240 lines
11 KiB
HTML
240 lines
11 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>Complex One-Dimensional DFTs (FFTW 3.3.10)</title>
|
||
|
|
|
||
|
|
<meta name="description" content="Complex One-Dimensional DFTs (FFTW 3.3.10)">
|
||
|
|
<meta name="keywords" content="Complex One-Dimensional DFTs (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="Complex-Multi_002dDimensional-DFTs.html" rel="next" title="Complex Multi-Dimensional DFTs">
|
||
|
|
<link href="Tutorial.html" rel="prev" title="Tutorial">
|
||
|
|
<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="Complex-One_002dDimensional-DFTs"></span><div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="Complex-Multi_002dDimensional-DFTs.html" accesskey="n" rel="next">Complex Multi-Dimensional DFTs</a>, Previous: <a href="Tutorial.html" accesskey="p" rel="prev">Tutorial</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="Complex-One_002dDimensional-DFTs-1"></span><h3 class="section">2.1 Complex One-Dimensional DFTs</h3>
|
||
|
|
|
||
|
|
<blockquote>
|
||
|
|
<p>Plan: To bother about the best method of accomplishing an accidental result.
|
||
|
|
[Ambrose Bierce, <cite>The Enlarged Devil’s Dictionary</cite>.]
|
||
|
|
<span id="index-Devil"></span>
|
||
|
|
</p></blockquote>
|
||
|
|
|
||
|
|
|
||
|
|
<p>The basic usage of FFTW to compute a one-dimensional DFT of size
|
||
|
|
<code>N</code> is simple, and it typically looks something like this code:
|
||
|
|
</p>
|
||
|
|
<div class="example">
|
||
|
|
<pre class="example">#include <fftw3.h>
|
||
|
|
...
|
||
|
|
{
|
||
|
|
fftw_complex *in, *out;
|
||
|
|
fftw_plan p;
|
||
|
|
...
|
||
|
|
in = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
|
||
|
|
out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
|
||
|
|
p = fftw_plan_dft_1d(N, in, out, FFTW_FORWARD, FFTW_ESTIMATE);
|
||
|
|
...
|
||
|
|
fftw_execute(p); /* <span class="roman">repeat as needed</span> */
|
||
|
|
...
|
||
|
|
fftw_destroy_plan(p);
|
||
|
|
fftw_free(in); fftw_free(out);
|
||
|
|
}
|
||
|
|
</pre></div>
|
||
|
|
|
||
|
|
<p>You must link this code with the <code>fftw3</code> library. On Unix systems,
|
||
|
|
link with <code>-lfftw3 -lm</code>.
|
||
|
|
</p>
|
||
|
|
<p>The example code first allocates the input and output arrays. You can
|
||
|
|
allocate them in any way that you like, but we recommend using
|
||
|
|
<code>fftw_malloc</code>, which behaves like
|
||
|
|
<span id="index-fftw_005fmalloc"></span>
|
||
|
|
<code>malloc</code> except that it properly aligns the array when SIMD
|
||
|
|
instructions (such as SSE and Altivec) are available (see <a href="SIMD-alignment-and-fftw_005fmalloc.html">SIMD alignment and fftw_malloc</a>). [Alternatively, we provide a convenient wrapper function <code>fftw_alloc_complex(N)</code> which has the same effect.]
|
||
|
|
<span id="index-fftw_005falloc_005fcomplex"></span>
|
||
|
|
<span id="index-SIMD"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>The data is an array of type <code>fftw_complex</code>, which is by default a
|
||
|
|
<code>double[2]</code> composed of the real (<code>in[i][0]</code>) and imaginary
|
||
|
|
(<code>in[i][1]</code>) parts of a complex number.
|
||
|
|
<span id="index-fftw_005fcomplex"></span>
|
||
|
|
</p>
|
||
|
|
<p>The next step is to create a <em>plan</em>, which is an object
|
||
|
|
<span id="index-plan-1"></span>
|
||
|
|
that contains all the data that FFTW needs to compute the FFT.
