## 2 Methodology and Principles  
### Principles of the VMD Algorithm  
Variational mode decomposition (VMD) is a self-adaptive and completely non-recursive method for mode decomposition and signal processing and can determine the number of modes. Its self-adaptivity is reflected by the fact that the number of modes of a given sequence is determined according to the actual situation, and VMD can adaptively match the center frequency and finite bandwidth of each mode in the subsequent searching and solving processes. It may achieve the effective separation of IMFs and the frequency-domain division of signals, further obtaining the effective components of a given signal after decomposition and finally acquiring the optimal solution of a variational problem. VMD can be divided into the construction and solution of variational problems, which involves three critical concepts: classical Wiener filtering, Hilbert transform and frequency mixing. 1. Construction of variational problems Assume that each \"mode\" is a finite bandwidth with a center frequency. The variational problem is described as a quest for \(k\) mode functions \(u_{k}(t)\), making the sum of the estimated bandwidth of each mode minimized. The constraint condition requires that the sum of all modes should be equal to the input signal \(f\). The specific construction steps are as follows: 1. Analytic signals of each mode function \(u_{k}(t)\) are obtained by Hilbert transform to acquire a one-sided spectrum. 2. Mix the analytic signals of each mode and predict the center frequency \(e^{-j\omega_{k}t}\); modulate the frequency spectrum of each mode to the corresponding fundamental frequency band. 3. Calculate the squared norm \(L^{2}\) of the gradient of the above demodulated signal, and estimate the signal bandwidth of each mode. The constrained variational problem is as follows: \[\left\{ \begin{array}{l}\underset{\{u_{k}\}}{\min}\Big{\{}\underset{\{\}}{ \Sigma}\Big{\|}\partial_{t}\Big{[}\delta(t)+\frac{j}{\pi t}\times u_{k}(t) \Big{]}\times e^{-j\omega_{k}t}\Big{\|}^{2}\Big{\}}\\ \text{s.t.}\underset{k}{\sum}u_{k}=f\end{array} \right.\] (1) where \(\{u_{k}\}=\{u_{1},\cdots,u_{K}\},\{\omega_{k}\}=\{\omega_{1},\cdots,\omega_{ K}\}\), \(\underset{k}{\sum}=\sum_{k-1}^{K}\)[16]. 2. Solution of variational problems By introducing the quadratic penalty factor \(\alpha\) and Lagrangian operator \(\lambda(t)\), the constrained variational problem is changed into an unconstrained variational problem. The quadratic penalty factor can ensure accurate signal reconstruction in the presence of Gaussian noise. The Lagrangian operator makes the constraints remain stringent. The extended Lagrangian expression is stated as follows: \[ \begin{array}{l}L(\{u_{k}\},\{\omega_{k}\},\lambda)=a\underset{\overline{k }}{\Sigma}\Big{\|}\partial_{t}\Big{[}\delta(t)+\frac{j}{\pi t}\times u_{k}(t) \Big{]}\times e^{-j\omega_{k}t}\Big{\|}^{2}\\ \hskip 14.226378 pt+\hskip-14.226378 pt\left\|f(t)-\underset{k}{\sum}u_{k}(t) \right\|^{2}_{2}+\bigg{\langle}\lambda(t),f(t)-\underset{k}{\sum}u_{k}(t) \bigg{\rangle}\end{array} \] (2)  
In VMD, the Alternating Direction Method of Multipliers (ADMM) for multiplication operators is used to solve the above variational problem. Alternately update \(u_{k}^{n+1}\), \(\omega_{k}^{n+1}\) and \(\lambda^{n+1}\) to seek the saddle point for extending the Lagrangian expression [16]. The calculation of \(u_{k}^{n+1}\) can be expressed as follows:  
\[ \begin{array}{l}u_{k}^{n+1}=\underset{u_{k}\in X}{\operatorname{argmin}} \bigg{\{}\alpha\big{\|}\partial_{t}\Big{[}\delta(t)+\frac{j}{\pi t}\times u_{k} (t)\Big{]}\times e^{-j\omega_{k}t}\Big{\|}^{2}_{2}\\ \hskip 14.226378 pt+\hskip-14.226378 pt\left\|f(t)-\underset{i}{\sum}u_{i}(t)+ \frac{\lambda(t)}{2}\right\|^{2}_{2}\bigg{\}}\end{array}  \tag{3}\]  
where \(\omega_{k}\) is equivalent to \(\omega_{k}^{n+1}\); \(\sum_{i}u_{i}(t)\) is equivalent to \(\sum_{i\
eq k}u_{i}(t)^{n+1}\)[16]. Fourier isometric transform (Parseval/Plancherel's Theorem) is used to convert Formula (3) into the frequency domain.