annotate src/fftw-3.3.8/doc/html/One_002dDimensional-DFTs-of-Real-Data.html @ 167:bd3cc4d1df30

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
parents
children
rev   line source
cannam@167 1 <!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
cannam@167 2 <html>
cannam@167 3 <!-- This manual is for FFTW
cannam@167 4 (version 3.3.8, 24 May 2018).
cannam@167 5
cannam@167 6 Copyright (C) 2003 Matteo Frigo.
cannam@167 7
cannam@167 8 Copyright (C) 2003 Massachusetts Institute of Technology.
cannam@167 9
cannam@167 10 Permission is granted to make and distribute verbatim copies of this
cannam@167 11 manual provided the copyright notice and this permission notice are
cannam@167 12 preserved on all copies.
cannam@167 13
cannam@167 14 Permission is granted to copy and distribute modified versions of this
cannam@167 15 manual under the conditions for verbatim copying, provided that the
cannam@167 16 entire resulting derived work is distributed under the terms of a
cannam@167 17 permission notice identical to this one.
cannam@167 18
cannam@167 19 Permission is granted to copy and distribute translations of this manual
cannam@167 20 into another language, under the above conditions for modified versions,
cannam@167 21 except that this permission notice may be stated in a translation
cannam@167 22 approved by the Free Software Foundation. -->
cannam@167 23 <!-- Created by GNU Texinfo 6.3, http://www.gnu.org/software/texinfo/ -->
cannam@167 24 <head>
cannam@167 25 <title>FFTW 3.3.8: One-Dimensional DFTs of Real Data</title>
cannam@167 26
cannam@167 27 <meta name="description" content="FFTW 3.3.8: One-Dimensional DFTs of Real Data">
cannam@167 28 <meta name="keywords" content="FFTW 3.3.8: One-Dimensional DFTs of Real Data">
cannam@167 29 <meta name="resource-type" content="document">
cannam@167 30 <meta name="distribution" content="global">
cannam@167 31 <meta name="Generator" content="makeinfo">
cannam@167 32 <meta http-equiv="Content-Type" content="text/html; charset=utf-8">
cannam@167 33 <link href="index.html#Top" rel="start" title="Top">
cannam@167 34 <link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index">
cannam@167 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
cannam@167 36 <link href="Tutorial.html#Tutorial" rel="up" title="Tutorial">
cannam@167 37 <link href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" rel="next" title="Multi-Dimensional DFTs of Real Data">
cannam@167 38 <link href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" rel="prev" title="Complex Multi-Dimensional DFTs">
cannam@167 39 <style type="text/css">
cannam@167 40 <!--
cannam@167 41 a.summary-letter {text-decoration: none}
cannam@167 42 blockquote.indentedblock {margin-right: 0em}
cannam@167 43 blockquote.smallindentedblock {margin-right: 0em; font-size: smaller}
cannam@167 44 blockquote.smallquotation {font-size: smaller}
cannam@167 45 div.display {margin-left: 3.2em}
cannam@167 46 div.example {margin-left: 3.2em}
cannam@167 47 div.lisp {margin-left: 3.2em}
cannam@167 48 div.smalldisplay {margin-left: 3.2em}
cannam@167 49 div.smallexample {margin-left: 3.2em}
cannam@167 50 div.smalllisp {margin-left: 3.2em}
cannam@167 51 kbd {font-style: oblique}
cannam@167 52 pre.display {font-family: inherit}
cannam@167 53 pre.format {font-family: inherit}
cannam@167 54 pre.menu-comment {font-family: serif}
cannam@167 55 pre.menu-preformatted {font-family: serif}
cannam@167 56 pre.smalldisplay {font-family: inherit; font-size: smaller}
cannam@167 57 pre.smallexample {font-size: smaller}
cannam@167 58 pre.smallformat {font-family: inherit; font-size: smaller}
cannam@167 59 pre.smalllisp {font-size: smaller}
cannam@167 60 span.nolinebreak {white-space: nowrap}
cannam@167 61 span.roman {font-family: initial; font-weight: normal}
cannam@167 62 span.sansserif {font-family: sans-serif; font-weight: normal}
cannam@167 63 ul.no-bullet {list-style: none}
cannam@167 64 -->
cannam@167 65 </style>
cannam@167 66
cannam@167 67
cannam@167 68 </head>
cannam@167 69
cannam@167 70 <body lang="en">
cannam@167 71 <a name="One_002dDimensional-DFTs-of-Real-Data"></a>
cannam@167 72 <div class="header">
cannam@167 73 <p>
cannam@167 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>
cannam@167 75 </div>
cannam@167 76 <hr>
cannam@167 77 <a name="One_002dDimensional-DFTs-of-Real-Data-1"></a>
cannam@167 78 <h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3>
cannam@167 79
cannam@167 80 <p>In many practical applications, the input data <code>in[i]</code> are purely
cannam@167 81 real numbers, in which case the DFT output satisfies the &ldquo;Hermitian&rdquo;
cannam@167 82 <a name="index-Hermitian"></a>
cannam@167 83 redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
cannam@167 84 possible to take advantage of these circumstances in order to achieve
cannam@167 85 roughly a factor of two improvement in both speed and memory usage.
cannam@167 86 </p>
cannam@167 87 <p>In exchange for these speed and space advantages, the user sacrifices
cannam@167 88 some of the simplicity of FFTW&rsquo;s complex transforms. First of all, the
cannam@167 89 input and output arrays are of <em>different sizes and types</em>: the
cannam@167 90 input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
cannam@167 91 complex numbers (the non-redundant outputs); this also requires slight
cannam@167 92 &ldquo;padding&rdquo; of the input array for
cannam@167 93 <a name="index-padding"></a>
cannam@167 94 in-place transforms. Second, the inverse transform (complex to real)
cannam@167 95 has the side-effect of <em>overwriting its input array</em>, by default.
