In signal processing, the discrete Fourier transform (DFT) takes a [[finite-duration discrete-time signal]] and converts it to a same-length sequence of equally-spaced samples of the [[discrete-time Fourier transform (DTFT)]]. A discrete Fourier transform (DFT) of size $N$ corresponds to [[sampling the DTFT]] at frequencies $\omega=\frac{2\pi}{N}k$, and can be represented as: $ x(n) \xrightarrow[{\cal DFT}_N]{} X(k) $ where $n, k, N \in \mathbb{Z}$. Both $x(n)$ and $X(k)$ are finite-duration sequences: $ x(n)=0, \; n \not\in \{0, \ldots,N-1\} \; \wedge \; X(k)=0, \; k \not\in \{0, \ldots,N-1\} $ The analysis and synthesis equations can be derived from the [[discrete Fourier series (DFS)]] of a periodic signal $\tilde{x}(n)$ created by a [[periodic repetition]] of $x(n)$ with a period greater or equal to $N$. The analysis equation is: $ X(k)=\sum_{n=0}^{N-1} x(n) e^{-j\frac{2\pi}{N}kn},\;0 \leq n \leq N-1 $ and the synthesis equation: $ x(n)=\frac{1}{N}\sum_{k=0}^{N-1} X(k) e^{j\frac{2\pi}{N}kn},\;0 \leq n \leq N-1 $