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Note #2.
ContinuousTime
δ
(
t
) and
u
(
t
)
ECSE2410 Signals & Systems (Wozny)
Fall 2006
1.
The
delta (or impulse) function
,
)
(
t
,
is motivated as follows:
Form a finite pulse
1
then let
0
→
∆
thus
)
(
lim
)
(
0
t
t
∆
→
∆
=
Note the characteristics
of
)
(
t
:
(1)
zero width
(Occurs instantaneously, i.e., in zero time!)
(2)
infinite magnitude
(Not a function in the classical sense.
Sometimes called a “generalized”
function)
(3)
finite area
(Area =1)
Symbol
“fires” when argument is zero
i.e.
,
0
0
=

t
t
, or when
0
t
t
=
.
“Sampling property” of delta functions
.
Now find the area
,
A
, of the product graph,
)
(
1
t
x
∆
, as
0
→
∆
.
First form
∫
∫
∫
∆
∆
∆
∆

∆

∆
∞
∞

∆
=
=
=
2
2
2
2
)
(
)
(
)
(
)
(
1
1
dt
t
x
dt
t
x
dt
t
t
x
A
,
and taking the limit,
)
0
(
)
0
(
lim
)
0
(
lim
)
(
lim
lim
1
0
1
0
1
0
0
2
2
2
2
x
x
dt
x
dt
t
x
A
=
=
=
=
∆
∆
→
∆

∆
→
∆

∆
→
∆
→
∆
∫
∫
∆
∆
∆
∆
.
But
∫
∫
∞
∞

∞
∞

∆
→
∆
→
∆
=
=
dt
t
x
t
dt
t
x
t
A
)
(
)
(
)
(
)
(
lim
lim
0
0
1
Other functions could be used.
Theory of Distributions.
The square pulse was used by Dirac.
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This note was uploaded on 04/10/2008 for the course ECSE 2410 taught by Professor Wozny during the Spring '07 term at Rensselaer Polytechnic Institute.
 Spring '07
 WOZNY

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