Mercurial > hg > sv-dependency-builds

annotate src/fftw-3.3.8/doc/html/The-Halfcomplex_002dformat-DFT.html @ 167:bd3cc4d1df30

Find changesets by keywords (author, files, the commit message), revision number or hash, or revset expression.
Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 19 Nov 2019 14:52:55 +0000
parents
children
rev   line source
cannam@167 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
cannam@167 2 <html>
cannam@167 3 <!-- This manual is for FFTW
cannam@167 4 (version 3.3.8, 24 May 2018).
cannam@167 5
cannam@167 6 Copyright (C) 2003 Matteo Frigo.
cannam@167 7
cannam@167 8 Copyright (C) 2003 Massachusetts Institute of Technology.
cannam@167 9
cannam@167 10 Permission is granted to make and distribute verbatim copies of this
cannam@167 11 manual provided the copyright notice and this permission notice are
cannam@167 12 preserved on all copies.
cannam@167 13
cannam@167 14 Permission is granted to copy and distribute modified versions of this
cannam@167 15 manual under the conditions for verbatim copying, provided that the
cannam@167 16 entire resulting derived work is distributed under the terms of a
cannam@167 17 permission notice identical to this one.
cannam@167 18
cannam@167 19 Permission is granted to copy and distribute translations of this manual
cannam@167 20 into another language, under the above conditions for modified versions,
cannam@167 21 except that this permission notice may be stated in a translation
cannam@167 22 approved by the Free Software Foundation. -->
cannam@167 23 <!-- Created by GNU Texinfo 6.3, http://www.gnu.org/software/texinfo/ -->
cannam@167 24 <head>
cannam@167 25 <title>FFTW 3.3.8: The Halfcomplex-format DFT</title>
cannam@167 26
cannam@167 27 <meta name="description" content="FFTW 3.3.8: The Halfcomplex-format DFT">
cannam@167 28 <meta name="keywords" content="FFTW 3.3.8: The Halfcomplex-format DFT">
cannam@167 29 <meta name="resource-type" content="document">
cannam@167 30 <meta name="distribution" content="global">
cannam@167 31 <meta name="Generator" content="makeinfo">
cannam@167 32 <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
cannam@167 33 <link href="index.html#Top" rel="start" title="Top">
cannam@167 34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
cannam@167 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
cannam@167 36 <link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="up" title="More DFTs of Real Data">
cannam@167 37 <link href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" rel="next" title="Real even/odd DFTs (cosine/sine transforms)">
cannam@167 38 <link href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" rel="prev" title="More DFTs of Real Data">
cannam@167 39 <style type="text/css">
cannam@167 40 <!--
cannam@167 41 a.summary-letter {text-decoration: none}
cannam@167 42 blockquote.indentedblock {margin-right: 0em}
cannam@167 43 blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
cannam@167 44 blockquote.smallquotation {font-size: smaller}
cannam@167 45 div.display {margin-left: 3.2em}
cannam@167 46 div.example {margin-left: 3.2em}
cannam@167 47 div.lisp {margin-left: 3.2em}
cannam@167 48 div.smalldisplay {margin-left: 3.2em}
cannam@167 49 div.smallexample {margin-left: 3.2em}
cannam@167 50 div.smalllisp {margin-left: 3.2em}
cannam@167 51 kbd {font-style: oblique}
cannam@167 52 pre.display {font-family: inherit}
cannam@167 53 pre.format {font-family: inherit}
cannam@167 54 pre.menu-comment {font-family: serif}
cannam@167 55 pre.menu-preformatted {font-family: serif}
cannam@167 56 pre.smalldisplay {font-family: inherit; font-size: smaller}
cannam@167 57 pre.smallexample {font-size: smaller}
cannam@167 58 pre.smallformat {font-family: inherit; font-size: smaller}
cannam@167 59 pre.smalllisp {font-size: smaller}
cannam@167 60 span.nolinebreak {white-space: nowrap}
cannam@167 61 span.roman {font-family: initial; font-weight: normal}
cannam@167 62 span.sansserif {font-family: sans-serif; font-weight: normal}
cannam@167 63 ul.no-bullet {list-style: none}
cannam@167 64 -->
cannam@167 65 </style>
cannam@167 66
cannam@167 67
cannam@167 68 </head>
cannam@167 69
cannam@167 70 <body lang="en">
cannam@167 71 <a name="The-Halfcomplex_002dformat-DFT"></a>
cannam@167 72 <div class="header">
cannam@167 73 <p>
cannam@167 74 Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</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@167 75 </div>
cannam@167 76 <hr>
cannam@167 77 <a name="The-Halfcomplex_002dformat-DFT-1"></a>
cannam@167 78 <h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
cannam@167 79
cannam@167 80 <p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
cannam@167 81 <a name="index-FFTW_005fR2HC"></a>
cannam@167 82 <a name="index-r2c-1"></a>
cannam@167 83 <a name="index-r2hc"></a>
cannam@167 84 (see <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data">One-Dimensional DFTs of Real Data</a>) but with &ldquo;halfcomplex&rdquo;
cannam@167 85 format output, and may sometimes be faster and/or more convenient than
cannam@167 86 the latter.
