Mercurial > hg > sv-dependency-builds
diff src/fftw-3.3.8/doc/html/One_002dDimensional-DFTs-of-Real-Data.html @ 82:d0c2a83c1364
Add FFTW 3.3.8 source, and a Linux build
| author | Chris Cannam |
|---|---|
| date | Tue, 19 Nov 2019 14:52:55 +0000 |
| parents | |
| children |
line wrap: on
line diff
--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.8/doc/html/One_002dDimensional-DFTs-of-Real-Data.html Tue Nov 19 14:52:55 2019 +0000 @@ -0,0 +1,184 @@ +<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd"> +<html> +<!-- This manual is for FFTW +(version 3.3.8, 24 May 2018). + +Copyright (C) 2003 Matteo Frigo. + +Copyright (C) 2003 Massachusetts Institute of Technology. + +Permission is granted to make and distribute verbatim copies of this +manual provided the copyright notice and this permission notice are +preserved on all copies. + +Permission is granted to copy and distribute modified versions of this +manual under the conditions for verbatim copying, provided that the +entire resulting derived work is distributed under the terms of a +permission notice identical to this one. + +Permission is granted to copy and distribute translations of this manual +into another language, under the above conditions for modified versions, +except that this permission notice may be stated in a translation +approved by the Free Software Foundation. --> +<!-- Created by GNU Texinfo 6.3, http://www.gnu.org/software/texinfo/ --> +<head> +<title>FFTW 3.3.8: One-Dimensional DFTs of Real Data</title> + +<meta name="description" content="FFTW 3.3.8: One-Dimensional DFTs of Real Data"> +<meta name="keywords" content="FFTW 3.3.8: One-Dimensional DFTs of Real Data"> +<meta name="resource-type" content="document"> +<meta name="distribution" content="global"> +<meta name="Generator" content="makeinfo"> +<meta http-equiv="Content-Type" content="text/html; charset=utf-8"> +<link href="index.html#Top" rel="start" title="Top"> +<link href="Concept-Index.html#Concept-Index" rel="index" title="Concept Index"> +<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents"> +<link href="Tutorial.html#Tutorial" rel="up" title="Tutorial"> +<link href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" rel="next" title="Multi-Dimensional DFTs of Real Data"> +<link href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" rel="prev" title="Complex Multi-Dimensional DFTs"> +<style type="text/css"> +<!-- +a.summary-letter {text-decoration: none} +blockquote.indentedblock {margin-right: 0em} +blockquote.smallindentedblock {margin-right: 0em; font-size: smaller} +blockquote.smallquotation {font-size: smaller} +div.display {margin-left: 3.2em} +div.example {margin-left: 3.2em} +div.lisp {margin-left: 3.2em} +div.smalldisplay {margin-left: 3.2em} +div.smallexample {margin-left: 3.2em} +div.smalllisp {margin-left: 3.2em} +kbd {font-style: oblique} +pre.display {font-family: inherit} +pre.format {font-family: inherit} +pre.menu-comment {font-family: serif} +pre.menu-preformatted {font-family: serif} +pre.smalldisplay {font-family: inherit; font-size: smaller} +pre.smallexample {font-size: smaller} +pre.smallformat {font-family: inherit; font-size: smaller} +pre.smalllisp {font-size: smaller} +span.nolinebreak {white-space: nowrap} +span.roman {font-family: initial; font-weight: normal} +span.sansserif {font-family: sans-serif; font-weight: normal} +ul.no-bullet {list-style: none} +--> +</style> + + +</head> + +<body lang="en"> +<a name="One_002dDimensional-DFTs-of-Real-Data"></a> +<div class="header"> +<p> +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> [<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> +</div> +<hr> +<a name="One_002dDimensional-DFTs-of-Real-Data-1"></a> +<h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3> + +<p>In many practical applications, the input data <code>in[i]</code> are purely +real numbers, in which case the DFT output satisfies the “Hermitian” +<a name="index-Hermitian"></a> +redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is +possible to take advantage of these circumstances in order to achieve +roughly a factor of two improvement in both speed and memory usage. +</p> +<p>In exchange for these speed and space advantages, the user sacrifices +some of the simplicity of FFTW’s complex transforms. First of all, the +input and output arrays are of <em>different sizes and types</em>: the +input is <code>n</code> real numbers, while the output is <code>n/2+1</code> +complex numbers (the non-redundant