
	Where results make sense

Topic: Fractional Fourier transform

Related Topics

Fourier transform

Discrete Fourier transform

Chirplet transform

Fourier analysis

Frequency spectrum

Fourier series

Rice

Battle of the Alamo

United Nations headquarters

Mathematics

Guatemala

	Fast Fourier Transform
		FFTs became popular after J. Cooley of IBM and J. Tukey of Princeton published a paper in 1965 reinventing the algorithm and describing how to perform it conveniently on a computer (including how to arrange for the output to be produced in the natural ordering).
		This process is an example of the general technique of divide and conquer algorithms; in many traditional implementations, however, the explicit recursion is avoided, and instead one traverses the computational tree in breadth-first fashion.
		There are further FFT specializations for the cases of real data that have even/odd symmetry, in which case one can gain another factor of ~2 in time/space and the DFT becomes the discrete cosine/sine transform(s) (DCT/DST).
	www.ebroadcast.com.au /lookup/encyclopedia/ff/FFT.html (2315 words)

	Fourier transform (Site not responding. Last check: 2007-11-03)
		The Fourier transform, named for Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i.e.
		Fourier transforms have many scientific applications — in physics, number theory, combinatorics, signal processing, probability theory, statistics, cryptography, acoustics, oceanography, optics, geometry, and other areas.
		These Fourier variants can also be generalized to Fourier transforms on arbitrary locally compact abelian topological groups, which are studied in harmonic analysis; there, one transforms from a group to its dual group.
	fourier-transform.kiwiki.homeip.net (1062 words)

	Fractional Fourier Transform (Site not responding. Last check: 2007-11-03)
		Of course, applying the Fourier operator a third time yields the Fourier transform of the time reversed signal, and applying it a fourth time yields the original signal.
		The easiest way to see this is to think of finite length signals and the Fourier transform as a matrix operation Mx=X, where the rows of the matrix M contain the Fourier basis.
		Shown below is a fractal signal (0.00), its first (1.00) and second-order (2.00) Fourier transform (magnitude) and the intermediate fractional Fourier transforms in increments of 0.2.
	www.cs.dartmouth.edu /~farid/research/fracFourier.html (193 words)

	[No title] (Site not responding. Last check: 2007-11-03)
		We have studied the properties of one-and two-dimensional functions (images) which remain invariant under the Fractional Fourier Transform (an extension of the ordinary Fourier Transform, parametrized by an angular quantity in the one-dimensional case (two angles in the two-dimensional case)).
		We also studied properties of the Radon-Wigner transform (the square of the fractional Fourier transform for a given angle, and the projection of the Wigner transform for that angle).
		We have analyzed the structure of the discrete Fourier transform of a particular class of self-similar discrete signals with elements in the complex number field.
	www.esat.kuleuven.ac.be /sista/yearreport98/nonlinear3.html (312 words)

	Fractional Fourier transform biography .ms (Site not responding. Last check: 2007-11-03)
		The fractional Fourier transform (FRFT) is a linear transformation generalizing the continuous Fourier transform, and 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 an intermediate domain between time and frequency.
		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).
		See also the chirplet transform for a related generalization of the Fourier transform.
	www.biography.ms /Fractional_Fourier_transform.html (358 words)

	CWI Tract (Site not responding. Last check: 2007-11-03)
		Fourier analysis is a widely used method to obtain an impression of which frequencies a given signal consists of.
		Recently the wavelet transform has been introduced as a transform that is able to analyse signals in time and scale, which can be compared with reciprocal frequencies.
		The wavelet transform is, however, also a decomposition of a given signal in translated and dilated wavelets.
	www.cwi.nl /publications/Abstracts_tracts/tr-130.html (346 words)

	Imam Samil Yetik -Research- (Site not responding. Last check: 2007-11-03)
		The fractional Fourier transform is a generalization of the ordinary Fourier transform, and hence using it can improve the performance of any system that uses Fourier transform.
		We used the fractional Fourier transform for the synthesis of mutual intensity distributions.
		A method to approximate the perspective projections using the fractional Fourier transform is presented, and performance analysis is performed that determine the parameters of perspective projection that result in acceptable approximations.
	www.bme.ucdavis.edu /~isyetik/frft.html (288 words)

	Fractional Fourier Transform -- from Wolfram MathWorld
		The linear fractional Fourier transform is a discrete Fourier transform in which the exponent is modified by the addition of a factor
		The quadratic fractional Fourier transform is defined in signal processing and optics.
		So-called fractional Fourier domains correspond to oblique axes in the time-frequency plane, and thus the fractional Fourier transform (sometimes abbreviated FRT) is directly related to the Radon transforms of the Wigner distribution and the ambiguity function.
	mathworld.wolfram.com /FractionalFourierTransform.html (229 words)

	Fractional Fourier Analysis of signals with different types of time-frequency symmetry
		The fractional Fourier transform is an extension of the ordinary Fourier transform and depends on a parameter a that can be interpreted as a rotation angle in the time-frequency plane.
		The Radon-Wigner transform (RWT) of a signal is the squared modulus of its fractional Fourier transform.
		It was found that the fractional Fourier transform of a periodic signal at some angles is the superposition of some of its scaled, weighted and shifted replicas, with an additional quadratic phase factor.
	www.esat.kuleuven.ac.be /sista/yearreport97/node21.html (333 words)

