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Abstract
To reduce the computational complexity of soft-decision (SD) forward error correction (FEC), we propose a polar coding method with a low-complexity successive cancellation decoder. Polar coding induces channel polarization in which two bit-channels with lower and higher reliabilities are polarized. Only the less-reliable bit-channels are protected by SD-FEC, whereas the more-reliable bit-channels are offloaded, reducing the complexity of SD-FEC decoding. The degradation of the bit error ratio (BER) performance can be suppressed by designing the polar encoder structures for the successive cancellation decoder. We numerically demonstrate that the proposed method manages to both reduce the computational complexity by half and suppress the BER performance degradation by less than 0.6 dB, compared with the conventional method using only the SD-FEC.
© 2024 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In coherent optical transmission systems, soft-decision (SD) forward error correction (FEC) is essential for recovering bit error ratio (BER) performance [1,2]. The low-density parity check (LDPC) code has been widely introduced in the real 100-Gbit/s systems [3], and is approaching performance limits based on generalized mutual information (GMI).
The SD-FEC decoder requires iterative processing [4]. Although the real-time implementation of LDPC decoders is possible using the sum-product algorithm (SPA), the complicated configuration of iterative decoding increases power consumption. Power consumption can be increased in proportion to the number of input bits in the SD-FEC decoder. By increasing the signal baudrate and using higher-order quadrature amplitude modulation (QAM) and probabilistic shaping, the power consumption of SD-FEC decoders becomes more severe because the processing bit number increases [5]. Complexity reduction of SD-FEC decoders is indispensable for feature ultra-high-bitrate transmission systems.
To reduce the complexity of SD-FEC decoders, it is effective to offload some of the information bits onto a low-complexity FEC and not process all the bits in the SD-FEC [6–11]. Although bit errors remain after the low-complexity FEC decoder, they are perfectly corrected by the outer hard-decision (HD) FEC. The most well-known offloading method is multilevel coding (MLC) [6–9]. The symbol set of the QAM constellation is divided into lower sets using set-partitioning bits. Only the set-partitioning bits are protected by the SD-FEC. Recently, another offloading method based on bitwise channel polarization was reported [10,11]. The exclusive-or (XOR) operations induce channel polarization with higher and lower reliabilities. Only the less-reliable bit-channels are protected by the SD-FEC. For both methods, the number of input bits to the SD-FEC is decreased by offloading onto another low-complexity FEC, thereby decreasing the complexity of the SD-FEC.
In the offloading method, an optimal BER performance is achieved only when using an SD-FEC with an appropriate overhead. Using a different SD-FEC with an inappropriate overhead, the degradation of the BER performance is increased due to a mismatch between the quality of the signal to be decoded by the outer HD-FEC and that decoded by the inner codes of the low-complexity FEC and SD-FEC. It is required to adjust the total FEC overhead to approximately 20%, which is typically installed in optical fiber communications. To adjust the total FEC overhead while maintaining the BER performance, a fine granularity of code rate tunability in the low-complexity FEC is required. Unfortunately, the MLC method limits the code rate tunability because the code rate is directly related to the QAM constellation size and the number of set-partitioning bits. Although the bitwise XOR method is not directly limited by the QAM constellation, the code rate is restricted to (N – 1)/N by the bit size N in the encoder, such as 1/2, 2/3, 3/4, and so on. Tuning the overhead of the outer HD-FEC also allows the BER performance to be optimized for the required total FEC overhead. However, it is often preferable to avoid designing and implementing multiple HD-FEC circuits in a real chip design. Although probabilistic shaping is the effective method for rate tuning, there is a case where the rate tunability using probabilistic shaping is not applicable when the additional implementation complexity is not allowed in a transponder design.
