## 3 Variational Specific Mode Extraction
### Algorithm  
In this section, we introduce the procedure of variational specific mode extraction algorithm in detail. To enhance the robustness of this algorithm, we introduce auxiliary variables \(\xi\), i.e., signals with noise. Then, we transform the problem in Equation (21) into:  
\[\min_{\{|u_{k}(t)\},|v_{k}(t)|,|f_{k}(t)|,\xi}\left\{\sum_{k=1}^{n}\left[\|u^{ \prime\prime}{}_{k}(t)\|_{2}^{2}+\|v^{\prime\prime}{}_{k}(t)\|_{2}^{2}\right] +\psi(\xi)\right\}. \tag{22}\]  
where we explicitly consider the influence of noise by adding a noise function \(\psi(\xi)\) in Equation (22), ensuring that this method can still be effective in noisy environment. To solve the above constrained optimization problem, augmented Lagrangian can be transformed into:  
\[ \begin{array}{rl}\mathrm{L}(\{u_{k}(t)\},\{v_{k}(t)\},f_{k}(t),\xi,\lambda)& =\sum\limits_{k}\left[\|u^{\prime\prime}{}_{k}(t)\|_{2}^{2}+\|v^{\prime\prime }{}_{k}(t)\|_{2}^{2}\right]+\lambda^{T}\!\!\left(\sum\limits_{k}\left[u^{ \prime\prime}{}_{k}(t)+v^{\prime\prime}{}_{k}(t)\right]+\xi-g_{k}(t)\right)\\ &+\frac{\pi}{2}\|\sum\limits_{k}\left[u^{\prime\prime}{}_{k}(t)+v^{\prime \prime}{}_{k}(t)\right]+\xi-g_{k}(t)\|_{2}^{2}+\psi(\xi)\end{array}  \tag{23}\]  
where \(\lambda\) is a Lagrangian multiplier, and \(\alpha\) is a quadratic penalty parameter. Then, we use alternating direction method of multipliers (ADMM), which is a distributed computing frame [37], to solve this problem. This method is suitable for solving distributed convex optimization problems, and it integrates the advantages of dual ascent method and method of multipliers. The purpose of ADMM is to decompose the original function and the extended function. When the objective function is complex, ADMM can divide the objective function into several sub-problems. The general expression of ADMM problem solving is:  
\[ \begin{array}{rl}\min(b_{1}(x_{1})+b_{2}(x_{2})+\cdots b_{m}(x_{m}))\\ s.t.b_{1}x_{1}+b_{2}x_{2}+\cdots+b_{m}x_{m}=B\end{array}  \tag{24}\]  
where \(b_{m}\) is the sub-function decomposed by ADMM, and \(x_{m}\) is its corresponding variable. For the improved ADMM to solve optimization details, we will describe in the following content. Firstly, VSME algorithm is to update the auxiliary variables, thus, the sub-problem to be solved is as follows:  
\[ \begin{array}{rl}\xi&=\underset{\xi}{\mathrm{argmin}}\mathrm{L}(\{u_{k}(t)\},\{v_{k}(t)\},\{f_{k}(t)\},\xi,\lambda)\\ &=\underset{\xi}{\mathrm{argmin}}\left\{\psi(\xi)+\frac{\alpha}{2}\|\sum \limits_{k}\left[u^{\prime\prime}{}_{k}(t)+v^{\prime\prime}{}_{k}(t)\right]+ \xi-g_{k}(t)+\frac{1}{\alpha}\lambda\|_{2}^{2}\right\}\end{array}  \tag{25}\]  
The second step is to update two orthogonal demodulation signals \(u_{k}(t)\) and \(v_{k}(t)\), as follow:  
\[\begin{array}{rl}u_{k}^{n+1}&=\underset{u_{k}}{\mathrm{argmin}}\mathrm{L}( \{u_{i}(t)\},\{v_{i}(t)\},\{f_{i}(t)\},\xi,\lambda)\\ &=\underset{u_{k}}{\mathrm{argmin}}\left\{\|\Lambda u_{i}\|_{2}^{2}+\frac{ \alpha}{2}\|\sum\limits_{i}\left[u^{\prime\prime}{}_{i}(t)+v\[v_{k}^{n+1} =\underset{u_{k}}{\operatorname{argmin}}\mathrm{L}([u_{i}(t)],[v_{i} (t)],[f_{i}(t)],\xi,\lambda) \tag{27}\] \[=\underset{u_{k}}{\operatorname{argmin}}\left\{\|\Lambda v_{i}\|_ {2}^{2}+\frac{g}{2}\|\underset{i}{\Sigma}\left[u^{\prime\prime}{}_{i}(t)+v^{ \prime\prime}{}_{i}(t)\right]+\xi-g_{i}(t)+\frac{1}{\lambda}A\|_{2}^{2}\right\}\]  
where \(\Lambda\) is the second-order difference operator. We use Equations (26) and (27) to update the corresponding mode components. Next, the instantaneous frequency increment is obtained by arc-tangent demodulation method, and it is updated by:  
\[f_{k}^{n+1}(t)=-\frac{1}{2\pi}\frac{d}{dt}\Bigg{(}\mathrm{arctan}\Bigg{(}\frac{ v_{k}^{n+1}(t)}{u_{k}^{n+1}(t)}\Bigg{)}\Bigg{)} \tag{28}\]  
Since the instantaneous frequency fluctuation is a smooth function, we can update the instantaneous frequency by:  
\[f_{k}^{n+1}=f_{k}^{n}+\chi\Delta f_{k}^{n+1} \tag{29}\]  
where \(0<\chi<1\) is scale factor for stabilizing algorithm. We let \(\chi=0.5\), the algorithm can provide good results.