	ACADEMIC
		You know that Hermite-Gaussians are eigenfuctions of the Fourier Transform, that is by expanding any finite energy function in terms of Hermite-Gaussians, you can easily find the Fourier Transform of that function by the multiplication of the corresponding eigenvalue of the eigenfunction.
		You know that Fourier Transform can be evaluated with a single lens, by putting image plane and input plane at the the focal length of a lens; if one illimunates one of the planes, one gets Fourier Transform of the input at the other plane.
		Note that Fourier Transform gradually develops between the planes, this gradual development is Fractional Fourier Transform.
	users.ece.gatech.edu /~candan/frt.htm (1127 words)

	Fast Fourier Transform -- from Wolfram MathWorld
		The fast Fourier transform (FFT) is a discrete Fourier transform algorithm which reduces the number of computations needed for
		Hartley transform (Bracewell 1999) gives a further increase in speed by approximately a factor of two.
		Fast Fourier transform algorithms generally fall into two classes: decimation in time, and decimation in frequency.
	mathworld.wolfram.com /FastFourierTransform.html (497 words)

	International Commission for Optics Newsletter - Jan. 1999
		With the development of the fractional Fourier transform, the common frequency domain is seen to be merely a special case of a continuum of so-called fractional domains, a concept which is elegantly related to the notion of space-frequency distributions.
		In every area in which Fourier transforms and frequency-domain concepts are used, there exists the potential for generalization and improvement by using the fractional transform.
		In the former category, apart from the development of the fractional Fourier transform and its applications, he has made several other contributions to general optics, information optics and optical signal processing, as well as digital signal and image processing.
	www.ico-optics.org /ico_jan99.html (3206 words)

	Fourier transform (Site not responding. Last check: 2007-11-03)
		The Fourier transform, named after Jean Baptiste Joseph Fourier, is an integral transform that re-expresses a function in terms of sinusoidal basis functions, i.e.
		When f(''t'') is an even or odd function, the sine or cosine terms disappear and one is left with the cosine transform or sine transform, respectively.
		Time-frequency transforms such as the short-time Fourier transform, wavelet transforms, chirplet transforms, and the fractional Fourier transform try to obtain frequency information from a signal as a function of time (or whatever the independent variable is), although the ability to simultaneously resolve frequency and time is limited by a (mathematical) uncertainty principle.
	fourier-transform.iqnaut.net (989 words)

	The Fractional Fourier Transform and Applications - Bailey, Swarztrauber (ResearchIndex)
		Abstract: This paper describes the "fractional Fourier transform", which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm.
		Whereas the discrete Fourier transform (DFT) is based on integral roots of unity e \Gamma2ßi=n, the fractional Fourier transform is based on fractional roots of unity e \Gamma2ßiff, where ff is arbitrary.
		The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing...
	citeseer.ist.psu.edu /36158.html (617 words)

	Uncertainty principles invariant under the fractional Fourier transform (Site not responding. Last check: 2007-11-03)
		Uncertainty principles invariant under the fractional Fourier transform
		The Fourier transform is imbedded in a continuous group of operators, the fractional Fourier transforms, but the Heisenberg measure of overall spread turns out not to be invariant under that group.
		A new family is developed of measures that are invariant under the group of fractional Fourier transforms and that obey associated uncertainty principles.
	anziamj.austms.org.au /V33/part2/Mustard.html (159 words)

	Fractional convolution (Site not responding. Last check: 2007-11-03)
		called fractional convolution operators and which includes those of multiplication and convolution as particular cases is constructed by means of the Condon-Bargmann fractional Fourier transform.
		A fractional convolution theorem generalizes the standard Fourier convolution theorems and a fractional unit distribution generalizes the unit and delta distributions.
		Some explicit double-integral formulas for the fractional convolution between two functions are given and the induced operation between their corresponding Wigner distributions is found.
	www.austms.org.au /Publ/Jamsb/V40P2/abs/1354/1354.html (78 words)

	Communications Technology Web page - John Wiley & Sons, Ltd.
		As a generalization of the Fourier transform, the fractional Fourier transform is only richer in theory and more flexible in applications - but not more costly in implementation.
		Wherever Fourier transforms are used, there exists the potential for generalization and improvement by using the fractional transform.
		Matrix algebra is employed in a unified manner for both wave and geometrical optics, leading to many important results, such as those on general Fourier transform planes and optical invariants.
	www.wiley.com /legacy/wileychi/commstech/ozaktas.html (317 words)

	A shattered survey of the Fractional Fourier Transform (Site not responding. Last check: 2007-11-03)
		Abstract: In this survey paper we introduce the reader to the notion of the fractional Fourier transform, which may be considered as a fractional power of the classical Fourier transform.
		Like the complex exponentials are the basic functions in Fourier analysis, the chirps (signals sweeping through all frequencies in a certain interval) are the building blocks in the fractional Fourier analysis.
		Part of its roots can be found in optics where the fractional Fourier transform can be physically realized.
	www.cs.kuleuven.ac.be /~nalag/papers/ade/frft (184 words)

