Editing Wavelet (section)

=== Discrete wavelet transforms (discrete shift and scale parameters, continuous in time) ===
It is computationally impossible to analyze a signal using all wavelet coefficients, so one may wonder if it is sufficient to pick a discrete subset of the upper halfplane to be able to reconstruct a signal from the corresponding wavelet coefficients. One such system is the [[Affine transformation|affine]] system for some real parameters ''a'' > 1, ''b'' > 0. The corresponding discrete subset of the halfplane consists of all the points (''a<sup>m</sup>'', ''nb a<sup>m</sup>'') with ''m'', ''n'' in '''Z'''. The corresponding ''child wavelets'' are now given as
<math display="block">\psi_{m,n}(t) = \frac1{\sqrt{a^m}}\psi\left(\frac{t - nba^m}{a^m}\right).</math>

A sufficient condition for the reconstruction of any signal ''x'' of finite energy by the formula
<math display="block"> x(t)=\sum_{m\in\Z}\sum_{n\in\Z}\langle x,\,\psi_{m,n}\rangle\cdot\psi_{m,n}(t)</math>
is that the functions <math>\{\psi_{m,n}:m,n\in\Z\}</math> form an [[orthonormal basis]] of ''L''<sup>2</sup>('''R''').