Mercurial > hg > sv-dependency-builds

annotate src/fftw-3.3.5/doc/html/The-1d-Real_002ddata-DFT.html @ 83:ae30d91d2ffe

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Replace these with versions built using an older toolset (so as to avoid ABI compatibilities when linking on Ubuntu 14.04 for packaging purposes)
author Chris Cannam
date Fri, 07 Feb 2020 11:51:13 +0000
parents 2cd0e3b3e1fd
children
rev   line source
Chris@42 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
Chris@42 2 <html>
Chris@42 3 <!-- This manual is for FFTW
Chris@42 4 (version 3.3.5, 30 July 2016).
Chris@42 5
Chris@42 6 Copyright (C) 2003 Matteo Frigo.
Chris@42 7
Chris@42 8 Copyright (C) 2003 Massachusetts Institute of Technology.
Chris@42 9
Chris@42 10 Permission is granted to make and distribute verbatim copies of this
Chris@42 11 manual provided the copyright notice and this permission notice are
Chris@42 12 preserved on all copies.
Chris@42 13
Chris@42 14 Permission is granted to copy and distribute modified versions of this
Chris@42 15 manual under the conditions for verbatim copying, provided that the
Chris@42 16 entire resulting derived work is distributed under the terms of a
Chris@42 17 permission notice identical to this one.
Chris@42 18
Chris@42 19 Permission is granted to copy and distribute translations of this manual
Chris@42 20 into another language, under the above conditions for modified versions,
Chris@42 21 except that this permission notice may be stated in a translation
Chris@42 22 approved by the Free Software Foundation. -->
Chris@42 23 <!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ -->
Chris@42 24 <head>
Chris@42 25 <title>FFTW 3.3.5: The 1d Real-data DFT</title>
Chris@42 26
Chris@42 27 <meta name="description" content="FFTW 3.3.5: The 1d Real-data DFT">
Chris@42 28 <meta name="keywords" content="FFTW 3.3.5: The 1d Real-data DFT">
Chris@42 29 <meta name="resource-type" content="document">
Chris@42 30 <meta name="distribution" content="global">
Chris@42 31 <meta name="Generator" content="makeinfo">
Chris@42 32 <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
Chris@42 33 <link href="index.html#Top" rel="start" title="Top">
Chris@42 34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
Chris@42 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
Chris@42 36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
Chris@42 37 <link href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" rel="next" title="1d Real-even DFTs (DCTs)">
Chris@42 38 <link href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029" rel="prev" title="The 1d Discrete Fourier Transform (DFT)">
Chris@42 39 <style type="text/css">
Chris@42 40 <!--
Chris@42 41 a.summary-letter {text-decoration: none}
Chris@42 42 blockquote.smallquotation {font-size: smaller}
Chris@42 43 div.display {margin-left: 3.2em}
Chris@42 44 div.example {margin-left: 3.2em}
Chris@42 45 div.indentedblock {margin-left: 3.2em}
Chris@42 46 div.lisp {margin-left: 3.2em}
Chris@42 47 div.smalldisplay {margin-left: 3.2em}
Chris@42 48 div.smallexample {margin-left: 3.2em}
Chris@42 49 div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
Chris@42 50 div.smalllisp {margin-left: 3.2em}
Chris@42 51 kbd {font-style:oblique}
Chris@42 52 pre.display {font-family: inherit}
Chris@42 53 pre.format {font-family: inherit}
Chris@42 54 pre.menu-comment {font-family: serif}
Chris@42 55 pre.menu-preformatted {font-family: serif}
Chris@42 56 pre.smalldisplay {font-family: inherit; font-size: smaller}
Chris@42 57 pre.smallexample {font-size: smaller}
Chris@42 58 pre.smallformat {font-family: inherit; font-size: smaller}
Chris@42 59 pre.smalllisp {font-size: smaller}
Chris@42 60 span.nocodebreak {white-space:nowrap}
Chris@42 61 span.nolinebreak {white-space:nowrap}
Chris@42 62 span.roman {font-family:serif; font-weight:normal}
Chris@42 63 span.sansserif {font-family:sans-serif; font-weight:normal}
Chris@42 64 ul.no-bullet {list-style: none}
Chris@42 65 -->
Chris@42 66 </style>
Chris@42 67
Chris@42 68
Chris@42 69 </head>
Chris@42 70
Chris@42 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
Chris@42 72 <a name="The-1d-Real_002ddata-DFT"></a>
Chris@42 73 <div class="header">
Chris@42 74 <p>
Chris@42 75 Next: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="n" rel="next">1d Real-even DFTs (DCTs)</a>, Previous: <a href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029" accesskey="p" rel="prev">The 1d Discrete Fourier Transform (DFT)</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>
Chris@42 76 </div>
Chris@42 77 <hr>
Chris@42 78 <a name="The-1d-Real_002ddata-DFT-1"></a>
Chris@42 79 <h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>
Chris@42 80
Chris@42 81 <p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
Chris@42 82 <em>Y</em> of the size <code>n</code> real array <em>X</em>, exactly as defined
Chris@42 83 above, i.e.
Chris@42 84 <center><img src="equation-dft.png" align="top">.</center>This output array <em>Y</em> can easily be shown to possess the
Chris@42 85 &ldquo;Hermitian&rdquo; symmetry
Chris@42 86 <a name="index-Hermitian-1"></a>
Chris@42 87 <i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,where we take <em>Y</em> to be periodic so that
Chris@42 88 <i>Y<sub>n</sub> = Y</i><sub>0</sub>.</p>
Chris@42 89 <p>As a result of this symmetry, half of the output <em>Y</em> is redundant
Chris@42 90 (being the complex conjugate of the other half), and so the 1d r2c
Chris@42 91 transforms only output elements <em>0</em>&hellip;<em>n/2</em> of <em>Y</em>
Chris@42 92 (<em>n/2+1</em> complex numbers), where the division by <em>2</em> is
Chris@42 93 rounded down.
Chris@42 94 </p>
Chris@42 95 <p>Moreover, the Hermitian symmetry implies that
Chris@42 96 <i>Y</i><sub>0</sub>and, if <em>n</em> is even, the
Chris@42 97 <i>Y</i><sub><i>n</i>/2</sub>element, are purely real. So, for the <code>R2HC</code> r2r transform, the
Chris@42 98 halfcomplex format does not store the imaginary parts of these elements.
Chris@42 99 <a name="index-r2r-2"></a>
Chris@42 100 <a name="index-R2HC"></a>
Chris@42 101 <a name="index-halfcomplex-format-2"></a>
Chris@42 102 </p>
Chris@42 103
Chris@42 104 <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
Chris@42 105 <em>complex</em> array <em>X</em> with Hermitian symmetry, stored in the
Chris@42 106 r2c/<code>R2HC</code> output formats, respectively, where the backward
Chris@42 107 transform is defined exactly as for the complex case:
Chris@42 108 <center><img src="equation-idft.png" align="top">.</center>The outputs <code>Y</code> of this transform can easily be seen to be purely
Chris@42 109 real, and are stored as an array of real numbers.
Chris@42 110 </p>
Chris@42 111 <a name="index-normalization-9"></a>
Chris@42 112 <p>Like FFTW&rsquo;s complex DFT, these transforms are unnormalized. In other
Chris@42 113 words, applying the real-to-complex (forward) and then the
Chris@42 114 complex-to-real (backward) transform will multiply the input by
Chris@42 115 <em>n</em>.
Chris@42 116 </p>
Chris@42 117
Chris@42 118
Chris@42 119
Chris@42 120 </body>
Chris@42 121 </html>