Journal
COMMUNICATIONS IN NONLINEAR SCIENCE AND NUMERICAL SIMULATION
Volume 39, Issue -, Pages 252-270Publisher
ELSEVIER
DOI: 10.1016/j.cnsns.2016.03.010
Keywords
Time-scale calculus; Exponentials; Generalised Laplace and Z transforms; Systems theory; Fractional derivatives
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Funding
- National Funds through the Foundation for Science and Technology of Portugal (FCT) [PEst-UID/EEA/00066/2013]
- CIDMA
- FCT [UID/MAT/04106/2013]
- government of Spain [MTM2013-41704-P]
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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. (C) 2016 Elsevier B.V. All rights reserved.
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