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Abstract
In this paper, a polar coded probabilistic shaping (PS) 8-ary pulse amplitude modulation (PAM8) based on many-to-one (MTO) mapping is investigated for short-reach optical interconnection. By ingeniously assigning parity bits to ambiguities positions, no extra PS redundancy and no complex distribution matcher are required in the scheme comparing to traditional probabilistic amplitude shaping (PAS). The noise distributions after different transmission distances are studied and an optimal clock recovery method for PS signal is proposed to degrade the impact of severe eye skew effect on BER performance. The experimental results show that up to 1.2 dB and 0.8 dB shaping gains are respectively achieved over back-to-back (BTB) and 2-km standard single mode fiber (SSMF) transmission. With the help of the proposed optimal clock recovery method in the PS scheme, the shaping gain is improved from 0.15 dB to 0.4 dB after 10-km transmission. Moreover, compared to low-density parity-check (LDPC) code, the polar coded PS-PAM8 can provide an additional coding gain of 2.2 dB with code length of 256, which proves the performance superiority of polar code in short code length. Therefore, the proposed polar coded PS-PAM8 with low complexity and satisfactory BER performance is believed to be an alternative solution for the cost-sensitive short-reach optical interconnection.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Driven by the upcoming services such as virtual reality (VR), 4K/8K video applications and artificial intelligence (AI), the demand for short-reach optical interconnection increases rapidly in fifth-generation wireless systems (5G) front haul, metro access and data-center (DC) communications [1]. Compared to coherent detection employed in backbone network, the intensity modulation with direct detection (IM/DD) are extensively used in short-reach transmission due to its cost efficiency and structure simplicity [2,3]. Among many advanced modulation formats such as discrete multi-tone (DMT) and carrierless amplitude and phase modulation (CAP), multi-level pulse amplitude modulation (PAMm) has received much attention from academia and industry alike [4–9]. Especially, PAM8 is believed as a promising candidate for next-generation 800-Gbps or 1-Tbps Ethernet [10–12]. However, the extended number of levels surely results in stricter requirement of signal-to-noise ratio (SNR) and the low-cost devices induce severe nonlinear distortion, which makes it difficult to achieve the adequate optical power budget.
As an alternative approach, probabilistic shaping (PS) technique is regarded as a promising method to enhance system performance with non-uniformed probability distribution [10]. By transmitting the low-amplitude symbols more frequently than high-amplitude ones, it can reduce the average power of the transmitted signal and improve receiver sensitivity. Meanwhile, advanced forward error correction (FEC) technique can be used for PS signals to further improve the overall optical power budget and guarantee reliable transmission [11]. However, an incompatible problem of amalgamating the PS technique with FEC coding is that the parity bits are uniformly distributed (UD) and the symbol distribution after FEC encoding is not maintained. An elegant solution is the probabilistic amplitude shaping (PAS) architecture [12] which commonly employs constant composition distribution matcher (CCDM) [13] combined with low-density parity-check (LDPC) coding, and it is widely developed in coherent optical communications [14–16]. Although the PAS scheme achieves the desired shaping gain for coherent quadrature amplitude modulation (QAM), the high complexity of CCDM is a great challenge in cost-sensitive short-reach IM/DD transmission. Secondly, there still exists a high redundancy in the PAS system because extra shaping codes are required for CCDM processing besides the LDPC encoding [17]. Thirdly, a single bit error in one CCDM word can affect the whole frame, which will lead to a sharp increase of the error rate after inverse CCDM [18].
Recently, the many-to-one (MTO) mapping [19–21] is known as an efficient way to realize PS approach with FEC technique. It has a relatively low implementation complexity and redundancy sharing characteristic without the complicated distribution matcher and extra shaping redundancy, which could be a promising candidate for the low-cost PAM system. In [22], probabilistic shaping for PAM4 and PAM8 modulation based on MTO mapping is demonstrated over vertical-cavity surface-emitting-laser (VCSEL) optical interconnection. The PS-PAM4 system with MTO mapping and LDPC coding using Mach-Zehnder modulator (MZM) is achieved with superior receiver power sensitivity and enhanced spectral efficiency [23]. However, the iterative decoding between de-mapper and decoder is commonly required to recover the ambiguities bits in the MTO scheme, which also brings high computation complexity and data latency.
