Definition of the wavelet transform

Firstly, to design a wavelet filterbank, the wavelet transform and the inverse wavelet transform are summarized as follows.

The integral wavelet transform for f(t) is defined by

$$\tilde{f}(a,b)=\frac{1}{\sqrt{\vert a\vert}}\int_{-\infty}^\infty f(t)\overline{\psi\left(\frac{t-b}{a}\right)}dt,$$ (9)

where a is the ``scale parameter,'' b is the ``shift parameter,'' and $\overline{\psi}$ is the conjugate of $\psi$. The integral basis function is $\psi(t)$ scale-transformed by the parameter a and is shifted by the parameter b. The selection of $\psi(t)$ allows much mathematically freedom; however, in general, $\psi(t)$ is determined to be an integrable function satisfied by the following admissibility condition [20]:

$$D_\psi := \int_{-\infty}^\infty \frac{\vert\hat{\psi}(\omega)\vert^2}{\vert\omega\vert}d\omega \ < \infty$$ (10)

where $\hat{\psi}(\omega)$ is the Fourier transform of $\psi$. If the above equation is satisfied, $\psi$ is called a ``basic wavelet,'' and the inverse transform exists uniquely as follows[20]:

$$f(t)=\frac{1}{D_\psi} \int_{-\infty}^\infty \int_{-\infty}^\i... ...tilde{f}(a,b\ )\psi\left(\frac{t-b}{a}\right)\frac{dadb}{a^2}.$$ (11)

If $\psi$ is absolutely integrable function, the admissibility condition implies $\hat{\psi}(0)=0$.

If the basic wavelet is defined on a complex plane, it is possible that the wavelet transform is represented by the amplitude spectrum $\vert\tilde{f}(a,b)\vert$ and the phase spectrum $\arg(\tilde{f}(a,b))$ as follows [21]:

$$\tilde{f}(a,b)=\vert\tilde{f}(a,b)\vert e^{j \arg(\tilde{f}(a,b))}.$$ (12)

To construct an auditory filterbank simulating the auditory system, we selected as a basic wavelet the gammatone filter to simulate the response of the basilar membrane.