The discrete time Fourier series representation of a signal $x[n]$ with period $N$ is written a $x\left [ n \right ] = \sum _{k=0}^{N-1 }\: a_{k}\:e^{j\left ( 2kn\pi/N \right )}$. A discrete time periodic signal with period $N =3$, has the non-zero Fourier series coefficients: $a_{-3} = 2 $ and $a_{4} =1 $. The signal is
- $2 + 2e^{-\left ( j\frac{2\pi}{6}n \right )} \cos \left ( \frac{2\pi}{6}n \right )$
- $1 + 2e^{\left ( j\frac{2\pi}{6}n \right )} \cos \left ( \frac{2\pi}{6}n \right )$
- $1 + 2e^{\left ( j\frac{2\pi}{3}n \right )} \cos \left ( \frac{2\pi}{6}n \right )$
- $2 + 2e^{\left ( j\frac{2\pi}{6}n \right )} \cos \left ( \frac{2\pi}{6}n \right )$