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Collaborative FFE-assisted simplified soft-output MLSE with LDPC for high-speed IM-DD transmissions

Xue Zhao, Jing Zhang, Jiahao Zhou, Zhengyu Ma, Shaohua Hu, Bo Xu, Qianwu Zhang, and Kun Qiu

Open Access Open Access

Abstract

In this paper, we propose a feed-forward equalizer (FFE)-assisted simplified soft-output MLSE (sMLSE) by collaborating the maximum likelihood sequence estimation (MLSE) with soft-decision low-density-parity-check (LDPC) decoding. The simplified sMLSE results in undetermined log-likelihood ratio (LLR) magnitudes when the reserved level is less than or equal to the half of modulation order. This severely degrades the performance of soft-decision forward error correction (SD-FEC) decoding. In the FFE-assisted simplified sMLSE, we use the LLRs calculated from pre-set FFE to replace these undetermined LLRs of simplified sMLSE. Thus, the proposed method eliminates the SD-FEC decoding performance degradation resulted from simplification. We conduct experiments to transmit 184-Gb/s PAM-4 or 255-Gb/s PAM-8 signal in IM-DD system at C-band to evaluate the performance of the proposed sMLSE. The results show that the proposed sMLSE can effectively compensate for the degradation of LLR quality. For 255-Gb/s PAM-8 signal transmissions, the FFE-assisted simplified sMLSE achieves almost the same SD-FEC decoding performance as the conventional sMLSE but with 85% complexity reduction.

© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

1. Introduction

Driven by the data-consuming applications, the requirement of data traffic is increasing exponentially in high-speed data center interconnects (DCIs) [1]. In recent years, the data traffic of intra-DCIs (East-West traffic) is significantly larger than that of the traffic of inter-data center interconnects (North-South traffic) [2]. Intra-DCIs that cover 2-km transmission distance are much sensitive to the cost and power consumption. The intensity modulation and direct detection (IM-DD) has been widely implemented in intra-DCIs owing to its cost effectiveness, lower consumption and simple architecture. Among various modulation formats, 4-ary pulse amplitude modulation (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 [3]. As data traffic increasing, larger than 200 Gb/s per lane is expected to scale up to 800 Gb/s or 1.6 Tb/s, which can reduce the requirement of more optical lanes and the complexity of integration for the next-generation Ethernet [4,5].

The inter-symbol interference (ISI) resulted from the bandwidth-limited effect of low-cost transceivers and fiber chromatic dispersion (CD) has become the main limitation of high-speed IM-DD system [6]. To deal with these problems, the common solution is to utilize the adaptive equalizers at the receiver, such as feed-forward equalizer (FFE), decision-feedback equalizer (DFE) and maximum likelihood sequence estimation (MLSE) [7–10]. Among these equalizers, the MLSE has been proved as the optimal signal detection to remove ISI distortions without noise enhancement. Moreover, the forward error correction (FEC) can significantly reduce the required optical signal-to-noise ratio (SNR) and guarantee the reliable communication [11]. Compared to the hard-decision forward error correction (HD-FEC), the soft-decision forward error correction (SD-FEC) can further increase the receiver sensitivity as it provides a greater net coding gain (NCG). The combination of equalizers and powerful SD-FEC becomes an effective solution to realize high-speed IM-DD transmission systems [12–14]. In the conventional MLSE, the receiver outputs the most likely transmitted symbol sequence after equalization, rather than the log-likelihood ratios (LLRs) of bits which is used for SD-FEC decoding. This means that there is no direct information interaction between equalizers and decoders. Some effective information from the equalizers have been discarded. To solve this problem, the soft-output MLSE (sMLSE) based on Bahl-Cocke-Jelinek-Raviv (BCJR) algorithm is proposed to calculate the LLR. Unfortunately, the sMLSE has ultra-high computational complexity, especially for higher-order modulations or systems having longer delay spread [15]. In our previous work [16], we have proposed a simplified sMLSE to reduce the complexity of sMLSE. However, we find that compressing the possible states in the simplified sMLSE results in some undetermined LLR calculations. Although we can pre-set theses undetermined LLR magnitudes, this degrades the decoding gain and requires prior information of the system [17]. Increasing the reserved level or adding a third recursion to produce backup LLRs can effectively improve the LLR quality, but both of them increase the computational complexity [18,19].

