Editing 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.&nbsp; E.g., [[#Harris|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".&nbsp; 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 <math>\left\{-\tfrac{N-1}{2} \le n \le \tfrac{N-1}{2}\right\},</math> which he calls the ''finite Fourier transform data window'', and a sequence on <math>\{0 \le n \le N-1\},</math> which is the DFT data window.
}}&nbsp; So [[#Cooley|J.Cooley]] (pp. 77–78) describes the implementation as  ''discrete finite Fourier transform''.

or

* another name for the [[Fourier series]] coefficients.<ref name=Bachman/>

or

* another name for one snapshot of a [[short-time Fourier transform]].<ref name=Morelli/>

==See also==
* [[Fourier transform]]

==Notes==
{{notelist-ua|1}}

==References==
{{reflist|refs=
<ref name=Bachman>
George Bachman, Lawrence Narici, and Edward Beckenstein, ''Fourier and Wavelet Analysis'' (Springer, 2004), p. 264
</ref>

<ref name=Morelli>
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).
</ref>
}}

{{refbegin}}
#<li value="3">{{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
}}

==Further reading==
*Rabiner, Lawrence R.; Gold, Bernard (1975). ''Theory and application of digital signal processing''. Englewood Cliffs, N.J.: Prentice-Hall. pp 65–67. {{ISBN|0139141014}}.
{{refend}}

[[Category:Transforms]]
[[Category:Fourier analysis]]
[[Category:Fourier series]]