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Roots of Unity

Since $ e^{j2\pi k}=1$ for every integer $ k$, we can write

$\displaystyle 1^{k/M} = e^{j2\pi k/M}, \quad k=0,1,2,3,\dots,M-1.

These are the $ M$th roots of unity. The special case $ k=1$ is called a primitive $ M$th root of unity, since integer powers of it give all of the others:

$\displaystyle e^{j2\pi k/M} = \left(e^{j2\pi/M}\right)^k

The $ M$th roots of unity are so frequently used that they are often given a special notation in the signal processing literature:

$\displaystyle W_M^k \isdef e^{j2\pi k/M}, \qquad k=0,1,2,\dots,M-1,

where $ W_M$ denotes a primitive $ M$th root of unity.3.7 We may also call $ W_M$ a generator of the mathematical group consisting of the $ M$th roots of unity and their products.

We will learn later that the $ N$th roots of unity are used to generate all the sinusoids used by the length-$ N$ DFT and its inverse. The $ k$th complex sinusoid used in a DFT of length $ N$ is given by

$\displaystyle W_N^{kn} = e^{j2\pi k n/N} \isdef e^{j\omega_k t_n}
= \cos(\omega_k t_n) + j \sin(\omega_k t_n),
\quad n=0,1,2,\dots,N-1,

where $ \omega_k \isdef 2\pi k/NT$, $ t_n \isdef nT$, and $ T$ is the sampling interval in seconds.

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``Mathematics of the Discrete Fourier Transform (DFT), with Music and Audio Applications'', by Julius O. Smith III, W3K Publishing, 2003, ISBN 0-9745607-0-7.
Copyright © 2007-02-02 by Julius O. Smith III
Center for Computer Research in Music and Acoustics (CCRMA),   Stanford University
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