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 -