Editing Fractional Fourier transform (section)

==Application==
Fractional Fourier transform can be used in time frequency analysis and [[Digital signal processing|DSP]].<ref>{{cite journal |last1= Sejdić |first1= Ervin |last2= Djurović |first2= Igor |last3= Stanković |first3= LJubiša |date= June 2011 |title= Fractional Fourier transform as a signal processing tool: An overview of recent developments |journal= Signal Processing |volume= 91 |number= 6 |pages= 1351–1369 |doi= 10.1016/j.sigpro.2010.10.008|bibcode= 2011SigPr..91.1351S |s2cid= 14203403 }}</ref> It is useful to filter noise, but with the condition that it does not overlap with the desired signal in the time–frequency domain. Consider the following example. We cannot apply a filter directly to eliminate the noise, but with the help of the fractional Fourier transform, we can rotate the signal (including the desired signal and noise) first. We then apply a specific filter, which will allow only the desired signal to pass. Thus the noise will be removed completely. Then we use the fractional Fourier transform again to rotate the signal back and we can get the desired signal.

[[Image:FracFT App by stevencys.jpg|thumb|center|600px|Fractional Fourier transform in DSP]]
Thus, using just truncation in the time domain, or equivalently [[low-pass filter]]s in the frequency domain, one can cut out any [[convex set]] in time–frequency space. In contrast, using time domain or frequency domain tools without a fractional Fourier transform would only allow cutting out rectangles parallel to the axes.

Fractional Fourier transforms also have applications in quantum physics. For example, they are used to formulate entropic uncertainty relations,<ref>{{cite journal |last1= Huang |first1= Yichen |date= 24 May 2011 |title= Entropic uncertainty relations in multidimensional position and momentum spaces |journal= Physical Review A |volume= 83 |issue= 5 |page= 052124 |doi= 10.1103/PhysRevA.83.052124 |s2cid= 119243096 |arxiv= 1101.2944 |bibcode= 2011PhRvA..83e2124H }}</ref> in high-dimensional quantum key distribution schemes with single photons,<ref>{{cite journal |last1= Walborn |first1= SP |last2= Lemelle |first2= DS |last3= Tasca |first3= DS |last4= Souto Ribeiro |first4= PH |date= 13 June 2008 |title=Schemes for quantum key distribution with higher-order alphabets using single-photon fractional Fourier optics |journal= Physical Review A |volume= 77 |issue= 6 |page= 062323 |doi= 10.1103/PhysRevA.77.062323|bibcode= 2008PhRvA..77f2323W }}</ref> and in observing spatial entanglement of photon pairs.<ref>{{cite journal |last1= Tasca |first1= DS |last2= Walborn |first2= SP |last3= Souto Ribeiro |first3= PH |last4= Toscano|first4= F |date= 8 July 2008 |title= Detection of transverse entanglement in phase space |journal= Physical Review A |volume= 78 |issue= 1 |page= 010304(R) |doi= 10.1103/PhysRevA.78.010304 |arxiv= 0806.3044 |bibcode= 2008PhRvA..78a0304T |s2cid= 118607762 }}</ref>

They are also useful in the design of optical systems and for optimizing holographic storage efficiency.<ref>{{cite journal |last1= Pégard |first1= Nicolas C. |last2= Fleischer |first2= Jason W. |date= 2011 |title= Optimizing holographic data storage using a fractional Fourier transform |journal= Optics Letters |volume= 36 |issue= 13 |pages= 2551–2553 |doi= 10.1364/OL.36.002551 |pmid= 21725476 |bibcode= 2011OptL...36.2551P |url= http://www.opticsinfobase.org/abstract.cfm?URI=ol-36-13-2551|url-access= subscription }}</ref><ref name="Jago">{{cite journal |last1=Jagoszewski |first1= Eugeniusz |date= 1998 |title= Fractional Fourier transform in optical setups |journal=Optica Applicata|volume= XXVIII |issue= 3 |pages=227–237| url= 
https://dbc.wroc.pl/Content/41401/PDF/optappl_2803p227.pdf}}</ref>