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<title>One-Dimensional DFTs of Real Data - FFTW 3.1.1</title>
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(version 3.1.1, 4 March 2006).
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Copyright (C) 2003 Matteo Frigo.
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<h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3>
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<p>In many practical applications, the input data <code>in[i]</code> are purely
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real numbers, in which case the DFT output satisfies the “Hermitian”
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<a name="index-Hermitian-45"></a>redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
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possible to take advantage of these circumstances in order to achieve
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roughly a factor of two improvement in both speed and memory usage.
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<p>In exchange for these speed and space advantages, the user sacrifices
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some of the simplicity of FFTW's complex transforms. First of all, the
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input and output arrays are of <em>different sizes and types</em>: the
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input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
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complex numbers (the non-redundant outputs); this also requires slight
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“padding” of the input array for
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<a name="index-padding-46"></a>in-place transforms. Second, the inverse transform (complex to real)
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has the side-effect of <em>destroying its input array</em>, by default.
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Neither of these inconveniences should pose a serious problem for
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users, but it is important to be aware of them.
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<p>The routines to perform real-data transforms are almost the same as
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those for complex transforms: you allocate arrays of <code>double</code>
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and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code>),
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create an <code>fftw_plan</code>, execute it as many times as you want with
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<code>fftw_execute(plan)</code>, and clean up with
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<code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
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differences are that the input (or output) is of type <code>double</code>
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and there are new routines to create the plan. In one dimension:
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<pre class="example"> fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
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fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
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<p><a name="index-fftw_005fplan_005fdft_005fr2c_005f1d-47"></a><a name="index-fftw_005fplan_005fdft_005fc2r_005f1d-48"></a>
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for the real input to complex-Hermitian output (<dfn>r2c</dfn>) and
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complex-Hermitian input to real output (<dfn>c2r</dfn>) transforms.
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<a name="index-r2c-49"></a><a name="index-c2r-50"></a>Unlike the complex DFT planner, there is no <code>sign</code> argument.
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Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
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always <code>FFTW_BACKWARD</code>.
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<a name="index-FFTW_005fFORWARD-51"></a><a name="index-FFTW_005fBACKWARD-52"></a>(For single/long-double precision
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<code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
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<code>float</code> and <code>long double</code>, respectively.)
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<a name="index-precision-53"></a>
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Here, <code>n</code> is the “logical” size of the DFT, not necessarily the
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physical size of the array. In particular, the real (<code>double</code>)
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array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
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array has <code>n/2+1</code> elements (where the division is rounded down).
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For an in-place transform,
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<a name="index-in_002dplace-54"></a><code>in</code> and <code>out</code> are aliased to the same array, which must be
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big enough to hold both; so, the real array would actually have
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<code>2*(n/2+1)</code> elements, where the elements beyond the first <code>n</code>
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are unused padding. The kth element of the complex array is
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exactly the same as the kth element of the corresponding complex
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DFT. All positive <code>n</code> are supported; products of small factors are
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most efficient, but an <i>O</i>(<i>n</i> log <i>n</i>) algorithm is used even for prime
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<p>As noted above, the c2r transform destroys its input array even for
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out-of-place transforms. This can be prevented, if necessary, by
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including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
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unfortunately some sacrifice in performance.
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<a name="index-flags-55"></a><a name="index-FFTW_005fPRESERVE_005fINPUT-56"></a>This flag is also not currently supported for multi-dimensional real
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<p>Readers familiar with DFTs of real data will recall that the 0th (the
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“DC”) and <code>n/2</code>-th (the “Nyquist” frequency, when <code>n</code> is
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even) elements of the complex output are purely real. Some
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implementations therefore store the Nyquist element where the DC
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imaginary part would go, in order to make the input and output arrays
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the same size. Such packing, however, does not generalize well to
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multi-dimensional transforms, and the space savings are miniscule in
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any case; FFTW does not support it.
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<p>An alternative interface for one-dimensional r2c and c2r DFTs can be
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found in the `<samp><span class="samp">r2r</span></samp>' interface (see <a href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>), with “halfcomplex”-format output that <em>is</em> the same size
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(and type) as the input array.
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<a name="index-halfcomplex-format-57"></a>That interface, although it is not very useful for multi-dimensional
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transforms, may sometimes yield better performance.
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