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Abstract
The severe band-limited effect resulted from the low-cost optical transceiver increases the channel memory length and the number of taps of the equalizers. Besides, the interaction of fiber dispersion and square-law detection introduce nonlinear distortions in intensity modulation and direct-detection (IM/DD) transmission systems. The serious band-limited effect and nonlinear distortions degrade the transmission performance and bring challenges to current equalizers for low-complexity implementation. In this paper, we propose a trellis-compression nonlinear maximum likelihood sequence estimation (TC-NL-MLSE) algorithm to compensate the linear and nonlinear distortions with lower complexity. In the TC-NL-MLSE, we introduce a polynomial nonlinear filter (PNLF) to partly compensate both the linear distortions and nonlinear distortions. Then, we establish a look-up-table (LUT) to calculate the nonlinear branch metric (BM). To simplify the calculation, two or three levels with the highest probabilities are selected according to decision thresholds for each symbol to compress the state-trellis graph (STG). This significantly reduces computational complexity on BM calculations especially for high-order modulations. We conduct experiments to transmit beyond the 200-Gb/s PAM-8 signal over 2-km standard single mode fiber (SSMF) at C-band. The TC-NL-MLSE outperforms the reduced-state MLSE with PNLF, and can reach the 7%-overhead hard-decision forward error correction threshold. Moreover, the TC-NL-MLSE reduces the complexity by 97% compared with standard LUT-MLSE, limiting the multipliers around 100 at the expense of only 0.2-dB receiver sensitivity penalty.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Driven by variously emerging internet services, there is an exponential increasing requirement of the data traffic in data center interconnects (DCIs). In recent years, the data traffic of intra-data center interconnects (East-West traffic) is typically significantly greater than that of the traffic of inter-data center interconnects (North-South traffic) [1]. Intra-data center interconnects (Intra-DCIs) that cover 2-km or 10-km transmission distance is much sensitive to cost and power consumption. The intensity modulation and direct-detection (IM/DD) system has been widely implemented in 100-Gb/s and 400-Gb/s intra-DCIs due to its simple architecture and low cost. The next-generation Ethernet will target at 800 Gb/s or 1.6 Tb/s [2–4]. Larger than 200 Gb/s per lane is expected to scale up to 800 Gb/s or 1.6 Tb/s to reduce the requirement of more optical lanes and the complexity of integration [5]. This high-speed transmission results in serious band-limited effect with low-cost transceivers [6,7]. Therefore, advanced modulation formats, such as M-ary pulse amplitude modulation (PAM-M) [8], discrete multi-tone (DMT) [9], and carrier-less amplitude and phase modulation (CAP) [10], have been used to alleviate the bandwidth limitation. Among these modulation formats, PAM-4 has been chosen as a standard format by IEEE P802.3bs Task Force for 400-G Ethernet due to its simpler architecture and lower energy consumption [11,12]. For data rates up to 200 Gb/s per lane, to continue use the low-cost optical transceivers, the implementation of PAM-8 has been put on the agenda [13].
The main challenge of the high-speed IM/DD system is that the channel memory length is increased due to the serious band-limited effect from the low-cost optical transceivers. Furthermore, the signals suffer from nonlinear distortions including the nonlinear response of optical transceivers and the signal-to-signal beating interference (SSBI) caused by the interaction of CD and square-law detection [14]. The popular solutions to deal with these distortions are to use electric equalizers at the receiver side. Feedforward equalizer (FFE) is a well-known equalizer to mitigate the inter-symbol-interference (ISI) with simple architecture. Whereas, it enhances the noise at the frequency where the channel response has power fading [15]. The Volterra nonlinear filter (VNLF) has been demonstrated its effectiveness to alleviate both the linear ISI and nonlinear SSBI in IM/DD optical transmission systems [16]. However, the number of taps in VNLF grows rapidly with the channel memory length. Thus, the high computational complexity of VNLF hinder a practical implementation for cost-sensitive IM/DD systems. Additionally, decision-feedback equalizer (DFE) is combined with FFE in order to deal with linear and nonlinear interferences simultaneously. However, FFE-DFE may lead to error propagation due to feeding incorrect decision symbols, especially when the signal-noise ratio (SNR) is low [17,18].
