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364 Fiber Optic Communications

I(t) = 2RA Re{[s(t) + n (t)]e *i(ω IF t + ▵φ) } + n d
c
Heterodyne Rx LO
front end

t = T b
Envelope
I1
r (T ) detector H (ω)
If r (T ) > r (T ) 1 b
1 b 0 b
Comparator
t = T
select ‘1’ b
Envelope
H (ω)
I0
r (T ) detector
0 b
Figure 8.18 A heterodyne receiver with envelope detectors and matched filters for FSK.
where Q (a, r ∕ ) is the generalized Marcum Q-function defined as [6]
1 T F
∞ ( a + x 2 )
2
Q (a, b)= x exp − I (ax) dx
1 ∫ 0
b 2
( 2 2 ) ∞ ( ) k
a + b ∑ a
= exp − I (ab), b > a > 0. (8.228)
k
2 b
k=0
Substituting Eqs. (8.226) and (8.217) in Eq (8.227), we obtain
√
( )
⎡ ⎛ het ⎞⎤
√ 4
P(0|1)= 1 − Q ⎜ 2 het , 1 + ⎟⎥ . (8.229)
⎢
1
⎢ ⎜ 2 het ⎟⎥
⎣ ⎝ ⎠⎦
Combining Eqs. (8.224) and (8.229), we find
1
P = [P(1|0)+ P(0|1)]
b
2
{ ( √ )}
[ het ( )]
1 4 √ het ( )
= exp − 1 + + 1 − Q 1 2 het , 1 + 4∕ het . (8.230)
2 4 het 2
8.4.5 FSK: Asynchronous Detection
The transmitted signals s (t) and s (t) are the same as those in Section 8.4.3 (Eq. (8.166)). The output of the
1 0
heterodyne receiver front end in the absence of noise may be written as
{
I (t) when s (t) is transmitted
1
1
I(t)= (8.231)
I (t) when s (t) is transmitted,
0 0
where
{ [( Δ ) ]
2RA LO A cos + t +Δ for 0 < t ≤ T b
IF
I (t)= 2 (8.232)
1
0 otherwise,