# 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 ${\displaystyle x(t)}$, but with integration only on a finite interval, usually taken to be the interval ${\displaystyle [0,T]}$.[3] Equivalently, it is the Fourier transform of a function ${\displaystyle x(t)}$ multiplied by a rectangular window function. That is, the finite Fourier transform ${\displaystyle X(\omega )}$ of a function ${\displaystyle x(t)}$ on the finite interval ${\displaystyle [0,T]}$ is given by:
${\displaystyle X(\omega )={\frac {1}{\sqrt {2\pi }}}\int _{0}^{T}x(t)e^{-i\omega t}\,dt}$

## 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).