Finite Fourier transform

In mathematics the finite Fourier transform may refer to either

• another name for the discrete Fourier transform

or

• another name for the Fourier series coefficients

or

• a transform based on a Fourier-transform-like integral applied to a function $x(t)$, but with integration only on a finite interval, usually taken to be the interval $[0,T]$. Equivalently, it is the Fourier transform of a function $x(t)$ multiplied by a rectangular window function. That is, the finite Fourier transform $X(\omega)$ of a function $x(t)$ on the finite interval $[0,T]$ is given by:

$X(\omega) = \frac{1}{\sqrt{2\pi}} \int_{0}^T x(t) e^{- i\omega t}\,dt$