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### 31.1 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.