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Current fftw source
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 18 Oct 2016 13:40:26 +0100
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25 <title>FFTW 3.3.5: 1d Real-odd DFTs (DSTs)</title>
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72 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></a>
73 <div class="header">
74 <p>
75 Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</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>
76 </div>
77 <hr>
78 <a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029-1"></a>
79 <h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>
80
81 <p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
82 forward (and backward) DFTs as defined above, where the input array
83 <em>X</em> of length <em>N</em> is purely real and is also <em>odd</em> symmetry. In
84 this case, the output is odd symmetry and purely imaginary.
85 <a name="index-real_002dodd-DFT-1"></a>
86 <a name="index-RODFT-1"></a>
87 </p>
88
89 <a name="index-RODFT00"></a>
90 <p>For the case of <code>RODFT00</code>, this odd symmetry means that
91 <i>X<sub>j</sub> = -X<sub>N-j</sub></i>,where we take <em>X</em> to be periodic so that
92 <i>X<sub>N</sub> = X</i><sub>0</sub>.Because of this redundancy, only the first <em>n</em> real numbers
93 starting at <em>j=1</em> are actually stored (the <em>j=0</em> element is
94 zero), where <em>N = 2(n+1)</em>.
95 </p>
96 <p>The proper definition of odd symmetry for <code>RODFT10</code>,
97 <code>RODFT01</code>, and <code>RODFT11</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 odd symmetry, however,
100 the cosine terms in the DFT all cancel and the remaining sine terms are
101 written explicitly below. This formulation often leads people to call
102 such a transform a <em>discrete sine transform</em> (DST), although it is
103 really just a special case of the DFT.
104 <a name="index-discrete-sine-transform-2"></a>
105 <a name="index-DST-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="RODFT00-_0028DST_002dI_0029"></a>
112 <h4 class="subsubheading">RODFT00 (DST-I)</h4>
113 <a name="index-RODFT00-1"></a>
114 <p>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
115 <center><img src="equation-rodft00.png" align="top">.</center></p>
116 <a name="RODFT10-_0028DST_002dII_0029"></a>
117 <h4 class="subsubheading">RODFT10 (DST-II)</h4>
118 <a name="index-RODFT10"></a>
119 <p>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
120 <center><img src="equation-rodft10.png" align="top">.</center></p>
121 <a name="RODFT01-_0028DST_002dIII_0029"></a>
122 <h4 class="subsubheading">RODFT01 (DST-III)</h4>
123 <a name="index-RODFT01"></a>
124 <p>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
125 <center><img src="equation-rodft01.png" align="top">.</center>In the case of <em>n=1</em>, this reduces to
126 <i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.</p>
127 <a name="RODFT11-_0028DST_002dIV_0029"></a>
128 <h4 class="subsubheading">RODFT11 (DST-IV)</h4>
129 <a name="index-RODFT11"></a>
130 <p>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
131 <center><img src="equation-rodft11.png" align="top">.</center></p>
132 <a name="Inverses-and-Normalization-1"></a>
133 <h4 class="subsubheading">Inverses and Normalization</h4>
134
135 <p>These definitions correspond directly to the unnormalized DFTs used
136 elsewhere in FFTW (hence the factors of <em>2</em> in front of the
137 summations). The unnormalized inverse of <code>RODFT00</code> is
138 <code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
139 of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
140 in the original array multiplied by <em>N</em>, where <em>N</em> is the
141 <em>logical</em> DFT size. For <code>RODFT00</code>, <em>N=2(n+1)</em>;
142 otherwise, <em>N=2n</em>.
143 <a name="index-normalization-11"></a>
144 </p>
145
146 <p>In defining the discrete sine transform, some authors also include
147 additional factors of
148 &radic;2(or its inverse) multiplying selected inputs and/or outputs. This is a
149 mostly cosmetic change that makes the transform orthogonal, but
150 sacrifices the direct equivalence to an antisymmetric DFT.
151 </p>
152 <hr>
153 <div class="header">
154 <p>
155 Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</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>
156 </div>
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