m06 - Properties of the Fourier Transform
1.1
2006/08/01 12:59:15.188 GMT-5
2006/09/16 22:41:15.716 GMT-5
C.
Sidney
Burrus
csb@rice.edu
C.
Sidney
Burrus
csb@rice.edu
Doug
Kochelek
kochelek@rice.edu
Fourier
Fourier Integral
Fourier Transform
properties
The Fourier Transform has properties similar to those of the Fourier series.
Properties of the Fourier Transform
The properties of the Fourier transform are important in applying it to signal
analysis and to interpreting it. The main properties are given here using the
notation that the FT of a real valued function
x
(
t
)
over all time
t
is given by
ℱ
{
x
}
=
X
(
ω
)
.
-
Linear:
ℱ
{
x
+
y
}
=
ℱ
{
x
}
+
ℱ
{
y
}
-
Even and Oddness: if
x
(
t
)
=
u
(
t
)
+
j
v
(
t
)
and
X
(
ω
)
=
A
(
ω
)
+
j
B
(
ω
)
then
u
v
A
B
|
X
|
θ
even
0
even
0
even
0
odd
0
0
odd
even
0
0
even
0
even
even
π
/
2
0
odd
odd
0
even
π
/
2
-
Convolution: If continuous convolution is defined
by:
y
(
t
)
=
h
(
t
)
*
x
(
t
)
=
∫
−
∞
∞
h
(
t
−
τ
)
x
(
τ
)
ⅆ
τ
=
∫
−
∞
∞
h
(
λ
)
x
(
t
−
λ
)
ⅆ
λ
then
ℱ
{
h
(
t
)
*
x
(
t
)
}
=
ℱ
{
h
(
t
)
}
ℱ
{
x
(
t
)
}
-
Multiplication:
ℱ
{
h
(
t
)
x
(
t
)
}
=
1
2
π
ℱ
{
h
(
t
)
}
*
ℱ
{
x
(
t
)
}
-
Parseval:
∫
−
∞
∞
|
x
(
t
)
|
2
ⅆ
t
=
1
2
π
∫
−
∞
∞
|
X
(
ω
)
|
2
ⅆ
ω
-
Shift:
ℱ
{
x
(
t
−
T
)
}
=
X
(
ω
)
e
−
j
ω
T
-
Modulate:
ℱ
{
x
(
t
)
e
j
2
π
K
t
}
=
X
(
ω
−
2
π
K
)
-
Derivative:
ℱ
{
ⅆ
x
ⅆ
t
}
=
j
ω
X
(
ω
)
-
Stretch:
ℱ
{
x
(
a
t
)
}
=
1
|
a
|
X
(
ω
/
a
)
-
Orthogonality:
∫
−
∞
∞
e
−
j
ω
1
t
e
j
ω
2
t
=
2
π
δ
(
ω
1
−
ω
2
)