Graphics Programs Reference
In-Depth Information
13.2. The Fourier Transform
The Fourier Transform (FT) of the signal
x
()
is
∞
∫
j
ω
t
Fx
({}
X
()
x
()
e
=
=
d
(13.6)
∞
or
∞
∫
j
2π
ft
Fx
({}
X
()
=
=
x
()
e
d
(13.7)
∞
and the Inverse Fourier Transform (IFT) is
∞
∫
1
2π
1
X
()
e
j
ω
t
------
F
{
X
()
}
=
x
()
=
d
ω
(13.8)
∞
or
∞
∫
1
X
()
e
j
2π
ft
F
X
({}
x
()
=
=
d
f
(13.9)
∞
where, in general, represents time, while and represent fre-
quency in radians per second and Hertz, respectively. In this topic we will use
both notations for the transform, as appropriate (i.e.,
t
ω
=
2π
f
f
X
()
X
()
and
).
A detailed table of the FT pairs is listed in Appendix 13A. The FT properties
are (the proofs are left as an exercise):
Linearity:
1.
Fa
1
x
1
{
()
a
2
x
2
+
()
}
=
a
1
X
1
()
a
2
X
2
()
+
(13.10)
Symmetry: If
Fx
({}
X
()
=
then
2.
∞
∫
j
ω
t
2π
X
ω
(
)
=
X
()
e
d
(13.11)
∞
Shifting: For any real time
t
0
3.
±
j
ω
t
0
Fxt t
0
{
(
±
)
}
=
e
X
()
(13.12)
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