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
Parameter estimation of linear and quadratic chirps by employing the fractional fourier transform and a generalized time frequency transform
Shishir B. Sahay,T Meghasyam,Rahul K Roy,Gaurav Pooniwala,Sasank Chilamkurthy,Vikram M. Gadre +5 more
TL;DR: In this article, the authors provide a brief tutorial exposure to the Fractional Fourier Transform, followed by a report on experiments performed by the authors on a Generalized Time Frequency Transform (GTFT) proposed by them in an earlier paper.
Book ChapterDOI
Brief Historical Introduction
Lokenath Debnath,Firdous A. Shah +1 more
TL;DR: In this paper, the idea of expansion of a function in terms of trigonometric series without any attention to rigorous mathematical analysis was first introduced by Joseph Fourier (1770-1830).
Journal ArticleDOI
Dispersion correction for optical coherence tomography by the stepped detection algorithm in the fractional Fourier domain
TL;DR: A new method for dispersion correction based on the FRFD stepped detection algorithm that is able to adaptively compensate the dispersion at all depths of the sample is addressed and verified to be effective through the stratified media of ZnSe.
Journal ArticleDOI
Analysis of Dirichlet, Generalized Hamming and Triangular window functions in the linear canonical transform domain
Navdeep Goel,Kulbir Singh +1 more
TL;DR: It has been shown that by using linear canonical transform, the author is able to obtain all window parameters successfully as compared to fractional Fourier transform.
Proceedings ArticleDOI
ISAR imaging of fast-moving target based on FRFT Range Compression
TL;DR: A new reconstruction algorithm is proposed, which use Fractional Fourier Transform (FRFT) instead of DFT to implement the range compression, and indicates that the new algorithm can avoid the range dispersion and achieve refocused ISAR image effectively.
References
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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.
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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.
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.