Fractional Fourier transform

The continuous Fourier transform $\backslash mathcal\{F\}$ of a function $f:\; \backslash mathbb\{R\}\; \backslash mapsto\; \backslash mathbb\{C\}$ is a unitary operator of [[Lp space|

$L^2$ space]] that maps the function $f$ to its frequential version $\backslash hat\{f\}$ (all expressions are taken in the $L^2$ sense, rather than pointwise):

$$\backslash hat\{f\}(\backslash xi)\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; f(x)\backslash \; e^\{-\; 2\backslash pi\; i\; x\; \backslash xi\}\backslash ,\backslash mathrm\{d\}x$$

and $f$ is determined by $\backslash hat\{f\}$ via the inverse transform $\backslash mathcal\{F\}^\{-1\}\backslash ,\; ,$

$$f(x)\; =\; \backslash int\_\{-\backslash infty\}^\{\backslash infty\}\; \backslash hat\{f\}(\backslash xi)\backslash \; e^\{2\; \backslash pi\; i\; \backslash xi\; x\}\backslash ,\backslash mathrm\{d\}\backslash xi\backslash ,\; .$$

Let us study its n-th iterated $\backslash mathcal\{F\}^\{n\}$ defined by

$\backslash mathcal\{F\}^\{n\}[f]\; =\; \backslash mathcal\{F\}[\backslash mathcal\{F\}^\{n-1\}[f]]$ and $\backslash mathcal\{F\}^\{-n\}\; =\; (\backslash mathcal\{F\}^\{-1\})^n$ when n is a non-negative integer, and $\backslash mathcal\{F\}^\{0\}[f]\; =\; f$. Their sequence is finite since $\backslash mathcal\{F\}$ is a 4-periodic automorphism: for every function $f$, $\backslash mathcal\{F\}^4\; [f]\; =\; f$.

More precisely, let us introduce the parity operator $\backslash mathcal\{P\}$ that inverts $x$, $\backslash mathcal\{P\}[f]\backslash colon\; x\; \backslash mapsto\; f(-x)$. Then the following properties hold:

$$\backslash mathcal\{F\}^0\; =\; \backslash mathrm\{Id\},\; \backslash qquad\; \backslash mathcal\{F\}^1\; =\; \backslash mathcal\{F\},\; \backslash qquad\; \backslash mathcal\{F\}^2\; =\; \backslash mathcal\{P\},\; \backslash qquad\; \backslash mathcal\{F\}^4\; =\; \backslash mathrm\{Id\}$$

$$\backslash mathcal\{F\}^3\; =\; \backslash mathcal\{F\}^\{-1\}\; =\; \backslash mathcal\{P\}\; \backslash circ\; \backslash mathcal\{F\}\; =\; \backslash mathcal\{F\}\; \backslash circ\; \backslash mathcal\{P\}.$$

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

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 {{mvar|α}}, the {{mvar|α}}-angle fractional Fourier transform of a function ƒ is denoted by $\backslash mathcal\{F\}\_\backslash alpha\; (u)$ and defined by:Formally, this formula is only valid when the input function is in a sufficiently nice space (such as L¹ or Schwartz space), and is defined via a density argument in the general case.{{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 }}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 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 $\backslash mathcal\{F\}^2(f)=f(-t)~,\; ~~\backslash mathcal\{F\}\_\{\backslash alpha\}\; ~\; (f)$

must be simply {{math|f(t)}} or {{math|f(−t)}} for {{mvar|α}} an even or odd multiple of {{mvar|π}} respectively.

{{Equation box 1

|indent =::

|equation =

$\backslash mathcal\{F\}\_\backslash alpha[f](u)\; =$

\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

|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 $x\_a$ denote the argument of $\backslash mathcal\{F\}\_\backslash alpha$.

Remark: with the angular frequency ω convention instead of the frequency one, the FRFT formula is the Mehler kernel,

$$\backslash mathcal\{F\}\_\backslash alpha(f)(\backslash omega)\; =$$

\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~.

The {{math|α}}-th order fractional Fourier transform operator, $\backslash mathcal\{F\}\_\backslash alpha$, has the properties:

For any real angles {{math|α, β}},

$$\backslash mathcal\{F\}\_\{\backslash alpha+\backslash beta\}\; =\; \backslash mathcal\{F\}\_\backslash alpha\; \backslash circ\; \backslash mathcal\{F\}\_\backslash beta\; =\; \backslash mathcal\{F\}\_\backslash beta\; \backslash circ\; \backslash mathcal\{F\}\_\backslash alpha.$$

$$\backslash mathcal\{F\}\_\backslash alpha\; \backslash left\; [\backslash sum\backslash nolimits\_k\; b\_kf\_k(u)\; \backslash right\; ]=\backslash sum\backslash nolimits\_k\; b\_k\backslash mathcal\{F\}\_\backslash alpha\; \backslash left\; [f\_k(u)\; \backslash right\; ]$$

