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Replace these with versions built using an older toolset (so as to avoid ABI compatibilities when linking on Ubuntu 14.04 for packaging purposes)
author Chris Cannam
date Fri, 07 Feb 2020 11:51:13 +0000
parents 37bf6b4a2645
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Chris@10 3 <title>Complex One-Dimensional DFTs - FFTW 3.3.3</title>
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Chris@10 13 This manual is for FFTW
Chris@10 14 (version 3.3.3, 25 November 2012).
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Chris@10 16 Copyright (C) 2003 Matteo Frigo.
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Chris@10 48 <div class="node">
Chris@10 49 <a name="Complex-One-Dimensional-DFTs"></a>
Chris@10 50 <a name="Complex-One_002dDimensional-DFTs"></a>
Chris@10 51 <p>
Chris@10 52 Next:&nbsp;<a rel="next" accesskey="n" href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs">Complex Multi-Dimensional DFTs</a>,
Chris@10 53 Previous:&nbsp;<a rel="previous" accesskey="p" href="Tutorial.html#Tutorial">Tutorial</a>,
Chris@10 54 Up:&nbsp;<a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a>
Chris@10 55 <hr>
Chris@10 56 </div>
Chris@10 57
Chris@10 58 <h3 class="section">2.1 Complex One-Dimensional DFTs</h3>
Chris@10 59
Chris@10 60 <blockquote>
Chris@10 61 Plan: To bother about the best method of accomplishing an accidental result.
Chris@10 62 [Ambrose Bierce, <cite>The Enlarged Devil's Dictionary</cite>.]
Chris@10 63 <a name="index-Devil-15"></a></blockquote>
Chris@10 64
Chris@10 65 <p>The basic usage of FFTW to compute a one-dimensional DFT of size
Chris@10 66 <code>N</code> is simple, and it typically looks something like this code:
Chris@10 67
Chris@10 68 <pre class="example"> #include &lt;fftw3.h&gt;
Chris@10 69 ...
Chris@10 70 {
Chris@10 71 fftw_complex *in, *out;
Chris@10 72 fftw_plan p;
Chris@10 73 ...
Chris@10 74 in = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
Chris@10 75 out = (fftw_complex*) fftw_malloc(sizeof(fftw_complex) * N);
Chris@10 76 p = fftw_plan_dft_1d(N, in, out, FFTW_FORWARD, FFTW_ESTIMATE);
Chris@10 77 ...
Chris@10 78 fftw_execute(p); /* <span class="roman">repeat as needed</span> */
Chris@10 79 ...
Chris@10 80 fftw_destroy_plan(p);
Chris@10 81 fftw_free(in); fftw_free(out);
Chris@10 82 }
Chris@10 83 </pre>
Chris@10 84 <p>You must link this code with the <code>fftw3</code> library. On Unix systems,
Chris@10 85 link with <code>-lfftw3 -lm</code>.
Chris@10 86
Chris@10 87 <p>The example code first allocates the input and output arrays. You can
Chris@10 88 allocate them in any way that you like, but we recommend using
Chris@10 89 <code>fftw_malloc</code>, which behaves like
Chris@10 90 <a name="index-fftw_005fmalloc-16"></a><code>malloc</code> except that it properly aligns the array when SIMD
Chris@10 91 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@10 92 <a name="index-fftw_005falloc_005fcomplex-17"></a><a name="index-SIMD-18"></a>
Chris@10 93
Chris@10 94 <p>The data is an array of type <code>fftw_complex</code>, which is by default a
Chris@10 95 <code>double[2]</code> composed of the real (<code>in[i][0]</code>) and imaginary
Chris@10 96 (<code>in[i][1]</code>) parts of a complex number.
Chris@10 97 <a name="index-fftw_005fcomplex-19"></a>
Chris@10 98 The next step is to create a <dfn>plan</dfn>, which is an object
Chris@10 99 <a name="index-plan-20"></a>that contains all the data that FFTW needs to compute the FFT.
Chris@10 100 This function creates the plan:
Chris@10 101
Chris@10 102 <pre class="example"> fftw_plan fftw_plan_dft_1d(int n, fftw_complex *in, fftw_complex *out,
Chris@10 103 int sign, unsigned flags);
Chris@10 104 </pre>
Chris@10 105 <p><a name="index-fftw_005fplan_005fdft_005f1d-21"></a><a name="index-fftw_005fplan-22"></a>
Chris@10 106 The first argument, <code>n</code>, is the size of the transform you are
Chris@10 107 trying to compute. The size <code>n</code> can be any positive integer, but
Chris@10 108 sizes that are products of small factors are transformed most
Chris@10 109 efficiently (although prime sizes still use an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>) algorithm).
Chris@10 110
Chris@10 111 <p>The next two arguments are pointers to the input and output arrays of
Chris@10 112 the transform. These pointers can be equal, indicating an
Chris@10 113 <dfn>in-place</dfn> transform.
Chris@10 114 <a name="index-in_002dplace-23"></a>
Chris@10 115
Chris@10 116 <p>The fourth argument, <code>sign</code>, can be either <code>FFTW_FORWARD</code>
Chris@10 117 (<code>-1</code>) or <code>FFTW_BACKWARD</code> (<code>+1</code>),
Chris@10 118 <a name="index-FFTW_005fFORWARD-24"></a><a name="index-FFTW_005fBACKWARD-25"></a>and indicates the direction of the transform you are interested in;
Chris@10 119 technically, it is the sign of the exponent in the transform.
