Journal ArticleDOI
The fractional Fourier transform and time-frequency representations
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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
Application of the fractional Fourier transform to moving target detection in airborne SAR
TL;DR: To solve the problem whereby weak targets are shadowed by the sidelobes of strong ones, a new implementation of the CLEAN technique is proposed based on filtering in the fractional Fourier domain, and strong moving targets and weak ones can be detected iteratively.
Journal ArticleDOI
On the relationship between the Fourier and fractional Fourier transforms
TL;DR: This letter shows that the fractional Fourier transform is nothing more than a variation of the standard Fouriertransform and, as such, many of its properties can be deduced from those of the Fourier Transform by a simple change of variable.
Journal ArticleDOI
The discrete fractional cosine and sine transforms
Soo-Chang Pei,Min-Hung Yeh +1 more
TL;DR: The computations of DFRFT for even or odd signals can be planted into the half-size DFRCT and DFRST calculations, which will reduce the computational load of the DFR FT by about one half.
Journal ArticleDOI
Sampling of linear canonical transformed signals
TL;DR: The well-known Shannon sampling theorem and previously developed sampling criteria for Fresnel and fractional Fourier transformed signals are shown to be a special cases of the theorem developed here.
Journal ArticleDOI
Fractional convolution and correlation via operator methods and an application to detection of linear FM signals
O. Akay,G.F. Boudreaux-Bartels +1 more
TL;DR: This work derives explicit definitions of fractional convolution and correlation operations in a systematic and comprehensive manner and provides alternative formulations of those fractional operations that suggest efficient algorithms for discrete implementation.
References
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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.
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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.