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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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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.
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cannam@167 8 Copyright (C) 2003 Massachusetts Institute of Technology.
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cannam@167 24 <head>
cannam@167 25 <title>FFTW 3.3.8: The 1d Real-data DFT</title>
cannam@167 26
cannam@167 27 <meta name="description" content="FFTW 3.3.8: The 1d Real-data DFT">
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cannam@167 35 <link href="index.html#SEC_Contents" rel="contents" title="Table of Contents">
cannam@167 36 <link href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" rel="up" title="What FFTW Really Computes">
cannam@167 37 <link href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" rel="next" title="1d Real-even DFTs (DCTs)">
cannam@167 38 <link href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029" rel="prev" title="The 1d Discrete Fourier Transform (DFT)">
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cannam@167 68 </head>
cannam@167 69
cannam@167 70 <body lang="en">
cannam@167 71 <a name="The-1d-Real_002ddata-DFT"></a>
cannam@167 72 <div class="header">
cannam@167 73 <p>
cannam@167 74 Next: <a href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" accesskey="n" rel="next">1d Real-even DFTs (DCTs)</a>, Previous: <a href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029" accesskey="p" rel="prev">The 1d Discrete Fourier Transform (DFT)</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</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="The-1d-Real_002ddata-DFT-1"></a>
cannam@167 78 <h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>
cannam@167 79
cannam@167 80 <p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
cannam@167 81 <em>Y</em> of the size <code>n</code> real array <em>X</em>, exactly as defined
cannam@167 82 above, i.e.
cannam@167 83 <center><img src="equation-dft.png" align="top">.</center>
cannam@167 84 This output array <em>Y</em> can easily be shown to possess the
cannam@167 85 &ldquo;Hermitian&rdquo; symmetry
cannam@167 86 <a name="index-Hermitian-1"></a>
cannam@167 87 <i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,
cannam@167 88 where we take <em>Y</em> to be periodic so that
cannam@167 89 <i>Y<sub>n</sub> = Y</i><sub>0</sub>.
cannam@167 90 </p>
cannam@167 91 <p>As a result of this symmetry, half of the output <em>Y</em> is redundant
cannam@167 92 (being the complex conjugate of the other half), and so the 1d r2c
cannam@167 93 transforms only output elements <em>0</em>&hellip;<em>n/2</em> of <em>Y</em>
cannam@167 94 (<em>n/2+1</em> complex numbers), where the division by <em>2</em> is
cannam@167 95 rounded down.
cannam@167 96 </p>
cannam@167 97 <p>Moreover, the Hermitian symmetry implies that
cannam@167 98 <i>Y</i><sub>0</sub>
cannam@167 99 and, if <em>n</em> is even, the
cannam@167 100 <i>Y</i><sub><i>n</i>/2</sub>
cannam@167 101 element, are purely real. So, for the <code>R2HC</code> r2r transform, the
cannam@167 102 halfcomplex format does not store the imaginary parts of these elements.
cannam@167 103 <a name="index-r2r-2"></a>
cannam@167 104 <a name="index-R2HC"></a>
cannam@167 105 <a name="index-halfcomplex-format-2"></a>
cannam@167 106 </p>
cannam@167 107
cannam@167 108 <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
cannam@167 109 <em>complex</em> array <em>X</em> with Hermitian symmetry, stored in the
cannam@167 110 r2c/<code>R2HC</code> output formats, respectively, where the backward
cannam@167 111 transform is defined exactly as for the complex case:
cannam@167 112 <center><img src="equation-idft.png" align="top">.</center>
cannam@167 113 The outputs <code>Y</code> of this transform can easily be seen to be purely
cannam@167 114 real, and are stored as an array of real numbers.
cannam@167 115 </p>
cannam@167 116 <a name="index-normalization-9"></a>
cannam@167 117 <p>Like FFTW&rsquo;s complex DFT, these transforms are unnormalized. In other
cannam@167 118 words, applying the real-to-complex (forward) and then the
cannam@167 119 complex-to-real (backward) transform will multiply the input by
cannam@167 120 <em>n</em>.
cannam@167 121 </p>
cannam@167 122
cannam@167 123
cannam@167 124
cannam@167 125 </body>
cannam@167 126 </html>