In this paper, we extend our previous work of the simplified sMLSE with additional explanations and propose a FFE-assisted simplified sMLSE to improve the SD-FEC decoding. The simplified sMLSE with a small reserved level results in undetermined LLR magnitudes, which degrades the SD-FEC decoding performance. In the FFE-assisted simplified sMLSE, we utilize the FFE before sMLSE to generate the backup LLRs. Once the simplified sMLSE leads to undetermined LLR magnitudes, we use the backup LLRs to replace the undetermined LLRs. Therefore, the optimized LLRs are used for decoding. Thus, the combination of simplified sMLSE method and the LLR from FFE can greatly decrease the computational complexity of conventional sMLSE and result in no SD-FEC decoding performance degradation. We conduct a 184-Gb/s PAM-4 and a 255-Gb/s PAM-8 IM-DD systems to evaluate the performance of the proposed sMLSE. The experiment results show that the proposed sMLSE can effectively alleviate the degradation of LLR quality resulted from simplification. For a 255-Gb/s PAM-8 transmission, the FFE-assisted simplified sMLSE results in no performance degradation in SD-FEC decoding with ∼85% computational complexity reduction.

2. Principle of FFE-assisted simplified sMLSE

The sMLSE is achieved by Max-log-BCJR algorithm with lower complexity and outputs the LLRs of bits. For a PAM-4 symbol with gray coding, the LLRs of the most significant bit (MSB) L1 and the lowest significant bit (LSB) L2 can be expressed as,

$$\begin{aligned} {L_1}({{u_l}} )&= \ln \frac{{P({{u_l} ={-} 3|y} )+ P({{u_l} ={-} 1|y} )}}{{P({{u_l} = 3|y} )+ P({{u_l} = 1|y} )}}\textrm{ }\\ {L_2}({{u_l}} )&= \ln \frac{{P({{u_l} ={-} 3|y} )+ P({{u_l} = 3|y} )}}{{P({{u_l} ={-} 1|y} )+ P({{u_l} = 1|y} )}} \end{aligned}$$
where ${u_l}$ is the transmitted symbol with the elements of {-3, -1, 1, 3}. $P({{u_l}|y} )$ is the probability of the transmitted symbol u at time l in the condition of the received symbol sequence y. We define ${\tilde{\alpha }_l}(s )= \ln {\alpha _l}(s )$, ${\tilde{\beta }_l}(s )= \ln {\beta _l}(s )$ and ${\tilde{\gamma }_l}({s^{\prime},s} )= \ln {\gamma _l}({s^{\prime},s} )$. The $P({{u_l}|y} )$ is derived as,
$$\ln P({{u_l}|y} )= \mathop {\max }\limits_{s \subset \{{state\textrm{ }with\textrm{ }{u_l}} \}} \{{{{\tilde{\alpha }}_l}(s )+ {{\tilde{\beta }}_l}(s )} \}$$

The forward metric ${\tilde{\alpha }_l}(s )$ and the backward metric ${\tilde{\beta }_{l - 1}}({s^{\prime}} )$ can be calculated by recursive formulation as,

$${\tilde{\alpha }_l}(s )= \mathop {\max }\limits_{s^{\prime}} \{{{{\tilde{\alpha }}_{l - 1}}({s^{\prime}} )+ {{\tilde{\gamma }}_l}({s^{\prime},s} )} \}$$
$${\tilde{\beta }_{l - 1}}({s^{\prime}} )= \mathop {\max }\limits_s \{{{{\tilde{\beta }}_l}(s )+ {{\tilde{\gamma }}_l}({s^{\prime},s} )} \}$$
where $s^{\prime}$ and s denote the states of the trellis graph at time l-1 and l. ${\tilde{\gamma }_l}({s^{\prime},s} )$ is the branch matric (BM). Figures 1(a) and (b) show the forward and backward recursion steps of sMLSE. As for PAM-4, four branches are merged into one state, and the highest probability is selected for ${\tilde{\alpha }_l}(s )$. The ${\tilde{\beta }_{l - 1}}({s^{\prime}} )$ is calculated by the same process in the backward recursion. In sMLSE, we establish a look-up table (LUT) to record the multi-symbol pattern and corresponding distorted samples to calculate the nonlinear BM [20]. Assuming that the constellation points have equal probabilities, the nonlinear BM based on the LUT is expressed as,
$${\tilde{r}_l}({s^{\prime},s} )={-} \frac{{{{({{y_l} - LUT(i)} )}^2}}}{{2{\delta ^2}}},1 \le i \le {M^{L + 1}}$$
where ${y_l}$ is the received symbol at time l, the LUT(i) represents the mean value recorded in the i-th entry of LUT, ${\delta ^2}$ is the noise variance, M is the modulation order, and L is the memory length of sMLSE.

 figure: Fig. 1.