cannam@167 87 <a name="index-halfcomplex-format-1"></a>
cannam@167 88 The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
cannam@167 89 <a name="index-FFTW_005fHC2R"></a>
cannam@167 90 <a name="index-hc2r"></a>
cannam@167 91 This consists of the non-redundant half of the complex output for a 1d
cannam@167 92 real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
cannam@167 93 numbers (<code>double</code>) in the format:
cannam@167 94 </p>
cannam@167 95 <p align=center>
cannam@167 96 r<sub>0</sub>, r<sub>1</sub>, r<sub>2</sub>, ..., r<sub>n/2</sub>, i<sub>(n+1)/2-1</sub>, ..., i<sub>2</sub>, i<sub>1</sub>
cannam@167 97 </p>
cannam@167 98
cannam@167 99 <p>Here,
cannam@167 100 r<sub>k</sub>
cannam@167 101 is the real part of the <em>k</em>th output, and
cannam@167 102 i<sub>k</sub>
cannam@167 103 is the imaginary part. (Division by 2 is rounded down.) For a
cannam@167 104 halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its
cannam@167 105 real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
cannam@167 106 the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
cannam@167 107 only if <code>n</code> is even)&mdash;in these two cases, the imaginary part is
cannam@167 108 zero due to symmetries of the real-input DFT, and is not stored.
cannam@167 109 Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
cannam@167 110 length <code>n</code>, and vice versa for hc2r.
cannam@167 111 <a name="index-normalization-2"></a>
cannam@167 112 </p>
cannam@167 113
cannam@167 114 <p>Aside from the differing format, the output of
cannam@167 115 <code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
cannam@167 116 the corresponding 1d r2c/c2r transform
cannam@167 117 (i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
cannam@167 118 Recall that these transforms are unnormalized, so r2hc followed by hc2r
cannam@167 119 will result in the original data multiplied by <code>n</code>. Furthermore,
cannam@167 120 like the c2r transform, an out-of-place hc2r transform will
cannam@167 121 <em>destroy its input</em> array.
cannam@167 122 </p>
cannam@167 123 <p>Although these halfcomplex transforms can be used with the
cannam@167 124 multi-dimensional r2r interface, the interpretation of such a separable
cannam@167 125 product of transforms along each dimension is problematic. For example,
cannam@167 126 consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
cannam@167 127 transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
cannam@167 128 FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
cannam@167 129 (of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
cannam@167 130 columns (of size <code>n0</code>). Half of these column transforms, however,
cannam@167 131 are of imaginary parts, and should therefore be multiplied by <em>i</em>
cannam@167 132 and combined with the r2hc transforms of the real columns to produce the
cannam@167 133 2d DFT amplitudes; FFTW&rsquo;s r2r transform does <em>not</em> perform this
cannam@167 134 combination for you. Thus, if a multi-dimensional real-input/output DFT
cannam@167 135 is required, we recommend using the ordinary r2c/c2r
cannam@167 136 interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>).
cannam@167 137 </p>
cannam@167 138 <hr>
cannam@167 139 <div class="header">
cannam@167 140 <p>
cannam@167 141 Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html#Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html#More-DFTs-of-Real-Data" accesskey="u" rel="up">More DFTs of Real Data</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@167 142 </div>
cannam@167 143
cannam@167 144
cannam@167 145
cannam@167 146 </body>
cannam@167 147 </html>