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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Proceedings ArticleDOI
FRFT based on joint estimation time delay and radial velocity of underwater target
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The continuous fractional wavelet transform on generalized weighted Sobolev spaces
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TL;DR: In this article, the continuous fractional wavelet transform is defined and boundedness results of this transform on generalized weighted Sobolev spaces are obtained, and characterisation of the range of the WFT transform is discussed.
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Velocity and acceleration estimation in video sequences by the local polynomial periodogram
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Fractional Wavelet Transform in Terms of Fractional Convolution
Akhilesh Prasad,Praveen Kumar +1 more
TL;DR: In this article, a relation between the wavelet transform and inverse fractional Fourier transform is established, and the FrWT of a test function space is investigated in terms of fractional convolution operator and its adjoint.
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Research on DOA Estimation of Multi-Component LFM Signals Based on the FRFT
TL;DR: A novel algorithm for the direction of arrival (DOA) estimation based on the fractional Fourier transform (FRFT) is proposed which perfects the method theoretically and simulation results are provided to show the validity of the method.
References
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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.