Page 389 - Fiber Optic Communications Fund

370 Fiber Optic Communications

When a bit ‘0’ is transmitted, if I > I , the bit ‘0’ is mistaken as the bit ‘1’ and this probability is
T
∞ 1 ∞ ( I )
P(1|0)= p (I)dI = exp − dI
0
∫ 2 ∫ 2 2
2
I T I T
( )
I T
= exp − . (8.266)
2 2
Using Eqs. (8.260) and (8.265), we obtain
[ DD ( )]
2
P(1|0)= exp − 1 + , (8.267)
2 DD
where DD is given by
E av
DD
= . (8.268)
ASE
For OOK, E = E ∕2. So
av
1
E 1
DD
= , (8.269)
2
ASE
When DD ≫ 1, P(1|0)≅ exp (− DD ∕2).
Similarly, the probability of mistaking bit ‘1’ as bit ‘0’ is
I T
P(0|1)= p (I)dI
1
∫
0
√
( 2 ) ⎛ IRE 2 ⎞
1 I T RE + I ⎜ 1 ⎟
1
= exp − I dI. (8.270)
2 ∫ 0 2 2 0 ⎜ ⎜ 2 ⎟
2
⎟
⎝ ⎠
Changing the variable of integration from I to x, where
I
2
x = , (8.271)
2
2
2
2
and letting a = RE ∕ , Eq. (8.270) becomes
1
√
I T ∕ ( a + x 2 )
2
P(0|1)= x exp − I (ax) dx
∫ 2 0
0
( √ )
I T
= 1 − Q 1 a, (8.272)

√
where Q (a, I ∕) is the generalized Marcum’s Q-function given by Eq. (8.228). Using Eqs. (8.260),
1 T
(8.265), and (8.268), Eq. (8.272) can be rewritten as
√
( )
⎛ ⎞
√ 2
P(0|1)= 1 − Q 1 ⎜ 2 DD , DD 1 + ⎟ (8.273)
⎜ DD ⎟
⎝ ⎠