	Research:Ashok Veeraraghavan (Site not responding. Last check: 2007-11-03)
		The fractional Fourier transform is a time-frequency distribution and an extension of the classical Fourier transform.
		There are several known applications of the fractional Fourier transform in the areas of signal processing, especially in signal restoration and noise removal.
		This is followed by the details of the implementation and a theoretical model for the fixed-point errors involved in the implementation of this algorithm.
	www.umiacs.umd.edu /users/vashok/site/Research.htm (1350 words)

	[No title]
		Fractional Fourier Transform (FRT) is one of the tools used in the T-F analysis field [9].
		FractioNal Fourier transform (FRT) The FRT is a generalization of the conventional Fourier transform and has a history in mathematical physics and in digital signal processing (The interested reader is referred to [9] for a comprehensive overview).
		On the other hand, this algorithm requires the selection of fractional order values and the windows' locations, but the performance is expected to be relatively more robust with respect to arbitrary selection of these parameters, since they have more of a "global" rather than "local" effect.
	www.eng.tau.ac.il /~sharon/Files/FRT_BSS.doc (3567 words)

	Fast Fourier Transform
		3 FFT algorithms specialized for real and/or symmetric data
		There are other FFT algorithms distinct from Cooley-Tukey.
		FFT algorithms specialized for real and/or symmetric data
	www.ebroadcast.com.au /lookup/encyclopedia/fa/Fast_Fourier_transform.html (2315 words)

	The Fractional Fourier Transform: with Applications in Optics and Signal Processing:0471963461:Haldun M. Ozaktas ...
		The Fractional Fourier Transform provide a comprehensive and widely accessible account of the subject covering both theory and applications.
		As a generalisation of the Fourier transform, the fractional Fourier transform is richer in theory and more flexible in applications but not more costly in implementation.
		Fractional Fourier transform parameters and Gaussian beam parameters
	www.ecampus.com /book/0471963461 (489 words)

	Publications
		"A class of fractional integral transforms: A generalization of the fractional Fourier transform," to appear in IEEE Transactions on Signal Processing.
		"Inversion of integral transforms associated with a class of perturbed heat equations," jointly with D. Haimo, the Journal of Mathematical Analysis and Applications, Vol.
		"Inversion of an integral transform related to a general form of heat equation" the Journal of Mathematical Analysis and Applications (jointly with D. Haimo), Vol.
	condor.depaul.edu /~azayed/publications.html (1667 words)

	The Fractional Fourier Transform (Site not responding. Last check: 2007-11-03)
		As a generalization of the ordinary Fourier transform, the fractional Fourier transform is only richer in theory and more flexible in applications--but not more costly in implementation.
		Properties and applications of the ordinary Fourier transform are special cases of those of the fractional Fourier transform.
		The ordinary frequency domain is a special case of the continuum of fractional Fourier domains, which are intimately related to time-frequency representations such as the Wigner distribution.
	www.ee.bilkent.edu.tr /~haldun/wileybook.html (889 words)

	Discrete Fourier Transform -- from Wolfram MathWorld
		Discrete Fourier transforms are extremely useful because they reveal periodicities in input data as well as the relative strengths of any periodic components.
		In general, the discrete Fourier transform of a real sequence of numbers will be a sequence of complex numbers of the same length.
		A suitably scaled plot of the complex modulus of a discrete Fourier transform is commonly known as a
	mathworld.wolfram.com /DiscreteFourierTransform.html (396 words)

	Viewing the 2-D Fractional Fourier Transform with a Discrete Optics System (Site not responding. Last check: 2007-11-03)
		The chirp type Fractional Fourier Transform (FRT) of a signal s(t) is given by [1]
		It is trivial to see that, with a = 1, Sa(f) is the ordinary Fourier Transform, whereas it is not so evident that, with a = 0, Sa(f) yields back the original signal.
		The above implies that, in order to see transforms with different fractions, either many lenses, each with a different focal length, or a variable focus (zoom) lens must be employed: for example, to vary a in the range [0,1], f must be allowed to vary from f1 to infinity.
	www.dei.unipd.it /~nil/vis-e.html (525 words)

	TM-2004-172 Two-Channel SAR-GMTI via Fractional Fourier Transform (Site not responding. Last check: 2007-11-03)
		In this paper, a relatively unknown yet powerful technique, the so-called fractional Fourier transform (FrFT), is applied to the SAR-ATI in order to estimate moving target parameters.
		By mapping a target's signal onto a fractional Fourier axis, the FrFT permits a constant-velocity target to be fully focused in a fractional Fourier domain affording orders of magnitude improvement in SCR.
		Then moving target velocity and position parameters are derived and expressed in terms of an optimum fractional angle alpha and a measured fractional Fourier position up, allowing a target to be accurately repositioned and its velocity components computed.
	www.ottawa.drdc-rddc.gc.ca /html/tm2004_172_e.html (239 words)

Try your search on: Qwika (all wikis)