fftw3: Multi-Dimensional DFTs of Real Data
2.4 Multi-Dimensional DFTs of Real Data
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Multi-dimensional DFTs of real data use the following planner routines:
fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
double *in, fftw_complex *out,
unsigned flags);
fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
double *in, fftw_complex *out,
unsigned flags);
fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
double *in, fftw_complex *out,
unsigned flags);
as well as the corresponding 'c2r' routines with the input/output
types swapped. These routines work similarly to their complex
analogues, except for the fact that here the complex output array is cut
roughly in half and the real array requires padding for in-place
transforms (as in 1d, above).
As before, 'n' is the logical size of the array, and the consequences
of this on the the format of the complex arrays deserve careful
attention. Suppose that the real data has dimensions n[0] x n[1] x n[2]
x ... x n[d-1] (in row-major order). Then, after an r2c transform, the
output is an n[0] x n[1] x n[2] x ... x (n[d-1]/2 + 1) array of
'fftw_complex' values in row-major order, corresponding to slightly over
half of the output of the corresponding complex DFT. (The division is
rounded down.) The ordering of the data is otherwise exactly the same
as in the complex-DFT case.
For out-of-place transforms, this is the end of the story: the real
data is stored as a row-major array of size n[0] x n[1] x n[2] x ... x
n[d-1] and the complex data is stored as a row-major array of size n[0]
x n[1] x n[2] x ... x (n[d-1]/2 + 1) .
For in-place transforms, however, extra padding of the real-data
array is necessary because the complex array is larger than the real
array, and the two arrays share the same memory locations. Thus, for
in-place transforms, the final dimension of the real-data array must be
padded with extra values to accommodate the size of the complex
data--two values if the last dimension is even and one if it is odd.
That is, the last dimension of the real data must physically contain 2 *
(n[d-1]/2+1) 'double' values (exactly enough to hold the complex data).
This physical array size does not, however, change the _logical_ array
size--only n[d-1] values are actually stored in the last dimension, and
n[d-1] is the last dimension passed to the plan-creation routine.
For example, consider the transform of a two-dimensional real array
of size 'n0' by 'n1'. The output of the r2c transform is a
two-dimensional complex array of size 'n0' by 'n1/2+1', where the 'y'
dimension has been cut nearly in half because of redundancies in the
output. Because 'fftw_complex' is twice the size of 'double', the
output array is slightly bigger than the input array. Thus, if we want
to compute the transform in place, we must _pad_ the input array so that
it is of size 'n0' by '2*(n1/2+1)'. If 'n1' is even, then there are two
padding elements at the end of each row (which need not be initialized,
as they are only used for output).
These transforms are unnormalized, so an r2c followed by a c2r
transform (or vice versa) will result in the original data scaled by the
number of real data elements--that is, the product of the (logical)
dimensions of the real data.
(Because the last dimension is treated specially, if it is equal to
'1' the transform is _not_ equivalent to a lower-dimensional r2c/c2r
transform. In that case, the last complex dimension also has size '1'
('=1/2+1'), and no advantage is gained over the complex transforms.)
Info Catalog
fftw3: One-Dimensional DFTs of Real Data
fftw3: Tutorial
fftw3: More DFTs of Real Data