|
||
|
|
This function creates the plan:
|
||
|
|
</p>
|
||
|
|
<div class="example">
|
||
|
|
<pre class="example">fftw_plan fftw_plan_dft_1d(int n, fftw_complex *in, fftw_complex *out,
|
||
|
|
int sign, unsigned flags);
|
||
|
|
</pre></div>
|
||
|
|
<span id="index-fftw_005fplan_005fdft_005f1d"></span>
|
||
|
|
<span id="index-fftw_005fplan"></span>
|
||
|
|
|
||
|
|
<p>The first argument, <code>n</code>, is the size of the transform you are
|
||
|
|
trying to compute. The size <code>n</code> can be any positive integer, but
|
||
|
|
sizes that are products of small factors are transformed most
|
||
|
|
efficiently (although prime sizes still use an <i>O</i>(<i>n</i> log <i>n</i>)
|
||
|
|
algorithm).
|
||
|
|
</p>
|
||
|
|
<p>The next two arguments are pointers to the input and output arrays of
|
||
|
|
the transform. These pointers can be equal, indicating an
|
||
|
|
<em>in-place</em> transform.
|
||
|
|
<span id="index-in_002dplace"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>The fourth argument, <code>sign</code>, can be either <code>FFTW_FORWARD</code>
|
||
|
|
(<code>-1</code>) or <code>FFTW_BACKWARD</code> (<code>+1</code>),
|
||
|
|
<span id="index-FFTW_005fFORWARD"></span>
|
||
|
|
<span id="index-FFTW_005fBACKWARD"></span>
|
||
|
|
and indicates the direction of the transform you are interested in;
|
||
|
|
technically, it is the sign of the exponent in the transform.
|
||
|
|
</p>
|
||
|
|
<p>The <code>flags</code> argument is usually either <code>FFTW_MEASURE</code> or
|
||
|
|
<span id="index-flags"></span>
|
||
|
|
<code>FFTW_ESTIMATE</code>. <code>FFTW_MEASURE</code> instructs FFTW to run
|
||
|
|
<span id="index-FFTW_005fMEASURE"></span>
|
||
|
|
and measure the execution time of several FFTs in order to find the
|
||
|
|
best way to compute the transform of size <code>n</code>. This process takes
|
||
|
|
some time (usually a few seconds), depending on your machine and on
|
||
|
|
the size of the transform. <code>FFTW_ESTIMATE</code>, on the contrary,
|
||
|
|
does not run any computation and just builds a
|
||
|
|
<span id="index-FFTW_005fESTIMATE"></span>
|
||
|
|
reasonable plan that is probably sub-optimal. In short, if your
|
||
|
|
program performs many transforms of the same size and initialization
|
||
|
|
time is not important, use <code>FFTW_MEASURE</code>; otherwise use the
|
||
|
|
estimate.
|
||
|
|
</p>
|
||
|
|
<p><em>You must create the plan before initializing the input</em>, because
|
||
|
|
<code>FFTW_MEASURE</code> overwrites the <code>in</code>/<code>out</code> arrays.
|
||
|
|
(Technically, <code>FFTW_ESTIMATE</code> does not touch your arrays, but you
|
||
|
|
should always create plans first just to be sure.)
|
||
|
|
</p>
|
||
|
|
<p>Once the plan has been created, you can use it as many times as you
|
||
|
|
like for transforms on the specified <code>in</code>/<code>out</code> arrays,
|
||
|
|
computing the actual transforms via <code>fftw_execute(plan)</code>:
|
||
|
|
</p><div class="example">
|
||
|
|
<pre class="example">void fftw_execute(const fftw_plan plan);
|
||
|
|
</pre></div>
|
||
|
|
<span id="index-fftw_005fexecute"></span>
|
||
|
|
|
||
|
|
<p>The DFT results are stored in-order in the array <code>out</code>, with the
|
||
|
|
zero-frequency (DC) component in <code>out[0]</code>.
|
||
|
|
<span id="index-frequency"></span>
|
||
|
|
If <code>in != out</code>, the transform is <em>out-of-place</em> and the input
|
||
|
|
array <code>in</code> is not modified. Otherwise, the input array is
|
||
|
|
overwritten with the transform.
