## 4 Results and Discussion  
This section should provide a concise and precise description of the experimental results, their interpretation, and the experimental conclusions that can be drawn. It is well known that the EMD-wavelet algorithm has been successfully used in removing the WGN from noisy signals. However, few studies on VMD-based denoising have been conducted. The EMD is highly dependent on methods of extremal point findings, which are sensitive to noise. The robustness of the algorithm may be reduced by the degrees of freedom. To overcome these limitations, a novel VMD decomposition scheme was used to obtain the recovered signal. The actual signal was processed to obtain the best decomposition number \(K\). The real experimental signal was processed for a quantity evaluation of the denoising ability for four decomposition methods: the EMD-wavelet, EMD-DFA, EMD-wavelet-changed threshold, and VMD-DFA. The details are shown in Cases 1 and 2. **Case 1:** Applying VMD to Mixed FBG Noisy Signals  
As discussed before, the predefined decomposition number \(K\) is key to the VMD method. Previous studies set the number \(K\) to coincide with the EMD decomposition number. VMD may have an overbinning decomposition. The DFA method can also be applied to the mode number \(K\) definition. This may cause substantial simulation and experiment work since the FBG signal has not been studied before. In this paper, we propose an experimental way of solving the decomposition number definition problem. In Case 1, we obtained the actual FBG noisy signal by adding different SNR levels of white Gaussian noise to the experimental FBG signal. To evaluate the de-noising effect, quantitative evaluating indexes, such as the mean absolute error (MAE), the root-mean-square error (RMSE), and the SNR, are proposed here to assess the de-noising methods [26; 27]. Additionally, the cross-correlation coefficient is proposed to estimate the correlation between the noisy signal and the denoising signal. \[\text{MAE}=\frac{1}{N}\sum_{i=1}^{N}\Bigl{(}s(i)-\ \widetilde{f}(i)\Bigr{)}^{2} \tag{12}\]  
\[\text{RMSE}=\sqrt{\frac{1}{N}\sum_{i=1}^{N}\Bigl{(}s(i)-\ \widetilde{f}(i) \Bigr{)}^{2}} \tag{13}\]  
\[\text{SNR}=10\text{lg}\Biggl{[}\frac{\sum_{i=1}^{N}s^{2}(i)}{\sum_{i=1}^{N} \Bigl{(}s(i)-\ \widetilde{f}(i)\Bigr{)}^{2}}\Biggr{]} \tag{14}\]  
\[\text{Coss correlation}=\frac{Cov\Bigl{(}s(i),\widetilde{f}(i)\Bigr{)}}{\sqrt {Var(s(i))Var\Bigl{(}\widetilde{f}(i)\Bigr{)}}} \tag{15}\]  
where \(\widetilde{f}(i)\) represents the de-noising FBG signal, and the \(s(i)\) values are the noisy FBG signals. \(Cov(*)\) indicates the covariance function, and \(Var(*)\) shows the variance equation. The value of the data-fidelity constraint balance parameter \(\alpha\) is inversely proportional to the bandwidth of the signal mode and closely related to anti-noise performance. The bandwidth mode widens as \(\alpha\) decreases, and the modes may have poor anti-noise performance. However, the center frequency mode will be less accurate according to the bandwidth mode, which decreases with a greater \(\alpha\). Based on a previous study [3], the veracity of the damage of the primary wavelength relies heavily on the center frequency of extracted modes. Therefore, it is necessary to find a proper value of \(\alpha\), by not only considering the center frequency accuracy but also taking the anti-noise performance into account. In order to find a balance point of \(\alpha\), the anti-noise performance under different \(\alpha\) values was analyzed with different SNRinput mixed signals. The central frequency of Mode 1 corresponds to thecentral wavelength considered the damage characteristic, and the primary wavelength shifting with different crack lengths was calculated with various values of \(\alpha\). The decomposition level was set as 6. The interval of \(\alpha\) was set within [1, 5000] according to research indicating that the mode separation error is intolerable when \(\alpha\) is over 5000 [28].