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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The Multiple-Parameter Discrete Fractional FourierTransform
Soo-Chang Pei,Wen-Liang Hsue +1 more
TL;DR: In this paper, a multiple-parameter discrete fractional Fourier transform (MPDFRFT) was proposed to enhance the security of the MPDFRFT domain for encrypting digital data.
Journal ArticleDOI
A novel fiber nonlinearity suppression method in DWDM optical fiber transmission systems with an all-optical pre-distortion module
TL;DR: In this article, a novel all-optical scheme is proposed to reduce fiber nonlinear distortions in high-speed dense wavelength division multiplexed (DWDM) optical fiber communication systems by applying time-domain fractional Fourier transform (FRFT) based on the time-space duality theory.
Proceedings ArticleDOI
Synchronization and matched filtering in time-frequency using the sunflower spiral
TL;DR: It is argued that partial spiral correlation can lead to a significant reduction in computational complexity necessary for synchronization, and generalizations and identities based on the fractional Fourier transform are provided which omit the need for fractional delay filters.
Journal ArticleDOI
Global Exponential Stability of Fractional Order Complex-Valued Neural Networks with Leakage Delay and Mixed Time Varying Delays
TL;DR: In this paper, the global exponential stability of fractional order complex-valued neural networks with leakage delay and mixed time varying delays was investigated and sufficient conditions were established to ensure global exponential stabilisation.
Journal ArticleDOI
Analysis of Maneuvering Targets with Complex Motions by Two-Dimensional Product Modified Lv's Distribution for Quadratic Frequency Modulation Signals.
TL;DR: An effective parameter estimation algorithm is proposed—referred to as the Two-Dimensional product modified Lv’s distribution (2D-PMLVD)—for QFM signals, which can achieve better anti-noise performance than the other three representative methods.
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.