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Discrete Fourier transform
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==References== {{reflist|1|refs= <ref name=Strang> {{cite journal |last=Strang |first=Gilbert |title=Wavelets |journal=American Scientist |date=MayโJune 1994 |volume=82 |issue=3 |pages=250โ255 |jstor=29775194 |bibcode=1994AmSci..82..250S |quote=This is the most important numerical algorithm of our lifetime...}} </ref> <ref name=Sahidullah> {{cite journal |last=Sahidullah |first=Md. |author2=Saha, Goutam |title=A Novel Windowing Technique for Efficient Computation of MFCC for Speaker Recognition |journal=IEEE Signal Processing Letters |date=Feb 2013 |volume=20 |issue=2 |pages=149โ152 |doi=10.1109/LSP.2012.2235067 |arxiv=1206.2437 |bibcode=2013ISPL...20..149S|s2cid=10900793 }} </ref> <ref name=Cooley> {{cite journal |author=[[James Cooley|J. Cooley]], P. Lewis, and P. Welch |title=The finite Fourier transform |journal=IEEE Transactions on Audio and Electroacoustics |volume=17 |issue=2 |pages=77โ85 |year=1969 |doi=10.1109/TAU.1969.1162036}} </ref> <ref name=mathworks> {{cite web |title=Shift zero-frequency component to center of spectrum โ MATLAB fftshift |url=http://www.mathworks.com/help/matlab/ref/fftshift.html|website=mathworks.com |publisher=The MathWorks, Inc. |access-date=10 March 2014 |location=Natick, MA 01760}} </ref> <ref name=ProakisManolakis> {{Citation |last1 =Proakis |first1 =John G. |last2 =Manolakis |first2 =Dimitri G. |title =Digital Signal Processing: Principles, Algorithms and Applications |place=Upper Saddle River, NJ |publisher=Prentice-Hall International |year=1996 |edition=3 |language=en |id=sAcfAQAAIAAJ |isbn=9780133942897 |bibcode=1996dspp.book.....P |url-access=registration |url=https://archive.org/details/digitalsignalpro00proa}} </ref> <ref name=Oppenheim> {{cite book |last1=Oppenheim |first1=Alan V. |author-link=Alan V. Oppenheim |last2=Schafer |first2=Ronald W. |author2-link=Ronald W. Schafer |last3=Buck |first3=John R. |title=Discrete-time signal processing |page=[https://archive.org/details/discretetimesign00alan/page/571 571] |year=1999 |publisher=Prentice Hall |location=Upper Saddle River, N.J. |isbn=0-13-754920-2 |edition=2nd |url-access=registration |url=https://archive.org/details/discretetimesign00alan }} </ref> <ref name=McGillem> {{cite book |last1=McGillem |first1=Clare D. |last2=Cooper |first2=George R. |pages=171โ172 |title=Continuous and Discrete Signal and System Analysis |publisher=Holt, Rinehart and Winston |edition=2 |date=1984 |isbn=0-03-061703-0}} </ref> <ref name=Massar> {{Cite journal |last1=Massar |first1=S. |last2=Spindel |first2=P. |doi=10.1103/PhysRevLett.100.190401 |title=Uncertainty Relation for the Discrete Fourier Transform |journal=Physical Review Letters |volume=100 |issue=19 |year=2008 |pmid=18518426 |arxiv=0710.0723 |bibcode=2008PhRvL.100s0401M |page=190401|s2cid=10076374 }} </ref> <ref name=DeBrunner> {{cite journal |last1=DeBrunner |first1=Victor |last2=Havlicek |first2=Joseph P. |last3=Przebinda |first3=Tomasz|last4=รzaydin |first4=Murad |year=2005 |title=Entropy-Based Uncertainty Measures for <math>L^2(\mathbb{R}^n),\ell^2(\mathbb{Z})</math>, and <math>\ell^2(\mathbb{Z}/N\mathbb{Z})</math> With a Hirschman Optimal Transform for <math>\ell^2(\mathbb{Z}/N\mathbb{Z})</math> |journal=IEEE Transactions on Signal Processing |volume=53 |issue=8 |page=2690 |doi=10.1109/TSP.2005.850329 |url=http://redwood.berkeley.edu/w/images/9/95/2002-26.pdf |access-date=2011-06-23 |bibcode=2005ITSP...53.2690D|s2cid=206796625 }} </ref> <ref name=Donoho> {{cite journal |last1=Donoho |first1=D.L. |last2=Stark |first2=P.B |s2cid=115142886 |year=1989 |title=Uncertainty principles and signal recovery |journal=SIAM Journal on Applied Mathematics |volume=49 |issue=3 |pages=906โ931 |doi=10.1137/0149053}} </ref> <ref name=Santhanam> Santhanam, Balu; Santhanam, Thalanayar S. [https://ieeexplore.ieee.org/document/4217977 "''Discrete Gauss-Hermite functions and eigenvectors of the centered discrete Fourier transform''"], Proceedings of the 32nd IEEE ''International Conference on Acoustics, Speech, and Signal Processing'' (ICASSP 2007, SPTM-P12.4), vol. III, pp. 1385-1388. </ref> <ref name=Akansu> Akansu, Ali N.; Agirman-Tosun, Handan [http://web.njit.edu/~akansu/PAPERS/AkansuIEEE-TSP2010.pdf "''Generalized Discrete Fourier Transform With Nonlinear Phase''"], IEEE ''Transactions on Signal Processing'', vol. 58, no. 9, pp. 4547โ4556, Sept. 2010. </ref> <ref name=Baraquin> {{cite journal | author1=Isabelle Baraquin | author2=Nicolas Ratier | title=Uniqueness of the discrete Fourier transform | journal= Signal Processing | volume = 209 | pages = 109041 | year=2023 | issn=0165-1684 | doi=10.1016/j.sigpro.2023.109041| doi-access=free | bibcode=2023SigPr.20909041B }} </ref> }}
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