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<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual is for FFTW
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(version 3.3.10, 10 December 2020).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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Permission is granted to make and distribute verbatim copies of this
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manual provided the copyright notice and this permission notice are
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preserved on all copies.
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Permission is granted to copy and distribute modified versions of this
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manual under the conditions for verbatim copying, provided that the
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entire resulting derived work is distributed under the terms of a
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permission notice identical to this one.
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Permission is granted to copy and distribute translations of this manual
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into another language, under the above conditions for modified versions,
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except that this permission notice may be stated in a translation
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approved by the Free Software Foundation. -->
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<!-- Created by GNU Texinfo 6.7, http://www.gnu.org/software/texinfo/ -->
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>Complex One-Dimensional DFTs (FFTW 3.3.10)</title>
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<meta name="description" content="Complex One-Dimensional DFTs (FFTW 3.3.10)">
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<meta name="keywords" content="Complex One-Dimensional DFTs (FFTW 3.3.10)">
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<link href="index.html" rel="start" title="Top">
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<link href="Concept-Index.html" rel="index" title="Concept Index">
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<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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<link href="Tutorial.html" rel="up" title="Tutorial">
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<link href="Complex-Multi_002dDimensional-DFTs.html" rel="next" title="Complex Multi-Dimensional DFTs">
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</head>
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<body lang="en">
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<span id="Complex-One_002dDimensional-DFTs"></span><div class="header">
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<p>
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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>
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</div>
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<hr>
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<span id="Complex-One_002dDimensional-DFTs-1"></span><h3 class="section">2.1 Complex One-Dimensional DFTs</h3>
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<blockquote>
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<p>Plan: To bother about the best method of accomplishing an accidental result.
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[Ambrose Bierce, <cite>The Enlarged Devil’s Dictionary</cite>.]
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<span id="index-Devil"></span>
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</p></blockquote>
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<p>The basic usage of FFTW to compute a one-dimensional DFT of size
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<code>N</code> is simple, and it typically looks something like this code:
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</p>
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<div class="example">
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<pre class="example">#include <fftw3.h>
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...
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{
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fftw_complex *in, *out;
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fftw_plan p;
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...
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in = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
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out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
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p = fftw_plan_dft_1d(N, in, out, FFTW_FORWARD, FFTW_ESTIMATE);
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...
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fftw_execute(p); /* <span class="roman">repeat as needed</span> */
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...
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fftw_destroy_plan(p);
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fftw_free(in); fftw_free(out);
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}
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</pre></div>
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<p>You must link this code with the <code>fftw3</code> library. On Unix systems,
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link with <code>-lfftw3 -lm</code>.
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</p>
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<p>The example code first allocates the input and output arrays. You can
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allocate them in any way that you like, but we recommend using
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<code>fftw_malloc</code>, which behaves like
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<span id="index-fftw_005fmalloc"></span>
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<code>malloc</code> except that it properly aligns the array when SIMD
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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.]
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<span id="index-fftw_005falloc_005fcomplex"></span>
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<span id="index-SIMD"></span>
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</p>
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<p>The data is an array of type <code>fftw_complex</code>, which is by default a
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<code>double[2]</code> composed of the real (<code>in[i][0]</code>) and imaginary
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(<code>in[i][1]</code>) parts of a complex number.
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<span id="index-fftw_005fcomplex"></span>
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</p>
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<p>The next step is to create a <em>plan</em>, which is an object
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<span id="index-plan-1"></span>
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that contains all the data that FFTW needs to compute the FFT.
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This function creates the plan:
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</p>
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<div class="example">
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<pre class="example">fftw_plan fftw_plan_dft_1d(int n, fftw_complex *in, fftw_complex *out,
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int sign, unsigned flags);
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</pre></div>
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<span id="index-fftw_005fplan_005fdft_005f1d"></span>
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<span id="index-fftw_005fplan"></span>
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<p>The first argument, <code>n</code>, is the size of the transform you are
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trying to compute. The size <code>n</code> can be any positive integer, but
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sizes that are products of small factors are transformed most
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efficiently (although prime sizes still use an <i>O</i>(<i>n</i> log <i>n</i>)
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algorithm).
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</p>
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<p>The next two arguments are pointers to the input and output arrays of
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the transform. These pointers can be equal, indicating an
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<em>in-place</em> transform.
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<span id="index-in_002dplace"></span>
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</p>
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<p>The fourth argument, <code>sign</code>, can be either <code>FFTW_FORWARD</code>
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(<code>-1</code>) or <code>FFTW_BACKWARD</code> (<code>+1</code>),
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<span id="index-FFTW_005fFORWARD"></span>
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<span id="index-FFTW_005fBACKWARD"></span>
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and indicates the direction of the transform you are interested in;
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technically, it is the sign of the exponent in the transform.
