## 3 The Principle of VMD-DWT-Based Denoising
### Brief Description of VMD  
The goal of VMD is to decompose the input signal into sub-signal modes, \(u_{k}\), which have specific sparsity properties. The basic modes \(K\) are most related to the center frequencies \(w_{k}\). A detailed description of VMD can be found in Reference [4]. In order to assess the modes \(u_{k}\) and the frequencies \(w_{k}\), the resulting constrained birational problem, described as follows, needs to be solved using VMD:  
\[\min_{\{u_{k}\},\{w_{k}\}}\Bigg{\{}\sum_{k}\Bigg{\|}\partial_{t}\bigg{[}\bigg{(} \delta(t)+\frac{j}{\pi t}\bigg{)}*u_{k}(t)e^{-jw_{k}t}\bigg{]}\|_{2}^{2} \Bigg{\}} \tag{1}\]  
\[\text{Subject to }\sum_{k}u_{k}=f \tag{2}\]  
in which the \(u_{k}\) and \(w_{k}\) are shorthand notations of the mode and its center frequency. The \(\delta\) denotes the Dirac distribution, \(\partial_{t}\) denotes the partial differential, and \(*\) indicates convolution.  
The reconstruction constraint can be addressed by making use of both a quadratic penalty term and Lagrangian multipliers \(\lambda\). The argument Lagrangian L is expressed as follows:  
\[\begin{array}{ll}\text{L}(\{u_{k}\},\{w_{k}\},\lambda)=&\alpha\sum_{k=1}^{ K}\Big{\|}\partial_{t}\Big{[}\Big{(}\delta(t)+\frac{j}{\pi t}\Big{)}*u_{k}(t) \Big{]}e^{-jw_{k}t}\|_{2}^{2}\\ &&+\big{\|}x(t)-\sum_{k=1}^{K}u_{k}(t)\big{\|}_{2}^{2}+\big{\langle}\lambda(t),\text{x}(t)-\sum_{k=1}^{K}u_{k}(t)\big{\rangle}\end{array} \tag{3}\]  
where \(\alpha\) represents the balance parameter in the data fidelity constraint and corresponds to the bandwidth of the mode. The solution to the problem is the alternate direction method of multipliers (ADMM). The Winner filtering in the Fourier domain is adopted to update the modes \(u_{k}\). According to Reference [4], the embedded Winner filtering in the VMD algorithm makes the modes more robust to sampling and noise. The sub-mode \(u_{k}\) in the time domain and the filtered analytic signal transformed by inverse Fourier can then be defined as follows:  
\[\hat{u}_{k}^{n+1}(w)=\frac{\hat{f}(w)-\sum_{i\
eq k}\hat{u}_{i}(w)+\frac{ \hat{\lambda}(w)}{2}}{1+2\alpha(w-w_{k})^{2}} \tag{4}\]  
\[\hat{u}_{k}(t)=R\{ifft(\hat{u}_{k}(w))\} \tag{5}\]in which the \(\hat{f}(w)\) shows the Fourier transform of the signal \(f(\mathrm{t})\). The \(R\) denotes the real analysis signal part, and the \(ifft(.)\) expresses the inverse Fourier transform of the signal.  
The optimal \(w_{k}\) by Fourier domain is given as follows:  
\[\mathrm{w}_{\mathrm{k}}^{n+1}=\frac{\int_{0}^{\infty}w|\hat{\mathrm{u}_{ \mathrm{k}}}(w)|^{2}\mathrm{d}w}{\int_{0}^{\infty}|\hat{\mathrm{u}_{\mathrm{k} }}(w)|^{2}\mathrm{d}w}. \tag{6}\]  
VMD was applied to the experimental FBG signal, shown in Figure 4a, and the BLIMFs from the VMD of the experimental signal with the equal decomposition number of the EMD and the balance parameter set to 400 can be seen in Figure 4c. Figure 4b depicts the IMFs extracted from the EMD of the FBG signal. It is obvious that the intrinsic modes \(u_{6}\) and \(u_{7}\) contained little information that the VMD decomposition was overbinning, and the results were redundant. Thus, the VMD predefinition method where the basic mode number \(K\) was equal to the EMD decomposition level was unreasonable, and a better use for the FBG experiment data can be seen in the following section.  
Figure 4: \(Cont.\)