## 3 Variational Specific Mode Extraction
### Performance Analysis  
In this subsection, a theoretical analysis of VSME performance is provided, including quantitative analysis of instantaneous frequency estimation error and convergence conditions.  
Firstly, the problem of instantaneous frequency quantization is discussed, \(\varphi_{(m)}(t)\) is the phase function of the original signal \(g(t)\) in step \(m\) of iteration, and its corresponding demodulation operator is \(f_{d_{j}(m)}(t,u)=e^{-i\Delta\varphi_{(m),u}(t)}\). where phase \(\Delta\varphi_{(m),u}(t)\) is the remainder of the first-order Taylor expansion of estimated phase \(\varphi_{(m)}(t)\) in the vicinity of \(u\), as follows:  
\[\begin{array}{ll}\Delta\varphi_{(m),u}(t)&=\varphi_{(m)}(t)-\left[\varphi_{( m)}(u)+r\varphi_{(m)}(u)(t-u)\right]\\ &=\sum_{k=2}^{\infty}\varphi_{(m)}^{(k)}(u)\frac{(t-u)^{k}}{k!}\end{array} \tag{31}\]  
The time-frequency energy density of a signal can be defined as:  
\[P_{(m)}(t,\omega)=T^{2}\sum_{n_{1}=-\infty}^{+\infty}\sum_{n_{2}=-\infty}^{+ \infty}\left[g_{d}(t+n_{1}T,t)\overline{g_{d}(t+n_{2}T,t)\times s(n_{1}T)s(n_ {2}T)e^{i\omega(n_{2}-n_{1})T}}\right] \tag{32}\]  
where \(T\) is the sampling interval, \(nT\) stands for the discrete sequence, \(s\) represents the Gaussian function, and \(\omega\) is the angular frequency.  
In the time-frequency dimension, the value of instantaneous angular frequency can be estimated as:  
\[\begin{array}{ll}\hat{\omega}_{(m)}(t)=\underset{\omega}{\text{argmax}}P_{ (m)}(t,\omega)\end{array} \tag{33}\]  
General expression of instantaneous angular frequency estimation error:  
\[\begin{array}{ll}\Delta\hat{\omega}_{(m)}(t)&=\omega_{(m)}(t)-\omega(t)\\ &=-\left[\left.\frac{\partial P_{(m)}(t,\omega)}{\partial\omega}\right|_{0}+ \left.\frac{\partial P_{(m)}(t,\omega)}{\partial\omega}\right|_{0}\delta_{ \Delta\phi,\Delta\varphi}+\left.\frac{\partial P_{(m)}(t,\omega)}{\partial \omega}\right|_{0}\delta_{\varepsilon}\right]/\left[\left.\frac{\partial^{2}P _{(m)}(t,\omega)}{\partial\omega^{2}}\right|_{0}\right]\end{array} \tag{34}\]  
where \(\delta_{\Delta\phi,\Delta\varphi}\) is the error caused by the residual error of deviation, and \(\left.\partial P_{(m)}(t,\omega)/\partial\omega\right|_{0}\delta_{\varepsilon}\) is caused by the noise.  
For the problem of convergence condition, the function expression of original signal phase \(\varphi(t)\) can be established by the determined parameter \(\alpha\), i.e., the phase function is \(\varphi(t,\alpha)\). The \(m-th\) phase function \(r_{(m)}(t)\) is recorded as \(\varphi_{(m)}(t,\alpha)\), and the deviation \(\Delta\hat{\omega}_{(M)}(t)\) of IF estimation error of the \(M-th\) component is smaller than other \(\Delta\hat{\omega}(t)\) values on the spectrum. Thus, the estimated instantaneous frequency \(r^{\prime}_{(M+1)}(t)=\varphi^{\prime}\big{(}t,\alpha_{(M+1)}\big{)}\) is closer to the true IF \(\varphi(t,\alpha)\), i.e., the estimated parameter \(\alpha_{(M+1)}\) of the IF function \(\varphi^{\prime}\big{(}t,\alpha_{(M+1)}\big{)}\) is closer to the actual parameter \(\alpha\) of the true IF function \(\varphi^{\prime}(t,\alpha)\). In short, the implementation of iterative procedure will gradually match the true IF, and eventually converge.  
In addition, increasing the variance of Gaussian function would reduce the variance of the IF estimation error, but would increase its bias. To solve this problem, the variance of Gaussian function is solved by the optimization problem of minimizing the mean square error of the IF estimation:  
\[\sigma(t)=\underset{\alpha}{\text{argmin}}E\Big{\{}(\Delta\hat{\omega}_{m}(t ))^{2}\Big{\}} \tag{35}\]  
where the variance \(\sigma(t)\) varies with the third-derivative of the phase function \(\varphi(t)\) and the second-derivative of estimated IF \(r_{(m)}(t)\). Since the difference \(\varphi^{(3)}(t)-r_{(m)}^{(2)}(t)\) is much smaller than the third-derivative \(\varphi^{(3)}(t)\) of phase function, thus a time invariant variance of the window function can minimize the estimation error of the IF.