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Current fftw source
| author | Chris Cannam <cannam@all-day-breakfast.com> |
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| date | Tue, 18 Oct 2016 13:40:26 +0100 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.5/doc/html/Complex-One_002dDimensional-DFTs.html Tue Oct 18 13:40:26 2016 +0100 @@ -0,0 +1,250 @@ +<!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 3.3.5, 30 July 2016). + +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 5.2, http://www.gnu.org/software/texinfo/ --> +<head> +<title>FFTW 3.3.5: Complex One-Dimensional DFTs</title> + +<meta name="description" content="FFTW 3.3.5: Complex One-Dimensional DFTs"> +<meta name="keywords" content="FFTW 3.3.5: Complex One-Dimensional DFTs"> +<meta name="resource-type" content="document"> +<meta name="distribution" content="global"> +<meta name="Generator" content="makeinfo"> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8"> +<link href="index.html#Top" rel="start" title="Top"> +<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> +<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> +<link href="Tutorial.html#Tutorial" rel="up" title="Tutorial"> +<link href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" rel="next" title="Complex Multi-Dimensional DFTs"> +<link href="Tutorial.html#Tutorial" rel="prev" title="Tutorial"> +<style type="text/css"> +<!-- +a.summary-letter {text-decoration: none} +blockquote.smallquotation {font-size: smaller} +div.display {margin-left: 3.2em} +div.example {margin-left: 3.2em} +div.indentedblock {margin-left: 3.2em} +div.lisp {margin-left: 3.2em} +div.smalldisplay {margin-left: 3.2em} +div.smallexample {margin-left: 3.2em} +div.smallindentedblock {margin-left: 3.2em; font-size: smaller} +div.smalllisp {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} +pre.smalldisplay {font-family: inherit; font-size: smaller} +pre.smallexample {font-size: smaller} +pre.smallformat {font-family: inherit; font-size: smaller} +pre.smalllisp {font-size: smaller} +span.nocodebreak {white-space:nowrap} +span.nolinebreak {white-space:nowrap} +span.roman {font-family:serif; font-weight:normal} +span.sansserif {font-family:sans-serif; font-weight:normal} +ul.no-bullet {list-style: none} +--> +</style> + + +</head> + +<body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000"> +<a name="Complex-One_002dDimensional-DFTs"></a> +<div class="header"> +<p> +Next: <a href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" accesskey="n" rel="next">Complex Multi-Dimensional DFTs</a>, Previous: <a href="Tutorial.html#Tutorial" accesskey="p" rel="prev">Tutorial</a>, Up: <a href="Tutorial.html#Tutorial" 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#Concept-Index" title="Index" rel="index">Index</a>]</p> +</div> +<hr> +<a name="Complex-One_002dDimensional-DFTs-1"></a> +<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>.] +<a name="index-Devil"></a> +</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 +<a name="index-fftw_005fmalloc"></a> +<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_005fmalloc">SIMD alignment and fftw_malloc</a>). [Alternatively, we provide a convenient wrapper function <code>fftw_alloc_complex(N)</code> which has the same effect.] +<a name="index-fftw_005falloc_005fcomplex"></a> +<a name="index-SIMD"></a> +</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. +<a name="index-fftw_005fcomplex"></a> +</p> +<p>The next step is to create a <em>plan</em>, which is an object +<a name="index-plan-1"></a> +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> +<a name="index-fftw_005fplan_005fdft_005f1d"></a> +<a name="index-fftw_005fplan"></a> + +<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. +<a name="index-in_002dplace"></a> +</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>), +<a name="index-FFTW_005fFORWARD"></a> +<a name="index-FFTW_005fBACKWARD"></a> +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 +<a name="index-flags"></a> +<code>FFTW_ESTIMATE</code>. <code>FFTW_MEASURE</code> instructs FFTW to run +<a name="index-FFTW_005fMEASURE"></a> +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 +<a name="index-FFTW_005fESTIMATE"></a> +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> +<a name="index-fftw_005fexecute"></a> + +<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>. +<a name="index-frequency"></a> +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> +<a name="index-execute-1"></a> +<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">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> +<a name="index-fftw_005fdestroy_005fplan"></a> +<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>. +<a name="index-fftw_005ffree"></a> +</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">What FFTW Really Computes</a>. +<a name="index-DFT-1"></a> +<a name="index-normalization"></a> +</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">Complex numbers</a>. +(The C++ <code><complex></code> template class may also be usable via a +typecast.) +<a name="index-C_002b_002b"></a> +</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. +<a name="index-precision"></a> +</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">FFTW Reference</a>). +<a name="index-FFTW_005fPATIENT"></a> +You can also save plans for future use, as described by <a href="Words-of-Wisdom_002dSaving-Plans.html#Words-of-Wisdom_002dSaving-Plans">Words of Wisdom-Saving Plans</a>. +</p> +<hr> +<div class="header"> +<p> +Next: <a href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" accesskey="n" rel="next">Complex Multi-Dimensional DFTs</a>, Previous: <a href="Tutorial.html#Tutorial" accesskey="p" rel="prev">Tutorial</a>, Up: <a href="Tutorial.html#Tutorial" 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#Concept-Index" title="Index" rel="index">Index</a>]</p> +</div> + + + +</body> +</html>