1. Introduction

Tremendous Internet traffic has been driving the improvement of the optical fiber transmission systems to meet various requirements from different applications [1], such as 5G, virtual reality, high-definition video, cloud computing and storage, etc. Specifically, major traffic flows over the data center related networks, the typical cost-effective networks with short transmission distance. There are many short-to-medium reach solutions focusing on both low cost and high data rate [2–4]. Among these solutions, intensity modulation and direct detection (IM/DD) dominates benefiting from less requirements of devices and simpler digital signal processing (DSP) algorithms compared to the conventional and simplified coherent solutions [5,6]. However, conventional IM/DD schemes suffer from serious frequency selective fading due to chromatic dispersion (CD). The single sideband (SSB) modulation is introduced to solve the problem [7,8]. Compared to the CD pre-compensated double sideband (DSB) modulation, the SSB modulation does not need CD estimation and the power loss from one sideband can be mitigated by the optimization of modulation index [8]. Besides, for IM/DD solutions, pulse amplitude modulation (PAM), carrier-less amplitude and phase modulation (CAP) and discrete multi-tone (DMT) are three representative high-order modulation schemes [2]. Due to the multi-carrier feature, DMT schemes are more flexible and effective to enhance the spectral efficiency (SE) in a fading channel. Each subcarrier of DMT can be individually loaded and is compatible with advanced modulation formats. By loading quadrature amplitude modulation (QAM) and power levels for DMT systems, classical bit and power loading (BPL) schemes efficiently enhance the data rate or the power margin based on gap theory [9] or greedy algorithm [10].

However, the conventional QAM has two main drawbacks, coarse modulation granularity and intrinsic gap to the Shannon limit. A simple method to refine the granularity is to mix different QAMs in the time domain to generate time domain hybrid QAM (TDHQ) [11,12] or in the frequency domain to generate frequency domain hybrid QAM (FDHQ) [13]. By selecting the partitions of conventional QAMs to enlarge the Euclidean distance, the set-partitioning QAM (SP-QAM) not only achieves fractional SE but also shows superior performance than the conventional QAMs [14]. Furthermore, to mitigate the gap to Shannon limit and enhance the granularity of SE, the probabilistic shaping is proposed by manipulating the probability distribution of conventional QAMs [15]. The most common approach to realize PS-QAM is to combine the probabilistic amplitude shaping (PAS) and constant composition distribution matcher (CCDM) due to its simple implementation and superior performance [15,16]. As demonstrated by the experiments, the PS-QAM achieves negligible gap to the Shannon capacity in the optical fiber transmission systems [17].

PS-QAM is identified by its probability distribution of constellation points, which can be measured by entropy, $S = - \sum\limits_i {{P_i}\log {P_i}}$, where ${P_i}$ is the probability of i^th constellation point [18]. By allocating the PS-QAM with different entropy to multi-carriers, higher mutual information (MI) is achieved in a colored SNR channel, which is called entropy loading (EL) referring to BPL [19]. The EL for DMT systems is first proposed in [20]. And EL outperforms BPL in an optical fiber communication system [21,22] and a visible light communication (VLC) system [23]. However, these schemes are hard to realize in practice considering the implementation of CCDM. The CCDM requires more than 10⁴ symbols to reach the expected entropy [24]. And DMT systems generally have hundreds of subcarriers to enhance the performance of one-tap equalization. Thus, the conventional EL schemes will produce large frame length, which occupies long time period. Thus, more training symbols are required to track channel variation. Furthermore, the potential hundreds of distribution matchers (DMs) consume intolerable resources. These factors dramatically increase the complexity. Another critical problem is the performance metric. The proposed EL schemes generally adopt MI, generalized mutual information (GMI) and/or the deduced achievable information rate (AIR) as the performance metric due to their rigor in theory [15,19–23]. However, considering the implementation penalty, they cannot precisely reflect the practical performance. To achieve the performance indicated by MI/GMI, the DM should be ideal and the FEC code should have infinite code length and adapt its code rate according to different shaping parameters [15,25]. It is almost impossible to realize in practice. Thus, the SE in information bits per symbol (IBPS) or information rate is more convincing [16,26]. The off-the-shelf FEC code and its constraint of normalized generalized mutual information (NGMI) are included in the calculation of IBPS. Besides, the implementation of CCDM is also considered to reflect shaping loss due to limited length of output symbol. Based on the constructed relationship between SNR and PS-256QAM with the maximum IBPS, we proposed a uniform entropy loading (UEL) algorithm [27], where all the subcarriers can be allocated with the same PS-QAM due to the SNR equalization of precoding operation [8]. Therefore, only 1 DM is needed for UEL.

