Fractional Fourier remodel – Wikipedia

2023-12-16 08:52:05

N-th energy Fourier remodel

In mathematics, within the space of harmonic analysis, the fractional Fourier remodel (FRFT) is a household of linear transformations generalizing the Fourier transform. It may be regarded as the Fourier remodel to the n-th energy, the place n needn’t be an integer — thus, it may well remodel a operate to any intermediate area between time and frequency. Its functions vary from filter design and signal analysis to phase retrieval and pattern recognition.

The FRFT can be utilized to outline fractional convolution, correlation, and different operations, and can be additional generalized into the linear canonical transformation (LCT). An early definition of the FRFT was launched by Condon,^[1] by fixing for the Green’s function for phase-space rotations, and likewise by Namias,^[2] generalizing work of Wiener^[3] on Hermite polynomials.

Nevertheless, it was not widely known in sign processing till it was independently reintroduced round 1993 by a number of teams.^[4] Since then, there was a surge of curiosity in extending Shannon’s sampling theorem^[5]^[6] for indicators that are band-limited within the Fractional Fourier area.

A very totally different which means for “fractional Fourier remodel” was launched by Bailey and Swartztrauber^[7] as primarily one other title for a z-transform, and specifically for the case that corresponds to a discrete Fourier transform shifted by a fractional quantity in frequency area (multiplying the enter by a linear chirp) and evaluating at a fractional set of frequency factors (e.g. contemplating solely a small portion of the spectrum). (Such transforms will be evaluated effectively by Bluestein’s FFT algorithm.) This terminology has fallen out of use in a lot of the technical literature, nonetheless, as opposed to the FRFT. The rest of this text describes the FRFT.

Introduction[edit]

The continual Fourier transform ${displaystyle {mathcal {F}}}$ of a operate ${displaystyle f:mathbb {R} mapsto mathbb {C} }$ is a unitary operator of
${displaystyle L^{2}}$ space that maps the operate ${displaystyle f}$ to its frequential model ${displaystyle {hat {f}}}$ (all expressions are taken within the ${displaystyle L^{2}}$ sense, quite than pointwise):

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

and ${displaystyle f}$ is set by ${displaystyle {hat {f}}}$ by way of the inverse remodel ${displaystyle {mathcal {F}}^{-1},,}$

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

Allow us to examine its n-th iterated ${displaystyle {mathcal {F}}^{n}}$ outlined by
${displaystyle {mathcal {F}}^{n}[f]={mathcal {F}}[{mathcal {F}}^{n-1}[f]]}$ and ${displaystyle {mathcal {F}}^{-n}=({mathcal {F}}^{-1})^{n}}$ when n is a non-negative integer, and ${displaystyle {mathcal {F}}^{0}[f]=f}$ . Their sequence is finite since ${displaystyle {mathcal {F}}}$ is a 4-periodic automorphism: for each operate ${displaystyle f}$ , ${displaystyle {mathcal {F}}^{4}[f]=f}$ .

Extra exactly, allow us to introduce the parity operator ${displaystyle {mathcal {P}}}$ that inverts ${displaystyle x}$ , ${displaystyle {mathcal {P}}[f]colon xmapsto f(-x)}$ . Then the next properties maintain:

{displaystyle {mathcal {F}}^{0}=mathrm {Id} ,qquad {mathcal {F}}^{1}={mathcal {F}},qquad {mathcal {F}}^{2}={mathcal {P}},qquad {mathcal {F}}^{4}=mathrm {Id} }

{displaystyle {mathcal {F}}^{3}={mathcal {F}}^{-1}={mathcal {P}}circ {mathcal {F}}={mathcal {F}}circ {mathcal {P}}.}

The FRFT offers a household of linear transforms that additional extends this definition to deal with non-integer powers ${displaystyle n=2alpha /pi }$ of the FT.

Definition[edit]

Word: some authors write the remodel when it comes to the “order $a$ ” as an alternative of the “angle $α$ “, through which case the $α$ is normally $a$ instances $π /2$ . Though these two types are equal, one should be cautious about which definition the writer makes use of.

