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Abstract
We propose a simple yet effective strategy to map forward error correction (FEC) encoded symbols to subcarriers in bandwidth-limited intensity-modulation and direct-detection (IM/DD) optical orthogonal frequency division multiplexing (OFDM) systems. The design exploits the reliability diversity of OFDM subcarriers in bandwidth-limited systems and allocates FEC encoded systematic symbols and parity-check symbols to high-reliability and low-reliability subcarriers, respectively. In contrast to adaptively-loaded OFDM or systems using multiple sets of FECs with different code rates for different subcarriers, the proposed design does not induce additional complexity and avoids the round-trip delay between transceivers. We investigate the performance of the proposed design in quasi-cyclic low-density parity-check (QC-LDPC) coded OFDM system under different decoding iterations. Experimental results of 120-Gb/s signals over 2 km and 100-Gb/s signals over 5 km show that the OFDM system using the proposed mapping strategy is superior to both OFDM and discrete-Fourier-transform spread (DFT-S) OFDM using conventional mappings without considering the subcarrier diversity or using pre-equalization, regardless of the received optical power, the FEC code rate, the length of the cyclic prefix, the transmission distance, the number of decoding iterations, and the degree distribution of the QC-LDPC optimized under either an infinite or a limited number of iterations. The proposed mapping is optimal when the systematic symbols and parity-check symbols are exactly loaded to high-reliability and low-reliability subcarriers respectively. The studies enable the proposed strategy promising for bandwidth-limited IM/DD OFDM systems such as low-latency data center interconnects.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Orthogonal frequency division multiplexing (OFDM) is a promising technique for bandwidth-limited intensity-modulation direct-detection (IM/DD) systems because this technology can exploit the performance diversity over the spectrum to maximize the system capacity. Adaptive bit/power loading and entropy loading have been proposed and widely investigated to make full use of the subcarrier diversity induced by limited bandwidth and/or fiber dispersion [1–3]. However, these techniques have high implementation complexity and the round-trip delay because the required channel state information (CSI) should be estimated at the receiver and fed back to the transmitter. Spread/precoding OFDM techniques and new multiplexing techniques such as discrete-Fourier-transform spread (DFT-S) OFDM [4–7] have been proposed to improve the performance of OFDM without the feedback of the CSI by spreading the noise over subcarriers, reducing the peak-to-average power ratio, and/or alleviating the impact of inter-symbol interference (ISI) and inter-carrier interference (ICI). All above techniques can be summarized to adopt the strategies of combining OFDM subcarriers and modulation formats, or manipulating the multiplexing/spreading matrices, but none of them consider the joint design of OFDM subcarriers and forward error correction (FEC).
FEC is an important technology to improve the performance and approach the capacity limit. Conventional low-cost IM/DD systems commonly use hard-decision FEC. Recently, with the drastically increasing traffic demands in applications such as data center interconnects (DCIs), high-capacity IM/DD systems are moving to the operation region with a pre-FEC bit error ratio (BER) of 10−2, in which soft-decision (SD) FEC should be employed [8–9]. This trend has been accelerated recently by the development of probabilistic shaping (PS), whose performance gain is obtained only in the low signal to noise ratio (SNR) region. Among SD-FEC, quasi-cyclic low-density parity-check (QC-LDPC) code is an important technology to guarantee the reliability and approach the Shannon limit due to the advantages of simple implementation, fast convergence and low error floor [10]. Conventional QC-LDPCs are commonly designed based on the degree distribution that is optimized under an infinite number of decoding iterations. This design results in the optimal performance when the number of decoding iterations is large but may exhibit a penalty in low-latency and low-power-consumption systems under a small number of iterations. In [11–12], a LDPC/QC-LDPC design with the degree distribution optimized under a limited number of iterations was proposed and investigated.
FEC-encoded OFDM has also been widely investigated and can increase the spectral efficiency, facilitate dispersion compensation and enhance the resistance to ISI. Earlier works are mainly based on coherent detection where the performance diversity of subcarriers is not severe and the mapping between the FEC bits/symbols and the OFDM subcarriers are not considered [13–15]. Recently, LDPC has also been investigated extensively in IM/DD OFDM systems for optical access, DCIs and radio fronthaul [16–21]. However, the LDPC in these works still operated in the same way as in a single-carrier system to correct errors or to facilitate the PS formats. On the other hand, modern high-capacity IM/DD systems can be severely bandwidth-limited due to low-cost devices and chromatic dispersion, and the SNR may vary significantly over subcarriers. To the best of our knowledge, there are few works that consider the joint design of the FEC and the OFDM subcarriers to make full use of the performance diversity over subcarriers. In [15], the authors took the reliability of subcarriers into consideration and proposed to use two sets of LDPCs for middle and edge subcarriers to improve the performance. However, this design inevitably increases the complexity.
In this paper, we propose a simple yet effective strategy to map FEC-coded bits/symbols to subcarriers in bandwidth-limited IM/DD OFDM systems. The proposed design allocates the systematic and parity-check bits/symbols according to the reliability of subcarriers without inducing additional complexity and latency. We investigate the proposed mapping strategy in a QC-LDPC encoded OFDM system. Experimental results show that the performance of this system is superior to those of OFDM and DFT-S-OFDM using conventional mappings without considering the subcarrier diversity or using pre-equalization, regardless of the received optical power, the code rate of the FEC, the length of the cyclic prefix (CP), the transmission distance, the number of decoding iterations, and the degree distribution optimized under either an infinite or a limited number of iterations. The study is based on QC-LDPC and OFDM but the strategy can also be migrated to other SD-FEC and multicarrier systems with subcarrier diversity.