In this paper, we further investigate the influence of constellations on IBPS under different SNR. By conducting Monte-Carlo simulation in an additive white Gaussian noise (AWGN) channel, we construct the relationship between SNR and the PS-QAM with maximum IBPS. We find that the PS-QAM with lower order constellation performs better at lower SNR. For a fading channel, various SNR exists and different constellations are required to perform optimal EL. Thus, we call our EL scheme hybrid constellation entropy loading (HCEL). Correspondingly, the EL scheme using identical constellation is called identical constellation entropy loading (ICEL) in this paper. For the HCEL, we proposed an algorithm, optimally partitioned precoding (OPP), to partition multiple precoding sets. And the equally partitioned precoding (EPP) is performed for comparisons. The simulation and experimental results show that 3 precoding sets are large enough to achieve a good performance. In other word, only 3 DMs are required, dramatically reducing the complexity compared to conventional EL. This paper is the extension of the conference paper [28]. Compared to [28], we introduce the HCEL schemes in detail, conduct abundant simulations and extend the experiments for different transmission distances to thoroughly evaluate the performance of HCEL in this paper.

2. Hybrid constellation entropy loading

2.1 Optimal constellations of probabilistic shaped quadrature amplitude modulation

To conduct the EL, the first step is to construct the relationship between the optimal PS-QAM with SNR. Then, based on the relationship, the PS-QAM with specific entropy can be loaded to each subcarrier according to its SNR. To evaluate the performance of PS-QAM, the first thing is to choose the performance metric. As explained in the introduction, we refer to the SE in IBPS as the evaluation index instead of MI or GMI. The SE in IBPS of conventional QAM is expressed by

Based on the equations, we perform Monte-Carlo simulation to search for the PS-QAM with the maximum IBPS for different SNR under the constraint of NGMI. The results are shown in Fig. 1. As we can see, all the PS-QAMs require less SNR than the conventional QAM under the same IBPS, namely shaping gain, except the PS-256QAM at the SNR region lower than 5.5 dB. Furthermore, the PS-QAM with lower order performs better at the lower SNR region. This is due to the existence of coding gap to ideal FEC and shaping gap to ideal distribution [25], which cannot be revealed using GMI or MI. We summarize the optimal constellations in different SNR region as shown in Table 1. Note that the shaping parameter varies according to the SNR for different constellation. Therefore, we can conduct the EL using different constellations, namely HCEL.

 

Fig. 1. The relationships between IBPS and SNR for PS-16QAM, PS-64QAM, PS-256QAM and conventional QAM using the concatenated code of 20% LDPC and 6.25% staircase code under the NGMI limit of 0.858 in an AWGN channel.

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Table 1. The optimal constellation for different SNR region

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2.2 Partition the precoding sets

To approach the expected entropy or reduce the shaping loss, the output block length of CCDM should be larger than 10⁴ symbols [24]. At the same time, DMT systems generally contain hundreds of subcarriers to achieve a flat channel for each subcarrier. Thus, each subcarrier can be equalized with one-tap equalization. If the PS-QAM is individually loaded to each subcarrier, a total frame would consist of larger than 10⁶ symbols, burdening the buffer. Moreover, hundreds of different DMs consume intolerable hardware resources. Therefore, it is not suitable for practical applications. Reducing the type of PS-QAM among all the subcarriers is an effective solution. We adopt the precoding technique as the key. As we demonstrated in [8], the SNR of subcarriers in one precoding set is equalized due to noise equalization effect. Thus, only one PS-QAM is required for one precoding set with many subcarriers. However, how to determine the number of precoding sets (NPS) and the distribution of subcarriers in each precoding set is still a problem. Here, we propose two algorithms, EPP and OPP, to deal with the problem. Without loss of generality, we take two precoding sets as an example. The procedures are shown in Fig. 2.

