## 3 Transfer Learning Applied to Fault Diagnosis Methods
### Signal Processing Method  
Some scholars use high-dimensional signals, similar to images, for fault diagnosis, converting one-dimensional vibration signals into high-dimensional representations. However, this process may result in the loss of signal features and increase the complexity of both the data and the model [20]. Vibration signals contain rich health information and are widely used for bearing fault diagnosis. Commonly used signals for fault diagnosis include time-domain, frequency-domain, and time-frequency domain signals, with this paper focusing on time-domain signals. In this paper, we propose an RB signal processing method that combines random convolution layers with blind deconvolution. The Random Convolution Layer (RCL) is an auxiliary method designed to enhance the generalization performance of fault diagnosis models [21]. The RCL module is integrated into the input of the fault diagnosis model as a supplementary component. During training, the randomly generated data are mixed with the original data and fed into the model. By increasing sample diversity, the model learns a broader range of features, enhancing its generalization ability in the target domain. During testing, the RCL module is not involved, and only the trained model is used for classification prediction. The RCL improves the model's generalization by introducing randomness. This method randomly alters the size of the convolution kernel during the operation while maintaining the data dimensions. The RCL consists of convolution weights \(\sigma\). \[\sigma\in R^{h\times C_{in}\times C_{out}} \tag{5}\]  
Here, \(C_{in}\) and \(C_{out}\) represent the number of input and output feature channels, respectively, and \(h\) is the height of the convolution weights. The input data are denoted as \(I_{in}\). \[\mathrm{I_{in}}\in\mathrm{R^{H\times C_{in}}} \tag{6}\]  
Here, H is the height of the input. The output expression of the RCL is as follows:  
\[\mathrm{I_{out}}=\mathrm{I_{in}}\times\sigma \tag{7}\]  
\[\sigma\backsim\eta(0,\frac{\mathrm{z}}{\mathrm{h}\times\mathrm{C_{in}}\times \mathrm{C_{out}}}) \tag{8}\]  
Here, \(\mathrm{I_{out}}\in\mathrm{R^{H\times C_{out}}}\), with \(\mathrm{C_{in}}=\mathrm{C_{out}}\). z is a constant; z can take the values 1, 2, 3, and k is the size of the kernel, so  
\[\sigma\backsim\eta(0,\frac{\mathrm{z}}{\mathrm{k}}) \tag{9}\]  
During the training phase, a single threshold \(\mathrm{m}\in(0,3)\) was set, and a value \(\mathrm{m}_{0}\in(0,3)\) was randomly sampled in each epoch. If \(\mathrm{m}_{0}>\mathrm{m}\), \(\mathrm{I_{out}}\) after the RCL operation was used as input to the model; if \(\mathrm{m}_{0}<\mathrm{m}\), \(\mathrm{I_{in}}\) was used directly as input to the model. In this paper, \(\mathrm{m}=2\), and with a probability of 1/3 at each step, \(\mathrm{I_{out}}\) is used as the input to the model. This paper focuses on bearing fault diagnosis using vibration signals. Blind deconvolution (BD) technology is effective for processing vibration signals collected from sensors, which contain rich fault information. However, fault features may be masked or distorted due to noise and other factors affecting the signal during transmission. Blind deconvolution helps remove interference signals, extract relevant fault information from complex signals, and enhance the detectability of fault features. The core of the blind deconvolution problem is to design an inverse filter g to extract the fault-related periodic component \(\xi_{1}\) from the fault signal x. The theoretical formula for applying blind deconvolution in bearing fault diagnosis is  
\[\xi=\mathrm{x}*\mathrm{g}\approx\xi_{1} \tag{10}\]  
Here, \(\xi\) is the estimated value of \(\xi_{1}\), and x represents the fault signal. The symbol \(*\) denotes the convolution operation; g represents the function of the filter.