## 3 Ultrasonic Guided Wave Signal Processing Methods
### Variational Mode Decomposition
### Advantages of ISSP

The synthesized UGW signals can be utilized to illustrate the advantages of ISSP, which were generated by PC_disp package. The synthesized L(0,1), F (\(n\),1), L(0,2) and F(\(n\),3) (1 \(\leq n\leq\) 3) UGW signals propagating 1 m in the time domain were plotted in Figure 5a and b. Since the group velocities of F(\(n\),1) and F(\(n\),3) UGWs decrease with the increase of \(n\) from the analysis in Section 2.2, their arrival times at the same propagation distance are gradually backward. It can be observed that the time-domain differences among L(0,1) and F(\(n\),1) UGWs are quite small compared with that among L(0,2) and F(\(n\),3) UGWs, which can also be explained by the difference of their group velocities. Moreover, F(\(n\),3) UGWs manifest significant dispersion at high \(n\), namely their wave packets become wider and time resolutions weaken. Superimposing the above UGWs in the time domain, and applying Chirplet transformation to the synthesized signal, creates the results shown in Figure 5c. This distinguishes L(0,2) UGW from synthesized signal by the time-frequency domain signal, but the resolution of L(0,2) UGW is inevitably influenced by other UGWs. In consideration of the real UGW testing, multiplying the amplitude of L(0,1) + F(\(n\),1) + L(0,2)+ F(\(n\),3) synthesized signal at 1 m by 0.5 (the defect echo), setting the L(0,1) + F(\(n\),1) + L(0,2) + F(\(n\),3) synthesized signal at 5 m as the end-reflected echo, and the synthesized UGW signals in time, creates the following time-frequency and frequency domains that are illustrated in Figure 6a-c.

Figure 4: (**a**) Comparison of CS and RC filters, frequency distributions of (**b**) RC, (**c**) FBR-RC, and (**d**) FSR-FBR-RC filter bands. SSP with different filter bands were used to process the synthesized UGW signal, and the results are shown in Figure 6d,e, where the filter parameters are as follows: \(B_{T}=B_{\rm 3 dB}\), \(B_{f}=B_{T}/10\), \(\Delta f=B_{T}/20\), \(B_{r}=B_{T}/20\), \(B_{t}=B_{T}/40\), \(FBR=f_{L}/(B_{r}+2B_{i})\), \(FSR=f_{L}/\Delta f_{0}\) (\(\Delta f_{0}=B/20\)). The signal-to-noise ratio gain (SNRG) and defect-to-noise ratio gain (DNRG) are introduced to quantitatively measure the effects of SSPs, and their calculation formulas are as follows:

\[SNRG=20\big{[}\log_{10}(S_{out}/N_{out})-\log_{10}(S_{in}/N_{in})\big{]} \tag{10}\]

\[DNRG=20\big{[}\log_{10}(D_{out}/N_{out})-\log_{10}(D_{in}/N_{in})\big{]} \tag{11}\]

where \(S\) is the maximum amplitude of the signal, \(D\) is the amplitude of the defect echo, \(N\) is the noise level calculated by the root mean square of the whole signal, and subscripts \(in\) and \(out\) represent before and after the SSP processing, respectively. Compared with the synthesized UGW signal, the sparse signals after applying SSPs show the features (defect and pipe end) obviously, and the coherent noise is successfully eliminated (Figure 6d,e). It can be observed that GS-SSP only retains the end-reflected echo while the defect echo vanishes, and RC-SSP completely retains both. Compared with RC-SSP, FBR-RC-SSP and FSR-FBR-RC-SSP further improve the signal resolution, of which SNRG increases by 29.05% and 68.29%, and DNRG increases by 32.81% and 61.45%, respectively, when PT is taken as the signal recombination method. Despite the

\[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]0 Signals of (**a**) L(0,1) + F(\(n\),1), (**b**) L(0,2) + F(\(n\),3) UGWs in the time domain, and signals of (**c**) synthesized UGW signal in time and frequency domains. \[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\]1 Synthesized UGW signals in the (**a**) time, (**b**) time-frequency, and (**c**) frequency domains, sparse signals processed by (**d**) SSP-PT and (**e**) SSP-PTM.