## 5 Experiment Data Analysis
### Compare with Existing Method  
(1) Compare with wavelet transform  
Wavelet transform is a common and effective method in extracting weak signals. The wavelet transform decomposes the signal into the scale domain, and through multi-resolution decomposition, the weak signal components in the original signal become prominent. First of all, the bearing vibration signal is decomposed and reconstructed by 11 layers using Haar wavelet. The reconstructed signal is shown in Figure 17a. The recombined signal is then subjected to power spectrum analysis, as shown in Figure 17b.  
Figure 16: The analysis of autocorrelation envelope spectrum: (**a**) improved autocorrelation envelope power spectrum analysis; (**b**) autocorrelation envelope power spectrum analysis; (**c**) cepstral analysis; and (**d**) autocorrelation envelope maximum entropy spectrum analysis.  
Analyzing Figures 17a and 11b, one can see that the time-domain diagram after wavelet transform recombination is deviated from the time-domain diagram of the original signal. The time-domain image reorganized by wavelet transform cannot fully reflect the vibration state. Analyzing the power spectrum of the wavelet-recombined signal of Figure 17b, there is a high frequency component of 6250 Hz, which shows that wavelet transform can realize the detection of bearing faults. Comparing Figure 17b with Figure 14c, it can be obtained that when the wavelet transform's recombined signal is analyzed for power spectrum, high-frequency components can be extracted, but the extraction of low-frequency components is not accurate.  
Compare with empirical mode decomposition  
Empirical mode decomposition (EMD) is often used to extract weak faults of bearings. It can decompose complex signals into a limited number of intrinsic mode functions. The decomposed IMF components contain local characteristic signals of different time scales of the original signal. The bearing signal is decomposed by EMD. The time-domain signal and spectrum of 1-5 IMFs are shown in Figure 18.  
According to Figure 18, time-domain signals of different IMFs are obtained through EMD decomposition. Different IMFs have different power spectrum. Among them, the power spectrum frequency of IMF4 is the purest, with little noise interference. The peak value is 5958 Hz, which is closing to the natural frequency of the sensor. Compared with Figure 14c, the maximum frequency in the power spectrum obtained after Kalman filter is 6157 Hz, which is close to the natural frequency of the sensor 6325 Hz.  
Finally, the signal to noise ratio (SNR) and root mean square error (RMSE) are analyzed, as shown in Table 3.  
The SNR value represents the ratio of the effective signal energy to the noise energy in the signal. The higher the value, the smaller the value of noise mixed in the signal. The RMSE value reflects the difference between the signal after noise reduction and the original signal. The lower the value, the closer the signal after noise reduction is to the original signal.  
Figure 17: Wavelet transform noise reduction signal analysis: (**a**) the time-domain curve of the reconstructed signal after wavelet decomposition; and (**b**) power spectrum analysis curve.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Noise Reduction Method** & **SNR** & **RMSE** \\ \hline Kalman filter & 5.6302 & 0.3833 \\ \hline Wavelet transform & 1.7621 & 0.3925 \\ \hline Empirical mode decomposition & 1.7039 & 0.5095 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Noise reduction signal quality evaluation index.  
According to the data in Table 3, the noise reduction effect of the Kalman filter is better than that of wavelet transform and Empirical mode decomposition.  
Figure 18: Empirical mode decomposition analysis: (**a**) the time-domain curves of the 1-5 IMFs; and (**b**) the power spectrum of the 1-5 IMFs.