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Current fftw source
author Chris Cannam
date Tue, 18 Oct 2016 13:40:26 +0100
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Chris@42 3 <!-- This manual is for FFTW
Chris@42 4 (version 3.3.5, 30 July 2016).
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Chris@42 6 Copyright (C) 2003 Matteo Frigo.
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Chris@42 24 <head>
Chris@42 25 <title>FFTW 3.3.5: Complex One-Dimensional DFTs</title>
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Chris@42 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
Chris@42 72 <a name="Complex-One_002dDimensional-DFTs"></a>
Chris@42 73 <div class="header">
Chris@42 74 <p>
Chris@42 75 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> &nbsp; [<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>
Chris@42 76 </div>
Chris@42 77 <hr>
Chris@42 78 <a name="Complex-One_002dDimensional-DFTs-1"></a>
Chris@42 79 <h3 class="section">2.1 Complex One-Dimensional DFTs</h3>
Chris@42 80
Chris@42 81 <blockquote>
Chris@42 82 <p>Plan: To bother about the best method of accomplishing an accidental result.
Chris@42 83 [Ambrose Bierce, <cite>The Enlarged Devil&rsquo;s Dictionary</cite>.]
Chris@42 84 <a name="index-Devil"></a>
Chris@42 85 </p></blockquote>
Chris@42 86
Chris@42 87
Chris@42 88 <p>The basic usage of FFTW to compute a one-dimensional DFT of size
Chris@42 89 <code>N</code> is simple, and it typically looks something like this code:
Chris@42 90 </p>
Chris@42 91 <div class="example">
Chris@42 92 <pre class="example">#include &lt;fftw3.h&gt;
Chris@42 93 ...
Chris@42 94 {
Chris@42 95 fftw_complex *in, *out;
Chris@42 96 fftw_plan p;
Chris@42 97 ...
Chris@42 98 in = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
Chris@42 99 out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
Chris@42 100 p = fftw_plan_dft_1d(N, in, out, FFTW_FORWARD, FFTW_ESTIMATE);
Chris@42 101 ...
Chris@42 102 fftw_execute(p); /* <span class="roman">repeat as needed</span> */
Chris@42 103 ...
Chris@42 104 fftw_destroy_plan(p);
Chris@42 105 fftw_free(in); fftw_free(out);
Chris@42 106 }
Chris@42 107 </pre></div>
Chris@42 108
Chris@42 109 <p>You must link this code with the <code>fftw3</code> library. On Unix systems,
Chris@42 110 link with <code>-lfftw3 -lm</code>.
Chris@42 111 </p>
Chris@42 112 <p>The example code first allocates the input and output arrays. You can
Chris@42 113 allocate them in any way that you like, but we recommend using
Chris@42 114 <code>fftw_malloc</code>, which behaves like
Chris@42 115 <a name="index-fftw_005fmalloc"></a>
Chris@42 116 <code>malloc</code> except that it properly aligns the array when SIMD
Chris@42 117 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.]
Chris@42 118 <a name="index-fftw_005falloc_005fcomplex"></a>
Chris@42 119 <a name="index-SIMD"></a>
Chris@42 120 </p>
Chris@42 121
Chris@42 122 <p>The data is an array of type <code>fftw_complex</code>, which is by default a
Chris@42 123 <code>double[2]</code> composed of the real (<code>in[i][0]</code>) and imaginary
Chris@42 124 (<code>in[i][1]</code>) parts of a complex number.
Chris@42 125 <a name="index-fftw_005fcomplex"></a>
Chris@42 126 </p>
Chris@42 127 <p>The next step is to create a <em>plan</em>, which is an object
Chris@42 128 <a name="index-plan-1"></a>
Chris@42 129 that contains all the data that FFTW needs to compute the FFT.
