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comparison fft/fftw/fftw-3.3.4/doc/html/The-1d-Real_002ddata-DFT.html @ 19:26056e866c29
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| author | Chris Cannam |
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| date | Tue, 06 Oct 2015 13:08:39 +0100 |
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| 18:8db794ca3e0b | 19:26056e866c29 |
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| 1 <html lang="en"> | |
| 2 <head> | |
| 3 <title>The 1d Real-data DFT - FFTW 3.3.4</title> | |
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| 5 <meta name="description" content="FFTW 3.3.4"> | |
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| 8 <link rel="up" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" title="What FFTW Really Computes"> | |
| 9 <link rel="prev" href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029" title="The 1d Discrete Fourier Transform (DFT)"> | |
| 10 <link rel="next" href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029" title="1d Real-even DFTs (DCTs)"> | |
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| 12 <!-- | |
| 13 This manual is for FFTW | |
| 14 (version 3.3.4, 20 September 2013). | |
| 15 | |
| 16 Copyright (C) 2003 Matteo Frigo. | |
| 17 | |
| 18 Copyright (C) 2003 Massachusetts Institute of Technology. | |
| 19 | |
| 20 Permission is granted to make and distribute verbatim copies of | |
| 21 this manual provided the copyright notice and this permission | |
| 22 notice are preserved on all copies. | |
| 23 | |
| 24 Permission is granted to copy and distribute modified versions of | |
| 25 this manual under the conditions for verbatim copying, provided | |
| 26 that the entire resulting derived work is distributed under the | |
| 27 terms of a permission notice identical to this one. | |
| 28 | |
| 29 Permission is granted to copy and distribute translations of this | |
| 30 manual into another language, under the above conditions for | |
| 31 modified versions, except that this permission notice may be | |
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| 47 <body> | |
| 48 <div class="node"> | |
| 49 <a name="The-1d-Real-data-DFT"></a> | |
| 50 <a name="The-1d-Real_002ddata-DFT"></a> | |
| 51 <p> | |
| 52 Next: <a rel="next" accesskey="n" href="1d-Real_002deven-DFTs-_0028DCTs_0029.html#g_t1d-Real_002deven-DFTs-_0028DCTs_0029">1d Real-even DFTs (DCTs)</a>, | |
| 53 Previous: <a rel="previous" accesskey="p" href="The-1d-Discrete-Fourier-Transform-_0028DFT_0029.html#The-1d-Discrete-Fourier-Transform-_0028DFT_0029">The 1d Discrete Fourier Transform (DFT)</a>, | |
| 54 Up: <a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a> | |
| 55 <hr> | |
| 56 </div> | |
| 57 | |
| 58 <h4 class="subsection">4.8.2 The 1d Real-data DFT</h4> | |
| 59 | |
| 60 <p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform | |
| 61 Y of the size <code>n</code> real array X, exactly as defined | |
| 62 above, i.e. | |
| 63 <center><img src="equation-dft.png" align="top">.</center>This output array Y can easily be shown to possess the | |
| 64 “Hermitian” symmetry | |
| 65 <a name="index-Hermitian-299"></a><i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,where we take Y to be periodic so that | |
| 66 <i>Y<sub>n</sub> = Y</i><sub>0</sub>. | |
| 67 | |
| 68 <p>As a result of this symmetry, half of the output Y is redundant | |
| 69 (being the complex conjugate of the other half), and so the 1d r2c | |
| 70 transforms only output elements 0<small class="dots">...</small>n/2 of Y | |
| 71 (n/2+1 complex numbers), where the division by 2 is | |
| 72 rounded down. | |
| 73 | |
| 74 <p>Moreover, the Hermitian symmetry implies that | |
| 75 <i>Y</i><sub>0</sub>and, if n is even, the | |
| 76 <i>Y</i><sub><i>n</i>/2</sub>element, are purely real. So, for the <code>R2HC</code> r2r transform, these | |
| 77 elements are not stored in the halfcomplex output format. | |
| 78 <a name="index-r2r-300"></a><a name="index-R2HC-301"></a><a name="index-halfcomplex-format-302"></a> | |
| 79 | |
| 80 <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the | |
| 81 <em>complex</em> array X with Hermitian symmetry, stored in the | |
| 82 r2c/<code>R2HC</code> output formats, respectively, where the backward | |
| 83 transform is defined exactly as for the complex case: | |
| 84 <center><img src="equation-idft.png" align="top">.</center>The outputs <code>Y</code> of this transform can easily be seen to be purely | |
| 85 real, and are stored as an array of real numbers. | |
| 86 | |
| 87 <p><a name="index-normalization-303"></a>Like FFTW's complex DFT, these transforms are unnormalized. In other | |
| 88 words, applying the real-to-complex (forward) and then the | |
| 89 complex-to-real (backward) transform will multiply the input by | |
| 90 n. | |
| 91 | |
| 92 <!-- =========> --> | |
| 93 </body></html> | |
| 94 |