On the other hand, as for soft-decision (SD) FEC technique, the polar code [24] with low encoding and decoding complexities develops rapidly and has been chosen as a channel coding scheme in the 5G standardization process [25]. Assisted by the successive cancellation (SC) decoding or successive cancellation list (SCL) decoding, it can avoid the inherent error floor problem in iterative decoding [26]. Thus, benefiting from low complexity and high capability, the polar code is a strong candidate to LDPC [27] for short-reach PAM transmission where the requirements of low cost and low power consumption are vital [28].
In this paper, for the first time, we introduce a polar-coded PS-PAM8 signal by mapping 16-point 4-bits/symbol PAM16 into 8-point PAM8 based on the MTO method. By ingeniously assigning parity bits to ambiguities positions, no iterative decoding are required between the de-mapper and the decoder. Using a 10-GHz commercial directly modulated laser (DML), the PS-PAM8 transmission at a net rate of 28 Gbit/s over 10-km standard single-mode fiber (SSMF) is experimentally demonstrated in C-band without dispersion compensation. The noise distributions after different transmission distances are studied and an optimal clock recovery method for PS signal is proposed to degrade the impact of severe eye skew effect on BER performance. Results show that, up to 1.2 dB and 0.8 dB shaping gains @BER=10−3 are achieved with the optimal SCL decoder over back-to-back (BTB) and 2-km SSMF transmission, respectively. Aided by the optimal clock recovery method, the shaping gain is increased from 0.15 dB to 0.4 dB after 10-km transmission. Moreover, comparing to LDPC coded PS-PAM8, the polar code can provide an additional coding gain of 2.2 dB @BER=10−3 with the code length of 256. Therefore, the proposed polar coded PS-PAM8 system would be an alternative solution for the cost-sensitive short-reach optical interconnection considering low complexity and satisfactory performance improvement.
The rest of this paper is organized as follows. Section 2 describes the principle of polar-coded PS-PAM8 based on MTO mapping. The experimental setup is illustrated detailedly in Section 3. Section 4 investigates the performance after different transmission distances. An optimal clock recovery method is proposed to counter the eye skew effect and improve the shaping gain in the PS scheme. The comparison between polar code and LDPC code for PS-PAM8 is also provided. Finally, the conclusion is drawn in Section 5.
2. Principle
The principle of polar coded PS-PAM8 system based on MTO mapping is illustrated in Fig. 1. The pseudo random binary sequence (PRBS) U as the input data stream is encoded by the systematic polar encoder [29] to form the codeword $C = ({{c_0},{c_1},\ldots ,{c_i},\ldots } )$. The method of Gaussian approximation [15] is used here for the construction of polar code to obtain the capacity of different bit channels [30]. Different from non-systematic polar encoder, the information bits are assigned to bit channels with high reliability at the output side to form the bits sequence ${Q_J}$, where J is the set of indices known as information bit indices. Then, the frozen bits are set to 0 and assigned to bit channels with low reliability at the input side to form the bits sequence ${Y_{Jc}}$, where $Jc$ is the set of indices called as frozen bit indices. In each code unit, the polar coded data ${C_N}$ can be defined as
Fig. 1. MTO mapping scheme for polar-coded PS-PAM8. (a) The PS-PAM8 probability distribution used in this paper.
