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Abstract
The combination of probabilistic shaping (PS) technology and forward error correction (FEC) technology can significantly boost the performance of a transmission system. In this paper, we propose a probabilistic shaping distribution matching algorithm employing uneven segmentation for data center optical networks, while keeping extremely low computational complexity for both encoding and decoding. Based on the proposed probabilistic shaping distribution matching algorithm, we develop a novel integrated scheme of PS and FEC coding that lifts the restrictions on the use of FEC technology and increases the use of interleaver. An experiment used to evaluate the probabilistically shaped data transmission is successfully conducted over a 25 km standard single-mode fiber (SSMF) with 16 quadrature amplitude modulation (16-QAM). Simultaneously, we use a simulation software to analyze the bit error rate performance at higher resolution. The results show that the joint coding scheme can achieve a 0.4dB performance improvement compared with the single FEC system.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Over the past decade, due to the explosive growth of Internet-based services such as cloud computing and stream video, the landscape of data transmission has been reshaped to have more and more data flows concentrated on the inter-data center or intra-data center, demanding an explosive growth on the capacity of the current short reach communication system [1–4]. However, the data center optical network has its unique characteristics to take into account the special requirements for power consumption and delay while pursuing high speed.
As an effective measure to bridge the gap to the Shannon limit, probabilistic shaping (PS) has attracted a lot of research interests recently. Communication channels often have non-uniform capacity-achieving input distributions [5], which is the main motivation for probabilistic shaping (PS). In other words, a practical transmission scheme using non-uniform distribution at the channel input was developed. In doing so, the constellation average power is reduced and PS gain is acquired. Different probability distributions tailored for transmission over the optical fiber were studied [6–9]. Many different PS schemes have been proposed in the literatures and their performance improvement has also been verified. For instance, the dyadic probabilistic shaping (PS) for pulse amplitude modulation (PAM) has been investigated and experimentally demonstrated in [10], and achieved SNR gains of 0.61 and 1.74 dB for PAM-4 and PAM-8, respectively. The rate flexibility and probabilistic shaping gain of 4-dimensional signaling was experimentally tested for short-reach, unrepeated transmission [11]. A PS-1024-QAM OFDM fiber transmission in a low-cost intensity modulation combined with direct detection (IM/DD) system is demonstrated, which obtained higher achievable-information-rate performance and stronger nonlinearity robustness [12]. In the literature [13], 106-Gbaud PS-PAM-8 (2-bit/symbol) signals is demonstrated with BER of 3.8×10−3 (7% HD-FEC threshold) over 1-km non-zero dispersion-shifted fiber (NZDSF) transmission based on pre-equalization (Pre-EQ) and clipping technique for dual-lane 400-GE short-reach applications.
Many transmission experiments have been carried out showing the superiority of PS in optical communications, but only few of them have included forward error correction (FEC). The only important milestone for combining PS with FEC was the invention of probabilistic amplitude shaping (PAS) [14], which concatenates a shaping outer code called distribution matcher [15] with an FEC inner code. Researchers have also carried out some improved based on the PAS scheme. For example, in the literature [16], the PAS scheme was applied to 4D modulation format and achieved good results. However, this solution has some limitations. Specifically, only a limited number of bits can be used as FEC check bits and the used type of FEC needs to meet the assumption of uniform distribution [17], for example, the Polar code that does not contain the original information sequence after encoding is not suitable for this scheme. As a matter of fact, the optical network of the data center is more sensitive to power consumption and delay, which requires PS technology with lower complexity and wider FEC technology compatibility (especially low-complexity hard-decision coding). Therefore, it is not surprising that we should focus on exploring new PS technologies and joint coding schemes to further unleash the enormous potential.
In this paper, we propose a probabilistic shaping distribution matching algorithm applied to data center optical networks with characteristic of low computational complexity. Based on this algorithm, we develop a new integrated scheme of PS and FEC coding, which lifts the restrictions on the use of FEC technology and increases the use of interleaver. Energy efficiency and bit error rate performance have been greatly enhanced by employing our proposed scheme. A 14 Gbaud 16-QAM probabilistic shaping data transmission over 25 km SSMF is successfully carried out in the experiment.