 

Fig. 2. The schematic diagrams of EPP algorithm and OPP algorithm. (a) The SNR variation of a fading channel. EPP: (b) The SNR variation after EPP. OPP: (c) The sorted SNR variation, (d) all the precoding patterns of two sets, (e) the total capacity versus the subcarriers in Set 1 for the patterns in (d), (f) the SNR variation after OPP which achieves the maximum total capacity with recovered subcarrier indices.

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Assume the SNR variation of a fading channel in frequency domain is like the one shown in Fig. 2(a). The EPP algorithm divides the subcarriers into two sets with the same number of subcarriers and the precoding operation is individually conducted for each set, see Fig. 2(b). And it is easy to extend to multiple precoding sets. For the OPP algorithm, the subcarriers should be sorted first in decreasing order of SNR, as shown in Fig. 2(c). Then, we traverse all the patterns of two continuous precoding sets for the sorted indices by moving the dividing point between two precoding sets, as shown in Fig. 2(d). The relationship between total capacity and the patterns represented by the subcarriers in Set 1 is shown in Fig. 2(e). Finally, we choose the pattern that achieves the maximum total capacity and recover its subcarrier indices to the frequency domain, as shown in Fig. 2(f). For each precoding set, we iteratively drop the subcarrier with the worst SNR to find the maximum total capacity. The procedure is the same with UEL but conducted individually for each set [27], which is not presented in Fig. 2. Furthermore, for the multiple precoding sets, the only difference is the increase of the dividing points. Finally, the PS-QAMs are loaded to the partitioned precoding sets. Note that the number of PS-QAM is the same as the NPS for both EPP and OPP.

3. Simulation and analysis

A typical SSB-DMT system is used to evaluate the performance of the proposed EL schemes and conventional BPL scheme, LC scheme. The simulation model of the SSB-DMT system is given in Fig. 3, which is conducted in the co-simulation environment of MATLAB and VPI TransmissionMaker 9.9.

 

Fig. 3. (a) The simulation setup for SSB-DMT systems, and (b) the digital signal processing flow of transmitter and receiver. Tx: transmitter, Rx: receiver, DSP: digital signal processing, CP: cyclic prefix, DAC: digital-to-analog converter, LPF: low-pass filter, EA: electrical amplifier, DD-MZM: Dual-driver Mach-Zehnder modulator, SSMF: standard single mode fiber, VOA: variable optical attenuator, EDFA: erbium doped fiber amplifier, OBPF: optical band-pass filter, PD: photodiode, ADC: analog-to-digital converter.

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The serial binary data is generated by the Mersenne Twister algorithm and converted to the parallel data according to the mapping rule. For the EL schemes, the uniformly distributed data in the same precoding set are shaped by a CCDM to the data with specific probability distribution. Then, the data is mapped to the QAM modulated signal, followed by the inverse fast Fourier transform (IFFT) operation with 512 points. Note that the 16 edge subcarriers in high frequency is left null to avoid strong filtering effect. Therefore, the maximum number of data subcarriers is 240 for all the SSB-DMT signal. And the subcarriers in each precoding set are individually precoded by a discrete Fourier transform (DFT) matrix. After cyclic prefix (CP) insertion with 20 samples and parallel-to-serial conversion, the digital signal processing (DSP) in transmitter is completed. Then, the obtained digital signal is output by a digital-to-analog converter (DAC) with 8-bit resolution and 40 GSa/s sampling rate. After that, the baseband signal passes a Butterworth filter with the 3-dB bandwidth of 15 GHz to emulate the bandwidth-limited scenario. Subsequently, the signal is amplified before driving the dual driver Mach-Zehnder modulator (DD-MZM) to generate optical SSB signal [7,8]. The modulation index is optimized for different configuration. The center frequency of the optical carrier is set to 1550.12 nm and the launch power is fixed to 4 dBm. After the standard single mode fiber (SSMF) transmission, the signal is pre-amplified by an erbium doped fiber amplifier (EDFA) with 20 dB gain and 0 dB noise figure. The optical signal-to-noise ratio (OSNR) in 0.1 nm reference bandwidth is precisely controlled by a SetOSNR model. The following optical band-pass filter (OBPF) aims to suppress out-of-band noise. And the variable optical attenuator (VOA) keeps the received power constant at 0 dBm before photodetector (PD). After detected by PD, the electric signal is also amplified and filtered. Before sent to the DSP in the receiver, the signal is sampled by an analog-to-digital converter (ADC) with 8-bit resolution and 80 GSa/s sampling rate. The procedures in the receiver is the inverse in the transmitter. Nonlinearity mitigation methods [30] are not used in the Rx DSP due to their high complexity and irrelevance for the comparisons of loading algorithms. Note that 1 training symbol (TS) is used for frame synchronization and 9 TSs are used for channel estimation. Including TSs, a total frame contains 170 DMT symbols.

