### 32.1 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.