Biograph: Current Position: Chairman and Professor, Department of Mathematical Sciences, DePaul University, Chicago, IL (2001- present). Professor and Associate Chair for Graduate Studies, Department of Mathematics, University of Central Florida, Orlando, Florida (1999-2001). Professor, University of Central Florida, Orlando, Florida (1990-2001). California Polytechnic State University, San Luis Obispo, California (1980-1990). Chair, SampTA Steering Committee, An International Group on Sampling Theory and its Applications, 2013-2017
Served on the Editorial Boards of 21 mathematics and engineering journals, including: Managing Editor: Journal of Sampling Theory in Image and Signal Processing (1999- 2005), International Journal of Fractional Calculus and Applied Analysis, (2000-present), The Journal of Integral Transforms and Special Functions (2012-present), The International Journal of Mathematics and Mathematical Sciences, (1999-present), Journal of Multidimensional Systems and Signal Processing (2002-2012). Inaugural College of Science and Health Award for Excellence in Research, 2012. Quality Matters Star Award for Excellence in Online Course Design, 2012. The Spirit of Inquiry Awards, DePaul University, September 2007. Research Incentive Award, University of Central Florida, 2001. College of Arts and Sciences Distinguished Researcher Award (2000), University of Central Florida.
Title: Sampling Theorem for Two-dimensional Fractional Fourier Transform
Abstract: The fractional Fourier transform, which is a generalization of the Fourier transform, has become the focus of many research papers in recent years because of its applications in signal processing and optics. The fractional Fourier transform has been extended to $n$ dimensions using tensor product of $n$ copies of the one-dimensional transform.
Recently, a new two-dimensional fractional Fourier transform that is not a tensor product of two one-dimensional transforms was introduced. The transform depends on two angles $\alpha$ and $\beta$ that are coupled so that the transform parameters depend on the average of the sum and the difference of the angles. The transform is not only important from a theoretical point of view, but it also has interesting applications as it is closely related to the ambiguity function and the Wigner distribution of two-dimensional signals.
The aim of this talk is to present sampling theorem for this new transform. Unlike the sampling theorem in the tensor product case, where the sampling function is a product of two Sinc functions, one in each of the transform variables, in the new sampling theorem the sampling function is a product of two Sinc functions whose arguments are not the variables of the transform but a weighted sum and a weighted difference of the transform variables.
Furthermore, the sample points depend on the average of the sum and the average of the difference of the transform angles, $\alpha, \beta ,$ which leads to a more interesting configuration and distribution of the sample points.