# Ambiguity about roots of unity in discrete Fourier transform

1. Apr 27, 2012

### CantorSet

Hi everyone, I have a question on the discrete Fourier transform. I already know its a change of basis operator on $C^N$ between the usual orthonormal basis and the "Fourier" basis, which are vectors consisting of powers of the $N$ roots of unity.

But if i recall correctly from complex analysis, the root of a complex number is not unique. So for example, if we look at the first entry of the first Fourier basis vector, it is $e^{\frac{2 \pi i }{N}}$. But there are N solutions here. Which one is the actual first entry in the first Fourier basis vector?