## 2 Acoustic Emission Method and Wavelet Transform  
The AE method has many advantages, of which the possibility of using the device during regular operation deserves particular attention, as well as the minimal, often a total lack of influence of the measuring system on the working signals. However, when constructing a measuring system, the conditions in which it will work should be taken into account. These conditions refer mainly to the registering device exposed to the environmental impact. This aspect is particularly important for recorded signals, i.e., special attention should be given to the possible recording of signals coming from outside the measurement system (disturbances) and the noise generated by the apparatus. In some cases, there may be a situation in which the signal-to-noise ratio is very low, and then an effective method of filtering the noise from the signal should be applied. The wider bandwidth signal further complicates the task or if this band is not known. Then a method is sought, which could be considered universal, i.e., effective in the whole band of the registered signal. The solution may be to use a wavelet transform that meets these requirements [21; 22]. Wavelets are functions that, by meeting specific mathematical requirements, create a representation of signals or data. Wavelet analysis divides input data into components at different frequencies, analyzing each element with a resolution matched to its scale. To illustrate the situation, one can say that due to its ability to adjust the scale, wavelet analysis allows simultaneous observation of both the whole picture and the smallest details. That is why wavelets are very useful for approximating functions (signal processing) with discontinuities or sharp peaks. In the presented work, the wavelet transform will be implemented as a tool to detect defects in engineering structures. A linear combination of wavelet functions is used to represent the signal (where it is any variable). A discussion on the mathematical foundations of wavelet transform can be found in extensive literature, e.g., [23; 24; 25]. Continuous wavelet transform of the signal f(t) is defined as:  
\[\mathrm{Wf(a,b)}=\int_{-\infty}^{\infty}\mathrm{f(t)}\;\overline{\psi}_{\mathrm{a,b}}(\mathrm{t})\mathrm{dt}, \tag{1}\]  
wherein the overbar denotes the complex conjugate of the function under it. The function (t) belongs to the space of measurable, square-integrable one-dimensional functions \(\mathrm{L^{2}(\mathbf{R})}\) and is an admissible basic (mother) wavelet if it fulfills the condition:  
\[\int_{0}^{\infty}\frac{\left|\Psi(\omega)\right|^{2}}{\omega}\mathrm{d}\omega<\infty, \tag{2}\]  
where \(\Psi(\omega)\) is the Fourier transform of function \(\psi(\mathrm{t})\):  
\[\Psi(\omega)=\int_{-\infty}^{\infty}\psi(\mathrm{t})\mathrm{e}^{-\mathrm{i} \omega\mathrm{t}}\mathrm{dt} \tag{3}\]  
The average value of the wavelet function is zero, which can be expressed by the relation:  
\[\int_{-\infty}^{\infty}\psi(\mathrm{t})\mathrm{dt}=0 \tag{4}\]  
For signal decomposition, copies of the wavelet function, representing the family of wavelets, are used. They are obtained by scaling and translating the function (t):  
\[\psi_{\mathrm{a,b}}=1/\sqrt{\mathrm{|a|}\cdot\psi}((\mathrm{t-b})/\mathrm{a}), \tag{5}\]  
where t is a time or space coordinate, a is a scaling parameter, b means a wavelet shift in the time/space domain. Parameters a and b belong to the set of real numbers (a, b \(\in\)\(\mathbf{R}\)) and a \(\
eq\) 0. The number \(\mathrm{|a|^{-1/2}}\) is a normalizing factor, ensuring constant wavelet energy, independent of the scale which means that the norm \(\left|\psi_{a,b}\right|\)\(\parallel\) = \(\left|\left|\left|\psi\right|\right|\right|\) = 1 [26]. In the analyses presented in this paper, the leading role plays the discrete wavelet transform (DWT).