cannam@95: Previous: Multi-Dimensional DFTs of Real Data, cannam@95: Up: Tutorial cannam@95:
FFTW supports several other transform types via a unified r2r
cannam@95: (real-to-real) interface,
cannam@95: so called because it takes a real (double) array and outputs a
cannam@95: real array of the same size. These r2r transforms currently fall into
cannam@95: three categories: DFTs of real input and complex-Hermitian output in
cannam@95: halfcomplex format, DFTs of real input with even/odd symmetry
cannam@95: (a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete
cannam@95: Hartley transforms (DHTs), all described in more detail by the
cannam@95: following sections.
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The r2r transforms follow the by now familiar interface of creating an
cannam@95: fftw_plan, executing it with fftw_execute(plan), and
cannam@95: destroying it with fftw_destroy_plan(plan). Furthermore, all
cannam@95: r2r transforms share the same planner interface:
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fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out, cannam@95: fftw_r2r_kind kind, unsigned flags); cannam@95: fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out, cannam@95: fftw_r2r_kind kind0, fftw_r2r_kind kind1, cannam@95: unsigned flags); cannam@95: fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2, cannam@95: double *in, double *out, cannam@95: fftw_r2r_kind kind0, cannam@95: fftw_r2r_kind kind1, cannam@95: fftw_r2r_kind kind2, cannam@95: unsigned flags); cannam@95: fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out, cannam@95: const fftw_r2r_kind *kind, unsigned flags); cannam@95:cannam@95:
cannam@95: Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional
cannam@95: transforms for contiguous arrays in row-major order, transforming (real)
cannam@95: input to output of the same size, where n specifies the
cannam@95: physical dimensions of the arrays. All positive n are
cannam@95: supported (with the exception of n=1 for the FFTW_REDFT00
cannam@95: kind, noted in the real-even subsection below); products of small
cannam@95: factors are most efficient (factorizing n-1 and n+1 for
cannam@95: FFTW_REDFT00 and FFTW_RODFT00 kinds, described below), but
cannam@95: an O(n log n) algorithm is used even for prime sizes.
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Each dimension has a kind parameter, of type
cannam@95: fftw_r2r_kind, specifying the kind of r2r transform to be used
cannam@95: for that dimension.
cannam@95: (In the case of fftw_plan_r2r, this is an array kind[rank]
cannam@95: where kind[i] is the transform kind for the dimension
cannam@95: n[i].) The kind can be one of a set of predefined constants,
cannam@95: defined in the following subsections.
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In other words, FFTW computes the separable product of the specified cannam@95: r2r transforms over each dimension, which can be used e.g. for partial cannam@95: differential equations with mixed boundary conditions. (For some r2r cannam@95: kinds, notably the halfcomplex DFT and the DHT, such a separable cannam@95: product is somewhat problematic in more than one dimension, however, cannam@95: as is described below.) cannam@95: cannam@95:
In the current version of FFTW, all r2r transforms except for the cannam@95: halfcomplex type are computed via pre- or post-processing of cannam@95: halfcomplex transforms, and they are therefore not as fast as they cannam@95: could be. Since most other general DCT/DST codes employ a similar cannam@95: algorithm, however, FFTW's implementation should provide at least cannam@95: competitive performance. cannam@95: cannam@95: cannam@95: cannam@95: