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On the Properties of the Twiddle Factor and their Applications to the DFT

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The DFT (Discrete Fourier Transform) is essentially a sequence of polynomials of the twiddle factor WkN, thus the relationship between the properties of twiddle factors WknN and algorithms for the DFT is very close. This paper intends to summarize and investigate the properties of WknN and explain how they are used in some efficient algorithms for DFT. Besides the periodicity and symmetry of WknN, the real coefficient pairs of WknN and the occurrence number of WiN (I = kn mod N) are presented and discussed. A new algorithm based on these properties is presented too.

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