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 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 of a function is a unitary operator of space that maps the function to its frequential version (all expressions are taken in the sense, rather than pointwise):
and is determined by via the inverse transform
Let us study its n-th iterated defined by
and when n is a non-negative integer, and . Their sequence is finite since is a 4-periodic automorphism: for every function , .
More precisely, let us introduce the parity operator that inverts , . Then the following properties hold:
The FRFT provides a family of linear transforms that further extends this definition to handle non-integer powers of the FT.
Definition
Note: some authors write the transform in terms of the "order a" instead of the "angle α", in which case the α is usually a times π/2. Although these two forms are equivalent, one must be careful about which definition the author uses.
For any realα, the α-angle fractional Fourier transform of a function ƒ is denoted by and defined by
For α = π/2, this becomes precisely the definition of the continuous Fourier transform, and for α = −π/2 it is the definition of the inverse continuous Fourier transform.
The FRFT argument u is neither a spatial one x nor a frequency ξ. We will see why it can be interpreted as linear combination of both coordinates (x,ξ). When we want to distinguish the α-angular fractional domain, we will let denote the argument of .
Remark: with the angular frequency ω convention instead of the frequency one, the FRFT formula is the Mehler kernel,
Properties
The α-th order fractional Fourier transform operator, , has the properties:
Define the shift and the phase shift operators as follows:
Then
that is,
Transform of a scaled function
Define the scaling and chirp multiplication operators as follows:
Then,
Notice that the fractional Fourier transform of cannot be expressed as a scaled version of . Rather, the fractional Fourier transform of turns out to be a scaled and chirp modulated version of where is a different order.[11]
Here again the special cases are consistent with the limit behavior when α approaches a multiple of π.
The FRFT has the same properties as its kernels :
symmetry:
inverse:
additivity:
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.[12][13] A quantum algorithm to implement a version of the discrete fractional Fourier transform in sub-polynomial time is described by Somma.[14]
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.[16] These have been generalized into a supersymmetric FRFT, and a supersymmetric Radon transform.[16] There is also a fractional Radon transform, a symplectic FRFT, and a symplectic wavelet transform.[17] 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.[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 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.
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.
Fractional Fourier transform can be used in time frequency analysis and DSP.[19] 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 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,[20] in high-dimensional quantum key distribution schemes with single photons,[21] and in observing spatial entanglement of photon pairs.[22]
They are also useful in the design of optical systems and for optimizing holographic storage efficiency.[23][24]
^Namias, V. (1980). "The fractional order Fourier transform and its application to quantum mechanics". IMA Journal of Applied Mathematics. 25 (3): 241–265. doi:10.1093/imamat/25.3.241.
^Wiener, N. (April 1929). "Hermitian Polynomials and Fourier Analysis". Journal of Mathematics and Physics. 8 (1–4): 70–73. doi:10.1002/sapm19298170.
^Tao, Ran; Deng, Bing; Zhang, Wei-Qiang; Wang, Yue (2008). "Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain". IEEE Transactions on Signal Processing. 56 (1): 158–171. Bibcode:2008ITSP...56..158T. doi:10.1109/TSP.2007.901666. S2CID7001222.
^Bailey, D. H.; Swarztrauber, P. N. (1991). "The fractional Fourier transform and applications". SIAM Review. 33 (3): 389–404. doi:10.1137/1033097. (Note that this article refers to the chirp-z transform variant, not the FRFT.)
^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.
^If α is an integer multiple of π, 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
must be simply f(t) or f(−t) for α an even or odd multiple of π respectively.
^An elementary recipe, using the contangent function, and its (multi-valued) inverse, for in terms of and exists.
^Somma, Rolando D. (2016). "Quantum simulations of one dimensional quantum systems". Quantum Information and Computation. 16: 1125–1168. arXiv:1503.06319v2.
^Shi, Jun; Zhang, NaiTong; Liu, Xiaoping (June 2012). "A novel fractional wavelet transform and its applications". Sci. China Inf. Sci. 55 (6): 1270–1279. doi:10.1007/s11432-011-4320-x. S2CID3772011.