2. Principle of the probabilistic shaping coding scheme
First, we conduct a brief review of the PAS probabilistic shaping system. In the probability shaping system, the shaping gain is obtained by artificially changing the constellation point probability to meet a specific distribution. Considering the N-dimensional vector xN transmitted on an additive white Gaussian noise (AWGN) channel, the input distribution that reaches the capacity limit is Gaussian distribution. In practice, the sent information sequence is a random sequence that satisfies a uniform distribution. A device called distribution matcher is used to convert the signal into a non-uniformly distributed shaping sequence. Distribution matcher needs to ensure that the output shaping sequence meets a certain target distribution. After that, the FEC technology transforms the shaping sequence into an error correction coding sequence and endows the signal with anti-noise ability. Finally, the whole information transmission is achieved through modulation, fiber channel, demodulation and de-shaping. The probability shaping transmission system of the traditional PAS scheme is shown in the Fig. 1.
As shown in the Fig. 1, PS-Encoder and PS-Decoder form a distribution matcher, which is the core of the probabilistic shaping system. In summary, the distribution matcher needs to meet the following characteristics: (1) provide a binary interface for the input source coding part of the digital communication system; (2) recover the input from the output (i.e. the mapping is reversible); (3) the output is approximately Gaussian distribution.
2.1. Multi-set shaping method
We propose a novel probabilistic shaping method which meet the above requirements based on multi-set mapping. The core of the method is to construct a one-to-one mapping applied to coding, and the principles are as follows.
For an M-QAM modulation system, each constellation point is represented by an m = log2(M) bits binary sequence, where M is the order of modulation. We build a series of code sets based on the constellation diagram, and the specific rules to determine the sets are as follows. First, define the set S1 of size n1=2m, and store the labels of all constellation points in the set. Then, select 2m-1 symbols with lower constellation point energy in S1 to construct another set S2, and define n2 = 2m-1. In particular, there may be cases where the same energy points are not completely selected, and the principle of maximum Euclidean distance is used to solve it. Repeat this process until the last set Sm-1 with nm-1=22=4 labels are selected from Sm-2. Finally, any number of sets can be selected according to the desired constellation distribution, and the number of selections is represented by S. The selected sets are combined to generate the mapping codewords, thereby performing a one-to-one mapping with the block sequence of the precoding sequence. After the mapping, the appearance probability of the constellation points will be divided into S levels and achieve a similar Gaussian distribution.
For example, we use the 16-QAM system for a more intuitive description. The constellation diagram is shown in Fig. 2, where m = log2(M) = 4. According to the above method, the set S1={0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111}, S2={1011 0011 1001 0001 1010 0111 0000 1101}, and S3={1011 0011 0001 1001} are determined. As mentioned earlier, the number of selected sets is not fixed. For example, S1 and S3 can be combined to construct coding sequences. Under this scheme, a mapping table with a block length of log2 (n1·n3) = 6 can be constructed. And because the selected numbers of set are 2, the sequence length after encoding is 2·m=8, which realizes the shaping coding of 6/8 code rate. Similarly, when all the sets are combined to construct coding sequences, 9/12 mapping coding can be realized. The constellation points probability after shaping under the two schemes are shown in the Fig. 3.
As can be seen, this method satisfies the three characteristics of shaping matching: (1) the input of the system is random binary sequences; (2) the system is reversible based on one-to-one mapping; and (3) the information distribution after shaping meet the Gaussian distribution.
2.2. Uneven segmentation distribution matching algorithm
The proposed method implements probabilistic constellation shaping by transmitting more points in small subsets. On this basis, we propose a low-complexity algorithm applied to the previous shaping method. The algorithm further simplifies the computational complexity of the system by constructing a specific relationship between constellation points and labels. In particular, this relationship simplifies the table lookup process required in the mapping to a simple shape-bit addition operation. We call this algorithm as the uneven segmentation distribution matching (UEDM) algorithm, and details of the algorithm are as follows:
The 16-QAM modulation is applied here as an example for analysis. The constellation diagram labeled by the algorithm is shown in the Fig. 4. Based on this, three sets are determined, namely S1={0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111}, S2={0000 0001 0010 0011 0100 0101 0110 0111}, and S3={0000 0001 0010 0011}. It can be seen that due to the special labels of the constellation points, the first bit of all symbols in S2 is 0, and the first two bits of symbols in S3 are both 0, which makes the generated codewords have a special structure. Therefore, the mapping can be realized only by adding 0 to the unevenly divided sequence. Among shaping sequences, the retained original sequences are called information bits, and the added redundant bits are called shaping bits.
Since the shaping sequence is composed of information bits and shaping bits, it also simplifies the PS-Decoder algorithm. At the receiving end, the original information sequence can be restored by deleting the corresponding shaping bits. We use the maximum number of retrieval times required to perform one decoding as an indicator of computational complexity analysis, and compare with the look-up table decoding method used in many representative technologies. Four types of sets selection conditions are specifically calculated, and the results are shown in Fig. 5.