The maximum likelihood sequence estimation (MLSE) has been proved as the optimal signal detection for removing ISI distortions without noise enhancement or error propagation [19]. Unfortunately, MLSE has ultra-high computational complexity, especially for higher order modulations or systems having longer delay spread. A straightforward solution to reduce the complexity of MLSE is to use the absolute value instead of the square of the distance in branch metric (BM) calculation. This saves the multiplication by 25% with no sensitivity penalty [20]. While, this kind of simplification is insufficient in cost-sensitive IM/DD system. A reduced-state MLSE based on channel estimation can reduce the complexity of MLSE at O-band by making a coarse pre-decision at the output of FFE and hence constrain the possible states in MLSE [21]. However, due to nonlinear device response and SSBI, conventional MLSE based on linear channel estimation cannot address nonlinear distortions. Therefore, the VNLF or deep neural network (DNN) has been used in traditional MLSE to compensate the nonlinear distortions or estimate the nonlinear channel response [22,23].
In this paper, we extend our previous work [24] of TC-MLSE with an additional explanation and propose to introduce the polynomial nonlinear filter (PNLF) to partly mitigate the nonlinear distortions, defined as TC-NL-MLSE, and improve its ability to combat nonlinear distortions. The nonlinear PNLF has two orders and only contains self-beating items to keep the low-complexity implementation. We make a coarse decision with a threshold detector to constrain the number of states fed into MLSE and reduce the complexity. In TC-NL-MLSE, we calculate the nonlinear BM by a look-up table (LUT). The LUT records the multi-symbol pattern and possible distorted samples for each training symbol, where the LUT is independent on the accuracy of channel estimation. Thus, the combination of PNLF and LUT is straightforward to handle pattern-dependent linear and nonlinear distortions. We conduct experiments to transmit over 200-Gb/s PAM-8 signal over 2-km standard single mode fiber (SSMF). We compare the proposed TC-NL-MLSE with the conventional MLSE and the nonlinear reduced-state MLSE with PNLF, noted as PNLF-MLSE. The experimental results show that only the BER performance of TC-NL-MLSE can reach the hard-decision FEC (HD-FEC) limitation of 3.8 × 10−3 so that the requirement of FEC and the corresponding consumption can be significantly reduced. Meanwhile, the multipliers in TC-NL-MLSE is limited around 100, which is meaningful for cost-sensitive beyond 200-Gb/s IM/DD transmission systems.
2. Principle of TC-NL-MLSE
2.1 Nonlinear PNLF-MLSE
The conventional MLSE is implemented by calculating the BM after estimating the linear channel response [25]. For such MLSE based on channel estimation, the BM is calculated by,
where Y is the received sequence, H is the channel response with L+1 coefficients and X is the data sequence {xk, xk-1, xk-2, …, xk-L}. Obviously, the performance of such MLSE scheme strongly depends on the accuracy of channel estimation. Nevertheless, the channel response is not linear in high-speed IM/DD system due to the device nonlinearities and SSBI so that the calculation of BM is inaccurate. This causes performance degradation for such MLSE. By a two-order PNLF, which is only using the diagonal Volterra kernels, the PNLF-MLSE can estimate the nonlinear channel response and calculate nonlinear BM to combat nonlinear distortions. The block diagram is shown in Fig. 1. The output Pn of a two-order PNLF can be expressed as,2.2 TC-NL-MLSE
