The prime-factor algorithm , also called the Good–Thomas algorithm, is a fast Fourier transform algorithm that re-expresses the discrete Fourier transform of a sizeN = N1N2 as a two-dimensional N1×N2 DFT, but only for the case where N1 and N2 are relatively prime. These smaller transforms of size N1 and N2 can then be evaluated by applying PFA recursively or by using some other FFT algorithm. PFA should not be confused with the mixed-radix generalization of the popular Cooley–Tukey algorithm, which also subdivides a DFT of size N = N1N2 into smaller transforms of size N1 and N2. The latter algorithm can use any factors, but it has the disadvantage that it also requires extra multiplications by roots of unity called twiddle factors, in addition to the smaller transforms. On the other hand, PFA has the disadvantages that it only works for relatively prime factors and that it requires a more complicated re-indexing of the data based on the Chinese remainder theorem. Note, however, that PFA can be combined with mixed-radix Cooley–Tukey, with the former factorizingN into relatively prime components and the latter handling repeated factors. PFA is also closely related to the nested Winograd FFT algorithm, where the latter performs the decomposed N1 by N2transform via more sophisticated two-dimensional convolution techniques. Some older papers therefore also call Winograd's algorithm a PFA FFT.
Algorithm
Recall that the DFT is defined by the formula: The PFA involves a re-indexing of the input and output arrays, which when substituted into the DFT formula transforms it into two nested DFTs.
Re-indexing
Suppose that N = N1N2, where N1 and N2 are relatively prime. In this case, we can define a bijective re-indexing of the input n and output k by: where N1−1 denotes the modular multiplicative inverse of N1moduloN2 and vice versa for N2−1; the indices ka and na run from 0, …, Na − 1. These inverses only exist for relatively prime N1 and N2, and that condition is also required for the first mapping to be bijective. This re-indexing of n is called the Ruritanian mapping, while this re-indexing of k is called the CRT mapping. The latter refers to the fact that k is the solution to the Chinese remainder problem k = k1 mod N1 and k = k2 mod N2. A great deal of research has been devoted to schemes for evaluating this re-indexing efficiently, ideally in-place, while minimizing the number of costly modulo operations.
DFT re-expression
The above re-indexing is then substituted into the formula for the DFT, and in particular into the productnk in the exponent. Because e2πi = 1, this exponent is evaluated modulo N: any N1N2 = N cross term in the nk product can be set to zero. The remaining terms give: The inner and outer sums are simply DFTs of size N2 and N1, respectively.
