## 2 Materials and Methods
### Multi-Scale Weight Fusion  
This paper employs an adaptive weight fusion method to perform feature-weighted fusion of acoustic radiation signals at different scales, taking into signal feature information. A lightweight channel attention mechanism called CBAM (convolutional block attention module) [33] is utilized for this purpose, with its specific structure illustrated in Figure 1.  
After processing the acoustic radiation signal using MFAVGH, we obtain multi-channel fault characteristic information. Considering that each channel contains distinct bearing fault characteristics, it is essential to assign a higher weight to channels with abundant bearing fault characteristic information and a lower weight to channels with weaker fault characteristic information. The processes of channel attention mechanism and spatial attention mechanism are illustrated in Figure 2a,b respectively. Global average pooling and global maximum pooling operations are respectively used to perform feature compression processing on the input signals; \(Y^{\lambda}_{\text{avg}}\) and \(Y^{\lambda}_{\text{max}}\) are obtained as follows:  
\[Y^{\lambda}_{\text{avg}}=\frac{1}{N}\sum_{n=1}^{N}EAVGH_{\lambda}(n) \tag{14}\]  
\[Y^{\lambda}_{\text{max}}=\max_{1\leq n\leq N}EAVGH_{\lambda}(n) \tag{15}\]  
where \(EAVGH_{\lambda}\) (1 \(\leq\lambda\leq\) M) represents the total number of input M structural element scales and N represents the length of input signal. The generated two feature maps are fed  
Figure 1: CBAM channel attention mechanism. Figure 2: Input signal is weighted: (**a**) channel attention mechanism, (**b**) spatial attention mechanism. into a double-layer neural network (MLP) with shared weights to learn the dependency relationship between channels; the dimensionality between the two neural layers is reduced by compression ratio. Finally, the MLP output features are carried out via sum-based operations, and sigmoid activation operations are performed to generate the final channel weight \(\omega_{\text{m}}\), the expression of which is defined as follows:  
\[ \begin{split}\omega_{\lambda}(EAVGH_{\lambda})=\sigma(MLP(AvgPool( EAVGH_{\lambda}))+MLP(MaxPool(EAVGH_{\lambda})))\\ =\sigma(W_{1}(W_{0}(Y_{\text{avg}}^{\lambda}))+W_{1}(W_{0}(Y_{ \text{max}}^{\lambda})))\end{split}  \tag{16}\]  
where \(W_{1}\) and \(W_{2}\) respectively represent the neural nodes of the two fully connected layers, \(\sigma(\bullet)\) represents the sigmoid function, and the final output signal \(\hat{Y}_{\lambda}\) is:  
\[\hat{Y}_{\lambda}=\omega_{\lambda}Y_{\lambda} \tag{17}\]  
After passing through the channel attention mechanism, the spatial attention mechanism takes \(\hat{Y}_{\lambda}\) as the input of this module. Global average pooling and global maximum pooling operations are then continuously used to obtain \(\hat{Y}_{\text{avg}}^{n}\) and \(\hat{Y}_{\text{max}}^{n}\):  
\[\hat{Y}_{\text{avg}}^{n}=\frac{1}{M}\sum_{\lambda=1}^{M}\hat{Y}_{n}(\lambda) \tag{18}\]  
\[\hat{Y}_{\text{max}}^{n}=\max_{1\leq\lambda\leq M}\hat{Y}_{n}(\lambda) \tag{19}\]  
The two generated feature graphs are concatenated based on the channel dimensions, followed by dimensionality reduction using convolutional check channels to obtain a single-channel feature graph. Finally, the dependency between spatial and channel dimensions is learned through a sigmoid function, generating the weight of the spatial dimension \(\omega_{n}\), defined as follows:  
\[\omega_{n}(\hat{Y}_{n})=\sigma(f([AvgPool(\hat{Y}_{n});MaxPool(\hat{Y}_{n})]))= \sigma(f([\hat{Y}_{\text{avg}}^{n};\hat{Y}_{\text{max}}^{n}])) \tag{20}\]  
where \(\sigma(\bullet)\) represents the sigmoid function, \(f(\bullet)\) represents the convolution operation, \([j]\) represents the concat operation, and the final output signal \(\hat{Y}\) is:  
\[\widetilde{Y}=\omega_{\text{n}}\hat{Y}_{\lambda} \tag{21}\]  
The weighted fusion mathematical morphological layer, as depicted in Figure 3, is utilized to process the acoustic array signal.