After studying this chapter you will be able to

• define discrete-time and continuous-time signals;

• measure the “size” of a discrete-time signal using norms;

• use the z-transform to convert discrete-time signals to the complex z-plane;

• use the discrete-time Fourier transform to convert discrete-time signals to the frequency domain;

• describe the properties of the z-transform and the discrete-time Fourier transform;

• define a discrete-time state-space system;

• determine properties of discrete-time systems such as stability, controllability, observability, time invariance, and linearity;

• approximate a nonlinear system in the neighborhood of a certain operating point by a linear time-invariant system;

• check stability, controllability, and observability for linear time-invariant systems;

• represent linear time-invariant systems in different ways; and

• deal with interactions between linear time-invariant systems.

Introduction

This chapter deals with two important topics: signals and systems. A signal is basically a value that changes over time. For example, the outside temperature as a function of the time of the day is a signal. More specifically, this is a continuous-time signal; the signal value is defined at every time instant. If we are interested in measuring the outside temperature, we will seldom do this continuously. A more practical approach is to measure the temperature only at certain time instants, for example every minute. The signal that is obtained in that way is a sequence of numbers; its values correspond to certain time instants. Such an ordered sequence is called a discrete-time signal.