Mercurial > hg > sv-dependency-builds

comparison src/fftw-3.3.8/doc/html/1d-Real_002dodd-DFTs-_0028DSTs_0029.html @ 167:bd3cc4d1df30

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 19 Nov 2019 14:52:55 +0000
parents
children
comparison
equal deleted inserted replaced
166:cbd6d7e562c7 167:bd3cc4d1df30
1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
2 <html>
3 <!-- This manual is for FFTW
4 (version 3.3.8, 24 May 2018).
5
6 Copyright (C) 2003 Matteo Frigo.
7
8 Copyright (C) 2003 Massachusetts Institute of Technology.
9
10 Permission is granted to make and distribute verbatim copies of this
11 manual provided the copyright notice and this permission notice are
12 preserved on all copies.
13
14 Permission is granted to copy and distribute modified versions of this
15 manual under the conditions for verbatim copying, provided that the
16 entire resulting derived work is distributed under the terms of a
17 permission notice identical to this one.
18
19 Permission is granted to copy and distribute translations of this manual
20 into another language, under the above conditions for modified versions,
21 except that this permission notice may be stated in a translation
22 approved by the Free Software Foundation. -->
23 <!-- Created by GNU Texinfo 6.3, http://www.gnu.org/software/texinfo/ -->
24 <head>
25 <title>FFTW 3.3.8: 1d Real-odd DFTs (DSTs)</title>
26
27 <meta name="description" content="FFTW 3.3.8: 1d Real-odd DFTs (DSTs)">
28 <meta name="keywords" content="FFTW 3.3.8: 1d Real-odd DFTs (DSTs)">
29 <meta name="resource-type" content="document">
30 <meta name="distribution" content="global">
31 <meta name="Generator" content="makeinfo">
32 <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
33 <link href="index.html#Top" rel="start" title="Top">
34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
37 <link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" rel="next" title="1d Discrete Hartley Transforms (DHTs)">
38 <link href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" rel="prev" title="1d Real-even DFTs (DCTs)">
39 <style type="text/css">
40 <!--
41 a.summary-letter {text-decoration: none}
42 blockquote.indentedblock {margin-right: 0em}
43 blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
44 blockquote.smallquotation {font-size: smaller}
45 div.display {margin-left: 3.2em}
46 div.example {margin-left: 3.2em}
47 div.lisp {margin-left: 3.2em}
48 div.smalldisplay {margin-left: 3.2em}
49 div.smallexample {margin-left: 3.2em}
50 div.smalllisp {margin-left: 3.2em}
51 kbd {font-style: oblique}
52 pre.display {font-family: inherit}
53 pre.format {font-family: inherit}
54 pre.menu-comment {font-family: serif}
55 pre.menu-preformatted {font-family: serif}
56 pre.smalldisplay {font-family: inherit; font-size: smaller}
57 pre.smallexample {font-size: smaller}
58 pre.smallformat {font-family: inherit; font-size: smaller}
59 pre.smalllisp {font-size: smaller}
60 span.nolinebreak {white-space: nowrap}
61 span.roman {font-family: initial; font-weight: normal}
62 span.sansserif {font-family: sans-serif; font-weight: normal}
63 ul.no-bullet {list-style: none}
64 -->
65 </style>
66
67
68 </head>
69
70 <body lang="en">
71 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></a>
72 <div class="header">
73 <p>
74 Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
75 </div>
76 <hr>
77 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029-1"></a>
78 <h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>
79
80 <p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
81 forward (and backward) DFTs as defined above, where the input array
82 <em>X</em> of length <em>N</em> is purely real and is also <em>odd</em> symmetry. In
83 this case, the output is odd symmetry and purely imaginary.
84 <a name="index-real_002dodd-DFT-1"></a>
85 <a name="index-RODFT-1"></a>
86 </p>
87
88 <a name="index-RODFT00"></a>
89 <p>For the case of <code>RODFT00</code>, this odd symmetry means that
90 <i>X<sub>j</sub> = -X<sub>N-j</sub></i>,
91 where we take <em>X</em> to be periodic so that
92 <i>X<sub>N</sub> = X</i><sub>0</sub>.
93 Because of this redundancy, only the first <em>n</em> real numbers
94 starting at <em>j=1</em> are actually stored (the <em>j=0</em> element is
95 zero), where <em>N = 2(n+1)</em>.
96 </p>
97 <p>The proper definition of odd symmetry for <code>RODFT10</code>,
98 <code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
99 because of the shifts by <em>1/2</em> of the input and/or output, although
100 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,
101 the cosine terms in the DFT all cancel and the remaining sine terms are
102 written explicitly below. This formulation often leads people to call
103 such a transform a <em>discrete sine transform</em> (DST), although it is
104 really just a special case of the DFT.
105 <a name="index-discrete-sine-transform-2"></a>
106 <a name="index-DST-2"></a>
107 </p>
108
109 <p>In each of the definitions below, we transform a real array <em>X</em> of
110 length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
111 </p>
112 <a name="RODFT00-_0028DST_002dI_0029"></a>
113 <h4 class="subsubheading">RODFT00 (DST-I)</h4>
114 <a name="index-RODFT00-1"></a>
115 <p>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
116 <center><img src="equation-rodft00.png" align="top">.</center>
117 </p>
118 <a name="RODFT10-_0028DST_002dII_0029"></a>
119 <h4 class="subsubheading">RODFT10 (DST-II)</h4>
120 <a name="index-RODFT10"></a>
121 <p>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
122 <center><img src="equation-rodft10.png" align="top">.</center>
123 </p>
124 <a name="RODFT01-_0028DST_002dIII_0029"></a>
125 <h4 class="subsubheading">RODFT01 (DST-III)</h4>
126 <a name="index-RODFT01"></a>
127 <p>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
128 <center><img src="equation-rodft01.png" align="top">.</center>
129 In the case of <em>n=1</em>, this reduces to
130 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
131 </p>
132 <a name="RODFT11-_0028DST_002dIV_0029"></a>
133 <h4 class="subsubheading">RODFT11 (DST-IV)</h4>
134 <a name="index-RODFT11"></a>
135 <p>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
136 <center><img src="equation-rodft11.png" align="top">.</center>
137 </p>
138 <a name="Inverses-and-Normalization-1"></a>
139 <h4 class="subsubheading">Inverses and Normalization</h4>
140
141 <p>These definitions correspond directly to the unnormalized DFTs used
142 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
143 summations). The unnormalized inverse of <code>RODFT00</code> is
144 <code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
145 of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
146 in the original array multiplied by <em>N</em>, where <em>N</em> is the
147 <em>logical</em> DFT size. For <code>RODFT00</code>, <em>N=2(n+1)</em>;
148 otherwise, <em>N=2n</em>.
149 <a name="index-normalization-11"></a>
150 </p>
151
152 <p>In defining the discrete sine transform, some authors also include
153 additional factors of
154 &radic;2
155 (or its inverse) multiplying selected inputs and/or outputs. This is a
156 mostly cosmetic change that makes the transform orthogonal, but
157 sacrifices the direct equivalence to an antisymmetric DFT.
158 </p>
159 <hr>
160 <div class="header">
161 <p>
162 Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
163 </div>
164
165
166
167 </body>
168 </html>