## 2 Wavelet Transform Algorithm Transmittance
### Identification of Algorithm Transmittance  
The determination of the algorithm's transmittance concerning the subsequent output quantities can be carried out using Equation (2) or by understanding the properties of the wavelet utilized by the analyzed algorithm. This process can also be executed for an existing implementation of the algorithm (e.g., implemented in MATLAB [31], GNU Octave [32], or the PyWavelets package [33]), employing a suitable identification algorithm [18]. Since the mentioned identification algorithm has been utilized in previous studies [16; 22], and its application is straightforward, the analytical approach will be elaborated below. The illustration pertains to the \"Daubechies\" [21] wavelet family, assuming that the WT algorithm will process \(N=8\) input quantities to produce \(M=8\) output quantities, utilizing the \"db2\" wavelet and undergoing \(K=2\) iterations of the signal decomposition process. In the case under analysis, the vector of input quantities, previously mentioned in Equation (1), can be represented as [5]  
\[\mathbf{X}^{T}=\Big{[}S_{2,0}\quad S_{2,1}\quad T_{2,0}\quad T_{2,1}\quad T_{1,0}\quad T_{1,1}\quad T_{1,2}\quad T_{1,3}\Big{]}, \tag{8}\]  
where the symbol \(S_{m,n}\) denotes the approximations, and the symbol \(T_{m,n}\) denotes the details of the signal for the scale number \(m\) and the time shift number \(n\), while the vector of output quantities can be described in the form  
\[\mathbf{x}^{T}=\Big{[}S_{0,0}\quad S_{0,1}\quad S_{0,2}\quad S_{0,3}\quad S_{ 0,A}\quad S_{0,5}\quad S_{0,6}\quad S_{0,7}\Big{]}. \tag{9}\]  
The signal decomposition process, enabling the calculation of the values of the quantities referenced in Equation (8) from the values of the quantities specified in Equation (9), can be recursively defined using the equations outlined in [5]:  
\[S_{m+1,n} =\frac{1}{\sqrt{2}}\sum_{k=0}^{N_{k}-1}c_{k}S_{m,2n+k}, \tag{10}\] \[T_{m+1,n} =\frac{1}{\sqrt{2}}\sum_{k=0}^{N_{k}-1}b_{k}S_{m,2n+k}. \tag{11}\]  
where the symbol \(c_{k}\) represents consecutive non-zero scaling factors, which will be elaborated on later in this paper, while the symbol \(b_{k}\) signifies the coefficients computed in accordance with the following equation:  
\[b_{k}=(-1)^{k}c_{N_{k}-k-1}, \tag{12}\]  
where \(N_{k}\) is the number of non-zero scaling factors resulting from the properties of the wavelet used. According to Equations (10) and (11), to determine the values of the quantities outlined in Equation (8), one must have knowledge of the number and values of consecutive scaling factors. In considering the assumptions regarding the analyzed wavelet \"db2\", this wavelet possesses \(N_{k}=4\) non-zero scaling factors: \(c_{0}\), \(c_{1}\), \(c_{2}\), and \(c_{3}\)[5; 21]. Furthermore, it is characterized by the following assumptions [5; 21]:  
\[\sum_{k=0}^{N_{k}-1}c_{k}=2, \tag{13}\] \[\sum_{k=0}^{N_{k}-1}(-1)^{k}c_{k}k^{m}=0,\] (14) \[\sum_{k=0}^{N_{k}-1}c_{k}c_{k+2k^{\prime}}= \begin{cases}2&\text{ where }k^{\prime}=0\\ 0&\text{in other cases}\end{cases} , \tag{15}\]  
where in Equation (14), \(m\in[0;\frac{N_{k}}{2}-1]\), and in Equation (15), \(k^{\prime}\in\mathbb{N}\). Based on the above assumptions, the system of equations can be written as follows:  
\[ \begin{cases}c_{0}+c_{1}+c_{2}+c_{3}=2&\text{according to (\ref{eq:c_1})}\\ c_{0}-c_{1}+c_{2}-c_{3}=0&\text{according to (\ref{eq:c_1}) for }m=0\\ -1c_{1}+2c_{2}-3c_{3}=0&\text{according to (\ref{eq:c_1}) for }m=1^{\prime}\\ c_{0}^{2}+c_{1}^{2}+c_{2}^{2}+c_{3}^{2}=2&\text{according to (\ref{eq:c_1})}\end{cases} \]  
and solving this system allows the values of the analyzed scaling factors to be determined:  
\[c_{0}=\frac{1+\sqrt{3}}{4},c_{1}=\frac{3+\sqrt{3}}{4},c_{2}=\frac{3-\sqrt{3}}{ 4},c_{3}=\frac{1-\sqrt{3}}{4}. \tag{16}\]  
Based on Equations (10) and (11), relationships describing the output values of the analyzed algorithm, indicated in Equation (8), are obtained:  
\[