## 2 Methodologies  
### Variational Mode Decomposition  
VMD is a brand-new adaptive time-frequency analysis algorithm based on the ideas of Wiener filtering, Hilbert transform and heterodyne demodulation [36]. This algorithm fulfills the adaptive decomposition of the signal in a completely non-recursive way. It has a strict mathematical theoretical foundation, good anti-noise effect and fast calculation speed [37]. VMD decomposes the signal \(x(t)\) into \(k\) orders of BLIMFs \(u_{k}(t)\). The center frequency of \(k\)th order BLIMF is \(\omega_{k}\). In EMD, the signal's component, whose mean value of the upper and lower envelopes determined by the local maximum and local minimum points equals zero, is defined as IMF, while VMD redefines BLIMF as an AM-FM signal to establish the variational model. The expression of BLIMF is as follows:  
\[u_{k}(t)=A_{k}(t)\cos(\varphi_{k}(t)) \tag{1}\]  
where \(\varphi_{k}^{\prime}(t)=\omega(t)=d\varphi_{k}(t)/dt\) is the instantaneous frequency and \(A_{k}(t)\) is the amplitude. During period \([t-\delta,t+\delta]\), where \(\delta=2\pi/\varphi_{k}^{\prime}(t)\), \(u_{k}(t)\) is regarded as a harmonic signal. The steps of establishing the variational model are as follows:  
1. Hilbert transform is performed on each BLIMF \(u_{k}(t)\). To obtain the unilateral spectrum of \(u_{k}(t)\), an analytical signal is constructed, i.e., \((\delta(t)+j/\pi t)*u_{k}(t)\), where \(\delta(t)\) represents Dirichlet function and \(*\) is the convolutional symbol. 2. The analytical signal of each BLIMF is mixed with a pre-estimated center frequency. Each BLIMF's spectrum is transferred to the fundamental frequency range, i.e., \([(\delta(t)+j/\pi t)*u_{k}(t)]e^{-j\omega_{k}t}\). 3. By demodulating the \(L_{2}\) norm of the signal gradient, the bandwidth of each BLIMF is estimated. 4. By introducing constraints, the following optimal variational model is constructed: \[ \begin{array}{l}\min\limits_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k }\|\,\partial_{t}\Big{[}\Big{(}\delta(t)+\frac{j}{\pi t}\Big{)}u_{k}(t)\Big{]} e^{-j\omega_{k}t}\,\|_{2}^{2}\right\}\\ \text{s.t.}\sum\limits_{k}u_{k}=f\end{array} \] (2) where \(u_{k}=\{u_{1},u_{2},\ldots,u_{k}\}\) are BLIMFs, and \(\omega_{k}=\{\omega_{1},\omega_{2},\ldots,\omega_{k}\}\) are center frequencies corresponding to each BLIMF. The quadratic penalty factor \(\alpha\) and Lagrange multiplication operator \(\lambda\) are utilized to build the extended Lagrange function \(L(\{u_{k}\},\{\omega_{k}\},\lambda)\) to arrive at the best solution for the variational model discussed above:  
\[ \begin{array}{l}L(\{u_{k}\},\{\omega_{k}\},\lambda)=\!\!\!\!\!\!\!\!\!\!\!\! \sum\limits_{k}\!\left\|\partial_{t}\Big{[}\Big{(}\delta(t)+\frac{j}{\pi t} \Big{)}u_{k}(t)\Big{]}e^{-j\omega_{k}t}\right\|_{2}^{2}\\ +\!\!\!\left\|f(t)-\sum\limits_{k}u_{k}(t)\right\|_{2}^{2}+\left\langle\lambda(t ),f(t)-\sum\limits_{k}u_{k}(t)\right\rangle\end{array}  \tag{3}\]  
Thus, the limited variational problem becomes an unconstrained problem. Among this unconstrained problem, the quadratic penalty factor ensures accuracy of signal reconstruction and the Lagrange multiplier ensures the strictness of model constraints. The main steps of the VMD algorithm are as follows:  
1. Input signal \(x(t)\);
2. Initialize \(\hat{u}_{k}^{1},\omega_{k}^{1},\hat{\lambda}_{k}^{1}\) and \(n\);(3) Update \(u_{k}\) and \(\omega_{k}\); \[\hat{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum\limits_{i\
eq k}\hat{u}_{i}( \omega)+\frac{\hat{\lambda}(\omega)}{2}}{1+2\alpha(\omega-\omega_{k})^{2}}\] (4) \[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega|\hat{u}_{k}(\omega)|^{2}d\omega }{\int_{0}^{\infty}|\hat{u}_{k}(\omega)|^{2}d\omega}\] (5) (4) Update \(\lambda\); \[\hat{\lambda}^{n+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau\left[\hat{f}(\omega )-\sum\limits_{k=1}^{k}\hat{u}_{k}^{n+1}(\omega)\right]\] (6) (5) The convergence is judged according to the following equation. If the convergence condition is met, the decomposition process ends.