Nth Roots of Unity

As introduced in §3.14, the complex numbers

$$W_N^k \isdef e^{j\omega_k T} \isdef e^{j k 2\pi (f_s/N) T} = e^{j k 2\pi/N}, \quad k=0,1,2,\ldots,N-1,$$

are called the $N$th roots of unity because each of them satisfies

$$\left[W_N^k\right]^N = \left[e^{j\omega_k T}\right]^N = \left[e^{j k 2\pi/N}\right]^N = e^{j k 2\pi} = 1.$$

In particular, $W_N$ is called the primitive $N$th root of unity.^6.2

The $N$th roots of unity are plotted in the complex plane in Fig. 6.1 for $N=8$. It is easy to find them graphically by dividing the unit circle into $N$ equal parts using $N$ points, with one point anchored at $z=1$, as indicated in Fig. 6.1. When $N$ is even, there will a point at $z=-1$ (corresponding to a sinusoid with frequency at exactly half the sampling rate), while if $N$ is odd, there is no point at $z=-1$.

Figure: The $N$ roots of unity for $N=8$.