We first evaluate the impact of the NPS or the number of DM for EL schemes. The net data rate (NDR) is used as the evaluation index, which has excluded all the redundancy from CP, TS, and FEC. As shown in Fig. 4, the NDR of all the EL schemes, ICEL with EPP, HCEL with EPP, and HCEL with OPP, increase with NPS and they tend to saturate when the NPS is larger than 3. Thus, we adopt 3 as the NPS for all the EL schemes to do the following comparisons in simulation. Note that when the NPS increases to the maximum, the number of data subcarriers (240 for this configuration), HCEL with EPP and HCEL with OPP will become the conventional EL using hybrid constellations and have the same NDR. In other words, the conventional EL can only achieve slight improvement of NDR at the expense of large frame and hundreds of DMs.

 

Fig. 4. Net data rate versus the number of precoding sets for different EL schemes.

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We change the channel state by varying the OSNR. Then, all the adaptive loading algorithms, LC algorithm [10] and three EL algorithms, are conducted for different optical fiber length. The relationships between NDR and OSNR are shown in Fig. 5. And the NGMI of all the points is larger than the FEC limit, 0.858. Thanks to the SSB modulation, the NDR almost increases linearly with the OSNR for different fiber length. Besides, regardless of fiber length, the HCEL with OPP scheme always achieves the largest NDR and the LC scheme always has the lowest NDR. Furthermore, the HCEL with OPP scheme generally has 2 dB OSNR gain than the LC scheme under the same NDR. Besides, with the help of HC, the performance of EPP algorithm is enhanced. And the NDR of HCEL with EPP scheme approaches that of HCEL with OPP. This is due to the specific SNR variation. Especially for the SNR variation when the fiber length is equal to 20 km, the partition of precoding sets with equal interval is almost the optimal. The detail is shown in Fig. 6.

 

Fig. 5. The net data rate versus OSNR for BPL and EL algorithms when the length of SSMF is (a) 0 km, (b) 20 km, (c) 80 km.

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Fig. 6. The SNR variation and relative theoretical capacity for different partitioned patterns during the searching procedure of OPP algorithm. (a) The SNR variation and (b) the relative theoretical capacity when the length of SSMF is 20 km and OSNR is 34 dB. (c) The SNR variation and (d) the relative theoretical capacity when the length of SSMF is 80 km and OSNR is 34 dB.

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The relative SNR variation does not change with the OSNR, so we choose 34 dB as the OSNR value to view the detail of partition. As depicted in Fig. 6(a), the SNR in dB of probe signal, 4QAM modulated DMT signal, almost decreases linearly with the subcarrier indices or the frequency. After sorting the subcarriers in the decreasing order of SNR as described in Fig. 2, we calculate the theoretical capacity (TC) of different partitioned patterns with 3 precoding sets, as shown in Fig. 6(b). Note that when the number of Set 1 and Set 2 are determined, the number of Set 3 is obtained due to the fixed number of total subcarriers. The 3 SNR ladders for EPP and OPP scheme can also be clearly observed in Fig. 6(a). As we can see in Fig. 6(b), the variation of TC is unimodal. And the precoding pattern achieves the maximum TC is selected to finish OPP algorithm. Besides, there exists a large area near the equally partitioned point to approach the maximum TC, which means many precoding patterns can realize near optimal performance for this SNR variation. We also label the TC of EPP near the color bar, which is quite closed to the optimal TC achieved by OPP. This is the reason why the HCEL with EPP has excellent performance when fiber length is 20 km. But the SNR variation becomes more complex when the fiber length is 80 km, as shown in Fig. 6(c). The subcarriers in the first and the second precoding set almost have the same SNR for the HCEL with EPP. For this SNR variation, the area to approach the optimal TC is smaller and away from the equally partitioned point as presented in Fig. 6(d). Thus, the HCEL with EPP performs worse than the HCEL with OPP, as shown in Fig. 5(c). The ability to adapt the precoding sets to the different SNR variation makes the HCEL with OPP scheme always achieve the optimal performance. To view the detail of loading results, we give the IBPS and power level of each subcarrier for all the adaptive loading algorithms in Fig. 7. In Figs. 7(a) and 7(c), for the EL schemes, there are three different IBPS corresponding to the NPS. The IBPS distribution of HCEL with OPP is similar to that of LC scheme. Though with the same precoding pattern, HCEL with EPP outperforms the ICEL with EPP in terms of IBPS thanks to low shaping loss.

