IEEE PHOTONICS TECHNOLOGY LETTERS, VOL. 25, NO. 8, APRIL 15, 2013 717
Analytical Evaluation of Practical DBP-Based
Intra-Channel Nonlinearity Compensators
Guanjun Gao, Jie Zhang, and Wanyi Gu
Abstract—We derive an analytical expression for the trans-
mission performance of intra-channel nonlinearity compensated
and dispersion unmanaged orthogonal frequency-divison multi-
plexing (OFDM) systems using practical digital back propagation
(DBP) with span-length step size. The analytical expression is
verified by numerical simulations with evaluation inaccuracy
less than 0.4 dB for 1000-km 16-QAM modulated polarization
division multiplexed-OFDM transmissions. Using the derived
theory, the dependence of performance and optimum nonlinear
compensation coefficient on system parameters is identified. The
limitation of processing bandwidth on the performance improve-
ment using practical intra-channel DBP is also investigated.
Analytical results show that for achieving 1-dB improvement on
maximum Q
2
factor, the processing signal bandwidth of span-
length step size DBP should be within 55 GHz for 2000-km SSMF
transmissions over 62.5-km spans.
Index Terms— Coherent, digital back propagation, fiber
nonlinearity compensation, optical fiber communication.
I. INTRODUCTION
W
ITH the elegant solution of linear impairments miti-
gation by using digital signal processing, fiber nonlin-
earity is viewed as the fundamental obstacle that limits the
fiber channel capacity. For overcoming this obstacle, digital
back propagation (DBP) based intra and inter-channel nonlin-
earity compensation have been extensively studied, exhibiting
promising benefits for further improving the transmission per-
formance [1]–[8]. Assuming fine enough propagation step size
and sufficient processing power, the effectiveness of ideal full-
field DBP is affected by nondeterministic influences involved
in the nonlinear propagation, including stochastic polarization
mode dispersion (PMD) effect and signal-noise interaction [1],
and [9]–[11], in which the stochastic PMD effect imposes a
fundamental limit for the system capacity [11]. Analytical
performance expressions have been derived for these ideal
DBP [10], [12] or non-DBP systems [13], [16]. However, in
practical applications as demonstrated by [3], [8], the step size
Manuscript received December 2, 2012, revised January 17, 2013, accepted
February 6, 2013. Date of publication February 25, 2013; date of current
version March 22, 2013. This work was supported in part by 973 Pro-
gram under Grant 2010CB328204, in part by 863 Program under Grant
2012AA011301, in part by the China Post-Doctoral Foundation Project under
Grant 2012M520207, in part by the NSFC Project under Grant 62171189,
Grant 61201154, Grant 60932004, and Grant 61271191, and in part by the
RFDP Project under Grant 20090005110013.
The authors are with the State Key Laboratory of Information Photonics
and Optical Communications, Beijing University of Posts and Telecom-
munications, Beijing 100876, China (e-mail: championgao@gmail.com;
lgr24@bupt.edu.cn; wyg@bupt.edu.cn).
Color versions of one or more of the figures in this letter are available
online at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/LPT.2013.2247753
Fig. 1. FWM generation and propagation for nonlinearity compensated
systems with DBP.
of DBP are likely to be equal to the span length or even larger,
and the nonlinear compensation parameters usually need to
be optimized empirically or measured in advance. Compared
to the ideal DBP or non-DBP cases, there is no analytical
performance evaluation method for systems with practical
DBP compensation using relatively few steps. In this letter,
we derive an analytical expression for the commonly used
span-length step size DBP for dispersion unmanaged OFDM
systems and demonstrate its excellent evaluation accuracy
through numerical simulations. Using the analytical expres-
sion, the dependence of the transmission performance and the
optimum nonlinear compensation coefficient on system config-
urations can be clearly identified. Theoretical results show that
differing with systems using ideal DBP compensation, even
for the deterministic channels, the performance improvement
of practical intra-channel DBP can be limited and largely
dependent on signal bandwidth. The benefit limitations of
practical intra-channel DBP on signal bandwidth are then
quantified analytically and numerically.
II. T
HEORETICAL MODELING
For systems with nonlinearity compensation using DBP
as shown in Fig. 1, signals are transmitted along N
s
spans
with fiber loss, chromatic dispersion, nonlinear coefficient and
span length denoted by α, β
2
, γ and L respectively. For
polarization division multiplexed (PDM) systems, γ = 8γ
0
/9
with γ
0
and γ denoting the nonlinear coefficient in fibers
with fixed or randomly varying birefringence eigen axis. At
the receiver side, DBP are implemented on N
t
virtual spans
(VS), with the counterparts of α, β
2
, γ and L denoted by
α
, β
2
, γ
and L
, satisfying N
t
L
= N
s
L, β
2
=−β
2
,
γ
=−γ and α
L
=−αL for N
t
< N
s
or α
=−α
for N
t
≥ N
s
. For ideal DBP with fine enough processing
step size, the transmission channel is completely invertible
in the absence of nondeterministic effects. Thus the received
signals would be unpolluted, as fresh as at the transmitter.
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