## 4 Selection of the Mother Wavelet  
To conduct an effective WPT analysis, it is essential to select the optimal mother wavelet. For this purpose, a procedure based on the approach presented in a previous study [25] is applied. Although that study focused on vibratory signals, it explored the Daubechies, Symlets, and Coiflets wavelet families, which are particularly well suited for discrete signal processing applications. These wavelet families share several important characteristics that justify their selection in this work. Firstly, all three families consist of compactly supported wavelets with orthogonal or nearly orthogonal properties, making them ideal for energy-preserving decomposition, which is essential when analyzing energy distribution in AE signals. Additionally, they  
Figure 3: WPT decomposition process for a decomposition level of 3.  
offer good time-frequency localization, allowing for the accurate identification of transient events such as burst-type acoustic emissions. The Daubechies wavelets are known for their ability to represent signals with sharp discontinuities or rapid changes, thanks to their high number of vanishing moments. The Symlets are a modified version of Daubechies wavelets with improved symmetry, which can reduce phase distortion, which is useful for analyzing signals where temporal alignment is important. The Coiflets provide an even higher number of vanishing moments for both the wavelet and scaling functions, enhancing their performance in capturing both smooth and detailed signal components. These characteristics are particularly relevant for acoustic emission signals, which often exhibit short-duration, high-frequency burst events that resemble wavelet shapes in time [38]. Therefore, the selected families offer the mathematical and practical advantages required for accurate and efficient signal decomposition in the context of damage detection via AE monitoring. Other wavelet families, such as Biorthogonal, Meyer, or Morlet, were not included in this analysis, as they are either not compactly supported, not orthogonal, or primarily designed for continuous wavelet analysis, making them less suitable for the discrete, energy-based WPT approach used in this work. For this reason, the same wavelet families will be used in this study. By selecting the appropriate mother wavelet, it will also be able to filter out noise from the signal, eliminating the need for clustering the signals. This approach is particularly advantageous because the damage waveforms are better suited to wavelet transformation. As a result, the detection process can be performed more quickly, a point demonstrated in previous works, which highlighted the similarity between the shape of the wavelet and the burst event generated by defects in the signal. These studies also focused on a maximum of ten vanishing moments. A comprehensive summary of this parameter can be found in a previous work [25], but in essence, it refers to the fact that within the mother wavelet families, there are wavelets of varying orders, each with distinct properties. For example, the Coiflets family ranges from Coif1 to Coif5, while the Daubechies family spans from db1 to db45. The aforementioned pre-selection is intended to study all wavelets under the same conditions, as prior research has shown that increasing the number of vanishing moments leads to longer computation times with minimal variation in results. As the aim of this application is condition monitoring of the system, this consideration is important for ensuring a fast and efficient procedure. For the same reason, the decomposition level will be set at 3. In this study, the decomposition level for the WPT was fixed at level 3. This choice is based on a trade-off between time-frequency resolution and computational cost, and is consistent with previous studies [25], that showed level 3 to be effective in capturing the frequency content relevant to different signals while maintaining manageable feature dimensionality.