The quick Fourier transform: an FFT based on symmetries

This paper looks at an approach that uses symmetric properties of the basis function to remove redundancies in the calculation of the discrete Fourier transform (DFT). We develop an algorithm called the quick Fourier transform (QFT) that reduces the number of floating-point operations necessary to compute the DFT by a factor of two or four over direct methods or Goertzel´s method for prime lengths. By further application of the idea to the calculation of a DFT of length-2^M , we construct a new O(NlogN) algorithm, with computational complexities comparable to the Cooley-Tukey algorithm. We show that the power-of-two QFT can be implemented in terms of discrete sine and cosine transforms. The algorithm can be easily modified to compute the DFT with only a subset of either input or output points and reduces by nearly half the number of operations when the data are real