Real-data DFT Array Format

d@0: d@0: d@0: Real-data DFT Array Format - FFTW 3.2.1 d@0: d@0: d@0: d@0: d@0: d@0: d@0: d@0: d@0: d@0: d@0: d@0: d@0: d@0:

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4.3.4 Real-data DFT Array Format

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d@0: The output of a DFT of real data (r2c) contains symmetries that, in d@0: principle, make half of the outputs redundant (see What FFTW Really Computes). (Similarly for the input of an inverse c2r transform.) In d@0: practice, it is not possible to entirely realize these savings in an d@0: efficient and understandable format that generalizes to d@0: multi-dimensional transforms. Instead, the output of the r2c d@0: transforms is slightly over half of the output of the d@0: corresponding complex transform. We do not “pack” the data in any d@0: way, but store it as an ordinary array of fftw_complex values. d@0: In fact, this data is simply a subsection of what would be the array in d@0: the corresponding complex transform. d@0: d@0:

Specifically, for a real transform of d (= rank) d@0: dimensions n₀ × n₁ × n₂ × … × n_d-1, the complex data is an n₀ × n₁ × n₂ × … × (n_d-1/2 + 1) array of d@0: fftw_complex values in row-major order (with the division rounded d@0: down). That is, we only store the lower half (non-negative d@0: frequencies), plus one element, of the last dimension of the data from d@0: the ordinary complex transform. (We could have instead taken half of d@0: any other dimension, but implementation turns out to be simpler if the d@0: last, contiguous, dimension is used.) d@0: d@0:

For an out-of-place transform, the real data is simply an array with d@0: physical dimensions n₀ × n₁ × n₂ × … × n_d-1 in row-major order. d@0: d@0:

For an in-place transform, some complications arise since the complex data d@0: is slightly larger than the real data. In this case, the final d@0: dimension of the real data must be padded with extra values to d@0: accommodate the size of the complex data—two extra if the last d@0: dimension is even and one if it is odd. That is, the last dimension of d@0: the real data must physically contain d@0: 2 * (n_d-1/2+1)double values (exactly enough to hold the complex data). This d@0: physical array size does not, however, change the logical array d@0: size—only d@0: n_d-1values are actually stored in the last dimension, and d@0: n_d-1is the last dimension passed to the planner. d@0: d@0: d@0: d@0: