## 3 Vibration Signal Preprocessing and Fault Feature Extraction
### Method Verification Experiments  
To demonstrate that the GWO-VMD method proposed in Section 3.1 can help signal reconstruction in noise elimination, contrast experiments were conducted and results are reported in this section. GWO-VMD and VMD were both applied to the same experimental signal for decomposition, then the correlation-based signal reconstruction proposed in Section 3.2 was applied to the two decomposition results to rebuild the signals; finally, four performance indicators--Kurtosis, SNR, RMSE, and permutation entropy--of the two reconstructed signals were compared to verify the signal reconstruction effects. The experimental procedure is shown in Figure 2.  
Like the method adopted by [PERSON] et al. (2018), experimental analytic signals were manually constructed to emulate fault vibration signals in a gearbox. The experimental signal, denoted as \(x(t)\), consisted of a periodic impulsive signal, denoted as \(s(t)\), and a Gaussian white noise with a Signal-Noise Ratio (SNR) of 5 dB, denoted as \(x_{n}(t)\). The signals lasted 1 s, the sampling frequency was \(f_{s}=2048\) Hz, and the sampling points were \(n_{p}=2048\). The \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Parameter** & **Notations** & **Mathematical** \\  & **Names** & & **Expressions** \\ \hline
1 & Maximum amplitude & \(v_{max}\) & \(v_{max}=\max(v(t))\) \\
2 & Minimum amplitude & \(v_{min}\) & \(v_{min}=\min(v(t))\) \\
3 & Peakedness 1 & \(v_{p}\) & \(v_{p}=\frac{1}{N_{c}}\sum v(t)^{4}\) \\
4 & Peak-to-peak value & \(v_{pp}=\max(v(t))-\min(v(t))\) \\
5 & Mean value & \(v_{nn}\) & \(v_{m}=\frac{1}{N_{c}}\sum v(t)\) \\
6 & Slant & \(v_{s}\) & \(v_{s}=\frac{1}{N_{c}}\sum v(t)^{3}\) \\
7 & Root mean square value & \(v_{rms}\) & \(v_{rms}=\sqrt{\frac{1}{N_{c}}\sum v(t)^{2}}\) \\
8 & Root square amplitude & \(v_{r}\) & \(v_{r}=\left(\frac{1}{N_{c}}\sum\sqrt{\left[v(t)\right]}\right)^{2}\) \\
9 & Mean of absolute value & \(v_{ma}\) & \(v_{ma}=\frac{1}{N_{c}}\sum\left[v(t)\right]\) \\
10 & Variance & \(v_{\sigma}\) & \(v_{\sigma}=\frac{1}{N_{c}}\sum\left(v(t)-v_{m}\right)^{2}\) \\
11 & Shape factor & \(v_{\text{\emph{sf}}}\) & \(v_{\text{\emph{sf}}}=\frac{v_{ma}}{N_{c}}\) \\
12 & Skewness & \(v_{sn}\) & \(v_{sn}=\frac{\frac{1}{N_{c}}\sum\left(v(t)-v_{m}\right)^{3}}{\frac{2}{3}}\) \\
13 & Impulse factor & \(v_{\text{\emph{sf}}}\) & \(v_{\text{\emph{rf}}}=\frac{v_{max}}{N_{c}}\) \\
14 & Crest factor & \(v_{\text{\emph{cf}}}\) & \(v_{\text{\emph{cf}}}=\frac{v_{max}}{N_{c}}\) \\
15 & Clearance factor & \(v_{\text{\emph{bf}}}\) & \(v_{\text{\emph{bf}}}=\frac{\frac{1}{N_{c}}\sum\left(v(t)-v_{m}\right)^{4}}{v^{ 2}}\) \\
16 & Kurtosis factor & \(v_{\text{\emph{bf}}}\) & \(v_{\text{\emph{bf}}}=\frac{\frac{1}{N_{c}}\sum\left(v(t)-v_{m}\right)^{4}}{v^{ 2}}\) \\
17 & Energy & \(v_{E}\) & \(v_{E}=\int_{-\infty}^{+\infty}\left|v(t)\right|^{2}dt\) \\
18 & Permutation entropy & \(v_{\text{\emph{pr}}}=-\frac{1}{\ln(\text{\emph{mf}})}\sum\limits_{i=1}^{q}p_{ i}lnp_{i}\) \\ \hline \hline \end{tabular}  
* In the calculation equation of Peakedness, \(N_{c}\) indicates the length of \(v^{\prime}(t)\).  
\end{table} 
Table 2: List of the 18 feature parameters. emulated feature frequency of a fault was \(f_{m}\) = 20 Hz, and the frequency of carrier signal was \(f_{1}\) = 180 Hz. The expressions of the experimental signals are as follows:  
\[\left\{ \begin{array}{c}s(t)=0.8\text{exp}(-100t_{0})\cos(2\pi f_{1}t)\\ x(t)=s(t)+x_{n}(t)\end{array} \right. \tag{13}\]  
where \(t_{0}\) = mod(\(n_{p}\)/\(f_{s}\), 1/\(f_{m}\)), and mod represents module operation. The waveforms of the experimental signals and their spectra are shown in Figure 3, the envelope spectra of \(s(t)\) and \(x(t)\) are shown in Figure 4.  
Figure 3: Waveforms of the experimental signals and their spectra. Figure 2: Contrast experiments of GWO-VMD-based and VMD-based signal decomposition.