## 3 Simulation
### Parameter Optimization  
In order to accurately compare with ACMD, it is proved that the method proposed in this paper optimizes the \(\alpha,\beta\) value in the ACMD. In this paper, the same simulation signal is constructed as ACMD (two signals with large IFs fluctuation):  
\[\left\{\begin{array}{l}S_{1}(t)=\exp(-0.3t)\times\cos(700\pi t+\cos(50\pi t)) \\ S_{2}(t)=\exp(-0.6t)\times\cos(500\pi t+\cos(40\pi t))\\ S(t)=S_{1}(t)+S_{2}(t)\end{array}\right. \tag{16}\]  
The two signals IFs are \(f_{1}(t)=350-25\sin(50\pi t)\) and \(f_{2}(t)=250-20\sin(40\pi t)\), respectively. The time-domain and frequency-domain diagrams of the signal are shown in Figure 6. The two main frequencies in the spectrum (250 and 350 Hz) correspond to the center frequencies of the IFs of the two signal components.  
\(\alpha\) and \(\beta\) of the ACMD processing simulation signal are set to \(1\times 10^{-3}\) and \(1\times 10^{-4}\), respectively. The simulated signals were processed by ACMD to obtain the final estimated signal IFs. The SNRs of IFs1 and S1 (t), and IFs2 and S2 (t) are calculated, respectively, and \(SNR1=23.970,SNR2=29.155\).  
An improved STA was used to optimize \(\alpha\) and \(\beta\) in ACMD, with the SNR as the Z-axis, and C and D as the X- and Y-axes, respectively. The results are shown in Figure 7. The range of the \(\alpha\) value optimization in this paper is \(1\times 10^{-5}-1\times 10^{-2}\), and the range of \(\beta\) value optimization in this paper is \(1\times 10^{-6}-1\times 10^{-3}\). In Figure 7a, \(SNR1=26.16,\alpha=3\times 10^{-5}\), \(\beta\)\(=8.61\times 10^{-4}\). In Figure 7b, \(SNR2=32.63,\alpha=6\times 10^{-5}\), and \(\beta=2.95\times 10^{-4}\). As there are \(10^{6}\) data in the optimization result, the color display of the graph is not obvious, so this paper extracts 400 data points around the largest data point to enlarge the graph. As shown in Figure 7a,b, the maximum point can be clearly observed in the figure.  
Figure 6: Simulation signal ((a) time-domain diagram, (b) frequency domain diagram).  
Figure 7: Instantaneous frequency(IF)s1 and IFs2 optimization process((a). SNR1 for different \(\alpha\) and \(\beta\); (b). SNR2 for different \(\alpha\) and \(\beta\)).  
The figure in this paper is the simulation display of Matlab2016a version, and the data in Figure 7a,b are the approximations obtained by rounding. This article exports 100 detailed data, as shown in Tables 2 and 3, where Table 2 is the IFs1 with different \(\alpha,\beta\) corresponding to SNR data. Table 3 is the IFs2 with different \(\alpha,\beta\) corresponding to SNR data. It can be observed from Table 2 that when \(SNR1=26.159\), the SNR of IFs1 is the largest. When \(SNR2=32.633\), the SNR of IFs2 is the largest in Table 3.  
Based on the above chart, we can obtain such a conclusion: When the SNR of different signals is the largest, they correspond to different \(\alpha\) and \(\beta\) values. Therefore, ACMD uses artificial settings of \(\alpha\) and \(\beta\), and using the same \(\alpha\) and \(\beta\) for different signals has drawbacks. Employing the method proposed in this paper can improve the SNR. The SNR improvement is as follows:  
\[R=\frac{SNR_{i}-SNR_{o}}{SNR_{o}}\times 100\% \tag{17}\]  
where \(R\) is the SNR improvement; \(SNR_{i}\) denotes the SNR of the method proposed in this article; \(SNR_{o}\) denotes the SNR obtained using the ACMD method.  
From Equation (17), we can see that the IFs1 and IFs2 obtained by the method proposed in this paper are 9.134% and 11.931% higher than those obtained by the ACMD method, respectively.