If {{math|α}} is an integer multiple of $\backslash pi\; /\; 2$, then:

$$\backslash mathcal\{F\}\_\backslash alpha\; =\; \backslash mathcal\{F\}\_\{k\backslash pi/2\}\; =\; \backslash mathcal\{F\}^k\; =\; (\backslash mathcal\{F\})^k$$

Moreover, it has following relation

$$\backslash begin\{align\}$$

\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}

$$(\backslash mathcal\{F\}\_\backslash alpha)^\{-1\}=\backslash mathcal\{F\}\_\{-\backslash alpha\}$$

$$\backslash mathcal\{F\}\_\{\backslash alpha\_1\}\backslash mathcal\{F\}\_\{\backslash alpha\_2\}=\backslash mathcal\{F\}\_\{\backslash alpha\_2\}\backslash mathcal\{F\}\_\{\backslash alpha\_1\}$$

$$\backslash left\; (\backslash mathcal\{F\}\_\{\backslash alpha\_1\}\backslash mathcal\{F\}\_\{\backslash alpha\_2\}\; \backslash right\; )\backslash mathcal\{F\}\_\{\backslash alpha\_3\}\; =\; \backslash mathcal\{F\}\_\{\backslash alpha\_1\}\; \backslash left\; (\backslash mathcal\{F\}\_\{\backslash alpha\_2\}\backslash mathcal\{F\}\_\{\backslash alpha\_3\}\; \backslash right\; )$$

$$\backslash int\; f(t)g^*(t)dt=\backslash int\; f\_\backslash alpha(u)g\_\backslash alpha^*(u)du$$

$$\backslash mathcal\{F\}\_\backslash alpha\backslash mathcal\{P\}=\backslash mathcal\{P\}\backslash mathcal\{F\}\_\backslash alpha$$

$$\backslash mathcal\{F\}\_\backslash alpha[f(-u)]=f\_\backslash alpha(-u)$$

{{see also|Generalizations of Pauli matrices#Construction: The clock and shift matrices}}

Define the shift and the phase shift operators as follows:

$$\backslash begin\{align\}$$

\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}

Then

$$\backslash begin\{align\}$$

\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}

that is,

$$\backslash begin\{align\}$$

\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}

Define the scaling and chirp multiplication operators as follows:

$$\backslash begin\{align\}$$

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}

Then,

$$\backslash begin\{align\}$$

\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}

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

The FRFT is an integral transform

$$\backslash mathcal\{F\}\_\backslash alpha\; f\; (u)\; =\; \backslash int\; K\_\backslash alpha\; (u,\; x)\; f(x)\backslash ,\; \backslash mathrm\{d\}x$$

where the α-angle kernel is

\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}

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: $K\_\backslash alpha~(u,\; u\text{'})=K\_\backslash alpha\; ~(u\text{'},\; u)$
• inverse: $K\_\backslash alpha^\{-1\}\; (u,\; u\text{'})\; =\; K\_\backslash alpha^*\; (u,\; u\text{'})\; =\; K\_\{-\backslash alpha\}\; (u\text{'},\; u)$
• additivity: $K\_\{\backslash alpha+\backslash beta\}\; (u,u\text{'})\; =\; \backslash int\; K\_\backslash alpha\; (u,\; u$) K_\beta (u, u')\,\mathrm{d}u''.

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.{{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.{{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}}
• The Fractional wavelet transform (FRWT) is a generalization of the classical wavelet transform in the fractional Fourier transform domains.{{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 }}
• The chirplet transform for a related generalization of the wavelet transform.

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.{{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 }} These have been generalized into a supersymmetric FRFT, and a supersymmetric Radon transform. There is also a fractional Radon transform, a symplectic FRFT, and a symplectic wavelet transform.{{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 }} Because quantum circuits are based on unitary operations, they are useful for computing integral transforms as the latter are unitary operators on a function space. A quantum circuit has been designed which implements the FRFT.{{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 }}

{{further|Linear canonical transformation}}

File:Rect turning into a sinc.webm

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

The fractional Fourier transform is a rotation operation on a time–frequency distribution. From the definition above, for α = 0, there will be no change after applying the fractional Fourier transform, while for α = π/2, the fractional Fourier transform becomes a plain Fourier transform, which rotates the time–frequency distribution with π/2. For other value of 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 α.

Image:FracFT Rotate by stevencys.jpg

{{See also|Time–frequency analysis}}

The diffraction of light can be calculated using integral transforms. The 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.{{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 }}{{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 }} When the angle $\backslash alpha$ becomes $\backslash pi/2$, the fractional Fourier transform is the standard Fourier transform and gives the far-field diffraction pattern. The near-field diffraction maps to values of $\backslash alpha$ between 0 and $\backslash pi/2$.

Fractional Fourier transform can be used in time frequency analysis and DSP.{{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 }} 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

Thus, using just truncation in the time domain, or equivalently low-pass filters 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,{{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 }} in high-dimensional quantum key distribution schemes with single photons,{{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 }} and in observing spatial entanglement of photon pairs.{{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 }}

They are also useful in the design of optical systems and for optimizing holographic storage efficiency.{{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 }}{{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}}

• 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

{{reflist}}

• {{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 |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 }}

• [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}}

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