Considering that the SNR is mainly constrained by the received optical power including the optical carrier, the PS-PAM8 should employ the asymmetric distribution (AD) where low-amplitude symbols have higher probabilities [31]. The dyadic approximation from the optimal distribution can be found by geometric Huffman coding. For simple implementation, a typical probabilistic distribution with {1/4, 1/4, 1/8, 1/8, 1/16, 1/16, 1/16, 1/16} in [22] is adopted for the MTO mapper. Based on the MTO rule, 4-bits sequences are assigned to eight amplitudes of PAM8 signals. For instance, the PAM8 symbol ‘7’ is mapped from single codeword ‘1001’, while symbol ‘0’ is mapped from four codewords, ‘0000’, ‘0001’, ‘0011’ and ‘0010’. Note that this mapping creates ambiguities in some bit positions as shown in Fig. 2(a) marked by ‘X’. These ambiguities bits can be distinguished by iterative information exchange between PS de-mapping and FEC decoding, which greatly increases the system complexity [32]. To depress the effect of ambiguities without additional complexity, we respectively assign the data bits D and the parity bits P to the first two bit positions and the last two bit positions of 4-bits sequences. The data bits are all deterministic and the ambiguities only occur on parity bits of polar code. These uncertain parity bits are implicitly punctured and their log-likelihood ratios (LLR) are set to zero in the polar decoder. Since the decoding performance of polar code will not be significantly affected by puncturing, the iterative decoding between PS de-mapper and polar decoder is not indispensable in the proposed scheme. The polar coded UD-PAM8 signal is also modulated according to [33] for comparison.
Fig. 2. Experimental setup. (a) The electrical spectra of the received signals with different transmission distances. Eye diagrams of (b) UD-PAM8 and (c) PS-PAM8 after back-to-back (BTB) transmission when received optical power is −19 dBm. (AWG: arbitrary waveform generator; DML: directly modulated laser; SSMF: standard single-mode fiber; VOA: variable optical attenuator; EDFA: erbium-doped fiber amplifier; OBPF: optical band-pass filter; PD: photodetector; DSO: digital sampling oscilloscope)
3. Experimental setup
To investigate the performance of polar coded PS-PAM8 based on MTO mapping, experiments have been carried out directly as illustrated in Fig. 2. At the transmitter, the systematic polar encoder is used to generate the coded bits, and a systematic LDPC encoder based on the progressive edge–growth (PEG) construction method [34] is also employed for comparison. After interleaving by a pseudo-random interleaver, the MTO mapper converts data bits and parity bits into PS-PAM8 signals according to the mapping rule. Then the offline signals are ported to an arbitrary waveform generator (AWG, Keysight 8195) with 65-GSa/s sampling rate and the output 14-Gbaud signals are amplified by an electrical amplifier. A commercial 10-GHz DML (NEL NLK1551SSC) operating at 1549.2-nm is used for modulation and a variable optical attenuator (VOA) is utilized to vary the received optical power after SSMF transmission. At the receiver, the optical signal is detected by a commercial photodetector (PD, Finisar XPDV2120R). Since no inline trans-impedance amplifier (TIA) is cascaded behind PD, the combination of erbium-doped fiber amplifier (EDFA) and an optical band-pass filter (OBPF) is inserted before PD to amplify the signal and remove out-of-band noise. Subsequently, the detected signals are captured by a real-time digital sampling oscilloscope (DSO, LeCroy LabMaster) with 80GSa/s sampling rate, and the offline digital signal processing (DSP) is carried out in MATLAB. The timing offset and jitter are eliminated by the clock recovery (CR) employing a Gardner phase detector [35] with transition selection. The equalization is conducted using a Volterra nonlinear equalizer (VNLE) [36] to depress the nonlinear distortion, where the 3-rd order kernels of the VNLE are optimized as (131,5,1). After the MTO de-mapper, the LLR are fed into the de-interleaver and the soft decoding is performed with the FEC decoder. The SCL decoding is used in the polar decoder, whereas the belief-propagation (BP) decoding algorithm is employed in the LDPC decoder. Finally, we evaluate the BER performance by direct error counting. Note that the UD-PAM8 signals are also demonstrated to investigate the shaping gain of the proposed scheme. The rates of polar codes are set to 2/3 and 1/2 for UD-PAM8 and PS-PAM8 signals, respectively, to obtain the same information rate of 28-Gbps (UD-PAM8: 14 Gbaud×(3×2/3) bits/symbol = 28 Gbps; PS-PAM8: 14 Gbaud×(4×1/2) bits/symbol=28 Gbps).
