Chris@19: Previous: 1d Discrete Hartley Transforms (DHTs), Chris@19: Up: What FFTW Really Computes Chris@19:
The multi-dimensional transforms of FFTW, in general, compute simply the Chris@19: separable product of the given 1d transform along each dimension of the Chris@19: array. Since each of these transforms is unnormalized, computing the Chris@19: forward followed by the backward/inverse multi-dimensional transform Chris@19: will result in the original array scaled by the product of the Chris@19: normalization factors for each dimension (e.g. the product of the Chris@19: dimension sizes, for a multi-dimensional DFT). Chris@19: Chris@19:
The definition of FFTW's multi-dimensional DFT of real data (r2c) Chris@19: deserves special attention. In this case, we logically compute the full Chris@19: multi-dimensional DFT of the input data; since the input data are purely Chris@19: real, the output data have the Hermitian symmetry and therefore only one Chris@19: 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@19: Y[k0, k1, ..., Chris@19: kd-1]has the symmetry: Chris@19: Y[k0, k1, ..., Chris@19: kd-1] = Y[n0 - Chris@19: k0, n1 - k1, ..., Chris@19: nd-1 - kd-1]*(where each dimension is periodic). Because of this symmetry, we only Chris@19: store the Chris@19: kd-1 = 0...nd-1/2+1elements of the last dimension (division by 2 is rounded Chris@19: down). (We could instead have cut any other dimension in half, but the Chris@19: last dimension proved computationally convenient.) This results in the Chris@19: peculiar array format described in more detail by Real-data DFT Array Format. Chris@19: Chris@19:
The multi-dimensional c2r transform is simply the unnormalized inverse Chris@19: of the r2c transform. i.e. it is the same as FFTW's complex backward Chris@19: multi-dimensional DFT, operating on a Hermitian input array in the Chris@19: peculiar format mentioned above and outputting a real array (since the Chris@19: DFT output is purely real). Chris@19: Chris@19:
We should remind the user that the separable product of 1d transforms
Chris@19: along each dimension, as computed by FFTW, is not always the same thing
Chris@19: as the usual multi-dimensional transform. A multi-dimensional
Chris@19: R2HC (or HC2R) transform is not identical to the
Chris@19: multi-dimensional DFT, requiring some post-processing to combine the
Chris@19: requisite real and imaginary parts, as was described in The Halfcomplex-format DFT. Likewise, FFTW's multidimensional
Chris@19: FFTW_DHT r2r transform is not the same thing as the logical
Chris@19: multi-dimensional discrete Hartley transform defined in the literature,
Chris@19: as discussed in The Discrete Hartley Transform.
Chris@19:
Chris@19:
Chris@19: