## 2 Variational Mode Decomposition  
VMD is a solution process for variation problems based on classical Weiner filtering, Hilbert transformation, and frequency mixture. Adaptive signal decomposition is implemented by identifying the optimal solution of the constrained variation model. The input signal is decomposed into multiple sparse mode components. Assume that each mode has limited bandwidth with a central frequency; the central frequency and bandwidth are updated continuously during the decomposition. VMD is to identify the mode function \(u_{k}(t)\) with the minimum sum of the \(K\) estimated bandwidths; the sum of the modes is the input signal \(f\). 1. Each mode function \(u_{k}(t)\) undergoes a Hilbert transformation to obtain the analytical signal of each intrinsic mode function and the unilateral frequency spectrum: \[\left[\delta(t)+\frac{j}{\pi t}\right]*u_{t}(t).\] (1)
2. The estimated central frequencies \(e^{-j\eta_{k}t}\) of all the analytical mode signals are merged, and the corresponding mode frequency spectrum is modulated in the respective base frequency band: \[\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{t}(t)\right]e^{-j\eta_{k}t}.\] (2)
3. The square norm \(L^{2}\) of the gradient of the demodulated signal is calculated, and the bandwidth of each mode component is estimated. The expression for the corresponding constrained variation model is as follows: \[\left\{ \begin{array}{ll}\min\limits_{\{u_{k}\},\{\omega_{k}\}}&\left\{ \sum\limits_{k}\|\partial_{t}\big{[}\left(\delta(t)+\frac{j}{\pi t}\right)*u_{ t}(t)\big{]}e^{-j\eta_{k}t}\big{\|}^{2}\right\}\\ s.t&\sum\limits_{k}u_{k}=f\end{array} \right.\] (3) where \(\{u_{k}\}\) represents the \(K\) IMF components after decomposition, \(\{u_{k}\}=\{u_{1},\cdots u_{k}\}\), and \(\{\omega_{k}\}\) represents central frequency of each component, \(\{\omega_{k}\}=\{\omega_{1},\cdots\omega_{k}\}\). The second-order penalty factor \(\partial\) and Lagrange multiplying operator \(\lambda(t)\) are introduced. The second-order penalty factor guarantees signal reconfiguration accuracy in environments with Gaussian noise. The Lagrange operator ensures the strictness of the constrained condition. The extended Lagrange expression is as follows:  
\[ \begin{array}{ll}L(\{u_{k}\},\{\omega_{k}\},\lambda):&=\alpha\sum\limits_{ k}\|\partial_{t}\Big{[}\left(\delta(t)+\frac{j}{\pi t}\right)*u_{t}(t)\Big{]}e^{-j \eta_{k}t}\big{\|}^{2}\\ &+\|f(t)-\sum\limits_{k}u_{k}(t)\|^{2}+\left\langle\lambda(t),f(t)-\sum\limits _{k}u_{k}(t)\right\rangle\end{array}  \tag{4}\]where, \(\alpha\) is the penalty factor and \(\lambda(t)\) is the Lagrange operator. The \"saddle point\" of extended Lagrange expression is calculated via the alternating direction method of multiplier (ADMM) algorithm. The detailed procedure is as follows:  
1. Initialize \(\left\{\widetilde{u}_{k}^{1}\right\}\), \(\left\{\omega_{k}^{1}\right\}\), \(\widetilde{\lambda}^{1}\), \(n\);
2. Execution cycle: \(n=n+1\);
3. For all \(\omega\geq 0\), update \(\widetilde{u}_{k}\) : \[\widetilde{u}_{k}^{n+1}(\omega)\leftarrow\frac{\widehat{f}(\omega)-\sum_{i<k }\widetilde{u}_{i}^{n+1}(\omega)-\sum_{i>k}\widetilde{u}_{i}^{n}(\omega)+ \frac{\widetilde{\lambda}^{n}(\omega)}{2}}{1+2\alpha(\omega-\omega_{k}^{n})^{ 2}},k\in\left\{1,K\right\}\] (5)
4. Update \(\omega_{k}\): \[\omega_{k}^{n+1}\leftarrow\frac{\int_{0}^{\infty}\omega\left| \widetilde{u}_{k}^{n+1}(\omega)\right|^{2}d\omega}{\int_{0}^{\infty} \left|\widetilde{u}_{k}^{n+1}(\omega)\right|^{2}d\omega},k\in\left\{1,K\right\}\] (6)
5. Update \(\lambda\): \[\widetilde{\lambda}^{n+1}(\omega)\leftarrow\widetilde{\lambda}^{n}(\omega)+ \tau\left(\widetilde{f}(\omega)-\sum_{k}\widetilde{u}_{k}^{n+1}(\omega)\right)\] (7)
6. Repeat steps (2) through (5) until the iteration stop condition is satisfied: \[\sum_{k}\left\|\widetilde{u}_{k}^{n+1}-\widetilde{u}_{k}^{n}\right\|_{2}^{2} /\left\|\widetilde{u}_{k}^{n}\right\|_{2}^{2}<\epsilon\] (8)
7. Stop iteration and obtain the \(K\) IMF components. Before VMD, the number of modes \(K\) should be determined.