Algebraic signal processing theory: Cooley-Tukey type algorithms for real DFTs
IEEE Transactions on Signal Processing
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One class of efficient algorithms for computing a discrete Fourier transform (DFT) is based on a recursive polynomial factorization of the polynomial 1-z-N. The Bruun algorithm is a typical example of such algorithms. Previously, the Bruun algorithm, which is applicable only when system lengths are powers of two in its original form, is generalized and modified to be applicable to the case when the length is other than a power of two. This generalized algorithm consists of transforms Td,f with prime d and real f in the range 0≤fd,0 computes residues X(z)mod(1-z-2) and X(z)mod(1-2 cos(πk/d)z-1+z-2), k=1, 2, ..., d-1, and Td,f (f ≠0) computes residues X(z)mod(1-2cos(2π(f+k)/d)z-1+z-2), k=0, 1, ..., d-1 for a given real signal X(z) of length 2d. The purpose of this paper is to find efficient algorithms for Td,f. First, polynomial factorization algorithms are derived for Td,0 and Td,14/. When f is neither 0 nor 1/4, it is not feasible to derive a polynomial factorization algorithm. Two different implementations of Td,f for such f are derived. One implementation realizes Td,f via a d-point DFT, for which a variety of fast algorithms exist. The other implementation realizes Td,f via Td, 14/, for which the polynomial factorization algorithm exists. Comparisons show that for d≥5, these implementations achieve better performance than computing each output of Td,f separately.