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Add FFTW3
author Chris Cannam
date Wed, 20 Mar 2013 15:35:50 +0000
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+<html lang="en">
+<head>
+<title>The 1d Real-data DFT - FFTW 3.3.3</title>
+<meta http-equiv="Content-Type" content="text/html">
+<meta name="description" content="FFTW 3.3.3">
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+<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)">
+<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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+<!--
+This manual is for FFTW
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+Copyright (C) 2003 Matteo Frigo.
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+Copyright (C) 2003 Massachusetts Institute of Technology.
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+<div class="node">
+<a name="The-1d-Real-data-DFT"></a>
+<a name="The-1d-Real_002ddata-DFT"></a>
+<p>
+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>,
+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>,
+Up:&nbsp;<a rel="up" accesskey="u" href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>
+<hr>
+</div>
+
+<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
+Y of the size <code>n</code> real array X, exactly as defined
+above, i.e. 
+<center><img src="equation-dft.png" align="top">.</center>This output array Y can easily be shown to possess the
+&ldquo;Hermitian&rdquo; symmetry
+<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
+<i>Y<sub>n</sub> = Y</i><sub>0</sub>.
+
+   <p>As a result of this symmetry, half of the output Y is redundant
+(being the complex conjugate of the other half), and so the 1d r2c
+transforms only output elements 0<small class="dots">...</small>n/2 of Y
+(n/2+1 complex numbers), where the division by 2 is
+rounded down.
+
+   <p>Moreover, the Hermitian symmetry implies that
+<i>Y</i><sub>0</sub>and, if n is even, the
+<i>Y</i><sub><i>n</i>/2</sub>element, are purely real.  So, for the <code>R2HC</code> r2r transform, these
+elements are not stored in the halfcomplex output format. 
+<a name="index-r2r-297"></a><a name="index-R2HC-298"></a><a name="index-halfcomplex-format-299"></a>
+
+   <p>The c2r and <code>H2RC</code> r2r transforms compute the backward DFT of the
+<em>complex</em> array X 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-300"></a>Like FFTW'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
+n.
+
+<!-- =========> -->
+   </body></html>
+