FFTW 3.3.5: More DFTs of Real Data

FFTW supports several other transform types via a unified r2r Chris@42: (real-to-real) interface, Chris@42: Chris@42: so called because it takes a real (double) array and outputs a Chris@42: real array of the same size. These r2r transforms currently fall into Chris@42: three categories: DFTs of real input and complex-Hermitian output in Chris@42: halfcomplex format, DFTs of real input with even/odd symmetry Chris@42: (a.k.a. discrete cosine/sine transforms, DCTs/DSTs), and discrete Chris@42: Hartley transforms (DHTs), all described in more detail by the Chris@42: following sections. Chris@42:

The r2r transforms follow the by now familiar interface of creating an Chris@42: fftw_plan, executing it with fftw_execute(plan), and Chris@42: destroying it with fftw_destroy_plan(plan). Furthermore, all Chris@42: r2r transforms share the same planner interface: Chris@42:

Chris@42:

fftw_plan fftw_plan_r2r_1d(int n, double *in, double *out,
Chris@42:                            fftw_r2r_kind kind, unsigned flags);
Chris@42: fftw_plan fftw_plan_r2r_2d(int n0, int n1, double *in, double *out,
Chris@42:                            fftw_r2r_kind kind0, fftw_r2r_kind kind1,
Chris@42:                            unsigned flags);
Chris@42: fftw_plan fftw_plan_r2r_3d(int n0, int n1, int n2,
Chris@42:                            double *in, double *out,
Chris@42:                            fftw_r2r_kind kind0,
Chris@42:                            fftw_r2r_kind kind1,
Chris@42:                            fftw_r2r_kind kind2,
Chris@42:                            unsigned flags);
Chris@42: fftw_plan fftw_plan_r2r(int rank, const int *n, double *in, double *out,
Chris@42:                         const fftw_r2r_kind *kind, unsigned flags);
Chris@42:

Just as for the complex DFT, these plan 1d/2d/3d/multi-dimensional Chris@42: transforms for contiguous arrays in row-major order, transforming (real) Chris@42: input to output of the same size, where n specifies the Chris@42: physical dimensions of the arrays. All positive n are Chris@42: supported (with the exception of n=1 for the FFTW_REDFT00 Chris@42: kind, noted in the real-even subsection below); products of small Chris@42: factors are most efficient (factorizing n-1 and n+1 for Chris@42: FFTW_REDFT00 and FFTW_RODFT00 kinds, described below), but Chris@42: an O(n log n) algorithm is used even for prime sizes. Chris@42:

Each dimension has a kind parameter, of type Chris@42: fftw_r2r_kind, specifying the kind of r2r transform to be used Chris@42: for that dimension. Chris@42: Chris@42: Chris@42: (In the case of fftw_plan_r2r, this is an array kind[rank] Chris@42: where kind[i] is the transform kind for the dimension Chris@42: n[i].) The kind can be one of a set of predefined constants, Chris@42: defined in the following subsections. Chris@42:

In other words, FFTW computes the separable product of the specified Chris@42: r2r transforms over each dimension, which can be used e.g. for partial Chris@42: differential equations with mixed boundary conditions. (For some r2r Chris@42: kinds, notably the halfcomplex DFT and the DHT, such a separable Chris@42: product is somewhat problematic in more than one dimension, however, Chris@42: as is described below.) Chris@42:

In the current version of FFTW, all r2r transforms except for the Chris@42: halfcomplex type are computed via pre- or post-processing of Chris@42: halfcomplex transforms, and they are therefore not as fast as they Chris@42: could be. Since most other general DCT/DST codes employ a similar Chris@42: algorithm, however, FFTW’s implementation should provide at least Chris@42: competitive performance. Chris@42:

• The Halfcomplex-format DFT:		Chris@42:
• Real even/odd DFTs (cosine/sine transforms):		Chris@42:
• The Discrete Hartley Transform:		Chris@42:

2.5 More DFTs of Real Data