Chris@42 130 This function creates the plan:
Chris@42 131 </p>
Chris@42 132 <div class="example">
Chris@42 133 <pre class="example">fftw_plan fftw_plan_dft_1d(int n, fftw_complex *in, fftw_complex *out,
Chris@42 134 int sign, unsigned flags);
Chris@42 135 </pre></div>
Chris@42 136 <a name="index-fftw_005fplan_005fdft_005f1d"></a>
Chris@42 137 <a name="index-fftw_005fplan"></a>
Chris@42 138
Chris@42 139 <p>The first argument, <code>n</code>, is the size of the transform you are
Chris@42 140 trying to compute. The size <code>n</code> can be any positive integer, but
Chris@42 141 sizes that are products of small factors are transformed most
Chris@42 142 efficiently (although prime sizes still use an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>) algorithm).
Chris@42 143 </p>
Chris@42 144 <p>The next two arguments are pointers to the input and output arrays of
Chris@42 145 the transform. These pointers can be equal, indicating an
Chris@42 146 <em>in-place</em> transform.
Chris@42 147 <a name="index-in_002dplace"></a>
Chris@42 148 </p>
Chris@42 149
Chris@42 150 <p>The fourth argument, <code>sign</code>, can be either <code>FFTW_FORWARD</code>
Chris@42 151 (<code>-1</code>) or <code>FFTW_BACKWARD</code> (<code>+1</code>),
Chris@42 152 <a name="index-FFTW_005fFORWARD"></a>
Chris@42 153 <a name="index-FFTW_005fBACKWARD"></a>
Chris@42 154 and indicates the direction of the transform you are interested in;
Chris@42 155 technically, it is the sign of the exponent in the transform.
Chris@42 156 </p>
Chris@42 157 <p>The <code>flags</code> argument is usually either <code>FFTW_MEASURE</code> or
Chris@42 158 <a name="index-flags"></a>
Chris@42 159 <code>FFTW_ESTIMATE</code>. <code>FFTW_MEASURE</code> instructs FFTW to run
Chris@42 160 <a name="index-FFTW_005fMEASURE"></a>
Chris@42 161 and measure the execution time of several FFTs in order to find the
Chris@42 162 best way to compute the transform of size <code>n</code>. This process takes
Chris@42 163 some time (usually a few seconds), depending on your machine and on
Chris@42 164 the size of the transform. <code>FFTW_ESTIMATE</code>, on the contrary,
Chris@42 165 does not run any computation and just builds a
Chris@42 166 <a name="index-FFTW_005fESTIMATE"></a>
Chris@42 167 reasonable plan that is probably sub-optimal. In short, if your
Chris@42 168 program performs many transforms of the same size and initialization
Chris@42 169 time is not important, use <code>FFTW_MEASURE</code>; otherwise use the
Chris@42 170 estimate.
Chris@42 171 </p>
Chris@42 172 <p><em>You must create the plan before initializing the input</em>, because
Chris@42 173 <code>FFTW_MEASURE</code> overwrites the <code>in</code>/<code>out</code> arrays.
Chris@42 174 (Technically, <code>FFTW_ESTIMATE</code> does not touch your arrays, but you
Chris@42 175 should always create plans first just to be sure.)
Chris@42 176 </p>
Chris@42 177 <p>Once the plan has been created, you can use it as many times as you
Chris@42 178 like for transforms on the specified <code>in</code>/<code>out</code> arrays,
Chris@42 179 computing the actual transforms via <code>fftw_execute(plan)</code>:
Chris@42 180 </p><div class="example">
Chris@42 181 <pre class="example">void fftw_execute(const fftw_plan plan);
Chris@42 182 </pre></div>
Chris@42 183 <a name="index-fftw_005fexecute"></a>
Chris@42 184
Chris@42 185 <p>The DFT results are stored in-order in the array <code>out</code>, with the
Chris@42 186 zero-frequency (DC) component in <code>out[0]</code>.
