## 4 Simulation Validation
### Case 2: AM-FM Superposition Signal  
To further prove the efficacy of the proposed IAVMD method, we analyze and investigate a numerical signal \(z(t)\) shown in Equation (19), which is composed of three AM-FM signals and a stochastic noise:  
\[\left\{\begin{array}{l}z(t)=q_{1}(t)+q_{2}(t)+q_{3}(t)+q_{4}(t)\\ q_{1}(t)=(0.8+0.8\sin(30\pi t))\cos(800\pi t+0.6\sin(24\pi t))\\ q_{2}(t)=(1+\sin(20\pi t))\cos(400\pi t+0.6\sin(16\pi t))\\ q_{3}(t)=(0.5+0.5\sin(10\pi t))\cos(200\pi t+0.6\sin(8\pi t))\end{array}\right. \tag{19}\]  
where three components \(q_{1}(t)\), \(q_{2}(t)\), and \(q_{3}(t)\) are AM-FM signals containing the main frequency of 400 Hz, 200 Hz, and 100 Hz, respectively. The component \(q_{4}(t)\) is a stochastic noise added in the numerical signal \(z(t)\), and its value is generated by the \(randn()\) function in MATLAB. The sampling  
Figure 12: The time-frequency representations obtained by different methods for simulation 1: (**a**) OVMD, (**b**) LMD, (**c**) EMD, and (**d**) WT.  
frequency and sampling number of the simulation signal \(z(t)\) are 3000 Hz and 3000, respectively. Figure 13 shows the simulation signal \(z(t)\) and its corresponding FFT spectrum.  
The proposed IAVMD method is adopted to analyze the simulation signal \(z(t)\). First, waveform the matching extension is used to preprocess the simulation signal \(z(t)\), after which the GWO method is employed to determine automatically the VMD parameters \(\hat{\alpha}=1023\) and \(K=3\). Finally, VMD containing the optimal parameters is adopted to analyze the simulation signal \(z(t)\), and the ultimate decomposition results are displayed in Figure 14. The corresponding time-frequency representations are shown in Figure 15. Figure 15a,b shows the two-dimensional and three-dimensional TFR based on IAVMD, respectively. It can be clearly seen from Figures 14 and 15 that three components obtained by the proposed method are very close to three real AM-FM components of the original signal, which indicates that the IAVMD method is effective in analyzing multi-component signals with modulation phenomena.  
As a contrast, four methods (i.e., OVMD, LMD, EMD, and WT) are employed to deal with the same simulation signal \(z(t)\). Note that to consolidate the results of the proposed method two key parameters of OVMD are optimized as \(\hat{\alpha}=751\) and \(K=3\). Besides, the implementation of the TFR of other three methods is the same as that of simulation 1. The TFR obtained by the four methods are plotted in Figure 16a-d, respectively. Seen from Figure 16a, in the first two components obtained by OVMD, there is some deviation from the true component. That is, the first two components have a low resolution in TFR based on OVMD. As shown in Figure 16b,c, the time-frequency trajectory is not clear in the TFR based on EMD and LMD, and their time-frequency aggregation is not as good as in the proposed method. As shown in Figure 16d, the time-frequency trajectories are scattered, and three frequency components of the original signal cannot be identified in TFR based on WT. Therefore, the above comparison results further validate the effectiveness of the proposed method in multi-component signal analysis.  
Figure 16: The time-frequency representations obtained by different methods for simulation 2: (**a**) OVMD, (**b**) LMD, (**c**) EMD, and (**d**) WT.  
Figure 15: The time-frequency representations obtained by IAVMD for simulation 2: (**a**) Two-dimensional and (**b**) three-dimensional.  
In short, the contribution of this section is that two simulation cases are adopted to verify the effectiveness of our proposed IAVMD method. Besides, the superiority of the IAVMD method is demonstrated via the comparisons of various similar methods (e.g., OVMD, LMD, EMD, and WT).