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Add FFTW 3.3.8 source, and a Linux build
author Chris Cannam <cannam@all-day-breakfast.com>
date Tue, 19 Nov 2019 14:52:55 +0000
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25 <title>FFTW 3.3.8: One-Dimensional DFTs of Real Data</title>
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71 <a name="One_002dDimensional-DFTs-of-Real-Data"></a>
72 <div class="header">
73 <p>
74 Next: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">Multi-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" accesskey="p" rel="prev">Complex Multi-Dimensional DFTs</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
75 </div>
76 <hr>
77 <a name="One_002dDimensional-DFTs-of-Real-Data-1"></a>
78 <h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3>
79
80 <p>In many practical applications, the input data <code>in[i]</code> are purely
81 real numbers, in which case the DFT output satisfies the &ldquo;Hermitian&rdquo;
82 <a name="index-Hermitian"></a>
83 redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
84 possible to take advantage of these circumstances in order to achieve
85 roughly a factor of two improvement in both speed and memory usage.
86 </p>
87 <p>In exchange for these speed and space advantages, the user sacrifices
88 some of the simplicity of FFTW&rsquo;s complex transforms. First of all, the
89 input and output arrays are of <em>different sizes and types</em>: the
90 input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
91 complex numbers (the non-redundant outputs); this also requires slight
92 &ldquo;padding&rdquo; of the input array for
93 <a name="index-padding"></a>
94 in-place transforms. Second, the inverse transform (complex to real)
95 has the side-effect of <em>overwriting its input array</em>, by default.
96 Neither of these inconveniences should pose a serious problem for
97 users, but it is important to be aware of them.
98 </p>
99 <p>The routines to perform real-data transforms are almost the same as
100 those for complex transforms: you allocate arrays of <code>double</code>
101 and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code> or
102 <code>fftw_alloc_complex</code>), create an <code>fftw_plan</code>, execute it as
103 many times as you want with <code>fftw_execute(plan)</code>, and clean up
104 with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
105 differences are that the input (or output) is of type <code>double</code>
106 and there are new routines to create the plan. In one dimension:
107 </p>
108 <div class="example">
109 <pre class="example">fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
110 unsigned flags);
111 fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
112 unsigned flags);
113 </pre></div>
114 <a name="index-fftw_005fplan_005fdft_005fr2c_005f1d"></a>
115 <a name="index-fftw_005fplan_005fdft_005fc2r_005f1d"></a>
116
117 <p>for the real input to complex-Hermitian output (<em>r2c</em>) and
118 complex-Hermitian input to real output (<em>c2r</em>) transforms.
119 <a name="index-r2c"></a>
120 <a name="index-c2r"></a>
121 Unlike the complex DFT planner, there is no <code>sign</code> argument.
122 Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
123 always <code>FFTW_BACKWARD</code>.
124 <a name="index-FFTW_005fFORWARD-1"></a>
125 <a name="index-FFTW_005fBACKWARD-1"></a>
126 (For single/long-double precision
127 <code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
128 <code>float</code> and <code>long double</code>, respectively.)
129 <a name="index-precision-1"></a>
130 </p>
131
132 <p>Here, <code>n</code> is the &ldquo;logical&rdquo; size of the DFT, not necessarily the
133 physical size of the array. In particular, the real (<code>double</code>)
134 array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
135 array has <code>n/2+1</code> elements (where the division is rounded down).
136 For an in-place transform,
137 <a name="index-in_002dplace-1"></a>
138 <code>in</code> and <code>out</code> are aliased to the same array, which must be
139 big enough to hold both; so, the real array would actually have
140 <code>2*(n/2+1)</code> elements, where the elements beyond the first
141 <code>n</code> are unused padding. (Note that this is very different from
142 the concept of &ldquo;zero-padding&rdquo; a transform to a larger length, which
143 changes the logical size of the DFT by actually adding new input
144 data.) The <em>k</em>th element of the complex array is exactly the
145 same as the <em>k</em>th element of the corresponding complex DFT. All
146 positive <code>n</code> are supported; products of small factors are most
147 efficient, but an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>)
148 algorithm is used even for prime sizes.
149 </p>
150 <p>As noted above, the c2r transform destroys its input array even for
151 out-of-place transforms. This can be prevented, if necessary, by
152 including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
153 unfortunately some sacrifice in performance.
154 <a name="index-flags-1"></a>
155 <a name="index-FFTW_005fPRESERVE_005fINPUT"></a>
156 This flag is also not currently supported for multi-dimensional real
157 DFTs (next section).
158 </p>
159 <p>Readers familiar with DFTs of real data will recall that the 0th (the
160 &ldquo;DC&rdquo;) and <code>n/2</code>-th (the &ldquo;Nyquist&rdquo; frequency, when <code>n</code> is
161 even) elements of the complex output are purely real. Some
162 implementations therefore store the Nyquist element where the DC
163 imaginary part would go, in order to make the input and output arrays
164 the same size. Such packing, however, does not generalize well to
165 multi-dimensional transforms, and the space savings are miniscule in
166 any case; FFTW does not support it.
167 </p>
168 <p>An alternative interface for one-dimensional r2c and c2r DFTs can be
169 found in the &lsquo;<samp>r2r</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
170 (and type) as the input array.
171 <a name="index-halfcomplex-format"></a>
172 That interface, although it is not very useful for multi-dimensional
173 transforms, may sometimes yield better performance.
174 </p>
175 <hr>
176 <div class="header">
177 <p>
178 Next: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">Multi-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" accesskey="p" rel="prev">Complex Multi-Dimensional DFTs</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</a> &nbsp; [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html#Concept-Index" title="Index" rel="index">Index</a>]</p>
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