Nth Roots of Unity

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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 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,

is called a primitive th root of unity.^6.2

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

**Figure 6.1:** The roots of unity for .
$\includegraphics[width=\textwidth]{eps/dftfreqs}$

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written by Julius Orion Smith III
Julius Smith's background is in electrical engineering (BS Rice 1975, PhD Stanford 1983). He is presently Professor of Music and Associate Professor (by courtesy) of Electrical Engineering at Stanford's Center for Computer Research in Music and Acoustics (CCRMA), teaching courses and pursuing research related to signal processing applied to music and audio systems. See http://ccrma.stanford.edu/~jos/ for details.

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Chapters

Mathematics of the DFT
    Derivation of the Discrete Fourier Transform (DFT)
       Orthogonality of Sinusoids
          Nth Roots of Unity