## 2 Materials and Methods
### Feature Engineering
#### 2.1.2 Frequency-Domain Signal Feature Engineering  
Wavelet analysis is a time-frequency analysis method [9] that can not only be used to obtain the power spectrum of a signal, but also to analyze and control the time resolution and frequency resolution [10; 11]. In this study, data features were extracted from the audio signal using wavelet scattering theory, which can be used to make a further classification assessment. The Morlet wavelet is based on the use of wavelet transformation for a series of signals:  
\[\psi(t)=e^{i\omega_{0}t}e^{\frac{-t^{2}}{2}} \tag{8}\]  
where \(\omega_{0}\) is the wave number. The Morlet wave function can be further dilated and translated:  
\[\psi_{(a,b)}(t)=\frac{1}{\sqrt{a}}\psi\left(\frac{t-b}{a}\right) \tag{9}\]  
where \(a\) is the extension (dilation) and \(b\) is the translation. As a result of this property, regardless of whether the frequency is high or low, different time and frequency resolutionscan be used for effective analysis. The signal analysis in the time and frequency domains can be obtained by wavelet transform:  
\[W_{x}=\int_{-\infty}^{\infty}x(t)\frac{1}{\sqrt{a}}\psi\bigg{(}\frac{t-b}{a} \bigg{)}dt \tag{10}\]  
Application of the short-time Fourier transform (STFT) to the divided signal can also increase the time resolution of the signal. However, the adjustment of the bandwidth and central frequency of the filter via scaling results in better time and frequency resolutions for the wavelet transform. Wavelet scattering transformation was proposed by [PERSON] and [PERSON] in 2014 [12]. The theory is that a wavelet-transformed time-series signal is analyzed using the Convolution Neural Network (CNN) framework, and a wavelet function (high-pass filter) is convolved with the scaling function of a Gabor function (low-pass filter). Equation (11a) is the Gabor function and Equation (11b) is the scaling function of the Gabor function. \[g(t)=\frac{1}{\sqrt{2\pi}\sigma^{c}}e^{\frac{-t^{2}}{2\sigma^{2}}}e^{i\theta x} \tag{11a}\] \[\varnothing_{J}(t)=2^{-2 JQ}g\Big{(}2^{-JQ}t\Big{)} \tag{11b}\]  
where \(\theta\) is the center frequency; when its value is zero, it is a Gaussian function. \(\sigma\) is the standard deviation of the data signal. \(J\) is the scale and \(Q\) is a number of wavelet functions per octave; for example, there are 12 notes in an octave for music. The values of \(J\) have several octaves for high-pass decomposition. Using different scales, convolutional signals, and wavelet functions, from low- to high-frequency information a locally deformed and scale-invariant wavelet conversion coefficient can be obtained:  
\[W_{x}=\left(x(t)\star\varnothing(t),x(t)\star\psi_{\lambda}(t)\right)_{t\in \mathcal{R},\lambda\in\Lambda} \tag{12}\]  
where \(\lambda\) is the center frequency of the wavelet function, and the mean value of the wavelet coefficients is zero. \(x(t)\) represents the original signal, \(\varnothing(t)\) is the scaling function, and \(\psi_{\lambda}(t)\) is the wavelet function. In order to obtain the characteristic information of the signal, the absolute value is added, which is the modulus. According to [12], the modulus provides the stability and translation stability features, i.e.,  
\[U_{\lambda}x(t)=\left|x(t)\star\psi_{\lambda}(t)\right| \tag{13}\]  
The modulus is used to compute a lower frequency envelope. In order to obtain stable wavelet coefficients, the scaling function (low-pass filter) is convolved, and we obtain the wavelet scattering characteristic coefficient of the signal:  
\[S_{1}x(t,\lambda_{1})=\left|x(t)\star\psi_{\lambda_{1}}(t)\right|\star \varnothing_{J}(t) \tag{14}\]  
In the expression of the wavelet scattering framework, a series of wavelet scattering coefficients are analyzed using the deep convolution network architecture. In order to obtain a low variance and stable features, an averaging function and modulus are applied after the wavelet convolution.