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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

Damage evolution and rupture prediction of 42CrMoA nitrided steel by acoustic emission and infrared thermography techniques

TL;DR: In this article, the acoustic and IR signals were experimentally obtained by the AE sensors and IR detector, respectively, with the sensors and detector distributed rationally during the fatigue test of sine amplitude loading.
Proceedings ArticleDOI

Reality-preserving image encryption associated with the generalized Hilbert transform

TL;DR: A novel image encryption technique is proposed associated with the generalized Hilbert transform and the pixel permutation matrix are as keys in the proposed method, which can improve the security of the image encryption.
Dissertation

First-order optical systems in information processing and optronic devices

TL;DR: A flexible optical system design corresponding to ortho-symplectic transformations; rotator, separable fractional Fourier transform (FRFT), and gyrator operation (GT) is developed on the basis of a matrix formalism and the main properties and applications of the GT operation are demonstrated, which is little-known by the optical community.
Proceedings ArticleDOI

Optimal waveform design oriented toward cognitive radar in fractional Fourier domain

TL;DR: In this article, the problem of waveform design using Fractional Fourier Transform (FRFT) in signal-dependent interference, as well as additive channel noise for stochastic extended target is investigated.
Journal ArticleDOI

Robust Image Compression Algorithm using Discrete Fractional Cosine Transform

TL;DR: The merit of effectiveness of proposed technique over existing technique is superior due to application of discrete fractional cosine transform by which higher compression ratio and PSNR are obtained without any artifacts in compressed images.
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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