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date Wed, 20 Mar 2013 15:35:50 +0000
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3 <title>One-Dimensional DFTs of Real Data - FFTW 3.3.3</title>
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49 <a name="One-Dimensional-DFTs-of-Real-Data"></a>
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51 <p>
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>,
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57
58 <h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3>
59
60 <p>In many practical applications, the input data <code>in[i]</code> are purely
61 real numbers, in which case the DFT output satisfies the &ldquo;Hermitian&rdquo;
62 <a name="index-Hermitian-46"></a>redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
63 possible to take advantage of these circumstances in order to achieve
64 roughly a factor of two improvement in both speed and memory usage.
65
66 <p>In exchange for these speed and space advantages, the user sacrifices
67 some of the simplicity of FFTW's complex transforms. First of all, the
68 input and output arrays are of <em>different sizes and types</em>: the
69 input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
70 complex numbers (the non-redundant outputs); this also requires slight
71 &ldquo;padding&rdquo; of the input array for
72 <a name="index-padding-47"></a>in-place transforms. Second, the inverse transform (complex to real)
73 has the side-effect of <em>overwriting its input array</em>, by default.
74 Neither of these inconveniences should pose a serious problem for
75 users, but it is important to be aware of them.
76
77 <p>The routines to perform real-data transforms are almost the same as
78 those for complex transforms: you allocate arrays of <code>double</code>
79 and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code> or
80 <code>fftw_alloc_complex</code>), create an <code>fftw_plan</code>, execute it as
81 many times as you want with <code>fftw_execute(plan)</code>, and clean up
82 with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
83 differences are that the input (or output) is of type <code>double</code>
84 and there are new routines to create the plan. In one dimension:
85
86 <pre class="example"> fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
87 unsigned flags);
88 fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
89 unsigned flags);
90 </pre>
91 <p><a name="index-fftw_005fplan_005fdft_005fr2c_005f1d-48"></a><a name="index-fftw_005fplan_005fdft_005fc2r_005f1d-49"></a>
92 for the real input to complex-Hermitian output (<dfn>r2c</dfn>) and
93 complex-Hermitian input to real output (<dfn>c2r</dfn>) transforms.
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.
95 Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
96 always <code>FFTW_BACKWARD</code>.
97 <a name="index-FFTW_005fFORWARD-52"></a><a name="index-FFTW_005fBACKWARD-53"></a>(For single/long-double precision
98 <code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
99 <code>float</code> and <code>long double</code>, respectively.)
100 <a name="index-precision-54"></a>
101
102 <p>Here, <code>n</code> is the &ldquo;logical&rdquo; size of the DFT, not necessarily the
103 physical size of the array. In particular, the real (<code>double</code>)
104 array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
105 array has <code>n/2+1</code> elements (where the division is rounded down).
106 For an in-place transform,
107 <a name="index-in_002dplace-55"></a><code>in</code> and <code>out</code> are aliased to the same array, which must be
108 big enough to hold both; so, the real array would actually have
109 <code>2*(n/2+1)</code> elements, where the elements beyond the first
110 <code>n</code> are unused padding. (Note that this is very different from
111 the concept of &ldquo;zero-padding&rdquo; a transform to a larger length, which
112 changes the logical size of the DFT by actually adding new input
113 data.) The kth element of the complex array is exactly the
114 same as the kth element of the corresponding complex DFT. All
115 positive <code>n</code> are supported; products of small factors are most
116 efficient, but an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>) algorithm is used even for prime sizes.
117
118 <p>As noted above, the c2r transform destroys its input array even for
119 out-of-place transforms. This can be prevented, if necessary, by
120 including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
121 unfortunately some sacrifice in performance.
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
123 DFTs (next section).
124
125 <p>Readers familiar with DFTs of real data will recall that the 0th (the
126 &ldquo;DC&rdquo;) and <code>n/2</code>-th (the &ldquo;Nyquist&rdquo; frequency, when <code>n</code> is
127 even) elements of the complex output are purely real. Some
128 implementations therefore store the Nyquist element where the DC
129 imaginary part would go, in order to make the input and output arrays
130 the same size. Such packing, however, does not generalize well to
131 multi-dimensional transforms, and the space savings are miniscule in
132 any case; FFTW does not support it.
133
134 <p>An alternative interface for one-dimensional r2c and c2r DFTs can be
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
136 (and type) as the input array.
137 <a name="index-halfcomplex-format-58"></a>That interface, although it is not very useful for multi-dimensional
138 transforms, may sometimes yield better performance.
139
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