## 2 Principles and Methods
### Variational Mode Decomposition  
VMD is an algorithm that adaptively decomposes a signal into multiple nearly orthogonal band-limited IMFs through quasi-orthogonal transformation. The core idea is to transform the signal decomposition problem into a constrained optimization problem [25]. The VMD algorithm can adaptively match the optimal center frequency and finite bandwidth of each mode during the search and solution processes, with the goal of reducing non-stationarity in time series with a high complexity and strong nonlinearity. The VMD algorithm is more robust with respect to sampling and noise, primarily involving two steps: establishing the variational constraint model and solving the model. First, a variational constraint model is established, transforming the signal decomposition problem into a constrained optimization problem, as expressed in Equation (1):  
\[\left\{ \begin{array}{l}\underset{\{\mu_{k}\}\{\{\omega_{k}\}\}}{\underset{k} {\sum}}\left\{\underset{k}{\sum}\bigg{\|}\partial_{t}\bigg{[}\delta(t)+\frac{ j}{\pi t}\mu_{k}(t)\bigg{]}e^{-j\omega_{k}t}t\bigg{\|}_{2}^{2}\right\}\\ s.t.\underset{k}{\sum}\mu_{k}(t)=f(t)\end{array} \right. \tag{1}\]  
where \(\{\mu_{k}\}\) represents the modal functions obtained from the decomposition; \(\{\omega_{k}\}\) denotes the center frequencies of the corresponding modes; \(\delta(t)\) is the Dirac distribution function; \(f(t)\) is the input signal; and \(t\) is time. Building on Equation (1), a quadratic penalty factor and an augmented Lagrangian function are introduced to transform the constrained problem into an unconstrained problem [26]. Equation (1) can be reformulated as follows:  
\[ \begin{array}{l}L(\{\mu_{k}\},\{\omega_{k}\},\lambda)=\\ \alpha\underset{k}{\sum}\bigg{\|}\partial_{t}\bigg{[}\bigg{(}\delta(t)+\frac{ j}{\pi t}\bigg{)}\mu_{k}(t)\bigg{]}e^{-j\omega_{k}}\bigg{\|}_{2}^{2}+\bigg{\|}f(t)- \underset{k}{\sum}\mu_{k}(t)\bigg{\|}_{2}^{2}+\\ \Big{\langle}\lambda(t),f(t)-\underset{k}{\sum}\mu_{k}(t)\bigg{\rangle}\end{array}  \tag{2}\]  
where \(\alpha\) is the penalty factor used to balance the signal-to-noise ratio and decomposition accuracy; \(\lambda\) is the Lagrange multiplier. Figure 1: Time-domain waveform of simulated strong cultural noise. Horizontal coordinates indicate sampling points and vertical coordinates indicate amplitude. (**a**) Clean signal; (**b**) simulated impulse noise signal; (**c**) simulated square noise signal; (**d**) simulated triangle noise signal; (**e**) simulated periodic noise signal. Next, the saddle point of Equation (2) was solved using the multiplier alternation method. After the iterative update, the frequency domain and center frequencies of the variational constraint model are obtained, leading to the optimal solution to the constrained variational problem and completing the optimization process. The updated expressions for the frequency domain and center frequencies of the variational constraint model are as follows:  
\[\mu_{k}^{n+1}=\text{arg min}\] \[\left\{\alpha\left\|\partial_{t}\left[\left(\delta(t)+\frac{j}{ \pi t}\right)\mu_{k}(t)\right]e^{-j\omega_{k}t}\right\|_{2}^{2}+\left\|f(t)- \sum\limits_{i}\mu_{i}(t)+\frac{\lambda(t)}{2}\right\|_{2}^{2}\right\} \tag{3}\]  
\[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\omega|\mu_{k}(\omega)|^{2}d\omega}{ \int_{0}^{\infty}|\mu_{k}(\omega)|^{2}d\omega} \tag{4}\]  
\[\lambda^{n+1}=\lambda^{n}+\rho\left(f(t)+\sum\limits_{k}\mu_{k}^{n+1}\right) \tag{5}\]  
where \(n\) represents the number of iterations; \(\rho\) is the update factor. The specific implementation process for decomposing the original signal \(f(t)\) into K IMF components in the VMD algorithm is as follows:  
1. Initialize \(\{\mu_{k}^{1}\}\), \(\{\omega_{k}^{1}\}\), \(\lambda_{1}\), and \(n\) to 0;
2. Update \(\mu_{k}\) and \(\omega_{k}\) according to Equations (3) and (4);
3. Update \(\lambda\) based on Equation (5);
4.