# A Design Method for Gammachirp Filterbank for Loudness Compensation in Hearing Aids
## 2 Design of Gammachirp Filterbank  
The basic structure of the M-channel filterbank applied in loudness compensation is shown in Figure 1. The input signal \(x(n)\) is first decomposed into different frequency bands to obtain subband signals. The \(G_{k}(z)\), \(\{k=0,1,\ldots,M-1\}\) represents the analysis filterbank, which carries out subband decomposition. Then, each band carries out loudness compensation in a specific frequency range. Some prescriptive procedures are used to set the amplifying characteristics of the hearing aids, such as NAL-NL2 [25], DSLv5 [26], and FIG6 [27], which are also called prescription formulas. The subsignal can be modified according to the required gain to satisfy the hearing comfort requirements of diverse patients. Lastly, the processed subsignals are combined into a reconstructed signal as the final output under the function of the synthesis filterbank \(F_{k}(z),\{k=0,1,\ldots,M-1\}\)[28]. The gammachirp filter is a kind of nonlinear filter, in accordance with the nonlinear response of the human ear basilar membrane to frequency [29], and its temporal impulse response function is as follows:  
\[g(t)=B^{n}t^{n-1}e^{-2\pi Bt}\cos(2\pi f_{r}t+c\ln t+\psi) \tag{1}\]  
where \(c\) is the rate of the frequency modulation, which is called the chirp factor with a value in the range of [\(-\)3, 3], \(\ln t\) is the natural logarithm of time, and \(\psi\) is the initial phase. The asymmetric parameter \(f_{r}\) is the center frequency of the gammachirp filter. Compared with the gammatone, the impulse response of gammachirp adds the \(c\ln t\). The \(f_{r}\) changes with the value of \(c\), and depends, to some extent, on the parameter \(B\) and the order \(n\). The value of parameter \(B\) satisfies the equation \(B=b\cdot ERB(f_{r})\) and \(ERB(f_{r})\) is the equivalent rectangular bandwidth of an auditory filter at \(f_{r}\). Typically, \(n=4,b=1.109\)[30]. Additionally, the chirp parameter \(c\) is linearly related to the intensity of the stimulus signal \(P_{s}\) (in dB SPL), which can be calculated by Equation (2). \[c=3.38-0.107P_{s} \tag{2}\]  
When \(P_{s}\) is greater than 30 dB SPL, the chirp factor \(c\) becomes negative, and the slope of the lower skirt of the frequency response changes more slowly than that of the upper skirt, which means that the change in the slope is steeper in the high frequency [30,31]. Conversely, the situation is reversed if \(c>0\). Figure 1: The basic structure of the M-channel filterbank applied in loudness compensation. The amplitude-frequency response of the gammachirp can be obtained by a Fourier transform of Equation (1) [32]:  
\[ \begin{array}{ll}G_{\mathrm{C}}(f)&=\frac{a\Gamma(n+jc)e^{i\phi} }{\left\{2\pi bERB(f_{r})+j2\pi(f-f_{r})\right\}^{n+jc}}=\frac{a\Gamma(n+jc)e^{ i\phi}}{\left\{2\pi\sqrt{\overline{b}^{2}+\left(f-f_{r}\right)^{2}}\cdot e^{i \phi}\right\}^{n+jc}}\\ &=\overline{a}\frac{1}{\left\{2\pi\sqrt{\overline{b}^{2}+\left(f-f_{r}\right) ^{2}}\right\}^{n}\cdot e^{j\mu\theta}}\frac{1}{\left\{2\pi\sqrt{\overline{b}^{ 2}+\left(f-f_{r}\right)^{2}}\right\}^{jc}\cdot e^{-c\theta}}\\ &\qquad\qquad\theta=\arctan\frac{f-f_{r}}{\overline{b}^{2}},\,\overline{b}= bERB(f_{r}),\,\overline{a}=a\Gamma(n+jc)e^{i\phi}\end{array}  \tag{3}\]  
where the first term \(\overline{a}\) is a constant, and the second multiplier term is the amplitude-frequency response of the gammatone filter, which is denoted by \(G_{\mathrm{T}}(f)\), while the last multiplier term represents the asymmetric function \(H_{A}(f)\). After amplitude normalization, Equation (3) can be written as:  
\[G_{\mathrm{C}}(f)=G_{T}(f)\cdot H_{A}(f) \tag{4}\]  
The corresponding amplitude spectrum can be expressed as follows:  
\[|G_{\mathrm{C}}(f)|=\frac{\overline{a}}{\left\{2\pi\sqrt{\overline{b}^{2}+ \left(f-f_{r}\right)^{2}}\right\}^{n}}\cdot e^{i\theta(f)}=|G_{\mathrm{T}}(f) |\cdot|H_{A}(f)| \tag{5}\]  
It can be seen from previous derivations that the gammachirp filter can be realized by cascading an asymmetric filter \(H_{A}(f)\) with a gammatone filter \(G_{T}(f)\).