ABSTRACT
This chapter is devoted to the study of the linear canonical transformations (LCT) which may be viewed as a group of unitary transformations acting on the space of square-integrable functions.
The linear canonical transforms are generalizations of several integral transforms, including the fractional Fourier transform. Unlike the fractional Fourier transform which depends on one angle, the linear canonical transformations depend on four parameters, a,b,c,d, with ad-bc=1. These transformations have been the focus of many research articles in recent years because of their applications in optics, radar system analysis, and signal processing.
The chapter introduces the linear canonical transforms and another variant of them known as the offset linear canonical transforms and discusses in detail some of their important properties, such as additivity, eigenfunctions, convolution, Poisson summation formula. The extension of the transforms to higher dimensions is accomplished through their metaplectic representations. The chapter is concluded with a presentation of a sampling theorem for signals bandlimited to a disk in the linear canonical transformation domain.
