Roots of Unity

Since, by Euler's identity, $e^{j2\pi}=\cos(2\pi)+j\sin(2\pi)=1$, we can write

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

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

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

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

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

where $W_M$ denotes the primitive $M$th root of unity.^3.5 We may also call $W_M$ the 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 DFT and its inverse. The $k$th sinusoid is given by

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

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