|
||
|
|
</p>
|
||
|
|
<span id="index-execute-1"></span>
|
||
|
|
<p>If you want to transform a <em>different</em> array of the same size, you
|
||
|
|
can create a new plan with <code>fftw_plan_dft_1d</code> and FFTW
|
||
|
|
automatically reuses the information from the previous plan, if
|
||
|
|
possible. Alternatively, with the “guru” interface you can apply a
|
||
|
|
given plan to a different array, if you are careful.
|
||
|
|
See <a href="FFTW-Reference.html">FFTW Reference</a>.
|
||
|
|
</p>
|
||
|
|
<p>When you are done with the plan, you deallocate it by calling
|
||
|
|
<code>fftw_destroy_plan(plan)</code>:
|
||
|
|
</p><div class="example">
|
||
|
|
<pre class="example">void fftw_destroy_plan(fftw_plan plan);
|
||
|
|
</pre></div>
|
||
|
|
<span id="index-fftw_005fdestroy_005fplan"></span>
|
||
|
|
<p>If you allocate an array with <code>fftw_malloc()</code> you must deallocate
|
||
|
|
it with <code>fftw_free()</code>. Do not use <code>free()</code> or, heaven
|
||
|
|
forbid, <code>delete</code>.
|
||
|
|
<span id="index-fftw_005ffree"></span>
|
||
|
|
</p>
|
||
|
|
<p>FFTW computes an <em>unnormalized</em> DFT. Thus, computing a forward
|
||
|
|
followed by a backward transform (or vice versa) results in the original
|
||
|
|
array scaled by <code>n</code>. For the definition of the DFT, see <a href="What-FFTW-Really-Computes.html">What FFTW Really Computes</a>.
|
||
|
|
<span id="index-DFT-1"></span>
|
||
|
|
<span id="index-normalization"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>If you have a C compiler, such as <code>gcc</code>, that supports the
|
||
|
|
C99 standard, and you <code>#include <complex.h></code> <em>before</em>
|
||
|
|
<code><fftw3.h></code>, then <code>fftw_complex</code> is the native
|
||
|
|
double-precision complex type and you can manipulate it with ordinary
|
||
|
|
arithmetic. Otherwise, FFTW defines its own complex type, which is
|
||
|
|
bit-compatible with the C99 complex type. See <a href="Complex-numbers.html">Complex numbers</a>.
|
||
|
|
(The C++ <code><complex></code> template class may also be usable via a
|
||
|
|
typecast.)
|
||
|
|
<span id="index-C_002b_002b"></span>
|
||
|
|
</p>
|
||
|
|
<p>To use single or long-double precision versions of FFTW, replace the
|
||
|
|
<code>fftw_</code> prefix by <code>fftwf_</code> or <code>fftwl_</code> and link with
|
||
|
|
<code>-lfftw3f</code> or <code>-lfftw3l</code>, but use the <em>same</em>
|
||
|
|
<code><fftw3.h></code> header file.
|
||
|
|
<span id="index-precision"></span>
|
||
|
|
</p>
|
||
|
|
|
||
|
|
<p>Many more flags exist besides <code>FFTW_MEASURE</code> and
|
||
|
|
<code>FFTW_ESTIMATE</code>. For example, use <code>FFTW_PATIENT</code> if you’re
|
||
|
|
willing to wait even longer for a possibly even faster plan (see <a href="FFTW-Reference.html">FFTW Reference</a>).
|
||
|
|
<span id="index-FFTW_005fPATIENT"></span>
|
||
|
|
You can also save plans for future use, as described by <a href="Words-of-Wisdom_002dSaving-Plans.html">Words of Wisdom-Saving Plans</a>.
|
||
|
|
</p>
|
||
|
|
<hr>
|
||
|
|
<div class="header">
|
||
|
|
<p>
|
||
|
|
Next: <a href="Complex-Multi_002dDimensional-DFTs.html" accesskey="n" rel="next">Complex Multi-Dimensional DFTs</a>, Previous: <a href="Tutorial.html" accesskey="p" rel="prev">Tutorial</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>
|