mirror of
https://github.com/tildearrow/furnace.git
synced 2024-11-06 12:55:05 +00:00
174 lines
9 KiB
HTML
174 lines
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>One-Dimensional DFTs of Real Data (FFTW 3.3.10)</title>
|
||
|
|
|
||
|
|
<meta name="description" content="One-Dimensional DFTs of Real Data (FFTW 3.3.10)">
|
||
|
|
<meta name="keywords" content="One-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="Multi_002dDimensional-DFTs-of-Real-Data.html" rel="next" title="Multi-Dimensional DFTs of Real Data">
|
||
|
|
<link href="Complex-Multi_002dDimensional-DFTs.html" rel="prev" title="Complex Multi-Dimensional DFTs">
|
||
|
|
<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="One_002dDimensional-DFTs-of-Real-Data"></span><div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html" accesskey="n" rel="next">Multi-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-Multi_002dDimensional-DFTs.html" accesskey="p" rel="prev">Complex Multi-Dimensional DFTs</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="One_002dDimensional-DFTs-of-Real-Data-1"></span><h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3>
|
||
|
|
|
||
|
|
<p>In many practical applications, the input data <code>in[i]</code> are purely
|
||
|
|
real numbers, in which case the DFT output satisfies the “Hermitian”
|
||
|
|
<span id="index-Hermitian"></span>
|
||
|
|
redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
|
||
|
|
possible to take advantage of these circumstances in order to achieve
|
||
|
|
roughly a factor of two improvement in both speed and memory usage.
|
||
|
|
</p>
|
||
|
|
<p>In exchange for these speed and space advantages, the user sacrifices
|
||
|
|
some of the simplicity of FFTW’s complex transforms. First of all, the
|
||
|
|
input and output arrays are of <em>different sizes and types</em>: the
|
||
|
|
input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
|
||
|
|
complex numbers (the non-redundant outputs); this also requires slight
|
||
|
|
“padding” of the input array for
|
||
|
|
<span id="index-padding"></span>
|
||
|
|
in-place transforms. Second, the inverse transform (complex to real)
|
||
|
|
has the side-effect of <em>overwriting its input array</em>, by default.
|
||
|
|
Neither of these inconveniences should pose a serious problem for
|
||
|
|
users, but it is important to be aware of them.
|
||
|
|
</p>
|
||
|
|
<p>The routines to perform real-data transforms are almost the same as
|
||
|
|
those for complex transforms: you allocate arrays of <code>double</code>
|
||
|
|
and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code> or
|
||
|
|
<code>fftw_alloc_complex</code>), create an <code>fftw_plan</code>, execute it as
|
||
|
|
many times as you want with <code>fftw_execute(plan)</code>, and clean up
|
||
|
|
with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
|
||
|
|
differences are that the input (or output) is of type <code>double</code>
|
||
|
|
and there are new routines to create the plan. In one dimension:
|
||
|
|
</p>
|
||
|
|
<div class="example">
|
||
|
|
<pre class="example">fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
|
||
|
|
unsigned flags);
|
||
|
|
fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
|
||
|
|
unsigned flags);
|
||
|
|
</pre></div>
|
||
|
|
<span id="index-fftw_005fplan_005fdft_005fr2c_005f1d"></span>
|
||
|
|
<span id="index-fftw_005fplan_005fdft_005fc2r_005f1d"></span>
|
||
|
|
|
||
|
|
<p>for the real input to complex-Hermitian output (<em>r2c</em>) and
|
||
|
|
complex-Hermitian input to real output (<em>c2r</em>) transforms.
|
||
|
|
<span id="index-r2c"></span>
|
||
|
|
<span id="index-c2r"></span>
|
||
|
|
Unlike the complex DFT planner, there is no <code>sign</code> argument.
|
||
|
|
Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
|
||
|
|
always <code>FFTW_BACKWARD</code>.
|
||
|
|
<span id="index-FFTW_005fFORWARD-1"></span>
|
||
|
|
<span id="index-FFTW_005fBACKWARD-1"></span>
|
||
|
|
(For single/long-double precision
|
||
|
|
<code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
|
||
|
|
<code>float</code> and <code>long double</code>, respectively.)
|
||
|
|
<span id="index-precision-1"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>Here, <code>n</code> is the “logical” size of the DFT, not necessarily the
|
||
|
|
physical size of the array. In particular, the real (<code>double</code>)
|
||
|
|
array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
|
||
|
|
array has <code>n/2+1</code> elements (where the division is rounded down).
|
||
|
|
For an in-place transform,
|
||
|
|
<span id="index-in_002dplace-1"></span>
|
||
|
|
<code>in</code> and <code>out</code> are aliased to the same array, which must be
|
||
|
|
big enough to hold both; so, the real array would actually have
|
||
|
|
<code>2*(n/2+1)</code> elements, where the elements beyond the first
|
||
|
|
<code>n</code> are unused padding. (Note that this is very different from
|
||
|
|
the concept of “zero-padding” a transform to a larger length, which
|
||
|
|
changes the logical size of the DFT by actually adding new input
|
||
|
|
data.) The <em>k</em>th element of the complex array is exactly the
|
||
|
|
same as the <em>k</em>th element of the corresponding complex DFT. All
|
||
|
|
positive <code>n</code> are supported; products of small factors are most
|
||
|
|
efficient, but an <i>O</i>(<i>n</i> log <i>n</i>)
|
||
|
|
algorithm is used even for prime sizes.
|
||
|
|
</p>
|
||
|
|
<p>As noted above, the c2r transform destroys its input array even for
|
||
|
|
out-of-place transforms. This can be prevented, if necessary, by
|
||
|
|
including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
|
||
|
|
unfortunately some sacrifice in performance.
|
||
|
|
<span id="index-flags-1"></span>
|
||
|
|
<span id="index-FFTW_005fPRESERVE_005fINPUT"></span>
|
||
|
|
This flag is also not currently supported for multi-dimensional real
|
||
|
|
DFTs (next section).
|
||
|
|
</p>
|
||
|
|
<p>Readers familiar with DFTs of real data will recall that the 0th (the
|
||
|
|
“DC”) and <code>n/2</code>-th (the “Nyquist” frequency, when <code>n</code> is
|
||
|
|
even) elements of the complex output are purely real. Some
|
||
|
|
implementations therefore store the Nyquist element where the DC
|
||
|
|
imaginary part would go, in order to make the input and output arrays
|
||
|
|
the same size. Such packing, however, does not generalize well to
|
||
|
|
multi-dimensional transforms, and the space savings are miniscule in
|
||
|
|
any case; FFTW does not support it.
|
||
|
|
</p>
|
||
|
|
<p>An alternative interface for one-dimensional r2c and c2r DFTs can be
|
||
|
|
found in the ‘<samp>r2r</samp>’ interface (see <a href="The-Halfcomplex_002dformat-DFT.html">The Halfcomplex-format DFT</a>), with “halfcomplex”-format output that <em>is</em> the same size
|
||
|
|
(and type) as the input array.
|
||
|
|
<span id="index-halfcomplex-format"></span>
|
||
|
|
That interface, although it is not very useful for multi-dimensional
|
||
|
|
transforms, may sometimes yield better performance.
|
||
|
|
</p>
|
||
|
|
<hr>
|
||
|
|
<div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html" accesskey="n" rel="next">Multi-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-Multi_002dDimensional-DFTs.html" accesskey="p" rel="prev">Complex Multi-Dimensional DFTs</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>
|