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Decimation-in-frequency Fast Fourier Transforms for the Symmetric Group a·r·t Eric J. Malm, Mathematics ’05 Thesis Advisor: Michael E. Orrison The Discrete Fourier Transform Group-Theoretic DFTs Different spaces have different symmetries: The DFT reveals structure that is invariant under symmetry. Space time domain sphere lists Suppose we want to analyze some periodic signal f : • We pick an arbitrary full time period of f • Take N samples f 0, f 1, . . . , f N −1 of f in this time period A B C f HtL (1 3) −−→ Symmetry time translations rotations about center permutations C B A (1 3 2) −−−→ B A C Group Z/NZ SO(3) Sn (2 3) −−→ B C A 2 We therefore want generalized DFTs that show us similar symmetry-invariant structure. We can write these symmetries abstractly as groups and define these new DFTs using tools from abstract algebra: t 1 Figure 1: N = 8 samples of a periodic signal f . For 0 ≤ k ≤ N − 1, define the Discrete Fourier Transform (DFT) to be fˆk = FFTs for the Symmetric Group Results For the symmetric group Sn, we select the subgroup chain We have computed sparse matrix factorizations of the DFT matrix for Sn for n = 3 to 6 using Mathematica. 1 = S1 < S2 < S3 < · · · < S n N −1 ∑ f j ω − jk where ω = e2πi/N . j =0 The fˆ0. fˆ1, . . . , fˆN −1 are the Fourier coefficients of the samples f 0, f 1, . . . , f N −1. fˆ0 f0 output input . . . −−→ DFTN −−−→ . f N −1 fˆN −1 Decomposition of original signal into “pure frequencies”: N −1 2πk 2πk ˆ f (t) ≈ ∑ fk cos t + i sin t N N k =0 “pure” frequency amplitude Our signal f above then decomposes as shown below: 1 f HtL t 1 2 = + 1 t 1 2 + 2 3 7→ 1 2 1 t 1 2 Figure 2: Fourier decomposition of f into three pure frequencies. Suppose we sample our signal over a different time period • The samples f 0, . . . , f N −1 could be much different • But the Fourier coefficients fˆk will not be • The DFT is invariant under translational symmetry ⊕ 14 112 966 9278 ⊗ 4 42 424 4631 tDIF n 2.7 5.3 8.8 13.8 tnM 12 n(n − 1) 2.7 3 5.4 6 9.1 10 13.6 15 Table 2: Operation counts for the evaluation of the decimation-indenotes the reduced complexity of our frequency FFT. Here, tDFT n decimation-in-frequency algorithm for Sn , while tnM denotes the reduced complexity of Maslen’s decimation-in-time algorithm. Paths through character graph index rows and columns of matrix algebra blocks: = {functions f : G → C} −−→ { f ∈ group algebra CG } Wedderburn’s Theorem The group algebra CG of a finite group G is isomorphic to an algebra of block diagonal matrices: CG ∼ = h M = Cdi × d i . i =1 For example, CS3 decomposes thus: • •• 1×1 2×2 1×1 ∼ CS3 = C ⊕ C ⊕ C = •• • di × di Every C-algebra-isomorphism D : CG → C is i =1 called a Discrete Fourier Transform (DFT) for G. The coefficients of the matrix D ( f ) are called the (generalized) Fourier coefficients of f . Lh The Problem Naı̈ve DFTs use N 2 operations 1 t n 3 4 5 6 Blocks in matrix algebra for CSn ↔ partitions of n: • • • • CS3 ∼ ⊕ ⊕ = •• Table 1: Some different spaces and their associated symmetries 1 har vey·mudd·college The Solution (for Z/NZ) The Cooley-Tukey Fast Fourier Transform (FFT) computes classical DFT in N log N operations Cooley-Tukey FFT uses factorization of the group Z/NZ: 1 < Z/p1Z < Z/p1 p2Z < · · · < Z/NZ Other groups G admit different subgroup chains: 1 = G0 < G1 < G2 < · · · < Gn = G Figure 3: Character graph for 1 < S2 < S3. Pair of paths shown indexes coefficient in first row and second column of second matrix block. = Decimation-in-frequency approach to FFT: • Partial paths in character graph give subspaces of CSn • At stage for Sk, we project onto these subspaces • Build sparse factor from projections Figure 4: Sparse factorizations of DFT matrix for Sn for n = 3 to 5. • By stage for Sn, we have full paths • Each pair of paths corresponds to a Fourier coefficient log10 Tn Conjecture The complexity of the evaluation of our decimationin-frequency FFT for Sn is O(n2n!). The complexity of the inverse transform is also O(n2n!). 13 11 9 7 5 Acknowledgments I would like to thank my thesis advisor, Michael Orrison, for his insight, support, and guidance on this project, my second reader, Shahriar Shahriari, for his helpful commentary, and Claire Connelly for her excellent thesis class and LATEX assistance. 3 3 5 7 9 10 n Figure 5: Comparison of costs for group algebra multiplication (red line) and FFT-based matrix algebra multiplication (green line). For n ≥ 5, the FFT-based multiplication is more efficient. Copyright © 2005, Harvey Mudd College Department of Mathematics.