## 3 Wavelet Packet Transform  
The Wavelet Transform (WT) is a mathematic tool that presents information in both time and frequency domains, unlike other traditional tools such as Fourier Transform, which only provides insights in the frequency domain [47]. The WT breaks down the signal into various frequency components by applying a translation (\(\tau\)) and expansion (s) process to a function known as the mother wavelet \(\psi(t)\), resulting in the wavelets \(\psi_{s,\tau}(t)\) as illustrated in Equation (1) [48]:  
\[\Psi_{s},\tau(t)=\frac{1}{\sqrt{s}}\Psi\bigg{(}\frac{1-\tau}{s}\bigg{)} \tag{1}\]  
Since the signals in this study are measured discretely, it is necessary to apply the Discrete Wavelet Transform (DWT). Using digital filters, which are associated with the mother wavelet, this method decomposes and divides the signal frequency resolution into two equal parts at each level of decomposition. One part corresponds to the approximation (low frequency), while the other represents the detail (high frequency). The technique applied in this work, the Wavelet Packet Transform (WPT) is essentially the DWT applied recursively up to a specified decomposition level \(k\), which determines the number of packets or frequency ranges into which the analyzed frequency is divided \(\left(2^{k}\right)\) (Figure 3), where \(W(k,j)\) represents the signal coefficients for each packet, with \(j\) indicating the packet's position within the decomposition level (Equation (2)). In this way, the WPT allows performing a filtering process more times on a given signal. This decomposition is achieved by passing the study signal \(x[k]\) through a low-pass filter \(g\) to obtain the approximation coefficients \(A\) with Equation (3), and a high pass filter \(h\) to obtain the detail coefficients \(D\) with Equation (4) [49], where \(n\) is each point of the signal after filtering. In these equations, \(i\) represents the coefficient index and \(N\) is the total number of coefficients.  
\[W(k,j)\ =\ \{w_{1}(k,j)\,\ldots,\ w_{N}(k,j)\}=\{w_{i}(k,j)\} \tag{2}\]  
Figure 2: Acoustic emission raw data collected during the test and time history of the applied loading conditions.  
Therefore, two key parameters must be selected in advance when using this tool: the mother wavelet and the decomposition level \(k\), both of which significantly influence the results. The importance of such a good selection is demonstrated in some other works [25; 50]. To analyze changes in signals corresponding to defects, it is beneficial to work in terms of energy, as this approach provides clearer and more interpretable information. The use of wavelet tools yields a large number of coefficients, making energy analysis particularly useful. Moreover, the energy of the packets has been proved to be an effective feature for detecting changes in the dynamical behavior of the system [50; 51; 52]. The energy \(E\)\((k,j)\), for each packet i at decomposition level \(k\), is computed as the sum of the squares of its coefficients, as described in Equation (5), with additional details provided in Equation (2):  
\[\text{E }(k,j){=}\sum_{i}\left[w_{i}(k,j)\right]^{2} \tag{5}\]  
Various families of mother wavelets are discussed in the literature, including classical functions such as Daubechies, Coiflet, Meyer, and others [49]. Each family has distinct characteristics that make it more suitable for certain applications; however, there is no consensus in the literature regarding the optimal choice of mother wavelet for a specific case. For discrete signals and tools, families Daubechies, Symlets, and Coiflets can be used and are the most common [25], and that is why these families will be examined in this work. This analysis will help identify the most appropriate mother wavelet for each case, enabling clearer differentiation between conditions by analyzing the energy of experimental AE signals and optimizing the success rate of an intelligent classification system.