The complex numbers $e^{j\frac{2\pi}{N}k}$ for $k \in \{0, \ldots\, N-1\}$ are equally spaced over the unit circle of the complex plane. For $N=8$: ![[circuni.svg|300]] It can easily be shown that the sum of the $N$ complex numbers is zero: $ \sum_{n=0}^{N-1} e^{j\frac{2\pi}{N}n}=0 $ Give the [[frequency periodicity of the discrete-time complex exponential signal]]: $ e^{j \frac{2\pi}{N} kn} = e^{j \frac{2\pi}{N} (k+lN)n} $ The previous result can be extended to: $\frac{1}{N} \sum_{n=0}^{N-1} e^{j \frac{2\pi}{N} kn} = \begin{cases} 1 & \text{if $k=mN$},\\ 0 & \text{otherwise}.\\ \end{cases}$ Next: [[determination of the Fourier series coefficients]]