# Peculiarities of Applying Partial Convolutions to the Computation of Reduced Numerical Convolutions
## 5 Results
### Increasing the Number of Digits in the Hybrid Number System
This allows us to preserve the character of sampling of the scales of the \"input\" and \"output\" functions. Let us obtain specific formulas that allow us to perform this operation. The initial range of variation of the functions under consideration, whose values are represented in the form (8), is limited by the number \(w_{2}=\prod_{1}^{N}p_{j}\). To fulfill the condition ensuring the fulfilment of formula (38), the prime number \(p_{N+1}\) corresponding to the additional digit must be given by the formula  
\[u_{i}(n+1)\equiv u_{i}(n)+1\,\,(p_{i}) \tag{26}\]8  
Examples of prime numbers \(p_{N+1}\) represented in the form (39) are given in \[u_{i}(n+1)\equiv u_{i}(n)+1\,\,(p_{i}) \tag{26}\]9  
As Table 2 emphasizes, the using even relatively small numbers \(p_{i}\) gives possibility to ensure the computation of partial convolutions of practical interest. Indeed, the range of variation of the numbers represented in the form under consideration is quite large (it is bounded by the product \(Mp_{N+1}\)). Proceeding from the above, we will consider the case when  
\[u_{i}\equiv U\bmod p_{i} \tag{27}\]0  
To return to the original sampling scale, operation (38) must be applied to the result of the computation using partial convolutions, which is represented in the form. \[u_{i}\equiv U\bmod p_{i} \tag{27}\]1  
This notation, in particular, emphasizes that the integer values of \(\widetilde{U}\) vary in the range \(0<\widetilde{U}<w_{1}w_{2}\). According to formula (33), for an arbitrary integer \(s\) it is true that  
\[u_{i}\equiv U\bmod p_{i} \tag{27}\]2  
From where  
\[u_{i}\equiv U\bmod p_{i} \tag{27}\]3  
since \(\left\lfloor\frac{s}{w_{1}w_{2}}\right\rfloor w_{1}w_{2}\) is divisible by \(w_{1}\). Similarly,  
\[u_{i}\equiv U\bmod p_{i} \tag{27}\]4  
There are only the values calculated by mod \(w_{2}\) or mod \(w_{1}\) in the right part of the formula (38). Consequently, as follows from formulas (43) and (44), instead of the value \(\widetilde{U}\) given by formula (41), i.e., providing for taking mod \(w_{1}w_{2}\), we can use the value \(U\) calculated according to the usual rules of addition and multiplication  
\[u_{i}\equiv U\bmod p_{i} \tag{27}\]5 \begin{table}
\begin{tabular}{c c c c c c c} \hline \(\boldsymbol{p_{4}}\) & \(\boldsymbol{p_{3}}\) & \(\boldsymbol{p_{2}}\) & \(\boldsymbol{p_{1}}\) & \(\boldsymbol{M=p_{1}p_{2}\cdots p_{s}}\) & \(\boldsymbol{p_{N+1}}\) & \(\boldsymbol{Mp_{N+1}}\) \\ \hline - & - & 3 & 2 & 6 & 7 & 42 \\ - & - & 5 & 2 & 10 & 11 & 110 \\ - & - & 11 & 2 & 22 & 23 & 506 \\ - & 5 & 3 & 2 & 30 & 31 & 930 \\ - & 7 & 3 & 2 & 42 & 43 & 1806 \\ - & 11 & 3 & 2 & 66 & 67 & 4422 \\ - & 7 & 5 & 2 & 70 & 71 & 4970 \\ - & 17 & 3 & 2 & 102 & 103 & 10,506 \\ - & 13 & 5 & 2 & 130 & 131 & 17,030 \\ - & 19 & 5 & 2 & 190 & 191 & 36,290 \\ 7 & 5 & 3 & 2 & 210 & 211 & 44,310 \\ - & 31 & 5 & 2 & 310 & 311 & 96,410 \\ 11 & 5 & 3 & 2 & 330 & 331 & 109,230 \\ - & 19 & 11 & 2 & 418 & 419 & 175,142 \\ \hline \end{tabular}
\end{table} 
Table 2: Multipliers \(p_{i}\) giving examples of numbers of the form (39). (The symbol “-” means that this multiplier is not included in M). Or, taking into account (40)  
\[U=w_{1}\Bigl{(}\sum_{i}u_{i}\widetilde{\alpha}_{i}\prod_{j\
eq 1}^{N}p_{j}\Bigr{)}+ e_{N+1}u_{N+1} \tag{46}\]  
Let us apply the mod \(w_{1}\) operation to expression (32). Then  
\[U(\text{mod}\ w_{1})\equiv(e_{N+1}u_{N+1})(\text{mod}\ w_{1})=u_{N+1} \tag{47}\]  
where it is taken into account that  
\[e_{i}\equiv 1\ \text{mod}\ p_{i} \tag{48}\]  
The identity (48) follows directly from (25). As a result, we obtain the following calculation formula, which solves the problem. \[\left\lfloor\frac{U}{w_{1}}\right\rfloor\equiv\Bigl{(}\sum_{i}u_{i}\widetilde {\alpha}_{i}\prod_{j\
eq 1}^{N}p_{j}-u_{N+1}\Bigr{)}\ \text{mod}\ w_{2} \tag{49}\]  
Let us consider how exactly the obtained formula can be applied to calculate digital convolutions.