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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Efficient Compressed Sensing Method for Moving-Target Imaging by Exploiting the Geometry Information of the Defocused Results
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Multiple-Parameter Discrete Fractional Transform and its Applications
Xuejing Kang,Ran Tao,Feng Zhang +2 more
TL;DR: Two separate theoretical frameworks called the type I and II MPDFRT that can include existing multiple-parameter transforms as special cases are proposed that can construct new types of transforms that may be useful in signal processing and information security.
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Fractional dual tree complex wavelet transform and its application to biometric security during communication and transmission
TL;DR: In this paper, biometric is chosen as the primary application and hence a new technique is proposed for securing biometrics during communication and transmission over insecure channel, i.e. the fractional dual tree complex wavelet transform (FrDT-CWT).
A shattered survey of the Fractional Fourier Transform
TL;DR: In this paper, the authors introduce the notion of fractional Fourier transform, which may be considered as a fractional power of the classical Fourier trans-form, and give an introduction to the denition, the properties and computational aspects of both the continuous and discrete Fourier transforms.
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Rhapsody in fractional
J. A. Tenreiro Machado,António M. Lopes,Fernando B. Duarte,Manuel Duarte Ortigueira,Raul T. Rato +4 more
TL;DR: In this paper, the concept of fractional sums, fractional powers of a square matrix, tolerant computing and FracSets, negative probabilities and fractional delay discrete-time linear systems are introduced.
References
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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.
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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.
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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.