak detection requirement.  
Another factor regarding the performance of the frequency selection algorithm is the analysis window length. There is a tradeoff between the frequency and time resolutions. A longer window enables detection of closer spectral peaks but blurs the location of a bifurcation (if present). Conversely, a change in time can be detected more precisely with a shorter window but with an expense of lost frequency resolution. Both cases are presented in Figure 5. During a consistently biphonic segment \(t\in(0.3,0.5)\), the shortest window length (\(N=50\)) fails to detect the second fundamental frequency consistently due to the insufficient spectral resolution. Meanwhile, during the segment \(t\in(0.6,1.0)\), in which the voice type switches rapidly, a wide window (\(N\geq 100\)) bridges over the transitions and reports biphonic frequencies throughout this segment. To determine the optimal window size for general use, further experimentation with additional signal samples is needed. Specifically, the algorithm and its configuration must be tested with a wider range of speaking fundamental frequencies, including male voice, with which biphonic frequencies are closer together by virtue of lower fundamental frequency.