Extended Bilateral transforms and their applications

Abstract

A generalisation of the Dirac-delta function and its family of derivatives recently proposed as a means of introducing impulses on the complex plane in Laplace and z transform domains is shown to extend the applications of Bilateral Laplace and z transforms. Transforms of two-sided signals and sequences are made possible by a extending the domain of distributions to cover generalized functions of complex variables. The domains of Bilateral Laplace and z transforms are shown to extend to two-sided exponentials and fast-rising functions, which, without such generalized impulses have no transform. Applications include generalized forms of the sampling theorem, a new type of spatial convolution on the s and z planes and solutions of differential and difference equations with two-sided infinite duration forcing functions and sequences.