## 2 Related Works  
### Variational Mode Extraction  
The VME algorithm is derived from VMD and shares a similar mathematical principle. However, VME extracts a single specific component, resulting in higher efficiency. In VME, the input signal s(t) is decomposed into two parts:  
\[\text{s(t)}{=}\ u_{\text{d}}(\text{t})+\text{s}_{\text{r}}(\text{t}) \tag{1}\]  
In Equation (1), \(\text{u}_{\text{d}}(\text{t})\) represents the desired mode component, and \(\text{s}_{\text{r}}(\text{t})\) represents the residual signal.  
Moreover, \(\text{u}_{\text{d}}(\text{t})\) needs to be compactly surrounded by the center frequency \(\omega\) after the Hilbert transform and have minimal overlap with \(\text{u}_{\text{d}}(\text{t})\) and \(\text{s}_{\text{r}}(\text{t})\). Therefore, the constraints need to be minimized to obtain the desired mode components, as follows:  
\[\text{J}_{1}=\left\|\partial_{\text{t}}\left[\left(\delta(\text{t})+\frac{ \text{j}}{\pi\text{t}}\right)*\text{u}_{\text{d}}(\text{t})\right]\text{e}^{- \text{j}\omega_{\text{d}}\text{t}}\right\|_{2}^{2} \tag{2}\]  
In Equation (2), \(\delta\) represents the Dirac distribution, \(\omega_{\text{d}}\) denotes the center frequency of the mode component \(\text{u}_{\text{d}}(\text{t})\), and \(*\) denotes the convolution operation.  
The spectral overlap between \(\text{u}_{\text{d}}(\text{t})\) and \(\text{s}_{\text{r}}(\text{t})\) is minimized, and a penalty function is introduced, as follows:  
\[\text{J}_{2}=\left\|\ \beta(\text{t})*\text{s}_{\text{r}}(\text{t})\ \right\|_{2}^{2} \tag{3}\]In Equation (3), \(\beta(t)\) represents the impulse response of the frequency response filter. Therefore, the problem of finding modes can be formulated as a problem of constrained minimization when Equations (2) and (3) are combined:  
\[\begin{array}{l}\min_{u_{d},\omega_{d},s_{r}}\{\omega_{d},s_{r}}\{\omega_{1}+J _{2}\}\\ \text{subject to }:u_{d}(t)+s_{r}(t)=s(t)\end{array} \tag{4}\]  
In Equation (4), \(\alpha\) denotes the parameter that balances \(J_{1}\) and \(J_{2}\).  
The previously described constrained optimization problem is transformed into an unconstrained format. This involves incorporating both a quadratic penalty term and Lagrange multipliers \(\lambda\), thereby forming the augmented Lagrange function:  
\[\begin{array}{l}\Gamma(u_{d},\omega_{d},\lambda)=\alpha\|j(\omega-\omega_{ d})[1+\text{sgn}(\omega)\hat{u}_{d}]\|_{2}^{2}+\|\hat{\beta}(\omega)*\hat{f}_{r}( \omega)\|_{2}^{2}\\ +\|\hat{f}(\omega)-(\hat{u}_{d}(\omega)+\hat{f}_{r}(\omega))\|_{2}^{2}+\langle \hat{\lambda}(\omega),\hat{f}(\omega)-(\hat{u}_{d}(\omega)+\hat{f}_{r}(\omega) )\rangle\end{array} \tag{5}\]  
The alternating direction method of multipliers is used to find the saddle point of the Lagrange function.  
The steps are as follows:  
1. Initialize \(\hat{u}_{d}^{1}\), \(\hat{\lambda}^{1}\), \(\hat{\omega}_{d}^{1}\), \(n=1\), and estimate the initial value of \(\hat{\omega}_{d}^{1}\).
2. According to Equation (6), update \(\hat{u}_{d}\): \[\hat{u}_{d}^{n+1}(\omega)=\frac{\xi(\omega)+\alpha^{2}(\omega-\omega_{d}^{n+1 })^{4}\cdot\hat{u}_{d}^{n}(\omega)+\hat{\lambda}(\omega)/2}{\left[1+\alpha^{2 }(\omega-\omega_{d}^{n+1})^{4}\right]\left[1+2\alpha(\omega-\omega_{d}^{n})^{ 2}\right]}\] (6)
3. According to Equation (7), update \(\omega_{d}\): \[\omega_{d}^{n+1}=\frac{\int_{0}^{\infty}\omega\left|u_{d}^{n+1}(\omega)\right| ^{2}\text{d}\omega}{\int_{0}^{\infty}\left|u_{d}^{n+1}(\omega)\right|^{2} \text{d}\omega}\] (7)
4. According to Equation (8), update the Lagrange multipliers for all \(\omega>0\): \[\hat{\lambda}^{n+1}=\hat{\lambda}^{n}+\tau\cdot\left[\frac{\hat{x}(\omega)- \hat{u}_{d}^{n+1}(\omega)}{1+\alpha^{2}(\omega-\omega_{d}^{n+1})^{4}}\right]\] (8)
5. Repeat steps 2\(\sim\)4 until the iteration stop condition is satisfied: \[\frac{\left\|\hat{u}_{d}^{n+1}-\hat{u}_{d}^{n}\right\|_{2}^{2}}{\|\hat{u}_{d}^ {n}\|_{2}^{2}}<\varepsilon\] (9)
6. End the loop and obtain the desired mode \(u_{d}(t)\).