Page 326 - Fiber Optic Communications Fund

Transmission System Design 307

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intensity noise (RIN), and imperfections in 90 hybrid. In the presence of shot noise, Eqs. (5.109) and (5.110)
are modified as (with = 0, Δ = ∕2)
IF
R 2 2
I = {A |s(t)| + |A LO | + 2A A Re{s(t)}}+ n shot +, (7.30)
r
r LO
+
2
R 2 2
I = {A |s(t)| + |A LO | − 2A A Re{s(t)}}+ n shot −, (7.31)
−
r LO
r
2
−
+
where n shot + and n shot − are the shot noise introduced by PD and PD , respectively. Subtracting Eq. (7.30)
from Eq. (7.31), we have
I = I − I = 2RA A Re{s(t)} + n shot + − n shot −. (7.32)
+
−
r LO
Let
n shot = n shot + − n shot −. (7.33)
Since n shot + and n shot − are statistically independent, the variance of n shot is the sum of the variances of n shot +
and n shot −:
2 = 2 + + 2 −
shot shot shot
= 2qI B + 2qI B . (7.34)
+ e
− e
Let
P = A 2 , (7.35)
LO LO
2 2
P (t)= A s (t). (7.36)
r
r
Here, P LO and P are the LO and receiver power, respectively. First consider the OOK. For bit ‘1’ (s(t)= 1
r
within the bit slot), the mean and variances of the current are
√
I = 2R P P , (7.37)
1 1r LO
2
P = A , (7.38)
1r r
2 = 2qI B + 2qI B , (7.39)
1,shot 1+ e 1− e
R { √ }
I 1+ = P + P LO + 2 P P , (7.40)
1r LO
1r
2
R { √ }
I = P + P − 2 P P . (7.41)
1− 1r LO 1r LO
2
Using Eqs. (7.40) and (7.41), Eq. (7.39) becomes
2
= 2qB R(P + P ). (7.42)
1,shot e 1r LO
The total noise variance of ‘1’ is
2
= 2 + 2
1 1,shot 1,thermal
= 2qB R(P + P LO )+ 4k TB ∕R . (7.43)
e
1r
L
e
B
Similarly, for bit ‘0’, we have
I = 0, (7.44)
0