# An Optimal Resonant Frequency Band Feature Extraction Method Based on Empirical Wavelet Transform
## 1 Introduction  
Rolling bearings are the most widely used components in mechanical systems [1]. In actual operation, the rotating machinery is mostly in complex environments such as those with high speeds, high loads, a large temperature difference and so on, which causes the probability of failure to be much higher than for other equipment [2]. With the increasing requirements of high precision, high speed and large loads of mechanical equipment, it is becoming more and more complicated in terms of structure and larger in size, accompanied by complex abnormal noise [3]. As a result, it becomes more and more difficult to identify potential faults from the machine, which makes the fault feature extraction of rotating mechanical vibration signals a huge challenge [4]. In practice, time-frequency analysis is widely used to detect sensitive features in vibration and sound signals [5]. The Empirical Mode Decomposition (EMD) method, developed by [PERSON] et al. [6], can adaptively decompose complex data sets into finite intrinsic modal functions (IMF) withoutgetting any prior knowledge. Especially in the case of intermittent signals, impulse interferences or strong noise, the method presents some shortcomings such as mode mixing effect and endpoint effect when decomposing signals [7]. To solve this problem, various improved methods have emerged and have achieved some improvement of the results in various applications [8; 9; 10]. An approach called \"empirical wavelet transform\" (EWT), which combines the adaptability of EMD with the theoretical framework of wavelet analysis, effectively solves the mode mixing phenomenon problem [11]. By adaptively dividing the Fourier spectrum of signals, EWT constructs an appropriate orthogonal wavelet filter bank to extract the AM-FM signal components with a compact support set of spectra [12]. Since each IMF has a limited bandwidth with different central frequencies, it is important to optimally segment the Fourier spectrum in this method. To solve the problem of an automatic division of the Fourier spectrum in the EWT, [PERSON] et al. proposed a new boundary segmentation method based on sparsity guidance [13]. First the characteristic frequencies of the extracted faults are calculated, and then applied to the sparsity function to find the cutoff frequencies and divide the spectrum boundaries. [PERSON] [14] constructed the trend components by analyzing the waveform characteristics of the spectrum. Then the minimum value of the trend component was used to divide the boundary, and a fast empirical wavelet method was proposed. [PERSON] [15] took the Harmonic Significance Index as the optimization object, and extracted the fault characteristics of bearing by quantitatively analyzing the sensitive frequency band. These methods solve the problem of spectrum segmentation by studying the characteristics of the Fourier spectrum. In order to achieve self-adaptability, [PERSON] applied the parameterless clustering scale-space histogram method to the EWT, which can achieve spectrum segmentation with a data-driven approach [16]. However, when the signal-to-noise ratio (SNR) of the signal is relatively low, the spectrum will be seriously disturbed, and the decomposition results will show a lot of noise. [PERSON] noted through an in-depth study of the method that when too fine of a frequency partition is set, many narrow band modes are extracted with two corresponding problems. On the one hand, several modes may show the same modulation information, resulting in unnecessary redundancy. On the other hand, too coarse of a frequency partition will mix different mono-components and may introduce artifacts [17]. To compensate for the over-decomposition of vibration modes, [PERSON] used priori knowledge to set the scale parameter as the predetermined value, aimed at solving the problem of signal distortion caused by the excessive frequency band. This approach provides a way to apply scale-space segmentation in a noisy background.