## 2 Methodology
### Decomposition Methods
#### 2.1.4 Variational Mode Decomposition  
Variational mode decomposition (VMD) [51], a newly non-recursive signal processing technique, is utilized to adaptively decompose a real valued signal into a discrete number of band-limited sub-signals, namely the modes \(y_{k}\), having specific sparsity properties. Each mode decomposed by the VMD approach can be compressed around a center pulsation \(w_{k}\), which is determined along with the decomposition process. To estimate the bandwidth of each mode, the following procedures should be considered: (1) as to each mode \(y_{k}\), applying the Hilbert transform to calculate the associated analytic signal so that a unilateral frequency spectrum can be obtained; (2) mixing with an exponential tuned to the respective estimated center frequency in order to shift the mode's frequency spectrum to baseband; and (3) estimating the bandwidth for each mode \(y_{k}\) by using the H1 Gaussian smoothness of the demodulated signal. Thus, the constrained variational problem can be presented as follows:  
\[\min_{\{y_{k}\},\{w_{k}\}}\left\{\sum\limits_{k=1}^{\text{K}}\|\partial_{t} \left[\left(\delta(t)+\frac{j}{\pi t}\right)\otimes y_{k}(t)\right]e^{-jn_{k} t}\|_{2}^{2}\right\}s.t.\sum\limits_{k=1}^{K}y_{k}=f(t) \tag{10}\]  
where \(f(t)\) is the original main signal, \(y_{k}\) is the \(k\)th component of the original signal; \(w_{k}\), \(\partial(t)\) and \(\otimes\) represent center frequency of \(y_{k}\), the Dirac distribution and convolution operator, respectively; \(k\) denotes the number of modes, while \(t\) is time script.  
Taking both penalty term and Lagrangian multipliers \(\lambda\) into consideration, the above constrained problem can be converted into an unconstrained one that can be addressed more easily, which is shown as follows:  
\[\begin{gathered} L(\{y_{k}\},\{w_{k}\},\lambda)=\\ \alpha\sum\limits_{k=1}^{K}\|\partial_{t}\left[\left(\delta(t)+ \frac{j}{\pi t}\right)\otimes y_{k}(t)\right]e^{-j\eta_{k}t}\|_{2}^{2}+\|f(t)- \sum\limits_{k=1}^{K}y_{k}(t)\|_{2}^{2}+\left\langle\lambda(t),f(t)-\sum \limits_{k=1}^{K}y_{k}(t)\right\rangle\end{gathered} \tag{11}\]  
where \(\alpha\) represents the balancing parameter of the data fidelity constraint.  
The augmented Lagrangian \(L\) is determined in Equation (11) and its saddle point in a sequence of iterative sub-optimizations can be found through using the alternate direction method of multipliers (ADMM). According to this ADMM optimization method, it is assumed that updating \(y_{k}\) and \(w_{k}\) in two directions helps to realize the analysis process of the VMD. The complete and detailed procedures of this algorithm are available in [51]. Consequently, solutions for \(y_{k}\) and \(w_{k}\) are described as follows [51]:  
\[\hat{y}_{k}^{n+1}=\frac{\hat{f}(w)-\sum\limits_{i\
eq k}\hat{y}_{i}(w)+\frac{ \hat{\lambda}(w)}{2}}{1+2\alpha(w-w_{k})^{2}} \tag{12}\]  
\[w_{k}^{n+1}=\frac{\int_{0}^{\infty}w\left|\hat{y}_{k}^{n+1}(w)\right|^{2}\!dw }{\int_{0}^{\infty}\!\left|\hat{y}_{k}^{n+1}(w)\right|^{2}\!dw} \tag{13}\]  
where \(\hat{f}(w)\), \(\hat{y}_{i}(w)\), \(\hat{\lambda}(w)\) and \(\hat{y}_{k}^{n+1}(w)\) denote the Fourier transforms of \(f(t)\), \(y_{i}(t)\), \(\lambda(t)\) and \(y_{k}^{n+1}(t)\) respectively, while \(n\) represents the number of iterations.