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<h3 class="section">2.2 Complex Multi-Dimensional DFTs</h3>
<p>Multi-dimensional transforms work much the same way as one-dimensional
transforms: you allocate arrays of <code>fftw_complex</code> (preferably
using <code>fftw_malloc</code>), create an <code>fftw_plan</code>, execute it as
many times as you want with <code>fftw_execute(plan)</code>, and clean up
with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>).
<p>FFTW provides two routines for creating plans for 2d and 3d transforms,
and one routine for creating plans of arbitrary dimensionality.
The 2d and 3d routines have the following signature:
<pre class="example"> fftw_plan fftw_plan_dft_2d(int n0, int n1,
fftw_complex *in, fftw_complex *out,
int sign, unsigned flags);
fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
fftw_complex *in, fftw_complex *out,
int sign, unsigned flags);
</pre>
<p><a name="index-fftw_005fplan_005fdft_005f2d-39"></a><a name="index-fftw_005fplan_005fdft_005f3d-40"></a>
These routines create plans for <code>n0</code> by <code>n1</code> two-dimensional
(2d) transforms and <code>n0</code> by <code>n1</code> by <code>n2</code> 3d transforms,
respectively. All of these transforms operate on contiguous arrays in
the C-standard <dfn>row-major</dfn> order, so that the last dimension has the
fastest-varying index in the array. This layout is described further in
<a href="Multi_002ddimensional-Array-Format.html#Multi_002ddimensional-Array-Format">Multi-dimensional Array Format</a>.
<p>FFTW can also compute transforms of higher dimensionality. In order to
avoid confusion between the various meanings of the the word
“dimension”, we use the term <em>rank</em>
<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
example, we say that a 2d transform has rank 2, a 3d transform has
rank 3, and so on. You can plan transforms of arbitrary rank by
means of the following function:
<pre class="example"> fftw_plan fftw_plan_dft(int rank, const int *n,
fftw_complex *in, fftw_complex *out,
int sign, unsigned flags);
</pre>
<p><a name="index-fftw_005fplan_005fdft-42"></a>
Here, <code>n</code> is a pointer to an array <code>n[rank]</code> denoting an
<code>n[0]</code> by <code>n[1]</code> by <small class="dots">...</small> by <code>n[rank-1]</code> transform.
Thus, for example, the call
<pre class="example"> fftw_plan_dft_2d(n0, n1, in, out, sign, flags);
</pre>
<p>is equivalent to the following code fragment:
<pre class="example"> int n[2];
n[0] = n0;
n[1] = n1;
fftw_plan_dft(2, n, in, out, sign, flags);
</pre>
<p><code>fftw_plan_dft</code> is not restricted to 2d and 3d transforms,
however, but it can plan transforms of arbitrary rank.
<p>You may have noticed that all the planner routines described so far
have overlapping functionality. For example, you can plan a 1d or 2d
transform by using <code>fftw_plan_dft</code> with a <code>rank</code> of <code>1</code>
or <code>2</code>, or even by calling <code>fftw_plan_dft_3d</code> with <code>n0</code>
and/or <code>n1</code> equal to <code>1</code> (with no loss in efficiency). This
pattern continues, and FFTW's planning routines in general form a
“partial order,” sequences of
<a name="index-partial-order-43"></a>interfaces with strictly increasing generality but correspondingly
greater complexity.
<p><code>fftw_plan_dft</code> is the most general complex-DFT routine that we
describe in this tutorial, but there are also the advanced and guru interfaces,
<a name="index-advanced-interface-44"></a><a name="index-guru-interface-45"></a>which allow one to efficiently combine multiple/strided transforms
into a single FFTW plan, transform a subset of a larger
multi-dimensional array, and/or to handle more general complex-number
formats. For more information, see <a href="FFTW-Reference.html#FFTW-Reference">FFTW Reference</a>.
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<hr>
<h4>Footnotes</h4><p class="footnote"><small>[<a name="fn-1" href="#fnd-1">1</a>]</small> The
term “rank” is commonly used in the APL, FORTRAN, and Common Lisp
traditions, although it is not so common in the C world.</p>
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