## 3 Diagnosis and Verification Analysis of Fault Signal Based on Acoustic Testing of MWA
The EMD method can decompose the complicated signal into a set of complete and almost orthogonal component intrinsic mode functions (IMFs). The EMD has numerous advantages, such as the automatic production of basic function, adaptive filter, adaptive multi-resolution, etc., which is suitable for processing nonlinear, non-steady signals and has been widely applied in the fault diagnosis of rolling bearings recently [27]. Therefore, this paper introduces the ensemble empirical mode decomposition method (EEMD), which is an improved EMD method [28], to analyze the bearing  
Figure 7: Acoustic and vibrational experiments of an MWA. fault acoustic signal. EEMD defines the true IMF components as the mean of an ensemble of trials, each of which consists of the signal plus a white noise of finite amplitude, and the algorithm is defined as follows:  
Step (1): Adding random Gaussian white noise series \(n_{m}(t)\) to the target signal \(x(t)\),  
\[x_{m}(t)=x(t)+kn_{m}(t) \tag{5}\]  
here, \(k\) is the amplitude coefficient of the inserted white noise;  
Step (2): The signal with the white noise is decomposed into a set of IMFs by EMD;  
Step (1) and step (2) are repeated, and different white noise series are added every time. Calculating the mean (ensemble) of the IMFs that have been obtained as the final result:  
\[\overline{c}_{i}=\sum_{m=1}^{N}c_{i,m}/N \tag{6}\]  
where \(N\) is the ensemble times of EMD, and \(c_{i,m}\) is the \(i\)th IMF caused by the \(m\)th EMD. This EEMD method for diagnosing bearing approach can be summarized as follows [21]:  
1. A set of IMF components is obtained from the test vibration signal from the EEMD. 2. The kurtosis of each IMF component is calculated, and all the IMFs which kurtosis values are greater than 3 are selected to reconstruct the signal. 3. The envelope signal is obtained for the reconstructed signal by Hilbert transform. 4. The envelope spectrum is calculated from the envelope signal by Fourier transform. 5. The envelope spectrum is analyzed to estimate the fault type of the rolling bearing based on the fault characteristic frequencies. In the light of the above approach, the envelope spectrum of the two MWAs can be achieved. Figures 8 and 9 show the time domain waveform and the corresponding envelope spectrum of the acoustic signal obtained from the normal MWA and the MWA with a wear retainer separately. It can be seen from Figures 8a and 9a that the time domain waveform of the normal MWA is quite similar to white Gaussian noise, while the MWA with a wear retainer displays an obvious fluctuation which indicates an unsteady state. According to formulas (1)-(4), the characteristic frequencies of a bearing with an outer ring's rotating speed of 3000 r/min produced by a wear retainer, a spitted ball, an outer ring, and an inner ring can be calculated as =30 Hz, =125 Hz, =244 Hz, =356 Hz, respectively. As can be seen from Figures 8b and 9b, no characteristic frequencies can be found out in the envelope spectrum of the normal MWA, but a characteristic frequency of a wear retainer (30.2 Hz) and its multiple frequencies (60 Hz, 90.2 Hz) can be identified that are consistent with the predicted frequencies (30 Hz, 60 Hz, and 90 Hz), which indicates a wear fault of a retainer in the bearings. The above studies demonstrate that the fault information is in good attachment with the acoustic signal which can be utilized to monitor the state of an MWA. Figure 8: The signal of a normal MWA: (**a**) time domain waveform; (**b**) envelope spectrum. Figure 6:0 The signal of the MWA with a cage fault: (**a**) time domain waveform; (**b**) envelope spectrum. Health Monitoring of the Rolling Bearings in MWAs Based on the Clustering Fusion of Normal Acoustic Parameters  
In the course of long-time operation of MWAs in the lab, the health status of MWAs was monitored in order to predict the probability of failure, which might provide instructions for fault diagnosis and precaution avoiding a severe failure. The study could guarantee the spacecraft operation performance and its lifetime.