## 4 Numerical Simulation  
To validate the analytical expressions derived here, and which can be used to determine the standard deviation of the estimated amplitude of a sinusoidal signal from a set of data points corrupted by jitter or phase noise, several numerical simulations have been carried out. These simulations consisted of numerically creating a sinusoidal signal with the desired parameters, extracting a set of data points affected by jitter and/or phase noise with the desired statistics, and fitting, in a least square sense, a sinusoidal model to estimate the signal amplitude, offset, and initial phase. In this work, it is considered that the frequency of the signal is known, and thus it does not have to be estimated from the data points. In Figure 1, one can see an example with 100 data points obtained from a sinusoid corrupted by phase noise with a standard deviation of 0.1 rad. By repeating this procedure many times, one obtains several estimates for the sinusoidal parameters. The focus here is on its amplitude. With all those values, one is able to compute their standard deviation and study how it varies with different settings like the number of samples used or the jitter/phase noise standard deviation, for example. In the following, the obtained results are shown. Figure 2 shows the standard deviation of the estimated amplitude as a function of jitter/phase noise standard deviation. The range of values of the injected phase noise goes from 0 (no phase noise created) to 1 rad. The average of the 2000 values obtained for the sinusoidal amplitude standard deviation, for each value of phase noise standard deviation, is plotted using solid filled circles. The vertical bars represent the confidence intervals for  
the estimated standard deviation considering a 99.9% confidence level and the case of a chi-squared distribution. The solid line depicts the analytical values given by (27). As observed, these values agree very well with the numerically simulated data, validating the analytical derivations presented here. There is a small discrepancy for higher values of jitter/phase noise standard deviation that deserves further study. This is carried out next. As seen with the numerical simulations presented, the approximations used are quite good if the phase noise standard deviation is lower than 1 rad. In order to justify the applicability of the current work to everyday applications, despite the approximations made, a short list with some typical values of phase noise encountered is presented here:  
* Ethernet (Gigabit Ethernet, 125 MHz): 4 mrad [19]. * PCIe Gen3 (8 GHz): 0.1 rad [20]. * Digital Audio (192 kHz DAC): 0.15 mrad [21]. * High-Speed ADC (500 MHz): 0.5 mrad [22]. * FPGA/ASIC Clock Signals (200 MHz): 40 mrad [23]. * HDMI 2.0 (3.4 GHz): 0.8 rad [24]. As can be seen in these random examples, the values of phase noise standard deviation are lower than 0.8 rad and typically much lower. There are, however, some applications where the amount of phase noise is higher, for example:  
Figure 1: Simulated data points (100 points) for a sinewave with an amplitude of 3 V, an initial phase of \(\pi\)/3 rad, and a frequency of 1 Hz (\(f_{x}\)), corrupted with phase noise/jitter, normally distributed, with null means and a standard deviation of 0.1 rad. The sampling frequency is \(f_{s}\). Figure 2: Standard deviation of the estimated sinewave amplitude as a function of phase noise standard deviation. The circles represent the values obtained with the Monte Carlo analysis. The confidence intervals for a confidence level of 99.9% are represented by the vertical bars. The solid line represents the value given by the theoretical expression (27). The number of repetitions carried out (R) was 2000. * 5G NR Base Stations (3.5 GHz): 7 rad [25]. * PLL in RF Transceivers (2.4 GHz): 5 rad [26]. Evidently, the current study is not applicable to these two example applications.