Graphics Programs Reference
In-Depth Information
The spectrum of the signal is determined from its complex envelope
. The complex exponential term in Eq. (3.34) introduces a frequency shift
about the center frequency
s
1
()
s
()
. Taking the FT of
yields
f
o
s
()
τ
---
∫
∞
∫
j
2πµ
t
2
2
-
e
j
πµ
t
2
t
j
ω
t
j
ω
t
S
()
Rect
=
e
d
=
exp
-----------------
e
d
(3.36)
∞
τ
---
Let
, and perform the change of variable
µ′
=
2πµ
=
2π
B
τ
⁄
µ′
-----
t
ω
µ′
µ′
-----
x
=
-----
;
dx
=
dt
(3.37)
Thus, Eq. (3.36) can be written as
x
2
∫
π
µ′
j
ω
2
e
j
π
x
2
⁄
2µ′
⁄
2
S
()
=
-----
e
d
(3.38)
x
1
x
2
∫
x
1
j
ω
2
e
j
π
x
2
e
j
π
x
2
π
µ′
⁄
2µ′
⁄
2
⁄
2
∫
S
()
=
-----
e
d
d
(3.39)
0
0
where
µ′
-----
τ
---
ω
µ′
B
τ
2
f
x
1
=
+
-----
=
------
1
+
----------
(3.40)
B
⁄
2
µ′
-----
τ
---
ω
µ′
B
τ
2
f
x
2
=
-----
=
------
1
----------
(3.41)
B
⁄
2
The Fresnel integrals, denoted by
and
, are defined by
C
()
S
()
x
∫
πυ
2
2
---------
C
()
=
cos
d
(3.42)
0
x
∫
πυ
2
2
S
()
=
sin
---------
d
(3.43)
0
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