In Fig. 5, the symbol (x. y) represents a block code that encodes x bits into y bits, so the code rate is x/y. The four cases are selection sets S2 and S3, selection sets S1 and S3, selection sets S1 and S2, and selection sets S1, S2 and S3. When the code rate is 9/12, the proposed algorithm only needs to perform three retrieval calculation operations to achieve mapping and demapping, while the look-up table method needs to be performed 512 times. Obviously, the proposed algorithm has significant advantages over traditional methods in terms of computational complexity. More importantly, as the block length increases, the reduction in computational complexity becomes more obvious.
What is more, the algorithm can be extended to high-order modulation formats. For example, in Polarization Division Multiplexing (PDM), a coherent transmitter modulates data into two polarizations, and in each polarization, the in-phase and quadrature components of the carrier are modulated separately. More specifically, in a polarization multiplexing system, it is only necessary to independently use algorithms for signals in a single polarization direction to achieve shaping coding.
2.3. Integration scheme of the shaping code and FEC
We observe that error propagation phenomena exist in traditional shaping codes. More specifically, the mapping encoding is one-to-one, and the Hamming distance between sequence before and after encoding may not be preserved. For example, when map the codeword “00000” to “0000 0000”, and map the codeword “00011” to “0000 0100”, that is, if these codewords are sent, the BER before decoding is 1/8, but the decoding BER is 2/5. The longer the code length increases, the more serious this issue becomes. We propose a joint coding scheme suitable for data center optical networks that can solve this problem.
Even in the traditional PAS scheme, the error propagation phenomenon still exists in the distribution matching part. However, since the shaping and decoding in PAS is completed after FEC decoding, the BER can be less than a specific threshold before and after shaping decoding to ensure reliable information transmission. This solution is not suitable for data center optical networks that are sensitive to complexity and cost, and the specific reasons are as follows. First, the code rate and code type of FEC are restricted. Specifically, the check bit can only be represented by a limited number of sign bits that do not affect the amplitude, so the constellation must have symmetry and the choice of code rate is limited. Second, the code type must meet the assumption of uniform distribution. In other words, the encoded codeword must retain the information sequence. Furthermore, most FEC technologies have poor effect on sequence that generates continuous errors. The solution to this problem mainly relies on the interleaving technology, and the position of the interleaver is generally deployed after the FEC module to take effect. However, it is impossible to deploy an interleaver in a solution that combines PAS-DM, because the interleaver will change the shaping sequence and the sequence distribution, making the distribution matcher ineffective.
In the distributed matching algorithm proposed in this paper, since the original information is stored in the coding sequence, the receiver can achieve decoding only by deleting the shaping bits. Thus, the error propagation does not occur during demapping. Based on this feature, we propose a joint coding scheme as shown in the Fig. 6. In this scheme, the original information sequence is first sent to the FEC module, and then to the interleaver to improve the signal resistance to continuous errors. Finally, the proposed UEDM distributed matcher is used to achieve probabilistic shaping to obtain shaping gain. Here, FEC usage restrictions are completely lifted, and any form of FEC technology can be well compatible. Moreover, the introduction of the interleaver further improves the overall anti-noise capability of the system. As far as we know, this is the first time that a joint coding scheme has been proposed in which the shaping code is followed by the FEC code.
3. Experiment setup and results
In the above, we have discussed the advantages of the proposed scheme over PAS in terms of computational complexity and FEC compatibility. Below we verify the shaping effect and joint coding effect provided by the proposed scheme under the special labeled constellation diagram where the PAS scheme becomes unavailable. The Fig. 7 shows an experimental device based on a coherent optical communication system. A 14Gbaud 16-QAM system on a 25km standard single-mode fiber is realized.
Fig. 7. Experimental setup. (ECL: external cavity laser; EA: electrical amplifier; PBS/PBC: Polarization Beam Combiner/Splitter; EDFA: Erbium Doped Fiber Application Amplifier; VOA: Variable Optical Attenuator;).
