# Diagnosis Method for Mechanical Faults Based on Rotation Synchrotexting Chirplet Transform
## 3 Numerical Experiments
Therefore, the RSECT gets better results than the GLCT and SET when dealing with multi-component complex non-stationary signals. The more concentrated energy of the TF results indicates a stronger localization ability and better time-frequency characteristics. Renyi entropy is an important index to evaluate TF aggregation. A lower value of Renyi entropy indicates more concentrated energy. Therefore, to further verify the superiority of the proposed method numerically, white noise with SNR from 1 dB to 30 dB is added to the simulation signal, respectively. The Renyi entropy of TF representation generated by the three different methods is shown in Figure 9. It can be found that the noise increases the Renyi entropy, indicating that the noise can derive the energy aggregation of the above methods. In addition, the GLCT is susceptible to noise interference. The RSECT has the lowest Renyi entropy value, indicating that the method has the best TF aggregation and noise immunity. The Renyi entropy value of the SET is between the two methods. To further test the effectiveness of the proposed method in practical applications, a more complex multi-component FM signal is constructed here, i.e.,  
\[f(t)=\cos(400\pi t)+\cos(2\pi(80t+150t^{2}+\frac{3}{\pi}\cos(4\pi t))) \tag{14}\]  
The signal is comprised of a constant-frequency signal and an FM signal, and the instantaneous frequency components of both of them are crossed. The sampling frequency is 1000 Hz, and the sampling time is 1 s. The time-domain waveform of the simulation signal is shown in Figure 10.  
The simulated signal is processed by the RSECT, SET, and GLCT, respectively. Figures 11-13 show the TF distributions obtained from the GLCT, SET, and RSECT, respectively. Comparing Figures 11-13, the frequency characteristics of the constant-frequency signal and FM signals can be basically extracted regardless of the GLCT, the SET, or the RSECT. However, the three methods have obvious differences in the TF resolution of the signal. The GLCT is obviously inferior to the SET and RSECT. Comparing Figures 12 and 13, the SET is significantly inferior to the RSECT when dealing with the instantaneous frequencies of cross-components. The RSECT combines the respective advantages of the GLCT and SET, which not only maintain  
Figure 4:0 Rényi entropies of the TF representations generated by three different methods. the high-precision TF resolution but also have certain robustness to the processing of cross frequencies. From the local zoomed-in diagram of the TF distribution, it can be seen that the GLCT has a certain anti-aliasing property, but its TF accuracy is not high enough at the cross of frequency components. The SET meets a certain time-frequency accuracy, but the anti-aliasing property is not enough. Only the RSECT satisfies both the anti-frequency aliasing performance and high TF accuracy. Therefore, the RSECT gets better results than the GLCT and SET in processing cross-frequency signals. Figure 4:1 TF distribution of \(f(t)\) based on GLCT. **(a)** TF distribution (b) Local enlarged figure. Figure 4:2 Time domain waveform of simulation signal.