**Acknowledgements**

*Wavelet Transform and Complexity*

**Table 5.**

with comparison to the true TOF.

waves generated by means of LASER excitation.

**5. Conclusion**

for **Figure 6a**–**c**, respectively. **Figure 7** and **Figure 8** represent the same wavelet echo analysis for Mexican hat and db4 mother wavelets respectively. **Table 2** show the resulting TOF from the internal defects at three scan points with reference to the different mother wavelets while **Table 3** shows the error of the resulting TOF compared to the calculated true TOF for each of the cases in (**Table 2**). It is found that the use of the Morlet mother wavelet gives the least estimation error and it is apparently the most accurate mother wavelet to use for this kind of analysis. For the case of the TFR that results from the application of the SSWT, it can be observed that by the accurate redistribution of the energy that compose to the signal, the obtained images achieve an improved depiction of their modal frequencies in comparison with the conventional CWT, aiding to superiorly identify the behavior of the phenomenon. Moreover, for the scope of application of this study, by identifying the first instant of time when the bi-dimensional manifold created by means of the contour mapping of the SSWT apparently becomes closed by connecting all the modal frequencies of the signal of interest, it is possible to determine the onset of said signal. As is well known, the accurate determination of this instant of time is critical for the TOF-related methods; hence, by means of this methodology, the required precision for the onset pick is achieved when only the signal waveform is used for this purpose. Synchrosqueezing wavelet transform contour map for the scan points of interest and TOF estimation are calculated **Figures 9**–**11**. **Tables 4**–**5**. show the resulting TOF for the three scan points and Table 5 shows the corresponding error

*Resulting time of flight error compared with the true time of flight, in microseconds.*

Synchrosqueezing transform 1.43 5.26 3.14

**Scan point 1 Scan point 2 Scan point 3**

Nevertheless, considerations must be taken in order to not analyze a very small signal, this with the aim to avoid the negative effects of the Cone of Influence (COI)

Indeed, the acoustic emission phenomena have been utilized as a powerful tool with the purpose to either detect, locate or assess damage for a wide range of applications. Derived from its monitoring, one of the major challenges in analyzing the resulting wavelet or synchrosqueezing transform signal is to identify and extract each generated AE event. Typically, this event detection is carried out by a thresholding approach over the raw signal. In this regard, the wavelet algorithm has resulted in a very useful and successful technique in detecting the time of flight of the acoustic emission echoes generated by defects at their corresponding frequencies. The accuracy of the algorithm was investigated experimentally using metallic structure. This algorithm is more powerful than the conventional Fourier transform algorithm. Various mother wavelets have been used to compare the correlation between the mother wavelet and the acquired A-scan signals. A mother wavelet with higher correlation would provide more accurate results. Thus, it is important to select the mother wavelet carefully to avoid misleading results. In regard with the synchrosqueezing transform, although improved resolution capabilities, the error in regard with the time of flight determination is not reduced. The Morlet wavelet is revealed as the most suitable wavelet dealing with such acoustic emission

of the CWT, since the SSWT still leads to inaccuracies for these areas.

**108**

This work was supported in part by the CONACyT scholarship grant number 411711, Mexico, and the Ministry of Economy and Competitiveness under the TRA2016-80472-R Research Project, Spain.
