The fractional Fourier transform: theory, implementation and error analysis
TL;DR: It is hoped that this implementation and fixed-point error analysis will lead to a better understanding of the issues involved in finite register length implementation of the discrete fractional Fourier transform and will help the signal processing community make better use of the transform.
About: This article is published in Microprocessors and Microsystems.The article was published on 2003-11-03. It has received 128 citations till now. The article focuses on the topics: Fractional Fourier transform & Non-uniform discrete Fourier transform.
Citations
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TL;DR: This paper is geared toward signal processing practitioners by emphasizing the practical digital realizations and applications of the FRFT, which is closely related to other mathematical transforms, such as time-frequency and linear canonical transforms.
335 citations
Book•
01 Jan 1998
TL;DR: Advances in Imaging and Electron Physics merges two long-running serials, Advances in Electronics and Electro-Magnetic Physics (AEEE and EEE), and advances in Optical and Electro Electron Microscopy (OEM) as mentioned in this paper.
Abstract: Advances in Imaging and Electron Physics merges two long-running serials, Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. The series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science, and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains. * Contains contributions from leading authorities on the subject matter* Informs and updates on all the latest developments in the field of imaging and electron physics* Provides practitioners interested in microscopy, optics, image processing, mathematical morphology, electromagnetic fields, electron, and ion emission with a valuable resource* Features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science, and digital image processing
225 citations
TL;DR: A robust copyright protection scheme based on fractional Fourier transform (FrFT) and visual cryptography (VC) and the experimental results show that the proposed scheme is strong enough to resist various signal processing operations.
131 citations
TL;DR: The present article collects the scattered tools into a dashboard view, briefly introduces their usage and algorithms, evaluates the accuracy, compares the performance, and provides informative comments for selection.
Abstract: In recent years, as fractional calculus becomes more and more broadly used in research across different academic disciplines, there are increasing demands for the numerical tools for the computation of fractional integration/differentiation, and the simulation of fractional order systems. Time to time, being asked about which tool is suitable for a specific application, the authors decide to carry out this survey to present recapitulative information of the available tools in the literature, in hope of benefiting researchers with different academic backgrounds. With this motivation, the present article collects the scattered tools into a dashboard view, briefly introduces their usage and algorithms, evaluates the accuracy, compares the performance, and provides informative comments for selection.
128 citations
References
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TL;DR: In this article, the authors consider filtering in fractional Fourier domains, which enables significant reduction of the error compared with ordinary Fourier domain filtering for certain types of degradation and noise, while requiring only O(N log N) implementation time.
Abstract: For time-invariant degradation models and stationary signals and noise, the classical Fourier domain Wiener filter, which can be implemented in O(N log N) time, gives the minimum mean-square-error estimate of the original undistorted signal. For time-varying degradations and nonstationary processes, however, the optimal linear estimate requires O(N/sup 2/) time for implementation. We consider filtering in fractional Fourier domains, which enables significant reduction of the error compared with ordinary Fourier domain filtering for certain types of degradation and noise (especially of chirped nature), while requiring only O(N log N) implementation time. Thus, improved performance is achieved at no additional cost. Expressions for the optimal filter functions in fractional domains are derived, and several illustrative examples are given in which significant reduction of the error (by a factor of 50) is obtained.
321 citations
Journal Article•
TL;DR: Simulation results indicate that the energy of LFM signal will be collected effectively when the fractional order is matching with its modulation slope and in weak signals detection of underwater acoustic domain, the authors can get high anti-Doppler performance using the Fractional fourier transform algorithm.
Abstract: Based on the concept of the fractional fourier transform, its digital computation is given through computer simulation. In terms of linear frequency modulation (LFM) signal, the relation between fractional order and modulation slope is analyzed and the performance comparison with matched filter is given. Moreover, the separation of LFM signal and noise is realized in low signal-to-noise ratio through simulation. Simulation results indicate that the energy of LFM signal will be collected effectively when the fractional order is matching with its modulation slope. In weak signals detection of underwater acoustic domain, we can get high anti-Doppler performance using the Fractional fourier transform algorithm.
243 citations
"The fractional Fourier transform: t..." refers background in this paper
...[7], which computes the samples of the continuous FRFT and hence is strictly not the discrete FRFT (DFRFT)....
[...]
...Therefore, many attempts have been made to find the discrete version of the FRFT [6,7]....
[...]
Book•
01 Jan 1998
TL;DR: Advances in Imaging and Electron Physics merges two long-running serials, Advances in Electronics and Electro-Magnetic Physics (AEEE and EEE), and advances in Optical and Electro Electron Microscopy (OEM) as mentioned in this paper.
Abstract: Advances in Imaging and Electron Physics merges two long-running serials, Advances in Electronics and Electron Physics and Advances in Optical and Electron Microscopy. The series features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science, and digital image processing, electromagnetic wave propagation, electron microscopy, and the computing methods used in all these domains. * Contains contributions from leading authorities on the subject matter* Informs and updates on all the latest developments in the field of imaging and electron physics* Provides practitioners interested in microscopy, optics, image processing, mathematical morphology, electromagnetic fields, electron, and ion emission with a valuable resource* Features extended articles on the physics of electron devices (especially semiconductor devices), particle optics at high and low energies, microlithography, image science, and digital image processing
225 citations
TL;DR: The calculation of the statistical mean-squared error at the output of the filter is discussed in detail and some of the approaches used in investigating them are reviewed.
Abstract: The accuracy of a digital filter is limited by the finite word length used in its implementation. Techniques have been developed to analyze this problem. Good agreement between the theoretical and experimental results has been reported. This paper discusses some of these accuracy problems and reviews some of the approaches used in investigating them. The calculation of the statistical mean-squared error at the output of the filter is discussed in detail.
217 citations
15 Mar 1999
TL;DR: This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions and supports confidence that it will be accepted as the definitive definition of this transform.
Abstract: We propose and consolidate a definition of the discrete fractional Fourier transform which generalizes the discrete Fourier transform (DFT) in the same sense that the continuous fractional Fourier transform (FRT) generalizes the continuous ordinary Fourier Transform. This definition is based on a particular set of eigenvectors of the DFT which constitutes the discrete counterpart of the set of Hermite-Gaussian functions. The fact that this definition satisfies all the desirable properties expected of the discrete FRT, supports our confidence that it will be accepted as the definitive definition of this transform.
210 citations
"The fractional Fourier transform: t..." refers methods in this paper
...In the recent years, the FRFT has gained great significance as a tool for signal processing [4 – 8]. Therefore, many attempts have been made to find the discrete version of the FRFT [ 6 ,7]....
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