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<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
“Hermitian” 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.
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