# Analysis of Electroencephalograms Based on the Phase Plane Method
## 3 Determination of the Signal-to-Noise Ratio Based on the Phase Portrait
In this case, they speak of a certain decrease in the level of the functional activity of the brain. One of the important aspects of the use of the EEG is the study of epilepsy. Data from modern research indicate that the brain in epilepsy is characterized by a number of functional rearrangements at the macro- and micro-structural level. Let us consider the implementation of the EEG of a person with epilepsy (Figure 8). The same filtering was performed here as in the previous case (Figure 6). This implementation differs from the implementation shown in Figure 6, even visually. It can be seen that it contains slow components. The spectrum of this implementation is shown in Figure 9. It contains powerful frequency components in the ranges of the \(\theta\) and \(\delta\) rhythms and practically does not contain the \(\alpha\) rhythm. Figure 8: Implementation of an EEG of a person with epilepsy. Figure 7: EEG amplitude spectrum of the awake adult shown in Figure 6.  
The features of the EEG signal characterize it as nonstationary and broadband. In the case of a nonstationary process, the Fourier transform does not provide any time information. This method allows seeing the frequencies contained in the signal, but does not allow determining at what time interval of the oscillations certain frequencies exist. Therefore, the Fourier transform can be used to analyze nonstationary signals when only frequency information is of interest, and the lifetime of the spectral components is not important. In addition, a frequency component that is small in amplitude and duration may not be detected in the spectral plot. The variety of factors influencing the statistical properties of physiological processes largely cause their nonstationarity, which leads to the need to search for adequate methods for the statistical analysis of the EEG. This determines the wide variety of methods for the statistical processing of the EEG [26; 27; 28]. The PP of this EEG implementation (Figure 10) has the form of an ellipse. This confirms the conclusion that the mathematical model of the EEG is an additive mixture of harmonic components and Gaussian noise [29]. But, in three cases, the PP of this EEG implementation does not have the form of an ellipse, due to the random character of the EEG. The SNR can be determined based on the phase portrait. The power of the sinusoidal oscillation is \(A^{2}/2\) (see  
Recently, the wavelet transform (WT) has been widely used in the study of biomedical signals, given their nonstationarity, and, in particular, in EEG processing [30; 31; 32; 33; 34]. This method makes it possible to numerically characterize the duration and change of the basic physiological rhythms, as well as to trace the change in frequency in time within each rhythm. In addition, it allows the distribution of power over the frequency ranges to be traced. The wavelet transform decomposes the original signal into its approximation and detail components, followed by refinement by the iterative method. Each step of such a refinement corresponds to a certain level of signal decomposition and reconstruction. To realize this possibility, there are series of orthogonal wavelets. They are created based on the representation of the signal space as a system of nested subspaces, which differ from each other only by changing the scale of the independent variable [31; 35]. This type of analysis is based on the following premises:  
* Signal space \(\
u\) can be divided into nested subspaces \(\
u_{j}\) that do not intersect;
* For any function \(s(t)\in\
u_{j}\), its compressed version belongs to the space \(\
u_{j-1}\);
* There is a function \(\phi(x)\in\
u_{0}\), the shifts of which \(\phi_{0,k}=\phi(t-k)\) create the orthonormal basis of space \(\
u_{0}\). Functions \(\phi_{j,k}=2^{\frac{-j}{2}}\phi(2^{-j}t-k)\) create the orthonormal basis of the space.