Fig. 1. (a) The forward recursion step and (b) the backward recursion step of sMLSE.

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The sMLSE consists of two Viterbi-Algorithms, one running forward through the trellis to compute the $\tilde{\alpha }$ and the other running backward through the trellis to compute the $\tilde{\beta }$[21]. The complexity of sMLSE is approximately twice of that of the conventional hard-output MLSE. We employ a threshold detector to reserve P levels with the highest probability from M levels for each symbol to compress the state-trellis graph, where P < M. Figure 2(a) depicts the block diagram of the proposed FFE-assisted simplified sMLSE. We first adopt FFE to partially compensate the linear distortions. A post filter (PF) with transfer function of H(z) = 1+α·z−1 is adopted after the FFE to suppress the noise enhancement. The tap coefficient is obtained by solving the Yule-Walker Equations based on the auto-regressive model. Then, the sMLSE outputs the LLRs to LDPC decoder. In the SD-FEC, the quality of LLR strongly affects the decoding performance. However, the trellis compression is achieved by abandoning the states with lower possibilities. When the reserved level P is less than or equal to M/2, either the numerator or denominator of L1 or L2 in Eq. (1) is equal to zero. In other words, the simplified sMLSE results in undetermined LLR magnitudes. This severely degrades the SD-FEC decoding performance. Table 1 shows the LLR states of simplified sMLSE with P = 2. In Tab. 1, ‘×’ represents that the LLR magnitude is undetermined, and ‘√’ represents determined LLR magnitude. For PAM-4 signal, the reserved levels pattern with P = 2 in simplified sMLSE contains {-3, -1}, {-1, 1} and {1, 3}. For example, if the reserved levels of the threshold detector are -3 and -1, the probabilities of levels one and three are zeros. Recalling Eq. (1), the denominator of L1 is zero. Therefore, there is no estimated of the L1 magnitude. Similarly, if the reserved levels are {-1, 1} or {1, 3}, the numerator of L2 or L1 is zero, and we cannot estimate the magnitude of L2 or L1. In this paper, we propose to use the LLRs calculated from FFE (noted as Lffe_n(ul), where n is the bit index) to replace these undetermined LLRs. Figure 2(b) shows the example of undetermined LLRs replacement. L1(ul) and L2(ul) are the LLRs of bits, which are calculated from simplified sMLSE. The LLR marked in red indicates undetermined LLR magnitude. In Fig. 2(b), L1(u1) is undetermined, we use the Lffe_1(u1) calculated from FFE to replace the L1(u1). Finally, the optimized LLRs consisted of LLRs from FFE and sMLSE are loaded into LDPC decoder. Note that, for determined LLR magnitudes like L2(u1), we don’t need to calculate Lffe_2(u1) as a substitution.

 figure: Fig. 2.

Fig. 2. (a) Block diagram of FFE-assisted simplified sMLSE. (b) Example of undetermined LLRs replacement.

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Tables Icon

Table 1. The LLR states of simplified sMLSE with P = 2

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We compare the complexity of conventional sMLSE and our proposed FFE-assisted simplified sMLSE as shown in Tab. 2. For K-tap FFE, it requires K multiplications. In terms of the conventional sMLSE, the LUT has M L + 1 entries. So, the BM calculation needs 2 M L + 1 multiplications. Thus, the number of multiplications can be noted as K + 2 M L + 1. As for the proposed FFE-assisted simplified sMLSE, we compress the likely states of each symbol from M to P. Therefore, the number of multiplications is reduced from K + 2 M L + 1 to K + 2P L + 1.

