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author Chris Cannam <cannam@all-day-breakfast.com>
date Mon, 02 Mar 2020 14:03:47 +0000
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cannam@127 3 <!-- This manual is for FFTW
cannam@127 4 (version 3.3.5, 30 July 2016).
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cannam@127 6 Copyright (C) 2003 Matteo Frigo.
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cannam@127 24 <head>
cannam@127 25 <title>FFTW 3.3.5: Complex Multi-Dimensional DFTs</title>
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cannam@127 69 </head>
cannam@127 70
cannam@127 71 <body lang="en" bgcolor="#FFFFFF" text="#000000" link="#0000FF" vlink="#800080" alink="#FF0000">
cannam@127 72 <a name="Complex-Multi_002dDimensional-DFTs"></a>
cannam@127 73 <div class="header">
cannam@127 74 <p>
cannam@127 75 Next: <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">One-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-One_002dDimensional-DFTs.html#Complex-One_002dDimensional-DFTs" accesskey="p" rel="prev">Complex One-Dimensional DFTs</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>
cannam@127 76 </div>
cannam@127 77 <hr>
cannam@127 78 <a name="Complex-Multi_002dDimensional-DFTs-1"></a>
cannam@127 79 <h3 class="section">2.2 Complex Multi-Dimensional DFTs</h3>
cannam@127 80
cannam@127 81 <p>Multi-dimensional transforms work much the same way as one-dimensional
cannam@127 82 transforms: you allocate arrays of <code>fftw_complex</code> (preferably
cannam@127 83 using <code>fftw_malloc</code>), create an <code>fftw_plan</code>, execute it as
cannam@127 84 many times as you want with <code>fftw_execute(plan)</code>, and clean up
cannam@127 85 with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>).
cannam@127 86 </p>
cannam@127 87 <p>FFTW provides two routines for creating plans for 2d and 3d transforms,
cannam@127 88 and one routine for creating plans of arbitrary dimensionality.
cannam@127 89 The 2d and 3d routines have the following signature:
cannam@127 90 </p><div class="example">
cannam@127 91 <pre class="example">fftw_plan fftw_plan_dft_2d(int n0, int n1,
cannam@127 92 fftw_complex *in, fftw_complex *out,
cannam@127 93 int sign, unsigned flags);
cannam@127 94 fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
cannam@127 95 fftw_complex *in, fftw_complex *out,
cannam@127 96 int sign, unsigned flags);
cannam@127 97 </pre></div>
cannam@127 98 <a name="index-fftw_005fplan_005fdft_005f2d"></a>
cannam@127 99 <a name="index-fftw_005fplan_005fdft_005f3d"></a>
cannam@127 100
cannam@127 101 <p>These routines create plans for <code>n0</code> by <code>n1</code> two-dimensional
cannam@127 102 (2d) transforms and <code>n0</code> by <code>n1</code> by <code>n2</code> 3d transforms,
cannam@127 103 respectively. All of these transforms operate on contiguous arrays in
cannam@127 104 the C-standard <em>row-major</em> order, so that the last dimension has the
cannam@127 105 fastest-varying index in the array. This layout is described further in
cannam@127 106 <a href="Multi_002ddimensional-Array-Format.html#Multi_002ddimensional-Array-Format">Multi-dimensional Array Format</a>.
