# Feature Extraction of Bearing Weak Fault Based on Sparse Coding Theory and Adaptive EWT
## 1 Introduction  
Rolling bearings are crucial but easily damaged components in rotating machinery. Statistics show that more than 30% of mechanical failures are related to bearings in rotating machinery, and the operating status of bearings directly determines whether the equipment can run normally. At present, the most effective and commonly used method of bearing fault diagnosis is vibration analysis. However, in practice, the low SNR and non-stationary nature of early fault vibration signals mean that it is hard to extract fault characteristic information from noise interference signals. How to analyze and process vibration data to obtain fault characteristic information is the main difficulty [1]. Therefore, it is necessary to develop effective signal-processing methods to achieve accurate fault detection. The sparse theory was proposed by [PERSON] and [PERSON] [2], and has been continuously developed and deepened to the point where it is a subject of intense study in image and signal processing [3; 4; 5; 6; 7], and compressed sensing [8]. The sparse representation of signals aims to find the sparsest expression in the redundant atomic dictionary, using the least number of atoms to represent the signal to capture and express the essence of the signal. [PERSON] et al. [9] proposed a method to construct different redundant dictionaries according to the different impact morphological characteristics of gear and bearing faults. The method realized the separation of composite faults from gears and bearings utilizing sparse representation. [PERSON] et al. [10] developed an adaptive matching pursuit algorithm to denoise the original signal and then utilized the fast spectrum correlation method to identify the fault category. A framework of sparse representation classification for fault diagnosis in planetary bearings has also been developed [11]. [PERSON] et al. [12] adopted the improved parametric of empirical wavelet transforms (EWT) and adaptive sparse coding shrinkage de-noising to carry out adaptive sparse de-noising for the signal of a fault bearing, which is beneficial in increasing the strong pulse features of the fault. [PERSON] et al. [13] constructed an overcomplete atomic library using the shock function of the second-order bearing damping that was able to identify the natural frequency through the correlation filtering method, and reduce the redundancy of the dictionary. The bearing fault signals were matched by the matching pursuit method improving the efficiency and accuracy of signal reconstruction under low SNR. [PERSON] et al. [14] utilized SISC to achieve feature extraction of early weak faults, verified by vibration signals of bearings and gears and achieved good results. [PERSON] et al. [15] proposed an algorithm based on the FSS and attenuated cosine dictionary to diagnose bearing weak faults, and compared their method with methods such as BPDN and WPT. EWT was first proposed by [PERSON] [16]. Its main idea is to extract the closely supported amplitude and frequency modulation components. Adaptive signal decomposition is then achieved by dividing the Fourier spectrum and constructing a series of orthogonal wavelet filter banks. [PERSON] et al. [17] combined EWT and an improved threshold function method to reduce the noise of a vibration signal, which was helpful in detecting bearing early faults by signal reconstruction. [PERSON] et al. [1] denoised the vibration signal of a wind turbine generator bearing using a wavelet spatial adjacency coefficient of a data-driven threshold. They then utilized EWT to diagnose weak and composite faults of the bearing. [PERSON] et al. [18] presented a combination algorithm based on the Pearson correlation coefficient to correct the over-decomposing tendency of EWT. [PERSON] et al. [19] improved EWT and then applied their improved EWT to fault diagnosis of a train-wheel hub bearing. [PERSON] et al.