4.3.3 Real-data DFTs

     fftw_plan fftw_plan_dft_r2c_1d(int n0,
                                    double *in, fftw_complex *out,
                                    unsigned flags);
     fftw_plan fftw_plan_dft_r2c_2d(int n0, int n1,
                                    double *in, fftw_complex *out,
                                    unsigned flags);
     fftw_plan fftw_plan_dft_r2c_3d(int n0, int n1, int n2,
                                    double *in, fftw_complex *out,
                                    unsigned flags);
     fftw_plan fftw_plan_dft_r2c(int rank, const int *n,
                                 double *in, fftw_complex *out,
                                 unsigned flags);

Plan a real-input/complex-output discrete Fourier transform (DFT) in zero or more dimensions, returning an fftw_plan (see Using Plans).

Once you have created a plan for a certain transform type and parameters, then creating another plan of the same type and parameters, but for different arrays, is fast and shares constant data with the first plan (if it still exists).

The planner returns NULL if the plan cannot be created. A non-NULL plan is always returned by the basic interface unless you are using a customized FFTW configuration supporting a restricted set of transforms, or if you use the FFTW_PRESERVE_INPUT flag with a multi-dimensional out-of-place c2r transform (see below).

Arguments

• rank is the rank of the transform (it should be the size of the array *n), and can be any non-negative integer. (See Complex Multi-Dimensional DFTs, for the definition of “rank”.) The ‘_1d’, ‘_2d’, and ‘_3d’ planners correspond to a rank of 1, 2, and 3, respectively. The rank may be zero, which is equivalent to a rank-1 transform of size 1, i.e. a copy of one real number (with zero imaginary part) from input to output.
• n0, n1, n2, or n[0..rank-1], (as appropriate for each routine) specify the size of the transform dimensions. They can be any positive integer. This is different in general from the physical array dimensions, which are described in Real-data DFT Array Format.
  • FFTW is best at handling sizes of the form 2^a 3^b 5^c 7^d 11^e 13^f,where e+f is either 0 or 1, and the other exponents are arbitrary. Other sizes are computed by means of a slow, general-purpose algorithm (which nevertheless retains O(n log n) performance even for prime sizes). (It is possible to customize FFTW for different array sizes; see Installation and Customization.) Transforms whose sizes are powers of 2 are especially fast, and it is generally beneficial for the last dimension of an r2c/c2r transform to be even.
• in and out point to the input and output arrays of the transform, which may be the same (yielding an in-place transform). These arrays are overwritten during planning, unless FFTW_ESTIMATE is used in the flags. (The arrays need not be initialized, but they must be allocated.) For an in-place transform, it is important to remember that the real array will require padding, described in Real-data DFT Array Format.
• flags is a bitwise OR (‘|’) of zero or more planner flags, as defined in Planner Flags.

The inverse transforms, taking complex input (storing the non-redundant half of a logically Hermitian array) to real output, are given by:

     fftw_plan fftw_plan_dft_c2r_1d(int n0,
                                    fftw_complex *in, double *out,
                                    unsigned flags);
     fftw_plan fftw_plan_dft_c2r_2d(int n0, int n1,
                                    fftw_complex *in, double *out,
                                    unsigned flags);
     fftw_plan fftw_plan_dft_c2r_3d(int n0, int n1, int n2,
                                    fftw_complex *in, double *out,
                                    unsigned flags);
     fftw_plan fftw_plan_dft_c2r(int rank, const int *n,
                                 fftw_complex *in, double *out,
                                 unsigned flags);

The arguments are the same as for the r2c transforms, except that the input and output data formats are reversed.

FFTW computes an unnormalized transform: computing an r2c followed by a c2r transform (or vice versa) will result in the original data multiplied by the size of the transform (the product of the logical dimensions). An r2c transform produces the same output as a FFTW_FORWARD complex DFT of the same input, and a c2r transform is correspondingly equivalent to FFTW_BACKWARD. For more information, see What FFTW Really Computes.