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author Chris Cannam
date Wed, 07 Sep 2016 10:40:32 +0100
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Chris@19 1 <html lang="en">
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Chris@19 3 <title>Complex Multi-Dimensional DFTs - FFTW 3.3.4</title>
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Chris@19 49 <a name="Complex-Multi-Dimensional-DFTs"></a>
Chris@19 50 <a name="Complex-Multi_002dDimensional-DFTs"></a>
Chris@19 51 <p>
Chris@19 52 Next:&nbsp;<a rel="next" accesskey="n" href="One_002dDimensional-DFTs-of-Real-Data.html#One_002dDimensional-DFTs-of-Real-Data">One-Dimensional DFTs of Real Data</a>,
Chris@19 53 Previous:&nbsp;<a rel="previous" accesskey="p" href="Complex-One_002dDimensional-DFTs.html#Complex-One_002dDimensional-DFTs">Complex One-Dimensional DFTs</a>,
Chris@19 54 Up:&nbsp;<a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a>
Chris@19 55 <hr>
Chris@19 56 </div>
Chris@19 57
Chris@19 58 <h3 class="section">2.2 Complex Multi-Dimensional DFTs</h3>
Chris@19 59
Chris@19 60 <p>Multi-dimensional transforms work much the same way as one-dimensional
Chris@19 61 transforms: you allocate arrays of <code>fftw_complex</code> (preferably
Chris@19 62 using <code>fftw_malloc</code>), create an <code>fftw_plan</code>, execute it as
Chris@19 63 many times as you want with <code>fftw_execute(plan)</code>, and clean up
Chris@19 64 with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>).
Chris@19 65
Chris@19 66 <p>FFTW provides two routines for creating plans for 2d and 3d transforms,
Chris@19 67 and one routine for creating plans of arbitrary dimensionality.
Chris@19 68 The 2d and 3d routines have the following signature:
Chris@19 69 <pre class="example"> fftw_plan fftw_plan_dft_2d(int n0, int n1,
Chris@19 70 fftw_complex *in, fftw_complex *out,
Chris@19 71 int sign, unsigned flags);
Chris@19 72 fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
Chris@19 73 fftw_complex *in, fftw_complex *out,
Chris@19 74 int sign, unsigned flags);
Chris@19 75 </pre>
Chris@19 76 <p><a name="index-fftw_005fplan_005fdft_005f2d-39"></a><a name="index-fftw_005fplan_005fdft_005f3d-40"></a>
Chris@19 77 These routines create plans for <code>n0</code> by <code>n1</code> two-dimensional
Chris@19 78 (2d) transforms and <code>n0</code> by <code>n1</code> by <code>n2</code> 3d transforms,
Chris@19 79 respectively. All of these transforms operate on contiguous arrays in
Chris@19 80 the C-standard <dfn>row-major</dfn> order, so that the last dimension has the
Chris@19 81 fastest-varying index in the array. This layout is described further in
Chris@19 82 <a href="Multi_002ddimensional-Array-Format.html#Multi_002ddimensional-Array-Format">Multi-dimensional Array Format</a>.
Chris@19 83
Chris@19 84 <p>FFTW can also compute transforms of higher dimensionality. In order to
Chris@19 85 avoid confusion between the various meanings of the the word
Chris@19 86 &ldquo;dimension&rdquo;, we use the term <em>rank</em>
Chris@19 87 <a name="index-rank-41"></a>to denote the number of independent indices in an array.<a rel="footnote" href="#fn-1" name="fnd-1"><sup>1</sup></a> For
Chris@19 88 example, we say that a 2d transform has rank&nbsp;2, a 3d transform has
Chris@19 89 rank&nbsp;3, and so on. You can plan transforms of arbitrary rank by
Chris@19 90 means of the following function:
Chris@19 91
Chris@19 92 <pre class="example"> fftw_plan fftw_plan_dft(int rank, const int *n,
Chris@19 93 fftw_complex *in, fftw_complex *out,
Chris@19 94 int sign, unsigned flags);
Chris@19 95 </pre>
Chris@19 96 <p><a name="index-fftw_005fplan_005fdft-42"></a>
Chris@19 97 Here, <code>n</code> is a pointer to an array <code>n[rank]</code> denoting an
Chris@19 98 <code>n[0]</code> by <code>n[1]</code> by <small class="dots">...</small> by <code>n[rank-1]</code> transform.
Chris@19 99 Thus, for example, the call
Chris@19 100 <pre class="example"> fftw_plan_dft_2d(n0, n1, in, out, sign, flags);
Chris@19 101 </pre>
Chris@19 102 <p>is equivalent to the following code fragment:
Chris@19 103 <pre class="example"> int n[2];
Chris@19 104 n[0] = n0;
Chris@19 105 n[1] = n1;
Chris@19 106 fftw_plan_dft(2, n, in, out, sign, flags);
Chris@19 107 </pre>
Chris@19 108 <p><code>fftw_plan_dft</code> is not restricted to 2d and 3d transforms,
Chris@19 109 however, but it can plan transforms of arbitrary rank.
Chris@19 110
Chris@19 111 <p>You may have noticed that all the planner routines described so far
Chris@19 112 have overlapping functionality. For example, you can plan a 1d or 2d
Chris@19 113 transform by using <code>fftw_plan_dft</code> with a <code>rank</code> of <code>1</code>
Chris@19 114 or <code>2</code>, or even by calling <code>fftw_plan_dft_3d</code> with <code>n0</code>
Chris@19 115 and/or <code>n1</code> equal to <code>1</code> (with no loss in efficiency). This
Chris@19 116 pattern continues, and FFTW's planning routines in general form a
Chris@19 117 &ldquo;partial order,&rdquo; sequences of
Chris@19 118 <a name="index-partial-order-43"></a>interfaces with strictly increasing generality but correspondingly
Chris@19 119 greater complexity.
Chris@19 120
Chris@19 121 <p><code>fftw_plan_dft</code> is the most general complex-DFT routine that we
Chris@19 122 describe in this tutorial, but there are also the advanced and guru interfaces,
Chris@19 123 <a name="index-advanced-interface-44"></a><a name="index-guru-interface-45"></a>which allow one to efficiently combine multiple/strided transforms
Chris@19 124 into a single FFTW plan, transform a subset of a larger
Chris@19 125 multi-dimensional array, and/or to handle more general complex-number
Chris@19 126 formats. For more information, see <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>.
Chris@19 127
Chris@19 128 <!-- -->
Chris@19 129 <div class="footnote">
Chris@19 130 <hr>
Chris@19 131 <h4>Footnotes</h4><p class="footnote"><small>[<a name="fn-1" href="#fnd-1">1</a>]</small> The
Chris@19 132 term &ldquo;rank&rdquo; is commonly used in the APL, FORTRAN, and Common Lisp
Chris@19 133 traditions, although it is not so common in the C&nbsp;world.</p>
Chris@19 134
Chris@19 135 <hr></div>
Chris@19 136
Chris@19 137 </body></html>
Chris@19 138