# Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions
## 6 Discussion
### Justification of the Constructiveness of the Proposed Approach
The presented simple example, among other things, clearly shows that the operation of reducing the convolution result to the original scale is indeed justified. Figure 3: Model function \(f_{1}\) (curve 1, red dots) and the result of applying the moving average calculation operation to it (curve 2, green dots). In many cases, this is also dictated by physical considerations. Figure 4 also presents the smoothed model curve obtained from the original function using the moving average method (curve 1). The same figure shows the result of calculations by formula (51) using partial convolutions. Specifically, the formula was used  
\[\left\langle f_{1}\right\rangle_{0}=\left(5\!\cdot\!\left\langle f_{1}^{1} \right\rangle+6\!\cdot\!\left\langle f_{1}^{2}\right\rangle-\left\langle f_{1} ^{3}\right\rangle\right)\!\mod 10 \tag{59}\]  
where  
\[\left\langle f_{1}^{1}\right\rangle=\left(\sum_{i=k-5}^{i=k+5}f_{1}(k),\text{ mod}2\right),\text{mod}2 \tag{60}\]  
\[\left\langle f_{1}^{2}\right\rangle=\left(\sum_{i=k-5}^{i=k+5}f_{1}(k),\text{ mod}5\right),\text{mod}5 \tag{61}\]  
\[\left\langle f_{1}^{3}\right\rangle=\left(\sum_{i=k-5}^{i=k+5}f_{1}(k),\text{ mod}11\right),\text{mod}11 \tag{62}\]  
Formulas (60)-(62) correspond to calculations in Galois fields. The values of the original function taken modulo the corresponding prime number \(p_{i}\) are summed up, and then the operation of taking modulo \(p_{i}\) is applied to the summation result again. It can be seen that the presented curves coincide with the accuracy up to rounding down, i.e., formula (49) really provides the desired result. It can also be seen that curve 2, Figure 4, which approximates the smoothed curve with downward rounding accuracy, can indeed be derived from calculations performed in Galois fields. We emphasize that Figure 4 shows that the result of calculations performed only in Galois fields (in particular, without using fractional values) coincides (with rounding accuracy) with the result obtained by formula (58). Thus, the used model example serves as a clear demonstration of the adequacy of the proposed methodology of bringing the convolution operation to calculations in Galois fields. The peculiarities of the methodology used are also vividly illustrated by Figure 5, which shows plots the results of intermediate calculations using formulas (60)-(62). Figure 4: Model function \(f_{1}\) smoothed using the moving average method (curve 1, green dots) and the result of calculating the analogue using formula (51), curve 2, red dots. As can be seen in Figure 5, all the functions presented in this figure, vary within the limits corresponding to a particular prime number \(p_{i}\). Besides, the behavior of obtained functions is not ordered, yet the utilization of formula (51) enables the desired outcome to be achieved. Figure 5: Plots of partial convolutions for three digits of number representation modulo 110; (**a–c**) correspond to partial convolutions in Galois fields \(\mathrm{GF}(2)\), \(\mathrm{GF}(5)\), and \(\mathrm{GF}(11)\), respectively. Accordingly, this figure serves as another visual illustration of the nature of the technique used. The result of the above convolution is computed using a set of functions, each of which varies within finite limits corresponding to a certain Galois field. Further, all operations for computing partial convolutions in accordance with the above are de facto performed in Galois fields. Consequently, the numerical analogue of the convolution theorem [15] applies to each of them. These functions are direct analogues of the transfer functions used in classical linear circuit theory. Formally, we can write at once  
\[F[U_{out}(i)]=\Big{(}\sum_{m}\,[K_{m}(i)]F[U_{m,in}(i)]\widetilde{\alpha}_{m} \prod_{j\
eq m}^{N}p_{j}-[K_{m+1}(i)]F[U_{m+1,in}(i)]\Big{)}(\text{mod }w_{2}) \tag{63}\]  
where \(F[f(i)]\) is a notation for the Galois Field Fourier Transform.