mirror of
https://github.com/tildearrow/furnace.git
synced 2024-12-04 18:27:25 +00:00
136 lines
6.9 KiB
HTML
136 lines
6.9 KiB
HTML
<!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 3.3.10, 10 December 2020).
|
|
|
|
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
|
|
preserved on all copies.
|
|
|
|
Permission is granted to copy and distribute modified versions of this
|
|
manual under the conditions for verbatim copying, provided that the
|
|
entire resulting derived work is distributed under the terms of a
|
|
permission notice identical to this one.
|
|
|
|
Permission is granted to copy and distribute translations of this manual
|
|
into another language, under the above conditions for modified versions,
|
|
except that this permission notice may be stated in a translation
|
|
approved by the Free Software Foundation. -->
|
|
<!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
|
|
<head>
|
|
<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
|
|
<title>The Halfcomplex-format DFT (FFTW 3.3.10)</title>
|
|
|
|
<meta name="description" content="The Halfcomplex-format DFT (FFTW 3.3.10)">
|
|
<meta name="keywords" content="The Halfcomplex-format DFT (FFTW 3.3.10)">
|
|
<meta name="resource-type" content="document">
|
|
<meta name="distribution" content="global">
|
|
<meta name="Generator" content="makeinfo">
|
|
<link href="index.html" rel="start" title="Top">
|
|
<link href="Concept-Index.html" rel="index" title="Concept Index">
|
|
<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
|
|
<link href="More-DFTs-of-Real-Data.html" rel="up" title="More DFTs of Real Data">
|
|
<link href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" rel="next" title="Real even/odd DFTs (cosine/sine transforms)">
|
|
<link href="More-DFTs-of-Real-Data.html" rel="prev" title="More DFTs of Real Data">
|
|
<style type="text/css">
|
|
<!--
|
|
a.summary-letter {text-decoration: none}
|
|
blockquote.indentedblock {margin-right: 0em}
|
|
div.display {margin-left: 3.2em}
|
|
div.example {margin-left: 3.2em}
|
|
div.lisp {margin-left: 3.2em}
|
|
kbd {font-style: oblique}
|
|
pre.display {font-family: inherit}
|
|
pre.format {font-family: inherit}
|
|
pre.menu-comment {font-family: serif}
|
|
pre.menu-preformatted {font-family: serif}
|
|
span.nolinebreak {white-space: nowrap}
|
|
span.roman {font-family: initial; font-weight: normal}
|
|
span.sansserif {font-family: sans-serif; font-weight: normal}
|
|
ul.no-bullet {list-style: none}
|
|
-->
|
|
</style>
|
|
|
|
|
|
</head>
|
|
|
|
<body lang="en">
|
|
<span id="The-Halfcomplex_002dformat-DFT"></span><div class="header">
|
|
<p>
|
|
Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html" accesskey="u" rel="up">More DFTs of Real Data</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>
|
|
</div>
|
|
<hr>
|
|
<span id="The-Halfcomplex_002dformat-DFT-1"></span><h4 class="subsection">2.5.1 The Halfcomplex-format DFT</h4>
|
|
|
|
<p>An r2r kind of <code>FFTW_R2HC</code> (<em>r2hc</em>) corresponds to an r2c DFT
|
|
<span id="index-FFTW_005fR2HC"></span>
|
|
<span id="index-r2c-1"></span>
|
|
<span id="index-r2hc"></span>
|
|
(see <a href="One_002dDimensional-DFTs-of-Real-Data.html">One-Dimensional DFTs of Real Data</a>) but with “halfcomplex”
|
|
format output, and may sometimes be faster and/or more convenient than
|
|
the latter.
|
|
<span id="index-halfcomplex-format-1"></span>
|
|
The inverse <em>hc2r</em> transform is of kind <code>FFTW_HC2R</code>.
|
|
<span id="index-FFTW_005fHC2R"></span>
|
|
<span id="index-hc2r"></span>
|
|
This consists of the non-redundant half of the complex output for a 1d
|
|
real-input DFT of size <code>n</code>, stored as a sequence of <code>n</code> real
|
|
numbers (<code>double</code>) in the format:
|
|
</p>
|
|
<p align=center>
|
|
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>
|
|
</p>
|
|
|
|
<p>Here,
|
|
r<sub>k</sub>
|
|
is the real part of the <em>k</em>th output, and
|
|
i<sub>k</sub>
|
|
is the imaginary part. (Division by 2 is rounded down.) For a
|
|
halfcomplex array <code>hc[n]</code>, the <em>k</em>th component thus has its
|
|
real part in <code>hc[k]</code> and its imaginary part in <code>hc[n-k]</code>, with
|
|
the exception of <code>k</code> <code>==</code> <code>0</code> or <code>n/2</code> (the latter
|
|
only if <code>n</code> is even)—in these two cases, the imaginary part is
|
|
zero due to symmetries of the real-input DFT, and is not stored.
|
|
Thus, the r2hc transform of <code>n</code> real values is a halfcomplex array of
|
|
length <code>n</code>, and vice versa for hc2r.
|
|
<span id="index-normalization-2"></span>
|
|
</p>
|
|
|
|
<p>Aside from the differing format, the output of
|
|
<code>FFTW_R2HC</code>/<code>FFTW_HC2R</code> is otherwise exactly the same as for
|
|
the corresponding 1d r2c/c2r transform
|
|
(i.e. <code>FFTW_FORWARD</code>/<code>FFTW_BACKWARD</code> transforms, respectively).
|
|
Recall that these transforms are unnormalized, so r2hc followed by hc2r
|
|
will result in the original data multiplied by <code>n</code>. Furthermore,
|
|
like the c2r transform, an out-of-place hc2r transform will
|
|
<em>destroy its input</em> array.
|
|
</p>
|
|
<p>Although these halfcomplex transforms can be used with the
|
|
multi-dimensional r2r interface, the interpretation of such a separable
|
|
product of transforms along each dimension is problematic. For example,
|
|
consider a two-dimensional <code>n0</code> by <code>n1</code>, r2hc by r2hc
|
|
transform planned by <code>fftw_plan_r2r_2d(n0, n1, in, out, FFTW_R2HC,
|
|
FFTW_R2HC, FFTW_MEASURE)</code>. Conceptually, FFTW first transforms the rows
|
|
(of size <code>n1</code>) to produce halfcomplex rows, and then transforms the
|
|
columns (of size <code>n0</code>). Half of these column transforms, however,
|
|
are of imaginary parts, and should therefore be multiplied by <em>i</em>
|
|
and combined with the r2hc transforms of the real columns to produce the
|
|
2d DFT amplitudes; FFTW’s r2r transform does <em>not</em> perform this
|
|
combination for you. Thus, if a multi-dimensional real-input/output DFT
|
|
is required, we recommend using the ordinary r2c/c2r
|
|
interface (see <a href="Multi_002dDimensional-DFTs-of-Real-Data.html">Multi-Dimensional DFTs of Real Data</a>).
|
|
</p>
|
|
<hr>
|
|
<div class="header">
|
|
<p>
|
|
Next: <a href="Real-even_002fodd-DFTs-_0028cosine_002fsine-transforms_0029.html" accesskey="n" rel="next">Real even/odd DFTs (cosine/sine transforms)</a>, Previous: <a href="More-DFTs-of-Real-Data.html" accesskey="p" rel="prev">More DFTs of Real Data</a>, Up: <a href="More-DFTs-of-Real-Data.html" accesskey="u" rel="up">More DFTs of Real Data</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>
|
|
</div>
|
|
|
|
|
|
|
|
</body>
|
|
</html>
|