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Fractional Fourier transforms and their optical implementation: II
 J. Opt. Soc. Am. A
"... Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Several mathematical properties ..."
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Cited by 56 (17 self)
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Fourier transforms of fractional order a are defined in a manner such that the common Fourier transform is a special case with order a = 1. An optical interpretation is provided in terms of quadratic graded index media and discussed from both wave and ray viewpoints. Several mathematical properties are derived. 1.
Digital computation of the fractional Fourier transform
 IEEE Trans. Signal Process
, 1996
"... AbstractAn algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with timebandwidth product N, the presented algorithm computes the fractional transform in O ( N log N) time. A definition for the discrete fractional Fourier transform that emerge ..."
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Cited by 36 (15 self)
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AbstractAn algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with timebandwidth product N, the presented algorithm computes the fractional transform in O ( N log N) time. A definition for the discrete fractional Fourier transform that emerges from our analysis is also discussed. I.
The Discrete Fractional Fourier Transform
, 2000
"... We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particula ..."
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Cited by 23 (4 self)
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We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite–Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform.
A discrete fractional random transform, Opt
 Comm
"... We propose a discrete fractional random transform based on a generalization of the discrete fractional Fourier transform with an intrinsic randomness. Such discrete fractional random transform inheres excellent mathematical properties of the fractional Fourier transform along with some fantastic fea ..."
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Cited by 2 (1 self)
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We propose a discrete fractional random transform based on a generalization of the discrete fractional Fourier transform with an intrinsic randomness. Such discrete fractional random transform inheres excellent mathematical properties of the fractional Fourier transform along with some fantastic features of its own. As a primary application, the discrete fractional random transform has been used for image encryption and decryption. Key words: fractional Fourier transform, discrete random transform, cryptography, image encryption and decryption PACS: 42.30.d, 42.40.i, 02.30.Uu 1
On the diagonalization of the discrete Fourier transform
, 2009
"... Dedicated to William Kahan and Beresford Parlett on the occasion of their 75th birthday Abstract. The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N = p is an odd prime number, we exhibit a c ..."
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Cited by 2 (2 self)
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Dedicated to William Kahan and Beresford Parlett on the occasion of their 75th birthday Abstract. The discrete Fourier transform (DFT) is an important operator which acts on the Hilbert space of complex valued functions on the ring Z/NZ. In the case where N = p is an odd prime number, we exhibit a canonical basis Φ of eigenvectors for the DFT. The transition matrix Θ from the standard basis to Φ defines a novel transform which we call the discrete oscillator transform (DOT for short). Finally, we describe a fast algorithm for computing Θ in certain cases. The discrete Fourier transform (DFT) is probably one of the most important operators in modern science. It is omnipresent in various fields of discrete mathematics and engineering, including combinatorics, number theory, computer science and, last but probably not least, digital signal processing. Formally, the DFT is
A shattered survey of the Fractional Fourier Transform
, 2002
"... 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. It has been intensely studied during the last decade, an attention it may have partially gained because of the vivid inter ..."
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Cited by 2 (1 self)
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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. It has been intensely studied during the last decade, an attention it may have partially gained because of the vivid interest in timefrequency analysis methods of signal processing, like wavelets. 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. We give an introduction to the definition, the properties and computational aspects of both the continuous and discrete fractional Fourier transforms. We include some examples of applications and some possible generalizations.
Fast algorithm for chirp transforms with zoomingin ability and its applications
 Journal of the Optical Society of America A,vol.17,no.4
, 2000
"... A general fast numerical algorithm for chirp transforms is developed by using two fast Fourier transforms and employing an analytical kernel. This new algorithm unifies the calculations of arbitrary realorder fractional Fourier transforms and Fresnel diffraction. Its computational complexity is bet ..."
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Cited by 1 (0 self)
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A general fast numerical algorithm for chirp transforms is developed by using two fast Fourier transforms and employing an analytical kernel. This new algorithm unifies the calculations of arbitrary realorder fractional Fourier transforms and Fresnel diffraction. Its computational complexity is better than a fast convolution method using Fourier transforms. Furthermore, one can freely choose the sampling resolutions in both x and u space and zoom in on any portion of the data of interest. Computational results are compared with analytical ones. The errors are essentially limited by the accuracy of the fast Fourier transforms and are higher than the order 10�12 for most cases. As an example of its application to scalar diffraction, this algorithm can be used to calculate nearfield patterns directly behind the aperture, 0 � z � d2 /�. It compensates another algorithm for Fresnel diffraction that is limited to z � d2 /�N [J. Opt. Soc. Am. A 15, 2111 (1998)]. Experimental results from waveguideoutput microcoupler diffraction are in good agreement with the calculations.
The Discrete Fractional Fourier Transform
"... Abstract—We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a ..."
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Abstract—We propose and consolidate a definition of the discrete fractional Fourier transform that generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform generalizes the continuous ordinary Fourier transform. This definition is based on a particular set of eigenvectors of the DFT matrix, which constitutes the discrete counterpart of the set of Hermite–Gaussian functions. The definition is exactly unitary, index additive, and reduces to the DFT for unit order. The fact that this definition satisfies all the desirable properties expected of the discrete fractional Fourier transform supports our confidence that it will be accepted as the definitive definition of this transform. Index Terms—Chirplets, discrete Wigner distributions, Hermite–Gaussian functions, time–frequency analysis.
Australian Optical Society NEWS A Beginner's Guide To The Fractional Fourier Transform
"... Just two years ago the phrase "fractional Fourier transform " would have triggered the headscratching reflex in 99.9 % of opticists. Meanwhile there has been a veritable explosion in the number of papers published on the subject of the fractional Fourier transform (referred to hereafter as the Frac ..."
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Just two years ago the phrase "fractional Fourier transform " would have triggered the headscratching reflex in 99.9 % of opticists. Meanwhile there has been a veritable explosion in the number of papers published on the subject of the fractional Fourier transform (referred to hereafter as the FractFT for want of a more elegant yet compact abbreviation). The majority of these papers have appeared in the leading optics journals of the United States of America and Europe. A quick count through the better known papers gives the following score: 1992 zero, 1993 five, 1994 twenty, 1995 six so far. The pervasive influence of the FractFT has been such that its mention only causes itchy scalp in a mere 99 % of opticists today. If this article must state its major objective, then it is to perplex 90 % of the readers (and thus inform the remaining 10%, which represents a tenfold increase in current awareness!) The beginner's guide will be presented in two parts. The first part, in this issue, is concerned mainly with the fascinating history of the FractFT and some speculation about the development of scientific ideas. The second part, to appear in the next issue, aims to make the main ideas behind the FractFT accessible and to discuss the numerous applications in optical systems.