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Prime Factor Algorithm
(PFA)
By the prime factorization theorem, every integer
can be uniquely
factored into a product of prime numbers
raised to an
integer power
:
As discussed above, a mixed-radix Cooley Tukey FFT can be used to
implement a length
DFT using DFTs of length
. However, for
factors of
that are mutually prime (such as
and
for
), 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
, 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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