Mercurial > hg > sv-dependency-builds

comparison src/fftw-3.3.5/doc/html/1d-Real_002deven-DFTs-_0028DCTs_0029.html @ 42:2cd0e3b3e1fd

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Current fftw source
author Chris Cannam
date Tue, 18 Oct 2016 13:40:26 +0100
parents
children
comparison
equal deleted inserted replaced
41:481f5f8c5634 42:2cd0e3b3e1fd
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 3.3.5, 30 July 2016).
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 5.2, http://www.gnu.org/software/texinfo/ -->
24 <head>
25 <title>FFTW 3.3.5: 1d Real-even DFTs (DCTs)</title>
26
27 <meta name="description" content="FFTW 3.3.5: 1d Real-even DFTs (DCTs)">
28 <meta name="keywords" content="FFTW 3.3.5: 1d Real-even DFTs (DCTs)">
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-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" rel="next" title="1d Real-odd DFTs (DSTs)">
38 <link href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" rel="prev" title="The 1d Real-data DFT">
39 <style type="text/css">
40 <!--
41 a.summary-letter {text-decoration: none}
42 blockquote.smallquotation {font-size: smaller}
43 div.display {margin-left: 3.2em}
44 div.example {margin-left: 3.2em}
45 div.indentedblock {margin-left: 3.2em}
46 div.lisp {margin-left: 3.2em}
47 div.smalldisplay {margin-left: 3.2em}
48 div.smallexample {margin-left: 3.2em}
49 div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
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.nocodebreak {white-space:nowrap}
61 span.nolinebreak {white-space:nowrap}
62 span.roman {font-family:serif; font-weight:normal}
63 span.sansserif {font-family:sans-serif; font-weight:normal}
64 ul.no-bullet {list-style: none}
65 -->
66 </style>
67
68
69 </head>
70
71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
72 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
73 <div class="header">
74 <p>
75 Next: <a href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" accesskey="p" rel="prev">The 1d Real-data DFT</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>
76 </div>
77 <hr>
78 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029-1"></a>
79 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
80
81 <p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
82 forward (and backward) DFTs as defined above, where the input array
83 <em>X</em> of length <em>N</em> is purely real and is also <em>even</em> symmetry. In
84 this case, the output array is likewise real and even symmetry.
85 <a name="index-real_002deven-DFT-1"></a>
86 <a name="index-REDFT-1"></a>
87 </p>
88
89 <a name="index-REDFT00"></a>
90 <p>For the case of <code>REDFT00</code>, this even symmetry means that
91 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,where we take <em>X</em> to be periodic so that
92 <i>X<sub>N</sub> = X</i><sub>0</sub>.Because of this redundancy, only the first <em>n</em> real numbers are
93 actually stored, where <em>N = 2(n-1)</em>.
94 </p>
95 <p>The proper definition of even symmetry for <code>REDFT10</code>,
96 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
97 because of the shifts by <em>1/2</em> of the input and/or output, although
98 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 even symmetry, however,
99 the sine terms in the DFT all cancel and the remaining cosine terms are
100 written explicitly below. This formulation often leads people to call
101 such a transform a <em>discrete cosine transform</em> (DCT), although it is
102 really just a special case of the DFT.
103 <a name="index-discrete-cosine-transform-2"></a>
104 <a name="index-DCT-2"></a>
105 </p>
106
107 <p>In each of the definitions below, we transform a real array <em>X</em> of
108 length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
109 </p>
110 <a name="REDFT00-_0028DCT_002dI_0029"></a>
111 <h4 class="subsubheading">REDFT00 (DCT-I)</h4>
112 <a name="index-REDFT00-1"></a>
113 <p>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
114 <center><img src="equation-redft00.png" align="top">.</center>Note that this transform is not defined for <em>n=1</em>. For <em>n=2</em>,
115 the summation term above is dropped as you might expect.
116 </p>
117 <a name="REDFT10-_0028DCT_002dII_0029"></a>
118 <h4 class="subsubheading">REDFT10 (DCT-II)</h4>
119 <a name="index-REDFT10"></a>
120 <p>An <code>REDFT10</code> transform (type-II DCT, sometimes called &ldquo;the&rdquo; DCT) in FFTW is defined by:
121 <center><img src="equation-redft10.png" align="top">.</center></p>
122 <a name="REDFT01-_0028DCT_002dIII_0029"></a>
123 <h4 class="subsubheading">REDFT01 (DCT-III)</h4>
124 <a name="index-REDFT01"></a>
125 <p>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
126 <center><img src="equation-redft01.png" align="top">.</center>In the case of <em>n=1</em>, this reduces to
127 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.Up to a scale factor (see below), this is the inverse of <code>REDFT10</code> (&ldquo;the&rdquo; DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the &ldquo;IDCT&rdquo;.
128 <a name="index-IDCT-3"></a>
129 </p>
130 <a name="REDFT11-_0028DCT_002dIV_0029"></a>
131 <h4 class="subsubheading">REDFT11 (DCT-IV)</h4>
132 <a name="index-REDFT11"></a>
133 <p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
134 <center><img src="equation-redft11.png" align="top">.</center></p>
135 <a name="Inverses-and-Normalization"></a>
136 <h4 class="subsubheading">Inverses and Normalization</h4>
137
138 <p>These definitions correspond directly to the unnormalized DFTs used
139 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
140 summations). The unnormalized inverse of <code>REDFT00</code> is
141 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
142 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
143 in the original array multiplied by <em>N</em>, where <em>N</em> is the
144 <em>logical</em> DFT size. For <code>REDFT00</code>, <em>N=2(n-1)</em> (note that
145 <em>n=1</em> is not defined); otherwise, <em>N=2n</em>.
146 <a name="index-normalization-10"></a>
147 </p>
148
149 <p>In defining the discrete cosine transform, some authors also include
150 additional factors of
151 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
152 mostly cosmetic change that makes the transform orthogonal, but
153 sacrifices the direct equivalence to a symmetric DFT.
154 </p>
155 <hr>
156 <div class="header">
157 <p>
158 Next: <a href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" accesskey="n" rel="next">1d Real-odd DFTs (DSTs)</a>, Previous: <a href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" accesskey="p" rel="prev">The 1d Real-data DFT</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>
159 </div>
160
161
162
163 </body>
164 </html>