## 3 Variational Specific Mode Extraction  
### Main Idea  
The proposed algorithm is based on the premise that the wide-band signal can be converted into narrow-band signal, and matching demodulation transform technique is used to modulate each component mode to its corresponding position. The IF of signal \(z_{k}(t)\) in Equation (3) is  
\[f_{i,g}(t)=f_{c}+\varphi^{\prime}(t)/2\pi \tag{13}\]  
Modulation source \(\varphi(t)\) in Equation (3) can be expressed as  
\[\varphi(t)=\varphi(u)+\varphi^{\prime}(u)(t-u)+\Delta\varphi_{u}(t) \tag{14}\]  
In the process of signal demodulation, a bivariate function of time \(t\) and time-shift \(u\) is introduced  
\[f_{d}(t,u)=e^{-i[\varphi(t)-\varphi(u)-\varphi^{\prime}(u)(t-u)]} \tag{15}\]  
as a backward MO to transform the mode component \(z_{k}(t)\) into a bivariate function:  
\[ \begin{array}{ll}z_{k}^{d}(t,u)&=z_{k}(t)\cdot f_{d}(t,u)\\ &=A(t)e^{i[2\pi f_{c}+\varphi(u)+\varphi^{\prime}(u)(t-u))}\end{array}  \tag{16}\]The one-dimensional signal, which only takes time as a function, is transformed into a two-dimensional bivariate function of time variable and time-shift variable. The IF of bivariate function can be expressed as follows:  
\[f_{i,g_{k}}(t)=\frac{1}{2\pi}\frac{\partial}{\partial t}\{2\pi f_{c}t+\varphi(u )+\varphi^{\prime}(u)(t-u)\} \tag{17}\]  
where \(f_{i,g_{k}}(t)\) is bivariate MO, and it suppresses the high order terms of \(\varphi(t)\). The instantaneous frequency of signal is the same as that of the corresponding time instant \(u\), i.e., \(f_{i,g_{k}}(t)=f_{i,g}(u)\). The demodulation process can be performed in two steps, as shown in Figure 1. In the first step, the original signal is demodulated to carrier signal \(A_{k}(t)e^{i2\pi/t}\) by using forward DO \(e^{-i\varphi(t)}\), i.e., the nonlinear instantaneous frequency is converted to the constant frequency. Therefore, the spectrum of demodulated signal is more concentrated than that of original signal. However, there is a problem in the demodulated spectrum, i.e., it cannot reflect the actual time-frequency characteristics of the original signal. Therefore, the demodulated signal needs to be further modulated into another pure carrier signal by the backward MO \(e^{-i[\varphi(t)-\varphi(u)-\varphi^{\prime}(u)(t-u)]}\) to get a bivariate function about time \(t\) and time-shift \(u\) in the second step. As a whole, the procedure can estimate instantaneous frequency of mode component, with which the demodulated baseband signal has the narrowest frequency band. To evaluate the baseband signal bandwidth, we calculate the 2-Norm of signal derivative, and express the mixed signal \(g(t)\) as follows:  
\[g(t)=\sum_{k=1}^{n}\left[u_{k}(t)\cos\left(2\pi\int_{0}^{t}\dot{f}_{k}(s)ds \right)+v_{k}(t)\cos\left(2\pi\int_{0}^{t}\dot{f}_{k}(s)ds\right)\right] \tag{18}\]  
where \(\dot{f}_{k}(t)\) stands for the frequency functions of demodulation operators, and \(u_{k}(t)\) and \(v_{k}(t)\) can be obtained as:  
\[u_{k}(t)=A_{k}(t)\cos\left(2\pi\int_{0}^{t}\left(f_{k}(s)-f_{k}(s)\right)ds+ \phi_{k}\right) \tag{19}\]  
\[v_{k}(t)=-A_{k}(t)\sin\left(2\pi\int_{0}^{t}\left(f_{k}(s)-f_{k}(s)\right)ds+ \phi_{k}\right) \tag{20}\]  
where the instantaneous amplitude can be given as \(A_{k}(t)=\sqrt{{u_{k}}^{2}(t)+{v_{k}}^{2}(t)}\). Figure 1: Description of demodulation procedure (Purple line: instantaneous frequency of the original FM signal; red line: instantaneous frequency of the forward demodulated signal; blue line: instantaneous frequency of the backward modulation signal.). Before demodulation, the spectrum of original signal is distributed around the frequency \(f_{c}+\varphi^{\prime}(u)/2\pi\). Furthermore, the evaluation of signal bandwidth is expressed as:  
\[\min_{\{|u_{k}(t)\},|v_{k}(t)\},|f_{k}(t)|}\left\{\sum_{k=1}^{n}\sqrt{\|u^{ \prime\prime}{}_{k}(t)\|_{2}^{2}+\|v^{\prime\prime}{}_{k}(t)\|_{2}^{2}}\right\} \tag{21}\]  
where the square root of 2-Norm of second-derivative is employed to evaluate the mode component bandwidth.