Exponentials and Laplace transforms on nonuniform time scales

• Published in: Communications in nonlinear science & numerical simulation 2016-10, Vol.39, p.252-270
• Main Authors: Ortigueira, Manuel D., Torres, Delfim F.M., Trujillo, Juan J.
• Format: Article
• Language: English
• Subjects: Difference equations; Exponentials; Fractional derivatives; Generalised Laplace and Z transforms; Laplace transforms; Mathematical analysis; Mathematical models; Sampling; Systems theory; Time; Time-scale calculus; Transforms
• ISSN: 1007-5704; 1878-7274
• DOI: 10.1016/j.cnsns.2016.03.010

•The paper deals with signals defined on non-uniformly spaced instants.•It presents a study of generalised exponentials.•With such exponentials new Laplace transforms are introduced.•This formulation allows the extending of fractional calculus to non-uniform time scales. 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.