Editing Fractional Fourier transform

{{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.

==Introduction==
The continuous [[Fourier transform]] <math>\mathcal{F}</math> of a function <math>f: \mathbb{R} \mapsto \mathbb{C}</math> is a [[unitary operator]] of [[Lp space|
<math>L^2</math> space]] that maps the function <math>f</math> to its frequential version <math>\hat{f}</math> (all expressions are taken in the <math>L^2</math> sense, rather than pointwise):

<math display=block>\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,\mathrm{d}x</math>

and <math>f</math> is determined by <math>\hat{f}</math> via the inverse transform <math>\mathcal{F}^{-1}\, ,</math>

<math display=block>f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)\ e^{2 \pi i \xi x}\,\mathrm{d}\xi\, .</math>

Let us study its [[iterated function|''n''-th iterated]] <math>\mathcal{F}^{n}</math> defined by 
<math>\mathcal{F}^{n}[f] = \mathcal{F}[\mathcal{F}^{n-1}[f]]</math> and <math>\mathcal{F}^{-n} = (\mathcal{F}^{-1})^n</math> when ''n'' is a non-negative integer, and <math>\mathcal{F}^{0}[f] = f</math>. Their sequence is finite since <math>\mathcal{F}</math> is a 4-periodic [[automorphism]]: for every function <math>f</math>, <math>\mathcal{F}^4 [f] = f</math>.

More precisely, let us introduce the '''parity operator''' <math>\mathcal{P}</math> that inverts <math>x</math>, <math>\mathcal{P}[f]\colon x \mapsto f(-x)</math>. Then the following properties hold:
<math display=block>\mathcal{F}^0 = \mathrm{Id}, \qquad \mathcal{F}^1 = \mathcal{F}, \qquad \mathcal{F}^2 = \mathcal{P}, \qquad \mathcal{F}^4 = \mathrm{Id}</math>
<math display=block>\mathcal{F}^3 = \mathcal{F}^{-1} = \mathcal{P} \circ \mathcal{F} = \mathcal{F} \circ \mathcal{P}.</math>

The FRFT provides a family of linear transforms that further extends this definition to handle non-integer powers <math>n = 2\alpha/\pi</math> of the FT.

==Definition==
Note: some authors write the transform in terms of the "order {{mvar|a}}" instead of the "angle {{mvar|α}}", in which case the {{mvar|α}} is usually {{mvar|a}} times {{math|''π''/2}}. Although these two forms are equivalent, one must be careful about which definition the author uses.

For any [[real number|real]] {{mvar|α}}, the {{mvar|α}}-angle fractional Fourier transform of a function ƒ is denoted by <math>\mathcal{F}_\alpha (u)</math> and defined by:<ref>Formally, this formula is only valid when the input function is in a sufficiently nice space (such as [[Lp space|L¹]] or [[Schwartz space]]), and is defined via a density argument in the general case.</ref><ref>{{Cite thesis |last= Missbauer |first= Andreas |title= Gabor Frames and the Fractional Fourier Transform |date= 2012 |type= MSc |publisher= [[University of Vienna]] |url=https://www.univie.ac.at/nuhag-php/bibtex/open_files/13586_Missbauer_Diplom_final.pdf |access-date=2018-11-03 |archive-url=https://web.archive.org/web/20181103131426/https://www.univie.ac.at/nuhag-php/bibtex/open_files/13586_Missbauer_Diplom_final.pdf |archive-date=2018-11-03 |url-status=dead }}</ref><ref>If {{mvar|α}} is an integer multiple of {{pi}}, then the [[cotangent]] and [[cosecant]] functions above diverge.  This apparent divergence can be handled by taking the [[limit of a function|limit]] in the sense of [[tempered distributions]], and leads to a [[Dirac delta function]] in the integrand. This approach is consistent with the intuition that, since <math>\mathcal{F}^2(f)=f(-t)~, ~~\mathcal{F}_{\alpha} ~ (f) </math> 
must be simply {{math|''f''(''t'')}} or {{math|''f''(−''t'')}} for  {{mvar|α}}  an [[Even and odd numbers|even or odd]] multiple of {{mvar|π}} respectively.</ref>
{{Equation box 1
|indent =::
|equation =  
<math>\mathcal{F}_\alpha[f](u) = 
\sqrt{1-i\cot(\alpha)} e^{i \pi \cot(\alpha) u^2} 
\int_{-\infty}^\infty 
e^{-2\pi i\left(\csc(\alpha) u x - \frac{\cot(\alpha)}{2} x^2\right)}
f(x)\, \mathrm{d}x
</math>
|cellpadding= 6
|border
|border colour = #0073CF
|bgcolor=#F9FFF7}}

