|
|
| Line 1: |
Line 1: |
| In [[mathematics]], in the area of [[harmonic analysis]], the '''fractional Fourier transform''' ('''FRFT''') is a family of [[linear transformation]]s generalizing the [[Fourier transform]]. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an [[integer]] — thus, it can transform a function to any ''intermediate'' domain between time and [[frequency]]. Its applications range from [[filter design]] and [[signal analysis]] to [[phase retrieval]] and [[pattern recognition]].
| | Hello and welcome. My name is Numbers Wunder. Playing baseball is the hobby he will by no means quit performing. For years he's been working as a meter reader and it's some thing he really enjoy. For a whilst I've been in South Dakota and my parents reside close by.<br><br>Here is my blog ... [http://Bit.ly/1pABYYJ meal delivery service] |
| | |
| The FRFT can be used to define fractional [[convolution]], [[correlation]], and other operations, and can also be further generalized into the [[linear canonical transformation]] (LCT). An early definition of the FRFT was introduced by [[Edward Condon|Condon]],<ref>E. U. Condon, "Immersion of the Fourier transform in a continuous group of functional transformations", ''Proc. Nat. Acad. Sci. USA'' '''23''', (1937) 158–164.</ref> by solving for the Green's function for phase-space rotations, and also by Namias,<ref>V. Namias, "The fractional order Fourier transform and its application to quantum mechanics," ''J. Inst. Appl. Math.'' '''25''', 241–265 (1980).</ref> generalizing work of [[Norbert Wiener|Wiener]]<ref>N. Wiener, "Hermitian Polynomials and Fourier Analysis", ''J. Mathematics and Physics'' '''8''' (1929) 70-73.</ref> on [[Hermite polynomials]].
| |
| | |
| However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by several groups.<ref>Luís B. Almeida, "The fractional Fourier transform and time-frequency representations," ''IEEE Trans. Sig. Processing'' '''42''' (11), 3084–3091 (1994).</ref> Since then, there has been a surge of interest in extending Shannon's sampling theorem<ref>Ran Tao, Bing Deng, Wei-Qiang Zhang and Yue Wang, "Sampling and sampling rate conversion of band limited signals in the fractional Fourier transform domain," ''IEEE Transactions on Signal Processing'', '''56''' (1), 158–171 (2008).</ref><ref>A. Bhandari and P. Marziliano, "Sampling and reconstruction of sparse signals in fractional Fourier domain," ''IEEE Signal Processing Letters'', '''17''' (3), 221–224 (2010).</ref> 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<ref>D. H. Bailey and P. N. Swarztrauber, "The fractional Fourier transform and applications," ''[[SIAM Review]]'' '''33''', 389-404 (1991). (Note that this article refers to the chirp-z transform variant, not the FRFT.)</ref> 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.
| |
| | |
| See also the [[chirplet transform]] for a related generalization of the [[Fourier transform]].
| |
| | |
| ==Introduction==
| |
| The continuous [[Fourier transform]] <math>\mathcal{F}</math> of a function {{nowrap|ƒ: '''R''' → '''C'''}} is an [[unitary operator]] of ''L''<sup>2</sup> that maps the function ƒ to its frequential version ƒ̂:
| |
| :<math>\hat{f}(\xi) = \int_{-\infty}^{\infty} f(x)\ e^{- 2\pi i x \xi}\,\mathrm{d}x</math>, for every [[real number]] <math>\xi</math>.
| |
| | |
| And ƒ is determined by ƒ̂ via the inverse transform <math>\mathcal{F}^{-1}</math>
| |
| | |
| :<math>f(x) = \int_{-\infty}^{\infty} \hat{f}(\xi)\ e^{2 \pi i \xi x}\,\mathrm{d}\xi, </math> for every real number ''x''.
| |
| | |
| Let us study its [[iterated function|''n''-th iterated]] <math>\mathcal{F}^{n}</math> defined by
| |
| <math>\mathcal{F}^{n}[f] = \mathcal{F}[\mathcal{F}^{n-1}[f]]</math> and <math>\mathcal{F}^{-n} = (\mathcal{F}^{-1})^n</math> when ''n'' is a non-negative integer, and <math>\mathcal{F}^{0}[f] = f</math>. Their sequence is finite since <math>\mathcal{F}</math> is a 4-periodic [[automorphism]]: for every function ƒ, <math>\mathcal{F}^4 [f] = f</math>.
| |
| | |
| More precisely, let us introduce the '''parity operator''' <math>\mathcal{P}</math> that inverts time, <math>\mathcal{P}[f]\colon t \mapsto f(-t)</math>. Then the following properties hold:
| |
| :<math>\mathcal{F}^0 = \mathrm{Id}, \qquad \mathcal{F}^1 = \mathcal{F}, \qquad \mathcal{F}^2 = \mathcal{P}, \qquad \mathcal{F}^4 = \mathrm{Id}</math>
| |
| :<math>\mathcal{F}^3 = \mathcal{F}^{-1} = \mathcal{P} \circ \mathcal{F} = \mathcal{F} \circ \mathcal{P}.</math>
| |
| | |
| The FrFT provides a family of linear transforms that further extends this definition to handle non-integer powers <math>n=2\alpha/\pi</math> of the FT.
