## 1 Introduction  
Rolling element bearings are extensively used in various industrial applications, such as aircrafts, marine vessels, coal mining machines, manufacturing, etc., and their failure is one of the most frequent reasons for machine breakdowns and accidents [1; 2; 3]. Consequently, the fault diagnosis of rolling element bearings is of great significance and has attracted increasing attention in recent years. Generally, a series of cyclic impacts are generated and the resonant frequencies of bearings and adjacent components are excited, triggering the modulation phenomenon in the incipient stage of bearing faults [4]. The collected vibration signal contains rich information of machines. Therefore, the vibration analysis is a preferred approach to diagnose bearing faults [5; 6; 7], and the fault characteristic frequency is often considered as the vital indicator of rolling element bearing faults [8]. However, noises inevitably exist and may inhibit the extraction of the fault feature [9]. Therefore, the extraction of the weak periodic feature, which can identify the fault type and the damage level at the same time, has gained a leading role in the fault diagnosis of rolling element bearings [10; 11]. Traditional approaches are based on the time domain or frequency domain analysis. In order to perceive and analyze the weak fault in rolling element bearings signals, state-of-the-art technologies, including the Wavelet Transform (WT) [12; 13; 14] and Empirical Mode Decomposition (EMD) [15; 16], are introduced to detect the early failure under noise interference. It is known that both WT and EMD attempt to decompose the processed signal into a series of distinct modes. By analyzing certain modes, the dominant or interesting features can be obtained and separated from other irrelevant modes. WT, including both Discrete Wavelet Transform (DWT) and Wavelet Packet Transform (WPT), is one of the most widely used methods for signal de-noising. The time and scale spectra information of the fault is always carried by its corresponding wavelet sub-band signal component. Therefore, it is able to understand the physical meaning of the fault detection process. However, the prescribed subdivision scheme and predetermined basis functions cannot adaptively process a signal based on the information contained in the signal [17; 18]. Correspondingly, EMD, proposed by [PERSON] et al. [19], is capable of decomposing a signal into a set of distinct modes adaptively. These modes represent natural oscillatory modes imbedded in the signal and behave as basis functions derived from the signal itself. However, the lack of mathematical theory often leads to mode mixing or over-estimation phenomenon [17; 20]. In 2013, by explicitly building an adaptive wavelet filter bank to segment the spectrum adaptively, [PERSON] [17] proposed a novel signal decomposition method and named it Empirical Wavelet Transform (EWT). Combining advantages of both WT and EMD, the EWT can construct the wavelet basis in an adaptive way and decompose a signal according to the contained information. Therefore, EWT has attracted much attention and been used in a variety of applications, including medicine, biology [21; 22], and machinery [23; 24; 25; 26; 27; 28; 29; 30; 31; 32; 33]. The traditional spectrum segmentation is achieved by detecting local maxima or minima of the spectrum. However, the amount of pseudo-maxima and pseudo-minima caused by the strong background noise always leads to an improper segmentation of the spectrum. In order to solve this problem, some modifications about the spectrum segmentation have been proposed [34; 35]. [PERSON] and [PERSON] [34] presented an adaptive multiple signal classification-empirical wavelet transform (MUSIC-EWT) methodology. Different from the traditional EWT, the MUSIC algorithm is used to estimate the frequencies contained in the signal and to build appropriate boundaries in the MUSIC-EWT. Comparatively, to ensure each mono-component is modulated by the fault characteristic frequency, [PERSON] et al.