DTFS

$x[n] = e^{j\omega_0n}$ is periodic only when $\omega_0 = 2\pi\frac{m}{N}$ . The fundamental period is $\frac{N}{gcd(m, N)}$ . $\omega_0$ is the value of phase at $n=1$

For $x[n] = e^{j2\pi\frac{n}{N}}$ , there are only $N$ distinct harmonics $\phi_k[n] = e^{j2\pi\frac{kn}{N}}$

$k=<N>$ indicates any period of duration $N$ and $\omega_0 = \frac{2\pi}{N}$

Signal can be represented as orthogonal signal set

\{\phi_k(t) = e^{j\omega_0kt}, k=<N>\}

x[n] = \sum_{k=<N>}a_ke^{jk\omega_0n} \newline a_k = \frac{1}{N} \sum_{k=<N>} x[n]e^{-jk\omega_0n}

There is no convergence issue since we are dealing with a finite number of harmonics.

DTFS Notion