The Cooley - Tukey Fast Fourier Transform Algorithm ∗

The publication by Cooley and Tukey [5] in 1965 of an e cient algorithm for the calculation of the DFT was a major turning point in the development of digital signal processing. During the ve or so years that followed, various extensions and modi cations were made to the original algorithm [6]. By the early 1970's the practical programs were basically in the form used today. The standard development presented in [21], [22], [1] shows how the DFT of a length-N sequence can be simply calculated from the two length-N/2 DFT's of the even index terms and the odd