## 3 Bispectrum Analysis of EEG Signals for an Emotion Recognition System
### Feature Extraction Based on Bispectrum Analysis and 3D Pyramid Filter  
#### 3.4.1 Bispectrum  
The power spectrum analysis, which has been widely used in biomedical signal processing, performs power distribution calculation using a function of frequency and pays no attention to the signal phase information. In the power spectral analysis, the signal is assumed to arise from a linear process, thus ignoring any possible interaction between components (phase coupling). However, the brain signal is part of the central nervous system with many nonlinear sources, so it is highly likely to have a phase coupling between signals [14]. The bispectrum analysis is superior to the power  
Figure 1: Block diagram of the EEG-based emotion recognition system. spectrum analysis because in its mathematical formula, there is a correlation calculation between the frequency components [46]; thus, the phase coupling components of the EEG signals could be revealed. Some characteristics of bispectrum analysis are the ability to extract deviations due to Gaussianity, to suppress the additive colored Gaussian noise of an unknown power spectrum and to detect nonlinearity properties [47]. Because of its superiority, bispectrum analysis is used in this research as the signal processing technique in the feature extraction step. We expect that by using bispectrum analysis, the recognition rate of the EEG-based automatic emotion recognition system will improve. For example, suppose there is a signal \(S_{3}\) that is a combination of two other signals, \(S_{1}\) and \(S_{2}\), where the frequencies are \(f_{1}\) = 20 Hz,\(f_{1}\) = 10 Hz and\(f_{3}\) = \(f_{1}\) + \(f_{2}\) = 30 Hz, and the phases are \(\phi_{1}\) = \(\pi\)/6, \(\phi_{2}\) = 5\(\pi\)/8 and \(\phi_{3}\) = \(\phi_{1}\) + \(\phi_{2}\), respectively. The signals \(S_{1}\), \(S_{2}\) and \(S_{3}\) are then defined as follows: \(S_{1}\) = 3 \(cos(2\pi f_{1}\) + \(\phi_{1}\)), \(S_{2}\) = 5 \(cos(2\pi f_{2}\) + \(\phi_{2}\)) and \(S_{3}\) = 8 \(cos(2\pi f_{3}\) + \(\phi_{3}\)). The resulting signal \(X(t)\) = \(S_{1}\) + \(S_{2}\) + \(S_{3}\) is then received by a sensor. With the sampling frequency of 100 Hz, signal \(X(t)\) is processed to reveal its power spectrum and its bispectrum values, and the result can be seen in Figure 2. The power spectrum (Figure 2a) produced using FFT showed that the dominant frequencies are 10, 20 and 30 Hz; however, it does not reveal the fact that the frequency 30 Hz is a resulting combination of frequencies 10 and 20 Hz. On the other hand, in bispectrum (Figure 2b), the pair of normalized frequencies 0.1 and 0.2 Hz (equal to the original frequencies 10 and 20 Hz), on coordinate (0.1,0.2) and (0.2,0.1), show a high spectrum, which means that they are strongly correlated because they produce the 30-Hz signal. Supposing that the signal \(S_{3}\) is noise, by using a bispectrum analysis, the noise signal will not be taken into consideration. Thus, we can conclude that through bispectrum analysis, the main frequency components are revealed while the other frequencies are eliminated. Because bispectrum analysis provides more information than the power spectrum analysis, it is expected that the use of bispectrum analysis in the EEG signals' feature extraction process will increase the recognition rate. The autocorrelation of a signal is the correlation between the signal and itself at a different time; for example, at time \(t\) and at time \(t+m\). The autocorrelation function of \(x(n)\) can be expressed as the expectation of stationary process, defined as:  
\[R_{xx}(m):=E\{x^{*}(n)x(n+m)\}. \tag{1}\]  
The higher order moments are a natural generalization of autocorrelation, and the cumulants are a nonlinear combination of moments. The first order cumulant (\(C_{1x}\)) from stationary process is the mean, \(C_{1x}=E\{x(t)\}\), with \(E\{.\}\) an expectation notation.