Page 364 - Fiber Optic Communications Fund

Performance Analysis 345

Substituting Eqs. (8.67) and (8.69) in Eq. (8.6), we find

A 2 ∞ − 2
u(T )=± a ∫ −∞ e 2a 2 d
b
2
√
2
=±2A T , (8.70)
0
√
where A 2 T is the pulse energy E .
0 1
8.2.2 Error Probability with an Arbitrary Receiver Filter
From Eqs. (8.27) and (8.28), we have
(√ )
1
P = erfc , (8.71)
b
2 8
[u (T )− u (T )] 2
b
b
1
0
= , (8.72)
2
2
where u (T ) and are given by Eqs. (8.6) and (8.7), respectively. Eqs. (8.71) and (8.72) are valid for arbitrary
j
b
filter shapes. From Eq. (7.8), we have
I − I 0
1
Q = . (8.73)
+ 0
1
Since the mean of bit ‘1’ (‘0’) is u (T )(u (T )), Eq. (8.72) becomes
1 b 0 b
( ) 2
I − I 0
1
= . (8.74)

When = ≡ , from Eqs. (8.74) and (8.73), we find
1
0
2
= 4Q , (8.75)
( )
1 Q
P = erfc . (8.76)
b √
2
2
8.3 Homodyne Receivers
Consider a fiber-optic transmission system with a homodyne balanced receiver, as shown in Fig. 8.9. Its
mathematical representation is shown in Fig. 8.10. Let the transmitted optical field distribution be
q = s(t) exp (−i t), (8.77)
s
c
where s(t) is the complex field envelope, is the angular frequency of the optical carrier. We make the
c
following assumptions to find the best achievable performance.
(1) Fiber dispersion, PMD, and nonlinearity are absent so that the fiber channel can be modeled as an AWGN
channel. In fact, fiber dispersion and PMD cause distortion, and nonlinearity causes both distortion and noise
enhancement, which will be discussed in Chapters 10 and 11.
(2) The frequency, phase, and polarization of the local oscillator are exactly aligned with the received signal.