# Feature Extraction of Bearing Weak Fault Based on Sparse Coding Theory and Adaptive EWT
## 5 Simulation and Experimental Verification
### Experimental Verification
The parameters of the correlation filtering method are resonance frequency  
Figure 11: RMS value trend graph and partially enlarged graph of bearing 3 full life data: (**a**) RMS value trend graph; and (**b**) Partially enlarged graph. Figure 12: The waveform of the 2056 th set of data. \(F=\{3500:20:7500\}\), damping ratio \(Z=\{0.01:0.005:0.2\}\), \(T=\{0:1/f_{s}:0.2\}\), and support interval \(w_{s}=68\). The result is shown in Figure 17. Taking some peak points for analysis, it is similar to the simulated result before the application of the algorithm, and, thus, it is still not easy to get the correct values of the atomic morphological parameters. Figure 16: Fast spectral kurtosis figure of the 2056 th set of data. Figure 10:0 Envelope spectrum of the 2056 th set of data. Figure 10:1 Schematic diagram of spectrum division of the envelope spectrum of the 2056 th set of data. Figure 10:2 CN and KSE values of each component of the envelope spectrum of the 2056 th set of data. The waveform of the 10 th component is shown in Figure 10:3 It can be seen that the SNR is still not high enough. The peak frequency 5134 Hz of the component frequency band is taken as the value of parameter \(f\), the damping ratio \(\varepsilon=0.07\), the support interval \(M=68\), and the time shift \(v=1/f_{\rm s}\). However, once the LWDFSS-AEWT algorithm is used to solve the coefficient and reconstruct the signal of the 10 th component signal, as Figure 19a,b show, the noise is largely removed. The fault characteristic frequency extracted, 294 Hz, is close to the theoretical fault characteristic frequency of the inner ring, which is 296.63 Hz, so it can be judged that the bearing inner ring is faulty. Figure 10:4 Correlation filtering result of the 2056 th set of data: (**a**) The correlation coefficient \(\kappa_{r}\) between the base atom and the original signal; (**b**) Resonance frequency of each point in \(\kappa_{r}\); and (**c**) Damping ratio of each point in \(\kappa_{r}\). Figure 10:5 The waveform of the 10 th component. As a comparison, results obtained with this set of data using the SISC algorithm based on a learning dictionary, the BPDN algorithm based on the Symlet8 wavelet packet, and the WPT Kurtogram algorithm [26] are also presented, as shown in Figures 20-22, respectively. Although the SISC algorithm based on a learning dictionary can get the base atom, the result is not ideal, as the matching degree between the base atom and the signal is not high. Although BPDN can extract the fault characteristic frequency and frequency doubling, the SNR is not high enough, and the reconstructed signal has poor results. The WPT Kurtogram can hardly distinguish the fault characteristic frequency. However, the LWDFSS-AEWT algorithm can obtain the ideal reconstructed signal and feature extraction results and is less affected by noise. Thus, the results show the advantages of frequency band division and the accuracy of frequency band selection by the AEWT method. Figure 10:6 The result of the 2056 th set of data processing based on LWDFSS-AEWT algorithm: (**a**) Waveform of reconstructed signal; (**b**) Envelope spectrum of the reconstructed signal. Figure 10:7 Results obtained with SISC based on a learning dictionary: (**a**) is the base atom obtained by learning; and (**b**) is the envelope spectrum of the reconstructed signal corresponding to the atom.