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fftw3: Complex Multi-Dimensional DFTs

 
 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').
 
    FFTW provides two routines for creating plans for 2d and 3d
 transforms, and one routine for creating plans of arbitrary
 dimensionality.  The 2d and 3d routines have the following signature:
      fftw_plan fftw_plan_dft_2d(int n0, int n1,
                                 fftw_complex *in, fftw_complex *out,
                                 int sign, unsigned flags);
      fftw_plan fftw_plan_dft_3d(int n0, int n1, int n2,
                                 fftw_complex *in, fftw_complex *out,
                                 int sign, unsigned flags);
 
    These routines create plans for 'n0' by 'n1' two-dimensional (2d)
 transforms and 'n0' by 'n1' by 'n2' 3d transforms, respectively.  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.
 
    FFTW can also compute transforms of higher dimensionality.  In order
 to avoid confusion between the various meanings of the the word
 "dimension", we use the term _rank_ to denote the number of independent
 indices in an array.(1)  For example, we say that a 2d transform has
 rank 2, a 3d transform has rank 3, and so on.  You can plan transforms
 of arbitrary rank by means of the following function:
 
      fftw_plan fftw_plan_dft(int rank, const int *n,
                              fftw_complex *in, fftw_complex *out,
                              int sign, unsigned flags);
 
    Here, 'n' is a pointer to an array 'n[rank]' denoting an 'n[0]' by
 'n[1]' by ... by 'n[rank-1]' transform.  Thus, for example, the call
      fftw_plan_dft_2d(n0, n1, in, out, sign, flags);
    is equivalent to the following code fragment:
      int n[2];
      n[0] = n0;
      n[1] = n1;
      fftw_plan_dft(2, n, in, out, sign, flags);
    'fftw_plan_dft' is not restricted to 2d and 3d transforms, however,
 but it can plan transforms of arbitrary rank.
 
    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 'n0' and/or 'n1' 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.
 
    ---------- Footnotes ----------
 
    (1) The term "rank" is commonly used in the APL, FORTRAN, and Common
 Lisp traditions, although it is not so common in the C world.
 

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