cannam@167 96 Neither of these inconveniences should pose a serious problem for
cannam@167 97 users, but it is important to be aware of them.
cannam@167 98 </p>
cannam@167 99 <p>The routines to perform real-data transforms are almost the same as
cannam@167 100 those for complex transforms: you allocate arrays of <code>double</code>
cannam@167 101 and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code> or
cannam@167 102 <code>fftw_alloc_complex</code>), create an <code>fftw_plan</code>, execute it as
cannam@167 103 many times as you want with <code>fftw_execute(plan)</code>, and clean up
cannam@167 104 with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
cannam@167 105 differences are that the input (or output) is of type <code>double</code>
cannam@167 106 and there are new routines to create the plan. In one dimension:
cannam@167 107 </p>
cannam@167 108 <div class="example">
cannam@167 109 <pre class="example">fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
cannam@167 110 unsigned flags);
cannam@167 111 fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
cannam@167 112 unsigned flags);
cannam@167 113 </pre></div>
cannam@167 114 <a name="index-fftw_005fplan_005fdft_005fr2c_005f1d"></a>
cannam@167 115 <a name="index-fftw_005fplan_005fdft_005fc2r_005f1d"></a>
cannam@167 116
cannam@167 117 <p>for the real input to complex-Hermitian output (<em>r2c</em>) and
cannam@167 118 complex-Hermitian input to real output (<em>c2r</em>) transforms.
cannam@167 119 <a name="index-r2c"></a>
cannam@167 120 <a name="index-c2r"></a>
cannam@167 121 Unlike the complex DFT planner, there is no <code>sign</code> argument.
cannam@167 122 Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
cannam@167 123 always <code>FFTW_BACKWARD</code>.
cannam@167 124 <a name="index-FFTW_005fFORWARD-1"></a>
cannam@167 125 <a name="index-FFTW_005fBACKWARD-1"></a>
cannam@167 126 (For single/long-double precision
cannam@167 127 <code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
cannam@167 128 <code>float</code> and <code>long double</code>, respectively.)
cannam@167 129 <a name="index-precision-1"></a>
cannam@167 130 </p>
cannam@167 131
cannam@167 132 <p>Here, <code>n</code> is the &ldquo;logical&rdquo; size of the DFT, not necessarily the
cannam@167 133 physical size of the array. In particular, the real (<code>double</code>)
cannam@167 134 array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
cannam@167 135 array has <code>n/2+1</code> elements (where the division is rounded down).
cannam@167 136 For an in-place transform,
cannam@167 137 <a name="index-in_002dplace-1"></a>
cannam@167 138 <code>in</code> and <code>out</code> are aliased to the same array, which must be
cannam@167 139 big enough to hold both; so, the real array would actually have
cannam@167 140 <code>2*(n/2+1)</code> elements, where the elements beyond the first
cannam@167 141 <code>n</code> are unused padding. (Note that this is very different from
cannam@167 142 the concept of &ldquo;zero-padding&rdquo; a transform to a larger length, which
cannam@167 143 changes the logical size of the DFT by actually adding new input
cannam@167 144 data.) The <em>k</em>th element of the complex array is exactly the
cannam@167 145 same as the <em>k</em>th element of the corresponding complex DFT. All
cannam@167 146 positive <code>n</code> are supported; products of small factors are most
cannam@167 147 efficient, but an <i>O</i>(<i>n</i>&nbsp;log&nbsp;<i>n</i>)
cannam@167 148 algorithm is used even for prime sizes.
cannam@167 149 </p>
cannam@167 150 <p>As noted above, the c2r transform destroys its input array even for
cannam@167 151 out-of-place transforms. This can be prevented, if necessary, by
cannam@167 152 including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
cannam@167 153 unfortunately some sacrifice in performance.
cannam@167 154 <a name="index-flags-1"></a>
cannam@167 155 <a name="index-FFTW_005fPRESERVE_005fINPUT"></a>
cannam@167 156 This flag is also not currently supported for multi-dimensional real
cannam@167 157 DFTs (next section).
cannam@167 158 </p>
cannam@167 159 <p>Readers familiar with DFTs of real data will recall that the 0th (the
cannam@167 160 &ldquo;DC&rdquo;) and <code>n/2</code>-th (the &ldquo;Nyquist&rdquo; frequency, when <code>n</code> is
cannam@167 161 even) elements of the complex output are purely real. Some
cannam@167 162 implementations therefore store the Nyquist element where the DC
cannam@167 163 imaginary part would go, in order to make the input and output arrays
cannam@167 164 the same size. Such packing, however, does not generalize well to
cannam@167 165 multi-dimensional transforms, and the space savings are miniscule in
cannam@167 166 any case; FFTW does not support it.
cannam@167 167 </p>
cannam@167 168 <p>An alternative interface for one-dimensional r2c and c2r DFTs can be
cannam@167 169 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
cannam@167 170 (and type) as the input array.
cannam@167 171 <a name="index-halfcomplex-format"></a>
cannam@167 172 That interface, although it is not very useful for multi-dimensional
cannam@167 173 transforms, may sometimes yield better performance.
cannam@167 174 </p>
cannam@167 175 <hr>
cannam@167 176 <div class="header">
cannam@167 177 <p>
cannam@167 178 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>
cannam@167 179 </div>
cannam@167 180
cannam@167 181
cannam@167 182
cannam@167 183 </body>
cannam@167 184 </html>