Page 536 - Fiber Optic Communications Fund

Digital Signal Processing 517

with constant fiber dispersion, nonlinear, and loss coefficients. The evolution of the field envelope in a fiber
is described by the NLSE (see Chapter 10),
q
=(N + D)q, (11.111)
z
where D denotes the fiber dispersion effect,
2
2
D =−i (11.112)
2 t 2
and N denotes the nonlinear and loss effects,

2
N(t, z)= i|q(t, z)| − . (11.113)
2
The formal solution of Eq. (11.111) can be obtained as follows:
dq
=(N + D), (11.114)
q
L
L
ln[q(t, z)]| = (N + D)dz, (11.115)
0 ∫
0
q(t, L)= Mq(t, 0), (11.116)
where
{ }
L
M = exp [N(t, z)+ D(t)]dz (11.117)
∫
0
and L is the fiber length. In general, q(t, L) can not be obtained in a closed form since N(t, z) has a term
2
proportional to |q(t, z)| which is unknown for z > 0. Eq. (11.116) is just another way of writing Eq. (11.111),
and numerical techniques have to be used to find q(t, L)[24]. Multiplying Eq. (11.116) by M −1 on both sides,
we find
−1
q(t, 0)= M q(t, L). (11.118)
In Eq. (11.118), q(t, L) represents the received field envelope which is distorted due to fiber dispersion and
−1
nonlinear effects. If we multiply the received field by the inverse fiber operator, M , distortions due to fiber
dispersion and nonlinear effects can be completely undone. Since
exp (̂x) exp (−̂x)= I, (11.119)

where I is an identity operator (Example 11.4), taking
L
̂ x = [N(t, z)+ D(t)]dz, (11.120)
∫
0
we find
[ ]
L
M −1 = exp − [N(t, z)+ D(t)]dz . (11.121)
∫
0
Eq. (11.118) with M −1 given by Eq. (11.121) is equivalent to solving the following partial differential
equation:
q b
=−[N + D]q , (11.122)
b
z