## 2 Methods
### Variational Mode Decomposition  
The variational mode decomposition method aims to decompose a real valued signal into a series of sub-signals with specific sparsity properties of its bandwidth in the spectral domain, while reproducing the input [17]. It is an adaptive and non-recursive signal decomposition approach in which each mode is assumed to be an amplitude modulation-frequency modulation (AM-FM) signal with a bandwidth and corresponding center frequency, i.e.:  
\[u_{k}(t)=A_{k}(t)\cos(\phi_{k}(t)), \tag{1}\]  
where, \(A_{k}(t)\) and \(\phi_{k}(t)\) are the instantaneous amplitude and phase of \(u_{k}(t)\) respectively. The whole process of VMD can be considered as the construction and solution of a constrained variational problem, as described by Equation (2):  
\[\begin{array}{l}\min\biggl{\{}\sum\limits_{k=1}^{K}\left\|\partial_{t}\left[ \left(\delta(t)+\frac{i}{nt}\right)\times u_{k}(t)\right]e^{-i\omega_{k}t} \right\|_{2}^{2}\biggr{\}},\\ s.t.\;\;\sum\limits_{k=1}^{K}u_{k}(t)=x(t)\;\;,\end{array} \tag{2}\]where \(u_{k}\) is the \(k\)th decomposed mode, \(\omega_{k}\) is the center frequency of the corresponding \(k\)th decomposed mode, \(\delta(t)\) is a Dirichlet function, \(x(t)\) is the input signal. The term \((\delta(t)+i/\pi t)\times u_{k}(t)\) is the Hilbert transform of \(u_{k}\).  
A quadratic penalty term \(\alpha\) and Lagrangian multipliers \(\lambda\) are adopted to render the variational problem unconstrained; the quadratic penalty term can guarantee better convergence properties, even in the presence of Gaussian noise, and the Lagrangian multiplier is used to enforce a strict constraint. The augmented Lagrangian is described as follows:  
\[\begin{split} L(u_{k},\omega_{k},\lambda)=\alpha\sum\limits_{k=1 }^{K}\|\partial_{t}\Big{[}\Big{(}\delta(t)+\frac{i}{\pi t}\Big{)}\times u_{k} (t)\Big{]}e^{-i\omega_{k}t}\|_{2}^{2}\\ +\|x(t)-\sum\limits_{k=1}^{K}u_{k}(t)\|_{2}^{2}+\left\langle \lambda,x(t)-\sum\limits_{k=1}^{K}u_{k}(t)\right\rangle\.\end{split} \tag{3}\]  
where \(\alpha\) is a parameter balancing the variational energy term and the fidelity term. The alternating direction method of the multipliers (ADMM) can be employed to solve Equation (3). The modes and the center frequencies are updated iteratively by Equations (4) and (5), respectively:  
\[\hat{u}_{k}^{n+1}=\frac{\hat{x}-\sum_{j\
eq k}\hat{u}_{j}+\big{(}\hat{\lambda}/ 2\big{)}}{1+2\alpha\big{(}\omega-\omega_{k}\big{)}^{2}}. \tag{4}\]  
\[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega\big{|}\hat{u}_{k}(\omega)\big{|} ^{2}d\omega}{\int_{0}^{\infty}\big{|}\hat{u}_{k}(\omega)\big{|}^{2}d\omega}. \tag{5}\]  
where \((\hat{\cdot})\) indicates a Fourier-transformed signal. The Lagrangian multiplier will also be updated according to the following equation after each epoch:  
\[\lambda^{n+1}=\lambda^{n}+\tau\Bigg{(}\hat{x}-\sum\limits_{k}\hat{u}_{k}^{n+1 }\Bigg{)}. \tag{6}\]  
Finally, the above iteration terminates when the convergence condition is satisfied, namely:  
\[\sum\limits_{k}\|\hat{u}_{k}^{n+1}-\hat{u}_{k}^{n}\|_{2}^{2}/\|\hat{u}_{k}^{n }\|_{2}^{2}<\varepsilon. \tag{7}\]  
According to the preceding descriptions, there are four parameters that must be specified in advance: the mode number \(K\), the balancing parameter \(\alpha\), the constraint intensity \(\tau\), and the tolerance of convergence criterion \(\varepsilon\). The detailed complete algorithm of VMD can be found in [17]. It is important to note that VMD decomposes the input signal trace by trace, the lateral continuity is not considered.