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Nth Roots of Unity

As introduced in §3.12, the complex numbers

$\displaystyle 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

$\displaystyle \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 a 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 be 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 6.1: The $ N$ roots of unity for $ N=8$.
\includegraphics[width=\twidth]{eps/dftfreqs}

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