## 2 Preliminary Knowledge  
### One-Dimensional Lightweight Convolution  
Convolutional neural networks (CNNs) are mostly used to solve image problems, so many fault diagnosis methods use 2D CNNs as a feature extraction method. Many researchers use various techniques to convert one-dimensional signals received by sensors into two-dimensional images for direct input into existing 1D CNNs for feature extraction. For example, convert one-dimensional sequence data into two-dimensional grayscale images [33; 34]; use continuous wavelet transform or fast Fourier transform methods to convert one-dimensional sequence data into time-frequency images [35; 36]. These conversion methods have achieved satisfactory results in fault diagnosis, but converting 1D sequence data to 2D images consumes additional computational cost, and the training cost of 2D CNNs is also higher than that of 1D CNNs. Therefore, this paper uses a lightweight 1D CNN to directly extract features from the original data. 1 DLCNN refers to the lightweight image classification network MobileNet V2 [37], which can reduce computing costs while ensuring accuracy [38]. Figure 2 shows two special convolutional blocks contained in lightweight convolutional neural networks: the depthwise separable convolution block and the inverted residual block. The fundamental ideas and tenets of each module in the lightweight CNNs are explained briefly in this section. The input of a standard 1D CNN is a vector and a convolution kernel, where the convolution kernel's size is often smaller than the input, and this configuration creates a local receptive field. Weight sharing refers to the fact that the filter's weights remain constant when each filter moves over the input map. However, the number of parameters grows with deeper convolutional layers, increasing the computing cost. Lightweight CNNs are based on depthwise separable convolutions, which are a form of factorized convolutions [39]. This form decomposes the standard convolution into a depthwise convolution (DW), which operates channel-wise, and a pointwise convolution (PW) with a \(1\times 1\) kernel. DW convolution differs from standard convolution in that it employs a single-channel convolution kernel. The feature maps generated through DW convolution have the same number of channels as the input. Then, the feature maps after DW convolution are combined with PW convolution to generate new feature maps. Figure 1 illustrates the computational process of depthwise separable convolution, which divides the conventional convolution operation into two sequential steps: filtering and combining. This approach  
Figure 2: Convolutional blocks in lightweight convolutional neural networks. (**a**) Depthwise separable convolution; (**b**) inverted residual block. effectively reduces the computational burden. Comparing this to a standard convolution can be expressed as follows:  
\[ \begin{array}{l}\frac{C_{1}\cdot K_{1}\cdot K_{2}\cdot F_{1}\cdot F_{2}+C_{1} \cdot C_{2}\cdot F_{1}\cdot F_{2}}{C_{1}\cdot C_{2}\cdot K_{1}\cdot K_{2}\cdot F _{1}\cdot F_{2}}\\ =\frac{1}{C_{0}}+\frac{1}{K_{1}\cdot K_{2}}\end{array}  \tag{1}\]  
where \(C_{1}\) denotes the count of input channels, \(C_{0}\) represents the number of output channels, \(K_{1}\times K_{2}\) is the size of the convolution kernel, and \(F_{1}\times F_{2}\) is the size of the feature map. The numerator is the computational cost of depthwise separable convolution, and the denominator is the computational cost of standard convolution. It can be seen from the simplified results that the depthwise separable convolution can save a lot of computational costs compared with the traditional convolution, and it also shows a good feature extraction ability in the model of this paper. The inverted residual structure can be seen as the inverse or opposite of the residual structure, and the \(1\times 1\) convolution kernel is employed to expand the dimension. Following that, DW convolution extracts features, and ultimately, the \(1\times 1\) convolution reduces the dimension.