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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

Some important fractional transformations for signal processing

TL;DR: This paper suggests a generalization of the Hartley transformation based on the fractional Fourier transform, coined it “fractional Hartley transform (FHT)” and additional useful transformations used for signal processing are discussed.
Journal ArticleDOI

Extracting Micro-Doppler Radar Signatures From Rotating Targets Using Fourier–Bessel Transform and Time–Frequency Analysis

TL;DR: The efficiency of the Fourier-Bessel transform and time-frequency (TF)-based method in conjunction with the fractional Fourier transform (FrFT), for extracting micro-Doppler radar signatures from the rotating targets is reported.
Journal ArticleDOI

Forecasting method of stock market volatility in time series data based on mixed model of ARIMA and XGBoost

TL;DR: The proposed DWT-ARIMA-GSXGB stock price prediction model has good approximation ability and generalization ability, and can fit the stock index opening price well.
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ISAR Imaging of Targets With Complex Motions Based on the Keystone Time-Chirp Rate Distribution

TL;DR: A new ISAR imaging algorithm based on the keystone time-chirp rate distribution (KTCRD) is proposed for the targets with complex motions that can estimate the parameters of multicomponent CPSs without searching procedures and can acquire high antinoise performance with a relatively low computational load.
Journal ArticleDOI

The discrete harmonic oscillator, Harper's equation, and the discrete fractional Fourier transform

TL;DR: In this article, the authors define a discrete fractional Fourier transform (FT) which is essentially the time-evolution operator of the discrete harmonic oscillator, and define its energy eigenfunctions as a discrete algebraic analogue of the Hermite-Gaussian functions.
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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