^Sejdić, Ervin; Djurović, Igor; Stanković, LJubiša (June 2011). "Fractional Fourier transform as a signal processing tool: An overview of recent developments". Signal Processing. 91 (6): 1351–1369. doi:10.1016/j.sigpro.2010.10.008. S2CID14203403.
Ding, Jian-Jiun (2007). Time frequency analysis and wavelet transform (Class notes). Taipei, Taiwan: Department of Electrical Engineering, National Taiwan University (NTU).
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.
Come leggere il tassoboxTetrapodi Rappresentanti dei gruppi esistenti di tetrapoda, (in senso orario dall'alto verso il basso): una rana (genere Pelophylax, un lissamphibio), un'hoatzin, uno scinco (due sauropsidi), e un topo (un sinapside) Classificazione scientifica Dominio Eukaryota Regno Animalia Phylum Chordata (clado) Craniata Subphylum Vertebrata Infraphylum Gnathostomata Clado Eugnathostomata Clado Teleostomi Superclasse Tetrapoda Gaffney, 1930 Sottogruppi Batrachomorpha / Amphibia Lissa…
Artikel ini tidak memiliki referensi atau sumber tepercaya sehingga isinya tidak bisa dipastikan. Tolong bantu perbaiki artikel ini dengan menambahkan referensi yang layak. Tulisan tanpa sumber dapat dipertanyakan dan dihapus sewaktu-waktu.Cari sumber: Samuel Morse – berita · surat kabar · buku · cendekiawan · JSTOR Samuel Morse (1791-1872) Untuk tokoh perkeretaapian Amerika Serikat, lihat Samuel Morse Fenton. Samuel Finley Breese Morse adalah seorang pel…
French poet, essayist, and editor (1873–1914) Charles PéguyPortrait of Charles Péguy, by Jean-Pierre Laurens, 1908BornCharles-Pierre Péguy(1873-01-07)7 January 1873Orléans, Third French RepublicDied5 September 1914(1914-09-05) (aged 41)Villeroy, FranceOccupationWriterAlma materÉcole Normale SupérieureSignature Charles Pierre Péguy (French: [ʃaʁl peɡi]; 7 January 1873 – 5 September 1914) was a French poet, essayist, and editor. His two main philosophies were social…
هذه المقالة عن المجموعة العرقية الأتراك وليس عن من يحملون جنسية الجمهورية التركية أتراكTürkler (بالتركية) التعداد الكليالتعداد 70~83 مليون نسمةمناطق الوجود المميزةالبلد القائمة ... تركياألمانياسورياالعراقبلغارياالولايات المتحدةفرنساالمملكة المتحدةهولنداالنمساأسترالياب…
Canadian filmmaker and artist Bruce LaBruceLaBruce in 2011Born (1964-01-03) January 3, 1964 (age 60)Southampton, Ontario, CanadaOccupation(s)Actor, writer, filmmaker, photographer, underground adult directorYears active1987–presentWebsitebrucelabruce.com Bruce LaBruce (born January 3, 1964)[1] is a Canadian artist,[2] writer, filmmaker, photographer, and underground director based in Toronto. Life and career LaBruce was born in Tiverton, Ontario.[3] He has cla…
Duomo Il mezzanino della stazione M3Stazione dellametropolitana di Milano GestoreATM Inaugurazione1964 (M1)1990 (M3) Statoin uso Linea Localizzazionepiazza del Duomo, Milano Zona tariffariaMi1 Tipologiastazione sotterranea, passante, con due binari in canna singola e due banchine laterali (M1)stazione sotterranea, passante, con due binari e relative banchine in due canne sovrapposte (M3) Interscambiotram urbaniautobus urbani DintorniDuomo di MilanoGalleria Vittorio Emanuele IIPalazzo Reale Acces…
