\( \mathbf{\mathcal{Z}} \) –Transformation in Discrete–Time LTI Systems

In this Chapter, we deal with the one-sided \( \mathcal{Z} \) –transformation, a very useful tool widely applied in digital filtering, for example, for solving difference input-output equations of discrete-time systems (see Chap. 3 ). We start with the definition and basic properties of the one-sided \( \mathcal{Z} \) –transformation. A table containing some useful \( \mathcal{Z} \) –transforms is also included. In the next part, using the convolution theorem, we introduce the transfer function of a digital filter and present formulas for transfer functions of FIR and IIR digital filters as well as different system interconnections presented in Chap. 3 . We also show the relation between the transfer function and the frequency response function defined in Chap. 7 . Then we present some basic criteria for the stability of digital filters.