Discrete-time differential systems

In this paper we formulate a coherent discrete-time signals and systems theory taking derivative concepts as basis. Two derivatives - nabla (forward) and delta (backward) - are defined and generalized to fractional orders, obtaining two formulations that are discrete versions of the well-known Grünwald-Letnikov derivatives. The eigenfunctions of such derivatives are the so-called nabla and delta exponentials. With these exponentials two generalized discrete-time Laplace transforms are deduced and their properties studied. These transforms are back compatible with the current Laplace and Z transforms. They are used to study the discrete-time linear systems defined by differential equations. These equations although discrete mimic the usual continuous-time equations that are uniformly approximated when the sampling interval becomes small. Impulse response and transfer function notions are introduced and obtained. The Fourier transform and the frequency response are also considered. This implies a unified mathematical framework that allows us to approximate the classic continuous-time case when the sampling rate is high or obtain the current discrete-time case based on difference equation.