outputs); this also requires slight +“padding” of the input array for +<a name="index-padding"></a> +in-place transforms. Second, the inverse transform (complex to real) +has the side-effect of <em>overwriting its input array</em>, by default. +Neither of these inconveniences should pose a serious problem for +users, but it is important to be aware of them. +</p> +<p>The routines to perform real-data transforms are almost the same as +those for complex transforms: you allocate arrays of <code>double</code> +and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code> or +<code>fftw_alloc_complex</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>). The only +differences are that the input (or output) is of type <code>double</code> +and there are new routines to create the plan. In one dimension: +</p> +<div class="example"> +<pre class="example">fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out, + unsigned flags); +fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out, + unsigned flags); +</pre></div> +<a name="index-fftw_005fplan_005fdft_005fr2c_005f1d"></a> +<a name="index-fftw_005fplan_005fdft_005fc2r_005f1d"></a> + +<p>for the real input to complex-Hermitian output (<em>r2c</em>) and +complex-Hermitian input to real output (<em>c2r</em>) transforms. +<a name="index-r2c"></a> +<a name="index-c2r"></a> +Unlike the complex DFT planner, there is no <code>sign</code> argument. +Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are +always <code>FFTW_BACKWARD</code>. +<a name="index-FFTW_005fFORWARD-1"></a> +<a name="index-FFTW_005fBACKWARD-1"></a> +(For single/long-double precision +<code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by +<code>float</code> and <code>long double</code>, respectively.) +<a name="index-precision-1"></a> +</p> + +<p>Here, <code>n</code> is the “logical” size of the DFT, not necessarily the +physical size of the array. In particular, the real (<code>double</code>) +array has <code>n</code> elements, while the complex (<code>fftw_complex</code>) +array has <code>n/2+1</code> elements (where the division is rounded down). +For an in-place transform, +<a name="index-in_002dplace-1"></a> +<code>in</code> and <code>out</code> are aliased to the same array, which must be +big enough to hold both; so, the real array would actually have +<code>2*(n/2+1)</code> elements, where the elements beyond the first +<code>n</code> are unused padding. (Note that this is very different from +the concept of “zero-padding” a transform to a larger length, which +changes the logical size of the DFT by actually adding new input +data.) The <em>k</em>th element of the complex array is exactly the +same as the <em>k</em>th element of the corresponding complex DFT. All +positive <code>n</code> are supported; products of small factors are most +efficient, but an <i>O</i>(<i>n</i> log <i>n</i>) + algorithm is used even for prime sizes. +</p> +<p>As noted above, the c2r transform destroys its input array even for +out-of-place transforms. This can be prevented, if necessary, by +including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with +unfortunately some sacrifice in performance. +<a name="index-flags-1"></a> +<a name="index-FFTW_005fPRESERVE_005fINPUT"></a> +This flag is also not currently supported for multi-dimensional real +DFTs (next section). +</p> +<p>Readers familiar with DFTs of real data will recall that the 0th (the +“DC”) and <code>n/2</code>-th (the “Nyquist” frequency, when <code>n</code> is +even) elements of the complex output are purely real. Some +implementations therefore store the Nyquist element where the DC +imaginary part would go, in order to make the input and output arrays +the same size. Such packing, however, does not generalize well to +multi-dimensional transforms, and the space savings are miniscule in +any case; FFTW does not support it. +</p> +<p>An alternative interface for one-dimensional r2c and c2r DFTs can be +found in the ‘<samp>r2r</samp>’ interface (see <a href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>), with “halfcomplex”-format output that <em>is</em> the same size +(and type) as the input array. +<a name="index-halfcomplex-format"></a> +That interface, although it is not very useful for multi-dimensional +transforms, may sometimes yield better performance. +</p> +<hr> +<div class="header"> +<p> +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> [<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> +</div> + + + +</body> +</html>