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Chris@82: <title>FFTW 3.3.8: One-Dimensional DFTs of Real Data</title>
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Chris@82: <a name="One_002dDimensional-DFTs-of-Real-Data"></a>
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Chris@82: Next: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">Multi-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" accesskey="p" rel="prev">Complex Multi-Dimensional DFTs</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</a> [<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>
Chris@82: </div>
Chris@82: <hr>
Chris@82: <a name="One_002dDimensional-DFTs-of-Real-Data-1"></a>
Chris@82: <h3 class="section">2.3 One-Dimensional DFTs of Real Data</h3>
Chris@82:
Chris@82: <p>In many practical applications, the input data <code>in[i]</code> are purely
Chris@82: real numbers, in which case the DFT output satisfies the “Hermitian”
Chris@82: <a name="index-Hermitian"></a>
Chris@82: redundancy: <code>out[i]</code> is the conjugate of <code>out[n-i]</code>. It is
Chris@82: possible to take advantage of these circumstances in order to achieve
Chris@82: roughly a factor of two improvement in both speed and memory usage.
Chris@82: </p>
Chris@82: <p>In exchange for these speed and space advantages, the user sacrifices
Chris@82: some of the simplicity of FFTW’s complex transforms. First of all, the
Chris@82: input and output arrays are of <em>different sizes and types</em>: the
Chris@82: input is <code>n</code> real numbers, while the output is <code>n/2+1</code>
Chris@82: complex numbers (the non-redundant outputs); this also requires slight
Chris@82: “padding” of the input array for
Chris@82: <a name="index-padding"></a>
Chris@82: in-place transforms. Second, the inverse transform (complex to real)
Chris@82: has the side-effect of <em>overwriting its input array</em>, by default.
Chris@82: Neither of these inconveniences should pose a serious problem for
Chris@82: users, but it is important to be aware of them.
Chris@82: </p>
Chris@82: <p>The routines to perform real-data transforms are almost the same as
Chris@82: those for complex transforms: you allocate arrays of <code>double</code>
Chris@82: and/or <code>fftw_complex</code> (preferably using <code>fftw_malloc</code> or
Chris@82: <code>fftw_alloc_complex</code>), create an <code>fftw_plan</code>, execute it as
Chris@82: many times as you want with <code>fftw_execute(plan)</code>, and clean up
Chris@82: with <code>fftw_destroy_plan(plan)</code> (and <code>fftw_free</code>). The only
Chris@82: differences are that the input (or output) is of type <code>double</code>
Chris@82: and there are new routines to create the plan. In one dimension:
Chris@82: </p>
Chris@82: <div class="example">
Chris@82: <pre class="example">fftw_plan fftw_plan_dft_r2c_1d(int n, double *in, fftw_complex *out,
Chris@82: unsigned flags);
Chris@82: fftw_plan fftw_plan_dft_c2r_1d(int n, fftw_complex *in, double *out,
Chris@82: unsigned flags);
Chris@82: </pre></div>
Chris@82: <a name="index-fftw_005fplan_005fdft_005fr2c_005f1d"></a>
Chris@82: <a name="index-fftw_005fplan_005fdft_005fc2r_005f1d"></a>
Chris@82:
Chris@82: <p>for the real input to complex-Hermitian output (<em>r2c</em>) and
Chris@82: complex-Hermitian input to real output (<em>c2r</em>) transforms.
Chris@82: <a name="index-r2c"></a>
Chris@82: <a name="index-c2r"></a>
Chris@82: Unlike the complex DFT planner, there is no <code>sign</code> argument.
Chris@82: Instead, r2c DFTs are always <code>FFTW_FORWARD</code> and c2r DFTs are
Chris@82: always <code>FFTW_BACKWARD</code>.
Chris@82: <a name="index-FFTW_005fFORWARD-1"></a>
Chris@82: <a name="index-FFTW_005fBACKWARD-1"></a>
Chris@82: (For single/long-double precision
Chris@82: <code>fftwf</code> and <code>fftwl</code>, <code>double</code> should be replaced by
Chris@82: <code>float</code> and <code>long double</code>, respectively.)
Chris@82: <a name="index-precision-1"></a>
Chris@82: </p>
Chris@82:
Chris@82: <p>Here, <code>n</code> is the “logical” size of the DFT, not necessarily the
Chris@82: physical size of the array. In particular, the real (<code>double</code>)
Chris@82: array has <code>n</code> elements, while the complex (<code>fftw_complex</code>)
Chris@82: array has <code>n/2+1</code> elements (where the division is rounded down).
Chris@82: For an in-place transform,
Chris@82: <a name="index-in_002dplace-1"></a>
Chris@82: <code>in</code> and <code>out</code> are aliased to the same array, which must be
Chris@82: big enough to hold both; so, the real array would actually have
Chris@82: <code>2*(n/2+1)</code> elements, where the elements beyond the first
Chris@82: <code>n</code> are unused padding. (Note that this is very different from
Chris@82: the concept of “zero-padding” a transform to a larger length, which
Chris@82: changes the logical size of the DFT by actually adding new input
Chris@82: data.) The <em>k</em>th element of the complex array is exactly the
Chris@82: same as the <em>k</em>th element of the corresponding complex DFT. All
Chris@82: positive <code>n</code> are supported; products of small factors are most
Chris@82: efficient, but an <i>O</i>(<i>n</i> log <i>n</i>)
Chris@82: algorithm is used even for prime sizes.
Chris@82: </p>
Chris@82: <p>As noted above, the c2r transform destroys its input array even for
Chris@82: out-of-place transforms. This can be prevented, if necessary, by
Chris@82: including <code>FFTW_PRESERVE_INPUT</code> in the <code>flags</code>, with
Chris@82: unfortunately some sacrifice in performance.
Chris@82: <a name="index-flags-1"></a>
Chris@82: <a name="index-FFTW_005fPRESERVE_005fINPUT"></a>
Chris@82: This flag is also not currently supported for multi-dimensional real
Chris@82: DFTs (next section).
Chris@82: </p>
Chris@82: <p>Readers familiar with DFTs of real data will recall that the 0th (the
Chris@82: “DC”) and <code>n/2</code>-th (the “Nyquist” frequency, when <code>n</code> is
Chris@82: even) elements of the complex output are purely real. Some
Chris@82: implementations therefore store the Nyquist element where the DC
Chris@82: imaginary part would go, in order to make the input and output arrays
Chris@82: the same size. Such packing, however, does not generalize well to
Chris@82: multi-dimensional transforms, and the space savings are miniscule in
Chris@82: any case; FFTW does not support it.
Chris@82: </p>
Chris@82: <p>An alternative interface for one-dimensional r2c and c2r DFTs can be
Chris@82: found in the ‘<samp>r2r</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
Chris@82: (and type) as the input array.
Chris@82: <a name="index-halfcomplex-format"></a>
Chris@82: That interface, although it is not very useful for multi-dimensional
Chris@82: transforms, may sometimes yield better performance.
Chris@82: </p>
Chris@82: <hr>
Chris@82: <div class="header">
Chris@82: <p>
Chris@82: Next: <a href="Multi_002dDimensional-DFTs-of-Real-Data.html#Multi_002dDimensional-DFTs-of-Real-Data" accesskey="n" rel="next">Multi-Dimensional DFTs of Real Data</a>, Previous: <a href="Complex-Multi_002dDimensional-DFTs.html#Complex-Multi_002dDimensional-DFTs" accesskey="p" rel="prev">Complex Multi-Dimensional DFTs</a>, Up: <a href="Tutorial.html#Tutorial" accesskey="u" rel="up">Tutorial</a> [<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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