## 8 Experimental Results  
The goal of the present study was to experimentally validate the analytical expression that relates to the standard deviation of the estimate of sinusoidal amplitude obtained by fitting a set of data points to the amount of jitter present in the sampling instants. The experimental setup described earlier was used to determine the precision of the amplitude estimation for different amounts of jitter. The standard deviation of the sampling jitter was controlled by phase-modulating the sampling clock with normally distributed random noise. This was achieved by using the functionality of the Agilent 33220A function generator, which has the capability to produce a rectangular signal with chosen frequency and amplitude. In the present work, a rectangular signal with 100 kHz frequency and an amplitude from 0 to 5 V was used. Furthermore, this generator has the capability of modulating several parameters of the main signal (like amplitude and phase) using a second internal generator. This secondary generator can produce a periodic signal with different shapes (sinusoidal, triangular, rectangular, etc.) as well as random noise. In the present study, random noise with a normal distribution was used. The phase of the main rectangular signal was thus phase-modulated by a second normally distributed random signal. This rectangular signal was then used in the data acquisition module to control the timing of the sample acquisition. The clock signal used was a rectangular wave phase-modulated by normally distributed random noise with a range of peak-to-peak values from 0 to 50\({}^{\circ}\) in order to control the amount of jitter present. For each value of jitter, the sine fitting test was carried out using a set of 100 points sampled from a sinewave produced by another function generator (Stanford Research DS 360) with an amplitude of 900 mV and a frequency of 5 kHz. For each value of phase noise setting, the sine fitting test was repeated 500 times, and the confidence interval for the estimated amplitude was computed. The estimate of the standard deviation was obtained from the square root of the sample variance using  
\[s=\frac{1}{\sqrt{n-1}}\sqrt{\sum_{i=1}^{n}(x_{i}-\overline{x})^{2}}, \tag{31}\]  
as per [13], Equations (9)-(13). Note that the variance of the population is unknown. The confidence interval for the standard deviation estimation is thus  
\[CI=\left]\frac{s\cdot\sqrt{n-1}}{X_{1-\frac{\delta}{2}}(n-1)};\frac{s\cdot \sqrt{n-1}}{X_{\frac{\delta}{2}-1}(n-1)}\right[ \tag{32}\]  
where the confidence level used (\(\gamma\)) was 99.9% and \(\delta=1-\gamma\). The number of data points (\(n\)) was 500. These confidence intervals were used to represent the error bars in the experimental data obtained, which are depicted in Figures 13-15. Note that the \"Phase Noise Setting [\"]\" is not the same as the jitter (or phase noise) standard deviation. The relationship between the two has to be determined. That is what is experimentally carried out next. To estimate the amount of jitter present, the test recommended in the IEEE Std. 1057-2017-IEEE Standard for Digitizing Waveform Recorders [27], namely the one in Section 4.9.2.2, was used. In this test, one applies a low frequency (\(f_{1}\)) sinewave to the waveform digitizer and computes the root-mean-square error from the residuals obtained from subtracting the data points from the least squares fitted sinewave, obtaining \(\sigma_{1}\). One then repeats the data acquisition for a high frequency compared to the waveform digitizer bandwidth (\(f_{2}\)) and computes a second value of root-mean-square error, obtaining \(\sigma_{2}\). The amount of jitter that is present is then estimated using\[\sigma_{t}=\frac{\sqrt{\sigma_{2}^{2}-\sigma_{1}^{2}}}{2\pi f_{2}A}, \tag{33}\]  
where \(A\) is the sinusoid amplitude. Figure 14: Representation of the relation between the measured values of phase noise standard deviation and the configured values of phase noise standard deviation. The number of data points used was 50, and the confidence level used was 99.9%.