# Diagnosis Method for Mechanical Faults Based on Rotation Synchrotexting Chirplet Transform
## 2 The Basic Theory of RSECT  
Generally, a nonstationary signal \(f(t)\) can be defined as  
\[f(t)=A(t)e^{i\int\varphi(t)dt} \tag{1}\]where \(A(t)\) and \(\varphi(t)\) denote the instantaneous amplitude and the phase function of the signal, respectively. Before illustrating the core ideal of RSECT, we first express the linear chirplet transform (LCT) result of the signal, i.e.,  
\[L(t,\omega,c)=\int_{-\infty}^{+\infty}f(t)h(\tau-t)e^{-j\omega t}e^{-ic(\tau-t )^{2}/2}dt \tag{2}\]  
where \(t\) is the time, \(\omega\) is the frequency, \(c\) is an optional variable, \(h(t)=e^{-(pi/0.32^{2})t^{2}}\) is a gaussian window function, \(e^{-ic(\tau-t)^{2}/2}\) is the discrete gaussian demodulation operator, and \(L(t,\omega,c)\) is a LCT coefficient. From Equation (2), LCT introduces the Gaussian demodulation operator on the basis of the STFT, the introduction of the Gaussian demodulation operator will make the kernel function of LCT with tilt characteristics. We can adjust the tilt angle of time-frequency ridge on time-frequency plane through optional variable c to find the best time-frequency energy. When the c value is appropriately selected, the time-frequency point in the neighborhood at a certain time will always make the Gaussian demodulation operator close to the demodulation part of the signal. At this time, the time-frequency energy located at the transient time-frequency has a high degree of aggregation, and the amplitude of the obtained coefficient \(|L(t,\omega,c)|\) of LCT will reach the maximum. The coefficient is calculated as follows. \[c_{m}=\text{argmax}|L(t,\omega,c)|\leq\left|\int_{-\infty}^{\infty}h(\tau-t)A (t)dt\right| \tag{3}\]  
Iterating each point in the TF plane, we can get \(c_{m}=\text{argmax}|L(t,\omega,c)|\) that maximizes \(|L(t,\omega,c)|\). According to the calculation formula of the two-dimensional instantaneous frequency spectrum, the two-dimensional instantaneous TF distribution of the signal can be estimated as follow. \[\varphi^{\prime}(t,\omega,c)=-j\frac{\partial_{t}L(t,\omega,c)}{2\pi L(t, \omega,c)} \tag{4}\]  
When \(L(t,\omega,c)\) is not 0, the synchroextracting operator (SEO) can be defined as follows  
\[\delta(\omega-\varphi^{\prime}(t,\omega,c))= \begin{cases}1&\omega=\varphi^{ \prime}(t,\omega,c)\\ 0&\omega\
eq\varphi^{\prime}(t,\omega,c)\end{cases}  \tag{5}\]  
The SEO satisfies the properties of a Dirac delta function and is designed to extract the TF coefficients of \(L(t,\omega,c)\) along with the curves \(\omega=\varphi^{\prime}(t,\omega,c)\), the synchroextracting LCT can be defined as  
\[T_{e}(t,\omega,c)=L_{s}(t,\omega,c)\cdot\delta(\omega-\varphi^{\prime}(t, \omega,c)) \tag{6}\]  
We rotate the TF result by adding a gaussian demodulation operator. The angle of rotation can be calculated as \(\arctan(-c)\). For a signal with unit sampling time \(t\in[0,T_{\rm s}]\) and sampling frequency \(F_{s}\), its inclination angle on the TF ridge should satisfy  
\[\alpha=\arctan(2\frac{T_{\rm s}}{F_{\rm s}}c)\ \ \ \alpha=(-\pi/2,\ \pi/2) \tag{7}\]  
For the entire time domain, the \(c\) can be calculated  
\[c=\tan(\alpha)\frac{F_{\rm s}}{2T_{\rm s}} \tag{8}\]  
\[\alpha=-\frac{\pi}{2}+\frac{\pi}{N_{c}+1},-\frac{\pi}{2}+\frac{2\pi}{N_{c}+1},\ \cdots,-\frac{\pi}{2}+\frac{N_{c}\pi}{N_{c}+1} \tag{9}\]where \(N_{c}\) is the number of discretization of the time-frequency interval. Then, the GLCT can be defined as  
\[L_{\rm s}(t,\omega,c)=\int_{-\infty}^{+\infty}f(t)h(\tau-t)e^{-|\omega t|}e^{-t \tan(\alpha)(\tau-t)^{2}/2}dt \tag{10}\]  
The parameter \(\alpha\) is discretized into \(N\) values, that is, the original time-frequency plane is divided into \(N+1\) time-frequency planes, i.e.,  
\[\alpha=-\pi/2+\pi/(N+1),-\pi/2+2\cdot\pi/(N+1)\ldots-\pi/2+N\cdot\pi/(N+1) \tag{11}\]  
Then the RSECT can be defined as  
\[T_{c}(t,\omega,N)=L_{\rm s}(t,\omega,N)\cdot\delta(\omega-\varphi^{\prime}(t,\omega,N)) \tag{12}\]  
When \(N=1\), Equation (12) will degenerate into SET.