Finite Fourier transform

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In mathematics the finite Fourier transform may refer to either

another name for discrete-time Fourier transform (DTFT) of a finite-length series. E.g., F.J.Harris (pp. 52–53) describes the finite Fourier transform as a "continuous periodic function" and the discrete Fourier transform (DFT) as "a set of samples of the finite Fourier transform". In actual implementation, that is not two separate steps; the DFT replaces the DTFT.{{efn-ua

|Harris' motivation for the distinction is to distinguish between an odd-length data sequence with the indices $\left\{-\tfrac{N-1}{2} \le n \le \tfrac{N-1}{2}\right\},$ which he calls the finite Fourier transform data window, and a sequence on $\{0 \le n \le N-1\},$ which is the DFT data window.

}} So J.Cooley (pp. 77–78) describes the implementation as discrete finite Fourier transform.

or

another name for the Fourier series coefficients.

or

another name for one snapshot of a short-time Fourier transform.

Notes

References

{{reflist|refs=

George Bachman, Lawrence Narici, and Edward Beckenstein, Fourier and Wavelet Analysis (Springer, 2004), p. 264

Morelli, E., "[http://citeseer.ist.psu.edu/morelli97high.html High accuracy evaluation of the finite Fourier transform using sampled data]," NASA technical report TME110340 (1997).

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{{cite journal |ref=Harris |doi=10.1109/PROC.1978.10837 |last=Harris |first=Fredric J. |title=On the use of Windows for Harmonic Analysis with the Discrete Fourier Transform |journal=Proceedings of the IEEE |volume=66 |issue=1 |pages=51–83 |date=Jan 1978 |url=http://web.mit.edu/xiphmont/Public/windows.pdf|citeseerx=10.1.1.649.9880

|s2cid=426548 }}

{{cite journal |ref=Cooley |last1=Cooley |first1=J. |last2=Lewis |first2=P. |last3=Welch |first3=P. |title=The finite Fourier transform |journal=IEEE Trans. Audio Electroacoustics |volume=17 |issue=2 |pages=77–85 |date=1969 |doi=10.1109/TAU.1969.1162036

}}

Finite Fourier transform

See also

Notes

References

Further reading