To manage both the complexity reduction of SD-FEC decoders and overhead tunability, we propose the use of a polar coding method. Although the polar code is a family of capacity-achieving error-correction codes [12], it is used for the complexity reduction of SD-FEC in this study. The polar code polarizes two bit-channels with higher and lower reliabilities. In the receiver, only the less-reliable bit-channels undergo the SD-FEC decoder, whereas the more-reliable bit-channels are offloaded onto a low-complexity successive cancellation (SC) decoder without iterative processing. The degradation of the BER performance can be suppressed by designing the polar encoder structures for the SC decoder, even with the use of a low-complexity SC decoder. Using different structures of the polar encoders, the total FEC overhead values are tuned to approximately 20% while suppressing the degradation of the BER performance. The numerical results show that the proposed method reduces the complexity of the SD-FEC decoders by less than half and suppresses the BER performance degradation by less than 0.6 dB, compared with the conventional method using only LDPC codes.
2. Concept of the proposed method based on polar codes
In the proposed method, small-size polar coding is used. Figure 1 illustrates the concept of the proposed method. To induce channel polarization, a polar encoder and low-complexity SC decoder without any iterative processes are introduced. The polar encoder performs channel combining, whereas the SC decoder performs channel splitting, inducing channel polarization in which bit-channels with lower and higher reliabilities are polarized. In the transmitter, only the less-reliable bit-channels are encoded for the SD-FEC, whereas the more-reliable bit-channels are not. In the receiver, the less-reliable bit-channels are decoded using an SD-FEC decoder, whereas the more-reliable bit-channels are offloaded onto a low-complexity SC decoder. The computational complexity is reduced by decreasing the number of the bits processed in the SD-FEC decoder. To improve the performance of the low-complexity SC decoder, the bits decoded by the SD-FEC decoder are used as “frozen” bits in the SC decoder.
Figure 2 shows the structures of the polar encoders used in the proposed method. Different FEC overheads are obtained using different encoder structures. Well-known polar encoders [12] with bit sizes of two and four are shown in Figs. 2(a) and 2(b), respectively. At the input of the encoders, the bits encoded by SD-FEC are assigned to the bit with the lowest index u1, whereas the offloaded bits are the higher bits (u2, u3, and u4). The code rate of polar encoder is defined as the bit number ratio of the offloaded bits to the total bits. Using only these two encoders, the code rate are limited to 1/2 and 3/4. We propose modified structures of polar encoders to improve the granularity of code rate tunability. By nesting typical encoders with sizes of two and four, a modified encoder with a bit size of five is constructed, as shown in Fig. 2(c). In this encoder, the code rate is tuned to 3/5 by assigning two SD-FEC-encoded bits to the lowest and second indices, u1 and u2, respectively. By nesting two size-4 encoders, an encoder with a bit size of seven is constructed, as shown in Fig. 2(d). In the encoder, the SD-FEC-encoded bits are assigned to u1 and u2, and the code rate is adjusted to 5/7. Their structures are effective for tuning the total FEC overhead to approximately 20% and for maintaining the performance of the SC decoder, as mentioned later.
In the SC decoder, the received bits are sequentially decoded in the order of the lower index of the bit-channel [12]. Before the SC decoding, the less-reliable bit-channels are decoded by the SD-FEC decoder, and the decoded bits are used as known bits called “frozen” bits in the SC decoder. For the size-4 encoder used in the transmitter, Fig. 3(a) shows the process decoding the information bit of the second index, u2, using the frozen bit u1. The log-likelihood ratio (LLR) value of u2, Lu2, can be calculated using the received LLRs and the HD value of the frozen bit u1, which is given by
The SC decoder is a single direction process and does not require any iterative processes, although it is based on the same hyperbolic tangent functions as the SPA. The computational complexity of the SC decoder is significantly lower than that of the SD-FEC decoder with the iteration process.
