"On the Coupled Fractional Fourier Transform and its Extensions"Zayed, AhmedThe Fourier transform, which has ubiquitous applications in mathematics, physics, and engineering, has been studied extensively for over a century. In 1980 a fractional version of the Fourier transform was introduced but did not received much attention until the early 1990s when it was found to have numerous applications in optics and time-frequency representation. The fractional Fourier transform, denoted by $F_\theta ,$ depends on a parameter $0\leq \theta \leq \pi/2,$ so that when $\theta=0,$ $F_0$ is the identity transformation and when $\theta=\pi/2,$ $F_{\pi/2}$ is the standard Fourier transform. In this talk we discuss a novel generalization of the fractional Fourier transform to two dimensions, which is called the coupled fractional Fourier transform and is denoted by $F_{\alpha, \beta}.$ This transform depends on two independent angles $\alpha$ and $\beta,$ with $ 0\leq \alpha, \beta \leq \pi/2,$ so that $F_{0,0}$ is the identity transformation and $F_{\pi/2, \pi/2},$ is the two-dimensional Fourier transform. For other values of $\alpha$ and $\beta ,$ we obtain other interesting configurations of the transform. One immediate application of this transform is in time-frequency representation because of its close relationship to the Wigner distribution function. After a brief introduction to the coupled fractional Fourier transform, we shall discuss some of its properties and extensions to different functions spaces. |
| http://univie.ac.at/projektservice-mathematik/e/talks/Zayed_2022-02_Abstract_Strobl_22.pdf |
« back