For  {{math|''α'' {{=}} ''π''/2}}, this becomes precisely the definition of the continuous Fourier transform, and for  {{math|''α'' {{=}} −''π''/2}} it is the definition of the inverse continuous Fourier transform.

The FRFT argument {{mvar|u}}  is neither a spatial one {{mvar|x}}  nor a frequency {{mvar|ξ}}. We will see why it can be interpreted as linear combination of both coordinates {{math|(''x'',''ξ'')}}. When we want to distinguish the {{mvar|α}}-angular fractional domain, we will let <math>x_a</math> denote the argument of <math>\mathcal{F}_\alpha</math>.

'''Remark:''' with the angular frequency ω convention instead of the frequency one, the FRFT formula is the [[Mehler kernel]],
<math display=block>\mathcal{F}_\alpha(f)(\omega) = 
\sqrt{\frac{1-i\cot(\alpha)}{2\pi}} 
e^{i \cot(\alpha) \omega^2/2} 
\int_{-\infty}^\infty 
e^{-i\csc(\alpha) \omega t + i \cot(\alpha) t^2/2}
f(t)\, dt~. 
</math>

===Properties===
The {{math|''α''}}-th order fractional Fourier transform operator, <math>\mathcal{F}_\alpha</math>, has the properties:

====Additivity====
For any real angles {{math|''α, β''}},
<math display=block>\mathcal{F}_{\alpha+\beta} = \mathcal{F}_\alpha \circ \mathcal{F}_\beta = \mathcal{F}_\beta \circ \mathcal{F}_\alpha.</math>

====Linearity====
<math display=block>\mathcal{F}_\alpha \left [\sum\nolimits_k b_kf_k(u)  \right ]=\sum\nolimits_k b_k\mathcal{F}_\alpha \left [f_k(u) \right ]</math>

====Integer Orders====
If {{math|''α''}} is an integer multiple of <math>\pi / 2</math>, then:
<math display=block>\mathcal{F}_\alpha = \mathcal{F}_{k\pi/2} = \mathcal{F}^k = (\mathcal{F})^k</math>

Moreover, it has following relation

<math display=block>\begin{align}
\mathcal{F}^2 &= \mathcal{P} && \mathcal{P}[f(u)]=f(-u)\\
\mathcal{F}^3 &= \mathcal{F}^{-1} = (\mathcal{F})^{-1} \\ 
\mathcal{F}^4 &= \mathcal{F}^0 = \mathcal{I} \\
\mathcal{F}^i &= \mathcal{F}^j && i \equiv j \mod 4
\end{align}</math>

====Inverse====
<math display=block>(\mathcal{F}_\alpha)^{-1}=\mathcal{F}_{-\alpha}</math>

====Commutativity====
<math display=block>\mathcal{F}_{\alpha_1}\mathcal{F}_{\alpha_2}=\mathcal{F}_{\alpha_2}\mathcal{F}_{\alpha_1}</math>

====Associativity====
<math display=block> \left (\mathcal{F}_{\alpha_1}\mathcal{F}_{\alpha_2} \right )\mathcal{F}_{\alpha_3} = \mathcal{F}_{\alpha_1} \left (\mathcal{F}_{\alpha_2}\mathcal{F}_{\alpha_3} \right )</math>

