Editing Fractional Fourier transform (section)

{{Short description|Mathematical operation}}
{{use dmy dates|date=September 2021}}

In [[mathematics]], in the area of [[harmonic analysis]], the '''fractional Fourier transform''' ('''FRFT''') is a family of [[linear transformation]]s generalizing the [[Fourier transform]]. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an [[integer]] &mdash; thus, it can transform a function to any ''intermediate'' domain between time and [[frequency]]. Its applications range from [[filter design]] and [[signal analysis]] to [[phase retrieval]] and [[pattern recognition]].

The FRFT can be used to define fractional [[convolution]], [[correlation]], and other operations, and can also be further generalized into the [[linear canonical transformation]] (LCT).  An early definition of the FRFT was introduced by [[Edward Condon|Condon]],<ref>{{cite journal |last= Condon |first= Edward U. |date= 1937 |title= Immersion of the Fourier transform in a continuous group of functional transformations |journal= [[Proc. Natl. Acad. Sci. USA]] |volume= 23 |issue= 3 |pages= 158–164 |doi= 10.1073/pnas.23.3.158 |pmid= 16588141 |author-link= Edward Condon|pmc= 1076889 |bibcode= 1937PNAS...23..158C |doi-access= free }}</ref> by solving for the [[Green's function]] for phase-space rotations, and also by Namias,<ref>{{cite journal |last= Namias |first= V. |date= 1980 |title= The fractional order Fourier transform and its application to quantum mechanics |journal= IMA Journal of Applied Mathematics |volume= 25 |issue= 3 |pages= 241–265 |doi= 10.1093/imamat/25.3.241}}</ref> generalizing work of [[Norbert Wiener|Wiener]]<ref>{{cite journal |last= Wiener |first= N. |date= April 1929 |title= Hermitian Polynomials and Fourier Analysis |journal= Journal of Mathematics and Physics |volume= 8 |issue= 1–4 |pages= 70–73 |doi= 10.1002/sapm19298170}}</ref> on [[Hermite polynomials]].

However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups.<ref>{{cite journal |last= Almeida |first= Luís B. |date= 1994 |title= The fractional Fourier transform and time–frequency representations |journal= IEEE Trans. Signal Process. |volume= 42 |number= 11 |pages= 3084–3091|doi= 10.1109/78.330368 |bibcode= 1994ITSP...42.3084A |s2cid= 29757211 }}</ref> Since then, there has been a surge of interest in extending Shannon's sampling theorem<ref>{{cite journal |last1= Tao |first1= Ran |last2= Deng |first2= Bing |last3= Zhang |first3= Wei-Qiang |last4= Wang |first4= Yue |date= 2008 |title= Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain |journal= IEEE Transactions on Signal Processing |volume= 56 |number= 1 |pages= 158–171|doi= 10.1109/TSP.2007.901666 |bibcode= 2008ITSP...56..158T |s2cid= 7001222 }}</ref><ref>{{cite journal |last1= Bhandari |first1= A. |last2= Marziliano |first2= P. |date= 2010 |title= Sampling and reconstruction of sparse signals in fractional Fourier domain |journal= IEEE Signal Processing Letters |volume= 17 |number= 3 |pages= 221–224|doi= 10.1109/LSP.2009.2035242 |bibcode= 2010ISPL...17..221B |hdl= 10356/92280 |s2cid= 11959415 |hdl-access= free }}</ref> for signals which are band-limited in the Fractional Fourier domain.

A completely different meaning for "fractional Fourier transform" was introduced by Bailey and Swartztrauber<ref>{{cite journal |last1= Bailey |first1= D. H. |last2= Swarztrauber |first2= P. N. |date= 1991 |title= The fractional Fourier transform and applications |journal= [[SIAM Review]] |volume= 33 |issue= 3 |pages= 389–404|doi= 10.1137/1033097 }} (Note that this article refers to the chirp-z transform variant, not the FRFT.)</ref> as essentially another name for a [[z-transform]], and in particular for the case that corresponds to a [[discrete Fourier transform]] shifted by a fractional amount in frequency space (multiplying the input by a linear [[chirp]]) and evaluating at a fractional set of frequency points (e.g. considering only a small portion of the spectrum).  (Such transforms can be evaluated efficiently by [[Bluestein's FFT algorithm]].) This terminology has fallen out of use in most of the technical literature, however, in preference to the FRFT. The remainder of this article describes the FRFT.