## 5 Conclusions  
To solve the problem of the original Morlet wavelet's inability to satisfy the admissible condition, which leads to the fuzzy and aliasing characteristics of time-frequency images, the implementation of a series of improvement measures is detailed in this paper. Firstly, the Morlet wavelet is improved such that its central frequency \(\omega_{0}=2\pi\) might satisfy the admissibility condition. By comparing the time-frequency characteristics of different versions of the Morlet wavelet, it becomes evident that the improved Morlet wavelet can effectively alleviate the feature ambiguity and aliasing problem. This shows that the improved Morlet wavelet possesses better expression and resolution abilities in extracting vibration signals' time-frequency features. Secondly, to further enhance feature learning, a fusion method of a Morlet wavelet and a residual network is proposed in this paper. As a deep learning model, the residual network can improve model performance by learning residual information. The improved Morlet wavelet is input to combine with the residual neural network, which can make full use of their advantages and improve the classification ability of vibration signals. In addition, the number of training samples is compared experimentally, and 600 samples for each class are selected as the number of training samples. By training different numbers of training samples, the relationship between sample size and classification performance can be explored. After the comparative analysis, 600 samples are selected as the number of training samples for each category to reduce training time and improve efficiency while fully learning features. The results show that the proposed method achieves the optimal outcome of a deep network after about 100 iterations, and the accuracy rate is 5.32% higher after 200 iterations. The convergence speed is faster, and the accuracy rate also achieves the learning effect of a deep network.  
In order to further illustrate this method's reliability, its performance is compared with three network structures under different signal-to-noise ratios regarding noise interferences. The results show that this method has better anti-noise performance than the network structures.  
Conceptualization, B.H. and Q.H.; methodology, B.H.; software, B.H. and Q.H.; validation, B.H.; formal analysis, B.H.; investigation, B.H. and Q.H.; resources, B.H., L.G. and Q.H.; data curation, B.H. and Q.H.; writing--original draft preparation, B.H.; writing--review and  
Figure 17: Comparison of results under different noise conditions for the XJTU-SY dataset.  
editing, B.H., L.G. and Q.H.; visualization, B.H. and Q.H.; supervision, L.G.; project administration, B.H. and L.G.; funding acquisition, L.G. All authors have read and agreed to the published version of the manuscript.
* [1]**Funding:** This work is supported by the National Natural Science Foundation of China (grant number 52077155). The authors acknowledge the anonymous reviewers for their helpful comments on the manuscript.
* [2]**Institutional Review Board Statement:** Not applicable.
* [3]**Informed Consent Statement:** Not applicable.
* [4]**Data Availability Statement:** The source code presented in this study is available upon request from the corresponding author.  
The authors would like to thank Tangong University for technical support and all members of our team for their contribution to the experiments.  
The authors declare no conflicts of interest.