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Mathematics of the DFT
    Proof of Euler's Identity
       Roots of Unity


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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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Previous: Back to Mth Roots
Next: Direct Proof of De Moivre's Theorem
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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