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    Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms


    Hennelly, Bryan M. and Sheridan, John T. (2005) Generalizing, optimizing, and inventing numerical algorithms for the fractional Fourier, Fresnel, and linear canonical transforms. Journal of the Optical Society of America A., 22 (5). pp. 917-927. ISSN 1084-7529

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    Abstract

    By use of matrix-based techniques it is shown how the space–bandwidth product (SBP) of a signal, as indicated by the location of the signal energy in the Wigner distribution function, can be tracked through any quadratic-phase optical system whose operation is described by the linear canonical transform. Then, applying the regular uniform sampling criteria imposed by the SBP and linking the criteria explicitly to a decomposition of the optical matrix of the system, it is shown how numerical algorithms (employing interpolation and decimation), which exhibit both invertibility and additivity, can be implemented. Algorithms appearing in the literature for a variety of transforms (Fresnel, fractional Fourier) are shown to be special cases of our general approach. The method is shown to allow the existing algorithms to be optimized and is also shown to permit the invention of many new algorithms.

    Item Type: Article
    Keywords: matrix-based techniques; space–bandwidth product (SBP); Wigner distribution function; numerical algorithms; fractional Fourier transform; Fresnel transform; linear canonical transform;
    Academic Unit: Faculty of Science and Engineering > Computer Science
    Faculty of Science and Engineering > Electronic Engineering
    Faculty of Science and Engineering > Research Institutes > Hamilton Institute
    Item ID: 8724
    Identification Number: https://doi.org/10.1364/JOSAA.22.000917
    Depositing User: Dr. Bryan Hennelly
    Date Deposited: 04 Sep 2017 14:35
    Journal or Publication Title: Journal of the Optical Society of America A.
    Publisher: Optical Society of America
    Refereed: Yes
    Funders: Enterprise Ireland, Science Foundation Ireland, Irish Research Council for Science Engineering and Technology (IRCSET)
    URI:

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