Why is the Linear Canonical Transform so little known?

Abstract

The linear canonical transform (LCT), is the name of a parameterized continuum of transforms which include, as particular cases, the most widely used linear transforms and operators in engineering and physics such as the Fourier transform, fractional Fourier transform (FRFT), Fresnel transform (FRST), time scaling, chirping, and others. Therefore the LCT provides a unified framework for studying the behavior of many practical transforms and system responses in optics and engineering in general. From the system‐engineering point of view the LCT provides a powerful tool for design and analysis of the characteristics of optical systems. Despite this fact only few authors take advantage of the powerful and general LCT theory for analysis and design of optical systems. In this paper we review some important properties about the continuous LCT and we present some new results regarding the discretization and computation of the LCT.