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The Cooley-Tukey FFT and Group Theory, Volume 48, Number 10 fea-maslen.qxp 10/15/01 10:03 AM Page 1151 The Cooley-Tukey FFT and Group Theory David K. Maslen and Daniel N. Rockmore Pure and Applied Mathematics—Two Sides of a rift in the mathematical community by showing the Coin ultimate unity of pure and applied mathematics. In November of 1979 there appeared in the Bulletin We will show that investigation of finite and fast of the AMS a paper by L. Auslander and R. Tolim- Fourier transforms continues to be a varied and in- ieri [3] with the delightful title “Is Computing with teresting direction of mathematical research. the Finite Fourier Transform Pure or Applied Math- Whereas Auslander and Tolimieri concentrated on ematics?” This rhetorical question was answered relations to nilpotent harmonic analysis and theta by showing that in fact the finite Fourier transform functions, we emphasize connections between the and the family of efficient algorithms used to com- famous Cooley-Tukey FFT and group representa- pute it (the Fast Fourier Transform (FFT), a pillar tion theory. In this way we hope to provide further of the world of digital signal processing) are of in- evidence of the rich interplay of ideas which can terest to both pure and applied mathematicians. Auslander had come of age as an applied math- be found at the nexus of pure and applied math- ematician at a time when pure and applied math- ematics. ematicians still received much the same training. Background The ends towards which these skills were then di- rected became a matter of taste. As Tolimieri retells The finite Fourier transform or discrete Fourier it,1 Auslander had become distressed at the de- transform (DFT) has several representation theo- velopment of a separate discipline of applied math- retic interpretations: either as an exact computa- ematics which had grown apart from much of core tion of the Fourier coefficients of a function on the mathematics. The effect of this development was cyclic group Z/nZ or a function of band-limit n on detrimental to both sides. On the one hand, applied the circle S1, or as an approximation to the Fourier mathematicians had fewer tools to bring to prob- transform of a function on the real line. For each lems, and, conversely, pure mathematicians were of these points of view there is a natural group- often ignoring the fertile bed of inspiration pro- theoretic generalization and also a corresponding vided by real-world problems. Auslander hoped set of efficient algorithms for computing the quan- their paper would help mend a growing perceived tities involved. These algorithms collectively make David K. Maslen is a mathematician at Susquehanna up the Fast Fourier Transform or FFT. International Group LLP. His e-mail address is Formally, the DFT is a linear transformation [email protected]. mapping any complex vector of length n, f = Daniel N. Rockmore is professor of mathematics and com- ( f (0),...,f(n − 1))t ∈ Cn, to its Fourier transform, puter science at Dartmouth College and on the external ∈ n faculty of the Santa Fe Institute. His e-mail address is f C . The k th component of f , the DFT of [email protected]. He is supported in part f at frequency k, is by NSF PFF Award DMS-9553134, AFOSR F49620-00-1- n−1 0280, and DOJ 2000-DT-CX-K001. He would also like to (1) f(k)= f (j)e2πijk/n, thank the Santa Fe Institute and the Courant Institute for j=0 their hospitality during some of the writing. Pieces of the √ introduction are similar to his paper “The FFT—an where i = −1, and the inverse Fourier transform algorithm the whole family can use”, which appeared is in Computing in Science & Engineering, January 2000, n−1 1 − pp. 62–67. (2) f (j)= f(k)e 2πijk/n. n 1Private communication. k=0 NOVEMBER 2001 NOTICES OF THE AMS 1151 fea-maslen.qxp 10/15/01 10:03 AM Page 1152 Thus, with respect to the standard basis, the DFT In the nonabelian case, ΘG admits an analo- can be expressed as the matrix-vector product gous factorization in terms of irreducible polyno- f = Fn · f, where Fn is the Fourier matrix of order mials of the form n whose j,k entry is equal to e2πijk/n. Computing a DFT directly would require n2 scalar operations.2 ΘD(G) = det D(g)xg , ∈ Instead, the FFT is a family of algorithms for com- g G puting the DFT of any f ∈ Cn in O(n log n) opera- where D is an irreducible matrix representation tions. Since inversion can be framed as the DFT of of G. The inner sum here is a generic Fourier trans- ˇ 1 − form over G. See [12] for a beautiful historical the function f (k)= n f ( k), the FFT also gives an efficient