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

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

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