Signals and Systems - Electrical Engineering

156 C H A P T E R 2: Continuous-Time Systems

FIGURE 2.18

Positive feedback system: the microphone picks

up input signalx(t)and the amplified and

delayed signalβy(t−τ), making the system

unstable.

+

×

x(t)

βy(t−τ)

Delay τ

y(t)

β

Solution

The input–output equation is

y(t)=x(t)+βy(t−τ)

If we use this expression to obtainy(t−τ), we get that

y(t−τ)=x(t−τ)+βy(t− 2 τ)

and replacing it in the input–output equation, we get

y(t)=x(t)+β[x(t−τ)+βy(t− 2 τ)]=x(t)+βx(t−τ)+β^2 y(t− 2 τ)

Repeating the above scheme, we will obtain the following expression fory(t)in terms of the input

y(t)=x(t)+βx(t−τ)+β^2 x(t− 2 τ)+β^3 x(t− 3 τ)+···

If we letx(t)=u(t)andβ=2, the corresponding output is

y(t)=u(t)+ 2 u(t− 1 )+ 4 u(t− 2 )+ 8 u(t− 3 )+···

which continuously grows as time increases. The output is clearly not a bounded signal, although

the input is bounded. Thus, the system is unstable, and the screeching sound from the speakers

will prove it—you need to separate the speakers and the microphone to avoid it. n

2.4 What Have We Accomplished? Where Do We Go from Here?....................

By now you should have begun to see the forest for the trees. In this chapter we connected signals

with systems. Especially, we initiated the study of linear time-invariant dynamic systems. As you will

learn throughout your studies, this model is of great use in representing systems in many engineering

applications. The appeal is its simplicity and mathematical structure. We also indicated some practi-

cal properties of systems such as causality and stability. Simple yet significant examples of systems,

ranging from the vocal system to simple RLC circuits, illustrate the use of the LTI model and point

to its practical application. At the same time, modulators also show that more complicated systems

need to be explored to be able to communicate wirelessly. Finally, you were given a system’s approach