Prime Factor Algorithm (PFA)

By the prime factorization theorem, every integer $ N$ can be uniquely factored into a product of prime numbers $ p_i$ raised to an integer power $ m_i\ge 1$:

$\displaystyle N = \prod_{i=1}^{n_p} p_i^{m_i}
$

As discussed above, a mixed-radix Cooley Tukey FFT can be used to implement a length $ N$ DFT using DFTs of length $ p_i$. However, for factors of $ N$ that are mutually prime (such as $ p_i^{m_i}$ and $ p_j^{m_j}$ for $ i\ne j$), a more efficient prime factor algorithm (PFA), also called the Good-Thomas FFT algorithm, can be used [26,80,35,43,10,83].A.4 The Chinese Remainder Theorem is used to re-index either the input or output samples for the PFA.A.5Since the PFA is only applicable to mutually prime factors of $ N$, it is ideally combined with a mixed-radix Cooley-Tukey FFT, which works for any integer factors.

It is interesting to note that the PFA actually predates the Cooley-Tukey FFT paper of 1965 [16], with Good's 1958 work on the PFA being cited in that paper [83].

The PFA and Winograd transform [43] are closely related, with the PFA being somewhat faster [9].


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Rader's FFT Algorithm for Prime Lengths
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Mixed-Radix Cooley-Tukey FFT