Journal ArticleDOI
The fractional Fourier transform and time-frequency representations
Reads0
Chats0
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
More filters
Journal ArticleDOI
Continuum quantum systems as limits of discrete quantum systems: II. State functions
TL;DR: In this article, pointwise convergence of a discrete state function to a continuous state function is shown to imply the algebraic criterion for convergence that was introduced in the prequel, and the normal approximation theorem and the convergence of the Kravchuk functions to the Hermite-Gaussians are expressed in terms of the notion of convergence.
Journal ArticleDOI
Joint timing/frequency offset estimation and correction based on FrFT encoded training symbols for PDM CO-OFDM systems.
TL;DR: The proposed TO/FO estimation algorithm performs robustly and accurately without any induced BER degradations and exhibits robust estimation result of timing offset with poor optical signal-to-noise ratio (OSNR) and nonlinear interference.
Journal ArticleDOI
Fractional Fourier transform of bandlimited periodic signals and its sampling theorems
TL;DR: It is shown that only 2K + 1 coefficients are sufficient to reconstruct anyFRFT domain bandlimited periodic signal, where K is the order of the highest nonzero harmonic component in the particular FRFT domain in which the signal is bandlimited.
Journal ArticleDOI
Shift-invariance of short-time Fourier transform in fractional Fourier domains
TL;DR: In this article, a new proof on the shift-invariance of linear time-frequency distributions on fractional Fourier domains is given, and it is shown that short-time Fourier transform (STFT) is the unique linear distribution satisfying magnitude-wise shift-consistency.
Journal ArticleDOI
Multiplicative filtering in the linear canonical transform domain
TL;DR: A model of multiplicative filtering for the band-limited signals in the LCT domain is presented by using the convolution theorem given in the literature and it is found from the simulation results that mean square error is minimum for different values of signal-to-noise ratio.
References
More filters
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.