Chris@10: Previous: Multi-Dimensional DFTs of Real Data, Chris@10: Up: Tutorial Chris@10:
FFTW supports several other transform types via a unified r2r
Chris@10: (real-to-real) interface,
Chris@10: so called because it takes a real (double) array and outputs a
Chris@10: real array of the same size. These r2r transforms currently fall into
Chris@10: three categories: DFTs of real input and complex-Hermitian output in
Chris@10: halfcomplex format, DFTs of real input with even/odd symmetry
Chris@10: (a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete
Chris@10: Hartley transforms (DHTs), all described in more detail by the
Chris@10: following sections.
Chris@10:
Chris@10:
The r2r transforms follow the by now familiar interface of creating an
Chris@10: fftw_plan, executing it with fftw_execute(plan), and
Chris@10: destroying it with fftw_destroy_plan(plan). Furthermore, all
Chris@10: r2r transforms share the same planner interface:
Chris@10:
Chris@10:
fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out, Chris@10: fftw_r2r_kind kind, unsigned flags); Chris@10: fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out, Chris@10: fftw_r2r_kind kind0, fftw_r2r_kind kind1, Chris@10: unsigned flags); Chris@10: fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2, Chris@10: double *in, double *out, Chris@10: fftw_r2r_kind kind0, Chris@10: fftw_r2r_kind kind1, Chris@10: fftw_r2r_kind kind2, Chris@10: unsigned flags); Chris@10: fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out, Chris@10: const fftw_r2r_kind *kind, unsigned flags); Chris@10:Chris@10:
Chris@10: Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional
Chris@10: transforms for contiguous arrays in row-major order, transforming (real)
Chris@10: input to output of the same size, where n specifies the
Chris@10: physical dimensions of the arrays. All positive n are
Chris@10: supported (with the exception of n=1 for the FFTW_REDFT00
Chris@10: kind, noted in the real-even subsection below); products of small
Chris@10: factors are most efficient (factorizing n-1 and n+1 for
Chris@10: FFTW_REDFT00 and FFTW_RODFT00 kinds, described below), but
Chris@10: an O(n log n) algorithm is used even for prime sizes.
Chris@10:
Chris@10:
Each dimension has a kind parameter, of type
Chris@10: fftw_r2r_kind, specifying the kind of r2r transform to be used
Chris@10: for that dimension.
Chris@10: (In the case of fftw_plan_r2r, this is an array kind[rank]
Chris@10: where kind[i] is the transform kind for the dimension
Chris@10: n[i].) The kind can be one of a set of predefined constants,
Chris@10: defined in the following subsections.
Chris@10:
Chris@10:
In other words, FFTW computes the separable product of the specified Chris@10: r2r transforms over each dimension, which can be used e.g. for partial Chris@10: differential equations with mixed boundary conditions. (For some r2r Chris@10: kinds, notably the halfcomplex DFT and the DHT, such a separable Chris@10: product is somewhat problematic in more than one dimension, however, Chris@10: as is described below.) Chris@10: Chris@10:
In the current version of FFTW, all r2r transforms except for the Chris@10: halfcomplex type are computed via pre- or post-processing of Chris@10: halfcomplex transforms, and they are therefore not as fast as they Chris@10: could be. Since most other general DCT/DST codes employ a similar Chris@10: algorithm, however, FFTW's implementation should provide at least Chris@10: competitive performance. Chris@10: Chris@10: Chris@10: Chris@10: