## 3 Materials and Methods
These filters were designed based on the specifications mentioned in [49]. The center frequency and the bandwidth of these eight filters are listed in Table 1. These eight filters were designed by using the third-order Butterworth prototype filters. It is demonstrated in the table that the bandwidth of the filters is logarithmically spaced from 265 to 1136 Hz, mimicking the frequency response of the basilar membrane (see Figure 2). The lowest center frequency is 394 Hz (the center frequency of the first filter) and the highest center frequency is 4871 Hz (the center frequency of the eighth bandpass filter). The magnitude spectrum of these eight bandpass filters is shown in Figure 6.  
Figure 5: The magnitude spectrum of the pre-emphasis filter with a cut-off frequency of 2000 Hz. The next signal processing steps included envelope detection and lowpass filtering. This work used a full-wave rectifier as an envelope detector, and an eighth-order finite-impulse response filter (FIR) was used as a lowpass filter. This lowpass filter was designed by using the Hamming window function. Several window functions, namely Hanning, Blackman, Bartlett, and Hamming, have been investigated in this work. The main advantages of these window functions are that they taper at their ends and avoid unnatural discontinuity in the speech segment. They also minimize the distortion in the underlying spectrum. Finally, the Hamming window function was selected as it provided the minimum passband ripple and maximum stopband attenuation compared to the other investigated window functions [50]. Finally, the detected signal envelope in each channel was used to modulate a biphasic pulse train. A non-linear mapping technique was used to produce the biphasic pulse train so that the interferences of the pulses in different channels were minimized. The eight filters (mentioned above) were replaced by eight gammatone filters in the second model while using the same other components. The pre-processed audio signals were divided into eight channels by using these eight gammatone filters. The name gammatone comes from the fact that the envelope of the impulse response of those filters is similar to the gamma function. Moreover, the fine structure of the impulse response is a tone at the center frequency of the filter, \(f_{0}\)[51; 52]. Those gammatone filters perform spectral analysis and convert an acoustic wave into the multichannel representation by mimicking the basilar membrane motion [53]. The gammatone filter has an impulse response that is similar to that of a cat's cochlea [54], and it is defined by:  
\[h(t)=ct^{n-1}e^{-2\pi bt}cos(2\pi f_{0}t+\varphi)u(t), \tag{1}\] \begin{table}
\begin{tabular}{c c} \hline \hline
**Bandwidth, Hz** & **Center Frequency, Hz** \\ \hline
265 & 394 \\
331 & 692 \\
431 & 1064 \\
516 & 1528 \\
645 & 2109 \\
805 & 2834 \\
1006 & 3740 \\
1136 & 4871 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The bandwidth and center frequencies of the eight filters. Figure 6: The magnitude response of the bandpass filterbank. where \(c\) is a constant, \(n\) is the filter order, \(b\) is the temporal decay coefficient, \(f_{0}\) is the center frequency of the filter, \(\varphi\) is the carrier phase, and \(u(t)\) is the unit step function. The filter order, \(n\), controls the relative shape of the envelope that becomes less skewed when \(n\) increases. The carrier phase, \(\varphi\), determines the relative position of the envelope. Let us assume that the carrier component is denoted by \(s(t)=cos(2\pi f_{0}t+\varphi)\) and the gammatone distribution function is defined by \(r(t)=t^{n-1}e^{-2\pi bt}u(t)\). Hence, the impulse response of the gammatone filter can be expressed as \(h(t)=cs(t)r(t)\). The parameter \(b\) determines the duration of the impulse response and hence determines the bandwidth of the gammatone filters, and the parameter \(n\) determines the tuning or quality factor (\(Q\)) of the filter. Figure 7 shows the impulse response of the gammatone filter with its constituent components.