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author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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3 <title>1d Real-odd DFTs (DSTs) - FFTW 3.3.3</title>
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49 <a name="1d-Real-odd-DFTs-(DSTs)"></a>
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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-312"></a><a name="index-RODFT-313"></a>
65
66 <p><a name="index-RODFT00-314"></a>For the case of <code>RODFT00</code>, this odd 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
69 starting at j=1 are actually stored (the j=0 element is
70 zero), where N = 2(n+1).
71
72 <p>The proper definition of odd symmetry for <code>RODFT10</code>,
73 <code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
74 because of the shifts by 1/2 of the input and/or output, although
75 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,
76 the cosine terms in the DFT all cancel and the remaining sine terms are
77 written explicitly below. This formulation often leads people to call
78 such a transform a <dfn>discrete sine transform</dfn> (DST), although it is
79 really just a special case of the DFT.
80 <a name="index-discrete-sine-transform-315"></a><a name="index-DST-316"></a>
81
82 <p>In each of the definitions below, we transform a real array X of
83 length n to a real array Y of length n:
84
85 <h5 class="subsubheading">RODFT00 (DST-I)</h5>
86
87 <p><a name="index-RODFT00-317"></a>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
88 <center><img src="equation-rodft00.png" align="top">.</center>
89
90 <h5 class="subsubheading">RODFT10 (DST-II)</h5>
91
92 <p><a name="index-RODFT10-318"></a>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
93 <center><img src="equation-rodft10.png" align="top">.</center>
94
95 <h5 class="subsubheading">RODFT01 (DST-III)</h5>
96
97 <p><a name="index-RODFT01-319"></a>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
98 <center><img src="equation-rodft01.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>.
100
101 <h5 class="subsubheading">RODFT11 (DST-IV)</h5>
102
103 <p><a name="index-RODFT11-320"></a>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
104 <center><img src="equation-rodft11.png" align="top">.</center>
105
106 <h5 class="subsubheading">Inverses and Normalization</h5>
107
108 <p>These definitions correspond directly to the unnormalized DFTs used
109 elsewhere in FFTW (hence the factors of 2 in front of the
110 summations). The unnormalized inverse of <code>RODFT00</code> is
111 <code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
112 of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
113 in the original array multiplied by N, where N is the
114 <em>logical</em> DFT size. For <code>RODFT00</code>, N=2(n+1);
115 otherwise, N=2n.
116 <a name="index-normalization-321"></a>
117
118 <p>In defining the discrete sine transform, some authors also include
119 additional factors of
120 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
121 mostly cosmetic change that makes the transform orthogonal, but
122 sacrifices the direct equivalence to an antisymmetric DFT.
123
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