## 3 Ultrasonic Guided Wave Signal Processing Methods
### Variational Mode Decomposition  
Variational mode decomposition (VMD) can separate harmonic signals with a relatively close frequency range without affecting the sampling frequency, and improve signal SNR by eliminating incoherent noise [32]. Compared with empirical mode decomposition (EMD), VMD has complete mathematical theory, which is more robust to sampling and noise, and shows better decomposition effect on complex non-stationary signals [33]. In VMD, inherent mode functions (IMFs) are defined as AM-FM signals, which are expressed as [34]:  
\[u_{k}(t)=A_{k}(t)\cos[\phi_{k}(t)] \tag{3}\]  
where \(A_{k}(t)\) is the instantaneous non-negative amplitude, \(\varphi_{k}(t)\) is the undiminished phase function, namely the instantaneous frequency \(\omega_{k}(t)=d\phi_{k}(t)/dt\geq 0\). The VMD model is expressed as follows [35]:  
\[\min_{\{u_{k}\},\{\omega_{k}\}}\left\{\sum\limits_{k}\left\|\hat{ \varrho}_{t}\left[\left(\delta(t)+\frac{j}{\pi t}\right)*u_{k}(t)\right]e^{-j \omega_{k}t}\right\|_{2}^{2}\right\} \tag{4}\] \[s.t.\sum\limits_{k}u_{k}=x \tag{5}\]  
where \(k\) is the number of modes, \(\{u_{k}\}\) and \(\{\omega_{k}\}\) are the \(k\)th IMF and the center frequency after decomposition, \(\delta(t)\) is the Dirac delta function, and \(*\) is the symbol of convolution operation. The VMD process (Algorithm 1) can be concluded as follows: [33; 34]. ```  Initialize \(\{a_{k}^{1}\}\), \(\{\omega_{k}^{1}\},\hat{\lambda}^{1}\), \(n\gets 0\) repeat \(n\gets n+1\) for\(k=1:K\)do  Update \(\hat{u}_{k}\) for all \(\omega\geq 0\) \(\hat{u}_{k}^{n+1}(\omega)\leftarrow\left\lceil\hat{x}(\omega)-\sum\limits_{i \
eq k}\hat{u}_{i}(\omega)+\hat{\lambda}(\omega)/2\right\rceil/\left[1+2\alpha (\omega-\omega_{k})^{2}\right]\)  Update \(\omega_{k}\) \(\omega_{k}^{n+1}\leftarrow\int_{0}^{\infty}\omega|\hat{u}_{k}(\omega)|^{2}d \omega/\int_{0}^{\infty}|\hat{u}_{k}(\omega)|^{2}d\omega\) endfor  Dual ascent for all \(\omega\geq 0\) \(\hat{\lambda}^{n+1}(\omega)\leftarrow\hat{\lambda}^{n}(\omega)+\gamma\left[ \hat{x}(\omega)-\sum\limits_{k}\hat{u}_{k}^{n+1}(\omega)\right]\) until convergence \(\sum\limits_{k}\left\|\hat{u}_{k}^{n+1}-\hat{u}_{k}^{n}\right\|_{2}^{2}/\left\| \hat{u}_{k}^{n}\right\|_{2}^{2}\leq\varepsilon\)  where \(\hat{u}_{k}^{n+1}(\omega)\), and \(\hat{\lambda}(\omega)\) are the frequency domain representations of the \(k\)th IMF \(u_{k}^{n+1}(t)\) signal to be decomposed to \(x(t)\) and the Lagrange operator \(\lambda(t)\), respectively, \(\alpha\) is the penalty factor, and \(\gamma\) is the noise tolerance, which are artificially set. When the number of modes \(K\) is set, the correct mode center frequency \(\omega_{k}\) may not be obtained due to the narrow bandwidth of IMF caused by large \(\alpha\), while low \(\alpha\) could lead to the incoherent noise having a higher level in IMF. The Lagrangian operator is directly affected by \(\gamma\), which can enhance the convergence performance in the case of low noise and should be zero in the case of high noise. ### Wavelet Transform

By adding window functions of different widths to the signal (wavelet transform (WT) applies a wider window function to observe the low frequency and a narrower window function to observe the high frequency), wavelet coefficients are obtained, and the signal can then be reconstructed by these coefficients [36]. The wavelet coefficient can be calculated as follows:

\[W=\frac{1}{\sqrt{S}}\int\limits_{-\infty}^{\infty}x(t)\psi\bigg{(}\frac{t-L}{ S}\bigg{)}dt \tag{6}\]

where, \(S\) and \(L\) are the scaling and translation factors, and \(\psi(t)\) is the mother wavelet functions. By selecting an appropriate threshold T, wavelet coefficients larger than the threshold are considered to be generated by useful feature signals and should be retained, while wavelet coefficients smaller than the threshold are considered to be generated by useless noise signals and are set to zero [37].