The high implementation complexity of MLSE is because the number of states expressed as M L+1 in state-trellis graph (STG) becomes enormous when the size of modulation levels M or the channel memory length L is too large. It is impractical for Viterbi algorithm to conduct BM calculation and find the survivor path with such enormous STG in practical implementations. In fact, the cumulative BM calculation of paths with lower probabilities in the STG have negligible contributions but resource consumption. We propose to compress the STG and neglect those paths with lower probabilities, so that the states and branches can be significantly reduced. Figure 2(a) depicts the schematic diagram of TC-NL-MLSE. The blue dotted line shows the process of STG generation with all states in standard MLSE. The red part indicates the generation of compressed STG for TC-NL-MLSE. We first adopt a two-order PNLF to partly mitigate the linear and nonlinear distortions and make a coarse pre-decision to the received signals. Thus, the residual impairments can be addressed by MLSE. A post filter (PF) with transfer function in z-transform of H(z) = 1+α·z-1 is used after the PNLF to whiten the colorful noise enhanced by PNLF. For the STG, with the help of the threshold detector, we choose N levels with highest probabilities from M levels of each symbol to construct states in STG. For example, if the pre-decision of a PAM-8 symbol is 1.4, the output of the threshold detector will be one and three if the level is compressed to two. Since N < M, the STG is compressed as shown in Fig. 2(c). Compared with STG for standard MLSE plotted in Fig. 2(b), the states and the branches of STG for TC-NL-MLSE are shrunk to N L+1 and N, respectively. Note that we can adjust the number of reserved levels, i.e., N, to balance the performance and complexity of TC-NL-MLSE. In the case of high SNR, N = 2 is sufficient for equalization. We can increase the N in the scenario of low SNR at the cost of higher complexity. As a kind of sequence detection, MLSE can be very effective in detecting signals with patter-dependent distortions, as long as the pattern-dependency characteristics can be obtained. The LUT can record the pattern information including linear and nonlinear impairments [26]. Therefore, we establish a LUT as shown in Fig. 2(d) for nonlinear BM calculation in TC-NL-MLSE. The LUT is indexed by the pattern of L+1 training consecutive symbols, and its entries record the samples processed by PNLF and PF corresponding to the indexing symbol patterns. For PAM-M symbol, the number of entries in LUT is M L+1. Each time a specific pattern appears, the corresponding recorded samples are accumulated at the corresponding entry in LUT. To adapt to channel variation, a forgetting factor µ is introduced in the accumulation of recorded sample to reduce the influence from previous samples [27]. Meanwhile, a counter N(i) is used to track the number of occurrences for each specific pattern. The recorded sample in the LUT is determined after average operation to mitigate the influence from noise. After training, we can calculate the nonlinear BM with the LUT by,
where the LUT(i) represents the mean value recorded in the i-th entry of LUT. Then, Viterbi algorithm in MLSE searches STG to find the most probable sequences. Overall, the combination of the PNLF and the LUT can compensate the linear and nonlinear distortions effectively. Also, for dynamic pattern-dependent distortions, the LUT can be updated by periodic training and/or decision feedback.
Fig. 2. (a) Block diagram of TC-NL-MLSE. State-trellis graph of (b) standard MLSE, (c) TC-NL-MLSE. (d) The setup of LUT model.
2.3 Complexity comparison
To evaluate the complexity reduction of our proposed MLSE, we compare the complexity of TC-NL-MLSE, TC-MLSE, LUT-MLSE and PNLF-MLSE as shown in Table 1. Since the multiplication complexity is much higher than addition complexity, we only record the number of multiplications. In terms of LUT based MLSE, the number of multiplications can be noted as K + M L+1, which increases exponentially with the memory length of L. As for PNLF-MLSE, considering the operations of H·X and y2, K+ (1 + K1+2K2) ·N max(K1, K2) multiplications are required. In regard to TC-NL-MLSE, the number of states is reduced from M L+1 to N L+1, so we can reduce the complexity to K1+2K2 + N L+1 multiplications. Thus, the complexity reduction is significant by the proposed MLSE especially for a large M. For comparison, the complexity of FFE, PNLF, reduced-state MLSE and VNLF-MLSE are also summarized in Table 1.