cannam@127 107 </p>
cannam@127 108 <p>FFTW can also compute transforms of higher dimensionality. In order to
cannam@127 109 avoid confusion between the various meanings of the the word
cannam@127 110 &ldquo;dimension&rdquo;, we use the term <em>rank</em>
cannam@127 111 <a name="index-rank"></a>
cannam@127 112 to denote the number of independent indices in an array.<a name="DOCF2" href="#FOOT2"><sup>2</sup></a> For
cannam@127 113 example, we say that a 2d transform has rank&nbsp;2, a 3d transform has
cannam@127 114 rank&nbsp;3, and so on. You can plan transforms of arbitrary rank by
cannam@127 115 means of the following function:
cannam@127 116 </p>
cannam@127 117 <div class="example">
cannam@127 118 <pre class="example">fftw_plan fftw_plan_dft(int rank, const int *n,
cannam@127 119 fftw_complex *in, fftw_complex *out,
cannam@127 120 int sign, unsigned flags);
cannam@127 121 </pre></div>
cannam@127 122 <a name="index-fftw_005fplan_005fdft"></a>
cannam@127 123
cannam@127 124 <p>Here, <code>n</code> is a pointer to an array <code>n[rank]</code> denoting an
cannam@127 125 <code>n[0]</code> by <code>n[1]</code> by &hellip; by <code>n[rank-1]</code> transform.
cannam@127 126 Thus, for example, the call
cannam@127 127 </p><div class="example">
cannam@127 128 <pre class="example">fftw_plan_dft_2d(n0, n1, in, out, sign, flags);
cannam@127 129 </pre></div>
cannam@127 130 <p>is equivalent to the following code fragment:
cannam@127 131 </p><div class="example">
cannam@127 132 <pre class="example">int n[2];
cannam@127 133 n[0] = n0;
cannam@127 134 n[1] = n1;
cannam@127 135 fftw_plan_dft(2, n, in, out, sign, flags);
cannam@127 136 </pre></div>
cannam@127 137 <p><code>fftw_plan_dft</code> is not restricted to 2d and 3d transforms,
cannam@127 138 however, but it can plan transforms of arbitrary rank.
cannam@127 139 </p>
cannam@127 140 <p>You may have noticed that all the planner routines described so far
cannam@127 141 have overlapping functionality. For example, you can plan a 1d or 2d
cannam@127 142 transform by using <code>fftw_plan_dft</code> with a <code>rank</code> of <code>1</code>
cannam@127 143 or <code>2</code>, or even by calling <code>fftw_plan_dft_3d</code> with <code>n0</code>
cannam@127 144 and/or <code>n1</code> equal to <code>1</code> (with no loss in efficiency). This
cannam@127 145 pattern continues, and FFTW&rsquo;s planning routines in general form a
cannam@127 146 &ldquo;partial order,&rdquo; sequences of
cannam@127 147 <a name="index-partial-order"></a>
cannam@127 148 interfaces with strictly increasing generality but correspondingly
cannam@127 149 greater complexity.
cannam@127 150 </p>
cannam@127 151 <p><code>fftw_plan_dft</code> is the most general complex-DFT routine that we
cannam@127 152 describe in this tutorial, but there are also the advanced and guru interfaces,
cannam@127 153 <a name="index-advanced-interface-1"></a>
cannam@127 154 <a name="index-guru-interface-1"></a>
cannam@127 155 which allow one to efficiently combine multiple/strided transforms
cannam@127 156 into a single FFTW plan, transform a subset of a larger
cannam@127 157 multi-dimensional array, and/or to handle more general complex-number
cannam@127 158 formats. For more information, see <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>.
cannam@127 159 </p>
cannam@127 160 <div class="footnote">
cannam@127 161 <hr>
cannam@127 162 <h4 class="footnotes-heading">Footnotes</h4>
cannam@127 163
cannam@127 164 <h3><a name="FOOT2" href="#DOCF2">(2)</a></h3>
cannam@127 165 <p>The
cannam@127 166 term &ldquo;rank&rdquo; is commonly used in the APL, FORTRAN, and Common Lisp
cannam@127 167 traditions, although it is not so common in the C&nbsp;world.</p>
cannam@127 168 </div>
cannam@127 169 <hr>
cannam@127 170 <div class="header">
cannam@127 171 <p>
cannam@127 172 Next: <a href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">One-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-One_002dDimensional-DFTs.html#Complex-One_002dDimensional-DFTs" accesskey="p" rel="prev">Complex One-Dimensional DFTs</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>
cannam@127 173 </div>
cannam@127 174
cannam@127 175
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