# An Improved Empirical Wavelet Transform and Its Applications in Rolling Bearing Fault Diagnosis
## 1 Introduction  
Mechanical automation occupies a dominant position in modern industrial systems. Rotary machinery and bearings are indispensable to high-speed rail, intelligent robots, manipulators, and large-scale equipment production lines. In many cases, bearings may be subjected to high speed and a heavy load and may also face complicated conditions such as high and low-temperature differences. Even if the bearing structure is very delicate, it is difficult to tolerate a too-complex working environment for a long time. Sometimes minor damage can spread undetected, causing damage to the bearing and thus affecting the operation of the equipment. Healthy bearings are the basis for the normal operation of the equipment. Bearings that are prone to damage and breakdown are also one of the greatest hidden dangers that lead to equipment failure [1]. It can be seen that it is necessary to monitor and diagnose bearing faults. With the popularization of mechanical automation and industrial intelligence, fewer people participate in the operation and management of equipment [2]. Real-time state-monitoring systems can compensate for the safety problems caused by manpower shortages. By monitoring the running state of mechanical equipment and collecting vibration acceleration signals of vulnerable parts, abnormalequipment can be found in time. When rotating machinery fails, the position deformation of the fault will lead to repeated collisions and periodic impact [3]. Extracting the components containing the impact information in the signal can improve the speed and accuracy of the diagnosis. Many scholars have studied methods of diagnosing vibration acceleration signals, such as short-time Fourier transform (STFT), Wigner-Ville distribution, and wavelet transform (WT) [4; 5]. The window function of STFT cannot be adaptively adjusted according to the characteristic frequency of the signal itself, Wigner-Ville distribution generates cross-terms when processing signals with multiple frequency components and wavelet analysis (including continuous wavelet transform, discrete wavelet transform, dual-tree complex wavelet transform, etc.) [4] requires manual selection of wavelet basis functions and lacks self-adaptability. Empirical mode decomposition (EMD), proposed by [PERSON] [6], can adaptively decompose a signal into a finite number of intrinsic mode functions (IMFs). These contain local features of different time scales in the original signal. However, the problems of modal aliasing, endpoint effect, and over envelope seriously affect the further development of EMD [7; 8]. In order to avoid modal aliasing and weaken the endpoint effect to a certain extent, [PERSON] [9] proposed extracting modal information from the spectrum via empirical wavelet transform (EWT). The method is adaptive and reliable in the mathematical derivation. First, the Fourier spectrum is divided adaptively. Then the wavelet filter banks are set up to extract multiple single components with tightly supported Fourier spectra. Finally, the Hilbert transform is used to get instantaneous frequency and amplitude for each component. With these advantages, the EWT method was used to extract fault information from rolling bearings [10; 11] and wind turbines in [12]. Real-time monitoring of power quality has also been widely promoted [13]. By comparing EMD, Ensemble EMD, EWT, and other methods, [PERSON] [14] and others came to the conclusion that the EWT method can effectively reveal the frequency and associated harmonics of faulty bearings. Some new problems were discovered in the process of using EWT to extract faults. For example, the setting of parameters is usually difficult and the calculation is time-consuming [15], the unreasonable division of the spectrum will lead to unsatisfactory noise reduction [16] or decomposition of too many invalid components [17], and so on. After consulting the literature, authors have divided the methods of optimizing EWT into three categories. The first category is processing the empirical modes (EMs) obtained.