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).