 

Fig. 7. The allocation results of different adaptive loading algorithms. (a) The IBPS and (b) signal power level versus subcarrier indices when the length of SSMF is 20 km and OSNR is 34 dB. (c) The IBPS and and (d) signal power level versus subcarrier indices when the length of SSMF is 80 km and OSNR is 34 dB.

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4. Experimental results and discussions

To evaluate the performance of the adaptive loading algorithms, we construct the experimental setup for SSB-DMT system. The setup is similar to the simulation model except the models are replaced by real instruments or devices, as shown in Fig. 8.

 

Fig. 8. (a) The experimental setup for SSB-DMT systems, and (b) the main system parameters. AWG: arbitrary waveform generator, DSO: digital storage oscilloscope, Sub.: subcarrier, Syn.: synchronization. CE: channel estimation.

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An arbitrary waveform generator (AWG, Keysight M8195A) is used to output the baseband signal in the transmitter. The optical carrier is generated by a tunable laser source (Coherent Solutions LaserBlade, with 100 kHz linewidth and 1549.67 nm center wavelength). The DD-MZM (Fujitsu FTM7937EZ611) is biased at the quadrature point for SSB modulation. The VOA1 in the front of an EDFA (Accelink OLP-EDFA-VGA, about 5 dB noise figure and 30 dB gain) is used to control the ROP. The OBPF aims to reduce out-of-band ASE noise from the EDFA and suppress the null band. The VOA2 keeps the optical power to the PD (New Focus Model 1014, with 3 dB bandwidth of about 40 GHz) constant at 2 dBm due to the poor sensitivity of the PD. And a digital storage oscilloscope (DSO, Lecroy 10-36Zi-A) samples the received signal in the receiver. More than 1 million sampling points are recorded for the NGMI and SNR calculation. The DSP is the same as that in simulation as depicted in Fig. 3(b).

For the SNR variation when the length of fiber is 80 km and the received optical power (ROP) is −15 dBm, as shown in Fig. 9(a), we give the results to find the optimal TC when the NPS is 2 and 3 in Figs. 9(b) and 9(c), respectively. It is clear that there only exists an optimal point for different NPS. The insets are the relative precoding patterns, where the optimal patterns of OPP are obviously different from the pattern of EPP.

 

Fig. 9. The results to find the optimal precoding pattern of HCEL with OPP scheme. (a) The SNR variation, (b) the theoretical capacity versus the number of subcarriers in precoding set 1 when NPS is 2, (c) the theoretical capacity variation with the number of subcarriers in precoding set 1 and 2 when NPS is 3. Insets: the allocated results of precoding sets. Different colors denote different sets. And the subscript f and g denote the subcarrier indices before and after sorting, respectively.

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The NPS is one of the most important parameters of the partitioned precoding scheme. Thus, we first investigate the impact of NPS as shown in Fig. 10. In accordance with the simulation results, the NDR increases with the NPS but tends to be flat when the NPS is larger than 3. Therefore, we still choose 3 as the NPS to evaluate the performance of EL schemes under different ROP. The results are presented in Fig. 10(b), where the NGMI of every point meets the constraint of FEC limit. It is obvious that the HCEL with OPP scheme achieves the largest NDR, followed by the HCEL with EPP. The third is the ICEL with EPP, and the worst scheme is the LC. Compared to the LC, the HCEL with OPP achieves 4.4 dB receiver sensitivity gain when NDR is 60 Gb/s as well as 9% NDR improvement when ROP is −15 dBm. Besides, thanks to the better precoding pattern, the HCEL with OPP has 1.2 dB receiver sensitivity gain over the HCEL with EPP when NDR is 60 Gb/s. The SNR variation for all adaptive loading schemes is shown in Fig. 10(c). The EPP schemes have three SNR ladders with equal interval. While the HCEL with OPP scheme has the similar SNR distribution with LC scheme. In addition, the constellations in the insets of Fig. 10(c) differ the EL schemes and the BPL scheme. That is, the constellation of LC scheme shows rectangular shape while those of EL schemes are circular. Furthermore, compared to the ICEL, the HCEL schemes have more compact constellations.

