# Intelligent Diagnosis of Bearing Failures Based on Recurrence Quantification and Energy Difference
## 2 Methodology
### Hilbert Transformation  
Hilbert transformation [14] is an effective analytical method for processing vibration and analytic signals. Considering a general continuous signal \(x(t)\) with its spectrum \(X(f)\), \(x(t)\) can be transformed as follows:  
\[x(t)=\int_{-\infty}^{+\infty}X(f)\mathrm{e}^{\mathrm{i}2\pi ft}\mathrm{d}f \tag{1}\]  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Mechanical Motions** & **Frequencies** \\ \hline Rotation frequency of inner race & \(f_{\mathrm{i}}=\frac{n}{60}\) \\ Slewing frequency of outer race & \(f_{\mathrm{o}}=0\) \\ Relative rotation frequency of inner and outer races & \(f_{\mathrm{r}}=f_{\mathrm{i}}-f_{\mathrm{o}}\) \\ Inner race fault & \(f_{\mathrm{f}}=\frac{1}{2}Z(1+\frac{d}{D}\cos\alpha)f_{\mathrm{r}}\) \\ Outer race fault & \(f_{\mathrm{O}}=\frac{1}{2}Z(1-\frac{d}{D}\cos\alpha)f_{\mathrm{r}}\) \\ Rolling element failure frequency & \(f_{\mathrm{b}}=\frac{D}{2}(1-\frac{d}{D^{2}}\cos^{2}\alpha)f_{\mathrm{r}}\) \\ Cage failure frequency & \(f_{\mathrm{P}}=\frac{1}{2}(1-\frac{d}{D}\cos\alpha)f_{\mathrm{r}}\) \\ \hline \hline \end{tabular}
\end{table}
Table 1: Frequencies of rolling bearings.  
\[x(t)=\int_{0}^{+\infty}X(f)\mathrm{e}^{\mathrm{j}2\pi ft}\mathrm{d}f+\int_{- \infty}^{0}X(f)\mathrm{e}^{\mathrm{j}2\pi ft}\mathrm{d}f \tag{2}\]  
It can be deduced that  
\[x(t)=\int_{0}^{+\infty}X(f)\mathrm{e}^{\mathrm{j}2\pi ft}\mathrm{d}f+\int_{0}^{ +\infty}X(f)\mathrm{e}^{-\mathrm{j}2\pi ft}\mathrm{d}f \tag{3}\]  
Since \(X(-f)=\overline{X(f)}\), the second term in the above equation can be changed to  
\[\int_{0}^{+\infty}X(f)\mathrm{e}^{-\mathrm{j}2\pi ft}\mathrm{d}f=\overline{ \int_{0}^{+\infty}X(f)\mathrm{e}^{\mathrm{j}2\pi ft}\mathrm{d}f} \tag{4}\]  
If the signal \(x(t)\) is represented as the real part of a complex signal consisting solely of positive frequency components, then \(x(t)\) can be transformed into the following form:  
\[x(t)=\mathrm{Re}\bigg{\{}\int_{0}^{+\infty}2X(f)^{\mathrm{j}2\pi ft}\mathrm{d} f\bigg{\}} \tag{5}\]  
Then, the complex signal of \(x(t)\) can be denoted as \(q(t)\):  
\[q(t)=\int_{0}^{+\infty}2X(f)\mathrm{e}^{\mathrm{j}2\pi ft}\mathrm{d}f \tag{6}\]  
The spectrum of the complex signal \(q(t)\) is denoted as \(Q(f)\):  
\[Q(f)=\left\{\begin{matrix}2X(f)&f>0\\ 0&f<0\end{matrix}\right. \tag{7}\]  
The spectrum \(Q(f)\) can be obtained by filtering \(X(f)\). Assuming the spectrum of the filter is \(H_{1}(f)\), its computation is as follows:  
\[H_{1}(f)=\left\{\begin{matrix}2&f>0\\ 0&f<0\end{matrix}\right. \tag{8}\]  
\[Q(f)=H_{1}(f)X(f) \tag{9}\]  
The time function \(h_{1}(t)\) corresponding to the spectrum \(H_{1}(f)\) is as follows:  
\[h_{1}(t)=\delta(t)+\mathrm{j}\frac{1}{\pi t} \tag{10}\]  
\[q(t)=h_{1}(t)*x(t)=\left(\delta(t)+\mathrm{j}\frac{1}{\pi t}\right)*x(t) \tag{11}\]  
The expression \(\widetilde{x}(t)\) is referred to as the Hilbert transformation of the signal \(x(t)\):  
\[\widetilde{x}(t)=\frac{1}{\pi t}*X(t)=\frac{1}{\pi}\int_{-\infty}^{+\infty} \frac{x(\tau)}{t-\tau}\mathrm{d}\tau \tag{12}\]  
It can be seen that a Hilbert transformation of a signal is equivalent to a filtering process, and the filtering factor \(h(t)\) is \(\frac{1}{\pi t}\).  
Appropriately applying Hilbert transformation enables the envelope analysis of fault signals, which in turn facilitates the calculation of the instantaneous phase and the examination of the instantaneous frequency. This approach provides valuable insights into the evolution of the entire fault signal.