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author Geogaddi\David <d.m.ronan@qmul.ac.uk>
date Thu, 09 Jul 2015 01:12:16 +0100
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1 <html lang="en">
2 <head>
3 <title>1d Real-even DFTs (DCTs) - FFTW 3.2.1</title>
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50 <a name="1d-Real-even-DFTs-(DCTs)"></a>
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52 Next:&nbsp;<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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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-291"></a><a name="index-REDFT-292"></a>
65 <a name="index-REDFT00-293"></a>For the case of <code>REDFT00</code>, this even symmetry means that
66 <i>X<sub>j</sub> = X<sub>N-j</sub></i>,where we take X to be periodic so that
67 <i>X<sub>N</sub> = X</i><sub>0</sub>. Because of this redundancy, only the first n real numbers are
68 actually stored, where N = 2(n-1).
69
70 <p>The proper definition of even symmetry for <code>REDFT10</code>,
71 <code>REDFT01</code>, and <code>REDFT11</code> transforms is somewhat more intricate
72 because of the shifts by 1/2 of the input and/or output, although
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,
74 the sine terms in the DFT all cancel and the remaining cosine terms are
75 written explicitly below. This formulation often leads people to call
76 such a transform a <dfn>discrete cosine transform</dfn> (DCT), although it is
77 really just a special case of the DFT.
78 <a name="index-discrete-cosine-transform-294"></a><a name="index-DCT-295"></a>
79 In each of the definitions below, we transform a real array X of
80 length n to a real array Y of length n:
81
82 <h5 class="subsubheading">REDFT00 (DCT-I)</h5>
83
84 <p><a name="index-REDFT00-296"></a>An <code>REDFT00</code> transform (type-I DCT) in FFTW is defined by:
85 <center><img src="equation-redft00.png" align="top">.</center>Note that this transform is not defined for n=1. For n=2,
86 the summation term above is dropped as you might expect.
87
88 <h5 class="subsubheading">REDFT10 (DCT-II)</h5>
89
90 <p><a name="index-REDFT10-297"></a>An <code>REDFT10</code> transform (type-II DCT, sometimes called &ldquo;the&rdquo; DCT) in FFTW is defined by:
91 <center><img src="equation-redft10.png" align="top">.</center>
92
93 <h5 class="subsubheading">REDFT01 (DCT-III)</h5>
94
95 <p><a name="index-REDFT01-298"></a>An <code>REDFT01</code> transform (type-III DCT) in FFTW is defined by:
96 <center><img src="equation-redft01.png" align="top">.</center>In the case of n=1, this reduces to
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> (&ldquo;the&rdquo; DCT), and so the <code>REDFT01</code> (DCT-III) is sometimes called the &ldquo;IDCT&rdquo;.
98 <a name="index-IDCT-299"></a>
99
100 <h5 class="subsubheading">REDFT11 (DCT-IV)</h5>
101
102 <p><a name="index-REDFT11-300"></a>An <code>REDFT11</code> transform (type-IV DCT) in FFTW is defined by:
103 <center><img src="equation-redft11.png" align="top">.</center>
104
105 <h5 class="subsubheading">Inverses and Normalization</h5>
106
107 <p>These definitions correspond directly to the unnormalized DFTs used
108 elsewhere in FFTW (hence the factors of 2 in front of the
109 summations). The unnormalized inverse of <code>REDFT00</code> is
110 <code>REDFT00</code>, of <code>REDFT10</code> is <code>REDFT01</code> and vice versa, and
111 of <code>REDFT11</code> is <code>REDFT11</code>. Each unnormalized inverse results
112 in the original array multiplied by N, where N is the
113 <em>logical</em> DFT size. For <code>REDFT00</code>, N=2(n-1) (note that
114 n=1 is not defined); otherwise, N=2n.
115 <a name="index-normalization-301"></a>
116 In defining the discrete cosine transform, some authors also include
117 additional factors of
118 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
119 mostly cosmetic change that makes the transform orthogonal, but
120 sacrifices the direct equivalence to a symmetric DFT.
121
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