Editing Fractional Fourier transform (section)

==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>