New signal representation based on the fractional Fourier transform: definitions

Mendlovic, David; Zalevsky, Zeev; Dorsch, Rainer G.; Bitran, Yigal; Lohmann, Adolf W.; Özaktaş, Haldun M.

doi:10.1364/josaa.12.002424

Cited by 52 publications

(20 citation statements)

References 12 publications

(9 reference statements)

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“…15,16 A number of algorithms are also specifically devised for FrFT's [17][18][19][20][21] whose properties have been intensively investigated both mathematically [3][4][5]22 and physically [23][24][25] in terms of their applications to optical beam propagation, 6 imaging, [24][25][26] diffraction, 27 and signal and image processing. [28][29][30][31][32][33][34][35][36] The relations among FnT's, DFT's, and FrFT's are also well established. 32,37,38 Our aim in this paper is to develop an efficient algorithm that unifies the evaluations of these formulas.…”

Section: Introductionmentioning

confidence: 94%

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Fast algorithm for chirp transforms with zooming-in ability and its applications

et al. 2000

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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 real-order 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 Ϫ12 for most cases. As an example of its application to scalar diffraction, this algorithm can be used to calculate near-field patterns directly behind the aperture, 0

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Section: Introductionmentioning

confidence: 94%

“…34,47 Another explicit form of the sampling condition will be derived in Subsection 3.B. Under this restriction, one may correctly calculate the numerical transform of a given function.…”

mentioning

confidence: 99%

Fast algorithm for chirp transforms with zooming-in ability and its applications

et al. 2000

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“…3 The WDF is especially important to optics because it is a powerful tool for designing and analyzing optical systems. 4 A nice example for an introduction to the WDF comes from the area of music. Neither the representation of music as a function of time nor its representation as a function of frequency is suitable for a musician.…”

Section: Background a Wigner Representationmentioning

confidence: 99%

“…This chart performs a Cartesian-to-polar coordinate transform of the (x, p) chart. 4 Here all the FRT orders of the function are drawn as angular vectors. Each FRT orders is drawn along the r axis in a specific angular orientation of ϭ p(/2), where p is the fractional order.…”

Section: B (R P) Chartmentioning

confidence: 99%

Wigner-related phase spaces for signal processing and their optical implementation

2000

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Phase spaces are different ways to represent signals. Owing to their properties, they are often used for signal compression and recognition with high discrimination abilities. We present several recently introduced Wigner-related sets of representations that have improved signal processing performance, and we introduce an optical implementation. This study deals with the generalized Wigner spaces, the fractional Fourier transform, and the x -p and the r -p representations. The optical implementations are demonstrated and discussed.

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“…The FRT modulus squared for varying x and p is known as (x±p) representation or Radon±Wigner transform of t 0 ¹ … † [11]. For xˆ0, it can be written as [10]:…”

Section: The Frt Approach To Strehl Ratiomentioning

confidence: 99%

Short Communication Analysis of the asymmetric apodization using the fractional Fourier transform

Granieri¹,

Zalvidea²,

Sicre³

1999

Journal of Modern Optics

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Abstract.The e ect of asymmetric apodization is analysed using the fractional Fourier transform. The focusing properties of an apodizing pupil mask are investigated by means of a simple display in a generalized phase-space (or x±p domain). Some comparative computer simulations are performed and curves displaying the Strehl ratio versus defocus are presented in order to illustrate the possibilities of our approach. IntroductionSeveral e orts to improve the quality of the point spread function (PSF) of an optical imaging system have been done by using apodizer apertures. The purpose of the apodization process is to produce the suppression of the secondary maxima, or side lobes, of a di raction pattern in order to enhance the resolving power of the optical system. It is a classical problem and it has been widely studied [1,2]. In the apodization method proposed by Cheng and Siu [3], the aperture is modi®ed in an asymmetric mode. This asymmetric aperture function produces the suppression of the secondary maxima at a side of the central peak of the di raction pattern at the cost of increasing the lobes at the opposite side [4].The fractional Fourier transform (FRT) was introduced in optical signal processing by Ozaktas and Mendlovic [5] and Lohmann [6]. Two di erent optical de®nitions of the FRT have been given. In the ®rst, the FRT was de®ned based on light propagation in quadratic graded index media. In the second de®nition, the FRT of fractional order p results from a phase-space rotation of the input Wigner distribution function by an angle of pº=2. The FRT is a generalization of the classical Fourier transform and the information content stored in the FRT changes from spatial to spectral as the fractional order varies from pˆ0 to pˆ1. The relation between a FRT of order p and the free-space di raction [7,8] allows the analysis of the tolerance of optical imaging systems to focus errors and/or aberrations. There are several criteria to analyse the performance of an optical imaging system based on the on-axis irradiance. The Strehl ratio (SR), de®ned as the intensity values on axis at the di raction focus conveniently normalized, is an important image quality parameter [9]. The purpose of this letter is to analyse the

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New signal representation based on the fractional Fourier transform: definitions

Cited by 52 publications

References 12 publications

Fast algorithm for chirp transforms with zooming-in ability and its applications

Fast algorithm for chirp transforms with zooming-in ability and its applications

Wigner-related phase spaces for signal processing and their optical implementation

Short Communication Analysis of the asymmetric apodization using the fractional Fourier transform

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