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<!-- This manual is for FFTW
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(version 3.3.10, 10 December 2020).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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Permission is granted to make and distribute verbatim copies of this
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manual provided the copyright notice and this permission notice are
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preserved on all copies.
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Permission is granted to copy and distribute modified versions of this
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manual under the conditions for verbatim copying, provided that the
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entire resulting derived work is distributed under the terms of a
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permission notice identical to this one.
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>1d Real-odd DFTs (DSTs) (FFTW 3.3.10)</title>
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<meta name="description" content="1d Real-odd DFTs (DSTs) (FFTW 3.3.10)">
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<link href="index.html" rel="start" title="Top">
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<link href="Concept-Index.html" rel="index" title="Concept Index">
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<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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<link href="What-FFTW-Really-Computes.html" rel="up" title="What FFTW Really Computes">
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<link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" rel="next" title="1d Discrete Hartley Transforms (DHTs)">
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<link href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" rel="prev" title="1d Real-even DFTs (DCTs)">
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<span id="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></span><div class="header">
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<p>
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Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<span id="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029-1"></span><h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>
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<p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
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forward (and backward) DFTs as defined above, where the input array
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<em>X</em> of length <em>N</em> is purely real and is also <em>odd</em> symmetry. In
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this case, the output is odd symmetry and purely imaginary.
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<span id="index-real_002dodd-DFT-1"></span>
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<span id="index-RODFT-1"></span>
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</p>
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<span id="index-RODFT00"></span>
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<p>For the case of <code>RODFT00</code>, this odd symmetry means that
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<i>X<sub>j</sub> = -X<sub>N-j</sub></i>,
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where we take <em>X</em> to be periodic so that
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<i>X<sub>N</sub> = X</i><sub>0</sub>.
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Because of this redundancy, only the first <em>n</em> real numbers
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starting at <em>j=1</em> are actually stored (the <em>j=0</em> element is
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zero), where <em>N = 2(n+1)</em>.
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</p>
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<p>The proper definition of odd symmetry for <code>RODFT10</code>,
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<code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
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because of the shifts by <em>1/2</em> of the input and/or output, although
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the corresponding boundary conditions are given in <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html">Real even/odd DFTs (cosine/sine transforms)</a>. Because of the odd symmetry, however,
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the cosine terms in the DFT all cancel and the remaining sine terms are
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written explicitly below. This formulation often leads people to call
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such a transform a <em>discrete sine transform</em> (DST), although it is
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really just a special case of the DFT.
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<span id="index-discrete-sine-transform-2"></span>
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<span id="index-DST-2"></span>
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</p>
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<p>In each of the definitions below, we transform a real array <em>X</em> of
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length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
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</p>
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<span id="RODFT00-_0028DST_002dI_0029"></span><h4 class="subsubheading">RODFT00 (DST-I)</h4>
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<span id="index-RODFT00-1"></span>
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<p>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
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<center><img src="equation-rodft00.png" align="top">.</center>
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</p>
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<span id="RODFT10-_0028DST_002dII_0029"></span><h4 class="subsubheading">RODFT10 (DST-II)</h4>
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<span id="index-RODFT10"></span>
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<p>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
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<center><img src="equation-rodft10.png" align="top">.</center>
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</p>
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<span id="RODFT01-_0028DST_002dIII_0029"></span><h4 class="subsubheading">RODFT01 (DST-III)</h4>
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<span id="index-RODFT01"></span>
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<p>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
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<center><img src="equation-rodft01.png" align="top">.</center>
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In the case of <em>n=1</em>, this reduces to
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<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.
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</p>
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<span id="RODFT11-_0028DST_002dIV_0029"></span><h4 class="subsubheading">RODFT11 (DST-IV)</h4>
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<span id="index-RODFT11"></span>
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<p>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
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<center><img src="equation-rodft11.png" align="top">.</center>
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</p>
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<span id="Inverses-and-Normalization-1"></span><h4 class="subsubheading">Inverses and Normalization</h4>
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<p>These definitions correspond directly to the unnormalized DFTs used
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elsewhere in FFTW (hence the factors of <em>2</em> in front of the
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summations). The unnormalized inverse of <code>RODFT00</code> is
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<code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
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of <code>RODFT11</code> is <code>RODFT11</code>. Each unnormalized inverse results
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in the original array multiplied by <em>N</em>, where <em>N</em> is the
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<em>logical</em> DFT size. For <code>RODFT00</code>, <em>N=2(n+1)</em>;
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otherwise, <em>N=2n</em>.
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<span id="index-normalization-11"></span>
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</p>
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<p>In defining the discrete sine transform, some authors also include
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additional factors of
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√2
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(or its inverse) multiplying selected inputs and/or outputs. This is a
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mostly cosmetic change that makes the transform orthogonal, but
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sacrifices the direct equivalence to an antisymmetric DFT.
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</p>
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<hr>
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<div class="header">
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<p>
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Next: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html" accesskey="n" rel="next">1d Discrete Hartley Transforms (DHTs)</a>, Previous: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" accesskey="p" rel="prev">1d Real-even DFTs (DCTs)</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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</body>
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</html>
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