====Unitarity====
<math display=block>\int f(t)g^*(t)dt=\int f_\alpha(u)g_\alpha^*(u)du</math>

====Time Reversal====
<math display=block>\mathcal{F}_\alpha\mathcal{P}=\mathcal{P}\mathcal{F}_\alpha</math>
<math display=block>\mathcal{F}_\alpha[f(-u)]=f_\alpha(-u)</math>

====Transform of a shifted function====
{{see also|Generalizations of Pauli matrices#Construction: The clock and shift matrices}}

Define the shift and the phase shift operators as follows:

<math display=block>\begin{align}
\mathcal{SH}(u_0)[f(u)] &= f(u+u_0) \\
\mathcal{PH}(v_0)[f(u)] &= e^{j2\pi v_0u}f(u)
\end{align}</math>

Then
<math display=block>\begin{align}
\mathcal{F}_\alpha \mathcal{SH}(u_0) &= e^{j\pi u_0^2 \sin\alpha \cos\alpha} \mathcal{PH}(u_0\sin\alpha) \mathcal{SH}(u_0\cos\alpha) \mathcal{F}_\alpha,
\end{align}</math>

that is,

<math display=block>\begin{align}
\mathcal{F}_\alpha [f(u+u_0)] &=e^{j\pi u_0^2 \sin\alpha \cos\alpha}  e^{j2\pi uu_0 \sin\alpha} f_\alpha (u+u_0 \cos\alpha)
\end{align}</math>

====Transform of a scaled function====
Define the scaling and chirp multiplication operators as follows:
<math display=block>\begin{align}
M(M)[f(u)] &= |M|^{-\frac{1}{2}} f \left (\tfrac{u}{M} \right) \\
Q(q)[f(u)] &= e^{-j\pi qu^2 } f(u)
\end{align}</math>

Then, 
<math display=block>\begin{align}
\mathcal{F}_\alpha  M(M) &= Q \left (-\cot \left (\frac{1-\cos^2 \alpha'}{\cos^2 \alpha}\alpha \right ) \right)\times M \left (\frac{\sin \alpha}{M\sin \alpha'} \right )\mathcal{F}_{\alpha'} \\ [6pt]
\mathcal{F}_\alpha \left [|M|^{-\frac{1}{2}} f \left (\tfrac{u}{M} \right)  \right ] &= \sqrt{\frac{1-j \cot\alpha}{1-jM^2  \cot\alpha}} e^{j\pi u^2\cot \left (\frac{1-\cos^2 \alpha'}{\cos^2 \alpha}\alpha \right )} \times f_a \left (\frac{Mu \sin\alpha'}{\sin\alpha} \right )
\end{align}</math>

Notice that the fractional Fourier transform of <math>f(u/M)</math> cannot be expressed as a scaled version of <math>f_\alpha (u)</math>. Rather, the fractional Fourier transform of <math>f(u/M)</math> turns out to be a scaled and chirp modulated version of <math>f_{\alpha'}(u)</math> where <math>\alpha\neq\alpha'</math> is a different order.<ref>An elementary recipe, using the contangent function, and its (multi-valued) inverse, for <math>\alpha'</math> in terms of <math>\alpha</math> and <math>M</math> exists.</ref>

===Fractional kernel===
The FRFT is an [[integral transform]]
<math display=block>\mathcal{F}_\alpha f (u) = \int K_\alpha (u, x) f(x)\, \mathrm{d}x</math>
where the α-angle kernel is
<math display=block>K_\alpha (u, x) = \begin{cases}\sqrt{1-i\cot(\alpha)} \exp \left(i \pi (\cot(\alpha)(x^2+ u^2) -2 \csc(\alpha) u x) \right) & \mbox{if } \alpha \mbox{ is not a multiple of }\pi, \\
\delta (u - x) & \mbox{if } \alpha \mbox{ is a multiple of } 2\pi, \\
\delta (u + x) & \mbox{if } \alpha+\pi \mbox{ is a multiple of } 2\pi, \\
\end{cases}</math>

Here again the special cases are consistent with the limit behavior when  {{mvar|α}}  approaches a multiple of {{mvar|π}}.

