## 2 Basic Theories
### Variational Mode Decomposition [11; 12; 13; 14; 15]  
Compared to the traditional Empirical Mode Decomposition (EMD) algorithm, Variational Mode Decomposition (VMD) is an adaptive and fully non-recursive method for mode variation and signal processing proposed by [PERSON] in 2014. Its adaptability allows for determining the number of Intrinsic Mode Function components based on the actual sequence. VMD decomposition overcomes the end-point effects and mode mixing problems of EMD decomposition. The corresponding mathematical equation is  
\[u_{k}(t)=A_{k}(t)\cos(\phi_{k}(t)) \tag{5}\]  
In the equation, \(\{u_{k}\}\) represents the instantaneous amplitude, and \(\varphi_{k}(t)\) represents the instantaneous phase. Through the Hilbert-Huang Transform (HHT), analytic signals of different modes are obtained. Subsequently, each Intrinsic Mode Function (IMF) signal is modulated to its corresponding baseband by multiplying with the estimated center frequency.  
By applying Gaussian smoothness, the bandwidth of each IMF component is estimated to yield a constrained variational model:\[\left\{\begin{array}{l}\underset{\{\{u_{k}\},\{\omega_{k}\}\}}{\min}\left\{ \underset{\{k\}}{\sum}\left\|\partial_{t}\left[\left(\delta(t)+\frac{j}{\pi t} \right)\times u_{k}(t)\right]e^{-j\omega_{k}t}\right\|^{2}\right\}\\ \underset{\{s.t.\sum}\underset{k}{\sum}u_{k}=f}{\sum}\\ \end{array}\right. \tag{6}\]  
In the equation, \(\{u_{k}\}\) represents the K IMFs of the decomposition, \(\{\omega_{k}\}\) denotes the central frequency component of each IMF, and \(f\) is the input signal.  
To find the optimal solution for the constrained variational problem, the Lagrange multiplier \(\tau(\text{t})\) and a second-order penalty factor (\(\alpha\)) are introduced to transform the constrained variational problem into an unconstrained variational problem. The \(\tau(\text{t})\) ensures the strictness of the constraints, while the \(\alpha\) ensures accurate signal reconstruction in Gaussian noise environments. The extended Lagrangian expression is as follows:  
\[\begin{array}{l}L(\{u_{k}\},\{\omega_{k}\},\lambda)=\alpha\underset{\overline {k}}{\sum}\left\|\partial_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right) \times u_{k}(t)\right]e^{-j\omega_{k}t}\right\|^{2}+\left\|f\left(t\right)- \underset{\overline{k}}{\sum}u_{k}(t)\right\|^{2}\\ +\left\langle\lambda(t),f(t)-\underset{\overline{k}}{\sum}u_{k}(t)\right\rangle \end{array} \tag{7}\]  
Then, utilizing the Alternating Direction Method of Multipliers (ADMM), we make a succession update of each component and central frequency to obtain the saddle point of the unconstrained model. Based on frequency-domain space, all components are derived as follows:  
\[\hat{u}_{k}^{n+1}(\omega)=\frac{\hat{f}(\omega)-\underset{\overline{i}\
eq k} {\sum}\hat{u}_{i}(\omega)+\frac{\hat{\lambda}(\omega)}{2}}{1+2\alpha(\omega- \omega_{k})^{2}} \tag{8}\]  
\[\left\{\begin{array}{l}\underset{\{u_{k}\},\{\omega_{k}\}}{\min}\left\{ \underset{\overline{k}}{\sum}\left\|\partial_{t}\left[\left(\delta(t)+\frac{j }{\pi t}\right)\times u_{k}(t)\right]e^{-j\omega_{k}t}\right\|^{2}\right\}\\ \underset{\{s.t.\sum}\underset{k}{\sum}u_{k}=f}{\sum}\\ \end{array}\right. \tag{9}\]