## 2 Fundamental Theories  
### Variational Mode Decomposition  
Variational mode decomposition (VMD) is a novel non-recursive signal decomposition method [38], whose essence is to solve a variational optimization problem. Setting a scale \(K\) in advance, a given signal can be decomposed into \(K\) band-limited intrinsic mode functions (IMFs), among which each sub-signal is related to a certain center frequency. The constrained variation problem can be described as follows:  
\[\begin{array}{l}\min\limits_{m_{k}\omega_{k}}\left\{\sum\limits_{k=1}^{K} \left\|\partial_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)\star m_{k}(t) \right]e^{-j\omega_{k}t}\right\|_{2}^{2}\right\}\\ s.t.\sum\limits_{k=1}^{K}m_{k}=f,\;\;k=1,2,\ldots,K\end{array} \tag{1}\]  
where \(m_{k}=\{m_{1},m_{2},\ldots,m_{k}\}\) represents the set of all mode functions, and \(\omega_{k}=\{\omega_{1},\omega_{2},\ldots,\omega_{k}\}\) represents the set of center frequencies, while \(\partial_{t}\) and \(\delta(t)\) are the partial derivative of time \(t\) for the function and unit pulse function, respectively. \(f\) is the given real valued input signal.  
Lagrange multipliers are introduced to obtain the optimal solution of above constrained variational problem. Then problem (1) can be specified as follows:  
\[\begin{array}{l}L(m_{k},\omega_{k},\beta) =\alpha\sum\limits_{k=1}^{K}\left\|\partial_{t}\left[\left(\delta( t)+\frac{j}{\pi t}\right)\star m_{k}(t)\right]e^{-j\omega_{k}t}\right\|_{2}^{2}\\ +\left\|f(t)-\sum\limits_{k}^{K}m_{k}(t)\right\|_{2}^{2}+\left(\beta(t),f(t)- \sum\limits_{k}^{K}m_{k}(t)\right)\end{array} \tag{2}\]  
where \(\alpha\) and \(\beta(t)\) represent the penalty factor and Lagrange multiplier respectively.  
Next, the alternate direction method of multipliers (ADMM) and iterative search are utilized to obtain the saddle point of Lagrange multipliers [39]. The optimization problems of \(m_{k}\) and \(\omega_{k}\) are formulated as (3) and (4) respectively.  
\[m_{k}^{n+1}=\min\left\{\begin{array}{c}\alpha\Big{\|}\partial_{t}\Big{[} \Big{(}\delta(t)+\frac{j}{\pi t}\Big{)}\ast m_{k}(t)\Big{]}e^{-j\omega_{k}t} \Big{\|}_{2}^{2}\\ +\Big{\|}f(t)-\sum\limits_{i}m_{i}(t)+\frac{\delta(t)}{2}\Big{\|}_{2}^{2}\end{array}\right\} \tag{3}\]  
\[\omega_{k}^{n+1}=\min\left\{\Big{\|}\partial_{t}\Bigg{[}\Bigg{(}\delta(t)+ \frac{j}{\pi t}\Big{)}\ast m_{k}(t)\Big{]}e^{-j\omega_{k}t}\Big{\|}_{2}^{2}\right\} \tag{4}\]  
Solving problems (3) and (4), the iterative equations are deduced as follows:  
\[\hat{m}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum_{i\
eq k}\hat{m}_{i}( \omega)+\frac{\hat{\beta}(\omega)}{2}}{1+2\alpha(\omega-\omega_{k})^{2}} \tag{5}\]  
\[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega\big{|}\hat{m}_{k}(\omega) \big{|}^{2}d\omega}{\int_{0}^{\infty}\big{|}\hat{m}_{k}(\omega)\big{|}^{2}d\omega} \tag{6}\]  
The Lagrange multipliers can be iterated with Equation (7).  
\[\hat{\beta}^{n+1}(\omega)=\hat{\beta}^{n}(\omega)+\tau(f(\omega)-\sum_{k} \hat{m}_{k}^{n+1}(\omega)) \tag{7}\]  
where \(\tau\) is an updating parameter.  
The main steps of VMD can be summarized as follows:  
Step 1: Initialize \(m_{k}^{1}\), \(\omega_{k}^{1}\), \(\beta^{1}\), \(n=1\);  
Step 2: Execute loop, \(n=n+1\);  
Step 3: Update \(m_{k}\) and \(\omega_{k}\) based on Equations (5) and (6);  
Step 4: Update \(\beta\) based on Equation (7);  
Step 5: If \(\sum\limits_{k}\Big{\|}\hat{m}_{k}^{n+1}-\hat{m}_{k}^{n}\Big{\|}_{2}^{2}/\| \hat{m}_{k}^{n}\|_{2}^{2}<\varepsilon\) loop end, else turn to step 2 for next iteration.