Databases Reference
In-Depth Information
Δ
t
F I GU R E 12 . 4
A function of time.
limit of this function as
t
goes to zero is called the
Dirac delta function
,or
impulse function
,
and is denoted by
δ(
t
)
:
t
rect
(
t
)
lim
=
δ(
t
)
(31)
t
t
→
0
Therefore,
L
[
f
(
t
)
]=
lim
t
→
0
L
[
f
S
(
t
)
]=
f
(τ )
L
[
δ(
t
−
τ)
]
d
τ
(32)
where
n
t
goes to zero. Denote the response of the system
L
to an impulse,
or the
impulse response
,by
h
t
goes to
τ
as
(
t
)
:
h
(
t
)
=
L
[
δ(
t
)
]
(33)
Then, if the system is also time invariant,
L
[
f
(
t
)
]=
f
(τ )
h
(
t
−
τ)
d
τ
(34)
Using the convolution theorem, we can see that the Fourier transform of the impulse response
h
.
The Dirac delta function is an interesting function. In fact, it is not clear that it is a function
at all. It has an integral that is clearly one, but at the only point where it is not zero, it is
undefined! One property of the delta function that makes it very useful is the
sifting
property:
t
2
(
t
)
is the transfer function
H
(ω)
f
(
t
0
)
t
1
t
0
t
2
f
(
t
)δ(
t
−
t
0
)
dt
=
(35)
0
otherwise
t
1
12.6.4 Filter
The linear systems of most interest to us will be systems that permit certain frequency com-
ponents of the signal to pass through, while attenuating all other components of the signal.
Such systems are called
filters
. If the filter allows only frequency components below a certain
