FFT-like factorizations using group theory
نویسندگان
چکیده
The Fast Fourier Transform (FFT) is based on an important factorization of the Fourier matrix Fn into a product of sparse matrices. In this paper, we demonstrate the existence of a set of FFT-like factorizations for an arbitrary Kronecker product of Fourier matrices F = Fn1 ⊗ . . . ⊗ Fnk . We show that there exists such a factorization for any chain of nested subgroups of the Abelian group Zn1× . . .×Znk . The classical FFT will then be a special case of this scheme. However, the construction of FFT-like factorizations will allow a lot of freedom. This will allow to construct factorizations which cannot be obtained by simply inserting the classical FFT of each of the Kronecker factors of F . It will also be shown that the FFT-like factorizations of the matrix F can be brought into correspondence with the partitions of F into nested grids of rank-one blocks.
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