For any real $α$ , the $α$ -angle fractional Fourier remodel of a operate ƒ is denoted by ${displaystyle {mathcal {F}}_{alpha }(u)}$ and outlined by

${displaystyle {mathcal {F}}_{alpha }[f](u)={sqrt {1-icot(alpha )}}e^{ipi cot(alpha )u^{2}}int _{-infty }^{infty }e^{-2pi ileft(csc(alpha )ux-{frac {cot(alpha )}{2}}x^{2}proper)}f(x),mathrm {d} x}$

Formally, this components is simply legitimate when the enter operate is in a sufficiently good area (comparable to L1 or Schwartz space), and is outlined by way of a density argument, in a method much like that of the extraordinary Fourier transform (see article), within the basic case.^[8]

If $α$ is an integer a number of of π, then the cotangent and cosecant capabilities above diverge. Nevertheless, this may be dealt with by taking the limit, and results in a Dirac delta function within the integrand. Extra immediately, since ${displaystyle {mathcal {F}}^{2}(f)=f(-t)~,~~{mathcal {F}}_{alpha }~(f)}$
should be merely $f (t)$ or $f (- t)$ for $α$ an even or odd a number of of $π$ respectively.

For $α = π /2$ , this turns into exactly the definition of the continual Fourier remodel, and for $α = - π /2$ it’s the definition of the inverse steady Fourier remodel.

The FRFT argument $u$ is neither a spatial one $x$ nor a frequency $ξ$ . We’ll see why it may be interpreted as linear mixture of each coordinates $(x, ξ)$ . Once we need to distinguish the $α$ -angular fractional area, we are going to let ${displaystyle x_{a}}$ denote the argument of ${displaystyle {mathcal {F}}_{alpha }}$ .

Comment: with the angular frequency ω conference as an alternative of the frequency one, the FRFT components is the Mehler kernel,

{displaystyle {mathcal {F}}_{alpha }(f)(omega )={sqrt {frac {1-icot(alpha )}{2pi }}}e^{icot(alpha )omega ^{2}/2}int _{-infty }^{infty }e^{-icsc(alpha )omega t+icot(alpha )t^{2}/2}f(t),dt~.}

Properties[edit]

The $α$ -th order fractional Fourier remodel operator, ${displaystyle {mathcal {F}}_{alpha }}$ , has the properties:

Additivity[edit]

For any actual angles $α, β$ ,

{displaystyle {mathcal {F}}_{alpha +beta }={mathcal {F}}_{alpha }circ {mathcal {F}}_{beta }={mathcal {F}}_{beta }circ {mathcal {F}}_{alpha }.}

Linearity[edit]

{displaystyle {mathcal {F}}_{alpha }left[sum nolimits _{k}b_{k}f_{k}(u)right]=sum nolimits _{okay}b_{okay}{mathcal {F}}_{alpha }left[f_{k}(u)right]}

Integer Orders[edit]

If $α$ is an integer a number of of ${displaystyle pi /2}$ , then:

{displaystyle {mathcal {F}}_{alpha }={mathcal {F}}_{kpi /2}={mathcal {F}}^{okay}=({mathcal {F}})^{okay}}

Furthermore, it has following relation

{displaystyle {start{aligned}{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}&&iequiv jmod 4end{aligned}}}

Inverse[edit]

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

Commutativity[edit]

{displaystyle {mathcal {F}}_{alpha _{1}}{mathcal {F}}_{alpha _{2}}={mathcal {F}}_{alpha _{2}}{mathcal {F}}_{alpha _{1}}}

Associativity[edit]

{displaystyle left({mathcal {F}}_{alpha _{1}}{mathcal {F}}_{alpha _{2}}proper){mathcal {F}}_{alpha _{3}}={mathcal {F}}_{alpha _{1}}left({mathcal {F}}_{alpha _{2}}{mathcal {F}}_{alpha _{3}}proper)}

Unitarity[edit]

{displaystyle int f(u)g^{*}(u)du=int f_{alpha }(u)g_{alpha }^{*}(u)du}

Time Reversal[edit]

{displaystyle {mathcal {F}}_{alpha }{mathcal {P}}={mathcal {P}}{mathcal {F}}_{alpha }}

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

Remodel of a shifted operate[edit]

Outline the shift and the section shift operators as follows:

{displaystyle {start{aligned}{mathcal {SH}}(u_{0})[f(u)]&=f(u+u_{0}){mathcal {PH}}(v_{0})[f(u)]&=e^{j2pi v_{0}u}f(u)finish{aligned}}}