Chris@42 187 <a name="index-frequency"></a>
Chris@42 188 If <code>in != out</code>, the transform is <em>out-of-place</em> and the input
Chris@42 189 array <code>in</code> is not modified. Otherwise, the input array is
Chris@42 190 overwritten with the transform.
Chris@42 191 </p>
Chris@42 192 <a name="index-execute-1"></a>
Chris@42 193 <p>If you want to transform a <em>different</em> array of the same size, you
Chris@42 194 can create a new plan with <code>fftw_plan_dft_1d</code> and FFTW
Chris@42 195 automatically reuses the information from the previous plan, if
Chris@42 196 possible. Alternatively, with the &ldquo;guru&rdquo; interface you can apply a
Chris@42 197 given plan to a different array, if you are careful.
Chris@42 198 See <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>.
Chris@42 199 </p>
Chris@42 200 <p>When you are done with the plan, you deallocate it by calling
Chris@42 201 <code>fftw_destroy_plan(plan)</code>:
Chris@42 202 </p><div class="example">
Chris@42 203 <pre class="example">void fftw_destroy_plan(fftw_plan plan);
Chris@42 204 </pre></div>
Chris@42 205 <a name="index-fftw_005fdestroy_005fplan"></a>
Chris@42 206 <p>If you allocate an array with <code>fftw_malloc()</code> you must deallocate
Chris@42 207 it with <code>fftw_free()</code>. Do not use <code>free()</code> or, heaven
Chris@42 208 forbid, <code>delete</code>.
Chris@42 209 <a name="index-fftw_005ffree"></a>
Chris@42 210 </p>
Chris@42 211 <p>FFTW computes an <em>unnormalized</em> DFT. Thus, computing a forward
Chris@42 212 followed by a backward transform (or vice versa) results in the original
Chris@42 213 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>.
Chris@42 214 <a name="index-DFT-1"></a>
Chris@42 215 <a name="index-normalization"></a>
Chris@42 216 </p>
Chris@42 217
Chris@42 218 <p>If you have a C compiler, such as <code>gcc</code>, that supports the
Chris@42 219 C99 standard, and you <code>#include &lt;complex.h&gt;</code> <em>before</em>
Chris@42 220 <code>&lt;fftw3.h&gt;</code>, then <code>fftw_complex</code> is the native
Chris@42 221 double-precision complex type and you can manipulate it with ordinary
Chris@42 222 arithmetic. Otherwise, FFTW defines its own complex type, which is
Chris@42 223 bit-compatible with the C99 complex type. See <a href="Complex-numbers.html#Complex-numbers">Complex numbers</a>.
Chris@42 224 (The C++ <code>&lt;complex&gt;</code> template class may also be usable via a
Chris@42 225 typecast.)
Chris@42 226 <a name="index-C_002b_002b"></a>
Chris@42 227 </p>
Chris@42 228 <p>To use single or long-double precision versions of FFTW, replace the
Chris@42 229 <code>fftw_</code> prefix by <code>fftwf_</code> or <code>fftwl_</code> and link with
Chris@42 230 <code>-lfftw3f</code> or <code>-lfftw3l</code>, but use the <em>same</em>
Chris@42 231 <code>&lt;fftw3.h&gt;</code> header file.
Chris@42 232 <a name="index-precision"></a>
Chris@42 233 </p>
Chris@42 234
Chris@42 235 <p>Many more flags exist besides <code>FFTW_MEASURE</code> and
Chris@42 236 <code>FFTW_ESTIMATE</code>. For example, use <code>FFTW_PATIENT</code> if you&rsquo;re
Chris@42 237 willing to wait even longer for a possibly even faster plan (see <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>).
Chris@42 238 <a name="index-FFTW_005fPATIENT"></a>
Chris@42 239 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>.
Chris@42 240 </p>
Chris@42 241 <hr>
Chris@42 242 <div class="header">
Chris@42 243 <p>
Chris@42 244 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> &nbsp; [<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>
Chris@42 245 </div>
Chris@42 246
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