The Dirac delta function, often referred to as
the unit impulse or delta function, is the function that
defines the idea of a unit impulse in continuoustime. Informally, this function is one that
is infinitesimally narrow, infinitely tall, yet integrates to
one. Perhaps the simplest way to visualize
this is as a rectangular pulse from
a−ε2
a
ε
2
to
a+ε2
a
ε
2
with a height of
1ε
1
ε
.
As we take the limit of this setup as
ε
ε approaches 0, we see that the width tends to zero and the height
tends to infinity as the total area remains constant at one.
The impulse function is often written as
δt
δt
.
∫−∞∞δtd
t
=1
t
δ
t
1
(1)Below is a brief list a few important properties of
the unit impulse without going into detail of their proofs.

δαt=1αδt
δ
α
t
1
α
δ
t

δt=δ−t
δ
t
δ
t

δt=dd
t
ut
δ
t
t
u
t
, where
ut
u
t
is the unit step.

f
(
t
)
δ
(
t
)
=
f
(
0
)
δ
(
t
)
f
(
t
)
δ
(
t
)
=
f
(
0
)
δ
(
t
)
The last of these is especially important as it gives rise to the sifting property of the dirac delta function, which selects the value of a function at a specific time and is especially important in studying the relationship of an operation called convolution to time domain analysis of linear time invariant systems. The sifting property is shown and derived below.
∫

∞
∞
f
(
t
)
δ
(
t
)
d
t
=
∫

∞
∞
f
(
0
)
δ
(
t
)
d
t
=
f
(
0
)
∫

∞
∞
δ
(
t
)
d
t
=
f
(
0
)
∫

∞
∞
f
(
t
)
δ
(
t
)
d
t
=
∫

∞
∞
f
(
0
)
δ
(
t
)
d
t
=
f
(
0
)
∫

∞
∞
δ
(
t
)
d
t
=
f
(
0
)
(2)
"My introduction to signal processing course at Rice University."