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2 <head>
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3 <title>1d Real-even DFTs (DCTs) - FFTW 3.2.1</title>
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8 <link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes">
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9 <link rel="prev" href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" title="The 1d Real-data DFT">
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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)">
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12 <!--
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13 This manual is for FFTW
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14 (version 3.2.1, 5 February 2009).
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15
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16 Copyright (C) 2003 Matteo Frigo.
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17
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18 Copyright (C) 2003 Massachusetts Institute of Technology.
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19
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20 Permission is granted to make and distribute verbatim copies of
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22 notice are preserved on all copies.
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24 Permission is granted to copy and distribute modified versions of
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25 this manual under the conditions for verbatim copying, provided
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26 that the entire resulting derived work is distributed under the
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27 terms of a permission notice identical to this one.
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29 Permission is granted to copy and distribute translations of this
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31 modified versions, except that this permission notice may be
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46 </head>
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47 <body>
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48 <div class="node">
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49 <p>
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50 <a name="1d-Real-even-DFTs-(DCTs)"></a>
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51 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
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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>,
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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>,
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54 Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
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55 <hr>
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56 </div>
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57
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58 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
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59
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60 <p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
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61 forward (and backward) DFTs as defined above, where the input array
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62 X of length N is purely real and is also <dfn>even</dfn> symmetry. In
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63 this case, the output array is likewise real and even symmetry.
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64 <a name="index-real_002deven-DFT-291"></a><a name="index-REDFT-292"></a>
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65 <a name="index-REDFT00-293"></a>For the case of <code>REDFT00</code>, this even symmetry means that
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66 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,where we take X to be periodic so that
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67 <i>X<sub>N</sub> = X</i><sub>0</sub>. Because of this redundancy, only the first n real numbers are
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68 actually stored, where N = 2(n-1).
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69
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70 <p>The proper definition of even symmetry for <code>REDFT10</code>,
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71 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
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72 because of the shifts by 1/2 of the input and/or output, although
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73 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,
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74 the sine terms in the DFT all cancel and the remaining cosine terms are
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75 written explicitly below. This formulation often leads people to call
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76 such a transform a <dfn>discrete cosine transform</dfn> (DCT), although it is
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77 really just a special case of the DFT.
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78 <a name="index-discrete-cosine-transform-294"></a><a name="index-DCT-295"></a>
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79 In each of the definitions below, we transform a real array X of
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80 length n to a real array Y of length n:
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81
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82 <h5 class="subsubheading">REDFT00 (DCT-I)</h5>
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83
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84 <p><a name="index-REDFT00-296"></a>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
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85 <center><img src="equation-redft00.png" align="top">.</center>Note that this transform is not defined for n=1. For n=2,
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86 the summation term above is dropped as you might expect.
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87
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88 <h5 class="subsubheading">REDFT10 (DCT-II)</h5>
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89
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90 <p><a name="index-REDFT10-297"></a>An <code>REDFT10</code> transform (type-II DCT, sometimes called “the” DCT) in FFTW is defined by:
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91 <center><img src="equation-redft10.png" align="top">.</center>
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92
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93 <h5 class="subsubheading">REDFT01 (DCT-III)</h5>
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94
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95 <p><a name="index-REDFT01-298"></a>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
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96 <center><img src="equation-redft01.png" align="top">.</center>In the case of n=1, this reduces to
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97 <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”.
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98 <a name="index-IDCT-299"></a>
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99
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100 <h5 class="subsubheading">REDFT11 (DCT-IV)</h5>
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101
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102 <p><a name="index-REDFT11-300"></a>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
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103 <center><img src="equation-redft11.png" align="top">.</center>
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104
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105 <h5 class="subsubheading">Inverses and Normalization</h5>
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106
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107 <p>These definitions correspond directly to the unnormalized DFTs used
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108 elsewhere in FFTW (hence the factors of 2 in front of the
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109 summations). The unnormalized inverse of <code>REDFT00</code> is
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110 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
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111 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
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112 in the original array multiplied by N, where N is the
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113 <em>logical</em> DFT size. For <code>REDFT00</code>, N=2(n-1) (note that
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114 n=1 is not defined); otherwise, N=2n.
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115 <a name="index-normalization-301"></a>
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116 In defining the discrete cosine transform, some authors also include
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117 additional factors of
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118 √2(or its inverse) multiplying selected inputs and/or outputs. This is a
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119 mostly cosmetic change that makes the transform orthogonal, but
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120 sacrifices the direct equivalence to a symmetric DFT.
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121
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122 <!-- =========> -->
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123 </body></html>
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124
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