# Analysis of Electroencephalograms Based on the Phase Plane Method
## 3 Determination of the Signal-to-Noise Ratio Based on the Phase Portrait  
The possibilities of the phase plane method are considered in the analysis of stochastic processes. First, we consider Gaussian noise with zero mean value and standard deviation \(\sigma=1\). The Gaussian noise PP is shown in Figure 3.  
In Figure 3, it is seen that the deviation from the mean practically does not exceed 3, i.e., the 3-sigma rule is carried out [17]. Along the ordinate axis, the derivative of the normal process has been placed. Let us define its mean value and dispersion using the statistics' basic formulas [18,19]. The obtained results are given in Table 1.  
The probability density function of the normal process derivative is shown in Figure 4.  
The resulting probability density function characterizes a new random process, which has zero mean value, and the values of the process are concentrated in the interval \(-4\sigma\leq x^{\prime}\leq 4\sigma\). This is shown in Table 2.  
One of the most-commonly used models of a received signal in radio engineering and communications is the additive mixture of useful signals and noise. In radio engineering and communications, the Gaussian random process is a mathematical model of active and passive interferences, atmospheric and cosmic noise, fading channels, and channels with multipath propagation. The receiver's fluctuation noises also have a normal law of \begin{table}
\begin{tabular}{c c c} \hline \hline
**Probability Density Function** & **Mean Value** & **Dispersion** \\ \hline \(\begin{cases}\frac{x}{2\sigma^{2}}\exp(\frac{-x^{2}}{2\sigma^{2}}),&\text{if }x \geq 0\\ \frac{x^{2}}{2\sigma^{2}}\exp(\frac{-x^{2}}{2\sigma^{2}}),&\text{if }x\leq 0\end{cases}\) & 0 & \(2\sigma^{2}=2\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Statistical parameters of the normal process derivative. Figure 3: Gaussian noise PP. Figure 2: PP of triangular oscillation. distribution [19]. The adequacy of this model for many real interferences and signals and its universality are explained by the central limit theorem of the probability theory. Let us consider the signal PP at different signal-to-noise ratios (SNRs) (Figure 5). The harmonic signal is a useful signal, and Gaussian noise is an interference. From the figures (Figure 5), it can be seen that the PP of the additive mixture of the harmonic signal and normal noise has an ellipse form and the ellipse line width is a function of the noise power. Since the differentiation operation is linear, it is possible to determine the ellipse line width. To do this, the probability of \(x^{\prime}\) should be defined in certain limits. The probability of \(x^{\prime}=y\) should be in the range \([-a,a]\) and \(P(-a\leq y\leq a)=\left\lfloor 1-\exp(-y^{2}/2\sigma^{2})\right\rfloor\). Let us determine this probability at different values of the parameter \(a\). The results are shown in Table 2.  
Therefore, the ellipse line thickness is determined by the value of \(8\sigma\). Obviously, for the phase portrait of the additive mixture of the harmonic signal and normal noise to have the form of an ellipse, the following condition should be satisfied:  
\[A\omega\sin(\omega t+\phi)/4\sigma>1\,. \tag{2}\]  
**4. Electroencephalogram Phase Portrait and Wavelet Analysis**  
Numerical methods in biomedical research belong to a rapidly developing field, which provides a state-of-the-art tool for biomedical research and applications. These methods are \begin{table}
\begin{tabular}{c c c c} \hline \hline \(a\) & \(2\sigma\) & \(3\sigma\) & \(4\sigma\) \\ \hline \(P\) (\(-a\leq y\leq a\)) & 0.8647 & 0.9889 & 0.9997 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Probability of \(x^{\prime}=y\) at different values of the parameter \(a\). Figure 4: Probability density function of the normal process derivative. Figure 5: PP of the sum of harmonic oscillation and Gaussian noise. currently applied in medicine, specifically in neuroscience, cardiology, and pathology.