Page 381 - Fiber Optic Communications Fund

362 Fiber Optic Communications

The total output of the matched filter is
( )
s (t)
F
I (t)+ n (t)= + n (t) cos + n (t) sin
F F FI FQ
2
√
[ ] 2
s (t)
F
= + n (t) + n 2 FQ (t) ⋅ cos ( − ), (8.208)
FI
2
⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏞⏟
envelope
where
= (t − T )+Δ, (8.209)
IF b
{ }
n FQ (t)
= tan −1 . (8.210)
s (t)∕2 + n (t)
F FI
After passing through the envelope detector, the output sample at t = T is proportional to the envelope:
b
√
2
r(T )= [s (T )∕2 + n (T )] + n 2 FQ (T ). (8.211)
b
b
FI
b
b
F
When a bit ‘0’ is transmitted, s (T )= 0. Therefore,
F b
√
2
r(T )= n (T )+ n 2 (T ). (8.212)
b FI b FQ b
For a narrow-band noise process, it can be shown that the variances of the in-phase component n (t) and
FI
the quadrature component n FQ (t) are the same as for the narrow-band noise n (t)[2]. Therefore, n (T ) and
F
FI
b
2
n (T ) are Gaussian random variables with variance given by Eq. (8.207). The pdf of the envelope when
FQ b F
‘0’ is transmitted is given by the Rayleigh distribution [6],
( )
r r 2
p(r|‘0’ sent)= exp − . (8.213)
2 2 2
F F
When a bit ‘1’ is transmitted, r(t) is an envelope of a cosine wave in the presence of Gaussian noise
(Eq. (8.208)), its amplitude s (T )∕2 = E ∕2 (see Eq. (8.193)) and, therefore, the pdf of r(t) is given by the
b
F
1
Rician distribution [6]
( 2 2 ) ( )
r r + E ∕4 rE 1
1
p(r|‘1’ sent)= exp − I 0 , (8.214)
2 2 2 2 2
F F F
where I (x) is the modified zero-order Bessel function of the first kind. The threshold is determined by the
0
intersection of two curves p(r|‘1’ sent) and p(r|‘0’ sent) (see Eq. (8.19)):
p(r|‘1’ sent)= p(r|‘0’ sent), (8.215)
( 2 ) ( )
E ∕4 r E
T 1
1
exp − I 0 = 1. (8.216)
2 2 2 2
F F