## 3 Ultrasonic Guided Wave Signal Processing Methods
### Variational Mode Decomposition
Then, the useless IMFs containing incoherent noise are discarded, and the useful IMFs are denoised by WT to obtain DIMFs. Finally, DIMFs are processed by ISSP to obtain \(y_{t}(t)\), which are summed to obtain \(y(t)\) containing only feature signals. Due to the significant influence of incoherent noise, it is difficult to distinguish the defect signal from the NUS 1-5 in the time and time-frequency domains (Figure 10a,b), while the spectrum may also be polluted (Figure 10c). After performing EMD and VMD on NUS 1-5, the resulting first six IMFs in time and frequency domains are shown in Figure 10d-g, and their correlation coefficients with NUS 1-5 are calculated (Figure 11a). It can be noticed that IMF 1 and IMF 2 of EMD show the highest similarity with NUS 1-5, of which the correlation coefficients are 0.79 and 0.41, and the features in IMF 1 and IMF 2 are the pipe end and the defect, and other IMFs can be considered as having no

\[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]5 (**a**) Pure and noisy UGW signals, sparse signals processed by (**b**) ISSP-PT and (**c**) ISSP-PTM. \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]6 Schematic diagram of VWISSP. practical significance. Only IMF 2 with the highest correlation coefficient (0.63) of VMD is concentrated around 100 kHz, indicating that all features can be revealed through IMF 2, and the rest of IMFs can be considered as the incoherent noise. \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]7 (**a**) Correlation coefficients of IMFs, processing results of (**b**) E/VMD + WT, (**c**) E/VMD + ISSP, and (**d**) E/VMD + WT + ISSP. \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]8 NUS 1–5 in the (**a**) time, (**b**) time-frequency, and (**c**) frequency domains; IMFs processed by EMD in the (**d**) time and (**e**) frequency domains; IMFs processed by VMD in the (**f**) time and (**g**) frequency domains. It can be found from Figure 11b that WT can enhance the resolution of IMFs and the features of DIMFs become more obvious, and the sparse signals of IMFs and DIMFs after applying ISSP are shown in Figure 11c,d. Compared with ISSP, E/VMD + ISSP can significantly reduce the misjudments and further improve the signal resolution. However, multiple peaks would appear in the defect region processed by EMD + ISSP, which confuses the accurate identification of defects, while VMD + ISSP has no such trouble. Thanks to the denoising of WT, the resolution of sparse signals processed by E/VMD + WT + ISSP are enhanced further, of which SNRs are 24.42 dB and 24.99 dB, and DNRs are 11.85 dB and 13.07 dB. The denoting results of the E/VMD + WT + ISSP as E/VWISSP, which were applied to NUS and PUS in Section 4, are illustrated in Figure 12, and the calculated SNRG, DNRG, and number of misjudments are shown in \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]9 Compared with ISSP, EWISSP and VWISSP can further improve the signal resolution, with the average SNRG increased by 13.76% and 25.73%, respectively, and the average DNRG increased by 12.43% and 23.95%, respectively. However, EWISSP cannot retain the features and eliminate noise completely.