## 3 Errors of the Measurement Channel with a Frequency Carrier of Information
The proposed method describes how to select the frequency of the clock signal to improve the information processing parameters in the measurement circuit with a frequency information carrier. Figure 6 shows examples of the quantization and averaging errors for the signal \(x(t)\) changing with the frequency of 1 Hz. It is evident that only one of the above-mentioned errors has the dominant value at the ends of the processing range. For the upper range, where the period tends to the minimum values, the quantization error has a significant impact on the total error, and the averaging error is negligible in this case. The reverse is true for \(T_{xi}\) reaching \(T_{max}\). Taking the above into account, it is possible to use the transformed formulas for the errors mentioned to determine the value of the period \(T_{xi}\) at which, and, thus, the value of \(N_{xi}\) representing it, the given value of the resultant error is obtained. Since each measurement of the \(T_{xi}\) period is performed as an independent measurement [14], the processing error is also determined as the maximum error of a single measurement. Taking into account the considerations presented above, it can be assumed for the processing error \(\delta_{\Sigma}\) in the upper measuring range with a satisfactory approximation that its value describes the Formula (6). Therefore, after transformation, it can be written that the numerical value representing \(T_{xi}\) at which a given error occurs can be calculated from the transformed dependence (6) taking into account Formula (3):  
\[N_{xi}=\frac{1}{\delta_{k}}100\%. \tag{9}\]  
In the lower part of the frequency conversion range, the value of the averaging error [17] described by relationship (7) prevails. Due to the problem of the direct determination of the dependence on the value of \(N_{xi}\) from the above-mentioned equation, the Taylor series expansion [32] of the sinus function was used for the transformation; only the first two unfolding elements were taken. As a result, after transformations and assuming  
Figure 6: Summary of sample quantization error and averaging waveforms for \(F\) = 1 Hz and \(f_{0}\) = 1 MHz. The red line curve shows the course of the averaging error; the black line curve shows the plot of the quantization error. the processing range of the X/f converter to be \(0\div X_{m}\), the dependence on the averaging error took the form:  
Figure 3.0  
The above dependence, after taking the Formula (3) into account, is used to obtain the formula for the value of \(N_{xi}\) obtained for the given value of the processing error:  
Figure 3.1  
The dependencies (9) and (11) allow to establish the \(N_{xi}\) threshold values at which the change of the clock signal frequency \(f_{0}\) in the f/N converter will improve the information processing parameters. Increasing \(f_{0}\) reduces the quantization error. The use of a programmable frequency multiplier [33] may enable an increase in the accuracy in the processing of high-frequency signals of a variable frequency signal [29]. In the low frequency range, the value of the frequency \(f_{0}\) influences the value of the number \(N_{xi}\). The appropriate reduction of \(f_{0}\) allows the information to be processed by the f/N converter with a reduced number of signal pulses with the frequency \(f_{0}\) registered by the f/N converter counter with the maximum averaging error \(\delta_{a}\) unchanged. Reducing the sum of pulses in the measurement of a given \(T_{xi}\) allows increasing the maximum length of the \(T_{xi}\) period [34] that can be measured by the f/N converter.