Chris@10 120
Chris@10 121 <p>The <code>flags</code> argument is usually either <code>FFTW_MEASURE</code> or
Chris@10 122 <a name="index-flags-26"></a><code>FFTW_ESTIMATE</code>. <code>FFTW_MEASURE</code> instructs FFTW to run
Chris@10 123 <a name="index-FFTW_005fMEASURE-27"></a>and measure the execution time of several FFTs in order to find the
Chris@10 124 best way to compute the transform of size <code>n</code>. This process takes
Chris@10 125 some time (usually a few seconds), depending on your machine and on
Chris@10 126 the size of the transform. <code>FFTW_ESTIMATE</code>, on the contrary,
Chris@10 127 does not run any computation and just builds a
Chris@10 128 <a name="index-FFTW_005fESTIMATE-28"></a>reasonable plan that is probably sub-optimal. In short, if your
Chris@10 129 program performs many transforms of the same size and initialization
Chris@10 130 time is not important, use <code>FFTW_MEASURE</code>; otherwise use the
Chris@10 131 estimate.
Chris@10 132
Chris@10 133 <p><em>You must create the plan before initializing the input</em>, because
Chris@10 134 <code>FFTW_MEASURE</code> overwrites the <code>in</code>/<code>out</code> arrays.
Chris@10 135 (Technically, <code>FFTW_ESTIMATE</code> does not touch your arrays, but you
Chris@10 136 should always create plans first just to be sure.)
Chris@10 137
Chris@10 138 <p>Once the plan has been created, you can use it as many times as you
Chris@10 139 like for transforms on the specified <code>in</code>/<code>out</code> arrays,
Chris@10 140 computing the actual transforms via <code>fftw_execute(plan)</code>:
Chris@10 141 <pre class="example"> void fftw_execute(const fftw_plan plan);
Chris@10 142 </pre>
Chris@10 143 <p><a name="index-fftw_005fexecute-29"></a>
Chris@10 144 The DFT results are stored in-order in the array <code>out</code>, with the
Chris@10 145 zero-frequency (DC) component in <code>out[0]</code>.
Chris@10 146 <a name="index-frequency-30"></a>If <code>in != out</code>, the transform is <dfn>out-of-place</dfn> and the input
Chris@10 147 array <code>in</code> is not modified. Otherwise, the input array is
Chris@10 148 overwritten with the transform.
Chris@10 149
Chris@10 150 <p><a name="index-execute-31"></a>If you want to transform a <em>different</em> array of the same size, you
Chris@10 151 can create a new plan with <code>fftw_plan_dft_1d</code> and FFTW
Chris@10 152 automatically reuses the information from the previous plan, if
Chris@10 153 possible. Alternatively, with the &ldquo;guru&rdquo; interface you can apply a
Chris@10 154 given plan to a different array, if you are careful.
Chris@10 155 See <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>.
Chris@10 156
Chris@10 157 <p>When you are done with the plan, you deallocate it by calling
Chris@10 158 <code>fftw_destroy_plan(plan)</code>:
Chris@10 159 <pre class="example"> void fftw_destroy_plan(fftw_plan plan);
Chris@10 160 </pre>
Chris@10 161 <p><a name="index-fftw_005fdestroy_005fplan-32"></a>If you allocate an array with <code>fftw_malloc()</code> you must deallocate
Chris@10 162 it with <code>fftw_free()</code>. Do not use <code>free()</code> or, heaven
Chris@10 163 forbid, <code>delete</code>.
Chris@10 164 <a name="index-fftw_005ffree-33"></a>
Chris@10 165 FFTW computes an <em>unnormalized</em> DFT. Thus, computing a forward
Chris@10 166 followed by a backward transform (or vice versa) results in the original
Chris@10 167 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@10 168 <a name="index-DFT-34"></a><a name="index-normalization-35"></a>
Chris@10 169
Chris@10 170 <p>If you have a C compiler, such as <code>gcc</code>, that supports the
Chris@10 171 C99 standard, and you <code>#include &lt;complex.h&gt;</code> <em>before</em>
Chris@10 172 <code>&lt;fftw3.h&gt;</code>, then <code>fftw_complex</code> is the native
Chris@10 173 double-precision complex type and you can manipulate it with ordinary
Chris@10 174 arithmetic. Otherwise, FFTW defines its own complex type, which is
Chris@10 175 bit-compatible with the C99 complex type. See <a href="Complex-numbers.html#Complex-numbers">Complex numbers</a>.
Chris@10 176 (The C++ <code>&lt;complex&gt;</code> template class may also be usable via a
Chris@10 177 typecast.)
Chris@10 178 <a name="index-C_002b_002b-36"></a>
Chris@10 179 To use single or long-double precision versions of FFTW, replace the
Chris@10 180 <code>fftw_</code> prefix by <code>fftwf_</code> or <code>fftwl_</code> and link with
Chris@10 181 <code>-lfftw3f</code> or <code>-lfftw3l</code>, but use the <em>same</em>
Chris@10 182 <code>&lt;fftw3.h&gt;</code> header file.
Chris@10 183 <a name="index-precision-37"></a>
Chris@10 184
Chris@10 185 <p>Many more flags exist besides <code>FFTW_MEASURE</code> and
Chris@10 186 <code>FFTW_ESTIMATE</code>. For example, use <code>FFTW_PATIENT</code> if you're
Chris@10 187 willing to wait even longer for a possibly even faster plan (see <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>).
Chris@10 188 <a name="index-FFTW_005fPATIENT-38"></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>.
Chris@10 189
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