Editing Wavelet (section)

=== Continuous wavelet transforms (continuous shift and scale parameters) ===
In [[continuous wavelet transform]]s, a given signal of finite energy is projected on a continuous family of frequency bands (or similar subspaces of the [[Lp space|''L<sup>p</sup>'']] [[function space]] ''L''<sup>2</sup>('''R''') ). For instance the signal may be represented on every frequency band of the form [''f'', 2''f''] for all positive frequencies ''f'' > 0. Then, the original signal can be reconstructed by a suitable integration over all the resulting frequency components.

The frequency bands or subspaces (sub-bands) are scaled versions of a subspace at scale 1. This subspace in turn is in most situations generated by the shifts of one generating function ψ in ''L''<sup>2</sup>('''R'''), the ''mother wavelet''. For the example of the scale one frequency band [1, 2] this function is
<math display="block">\psi(t)=2\,\operatorname{sinc}(2t)-\,\operatorname{sinc}(t)=\frac{\sin(2\pi t)-\sin(\pi t)}{\pi t}</math>
with the (normalized) [[sinc function]]. That, Meyer's, and two other examples of mother wavelets are:

{|
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| [[File:MeyerMathematica.svg|thumb|360px|[[Meyer wavelet|Meyer]]]]
|}
{|
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| [[File:MorletWaveletMathematica.svg|thumb|360px|[[Morlet wavelet|Morlet]]]]
|}
{|
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| [[File:MexicanHatMathematica.svg|thumb|360px|[[Mexican hat wavelet|Mexican hat]]]]
|}

The subspace of scale ''a'' or frequency band [1/''a'', 2/''a''] is generated by the functions (sometimes called ''child wavelets'')
<math display="block">\psi_{a,b} (t) = \frac1{\sqrt a }\psi \left( \frac{t - b}{a} \right),</math>
where ''a'' is positive and defines the scale and ''b'' is any real number and defines the shift. The pair (''a'', ''b'') defines a point in the right halfplane '''R'''<sub>+</sub> × '''R'''.

The projection of a function ''x'' onto the subspace of scale ''a'' then has the form
<math display="block">x_a(t)=\int_\R WT_\psi\{x\}(a,b)\cdot\psi_{a,b}(t)\,db</math>
with ''wavelet coefficients''
<math display="block">WT_\psi\{x\}(a,b)=\langle x,\psi_{a,b}\rangle=\int_\R x(t){\psi_{a,b}(t)}\,dt.</math>

For the analysis of the signal ''x'', one can assemble the wavelet coefficients into a [[scaleogram]] of the signal.

See a list of some [[Continuous wavelets]].