Chris@10: Previous: 1d Discrete Hartley Transforms (DHTs), Chris@10: Up: What FFTW Really Computes Chris@10:
The multi-dimensional transforms of FFTW, in general, compute simply the Chris@10: separable product of the given 1d transform along each dimension of the Chris@10: array. Since each of these transforms is unnormalized, computing the Chris@10: forward followed by the backward/inverse multi-dimensional transform Chris@10: will result in the original array scaled by the product of the Chris@10: normalization factors for each dimension (e.g. the product of the Chris@10: dimension sizes, for a multi-dimensional DFT). Chris@10: Chris@10:
The definition of FFTW's multi-dimensional DFT of real data (r2c) Chris@10: deserves special attention. In this case, we logically compute the full Chris@10: multi-dimensional DFT of the input data; since the input data are purely Chris@10: real, the output data have the Hermitian symmetry and therefore only one Chris@10: non-redundant half need be stored. More specifically, for an n0 × n1 × n2 × … × nd-1 multi-dimensional real-input DFT, the full (logical) complex output array Chris@10: Y[k0, k1, ..., Chris@10: kd-1]has the symmetry: Chris@10: Y[k0, k1, ..., Chris@10: kd-1] = Y[n0 - Chris@10: k0, n1 - k1, ..., Chris@10: nd-1 - kd-1]*(where each dimension is periodic). Because of this symmetry, we only Chris@10: store the Chris@10: kd-1 = 0...nd-1/2+1elements of the last dimension (division by 2 is rounded Chris@10: down). (We could instead have cut any other dimension in half, but the Chris@10: last dimension proved computationally convenient.) This results in the Chris@10: peculiar array format described in more detail by Real-data DFT Array Format. Chris@10: Chris@10:
The multi-dimensional c2r transform is simply the unnormalized inverse Chris@10: of the r2c transform. i.e. it is the same as FFTW's complex backward Chris@10: multi-dimensional DFT, operating on a Hermitian input array in the Chris@10: peculiar format mentioned above and outputting a real array (since the Chris@10: DFT output is purely real). Chris@10: Chris@10:
We should remind the user that the separable product of 1d transforms
Chris@10: along each dimension, as computed by FFTW, is not always the same thing
Chris@10: as the usual multi-dimensional transform. A multi-dimensional
Chris@10: R2HC (or HC2R) transform is not identical to the
Chris@10: multi-dimensional DFT, requiring some post-processing to combine the
Chris@10: requisite real and imaginary parts, as was described in The Halfcomplex-format DFT. Likewise, FFTW's multidimensional
Chris@10: FFTW_DHT r2r transform is not the same thing as the logical
Chris@10: multi-dimensional discrete Hartley transform defined in the literature,
Chris@10: as discussed in The Discrete Hartley Transform.
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