Roots of Unity

Since $e^{j2\pi}=1$ , we can write

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

where $M$ is any positive integer. The $M$ complex numbers $\{e^{j2\pi k/M}\}_{k=0}^{M-1}$ are called 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:

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

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

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

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 under ordinary complex multiplication.

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

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

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