# Electronic Fourier-Galois Spectrum Analyzer for the Field \(Gf(31)\)
## 2 Methods  
The basis corresponding to the Fourier-Galois transform is formed by the following sequences [26]:  
\[\overrightarrow{w}_{m}=\left(1,\theta^{m},\theta^{2m},\theta^{3m},\ldots, \theta^{(p-2)m}\right) \tag{1}\]  
where \(\theta\) is a primitive element whose degrees up to \((n-1)\)-th inclusive give all non-zero elements of the used Galois field \(GF(p)\), \(m=0,1,\ldots,p-2\). We emphasize that the sequences (1) can also be viewed as piecewise constant functions taking values in the field of \(GF(p)\). They are direct \"digital\" analogs of harmonic functions; in particular, they form a complete basis and are orthogonal in the following sense [26]:  
\[\sum_{j=1}^{j=p-1}w_{k_{1}}^{(j)}w_{k_{2}}^{(j)}=\left\{ \begin{array}{l}n,k_{ 1}+k_{2}\equiv 0(mod\ p-1)\\ 0,k_{1}+k_{2}\
ot\equiv 0(mod\ p-1)\end{array} \right. \tag{2}\]  
where \(n\) is the result of reducing the number of summands in the sum standing on the right-hand side to the value in the considered Galois field. The operations of addition and multiplication in Formula (2) are performed in the sense of the Galois field to which the sequences (1) belong. Note that in the case of the simplest Galois field \(GF(p)\), it means that addition and multiplication are performed modulo a prime number \(p\). The condition appearing in the right part of Formula (2) is an analog of the following equation for harmonic functions:  
\[\int_{0}^{2\pi}\exp\left(-i\frac{2\pi mt}{T_{0}}\right)\exp\left(i\frac{2\pi mt }{T_{0}}\right)\ \sim\delta_{m,n} \tag{3}\]  
Ratio (2) allows us to immediately pass to the spectral representation of the signal in the form of:  
\[\overrightarrow{u}=\sum_{j=1}^{j=p-1}z_{j}\overrightarrow{w}_{j}=\sum_{j=0}^{ j=p-2}z_{j}\left( \begin{array}{c}w_{1j}\\ w_{2j}\\ \vdots\\ w_{(p-1)j}\end{array} \right) \tag{4}\]  
where spectral components can be calculated as follows:  
\[z_{j}=q\left(\overrightarrow{u,w}_{j}^{T}\right)=q\sum_{i=1}^{i=p-1}u_{i}w_{ ij}^{T} \tag{5}\]  
where the value of \(q\) is determined by the condition \(q\left(\overrightarrow{w}_{j},\overrightarrow{w}_{j}^{T}\right)=1\). It is essential that the spectrum defined by Formula (5) describes a piecewise constant function (or sequence) \(\overrightarrow{u}=\left(u_{1},u_{2},\ldots u_{p-1}\right)\) with exhaustive completeness, provided that this function is periodic and its period contains exactly \(p-1\) cycles. This condition is automatically fulfilled when we consider, for example, a circular image divided into \(p-1\) pixels (in the simplest case, it is assumed that the \"width\" of such an image is exactly one pixel). The analogy between the sequences (1) and the harmonic functions used in classical Fourier analysis also consists of the fact that the following relation takes place [25]:  
\[\left(\stackrel{{\rightarrow}}{{w}}_{k},\stackrel{{ \rightarrow}}{{f}}(m)\right)=\theta^{mk}\left(\stackrel{{ \rightarrow}}{{w}}_{k},\stackrel{{\rightarrow}}{{f}}(0)\right) \tag{6}\]  
i.e., if we calculate the Fourier-Galois spectrum of a function shifted relative to the original function by \(m\) cycles, it will differ from the spectrum of the original function by the phase multiplier \(\theta^{mk}\). This relation is completely analogous to the well-known relation from classical Fourier analysis:  
\[F[u(t-t_{0})]=e^{-i\omega t_{0}}\int e^{-i\omega t_{1}}u(t)dt=e^{-i\omega t_{0 }}F[u(t)] \tag{7}\]  
Relationship (6) allows, among other things, to realize the following approach to measuring the rotation frequency of an arbitrary annular image [32]. Let us assume that the circular image is divided into \(p-1\) pixels, where \(p\) is a prime number. Then, let us put in correspondence to the given image a set of \(p-1\) digital recorders, i.e., recorders that perform analog-to-digital conversion of the signal arriving at their input.