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Journal ArticleDOI

The fractional Fourier transform and time-frequency representations

Luís B. Almeida
- 01 Nov 1994 - 
- Vol. 42, Iss: 11, pp 3084-3091
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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. >

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Citations
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Journal ArticleDOI

Generalized analytic signal associated with linear canonical transform

TL;DR: In this paper, the generalized analytic signal can suppress the negative frequency components in the linear canonical transform domain, which is the generalization of many famous linear integral transform, such as Fourier transform, fractional Fourier Transform and Fresnel transform.
Journal ArticleDOI

Image super-resolution reconstruction using the high-order derivative interpolation associated with fractional filter functions

Deyun Wei
TL;DR: Compared with many methods, extensive experimental results validate that the proposed method can obtain the better-edge characteristic, less blur and less aliasing of the SISR reconstruction.
Journal ArticleDOI

Research progress on discretization of fractional Fourier transform

TL;DR: A summary of discretizations of the fractional Fourier transform developed in the last nearly two decades is presented and it is hoped to offer a doorstep for the readers who are interested in the fractionsal Fouriers transform.
Journal ArticleDOI

Generalization of the Fractional Hilbert Transform

TL;DR: In this letter, the fractional Hilbert transform of a real signal is generalized to get an analytic version which contains no negative spectrum while maintaining the essential information of the real signal.
Journal ArticleDOI

Simplified fractional Fourier transforms

TL;DR: This study introduces several types of simplified fractional Fourier transform (SFRFT) that are simpler than the original FRFT in terms of digital computation, optical implementation, implementation of gradient-index media, and implementation of radar systems.
References
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Journal ArticleDOI

Time-frequency distributions-a review

TL;DR: A review and tutorial of the fundamental ideas and methods of joint time-frequency distributions is presented with emphasis on the diversity of concepts and motivations that have gone into the formation of the field.
Journal ArticleDOI

Linear and quadratic time-frequency signal representations

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.
Journal ArticleDOI

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.
Journal ArticleDOI

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.
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