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author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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3 <title>The 1d Real-data DFT - FFTW 3.3.3</title>
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49 <a name="The-1d-Real-data-DFT"></a>
50 <a name="The-1d-Real_002ddata-DFT"></a>
51 <p>
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>,
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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 &ldquo;Hermitian&rdquo; symmetry
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
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-297"></a><a name="index-R2HC-298"></a><a name="index-halfcomplex-format-299"></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-300"></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
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