Editing Wavelet (section)

== Definition of a wavelet ==
A wavelet (or a wavelet family) can be defined in various ways:

=== Scaling filter ===
An orthogonal wavelet is entirely defined by the scaling filter – a low-pass [[finite impulse response]] (FIR) filter of length 2''N'' and sum 1. In [[Biorthogonal system|biorthogonal]] wavelets, separate decomposition and reconstruction filters are defined.

For analysis with orthogonal wavelets the high pass filter is calculated as the [[quadrature mirror filter]] of the low pass, and reconstruction filters are the time reverse of the decomposition filters.

Daubechies and Symlet wavelets can be defined by the scaling filter.

=== Scaling function ===
Wavelets are defined by the wavelet function ψ(''t'') (i.e. the mother wavelet) and scaling function φ(''t'') (also called father wavelet) in the time domain.

The wavelet function is in effect a band-pass filter and scaling that for each level halves its bandwidth. This creates the problem that in order to cover the entire spectrum, an infinite number of levels would be required. The scaling function filters the lowest level of the transform and ensures all the spectrum is covered. See<ref>{{cite web|url=http://www.polyvalens.com/blog/?page_id=15#7.+The+scaling+function+%5B7%5D|title=A Really Friendly Guide To Wavelets – PolyValens|website=www.polyvalens.com}}</ref> for a detailed explanation.

For a wavelet with compact support, φ(''t'') can be considered finite in length and is equivalent to the scaling filter ''g''.

Meyer wavelets can be defined by scaling functions

=== Wavelet function ===
The wavelet only has a time domain representation as the wavelet function ψ(''t'').

For instance, [[Mexican hat wavelet]]s can be defined by a wavelet function. See a list of a few [[continuous wavelets]].