## 5 Note on Incoherent Sampling and Different Estimators  
Recall that the least squares estimator given in (11) is not the only estimator that can be used to estimate the amplitude of a sinewave from a set of data points. The most rudimentary one would be to find in the data points the highest and the lowest ones and subtract their values:  
\[\hat{A}=\frac{z_{max}-z_{min}}{2}. \tag{28}\]  
One can also compute the root-mean-square value of the data points and multiply by the square root of 2:  
\[\hat{A}=\sqrt{2}\cdot\sqrt{\frac{1}{M}{\sum_{i}z_{i}^{2}}}. \tag{29}\]  
Another one would be to compute the Discrete Fourier Transform of the data points, \(Z(f)\), and look for the highest coefficient in the absolute value, \(|Z(f)|_{max}\):  
\[\hat{A}=\frac{2}{N}|Z(f)|_{max}. \tag{30}\]  
There are also other possibilities like estimating the envelope of the Hilbert Transform, which is good for modulated signals but computationally complex, or looking for the  
Figure 3: Standard deviation of the estimated sinewave amplitude as a function of the number of samples. 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 approximate expression (27). slope of the data points at zero crossings, which is very sensitive to noise. In cases where the frequency is unknown, one can also use a least squares estimation (four-parameter sine fitting), but in this case it has to be an iterative procedure that tries different possible frequencies, so it is not a closed-form estimator like the three-parameter estimator. This estimator, which is the subject of this work, has the advantages of being simple to compute and uses information from all data points, which reduces estimation uncertainty. It is thus a commonly used estimator in applications where the signal frequency can be assumed to be known, for example, in testing and characterizing systems and devices. In those situations, one typically creates a sinusoidal stimulus signal with known parameters, including frequency, that is applied to the circuit under test. The response from the circuit or system is then sampled and often quantized, leading to a set of samples from which different parameters and figures of merit can be extracted using, among others, least square types of estimations. The user must therefore choose the number of samples to acquire from the signal, and that is achieved in a way that those samples cover an integer number of periods of the sinewave, which is called \"coherent sampling\". To illustrate what happens when this condition is not met, a simple numerical simulation was carried out, whose results can be seen next. In Figure 4, one can see a ser of 100 data points (circles), obtained at a rate of 100 Hz, that came from a sinewave with a frequency of 1.1 Hz. The data points hence do not cover an integer number of periods of the sinewave. The sinewave fitted to the data points using the three-parameter least squares algorithm led to the solid line in the figure, which clearly does not fit well to the data points. This illustrates how inadequate the procedure is when the signal frequency is not known. Recall, however, that the purpose of this work is to study the uncertainty of the sinusoidal amplitude estimation in the presence of phase noise or jitter when the three-parameter sine fitting is used; the purpose is not to compare which procedure is best, nor is it to study what happens when the frequency used in the sine fitting is in error. In Figure 5, one can observe the expected value of the sinewave amplitude when the data points cover one to three periods of the sinewave. It is easily observed how severely biased the estimator is in these extreme conditions. The correct value that should be estimated is represented by the solid line (3 V). Once again, one can see how poorly the three-parameter behaves when the chosen frequency is in error.