|
cannam@95
|
1 <html lang="en">
|
|
cannam@95
|
2 <head>
|
|
cannam@95
|
3 <title>1d Real-even DFTs (DCTs) - FFTW 3.3.3</title>
|
|
cannam@95
|
4 <meta http-equiv="Content-Type" content="text/html">
|
|
cannam@95
|
5 <meta name="description" content="FFTW 3.3.3">
|
|
cannam@95
|
6 <meta name="generator" content="makeinfo 4.13">
|
|
cannam@95
|
7 <link title="Top" rel="start" href="index.html#Top">
|
|
cannam@95
|
8 <link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
|
|
cannam@95
|
9 <link rel="prev" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" title="The 1d Real-data DFT">
|
|
cannam@95
|
10 <link rel="next" href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029" title="1d Real-odd DFTs (DSTs)">
|
|
cannam@95
|
11 <link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage">
|
|
cannam@95
|
12 <!--
|
|
cannam@95
|
13 This manual is for FFTW
|
|
cannam@95
|
14 (version 3.3.3, 25 November 2012).
|
|
cannam@95
|
15
|
|
cannam@95
|
16 Copyright (C) 2003 Matteo Frigo.
|
|
cannam@95
|
17
|
|
cannam@95
|
18 Copyright (C) 2003 Massachusetts Institute of Technology.
|
|
cannam@95
|
19
|
|
cannam@95
|
20 Permission is granted to make and distribute verbatim copies of
|
|
cannam@95
|
21 this manual provided the copyright notice and this permission
|
|
cannam@95
|
22 notice are preserved on all copies.
|
|
cannam@95
|
23
|
|
cannam@95
|
24 Permission is granted to copy and distribute modified versions of
|
|
cannam@95
|
25 this manual under the conditions for verbatim copying, provided
|
|
cannam@95
|
26 that the entire resulting derived work is distributed under the
|
|
cannam@95
|
27 terms of a permission notice identical to this one.
|
|
cannam@95
|
28
|
|
cannam@95
|
29 Permission is granted to copy and distribute translations of this
|
|
cannam@95
|
30 manual into another language, under the above conditions for
|
|
cannam@95
|
31 modified versions, except that this permission notice may be
|
|
cannam@95
|
32 stated in a translation approved by the Free Software Foundation.
|
|
cannam@95
|
33 -->
|
|
cannam@95
|
34 <meta http-equiv="Content-Style-Type" content="text/css">
|
|
cannam@95
|
35 <style type="text/css"><!--
|
|
cannam@95
|
36 pre.display { font-family:inherit }
|
|
cannam@95
|
37 pre.format { font-family:inherit }
|
|
cannam@95
|
38 pre.smalldisplay { font-family:inherit; font-size:smaller }
|
|
cannam@95
|
39 pre.smallformat { font-family:inherit; font-size:smaller }
|
|
cannam@95
|
40 pre.smallexample { font-size:smaller }
|
|
cannam@95
|
41 pre.smalllisp { font-size:smaller }
|
|
cannam@95
|
42 span.sc { font-variant:small-caps }
|
|
cannam@95
|
43 span.roman { font-family:serif; font-weight:normal; }
|
|
cannam@95
|
44 span.sansserif { font-family:sans-serif; font-weight:normal; }
|
|
cannam@95
|
45 --></style>
|
|
cannam@95
|
46 </head>
|
|
cannam@95
|
47 <body>
|
|
cannam@95
|
48 <div class="node">
|
|
cannam@95
|
49 <a name="1d-Real-even-DFTs-(DCTs)"></a>
|
|
cannam@95
|
50 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
|
|
cannam@95
|
51 <p>
|
|
cannam@95
|
52 Next: <a rel="next" accesskey="n" href="1d-Real_002dodd-DFTs-_0028DSTs_0029.html#g_t1d-Real_002dodd-DFTs-_0028DSTs_0029">1d Real-odd DFTs (DSTs)</a>,
|
|
cannam@95
|
53 Previous: <a rel="previous" accesskey="p" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT">The 1d Real-data DFT</a>,
|
|
cannam@95
|
54 Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
|
|
cannam@95
|
55 <hr>
|
|
cannam@95
|
56 </div>
|
|
cannam@95
|
57
|
|
cannam@95
|
58 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
|
|
cannam@95
|
59
|
|
cannam@95
|
60 <p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
|
|
cannam@95
|
61 forward (and backward) DFTs as defined above, where the input array
|
|
cannam@95
|
62 X of length N is purely real and is also <dfn>even</dfn> symmetry. In
|
|
cannam@95
|
63 this case, the output array is likewise real and even symmetry.
