The unit impulse is very useful in the analysis of signals, linear systems, and sampling. Consider the plot of a rectangular pulse in Figure 1. Note the height of the pulse is 1/τ1/τ and the width of the pulse is ττ. So we can write
∫
-
∞
∞
x
p
(
t
)
d
t
=
1
∫
-
∞
∞
x
p
(
t
)
d
t
=
1
(1)As we let ττ get small, then the width of the pulse gets successively narrower and its height gets progressively higher. In the limit as ττ approaches zero, we have a pulse which has infinite height, and zero width, yet its area is still one. We define the unit impulse function as
δ
(
t
)
≡
lim
τ
→
0
x
p
(
t
)
δ
(
t
)
≡
lim
τ
→
0
x
p
(
t
)
(2)The area under δ(t)δ(t) is one, and so we can write
∫
-
∞
∞
δ
(
t
-
τ
)
d
t
=
1
∫
-
∞
∞
δ
(
t
-
τ
)
d
t
=
1
(3)If we multiply the unit impulse by a constant, KK, its area is now equal to that constant, i.e.
∫
-
∞
∞
K
δ
(
t
-
τ
)
d
t
=
K
∫
-
∞
∞
K
δ
(
t
-
τ
)
d
t
=
K
(4)The area of the unit impulse is usually indicated by the number shown next to the arrow as seen in Figure 2.
Suppose we multiply the signal x(t)x(t) with a time-shifted unit impulse, δ(t-τ)δ(t-τ). The product is a unit impulse, having an area of x(τ)x(τ). This is illustrated in Figure 3.
In other words,
∫
-
∞
∞
x
(
t
)
δ
(
t
-
τ
)
d
t
=
x
(
τ
)
∫
-
∞
∞
x
(
t
)
δ
(
t
-
τ
)
d
t
=
x
(
τ
)
(5)Equation Equation 5 is called the sifting property of the unit impulse. As we will see, the sifting property of the unit impulse will be very useful.