Then

{displaystyle {start{aligned}{mathcal {F}}_{alpha }{mathcal {SH}}(u_{0})&=e^{jpi u_{0}^{2}sin alpha cos alpha }{mathcal {PH}}(u_{0}sin alpha ){mathcal {SH}}(u_{0}cos alpha ){mathcal {F}}_{alpha },finish{aligned}}}

that’s,

{displaystyle {start{aligned}{mathcal {F}}_{alpha }[f(u+u_{0})]&=e^{jpi u_{0}^{2}sin alpha cos alpha }e^{j2pi uu_{0}sin alpha }f_{alpha }(u+u_{0}cos alpha )finish{aligned}}}

Remodel of a scaled operate[edit]

Outline the scaling and chirp multiplication operators as follows:

{displaystyle {start{aligned}M(M)[f(u)]&=|M|^{-{frac {1}{2}}}fleft({tfrac {u}{M}}proper)Q(q)[f(u)]&=e^{-jpi qu^{2}}f(u)finish{aligned}}}

Then,

{displaystyle {start{aligned}{mathcal {F}}_{alpha }M(M)&=Qleft(-cot left({frac {1-cos ^{2}alpha ‘}{cos ^{2}alpha }}alpha proper)proper)instances Mleft({frac {sin alpha }{Msin alpha ‘}}proper){mathcal {F}}_{alpha ‘}[6pt]{mathcal {F}}_{alpha }left[|M|^{-{frac {1}{2}}}fleft({tfrac {u}{M}}right)right]&={sqrt {frac {1-jcot alpha }{1-jM^{2}cot alpha }}}e^{jpi u^{2}cot left({frac {1-cos ^{2}alpha ‘}{cos ^{2}alpha }}alpha proper)}instances f_{a}left({frac {Musin alpha ‘}{sin alpha }}proper)finish{aligned}}}

Discover that the fractional Fourier remodel of ${displaystyle f(u/M)}$ can’t be expressed as a scaled model of ${displaystyle f_{alpha }(u)}$ . Moderately, the fractional Fourier remodel of ${displaystyle f(u/M)}$ seems to be a scaled and chirp modulated model of ${displaystyle f_{alpha ‘}(u)}$ the place ${displaystyle alpha neq alpha ‘}$ is a special order.

Fractional kernel[edit]

The FRFT is an integral transform

{displaystyle {mathcal {F}}_{alpha }f(u)=int K_{alpha }(u,x)f(x),mathrm {d} x}

the place the α-angle kernel is

{displaystyle K_{alpha }(u,x)={start{circumstances}{sqrt {1-icot(alpha )}}exp left(ipi (cot(alpha )(x^{2}+u^{2})-2csc(alpha )ux)proper)&{mbox{if }}alpha {mbox{ just isn’t a a number of of }}pi ,delta (u-x)&{mbox{if }}alpha {mbox{ is a a number of of }}2pi ,delta (u+x)&{mbox{if }}alpha +pi {mbox{ is a a number of of }}2pi ,finish{circumstances}}}

Right here once more the particular circumstances are according to the restrict habits when $α$ approaches a a number of of $π$ .

The FRFT has the identical properties as its kernels :

symmetry: ${displaystyle K_{alpha }~(u,u’)=K_{alpha }~(u’,u)}$
inverse: ${displaystyle K_{alpha }^{-1}(u,u’)=K_{alpha }^{*}(u,u’)=K_{-alpha }(u’,u)}$
additivity: ${displaystyle K_{alpha +beta }(u,u’)=int K_{alpha }(u,u”)K_{beta }(u”,u’),mathrm {d} u”.}$

Associated transforms[edit]

There additionally exist associated fractional generalizations of comparable transforms such because the discrete Fourier transform.

Generalizations[edit]

The Fourier remodel is actually bosonic; it really works as a result of it’s according to the superposition precept and associated interference patterns. There’s additionally a fermionic Fourier remodel.^[13] These have been generalized right into a supersymmetric FRFT, and a supersymmetric Radon transform.^[13] There’s additionally a fractional Radon remodel, a symplectic FRFT, and a symplectic wavelet transform.^[14] As a result of quantum circuits are based mostly on unitary operations, they’re helpful for computing integral transforms because the latter are unitary operators on a function space. A quantum circuit has been designed which implements the FRFT.^[15]

Interpretation[edit]

A rect operate turns right into a sinc operate because the order of the fractional Fourier remodel turns into 1

The same old interpretation of the Fourier remodel is as a metamorphosis of a time area sign right into a frequency area sign. However, the interpretation of the inverse Fourier remodel is as a metamorphosis of a frequency area sign right into a time area sign. Fractional Fourier transforms remodel a sign (both within the time area or frequency area) into the area between time and frequency: it’s a rotation within the time–frequency domain. This angle is generalized by the linear canonical transformation, which generalizes the fractional Fourier remodel and permits linear transforms of the time–frequency area aside from rotation.