| |
| | |
| ==Definition==
| |
| | |
| For any [[real number|real]] α, the α-angle fractional Fourier transform of a function ƒ is denoted by <math>\mathcal{F}_\alpha (u)</math> and defined by | |
| :<math>\mathcal{F}_\alpha[f](u) =
| |
| \sqrt{1-i\cot(\alpha)} e^{i \pi \cot(\alpha) u^2}
| |
| \int_{-\infty}^\infty
| |
| e^{-i2\pi (\csc(\alpha) u x - \frac{\cot(\alpha)}{2} x^2)}
| |
| f(x)\, \mathrm{d}x.
| |
| </math>
| |
| | |
| (the square root is defined such that the argument of result lies in the interval <math>[-\pi/2, \pi/2]</math>)
| |
| | |
| If α is an integer multiple of π, then the [[cotangent]] and [[cosecant]] functions above diverge. However, this can be handled by taking the [[limit of a function|limit]], and leads to a [[Dirac delta function]] in the integrand. More easily, since <math>\mathcal{F}^2(f)=f(-t)</math>, <math>\mathcal{F}_\alpha(f)</math> must be simply <math>f(t)</math> or <math>f(-t)</math> for α an [[Even and odd numbers|even or odd]] multiple of <math>\pi</math>, respectively.
| |
| | |
| Note that, for <math>\alpha=\pi/2</math>, this becomes precisely the definition of the continuous Fourier transform, and for <math>\alpha=-\pi/2</math> 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 <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
| |
| :<math>\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 operator <math>\mathcal{F}_\alpha</math> has the properties :
| |
| * '''coherence''' with the FT powers: if when <math>\alpha \equiv \pi k\, [2\pi]</math>, where ''k'' is an integer, then
| |
| :<math>\mathcal{F}_\alpha = \mathcal{F}^{k}</math>
| |
| | |
| * '''additivity''': for any real angles α, β,
| |
| :<math>\mathcal{F}_{\alpha+\beta} = \mathcal{F}_\alpha \circ \mathcal{F}_\beta = \mathcal{F}_\beta \circ \mathcal{F}_\alpha.</math>
| |
| | |
| ===Fractional kernel===
| |
| The FrFT is an [[integral transform]]
| |
| :<math>\mathcal{F}_\alpha f (u) = \int K_\alpha (u, x) f(x)\, \mathrm{d}x</math>
| |
| where the α-angle kernel is
| |
| :<math>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>
| |
| | |
| (the square root is defined such that the argument of result lies in the interval <math>[-\pi/2, \pi/2]</math>).
| |
| | |
| 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: <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]] in
| |
| {{Harv|Candan|Kutay|Ozaktas|2000}} and
| |
| {{Harv|Ozaktas|Zalevsky|Kutay|2001|loc=Chapter 6}}.
| |
| | |
| Fractional wavelet transform (FRWT):<ref>J. Shi, N.-T. Zhang, and X.-P. Liu, "A novel fractional wavelet transform and its applications," Sci. China Inf. Sci. vol. 55, no. 6, pp. 1270-1279, June 2012. URL: http://www.springerlink.com/content/q01np2848m388647/</ref> A generalization of the classical wavelet transform (WT) in the fractional Fourier transform (FRFT) domains. The FRWT is proposed in order to rectify the limitations of the WT and the FRFT. This transform not only inherits the advantages of multiresolution analysis of the WT, but also has the capability of signal representations in the fractional domain which is similar to the FRFT. Compared with the existing FRWT, the FRWT (defined by Shi, Zhang, and Liu 2012) can offer signal representations in the time-fractional-frequency plane.
| |
| | |
| ===Generalization===
| |
| 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">Hendrik De Bie, ''Fourier transform and related integral transforms in superspace (2008)'',
| |
| http://www.arxiv.org/abs/0805.1918</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>Hong-yi Fan and Li-yun Hu, ''Optical transformation from chirplet to fractional Fourier transformation kernel (2009)'', http://www.arxiv.org/abs/0902.1800
| |
| </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>Andreas Klappenecker and Martin Roetteler, ''Engineering Functional Quantum Algorithms (2002)'', http://www.arxiv.org/abs/quant-ph/0208130</ref>
| |
| | |
| == Interpretation of the fractional Fourier transform ==
| |
| {{further2|[[Linear canonical transformation]]}}
| |
| 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. Apparently, fractional Fourier transforms can 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 below figure as an example. If the signal in the time domain is rectangular (as below), it will become a [[sinc function]] in the frequency domain. But if we apply 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|thumb|center|600px|Fractional Fourier transform]]
| |
| <br /> | |
| Actually, fractional Fourier transform is a rotation operation on the time frequency distribution. From the definition above, for ''α'' = 0, there will be no change after applying fractional Fourier transform, and for ''α'' = ''π''/2, fractional Fourier transform becomes a Fourier transform, which rotates the time frequency distribution with ''π''/2. For other value of ''α'', 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|thumb|center|600px|Time/frequency distribution of fractional Fourier transform.]]