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Table 2. Complexity comparison

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3. Experimental setup and results

3.1 Experimental setup

Figure 3(a) shows the experiment setup of IM-DD system. At the transmitter, a pseudo-random bit sequence (PRBS) was encoded by the LDPC code with block length of 64800 bits. Then, the encoded bit-streams are mapped into PAM-4/8 symbols with gray coding then shaped by a root raised cosine (RRC) filter with 0.01 roll-off factor. Then, 184-Gb/s PAM-4 or 255-Gb/s PAM-8 signal are generated via re-sampling where the arbitrary wave generator (AWG) is operating at 120 GSa/s. After amplified by an electrical amplifier (EA), a single-drive mode Mach-Zehnder modulator (MZM) with 40 GHz is used for Electro/Optic conversion. Then, a continuous-wave optical carrier at 1550 nm with 12-dBm optical power is launched into the MZM. Next, the generated optical PAM-4/PAM-8 signal is fed into 2/1-km standard single mode fiber (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). Given that our receiver employs a photodiode (PD) without a trans-impedance amplifier (TIA), we apply an Erbium doped fiber amplifier (EDFA) to boost the optical signal. Then the optical signal is detected via a TIA-free single-ended PD with 3-dB bandwidth of 40 GHz and captured by a real-time oscilloscope (RTO) operating at 256 GSa/s. Subsequently, the received signal is processed in offline DSP, including re-sampling, matched filter, FFE, sMLSE, LDPC decoding and bit error ratio (BER)/normalized generalized mutual information rate (NGMI) calculation. We use the NGMI as the evaluation metric for the SD-FEC decoding performance. As for equalizations, FFE, conventional sMLSE, the simplified sMLSE [16] and the proposed FFE-assisted simplified sMLSE with LDPC are employed and compared. Figure 3(b) depicts the frequency response of the optical back-to-back (OBTB) transmission. It shows that the 10-dB bandwidth of the system is approximately 26.2 GHz. Moreover, the frequency response fades rapidly when the frequency is beyond 40 GHz, which leads to a great degradation of transmission performance.

 figure: Fig. 3.

Fig. 3. (a) Experimental setup of IM-DD system. (b) The frequency response of the OBTB transmission.

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3.2 Results and analysis

We first investigate the performance of the proposed FFE-assisted simplified sMLSE for 184-Gb/s PAM-4 signal transmission over 2-km SSMF with the LDPC code rate of 9/10. Figure 4(a) shows the pre-FEC BER performance of 184-Gb/s PAM-4 signal after 2-km SSMF transmission. For equalizations, we use a 121-tap FFE, a sMLSE with memory length L of two, the simplified sMLSE (L = 2, P = 3) and the FFE-assisted simplified sMLSE (L = 2, P = 2) for comparisons. In Fig. 4(a), the FFE cannot reach the 11% SD-FEC threshold of 8.33 × 10−3 due to the noise enhancement. However, the sMLSE can alleviate the noise enhancement, and thus improving the pre-FEC BER performance significantly. Compared with the conventional sMLSE, the simplified sMLSE with P = 3 and the FFE-assisted simplified sMLSE with P = 2 both can achieve the same pre-FEC BER performance. Figure 4(b) shows the NGMI performance with different equalizations. In Fig. 4(b), the sMLSE can significantly improve the NGMI compared to FFE. The simplified sMLSE with P = 3 exhibit the same NGMI performance as the conventional sMLSE. However, when P is reduced to two, undetermined LLRs are generated so that the simplified sMLSE has poor NGMI performance. The proposed FFE-assisted simplified sMLSE with P = 2 can solve the undetermined LLR problem and achieve the same NGMI performance as the conventional sMLSE. There is no pre-FEC BER or NGMI performance penalty for the FFE-assisted simplified sMLSE with P = 2, while the complexity is reduced by ∼22% compared with simplified sMLSE with P = 3. Note that there may be performance penalty for a too small reserving level when the performance of FFE is severely degraded since the LLR accuracy is degraded accordingly. In general, the FFE output can be used to substitute the calculation of undetermined LLR. We also summarize the tap numbers and computational complexity of different equalization schemes in Tab. 3. Compared with the conventional sMLSE, the FFE-assisted simplified sMLSE with P = 2 reduces the number of multiplications by ∼45% according to Tab. 3 in the 184-Gb/s PAM-4 transmission.

 figure: Fig. 4.

