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Yevgen Voronenko and Markus Püschel (IEEE Transactions on Signal Processing, Vol. 57, No. 1, pp. 205-222, 2009)
Algebraic Signal Processing Theory: Cooley-Tukey Type Algorithms for Real DFTs
Preprint (324 KB)
Published paper (link to publisher)
In this paper we systematically derive a large class of fast general-radix algorithms for various types of real discrete Fourier transforms (real DFTs) including the discrete Hartley transform (DHT) based on the algebraic signal processing theory. This means that instead of manipulating the transform definition, we derive algorithms by manipulating the polynomial algebras underlying the transforms using one general method. The same method yields the well-known Cooley-Tukey fast Fourier transform (FFT) as well as general radix discrete cosine and sine transform algorithms. The algebraic approach makes the derivation concise, unifies and classifies many existing algorithms, yields new variants, enables structural optimization, and naturally produces a human-readable structural algorithm representation based on the Kronecker product formalism. We show, for the first time, that the general-radix Cooley-Tukey and the lesser known Bruun algorithms are instances of the same generic algorithm. Further, we show that this generic algorithm can be in stantiated for all four types of the real DFT and the DHT.Keywords: Algorithm theory and analysis, Discrete/fast Fourier transform, Algebraic signal processing theory, Fast algorithms