# Feature Extraction of Bearing Weak Fault Based on Sparse Coding Theory and Adaptive EWT
## 5 Simulation and Experimental Verification  
### Simulation Verification  
The simulation signal \(y\left(t\right)\) is constructed according to the characteristics of the internal ring of the rolling element and the outer ring of the rolling bearings when the peeling failure occurs. Its expression is [23; 24]:  
\[\text{y}(t)=\sum_{i}^{M}A_{i}s(t-iT-\tau_{i})+n(t) \tag{14}\]  
\[\left\{ \begin{aligned} A_{i}&=A_{0}\cos(2\pi Qt+ \varphi_{A})+C_{A}\\ s(t)&=e^{-Bl}\cos(2\pi ft+\varphi_{w})\end{aligned} \right. \tag{15}\]  
where \(Ai\) is the modulation amplitude, whose period is \(1/Q\), \(Q\) is the rotation period (frequency \(f\)\(=\)\(1/Q\)), \(C_{A}\) is the amplitude random constant (\(C_{A}\)\(>\)\(A_{0}\)), \(s(t)\) is the impact caused by pitting failure, \(T\) is the impact period, \(B\) is the attenuation coefficient, \(f\) is the resonance frequency of the system, the random variable \(\tau_{i}\) is the small sliding of the ith impact relative to the average period \(T\), and \(n(t)\) represents the random noise. The parameters of the outer ring fault simulation signal \(y_{o}(t)\) are set as: amplitude \(A_{o}\) = 2, sampling frequency \(f_{\text{s}}\) = 25,600 Hz, outer ring fault resonance frequency \(f\) = 2000 Hz, attenuation coefficient 1000, fault characteristic frequency \(f_{p}\) = 54 Hz. The SNR of the added white Gaussian noise was \(-17\) dB. The simulation fault signal with noise is shown in Figure 3.  
Figure 3: Simulated fault signal with noise. The envelope spectrum of the simulated signal with noise is shown in Figure 4. The frequency component of the signal is very complex, and the fault characteristic frequency is submerged in the noise, making it difficult to distinguish. FFT is performed on the simulation signal to obtain the Fourier spectrum, and then EWT is performed based on the scale space method to obtain the spectrum segmentation, as shown in Figure 5. After decomposition, 30 empirical mode components are obtained in total, and their corresponding kurtosis values of the square envelope spectrum and cross-correlation coefficients with the original signal are calculated, respectively. Here, the KSES value is normalized for comparative analysis with the CN value, and the results are shown in Figure 6.  
From the KSES and CN values, we can conclude that the fifth component is the one we need. From Figure 5 it can be seen that the frequency band of this component is positioned at 2000 Hz, which is consistent with the preset. At the same time, the frequency band of this component is not divided, and there is no need to merge other components. To save calculation time and improve efficiency, the parameter range of correlation filtering is set here according to the decomposition result of EWT. The base atom also uses the Laplace wavelet, as does the subsequent verification. Its parameters are set as resonance frequency \(F=\{1000:20:3000\}\), damping ratio \(Z=\{0.01:0.005:0.2\}\), \(T=\{0:1/f_{s}:0.2\}\), and support interval \(w_{s}=470\). The result of the correlation filtering is shown in Figure 7. Looking at the correlation coefficient \(\kappa_{r}\) between the simulated signal of the fault and the base atom, the higher peak point does not appear periodically, and it is not easy to judge whether it is the peak point corresponding to the fault shock signal. In  
Figure 4: Envelope spectrum of the simulated outer ring fault signal with SNR \(=-17\) dB. Figure 5: Adaptive spectrum segmentation. Figure 6: CN and KSES values of each component. this paper, the resonance frequency and damping ratio of three peak points with a higher \(\kappa_{r}\) value are used to study this further. It can be seen that the resonance frequency \(f\) values of the three points are quite different, it is not easy to select the optimal value, and the optimal value deviates greatly from the set 2000 Hz. At the same time, the damping ratio tends to a lower value, which is inconsistent with the preset value. It shows that for a low signal-to-noise ratio signal, it cannot identify the correct damping ratio.