Discrete-time fractional signals and systems

In this chapter, we formulate a coherent theory for discrete-time signals and systems taking two derivatives, namely the nabla (forward) and delta (backward), as basis. The eigenfunctions of such derivatives are the nabla and delta exponentials. With these eigenfunctions, two discrete-time Laplace transforms are introduced and their properties studied. These transforms are used to study the discrete-time linear systems defined by differential equations. The notions of impulse response and transfer function are introduced and discussed. Moreover, the Fourier transform and the frequency response are also considered. The framework is compatible with classic discrete-time signals and systems and allow for a uniform approximation of continuous systems when the sampling interval is reduced to zero.