An algorithm for computing the mixed radix fast Fourier transform

This paper presents an algorithm for computing the fast Fourier transform, based on a method proposed by Cooley and Tukey. As in their algorithm, the dimension
n
of the transform is factored (if possible), and
n/p
elementary transforms of dimension
p
are computed for each factor
p
of
n
. An improved method of computing a transform step corresponding to an odd factor of
n
is given; with this method, the number of complex multiplications for an elementary transform of dimension
p
is reduced from
(p-1)^{2}
to
(p-1)^{2}/4
for odd
p
. The fast Fourier transform, when computed in place, requires a final permutation step to arrange the results in normal order. This algorithm includes an efficient method for permuting the results in place. The algorithm is described mathematically and illustrated by a FORTRAN subroutine.