## 3 Synthesized UGW Signal Analysis
### UGW Signal Time-Frequency Analysis  
The chirplet transform, a generalization of both the wavelet and the short-time Fourier transform, can extract signal components with specific instantaneous frequencies and group delays and overcome the shortcomings of conventional time-frequency representation solutions such as the short-time Fourier transform and the Wigner-Ville distribution, which may not achieve good UGW mode separation results, whereas multiple modes usually occupy the same frequency band and intersect each other in time. The standard definition of the chirplet transform is given by the inner product of a basis function \(g(t)\) and the signal \(x(t)\)[26]:  
\[CT=\langle x(t),g(t)\rangle=\int_{-\infty}^{\infty}x(t)g^{*}(t)dt \tag{8}\]  
\[g(t)=T_{t0}F_{u0}S_{s}Q_{q}P_{p}c(t) \tag{9}\]  
where * denotes the complex conjugate, and \(g(t)\) and \(G(\omega)\) are basis function and the Fourier transform of basis function. \(T_{t0}\), \(F_{u0}\), \(S_{s}\), \(Q_{q}\), and \(P_{p}\) are the time shift, frequency shift, scaling, frequency shear and time shear operators, and the details are shown in Table 1. \(c(t)\) is one of the chirp signal families. The basis function can be adjusted by the operators of shifting, scaling, and shearing, leading to the generation of five-dimensional space, which combines the projections of the short-time Fourier transform (time and frequency shift) and a wavelet transform (time shift and scaling). The L(0,1), F(\(n\),1), L(0,2) and F(\(n\),3) (\(1\leq n\leq 4\)) UGW synthesized signals propagating 0.8 m in time and frequency domains were plotted in Figure 5, of which the spectrums were plotted by the chirplet transform. Although the cutoff frequencies of F(3,3) and F(4,3) are 138.474 kHz and 180.279 kHz, which are larger than the center frequency of 110 kHz, their excitation is still possible, since their cutoff frequencies are in the excitation frequency \begin{table}
\begin{tabular}{c c c} \hline \hline
**Operation** & **Transformation** \\ \hline Time shift \(T_{t0}\) & \(T_{t0}c(t)=c(t-t_{0})\) & \(t=t-t_{0}\) & \(w=w\) \\ Frequency shift \(F_{u0}\) & \(F_{u0}c(t)=c(t)\exp(iu\eta t)\) & \(t=t\) & \(w=w-w_{0}\) \\ Scaling \(S_{s}\) & \(S_{s}c(t)=c(t/s)/\sqrt{s}\) & \(t=t/s\) & \(w=sw\) \\ Frequency shear \(Q_{q}\) & \(Q_{q}c(t)=c(t)\exp(iu/2t^{2})\) & \(t=t\) & \(w=w-qt\) \\ Time shear \(P_{p}\) & \(P_{p}c(t)=c(t)\exp(it^{*}/2p)/\sqrt{ip}\) & \(t=t-pw\) & \(w=w\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Operations for the chirplet transform operators. Figure 4: Normalized (**a**) axial, (**b**) circumferential and (**c**) radial displacement distributions of the L(0,1), F(\(n\),1), L(0,2) and F(\(n\),3) UGWs (\(1\leq n\leq 4\)) at 110 kHz. band. Therefore, the frequencies of synthesized F(3,3) and F(4,3) are set as their cutoff frequencies. It can be observed that the wave packets of UGWs become wider while the propagation times become later with higher order \(n\), which may be attributed to the increase in phase velocity and decrease in group velocity. In addition, it can also be found that the time/frequency resolution of the synthesized signal composed with L(0,1) and F(\(n\),1) is poorer than that of the L(0,2) and F(\(n\),3) synthesized signal. Synthesized L(0,1), L(0,2), F(\(n\),1) and F(\(n\),3) UGW signals propagating 0.8 m in the time-frequency domain are shown in Figure 6a. It can be found that it would be difficult to distinguish L(0,2) from others only by the time domain signal, and the frequency domain signal may be easier to distinguish, but the spectrum obtained by the chirplet transform is based on a priori knowledge of dispersion and propagation distance. Figure 6b shows the synthesized signals at different propagation distances. It can be observed that the UGW signal with too large of a propagation distance or too long of a propagation time should be avoided on account of the wide wave packet and weak wave energy.