## 3 Synthetic Examples
### Simulation of Non-Stationary Signals  
For stationary signals with or without noise, both the traditional correlation analysis and the SSST-TD TF correlation analysis can identify the correlation between the signals. However, the SSST-TD TF correlation method can better identify the correlation at different times. Practical signals are often non-stationary, so the following signals were designed to analyze the advantages of the proposed method. They consisted of cosine, cosine frequency modulation (FM), and shock. We also added Gaussian noise to both with a mean value of 0 and a standard variance of 1. The time resolution was 1 ms, and the sampling points were 4096. The time domain signals are shown in Figure 10. The traditional correlation coefficient is shown in Figure 11. Obviously, it is impossible to distinguish the correlation between \(x\) and \(y\). For the frequency 30 Hz, it occurred at the second and third stage in \(x\) and only the third stage in \(y\), so the  
Figure 8: The correlation coefficients between \(x\) and \(y_{1}\) of (**A**) the traditional correlation and (**B**) the SSST-TD TF correlation. Figure 9: The correlation coefficient between \(x\) and \(y_{1}\) at different times. correlation coefficients was expected to change and not be constant. Thus, the correlation coefficient was not accurate. \[\left\{ \begin{array}{l}x_{1}\ =\ 2 exp(-2\pi t)\times\cos(300\times 2\pi t)\\ x_{2}\ =\ \cos(30\times 2\pi t+2\cos(100\times 2\pi t-2\cos(10\times 2\pi t)))\\ x_{3}\ =\ \cos(30\times 2\pi t)\\ \end{array} \right. \tag{18}\] \[\left\{ \begin{array}{l}y_{1}\ =\ 2 exp(-2\pi t)\times\cos(300 \times 2\pi t)\\ y_{2}\ =\ 2\cos(100\times 2\pi t-2\cos(10\times 2\pi t))\\ y_{3}\ =\ \cos(30\times 2\pi t)\\ \end{array} \right..\]  
The SSST of \(x\) and \(y\) is shown in Figure 12. It is clear that the TF distribution of \(x\) and \(y\) can be attained by the SSST, and the energy is gathered at the actual frequency. Moreover, the SSST can obtain the time-varying distribution more accurately. The correlation coefficient obtained by the SSST-TD TF correlation analysis is shown in Figure 13A. Comparing the SSST spectrum with the correlation coefficient, the signals were correlated in the first and third stages. The 30 Hz uncorrelation was also obtained in the second stage. The correlation coefficient of the frequency modulation was obtained as well, as can be seen in the local, enlarged detail in Figure 13B. The outline was clear, and this cannot be attained by traditional correlation analysis. For the non-stationary signals, traditional correlation analysis cannot obtain the accurate correlation. Similarly, the correlation coefficients were affected by noise. The SSST-TD TF correlation analysis not only obtained the correlation between the signals, but also the correlation as it changed with time and frequency. Thus, by analyzing the correlation in the TF plane, the correlations between signals are more accurate and specific. Figure 11: The traditional correlation coefficient between \(x\) and \(y\). Figure 12: The SSST of signals (**A**) \(x\) and (**B**) \(y\). Figure 10: The time domain signals of \(x\) and \(y\). Through a time series decomposition algorithm derived from the SSST, the SSST-TD TF correlation was carried out in the TF plane, and the local correlation between the signals was attained. Compared with the traditional correlation analysis method using sinusoidal signals, sinusoidal signals with noise, and non-stationary signals, the proposed method was validated by stationary and non-stationary signals simulation. The simulation results show that the traditional method can obtain the overall correlation between the signals but cannot reflect the local time and frequency correlations, especially the correlations between the non-stationary signals. The proposed method not only obtained the overall correlations between the signals, but also accurately identified the correlations between the non-stationary signals, revealing more specific and essential correlations between the signals.