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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">
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3 <title>1d Real-odd DFTs (DSTs) - FFTW 3.2.1</title>
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57
58 <h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>
59
60 <p>The Real-odd 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>odd</dfn> symmetry. In
63 this case, the output is odd symmetry and purely imaginary.
64 <a name="index-real_002dodd-DFT-302"></a><a name="index-RODFT-303"></a>
65 <a name="index-RODFT00-304"></a>For the case of <code>RODFT00</code>, this odd 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
68 starting at j=1 are actually stored (the j=0 element is
69 zero), where N = 2(n+1).
70
71 <p>The proper definition of odd symmetry for <code>RODFT10</code>,
72 <code>RODFT01</code>, and <code>RODFT11</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 odd symmetry, however,
75 the cosine terms in the DFT all cancel and the remaining sine terms are
76 written explicitly below. This formulation often leads people to call
77 such a transform a <dfn>discrete sine transform</dfn> (DST), although it is
78 really just a special case of the DFT.
79 <a name="index-discrete-sine-transform-305"></a><a name="index-DST-306"></a>
80 In each of the definitions below, we transform a real array X of
81 length n to a real array Y of length n:
82
83 <h5 class="subsubheading">RODFT00 (DST-I)</h5>
84
85 <p><a name="index-RODFT00-307"></a>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
86 <center><img src="equation-rodft00.png" align="top">.</center>
87
88 <h5 class="subsubheading">RODFT10 (DST-II)</h5>
89
90 <p><a name="index-RODFT10-308"></a>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
91 <center><img src="equation-rodft10.png" align="top">.</center>
92
93 <h5 class="subsubheading">RODFT01 (DST-III)</h5>
94
95 <p><a name="index-RODFT01-309"></a>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
96 <center><img src="equation-rodft01.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>.
98
99 <h5 class="subsubheading">RODFT11 (DST-IV)</h5>
100
101 <p><a name="index-RODFT11-310"></a>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
102 <center><img src="equation-rodft11.png" align="top">.</center>
103
104 <h5 class="subsubheading">Inverses and Normalization</h5>
105
106 <p>These definitions correspond directly to the unnormalized DFTs used
107 elsewhere in FFTW (hence the factors of 2 in front of the
108 summations). The unnormalized inverse of <code>RODFT00</code> is
109 <code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
110 of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
111 in the original array multiplied by N, where N is the
112 <em>logical</em> DFT size. For <code>RODFT00</code>, N=2(n+1);
113 otherwise, N=2n.
114 <a name="index-normalization-311"></a>
115 In defining the discrete sine transform, some authors also include
116 additional factors of
117 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
118 mostly cosmetic change that makes the transform orthogonal, but
119 sacrifices the direct equivalence to an antisymmetric DFT.
120
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