Signals and Systems - Electrical Engineering

360 C H A P T E R 6: Application to Control and Communications

frequency, bandwidth, spectrum, and modulation developed by means of the Fourier transform are

fundamental in the analysis and design of communication systems.

The aim of this chapter is to serve as an introduction to problems in classical control and commu-

nications and to link them with the Laplace and Fourier analyses. More in-depth discussion of these

topics can be found in many excellent texts in control and communications.

The other topic covered in this chapter is an introduction to analog filter design. Filtering is a very

important application of LTI systems in communications, control, and digital signal processing.

The material in this chapter will be complemented by the design of discrete filters in Chapter 11.

Important issues related to signals and system are illustrated in the design and implementation of

filters.

6.2 System Connections and Block Diagrams

Control and communication systems consist of interconnection of several subsystems. As we

indicated in Chapter 2, there are three important connections of LTI systems:

n Cascade

n Parallel

n Feedback

Cascade and parallel result from properties of the convolution integral, while the feedback con-

nection relates the output of the overall system to its input. With the background of the Laplace

transform we present now a transform characterization of these connections that can be related to

the time-domain characterizations given in Chapter 2.

The connection of two LTI continuous-time systems with transfer functionsH 1 (s)andH 2 (s)(and correspond-

ing impulse responsesh 1 (t)andh 2 (t)) can be done in:

n Cascade (Figure 6.1(a)): Provided that the two systems are isolated, the transfer function of the overall

system is

H(s)=H 1 (s)H 2 (s) (6.1)

n Parallel (Figure 6.1(b)): The transfer function of the overall system is

H(s)=H 1 (s)+H 2 (s) (6.2)

n Negative feedback (Figure 6.4): The transfer function of the overall system is

H(s)=

H 1 (s)

1 +H 2 (s)H 1 (s)

(6.3)

n Open-loop transfer function:Ho`(s)=H 1 (s).

n Closed-loop transfer function:Hc`(s)=H(s).