Table 1. Complexity comparison of different equalizers
3. Experimental setup and results
3.1 Experimental setup
Figure 3(a) shows the experimental setup of the IM/DD system. At the transmitter, a pseudo-random bit sequences (PRBS) with 216 bits is used for PAM-8 symbol mapping and up-sampled to 2 samples-per-symbol for root raised cosine (RRC) filter shaping with 0.01 roll-off factor. Then, 67- or 70-Gbaud PAM-8 signal with 33.84- or 35.35 GHz bandwidth is generated via fractional sampling when the 32-GHz bandwidth arbitrary waveform generator (AWG) is operating at 92 GSa/s. After amplified by an electrical driver, a single-drive mode Mach-Zehnder modulator (MZM) with 30 GHz is used for E/O conversion. Then a continuous-wave optical carrier at 1549.5 nm with 12.9-dBm optical power is launched into the MZM and the output power of the MZM is 5.3-dBm. Next, the generated optical PAM-8 signal is fed into 2-km SSMF with 0.2-dB/km fiber loss. At the receiver side, a variable optical attenuator (VOA) is employed to adjust the received optical power (ROP). Then the optical signal is directly detected via a trans-impedance amplifier-free (TIA-free) single-ended photodiode (PD) with 3-dB bandwidth of 40 GHz and captured by a digital phosphor oscilloscope (DPO) operating at 200 GSa/s. Subsequently, the received signal is processed in offline DSP, including resampling, matched filter, equalization, symbol demodulation and BER calculation. For equalization, we use six different equalization schemes, i.e., FFE, PNLF, the previous TC-MLSE, the standard LUT-MLSE and the proposed TC-NL-MLSE. For fair comparisons, we also introduce the PNLF into reduced-state MLSE, i.e., PNLF-MLSE to combat the nonlinear distortions. Figures 3(b) and (c) show the amplitude histograms of received 201-Gb/s PAM-8 signal at the ROP of 8 dBm with and without PNLF for equalization, respectively. Due to the severe ISI, the signal before PNLF equalization cannot be distinguished as shown in Fig. 3(b). After elimination of ISI and part of nonlinear distortions by PNLF, the histogram in Fig. 3(c) can be separated to eight levels. However, there exists the residual ISI and nonlinear distortions that will cause decision errors. Therefore, we select the reserved levels N = 2 to perform the TC-NL-MLSE.
Fig. 3. (a) Experimental setup of PAM-8 IM/DD system. The histograms of the 201-Gb/s PAM-8 signal (b) before, (c) after processed by PNLF.
3.2 Results and discussion
Since the filter length affects both the performance and complexity, we first optimize the tap length of PNLF in TC-NL-MLSE for the 201-Gb/s PAM-8 signal at ROP of 10 dBm as shown in Fig. 4(a). In Fig. 4(a), the performance tends to be saturated when the tap length of PNLF is identified to be (71, 3), however, the performance still fails to reach the 20% SD-FEC threshold. Then, we optimize the forgetting factor µ of LUT model and the tap coefficient α in PF. In Figs. 4(b) and (c), the optimal tap coefficient and the forgetting factor are 0.7 and 0.93 at 10-dBm ROP, respectively. For fair comparisons, we also adopt a PNLF (3, 3) in PNLF-MLSE to estimate the nonlinear channel response. Figures 5(a) and (b) show the first- and second-order PNLF (3, 3) kernels in PNLF-MLSE after training. In Fig. 5(a), the first and last tap is close to zero, indicating that there is little linear ISI in the signal after going through a 71-tap FFE. Figure 5(c) shows the BER performance of PNLF-MLSE and VNLF-MLSE which has non-diagonal kernels. The results show that, for PNLF-MLSE (3, 3) and PNLF-MLSE (1, 3), there is 0.2-dB and 0.7-dB receiver sensitivity penalty compared with VNLF-MLSE (3, 3), respectively. The complexity of the three schemes are 1096, 152 and 136, respectively. Therefore, we select the optimized PNLF for the implementation FFE + PNLF-MLSE.
Fig. 4. Measured BER performance vs. (a) memory length of PNLF, (b) the tap coefficient α in PF, (c) the forgetting factor µ in LUT setup for 200-Gb/s PAM-8 signal at ROP of 10 dBm.
