## 2 Fault Feature Extraction Based on Improved Fast Spectral Correlation  
### Brief Introduction of Fast Spectral Correlation  
Let \(x(t_{n})\) be a cyclostationary signal, its spectral correlation is defined as:  
\[SC_{x}(\alpha,f)=\frac{1}{F_{s}^{2}}\sum\limits_{n=-\infty}^{\infty}\sum \limits_{\tau=-\infty}^{\infty}R_{x}(t_{n},\tau)e^{-j2\pi\alpha n\frac{1}{F_{ s}}}e^{-j2\pi f\tau\frac{1}{F_{s}}}, \tag{1}\]  
where \(F_{s}\) is the sampling frequency, \(t_{n}\) denotes the time instants which can be calculated as \(t_{n}=n/F_{s}\), \(R_{x}(t_{n},\tau)\) represents the cyclic autocorrelation function of \(x(t_{n})\), \(\tau\) indicates the time delay, \(\alpha\) denotes the cyclic frequency and \(f\) represents the frequency. The spectral correlation is a two-dimensional Fourier transform of two frequency variables, which reflects the power distribution of the signal relative to the frequency and the cyclic frequency. The power corresponding to the cyclic frequency \(\alpha\) distributed along the line of the parallel frequency \(f\). In order to reduce the computational cost and improve the efficiency of spectral correlation, a fast spectral correlation method based on short time Fourier transform was proposed in Reference [28]. The short time Fourier transform (STFT) of the signal \(x(t_{n})\) is described as follows:  
\[X_{STFT}(i,f_{k})=\sum\limits_{n=0}^{N_{w}-1}x[iR+n]w[n]e^{-j2\pi n\frac{f_{k} }{F_{s}}}, \tag{2}\]  
where \(N_{w}\) represents the window length of STFT, \(R\) represents the block shift in STFT; \(w[n]\) is the function of time index \(n\); \(x[n]\) is the abbreviated form of \(x(t_{n}),f_{k}\) denotes the \(k\)-th discrete frequency and \(f_{k}=k\Delta f\), \(\Delta f\) represents the frequency resolution, which has the expression of \(\Delta f=F_{s}/N_{w}\). Equation (3) shows the phase-corrected STFT [28]:  
\[ \begin{array}{ll}X_{w}(i,f_{k})&=\sum\limits_{n=0}^{L-1}x[n]w[n-iR]e^{-j2 \pi n\frac{f_{k}}{F_{s}}}\\ &=X_{STFT}(i,f_{k})e^{-j2\pi iR\frac{f_{k}}{F_{s}}}\end{array} , \tag{3}\]  
where \(X_{w}\) (\(i,f_{k}\)) denotes the complex envelope of signal \(x(t_{n})\) at \(iR/F_{s}\), whose center is \(f_{k}\) and bandwidth is \(\Delta f\). \(|X_{w}(i,f_{k})|^{2}\) represents the energy flow in the frequency band. Then the cyclic spectrum based on STFT is defined as [28]:  
\[S_{x}(f,\alpha)=\frac{F_{s}}{L}X_{w}(f)X_{w}(f-\alpha)^{*}, \tag{4}\]  
where \(L\) is the length of signal \(x(t_{n})\). Assume that\(f=f_{k}=k\Delta f\) and \(\alpha=p\Delta f+\delta\), it can be deduced that\(f-\alpha=f_{k}-\alpha\approx f_{k-p}\) and \(\alpha\approx p\Delta f\). Applying these results to Equation (3), it comes that [28]:  
\[X_{w}(i,f_{k}-\alpha)\approx X_{w}(i,f_{k-p})e^{j2\pi(\frac{\alpha}{F_{s}}-p \frac{\Delta f}{F_{s}})(iR+N_{0})}. \tag{5}\]  
where \(p\) is index of STFT frequency closest to a given cyclic frequency \(\alpha\), \(\delta\) is residue. Insert Equations (5) and (3) into Equation (4), the following Equation is obtained [28]:  
\[ \begin{array}{ll}S_{x}(\alpha,f_{k};p)&=\frac{1}{K\|w\|^{2}F_{k}}\sum\limits_{i= 0}^{K-1}X_{w}(i,f_{k})X_{w}(i,f_{k-p})^{*}e^{-j2\pi(\frac{k}{B}-\frac{p}{N_{w}} )(iR+N_{0})}\\ &=\frac{1}{K\|w\|^{2}F_{k}}\underset{i\to\alpha}{D\!F\!T}\Big{\{}X_{STT}(i,f_{ k})X_{STT}(i,f_{k-p})^{*}\Big{\}}e^{-j2\pi N_{0}(\frac{k}{B}-\frac{p}{N_{w}})}\,\end{array}  \tag{6}\]  
When \(p=0\), the period of \(x(t_{n})\) is \(T\) and its frequency \(\alpha\) is \(1/T\). The energy flow will flow periodically in the band \([f_{k}-\Delta f/2,f_{k}+\Delta f/2]\). When \(p\
eq 0\), \(X_{w}(i,f_{k})X_{w}(i,f_{k-p})^{*}\) represents the energy flow between band \([f_{k}-\Delta f/2,f_{k}+\Delta f/2]\) and band \([f_{k-p}-\Delta f/2,f_{k-p}+\Delta f/2]\).