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| | In [[mathematics]] the '''finite Fourier transform''' may refer to either |
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| | * another name for the [[discrete Fourier transform]]<ref>J. Cooley, P. Lewis, and P. Welch, "The finite Fourier transform," ''IEEE Trans. Audio Electroacoustics'' '''17''' (2), 77-85 (1969).</ref> |
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| | * another name for the [[Fourier series]] coefficients<ref>George Bachman, Lawrence Narici, and Edward Beckenstein, ''Fourier and Wavelet Analysis'' (Springer, 2004), p. 264.</ref> |
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| | * a transform based on a Fourier-transform-like integral applied to a function <math>x(t)</math>, but with integration only on a finite interval, usually taken to be the interval <math>[0,T]</math>.<ref>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).</ref> Equivalently, it is the [[Fourier transform]] of a function <math>x(t)</math> multiplied by a rectangular [[window function]]. That is, the finite Fourier transform <math>X(\omega)</math> of a function <math>x(t)</math> on the finite interval <math>[0,T]</math> is given by: |
| | :<math> X(\omega) = \frac{1}{\sqrt{2\pi}} \int_{0}^T x(t) e^{- i\omega t}\,dt </math> |
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| | ==References== |
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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 , but with integration only on a finite interval, usually taken to be the interval .[3] Equivalently, it is the Fourier transform of a function multiplied by a rectangular window function. That is, the finite Fourier transform of a function on the finite interval is given by:
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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