## 3 The Principle of VMD-DWT-Based Denoising
### The Improved VMD-DWT Denoising Algorithm  
The thresholding value could be self-adapted with spectral characteristics under different signal models resulting from VMD. An improved wavelet-VMD algorithm based on the standard wavelet-VMD is proposed for the spectrum signal denoising. Three key techniques were adopted in this section: one is wavelet thresholding, another is the data-fidelity constraint parameter, and the third is the decomposition number. Compared with the traditional DWT denoising algorithm, the improved wavelet thresholding in the proposed algorithm considers the energy in each basic mode, and the denoising result is superior. Furthermore, in the traditional VMD algorithm, the decomposition number \(K\) is required to be predefined, and its value has a predictable impact on the efficiency of filtering. In the literature, the basic mode number \(K\) is usually defined with the same EMD decomposition number and the wavelet packet number or is selected by calculating the scaling exponent \(\alpha\) of the input signal. However, the technologies above have a limitation in the experiment signal for the complicated frequency components. Therefore, the number \(K\) was selected based on experience. In addition, the value of the data-fidelity constraint parameter \(\alpha\) can be estimated by analyzing the anti-noise performance at different \(\alpha\) values for the mixed signal of various SNRinput. The flow chart of the VMD-DWT denoising algorithm is shown in Figure 5.  
The real FBG signal \(f(i)\) with finite length \(N\) is described as follows:  
\[f(i)=s(\mathrm{i})+n(\mathrm{i})\ i=0,1,2,\ldots,N-1 \tag{7}\]  
where \(s(i)\) is the FBG noise free signal, and \(n(i)\) denotes random noise, which is associated with environment. The purpose of the denoising method is to find an estimated \(\widetilde{f}(i)\) of the signal \(f(i)\) with small error, i.e., completely remove the uncertain external disturbance noises from the experimental signals. In order to simplify the operation, we assumed that the SNRout denotes the signal/noise ratio (SNR) of the output signal and the SNRinput indicates the SNR of the input signal. Based on the aforementioned analysis, the process of the VMD-DWT signal de-noising algorithm is shown as follows:  
Figure 4: Decomposition of FBG reflectivity spectrum signals via EMD and VMD: (**a**) The FBG reflectivity spectrum signal, (**b**) the IMFs of the EMD via decomposition the reflectivity spectrum signal, and (**c**) the BLIMFs of the VMD via decomposition the reflectivity spectrum signal. Step 1: The decomposition number \(K\) of the experiment signal is estimated. As discussed above, the decomposition number is not universally predefined. Thus, the number \(K\) is defined based on experience. Considering the difficulties of calculating quantitative evaluation indexes, the mixed FBG signals by adding various \(\text{SNR}_{\text{input}}\) Gaussian white noise to the experimental signals has been used in this paper to find the relationship between the decomposition number \(K\) and the denoising results. In addition, the anti-noise performance of the proposed algorithm at different \(\alpha\) values is analyzed  
Figure 5: Flow chart of the wavelet-VMD based denoising algorithm. with mixed signals under various SNRinput values. The best decomposition number \(K\) and the balance parameter \(\alpha\) for the realistic experiment signal are estimated based on the denoising results. Details can be found in Section 4, Case 1. Step 2: The best decomposition number \(K\) is determined. An input signal with length \(N\) is decomposed into \(K\) band-limited intrinsic mode functions (BLIMFs) \(u_{k}\) by VMD, with respect to the sub-energy of the signal. The signal of length \(N\) in the time domain is extended \(2N\) to address the frequency domain. Step 3: In wavelet decomposition, the predetermined and fixed equivalent filter-bank structure is not possible. A translate wavelet is performed by using thresholding in each BLIMF.