mirror of
https://github.com/tildearrow/furnace.git
synced 2024-11-18 02:25:11 +00:00
168 lines
8.5 KiB
HTML
168 lines
8.5 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>Multi-Dimensional DFTs of Real Data (FFTW 3.3.10)</title>
|
||
|
|
|
||
|
|
<meta name="description" content="Multi-Dimensional DFTs of Real Data (FFTW 3.3.10)">
|
||
|
|
<meta name="keywords" content="Multi-Dimensional DFTs of Real Data (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="Tutorial.html" rel="up" title="Tutorial">
|
||
|
|
<link href="More-DFTs-of-Real-Data.html" rel="next" title="More DFTs of Real Data">
|
||
|
|
<link href="One_002dDimensional-DFTs-of-Real-Data.html" rel="prev" title="One-Dimensional 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="Multi_002dDimensional-DFTs-of-Real-Data"></span><div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="More-DFTs-of-Real-Data.html" accesskey="n" rel="next">More DFTs of Real Data</a>, Previous: <a href="One_002dDimensional-DFTs-of-Real-Data.html" accesskey="p" rel="prev">One-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html" accesskey="u" rel="up">Tutorial</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="Multi_002dDimensional-DFTs-of-Real-Data-1"></span><h3 class="section">2.4 Multi-Dimensional DFTs of Real Data</h3>
|
||
|
|
|
||
|
|
<p>Multi-dimensional DFTs of real data use the following planner routines:
|
||
|
|
</p>
|
||
|
|
<div class="example">
|
||
|
|
<pre class="example">fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
|
||
|
|
double *in, fftw_complex *out,
|
||
|
|
unsigned flags);
|
||
|
|
fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
|
||
|
|
double *in, fftw_complex *out,
|
||
|
|
unsigned flags);
|
||
|
|
fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
|
||
|
|
double *in, fftw_complex *out,
|
||
|
|
unsigned flags);
|
||
|
|
</pre></div>
|
||
|
|
<span id="index-fftw_005fplan_005fdft_005fr2c_005f2d"></span>
|
||
|
|
<span id="index-fftw_005fplan_005fdft_005fr2c_005f3d"></span>
|
||
|
|
<span id="index-fftw_005fplan_005fdft_005fr2c"></span>
|
||
|
|
|
||
|
|
<p>as well as the corresponding <code>c2r</code> routines with the input/output
|
||
|
|
types swapped. These routines work similarly to their complex
|
||
|
|
analogues, except for the fact that here the complex output array is cut
|
||
|
|
roughly in half and the real array requires padding for in-place
|
||
|
|
transforms (as in 1d, above).
|
||
|
|
</p>
|
||
|
|
<p>As before, <code>n</code> is the logical size of the array, and the
|
||
|
|
consequences of this on the the format of the complex arrays deserve
|
||
|
|
careful attention.
|
||
|
|
<span id="index-r2c_002fc2r-multi_002ddimensional-array-format"></span>
|
||
|
|
Suppose that the real data has dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
|
||
|
|
(in row-major order).
|
||
|
|
Then, after an r2c transform, the output is an n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × (n<sub>d-1</sub>/2 + 1)
|
||
|
|
array of
|
||
|
|
<code>fftw_complex</code> values in row-major order, corresponding to slightly
|
||
|
|
over half of the output of the corresponding complex DFT. (The division
|
||
|
|
is rounded down.) The ordering of the data is otherwise exactly the
|
||
|
|
same as in the complex-DFT case.
|
||
|
|
</p>
|
||
|
|
<p>For out-of-place transforms, this is the end of the story: the real
|
||
|
|
data is stored as a row-major array of size n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>
|
||
|
|
and the complex
|
||
|
|
data is stored as a row-major array of size n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × (n<sub>d-1</sub>/2 + 1)
|
||
|
|
.
|
||
|
|
</p>
|
||
|
|
<p>For in-place transforms, however, extra padding of the real-data array
|
||
|
|
is necessary because the complex array is larger than the real array,
|
||
|
|
and the two arrays share the same memory locations. Thus, for
|
||
|
|
in-place transforms, the final dimension of the real-data array must
|
||
|
|
be padded with extra values to accommodate the size of the complex
|
||
|
|
data—two values if the last dimension is even and one if it is odd.
|
||
|
|
<span id="index-padding-1"></span>
|
||
|
|
That is, the last dimension of the real data must physically contain
|
||
|
|
2 * (n<sub>d-1</sub>/2+1)
|
||
|
|
<code>double</code> values (exactly enough to hold the complex data).
|
||
|
|
This physical array size does not, however, change the <em>logical</em>
|
||
|
|
array size—only
|
||
|
|
n<sub>d-1</sub>
|
||
|
|
values are actually stored in the last dimension, and
|
||
|
|
n<sub>d-1</sub>
|
||
|
|
is the last dimension passed to the plan-creation routine.
|
||
|
|
</p>
|
||
|
|
<p>For example, consider the transform of a two-dimensional real array of
|
||
|
|
size <code>n0</code> by <code>n1</code>. The output of the r2c transform is a
|
||
|
|
two-dimensional complex array of size <code>n0</code> by <code>n1/2+1</code>, where
|
||
|
|
the <code>y</code> dimension has been cut nearly in half because of
|
||
|
|
redundancies in the output. Because <code>fftw_complex</code> is twice the
|
||
|
|
size of <code>double</code>, the output array is slightly bigger than the
|
||
|
|
input array. Thus, if we want to compute the transform in place, we
|
||
|
|
must <em>pad</em> the input array so that it is of size <code>n0</code> by
|
||
|
|
<code>2*(n1/2+1)</code>. If <code>n1</code> is even, then there are two padding
|
||
|
|
elements at the end of each row (which need not be initialized, as they
|
||
|
|
are only used for output).
|
||
|
|
</p>
|
||
|
|
<p>The following illustration depicts the input and output arrays just
|
||
|
|
described, for both the out-of-place and in-place transforms (with the
|
||
|
|
arrows indicating consecutive memory locations):
|
||
|
|
<img src="rfftwnd-for-html.png" alt="rfftwnd-for-html">
|
||
|
|
</p>
|
||
|
|
<p>These transforms are unnormalized, so an r2c followed by a c2r
|
||
|
|
transform (or vice versa) will result in the original data scaled by
|
||
|
|
the number of real data elements—that is, the product of the
|
||
|
|
(logical) dimensions of the real data.
|
||
|
|
<span id="index-normalization-1"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>(Because the last dimension is treated specially, if it is equal to
|
||
|
|
<code>1</code> the transform is <em>not</em> equivalent to a lower-dimensional
|
||
|
|
r2c/c2r transform. In that case, the last complex dimension also has
|
||
|
|
size <code>1</code> (<code>=1/2+1</code>), and no advantage is gained over the
|
||
|
|
complex transforms.)
|
||
|
|
</p>
|
||
|
|
<hr>
|
||
|
|
<div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="More-DFTs-of-Real-Data.html" accesskey="n" rel="next">More DFTs of Real Data</a>, Previous: <a href="One_002dDimensional-DFTs-of-Real-Data.html" accesskey="p" rel="prev">One-Dimensional DFTs of Real Data</a>, Up: <a href="Tutorial.html" accesskey="u" rel="up">Tutorial</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>
|