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cannam@127:Multi-dimensional DFTs of real data use the following planner routines: cannam@127:
cannam@127:fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1, cannam@127: double *in, fftw_complex *out, cannam@127: unsigned flags); cannam@127: fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2, cannam@127: double *in, fftw_complex *out, cannam@127: unsigned flags); cannam@127: fftw_plan fftw_plan_dft_r2c(int rank, const int *n, cannam@127: double *in, fftw_complex *out, cannam@127: unsigned flags); cannam@127:
as well as the corresponding c2r routines with the input/output
cannam@127: types swapped. These routines work similarly to their complex
cannam@127: analogues, except for the fact that here the complex output array is cut
cannam@127: roughly in half and the real array requires padding for in-place
cannam@127: transforms (as in 1d, above).
cannam@127:
As before, n is the logical size of the array, and the
cannam@127: consequences of this on the the format of the complex arrays deserve
cannam@127: careful attention.
cannam@127:
cannam@127: Suppose that the real data has dimensions n0 × n1 × n2 × … × nd-1 (in row-major order).
cannam@127: Then, after an r2c transform, the output is an n0 × n1 × n2 × … × (nd-1/2 + 1) array of
cannam@127: fftw_complex values in row-major order, corresponding to slightly
cannam@127: over half of the output of the corresponding complex DFT. (The division
cannam@127: is rounded down.) The ordering of the data is otherwise exactly the
cannam@127: same as in the complex-DFT case.
cannam@127:
For out-of-place transforms, this is the end of the story: the real cannam@127: data is stored as a row-major array of size n0 × n1 × n2 × … × nd-1 and the complex cannam@127: data is stored as a row-major array of size n0 × n1 × n2 × … × (nd-1/2 + 1). cannam@127:
cannam@127:For in-place transforms, however, extra padding of the real-data array
cannam@127: is necessary because the complex array is larger than the real array,
cannam@127: and the two arrays share the same memory locations. Thus, for
cannam@127: in-place transforms, the final dimension of the real-data array must
cannam@127: be padded with extra values to accommodate the size of the complex
cannam@127: data—two values if the last dimension is even and one if it is odd.
cannam@127:
cannam@127: That is, the last dimension of the real data must physically contain
cannam@127: 2 * (nd-1/2+1)double values (exactly enough to hold the complex data).
cannam@127: This physical array size does not, however, change the logical
cannam@127: array size—only
cannam@127: nd-1values are actually stored in the last dimension, and
cannam@127: nd-1is the last dimension passed to the plan-creation routine.
cannam@127:
For example, consider the transform of a two-dimensional real array of
cannam@127: size n0 by n1. The output of the r2c transform is a
cannam@127: two-dimensional complex array of size n0 by n1/2+1, where
cannam@127: the y dimension has been cut nearly in half because of
cannam@127: redundancies in the output. Because fftw_complex is twice the
cannam@127: size of double, the output array is slightly bigger than the
cannam@127: input array. Thus, if we want to compute the transform in place, we
cannam@127: must pad the input array so that it is of size n0 by
cannam@127: 2*(n1/2+1). If n1 is even, then there are two padding
cannam@127: elements at the end of each row (which need not be initialized, as they
cannam@127: are only used for output).
cannam@127:
The following illustration depicts the input and output arrays just
cannam@127: described, for both the out-of-place and in-place transforms (with the
cannam@127: arrows indicating consecutive memory locations):
cannam@127:
cannam@127:
These transforms are unnormalized, so an r2c followed by a c2r cannam@127: transform (or vice versa) will result in the original data scaled by cannam@127: the number of real data elements—that is, the product of the cannam@127: (logical) dimensions of the real data. cannam@127: cannam@127:
cannam@127: cannam@127:(Because the last dimension is treated specially, if it is equal to
cannam@127: 1 the transform is not equivalent to a lower-dimensional
cannam@127: r2c/c2r transform. In that case, the last complex dimension also has
cannam@127: size 1 (=1/2+1), and no advantage is gained over the
cannam@127: complex transforms.)
cannam@127:
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