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2.2 Complex Multi-Dimensional DFTs

Multi-dimensional transforms work much the same way as one-dimensional transforms: you allocate arrays of fftw_complex (preferably using fftw_malloc), create an fftw_plan, execute it as many times as you want with fftw_execute(plan), and clean up with fftw_destroy_plan(plan) (and fftw_free). The only difference is the routine you use to create the plan:

fftw_plan fftw_plan_dft_2d(int nx, int ny,
                           fftw_complex *in, fftw_complex *out,
                           int sign, unsigned flags);
fftw_plan fftw_plan_dft_3d(int nx, int ny, int nz,
                           fftw_complex *in, fftw_complex *out,
                           int sign, unsigned flags);
fftw_plan fftw_plan_dft(int rank, const int *n,
                        fftw_complex *in, fftw_complex *out,
                        int sign, unsigned flags);

These routines create plans for nx by ny two-dimensional (2d) transforms, nx by ny by nz 3d transforms, and arbitrary rank-dimensional transforms, respectively. In the third case, n is a pointer to an array n[rank] denoting an n[0] by n[1] by ... by n[rank-1] transform. All of these transforms operate on contiguous arrays in the C-standard row-major order, so that the last dimension has the fastest-varying index in the array. This layout is described further in Multi-dimensional Array Format.

You may have noticed that all the planner routines described so far have overlapping functionality. For example, you can plan a 1d or 2d transform by using fftw_plan_dft with a rank of 1 or 2, or even by calling fftw_plan_dft_3d with nx and/or ny equal to 1 (with no loss in efficiency). This pattern continues, and FFTW's planning routines in general form a "partial order," sequences of interfaces with strictly increasing generality but correspondingly greater complexity.

fftw_plan_dft is the most general complex-DFT routine that we describe in this tutorial, but there are also the advanced and guru interfaces, which allow one to efficiently combine multiple/strided transforms into a single FFTW plan, transform a subset of a larger multi-dimensional array, and/or to handle more general complex-number formats. For more information, see FFTW Reference.