## 2 Materials and Methods
### Convolution-Autocovariance Calculation  
When acquiring a periodic signal \(x(t)\), the noisy observed signal \(a(t)\) is described as:  
\[a(t)=x(t)+n(t)=\ \sum_{k=1}^{N}A_{k}\ \cos(w_{k}t+\phi_{k})+n(t) \tag{6}\]  
where and \(n(t)\) are the input signal and an additive stationary noise. Furthermore, \(A_{k}\), \(f_{k}\), and \(\phi_{k}\)for \(k=1,\dots,\ N\), are the amplitude, frequency, and phase for every single harmony component. Using this representation, we obtain:  
\[C_{2}^{a}(t)=C_{2}^{x}(t)+C_{2}^{a}(t)=E(x(ts)x(s+t))+E((n(s)n(s+t)) \tag{7}\]  
The autocorrelation of a white noise \(n(t)\) with zero mean and variance \(\sigma_{n}{}^{2}\) is a Dirac distribution \(\delta_{0}(t)\). If the signal is harmonic, it returns a harmonic signal with null phase. Then from (4) and (6), we obtain:  
\[C_{2}^{a}(t)=\sum_{k=1}^{N}\frac{{A_{k}}^{2}}{2}\ \cos(w_{k}t)+\sigma_{n}{}^{2} \delta_{0}(t) \tag{8}\]  
Figure 2: Representation of the noise-reduction algorithm workflow. In turn, Figure 6 shows a comparison of the result once the harmonic signal taken as a sample has been processed. This permits us to test the influence of the signal-to-noise ratio in the proposed algorithm. In Figure 6, it can be observed that after processing, the desired spectral components are visible and the noise was reduced. Moreover, with our proposed algorithm, the use of a signal decomposition mode is not necessary, unlike the algorithm proposed in [23]. It only needs the amplitude, frequency, and phase of the input signal. As can be seen in Figure 6, the amplitude of the signal at the output is affected during the noise-reduction process, and even more if it has a low SNR ratio at the input. However, the amplitude adjustment process can be improved by applying the result shown in Equation (3) as a non-linear amplitude adjustment factor. Since the statistical characteristics are based on the expected value operator (continuous model), and in practice a finite value of samples is used, the results will be more similar in amplitude if the number of samples to be processed increases (see Figure 7). In addition, the SNR ratio at the input also influences since the method significantly reduces noise, but at the output there are always samples of residual noise that is not eliminated and influence the amplitude of the signal. Figure 6: The spectrum of the original signal (blue), the spectrum of the noisy signal (black), and the spectrum obtained after applying our algorithm for noise reduction without additional information of the input signal (red); 16,000 processed samples. In relation to other classic noise reduction methods for the same experimental case and using the same harmonic signal, a wavelet-based method was used. This method uses soft heuristic [PERSON]'s unbiased risk estimate (SURE) thresholding to obtain the actual threshold, to reduce noise. Table 2 summarizes the results obtained in comparing the proposed method with that based on wavelets. As can be seen in Table 2, one of the most significant disadvantages of wavelet-based methods is a priori ignorance of the calculation of the noise threshold to be eliminated. This calculation is based on knowing the noise power to reduce the random signal, but in real situations, this parameter is generally unknown An example of application is provided in Figure 8.  
In general, the choice of one method or another to reduce noise depends mainly on the experimental situation required to process the information to reduce the random signal. There are different methods, each of which has specific characteristics. The effectiveness of one or the other depends on the characteristics of the process. A combination of methods based on statistical characteristics, as in our case, has the advantage of basing their analysis on the analytical foundations of the random process itself.