## 3 Ultrasonic Guided Wave Signal Processing Methods
### Variational Mode Decomposition
The new wavelet coefficients obtained by thresholding are:

\[W_{new}=\text{sgn}(W)(|W|-T)_{+} \tag{7}\]

### Chirplet Transform

As a generalization of wavelet transform and short-time Fourier transform, Chirplet transform can extract signal components with a specific instantaneous frequency and group delay, and overcome the shortcomings of conventional time-frequency representation solutions, such as short-time Fourier transform and Wigner-Ville distribution, that may not achieve good UGW mode separation results. The standard definition of the Chirplet transform is given by the inner product of a basis function \(g(t)\) and the signal \(x(t)\)[38]:

\[CT=\langle x(t),g(t)\rangle=\int_{-\infty}^{\infty}x(t)g^{*}(t)dt \tag{8}\]

\[g(t)=T_{t0}F_{u0}S_{s}Q_{q}P_{p}c(t) \tag{9}\]

where * denotes the complex conjugate, \(g(t)\) is the basic function and the Fourier transform of the basic function, \(T_{t0}\), \(F_{\omega 0}\), \(S_{s}\), \(Q_{q}\), and \(P_{p}\) are the time shift, frequency shift, scaling, frequency shear, and time shear operators, respectively, and \(c(t)\) is one of the chirp signal family. ## 4 Improved Split Spectrum Processing

At present, the research of SSP mainly includes filter bank design, filter parameters optimization and signal recombination methods. Since SSP is very sensitive to filter parameters (total bandwidth, filter bandwidth, and filter spacing), the identical optimal filter parameters corresponding to UGW signals may not be determined, of which brute force search method was introduced [10]. The signal recombination methods have been analyzed in Section 3.1, so the improvement for SSP in this paper is aimed at the design of filter bank. ### Filter Bank Design

Gaussian bandpass filters are commonly used in SSP for the ease of implementation. On the one hand, the non-overlapping filters should be evitable for the \"picket fence\" effect wherein substantial feature information would be lost, which is not conducive to the signal interpretation. On the other hand, the overlapping region of adjacent filters should be controlled to be as small as possible for better coherent noise elimination [39]. Hence, the filter bank design should take into account the above requirements. In addition to Gaussian (GS) filters, raised cosine (RC) filters can also be an alternative (Figure 4a), who's transition bands are in the shape of truncated raised cosine cycle, and that guarantee the energy contained is mostly concentrated around the center frequency (the flat top) [40]. Given the same filter spacing, it is easy to foresee that RC filters have better effects on eliminating coherent noise and splitting substantial feature signals, considering that the overlapping of RC filters is lower than that of GS filters. The filter parameters shared by GS and RC filters are total bandwidth \(B_{T}\), filter bandwidth \(B_{f}\), and filter spacing \(\Delta f\), of which \(B_{T}\) is the difference between high \(f_{H}\) and low \(f_{L}\) cut-off frequencies, and \(B_{\beta RC}\) can be replaced by the transition bandwidth \(B_{t}\) and the flat top width of \(B_{t}\). On the premise of eliminating coherent noise and extracting feature signals, the filters with small bandwidths would be the unavoidable choice, which may lead to the poor time resolution of the processed signal. In view of the above problem, this paper adopts the RC filters with variable bandwidths of which the ratio of center frequency to filter bandwidth (FBR =\(f_{c}/B_{f}\)) is constant, and their frequency distributions are shown in Figure 4c. As the center frequency of FBR-RC filters increases, it is beneficial to increase filter spacing in case of large overlapping between adjacent filters, which can be realized by introducing the ratio of center frequency to filter spacing (FSR =\(f_{c}/\Delta f\)). According to the analysis, FSR-FBR-RC filter bank (Figure 4d) would be the better option, and the SSP using it can be denoted as the improved SSP (ISSP).