# Effects of Sinusoidal Model on Non-Parallel Voice Conversion with Adversarial Learning
## 5 Evaluation and Discussion
### Statistical Evaluations  
To show that our model is able to convert speaker characteristics, several performance metrics are used. While the spectral distortion is designed to compute the distance between two power spectra, the root mean square (RMS) log spectral distance (LSD) metric is suggested here to carry out the evaluation as  
\[LSD_{RMS}=\sqrt{\frac{1}{N}\sum_{k=1}^{N}\left[logP(f_{k})-log\hat{P}(f_{k}) \right]^{2}} \tag{24}\]  
where \(k\) is the frequency bin, \(P(f)\) is the spectral power magnitude of the real speech, while \(\hat{P}(f)\) is the spectral power magnitude of the converted speech, and both are defined at \(N\) frequency points. The optimal value of LSD\({}_{\text{RMS}}\) is zero, which indicates a matching frequency content. Here, we average the LSD\({}_{\text{RMS}}\) values for the whole set of tested inter-gender examples (female-to-male and male-to-female) separately as  
\[Average\ LSD_{RMS}=\frac{1}{M}\sum_{i=1}^{M}LSD_{RMSi} \tag{25}\]  
where \(M\) is the total number of examined examples, which is equal to 10 in this study. We also use a set of example spectrograms for the converted voices, to see whether voice conversion results capture the target speaker. The results are shown in Figure 5, and we found that the proposed framework has lower LSD\({}_{\text{RMS}}\) values equal to 1.7 dB (female to male) and 2 dB (male to female) that are closer to the target speech spectrogram than the StarGAN model. Consequently, our proposed system introduces a smaller distortion to the sound quality and approaches a correct spectral criterion. In addition, the empirical cumulative distribution function (ECDF) [39] of the phase distortion mean (PDM) [40] is computed and presented in Figure 6. The reason to compute this function is to see whether these conversion methods can be normally distributed and  
Figure 5: Example of the spectrogram for source, target, and converted speech. The LSD values are averages for the whole set of tested examples. if we can compare it to the natural target speech. PDM is estimated in this experiment at a 5 ms frameshift by ([[http://covarep.github.io/covarep/](http://covarep.github.io/covarep/)]([http://covarep.github.io/covarep/](http://covarep.github.io/covarep/)) accessed on 24 December 2016):  
\[PDM=\sigma_{i}(f)=\sqrt{-2 log\left|\frac{1}{N}\sum_{n\in C}e^{j(PD_{n}(f)-\mu_{n} (f))}\right|} \tag{26}\]  
\[\mu_{n}(f)=\left(\frac{1}{N}\sum_{n\in C}e^{jPD_{n}(f)}\right) \tag{27}\]  
where \(C=\left\{i-\frac{N-1}{2},\ldots,i+\frac{N-1}{2}\right\}\), \(N\) is the number of frames, PD is the phase difference between two consecutive frequency components, and we denote the phase by \(\angle\). The standard deviation in Equation (26) is supposed to represent the noisiness of the voice source, which allows for obtaining a more robust estimate of the source shape in transients. Additionally, conventional systems had issues with modeling the high frequency voiced/voiceless contents, so we wanted to show that our system is better in this sense. Figure 6: Empirical cumulative distribution function of PDMs using 4 types of VC pairs evaluated with the natural target speech. As we wanted to quantify the noisiness in the higher frequency bands only, we zero out the PDD values below the MVF contour. The ECDF \(F_{n}(x)\) is expressed as  
\[F_{n}(x)=\frac{\#\{X_{i}:X_{i}\leq x\}}{n}=\frac{1}{n}\sum_{i=1}^{n}I_{X_{i}\leq x }(X_{i}) \tag{28}\]  
where \(X_{1},\ldots,X_{n}\) are the PDM variables, \(n\) is the number of experimental observations, \(\#A\) represents the number of elements in the sample \(X\leq x\), \(I\) is the indicator function  
\[I(x)=\left\{ \begin{array}{ll}1,&x\in A\\ 0,&x\
otin A\end{array} \right. \tag{29}\]  
In the positive \(x\)-axis distribution shown in Figure 6, the proposed system is better reconstructed than that of the StarGAN approach. These outcomes can be supported by the fact that developing a sinusoidal VC system is beneficial and can substantially transform the source speaker into a particular target without parallel corpora.