GNU Scientific Library Reference Manual - Third Edition (v1.12)by M. Galassi, J. Davies, J. Theiler, B. Gough, G. Jungman, P. Alken, M. Booth, F. Rossi Paperback (6"x9"), 592 pages, 60 figures ISBN 0954612078 RRP £24.95 ($39.95) |

## 30.1 Definitions

The continuous wavelet transform and its inverse are defined by the relations,

w(s,\tau) = \int f(t) * \psi^*_{s,\tau}(t) dt

and,

f(t) = \int \int_{-\infty}^\infty w(s, \tau) * \psi_{s,\tau}(t) d\tau ds

where the basis functions
\psi_{s,\tau} are obtained by scaling
and translation from a single function, referred to as the *mother
wavelet*.

The discrete version of the wavelet transform acts on equally-spaced
samples, with fixed scaling and translation steps (s,
\tau). The frequency and time axes are sampled *dyadically*
on scales of 2^j through a level parameter j.
The resulting family of functions
{\psi_{j,n}}
constitutes an orthonormal
basis for square-integrable signals.

The discrete wavelet transform is an O(N) algorithm, and is also
referred to as the *fast wavelet transform*.

ISBN 0954612078 | GNU Scientific Library Reference Manual - Third Edition (v1.12) | See the print edition |