## 3 Synthetic Examples  
### Simulation of Cosine Signals  
To verify the SSST-TD TF correlation method, a simulation was conducted. Because stationary signals, particularly the cosine ones, are a special case of the signals, analyzing stationary signalscan show the advantage and stability of the proposed method. Thus, we took two cosine signals. Signal \(x\) was cosine with frequency 30 Hz, and signal \(y_{1}\) was the sum of 30 Hz and 100 Hz. The time resolution was 0.1 ms, and the sampling points were 4096. The traditional correlation coefficient (directly computed in the time domain) between \(x\) and \(y_{1}\) is shown in Figure 1.  
\[\left\{\begin{array}{l}x=\cos(30\times 2\pi t)\\ y_{1}=\ \cos(30\times 2\pi t)+\cos(100\times 2\pi t)\end{array}\right. \tag{16}\]  
Due to the existence of a window function and the time resolution, a correlation coefficient has a certain bandwidth. Here, the correlation coefficient was 1 near 30 Hz, which means the two signals were correlated. Moreover, it was below 0.2 near 100 Hz, which is considered uncorrelated. At other frequencies, the correlation coefficient fluctuated greatly.  
Then, we applied the ST and the SSST to the signals \(x\) and \(y_{1}\). The magnitude of the TF spectrum is shown in Figure 2. Both the ST and the SSST obtained signal distributions in the TF domain. Moreover, TF representations gave an intuitive display of the signals. Comparing Figure 2A-D, we can conclude that the TF energy bandwidth of the ST was much wider than that of the SSST. The existence of the false spectral energies is likely to cause mistakes in interpreting the results of the spectral decomposition, especially in the wide energy bandwidth around 100 Hz. Using the SSST, the TF resolution was highly improved, and the energy concentrated where the actual frequencies were located.  
Next, we decomposed the TF spectrum of the ST and the SSST by the TD, that is, the ST-TD and the SSST-TD, and carried out a correlation analysis. The correlation coeffici  
Figure 1: The traditional correlation coefficient between \(x\) and \(y_{1}\).  
Figure 2: The time-frequency (TF) representations of the signals: (**A**) the S transform (ST) of \(x\), (**B**) the ST of \(y_{1}\), (**C**) the synchrosqueezed S transform (SSST) of \(x\), and (**D**) the SSST of \(y_{1}\).  
Obviously, near 30 Hz, both the ST-TD and the SSST-TD detected the frequency correlation between the signals. However, the bandwidth of the ST-TD correlation was wider than that of the SSST-TD, even worse than the traditional method. Near 100 Hz, the result of the ST was inaccurate due to the low TF resolution; that is, the ST was suitable for acquiring the TF distribution of signals but not suitable for calculating the correlation coefficient. Thus, the SSST-TD TF correlation analysis not only obtained the same result as the traditional method, but also obtained the TF correlation distribution.  
Figure 4 shows three SSST-TD TF correlation coefficients at different times. It is obvious that for the cosine signals, the overall trend of the SSST-TD TF correlation coefficient is consistent with that of the traditional method with small differences at the different frequencies. The main difference is that the SSST-TD TF correlation coefficient changes with time.