Take the determine under for example. If the sign within the time area is rectangular (as under), it turns into a sinc function within the frequency area. But when one applies the fractional Fourier remodel to the oblong sign, the transformation output shall be within the area between time and frequency.

The fractional Fourier remodel is a rotation operation on a time–frequency distribution. From the definition above, for α = 0, there shall be no change after making use of the fractional Fourier remodel, whereas for α = π/2, the fractional Fourier remodel turns into a plain Fourier remodel, which rotates the time–frequency distribution with π/2. For different worth of α, the fractional Fourier remodel rotates the time–frequency distribution in keeping with α. The next determine exhibits the outcomes of the fractional Fourier remodel with totally different values of α.

Time/frequency distribution of fractional Fourier remodel

Software[edit]

Fractional Fourier remodel can be utilized in time frequency evaluation and DSP.^[16] It’s helpful to filter noise, however with the situation that it doesn’t overlap with the specified sign within the time–frequency area. Contemplate the next instance. We can’t apply a filter on to eradicate the noise, however with the assistance of the fractional Fourier remodel, we are able to rotate the sign (together with the specified sign and noise) first. We then apply a selected filter, which can enable solely the specified sign to cross. Thus the noise shall be eliminated utterly. Then we use the fractional Fourier remodel once more to rotate the sign again and we are able to get the specified sign.

Thus, utilizing simply truncation within the time area, or equivalently low-pass filters within the frequency area, one can reduce out any convex set in time–frequency area. In distinction, utilizing time area or frequency area instruments with no fractional Fourier remodel would solely enable reducing out rectangles parallel to the axes.

Fractional Fourier transforms even have functions in quantum physics. For instance, they’re used to formulate entropic uncertainty relations,^[17] in high-dimensional quantum key distribution schemes with single photons,^[18] and in observing spatial entanglement of photon pairs.^[19]

They’re additionally helpful within the design of optical techniques and for optimizing holographic storage effectivity.^[20]

See additionally[edit]

Different time–frequency transforms:

References[edit]

