Digital Signal Processing Reference
In-Depth Information
h
(
t
)=
Π
3
(
t
−1.5)
x
(
t
)=3 exp(−2
t
)
u
(
t
)
3
1
t
, sec
t
, sec
−5
0
3
5
−5
0
3
5
A plot of signals h(t) = P
3
(t - 1.5) and x(t) = 3e
-2t
Fig. D.1
u(t)
Task 2
Write a code to invert the function x(t) around the y-axis to get x(- t). Then a
code to find the shifted versions x(t + t
o
) and x(- t + t
o
). Take t
o
= 3 and -3. Plot
all functions on the same graph. Comment on these operations.
Task 3
The Dirac delta function d(t) is an important tool in signal processing. Roughly
speaking, it is equivalent to a pulse of infinite height and very narrow width
(approaching zero). It is defined by the following integral:
Z
1
g
ð
t
Þ
d
ð
t
t
o
Þ
dt
¼
g
ð
t
o
Þ
1
where g(t) is a continuous function, t
o
is a constant. The delta function has the
following properties:
P1:
R
1
1
d
ð
t
Þ
dt
¼
1 (unit area),
P2: d(t) = d(-t) (even),
P3:
x(t)*d(t) = x(t),
or,
generally,
x(t)*d(t - t
o
) = x(t - t
o
),
where
t
o
is
a
constant.
The Dirac delta function can also be defined as the limit of several even
functions that can satisfy the above properties in the limit. These definitions
include:
1. Limit of the weighted rectangular pulse (box), P
2a
ð
t
Þ
(see Fig.
D.2
, left):
1
2a
P
2a
ð
t
Þ¼
lim
1
2a
1
; j
t
j
a
0
; j
t
j
[ 0
d
ð
t
Þ¼
lim
a
!
0
ð
a
!
0
Þ
2. Limit of the weighted absolutely-decaying exponential:
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