## 3 Materials and Methods
In the plot, the factor \(c\) was set to \(\frac{b^{n}}{(n-1)! }\), to make the area under the curve of gamma distribution equal to one [26]. The temporal decay coefficient \(b\) was set to 125, and the carrier frequency, \(f_{0}\), was chosen to be 1000 Hz. The shape of the magnitude characteristic of the gammatone filters with order 4 is very similar to that of the _roex_ function [55] that is commonly used to represent the magnitude response of the human auditory filter [56; 57]. The Fourier transform of the \(h(t)\) is given by \(H(f)\) and can be expressed as:  
\[H(f)=\frac{c}{2}(n-1)!(2\pi b)^{-n}\bigg{[}e^{i\varphi}\left(1+j\frac{(f-f_{0} )}{b}\right)^{-n}\bigg{]}+\frac{c}{2}(n-1)!(2\pi b)^{-n}\bigg{[}e^{-i\varphi} \left(1+j\frac{(f+f_{0})}{b}\right)^{-n}\bigg{]}. \tag{2}\]  
A complete derivation of the \(H(f)\) can be found in Appendix A. The impulse response, \(h(t)\), and the transfer function, \(H(f)\), of the gammatone filter with varying \(f_{0}/b\) are plotted in Figure 8, which shows that the two frequency components of the gammatone filters do not interfere with each other when \(f_{0}/b>8\). In this work, we selected \(f_{0}/b=9\). Another advantage of selecting \(f_{0}/b=9\) is that the bandwidth becomes proportional to \(b\), and it is claimed in [58] that the bandwidth (equivalent rectangular bandwidth) becomes independent of \(f_{0}\) when \(f_{0}/b>3\). The detailed proof is shown in Appendix B. The center frequency and the bandwidth of the gammatone filters are listed in Table 2, while the magnitude spectrum of the gammatone filterbank is shown in Figure 9. The filters are logarithmically spaced in frequency resolution similar to the basil  
Figure 7: The components of a gammatone filter: (**a**) gammatone distribution function, (**b**) the carrier tone, and (**c**) impulse response. Another main system component is the classifier, as shown in the proposed system's last sub-system presented in Figure 4. The processed signal from the cochlear model is applied to the input of a classifier for binary classification. In this work, a CNN was employed for this purpose. The CNN presented in [59] was adopted and optimized to the best of the classification. The CNN was optimized to the best of the classification. \begin{table}
\begin{tabular}{c c} \hline \hline
**Bandwidth, Hz** & **Center Frequency, Hz** \\ \hline
158 & 50 \\
173 & 186 \\
276 & 389 \\
478 & 690 \\
788 & 1139 \\
1249 & 1807 \\
1936 & 2802 \\
2960 & 4282 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: The center frequency and the bandwidth of the gammatone filters. Figure 8: The filter impulse responses, \(h(t)\), and their corresponding spectrums, \(H(f)\), for: (**a**) \(f_{0}/b=2\), (**b**) \(f_{0}/b=4\), (**c**) \(f_{0}/b=8\), and (**d**) \(f_{0}/b=10\). Figure 9: The magnitude spectrum of the gammatone filterbank. to implement the proposed system. The CNN includes feature extraction and classifier networks. The feature extractor produces a feature map based on the input data. The feature map accentuates the unique features from the original data. Consequently, the extracted feature map was applied to the classification neural network. The classification neural network operates on the feature map and performs classification functions. The feature extractor network consists of a special kind of neural network, of which the synaptic weights are determined via the training process. Usually, the feature extractor network consists of piles of convolutional layer and pooling layer pairs, as shown in Figure 4. It is widely accepted that pattern recognition algorithms perform better when the feature extractor network contains more layers. However, it is always challenging to train them as it incurs a substantial computational burden on the system [60]. Considering this limitation, this work used one convolutional layer as a feature extractor network. Unlike other conventional neural networks, no connection weights or a weighted sum are employed in the convolutional layer.