## 2 Theoretical Background  
### Variational Mode Decomposition  
Different from EMD, VMD is an adaptive and non-recursive method that can analyze both nonstationary and nonlinear signals. The scope of VMD decomposes a signal into band-limited sub-signals by using a non-recursive calculation method. Each decomposed sub-signal has certain sparse properties in the frequency domain. Additionally, the decomposed sub-signal is defined as the mode as well as EMD. Moreover, each sub-signal is assumed to be concentrated around a corresponding frequency centre. Thus, the bandwidth of each sub-signal can be chosen by utilizing the \(H1\) Gaussian smoothness for the transformed signal.  
VMD first utilizes the Hilbert transform to convert each mode \(u_{k}\) into an analytical expression \(u_{k}{}^{+}\) in a single-sided spectral domain to obtain the bandwidth of each mode function:  
\[u_{k}^{+}(t)=\left(\delta(t)+\frac{j}{\pi t}\right)\ast u_{k}(t) \tag{1}\]  
After the Hilbert transformation, the frequency spectrum of each mode is shifted to the baseband and the corresponding estimated centre frequency \(\omega_{k}\) is adjusted by using an exponential tuned term. Subsequently, the bandwidth is estimated according to the Gaussian smoothness of the demodulated signal by utilizing the squared L2-norm of the gradient [25]. Thus, the VMD process is realized by solving a constrained variational problem [26]:  
\[\min_{\{u_{k}\},\omega_{k}\}\left\{\sum_{k=1}^{K}\left\|\partial\left[\left( \delta(t)+\frac{j}{\pi t}\right)\ast u_{k}(t)\right]e^{-j\omega_{k}t}\right\| ^{2}_{2}\right\},\text{ {subject to}}\sum_{k=1}^{K}u_{k}(t)=f(t) \tag{2}\]  
where \(f(t)\) is the target signal, \(\{u_{k}\}:=\{u_{1},\ldots,u_{K}\}\) represents the set of the decomposed modes, and \(\{\omega_{k}\}:=\{\omega_{1},\ldots,\omega_{K}\}\) represents the respective centre frequencies, respectively. Subsequently, the constraint optimization problem in Equation (2) can be converted into an unconstrained problem via a quadratic penalty term and Lagrangian multipliers, as below:  
\[\mathcal{L}(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\sum\limits_{k=1}^{K}{\left| \partial\left[\left(\delta(t)+\frac{j}{m}\right)\ast u_{k}(t)\right]\right|^{2} +\left\|f(t)-\sum\limits_{k=1}^{K}{u_{k}(t)}\right\|^{2}_{2}+\left\langle \lambda(t),f(t)-\sum\limits_{k=1}^{K}{u_{k}(t)}\right\rangle} \tag{3}\]  
To solve the original minimization problem, the alternate direction method of multipliers (ADMM) is adopted to determine the saddle point of the augmented Lagrangian in a sequence of iterative suboptimizations. [21] summarizes the details of the VMD algorithm. By this process, all of the mode functions are obtained and updated by Wiener filtering to tune the centre frequency in the spectral domain:  
\[\begin{array}{l}\hat{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\sum_{i\in k }{\hat{u}_{i}^{n+1}(\omega)-\sum_{j\in k}{\hat{u}_{i}^{n}(\omega)}+\frac{1^{2 }(\omega)}{2}}}{1+2\alpha\left(\omega-\omega_{k}^{n}\right)^{2}}\quad(\omega>0 )\\ \omega_{k}^{n+1}=\frac{\sum_{i\in k}^{\infty}{\omega}{\left|{\hat{u}_{k}^{n+1}( \omega)}\right|^{2}}d\omega}{\int_{0}^{\infty}{\left|{\hat{u}_{k}^{n+1}(\omega )}\right|^{2}}d\omega}\end{array} \tag{4}\]  
The centre frequency \({\omega_{k}}^{n}\) is calculated from the weighted centre of each mode in the spectral domain: the centre represents the frequency of the least squares linear regression of the instantaneous phase. Figure 1 presents the VMD algorithm as well as Appendix A.