Appendix B: Wavelet transform

First, we summarize the wavelet transform and the inverse wavelet transform[Chui1992] for designing a constant Q filterbank.

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

$$\tilde{f}(a,b)=\frac{1}{\sqrt{\vert a\vert}}\int_{-\infty}^\infty f(t)\overline{\psi\left(\frac{t-b}{a}\right)}dt,$$ (42)

where a is the ``scale parameter,'' b is the ``shift parameter,'' $a,b \in {\bf {R}}$ with $a\not=0$, and $\overline{\psi}$ is the conjugate of $\psi$. The integral basis function is $\psi (t)$ scale-transformed by parameter a and shifted by parameter b. The selection of $\psi (t)$ allows much mathematical freedom; however, in general, $\psi (t)$ is determined to be an integrable function that satisfies the following ``admissibility condition'':

$$G_{\psi}:=\int_{-\infty}^{\infty} \frac{\vert\hat{\psi}(\omega)\vert^2}{\vert\omega\vert}d\omega < \infty,$$ (43)

where $\hat{\psi}$ is the Fourier transform of $\psi$, It follows that $\hat{\psi}$ is a continuous function, so that the finiteness of $G_{\psi}$ in Eq. (43) implies that $\hat{\psi}(0)=0$, or equivalently, $\int_{-\infty}^{\infty} \psi(t)dt=0$. If the above equation is satisfied, $\psi$ is called a ``basic wavelet,'' and a unique inverse transform exists as follows:

$$f(t)=\frac{1}{G_\psi} \int_{-\infty}^\infty \int_{-\infty}^\infty \tilde{f}(a,b)\psi\left(\frac{t-b}{a}\right)\frac{dadb}{a^2}$$ (44)

Since the analyzing wavelet $\psi (t)$ in Eq. (20) approximately satisfies the admissibility condition because $\vert\psi(0)\vert\approx 0$, it can be considered that this analyzing wavelet is the basic wavelet.

Equations. (42)-(44) are a continuous wavelet transform. The discrete wavelet transform corresponding to these equations is represented by

$$\psi_{p,q}(t):=\alpha^{-p/2}\psi \left(\frac{t-q\cdot b}{\alpha^p}\right),$$ (45)

$$\tilde{f}_{p,q}:=\tilde{f}(\alpha^p,q/f_s)=\int_{-\infty}^\infty f(t)\overline{\psi}_{p,q}(t)dt,$$ (46)

and

$$f(t)=\frac{1}{G_{p,q}}\sum_p\sum_q \tilde{f}_{p,q}\psi_{p,q}(t),$$ (47)

where p and q are integer parameters.