MASSACHUSETTS INSTITUTE OF TECHNOLOGY
DEPARTMENT OF MECHANICAL ENGINEERING
2.14 Analysis and Design of Feedback Control Sysytems
The Dirac Delta Function and Convolution
1
The Dirac Delta (Impulse) Function
The Dirac delta function is a nonphysical, singularity function with the following deFnition
δ
(
x
)=
(
0f
o
r
x
6
=0
undeFned
at
x
(1)
but with the requirement that
Z
∞
−∞
δ
(
x
)
dx
=1
,
(2)
that is, the function has unit area.
0 TT
T
T
1 2
3
4
a) Unit pulses of different extents
b) The impulse function
1/T
1
1/T
2
1/T
3
1/T
4
t
d
(t)
0
t
d
(x)
X
±igure 1: Unit pulses and the Dirac delta function.
±igure 1 shows a
unit pulse
function
δ
T
(
t
), that is a brief rectangular pulse function of duration
T
, deFned to have a constant amplitude 1
/T
over its extent, so that the area
T
×
1
/T
under the
pulse is unity:
δ
T
(
t
o
r
t
≤
0
1
/T
0
<t
≤
T
o
r
t>
0.
(3)
The Dirac delta function (also known as the impulse function) can be deFned as the limiting form
of the unit pulse
δ
T
(
t
) as the duration
T
approaches zero. As the duration
T
of
δ
T
(
t
) decreases,
the amplitude of the pulse increases to maintain the requirement of unit area under the function,
and
δ
(
t
) = lim
T
→
0
δ
T
(
t
)
.
(4)
The impulse is therefore deFned to exist only at time
t
= 0, and although its value is strictly
undeFned at that time, it must tend toward inFnity so as to maintain the property of unit area in
the limit. The
strength
of a scaled impulse
Kδ
(
t
) is deFned by its area
K
.
The limiting form of many other functions may be used to approximate the impulse. Common
functions include triangular, gaussian, and sinc (sin(
x
)/
x
) functions.
1
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View Full DocumentThe impulse function is used extensively in the study of linear systems, both spatial and tem
poral. Although true impulse functions are not found in nature, they are approximated by short
duration, high amplitude phenomena such as a hammer impact on a structure, or a lightning strike
on a radio antenna. As we will see below, the response of a causal linear system to an impulse
deFnes its response to all inputs.
An impulse occurring at
t
=
a
is
δ
(
t
−
a
).
1.1
The “Sifting” Property of the Impulse
When an impulse appears in a product within an integrand, it has the property of ”sifting” out
the value of the integrand at the point of its occurrence:
Z
∞
−∞
f
(
t
)
δ
(
t
−
a
)
dt
=
f
(
a
)(
5
)
This is easily seen by noting that
δ
(
t
−
a
) is zero except at
t
=
a
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 Spring '10
 WPP
 Digital Signal Processing, LTI system theory, Impulse response, Dirac delta function

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