The measured power spectrum density (PSD) of received signals with different transmission distances are shown in Fig. 4(a). Compared to the BTB condition, in the high frequency, the PSD after 2-km SSMF transmission degrades while the PSD after 10-km SSMF transmission increases. It is because that in DML-based transmission system, the T-fading occupies the leading role with short distance while the A-fading is the dominant factor with relatively long distance [37]. The eye diagrams of UD-PAM8 and PS-PAM8 after BTB transmission are respectively depicted in Fig. 4(b) and 4(c). It can be observed that the average value of PS-PAM8 is located close to symbol ‘2’ instead of the middle of symbols ‘3’ and ‘4’ as in UD-PAM8, which reveals the superiority of the PS-PAM8 signal in power consumption. For a fair comparison, the bias current of the laser diode is adjusted to make the same extinction ratio of both signals according to [31]. As a result, the average output power of DML for the PS signal is reduced to 6.4 dBm (from 7 dBm for the UD signal).
Fig. 3. The eye diagrams of the received signals (a) after BTB transmission; (b) after 2-km transmission; (c) after 10-km transmission. The probability density distributions (d) after BTB transmission; (e) after 2-km transmission; (f) after 10-km transmission.
Fig. 4. The BER performance for UD-PAM8 and PS-PAM8 with different clock offsets after 10-km transmission when received optical power is −20 dBm. The symbol period is noted as 2 T and the estimated noise standard deviations [${\sigma _1}$, ${\sigma _2}$, ${\sigma _3}$, ${\sigma _4}$, ${\sigma _5}$, ${\sigma _6}$] = [0.0572, 0.0542, 0.0540, 0.0710, 0.0750, 0.0808].
4. Results and discussion
4.1 Pre-FEC performance and optimal clock recovery method
Figure 3 shows the eye diagrams of received signals and the probability density distributions with different transmission distances. For the BTB case, the slight eye skewing effect is observed in Fig. 3(a) due to the chirp of DML. After clock recovery and equalization, the residual nonlinear impairments can be seen from Fig. 3(d), where the noise power induced by shot noise and relative intensity noise increases gradually with the increase of signal amplitude. For the case of 2-km transmission, the eye skewing effect becomes more obvious and the SNR decreases slightly as shown in Fig. 3(b) and 3(e). Figure 3(c) depicts severe eye skewing effect after 10-km transmission due to the interplay of the DML frequency chirp and chromatic dispersion [38]. Even after nonlinear equalization, the residual nonlinearity caused by eye skew still exists, which brings high noise power to both sides of the amplitude as shown in the probability distribution of Fig. 3(f).
In coherent optical system, the power sensitivity gain of PS signal is mainly contributed by two aspects: 1) reduced optical power with fixed Euclidean distance and 2) higher tolerance to nonlinearity due to low transmitted probabilities of high amplitudes. These contributions are also applicable to the PAM8 signals after BTB and 2-km transmission, where the PS-PAM8 with asymmetric distribution can reduce the output power of the transmitter and the probability of bit error at high amplitudes. However, in the case of 10-km transmission, the AD in the proposed PS-PAM8 become incompatible with the eye skew induced nonlinearity. The high noise power in the low amplitudes like ‘0’ and ‘1’ will produce more error bits and degrade the shaping gain. To deal with the nonlinearity, one method is to use the symmetric distribution (SD) with high probabilities at the middle-amplitude symbols [39]. Nevertheless, the signal average power will become larger and the shaping gain generated by the optical power reduction in the first aspect will be significantly reduced. Moreover, the coded modulation needs to change adaptively when transmission distances vary, which is complicated in terms of implementation complexity. In this paper, as an alternative solution, an optimal clock recovery method is proposed to reduce the noise power in low amplitudes by introducing a fixed clock offset. As shown in Fig. 4, when the clock offset increases from 0 to 0.2T (where T is the half symbol period), the pre-BER of UD signals keeps increasing. Thus, the signals after conventional clock recovery are ideal sampled and the optimal clock offset for UD is set to 0. However, in the case of PS-PAM8 signals, the pre-BER decreases firstly and then increases. The decrease part is due to the reduced noise power in low amplitudes as presented in Fig. 4(a) and Fig. 4(b). Taking the symbol ‘0’ as an example, the estimated noise standard deviation decreases from ${\sigma _1}\textrm{ = }0.0572$ to ${\sigma _2}\textrm{ = }0.0542$ as the clock offset increases from 0 to 0.1T. Thus, the performance of PS distribution with high transmitted probability in low amplitudes will be improved. When the clock offset is greater than 0.1T, the BER increases instead. The reason is that the large clock offset will increase noise power in high amplitudes as depicted in Fig. 4(c). Taking the symbol ‘7’ as an example, the estimated noise standard deviation increases from ${\sigma _5}\textrm{ = }0.075$ to ${\sigma _6}\textrm{ = }0.0808$ as the clock offset increases from 0.1T to 0.2T. Therefore, the optimal clock offset for PS scheme is found to be 0.1T and the optimal sampling instant is shown by the solid line in the eye diagram of received PS-PAM8 in insert of Fig. 4.
