Page 492 - Fiber Optic Communications Fund

Nonlinear Effects in Fibers 473

where n = Re{n(T)} and n = Im{n(T)}. Eq. (10.347) can be written as
i
r
s (T)= A(T) exp [i(T)], (10.363)
out
where
2
2
A(T)={[s (T)+ n (T)] + n (T)} 1∕2 , (10.364)
in
r
i
{ }
n (T)
i
(T)= tan −1 .
s (T)+ n (T)
in
r
n (T)
i
≈ . (10.365)
s (T)
in
2
2
In Eq. (10.365), we have ignored higher-order terms such as n and n . Using Eqs. (10.352), (10.353),
r
i
(10.360), and (10.361) in Eq. (10.365), we obtain
J−1
n 0i ∑ n F (T)
ji j
(T)= √ + √ , (10.366)
E j=1 p(T) E
where n = Re{n } and n = Im{n }. From Eqs. (10.366) and (10.356), it follows that
j
jr
j
ji
<(T) >= 0. (10.367)
Squaring and averaging Eq. (10.366) and using Eqs. (10.357) and (10.358), we obtain the variance of phase
noise as
J−1 2
∑ F (T)
2 2 m
=< >= + . (10.368)
lin 2E 2E 2
j=1 F (T)
0
Next, let us consider the impact of a matched filter on the phase noise. When a matched filter is used, the
received signal is
∞
∗
r = s (T)F (T)dT. (10.369)
∫ 0
out
−∞
Substituting Eq. (10.361) in Eq. (10.369) and using Eq. (10.354), we obtain
√
r =( E + n ). (10.370)
0
Note that the higher-order noise components given by the second term on the right-hand side of Eq. (10.361)
do not contribute because of the orthogonality of basis functions. Now, Eq. (10.368) reduces to
2
< n >
2 0i
= = . (10.371)
lin E 2E
From Eq. (10.370), we see that when a matched filter is used, the noise field is fully described by two degrees
of freedom, namely, the in-phase component n and the quadrature component n . The other degrees of
0r 0i
freedom are orthogonal to the signal and do not contribute after the matched filter. From Eq. (10.371), we see
that the quadrature component n is responsible for the linear phase noise.
0i