Probabilistic shaping: a step closer to the Shannon limit in channel capacity for optical communications

Abstract

Owing to the hypergrowth of cloud computing and video on demand, further promotion of data capacity among data centre interconnects is placed on the agenda. Generally, multiplexing and modulation techniques with spectral efficiency (SE) improvement are two common ways to increase the aggregate data rate. Up to now, multiplexing techniques have been successfully implemented in the dimensions of polarization, wavelength, fibre cores and modes, reaching a transmission rate over Pb/s [1 ]. As for the single lane (i.e., single fibre, single wavelength, single polarization) transmission, increasing the SE is preferable. By adopting higher order modulation formats, such as 4096 QAM, the SE can attain a goal of more than 10 bits/symbol [2 ]. Consequently, it is not surprising that the single channel capacity can exceed hundreds of Gb/s. However, the introduction of higher order modulation formats will make the system performance more sensitive to the signal to noise ratio (SNR) as the Euclidean distance between neighbouring constellation points becomes shorter under a fixed total power. In recent years, extensive research related to constellation shaping has been carried out to explore its applications in optical communications [3 ]. For an additive white Gaussian noise (AWGN) channel, the constellation shaped signals show performance closer to the Shannon limit compared to the conventional QAM signals, bringing a SNR gain up to 1.53 dB [4 ]. The key concept of constellation shaping is to mimic Gaussian signalling. Its implementation can be classified into two categories. One can adjust the locations of the constellation points on the complex plane to realize the Gaussian distribution as shown in Fig. 1 (a), which is termed as geometric shaping (GS). However, GS faces many practical problems. For arbitrary channels, no typical solution has been proposed for optimizing the locations of the constellation points in GS. Besides, the computational complexity of digital signal processing (DSP) increases owing to the irregular locations of constellation points. As a result, the other method of constellation shaping, named as probabilistic shaping (PS), is preferable. Unlike GS, PS changes the frequency of occurrence of the constellation points rather than their locations. Consequently, the DSP of PS signals is compatible with that of the conventional QAM signals. In addition, the optimization of probability distribution can be realized by tuning a simple parameter [3 ]. Types of constellation shaping a geometric shaping b probabilistic shaping [3 ] To verify the superiority of PS signals to the conventional QAM signals in optical communications, many research works have been reported on the comparison between the two signals. As mentioned above, up to 1.53 dB SNR gain can be achieved by PS, implying that a lower SNR is required at the receiver side. This will relax the requirement of high launched power. Subsequently, the accumulated nonlinear phase noise (NLPN) of PS signals will be weaker than that of conventional QAM signals in long-haul transmission. In [5 ], the performance of PS-64 QAM is compared with those of uniform 16 QAM and 32 QAM in coherent optical communication systems, where the net data rate is kept unchanged. For the transmission in pure silica-core fibre (PSCF), the maximum reach can be improved by 15.5% to 34% by applying the PS. Higher order modulation format should be adopted for PS signals to obtain the same data rate as uniformly shaped (US) signals. Therefore, PS signals are more sensitive to laser linewidth as their phase margin is smaller. In [6 ], the authors investigated the phase noise tolerance of both PS and conventional signals. The normalized generalized mutual information (NGMI) is selected as the evaluation metric as it can predict the post forward error correction (post-FEC) performance accurately [3 ]. PS-1024 QAM outperforms the US-256 QAM at a laser linewidth of 0.1 kHz. In this scenario, the phase noise is negligible and the shaping gain dominates. However, US-256 QAM has a better performance than PS-1024 QAM when the laser linewidth is broadened to 40 kHz. The benefit from shaping gain diminishes as the laser phase noise plays a more important role in determining the overall performance. In our recent work, we investigate the optimum constellation size by considering the effects from both shaping gain and laser linewidth. The results are shown in Fig. 2. The entropy of the four modulation formats is set at 7 bits/symbol. In the scenario of no laser phase noise, PS-256 QAM performs best as it has the largest shaping gain among the four modulation formats. The SNR gap between PS-256 QAM and PS-144 QAM becomes smaller when 10-kHz laser linewidth is presented. By increasing the laser linewidth to 400 kHz, the laser phase noise dominates over the AWGN. Consequently, PS-144 QAM is superior to the other modulation formats owing to the benefits resulted from phase margin and constellation shaping. NGMI against SNR for different modulation formats under laser linewidth of a 0 kHz; b 10 kHz and c 400 kHz Apart from smaller phase margin, the adoption of higher cardinality modulation in PS also gives rise to a higher peak-to-average power ratio (PAPR). Thus, the PS signals are more sensitive to the resolutions of digital-to-analog converter (DAC) and analog-to-digital converter (ADC), expressed as the effective number of bits (ENOB). In [7 ], we propose that it is not necessary to set the constellation size at 2N for PS signals, where N is an integer. For instance, PS-44 QAM can also be a replacement of US-32 QAM. In Fig. 3, we compare the performances of PS-64 QAM, PS-44 QAM and US-32 QAM under the constraint of different ENOB values. When the ENOB is high, such as 5.4, PS-64 QAM performs the best owing to the largest shaping gain. In this scenario, the limitation from ADC resolution is negligible. However, PS-44 QAM is superior to the other two formats if ENOB is reduced to 3.2. Compared to PS-64 QAM, PS-44 QAM has a lower PAPR, suffering less quantization distortions when the ADC has a limited resolution. Meanwhile, PS-44 QAM offers the advantage of shaping gain over US-32 QAM. NGMI against OSNR at ADC ENOB of a 5.4 and b 3.2 [7 ] Probabilistic shaping is a powerful technique for optical communications, providing both continuous data rate adaption and shaping gain. PS signals can reach higher overall data capacity and longer transmission distance compared to the conventional QAM signals. However, the adoption of higher cardinality demanded by PS signals will bring additional issues under certain circumstances such as poor laser linewidth and limited DAC and ADC resolutions. Optimum choice of the modulation format should take into account the consideration of practical resources that are available.

Topics & Concepts