## 5 Numerical Simulation and Verification
### Numerical Investigation of the Proposed Method
This situation is even more common in the guided-wave inspection, but it is more difficult to judge the character of the reflector from the measured signals. The specific parameters of the Gabor pulses \(g_{1}\)(t) and \(g_{2}\)(t) were tabulated in Table 2. The reflector is set as a defect. The sampling frequency is assumed to be \(f_{s}\) = 3000 kHz. The Gabor excitation pulse \(S_{i}\) is (\(\sigma\), \(\mu\), \(\theta_{inv}\),\(f_{c}\)) = (\(4\times 10^{-6}\) s, \(2\times 10^{-6}\) s, \(\pi\)/2 rad, 64 kHz). After calculating in MATLAB, Figure 9 indicates the numerical signals with different SNRs and comparisons of the results by the proposed method and the PCCM. For \(\hat{x}_{1}(t)\) with SNR = 1 dB in Figure 9a, the minimum values of the distance curves in Figure 9b are \(D_{1}\) = min[\(D(S_{i},\hat{x}_{1})\)] = 98.49 and \(D_{2}\) = min[\(D\overline{\overline{(S_{i},\hat{x}_{1})}}\)] = 115.12, respectively. \(D_{1}\) \(<\)\(D_{2}\), therefore the front-edge of the reflector is a step down, indicating the reflector as a defect feature. The value of the \(RICR\) is equal to 12.21 (\(RICR\)\(>\)1.05), thus the judgement is reliable. For \(\hat{x}_{2}(t)\) with SNR = 0.1 dB in Figure 9c, the minimum values in Figure 9d are \(D_{1}\) = min[\(D(S_{i},\hat{x}_{2})\)] = 94.1 and \(D_{2}\) = min[\(D\overline{\overline{(S_{i},\hat{x}_{2})}}\)] = 109.29, respectively. \(D_{1}\) \(<\)\(D_{2}\), therefore the front-edge of the reflector is a step down, indicating the reflector is a defect feature. The value of the \(RICR\) is equal to 11.82 (\(RICR\)\(>\)1.05), as a result, the judgement is reliable. The PCCM found difficult to determine and evaluate the front-edge and the type of the reflector, as shown in Figure 9e,f. It should be noted that, due to the differences in amplitudes, the front-edge (\(x_{1}(t)\)) in the proposed method was more distinct than that in the PCCM, which is more convenient to identify the change and type of the front-edge as well as the reflector. \begin{table}
\begin{tabular}{c c c c c c} \hline
**Pulses** & \(A_{i}\) & \(\sigma\) & \(u\) (s) & \(f_{c}\) (kHz) & \(\theta\) (rad) \\ \hline \(x_{1}(t)\) & 1 & \(4.0\times 10^{-6}\) & \(3.8\times 10^{-4}\) & 64 & \(\pi/2\) \\ \(x_{2}(t)\) & 0.5 & \(4.0\times 10^{-6}\) & \(4.0\times 10^{-4}\) & 64 & \(-\pi/2\) \\ \hline \end{tabular}
\end{table} 
Table 2: Parameters of Gabor pulses \(x_{1}(t)\) and \(x_{2}(t)\). Figure 8: Numerical signals with different signal-to noise ratios (SNRs) and comparisons of the results by the proposed method and correlation coefficient method; (**a**) \(\hat{x}_{1}(t)\) with SNR = 1 dB; (**b**) Distance curves \(\Psi_{1}\) (**Up**) and \(\Psi_{2}\) (**Down**) of \(\hat{x}_{1}(t)\); (**c**) \(\hat{x}_{2}(t)\) with SNR = 0.1 dB; (**d**) Distance curves \(\Psi_{1}\) (**Up**) and \(\Psi_{2}\) (**Down**) of \(\hat{x}_{2}(t)\); (e**) [PERSON]’s correlation coefficient method (PCCM) curves \(\Psi_{1}\) (**Up**) and \(\Psi_{2}\) (**Down**) of \(\hat{x}_{1}(t)\); (**f**) PCCM curves \(\Psi_{1}\) (**Up**) and \(\Psi_{2}\) (**Down**) of \(\hat{x}_{2}(t)\). Figure 9: Numerical signals with different SNRs and comparisons of the results by the proposed method and correlation coefficient method; (**a**) \(\hat{x}_{1}(t)\) with SNR = 1 dB; (**b**) Distance curves \(\Psi_{1}\) (**Up**) and \(\Psi_{2}\) (**Down**) of \(\hat{x}_{1}(t)\); (**c**) \(\hat{x}_{2}(t)\) with SNR = 0.1 dB; (**d**) Distance curves \(\Psi_{1}\) (**Up**) and \(\Psi_{2}\) (**Down**) of \(\hat{x}_{2}(t)\); (**e**) PCCM curves \(\Psi_{1}\) (**Up**) and \(\Psi_{2}\) (**Down**) of \(\hat{x}_{2}(t)\).