Journal ArticleDOI
The fractional Fourier transform and time-frequency representations
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TLDR
The authors briefly introduce the functional Fourier transform and a number of its properties and present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time- frequencies such as the Wigner distribution, the ambiguity function, the short-time Fouriertransform and the spectrogram.Abstract:
The functional Fourier transform (FRFT), which is a generalization of the classical Fourier transform, was introduced a number of years ago in the mathematics literature but appears to have remained largely unknown to the signal processing community, to which it may, however, be potentially useful. The FRFT depends on a parameter /spl alpha/ and can be interpreted as a rotation by an angle /spl alpha/ in the time-frequency plane. An FRFT with /spl alpha/=/spl pi//2 corresponds to the classical Fourier transform, and an FRFT with /spl alpha/=0 corresponds to the identity operator. On the other hand, the angles of successively performed FRFTs simply add up, as do the angles of successive rotations. The FRFT of a signal can also be interpreted as a decomposition of the signal in terms of chirps. The authors briefly introduce the FRFT and a number of its properties and then present some new results: the interpretation as a rotation in the time-frequency plane, and the FRFT's relationships with time-frequency representations such as the Wigner distribution, the ambiguity function, the short-time Fourier transform and the spectrogram. These relationships have a very simple and natural form and support the FRFT's interpretation as a rotation operator. Examples of FRFTs of some simple signals are given. An example of the application of the FRFT is also given. >read more
Citations
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Journal ArticleDOI
The multiple-parameter discrete fractional Fourier transform
Soo-Chang Pei,Wen-Liang Hsue +1 more
TL;DR: The proposed multiple-parameter discrete fractional Fourier transform (MPDFRFT) is shown to have all of the desired properties for fractional transforms and the double random phase encoding in the MPDFRFT domain significantly enhances data security.
Journal ArticleDOI
Diagnosis of Induction Motor Faults in the Fractional Fourier Domain
Manuel Pineda-Sanchez,Martin Riera-Guasp,Jose A. Antonino-Daviu,J. Roger-Folch,J. Perez-Cruz,Ruben Puche-Panadero +5 more
TL;DR: The use of the fractional FT (FrFT) instead of the FT to perform TMCSA and the optimization of the FrFT to generate a spectrum where the frequency-varying fault harmonics appear as single spectral lines and, therefore, facilitate the diagnostic process.
Journal ArticleDOI
Sampling of compact signals in offset linear canonical transform domains
TL;DR: The sampling theorem for OLCT signals presented here serves as a unification and generalization of previously developed sampling theorems.
Journal ArticleDOI
Two dimensional discrete fractional Fourier transform
Soo-Chang Pei,Min-Hung Yeh +1 more
TL;DR: This paper develops a 2D DFRFT which can preserve the rotation properties and provide similar results to continuous FRFT.
Journal ArticleDOI
Implementation of quantum and classical discrete fractional Fourier transforms.
Steffen Weimann,Armando Perez-Leija,Maxime Lebugle,Robert Keil,Malte C. Tichy,Markus Gräfe,René Heilmann,Stefan Nolte,Héctor M. Moya-Cessa,Gregor Weihs,Demetrios N. Christodoulides,Alexander Szameit +11 more
TL;DR: In the context of classical optics, this work implements discrete fractional Fourier transforms of exemplary wave functions and experimentally demonstrate the shift theorem and applies this approach in the quantum realm to Fourier transform separable and path-entangled biphoton wave functions.
References
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TL;DR: A tutorial review of both linear and quadratic representations is given, and examples of the application of these representations to typical problems encountered in time-varying signal processing are provided.
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The Fractional Order Fourier Transform and its Application to Quantum Mechanics
TL;DR: In this article, a generalized operational calculus is developed, paralleling the familiar one for the ordinary transform, which provides a convenient technique for solving certain classes of ordinary and partial differential equations which arise in quantum mechanics from classical quadratic hamiltonians.
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Image rotation, Wigner rotation, and the fractional Fourier transform
TL;DR: In this article, the degree p = 1 is assigned to the ordinary Fourier transform and the degree P = 1/2 to the fractional transform, where p is the degree of the optical fiber.
Journal ArticleDOI
Time-frequency representation of digital signals and systems based on short-time Fourier analysis
TL;DR: In this article, the authors developed a representation for discrete-time signals and systems based on short-time Fourier analysis and showed that a class of linear-filtering problems can be represented as the product of the time-varying frequency response of the filter multiplied by the short time Fourier transform of the input signal.