MFTA: The Prime Case
For transform size p, p a prime, Rader  developed an FT algorithm based on the multiplicative structure of the indexing set. The main idea is as follows. For a prime p, Z/p is a field and the unit group U(p) is cyclic. Reordering input and output data relative to a generator of U(p), the p-point FT becomes essentially a (p-1) x (p-1) skew-circulant matrix action. We require 2(p-1) additions to make this change. Rader computes this skew-circulant action by the convolution theorem that returns the computation to an FT computation. Since the size (p-1) is a composite number, the (p-1)-point FT can be implemented by Cooley-Tukey FFT algorithms. The Winograd algorithm for small convolutions also can be applied to the skew-circulant action. (See problems 3, 4 and 5 for basic properties of skew-circulant matrices.)
KeywordsDiscrete Fourier Transform Fundamental Factorization Permutation Matrix Real Multiplication Additive Stage
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