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cannam@127:The multi-dimensional transforms of FFTW, in general, compute simply the cannam@127: separable product of the given 1d transform along each dimension of the cannam@127: array. Since each of these transforms is unnormalized, computing the cannam@127: forward followed by the backward/inverse multi-dimensional transform cannam@127: will result in the original array scaled by the product of the cannam@127: normalization factors for each dimension (e.g. the product of the cannam@127: dimension sizes, for a multi-dimensional DFT). cannam@127:
cannam@127: cannam@127: cannam@127:The definition of FFTW’s multi-dimensional DFT of real data (r2c) cannam@127: deserves special attention. In this case, we logically compute the full cannam@127: multi-dimensional DFT of the input data; since the input data are purely cannam@127: real, the output data have the Hermitian symmetry and therefore only one cannam@127: 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 cannam@127: Y[k0, k1, ..., cannam@127: kd-1]has the symmetry: cannam@127: Y[k0, k1, ..., cannam@127: kd-1] = Y[n0 - cannam@127: k0, n1 - k1, ..., cannam@127: nd-1 - kd-1]*(where each dimension is periodic). Because of this symmetry, we only cannam@127: store the cannam@127: kd-1 = 0...nd-1/2+1elements of the last dimension (division by 2 is rounded cannam@127: down). (We could instead have cut any other dimension in half, but the cannam@127: last dimension proved computationally convenient.) This results in the cannam@127: peculiar array format described in more detail by Real-data DFT Array Format. cannam@127:
cannam@127:The multi-dimensional c2r transform is simply the unnormalized inverse cannam@127: of the r2c transform. i.e. it is the same as FFTW’s complex backward cannam@127: multi-dimensional DFT, operating on a Hermitian input array in the cannam@127: peculiar format mentioned above and outputting a real array (since the cannam@127: DFT output is purely real). cannam@127:
cannam@127:We should remind the user that the separable product of 1d transforms
cannam@127: along each dimension, as computed by FFTW, is not always the same thing
cannam@127: as the usual multi-dimensional transform. A multi-dimensional
cannam@127: R2HC (or HC2R) transform is not identical to the
cannam@127: multi-dimensional DFT, requiring some post-processing to combine the
cannam@127: requisite real and imaginary parts, as was described in The Halfcomplex-format DFT. Likewise, FFTW’s multidimensional
cannam@127: FFTW_DHT r2r transform is not the same thing as the logical
cannam@127: multi-dimensional discrete Hartley transform defined in the literature,
cannam@127: as discussed in The Discrete Hartley Transform.
cannam@127:
cannam@127: Previous: 1d Discrete Hartley Transforms (DHTs), Up: What FFTW Really Computes [Contents][Index]
cannam@127: