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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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3 <title>1d Real-even DFTs (DCTs) - FFTW 3.3.3</title>
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49 <a name="1d-Real-even-DFTs-(DCTs)"></a>
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51 <p>
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57
58 <h4 class="subsection">4.8.3 1d Real-even DFTs (DCTs)</h4>
59
60 <p>The Real-even symmetry DFTs in FFTW are exactly equivalent to the unnormalized
61 forward (and backward) DFTs as defined above, where the input array
62 X of length N is purely real and is also <dfn>even</dfn> symmetry. In
63 this case, the output array is likewise real and even symmetry.
64 <a name="index-real_002deven-DFT-301"></a><a name="index-REDFT-302"></a>
65
66 <p><a name="index-REDFT00-303"></a>For the case of <code>REDFT00</code>, this even symmetry means that
67 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,where we take X to be periodic so that
68 <i>X<sub>N</sub> = X</i><sub>0</sub>. Because of this redundancy, only the first n real numbers are
69 actually stored, where N = 2(n-1).
70
71 <p>The proper definition of even symmetry for <code>REDFT10</code>,
72 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
73 because of the shifts by 1/2 of the input and/or output, although
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,
75 the sine terms in the DFT all cancel and the remaining cosine terms are
76 written explicitly below. This formulation often leads people to call
77 such a transform a <dfn>discrete cosine transform</dfn> (DCT), although it is
78 really just a special case of the DFT.
79 <a name="index-discrete-cosine-transform-304"></a><a name="index-DCT-305"></a>
80
81 <p>In each of the definitions below, we transform a real array X of
82 length n to a real array Y of length n:
83
84 <h5 class="subsubheading">REDFT00 (DCT-I)</h5>
85
86 <p><a name="index-REDFT00-306"></a>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
87 <center><img src="equation-redft00.png" align="top">.</center>Note that this transform is not defined for n=1. For n=2,
88 the summation term above is dropped as you might expect.
89
90 <h5 class="subsubheading">REDFT10 (DCT-II)</h5>
91
92 <p><a name="index-REDFT10-307"></a>An <code>REDFT10</code> transform (type-II DCT, sometimes called &ldquo;the&rdquo; DCT) in FFTW is defined by:
93 <center><img src="equation-redft10.png" align="top">.</center>
94
95 <h5 class="subsubheading">REDFT01 (DCT-III)</h5>
96
97 <p><a name="index-REDFT01-308"></a>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
98 <center><img src="equation-redft01.png" align="top">.</center>In the case of n=1, this reduces to
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> (&ldquo;the&rdquo; DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the &ldquo;IDCT&rdquo;.
100 <a name="index-IDCT-309"></a>
101
102 <h5 class="subsubheading">REDFT11 (DCT-IV)</h5>
103
104 <p><a name="index-REDFT11-310"></a>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
105 <center><img src="equation-redft11.png" align="top">.</center>
106
107 <h5 class="subsubheading">Inverses and Normalization</h5>
108
109 <p>These definitions correspond directly to the unnormalized DFTs used
110 elsewhere in FFTW (hence the factors of 2 in front of the
111 summations). The unnormalized inverse of <code>REDFT00</code> is
112 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
113 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
114 in the original array multiplied by N, where N is the
115 <em>logical</em> DFT size. For <code>REDFT00</code>, N=2(n-1) (note that
116 n=1 is not defined); otherwise, N=2n.
117 <a name="index-normalization-311"></a>
118
119 <p>In defining the discrete cosine transform, some authors also include
120 additional factors of
121 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
122 mostly cosmetic change that makes the transform orthogonal, but
123 sacrifices the direct equivalence to a symmetric DFT.
124
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