Mercurial > hg > batch-feature-extraction-tool

comparison Lib/fftw-3.2.1/doc/html/.svn/text-base/1d-Real_002dodd-DFTs-_0028DSTs_0029.html.svn-base @ 15:585caf503ef5 tip

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Tidy up for ROLI
author Geogaddi\David <d.m.ronan@qmul.ac.uk>
date Tue, 17 May 2016 18:50:19 +0100
parents 636c989477e7
children
comparison
equal deleted inserted replaced
14:636c989477e7 15:585caf503ef5
1 <html lang="en">
2 <head>
3 <title>1d Real-odd DFTs (DSTs) - FFTW 3.2.1</title>
4 <meta http-equiv="Content-Type" content="text/html">
5 <meta name="description" content="FFTW 3.2.1">
6 <meta name="generator" content="makeinfo 4.8">
7 <link title="Top" rel="start" href="index.html#Top">
8 <link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
9 <link rel="prev" href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" title="1d Real-even DFTs (DCTs)">
10 <link rel="next" href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" title="1d Discrete Hartley Transforms (DHTs)">
11 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
12 <!--
13 This manual is for FFTW
14 (version 3.2.1, 5 February 2009).
15
16 Copyright (C) 2003 Matteo Frigo.
17
18 Copyright (C) 2003 Massachusetts Institute of Technology.
19
20 Permission is granted to make and distribute verbatim copies of
21 this manual provided the copyright notice and this permission
22 notice are preserved on all copies.
23
24 Permission is granted to copy and distribute modified versions of
25 this manual under the conditions for verbatim copying, provided
26 that the entire resulting derived work is distributed under the
27 terms of a permission notice identical to this one.
28
29 Permission is granted to copy and distribute translations of this
30 manual into another language, under the above conditions for
31 modified versions, except that this permission notice may be
32 stated in a translation approved by the Free Software Foundation.
33 -->
34 <meta http-equiv="Content-Style-Type" content="text/css">
35 <style type="text/css"><!--
36 pre.display { font-family:inherit }
37 pre.format { font-family:inherit }
38 pre.smalldisplay { font-family:inherit; font-size:smaller }
39 pre.smallformat { font-family:inherit; font-size:smaller }
40 pre.smallexample { font-size:smaller }
41 pre.smalllisp { font-size:smaller }
42 span.sc { font-variant:small-caps }
43 span.roman { font-family:serif; font-weight:normal; }
44 span.sansserif { font-family:sans-serif; font-weight:normal; }
45 --></style>
46 </head>
47 <body>
48 <div class="node">
49 <p>
50 <a name="1d-Real-odd-DFTs-(DSTs)"></a>
51 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></a>
52 Next:&nbsp;<a rel="next" accesskey="n" href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029">1d Discrete Hartley Transforms (DHTs)</a>,
53 Previous:&nbsp;<a rel="previous" accesskey="p" href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029">1d Real-even DFTs (DCTs)</a>,
54 Up:&nbsp;<a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
55 <hr>
56 </div>
57
58 <h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>
59
60 <p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
61 forward (and backward) DFTs as defined above, where the input array
62 X of length N is purely real and is also <dfn>odd</dfn> symmetry. In
63 this case, the output is odd symmetry and purely imaginary.
64 <a name="index-real_002dodd-DFT-302"></a><a name="index-RODFT-303"></a>
65 <a name="index-RODFT00-304"></a>For the case of <code>RODFT00</code>, this odd symmetry means that
66 <i>X<sub>j</sub> = -X<sub>N-j</sub></i>,where we take X to be periodic so that
67 <i>X<sub>N</sub> = X</i><sub>0</sub>. Because of this redundancy, only the first n real numbers
68 starting at j=1 are actually stored (the j=0 element is
69 zero), where N = 2(n+1).
70
71 <p>The proper definition of odd symmetry for <code>RODFT10</code>,
72 <code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
73 because of the shifts by 1/2 of the input and/or output, although
74 the corresponding boundary conditions are given in <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029">Real even/odd DFTs (cosine/sine transforms)</a>. Because of the odd symmetry, however,
75 the cosine terms in the DFT all cancel and the remaining sine terms are
76 written explicitly below. This formulation often leads people to call
77 such a transform a <dfn>discrete sine transform</dfn> (DST), although it is
78 really just a special case of the DFT.
79 <a name="index-discrete-sine-transform-305"></a><a name="index-DST-306"></a>
80 In each of the definitions below, we transform a real array X of
81 length n to a real array Y of length n:
82
83 <h5 class="subsubheading">RODFT00 (DST-I)</h5>
84
85 <p><a name="index-RODFT00-307"></a>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
86 <center><img src="equation-rodft00.png" align="top">.</center>
87
88 <h5 class="subsubheading">RODFT10 (DST-II)</h5>
89
90 <p><a name="index-RODFT10-308"></a>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
91 <center><img src="equation-rodft10.png" align="top">.</center>
92
93 <h5 class="subsubheading">RODFT01 (DST-III)</h5>
94
95 <p><a name="index-RODFT01-309"></a>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
96 <center><img src="equation-rodft01.png" align="top">.</center>In the case of n=1, this reduces to
97 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
98
99 <h5 class="subsubheading">RODFT11 (DST-IV)</h5>
100
101 <p><a name="index-RODFT11-310"></a>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
102 <center><img src="equation-rodft11.png" align="top">.</center>
103
104 <h5 class="subsubheading">Inverses and Normalization</h5>
105
106 <p>These definitions correspond directly to the unnormalized DFTs used
107 elsewhere in FFTW (hence the factors of 2 in front of the
108 summations). The unnormalized inverse of <code>RODFT00</code> is
109 <code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
110 of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
111 in the original array multiplied by N, where N is the
112 <em>logical</em> DFT size. For <code>RODFT00</code>, N=2(n+1);
113 otherwise, N=2n.
114 <a name="index-normalization-311"></a>
115 In defining the discrete sine transform, some authors also include
116 additional factors of
117 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
118 mostly cosmetic change that makes the transform orthogonal, but
119 sacrifices the direct equivalence to an antisymmetric DFT.
120
121 <!-- =========> -->
122 </body></html>
123