## 4 Results and Analysis
### Method Validation  
In this subsection, we used VSME to analyze a artificial signal, which is a intrawave amplitude modulated and frequency modulated signal. Artificial data generated from mathematical expression were calculated to evaluate the performance of VSME. All the calculation work was running in MATLAB R2016b. The artificial signal contains a signal whose frequency increases linearly with time, a signal whose frequency decreases linearly with time, and a signal with time-varying frequency and amplitude, as shown in Equation (36). \[f(t)=f_{ig1}(t)+f_{ig2}(t)+f_{ig3}(t) \tag{36}\]  
\[f_{ig1}(t)=\cos\!\left(2\pi\!\left(0.3t^{2}+12t+0.6\right)\right) \tag{37}\]  
\[f_{ig2}(t)=\left(0.4\cos\!\left(0.2\pi t\right)-1\right)\times\cos\!\left(2 \pi\!\left(5\!\left(\sin\frac{\pi t}{4}\right)\right)+0.8t^{2}+5t\right) \tag{38}\]  
\[f_{ig3}(t)=\cos\!\left(2\pi\!\left(-0.9t^{2}+30t+0.55\right)\right) \tag{39}\]  
For signal frequency extraction, we used VSME and RPRG (Ridge Path Regrouping) [42] algorithm to compare the extraction effect. RPRG method is a novel non-parametric algorithm to extract IF from signal with overlapping components in time-frequency representation. Figure 2a is an image of a mixed signal \(f(t)\), and Figure 2b shows the IF of original component signal \(f_{ig1}(t)\), \(f_{ig2}(t)\) and \(f_{ig3}(t)\). The IF of component signal extracted by RPRG is shown in Figure 2e. It is observed that at the intersection of IF of three component signals, an abnormal shape is generated. Using VSME to extract the IF of mode component, we can get a better result than RPRG, as shown in Figure 2c. To quantify the separation effect of VSME and RPRG on mixed signals \(f(t)\), Figure 2d,f show the absolute error between the estimated IF extracted by the two methods and the true IF. Figure 2: Frequency analysis of estimated component signal. (**a**) Original artificial signal. (**b**) Frequency of original component signal in time domain. (**c**) Frequency of estimated component signal extracted by RPRG. (**d**) Frequency of estimated component signal extracted by VSME. (**e**) Absolute error of estimated component signal frequency extracted by VSME. (**f**) Absolute error of estimated component signal frequency extracted by RPRG. To compare with different methods, we employed VSME and VMD to extract the mode components of original signal \(f(t)\). The red line shown in Figure 3a represents the estimated three mode components, and the blue line represents the absolute error between the estimated modes and the original modes in the vicinity of plane \(y=0\). The absolute error range is less than 0.05, and the relative error is not more than 1%. By comparing Figure 3a,b, we can conclude that the VMD algorithm has a poor separation result on mixed signals \(f(t)\). In this paper, the performance of VSME is studied by adding white noise to the artificial signal \(f(t)\). The signal-to-noise ratio (SNR) is defined as:  
\[SNR=10\log_{10}(P_{x}/P_{x}) \tag{40}\]  
where \(P_{x}\) represents the energy of noiseless signal, and \(P_{x}\) is the energy of the noise signal. The mean value of \(\varepsilon(t)\) is 0, and its variance \(\sigma_{\varepsilon}^{2}=1\). Figures 4 and 5 analyze the mode extraction performance of VSME algorithm for mixed signal \(f(t)\), when the SNR is 12 dB and 18 dB respectively. The results show that the algorithm is effective when the SNR is 18 dB; when the SNR is lower than 12 dB, the IF has obvious fluctuation effect in the time domain, as shown in Figure 4c. We will continue to optimize the algorithm in the future work, aiming at extract mode components more accurately in low SNR condition. The root mean squared error (RMSE) is used to measure the filtered signal error under different SNR levels, which is defined as:  
\[RMSE=\sqrt{\frac{1}{m}\sum_{1}^{m}(\hat{g}_{k}(t)-g_{k}(t))} \tag{41}\]  
where \(\hat{g}_{k}(t)\) is \(k-th\) estimated signal component, and \(g_{k}(t)\) is the true signal.