Exponentials and Laplace transforms on nonuniform time scales

TL;DR

This paper develops a unified framework for signals and systems on nonuniform time scales using nabla and delta derivatives, introducing generalized Laplace transforms compatible with classical transforms, and analyzing discrete-time systems with difference equations.

Contribution

It introduces a coherent approach to signals and systems on nonuniform time scales, including generalized Laplace transforms and eigenfunctions, bridging continuous and discrete-time theories.

Findings

Generalized Laplace transforms compatible with classical ones.

Eigenfunctions called nabla and delta exponentials.

Unified framework for continuous and discrete-time systems.

Abstract

We formulate a coherent approach to signals and systems theory on time scales. The two derivatives from the time-scale calculus are used, i.e., nabla (forward) and delta (backward), and the corresponding eigenfunctions, the so-called nabla and delta exponentials, computed. With these exponentials, two generalised discrete-time Laplace transforms are deduced and their properties studied. These transforms are compatible with the standard Laplace and $Z$ transforms. They are used to study discrete-time linear systems defined by difference equations. These equations mimic the usual continuous-time equations that are uniformly approximated when the sampling interval becomes small. Impulse response and transfer function notions are introduced. This implies a unified mathematical framework that allows us to approximate the classic continuous-time case when the sampling rate is high or to obtain the standard discrete-time case, based on difference equations, when the time grid becomes uniform.