4.8.6 Multi-dimensional Transforms

The multi-dimensional transforms of FFTW, in general, compute simply the separable product of the given 1d transform along each dimension of the array. Since each of these transforms is unnormalized, computing the forward followed by the backward/inverse multi-dimensional transform will result in the original array scaled by the product of the normalization factors for each dimension (e.g. the product of the dimension sizes, for a multi-dimensional DFT).

The definition of FFTW’s multi-dimensional DFT of real data (r2c) deserves special attention. In this case, we logically compute the full multi-dimensional DFT of the input data; since the input data are purely real, the output data have the Hermitian symmetry and therefore only one non-redundant half need be stored. More specifically, for an n₀ × n₁ × n₂ × … × n_d-1 multi-dimensional real-input DFT, the full (logical) complex output array Y[k₀, k₁, ..., k_d-1] has the symmetry: Y[k₀, k₁, ..., k_d-1] = Y[n₀ - k₀, n₁ - k₁, ..., n_d-1 - k_d-1]^* (where each dimension is periodic). Because of this symmetry, we only store the k_d-1 = 0...n_d-1/2+1 elements of the last dimension (division by 2 is rounded down). (We could instead have cut any other dimension in half, but the last dimension proved computationally convenient.) This results in the peculiar array format described in more detail by Real-data DFT Array Format.

The multi-dimensional c2r transform is simply the unnormalized inverse of the r2c transform. i.e. it is the same as FFTW’s complex backward multi-dimensional DFT, operating on a Hermitian input array in the peculiar format mentioned above and outputting a real array (since the DFT output is purely real).

We should remind the user that the separable product of 1d transforms along each dimension, as computed by FFTW, is not always the same thing as the usual multi-dimensional transform. A multi-dimensional R2HC (or HC2R) transform is not identical to the multi-dimensional DFT, requiring some post-processing to combine the requisite real and imaginary parts, as was described in The Halfcomplex-format DFT. Likewise, FFTW’s multidimensional FFTW_DHT r2r transform is not the same thing as the logical multi-dimensional discrete Hartley transform defined in the literature, as discussed in The Discrete Hartley Transform.