The FRFT has the same properties as its kernels :
* symmetry: <math>K_\alpha~(u, u')=K_\alpha ~(u', u)</math>
* inverse: <math>K_\alpha^{-1} (u, u') = K_\alpha^* (u, u') = K_{-\alpha} (u', u) </math>
* additivity: <math>K_{\alpha+\beta} (u,u') = \int K_\alpha (u, u'') K_\beta (u'', u')\,\mathrm{d}u''.</math>

===Related transforms===
There also exist related fractional generalizations of similar transforms such as the [[discrete Fourier transform]].

* The '''discrete fractional Fourier transform''' is defined by [[Zeev zalevsky|Zeev Zalevsky]].{{sfn|Candan|Kutay|Ozaktas|2000}}{{sfn|Ozaktas|Zalevsky|Kutay|2001|loc=Chapter 6}} A quantum algorithm to implement a version of the discrete fractional Fourier transform in sub-polynomial time is described by Somma.<ref>{{cite journal |last= Somma |first= Rolando D. |date= 2016 |title= Quantum simulations of one dimensional quantum systems |journal= Quantum Information and Computation |volume= 16 |pages= 1125–1168 |arxiv= 1503.06319v2}}</ref>
* The [[Fractional wavelet transform]] (FRWT) is a generalization of the classical [[wavelet transform]] in the fractional Fourier transform domains.<ref>{{cite journal |last1= Shi |first1= Jun |last2= Zhang |first2= NaiTong |last3= Liu |first3= Xiaoping |date= June 2012 |title= A novel fractional wavelet transform and its applications |journal= Sci. China Inf. Sci. |volume= 55 |number= 6 |pages= 1270–1279 |doi= 10.1007/s11432-011-4320-x|s2cid= 3772011 }}</ref>
* The [[chirplet transform]] for a related generalization of the [[wavelet transform]].

===Generalizations===
The Fourier transform is essentially [[bosonic]]; it works because it is consistent with the superposition principle and related interference patterns.  There is also a [[fermionic]] Fourier transform.<ref name = "xyz">{{cite journal |last= De Bie |first= Hendrik |date= 1 September 2008 |title= Fourier transform and related integral transforms in superspace |journal= Journal of Mathematical Analysis and Applications |volume= 345 |issue= 1 |pages= 147–164 |doi= 10.1016/j.jmaa.2008.03.047 |arxiv= 0805.1918 |bibcode= 2008JMAA..345..147D |s2cid= 17066592 }}</ref> These have been generalized into a [[supersymmetric]] FRFT, and a supersymmetric [[Radon transform]].<ref name = "xyz" />  There is also a fractional Radon transform, a [[time–frequency analysis|symplectic]] FRFT, and a symplectic [[wavelet transform]].<ref>{{cite journal |surname1= Fan |given1= Hong-yi |surname2= Hu |given2= Li-yun |title= Optical transformation from chirplet to fractional Fourier transformation kernel |date= 2009 |journal= Journal of Modern Optics |volume= 56 |issue= 11 |pages= 1227–1229 |doi= 10.1080/09500340903033690 |arxiv= 0902.1800 |bibcode= 2009JMOp...56.1227F |s2cid= 118463188 }}</ref>  Because [[quantum circuit]]s are based on [[unitary operation]]s, they are useful for computing [[integral transform]]s as the latter are unitary operators on a [[function space]].  A quantum circuit has been designed which implements the FRFT.<ref>{{cite journal |last1= Klappenecker |first1= Andreas |last2= Roetteler |first2=  Martin |date= January 2002 |title= Engineering Functional Quantum Algorithms |journal= Physical Review A |volume= 67 |issue= 1 |pages= 010302 |doi= 10.1103/PhysRevA.67.010302 |arxiv= quant-ph/0208130 |s2cid= 14501861 }}</ref>

