Signals and Systems - Electrical Engineering

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

FIGURE 2.11

Graphical convolution for a

unit-step input into an

averager withT= 1.

h(t−τ)

τ

0

1

t− 1 t

y(t)

t

01

1

x(τ)

τ

0

1

n Fort≥1, the overlap ofh(t−τ)andx(τ)remains constant, and as such the integral is unity

from then on, ory(t)=1 fort≥1. The response fort≥1 has attained steady state. Thus, the

complete response is given as

y(t)=r(t)−r(t− 1 )

wherer(t)=tu(t), the ramp function. n

nExample 2.13

Consider the graphical computation of the convolution integral of two pulses of the same duration

(see Figure 2.12).

FIGURE 2.12

Graphical convolution of two equal

pulses—that is, a system with input

x(t)=u(t)−u(t− 1 )and impulse

responseh(t)=x(t).

y(t)

t

01 2

1

h(t−τ)

τ

0

1

t− 1 t

x(τ)

τ

01

1