## 3 Digital Filters in the FFA  
Here we describe an implementation of the converter of position code to the FFA code. An opportunity to exclude an influence of the errors of analog-to-digital converter on the filter output sample is considered. Analysis of the modern implementations of computational algorithms in the FFA codes allows concluding that the time of reverse conversion to positional representation in them takes more than 50% of the common time of calculations [11; 12]. Figure 1: The variant of simplified structural diagram of digital filters (DF) operating in the finite field algebra (FFA) codes. Insufficient development of the theoretical foundations of construction of code converters and, following this, the insufficient development of methods and means for their implementation becomes a critical place in the entire cycle of development and implementation of computing devices of DF operating on the basis of the FFA. It leads to loss of its advantages. In [13; 16; 17] the algorithms of operation and hardware implementation of the devices for interfacing the positioning and computing FFA devices are reviewed. The devices in which the determination of residue is based on the use of the property of residues' cyclical nature are considered. Following the Equation (1)  
\[|\alpha_{1}|_{p_{n}}^{+}\in\ \{0\ldots p_{n}-1\}, \tag{12}\]  
the residue \(\alpha_{i}\) will repeat \(d\) times in the range of convertible numbers. The cycle of its reiteration will depend on the value of module \(p_{i}\). In other words, the value \(d\) can be determined from the Equations (13) and (14). \[d=|\frac{2^{s}}{p_{i}}[\,\ for\ \alpha_{i}\ \in\{0,\ldots,p_{i}-1\} \tag{13}\]  
\[d=|\frac{2^{s}}{p_{i}}[+1,\ for\ \alpha_{i}\ \in\{0,\ldots,g\} \tag{14}\]  
where \(g\)--is the residue of dividing the number \(2^{s}\) by \(p_{i}\); \(s\)--is the bit width of the number being converted in the FFA code. In accordance with the Equations (13) and (14) the residue \(\alpha_{i}\) corresponds to \(d\) numbers from the range of \(2^{s}\). To calculate the residue \(\alpha_{i}\) of initial number \(A\) it should be uniquely determined that the number \(A\in\{d\}\), i.e.,  
\[A\in d\ for\ d=|\frac{2^{s}}{p_{i}}[\,\ for\ \alpha_{i}\in\{0,\ldots,p_{i}-1\}\]  
\[A\in d\ for\ d=|\frac{2^{s}}{p_{i}}[+1,\ for\ \alpha_{i}\in\{0,\ldots,g\}\]  
For this goal the range of binary numbers presented in the FFA code can be divided into the subranges. The number of subranges and the values of numbers in them are determined by the value \(E=2^{s/2}\). The values of numbers in subranges will be within the numbers' intervals with a step equal to one. They will be determined by the following expressions:  
\[ \begin{array}{c}E_{1}=0,1,\ldots,2^{s/2}-1;\\ E_{2}=2^{s/2},2^{s/2}+1,\ldots,2^{\frac{s}{2}+1}-1;\\ \ldots\\ E_{k}=2^{(\frac{s}{2})+(\frac{s}{2}-1)},2^{(\frac{s}{2})+(\frac{s}{2}-1)}+1, \ldots,\ \ \ 2^{s}-1.\end{array}  \tag{15}\]  
From the expression (15) it follows that the upper \(s/2\) bits of number unambiguously determine the number of subrange \(E_{i}\) in which the number is. While the lower \(s/2\) bits of number \(A\) determines the index of number in the subrange. Thus, based on the cyclicity property of residues modulo, the finding of residue for the number \(A\) will include determination of subrange number and reading the residue from the memory device for each subrange. Considering the Equations (13)-(15), the algorithm of residue finding on the basis of subrange determination will include the following procedures:  
1. Determining the numbers subrange in the positional numeral system (PNS) \(D_{p}=2^{s}\)
2. Splitting the range \(D_{p}\) into subranges (15). 3. Determining the subrange by the \(s/2\) upper bits of initial number \(A\). 4. Getting the residue \(\alpha_{i}\) from the memory device using the specific address by the \(s/2\) lower bits of initial number \(A\).