From infinitely continuous to finite discrete

Abstract

In previous chapters we developed concepts which describe the properties of continuous-time LTI systems, such as is shown in the analog part of Fig. 6.16. On the other hand, when we used PITSA to demonstrate these properties we were, of course, working with sequences of numbers. This was acceptable in the context of the examples we used. In a general context, however, we must be well aware of several important differences between discrete and continuous systems. Our intuitive grasp of the more important system properties must by now be firm enough that we can extend our view using a more formal approach. In this fashion we will not only acquire additional tools for data processing, but also we will gain some insight into the links between systems defined for infinite continuous-time signals and systems defined for finite discrete-time signals. The focus of this chapter will be on consequences of this transition for the concepts which we developed so far. We will see for example that the transition from continuous-time to discrete-time systems corresponds to a transition from aperiodic to periodic Fourier spectra. This property for example will allow us to take a new look at the aliasing problem. Furthermore, we will meet the z-transform for discrete sequences, the discrete counterpart of the Laplace transform. We will see that the concepts introduced for continuous-time systems, such as the transfer function or the concept of poles and zeros, remains valid for discrete-time systems as well. Even more important, we will see that most of the relevant system properties can intuitively be understood from their analog counterparts. In the following the sequence notation of Oppenheim and Schafer (1989) will be used.