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Chris@10: <h4 class="subsection">4.3.4 Real-data DFT Array Format</h4>
Chris@10:
Chris@10: <p><a name="index-r2c_002fc2r-multi_002ddimensional-array-format-199"></a>
Chris@10: The output of a DFT of real data (r2c) contains symmetries that, in
Chris@10: principle, make half of the outputs redundant (see <a href="What-FFTW-Really-Computes.html#What-FFTW-Really-Computes">What FFTW Really Computes</a>). (Similarly for the input of an inverse c2r transform.) In
Chris@10: practice, it is not possible to entirely realize these savings in an
Chris@10: efficient and understandable format that generalizes to
Chris@10: multi-dimensional transforms. Instead, the output of the r2c
Chris@10: transforms is <em>slightly</em> over half of the output of the
Chris@10: corresponding complex transform. We do not “pack” the data in any
Chris@10: way, but store it as an ordinary array of <code>fftw_complex</code> values.
Chris@10: In fact, this data is simply a subsection of what would be the array in
Chris@10: the corresponding complex transform.
Chris@10:
Chris@10: <p>Specifically, for a real transform of d (= <code>rank</code>)
Chris@10: dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub>, the complex data is an n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × (n<sub>d-1</sub>/2 + 1) array of
Chris@10: <code>fftw_complex</code> values in row-major order (with the division rounded
Chris@10: down). That is, we only store the <em>lower</em> half (non-negative
Chris@10: frequencies), plus one element, of the last dimension of the data from
Chris@10: the ordinary complex transform. (We could have instead taken half of
Chris@10: any other dimension, but implementation turns out to be simpler if the
Chris@10: last, contiguous, dimension is used.)
Chris@10:
Chris@10: <p><a name="index-out_002dof_002dplace-200"></a>For an out-of-place transform, the real data is simply an array with
Chris@10: physical dimensions n<sub>0</sub> × n<sub>1</sub> × n<sub>2</sub> × … × n<sub>d-1</sub> in row-major order.
Chris@10:
Chris@10: <p><a name="index-in_002dplace-201"></a><a name="index-padding-202"></a>For an in-place transform, some complications arise since the complex data
Chris@10: is slightly larger than the real data. In this case, the final
Chris@10: dimension of the real data must be <em>padded</em> with extra values to
Chris@10: accommodate the size of the complex data—two extra if the last
Chris@10: dimension is even and one if it is odd. That is, the last dimension of
Chris@10: the real data must physically contain
Chris@10: 2 * (n<sub>d-1</sub>/2+1)<code>double</code> values (exactly enough to hold the complex data). This
Chris@10: physical array size does not, however, change the <em>logical</em> array
Chris@10: size—only
Chris@10: n<sub>d-1</sub>values are actually stored in the last dimension, and
Chris@10: n<sub>d-1</sub>is the last dimension passed to the planner.
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