|
|
cannam@95
|
64 <a name="index-real_002deven-DFT-301"></a><a name="index-REDFT-302"></a>
|
|
cannam@95
|
65
|
|
cannam@95
|
66 <p><a name="index-REDFT00-303"></a>For the case of <code>REDFT00</code>, this even symmetry means that
|
|
cannam@95
|
67 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,where we take X to be periodic so that
|
|
cannam@95
|
68 <i>X<sub>N</sub> = X</i><sub>0</sub>. Because of this redundancy, only the first n real numbers are
|
|
cannam@95
|
69 actually stored, where N = 2(n-1).
|
|
cannam@95
|
70
|
|
cannam@95
|
71 <p>The proper definition of even symmetry for <code>REDFT10</code>,
|
|
cannam@95
|
72 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
|
|
cannam@95
|
73 because of the shifts by 1/2 of the input and/or output, although
|
|
cannam@95
|
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 even symmetry, however,
|
|
cannam@95
|
75 the sine terms in the DFT all cancel and the remaining cosine terms are
|
|
cannam@95
|
76 written explicitly below. This formulation often leads people to call
|
|
cannam@95
|
77 such a transform a <dfn>discrete cosine transform</dfn> (DCT), although it is
|
|
cannam@95
|
78 really just a special case of the DFT.
|
|
cannam@95
|
79 <a name="index-discrete-cosine-transform-304"></a><a name="index-DCT-305"></a>
|
|
cannam@95
|
80
|
|
cannam@95
|
81 <p>In each of the definitions below, we transform a real array X of
|
|
cannam@95
|
82 length n to a real array Y of length n:
|
|
cannam@95
|
83
|
|
cannam@95
|
84 <h5 class="subsubheading">REDFT00 (DCT-I)</h5>
|
|
cannam@95
|
85
|
|
cannam@95
|
86 <p><a name="index-REDFT00-306"></a>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
|
|
cannam@95
|
87 <center><img src="equation-redft00.png" align="top">.</center>Note that this transform is not defined for n=1. For n=2,
|
|
cannam@95
|
88 the summation term above is dropped as you might expect.
|
|
cannam@95
|
89
|
|
cannam@95
|
90 <h5 class="subsubheading">REDFT10 (DCT-II)</h5>
|
|
cannam@95
|
91
|
|
cannam@95
|
92 <p><a name="index-REDFT10-307"></a>An <code>REDFT10</code> transform (type-II DCT, sometimes called “the” DCT) in FFTW is defined by:
|
|
cannam@95
|
93 <center><img src="equation-redft10.png" align="top">.</center>
|
|
cannam@95
|
94
|
|
cannam@95
|
95 <h5 class="subsubheading">REDFT01 (DCT-III)</h5>
|
|
cannam@95
|
96
|
|
cannam@95
|
97 <p><a name="index-REDFT01-308"></a>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
|
|
cannam@95
|
98 <center><img src="equation-redft01.png" align="top">.</center>In the case of n=1, this reduces to
|
|
cannam@95
|
99 <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> (“the” DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the “IDCT”.
|
|
cannam@95
|
100 <a name="index-IDCT-309"></a>
|
|
cannam@95
|
101
|
|
cannam@95
|
102 <h5 class="subsubheading">REDFT11 (DCT-IV)</h5>
|
|
cannam@95
|
103
|
|
cannam@95
|
104 <p><a name="index-REDFT11-310"></a>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
|
|
cannam@95
|
105 <center><img src="equation-redft11.png" align="top">.</center>
|
|
cannam@95
|
106
|
|
cannam@95
|
107 <h5 class="subsubheading">Inverses and Normalization</h5>
|
|
cannam@95
|
108
|
|
cannam@95
|
109 <p>These definitions correspond directly to the unnormalized DFTs used
|
|
cannam@95
|
110 elsewhere in FFTW (hence the factors of 2 in front of the
|
|
cannam@95
|
111 summations). The unnormalized inverse of <code>REDFT00</code> is
|
|
cannam@95
|
112 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
|
|
cannam@95
|
113 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
|
|
cannam@95
|
114 in the original array multiplied by N, where N is the
|
|
cannam@95
|
115 <em>logical</em> DFT size. For <code>REDFT00</code>, N=2(n-1) (note that
|
|
cannam@95
|
116 n=1 is not defined); otherwise, N=2n.
|
|
cannam@95
|
117 <a name="index-normalization-311"></a>
|
|
cannam@95
|
118
|
|
cannam@95
|
119 <p>In defining the discrete cosine transform, some authors also include
|
|
cannam@95
|
120 additional factors of
|
|
cannam@95
|
121 √2(or its inverse) multiplying selected inputs and/or outputs. This is a
|
|
cannam@95
|
122 mostly cosmetic change that makes the transform orthogonal, but
|
|
cannam@95
|
123 sacrifices the direct equivalence to a symmetric DFT.
|
|
cannam@95
|
124
|
|
cannam@95
|
125 <!-- =========> -->
|
|
cannam@95
|
126 </body></html>
|
|
cannam@95
|
127
|
