Signals and Systems - Electrical Engineering

88 C H A P T E R 1: Continuous-Time Signals

1.4.2 Unit-Step, Unit-Impulse, and Ram Signals

Unit-Step and Unit-Impulse Signals

Consider a rectangular pulse of duration 1 and unit area

p 1 (t)=







1

1

−1/ 2 ≤t≤1/ 2

0 t<−1/2 andt> 1/ 2

(1.22)

Its integral is

u 1 (t)=

∫t

−∞

p 1 (τ)dτ=











1 t> 1/ 2

1

1

(

t+

1

2

)

−1/ 2 ≤t≤1/ 2

0 t<−1/ 2

(1.23)

The pulsep 1 (t)and its integralu 1 (t)are shown in Figure 1.6.

Suppose that 1 →0, then

n The pulsep 1 (t)still has a unit area but is an extremely narrow pulse. We will call the limit the

unit-impulsesignal,

δ(t)=lim

1 → 0

p 1 (t) (1.24)

which is zero for all values oftexcept att=0 when its value is not defined.

n The integralu 1 (t), as 1 →0 has a left-side limit ofu 1 (−)→0 and a right-side limit ofu 1 ()→

1, for some infinitesimal >0, and att=0 it is 1/2. Thus, the limit is

lim

1 → 0

u 1 (t)=







1 t> 0

1 / 2 t= 0

0 t< 0

(1.25)

FIGURE 1.6

Generation ofδ(t)andu(t)from limit as 1 → 0 of a

pulsep 1 (t)and its integralu 1 (t).

pΔ(t)

−Δ/2 Δ/2

1/Δ

t t

t t

1

0.5

δ(t) u(t)

1

(1)

uΔ(t)

−Δ/2 Δ/2