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3 <title>Complex Multi-Dimensional DFTs - FFTW 3.3.3</title>
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51 <p>
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>,
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57
58 <h3 class="section">2.2 Complex Multi-Dimensional DFTs</h3>
59
60 <p>Multi-dimensional transforms work much the same way as one-dimensional
61 transforms: you allocate arrays of <code>fftw_complex</code> (preferably
62 using <code>fftw_malloc</code>), create an <code>fftw_plan</code>, execute it as
63 many times as you want with <code>fftw_execute(plan)</code>, and clean up
64 with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>).
65
66 <p>FFTW provides two routines for creating plans for 2d and 3d transforms,
67 and one routine for creating plans of arbitrary dimensionality.
68 The 2d and 3d routines have the following signature:
69 <pre class="example"> fftw_plan fftw_plan_dft_2d(int n0, int n1,
70 fftw_complex *in, fftw_complex *out,
71 int sign, unsigned flags);
72 fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
73 fftw_complex *in, fftw_complex *out,
74 int sign, unsigned flags);
75 </pre>
76 <p><a name="index-fftw_005fplan_005fdft_005f2d-39"></a><a name="index-fftw_005fplan_005fdft_005f3d-40"></a>
77 These routines create plans for <code>n0</code> by <code>n1</code> two-dimensional
78 (2d) transforms and <code>n0</code> by <code>n1</code> by <code>n2</code> 3d transforms,
79 respectively. All of these transforms operate on contiguous arrays in
80 the C-standard <dfn>row-major</dfn> order, so that the last dimension has the
81 fastest-varying index in the array. This layout is described further in
82 <a href="Multi_002ddimensional-Array-Format.html#Multi_002ddimensional-Array-Format">Multi-dimensional Array Format</a>.
83
84 <p>FFTW can also compute transforms of higher dimensionality. In order to
85 avoid confusion between the various meanings of the the word
86 &ldquo;dimension&rdquo;, we use the term <em>rank</em>
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
88 example, we say that a 2d transform has rank&nbsp;2, a 3d transform has
89 rank&nbsp;3, and so on. You can plan transforms of arbitrary rank by
90 means of the following function:
91
92 <pre class="example"> fftw_plan fftw_plan_dft(int rank, const int *n,
93 fftw_complex *in, fftw_complex *out,
94 int sign, unsigned flags);
95 </pre>
96 <p><a name="index-fftw_005fplan_005fdft-42"></a>
97 Here, <code>n</code> is a pointer to an array <code>n[rank]</code> denoting an
98 <code>n[0]</code> by <code>n[1]</code> by <small class="dots">...</small> by <code>n[rank-1]</code> transform.
99 Thus, for example, the call
100 <pre class="example"> fftw_plan_dft_2d(n0, n1, in, out, sign, flags);
101 </pre>
102 <p>is equivalent to the following code fragment:
103 <pre class="example"> int n[2];
104 n[0] = n0;
105 n[1] = n1;
106 fftw_plan_dft(2, n, in, out, sign, flags);
107 </pre>
108 <p><code>fftw_plan_dft</code> is not restricted to 2d and 3d transforms,
109 however, but it can plan transforms of arbitrary rank.
110
111 <p>You may have noticed that all the planner routines described so far
112 have overlapping functionality. For example, you can plan a 1d or 2d
113 transform by using <code>fftw_plan_dft</code> with a <code>rank</code> of <code>1</code>
114 or <code>2</code>, or even by calling <code>fftw_plan_dft_3d</code> with <code>n0</code>
115 and/or <code>n1</code> equal to <code>1</code> (with no loss in efficiency). This
116 pattern continues, and FFTW's planning routines in general form a
117 &ldquo;partial order,&rdquo; sequences of
118 <a name="index-partial-order-43"></a>interfaces with strictly increasing generality but correspondingly
119 greater complexity.
120
121 <p><code>fftw_plan_dft</code> is the most general complex-DFT routine that we
122 describe in this tutorial, but there are also the advanced and guru interfaces,
123 <a name="index-advanced-interface-44"></a><a name="index-guru-interface-45"></a>which allow one to efficiently combine multiple/strided transforms
124 into a single FFTW plan, transform a subset of a larger
125 multi-dimensional array, and/or to handle more general complex-number
126 formats. For more information, see <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>.
127
128 <!-- -->
129 <div class="footnote">
130 <hr>
131 <h4>Footnotes</h4><p class="footnote"><small>[<a name="fn-1" href="#fnd-1">1</a>]</small> The
132 term &ldquo;rank&rdquo; is commonly used in the APL, FORTRAN, and Common Lisp
133 traditions, although it is not so common in the C&nbsp;world.</p>
134
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