## 1 Introduction  
Rolling bearings are a core component of rotating mechanical systems. Bearing failure results in safety issues [1]. Therefore, the development of accurate and effective bearing fault diagnosis techniques is crucial for ensuring the operational safety of rotating machinery. In traditional methods used to extract vibration signal features, time-frequency analysis methods are commonly implemented, with examples of such methods including frequency domain statistical feature analysis [2], fast Fourier transform [3], wavelet transform [4], and time-frequency map analysis [5]. However, traditional manual extraction methods have obvious shortcomings. First of all, they require artificial feature selection, design, and extraction, all of which consume a great deal of time. Secondly, traditional feature extraction methods are based on artificial assumptions and simplified models, which may not be able to capture nonlinear relationships in complex fault modes. They also demonstrate insufficient adaptability when dealing with new types of faults. Therefore, it is necessary to reduce dependence on empirical knowledge and improve the recognition rate of complex faults [6]. With the recent developments in deep learning, bearing fault diagnosis based on deep learning has been applied extensively. This method is driven by original signal data and uses a neural network model to extract features and complete classification tasks [7]. However, in cases where the provided training dataset is unbalanced, with limited samples for certain fault labels, the performance of CNNs tends to decline. To address the issue of dataset imbalance, the preferred approach is to employ a generative adversarial network (GAN) for data generation. However, in existing research studies, GANs primarily focus on the overall similarity of the generated data to the original measurements while overlooking the similarity in fault characteristics. This oversight results in a loss of crucial information for fault diagnosis. To address this issue, [PERSON] et al. [8] proposed two modifications for the general GAN. [PERSON] et al. [9] solved the problem of gradient disappearance in the training process of deep neural networks by introducing residual joining, which further improved the classification performance. Regarding development based on convolutional neural networks, [PERSON] et al. [10] established a two-dimensional CNN model for rolling bearing fault diagnosis. [PERSON] et al. [11] added a variable dynamic weight layer to a residual neural network model to improve its diagnosis accuracy. CNNs perform well in actual diagnosis tasks, but there are still several problems: First, when the number of recognition network layers increases, the gradient disappears, which makes training more difficult. Second, the ability of CNNs to extract fault features and generalize them is weakened under multi-operating conditions and when there is complex environmental noise [12; 13]. To solve this problem, a model based on wavelet transform and a residual neural network is proposed and applied to bearing fault diagnosis under multiple working conditions. First, the original vibration signal is transformed into a two-dimensional time-frequency signal via wavelet transform, and then the obtained two-dimensional signal is used as the input for residual network feature extraction. By inputting the wavelet-transformed signal into the residual network, the multi-scale and frequency information extracted via wavelet transform can be used to combine the advantages of the residual network, thus achieving better feature learning. The main innovations of this paper include: (1) by setting the central frequency \(\omega_{0}=2\pi\) and the parameter \(\beta=1.82\) in the modified Morlet wavelet, it approximates the admissible conditions, thereby improving the characteristic issue of time-frequency graph aliasing. (2) The improved Morlet wavelet transform and neural network are combined and applied to bearing fault diagnosis.