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| author | Chris Cannam |
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| date | Wed, 20 Mar 2013 15:35:50 +0000 |
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--- /dev/null Thu Jan 01 00:00:00 1970 +0000 +++ b/src/fftw-3.3.3/doc/html/One_002dDimensional-DFTs-of-Real-Data.html Wed Mar 20 15:35:50 2013 +0000 @@ -0,0 +1,142 @@ +<html lang="en"> +<head> +<title>One-Dimensional DFTs of Real Data - FFTW 3.3.3</title> +<meta http-equiv="Content-Type" content="text/html"> +<meta name="description" content="FFTW 3.3.3"> +<meta name="generator" content="makeinfo 4.13"> +<link title="Top" rel="start" href="index.html#Top"> +<link rel="up" href="Tutorial.html#Tutorial" title="Tutorial"> +<link rel="prev" href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" title="Complex Multi-Dimensional DFTs"> +<link rel="next" href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" title="Multi-Dimensional DFTs of Real Data"> +<link href="http://www.gnu.org/software/texinfo/" rel="generator-home" title="Texinfo Homepage"> +<!-- +This manual is for FFTW +(version 3.3.3, 25 November 2012). + +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. + --> +<meta http-equiv="Content-Style-Type" content="text/css"> +<style type="text/css"><!-- + pre.display { font-family:inherit } + pre.format { font-family:inherit } + pre.smalldisplay { font-family:inherit; font-size:smaller } + pre.smallformat { font-family:inherit; font-size:smaller } + pre.smallexample { font-size:smaller } + pre.smalllisp { font-size:smaller } + span.sc { font-variant:small-caps } + span.roman { font-family:serif; font-weight:normal; } + span.sansserif { font-family:sans-serif; font-weight:normal; } +--></style> +</head> +<body> +<div class="node"> +<a name="One-Dimensional-DFTs-of-Real-Data"></a> +<a name="One_002dDimensional-DFTs-of-Real-Data"></a> +<p> +Next: <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>, +Previous: <a rel="previous" accesskey="p" href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs">Complex Multi-Dimensional DFTs</a>, +Up: <a rel="up" accesskey="u" href="Tutorial.html#Tutorial">Tutorial</a> +<hr> +</div> + +<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-46"></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>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-47"></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>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: + +<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> + <p><a name="index-fftw_005fplan_005fdft_005fr2c_005f1d-48"></a><a name="index-fftw_005fplan_005fdft_005fc2r_005f1d-49"></a> +for the real input to complex-Hermitian output (<dfn>r2c</dfn>) and +complex-Hermitian input to real output (<dfn>c2r</dfn>) transforms. +<a name="index-r2c-50"></a><a name="index-c2r-51"></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-52"></a><a name="index-FFTW_005fBACKWARD-53"></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-54"></a> + + <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-55"></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 kth element of the complex array is exactly the +same as the kth 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>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-56"></a><a name="index-FFTW_005fPRESERVE_005fINPUT-57"></a>This flag is also not currently supported for multi-dimensional real +DFTs (next section). + + <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>An alternative interface for one-dimensional r2c and c2r DFTs can be +found in the ‘<samp><span class="samp">r2r</span></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-58"></a>That interface, although it is not very useful for multi-dimensional +transforms, may sometimes yield better performance. + +<!-- --> + </body></html> +