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</p>
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<p>The <code>flags</code> argument is usually either <code>FFTW_MEASURE</code> or
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<span id="index-flags"></span>
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<code>FFTW_ESTIMATE</code>. <code>FFTW_MEASURE</code> instructs FFTW to run
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<span id="index-FFTW_005fMEASURE"></span>
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and measure the execution time of several FFTs in order to find the
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best way to compute the transform of size <code>n</code>. This process takes
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some time (usually a few seconds), depending on your machine and on
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the size of the transform. <code>FFTW_ESTIMATE</code>, on the contrary,
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does not run any computation and just builds a
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<span id="index-FFTW_005fESTIMATE"></span>
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reasonable plan that is probably sub-optimal. In short, if your
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program performs many transforms of the same size and initialization
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time is not important, use <code>FFTW_MEASURE</code>; otherwise use the
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estimate.
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</p>
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<p><em>You must create the plan before initializing the input</em>, because
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<code>FFTW_MEASURE</code> overwrites the <code>in</code>/<code>out</code> arrays.
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(Technically, <code>FFTW_ESTIMATE</code> does not touch your arrays, but you
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should always create plans first just to be sure.)
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</p>
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<p>Once the plan has been created, you can use it as many times as you
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like for transforms on the specified <code>in</code>/<code>out</code> arrays,
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computing the actual transforms via <code>fftw_execute(plan)</code>:
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</p><div class="example">
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<pre class="example">void fftw_execute(const fftw_plan plan);
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</pre></div>
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<span id="index-fftw_005fexecute"></span>
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<p>The DFT results are stored in-order in the array <code>out</code>, with the
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zero-frequency (DC) component in <code>out[0]</code>.
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<span id="index-frequency"></span>
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If <code>in != out</code>, the transform is <em>out-of-place</em> and the input
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array <code>in</code> is not modified. Otherwise, the input array is
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overwritten with the transform.
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</p>
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<span id="index-execute-1"></span>
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<p>If you want to transform a <em>different</em> array of the same size, you
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can create a new plan with <code>fftw_plan_dft_1d</code> and FFTW
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automatically reuses the information from the previous plan, if
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possible. Alternatively, with the “guru” interface you can apply a
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given plan to a different array, if you are careful.
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See <a href="FFTW-Reference.html">FFTW Reference</a>.
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</p>
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<p>When you are done with the plan, you deallocate it by calling
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<code>fftw_destroy_plan(plan)</code>:
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</p><div class="example">
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<pre class="example">void fftw_destroy_plan(fftw_plan plan);
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</pre></div>
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<span id="index-fftw_005fdestroy_005fplan"></span>
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<p>If you allocate an array with <code>fftw_malloc()</code> you must deallocate
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it with <code>fftw_free()</code>. Do not use <code>free()</code> or, heaven
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forbid, <code>delete</code>.
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<span id="index-fftw_005ffree"></span>
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</p>
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<p>FFTW computes an <em>unnormalized</em> DFT. Thus, computing a forward
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followed by a backward transform (or vice versa) results in the original
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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>.
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<span id="index-DFT-1"></span>
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<span id="index-normalization"></span>
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</p>
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<p>If you have a C compiler, such as <code>gcc</code>, that supports the
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C99 standard, and you <code>#include <complex.h></code> <em>before</em>
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<code><fftw3.h></code>, then <code>fftw_complex</code> is the native
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double-precision complex type and you can manipulate it with ordinary
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arithmetic. Otherwise, FFTW defines its own complex type, which is
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bit-compatible with the C99 complex type. See <a href="Complex-numbers.html">Complex numbers</a>.
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(The C++ <code><complex></code> template class may also be usable via a
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typecast.)
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<span id="index-C_002b_002b"></span>
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</p>
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<p>To use single or long-double precision versions of FFTW, replace the
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<code>fftw_</code> prefix by <code>fftwf_</code> or <code>fftwl_</code> and link with
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<code>-lfftw3f</code> or <code>-lfftw3l</code>, but use the <em>same</em>
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<code><fftw3.h></code> header file.
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<span id="index-precision"></span>
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</p>
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<p>Many more flags exist besides <code>FFTW_MEASURE</code> and
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<code>FFTW_ESTIMATE</code>. For example, use <code>FFTW_PATIENT</code> if you’re
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willing to wait even longer for a possibly even faster plan (see <a href="FFTW-Reference.html">FFTW Reference</a>).
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<span id="index-FFTW_005fPATIENT"></span>
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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>.
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</p>
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<hr>
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<div class="header">
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<p>
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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>
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</div>
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</body>
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</html>
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