Fig. 5. The PNLF kernels of (a) first order, (b) second order. (c) The BER performance comparison of PNLF-MLSE and VNLF-MLSE.
Due to the severe bandwidth limitation, the PAM-8 signal suffers from serious power degradation in high-frequency region. Figure 6(a) shows the BER performance of the 201-Gb/s PAM-8 signal after 2-km SSMF transmission employing LUT-MLSE, PNLF-MLSE and TC-NL-MLSE. For comparisons, we also employ a 71-tap FFE and a two-order PNLF (71, 3) equalizations. We also summarize the tap numbers and computational complexity of different equalization schemes in Table 2. In Fig. 6(a), the FFE and PNLF fail to meet the transmission performance. While, all MLSE based equalization schemes outperform PNLF due to mitigation of the residual ISI and nonlinear distortions without noise enhancement. Among those MLSE schemes, the proposed TC-NL-MLSE exhibits around one-order magnitude improvement in BER performance than FFE + PNLF-MLSE based on channel estimation. Besides, employing TC-NL-MLSE can achieve 2-dB receiver sensitivity improvement at SD-FEC threshold of 2.7 × 10−2 compared with FFE + PNLF-MLSE. Moreover, the complexity of the proposed TC-NL-MLSE is increased by six compared with the previous TC-MLSE as shown in Table 2, however, the BER performance of TC-NL-MLSE can be below the 7% overhead HD-FEC limitation of 3.8 × 10−3. This can relax the requirement of FEC and the corresponding consumption. Note that, our proposed TC-NL-MLSE with N = 2 exhibits similar performance to standard LUT-MLSE schemes. However, TC-NL-MLSE reduces the number of multiplications from 590 to 86 at the expense of only 0.2-dB receiver sensitivity penalty at FEC limitation of 3.8 × 10−3.
Table 2. Complexity comparison of different equalizers
We further increase the bit rate of PAM-8 signal to 210 Gb/s to demonstrate the performance of the proposed TC-NL-MLSE. The bit rate of 210 Gb/s leads to more severe power fading at high-frequency region of signal spectrum, where L = 2 no longer meets the performance requirements. Therefore, we increase the first-order tap length of PNLF from 71 to 111 and the memory length of MLSE from two to three to combat the channel distortions. Figure 6(b) depicts the measured BER performance of the 210-Gb/s PAM-8 signal transmission over 2-km SSMF. In Fig. 6(b), the proposed TC-NL-MLSE shows BER advantage than other equalization schemes. With the help of TC-NL-MLSE, the receiver sensitivity considering the 20% overhead SD-FEC can be improved by about 1.9 dB compared with FFE + PNLF-MLSE. Compared with the TC-MLSE [24], the BER of TC-NL-MLSE can reach the HD-FEC limit of 3.8 × 10−3 at the cost of extra six multiplications as shown in Table 2. Moreover, the TC-NL-MLSE has almost the same performance as standard LUT-MLSE. While, for TC-NL-MLSE, the number of multiplications reduce from 4214 to 134 for 210-Gb/s transmission, which is ∼97% computational complexity reduction. The results further prove the effectivity of the proposed TC-NL-MLSE in eliminating residual linear and nonlinear distortions with a lower computational complexity.
4. Conclusion
We have proposed and experimentally demonstrated a simplified TC-NL-MLSE scheme in the beyond 200-Gb/s PAM-8 signal transmission IM/DD systems. The experimental results show that TC-NL-MLSE can compensate residual linear and nonlinear distortions effectively so that the BER performance can be below the 7% overhead HD-FEC limitation. There is only 0.2-dB penalty in receiver sensitivity compared with a standard LUT-MLSE but the number of multiplications is reduced by ∼97%.
Funding
National Key Research and Development Program of China (2018YFB2201803); National Natural Science Foundation of China (61871082, 62111530150); Open Fund of IPOC(BUPT) (No. IPOC2020A011); Science and Technology Commission of Shanghai Municipality (SKLSFO2021-01); Fundamental Research Funds for the Central Universities (ZYGX2019J008, ZYGX2020ZB043).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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