## 2 Materials and Methods
### Proposed Model
#### 2.3.4 Hybrid Feature Extraction  
1. Short-Time Signal Energy  
Short-time energy represents the dominant and most natural characteristic utilized. From a physical aspect, energy constitutes a measure of how much signal there is at any moment. The calculation of a signal's energy is generally performed on a short-term basis as a result of windowing the signal at a specific time, squaring samples, and obtaining the average [46; 47; 48]. The square root of the signal's energy is known as the root mean square (RMS) value. The short-time energy function (\(E_{n}\)) of a speech frame having length \(N\) is described in Equation (9):  
\[E_{n}=\frac{1}{N}\sum_{m=-\infty}^{\infty}\left[x(m)w(n-m)\right]^{2}. \tag{9}\]  
The RMS energy (\(E_{n(\text{RMS})}\)) of the frame in question can be presented as in Equation (10):  
\[E_{n(\text{RMS})}=\sqrt{\sum_{m=-\infty}^{\infty}\left[x(m)w(n-m)\right]^{2}}, \tag{10}\]  
where \(x(m)\) denotes the discrete-time audio signal, \(n\) is the time index of the short-time energy, and \(w(m)\) denotes the window function.  
2. Short-Time Zero Crossing Rate  
The ZCR denotes the number of times speech samples alter the algebraic sign in a specific frame. The rate indicating the occurrence of zero crossings represents a simple measure of the signal's frequency content. It is also a measure of the number of times in a particular time interval/frame when the amplitude of speech signals passes through a zero value [46; 47; 48].  
The short-time zero crossing (\(Z_{n}\)) of a speech frame and \(sgn[x(n)]\) are described in Equations (11) and (12), respectively:  
\[Z_{n}=\frac{1}{2}\sum_{m=-\infty}^{\infty}\big{|}sgn[x(n-m)]-sgn[x(n-m-1)] \big{|}w(m), \tag{11}\]  
where:  
\[sgn[x(n)]=\left\{\begin{array}{cc}1,&x(n)\geq 0\\ -1,&x(n)<0\end{array}\right., \tag{12}\]  
and \(w(m)\) refers to a window with length \(n\), given in the equation.  
3. Mel Frequency Cepstral Coefficient (MFCC)  
MFCC is one of the most widely used and most effective feature extraction methods in most speech processing systems. One of its significant advantages is that the same words are not affected too much by changes that occur during vocalization.  
MFCC imitates the perception of human ears and is calculated by using a fast Fourier transform (FFT), which is a fast algorithm used to implement the discrete Fourier transforms. The discrete Fourier transform (DFT) is expressed as in Equation (13) for an \(N\)-sample frame:  
\[f(n)=\sum_{k=0}^{N-1}y_{k}e^{-\frac{2\pi j\text{bin}}{N}},\ 0\leq n\leq N-1, \tag{13}\]  
where \(f(n)\) denotes the DFT, \(n\) is the sample index, and \(y\) refers to the signal windowing results.  
Since the human hearing system perceives frequency values up to 1 kHz linearly and frequency values higher than 1 kHz logarithmically, there was a need for a unit that takes the hearing system as amodel [49]. This unit is called the Mel frequency, where linear frequency values are converted into Mel frequency values using Equation (14), where \"\(f\)\" refers to the frequency in Hz, and \"Mel (\(f\))\" refers to the Mel frequency:  
\[\text{Mel}(f)=2595\times log\bigg{(}1+\frac{f}{700}\bigg{)}. \tag{14}\]  
Finally, the Mel spectrum, of which the logarithm has been taken, is converted back to the time domain.  
Delta-MFCC (D-MFCC) and the delta-delta-MFCC (DD-MFCC) are the first-order and the second-order derivatives of the MFCC, respectively [50]. D-MFCC and DD-MFCC are also known under the name of differential and acceleration coefficients.  
The D-MFCC coefficients are calculated using Equation (15):  
\[d_{t}=\frac{\sum_{\theta=1}^{\Theta}\theta(c_{t+\theta}-c_{t-\theta})}{2\sum_{ \theta=1}^{\Theta}\theta^{2}}, \tag{15}\]  
where \(d_{t}\) refers to the delta coefficient at time \(t\), calculated with regard to the corresponding static coefficients \(c_{t-\theta}\) to \(c_{t+\theta}\), and \(\Theta\) refers to the size of the delta window.  
The DD-MFCC coefficients are computed by taking derivative of Equation (15).