Mercurial > hg > sv-dependency-builds

annotate src/fftw-3.3.5/doc/html/The-1d-Real_002ddata-DFT.html @ 169:223a55898ab9 tip default

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Add null config files
author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
parents 7867fa7e1b6b
children
rev   line source
cannam@127 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
cannam@127 2 <html>
cannam@127 3 <!-- This manual is for FFTW
cannam@127 4 (version 3.3.5, 30 July 2016).
cannam@127 5
cannam@127 6 Copyright (C) 2003 Matteo Frigo.
cannam@127 7
cannam@127 8 Copyright (C) 2003 Massachusetts Institute of Technology.
cannam@127 9
cannam@127 10 Permission is granted to make and distribute verbatim copies of this
cannam@127 11 manual provided the copyright notice and this permission notice are
cannam@127 12 preserved on all copies.
cannam@127 13
cannam@127 14 Permission is granted to copy and distribute modified versions of this
cannam@127 15 manual under the conditions for verbatim copying, provided that the
cannam@127 16 entire resulting derived work is distributed under the terms of a
cannam@127 17 permission notice identical to this one.
cannam@127 18
cannam@127 19 Permission is granted to copy and distribute translations of this manual
cannam@127 20 into another language, under the above conditions for modified versions,
cannam@127 21 except that this permission notice may be stated in a translation
cannam@127 22 approved by the Free Software Foundation. -->
cannam@127 23 <!-- Created by GNU Texinfo 5.2, http://www.gnu.org/software/texinfo/ -->
cannam@127 24 <head>
cannam@127 25 <title>FFTW 3.3.5: The 1d Real-data DFT</title>
cannam@127 26
cannam@127 27 <meta name="description" content="FFTW 3.3.5: The 1d Real-data DFT">
cannam@127 28 <meta name="keywords" content="FFTW 3.3.5: The 1d Real-data DFT">
cannam@127 29 <meta name="resource-type" content="document">
cannam@127 30 <meta name="distribution" content="global">
cannam@127 31 <meta name="Generator" content="makeinfo">
cannam@127 32 <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
cannam@127 33 <link href="index.html#Top" rel="start" title="Top">
cannam@127 34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
cannam@127 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
cannam@127 36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
cannam@127 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)">
cannam@127 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)">
cannam@127 39 <style type="text/css">
cannam@127 40 <!--
cannam@127 41 a.summary-letter {text-decoration: none}
cannam@127 42 blockquote.smallquotation {font-size: smaller}
cannam@127 43 div.display {margin-left: 3.2em}
cannam@127 44 div.example {margin-left: 3.2em}
cannam@127 45 div.indentedblock {margin-left: 3.2em}
cannam@127 46 div.lisp {margin-left: 3.2em}
cannam@127 47 div.smalldisplay {margin-left: 3.2em}
cannam@127 48 div.smallexample {margin-left: 3.2em}
cannam@127 49 div.smallindentedblock {margin-left: 3.2em; font-size: smaller}
cannam@127 50 div.smalllisp {margin-left: 3.2em}
cannam@127 51 kbd {font-style:oblique}
cannam@127 52 pre.display {font-family: inherit}
cannam@127 53 pre.format {font-family: inherit}
cannam@127 54 pre.menu-comment {font-family: serif}
cannam@127 55 pre.menu-preformatted {font-family: serif}
cannam@127 56 pre.smalldisplay {font-family: inherit; font-size: smaller}
cannam@127 57 pre.smallexample {font-size: smaller}
cannam@127 58 pre.smallformat {font-family: inherit; font-size: smaller}
cannam@127 59 pre.smalllisp {font-size: smaller}
cannam@127 60 span.nocodebreak {white-space:nowrap}
cannam@127 61 span.nolinebreak {white-space:nowrap}
cannam@127 62 span.roman {font-family:serif; font-weight:normal}
cannam@127 63 span.sansserif {font-family:sans-serif; font-weight:normal}
cannam@127 64 ul.no-bullet {list-style: none}
cannam@127 65 -->
cannam@127 66 </style>
cannam@127 67
cannam@127 68
cannam@127 69 </head>
cannam@127 70
cannam@127 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
cannam@127 72 <a name="The-1d-Real_002ddata-DFT"></a>
cannam@127 73 <div class="header">
cannam@127 74 <p>
cannam@127 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>
cannam@127 76 </div>
cannam@127 77 <hr>
cannam@127 78 <a name="The-1d-Real_002ddata-DFT-1"></a>
cannam@127 79 <h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>
cannam@127 80
cannam@127 81 <p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
cannam@127 82 <em>Y</em> of the size <code>n</code> real array <em>X</em>, exactly as defined
cannam@127 83 above, i.e.
cannam@127 84 <center><img src="equation-dft.png" align="top">.</center>This output array <em>Y</em> can easily be shown to possess the
cannam@127 85 &ldquo;Hermitian&rdquo; symmetry
cannam@127 86 <a name="index-Hermitian-1"></a>
cannam@127 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
cannam@127 88 <i>Y<sub>n</sub> = Y</i><sub>0</sub>.</p>
cannam@127 89 <p>As a result of this symmetry, half of the output <em>Y</em> is redundant
cannam@127 90 (being the complex conjugate of the other half), and so the 1d r2c
cannam@127 91 transforms only output elements <em>0</em>&hellip;<em>n/2</em> of <em>Y</em>
cannam@127 92 (<em>n/2+1</em> complex numbers), where the division by <em>2</em> is
cannam@127 93 rounded down.
cannam@127 94 </p>
cannam@127 95 <p>Moreover, the Hermitian symmetry implies that
cannam@127 96 <i>Y</i><sub>0</sub>and, if <em>n</em> is even, the
cannam@127 97 <i>Y</i><sub><i>n</i>/2</sub>element, are purely real. So, for the <code>R2HC</code> r2r transform, the
cannam@127 98 halfcomplex format does not store the imaginary parts of these elements.
cannam@127 99 <a name="index-r2r-2"></a>
cannam@127 100 <a name="index-R2HC"></a>
cannam@127 101 <a name="index-halfcomplex-format-2"></a>
cannam@127 102 </p>
cannam@127 103
cannam@127 104 <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
cannam@127 105 <em>complex</em> array <em>X</em> with Hermitian symmetry, stored in the
cannam@127 106 r2c/<code>R2HC</code> output formats, respectively, where the backward
cannam@127 107 transform is defined exactly as for the complex case:
cannam@127 108 <center><img src="equation-idft.png" align="top">.</center>The outputs <code>Y</code> of this transform can easily be seen to be purely
cannam@127 109 real, and are stored as an array of real numbers.
cannam@127 110 </p>
cannam@127 111 <a name="index-normalization-9"></a>
cannam@127 112 <p>Like FFTW&rsquo;s complex DFT, these transforms are unnormalized. In other
cannam@127 113 words, applying the real-to-complex (forward) and then the
cannam@127 114 complex-to-real (backward) transform will multiply the input by
cannam@127 115 <em>n</em>.
cannam@127 116 </p>
cannam@127 117
cannam@127 118
cannam@127 119
cannam@127 120 </body>
cannam@127 121 </html>