### 16.4 Mixed-radix FFT routines for complex data

This section describes mixed-radix FFT algorithms for complex data. The mixed-radix functions work for FFTs of any length. They are a reimplementation of Paul Swarztrauber’s Fortran FFTPACK library. The theory is explained in the review article Self-sorting Mixed-radix FFTs by Clive Temperton. The routines here use the same indexing scheme and basic algorithms as FFTPACK.

The mixed-radix algorithm is based on sub-transform modules—highly optimized small length FFTs which are combined to create larger FFTs. There are efficient modules for factors of 2, 3, 4, 5, 6 and 7. The modules for the composite factors of 4 and 6 are faster than combining the modules for 2*2 and 2*3.

For factors which are not implemented as modules there is a fall-back to a general length-n module which uses Singleton’s method for efficiently computing a DFT. This module is O(n^2), and slower than a dedicated module would be but works for any length n. Of course, lengths which use the general length-n module will still be factorized as much as possible. For example, a length of 143 will be factorized into 11*13. Large prime factors are the worst case scenario, e.g. as found in n=2*3*99991, and should be avoided because their O(n^2) scaling will dominate the run-time (consult the document GSL FFT Algorithms included in the GSL distribution if you encounter this problem).

The mixed-radix initialization function `gsl_fft_complex_wavetable_alloc` returns the list of factors chosen by the library for a given length n. It can be used to check how well the length has been factorized, and estimate the run-time. To a first approximation the run-time scales as n \sum f_i, where the f_i are the factors of n. For programs under user control you may wish to issue a warning that the transform will be slow when the length is poorly factorized. If you frequently encounter data lengths which cannot be factorized using the existing small-prime modules consult GSL FFT Algorithms for details on adding support for other factors.

All the functions described in this section are declared in the header file `gsl_fft_complex.h`.

Function: gsl_fft_complex_wavetable * gsl_fft_complex_wavetable_alloc (size_t n)

  This function prepares a trigonometric lookup table for a complex FFT of length n. The function returns a pointer to the newly allocated `gsl_fft_complex_wavetable` if no errors were detected, and a null pointer in the case of error. The length n is factorized into a product of subtransforms, and the factors and their trigonometric coefficients are stored in the wavetable. The trigonometric coefficients are computed using direct calls to `sin` and `cos`, for accuracy. Recursion relations could be used to compute the lookup table faster, but if an application performs many FFTs of the same length then this computation is a one-off overhead which does not affect the final throughput.

  The wavetable structure can be used repeatedly for any transform of the same length. The table is not modified by calls to any of the other FFT functions. The same wavetable can be used for both forward and backward (or inverse) transforms of a given length.

Function: void gsl_fft_complex_wavetable_free (gsl_fft_complex_wavetable * wavetable)

  This function frees the memory associated with the wavetable wavetable. The wavetable can be freed if no further FFTs of the same length will be needed.

These functions operate on a `gsl_fft_complex_wavetable` structure which contains internal parameters for the FFT. It is not necessary to set any of the components directly but it can sometimes be useful to examine them. For example, the chosen factorization of the FFT length is given and can be used to provide an estimate of the run-time or numerical error. The wavetable structure is declared in the header file `gsl_fft_complex.h`.

Data Type: gsl_fft_complex_wavetable

  This is a structure that holds the factorization and trigonometric lookup tables for the mixed radix fft algorithm. It has the following components:

  `size_t n`

    This is the number of complex data points

  `size_t nf`

    This is the number of factors that the length `n` was decomposed into.

  `size_t factor[64]`

    This is the array of factors. Only the first `nf` elements are used.

  `gsl_complex * trig`

    This is a pointer to a preallocated trigonometric lookup table of `n` complex elements.

  `gsl_complex * twiddle[64]`

    This is an array of pointers into `trig`, giving the twiddle factors for each pass.

The mixed radix algorithms require additional working space to hold the intermediate steps of the transform.

Function: gsl_fft_complex_workspace * gsl_fft_complex_workspace_alloc (size_t n)

  This function allocates a workspace for a complex transform of length n.

Function: void gsl_fft_complex_workspace_free (gsl_fft_complex_workspace * workspace)

  This function frees the memory associated with the workspace workspace. The workspace can be freed if no further FFTs of the same length will be needed.

The following functions compute the transform,

Function: int gsl_fft_complex_forward (gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)
Function: int gsl_fft_complex_transform (gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work, gsl_fft_direction sign)
Function: int gsl_fft_complex_backward (gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)
Function: int gsl_fft_complex_inverse (gsl_complex_packed_array data, size_t stride, size_t n, const gsl_fft_complex_wavetable * wavetable, gsl_fft_complex_workspace * work)

  These functions compute forward, backward and inverse FFTs of length n with stride stride, on the packed complex array data, using a mixed radix decimation-in-frequency algorithm. There is no restriction on the length n. Efficient modules are provided for subtransforms of length 2, 3, 4, 5, 6 and 7. Any remaining factors are computed with a slow, O(n^2), general-n module. The caller must supply a wavetable containing the trigonometric lookup tables and a workspace work. For the `transform` version of the function the sign argument can be either `forward` (-1) or `backward` (+1).

  The functions return a value of `0` if no errors were detected. The following `gsl_errno` conditions are defined for these functions:

  `GSL_EDOM`

    The length of the data n is not a positive integer (i.e. n is zero).

  `GSL_EINVAL`

    The length of the data n and the length used to compute the given wavetable do not match.

Here is an example program which computes the FFT of a short pulse in a sample of length 630 (=2*3*3*5*7) using the mixed-radix algorithm.

    #include <stdio.h>
    #include <math.h>
    #include <gsl/gsl_errno.h>
    #include <gsl/gsl_fft_complex.h>

    #define REAL(z,i) ((z)[2*(i)])
    #define IMAG(z,i) ((z)[2*(i)+1])

    int
    main (void)
    {
      int i;
      const int n = 630;
      double data[2*n];

      gsl_fft_complex_wavetable * wavetable;
      gsl_fft_complex_workspace * workspace;

      for (i = 0; i < n; i++)
        {
          REAL(data,i) = 0.0;
          IMAG(data,i) = 0.0;
        }

      data[0] = 1.0;

      for (i = 1; i <= 10; i++)
        {
          REAL(data,i) = REAL(data,n-i) = 1.0;
        }

      for (i = 0; i < n; i++)
        {
          printf ("%d: %e %e\n", i, REAL(data,i),
                                    IMAG(data,i));
        }
      printf ("\n");

      wavetable = gsl_fft_complex_wavetable_alloc (n);
      workspace = gsl_fft_complex_workspace_alloc (n);

      for (i = 0; i < (int) wavetable->nf; i++)
        {
           printf ("# factor %d: %zu\n", i,
                   wavetable->factor[i]);
        }

      gsl_fft_complex_forward (data, 1, n,
                               wavetable, workspace);

      for (i = 0; i < n; i++)
        {
          printf ("%d: %e %e\n", i, REAL(data,i),
                                    IMAG(data,i));
        }

      gsl_fft_complex_wavetable_free (wavetable);
      gsl_fft_complex_workspace_free (workspace);
      return 0;
    }

Note that we have assumed that the program is using the default `gsl` error handler (which calls `abort` for any errors). If you are not using a safe error handler you would need to check the return status of all the `gsl` routines.