## 3 Digital Filters in the FFA
Moreover, the common model of calculation accuracy for an output sample in the positioning DF considering that the intermediate results will be truncated can be represented as follows:  
\[e_{er\ com}(nT)=Q_{ADC}(nT)+\ e_{q\ coef}(nT)+\ e_{q\ is}(nT)+e_{tr\ out}(nT)+ \ e_{add}(nT) \tag{20}\]  
where \(e_{tr\ out}(nT)\)--the error introduced by truncation of intermediate calculation results in the filter links. In the computational device of DF, operating on the basis of FFA, there is no such disadvantages as the operations of truncation (rounding off) of intermediate calculation results, the additional errors, the errors of quantification of input data and filter coefficients. Therefore, there is no accumulation of errors in the filter when calculating the output sample. Then the accuracy model of output sample calculation for the DF operating on the basis of FFA can be represented as follows [14]:  
\[e_{er\ com}(nT)=Q_{ADC}(nT). \tag{21}\]  
Analysis of Equations (19)-(21) allows concluding that the accuracy of DF operating on the basis of FFA is significantly higher than the accuracy of positioning filters. Analysis of the influence of errors that occur when calculating the output sample of the FFA filter during signal processing in radio channels showed that the signal-to-noise ratio (SNR) is about 68-70 dB. If the amplitude of the input signal is less than a half of the voltage of the full scale of ADC, then there is an additional attenuation about \(-20\) dB. In this case, the SNR value is calculated as follows: \(\text{SNR}=74\ \text{dB}-20\ \text{dB}=54\ \text{dB}\). However, as soon as radio channels are subject to significant fading, it can be argued that the SNR value will be 40-50 dB. Thus, it can be concluded that in case of exclusion of ADC from the signal processing path, for example, from the digital receiver path (Figures 3 and 4) the value of SNR can be increased. As soon as while building the digital receiver the range of data being processed in the digital signal processing device (Figures 3 and 4) is known, namely, the bit width of used ADC is known, then there is an opportunity to represent input data being processed in the filters' computational channel in the FFA codes without ADC and the device of their conversion. Considering Equation (12), the values of residues for any module of the selected base system do not exceed the value of the module and their count is equal to the module value. Then as a source of input data for the DF's computational channel a memory device can be used. It records the values of residuals for the selected module. Therefore, the FFA ring property allows one to significantly simplify the hardware implementation of the computing device of the filter, to increase its performance and to exclude the error of output sample calculation. Thus  
\[e_{er\ com}(nT)=0. \tag{22}\]In the case of such calculations, the error introduced by analog-to-digital conversion of input data is excluded from the channel of FFA digital filter. This further increases the accuracy of the calculation of the filter output sample. The structure of the computational channel of DF operating based on the FFA is shown in Figure 5.  
Thus, the data converter from positional representation to the FFA code can be excluded from the DF structure. The ADC can be used as a control device for the selection of the required residue from the memory device of the output sample. In this case, one ADC can be used for all computational channels of the FFA DF.