Inmathematics, in the area ofharmonic analysis, thefractional Fourier transform (FRFT) is a family oflinear transformations generalizing theFourier transform. It can be thought of as the Fourier transform to then-th power, wheren need not be aninteger — thus, it can transform a function to anyintermediate domain between time andfrequency. Its applications range fromfilter design andsignal analysis tophase retrieval andpattern recognition.

The FRFT can be used to define fractionalconvolution,correlation, and other operations, and can also be further generalized into thelinear canonical transformation (LCT). An early definition of the FRFT was introduced byCondon,^[1] by solving for theGreen's function for phase-space rotations, and also by Namias,^[2] generalizing work ofWiener^[3] onHermite polynomials.

However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups.^[4] Since then, there has been a surge of interest in extending Shannon's sampling theorem^[5][6] 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^[7] as essentially another name for az-transform, and in particular for the case that corresponds to adiscrete Fourier transform shifted by a fractional amount in frequency space (multiplying the input by a linearchirp) 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 byBluestein'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.

The continuousFourier transform${\displaystyle \mathcal{F}}$ of a function${\displaystyle f:\mathbb{R}\mapsto \mathbb{C}}$ is aunitary operator of${\displaystyle L^2}$ space that maps the function${\displaystyle f}$ to its frequential version${\displaystyle \hat{f}}$ (all expressions are taken in the${\displaystyle L^2}$ sense, rather than pointwise):

$${\displaystyle \hat{f}(\xi )={\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}f(x)e^{-2\pi ix\xi }\mathrm{d}x}$$

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

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

Let us study itsn-th iterated${\displaystyle {\mathcal{F}}^n}$ defined by${\displaystyle {\mathcal{F}}^n[f]=\mathcal{F}[{\mathcal{F}}^{n-1}[f]]}$ and${\displaystyle {\mathcal{F}}^{-n}=({\mathcal{F}}^{-1}{)}^n}$ whenn is a non-negative integer, and${\displaystyle {\mathcal{F}}^0[f]=f}$. Their sequence is finite since${\displaystyle \mathcal{F}}$ is a 4-periodicautomorphism: for every function${\displaystyle f}$,${\displaystyle {\mathcal{F}}^4[f]=f}$.

More precisely, let us introduce theparity operator${\displaystyle \mathcal{P}}$ that inverts${\displaystyle x}$,${\displaystyle \mathcal{P}[f]:x\mapsto f(-x)}$. Then the following properties hold:$${\displaystyle {\mathcal{F}}^0=\mathrm{I}\mathrm{d},{\mathcal{F}}^1=\mathcal{F},{\mathcal{F}}^2=\mathcal{P},{\mathcal{F}}^4=\mathrm{I}\mathrm{d}}$$$${\displaystyle {\mathcal{F}}^3={\mathcal{F}}^{-1}=\mathcal{P}\circ \mathcal{F}=\mathcal{F}\circ \mathcal{P}.}$$

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

Note: some authors write the transform in terms of the "ordera" instead of the "angleα", in which case theα is usuallya timesπ/2. Although these two forms are equivalent, one must be careful about which definition the author uses.

For anyrealα, theα-angle fractional Fourier transform of a function ƒ is denoted by${\displaystyle {\mathcal{F}}_{\alpha }(u)}$ and defined by:^[8][9][10]

Forα =π/2, this becomes precisely the definition of the continuous Fourier transform, and forα = −π/2 it is the definition of the inverse continuous Fourier transform.

The FRFT argumentu is neither a spatial onex nor a frequencyξ. We will see why it can be interpreted as linear combination of both coordinates(x,ξ). When we want to distinguish theα-angular fractional domain, we will let${\displaystyle x_a}$ denote the argument of${\displaystyle {\mathcal{F}}_{\alpha }}$.

Remark: with the angular frequency ω convention instead of the frequency one, the FRFT formula is theMehler kernel,$${\displaystyle {\mathcal{F}}_{\alpha }(f)(\omega )=\sqrt{\frac{1-i\cot (\alpha )}{2\pi }}{e}^{i\cot (\alpha ){\omega }^2/2}{\int }_{-\mathrm{\infty }}^{\mathrm{\infty }}{e}^{-i\csc (\alpha )\omega t+i\cot (\alpha )t^2/2}f(t)dt.}$$

Theα-th order fractional Fourier transform operator,${\displaystyle {\mathcal{F}}_{\alpha }}$, has the properties:

For any real anglesα, β,$${\displaystyle {\mathcal{F}}_{\alpha +\beta }={\mathcal{F}}_{\alpha }\circ {\mathcal{F}}_{\beta }={\mathcal{F}}_{\beta }\circ {\mathcal{F}}_{\alpha }.}$$

$${\displaystyle {\mathcal{F}}_{\alpha }\left[{\sum }_k{b}_k{f}_k(u)\right]={\sum }_k{b}_k{\mathcal{F}}_{\alpha }\left[f_k(u)\right]}$$

Ifα is an integer multiple of${\displaystyle \pi /2}$, then:$${\displaystyle {\mathcal{F}}_{\alpha }={\mathcal{F}}_{k\pi /2}={\mathcal{F}}^k=(\mathcal{F}{)}^k}$$

Moreover, it has following relation

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

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

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

$${\displaystyle \int f(t)g^{\ast }(t)dt=\int f_{\alpha }(u)g_{\alpha }^{\ast }(u)du}$$

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

Define the shift and the phase shift operators as follows:

Then

that is,

Define the scaling and chirp multiplication operators as follows:

Then,

Notice that the fractional Fourier transform of${\displaystyle f(u/M)}$ cannot be expressed as a scaled version of${\displaystyle f_{\alpha }(u)}$. Rather, the fractional Fourier transform of${\displaystyle f(u/M)}$ turns out to be a scaled and chirp modulated version of${\displaystyle f_{{\alpha }^{\prime }}(u)}$ where${\displaystyle \alpha \ne {\alpha }^{\prime }}$ is a different order.^[11]

The FRFT is anintegral transform$${\displaystyle {\mathcal{F}}_{\alpha }f(u)=\int K_{\alpha }(u,x)f(x)\mathrm{d}x}$$where the α-angle kernel is

Here again the special cases are consistent with the limit behavior whenα approaches a multiple ofπ.

