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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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+<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
+<html>
+<!-- This manual is for FFTW
+(version 3.3.5, 30 July 2016).
+
+Copyright (C) 2003 Matteo Frigo.
+
+Copyright (C) 2003 Massachusetts Institute of Technology.
+
+Permission is granted to make and distribute verbatim copies of this
+manual provided the copyright notice and this permission notice are
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+
+Permission is granted to copy and distribute modified versions of this
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+permission notice identical to this one.
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+approved by the Free Software Foundation. -->
+<!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ -->
+<head>
+<title>FFTW 3.3.5: 1d Real-odd DFTs (DSTs)</title>
+
+<meta name="description" content="FFTW 3.3.5: 1d Real-odd DFTs (DSTs)">
+<meta name="keywords" content="FFTW 3.3.5: 1d Real-odd DFTs (DSTs)">
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+<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
+<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
+<link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
+<link href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" rel="next" title="1d Discrete Hartley Transforms (DHTs)">
+<link href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" rel="prev" title="1d Real-even DFTs (DCTs)">
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+<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
+<a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029"></a>
+<div class="header">
+<p>
+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>
+</div>
+<hr>
+<a name="g_t1d-Real_002dodd-DFTs-_0028DSTs_0029-1"></a>
+<h4 class="subsection">4.8.4 1d Real-odd DFTs (DSTs)</h4>
+
+<p>The Real-odd symmetry DFTs in FFTW are exactly equivalent to the unnormalized
+forward (and backward) DFTs as defined above, where the input array
+<em>X</em> of length <em>N</em> is purely real and is also <em>odd</em> symmetry.  In
+this case, the output is odd symmetry and purely imaginary.
+<a name="index-real_002dodd-DFT-1"></a>
+<a name="index-RODFT-1"></a>
+</p>
+
+<a name="index-RODFT00"></a>
+<p>For the case of <code>RODFT00</code>, this odd symmetry means that
+<i>X<sub>j</sub> = -X<sub>N-j</sub></i>,where we take <em>X</em> to be periodic so that
+<i>X<sub>N</sub> = X</i><sub>0</sub>.Because of this redundancy, only the first <em>n</em> real numbers
+starting at <em>j=1</em> are actually stored (the <em>j=0</em> element is
+zero), where <em>N = 2(n+1)</em>.
+</p>
+<p>The proper definition of odd symmetry for <code>RODFT10</code>,
+<code>RODFT01</code>, and <code>RODFT11</code> transforms is somewhat more intricate
+because of the shifts by <em>1/2</em> of the input and/or output, although
+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,
+the cosine terms in the DFT all cancel and the remaining sine terms are
+written explicitly below.  This formulation often leads people to call
+such a transform a <em>discrete sine transform</em> (DST), although it is
+really just a special case of the DFT.
+<a name="index-discrete-sine-transform-2"></a>
+<a name="index-DST-2"></a>
+</p>
+
+<p>In each of the definitions below, we transform a real array <em>X</em> of
+length <em>n</em> to a real array <em>Y</em> of length <em>n</em>:
+</p>
+<a name="RODFT00-_0028DST_002dI_0029"></a>
+<h4 class="subsubheading">RODFT00 (DST-I)</h4>
+<a name="index-RODFT00-1"></a>
+<p>An <code>RODFT00</code> transform (type-I DST) in FFTW is defined by:
+<center><img src="equation-rodft00.png" align="top">.</center></p>
+<a name="RODFT10-_0028DST_002dII_0029"></a>
+<h4 class="subsubheading">RODFT10 (DST-II)</h4>
+<a name="index-RODFT10"></a>
+<p>An <code>RODFT10</code> transform (type-II DST) in FFTW is defined by:
+<center><img src="equation-rodft10.png" align="top">.</center></p>
+<a name="RODFT01-_0028DST_002dIII_0029"></a>
+<h4 class="subsubheading">RODFT01 (DST-III)</h4>
+<a name="index-RODFT01"></a>
+<p>An <code>RODFT01</code> transform (type-III DST) in FFTW is defined by:
+<center><img src="equation-rodft01.png" align="top">.</center>In the case of <em>n=1</em>, this reduces to
+<i>Y</i><sub>0</sub> = <i>X</i><sub>0</sub>.</p>
+<a name="RODFT11-_0028DST_002dIV_0029"></a>
+<h4 class="subsubheading">RODFT11 (DST-IV)</h4>
+<a name="index-RODFT11"></a>
+<p>An <code>RODFT11</code> transform (type-IV DST) in FFTW is defined by:
+<center><img src="equation-rodft11.png" align="top">.</center></p>
+<a name="Inverses-and-Normalization-1"></a>
+<h4 class="subsubheading">Inverses and Normalization</h4>
+
+<p>These definitions correspond directly to the unnormalized DFTs used
+elsewhere in FFTW (hence the factors of <em>2</em> in front of the
+summations).  The unnormalized inverse of <code>RODFT00</code> is
+<code>RODFT00</code>, of <code>RODFT10</code> is <code>RODFT01</code> and vice versa, and
+of <code>RODFT11</code> is <code>RODFT11</code>.  Each unnormalized inverse results
+in the original array multiplied by <em>N</em>, where <em>N</em> is the
+<em>logical</em> DFT size.  For <code>RODFT00</code>, <em>N=2(n+1)</em>;
+otherwise, <em>N=2n</em>.
+<a name="index-normalization-11"></a>
+</p>
+
+<p>In defining the discrete sine transform, some authors also include
+additional factors of
+&radic;2(or its inverse) multiplying selected inputs and/or outputs.  This is a
+mostly cosmetic change that makes the transform orthogonal, but
+sacrifices the direct equivalence to an antisymmetric DFT.
+</p>
+<hr>
+<div class="header">
+<p>
+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>
+</div>
+
+
+
+</body>
+</html>