Characteristic of the gammatone filter

The gammatone filter is an auditory filter designed by Patterson[22], and is known to simulate well the response of the basilar membrane. The impulse response of the gammatone filter is defined by

$$gt(t)=At^{N-1}e^{-2\pi b_ft}\cos(2\pi f_0 t),\quad t\geq 0,$$ (13)

where $At^{N-1}e^{-2\pi b_f t}$ is the amplitude term represented by the Gamma distribution and f₀ is the center frequency. The amplitude characteristics of the gammatone filter are represented, approximately, by

$$GT(f) \approx \left[1+\frac{j(f-f_0)}{b_f}\right]^{-N},\quad 0<f<\infty,$$ (14)

where GT(f) is the Fourier transform of gt(t) and represents bandpass filtering with a center frequency of f₀. Characteristics of impulse response and amplitude of the gammatone filter are shown in Fig. . Since it is clear from this figure that $GT(f)\approx 0$, the gammatone filter satisfies approximately the admissibility condition, meaning it can be used sufficiently as the basic wavelet.