TY - JOUR

T1 - Exponentials and Laplace transforms on nonuniform time scales

AU - Ortigueira, Manuel D.

AU - Torres, Delfim F M

AU - Trujillo, Juan J.

N1 - Foundation for Science and Technology (PEst-UID/EEA/00066/2013 ; UID/MAT/04106/2013)
government of Spain (MTM2013-41704-P)

PY - 2016/10/1

Y1 - 2016/10/1

N2 - 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.

AB - 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.

KW - Exponentials

KW - Fractional derivatives

KW - Generalised Laplace and Z transforms

KW - Systems theory

KW - Time-scale calculus

UR - http://www.scopus.com/inward/record.url?scp=84963705035&partnerID=8YFLogxK

U2 - 10.1016/j.cnsns.2016.03.010

DO - 10.1016/j.cnsns.2016.03.010

M3 - Article

AN - SCOPUS:84963705035

SN - 1007-5704

VL - 39

SP - 252

EP - 270

JO - Communications In Nonlinear Science And Numerical Simulation

JF - Communications In Nonlinear Science And Numerical Simulation

ER -