## 2 Wavelet Transform Algorithm Transmittance
### Identification of Algorithm Transmittance
\begin{split} S_{2,0}=&\ \frac{1}{\sqrt{2}}\big{(}c_{0}S_{1,0}+c_{1}S_{1,1 }+c_{2}S_{1,2}+c_{3}S_{1,3}\big{)}=\\ &\ \frac{1}{\sqrt{2}}\frac{c_{0}}{\sqrt{2}}\big{(}c_{0}S_{0,0}+c_{ 1}S_{0,1}+c_{2}S_{0,2}+c_{3}S_{0,3}\big{)}+\\ &\ \frac{1}{\sqrt{2}}\frac{c_{1}}{\sqrt{2}}\big{(}c_{0}S_{0,2}+c_{ 1}S_{0,3}+c_{2}S_{0,4}+c_{3}S_{0,5}\big{)}+,\\ &\ \frac{1}{\sqrt{2}}\frac{c_{2}}{\sqrt{2}}\big{(}c_{0}S_{0,4}+c_{ 1}S_{0,5}+c_{2}S_{0,6}+c_{3}S_{0,7}\big{)}+\\ &\ \frac{1}{\sqrt{2}}\frac{c_{3}}{\sqrt{2}}\big{(}c_{0}S_{0,6}+c_{ 1}S_{0,7}+c_{2}S_{0,0}+c_{3}S_{0,1}\big{)}\\ S_{2,1}=&\ \frac{1}{\sqrt{2}}\big{(}c_{0}S_{1,2}+c_{ 1}S_{1,3}+c_{2}S_{1,4}+c_{3}S_{1,5}\big{)}=\\ &\ \frac{1}{\sqrt{2}}\frac{c_{0}}{\sqrt{2}}\big{(}c_{0}S_{0,4}+c_{ 1}S_{0,5}+c_{2}S_{0,6}+c_{3}S_{0,7}\big{)}+\\ &\ \frac{1}{\sqrt{2}}\frac{c_{1}}{\sqrt{2}}\big{(}c_{0}S_{0,6}+c_{ 1}S_{0,7}+c_{2}S_{0,0}+c_{3}S_{0,1}\big{)}+,\\ &\ \frac{1}{\sqrt{2}}\frac{c_{2}}{\sqrt{2}}\big{(}c_{0}S_{0,0}+c_{ 1}S_{0,1}+c_{2}S_{0,2}+c_{3}S_{0,3}\big{)}+\\ &\ \frac{1}{\sqrt{2}}\frac{c_{3}}{\sqrt{2}}\big{(}c_{0}S_{0,2}+c_{ 1}S_{0,3}+c_{2}S_{0,4}+c_{3}S_{0,5}\big{)}\end{split}  \tag{18}\]  
\[\mathbf{A}= \begin{bmatrix}\frac{5-\sqrt{3}}{16}&\frac{5+\sqrt{3}}{16}&\frac{3+3 \sqrt{3}}{16}&\frac{5+3\sqrt{3}}{16}&\frac{3+\sqrt{3}}{16}&\frac{3-\sqrt{3}}{16 }&\frac{5-3\sqrt{3}}{16}&\frac{3-3\sqrt{3}}{16}\\ \frac{3+\sqrt{3}}{16}&\frac{3-\sqrt{3}}{16}&\frac{5-3\sqrt{3}}{16}&\frac{3-3 \sqrt{3}}{16}&\frac{5-\sqrt{3}}{16}&\frac{5+\sqrt{3}}{16}&\frac{5+3\sqrt{3}}{16 }\\ -\frac{1+\sqrt{3}}{16}&\frac{1-\sqrt{3}}{16}&\frac{3-3\sqrt{3}}{16}&-\frac{1+ \sqrt{3}}{16}&\frac{3-5\sqrt{3}}{16}&\frac{1-\sqrt{3}}{16}&\frac{5+3\sqrt{3}}{ 16}\\ -\frac{1-\sqrt{3}}{16}&\frac{1-\sqrt{3}}{16}&\frac{3-3\sqrt{3}}{16}&-\frac{1+ \sqrt{3}}{16}&\frac{1-\sqrt{3}}{16}&\frac{3-\sqrt{3}}{16}&-\frac{1+\sqrt{3}}{16 }\\ \frac{1-\sqrt{3}}{4\sqrt{2}}&\frac{3-\sqrt{3}}{4\sqrt{2}}&\frac{3+\sqrt{3}}{4 \sqrt{2}}&\frac{-1+\sqrt{3}}{4\sqrt{2}}&0&0&0\\ 0&0&\frac{1-\sqrt{3}}{4\sqrt{2}}&-\frac{3-\sqrt{3}}{4\sqrt{2}}&\frac{3+\sqrt{ 3}}{4\sqrt{2}}&-\frac{1+\sqrt{3}}{4\sqrt{2}}&0&0\\ 0&0&0&\frac{1-\sqrt{3}}{4\sqrt{2}}&-\frac{3-\sqrt{3}}{4\sqrt{2}}&\frac{3+ \sqrt{3}}{4\sqrt{2}}&\frac{3+\sqrt{3}}{4\sqrt{2}}&-\frac{1+\sqrt{3}}{4\sqrt{2} }\\ \frac{3+\sqrt{3}}{4\sqrt{2}}&-\frac{1+\sqrt{3}}{4\sqrt{2}}&0&0&0&\frac{1- \sqrt{3}}{4\sqrt{2}}&-\frac{3-\sqrt{3}}{4\sqrt{2}}\end{bmatrix} . \tag{33}\]  
Knowing the values of the subsequent transformation coefficients \(a_{i,j}\) allows us, in accordance with Equation (2), to determine the transfer functions related to the subsequent output quantities of the analyzed algorithm. A certain regularity can be noticed here--the transmittances related to the output quantities with the same scale number will be identical, regardless of the time shift number (this applies separately to details and signal approximation). Hence,  
\[H_{S_{\text{m,r}}}(z) =H_{S_{\text{m,r^{\prime}}}}(z), \tag{34}\] \[H_{T_{\text{m,r}}}(z) =H_{T_{\text{m,r^{\prime}}}}(z), \tag{35}\]  
for \(n^{\prime}\in\mathbb{N}\). This property significantly simplifies the analysis of the metrological characteristics of WT algorithms, especially considering that the quantity of output values generated by these algorithms typically exceeds the number of scales analyzed, stemming from the iterations in the input signal decomposition process [3,16]. It is essential to highlight that the analysis presented can also be conducted for other wavelet families, varying numbers of decomposition process iterations, and different quantities of algorithm input. In instances involving wavelet families with explicitly defined mother wavelet equations (instead of utilizing appropriate relations as shown in the example), it becomes feasible to determine the algorithm's transfer function through suitable transformations [26]. Nonetheless, the connection between the algorithm's transfer function and the transformation matrix coefficient values is consistently described by Equation (2).