# Effects of Sinusoidal Model on Non-Parallel Voice Conversion with Adversarial Learning
## 3 Proposed Continuous Sinusoidal Model
### Analysis Phase: Feature Extraction  
Here, we have designed our continuous sinusoidal model (i.e., in which all parameters are continuous) using three acoustic parameters (contF0, maximum voiced frequency and mel-generalized cepstrum). We use a continuous F0 (not a discontinuous F0 such as Swipe, Yin, DIO, etc.) tracker which takes non-zero pitch values even when voicing is not present and does not apply a strict voiced/unvoiced decision. We choose the continuous modeling proposed in [28] as it can be more effective in achieving natural synthesized speech. Another excitation continuous parameter is the maximum voiced frequency (MVF) [29], which was recently shown to result in a major improvement in the quality of synthesized speech. During the synthesis of various sounds, the MVF parameter can be used as a boundary frequency to separate the voiced and unvoiced components. Therefore, contF0, MVF, and MGC [30] parameter streams are calculated during the analysis phase to be converted. The advantage of this CSM is that it is relatively simple; it has only two one-dimensional parameters for modeling excitation (contF0 and MVF) and the synthesis part is computationally feasible, therefore speech generation can be performed in real time. It should be noted that contF0 is still sensitive to additive noise in speech signals and suffers from short-term errors (when it changes rather quickly over time). Moreover, it can cause some tracking errors when the speech signal amplitude is low, voice is creaky, or there is a low harmonic-to-noise ratio (HNR). To mitigate these issues, an instantaneous frequency-based (i.e., the time-derivative of the phase) method is employed. In our im  
Figure 1: Overview of the developed system. CSM consists of three analysis algorithms (for determining the contF0, spectral envelope, and MVF) and a synthesis algorithm incorporating these parameters. plementation, the formula for computing the instantaneous frequency \(IF(\tau)\) is given by [PERSON]'s equation [31]:  
\[IF(\tau)=\frac{a\frac{db}{d\tau}-b\frac{da}{d\tau}}{a^{2}+b^{2}} \tag{1}\]  
where \(a\) and \(b\) are, respectively, the real and imaginary parts of the spectrum of a waveform \(S(w)\). Consequently, contF0 can be further corrected by  
\[Corrected\ contF0=\frac{\sum_{k=1}^{K}\lvert S(kw_{0},\ t)\rvert\ IF(kw_{0},t)}{ \sum_{k=1}^{K}\lvert S(kw_{0},\ t)\rvert} \tag{2}\]  
where \(K\) represents the harmonics number used for refining (we set K = 6), \(w_{0}\) denotes the angular frequency of the contF0 at a temporal position \(t\). The new corrected contF0 performance can be compared with the performance of a reference pitch contour (f0_egg, estimated from the electroglottograph) as shown in Figure 2. We can see that there is almost equivalent performance between the refined contF0 and the f0_egg, and much better than the baseline [28]. It can be also observed in the unvoiced region (frames from 170 to 202 in Figure 2) that the new contF0 is reduced more significantly than for the baseline. In general, the spectral envelope must be characterized using a small number of parameters, either for reaching a high compression rate or for statistical modeling. Preferably, a mel-generalized cepstral (MGC) analysis is implemented on the speech signals, as these features have shown their efficiency to accurately and robustly capture the spectral envelope. In this study, we followed the CheapTrick algorithm proposed by [PERSON] [30]. Thus, we found 36 coefficients are enough to run our synthesized converted speech. As a result, Figure 3 shows three streams of acoustic information extracted from a speech frame: F0, MVF, and the spectral envelope. We then had an acoustic vector with 36 M  
Figure 2: Examples of F0 contours from a female speaker extracted by the baseline, EGG, and corrected methods. \[\beta_{a}\Big{(}w_{0}^{i}\Big{)}=tan^{-1}\frac{\left(1-\alpha^{2}\right)\sin w_{0} ^{i}}{\left(1+\alpha^{2}\right)\cos w_{0}^{i}-2\alpha} \tag{9}\]  
where \(\alpha\) is the all-pass factor that takes 0.42 for \(F_{s}=16\;kHz\).