Prime-factor FFT algorithm explained

The prime-factor algorithm (PFA), also called the Good–Thomas algorithm (1958/1963), is a fast Fourier transform (FFT) algorithm that re-expresses the discrete Fourier transform (DFT) of a size N = 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 (not necessarily relatively prime), 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 (e.g. it is useless for power-of-two sizes) and that it requires more complicated re-indexing of the data based on the additive group isomorphisms. Note, however, that PFA can be combined with mixed-radix Cooley–Tukey, with the former factorizing N 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 N₂ transform via more sophisticated two-dimensional convolution techniques. Some older papers therefore also call Winograd's algorithm a PFA FFT.

(Although the PFA is distinct from the Cooley–Tukey algorithm, Good's 1958 work on the PFA was cited as inspiration by Cooley and Tukey in their 1965 paper, and there was initially some confusion about whether the two algorithms were different. In fact, it was the only prior FFT work cited by them, as they were not then aware of the earlier research by Gauss and others.)

Algorithm

Let

a(x) be a polynomial and \omega _{n} be a principal n -th root of unity. We define the DFT of a(x) as the n -tuple (\hat{a} _{j)} = (a(\omega ^{j))} _{n} .In other words,$$\backslash hat\_j\; =\; \backslash sum\_^\; a\_i\; \backslash omega\_n^\; \backslash quad\; \backslash text\; j\; =\; 0,\; 1,\; \backslash dots,\; n\; -\; 1.$$For simplicity, we denote the transformation as DFT _{\omega _{n}} .

The PFA relies on a coprime factorization of $n\; =\; \backslash prod\_^\; n\_d$ and turns

DFT _{\omega _{n}} into $\backslash bigotimes\_d\; \backslash text\_$ for some choices of \omega _{n _{d}} 's where $\backslash bigotimes$ is the tensor product.

Mapping Based on CRT

m \mapsto (m \bmod n _{d)} from Z _{n} to $\backslash prod\_^\; \backslash mathbb\_$ with $(m\_d)\; \backslash mapsto\; \backslash sum\_^\; e\_d\; m\_d$ as its inverse where e _{d} 's are the central orthogonal idempotent elements with $\backslash sum\_^\; e\_d\; =\; 1\; \backslash pmod$.Choosing \omega _{n _{d}} = \omega ^{e _{d}} _{n} (therefore, $\backslash prod\_^\; \backslash omega\_\; =\; \backslash omega\_n^\; =\; \backslash omega\_n$), we rewrite DFT _{\omega _{n}} as follows:$$\backslash hat\_j\; =\; \backslash sum\_^\; a\_i\; \backslash omega\_n^\; =\; \backslash sum\_^\; a\_i\; \backslash left(\backslash prod\_^\; \backslash omega\_\; \backslash right)^\; =\; \backslash sum\_^\; a\_i\; \backslash prod\_^\; \backslash omega\_^\; =\; \backslash sum\_^\; \backslash cdots\; \backslash sum\_^\; a\_\; \backslash prod\_^\; \backslash omega\_^\; .$$Finally, define a _{i _{0,} ... , i _{D}} = a _{\sum ^{D -1} _{d =0} i _{d} e _{d}} and \hat{a} _{j _{0,} ... , j _{D}} = \hat{a} _{\sum ^{D -1} _{d =0} j _{d} e _{d}} , we have $$\backslash hat\_\; =\; \backslash sum\_^\; \backslash cdots\; \backslash sum\_^\; a\_\; \backslash prod\_^\; \backslash omega\_^\; .$$Therefore, we have the multi-dimensional DFT, ⊗ ^{D -1} _{d =0} DFT _{\omega _{n _{d}}} .

As Algebra Isomorphisms

PFA can be stated in a high-level way in terms of algebra isomorphisms.We first recall that for a commutative ring

R and a group isomorphism from G to $\backslash prod\_d\; G\_d$,we have the following algebra isomorphism$$R[G]\; \backslash cong\; \backslash bigotimes\_d\; R[G\_d]$$ where o times refers to the tensor product of algebras.

