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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An introduction to the Fractional Fourier Transform and friends
TL;DR: In this paper, the authors introduce the notion of the fractional Fourier transform (FFT), which may be considered as a fractional power of the classical FFT, and give an introduction to the denition, the properties and approaches to the continuous FFT.
Proceedings ArticleDOI
Biomedical signal detection based on fractional fourier transform
TL;DR: In this paper, the second-order fractional Fourier transform moments and estimation were used for the multi-component chirps and interference situation, and the results indicate the great advantage on biomedical signal detection by FrFT.
Proceedings ArticleDOI
Direction of arrival estimation of coherent wideband LFM signals in multipath environment
Yue Cui,Kaihua Liu,Junfeng Wang +2 more
TL;DR: In this paper, a novel direction of arrival (DOA) estimation method is proposed for coherent wideband linear frequency modulated (LFM) signals in multipath environment, in which wideband LFM signals are first transformed into fractional Fourier (FRF) domain to construct the array model.
Journal ArticleDOI
Uncertainty Principles Associated with the Offset Linear Canonical Transform
Haiye Huo,Wenchang Sun,Li Xiao +2 more
TL;DR: In this article, the authors generalized the uncertainty principle for the offset linear canonical transform (OLCT) to the short-time OLCT and gave a lower bound for its essential support.
Journal ArticleDOI
Logarithmic uncertainty principle, convolution theorem related to continuous fractional wavelet transform and its properties on a generalized Sobolev space
Mawardi Bahri,Ryuichi Ashino +1 more
TL;DR: The Riemann–Lebesgue lemma for the Fr FT is derived, and the CFrWT in terms of the FrFT is introduced, and a different proof of the inner product relation and the inversion formula of the C FrWT are provided.
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.