# MTC-GAN Bearing Fault Diagnosis for Small Samples and Variable Operating Conditions
## 2 Theoretical Background
### Construction of Vibration Images
Based on the above two conditions, it is determined that the sample length \(L_{s}\) should satisfy the following equation:  
\[L_{s}=\max\bigl{(}{L_{s}}^{\prime},L_{s}^{\prime\prime}\bigr{)} \tag{7}\]  
After determining the sample length \(L_{s}\), it is necessary to further define the image height \(H\) and width \(W\). Traditional methods typically reshape one-dimensional acceleration data into a \(W\times W\) square matrix. Although this method is simple, it overlooks the physical characteristics inherent in the data, which may result in significant texture changes in the images under different rotational speeds. As indicated in the literature [15], selecting a higher aspect ratio (i.e., rectangular images) can mitigate the impact of speed variations on texture features. The key to this approach is that rectangular images can present the vibration signals more consistently across different rotational speeds. By capturing signal segments over a longer time axis, rectangular images ensure that texture features exhibit minimal variation under different speeds, thereby maintaining texture stability. This consistency in texture allows for accurate fault classification by the model, even under varying operating conditions. Additionally, rectangular images not only preserve the shape and characteristic information of the original signal but also better consolidate critical fault features, preventing data splitting at key feature points and enhancing the effectiveness of feature extraction. For the different fault eigenfrequencies, the common frequency term is the rotation frequency of the shaft,\(f_{r}\). The period of the envelope spectra of the inner ring and rolling element faults is determined by \(1/f_{r}\), and there are sidebands related to \(f_{r}\) in their envelope spectra. While outer ring faults and cage faults are represented differently in the envelope spectrum, their eigenfrequencies are still related to \(f_{r}\). All the characteristic information will be reflected in the vibration signal within one shaft rotation cycle. Therefore, the width of the 2D image sample is determined by the number of measurement points obtained during one shaft rotation, which is calculated as follows:  
\[W=\frac{L_{s}}{f_{r}} \tag{8}\]The sample height is then determined based on the sample width, which is calculated using the following formula:  
\[H=\frac{L_{s}}{W} \tag{9}\]  
The above image construction method not only maintains the consistency of texture features under different rotational speed conditions, thereby ensuring the accuracy of the classification model under varying operating conditions, but also effectively preserves the physical characteristics of the original signal. This is achieved by reasonably designing the sample length and the aspect ratio of the image, preventing data splitting at critical feature points, better concentrating key fault features, enhancing the ability to extract critical fault characteristics, and ultimately optimizing the efficiency and effectiveness of model training.