## 4 Theoretical Background  
### Continuous Wavelet Transform  
In this section, we briefly review continuous wavelet transformation (CWT), which is a method of analyzing the time and frequency domains simultaneously via wavelet transform. Wavelet transform (WT) is a method of decomposing arbitrary signals into functions defined as wavelets. When a signal obtained from equipment is subjected to CWT, time and frequency characteristics are generated as an image; for detecting equipment abnormalities, images are used to feed convolutional neural networks [27; 28; 29; 30]. The wavelet function, \(\Psi_{a,b}(t)\), can be obtained by scaling and translating the parent wavelet function as shown in Equation (1):  
\[\Psi_{a,b}(t)=\frac{1}{\sqrt{a}}\Psi(\frac{t-b}{a}),\;a>0,\;a,b\in R \tag{1}\]  
where \(a\) is a scale parameter related to the time resolution of the wavelet function, \(b\) is a translation parameter that determines the position of the wavelet function on the time axis, and \(\Psi_{a,b}(t)\) represents the mother wavelet. The wavelet transform is the inner product between the given function and the wavelet function, as shown in Equation (2). For any signal function given, \(x(t)\) can be integrated over the entire interval, and the CWT is defined mathematically as follows:  
\[W_{\Psi}(a,b)=<x(t),\Psi_{a,b}(t)>=\int_{-\infty}^{+\infty}\;x(t)\;\Psi_{a,b}^{ *}(t)\;dt \tag{2}\]  
Here, \(\Psi_{a,b}^{*}(t)\) is the complex conjugate function of the mother wavelet function \(\Psi_{a,b}(t)\). Based on the above operation, signal \(x(t)\) is transformed and projected into two dimensions (time and scale), and the one-dimensional time series is converted into a two-dimensional time-frequency image. The choice of the mother wavelet function influences classification accuracy significantly. Depending on the mother wavelet, there are pros and cons. In continuous wavelet transform, Bump, Mexican hat, Morlet, Paul mother wavelet, etc., are commonly used. Mexican hat and Paul mother wavelets are known for their advantages in time domain analysis. The Morlet mother wavelet, however, has advantages in frequency analysis. The bump wavelet has a band shape and is suitable for analyzing frequencies locally. In this paper, Molet wavelets were used to analyze gearbox vibration data, which has the advantage of extracting features in the frequency domain that are difficult to identify in the time domain. The Morlet wavelet function is  
\[\Psi(t)=\frac{1}{\sqrt{\pi f_{b}}}e^{2\pi if_{b}t}e^{\frac{t^{2}}{f_{b}}}\  
where \(X_{j}^{l}\) represents the \(j^{\prime}\)th element of the \(l^{\prime}\)th layer, \(M_{j}\) represents the \(j^{\prime}\)th convolution region of the \((l-1)\)th layer, \(X_{i}^{l-1}\) represents the element, \(\omega_{ij}^{l}\) is the weight matrix, \(b_{j}^{l}\) represents bias, and \(f(\cdot)\) is the activation function. A rectified linear unit (ReLU) is the most popular activation function and is given by  
\[f(x)=\max(0,x) \tag{5}\]  
Pooling layers often follow convolution layers to realize the dimension reduction of feature maps and to keep the invariance of characteristic scales. The most commonly used pooling approaches are maxpooling, mean pooling, and stochastic pooling. In general, pooling layers are used only to implement a decrease of dimension without updating weights. After input images are alternately propagated by convolution and pooling steps, the extracted information and features are classified using the fully connected layer net. On FC layers, the input data are weighted summations of 1-D feature vectors:  
\[y^{n}=f(\omega^{n}x^{n-1}+b^{n}) \tag{6}\]  
where \(n\) is the sequence number of net layers, \(y^{n}\) is the output of the fully connected layer, \(x^{n-1}\) is the 1-D feature vectors, \(\omega^{n}\) is a weight coefficient, and \(b^{n}\) represents biases. The commonly used activation function \(f(\cdot)\) is SoftMax.