## 3 Theoretical Background
### Mel-Frequency Cepstrum Coefficients Method  
As valid acoustic features, MFCCs are widely used in civil engineering [26; 27; 28; 29; 30; 31]. The MFCCs represent the cosine transformation result of the real logarithm of the short-term energy spectrum on a frequency scale. The relationship between the frequencies on the Mel frequency and Hertzian scales is as follows:  
\[Mel(f)=2595\cdot\lg(1+\frac{f}{700}) \tag{5}\]  
Figure 5 shows the general process for determining MFCCs. There are four steps: preprocessing, fast Fourier transform (FFT), Meier frequency filtering, and discrete cosine transform (DCT).  
Figure 4: Support vectors and optimal hyperplane of a linearly separable case.  
Step 1: The preprocessing step is divided into two parts: frame splitting and windowing. Frame splitting can be completed by continuous segmentation; however, the overlapping segmentation method is generally used to make the transition between frames smooth and consistent; that is, the end of each frame overlaps with the head of the next frame. We need to multiply the speech frame by the window function to reduce the truncation effect of the speech frame and reduce the slope of the two ends of the frame so that the two ends of the speech frame do not cause sharp changes and smooth transition to 0. Let the frame signal be \(x(n)\) and the window function be \(w(n)\). Then, signal \(y(n)\) after adding the window is  
\[y(n)=x(n)\cdot w(n), \tag{6}\]  
where \(0\leq n\leq N-1\); \(N\) is the number of sampling points for each frame. The most commonly used window function is the Hamming window.  
\[w(n)=0.54-0.46 cos\frac{2\pi n}{N-1}\ 0\leq n\leq N-1 \tag{7}\]  
Step 2: Because discrete Fourier transform involves a large amount of calculation, an efficient FFT can be used to transform the speech frame from the time domain to the frequency domain.  
Step 3: The discrete spectrum obtained from the FFT is filtered using a sequential triangular filter to obtain a set of coefficients \(m_{1},m_{2},\ldots\). The center frequency \(f(i)\) of each triangular filter in this filter set is equally spaced on the Mel frequency axis, and its span is also equal on the Mel frequency scale. The number of filters \(p\) is determined by the cutoff frequency of the signal, and all filters cover, in general, from 0 Hz to the Nyquist frequency, that is, one-half of the sampling rate.  
The equation for calculating \(m_{i}\) is as follows:  
\[m_{i}=\ln\left(\sum_{k=0}^{N-1}|x(k)|\cdot H_{i}(k)\right)\ i=1,\ 2,\ldots,\ p \tag{8}\]  
The calculation of \(H_{i}(k)\) is carried out as follows.  
\[H_{i}(k)=\begin{cases}0&k<f[i-1]\ or\ k>f[i+1]\\ \frac{2(k-f[i-1])}{(f[i+1]-f[i-1])-(f[i-1])}&f[i-1]\leq k\leq f[i]\\ \frac{2(f[i+1]-f[i-1])}{(f[i+1]-f[i-1])-(f[i+1]-f[i])}&f[i]\leq k\leq f[i+1]\\ \end{cases} \tag{9}\]  
Here, \(y[i]\) is the center frequency of the triangular filter, which satisfies the following equation:  
\[Mel(f[i+1])-Mel(f[i])=Mel(f[i])-Mel(f[i-1]). \tag{10}\]  
Step 4: The Mel spectrum obtained in the previous step is transformed into the time domain, and the result is the MFCCs. Because MFCCs are real numbers, they can be transformed into the time domain using a DCT. The MFCC equation is as follows.  
\[c_{i}=\sqrt{\frac{2}{N}}\sum_{j=1}^{p}m_{j}cos\left[\frac{\pi i}{\mathrm{P}} (j-0.5)\right] \tag{11}\]  
Figure 5: Flowchart of MFCCs.  
The normalized energy was calculated as the 13 th dimensional component of the feature vector for all scoring frames in the speech signal or speech signal file. In this study, 13 dimensions were considered, and the average value of all frames in each dimension was considered as the acoustic feature.