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Note #2.
ContinuousTime
δ
(
t
) and
u
(
t
)
ECSE2410 Signals & Systems (Wozny)
1.
The
delta distribution (impulse “function”)
,
)
(
t
. Motivation: the limit of a sequence of functions:
Note#2.
p.1
Form a finite pulse
1
then let
0
→
Δ
thus
)
(
lim
)
(
0
t
t
Δ
→
Δ
=
0
)
(
t
t
)
(
t
Δ
0
2
Δ
2
Δ
−
Δ
1
1
t
)
(
t
Δ
0
Δ
1
t
Note the characteristics
of
)
(
t
:
(1)
zero width
(Occurs instantaneously, i.e., in zero time!)
(2)
infinite magnitude
(Consequently, it is
not
a function according to the classical definition of a
function.
However, engineers still call it a function, or sometimes called a “generalized”
function)
(3)
finite area
(Area =1)
Symbol for a “shifted” delta distribution with area=1
:
“Sampling property” of delta distribution
.
Now find the area
,
A
, of the product graph,
)
(
)
(
)
(
1
t
x
t
x
t
Δ
Δ
=
⋅
, as
0
→
Δ
.
First form
∫
∫
Δ
Δ
−
Δ
∞
∞
−
Δ
=
=
2
2
)
(
)
(
)
(
1
dt
t
x
dt
t
t
x
A
,
and taking the limit,
()
)
0
(
)
0
(
lim
)
(
lim
lim
1
0
1
0
0
2
2
x
x
dt
t
x
A
=
=
=
Δ
Δ
→
Δ
−
Δ
→
Δ
→
Δ
∫
Δ
Δ
.
But
∫
∫
∫
∞
∞
−
∞
∞
−
Δ
→
Δ
∞
∞
−
Δ
→
Δ
→
Δ
=
=
=
dt
t
x
t
dt
t
x
t
dt
t
x
t
A
)
(
)
(
)
(
)
(
lim
)
(
)
(
lim
lim
0
0
0
1
Functions other than pulses will also work, as shown later.
The pulse function was originally used by Dirac.
Rigorous proof
of the delta distribution led to a mathematical theory of distributions.
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This note was uploaded on 05/03/2009 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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