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
Theory and applications of short-time linear canonical transform
Deyun Wei,Huimin Hu +1 more
TL;DR: In this article, the authors used the short-time linear canonical transform (STLCT) to solve the problem of nonstationary signals whose linear canonical frequencies change over time due to the lack of time localization information.
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Flexible OFDM-based access systems with intrinsic function of chromatic dispersion compensation
Tsuyoshi Konishi,T. Murakawa,Tomotaka Nagashima,Makoto Hasegawa,Satoshi Shimizu,Kuninori Hattori,Masayuki Okuno,Shinji Mino,Akira Himeno,Hiroyuki Uenohara,Naoya Wada,Gabriella Cincotti +11 more
TL;DR: In this article, the intrinsic chromatic dispersion compensation functionality of optical fractional orthogonal frequency division multiplexing is discussed and experimentally demonstrated using dispersion-tunable transmitter and receiver based on wavelength selective switching devices.
Journal ArticleDOI
The Fractional S-Transform on Spaces of Type
TL;DR: In this article, the fractional S-transform is shown to be a continuous linear map of the space of type defined on the input space into the space on the output space.
Journal ArticleDOI
Sampling theorem for two dimensional fractional Fourier transform
TL;DR: A new two dimensional fractional Fourier transform that is not a tensor product of two one-dimensional transforms was introduced, and the aim of this paper is to derive sampling theorem for this new transform.
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Riesz fractional order derivative in Fractional Fourier Transform domain: An insight
TL;DR: The design flexibility of the proposed approach is confirmed due to the fact that it provides an optimal value of performance metrics corresponding to the variation of the fractional order of Riesz derivative and fractional parameter in the rotation angle of Fractional Fourier Transform (FrFT).
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.
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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.