## 2 Methods  
The paper considers the mapping of a signal in the form of a function that takes values in a finite algebraic ring containing zero divisors. The choice of this approach is due to the following considerations.  
For many problems of practical interest, the result of performing the convolution of two digital functions lies in a limited range of values. This paper considers such convolution operations whose values do not exceed a certain number \(P\), which is the product of prime numbers.  
\[P=p_{1}p_{2}\ldots p_{S} \tag{1}\]  
When this condition is satisfied, the result of the convolution in calculations modulo the number \(P\), coinciding with the result of calculations in the sense of ordinary arithmetic operations.  
The transition to residue classes modulo the numbers \(p_{i}\) allows us to consider any number \(u<P\), which can be included as an element of the algebraic ring of residue classes modulo \(P\). Specifically,  
\[u\equiv e_{1}u_{1}+e_{2}u_{2}+\cdots+e_{N}u_{N}(P) \tag{2}\]  
where \(e_{i}\) are idempotent mutually canceling elements, and \(u_{i}=0,1,2,\ldots,p_{i}-1\).  
The idempotent elements are formed according to the rule  
\[e_{i}=\alpha_{i}{\prod_{i\
eq j}^{N}p_{j}} \tag{3}\]  
where \(\alpha_{i}\) is an integer. The choice of these numbers is based on the condition  
\[e_{i}e_{i}=1 \tag{4}\]  
It is obvious that by construction we have [56]  
\[e_{i}p_{i}\equiv 0(P) \tag{5}\]  
since any product of the form (5) contains a factor \(P=p_{1}p_{2}\ldots p_{N}\).  
With the choice of integers \(\alpha_{i}\) made, it also holds.  
\[e_{1}+e_{2}+\cdots+e_{N}\equiv 1(P) \tag{6}\]  
Consider the product of two numbers written in the form (2). Since \(e_{i}\) are mutually annihilating idempotent elements, we have  
\[u^{(1)}u^{(2)}=e_{1}u_{1}^{(1)}u_{1}^{(2)}+e_{2}u_{2}^{(1)}u_{2}^{(2)}+\cdots+ e_{N}u_{N}^{(1)}u_{N}^{(2)} \tag{7}\]In this case, the calculation of the products \(u_{i}^{(1)}u_{i}^{(2)}\) is actually performed in the sense of the multiplication operation in the Galois field \(GF(p_{i})\), since the multiplication operation is performed modulo \(p_{i}\). This conclusion is also true for the addition operation [56].  
Let us consider the convolution written in the form  
\[U_{out}(i)=\sum_{j}K(j)\ U_{in}(i-j) \tag{8}\]  
where \(K(j)\) is the kernel of the convolution operator, and \(U_{out}(i)\) and \(U_{in}(i)\) are functions that we will treat as \"output\" and \"input\", respectively.  
If the condition is satisfied  
\[U_{out}(i)<p_{1}p_{2}\ldots p_{N}, \tag{9}\]  
then, in Formula (9) we can substitute the expansion through idempotent elements (2), both for \(K(j)\) and for \(U_{in}(i)\). We have:  
\[U_{out}(i)=\sum_{j}\Big{(}\sum_{m}e_{m}K_{m}(j)\Big{)}\Big{(}\sum_{m}e_{m}U_{ m,in}(i-j)\Big{)} \tag{10}\]  
By swapping the summation signs, we obtain [56]  
\[U_{out}(i)=\sum_{m}e_{m}\Big{(}\sum_{j}K_{m}(j)U_{m,in}(i-j)\Big{)} \tag{11}\]  
It can be seen that the convolution operation is factorized. Each component of the functions used can be operated on in a completely independent manner.  
Accordingly, in this case it is permissible to speak of partial convolutions, each of which corresponds to a certain digit in the hybrid number system. We have:  
\[U_{out,m}(i)=\sum_{j}K_{m}(j)U_{m,in}(i-j) \tag{12}\]