Exponentials and Laplace transforms on nonuniform time scales

Manuel D. Ortigueira, Delfim F M Torres, Juan J. Trujillo

Research output: Contribution to journalArticlepeer-review

18 Citations (Scopus)


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.

Original languageEnglish
Pages (from-to)252-270
Number of pages19
JournalCommunications In Nonlinear Science And Numerical Simulation
Publication statusPublished - 1 Oct 2016


  • Exponentials
  • Fractional derivatives
  • Generalised Laplace and Z transforms
  • Systems theory
  • Time-scale calculus


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