At the transmitting end, an external cavity laser (ECL) with a line width less than 100 kHz is used to generate a continuous light wave with a center wavelength of 1550 nm. Then two I/Q modulators are used to modulate the orthogonal light waves output by the polarization beam splitter, and the driving signals of each channel are obtained by using 25 GSa/s arbitrary waveform generator (AWG) and offline MATLAB program. The power of the received signal is adjusted by an optical attenuator in order to obtain different system optical signal to noise ratios (OSNR). At the receiving end, the optical signal is sent into an integrated coherent receiver (ICR) to beat with the local oscillator (LO) signal. Offline digital signal processing is applied to recover the transmitted signal, including low-pass filtering, Gram-Schmidt orthogonalizing process (GSOP), clock recovery, frequency offset estimation (FOE), CPR, and BER calculation.
To verify the advantages of the proposed UEDM algorithm, we use four different coding rules with different parameters in Table 1 to conduct experiments. In order to achieve a fair comparison, we selected a different baud rate for each case to have the same information rate.
Table 1. Modulation format parameters
The successful transmission of the UEDM-LDPC-16QAM signal means that our novel scheme is reasonable and achievable. The entire constellations of the received signals are shown in Fig. 8. At the same time, the probability distribution of the constellation points achieves the goals of dense low-energy point and sparse high-energy point.
Fig. 8. Constellation of (a) 6/8UEDM-16-QAM, (b) 9/12UEDM-16-QAM, (c) LDPC-16-QAM, (d) PS-LDPC-16-QAM.
Through experiments, we compare the transmission performance of the proposed scheme under different encoding formats in detail. The distribution matching algorithm used in the experiment is the UEDM algorithm proposed in this paper, and the shaping codes of 6/8 and 9/12 are constructed from sets 1, 3 and sets 1, 2, and 3 respectively. At the same time, we use DVB-S2 standard low-density parity check code (LDPC) for experimental verification [18], where the block length is 64800. In Fig. 9, the decoding BER is shown as a function of optical signal to noise ratio (OSNR). Firstly, we can see that compared with the traditional 16-QAM system, in the OSNR range of interest, the system that have undergone PS achieve better BER performance than tradition 16-QAM. When the OSNR is about 18dB, the error rate of the probabilistic shaping system reaches the hard decision FEC threshold. Therefore, reliable information transmission can be achieved by using standard hard decision FEC in the proposed integration scheme. On the contrary, under the condition of the same optical signal-to-noise ratio, the traditional 16QAM system has a bit error rate of about 0.01, which cannot meet the standard of reliable transmission. It is worthwhile to mention that the UEDM systems under the two code rates show similar BER performance. This is because the signal constellation points of two codes have similar average energy and obtain similar shaping gains. Secondly, we experiment and verify the superiority of the combination of PS and soft decision FEC technology in proposed joint coding scheme. In the range of resolution that the experiment can provide, the combined scheme shows a performance gain of about 0.4dB compared to the single FEC scheme. The joint coding scheme achieves error-free transmission under an optical signal-to-noise ratio of 5.9dB, while a single FEC scheme needs at about 6.3dB to achieve the same performance.
Moreover, in order to show the performance of the joint coding scheme when using soft decision FEC better, we use simulation software to further compare the bit error rate performance at higher resolution, and the result is shown in Fig. 10. It can be seen that all codes enter the waterfall zone when the signal-to-noise ratio (SNR) is 7 dB and then enter the error floor zone when the SNR is 8 dB. However, the difference is that the joint coding scheme reduces the error floor. In the joint coding scheme, the performance improvement brought by 9/12UEDM becomes more obvious, making the system reach a reliable transmission under the condition of a SNR of 11 dB. What is more, under the premise that the joint coding scheme maintains the same overall coding rate, due to the use of low-complexity shaping technology, the overall computational complexity of the system is lower too. The above data strongly proves the superiority of the proposed scheme.
4. Conclusions
We propose a probabilistic shaping coding algorithm called UEDM, which has extremely low coding and decoding complexity and is suitable for data center optical networks. In addition, the proposed algorithm has a special advantage, that is, the original information sequence is preserved in the shaping sequence. Based on this feature, a new integrated scheme of PS and FEC coding is constructed, which eliminates the restriction on the use of FEC technology in the joint coding technology and increases the use of interleaver. An experiment has been successfully carried out, and 14 Gbaud probability shaping data transmission has been successfully completed on 25 km SSMF. By adopting our method, energy efficiency and bit error rate performance can be greatly improved. The experiment and simulation results suggest that our proposed scheme will have a promising and future-proof application in next generation data-center system.
Funding
National Key Research and Development Program of China (2018YFB1801705); National Natural Science Foundation of China (61605013, 61727817, 61835005, 61875016, 62021005).
Disclosures
The authors declare no conflicts of interest.
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