2.5: Roots of Unity and Related Topics

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The complex number \(z=e^{j2π/N}\) is illustrated in Figure \(\PageIndex{1}\). It lies on the unit circle at angle \(θ=2π/N\). When this number is raised to the n^th power, the result is \(z^n=e^{j2πn/N}\). This number is also illustrated in Figure \(\PageIndex{1}\). When one of the complex numbers \(e^{j2πn/N}\) is raised to the N^th power, the result is

\[(e^{j2πn/N})^N=e^{j2πn}=1 \nonumber \]

We say that \(e^{j2πn/N}\) is one of the \(N\)^th roots of unity, meaning that \(e^{j2πn/N}\) is one of the values of \(z\) for which

\[z^N−1=0 \nonumber \]

There are \(N\) such roots, namely,

\[e^{j2πn/N},n=0,1,...,N−1 \nonumber \]

As illustrated in the Figure \(\PageIndex{2}\), the 12^th roots of unity are uniformly distributed around the unit circle at angles \(2πn/12\). The sum of all of the \(N\)^th roots of unity is zero:

\[S_N=\sum_{n=0}^{N−1}e^{j2πn/N}=0 \nonumber \]

This property, which is obvious from the Figure, is illustrated in Figure, where the partial sums \(S_k=\sum_{n=0}^{k−1} e^{j2πn/N}\) are plotted for \(k=1,2,...,N\).

These partial sums will become important to us in our study of phasors and light diffraction in "Phasors" and in our discussion of filters in "Filtering".

Geometric Sum Formula

It is natural to ask whether there is an analytical expression for the partial sums of roots of unity:

\[S_k=\sum_{n=0}^{k−1}e^{j2πn/N} \nonumber \]

We can imbed this question in the more general question, is there an analytical solution for the “geometric sum”

\[S_k=\sum_{n=0}^{k−1}z^n? \nonumber \]

The answer is yes, and here is how we find it. If \(z=1\), the answer is \(S_k=k\). If \(z≠1\), we can premultiply \(S_k\) by \(z\) and proceed as follows:

\[\begin{align*} zS_k &=\sum^{k−1}_{n=0}z^{n+1}=\sum^k_{m=1}z^m \\[4pt] &=\sum^{k−1}_{m=0}z^m+z^k−1 \\[4pt] &=S_k+z^k−1 \end{align*} \nonumber \]

From this formula we solve for the geometric sum:

\[S_k=\begin{matrix}\frac {1−z^k} {1−z} & z≠1\\k & z=1\end{matrix} \nonumber \]

This basic formula for the geometric sum \(S_k\) is used throughout electromagnetic theory and system theory to solve problems in antenna design and spectrum analysis. Never forget it.

Exercise \(\PageIndex{1}\)

Find formulas for \(S_k=\sum_{n=0}^{k−1}e^{jnθ}\) and for \(S_k=\sum_{n=0}^{k−1}e^{j2π/Nn}\).

Exercise \(\PageIndex{2}\)

Prove \(\sum_{n=0}^{N−1}e^{j2πn/N}=0\).

Exercise \(\PageIndex{3}\)

Find formulas for the magnitude and phase of the partial sum \(S_k=\sum_{n=0}^{k−1}e^{j2πn/N}\).

Exercise \(\PageIndex{4}\)

(MATLAB) Write a MATLAB program to compute and plot the partial sum \(S_k=\sum_{n=0}^{k−1}e^{j2πn/N}\) for \(k=1,2,...,N\). You should observe the last Figure.

Exercise \(\PageIndex{5}\)

Find all roots of the equation \(z^3+z^2+3z−15=0\).

Exercise \(\PageIndex{6}\)

Find \(c\) so that \((1+j)\) is a root of the equation \(z^{17}+2z^{15}−c=0\).