Page 202 - Fiber Optic Communications Fund

Optical Modulators and Modulation Schemes 183

Since
1
⟨b b ⟩ = if k = l
k l
4
= 0 otherwise (4.131)
and
⟨b ⟩ = 0, (4.132)
k
Eq. (4.130) reduces to
{ }
L L
1 1 ∑ ∑
2 2 i2f(l−k)T b
m
(f)= A |̃p(f)| + lim e . (4.133)
0
4T b L→∞ 4(2L + 1)T b l=−L k=−L
Using the following identities:
L L
∑ 1 ∑ l
lim e i2flT b = lim (f − ),
L→∞ L→∞ T T
l=−L b l=−L b
L
1 ∑
lim e −i2fkT b = 1, (4.134)
L→∞ 2L + 1
k=−L
Eq. (4.133) reduces to
2 2 { ∞ }
A |̃p(f)| 1 ∑ l
0
(f)= 1 + (f − ) . (4.135)
m
4T b T b l=−∞ T b

Example 4.6
A raised-cosine pulse is defined as
[ ( )] ( )
1 t t
p(t)= 1 + cos rect . (4.136)
2 T B 2T B
In a polar signaling scheme, raised-cosine pulses are used. Find the PSD.
Solution:
First let us calculate the Fourier transform of p(t).
[ ( )]
t
 rect = 2T sinc (2fT ), (4.137)
b
b
2T b
[ ( )] [ ]
t exp (i2f t)+ exp (−i2f t)
0
0
 x(t) cos =  x(t)
T 2
b
̃ x(f − f )+ ̃x(f + f )
0
0
= , (4.138)
2