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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 [27,83,36,45,10,86].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 [17], with Good's 1958 work on the PFA being cited in that paper [86].

The PFA and Winograd transform [45] are closely related, with the PFA being somewhat faster [9].

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``Mathematics of the Discrete Fourier Transform (DFT), with Audio Applications --- Second Edition'', by Julius O. Smith III, W3K Publishing, 2007, ISBN 978-0-9745607-4-8
Copyright © 2024-04-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University