Digital Signal Processing Reference
In-Depth Information
2.
P2
: d(t) = d(-t) (even).
3.
P3
: x(t)*d(t) = x(t), or, more generally, x(t)*d(t - t
o
) = x(t - t
o
), where t
o
is a constant. That is, the convolution of a function x(t) with d(t) yields x(t).
Alternative Representations of the Delta Function
The Dirac delta function can also be defined as the limit of several even functions
that satisfy the above properties in the limit. These definitions include:
1. The limit of the weighted rectangular pulse (box), P
2a
(t) (see Fig.
1.8
, left):
1
2a
P
2a
ð
t
Þ¼
lim
1
2a
1
; j
t
j
a
0
; j
t
j
[ a
:
d
ð
t
Þ¼
lim
a
!
0
a
!
0
2. The limit of the weighted absolutely-decaying exponential:
1
2a
e
j
x
a
:
3. The limit of the weighted triangular pulse (see Fig.
1.8
, right):
d
ð
x
Þ¼
lim
a
!
0
1
2a
K
2a
ð
t
Þ¼
lim
1
a
1
j
t
j
a
; j
t
j
a
0
; j
t
j
[ a
:
d
ð
t
Þ¼
lim
a
!
0
a
!
0
1.2.1.3 The Unit Step Function
The unit step function is defined as:
u
ð
t
Þ¼
1
;
t [ 0
0
;
t\0
:
[1/(2a)]
Π
2a
(
t
)
(1/a)
Λ
2a
(
t
)
1/(2a)
1/a
t
t
− a
a
− a
a
Fig. 1.8
Weighted rectangular and triangular pulses
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