Page 385 - Fiber Optic Communications Fund

366 Fiber Optic Communications

where n jFI and n jFQ are the in-phase and quadrature components of the noise output of the matched filter,
H (), respectively. The variance of n jFI and n jFQ , j = 1, 0 is given by Eq. (8.207)
Ij
het
N E
2 2 2 0 1
≡ = = , j = 0, 1. (8.242)
F jFI jFQ 4
The pdf of the envelope r (T ) when s (t) is transmitted is given by the Rayleigh distribution
0 b 1
( 2 )
r 0 r 0
p (r |‘1’ sent)= exp − . (8.243)
r0 0 2 2
2
F F
The pdf of the envelope r (T ) is given by the Rician distribution
b
1
( 2 2 ) ( )
r 1 r + E ∕4 rE 1
1
p (r |‘1’ sent)= 2 exp − 2 I 0 2 . (8.244)
1
r1
2 2
F F F
If r (T ) > r (T ), it will be decided that ‘1’ is transmitted. Therefore, an error is made if r (T ) < r (T )
1
b
1
b
0
0
b
b
when s (t) is transmitted. So, the probability of mistaking ‘1’ as ‘0’ is
1
P(0|‘1’ sent)= P(r (T ) < r (T )|‘1’ sent). (8.245)
0
b
b
1
The probability that r (T ) < r (T ) can be found as follows. Since r (T ) and r (T ) are independent random
0
b
b
1
0
1
b
b
variables, the joint pdf of r (T ) and r (T ) can be written as
b
1
b
0
p (r , r |‘1’ sent)= p (r |‘1’ sent)p (r |‘1’ sent). (8.246)
r 1 r 0 1 0 r 0 0 r 1 1
The chance that r (T ) < r (T ) is the same as that r (T ) has a value r in the range 0 < r < ∞ and r (T )
1 b 0 b 1 b 1 1 0 b
has a value greater than r ,
1
∞ { ∞ }
( )
P(r (T ) < r (T )|‘1’ sent)= ∫ ∫ p r 1 r 0 r , r |‘1’ sent dr 1 dr . (8.247)
b
0
0
1
b
0
1
r 1 0
Using Eqs. (8.243), (8.244), and (8.246), Eq. (8.247) can be simplified as
∞ { ∞ }
P(0|‘1’ sent)= p (r ) p (r )dr 0 dr 1
1
∫ r 1 ∫ r 0
0
0 r 1
( )
∞ r 2
= p (r ) exp − 1 dr 1
1
∫ r 1 2
0 2
F
( ) ( )
∞ 2 2
1 r + E ∕8 r E
1 1
1
1
= r exp − I 0 dr . (8.248)
1
1
2 ∫ 0 2 2 2
F F F
Let
√
′
r = r 2, (8.249)
1 1
√
′
E = E ∕ 2. (8.250)
1
1