# Electronic Fourier-Galois Spectrum Analyzer for the Field \(Gf(31)\)
## 3 Results
### Algorithm of the Fourier-Galois Spectrum Analyzer
For example, in the case \(k=4\), according to Table 2, Formula (40) goes to:  
\[Z_{k}^{(6)}=U_{6}U_{12}U_{18}U_{24}U_{30} \tag{41}\]  
Similarly, for other groups of five summands each, we can write the following expressions:  
\[Z_{k}^{(5)}=g_{1}^{5k}g_{2}^{5k}\left(U_{5}g_{3}^{5k}+U_{11}g_{3}^{11k}+U_{17} g_{3}^{17k}+U_{23}g_{3}^{2k}+U_{29}g_{3}^{29k}\right) \tag{42}\]  
\[Z_{k}^{(4)}=g_{1}^{4k}g_{2}^{4k}\left(U_{4}g_{3}^{4k}+U_{10}g_{3}^{10k}+U_{16} g_{3}^{16k}+U_{22}g_{3}^{22k}+U_{28}g_{3}^{28k}\right) \tag{43}\]  
\[Z_{k}^{(3)}=g_{1}^{3k}g_{2}^{3k}\left(U_{3}g_{3}^{3k}+U_{9}g_{3}^{9k}+U_{15} g_{3}^{15k}+U_{21}g_{3}^{21k}+U_{27}g_{3}^{27k}\right) \tag{44}\]  
\[Z_{k}^{(2)}=g_{1}^{2k}g_{2}^{2k}\left(U_{2}g_{3}^{2k}+U_{8}g_{3}^{8k}+U_{14} g_{3}^{14k}+U_{20}g_{3}^{20k}+U_{26}g_{3}^{26k}\right) \tag{45}\]  
\[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]0  
Using Formula (25), these equations can be simplified. We then have the following:  
\[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]1  
\[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]2  
\[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]3  
\[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]4  
\[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]5  
Formulas (47)-(51) show that, as in the simplest case considered above, the separated groups of five summands each form a certain binary number. This follows from the fact that on the right-hand side of Formulas (47)-(51), different degrees of the element \(g_{3}\) appear,i.e., the degrees of two. It can also be seen that each of these numbers contains five binary digits. We also emphasize that these formulas are valid for the nondegenerate case. For the degenerate case, the calculations should be carried out using Formula (37), but here also the sum is divided into groups of five elements. Further, depending on the value of \(k\) function \(g_{1}^{k}\) can take only two values, either \(1\) or \(30\), and in the latter case, as follows from Formula (33), multiplication by \(g_{1}^{k}\) means the inversion of the logical variables of the binary number formed by the sum in brackets. Functions \(g_{2}^{k}\) can take only three values--1, 5, and 25. Consequently, the calculation of any component of the spectrum in the considered case is reduced to the summation of six numbers, which in binary representation are given by Formulas (47)-(51). This fact justifies the choice of the Galois field used. The calculation of the spectral component using the proposed algorithm is illustrated in \[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]6 The first column of this table contains the numbers \(i\), which play the role of the current discrete variable for the functions under consideration. In the second column are the values of the model function \(f\) used to test the proposed algorithm. The model function is presented in Figure 2, and its Fourier-Galois spectrum is in \[Z_{k}^{(6)}=U_{6}g_{3}^{k}+U_{12}g_{3}^{2k}+U_{18}g_{3}^{3k}+U_{24}g_{3}^{4k}+U_{ 30} \tag{38}\]7 \begin{table}
\begin{tabular}{c c c c c c c c c} \hline \(i\) & \(f\) & \(F_{2}\) & \(F_{1}\) & \(F_{3}\)(6) & \(F_{3}\)(5) & \(F_{3}\)(4) & \(F_{3}\)(5) & \(F_{3}\)(2) & \(F_{3}\)(1) \\ \hline  
& & & & & \\ \(\Sigma\) & & & & & 10 & 6 & 6 & 17 & 17 & 9 \\ \(F_{1}\) & & & & & 1 & 30 & 1 & 30 & 1 & 30 \\ \(F_{1}\Sigma\) & & & & & 10 & 25 & 6 & 14 & 17 & 22 \\ \(F_{2}\) & & & & & 1 & 25 & 5 & 1 & 25 & 5 \\ \(F_{2}F_{1}\Sigma\) & & & & 10 & 5 & 30 & 14 & 22 & 17 \\ \(Z_{1}\) & & & & & & & & & 5 \\ \hline \end{tabular}
\end{table} 
Table 4: An illustration of the spectral component calculation \(k=1\). In the third and fourth columns stand the values of the functions \(F_{2}\) and \(F_{1}\), respectively.