Signals and Systems - Electrical Engineering

2.3 LTI Continuous-Time Systems 155

FIGURE 2.17

Echo system with two paths.

×

×

+

x(t)

Delay τ 1

Delay τ 2

α 2

α 1

y(t)

wherex(t)is the input, andαi,τi>0, fori=1 and 2, are attenuation factors and delays. Thus,

the output is the superposition of attenuated and delayed versions of the input. Typically, the

attenuation factors are less than unity. Is this system causal and BIBO stable?

Solution

Since the output depends only on past values of the input, the echo system is causal. To determine

if the system is BIBO stable we consider a bounded input signalx(t), and determine if the output

is bounded. Supposex(t)is bounded by a finite valueM, or|x(t)|<M<∞, for all times, which

means that the value ofx(t)cannot exceed an envelope [−M,M] at all times. This would also hold

when we shiftx(t)in time, so that

|y(t)|≤|α 1 ||x(t−τ 1 )|+|α 2 ||x(t−τ 2 )|<[|α 1 |+|α 2 |]M

so the corresponding output is bounded. The system is BIBO stable.

We can also find the impulse responseh(t)of the echo system, and show that it satisfies the abso-

lutely integrable condition of BIBO stability. Indeed, if we let the input of the echo system be

x(t)=δ(t)the output is

y(t)=h(t)=α 1 δ(t−τ 1 )+α 2 δ(t−τ 2 )

and the integral is

∫∞

−∞

|h(t)|dt=|α 1 |

∫∞

−∞

δ(t−τ 1 )dt+|α 2 |

∫∞

−∞

δ(t−τ 2 )dt=|α 1 |+|α 2 |<∞

n

nExample 2.18

Consider a positive feedback system created by a microphone close to a set of speakers that are

putting out an amplified acoustic signal (see Figure 2.18). The microphone picks up the input

signalx(t)as well as the amplified and delayed signalβy(t−τ),|β|≥1. Find the equation that

connects the inputx(t)and the outputy(t)and recursively from it obtain an expression fory(t)in

terms of past values of the input. Determine if the system is BIBO stable or not—usex(t)=u(t),

β=2, andτ=1 in doing so.