3
<title>DWT Definitions - GNU Scientific Library -- Reference Manual</title>
4
<meta http-equiv="Content-Type" content="text/html">
5
<meta name="description" content="GNU Scientific Library -- Reference Manual">
6
<meta name="generator" content="makeinfo 4.8">
7
<link title="Top" rel="start" href="index.html#Top">
8
<link rel="up" href="Wavelet-Transforms.html#Wavelet-Transforms" title="Wavelet Transforms">
9
<link rel="next" href="DWT-Initialization.html#DWT-Initialization" title="DWT Initialization">
10
<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
12
Copyright (C) 1996, 1997, 1998, 1999, 2000, 2001, 2002, 2003, 2004, 2005, 2006 The GSL Team.
14
Permission is granted to copy, distribute and/or modify this document
15
under the terms of the GNU Free Documentation License, Version 1.2 or
16
any later version published by the Free Software Foundation; with the
17
Invariant Sections being ``GNU General Public License'' and ``Free Software
18
Needs Free Documentation'', the Front-Cover text being ``A GNU Manual'',
19
and with the Back-Cover Text being (a) (see below). A copy of the
20
license is included in the section entitled ``GNU Free Documentation
23
(a) The Back-Cover Text is: ``You have freedom to copy and modify this
24
GNU Manual, like GNU software.''-->
25
<meta http-equiv="Content-Style-Type" content="text/css">
26
<style type="text/css"><!--
27
pre.display { font-family:inherit }
28
pre.format { font-family:inherit }
29
pre.smalldisplay { font-family:inherit; font-size:smaller }
30
pre.smallformat { font-family:inherit; font-size:smaller }
31
pre.smallexample { font-size:smaller }
32
pre.smalllisp { font-size:smaller }
33
span.sc { font-variant:small-caps }
34
span.roman { font-family:serif; font-weight:normal; }
35
span.sansserif { font-family:sans-serif; font-weight:normal; }
41
<a name="DWT-Definitions"></a>
42
Next: <a rel="next" accesskey="n" href="DWT-Initialization.html#DWT-Initialization">DWT Initialization</a>,
43
Up: <a rel="up" accesskey="u" href="Wavelet-Transforms.html#Wavelet-Transforms">Wavelet Transforms</a>
47
<h3 class="section">30.1 Definitions</h3>
49
<p><a name="index-DWT_002c-mathematical-definition-2046"></a>
50
The continuous wavelet transform and its inverse are defined by
53
where the basis functions <!-- {$\psi_{s,\tau}$} -->
54
\psi_{s,\tau} are obtained by scaling
55
and translation from a single function, referred to as the <dfn>mother
58
<p>The discrete version of the wavelet transform acts on equally-spaced
59
samples, with fixed scaling and translation steps (s,
60
\tau). The frequency and time axes are sampled <dfn>dyadically</dfn>
61
on scales of 2^j through a level parameter j.
62
<!-- The wavelet @math{\psi} -->
63
<!-- can be expressed in terms of a scaling function @math{\varphi}, -->
65
<!-- \beforedisplay -->
67
<!-- \psi(2^{j-1},t) = \sum_{k=0}^{2^j-1} g_j(k) * \bar{\varphi}(2^j t-k) -->
69
<!-- \afterdisplay -->
73
<!-- \psi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} g_j(k) * \bar@{\varphi@}(2^j t-k) -->
79
<!-- \beforedisplay -->
81
<!-- \varphi(2^{j-1},t) = \sum_{k=0}^{2^j-1} h_j(k) * \bar{\varphi}(2^j t-k) -->
83
<!-- \afterdisplay -->
87
<!-- \varphi(2^@{j-1@},t) = \sum_@{k=0@}^@{2^j-1@} h_j(k) * \bar@{\varphi@}(2^j t-k) -->
91
<!-- The functions @math{\psi} and @math{\varphi} are related through the -->
93
<!-- @c{$g_{n} = (-1)^n h_{L-1-n}$} -->
94
<!-- @math{g_@{n@} = (-1)^n h_@{L-1-n@}} -->
95
<!-- for @c{$n=0 \dots L-1$} -->
96
<!-- @math{n=0 ... L-1}, -->
97
<!-- where @math{L} is the total number of coefficients. The two sets of -->
98
<!-- coefficients @math{h_j} and @math{g_i} define the scaling function and -->
100
The resulting family of functions <!-- {$\{\psi_{j,n}\}$} -->
102
constitutes an orthonormal
103
basis for square-integrable signals.
105
<p>The discrete wavelet transform is an O(N) algorithm, and is also
106
referred to as the <dfn>fast wavelet transform</dfn>.
added
removed
DWT-Definitions.html
