Finite Fourier transform

In mathematics the finite Fourier transform may refer to either

• another name for the discrete Fourier transform^[1]

or

• another name for the Fourier series coefficients^[2]

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).