A technical issue with the proposed method is the appropriate design of the encoder structure for an SC decoder. The encoder structure determines the bit-channel capacity, which can be approximately estimated as a binary erasure channel using previously-decoded bits as frozen bits [12]. Figure 4 shows the bit-channel capacity of the proposed encoders calculated when the erasure probability is 0.5. The capacity increases without obvious decrease, especially over the lower bit-channel index, although those above 0.5 fluctuate slightly over the higher index. It is very effective in maintaining the performance of the SC decoder because the decoding is sequentially performed in the order of the bit-channel index. For comparison, Figs. 5(a), (b), (c), and (d) show the calculated capacity in other structures of typical polar encoders with bit sizes of 8, 16, 64, and 128, respectively, which are well-known in the research field of wireless communications [12]. As the encoder size increases, the function becomes more complicated. The large-size encoders are suitable for applications in wireless systems because their channel polarization is enhanced. By assigning information bits to bit-channels with capacities of approximately one, and frozen bits to others, the capacity-approaching performance is obtained. However, the large-size encoders are not suitable for the SC decoder. This is because the zigzagging change in the capacity over the bit-channel index significantly degrades the performance of the SC decoder. In particular, the capacity alternation between the less-reliable bit-channels and more-reliable bit-channels for the lower index severely degrades the performance of the SC decoder. The nested structure, which has a similarity to convolutional polar codes [13], is effective in increasing the number of frozen bits while avoiding the zigzagging change in the capacity, which also improves the granularity of code rate tunability. In the nested encoder as shown in Fig. 2(c), the second bit-channel is further connected to the information bit and more polarized. As a result, the capacity of the second index becomes almost the same as that of the lowest index, which is appropriate to maintain the performance of the SC decoder, even though the frozen bits are increased.
Fig. 4. Calculated bit-channel capacity in the proposed encoders with bit sizes of (a) two, (b) four, (c) five, and (d) seven. The erasure probability is 0.5.
Fig. 5. Calculated bit-channel capacity in typical polar encoders with bit sizes of (a) 8, (b) 16, (c) 64, and (d) 128.
Before the SD-FEC decoder, a soft-decision operation, which performs the reverse polar encoding process, is required to recover the SD-FEC-encoded bits. For the size-4 encoder, the LLR values of bits encoded by the SD-FEC, Lu1, are calculated as
Using XOR operators in the encoders is in common with the proposed method and previously reported XOR method [10,11]. In particular, the size-2 encoder structures in the proposed and the XOR methods are identical. In the XOR method, only one bit-channel is polarized by the XOR operators in the encoder, and it is protected by the SD-FEC, resulting in an effective reduction of the decoder complexity. Because the encoder structure is uniquely determined, the granularity of code rate tunability is limited. Conversely, the proposed polar encoders polarize most bit-channels, improving the granularity of code rate tunability. In addition, we designed encoder structures to reduce the zigzagging change of the bit-channel capacity, which is suitable for the SC decoder to suppress the degradation of the BER performance. There could be a variety of polar encoder structures designed for the SC decoder, although the nested structures of typical polar encoders are shown in this work.
3. Simulation of the performance evaluation
Figure 6 shows the simulation model used to evaluate the performance of the proposed method. We assume that input bits are encoded by an outer HD-FEC, whose overhead and threshold for the pre-FEC BER are 6.25% and 4.5 × 10−3, respectively [14]. These are divided into two streams. Only one is encoded using a DVB-S2 LDPC encoder, whereas the other is offloaded. The offloaded and LDPC-encoded bits are encoded using the proposed polar encoder. The LDPC overhead values are determined to attain the optimal BER characteristic where the output error floor, which is due to the more-reliable bit-channels, becomes close to the BER threshold of the outer HD-FEC, although it would be practical to maintain some margin between them. After bit interleaving, the polar-coded bits are mapped onto 64QAM symbols with Gray code. The QAM samples are converted linearly into the optical domain.
After the optical QAM signal suffers from additive white Gaussian noise (AWGN) with a specified signal-to-noise ratio (SNR), it is received based on optical coherent reception, in which the laser phase noise is neglected in the simulation. In the receiver, the LLR values of the received bits are calculated by SD symbol demapping, followed by a bit deinterleaver. As described by Eq. (4), the LLR values of the LDPC-encoded bits are recovered by a soft-decision operation to reverse the polar encoder, and are decoded by an LDPC decoder in which the iteration is performed 20 times. The remaining bits are decoded by an SC decoder using the LLRs decoded by the LDPC decoder as frozen bits. By HD of all the obtained LLR values, the BER values are calculated.
