# Enhancing Impact Localization from Fluid-Pipe Coupled Vibration under Noisy Environment
## 2 Materials and Methods
### Continuous Wavelet Transform for a Coupled Wave  
The wavelet transform is used for time-frequency (scale) analysis with high resolution on both domains. A wavelet transform is categorized as two transformations: one is the continuous wavelet transform (CWT) and the other is the discrete wavelet transform (DWT). CWT was first introduced as a mathematically initiated multi-resolution analysis [33]. DWT was just used to obtain transformation results at the scales sampled along the dyadic sequence. In this study, CWT was employed to detect coupled wave signals and to distinguish them from noise. CWT is formulated as follows:  
\[W_{x}(a,b)=x*\psi=\frac{1}{\sqrt{a}}\int x(t)\psi\left(\frac{t-b}{a}\right)dt \tag{2}\]  
where \(x(t)\) is a time signal, \(a\) is a scale parameter, \(b\) is a shifting parameter, and \(\psi(t)\) stands for wavelet. The wavelet is defined by scaling and shifting a scaling function, as chosen in the dilation equation below:  
\[\psi_{k,j}(t)=2^{k/2}\phi(2^{k}t-j) \tag{3}\]  
where \(\phi(t)\) is the scaling function, \(\psi_{k,j}(t)\) is the wavelet generated with respect to scale parameter \(k\) and shifting parameter \(j\). The performance of the wavelet transform largely depends on the mother wavelet, and therefore, the similarity between the mother wavelet and signals is of particular importance. Various wavelets have been theoretically taken into account and employed to detect damage in beams and plates [34; 35; 36] and propagating waves [37; 38]. The coupled wave induced by an impact is a quasi-longitudinal wave, being continuous to its derivatives, and this means that wavelet modulus maxima appear in every vanishing moment [34]. [PERSON] et al. [35] mentioned that wavelets with a higher number of vanishing moments give more stable performance, but the number of vanishing moments should be considered with the support size of the wavelets since wavelets with a higher number of vanishing moments have longer supports. The aim of CWT here is to estimate arrival time of the coupled wave with accelerometers; [PERSON] and [PERSON] [39] showed that one vanishing moment is enough to detect singularities in acceleration signals. Therefore, it must be useful to select a simple and basic mother wavelet to detect the arrival time of the fluid-pipe coupled wave, which has its own frequency band. The simplest and most basic wavelet could be the Haar wavelet, which has one vanishing moment and a unity support size. The wavelet function of the Haar wavelet is formulated as  
\[\psi(t)=\left\{ \begin{array}{rl}&1\ \left(0\leq t<\frac{1}{2}\right)\\ &-1\ \left(\frac{1}{2}\leq t<1\right)\\ &0\ \ \ \ otherwise\end{array} \right. \tag{4}\]  
and scaling function is also formulated as  
\[\phi(t)=\left\{ \begin{array}{rl}&1\ \left(0\leq t<1\right)\\ &0\ \ \ otherwise\end{array} \right. \tag{5}\]  
Figure 2 shows the wavelet function and the scaling function. The abrupt change of the Haar wavelet is a good feature that can make peaks in CWT stretch straight along the scale as shown in Figure 3b. It is noteworthy that the localized property of the Haar wavelet is indispensable for estimating an accurate time. Moreover, the narrow and straight extension of peaks of the Haar wavelet in CWT is also essential for the time-delay-estimation algorithm proposed in Section 2.4. The Haar wavelet shows a more straight and narrower feature than other wavelets as shown in Appendix A. Every pipeline shares some common parts (for example, the structural material, coating condition, etc. ), but most conditions (such as boundary conditions and environments) are different. Being within the same area, they could share similar features in frequency domain. The frequency response is associated with dispersive characteristics and structural resonance, and the response should be clearly identified for each pipeline before monitoring. Therefore, it is worthwhile observing how the coupled wave could be described in CWT, using the pipe's resonance.