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<a name="The-1d-Real_002ddata-DFT"></a>
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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>
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<a name="The-1d-Real_002ddata-DFT-1"></a>
<h4 class="subsection">4.8.2 The 1d Real-data DFT</h4>

<p>The real-input (r2c) DFT in FFTW computes the <em>forward</em> transform
<em>Y</em> of the size <code>n</code> real array <em>X</em>, exactly as defined
above, i.e.
<center><img src="equation-dft.png" align="top">.</center>This output array <em>Y</em> can easily be shown to possess the
&ldquo;Hermitian&rdquo; symmetry
<a name="index-Hermitian-1"></a>
<i>Y<sub>k</sub> = Y<sub>n-k</sub></i><sup>*</sup>,where we take <em>Y</em> to be periodic so that
<i>Y<sub>n</sub> = Y</i><sub>0</sub>.</p>
<p>As a result of this symmetry, half of the output <em>Y</em> is redundant
(being the complex conjugate of the other half), and so the 1d r2c
transforms only output elements <em>0</em>&hellip;<em>n/2</em> of <em>Y</em>
(<em>n/2+1</em> complex numbers), where the division by <em>2</em> is
rounded down.
</p>
<p>Moreover, the Hermitian symmetry implies that
<i>Y</i><sub>0</sub>and, if <em>n</em> is even, the
<i>Y</i><sub><i>n</i>/2</sub>element, are purely real.  So, for the <code>R2HC</code> r2r transform, the
halfcomplex format does not store the imaginary parts of these elements.
<a name="index-r2r-2"></a>
<a name="index-R2HC"></a>
<a name="index-halfcomplex-format-2"></a>
</p>

<p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
<em>complex</em> array <em>X</em> with Hermitian symmetry, stored in the
r2c/<code>R2HC</code> output formats, respectively, where the backward
transform is defined exactly as for the complex case:
<center><img src="equation-idft.png" align="top">.</center>The outputs <code>Y</code> of this transform can easily be seen to be purely
real, and are stored as an array of real numbers.
</p>
<a name="index-normalization-9"></a>
<p>Like FFTW&rsquo;s complex DFT, these transforms are unnormalized.  In other
words, applying the real-to-complex (forward) and then the
complex-to-real (backward) transform will multiply the input by
<em>n</em>.
</p>


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author	Chris Cannam <cannam@all-day-breakfast.com>
date	Mon, 02 Mar 2020 14:03:47 +0000
parents	7867fa7e1b6b
children