^ Condon, Edward U. (1937). “Immersion of the Fourier transform in a continuous group of functional transformations”. Proc. Natl. Acad. Sci. USA. 23 (3): 158–164. Bibcode:1937PNAS…23..158C. doi:10.1073/pnas.23.3.158. PMC 1076889. PMID 16588141.
^ Namias, V. (1980). “The fractional order Fourier remodel and its software to quantum mechanics”. IMA Journal of Utilized Arithmetic. 25 (3): 241–265. doi:10.1093/imamat/25.3.241.
^ Wiener, N. (April 1929). “Hermitian Polynomials and Fourier Evaluation”. Journal of Arithmetic and Physics. 8 (1–4): 70–73. doi:10.1002/sapm19298170.
^ Almeida, Luís B. (1994). “The fractional Fourier remodel and time–frequency representations”. IEEE Trans. Sign Course of. 42 (11): 3084–3091. Bibcode:1994ITSP…42.3084A. doi:10.1109/78.330368. S2CID 29757211.
^ Tao, Ran; Deng, Bing; Zhang, Wei-Qiang; Wang, Yue (2008). “Sampling and sampling price conversion of band restricted indicators within the fractional Fourier remodel area”. IEEE Transactions on Sign Processing. 56 (1): 158–171. Bibcode:2008ITSP…56..158T. doi:10.1109/TSP.2007.901666. S2CID 7001222.
^ Bhandari, A.; Marziliano, P. (2010). “Sampling and reconstruction of sparse indicators in fractional Fourier area”. IEEE Sign Processing Letters. 17 (3): 221–224. Bibcode:2010ISPL…17..221B. doi:10.1109/LSP.2009.2035242. hdl:10356/92280. S2CID 11959415.
^ Bailey, D. H.; Swarztrauber, P. N. (1991). “The fractional Fourier remodel and functions”. SIAM Review. 33 (3): 389–404. doi:10.1137/1033097. (Word that this text refers back to the chirp-z remodel variant, not the FRFT.)
^ Missbauer, Andreas (2012). Gabor Frames and the Fractional Fourier Transform (PDF) (MSc). University of Vienna. Archived from the original (PDF) on 3 November 2018. Retrieved 3 November 2018.
^ Candan, Kutay & Ozaktas 2000.
^ Somma, Rolando D. (2016). “Quantum simulations of 1 dimensional quantum techniques”. Quantum Data and Computation. 16: 1125–1168. arXiv:1503.06319v2.
^ Shi, Jun; Zhang, NaiTong; Liu, Xiaoping (June 2012). “A novel fractional wavelet remodel and its functions”. Sci. China Inf. Sci. 55 (6): 1270–1279. doi:10.1007/s11432-011-4320-x. S2CID 3772011.
^ ^a ^b De Bie, Hendrik (1 September 2008). “Fourier remodel and associated integral transforms in superspace”. Journal of Mathematical Evaluation and Functions. 345 (1): 147–164. arXiv:0805.1918. Bibcode:2008JMAA..345..147D. doi:10.1016/j.jmaa.2008.03.047. S2CID 17066592.
^ Fan, Hong-yi; Hu, Li-yun (2009). “Optical transformation from chirplet to fractional Fourier transformation kernel”. Journal of Trendy Optics. 56 (11): 1227–1229. arXiv:0902.1800. Bibcode:2009JMOp…56.1227F. doi:10.1080/09500340903033690. S2CID 118463188.
^ Klappenecker, Andreas; Roetteler, Martin (January 2002). “Engineering Useful Quantum Algorithms”. Bodily Evaluation A. 67 (1): 010302. arXiv:quant-ph/0208130. doi:10.1103/PhysRevA.67.010302. S2CID 14501861.
^ Sejdić, Ervin; Djurović, Igor; Stanković, LJubiša (June 2011). “Fractional Fourier remodel as a sign processing instrument: An outline of latest developments”. Sign Processing. 91 (6): 1351–1369. doi:10.1016/j.sigpro.2010.10.008. S2CID 14203403.
^ Huang, Yichen (24 Could 2011). “Entropic uncertainty relations in multidimensional place and momentum areas”. Bodily Evaluation A. 83 (5): 052124. arXiv:1101.2944. Bibcode:2011PhRvA..83e2124H. doi:10.1103/PhysRevA.83.052124. S2CID 119243096.
^ Walborn, SP; Lemelle, DS; Tasca, DS; Souto Ribeiro, PH (13 June 2008). “Schemes for quantum key distribution with higher-order alphabets utilizing single-photon fractional Fourier optics”. Bodily Evaluation A. 77 (6): 062323. doi:10.1103/PhysRevA.77.062323.
^ Tasca, DS; Walborn, SP; Souto Ribeiro, PH; Toscano, F (8 July 2008). “Detection of transverse entanglement in section area”. Bodily Evaluation A. 78 (1): 010304(R). arXiv:0806.3044. doi:10.1103/PhysRevA.78.010304. S2CID 118607762.
^ Pégard, Nicolas C.; Fleischer, Jason W. (2011). “Optimizing holographic data storage using a fractional Fourier transform”. Optics Letters. 36 (13): 2551–2553. Bibcode:2011OptL…36.2551P. doi:10.1364/OL.36.002551. PMID 21725476.

Bibliography[edit]

Candan, C.; Kutay, M. A.; Ozaktas, H. M. (Could 2000). “The discrete fractional Fourier transform” (PDF). IEEE Transactions on Sign Processing. 48 (5): 1329–1337. Bibcode:2000ITSP…48.1329C. doi:10.1109/78.839980. hdl:11693/11130.
Ding, Jian-Jiun (2007). Time frequency evaluation and wavelet remodel (Class notes). Taipei, Taiwan: Division of Electrical Engineering, Nationwide Taiwan College (NTU).
Lohmann, A. W. (1993). “Picture rotation, Wigner rotation and the fractional Fourier remodel”. J. Choose. 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. Collection in Pure and Utilized 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 functions”. IEEE Trans. Sign Course of. 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 from the original (PDF) on 16 July 2011.

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