Computational complexity is calculated based on the number of multiplexers used in all decoders. The complexity of LDPC decoder per a decoded bit is given by [15]
Using the size-2 encoder, we adjust the total overhead to 28% and 21% using different LDPC codes with overheads of 50% and 33%, respectively. The results are indicated by red circles in Figs. 7(a) and 7(b). For the overhead of 28% and 21%, the SNR values required for BER of less than 4.5 × 10−3, which is the BER threshold before the HD-FEC [14], are estimated as 16.4 and 17.3 dB, respectively. The computational complexity is calculated to be 41. For comparison, the blue circles in Figs. 7(a) and 7(b) represent the results of the conventional method using only an LDPC code. At a 28% total overhead, the required SNR value and complexity are 16.9 dB and 88, respectively, and 18.1 dB and 68, respectively, at a 20% overhead. When using the size-2 encoders, the required SNR performance is improved by more than 0.5 dB, even though the complexity is reduced by approximately half.
Fig. 7. Calculated BER as a function of SNR. (a) size-2 polar encoder with 50% overhead LDPC, (b) size-2 polar encoder with 33% overhead LDPC, and (c) size-4 polar encoder with 33% overhead LDPC.
When using the size-4 encoder with the 33% overhead LDPC code, the total overhead is tuned to 13%. The calculated BERs are plotted as red circles in Fig. 7(c), where a required SNR of 19 dB is observed. The complexity is calculated as 20. The blue circles represent those of the conventional method using only the LDPC code. The required SNR and complexity are 18.4 dB and 67, respectively, and the total overhead is 18%. In the proposed method, even with their overhead difference of 5%, the degradation of the BER performance is suppressed by 0.6 dB, whereas the complexity is reduced by less than 1/3.
Using the modified size-5 encoder, the total overhead is tuned to 27% and 23% with LDPC codes having overheads of 67% and 50%, respectively. The BER performance at a total overhead of 27% is indicated by red circles in Fig. 8(a). The required SNR and complexity are 17.4 dB and 43, respectively. Compared with the conventional method with a total overhead of 28%, which is indicated by the blue circles, the SNR penalty can be suppressed by 0.5 dB even with the complexity reduction by half. The red circles in Fig. 8(b) indicate the BER performance with a total overhead of 23%. The required SNR and complexity are calculated as 18.0 dB and 31.8, respectively. Compared with the conventional method with a total overhead of 20%, as shown by the blue circles in Fig. 8(b), the BER performance is comparable, whereas the complexity is approximately half.
Fig. 8. Calculated dependence of BER on SNR for the proposed method using size-5 nested polar encoder with LDPC codes having (a) 67% and (b) 50% overheads.
When using the size-7 polar encoder with LDPC codes with overheads of 67% and 50%, the total overhead values are tuned to 20% and 18%, respectively. The calculated BER performance at the total overhead of 20% is indicated by red circles in Fig. 9(a). The required SNR and complexity are estimated as 18.0 dB and 30, respectively. The blue circles represent those of the conventional method using only the LDPC code with the same overhead. The SNR penalty in the proposed method is negligible, whereas the complexity is reduced by half. Figure 9(b) shows the BER performance with the total overhead of 18%. The required SNR and complexity are 18.6 dB and 22, respectively. Compared with the conventional method, the SNR penalty is suppressed by less than 0.3 dB, even though the complexity is reduced by approximately 1/3.
Fig. 9. Calculated BERs for the proposed method using size-7 nested polar encoder with LDPC codes having (a) 67% and (b) 50% overheads.
