Finite Fourier transform

In mathematics the finite Fourier transform may refer to either

* another name for the discrete Fourier transform [J. Cooley, P. Lewis, and P. Welch, "The finite Fourier transform," "IEEE Trans. Audio Electroacoustics" 17 (2), 77-85 (1969).]

or

* another name for the Fourier series coefficients [George Bachman, Lawrence Narici, and Edward Beckenstein, "Fourier and Wavelet Analysis" (Springer, 2004), p. 264.]

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 $[0,T]$. [M. Eugene, " [http://citeseer.ist.psu.edu/morelli97high.html High accuracy evaluation of the finite Fourier transform using sampled data] ," NASA technical report TME110340 (1997).] 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::$X(omega)\; =\; frac\{1\}\{sqrt\{2pi\; int\_\{0\}^T\; x(t)\; e^\{-\; iomega\; t\},dt$

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