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<title>GNU Scientific Library – Reference Manual: Overview of real data FFTs</title>
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<a name="Overview-of-real-data-FFTs"></a>
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<p>
Next: <a href="Radix_002d2-FFT-routines-for-real-data.html#Radix_002d2-FFT-routines-for-real-data" accesskey="n" rel="next">Radix-2 FFT routines for real data</a>, Previous: <a href="Mixed_002dradix-FFT-routines-for-complex-data.html#Mixed_002dradix-FFT-routines-for-complex-data" accesskey="p" rel="previous">Mixed-radix FFT routines for complex data</a>, Up: <a href="Fast-Fourier-Transforms.html#Fast-Fourier-Transforms" accesskey="u" rel="up">Fast Fourier Transforms</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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<a name="Overview-of-real-data-FFTs-1"></a>
<h3 class="section">16.5 Overview of real data FFTs</h3>
<a name="index-FFT-of-real-data"></a>
<p>The functions for real data are similar to those for complex data.
However, there is an important difference between forward and inverse
transforms. The Fourier transform of a real sequence is not real. It is
a complex sequence with a special symmetry:
A sequence with this symmetry is called <em>conjugate-complex</em> or
<em>half-complex</em>. This different structure requires different
storage layouts for the forward transform (from real to half-complex)
and inverse transform (from half-complex back to real). As a
consequence the routines are divided into two sets: functions in
<code>gsl_fft_real</code> which operate on real sequences and functions in
<code>gsl_fft_halfcomplex</code> which operate on half-complex sequences.
</p>
<p>Functions in <code>gsl_fft_real</code> compute the frequency coefficients of a
real sequence. The half-complex coefficients <em>c</em> of a real sequence
<em>x</em> are given by Fourier analysis,
Functions in <code>gsl_fft_halfcomplex</code> compute inverse or backwards
transforms. They reconstruct real sequences by Fourier synthesis from
their half-complex frequency coefficients, <em>c</em>,
The symmetry of the half-complex sequence implies that only half of the
complex numbers in the output need to be stored. The remaining half can
be reconstructed using the half-complex symmetry condition. This works
for all lengths, even and odd—when the length is even the middle value
where <em>k=n/2</em> is also real. Thus only <var>n</var> real numbers are
required to store the half-complex sequence, and the transform of a real
sequence can be stored in the same size array as the original data.
</p>
<p>The precise storage arrangements depend on the algorithm, and are
different for radix-2 and mixed-radix routines. The radix-2 function
operates in-place, which constrains the locations where each element can
be stored. The restriction forces real and imaginary parts to be stored
far apart. The mixed-radix algorithm does not have this restriction, and
it stores the real and imaginary parts of a given term in neighboring
locations (which is desirable for better locality of memory accesses).
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Next: <a href="Radix_002d2-FFT-routines-for-real-data.html#Radix_002d2-FFT-routines-for-real-data" accesskey="n" rel="next">Radix-2 FFT routines for real data</a>, Previous: <a href="Mixed_002dradix-FFT-routines-for-complex-data.html#Mixed_002dradix-FFT-routines-for-complex-data" accesskey="p" rel="previous">Mixed-radix FFT routines for complex data</a>, Up: <a href="Fast-Fourier-Transforms.html#Fast-Fourier-Transforms" accesskey="u" rel="up">Fast Fourier Transforms</a> [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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