The FRFT has the same properties as its kernels :

• symmetry:${\displaystyle K_{\alpha }(u,u^{\prime })=K_{\alpha }(u^{\prime },u)}$
• inverse:${\displaystyle K_{\alpha }^{-1}(u,u^{\prime })=K_{\alpha }^{\ast }(u,u^{\prime })=K_{-\alpha }(u^{\prime },u)}$
• additivity:${\displaystyle K_{\alpha +\beta }(u,u^{\prime })=\int K_{\alpha }(u,u^{″})K_{\beta }(u^{″},u^{\prime })\mathrm{d}{u}^{″}.}$

There also exist related fractional generalizations of similar transforms such as thediscrete Fourier transform.

• Thediscrete fractional Fourier transform is defined byZeev Zalevsky.^[12][13] A quantum algorithm to implement a version of the discrete fractional Fourier transform in sub-polynomial time is described by Somma.^[14]
• TheFractional wavelet transform (FRWT) is a generalization of the classicalwavelet transform in the fractional Fourier transform domains.^[15]
• Thechirplet transform for a related generalization of thewavelet transform.

The Fourier transform is essentiallybosonic; it works because it is consistent with the superposition principle and related interference patterns. There is also afermionic Fourier transform.^[16] These have been generalized into asupersymmetric FRFT, and a supersymmetricRadon transform.^[16] There is also a fractional Radon transform, asymplectic FRFT, and a symplecticwavelet transform.^[17] Becausequantum circuits are based onunitary operations, they are useful for computingintegral transforms as the latter are unitary operators on afunction space. A quantum circuit has been designed which implements the FRFT.^[18]

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 thetime–frequency domain. This perspective is generalized by thelinear 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 asinc 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.

The fractional Fourier transform is a rotation operation on atime–frequency distribution. From the definition above, forα = 0, there will be no change after applying the fractional Fourier transform, while forα = 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.

The diffraction of light can be calculated using integral transforms. TheFresnel 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 forFraunhofer diffraction. The fractional Fourier transform is equivalent to the Fresnel diffraction equation.^[19][20] When the angle${\displaystyle \alpha }$ becomes${\displaystyle \pi /2}$, the fractional Fourier transform is the standard Fourier transform and gives the far-field diffraction pattern. The near-field diffraction maps to values of${\displaystyle \alpha }$ between 0 and${\displaystyle \pi /2}$.

Fractional Fourier transform can be used in time frequency analysis andDSP.^[21] 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.

Thus, using just truncation in the time domain, or equivalentlylow-pass filters in the frequency domain, one can cut out anyconvex 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,^[22] in high-dimensional quantum key distribution schemes with single photons,^[23] and in observing spatial entanglement of photon pairs.^[24]

They are also useful in the design of optical systems and for optimizing holographic storage efficiency.^[25]

• 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

• Candan, C.; Kutay, M. A.; Ozaktas, H. M. (May 2000)."The discrete fractional Fourier transform"(PDF).IEEE Transactions on Signal Processing.48 (5):1329–1337.Bibcode:2000ITSP...48.1329C.doi:10.1109/78.839980.hdl:11693/11130.
• Ding, Jian-Jiun (2007).Time frequency analysis and wavelet transform (Class notes). Taipei, Taiwan: Department of Electrical Engineering, National Taiwan University (NTU).
• Lohmann, A. W. (1993). "Image rotation, Wigner rotation and the fractional Fourier transform".J. Opt. Soc. Am.A (10):2181–2186.Bibcode:1993JOSAA..10.2181L.doi:10.1364/JOSAA.10.002181.
• Ozaktas, Haldun M.; Zalevsky, Zeev; Kutay, M. Alper (2001).The Fractional Fourier Transform with Applications in Optics and Signal Processing. Series in Pure and Applied Optics. John Wiley & Sons.ISBN 978-0-471-96346-2.
• Pei, Soo-Chang; Ding, Jian-Jiun (2001). "Relations between fractional operations and time–frequency distributions, and their applications".IEEE Trans. Signal Process.49 (8):1638–1655.Bibcode:2001ITSP...49.1638P.doi:10.1109/78.934134.
• Saxena, Rajiv; Singh, Kulbir (January–February 2005)."Fractional Fourier transform: A novel tool for signal processing"(PDF).J. Indian Inst. Sci.85:11–26. Archived fromthe original(PDF) on 16 July 2011.

• DiscreteTFDs -- software for computing the fractional Fourier transform and time–frequency distributions
• "Fractional Fourier Transform" by Enrique Zeleny,The Wolfram Demonstrations Project.
• Dr YangQuan Chen's FRFT (Fractional Fourier Transform) Webpages
• LTFAT - A free (GPL) Matlab / Octave toolbox Contains several version of thefractional Fourier transformArchived 4 March 2016 at theWayback Machine.

• Fourier analysis
• Time–frequency analysis
• Integral transforms

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