We compare the BER performance of the proposed method with that of the MLC and XOR methods when the overhead is adjusted to the various targets. The QAM constellation size is fixed at 64 for all methods. As a baseline, the results of the conventional method using only LDPC codes are shown in Fig. 10(d). The MLC method achieves the optimal BER performance using LDPC codes with overhead values of 50% and 300% for 1-bit and 2-bit set-partitioning cases, respectively, as shown by red circles and blue circles in Fig. 10(a), respectively. Using LDPC codes with the appropriate overheads, the output error floor becomes close to the BER threshold of the outer HD-FEC, thus optimizing the BER characteristic. The total FEC overhead values for the 1-bit and 2-bit set-partitioning cases are 13% and 43%, respectively. To achieve a total overhead of approximately 20%, the different LDPC codes with overheads of 100%, 67%, and 50% are used for the 2-bit set-partitioning case. In Fig. 10(a), blue triangles, blue squares, and blue crosses indicate the calculated results with total FEC overheads of 28%, 23%, and 20%, respectively. As the LDPC overhead decreases, the discrepancy between the error floors and the pre-HD-FEC BER threshold becomes larger, which deviates from the optimal BER characteristic.
Fig. 10. Calculated BER performance with (a) MLC, (b) XOR, (c) proposed methods, and (d) conventional method using only LDPC codes. The numbers indicate the total FEC overhead.
Figure 10(b) shows the calculated results in the XOR method. Using the nearly optimized LDPC codes for the encoder size N of 2, 3, 6, and 12, the BER performance is maintained. For N = 2, the calculated BERs using LDPC codes with overheads of 50% and 33% are indicated by red circles and red triangles in Fig. 10(b), respectively, and the total FEC overhead values are 28% and 21%, respectively. The blue circles and green circles indicate the calculated BERs for N = 3 and 6, respectively, and the total FEC overhead values are tuned to 14% and 10%, respectively. For N = 12, the results are shown by orange circles, and the total FEC overhead is tuned to 9%. The total FEC overhead can be tuned by adjusting the code rate of the XOR method, even if an appropriate LDPC code is used. As the encoder size N is increased, the total FEC overhead decreases steeply, deviating from the target of approximately 20%. For the cases of N ≥ 3, the total overhead of approximately 20% cannot be achieved. Although the total FEC overhead of approximately 20% can be achieved only at N = 2, the granularity of code rate tunability is poor.
The BER performance of the proposed method is shown in Fig. 10(c). The results for the size-2 polar encoder using LDPC codes with overheads of 50% and 33% are indicated by red circles and red triangles, respectively, and those for the size-4 encoder using the LDPC code with overhead of 33% are indicated by blue triangles. The total FEC overhead values are 28%, 21%, and 13%, respectively. The green circles and green triangles show the results for the size-5 encoder using nearly optimized LDPC codes with 67% and 50% overheads, respectively. The total FEC overheads are 27% and 23%, respectively. The orange circles and orange triangles indicate the results for the size-7 encoder using nearly optimized LDPC codes with 67% and 50% overheads, respectively, and the total overhead values are 20% and 18%, respectively. Using different polar encoder structures improves the granularity of the FEC overhead tunability while maintaining the BER performance.
Finally, we summarize the relationships between the required SNR, complexity, and total FEC overhead values in the proposed method. In Fig. 11(a), the total overhead and required SNR values for the proposed method are plotted using red open circles. For comparison, blue squares and green triangles represent those in the XOR and MLC methods, respectively. The crosses indicate the results of the conventional method using only LDPC codes. The results obtained using the size-2 encoder are indicated as overlapping plots because the proposed and XOR methods become identical in that case. Although the BER performance of all methods are comparable, only the proposed method can finely tune the total overhead from 13% to 28%, which is not possible with the MLC and XOR methods. Consequently, the proposed method can achieve fine tunability of the total FEC overhead and maintain the degradation of the BER performance.
Fig. 11. Calculated relationship (a) between the required SNR and total overhead values and (b) between the complexity and the total overhead values.
Figure 11(b) shows the relationship between the computational complexity and total overhead. The mark definitions are the same as those shown in Fig. 11(a). Although the MLC method is advantageous for complexity reduction, the tunability of the total FEC overhead is limited. Although the complexity of the proposed method is comparable to or slightly less than that of the XOR method, it is approximately half compared with that of the conventional method.
