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Replace these with versions built using an older toolset (so as to avoid ABI compatibilities when linking on Ubuntu 14.04 for packaging purposes)
author Chris Cannam
date Fri, 07 Feb 2020 11:51:13 +0000
parents 37bf6b4a2645
children
rev   line source
Chris@10 1 <html lang="en">
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Chris@10 3 <title>The 1d Real-data DFT - FFTW 3.3.3</title>
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Chris@10 49 <a name="The-1d-Real-data-DFT"></a>
Chris@10 50 <a name="The-1d-Real_002ddata-DFT"></a>
Chris@10 51 <p>
Chris@10 52 Next:&nbsp;<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>,
Chris@10 53 Previous:&nbsp;<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>,
Chris@10 54 Up:&nbsp;<a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
Chris@10 55 <hr>
Chris@10 56 </div>
Chris@10 57
Chris@10 58 <h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>
Chris@10 59
Chris@10 60 <p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
Chris@10 61 Y of the size <code>n</code> real array X, exactly as defined
Chris@10 62 above, i.e.
Chris@10 63 <center><img src="equation-dft.png" align="top">.</center>This output array Y can easily be shown to possess the
Chris@10 64 &ldquo;Hermitian&rdquo; symmetry
Chris@10 65 <a name="index-Hermitian-296"></a><i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,where we take Y to be periodic so that
Chris@10 66 <i>Y<sub>n</sub> = Y</i><sub>0</sub>.
Chris@10 67
Chris@10 68 <p>As a result of this symmetry, half of the output Y is redundant
Chris@10 69 (being the complex conjugate of the other half), and so the 1d r2c
Chris@10 70 transforms only output elements 0<small class="dots">...</small>n/2 of Y
Chris@10 71 (n/2+1 complex numbers), where the division by 2 is
Chris@10 72 rounded down.
Chris@10 73
Chris@10 74 <p>Moreover, the Hermitian symmetry implies that
Chris@10 75 <i>Y</i><sub>0</sub>and, if n is even, the
Chris@10 76 <i>Y</i><sub><i>n</i>/2</sub>element, are purely real. So, for the <code>R2HC</code> r2r transform, these
Chris@10 77 elements are not stored in the halfcomplex output format.
Chris@10 78 <a name="index-r2r-297"></a><a name="index-R2HC-298"></a><a name="index-halfcomplex-format-299"></a>
Chris@10 79
Chris@10 80 <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
Chris@10 81 <em>complex</em> array X with Hermitian symmetry, stored in the
Chris@10 82 r2c/<code>R2HC</code> output formats, respectively, where the backward
Chris@10 83 transform is defined exactly as for the complex case:
Chris@10 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
Chris@10 85 real, and are stored as an array of real numbers.
Chris@10 86
Chris@10 87 <p><a name="index-normalization-300"></a>Like FFTW's complex DFT, these transforms are unnormalized. In other
Chris@10 88 words, applying the real-to-complex (forward) and then the
Chris@10 89 complex-to-real (backward) transform will multiply the input by
Chris@10 90 n.
Chris@10 91
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