Fig. 4. (a) Pre-FEC BER, (b) NGMI performance of 184-Gb/s PAM-4 signal after 2-km SSMF transmission.

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Table 3. Complexity comparisons

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We further introduce the FFE-assisted simplified sMLSE into PAM-8 signal transmission to study its performance. We reserve two levels to compress the state-trellis graph for the FFE- assisted simplified sMLSE. We choose the LDPC with code rate of 5/6 to evaluate the decoding performance. Figures 5(a) and (b) depict the pre-FEC BER and NGMI performance of 255-Gb/s PAM-8 signal transmission over 1-km SSMF employing FFE, sMLSE and FFE-assisted simplified sMLSE. In Fig. 5(a), the 161-tap FFE is failed to reach the 20% SD-FEC threshold. While, the pre-FEC BER of sMLSE and FFE-assisted simplified sMLSE both are below the 20% SD-FEC threshold. The results also show that the proposed FFE-assisted simplified sMLSE can achieve ∼85% computational complexity reduction compared to the conventional MLSE with the same pre-FEC BER and NGMI performance.

 figure: Fig. 5.

Fig. 5. (a) Pre-FEC BER, (b) NGMI performance of 255-Gb/s PAM-8 signal after 1-km SSMF transmission.

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4. Conclusion

We have proposed and experimentally demonstrated a FFE-assisted simplified sMLSE to use the LLRs calculated from FFE to replace the undetermined LLRs of simplified sMLSE for high-speed IM-DD transmission systems. The experiment results show that the FFE-assisted simplified sMLSE can effectively compensate for the degradation of LLR quality resulted from simplification and provide the same SD-FEC decoding performance compared to the conventional sMLSE with large complexity reduction.

Funding

National Natural Science Foundation of China (62301128, U22A2086); Science and Technology Commission of Shanghai Municipality (SKLSFO2021-01); Open Fund of IPOC (BUPT) (No. IPOC2020A011); Fundamental Research Funds for the Central Universities (ZYGX2019J008, ZYGX2020ZB043).

Acknowledgments

This work was supported by National Natural Science Foundation of China (NSFC) (62301128 and U22A2086), STCSM (SKLSFO2021-01), Open Fund of IPOC (BUPT) (No. IPOC2020A011), Fundamental Research Funds for the Central Universities (ZYGX2020ZB043 and ZYGX2019J008).

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.

References

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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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Figures (5)
Fig. 1.
Fig. 1. (a) The forward recursion step and (b) the backward recursion step of sMLSE.

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Fig. 2.
Fig. 2. (a) Block diagram of FFE-assisted simplified sMLSE. (b) Example of undetermined LLRs replacement.

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Fig. 3.
Fig. 3. (a) Experimental setup of IM-DD system. (b) The frequency response of the OBTB transmission.

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Fig. 4.
Fig. 4. (a) Pre-FEC BER, (b) NGMI performance of 184-Gb/s PAM-4 signal after 2-km SSMF transmission.

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Fig. 5.
Fig. 5. (a) Pre-FEC BER, (b) NGMI performance of 255-Gb/s PAM-8 signal after 1-km SSMF transmission.

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Tables (3)

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Table 1. The LLR states of simplified sMLSE with P = 2

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Table 2. Complexity comparison

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Table 3. Complexity comparisons

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Equations (5)

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L 1 ( u l ) = ln ⁡ P ( u l = − 3 | y ) + P ( u l = − 1 | y ) P ( u l = 3 | y ) + P ( u l = 1 | y )   L 2 ( u l ) = ln ⁡ P ( u l = − 3 | y ) + P ( u l = 3 | y ) P ( u l = − 1 | y ) + P ( u l = 1 | y )
ln ⁡ P ( u l | y ) = max s ⊂ { s t a t e   w i t h   u l } ⁡ { α ~ l ( s ) + β ~ l ( s ) }
α ~ l ( s ) = max s ′ ⁡ { α ~ l − 1 ( s ′ ) + γ ~ l ( s ′ , s ) }
β ~ l − 1 ( s ′ ) = max s ⁡ { β ~ l ( s ) + γ ~ l ( s ′ , s ) }
r ~ l ( s ′ , s ) = − ( y l − L U T ( i ) ) 2 2 δ 2 , 1 ≤ i ≤ M L + 1
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