## 1 Introduction  
Rotating machinery is a type of equipment that is widely used in various industries, the safety and reliable operation of which have become increasingly important [1]. As a significant component of rotating machinery, rolling bearings often operate under very harsh environments, such as high temperatures, high speeds and heavy loads, and are thus prone to failure [2]. Therefore, it is of great significance to make timely and accurate diagnoses of rolling bearing failures [3]. In most cases, rolling bearing faults are often compound because of the mutual contact between various components, and multiple fault features are superimposed and interfere with each other, thereby complicating the fault features and creating difficulties in bearing fault diagnosis [4; 5]. Bearing fault diagnosis is often based upon vibration signals collected by acceleration sensors, but the sensor positions and numbers are limited in some special cases, e.g., nuclear power equipment monitoring [6]. When acoustic sensors can be positioned in a non-intrusive and non-contacting way, the use of a small number of microphone sensors to collect device status signals is an appropriate option for such special cases. However, acoustic signals are easily interfered with environmental noise, which makes it difficult to diagnose compound weak fault features [7]. In recent decades, researchers have proposed a number of methods for weak bearing fault diagnosis, a popular one among which is the adaptive signal decomposition method [8]. Empirical wavelet transform (EWT) is a commonly used method that mainly solves already-existing problems in the widely-used empirical mode decomposition (EMD) method, e.g., mode aliasing, endpoint effects, and unclear physical meaning [9; 10]. EWT first segments the Fourier spectrum of a signal, and then constructs a series of empirical wavelet-based filter banks to extract the intrinsic mode of the original signal. EWT has been extensively used in many fields, including bearing fault diagnosis [11; 12; 13]. [PERSON] [14] applied EWT to mechanical fault diagnosis. [PERSON] [15] proposed a bearing feature extraction and classification method based on EWT and compared it with ensemble empirical mode decomposition (EEMD), whose findings indicated that EWT has more adaptive feature extraction and fault classification characteristics than EEMD. [PERSON] [16] made a fault diagnosis of a generator bearing for a wind turbine by using EWT with vibration signals. [PERSON] [17] applied EWT to the bearing fault diagnosis by combining EWT and operational modal analysis (OMA), which improved the detection accuracy of fault diagnosis. [PERSON] [18] carried out research on bearing fault diagnosis based on order analysis and EWT, revealing that the combination of order analysis and EWT can be used to accurately identify bearing fault features under non-steady working conditions. However, EWT still has some problems when dealing with compound weak bearing faults that must be solved [19]. If the input signal is composed of two chirps that overlap in both time and frequency domains, EWT will not be able to separate them. Additionally, the adaptive detection of the number of signal modes is also a problem that must be further studied [9]. EWT boundary detection is based upon the Fourier spectrum, and because the characteristics of weak faults are not obvious, and the noise of the acoustic signal is relatively high, it is difficult to effectively identify the fault impact. In addition, due to the mutual influence of bearing faults, multiple fault features are superimposed and interfere with each other, thereby complicating the fault features and bringing difficulties to bearing fault diagnosis [20]. The bearing signal also exhibits a modulation phenomenon, and thus the bearing fault frequency cannot be accurately detected in the Fourier spectrum, especially in the case of compound faults [21]. Hence, boundary detection is not well performed in the Fourier spectrum.