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115 lines
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115 lines
5.1 KiB
HTML
<!DOCTYPE html PUBLIC "-//W3C//DTD HTML 4.01 Transitional//EN" "http://www.w3.org/TR/html4/loose.dtd">
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<html>
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<!-- This manual is for FFTW
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(version 3.3.10, 10 December 2020).
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Copyright (C) 2003 Matteo Frigo.
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Copyright (C) 2003 Massachusetts Institute of Technology.
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Permission is granted to make and distribute verbatim copies of this
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manual provided the copyright notice and this permission notice are
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preserved on all copies.
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Permission is granted to copy and distribute modified versions of this
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manual under the conditions for verbatim copying, provided that the
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entire resulting derived work is distributed under the terms of a
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permission notice identical to this one.
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Permission is granted to copy and distribute translations of this manual
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into another language, under the above conditions for modified versions,
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except that this permission notice may be stated in a translation
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approved by the Free Software Foundation. -->
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<head>
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<meta http-equiv="Content-Type" content="text/html; charset=utf-8">
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<title>The 1d Real-data DFT (FFTW 3.3.10)</title>
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<meta name="description" content="The 1d Real-data DFT (FFTW 3.3.10)">
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<link href="index.html" rel="start" title="Top">
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<link href="Concept-Index.html" rel="index" title="Concept Index">
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<link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
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<link href="What-FFTW-Really-Computes.html" rel="up" title="What FFTW Really Computes">
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<link href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" rel="next" title="1d Real-even DFTs (DCTs)">
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<link href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html" rel="prev" title="The 1d Discrete Fourier Transform (DFT)">
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</head>
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<body lang="en">
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<span id="The-1d-Real_002ddata-DFT"></span><div class="header">
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<p>
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Next: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html" accesskey="n" rel="next">1d Real-even DFTs (DCTs)</a>, Previous: <a href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html" accesskey="p" rel="prev">The 1d Discrete Fourier Transform (DFT)</a>, Up: <a href="What-FFTW-Really-Computes.html" accesskey="u" rel="up">What FFTW Really Computes</a> [<a href="index.html#SEC_Contents" title="Table of contents" rel="contents">Contents</a>][<a href="Concept-Index.html" title="Index" rel="index">Index</a>]</p>
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</div>
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<hr>
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<span id="The-1d-Real_002ddata-DFT-1"></span><h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>
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<p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
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<em>Y</em> of the size <code>n</code> real array <em>X</em>, exactly as defined
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above, i.e.
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<center><img src="equation-dft.png" align="top">.</center>
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This output array <em>Y</em> can easily be shown to possess the
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“Hermitian” symmetry
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<span id="index-Hermitian-1"></span>
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<i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,
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where we take <em>Y</em> to be periodic so that
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<i>Y<sub>n</sub> = Y</i><sub>0</sub>.
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</p>
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<p>As a result of this symmetry, half of the output <em>Y</em> is redundant
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(being the complex conjugate of the other half), and so the 1d r2c
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transforms only output elements <em>0</em>…<em>n/2</em> of <em>Y</em>
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(<em>n/2+1</em> complex numbers), where the division by <em>2</em> is
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rounded down.
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</p>
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<p>Moreover, the Hermitian symmetry implies that
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<i>Y</i><sub>0</sub>
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and, if <em>n</em> is even, the
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<i>Y</i><sub><i>n</i>/2</sub>
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element, are purely real. So, for the <code>R2HC</code> r2r transform, the
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halfcomplex format does not store the imaginary parts of these elements.
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<span id="index-r2r-2"></span>
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<span id="index-R2HC"></span>
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<span id="index-halfcomplex-format-2"></span>
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</p>
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<p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
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<em>complex</em> array <em>X</em> with Hermitian symmetry, stored in the
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r2c/<code>R2HC</code> output formats, respectively, where the backward
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transform is defined exactly as for the complex case:
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<center><img src="equation-idft.png" align="top">.</center>
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The outputs <code>Y</code> of this transform can easily be seen to be purely
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real, and are stored as an array of real numbers.
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</p>
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<span id="index-normalization-9"></span>
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<p>Like FFTW’s complex DFT, these transforms are unnormalized. In other
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words, applying the real-to-complex (forward) and then the
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complex-to-real (backward) transform will multiply the input by
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<em>n</em>.
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</p>
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</body>
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</html>
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