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cannam@127: <title>FFTW 3.3.5: Multi-dimensional Transforms</title>
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cannam@127: <a name="Multi_002ddimensional-Transforms"></a>
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cannam@127: <p>
cannam@127: Previous: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="p" rel="prev">1d Discrete Hartley Transforms (DHTs)</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</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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cannam@127: <hr>
cannam@127: <a name="Multi_002ddimensional-Transforms-1"></a>
cannam@127: <h4 class="subsection">4.8.6 Multi-dimensional Transforms</h4>
cannam@127:
cannam@127: <p>The multi-dimensional transforms of FFTW, in general, compute simply the
cannam@127: separable product of the given 1d transform along each dimension of the
cannam@127: array. Since each of these transforms is unnormalized, computing the
cannam@127: forward followed by the backward/inverse multi-dimensional transform
cannam@127: will result in the original array scaled by the product of the
cannam@127: normalization factors for each dimension (e.g. the product of the
cannam@127: dimension sizes, for a multi-dimensional DFT).
cannam@127: </p>
cannam@127:
cannam@127: <a name="index-r2c-3"></a>
cannam@127: <p>The definition of FFTW’s multi-dimensional DFT of real data (r2c)
cannam@127: deserves special attention. In this case, we logically compute the full
cannam@127: multi-dimensional DFT of the input data; since the input data are purely
cannam@127: real, the output data have the Hermitian symmetry and therefore only one
cannam@127: non-redundant half need be stored. More specifically, for an n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub> multi-dimensional real-input DFT, the full (logical) complex output array
cannam@127: <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
cannam@127: <i>k</i><sub><i>d-1</i></sub>]has the symmetry:
cannam@127: <i>Y</i>[<i>k</i><sub>0</sub>, <i>k</i><sub>1</sub>, ...,
cannam@127: <i>k</i><sub><i>d-1</i></sub>] = <i>Y</i>[<i>n</i><sub>0</sub> -
cannam@127: <i>k</i><sub>0</sub>, <i>n</i><sub>1</sub> - <i>k</i><sub>1</sub>, ...,
cannam@127: <i>n</i><sub><i>d-1</i></sub> - <i>k</i><sub><i>d-1</i></sub>]<sup>*</sup>(where each dimension is periodic). Because of this symmetry, we only
cannam@127: store the
cannam@127: <i>k</i><sub><i>d-1</i></sub> = 0...<i>n</i><sub><i>d-1</i></sub>/2+1elements of the <em>last</em> dimension (division by <em>2</em> is rounded
cannam@127: down). (We could instead have cut any other dimension in half, but the
cannam@127: last dimension proved computationally convenient.) This results in the
cannam@127: peculiar array format described in more detail by <a href="Real_002ddata-DFT-Array-Format.html#Real_002ddata-DFT-Array-Format">Real-data DFT Array Format</a>.
cannam@127: </p>
cannam@127: <p>The multi-dimensional c2r transform is simply the unnormalized inverse
cannam@127: of the r2c transform. i.e. it is the same as FFTW’s complex backward
cannam@127: multi-dimensional DFT, operating on a Hermitian input array in the
cannam@127: peculiar format mentioned above and outputting a real array (since the
cannam@127: DFT output is purely real).
cannam@127: </p>
cannam@127: <p>We should remind the user that the separable product of 1d transforms
cannam@127: along each dimension, as computed by FFTW, is not always the same thing
cannam@127: as the usual multi-dimensional transform. A multi-dimensional
cannam@127: <code>R2HC</code> (or <code>HC2R</code>) transform is not identical to the
cannam@127: multi-dimensional DFT, requiring some post-processing to combine the
cannam@127: requisite real and imaginary parts, as was described in <a href="The-Halfcomplex_002dformat-DFT.html#The-Halfcomplex_002dformat-DFT">The Halfcomplex-format DFT</a>. Likewise, FFTW’s multidimensional
cannam@127: <code>FFTW_DHT</code> r2r transform is not the same thing as the logical
cannam@127: multi-dimensional discrete Hartley transform defined in the literature,
cannam@127: as discussed in <a href="The-Discrete-Hartley-Transform.html#The-Discrete-Hartley-Transform">The Discrete Hartley Transform</a>.
cannam@127: </p>
cannam@127: <hr>
cannam@127: <div class="header">
cannam@127: <p>
cannam@127: Previous: <a href="1d-Discrete-Hartley-Transforms-_0028DHTs_0029.html#g_t1d-Discrete-Hartley-Transforms-_0028DHTs_0029" accesskey="p" rel="prev">1d Discrete Hartley Transforms (DHTs)</a>, Up: <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes" accesskey="u" rel="up">What FFTW Really Computes</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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