==Interpretation==
===Time-frequency analysis===
{{further|Linear canonical transformation}}
[[File:Rect turning into a sinc.webm|thumb|A rect function turns into a sinc function as the order of the fractional Fourier transform becomes 1]]
The usual interpretation of the Fourier transform is as a transformation of a time domain signal into a frequency domain signal. On the other hand, the interpretation of the inverse Fourier transform is as a transformation of a frequency domain signal into a time domain signal. Fractional Fourier transforms  transform a signal (either in the time domain or frequency domain) into the domain between time and frequency: it is a rotation in the [[time–frequency domain]]. This perspective is generalized by the [[linear canonical transformation]], which generalizes the fractional Fourier transform and allows linear transforms of the time–frequency domain other than rotation.

Take the  figure below as an example. If the signal in the time domain is rectangular (as below), it   becomes a [[sinc function]] in the frequency domain. But if  one applies the fractional Fourier transform to the rectangular signal, the transformation output will be in the domain between time and frequency.
[[Image:FracFT Rec by stevencys.jpg|thumb|center|600px|Fractional Fourier transform]]

The fractional Fourier transform is a rotation operation on a [[time–frequency representation|time–frequency distribution]]. From the definition above, for ''α''&nbsp;=&nbsp;0, there will be no change after applying the fractional Fourier transform, while for ''α''&nbsp;=&nbsp;''π''/2, the fractional Fourier transform becomes a plain Fourier transform, which rotates the time–frequency distribution with&nbsp;''π''/2. For other value of&nbsp;''α'', the fractional Fourier transform rotates the time–frequency distribution according to α. The following figure shows the results of the fractional Fourier transform with different values of&nbsp;''α''.
[[Image:FracFT Rotate by stevencys.jpg|thumb|center|600px|Time/frequency distribution of fractional Fourier transform]]

{{See also|Time–frequency analysis}}

===Fresnel and Fraunhofer diffraction===
The diffraction of light can be calculated using integral transforms. The [[Fresnel diffraction|Fresnel diffraction integral]] is used to find the near field diffraction pattern. In the far-field limit this equation becomes a Fourier transform to give the equation for [[Fraunhofer diffraction]].  The fractional Fourier transform is equivalent to the Fresnel diffraction equation.<ref>{{cite journal |last1= Pellat-Finet |first1= Pierre |date= 15 September 1994 |title= Fresnel diffraction and the fractional-order Fourier transform |journal= Optics Letters |volume= 19 |issue= 18 |pages= 1388–1390 |doi= 10.1364/OL.19.001388|pmid= 19855528 |bibcode= 1994OptL...19.1388P }}</ref><ref>{{cite journal |last1= Pellat-Finet |last2= Bonnet |first1= Pierre |first2= Georges |date= 15 September 1994 |title= Fractional order Fourier transform and Fourier optics |journal= Optics Communications |volume= 111 |issue= 1–2 |page= 141 |doi= 10.1016/0030-4018(94)90154-6|bibcode= 1994OptCo.111..141P }}</ref> When the angle <math>\alpha</math> becomes <math>\pi/2</math>, the fractional Fourier transform is the standard Fourier transform and gives the far-field diffraction pattern. The near-field diffraction maps to values of <math>\alpha</math> between 0 and <math>\pi/2</math>.

