# MixDiff-TTS: Mixture Alignment and Diffusion Model for Text-to-Speech
## 3 Results
### Evaluation  
#### 3.3.1 Evaluation Metrics  
For evaluation, we adopt four objective metrics: SSIM [35], MCD [36], fundamental frequency (\(F_{0}\)) root mean squared error (RMSE), and Frechet Audio Distance (FAD) [37]. In speech synthesis, SSIM is commonly used to measure the structural similarity between real and synthesized mel-spectrograms. The mathematical formulation of SSIM is as follows:  
\[\text{SSIM}=\frac{(2\mu_{x}\mu_{y}+C_{1})(2\sigma_{xy}+C_{2})}{(\mu_{x}^{2}+ \mu_{y}^{2}+C_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+C_{2})} \tag{12}\]  
where \(x\) and \(y\) are the elements of the real and synthesized spectrograms, respectively. \(\mu_{x}\) and \(\mu_{y}\) denote their respective means, \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) represent their respective variances, \(\sigma_{xy}\) is their covariance, and \(C_{1}\) and \(C_{2}\) are constants that prevent the denominator from becoming zero. An SSIM value closer to 1 signifies greater structural similarity between synthesized and real mel-spectrograms.  
MCD is widely adopted to quantify spectral shape distortion between real and synthesized speech. The formula of MCD is shown below:  
\[\text{MCD}=\frac{10}{\ln 10}\sqrt{2\sum_{m=1}^{M}(c_{m}-\hat{c}_{m})^{2}} \tag{13}\]  
where \(M\) represents the dimensionality of Mel-Cepstral Coefficients (MCCs). Specifically, \(c_{m}\) denotes the \(m\)-th dimensional MCC of natural speech, while \(\hat{c}_{m}\) corresponds to the \(m\)-th dimensional MCC of synthesized speech. As a key spectral feature in speech processing, MCCs are widely used in academic research to characterize the overall spectral contour of speech. A lower MCD value indicates smaller cepstral distortion, implying that the spectral characteristics of synthesized speech more closely match those of natural speech.  
\(F_{0}\) RMSE is a commonly used objective metric for assessing the accuracy of \(F_{0}\) predictions. Specifically, its role is to quantify the discrepancy between predicted and true \(F_{0}\) values. The formula for \(F_{0}\) RMSE is as follows:  
\[F_{0}\,\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}(F_{i}-\hat{F}_{i})^{2}} \tag{14}\]  
where \(F_{i}\) denotes the real fundamental frequency of the \(i\)-th speech frame, while \(\hat{F}_{i}\) denotes its corresponding predicted fundamental frequency. A smaller \(F_{0}\) RMSE signifies that the fundamental frequency contours of synthesized and real speech are more closely matched, thereby reflecting more consistent time-domain dynamic characteristics of fundamental frequency.  
FAD, which is designed as a metric to quantify the quality of generated audio, measures the similarity of real to generated audio by calculating the distributional distance in their feature spaces. It relies on the VGGish audio feature extractor to map audio into a perceptual feature space and then computes the Frechet distance between Gaussian distributions in this space. Its calculation formula is as follows:  
\[\text{FAD}=\left\|\mu_{a}-\mu_{b}\right\|^{2}+tr(\sum_{a}+\sum_{b}-2\sqrt{ \sum_{a}\sum_{b}}) \tag{15}\]where \(a\) and \(b\) denote the real and synthesized speech, respectively. \(\mu\) represents the mean vector of speech features, \(\sum\) denotes the covariance matrix, and \(tr\) indicates the matrix trace. A smaller FAD value implies a closer match of the synthesized speech distribution with that of real speech, thereby indicating a higher degree of similarity between the two.  
We adopt two subjective evaluation metrics: MOS [38] and Comparative Mean Opinion Score (CMOS) [39], both of which are based on human ratings. Additionally, the Real-Time Factor (RTF) is employed to measure the model's audio generation speed. An RTF value below 1.0 indicates that the model generates audio quickly enough for real-time applications, and lower RTF values indicate better real-time performance.