## 2 Fault Feature Extraction Based on Improved Fast Spectral Correlation
### Improved Fast Spectral Correlation  
The signal of a faulty bearing is characterized by periodic impacts, which contains multiple harmonics in the spectrum and has a wide frequency characteristic. When the raw vibration signal is directly analyzed by fast spectral correlation, there may be many interference components in the obtained enhanced envelope spectrum around the characteristic cyclic frequencies. In order to extract the modulation information effectively, the kurtosis weighting was introduced to the fast spectral correlation analysis. Kurtosis is the fourth order of amplitude. It is sensitive to signals with larger amplitude and can play a role of energy amplification. The kurtosis was calculated for each column of the fast spectral correlation matrix and a greater weight was given for a bigger kurtosis. The modulation capacity of the cyclic frequency corresponding to the column can be effectively estimated. Therefore, a fast spectral correlation method based on kurtosis weight was proposed to enhance the fault features of bearings. The specific steps of the improved fast spectral correlation method are as follows. (1) Calculate the fast spectral correlation \(S_{x}^{Fast}(\alpha,f)\) of the original signal and calculate the kurtosis values of each column of \(S_{x}^{Fast}(\alpha,f)\)[45]:  
\[\text{KA}(\alpha)=\text{mean}\Big{\{}\big{[}S_{x}^{Fast}(\alpha,f)\big{]}^{4} \Big{\}}. \tag{10}\]  
(2) Calculate the weighting factor \(\text{KA}^{\prime}(\alpha)\) based on Equation (11). The weighting factor reflects modulation ability of the modulation frequency. If the factor is greater, the modulation effect corresponding to this frequency in the original signal is stronger. Conversely, the modulation effect is weaker. \[\text{KA}^{\prime}(\alpha)=\frac{\text{KA}(\alpha)-\text{min}[\text{KA}( \alpha)]}{\text{max}[\text{KA}(\alpha)]-\text{min}[\text{KA}(\alpha)]}. \tag{11}\](3) By weighting the above factors, the improved fast spectral correlation \(S_{x}^{Fast^{\prime}}(\alpha,f)\) is obtained as shown in Equation (11). The improved fast spectral correlation can eliminate or weaken the high frequency doubling of the cyclic frequency with weak modulation capability. \[S_{x}^{Fast^{\prime}}(\alpha,f)=\text{KA}^{\prime}(\alpha)\times S_{x}^{Fast}( \alpha,f). \tag{12}\]  
In order to verify the performance of the improved fast spectral correlation in extracting bearing fault information, the multi-component superposition signal \(x(t)\) shown in Equation (13) was investigated [46]. \[\left. \begin{array}{l}x(t)=x_{1}(t)+x_{2}(t)+x_{3}(t)\\ x_{1}(t)=\left[\sum\limits_{i=0}^{M-1}Dh(t-iT_{o})\right]*[Ae^{-\xi 2\pi f_{n}t} \cos(2\pi f_{d}t)]\\ x_{2}(t)=6\sin(2\pi f_{1}t)\\ x_{3}(t)=3 rand(1,N)\end{array} \right\}, \tag{13}\]  
where \(x_{1}(t)\) is the outer-race impact signal designed based on the rolling bearing outer-race fault model in Reference [47], the natural frequency \(f_{d}=f_{n}\sqrt{1-\xi^{2}}\), \(M=30\), \(D=1\), \(A=2\), \(\xi=0.05,f_{n}=3000\) Hz and the outer race fault frequency \(f_{o}=120\) Hz. \(x_{2}(t)\) is the fundamental frequency interference signal of the frequency \(f_{1}=20\) Hz. \(x_{3}(t)\) represents random noise signal. The sampling frequency of the simulated signal is 8192 Hz. Figure 1 shows the time domain waveforms of \(x_{1}(t)\), \(x_{2}(t)\), \(x_{3}(t)\), \(x(t)\) and the envelope spectrum of \(x(t)\) respectively. Figures 2 and 3 show the analysis results obtained by using fast spectral correlation and improved fast spectral correlation, respectively. Figure 2a shows the fast spectral correlation spectrum of \(x(t)\) obtained via the fast spectral correlation method, from which it is hard to identify the fault characteristics. The corresponding enhanced envelope spectrum shown in Figure 2b has peaks at \(f_{o}\), \(2f_{o}\), \(3f_{o}\) and \(4f_{o}\), but many noise components can also be noticed. As shown in Figure 3a, the spectral lines can be identified at \(f_{o}\), \(2f_{o}\), \(3f_{o}\) and \(4f_{o}\), which are more obvious than that reflected in Figure 2a.