Notes on Fast Fourier Transform Algorithms & Data Structures 2004 ( updated 2007 )

In this set of lecture notes we focus on the point-value representation obtained by looking at a particular set of points, the nth roots of unity. In the field of complex numbers C, there are exactly n different solutions to the equation x = 1. We call these solutions the n-th roots of unity. One of these roots of unity is ωn = cos(2π/n)+ i sin(2π/n), and this is called the principal nth root of unity. It is not too difficult to show that ωn generates the entire set of nth roots of unity as follows, for any n: 1,ωn,ω 2 n,ω 3 n, . . . ,ω n−1 n .