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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
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25 <title>FFTW 3.3.8: 1d Real-even DFTs (DCTs)</title>
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71 <a name="g_t1d-Real_002deven-DFTs-_0028DCTs_0029"></a>
72 <div class="header">
73 <p>
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> &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_002deven-DFTs-_0028DCTs_0029-1"></a>
78 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
79
80 <p>The Real-even 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>even</em> symmetry. In
83 this case, the output array is likewise real and even symmetry.
84 <a name="index-real_002deven-DFT-1"></a>
85 <a name="index-REDFT-1"></a>
86 </p>
87
88 <a name="index-REDFT00"></a>
89 <p>For the case of <code>REDFT00</code>, this even 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 are
94 actually stored, where <em>N = 2(n-1)</em>.
95 </p>
96 <p>The proper definition of even symmetry for <code>REDFT10</code>,
97 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
98 because of the shifts by <em>1/2</em> of the input and/or output, although
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,
100 the sine terms in the DFT all cancel and the remaining cosine terms are
101 written explicitly below. This formulation often leads people to call
102 such a transform a <em>discrete cosine transform</em> (DCT), although it is
103 really just a special case of the DFT.
104 <a name="index-discrete-cosine-transform-2"></a>
105 <a name="index-DCT-2"></a>
106 </p>
107
108 <p>In each of the definitions below, we transform a real array <em>X</em> of
109 length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
110 </p>
111 <a name="REDFT00-_0028DCT_002dI_0029"></a>
112 <h4 class="subsubheading">REDFT00 (DCT-I)</h4>
113 <a name="index-REDFT00-1"></a>
114 <p>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
115 <center><img src="equation-redft00.png" align="top">.</center>
116 Note that this transform is not defined for <em>n=1</em>. For <em>n=2</em>,
117 the summation term above is dropped as you might expect.
118 </p>
119 <a name="REDFT10-_0028DCT_002dII_0029"></a>
120 <h4 class="subsubheading">REDFT10 (DCT-II)</h4>
121 <a name="index-REDFT10"></a>
122 <p>An <code>REDFT10</code> transform (type-II DCT, sometimes called &ldquo;the&rdquo; DCT) in FFTW is defined by:
123 <center><img src="equation-redft10.png" align="top">.</center>
124 </p>
125 <a name="REDFT01-_0028DCT_002dIII_0029"></a>
126 <h4 class="subsubheading">REDFT01 (DCT-III)</h4>
127 <a name="index-REDFT01"></a>
128 <p>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
129 <center><img src="equation-redft01.png" align="top">.</center>
130 In the case of <em>n=1</em>, this reduces to
131 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
132 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;.
133 <a name="index-IDCT-3"></a>
134 </p>
135 <a name="REDFT11-_0028DCT_002dIV_0029"></a>
136 <h4 class="subsubheading">REDFT11 (DCT-IV)</h4>
137 <a name="index-REDFT11"></a>
138 <p>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
139 <center><img src="equation-redft11.png" align="top">.</center>
140 </p>
141 <a name="Inverses-and-Normalization"></a>
142 <h4 class="subsubheading">Inverses and Normalization</h4>
143
144 <p>These definitions correspond directly to the unnormalized DFTs used
145 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
146 summations). The unnormalized inverse of <code>REDFT00</code> is
147 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
148 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
149 in the original array multiplied by <em>N</em>, where <em>N</em> is the
150 <em>logical</em> DFT size. For <code>REDFT00</code>, <em>N=2(n-1)</em> (note that
151 <em>n=1</em> is not defined); otherwise, <em>N=2n</em>.
152 <a name="index-normalization-10"></a>
153 </p>
154
155 <p>In defining the discrete cosine transform, some authors also include
156 additional factors of
157 &radic;2
158 (or its inverse) multiplying selected inputs and/or outputs. This is a
159 mostly cosmetic change that makes the transform orthogonal, but
160 sacrifices the direct equivalence to a symmetric DFT.
161 </p>
162 <hr>
163 <div class="header">
164 <p>
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> &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>
166 </div>
167
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169
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