A unified framework for the fractional Fourier transform

@article{Cariolaro1998AUF,
  title={A unified framework for the fractional Fourier transform},
  author={Gianfranco Cariolaro and Tomaso Erseghe and Peter Kraniauskas and Nicola Laurenti},
  journal={IEEE Trans. Signal Process.},
  year={1998},
  volume={46},
  pages={3206-3219}
}
The paper investigates the possibility for giving a general definition of the fractional Fourier transform (FRT) for all signal classes [one-dimensional (1-D) and multidimensional, continuous and discrete, periodic and aperiodic]. Since the definition is based on the eigenfunctions of the ordinary Fourier transform (FT), the preliminary conditions is that the signal domain/periodicity be the same as the FT domain/periodicity. Within these classes, a general FRT definition is formulated, and the… 
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References

SHOWING 1-10 OF 36 REFERENCES
Multiplicity of fractional Fourier transforms and their relationships
TLDR
A general FRT definition is generated, based on eigenfunctions and eigenvalues of the ordinary Fourier transform, which allows us to generate all possible definitions and gives explicit relationships between the different FRTs.
Generalized fractional Fourier transforms
We generalize the definition of the fractional Fourier transform (FRT) by expanding the new definition proposed by Shih to the original definition. The generalized FRT is shown to have k-periodic
The fractional Fourier transform and time-frequency representations
  • L. B. Almeida
  • Mathematics, Computer Science
    IEEE Trans. Signal Process.
  • 1994
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.
The Fractional Fourier Transform and Applications
TLDR
The fractional Fourier transform and the corresponding fast algorithm are useful for such applications as computing DFTs of sequences with prime lengths, computing D FTs of sparse sequences, analyzing sequences with noninteger periodicities, performing high-resolution trigonometric interpolation, detecting lines in noisy images, and detecting signals with linearly drifting frequencies.
The Fractional Order Fourier Transform and its Application to Quantum Mechanics
We introduce the concept of Fourier transforms of fractional order, the ordinary Fourier transform being a transform of order 1. The integral representation of this transform can be used to construct
On the relationship between the Fourier and fractional Fourier transforms
  • A. Zayed
  • Computer Science, Mathematics
    IEEE Signal Processing Letters
  • 1996
TLDR
This letter shows that the fractional Fourier transform is nothing more than a variation of the standard Fouriertransform and, as such, many of its properties can be deduced from those of the Fourier Transform by a simple change of variable.
Fractionalization of Fourier transform
The conventional definition of fractional-order Fourier transform is demonstrate to be not unique. The same rules can be applied to create a new type of fractional-order Fourier transform which
Properties of the fractionalization of a Fourier transform
Abstract The new fractional Fourier transform, proposed by Shih [Optics Comm. 118 (1995) 495], is shown to differ from the conventional fractional Fourier transform by its four periodic eigenvalues
Fractional Fourier transforms and their optical implementation: I
It is often the case that an operation originally defined for integer orders can be generalized to fractional or even complex orders in a meaningful and useful way. A basic example is the power
Convolution, filtering, and multiplexing in fractional Fourier domains and their relation to chirp and wavelet transforms
TLDR
Convolution, filtering, and multiplexing of signals in fractional domains are discussed, revealing that under certain conditions one can improve on the special cases of these operations in the conventional space and frequency domains.
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