The pre-FEC performance after 10-km transmission with different clock offsets is depicted in Fig. 5. It can be noticed that the BER of PS-PAM8 decreases with the optimal clock recovery method under all received optical power. Although the decrease of pre-FEC BER is not particularly obvious, it will also bring a significant improvement after polar decoder, which will be illustrated as following.
Fig. 5. The pre-FEC performance for UD and PS signals after 10-km transmission with different clock offsets.
4.2 Performance of polar coded UD-PAM8 and PS-PAM8
To further investigate the post-FEC performance of the demonstrated system, the SC decoder and SCL decoder with different list sizes for UD-PAM8 and PS-PAM8 are respectively analyzed. Taking the BTB transmission as an example, the BER performance as a function of received optical power is presented in Fig. 6. Obviously, the performance of SCL decoder is much better than the SC decoder (L=1). Moreover, with the increasing decoding list size, the system performance becomes better at the cost of larger computational complexity. Interestingly, the optimal list sizes required for UD-PAM8 and PS-PAM8 are different. Note that an outer Bose–Chaudhuri–Hocquenghem (BCH) code having a threshold at 1 × 10−3 to realize the final BER of 1 × 10−15 is assumed here [40]. From Fig. 6(a), we notice that only 0.1 dB sensitivity gain is achieved @BER=10−3 when L increases from 8 to 16 and there is almost no BER difference between L=16 and L=32. However, in Fig. 6(b), about 0.5 dB power budget is obtained @BER=10−3 when L increases from 8 to 16 and more than 0.2 dB sensitivity gain can be further achieved using list size of 32. The reason is that some parity bits are punctured for MTO mapping in PS-PAM8 and more lists are required to eliminate the degradation of decoding performance caused by puncturing. Note that the maximum list size is set to 32 in this paper because its performance is very close to that of maximum likelihood decoding [26]. Therefore, the list size 16 and 32 are respectively chosen for UD-PAM8 and PS-PAM8 systems considering the tradeoff between computational complexity and system performance in the following investigation.
Fig. 6. The BER performance with SC/SCL decoding of different list sizes L for polar coded (a) UD-PAM8 and (b) PS-PAM8.
Next, the BER performance of PS-PAM8 and UD-PAM8 after transmission is studied. As shown in Fig. 7(a), in the case of BTB transmission, it can be observed that the PS-PAM8 achieves 1.2 dB shaping gain at the BER of 10−3 after polar decoding. The lower noise power and more distinct amplitudes for PS-PAM8 are observed in the probability distribution presented in insert (II) of Fig. 7(a). For the case of 2-km transmission, the fiber dispersion leads to an obvious decline in the overall performance both for UD-PAM8 and PS-PAM8. Meanwhile, the shaping gain after polar decoding decreases to 0.8 dB due to the influence of dispersion on noise distribution.
Fig. 7. The BER performance for polar coded PS-PAM8 and UD-PAM8 after (a) BTB transmission and (b) 2-km transmission. The probability distributions of (I) UD-PAM8 and (II) PS-PAM8 after equalization when received optical power is −20 dBm.
