Page 384 - Fiber Optic Communications Fund

Performance Analysis 365

{ [( Δ ) ]
2RA LO A cos − t +Δ for 0 < t ≤ T b
IF
I (t)= 2 (8.233)
0
0 otherwise.
As before, we ignore the scaling factor 2RA LO . The matched filter H () is matched to I (t), j = 0, 1 except
Ij
j
for the phase factor as in Section 8.4.4 (See Fig. 8.18). Suppose s (t) is transmitted so that I(t)= I (t). H ()
Ij
1
1
can be realized as correlators and their outputs in the absence of noise can be written as
T b [( Δ ) ]
I (t)= A 2 ∫ I () cos + ( + T − t) d, (8.234)
1F
IF
1
b
0 2
T b [( Δ ) ]
I (t)= A 2 ∫ I () cos − ( + T − t) d. (8.235)
1
IF
0F
b
0 2
Eq. (8.234) is similar to Eq. (8.188). Ignoring the frequency component centered around 2 and simplifying
IF
Eq. (8.234), we obtain
E 1
I (t)= , (8.236)
1F
2
2
where E = A T . Similarly, from Eq. (8.235), we obtain
b
1
A 2 T b [ ( Δ ) ]
I (t)= 2 ∫ 0 cos Δ + − 2 (t − T )+Δ d
0F
IF
b
A 2 T b
= [cos (Δ) cos (t)− sin (Δ) sin (t)] d
2 ∫ 0
[ ( )] [ ( ) ]
A 2 sin ΔT b cos ΔT b − 1
= cos (t)− sin (t), (8.237)
2 Δ Δ
where
( )
Δ
(t)= − (t − T )+Δ. (8.238)
IF b
2
If
2ΔfT = 2n, n = 1, 2, … ,
b
n
Δf = , (8.239)
T b
from Eq. (8.237) we find that I (t)= 0. The signals are orthogonal for asynchronous detection if the output
0F
of the filter H () is zero when s , k ≠ j is transmitted. Comparing Eqs. (8.178) and (8.239), we find that
Ij k
the minimum frequency difference to achieve orthogonality for asynchronous detection is twice that for syn-
chronous detection. In this section, we assume that Δf = 1∕T so that the output of the filter H () is zero
b I0
(ignoring noise) when s (t) is transmitted. In this case, the outputs of the envelope detectors can be written as
1
√
[ ] 2
E 1
r (T )= + n (T ) + n 2 (T ), (8.240)
1 b 1FI b 1FQ b
2
√
r (T )= n 2 0FI (T )+ n 2 0FQ (T ), (8.241)
b
0
b
b