# A Feature Extraction Method Using Auditory Nerve Response for Collapsing Coal-Gangue Recognition
## 3 Feature Extraction and Selection  
### AN Response Model  
The AN response model is a phenomenological model that can simulate the human auditory system. Each section provides a phenomenological description of the primary functional parts of the auditory periphery, from the middle ear to the AN. In this paper, we used [PERSON] et al.'s model of the auditory periphery [15], which is one of the most comprehensive auditory models, to calculate the AN response. Briefly, four sections of this auditory model are mainly used, i.e., the middle ear filter, C1 and C2 filter, outer hair cell section, and inner hair cell section. The schematic diagram of the model is shown in Figure 5.  
Figure 4: Acoustic pressure signal. (**a**) Coal acoustic pressure signal; (**b**) Gangue acoustic pressure signal.  
Figure 3: Coal and gangue vibrational signal. (**a**) Coal vibrational signal; (**b**) Gangue vibrational signal.  
The stimulus' instantaneous pressure signals (in Pa) are firstly input into the middle ear (ME) filter. The model's ME section is implemented by a series of three digital filters [16]:  
\[\left\{\begin{array}{l}ME_{1}(z)=0.0127(\frac{1.0000+1.0000z^{-1}}{1.0000-0.9986 z^{-1}})\\ ME_{2}(z)=0.0127(\frac{1.0000+1.0000z^{-1}}{1.0000-0.9986z^{-1}})\\ ME_{3}(z)=\frac{1.0000-1.9943z^{-1}+0.9973z^{-2}}{1.0000-1.9856z^{-1}+0.9892z^{- 2}}\end{array}\right. \tag{2}\]  
Three correlated filter paths, the C1, C2 filters in the signal path simulating frequency selection characteristics of the basilar membrane, and the broadband filter in the feedforward control path, follow the ME filter. There are two transduction functions following the C1 and C2 filters [17; 18].  
The transduction function of C1 filter:  
\[\begin{array}{l}V_{\text{ihc,CI}}=A_{\text{ihc}}[P_{\text{CI}}]\log_{10}(1+B _{\text{ihc}}|P_{\text{CI}}|)\\ A_{\text{ihc}}[P_{\text{C1}}]=\left\{\begin{array}{l}A_{\text{ihc0}}&\text{ for }P_{\text{CI}}>0\\ -\frac{|P_{\text{CI}}|^{\text{ihc}}+D_{\text{hc}}}{3|P_{\text{CI}}(t)|^{\text{ ihc}}+D_{\text{hc}}}A_{\text{ihc0}}&\text{ for }P_{\text{CI}}<0\end{array}\right.\end{array} \tag{3}\]  
where \(P_{\text{C1}}\) denotes the output of the signal path C1 filter and \(A_{\text{ihc0}}\), \(B_{\text{ihc}}\), \(C_{\text{ihc}}\), and \(D_{\text{ihc}}\) represent parameters which are equal to 0.1, 2000, 1.74, \(6.87\times 10^{-9}\), respectively.  
The transduction function of C2 filter:  
\[\begin{array}{l}V_{\text{ihc,C2}}=\text{signum}(-P_{\text{C2}})\times 2 \times A_{\text{ihc0}}\times\log_{10}(1.0+5\times 10^{-5}B_{\text{ihc}} \ \times(P_{\text{C2}}\cdot\text{CF})^{2})\\ \text{signum}(-P_{\text{C2}})=\left\{\begin{array}{l}-1&\text{ for }P_{\text{C2}}\geq 0\\ 1&\text{ for }P_{\text{C2}}<0\end{array}\right.\end{array} \tag{4}\]  
Figure 5: Schematic diagram of the AN response model [15]. Abbreviations: inner hair cell (IHC), outer hair cell (OHC), low-pass (LP) filter, static nonlinearity (NL), characteristic frequency (CF), and inverting nonlinearity (INV).  
where \(P_{\rm C2}\) represents the signal path C2 filter's output; CF represents the characteristic frequency; and \(A_{\rm iIkc0}\) and \(B_{\rm iIkc}\) denote the parameters which are equal to 0.1 and 2000, respectively.  
The feedforward control path is composed of broadband filter and outer hair cell section including low-pass filter, static nonlinearity, and scale zooming, which can adjust the gain and bandwidth of the C1 filter to reflect several level-dependent characteristics in the cochlea [16; 19].  
The two transduction functions' joint responses after the C1 and C2 filters are input into a seventh-order IHC low-pass filter to obtain the result of the inner hair cell section with a single characteristic frequency [17; 18]. Finally, a series of the inner hair cell model's outputs corresponding to all characteristic frequencies can be combined to form the auditory spectrum.  
Detailed information of the model segments can be found in [PERSON] and [PERSON] et al. [15; 17; 20].