Databases Reference
In-Depth Information
12.5.3 Convolution Theorem
When we examine the relationships between the input and output of linear systems, we will
encounter integrals of the following forms:
∞
f
(
t
)
=
f
1
(τ )
f
2
(
t
−
τ)
d
τ
−∞
or
∞
f
(
t
)
=
f
1
(
t
−
τ)
f
2
(τ )
d
τ
−∞
These are called convolution integrals. The convolution operation is often denoted as
f
(
t
)
=
f
1
(
t
)
⊗
f
2
(
t
)
The convolution theorem states that if
F
(ω)
=
F
[
f
(
t
)
]=
F
[
f
1
(
t
)
⊗
f
2
(
t
)
]
,
F
1
(ω)
=
F
[
f
1
(
t
)
]
, and
F
2
(ω)
=
F
[
f
2
(
t
)
]
, then
F
(ω)
=
F
1
(ω)
F
2
(ω)
We can also go in the other direction. If
F
(ω)
=
F
1
(ω)
⊗
F
2
(ω)
=
F
1
(σ )
F
2
(ω
−
σ)
d
σ
then
f
(
t
)
=
f
1
(
t
)
f
2
(
t
)
As mentioned earlier, this property of the Fourier transform is important because the
convolution integral relates the input and output of linear systems, which brings us to one
of the major reasons for the popularity of the Fourier transform. We have claimed that the
Fourier series and Fourier transformprovide us with an alternative frequency profile of a signal.
Although sinusoids are not the only basis set that can provide us with a frequency profile, they
do have an important property that helps us study linear systems, which we describe in the
next section.
12.6 Linear Systems
A linear system is a system that has the following two properties:
Homogeneity:
Suppose we have a linear system
L
with input
f
(
t
)
and output
g
(
t
)
:
g
(
t
)
=
L
[
f
(
t
)
]
If we have two inputs,
f
1
(
t
)
and
f
2
(
t
)
, with corresponding outputs,
g
1
(
t
)
and
g
2
(
t
)
, then
the output of the sum of the two inputs is simply the sum of the two outputs:
L
[
f
1
(
t
)
+
f
2
(
t
)
]=
g
1
(
t
)
+
g
2
(
t
)