4. Summary
We showed the method for reducing the SD-FEC decoder complexity with a fine tunability of the total FEC overhead to approximately 20% while maintaining the BER performance. Using a polar code with a low-complexity SC decoder, only the less-reliable bit-channels are protected by the SD-FEC, whereas the more-reliable bit-channels are offloaded, reducing the decoder complexity. To maintain BER performance, we proposed encoder structures designed for the SC decoder. In the proposed method, the degradation of the BER performance is suppressed by less than 0.6 dB, even though the complexity is reduced by approximately half, compared with the conventional method using only LDPC codes.
Funding
Japan Society for the Promotion of Science (21H01391).
Acknowledgment
The authors express their gratitude to Professor K. Inoue of Osaka University and Dr. T. Takahashi of Osaka University.
Disclosures
The authors declare no conflicts of interests.
Data availability
The 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
1. B. P. Smith and F. R. Kschischang, “Future prospects for FEC in fiber-optic communications,” IEEE J. Sel. Top. Quantum Electron. 16(5), 1245 (2010). [CrossRef]
2. A. Leven and L. Schmalen, “Status and recent advances on forward error correction technologies for lightwave systems,” J. Lightwave Technol. 32(16), 2735–2750 (2014). [CrossRef]
3. E. Yamazaki, S. Yamanaka, Y. Kisaka, et al., “Fast optical channel recovery in field demonstration of 100-GBit/s ethernet over OTN using real-time DSP,” Opt. Express 19(14), 13179–13184 (2011). [CrossRef]
4. I. B. Djordjevic, M. Arabeci, and L. L. Minkov, “Next generation FEC for high-capacity communication in optical transport networks,” J. Lightwave Technol. 27(16), 3518–3530 (2009). [CrossRef]
5. T. Kupfer, A. Bisplinghof, T. Duthel, et al., “Optimizing power consumption of a coherent DSP for metro and data center interconnects,” OFC, Th3 G.2 (2017).
6. U. Wachsmann, R. F. H. Fischer, and J. B. Huber, “Multilevel codes: Theoretical concepts and practical design rules,” IEEE Trans. Inf. Theory 45(5), 1361–1391 (1999). [CrossRef]
7. A. Bisplinghoff, S. Langenbach, and T. Kupfer, “Low-power, phase-slip tolerant, multilevel coding for M-QAM,” J. Lightwave Technol. 35(4), 1006–1014 (2017). [CrossRef]
8. Y. Koganei, T. Oyama, K. Sugitani, et al., “Multilevel coding with spatially coupled repeat-accumulate codes for high-order QAM optical transmission,” J. Lightwave Technol. 37(2), 486–492 (2019). [CrossRef]
9. M. Barakatain, D. Lentner, G. Böecherer, et al., “Performance-complexity tradeoffs of concatenated FEC for higher-order modulation,” J. Lightwave Technol. 38(11), 1 (2020). [CrossRef]
10. T. Kakizaki, M. Nakamura, F. Hamaoka, et al., “Low-complexity channel polarized multilevel coding for modulation-format-independent forward error correction,” ECOC, Th1 G.3 (2021).
11. T. Kakizaki, M. Nakamura, F. Hamaoka, et al., “Low-complexity channel-polarized multilevel coding for probabilistic amplitude shaping,” OFC, W3H4 (2022).
12. E. Arikan, “Channel polarization: a method for constructing capacity-achieving codes for symmetric binary-input memoryless channels,” IEEE Trans. Inf. Theory 55(7), 3051–3073 (2009). [CrossRef]
13. A. J. Ferris, C. Hirche, and D. Poulin, “Convolutional polar code,” arXiv, arXiv:1704.00715 (2017). [CrossRef]
14. L. M. Zhang and F. R. Kschischang, “Staircase codes with 6% to 33% overhead,” J. Lightwave Technol. 32(10), 1999–2002 (2014). [CrossRef]
15. M. Barakatain and F. R. Kschischang, “Low-complexity concatenated LDPC-staircase codes,” J. Lightwave Technol. 36(12), 2443–2449 (2018). [CrossRef]