==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>
==See also==
* [[Least-squares spectral analysis]]
* [[Fractional calculus]]
* [[Mehler kernel]]
Other time–frequency transforms:
* [[Linear canonical transformation]]
* [[Short-time Fourier transform]]
* [[Wavelet transform]]
* [[Chirplet transform]]
* [[Cone-shape distribution function]]
* [[Quadratic Fourier transform]]
* [[Chirp Z-transform]]

==References==
{{reflist}}

==Bibliography==
* {{cite journal |last1= Candan |first1= C. |last2= Kutay |first2= M. A. |last3= Ozaktas |first3= H. M. |date= May 2000 |title= The discrete fractional Fourier transform |journal= IEEE Transactions on Signal Processing |volume= 48 |issue= 5 |pages= 1329–1337 |doi= 10.1109/78.839980 |bibcode= 2000ITSP...48.1329C |url= http://repository.bilkent.edu.tr/bitstream/11693/11130/1/10.1109-78.839980.pdf |hdl= 11693/11130 |hdl-access= free}}
* {{cite book |last= Ding |first= Jian-Jiun |date= 2007 |title= Time frequency analysis and wavelet transform |type= Class notes |publisher= Department of Electrical Engineering, National Taiwan University (NTU) |location= Taipei, Taiwan}}
* {{cite journal |last= Lohmann |first= A. W. |date= 1993 |title= Image rotation, Wigner rotation and the fractional Fourier transform |journal= J. Opt. Soc. Am. |volume= A |number= 10 |pages= 2181–2186|doi= 10.1364/JOSAA.10.002181 |bibcode= 1993JOSAA..10.2181L }}
* {{cite book |last1= Ozaktas |first1= Haldun M. |last2= Zalevsky |first2= Zeev |last3= Kutay |first3= M. Alper |date= 2001 |title= The Fractional Fourier Transform with Applications in Optics and Signal Processing |publisher= John Wiley & Sons |series= Series in Pure and Applied Optics |url= http://www.ee.bilkent.edu.tr/~haldun/wileybook.html <!-- support page--> |isbn= 978-0-471-96346-2 |author1-link= Haldun Ozaktas}}
* {{cite journal |last1= Pei |first1= Soo-Chang |last2= Ding |first2= Jian-Jiun |date= 2001 |title= Relations between fractional operations and time–frequency distributions, and their applications |journal= IEEE Trans. Signal Process. |volume= 49 |number= 8 |pages= 1638–1655|doi= 10.1109/78.934134 |bibcode= 2001ITSP...49.1638P }}
* {{cite journal |last1= Saxena |first1= Rajiv |last2= Singh |first2= Kulbir |title= Fractional Fourier transform: A novel tool for signal processing |journal= J. Indian Inst. Sci. |date= January–February 2005 |volume= 85 |pages= 11–26 |url= http://journal.library.iisc.ernet.in/vol200501/paper2/11.pdf|archive-url= https://web.archive.org/web/20110716112239/http://journal.library.iisc.ernet.in/vol200501/paper2/11.pdf |archive-date= 16 July 2011 }}

==External links==
* [http://tfd.sourceforge.net/ DiscreteTFDs -- software for computing the fractional Fourier transform and time–frequency distributions]
*"[http://demonstrations.wolfram.com/FractionalFourierTransform/ Fractional Fourier Transform]" by Enrique Zeleny, [[The Wolfram Demonstrations Project]].
* [https://web.archive.org/web/20090215081459/http://mechatronics.ece.usu.edu/foc/FRFT/ Dr YangQuan Chen's FRFT (Fractional Fourier Transform) Webpages]
* [http://ltfat.sourceforge.net/ LTFAT - A free (GPL) Matlab / Octave toolbox] Contains several version of the [http://ltfat.sourceforge.net/doc/fourier/ffracft.php fractional Fourier transform] {{Webarchive|url=https://web.archive.org/web/20160304115744/http://ltfat.sourceforge.net/doc/fourier/ffracft.php |date=4 March 2016 }}.

{{DEFAULTSORT:Fractional Fourier Transform}}
[[Category:Fourier analysis]]
[[Category:Time–frequency analysis]]
[[Category:Integral transforms]]
[[Category:Articles containing video clips]]