 

Fig. 10. The results after 80km SSMF transmission: (a) The relationships between the NDR and the NPS for EL schemes, (b) the net data rate versus ROP where the NPS of EL schemes are 3, (c) the SNR variation. insets: relative constellations.

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We also evaluate the adaptive transmission schemes with the SSMF of different length. The Figs. 11(a) and 11(b) show the results for OB2B and 15 km transmission scenarios, respectively. The EL schemes hold the advantage over the BPL scheme. Compared to the LC, the HCEL with OPP achieves 4.1 dB receiver sensitivity gain in OB2B scenario when NDR is 84.0 Gb/s and 3.4 dB receiver sensitivity gain after 15 km transmission when NDR is 73.5 Gb/s. And the difference between HCEL with OPP and HCEL with EPP becomes small. This is due to the approximately linear SNR variation as shown in Fig. 11(c). For this SNR variation, the equally partitioned precoding pattern approaches the optimal precoding pattern, thus the EPP algorithm has the reduced gap to the OPP algorithm, which has been explained in Section 3.

 

Fig. 11. (a) The relationships between the NDR and the NPS for EL schemes in OB2B scenario, (b) the relationships between the NDR and the NPS for EL schemes after 15 km SSMF transmission, (c) the SNR variations when length of SSMF is 15 km and ROP is −19 dBm.

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The complexity for the loading schemes is analyzed as follows. The HCEL schemes, OPP and EPP, achieve much lower complexity than conventional EL by reducing the type of PS-QAM. In this work, we use 240 data subcarriers, which means maximal 240 pairs of DM and inverse DM are required for conventional EL. However, the HCEL schemes can achieve good enough performance when the number of pairs is 3, 1/80 of conventional EL. Since the OPP requires the searching procedure to find the optimal subcarrier pattern, it has higher complexity than EPP. Though the BPL individually allocates modulation format for each subcarrier, no need for DMs and inverse DMs makes it the loading algorithm with the lowest complexity in this work. Thus, the complexity relationship can be summarized as conventional EL >> HCEL with OPP > HCEL with EPP > BPL.

5. Conclusion

From the perspective of practical applications, we adopt the IBPS as the performance metric considering the shaping loss from the off-the-shelf FEC code with fixed code rate and the CCDM with finite output length. Under the constraint of NGMI as the FEC limit, the relationships between the SNR and the PS-QAM with the maximum IBPS are obtained by Monte-Carlo simulation in an AWGN channel. We found that the PS-QAM with lower order has higher IBPS and gave the optimal constellation for different SNR region with the concatenated code of 20% LDPC code and 6.25% staircase code. Based on the result, we proposed the HCEL scheme for the partitioned SSB-DMT systems. Two partitioning methods, EPP and OPP, are proposed and evaluated. As demonstrated by the simulation and experiments, more precoding sets can achieve larger NDR and 3 precoding sets are large enough to realize a good performance. Compared to the general hundreds of DMs adopted by the conventional EL, only 3 DMs are required for the proposed HCEL with adaptively partitioned precoding. In addition, the HCEL with OPP scheme always achieves the largest NDR compared to HCEL with EPP scheme and LC scheme. In experiments when the length of SSMF is 80 km and NDR is 60 Gb/s, HCEL with OPP scheme achieves 4.4 dB receiver sensitivity gain compared to LC scheme and 1.2 dB receiver sensitivity gain compared to HCEL with EPP scheme. Therefore, the proposed HCEL scheme is a promising solution for the practical application of EL to enhance the performance of short-to-medium reach transmission systems.