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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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Chris@10 3 <title>One-Dimensional DFTs of Real Data - FFTW 3.3.3</title>
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Chris@10 49 <a name="One-Dimensional-DFTs-of-Real-Data"></a>
Chris@10 50 <a name="One_002dDimensional-DFTs-of-Real-Data"></a>
Chris@10 51 <p>
Chris@10 52 Next:&nbsp;<a rel="next" accesskey="n" href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data">Multi-Dimensional DFTs of Real Data</a>,
Chris@10 53 Previous:&nbsp;<a rel="previous" accesskey="p" href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs">Complex Multi-Dimensional DFTs</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">2.3 One-Dimensional DFTs of Real Data</h3>
Chris@10 59
Chris@10 60 <p>In many practical applications, the input data <code>in[i]</code> are purely
Chris@10 61 real numbers, in which case the DFT output satisfies the &ldquo;Hermitian&rdquo;
Chris@10 62 <a name="index-Hermitian-46"></a>redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
Chris@10 63 possible to take advantage of these circumstances in order to achieve
Chris@10 64 roughly a factor of two improvement in both speed and memory usage.
Chris@10 65
Chris@10 66 <p>In exchange for these speed and space advantages, the user sacrifices
Chris@10 67 some of the simplicity of FFTW's complex transforms. First of all, the
Chris@10 68 input and output arrays are of <em>different sizes and types</em>: the
Chris@10 69 input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
Chris@10 70 complex numbers (the non-redundant outputs); this also requires slight
Chris@10 71 &ldquo;padding&rdquo; of the input array for
Chris@10 72 <a name="index-padding-47"></a>in-place transforms. Second, the inverse transform (complex to real)
Chris@10 73 has the side-effect of <em>overwriting its input array</em>, by default.
Chris@10 74 Neither of these inconveniences should pose a serious problem for
Chris@10 75 users, but it is important to be aware of them.
Chris@10 76
Chris@10 77 <p>The routines to perform real-data transforms are almost the same as
Chris@10 78 those for complex transforms: you allocate arrays of <code>double</code>
Chris@10 79 and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code> or
Chris@10 80 <code>fftw_alloc_complex</code>), create an <code>fftw_plan</code>, execute it as
Chris@10 81 many times as you want with <code>fftw_execute(plan)</code>, and clean up
Chris@10 82 with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
Chris@10 83 differences are that the input (or output) is of type <code>double</code>
Chris@10 84 and there are new routines to create the plan. In one dimension:
Chris@10 85
Chris@10 86 <pre class="example"> fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
Chris@10 87 unsigned flags);
Chris@10 88 fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
Chris@10 89 unsigned flags);
Chris@10 90 </pre>
Chris@10 91 <p><a name="index-fftw_005fplan_005fdft_005fr2c_005f1d-48"></a><a name="index-fftw_005fplan_005fdft_005fc2r_005f1d-49"></a>
Chris@10 92 for the real input to complex-Hermitian output (<dfn>r2c</dfn>) and
Chris@10 93 complex-Hermitian input to real output (<dfn>c2r</dfn>) transforms.
Chris@10 94 <a name="index-r2c-50"></a><a name="index-c2r-51"></a>Unlike the complex DFT planner, there is no <code>sign</code> argument.
Chris@10 95 Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
Chris@10 96 always <code>FFTW_BACKWARD</code>.
Chris@10 97 <a name="index-FFTW_005fFORWARD-52"></a><a name="index-FFTW_005fBACKWARD-53"></a>(For single/long-double precision
Chris@10 98 <code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
Chris@10 99 <code>float</code> and <code>long double</code>, respectively.)
Chris@10 100 <a name="index-precision-54"></a>
Chris@10 101
Chris@10 102 <p>Here, <code>n</code> is the &ldquo;logical&rdquo; size of the DFT, not necessarily the
Chris@10 103 physical size of the array. In particular, the real (<code>double</code>)
Chris@10 104 array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
Chris@10 105 array has <code>n/2+1</code> elements (where the division is rounded down).
Chris@10 106 For an in-place transform,
Chris@10 107 <a name="index-in_002dplace-55"></a><code>in</code> and <code>out</code> are aliased to the same array, which must be
Chris@10 108 big enough to hold both; so, the real array would actually have
Chris@10 109 <code>2*(n/2+1)</code> elements, where the elements beyond the first
Chris@10 110 <code>n</code> are unused padding. (Note that this is very different from
Chris@10 111 the concept of &ldquo;zero-padding&rdquo; a transform to a larger length, which
Chris@10 112 changes the logical size of the DFT by actually adding new input
Chris@10 113 data.) The kth element of the complex array is exactly the
Chris@10 114 same as the kth element of the corresponding complex DFT. All
Chris@10 115 positive <code>n</code> are supported; products of small factors are most
Chris@10 116 efficient, but an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>) algorithm is used even for prime sizes.
Chris@10 117
Chris@10 118 <p>As noted above, the c2r transform destroys its input array even for
Chris@10 119 out-of-place transforms. This can be prevented, if necessary, by
Chris@10 120 including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
Chris@10 121 unfortunately some sacrifice in performance.
Chris@10 122 <a name="index-flags-56"></a><a name="index-FFTW_005fPRESERVE_005fINPUT-57"></a>This flag is also not currently supported for multi-dimensional real
Chris@10 123 DFTs (next section).
Chris@10 124
Chris@10 125 <p>Readers familiar with DFTs of real data will recall that the 0th (the
Chris@10 126 &ldquo;DC&rdquo;) and <code>n/2</code>-th (the &ldquo;Nyquist&rdquo; frequency, when <code>n</code> is
Chris@10 127 even) elements of the complex output are purely real. Some
Chris@10 128 implementations therefore store the Nyquist element where the DC
Chris@10 129 imaginary part would go, in order to make the input and output arrays
Chris@10 130 the same size. Such packing, however, does not generalize well to
Chris@10 131 multi-dimensional transforms, and the space savings are miniscule in
Chris@10 132 any case; FFTW does not support it.
Chris@10 133
Chris@10 134 <p>An alternative interface for one-dimensional r2c and c2r DFTs can be
Chris@10 135 found in the &lsquo;<samp><span class="samp">r2r</span></samp>&rsquo; interface (see <a href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>), with &ldquo;halfcomplex&rdquo;-format output that <em>is</em> the same size
Chris@10 136 (and type) as the input array.
Chris@10 137 <a name="index-halfcomplex-format-58"></a>That interface, although it is not very useful for multi-dimensional
Chris@10 138 transforms, may sometimes yield better performance.
Chris@10 139
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