An auditory-motivated filterbank

In this investigation, a filterbank is implemented considering two points: (1) consideration of the properties of the auditory system and, (2) detection of a discontinuous point dealing with the complex spectrum. In order to construct a constant Q filterbank, this paper uses the gammatone filter as an analyzing wavelet.

The gammatone filter, which is an auditory filter designed by Patterson [Patterson et al.1995], simulates the response of the basilar membrane. Its impulse response is given by

$$gt(t)=At^{N-1}\exp({-2\pi b_f {\rm {ERB}}(f_0) t}) \cos(2\pi f_0 t), \qquad t \geq 0,$$ (18)

where A, b_f, and N are parameters, and $At^{N-1}\exp({-2\pi b_f {\rm {ERB}}(f_0) t})$ is the amplitude term represented by the Gamma distribution, f₀ is the center frequency, and ${\rm {ERB}}(f_0)$ is an equivalent rectangular bandwidth in f₀(t). In addition, amplitude characteristics of the gammatone filter are represented approximately by

$$GT(f)\approx \left[1+\frac{j(f-f_0)}{b_f{\rm {ERB}}(f_0)}\right]^{-N},\qquad 0 < f < \infty,$$ (19)

where GT(f) is the Fourier transform of gt(t).

In order to determine phase information, let the analyzing wavelet be the extend gammatone filter in Eq. (18) using the Hilbert transform. This analyzing wavelet is represented by

$$\psi(t)=At^{N-1}\exp({j2\pi f_0 t-2\pi b_f {\rm {ERB}}(f_0) t}),$$ (20)

where f₀=600 Hz, N=4, and b_f=0.25. Here, the bandwidth of the $\psi (t)$ becomes a quarter of the bandwidth of the auditory filter (about 1/4 ERB). The characteristics of the analyzing wavelet $\psi (t)$ of Eq. (20) are shown in Fig. 4 .

Next, the wavelet filterbank is designed using the wavelet transform (see appendix B). Here, let the wavelet transform of f(t) be

$$\tilde{f}(a,b)=\vert\tilde{f}(a,b)\vert\exp({j \arg(\tilde{f}(a,b))}),$$ (21)

where $\vert\tilde{f}(a,b)\vert$ is the amplitude spectrum and $\arg(\tilde{f}(a,b))$ is the phase spectrum; a is the scale parameter and b is the shift parameter.

Finally, an auditory-motivated filterbank is designed with the conditions shown in Table. III. The frequency characteristics of the wavelet filterbank are shown in Fig. 5.