Prime Factor Algorithm (PFA) Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

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 [24,78,81].A.5The Chinese Remainder Theorem is used to re-index either the input or output samples for the PFA.A.6Since 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 [14], with Good's 1958 work on the PFA being cited in that paper [81].

Next  |  Prev  |  Up  |  Top  |  Index  |  JOS Index  |  JOS Pubs  |  JOS Home  |  Search

[How to cite this work] [Order a printed hardcopy]

``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
CCRMA  [Automatic-links disclaimer]