Finite Fourier transform

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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[edit]

  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).