To see how PFA works, we choose

G = ( Z _{n,} + , 0 ) and G _{d} = ( Z _{n _{d}} , + , 0 ) be additive groups.We also identify R[G] as $\backslash frac$ and R[G _{d]} as $\backslash frac$.Choosing η = a \mapsto (a \bmod n _{d)} as the group isomorphism $G\; \backslash cong\; \backslash prod\_d\; G\_d$, we have the algebra isomorphism $\backslash eta^*\; :\; R[G]\; \backslash cong\; \backslash bigotimes\_d\; R[G\_d]$, or alternatively,$$\backslash eta^*\; :\; \backslash frac\; \backslash cong\; \backslash bigotimes\_d\; \backslash frac\; .$$Now observe that DFT _{\omega _{n}} is actually an algebra isomorphism from $\backslash frac$ to $\backslash prod\_i\; \backslash frac$ and each DFT _{\omega _{n _{d}}} is an algebra isomorphism from $\backslash frac$ to $\backslash prod\_\; \backslash frac$,we have an algebra isomorphism η ' from $\backslash bigotimes\_d\; \backslash prod\_\; \backslash frac$ to $\backslash prod\_i\; \backslash frac$.What PFA tells us is that $\backslash text\_\; =\; \backslash eta\text{'}\; \backslash circ\; \backslash bigotimes\_d\; \backslash text\_\; \backslash circ\; \backslash eta^*$ where η ^{*} and η ' are re-indexing without actual arithmetic in R .

Counting the Number of Multi-Dimensional Transformations

Notice that the condition for transforming

DFT _{\omega _{n}} into $\backslash eta\text{'}\; \backslash circ\; \backslash bigotimes\_d\; \backslash text\_\; \backslash circ\; \backslash eta^*$ relies on "an" additive group isomorphism η from ( Z _{n,} + , 0 ) to $\backslash prod\_d\; (\backslash mathbb\_,\; +,\; 0)$.Any additive group isomorphism will work.To count the number of ways transforming DFT _{\omega _{n}} into $\backslash eta\text{'}\; \backslash circ\; \backslash bigotimes\_d\; \backslash text\_\; \backslash circ\; \backslash eta^*$,we only need to count the number of additive group isomorphisms from ( Z _{n,} + , 0 ) to $\backslash prod\_d\; (\backslash mathbb\_,\; +,\; 0)$, or alternative, the number of additive group automorphisms on ( Z _{n,} + , 0 ) .Since ( Z _{n,} + , 0 ) is cyclic, any automorphism can be written as 1 \mapsto g where g is a generator of ( Z _{n,} + , 0 ) .By the definition of ( Z _{n,} + , 0 ) , g 's are exactly those coprime to n .Therefore, there are exactly \varphi(n) many such maps where \varphi is the Euler's totient function.The smallest example is n = 6 where \varphi(n) = 2 , demonstrating the two maps in the literature: the "CRT mapping" and the "Ruritanian mapping".

See also

• Bluestein's FFT algorithm
• Rader's FFT algorithm

References

• I. J. . Good . The interaction algorithm and practical Fourier analysis . Journal of the Royal Statistical Society, Series B . 20 . 2 . 361–372 . 1958 . 2983896 . Addendum, ibid. 22 (2), 373-375 (1960) .
• Book: Thomas, L. H. . Using a computer to solve problems in physics . Applications of Digital Computers . Ginn . Boston . 1963 .
• P. . Duhamel . M. . Vetterli . Fast Fourier transforms: a tutorial review and a state of the art . Signal Processing . 19 . 4 . 259–299 . 1990 . 10.1016/0165-1684(90)90158-U .
• S. C. . Chan . K. L. . Ho . On indexing the prime-factor fast Fourier transform algorithm . IEEE Transactions on Circuits and Systems. 38 . 8 . 951–953 . 1991 . 10.1109/31.85638 .
• I. J. . Good . The relationship between two fast Fourier transforms . IEEE Transactions on Computers . 100 . 3 . 310–317 . 1971 . 10.1109/T-C.1971.223236 . 585818 .