Then, the BER performance of polar coded UD-PAM8 and PS-PAM8 is investigated after 10-km transmission as shown in Fig. 8. It can be noticed that comparing to UD-PAM8, a slight degradation of pre-FEC BER is achieved using PS-PAM8. Due to the puncturing by MTO mapping, only 0.15 dB power budget appears after polar decoding for PS-PAM8. Using the proposed clock recovery method, the shaping gain of 0.4 dB is achieved after polar decoding, which reveals that the proposed clock recovery method is an efficient way to lessen the effect of eye skew in the PS scheme.
Fig. 8. The BER performance for polar coded UD-PAM8, PS-PAM8 and PS-PAM8 with clock offset after 10-km transmission. The probability distributions of (I) UD-PAM8 and (II) PS-PAM8 after equalization when received optical power is −20 dBm.
4.3 Performance of LDPC code and polar code for PS-PAM8
The measured BER performance for LDPC coded PS-PAM8 with BP decoding of different iteration numbers is shown in Fig. 9. As the iteration number I increases from 1 to 20, the BER after decoding decreases significantly. When the number of iterations is further increased to 30, the decoding performance is basically unchanged compared to that with the iteration number of 20. Thus, the BP decoding with iteration number of 20 is the best choice here.
Fig. 9. The BER performance with BP decoding of different iteration numbers I for LDPC coded PS-PAM8.
For comparison, the post-FEC performance of polar code and LDPC code is further explored for the MTO mapping based PS-PAM8 system. As shown in Fig. 10, using polar code instead of LDPC code, about 2.2 dB, 1.4 dB and 1 dB sensitivity gains are obtained in BTB case when the code lengths are 256, 512 and 1024, respectively. Similarly, the same trend can be noticed as code length increases after 2-km transmission. The reason is that the LDPC code does not perform well with limited code length compared to polar code and the performance gap decreases with the increase of code length [41].
Fig. 10. Measured BER performance for polar coded PS-PAM8 and LDPC coded PS-PAM8 with code lengths of (a) 256, (b) 512 and (c) 1024.
Finally, the computational complexity of the employed polar decoder and the LDPC decoder is investigated. Considering the higher cost of a multiplication compared to that of an addition, the required number of the real multiplications is used to represent the complexity of the decoder. The statistical results are described in Table 1, where q represents the column weight of LDPC code and equals to 3 here. Obviously, compared with polar decoder, LDPC decoder needs to increase the number of multiplications respectively by 53%, 38%, and 25% to approach the optimal performance when the code length is 256, 512, and 1024, which reveals the superiority in terms of decoding computational complexity for polar code. These results suggest that the polar code is a better candidate than LDPC code for the cost-sensitive PS-PAM8 system considering both high performance and low decoding computational complexity.
Table 1. Computational complexity for the employed polar decoder and LDPC decoder
5. Conclusion
In this work, for the first time, a polar coded PS-PAM8 signal based on MTO method is proposed and experimental demonstrated over 10 GHz DML based short-reach optical interconnection. With the optimal SCL decoder, up to 1.2 dB and 0.8 dB shaping gains @ BER=10−3 are obtained over BTB and 2-km SSMF transmission, respectively. However, due to the eye skew induced nonlinear distortion, the sensitivity gain of only 0.15 dB is achieved employing PS after 10-km transmission. Aided by the proposed optimal clock recovery method, the noise power in low amplitudes decreases and the shaping gain can be improved to 0.4 dB. On the other hand, thanks to the polar code, an additional coding gain of 2.2 dB @BER=10−3 is provided comparing to LDPC coded PS-PAM8 with code length of 256. This paper proves that the proposed polar coded PS-PAM8 with low complexity and satisfactory BER performance meets the requirement of low cost and low latency transmission, which could provide an efficient and practical way to improve the performance for the future short-reach optical interconnection.
Funding
National Natural Science Foundation of China (61675034, 61875019); National Key Research and Development Program of China (2019YFB1803601).
Disclosures
The authors declare no conflicts of interest.
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