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1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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2 <html>
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3 <!-- This manual is for FFTW
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4 (version 3.3.8, 24 May 2018).
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5
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6 Copyright (C) 2003 Matteo Frigo.
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7
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8 Copyright (C) 2003 Massachusetts Institute of Technology.
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9
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10 Permission is granted to make and distribute verbatim copies of this
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11 manual provided the copyright notice and this permission notice are
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13
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14 Permission is granted to copy and distribute modified versions of this
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15 manual under the conditions for verbatim copying, provided that the
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16 entire resulting derived work is distributed under the terms of a
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17 permission notice identical to this one.
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19 Permission is granted to copy and distribute translations of this manual
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22 approved by the Free Software Foundation. -->
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23 <!-- Created by GNU Texinfo 6.3, http://www.gnu.org/software/texinfo/ -->
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24 <head>
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25 <title>FFTW 3.3.8: 1d Real-even DFTs (DCTs)</title>
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26
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27 <meta name="description" content="FFTW 3.3.8: 1d Real-even DFTs (DCTs)">
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28 <meta name="keywords" content="FFTW 3.3.8: 1d Real-even DFTs (DCTs)">
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33 <link href="index.html#Top" rel="start" title="Top">
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34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
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35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
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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)">
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38 <link href="The-1d-Real_002ddata-DFT.html#The-1d-Real_002ddata-DFT" rel="prev" title="The 1d Real-data DFT">
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64 -->
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65 </style>
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66
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67
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68 </head>
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69
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70 <body lang="en">
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71 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
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72 <div class="header">
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73 <p>
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74 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> [<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>
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75 </div>
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76 <hr>
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77 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029-1"></a>
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78 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
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79
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80 <p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
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81 forward (and backward) DFTs as defined above, where the input array
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82 <em>X</em> of length <em>N</em> is purely real and is also <em>even</em> symmetry. In
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83 this case, the output array is likewise real and even symmetry.
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84 <a name="index-real_002deven-DFT-1"></a>
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85 <a name="index-REDFT-1"></a>
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86 </p>
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87
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88 <a name="index-REDFT00"></a>
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89 <p>For the case of <code>REDFT00</code>, this even symmetry means that
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90 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,
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91 where we take <em>X</em> to be periodic so that
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92 <i>X<sub>N</sub> = X</i><sub>0</sub>.
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93 Because of this redundancy, only the first <em>n</em> real numbers are
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94 actually stored, where <em>N = 2(n-1)</em>.
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95 </p>
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96 <p>The proper definition of even symmetry for <code>REDFT10</code>,
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97 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
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98 because of the shifts by <em>1/2</em> of the input and/or output, although
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99 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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100 the sine terms in the DFT all cancel and the remaining cosine terms are
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101 written explicitly below. This formulation often leads people to call
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102 such a transform a <em>discrete cosine transform</em> (DCT), although it is
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103 really just a special case of the DFT.
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104 <a name="index-discrete-cosine-transform-2"></a>
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105 <a name="index-DCT-2"></a>
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106 </p>
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107
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108 <p>In each of the definitions below, we transform a real array <em>X</em> of
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109 length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
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110 </p>
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111 <a name="REDFT00-_0028DCT_002dI_0029"></a>
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112 <h4 class="subsubheading">REDFT00 (DCT-I)</h4>
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113 <a name="index-REDFT00-1"></a>
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114 <p>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
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115 <center><img src="equation-redft00.png" align="top">.</center>
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116 Note that this transform is not defined for <em>n=1</em>. For <em>n=2</em>,
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117 the summation term above is dropped as you might expect.
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118 </p>
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119 <a name="REDFT10-_0028DCT_002dII_0029"></a>
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120 <h4 class="subsubheading">REDFT10 (DCT-II)</h4>
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121 <a name="index-REDFT10"></a>
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122 <p>An <code>REDFT10</code> transform (type-II DCT, sometimes called “the” DCT) in FFTW is defined by:
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123 <center><img src="equation-redft10.png" align="top">.</center>
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124 </p>
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125 <a name="REDFT01-_0028DCT_002dIII_0029"></a>
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126 <h4 class="subsubheading">REDFT01 (DCT-III)</h4>
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127 <a name="index-REDFT01"></a>
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128 <p>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
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129 <center><img src="equation-redft01.png" align="top">.</center>
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130 In the case of <em>n=1</em>, this reduces to
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131 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
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132 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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133 <a name="index-IDCT-3"></a>
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134 </p>
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135 <a name="REDFT11-_0028DCT_002dIV_0029"></a>
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136 <h4 class="subsubheading">REDFT11 (DCT-IV)</h4>
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137 <a name="index-REDFT11"></a>
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138 <p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
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139 <center><img src="equation-redft11.png" align="top">.</center>
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140 </p>
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141 <a name="Inverses-and-Normalization"></a>
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142 <h4 class="subsubheading">Inverses and Normalization</h4>
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143
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144 <p>These definitions correspond directly to the unnormalized DFTs used
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145 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
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146 summations). The unnormalized inverse of <code>REDFT00</code> is
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147 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
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148 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
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149 in the original array multiplied by <em>N</em>, where <em>N</em> is the
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150 <em>logical</em> DFT size. For <code>REDFT00</code>, <em>N=2(n-1)</em> (note that
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151 <em>n=1</em> is not defined); otherwise, <em>N=2n</em>.
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152 <a name="index-normalization-10"></a>
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153 </p>
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154
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155 <p>In defining the discrete cosine transform, some authors also include
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156 additional factors of
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157 √2
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158 (or its inverse) multiplying selected inputs and/or outputs. This is a
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159 mostly cosmetic change that makes the transform orthogonal, but
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160 sacrifices the direct equivalence to a symmetric DFT.
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161 </p>
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162 <hr>
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163 <div class="header">
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164 <p>
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165 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> [<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>
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166 </div>
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167
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168
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169
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170 </body>
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171 </html>
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