### 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.

Original language | English |
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Pages (from-to) | 252-270 |

Number of pages | 19 |

Journal | Communications In Nonlinear Science And Numerical Simulation |

Volume | 39 |

DOIs | |

Publication status | Published - 1 Oct 2016 |

### Keywords

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

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## Cite this

*Communications In Nonlinear Science And Numerical Simulation*,

*39*, 252-270. https://doi.org/10.1016/j.cnsns.2016.03.010