# Finite Fourier transform

In mathematics the finite Fourier transform may refer to either

or

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 $[0,T]$.[3] 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:
$X(\omega) = \frac{1}{\sqrt{2\pi}} \int_{0}^T x(t) e^{- i\omega t}\,dt$

## References

1. ^ J. Cooley, P. Lewis, and P. Welch, "The finite Fourier transform," IEEE Trans. Audio Electroacoustics 17 (2), 77-85 (1969).
2. ^ George Bachman, Lawrence Narici, and Edward Beckenstein, Fourier and Wavelet Analysis (Springer, 2004), p. 264.
3. ^ M. Eugene, "High accuracy evaluation of the finite Fourier transform using sampled data," NASA technical report TME110340 (1997).