inverse Fourier transform. exposition of these ideas. One of the main practical implications of the FFT Gauss’s interests ranged over all areas of math- is that it allows any cyclically invariant linear op- ematics and its applications, so it is perhaps not erator to be applied to a vector in only O(n log n) surprising that the first appearance of an FFT can also scalar operations. Indeed, the DFT diagonalizes be traced back to him [10]. Gauss was interested in any cyclic group-invariant operator, making pos- certain astronomical calculations, a recurrent area sible the following algorithm: (1) Compute the of application of the FFT, needed for interpolation Fourier transform (DFT). (2) Multiply the DFT by of asteroidal orbits from a finite set of equally spaced the eigenvalues of the operator, which are also observations. Surely the prospect of a huge labori- found using the Fourier transform. (3) Compute the ous hand calculation was good motivation for the de- inverse Fourier transform of the result. This tech- velopment of a fast algorithm. Making fewer hand nique is the basis of efficient digital filter (i.e., calculations also implies less opportunity for error convolution) and is also used for the efficient nu- and hence increased numerical stability! merical solution of partial differential equations. Gauss wanted to compute the Fourier coeffi- Some History cients ak and bk of a function represented by a Since the Fourier matrix is effectively the character Fourier series of bandwidth n, table of a cyclic group, it is not surprising that some m m of its earliest appearances are in number theory, the (5) f (x)= ak cos 2πkx+ bk sin 2πkx, subject which gave birth to character theory. Con- k=0 k=1 sideration of the Fourier matrix goes back at least as − far as to Gauss, who was interested in its connec- where m =(n 1)/2 for n odd and m = n/2 for n tions to quadratic reciprocity. In particular, Gauss even. He first observed that the Fourier coeffi- showed that for odd primes p and q, cients can be computed by a DFT of length n using the values of f at equispaced sample points. p q Trace(Fpq) Gauss then went on to show that if n = n1n2, this (3) = , q p Trace(Fp)Trace(Fq) DFT can in turn be reduced to first computing n1 DFTs of length n2, using equispaced subsets of p where q denotes the Legendre symbol. Gauss the sample points, i.e., a subsampled DFT, and also established a formula for the quadratic Gauss then combining these shorter DFTs using various sum Trace(Fn), which is discussed in detail in [3]. trigonometric identities. This is the basic idea Another early appearance of the DFT occurs in underlying the Cooley-Tukey FFT. the origins of representation theory in the work of Unfortunately, this reduction never appeared Dedekind and Frobenius on the group determi- outside of Gauss’s collected works. Similar ideas, nant. For a finite group G, the group determinant usually for the case n1 =2, were rediscovered in- ΘG is defined as the homogeneous polynomial in termittently over the succeeding years. Notable the variables xg (for each g ∈ G) given by the de- among these is the doubling trick of Danielson and terminant of the matrix whose rows and columns Lanczos (1942), performed in the service of x-ray are indexed by the elements of G with g,h-entry crystallography, another frequent employer of FFT equal to xgh−1 . Frobenius showed that when G is technology. Nevertheless, it was not until the pub- abelian, Θ admits the factorization lication of Cooley and Tukey’s famous paper [7] G that the algorithm gained any notice. The story of (4) ΘG = χ(g)xg , Cooley and Tukey’s collaboration is an interesting ∈ χ∈G g G one. Tukey arrived at the basic reduction while in a meeting of President Kennedy’s Science Advisory where G is the set of characters of G. The linear Committee, where among the topics of discus- form defined by the inner sum in (4) is a “generic” sions were techniques for offshore detection of DFT at the frequency χ. nuclear tests in the Soviet Union. Ratification of a 2At this point we must come clean about how we count proposed United States/Soviet Union nuclear test operations. Our count is either the number of complex ad- ban depended upon the development of a method ditions or the number of complex multiplications, for detecting the tests without actually visiting whichever is greater. the Soviet nuclear facilities. One idea required the 1152 NOTICES OF THE AMS VOLUME 48, NUMBER 10 fea-maslen.qxp 10/15/01 10:03 AM Page 1153 analysis of seismological time series obtained from (7) f (ρij)= f (x)ρij(x).
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