American economist This biography of a living person relies too much on references to primary sources. Please help by adding secondary or tertiary sources. Contentious material about living persons that is unsourced or poorly sourced must be removed immediately, especially if potentially libelous or harmful.Find sources: John J. Siegfried – news · newspapers · books · scholar · JSTOR (January 2019) (Learn how and when to remove this message) John J. Siegf…
Halaman ini berisi artikel tentang kota di Sumatera Utara. Untuk tumbuhan, lihat Binjai. Untuk kegunaan lain, lihat Binjai (disambiguasi). Kota BinjaiKotaTranskripsi bahasa daerah • Abjad Jawiكوتا بنجايGapura Selamat Datang di Kota Binjai LambangJulukan: Kota RambutanPetaKota BinjaiPetaTampilkan peta SumatraKota BinjaiKota Binjai (Indonesia)Tampilkan peta IndonesiaKoordinat: 3°36′00″N 98°29′07″E / 3.6°N 98.4853°E / 3.6; 98.4853Ne…
عمري شارون معلومات شخصية الميلاد 19 أغسطس 1964 (60 سنة)[1] إسرائيل[1] مواطنة إسرائيل الأب أرئيل شارون مناصب عضو الكنيست[2] عضو خلال الفترة17 فبراير 2003 – 23 نوفمبر 2005 فترة برلمانية دورة الكنيست السادسة عشر [لغات أخرى] عضو الكنيست[3] …
This article needs additional citations for verification. Please help improve this article by adding citations to reliable sources. Unsourced material may be challenged and removed.Find sources: 17th Cavalry Regiment – news · newspapers · books · scholar · JSTOR (December 2012) (Learn how and when to remove this message) 17th Cavalry Regiment17th Cavalry Regiment coat of armsActive1916-19211957-Country United StatesBranch United States ArmyTypeC…
For other uses, see One Last Ride (disambiguation). 12th and 13th episodes of the 7th season of Parks and Recreation One Last RideParks and Recreation episodesEpisode nos.Season 7Episodes 12, 13Directed byMichael SchurWritten byMichael SchurAmy PoehlerProduction code712/713Original air dateFebruary 24, 2015 (2015-02-24)Guest appearances Rashida Jones as Ann Perkins Rob Lowe as Chris Traeger Ben Schwartz as Jean-Ralphio Saperstein Christie Brinkley as Gayle Gergich Alison Beck…
Southeast Asian salafist organization founded in 1993 For the Egyptian organization of the same name, see Al-Jama'a al-Islamiyya. For other uses, see Al-Jama'a al-Islamiyya (disambiguation). Not to be confused with Jamaat-e-Islami (disambiguation). Jemaah IslamiyahJihadist flag used by Jemaah IslamiyahLeader Abdullah Sungkar (1993–1999) †[1] Abu Bakar Baasyir (1999–2003) (Imprisoned, later released)[1] Abu Rusdan (2003–2004) POW[1] Adung (2004–2005) POW…
Artikel ini membutuhkan rujukan tambahan agar kualitasnya dapat dipastikan. Mohon bantu kami mengembangkan artikel ini dengan cara menambahkan rujukan ke sumber tepercaya. Pernyataan tak bersumber bisa saja dipertentangkan dan dihapus.Cari sumber: Gunung Rinjani – berita · surat kabar · buku · cendekiawan · JSTOR (Mei 2024)Gunung RinjaniGunung Rinjani dan sabana di kaki gunung.Titik tertinggiKetinggian3.726 m (12.224 kaki)[1]Masuk dalam daftarUltr…
Dr. H.Abbas SaidS.H., M.H. Wakil Ketua Komisi Yudisial Republik IndonesiaMasa jabatan1 Juli 2013 – 18 Desember 2015PendahuluImam Anshori SalehPenggantiSukma ViolettaAnggota Komisi Yudisial Republik IndonesiaMasa jabatan20 Desember 2010 – 18 Desember 2015[1]Hakim Mahkamah Agung Republik IndonesiaMasa jabatan14 September 2004 – 20 Desember 2012 Informasi pribadiLahir3 April 1944 (umur 80) Kabupaten Kolaka, Sulawesi TenggaraSuami/istriSyarifah MasnonAna…