CHAPTER 5

REPRESENTING SIGNALS AND SYSTEMS

5.1 CHAPTER OBJECTIVES

On completion of this chapter, the reader should be able to:

1. explain how sampled signals are generated, and be able to generate sampled waveforms using an equation.

2. explain the relationship between continuous and discrete frequency.

3. derive the z transform of a signal.

4. convert from a z transform to a difference equation.

5. determine the stability of a given system.

6. explain the relationship between a pole–zero plot and a system’s frequency response.

7. explain what is meant by convolution, and how to calculate a system’s response using convolution.

5.2 INTRODUCTION

We now turn our attention from analyzing a given sampled signal to synthesizing (generating) a sampled signal. Many applications depend on the ability to generate a signal: One important application area is to generate signals for communication systems. Once the theory for generating signals is introduced, it will be possible to analyze systems which alter a signal as it passes through a system. These two operations—analysis and synthesis—may be viewed as the inverse of each other, and much of the same theory applies equally well to either.

5.3 DISCRETE-TIME WAVEFORM GENERATION

Consider a sine wave as a point on a circle, sampled at discrete angular increments as shown in Figure 5.1. The path traced out on the horizontal axis is sinusoidal. As will be seen in Chapter 7, a fundamental principle of signal processing called “Fourier’s ...