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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*.