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The Fractional Fourier Transform and Applications

SIAM Review, 1991
This paper describes the “fractional Fourier transform,” which admits computation by an algorithm that has complexity proportional to the fast Fourier transform algorithm. Whereas the discrete Fourier transform (DFT) is based on integral roots of unity $e^{{{ - 2\pi i} / n}} $, the fractional Fourier transform is based on fractional roots of unity $e^{
David H. Bailey, Paul N. Swarztrauber
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Hilbert transform associated with the fractional Fourier transform

IEEE Signal Processing Letters, 1998
The analytic part of a signal f(t) is obtained by suppressing the negative frequency content of f, or in other words, by suppressing the negative portion of the Fourier transform, f/spl circ/, of f. In the time domain, the construction of the analytic part is based on the Hilbert transform f/spl circ/ of f(t).
Zayed, Ahmed I.
core   +4 more sources

Digital computation of the fractional Fourier transform [PDF]

IEEE Transactions on Signal Processing, 1996
An algorithm for efficient and accurate computation of the fractional Fourier transform is given. For signals with time-bandwidth product N, the presented algorithm computes the fractional transform in O(NlogN) time.
H M Ozaktas, Orhan Arikan
exaly   +2 more sources

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Fractional Fourier Transform Meets Transformer Encoder

IEEE Signal Processing Letters, 2022
Utilizing signal processing tools in deep learning models has been drawing increasing attention. Fourier transform (FT), one of the most popular signal processing tools, is employed in many deep learning models. Transformer-based sequential input processing models have also started to make use of FT.
Furkan Sahinuç, Aykut Koç
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Fractional finite Fourier transform

Journal of the Optical Society of America A, 2004
We show that a fractional version of the finite Fourier transform may be defined by using prolate spheroidal wave functions of order zero. The transform is linear and additive in its index and asymptotically goes over to Namias's definition of the fractional Fourier transform.
Kedar, Khare, Nicholas, George
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The fractional Fourier transform on graphs

2017 Asia-Pacific Signal and Information Processing Association Annual Summit and Conference (APSIPA ASC), 2017
The emerging field of signal processing on graphs merges algebraic or spectral graph theory with discrete signal processing techniques to process signals on graphs. In this paper, a definition of the fractional Fourier transform on graphs (GFRFT) is proposed and consolidated, which extends the discrete fractional Fourier transform (DFRFT) in the same ...
Yi-Qian Wang, Bing-Zhao Li 0001, Qi-Yuan Cheng   +2 more
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Random fractional Fourier transform

Optics Letters, 2007
We propose a novel random fractional Fourier transform by randomizing the transform kernel function of the conventional fractional Fourier transform. The random fractional Fourier transform inherits the excellent mathematical properties from the fractional Fourier transform and can be easily implemented in optics.
Zhengjun, Liu, Shutian, Liu
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On Namias's Fractional Fourier Transforms

IMA Journal of Applied Mathematics, 1987
\textit{V. Namias} [J. Inst. Math. Appl. 25, 241-265 (1980; Zbl 0434.42014)] developed a theory of fractional powers for the Fourier transform and obtained a number of fractional formulae which he used to solve several types of Schrödinger equation. In this paper the authors attempt to provide the necessary mathematical framework for Namias' idea in ...
McBride, A. C., Kerr, F. H.
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Trainable Fractional Fourier Transform

IEEE Signal Processing Letters
Recently, the fractional Fourier transform (FrFT) has been integrated into distinct deep neural network (DNN) models such as transformers, sequence models, and convolutional neural networks (CNNs). However, in previous works, the fraction order $\boldsymbol{a}$ is merely considered a hyperparameter and selected heuristically or tuned manually to find ...
Emirhan Koç, Tuna Alikasifoglu, Arda Can Aras, Aykut Koç   +3 more
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Fractional discrete Fourier transforms

Optics Letters, 1996
Direct calculation of fractional Fourier transforms from the expressions derived for their optical implementation is laborious. An extension of the discrete Fourier transform would have only O(N(2)) computational complexity. We define such a system, offer a general way to compute the fractional discrete Fourier transform matrix, and numerically ...
Z T, Deng, H J, Caulfield, M, Schamschula   +2 more
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