The discrete Fourier series (DFS) is a [[frequency representation of periodic discrete-time signals]]. A discrete-time [[periodic signal]], $\tilde{x}(n)$, can be represented as a sum of [[discrete-time complex exponential signal|discrete-time complex exponentials]]: $ \tilde{x}(n)=\frac{1}{N}\sum_{k=0}^{N-1} \tilde{X}(k) e^{j\frac{2\pi}{N}kn} $ This equation is called the synthesis equation of the DFS. The $\tilde{X}(k)$ coefficients can be computed with: $ \tilde{X}(k)=\sum_{n=0}^{N-1} \tilde{x}(n) e^{-j\frac{2\pi}{N}kn} $ This equation is called the analysis equation of the DFS. The periodic sequences $\tilde{x}(n)$ and $\tilde{X}(k)$ are called a DFS pair: $\tilde{x}(n) \xrightarrow[\cal DFS]{}\tilde{X}(k)$