## 2 Basic Theory  
### VMD Decomposition Principle  
#### 2.1.1 Basic Principle  
The VMD algorithm is improved by [PERSON] and [PERSON] in 2014 [18]. The VMD algorithm evolves from the EMD algorithm, with both fundamentally decomposing signals into various modes [19]. Specifically, VMD seeks to decompose a real-valued input signal into a predetermined number of modes, denoted as \(K\), with each mode identified as \(\mu_{k}\). These modes are designed with finite bandwidths, ensuring that each extracted mode occupies a specific frequency range without overlapping with others. The primary goal of VMD is to ensure that the sum of these \(K\) modes, \(\mu_{k}\), precisely reconstructs the original input signal. The process involves a constrained optimization problem that aims to minimize the bandwidth of each mode while adhering to this reconstruction constraint. This optimization challenge is addressed through a variational approach, seeking the optimal set of modes that meet these criteria. This constrained optimization variational problem is expressed as follows:  
\[\min_{\{\mu_{k}\}\{\omega_{k}\}}\left\{\sum_{k=1}^{K}\left\|\partial_{t}\left[ \left(\delta(t)+\frac{j}{\pi\mathrm{t}}\right)\cdot\mu_{k}\right]e^{-j\omega_{ k}t}\right\|_{2}^{2}\right\}\quad\text{s.t.}\;\sum_{k=1}^{K}\mu_{k}(t)=x(t) \tag{1}\]  
where \(x(t)\) is the original input signal, \(\mu_{k}(t)\) and \(\omega_{k}\) are the Band-Limited Intrinsic Mode Functions (BLIMFs) and their central frequency, \(\delta(t)\) is the Fermi-Dirac distribution function, and \((\delta(t)+j/\pi\mathrm{t})\mu_{k}\) is the analytic signal obtained by convolution of each mode \(\mu_{k}(t)\) through Hilbert transform. The exponential term \(e^{(-j\omega_{k}t)}\) indicates that the analytic signal is modulated to the baseband, and the BLIMF bandwidth can be estimated by calculating the \(L^{2}\) norm.  
Then, by introducing the augmented Lagrangian \(L\) and quadratic penalty factor \(\alpha\), this constraint problem can be changed into an unconstraint problem as follows:  
\[\begin{split} L(\{\mu_{k}\},\{\omega_{k}\},\lambda)=\alpha\sum_{k= 1}^{K}\left\|\partial_{t}\bigg{[}\bigg{(}\delta(t)+\frac{j}{\tau\up t}\bigg{)} \cdot\mu_{k}\bigg{]}e^{-j\omega_{k}t}\right\|_{2}^{2}\\ \left\|x(t)-\sum_{k=1}^{K}u_{k}(t)\right\|_{2}^{2}+<\lambda(t),x (t)-\sum_{k=1}^{K}u_{k}(t)>\end{split} \tag{2}\]  
Lastly, it is necessary to find the saddle point of the augmented Lagrangian L in a sequence of iterative sub-optimizations, denoted as the Alternate Direction Method of Multipliers (ADMM). In this process, it must initialize \(\mu_{k}\), \(\omega_{k}\), and \(\lambda\) continuously as follows:  
\[\hat{\mu}_{k}^{n+1}(\omega)=\frac{\delta(\omega)-\sum_{i<k}\hat{\mu}_{i}^{n+1 }(\omega)-\sum_{i>k}\hat{\mu}_{i}^{n}(\omega)+\frac{\hat{\lambda}(\omega)}{2 }}{1+2\alpha\big{(}\omega-\omega_{k}^{n}\big{)}^{2}} \tag{3}\]  
\[\omega_{k}^{n+1}=\frac{\int_{0}^{\infty}\left|\hat{\mu}_{i}^{n+1}(\omega) \right|^{2}\text{d}\omega}{\int_{0}^{\infty}\left|\hat{\mu}_{i}^{n+1}(\omega) \right|^{2}\text{d}\omega} \tag{4}\]  
\[\hat{\lambda}^{n+1}(\omega)=\hat{\lambda}^{n}(\omega)+\tau\bigg{(}\delta( \omega)-\sum_{k=1}^{K}\hat{\mu}^{n+1}(\omega)\bigg{)} \tag{5}\]  
In Formulas (3)-(5), \(\wedge\) is a Fourier transformation, and \(\tau\) is the time step of dual rise method.  
It can set a discrimination accuracy \(\varepsilon\) when it satisfies the following formula:  
\[\frac{\sum_{k=1}^{K}\left\|\hat{\mu}_{k}^{n+1}-\hat{\mu}_{k}^{n}\right\|_{2}^ {2}}{\left\|\hat{\mu}_{k}^{n}\right\|_{2}^{2}}<\varepsilon \tag{6}\]  
The iteration stops, obtaining \(K\) modal functions satisfied by Formula (1) by inverse Fourier transform.