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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Journal ArticleDOI
Time–frequency analysis method based on affine Fourier transform and Gabor transform
TL;DR: This study obtains an affine relation between the AFT and the modified GT (MGT) and demonstrates that the A FT also has the affine relationship with other TFRs, such as the Gabor–Wigner transform and the general class of quadratic distribution.
Journal ArticleDOI
Adaptive waveform selection for maneuvering target tracking in cognitive radar
TL;DR: The factors influencing the target tracking accuracy are analyzed, and the maneuvering target tracking framework in cognitive radar is proposed and the superior performance of the adaptive waveform over the fixed waveform is illustrated with simulation examples.
Journal ArticleDOI
Fractional Spectrum Analysis for Nonuniform Sampling in the Presence of Clock Jitter and Timing Offset
TL;DR: This paper focuses on analyzing the fractional spectrum of nonuniform sampling which is affected by clock jitter and timing offset, and develops optimal filters that minimize the mean square error between the original and the compensated fractional spectra for both cases.
Proceedings Article
A new method of wavelet domain watermark embedding and extraction using Fractional Fourier Transform
Olcay Duman,Olcay Akay +1 more
TL;DR: A new watermark embedding and detecting method for blind and robust digital watermarking of images are presented that uses DWT domain to embed the watermark into the original image in such a way that it is imperceptible by the human visual system.
DissertationDOI
Time-frequency based methods for nonstationary signal analysis with application to EEG signals
TL;DR: Three new seizure detection algorithms that can classify seizure from nonseizure data with high accuracy are proposed and it was demonstrated that the proposed algorithms gave either equivalent or superior performance when compared against several other brain seizure algorithms previously reported in the literature.
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.