| 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 |