## 3 Materials and Methods  
### Background  
#### 3.1.1 Wavelet Transform-Synchrosqueezing Transform (WT-SST)  
To overcome the problems associated with time-domain approaches in analyzing biomedical signals, Discrete wavelet transform (DWT) has been deployed by many researchers in analyzing physiological signals, including EEG. This is due to its ability to present a signal in the time-frequency domain suitable for analyzing non-stationary signals [31; 32]. In this approach, focal and non-focal EEG signals \(x(n)\) are decomposed into sub-bands consisting of lower and higher frequency components using longer and shorter windows. These longer and shorter windows were achieved by using scaling and shifting functions on the signal. The scaling and shifting function corresponds to the low pass and high pass filters denoted as g[n] and h[n], respectively [33; 34]. These low pass and high pass filters produced the details and approximation coefficients, respectively. The process of filtering and decimation of EEG signals at each level continued until it reached the last level. Figure 1 demonstrates the three levels decomposition process. At each level, the decimation is achieved by downsampling the EEG signal into half to increase the frequency resolution of the signal.  
Generally, wavelet functions is defined in Equation (1) as:  
\[\Psi_{s,\tau}=\frac{1}{\sqrt{s}}\Psi\bigg{(}\frac{t-\tau}{s}\bigg{)} \tag{1}\]  
where \(\Psi\) is the mother wavelet, \(s\) is called the scale parameter and \(\tau\) shift parameter.  
From Equation (2), Wavelet transform is given as:  
\[\gamma(s,\tau)=\int f(t)\Psi_{s,\tau}^{*}(t)dt \tag{2}\]
To improve and sharpen the resolution of time-frequency representations in wavelet transform, a recent special case of an improved version of the real-location approach was proposed by [35]. This approach reallocates the values in a time-frequency plane from a point \(\mathfrak{R}(\mathbf{\beta},\mathbf{\hat{u}})\) to a different point \(\mathfrak{R}(\mathbf{\beta}^{\prime},\mathbf{\hat{u}}^{\prime})\) which is based on the local behavior of \(\mathfrak{R}(\mathbf{\beta},\mathbf{\hat{u}})\) around \((\mathbf{\beta},\mathbf{\hat{u}})\). The reallocation of coefficients results from a discrete-time continuous wavelet transform to obtain a compact time-frequency representation.  
The continuous wavelet transform can be defined as in Equation (4):  
\[\gamma_{x}^{\Psi}(\tau,s)=\frac{1}{s}\sum_{n=0}^{N-1}x(n)\Psi^{*} \left(\frac{n-\tau}{s}\right) \tag{4}\]  
where \(\gamma_{x}^{\Psi}(\tau,s)\) is the element of scalogram matrix while \(\Psi^{*}\) represents the mother wavelet's complex conjugate. The instantaneous frequency estimation is used to sharpen our EEG signals time scale representations and its described as in Equation (5) [36; 37].  
\[\widehat{\mathcal{O}}(\tau,s)=Re\left[\frac{1}{2\pi i}\frac{\gamma_{x}^{\Psi} (\tau+1,s)}{\gamma_{x}^{\Psi}(\tau,s)}\right] \tag{5}\]  
\(Re[.]\) is the real part of EEG signals. Therefore, the WT-SST \(\varpi^{\lambda}(\tau,s)\) can be described as in Equation (6):  
\[\varpi^{\lambda}(\tau,s)=\sum_{s=0\mid\gamma_{x}^{\Psi}(\tau,s)\mid> \lambda}^{S}\frac{1}{s}\gamma_{x}^{\Psi}(\tau,s)\delta[k-\widehat{\mathcal{O} }(\tau,s)] \tag{6}\]  
where \(\lambda\) is the value of the threshold. In this work, we used a Morlet wavelet transform in the proposed WT-SST time-frequency.