| |
| | |
| == Application ==
| |
| | |
| Fractional Fourier transform can be used in time frequency analysis and [[Digital signal processing|DSP]]. 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.
| |
| | |
| Fractional Fourier transforms are also used to design optical systems and optimize holographic storage efficiency.<ref>N. C. Pégard and J. W. Fleischer, "Optimizing holographic data storage using a fractional Fourier transform," Opt. Lett. 36, 2551-2553 (2011) [http://www.opticsinfobase.org/abstract.cfm?URI=ol-36-13-2551]</ref>
| |
| | |
| [[Image:FracFT App by stevencys.jpg|thumb|center|600px|Fractional Fourier transform in DSP.]]
| |
| Thus, using just truncation in the time domain, or equivalently [[low-pass filter]]s in the frequency domain, one can cut out any [[convex set]] in time-frequency space; just using time domain or frequency domain methods without fractional Fourier transforms only allow cutting out rectangles parallel to the axes.
| |
| <!--
| |
| I have read this artical a few times and this graphic is confusing. the top left pane shows the spectrum of signal with noise (how can you know that the noise is a nice diagonal line in that rep anyhow? what would generate this?) then rotated until the noise is vertical in the graph. I does not indicate why the original frequency rep cant be filtered directly in either the freq or time domain. Call me stupid if you like but one sentence could clear this up. Chris Dow-->
| |
| | |
| ==See also==
| |
| | |
| Other time-frequency transforms:
| |
| * [[Linear canonical transformation]]
| |
| * [[short-time Fourier transform]]
| |
| * [[wavelet transform]]
| |
| * [[chirplet transform]]
| |
| | |
| ==References==
| |
| {{reflist}}
| |
| | |
| ==External links==
| |
| * [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]].
| |
| * [http://mechatronics.ece.usu.edu/foc/FRFT/ Dr YangQuan Chen's FRFT (Fractional Fourier Transform) Webpages]
| |
| | |
| ==Bibliography==
| |
| | |
| * {{citation
| |
| |first1=Haldun M.
| |
| |last1=Ozaktas
| |
| |first2=Zeev
| |
| |last2=Zalevsky
| |
| |first3=M. Alper
| |
| |last3=Kutay
| |
| |title=The Fractional Fourier Transform with Applications in Optics and Signal Processing
| |
| |publisher=John Wiley & Sons
| |
| |year=2001
| |
| |series=Series in Pure and Applied Optics
| |
| |url=http://www.ee.bilkent.edu.tr/~haldun/wileybook.html <!-- support page-->
| |
| |isbn=0-471-96346-1
| |
| }}
| |
| * {{citation
| |
| |title=The discrete fractional Fourier transform
| |
| |last1=Candan
| |
| |first1=C.
| |
| |last2=Kutay
| |
| |first2=M.A.
| |
| |last3=Ozaktas
| |
| |first3=H.M.
| |
| |journal=IEEE Transactions on Signal Processing
| |
| |volume=48
| |
| |issue=5
| |
| |date=May 2000
| |
| |pages=1329–1337
| |
| |doi=10.1109/78.839980
| |
| }}
| |
| * A. W. Lohmann, "Image rotation, Wigner rotation and the fractional Fourier transform," ''J. Opt. Soc. Am.'' '''A 10''', 2181–2186 (1993).
| |
| * Soo-Chang Pei and Jian-Jiun Ding, "Relations between fractional operations and time-frequency distributions, and their applications," ''IEEE Trans. Sig. Processing'' '''49''' (8), 1638–1655 (2001).
| |
| * Jian-Jiun Ding, Time frequency analysis and wavelet transform class notes, the Department of Electrical Engineering, National Taiwan University (NTU), Taipei, Taiwan, 2007.
| |
| | |
| * Saxena, R., Singh, K., (2005) ''Fractional Fourier transform: A novel tool for signal processing'', J. Indian Inst. Sci., Jan.–Feb. 2005, 85, 11–26. http://journal.library.iisc.ernet.in/vol200501/paper2/11.pdf.
| |
| | |
| {{DEFAULTSORT:Fractional Fourier Transform}}
| |
| [[Category:Fourier analysis]]
| |
| [[Category:Time–frequency analysis]]
| |
| [[Category:Integral transforms]]
| |