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author Chris Cannam
date Wed, 07 Sep 2016 10:40:32 +0100
parents 26056e866c29
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Chris@19 1 <html lang="en">
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Chris@19 3 <title>The 1d Real-data DFT - FFTW 3.3.4</title>
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Chris@19 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)">
Chris@19 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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Chris@19 48 <div class="node">
Chris@19 49 <a name="The-1d-Real-data-DFT"></a>
Chris@19 50 <a name="The-1d-Real_002ddata-DFT"></a>
Chris@19 51 <p>
Chris@19 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@19 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@19 54 Up:&nbsp;<a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
Chris@19 55 <hr>
Chris@19 56 </div>
Chris@19 57
Chris@19 58 <h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>
Chris@19 59
Chris@19 60 <p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
Chris@19 61 Y of the size <code>n</code> real array X, exactly as defined
Chris@19 62 above, i.e.
Chris@19 63 <center><img src="equation-dft.png" align="top">.</center>This output array Y can easily be shown to possess the
Chris@19 64 &ldquo;Hermitian&rdquo; symmetry
Chris@19 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
Chris@19 66 <i>Y<sub>n</sub> = Y</i><sub>0</sub>.
Chris@19 67
Chris@19 68 <p>As a result of this symmetry, half of the output Y is redundant
Chris@19 69 (being the complex conjugate of the other half), and so the 1d r2c
Chris@19 70 transforms only output elements 0<small class="dots">...</small>n/2 of Y
Chris@19 71 (n/2+1 complex numbers), where the division by 2 is
Chris@19 72 rounded down.
Chris@19 73
Chris@19 74 <p>Moreover, the Hermitian symmetry implies that
Chris@19 75 <i>Y</i><sub>0</sub>and, if n is even, the
Chris@19 76 <i>Y</i><sub><i>n</i>/2</sub>element, are purely real. So, for the <code>R2HC</code> r2r transform, these
Chris@19 77 elements are not stored in the halfcomplex output format.
Chris@19 78 <a name="index-r2r-300"></a><a name="index-R2HC-301"></a><a name="index-halfcomplex-format-302"></a>
Chris@19 79
Chris@19 80 <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
Chris@19 81 <em>complex</em> array X with Hermitian symmetry, stored in the
Chris@19 82 r2c/<code>R2HC</code> output formats, respectively, where the backward
Chris@19 83 transform is defined exactly as for the complex case:
Chris@19 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@19 85 real, and are stored as an array of real numbers.
Chris@19 86
Chris@19 87 <p><a name="index-normalization-303"></a>Like FFTW's complex DFT, these transforms are unnormalized. In other
Chris@19 88 words, applying the real-to-complex (forward) and then the
Chris@19 89 complex-to-real (backward) transform will multiply the input by
Chris@19 90 n.
Chris@19 91
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Chris@19 94