  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.
