# Legendre Multiwavelet Transform and Its Application in Bearing Fault Detection
## 1 Introduction  
In industrial environments, rotating machinery has special importance due to its wide applications. Rolling bearings constitute one of the most widely used yet vulnerable components in rotating machinery, and their health deteriorates gradually over time. Bearing faults can lead to sudden system failure, resulting in incalculable economic losses and posing personal safety hazards [1, 2]. Moreover, owing to the complexity of equipment and the interrelation of its structure, compound faults often occur simultaneously, coupling the features of individual faults to form what are known as compound faults [3, 4]. To mitigate the issues arising from bearing faults such as equipment downtime and maintenance costs and to enhance equipment operating efficiency and safety, fault detection holds paramount importance. Developing a method with a simple model design, robust noise resistance (especially under strong noise interference), and high fault recognition accuracy bears engineering significance [5, 6]. Time-frequency analysis methods are widely employed in processing bearing signals due to their ability to provide signal analysis across both time and frequency domains. This effectively enhances the accuracy and reliability of fault diagnosis [7, 8, 9, 10, 11, 12, 13, 14, 15]. A common approach to fault diagnosis involves extracting fault features using time-frequency analysis methods, followed by the utilization of various classifiers. For instance, [PERSON] et al. [7] combined wavelet packet decomposition with maximum impact frequency band-based feature extraction technology to propose a new time-frequency method for bearing fault diagnosis. [PERSON] et al. [8] addressed the non-stationary and non-Gaussian issues in bearing signals by combining empirical wavelet transform with fuzzy correlation classification for fault diagnosis. [PERSON] et al. [9] proposed an approach that combines multiple classifiers, including the nearest mean classifier, nearest neighbor classifier, and Gaussian mixturemodel, to analyze and diagnose rolling bearing faults. [PERSON] et al. [10] addressed the complexities associated with weak and noise-prone compound fault features in rolling bearings. Their method involved utilizing adaptive local iterative filtering decomposition and the Teager-Kaiser energy operator to effectively extract diverse frequency components from vibration signals related to bearing faults, thereby enhancing the diagnosis of rolling bearing issues. Despite their ability to extract various frequency components from bearing fault vibration signals, time-frequency analysis methods often encounter challenges in adaptively extracting signal features across different environmental and operational conditions. Lately, researchers have increasingly adopted a trend of combining wavelet analysis and deep neural networks for fault diagnosis [16; 17; 18; 19; 20; 21; 22]. [PERSON] et al. [16] proposed a bearing fault diagnosis method that combines the advantages of dual-tree complex wavelet packets and deep belief networks. [PERSON] et al. [17] combined fast empirical wavelet transform (FEWT) with negative entropy spectrum decomposition (NSD) to construct an information graph. FEWT cyclically extracted vibration signals, obtaining envelope spectra for each component to diagnose compound bearing faults. [PERSON] et al. [18] applied conventional convolutional neural networks to perform multi-label classification on vibration signals that underwent wavelet analysis transformation. Their approach aimed at enabling compound fault diagnosis specifically for gearboxes. Experimental findings suggest that amalgamating traditional time-frequency analysis methods with deep learning can yield enhanced diagnostic accuracy and stability. Traditional wavelet analysis is simple and easy to implement, and it can analyze signals locally and extract features of different frequencies [22; 23; 24; 25; 26].