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<p>A plethora of research activities have taken place in the development of the Discrete Fractional Fourier Transform (DFRFT) as the discrete-time counterpart of the continuous-time Fractional Fourier transform (FRFT), which is a generalization of the Fourier transform (FT). The development depends heavily on generating appropriate orthonormal eigenvectors of the kernel matrix <b>F</b> of the discrete Fourier transform (DFT). A key requirement in generating those eigenvectors is being as close as possible to vectors formed by <i>samples</i> of the Hermite-Gaussian functions, which are the eigenfunctions of the FRFT. The widely used <i>sampling scheme</i> is <i>uniform</i>, where <i>all functions</i> are sampled at the <i>same</i> set of <i>uniformly spaced</i> points. The main focus of the present work is the development of a better <i>representative sampling scheme</i> where <i>different functions</i> are sampled at <i>different</i> sets of <i>non-uniformly</i> spaced points lying on the nonnegative time axis to capture their behaviors better. The samples include the extrema of the functions at their optimizers, zero values at their zeros, and possibly some other samples taken beyond the last maximizers. The simulation results show that the contributed <i>nonuniform</i> sampling scheme outperforms the <i>uniform</i> one since its corresponding eigenvectors have a <i>smaller approximation error</i>. The approximation error is the meaningful comparison criterion of the sampling schemes. A major part of this research project has been the development of several MATLAB functions (in the form of an experimental toolbox). To overcome the limitation on the order of the kernel matrix imposed by the numeric computation track of the sole MATLAB, one has to resort to the variable precision arithmetic (vpa) available in the Symbolic Math Toolbox, allowing a successful execution for orders of the square kernel matrix of the transform up to several hundreds. Therefore, it has been possible to overcome the sole problem hindering the adoption of the contributed <i>superior nonuniform sampling scheme</i> in practical applications.</p>

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A Representative Sampling Scheme of the Hermite-Gaussian Functions

  • 期刊论文

  • 发表日期：

    2026年5月6日

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  • Magdy Tawfik Hanna

摘要

A plethora of research activities have taken place in the development of the Discrete Fractional Fourier Transform (DFRFT) as the discrete-time counterpart of the continuous-time Fractional Fourier transform (FRFT), which is a generalization of the Fourier transform (FT). The development depends heavily on generating appropriate orthonormal eigenvectors of the kernel matrix F of the discrete Fourier transform (DFT). A key requirement in generating those eigenvectors is being as close as possible to vectors formed by samples of the Hermite-Gaussian functions, which are the eigenfunctions of the FRFT. The widely used sampling scheme is uniform, where all functions are sampled at the same set of uniformly spaced points. The main focus of the present work is the development of a better representative sampling scheme where different functions are sampled at different sets of non-uniformly spaced points lying on the nonnegative time axis to capture their behaviors better. The samples include the extrema of the functions at their optimizers, zero values at their zeros, and possibly some other samples taken beyond the last maximizers. The simulation results show that the contributed nonuniform sampling scheme outperforms the uniform one since its corresponding eigenvectors have a smaller approximation error. The approximation error is the meaningful comparison criterion of the sampling schemes. A major part of this research project has been the development of several MATLAB functions (in the form of an experimental toolbox). To overcome the limitation on the order of the kernel matrix imposed by the numeric computation track of the sole MATLAB, one has to resort to the variable precision arithmetic (vpa) available in the Symbolic Math Toolbox, allowing a successful execution for orders of the square kernel matrix of the transform up to several hundreds. Therefore, it has been possible to overcome the sole problem hindering the adoption of the contributed superior nonuniform sampling scheme in practical applications.