Keywords
Fiber nonlinearity, digital backpropagation, split-step Fourier method,
long-haul transmission, optical fiber communication, fiber impairment
compensation
Introduction
The demand for data traffic has surged in the past decade, owing to the
development of bandwidth-consuming applications such as big data, video
over IP applications, Internet Protocol Television (IPTV), and cloud
computing, among others. The proliferation of ‘smart devices’ supported
by the Internet of Things (IoT) technology also contributes
substantially to the increasing demand for data traffic. The
International Telecommunication Union (ITU) has predicted data traffic
to exceed 55 zettabytes (ZB) per month by 2030 (1). This expected growth
has driven research efforts in recent years toward protracting the data
rate of next-generation transmissions to 100 Gbit/s and beyond. While
there has been significant research into extending the total capacity of
wavelength division multiplexing (WDM) in the region of a few terabits
per second (2), a more trending area of research is to design optical
transceivers capable of employing high-order modulation formats and
symbol rates to achieve high data rate per channel. The 6th generation
(6G) technology has recently been investigated as a key enabler for
ultra-high-speed and reliable communication in the range of 100 Gbit/s
– 1 Tbit/s (3,4). Ultra-high data rate in 6G communication networks is
achieved by exploiting the sub-terahertz (THz) and THz spectrum (0.1 –
10 THz), which uses advanced optical fiber communication technologies
such as optical heterodyning and high-order modulation schemes for
faster and high-capacity data transfer (5,6). Again, the benefits of
radio-over-fiber (RoF) technology has been demonstrated for scaling 5G
and 6G networks by implementing their full integration with existing
access network technologies (3,7).
However, high and ultra-high data rate optical transmissions employing
high-order modulation schemes are critically affected by optical fiber
impairments, limiting their reach (8,9). Digital signal processing (DSP)
techniques and coherent detection have become popular for compensating
fiber impairments and promise to be a potential solution for designing
high data rate transmissions. Subsequently, several techniques presented
for nonlinear impairments compensation include digital backpropagation
(DBP) (10,11), Volterra series nonlinear equalizer (VSNLE) (12),
phase-conjugate twin waves (PCTW) (13), machine learning-based
techniques (14), nonlinear Fourier transform (NFT) (De Koster & Wahls,
2020), among others. The DBP technique has been demonstrated to be more
promising and effective for jointly mitigating fiber dispersion and
nonlinear distortions, becoming a benchmark for evaluating other
compensation techniques (16). This technique is implemented by using the
split-step Fourier method (SSFM) to determine a numerical solution of
the inverse nonlinear Schrödinger equation (NLSE) (10). In DBP, a signal
is propagated through a virtual fiber divided into small-sized steps per
fiber span and characterized by the inverted properties of a typical
fiber. However, the effective implementation of DBP depends considerably
on the number of steps and the selection of step sizes, as it can become
computationally complex with an increasing number of steps per span
(17). Thus, there is a trade-off between accuracy and computational
cost. Several advanced DBP techniques with low implementation
computational overhead have been proposed (18–21).
Notably, the low complexity correlated DBP (C-DBP) has been proposed
based on perturbation-based techniques to reduce the number of steps in
a 3200 km 28 GBaud 32-QAM transmission system by a factor of 10 (21). A
60% reduction in complexity has been realized for a 21-channel 32 GBaud
WDM PM-16QAM by the time-domain DBP (TD-DBP) using a randomly
distributed step size (18). The work reported that sensitivity to
quantization noise from FIR filter coefficients was significantly
improved by randomizing the step sizes. In (19), an 80% reduction in
required steps per fiber span is reported by the weighted DBP (WDBP),
which uses a weighted-average approach to consider the correlation
between adjacent symbols as well as an optimization of the position of
the nonlinear operator calculation point. The algorithm is demonstrated
in a 1600 km 112 Gbit/s PM-QPSK transmission link. Also, a low
complexity DBP using a single step DBP per fiber span is experimentally
demonstrated in a 112 Gbit/s PM-QPSK over 3200 km, reducing power
consumption and complexity by a factor of 16 and the computation time by
a factor of 20 (20).
In most of these techniques, the constant step size is used for DBP
backpropagation. However, this does not provide an optimal solution in
the presence of fiber attenuation. The nonlinear phase shift accumulated
in each step decreases exponentially with distance due to fiber
attenuation (22). Subsequently, various adaptive step size distribution
DBP algorithms such as the local error method (23), logarithmic step
size distribution method (24), and nonlinear phase-rotation method (25)
have been presented.
In this work, we propose a step size optimization scheme of the
split-step Fourier method for effective and low-complexity
implementation of the DBP technique for fiber impairment compensation.
The conventional logarithmic step size distribution scheme is modified
using the binary logarithm to generate optimum adaptive step sizes for
accurate DBP calculation. The algorithm is demonstrated in a 112 Gbit/s
DP-16QAM system over a 2400 km transmission link. The simulation results
show that implementing the binary logarithmic step size distribution for
step size optimization outperforms traditional logarithmic and constant
step size distribution techniques.
Variable step size
When a signal is launched into an optical fiber, the signal power
decreases exponentially along the fiber span in the presence of fiber
attenuation. As the instantaneous power determines the nonlinear phase
shift, its accumulation similarly follows an exponential profile. The
power along a fiber of length L is given by 1:
11\* MERGEFORMAT
()
where α is the attenuation coefficient. The accumulated nonlinear
phase shift is expressed as:
22\* MERGEFORMAT ()
where is the nonlinear coefficient of the virtual link. The nonlinear
impairment is observed to be larger at the beginning of the propagation
through the span. For backward propagation along the virtual fiber, this
works reversely. Since the nonlinear impairment varies along a span, a
smaller step size is required at larger powers for accurate DBP
estimation of signal distortion, as illustrated by Fig. 1(c).