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An integral split-step fourier method for digital back propagation

to compensate ﬁber nonlinearity

Jie Yang, Song Yu

, Minliang Li, Zhixiao Chen, Yi Han, Wanyi Gu

State Key Laboratory of Information Photonics and Optical Communications, Beijing University of Posts and Telecommunications, Beijing 100876, China

article info

Article history:

Received 4 June 2013

Received in revised form

7 August 2013

Accepted 4 September 2013

Available online 19 September 2013

Keywords:

Fiber nonlinearity

Digital back propagation (DBP)

Split-step Fourier method (SSFM)

Nonlinearity calculation position (NLCP)

Lagrange's Integral Mean Value Theorem

(LIMVT)

abstract

In optical ﬁber transmission systems, the split-step Fourier method (SSFM) has been widely used in

digital back propagation (DBP) to compensate ﬁber nonlinearity. In this paper, by using the Lagrange's

Integral Mean Value Theorem (LIMVT), we derive an analytical expression to calculate the optimal value

of the nonlinearity calculation position (NLCP) for different systems and we propose an integral SSFM

(I-SSFM) based on the expression. The I-SSFM can be performed more accurately and efﬁciently without

parameter optimization. Simulations of various transmission links show that the I-SSFM outperforms the

conventional asymmetric SSFM (A-SSFM) and the symmetric SSFM (S-SSFM) signiﬁcantly, especially

when we employ less amount of steps to ensure computation ef ﬁ ciency. The computation effort of the

I-SSFM reaches as low as 50% of that of the S-SSFM.

1. Introduction

In recent years, real-time 40 Gb/s [1] and 1 00 Gb/s [2,3] coherent

optical transmission systems have been demonstrated. To achieve

even higher bit-rate and spectral efﬁciency , multilevel modulation

formats such as 1 6-level quadrature amplitude modulation (1 6QAM)

and 64QAM are to be employed. Howev er, systems employing higher -

level modulation formats req uire higher SNR and hence higher launch

powers and this can signiﬁcantl y r educe the possible transmission

distance and capacity due to ﬁber nonlinear impairments [4,5].

Therefore, mitigation or compensation of ﬁber nonlinearity becomes

signiﬁcant and has been widely studied, among which the digital

signal processing (DSP) is a promising solution.

Of all the approaches employing DSP techniques to suppress

the ﬁber nonlinearity, the digital back propagation (DBP) [6–9]

shows the best performance and has become the benchmark for

ﬁber nonlinearity compensation. By employing the split-step

Fourier method (SSFM), DBP can solve the nonlinear Schrödinger

equation (NLSE) inversely and hence the optical signals can be

reconstructed at the receiver side.

Generally, there are 2 ways to realize the SSFM, i.e., the

asymmetric SSFM (A-SSFM) and the symmetric SSFM (S-SSFM).

The A-SSFM performs one linear and then one nonlinear operation

sequentially in each step and is not iterative. But to enhance the

calculation accuracy, the S-SSFM calculates half a linear and one

nonlinear followed by the rest half a linear operation sequentially

in each step and often utilizes two additional iterations [6,11 ]. The

non-iterative A-SSFM saves computation efforts, but the iterative

S-SSFM shows a better performance at a cost of higher imple-

mentation complexity.

To improve the computation efﬁciency without signiﬁcant

performance degradation, a modiﬁed non-iterative SSFM (M-SSFM)

for DBP was proposed in [10]. This proposal improves the perfor-

mance of DBP by shifting the nonlinearity calculation position

(NLCP) without additional iterations [10]. The M-SSFM outperforms

the A-SSFM and is less complex than the S-SSFM. However, to the

best of our knowledge, no analytical expression for the NLCP has

been reported, which means that it is difﬁcult to determine the

optimal value of NLCP in actual systems with various parameters,

like the dispersion and launch power. In [10] the optimum of NLCP

is obtained by enumerating and testing different values of NLCP.

Still, the M-SSFM is often combined with parameter optimization

[10], bringing extra work in obtaining the optimized parameters for

different systems.

In this paper, we reveal why better performance can be achieved

by shifting the NLCP and we derive an analytical expression to

calculate the optimal value of NLCP. This expression is derived with

the Lagrange's Integral Mean Value Theorem (LIMVT) and is

applicable to systems with different parameters, like the dispersion,

transmission length and launch power. And based on the expres-

sion, we propose an integral SSFM (I-SSFM). Unlike the M-SSFM, the

resulting expression allows us to obtain the optimum of NLCP

Contents lists available at ScienceDirect

journal homepage: www.elsevier.com/locate/optcom

Optics Communications

http://dx.doi.org/10.1016/j.optcom.2013.09.015

Corresponding author. Tel.: þ 86 01061198110.

E-mail address: yusong@bupt.edu.cn (S. Yu).

Optics Communications 312 (2014) 80–84

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