## 3 Errors of the Measurement Channel with a Frequency Carrier of Information  
The accuracy of the direct conversion of the frequency signal periods into numerical values in the frequency-to-code converter is mainly limited by two components: the quantization error and the averaging error, which are described in [14]. The quantization error in measuring the period of a variable frequency signal depends on the value of the converted period of the signal [29]. The quantization error \(\delta_{k}\) can be calculated from the dependence:  
\[\delta_{k}=\frac{T_{0}}{T_{xi}}100\%. \tag{6}\]  
An example of the quantization error curve is shown in Figure 3.  
The averaging error arises as a result of the integral action of the input quantity by the physical-quantity-to-frequency converter [14]. It was assumed that the test input signal \(x(t)\) has a sinusoidal shape, the amplitude and the constant component of which are equal to each other, and the variability range is between \(-X_{m}\) and \(+X_{m}\). Taking the above into account, the relative error of averaging is calculated from the dependence [14]:  
\[\delta_{a}=0.5\bigg{(}1-\frac{\sin\pi FT_{xi}}{\pi FT_{xi}}\bigg{)}*100. \tag{7}\]  
where \(F\) is the frequency of the signal \(x(t)\). This relationship shows that the averaging error depends on the value of the period \(T_{xi}\) and the frequency \(F\). The error value increases with the increase of both of these parameters. The analysis of the information processing error in the measurement channel with the frequency-to-code converter, after adopting the ideal X/f conversion (the work concerns the modification of the processing method in the f/N converter), allowed to determine the formula for the maximum total error [14] as:  
\[\delta_{\Sigma}=\frac{T_{g}}{T_{xi}}+0.5\bigg{(}1-\frac{\sin\pi FT_{xi}}{\pi FT _{xi}}\bigg{)}. \tag{8}\]  
Formula (8) shows that the increase of the \(T_{xi}\) period causes the increase of the averaging error, while its decrease causes the increase of the quantization error (Figure 4). An exemplary plot of the total error \(\delta_{\Sigma}\) is shown in Figure 5.  
Figure 3: An example of the quantization error curve for the frequency \(f_{0}=10\) MHz and the \(T_{xi}\) periods values of the quantized waveform varying from \(10\) μs to \(100\) μs. There is a certain minimum between the values causing the increase of the above-mentioned errors at which the measuring system works with the highest accuracy [2]. The frequency of the clock frequency generator \(f_{0}\) has an impact both on the quantization error \(\delta_{k}\) and indirectly, by influencing the measurement of successive periods \(T_{xi}\) of the frequency signal, also on the averaging error \(\delta_{n}\)[14]. The increase in the frequency of the clock generator reduces the quantization error and increases the resolution of the period measurement of the frequency signal [30]. Unfortunately, processing a larger number of pulses requires counters with greater capacity [31]. The opposite situation reduces the demand for the counter capacity but deteriorates the accuracy of the measurement [14]. The conclusion is that it is possible to choose such an output frequency value of the clock generator of the f/N converter that will allow to achieve an acceptable compromise between the capacity of the counter and the value of the clock signal frequency. Of course, it should be remembered that the frequency of the clock signal will produce different error values  
Figure 4: An example of the averaging error for the period of the tested signal changing in the range from 0 to 2 ms and the frequency \(F\) of the \(x(t)\) waveform varying in the range from 0 to 50 Hz. Figure 5: An example of the plot of the error of the conversion of information in the measurement channel with a frequency information carrier. depending on the frequency of the input signal of the f/N converter [29].