MFTA: The Prime Case
For transform size p, p a prime, Rader introduced an approach to construct algorithms which depends on the multiplicative structure of indexing set. In fact, for a prime p, Z/p is a field and the unit group U(p) is cyclic. Reordering input and output data corresponding to a generator of U(p), the p-point FFT becomes essentially a (p−1) × p−1) skew-circulant matrix. We require 2(p−1) additions to make this change. Rader computes this skew-circulant action by the convolution theorem which returns the computation to an FFT computation. Since the size (p−1) is a composite number, the (p−1)-point FT can be handled by Cooley-Tukey FFT algorithms. The Winograd algorithm for small convolutions can also be applied to the skew-circulant action.
KeywordsDiscrete Fourier Transform Unit Group Fundamental Factorization Permutation Matrix Real Multiplication
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