## 3 Simulation Analysis  
In this section, a simulated bearing fault signal is analyzed using the proposed method, to prove its effectiveness. Through the analysis of the failure mechanism, the bearing fault signal can be constructed as  
\[y(t)=\sum_{K}\left[Ae^{\frac{-\zeta}{\sqrt{1}-\zeta^{2}}2\pi f/(t-\tau-KT)} \times 2\pi f(t-\tau-KT)\right], \tag{31}\]  
where \(A\) denotes amplitude, \(\xi\) denotes damping ratio, \(f\) denotes frequency, \(\tau\) denotes time index, \(K\) denotes the number of impulses and \(T\) denotes cyclic period [27]. The noisy signal is created by adding Gaussian noise to the simulated signal. We used the discrete Daubechies 5 wavelet, which is an orthogonal wavelet transform, with three decomposition levels, and detail coefficients at each level were used for the estimation of parameters \(k_{1}\), \(k_{2}\), \(\sigma_{1}\) and \(\sigma_{2}\) utilizing the EM algorithm. The corresponding results are shown in Figure 2.  
The waveform of the simulated bearing fault signal in the time domain is shown in Figure 2a, where amplitude \(A=1.5\), frequency \(f=1500\) Hz, damping ratio \(\xi=0.080\), cyclic period \(T=0.007\) s and the corresponding theoretical fault feature frequency was \(f_{s}=135.1\) Hz. The length of this signal was 51,200 and the sampling frequency was set at 51.2 kHz. The waveform of the noisy signal in the time domain is shown in Figure 2b. The standard deviation of the additive Gaussian white noise was \(\sigma_{n}=2.0\).  
The signal-to-noise ratio (SNR) is introduced to describe the influence of noisy components, which is defined as  
\[\text{SNR}=10\log_{10}\frac{\|x\|_{2}^{2}}{\|n\|_{2}^{2}}. \tag{32}\]  
The SNR of the simulated signal was \(-16.082\) dB. The envelope spectrum analysis of the noisy bearing fault signal is shown in Figure 2c. It is obvious that the fault feature frequency cannot be obtained directly. The time domain waveform of the estimated signal is shown in Figure 2d. The SNR  
Figure 1: Flowchart of the proposed method. MAD; median absolute deviation.  
of the estimated signal was \(-4.526\) dB. We can see that the SNR improved significantly. The root mean square error (RMSE) was also used to evaluate our method. The RMSE is defined as  
\[\text{RMSE}=\sqrt{\frac{1}{N}\|x-\hat{x}\|_{2}^{2}}, \tag{33}\]  
where \(x\) denotes the noise-free simulated signal, \(\hat{x}\) denotes the estimated signal and \(N\) denotes the length of the signal. The RMSE of the estimated signal is \(0.5328\). The noise level is reduced successfully.  
The envelope spectrum analysis of the estimated bearing fault signal is shown in Figure 2e. Through the envelope analysis of the estimated signal, the fault feature frequency was \(f_{s}=135.1\) Hz, which is equal to the simulated value (135.1 Hz). The comparison of Figure 2c,e shows that the proposed method can effectively extract the fault feature frequency from the noisy signal, even if the transient component is drowned by heavy noise.  
The verification of the distribution of transient components and noisy bearing fault signals is shown in Figure 3. The distribution of transient components data is shown in Figure 3a. It is shown that the pdf of noise-free transient component was a mixture of two Laplace pdfs, where the estimated parameters were \(k_{1}=0.416\), \(k_{2}=0.584\), \(\sigma_{1}=0.488\) and \(\sigma_{2}=0.049\). The distribution of noisy data  
Figure 2: Results of simulated signal.  
value is shown in Figure 2(b). It can be found that the pdf of noisy value was a mixture of two LapGauss pdfs, where the estimated parameters \(k_{1}=0.250\), \(k_{2}=0.750\), \(\sigma_{1}=0.447\) and \(\sigma_{2}=0.186\).