2. Principle
2.1 QC-LDPC with a limited number of decoding iterations
The investigation in this paper is based on QC-LDPC especially using a limited number of decoding iterations to meet the requirements of low latency and low power consumption. A (J, L) QC-LDPC code is defined by a parity-check matrix of:
Conventionally, the optimal degree distribution is obtained by minimizing the decoding threshold, which can be obtained by the EXIT chart [22]. The EXIT chart is based on the transition of mutual information (MI) between the variable node decoder (VND) and the check node decoder (CND) of QC-LDPC:
Fig. 1. EXIT chart of two irregular QC-LDPC codes with a code rate of 0.8 based on (a) the decoding threshold and (b) the minimal required SNR for error-free decoding under 8 iterations. The degree distributions of code I are λ(x) = 0.2588x2+0.1058x3+0.04705x4+0.5882x5 and ρ(x) = 1.0x17; The degree distributions of code II are λ(x) = 0.5249x3+0.4705x4 and ρ(x) = 1.0x17.
Instead, in this paper, we find the optimal degree distribution by tracing the decoding trajectory [12]. In this case, the decoding threshold is the minimal required SNR for error-free decoding under a pre-set maximal number of decoding iterations. This takes the limited number of decoding iterations into consideration. Figure 1(b) shows the EXIT chart of the two irregular QC-LDPC codes at an SNR of 4.22 dB and 3.41 dB, respectively. These two SNRs are the minimal ones at which the decoding trajectories reach the MI of 0.999 after 8 iterations. This means that code II can have better performance than code I when the maximal number of decoding iterations is 8. Note that the optimal degree distribution considering a limited number of iterations may vary as the maximal decoding iteration changes.
After obtaining the optimal degree distribution, we use a hill-climbing algorithm to construct the check matrix of QC-LDPC in Eq. (1) [23]. For fair comparison, all QC-LDPCs adopted in this paper have the same sparsity and the same girth (set as 8).
2.2 Proposed subcarrier mapping strategy of FEC encoded symbols
The encoded QC-LDPC bits are used to construct quadrature amplitude modulation (QAM) symbols, which are mapped to OFDM subcarriers. To facilitate the interpretation, we define the QAM symbols constructed by systematic bits and parity-check bits as systematic symbols and parity-check symbols, respectively. Conventionally, there are two mapping/allocation strategies, as illustrated in Fig. 2(a)&(b). In mapping I, as shown in Fig. 2(a), the QAM symbols constructed by each QC-LDPC block are mapped to the same subcarrier. In this case, the systematic symbols and parity-check symbols of each block exist in the same subcarrier. When there is performance diversity over subcarriers as discussed later, different QC-LDPC blocks would exhibit different BERs. In conventional mapping II, as shown in Fig. 2(b), the symbols from each QC-LDPC block are mapped to all subcarriers in sequence and different blocks are mapped in the same way successively. For each QC-LDPC block, the systematic symbols and parity-check symbols are spread over subcarriers statistically. To enrich the comparison, we also implement interleaved QC-LDPC and map the interleaved symbols to subcarriers using the conventional mapping II, as shown in Fig. 2(c). In this case, the systematic symbols and the parity-check symbols also traverse all subcarriers. In both Fig. 2(b) and Fig. 2(c), the errors are evenly distributed over not only different FEC blocks but also the systematic and parity-check symbols within a block, even when different subcarriers have different BERs. Finally, it is also possible to use QC-LDPC with different code rates in different subcarriers. However, it requires multiple sets of FEC hardware at transceivers and is not suitable to practical implementation. Therefore, this mapping design is not considered in this paper.
Fig. 2. (a)&(b) Two conventional mapping strategies I and II; (c) Conventional mapping II with interleaved FEC; (d)&(e) Two implementations of the proposed mapping strategy. In the figures, “BN” represents the Nth QC-LDCP block. The systematic and parity-check symbols of the same block are marked with the same stripe but in different colors. Different blocks are marked with different stripes. Figures 2(b)–(e) are only for illustration. The numbers of systematic and parity-check symbols of a block are usually larger than the subcarrier number N but do not need to be an integer multiple of N. In Fig. 2(c), two blocks are interleaved as an example.
In bandwidth-limited OFDM systems, such as DCIs using low-cost devices to achieve high data rates, different subcarriers have different SNRs due to the frequency-selective channel response. Figure 3 depict the experimental results of (a) the SNR distribution over subcarriers and (b) the corresponding BER of a 16QAM OFDM signal before the FEC decoding. For comparison, we also show the results of DFT-S-OFDM in the figure. The numbers in the bracket are the received optical power (ROP). The experimental setup is the same as that in Section 3.1. The SNR is calculated based on the method in [6]. In OFDM, it is evident that the SNR is high for low-frequency subcarriers but degrades rapidly at high frequency due to the limited bandwidth. The SNR increases as the ROP increases. On the other hand, DFT-S-OFDM spreads the noise over subcarriers, resulting in a flat SNR profile. In Fig. 3(b), as expected, the bit errors in OFDM are mainly from the high-frequency subcarriers and decrease as the ROP increases. At high-frequency subcarriers, the BER can be higher than 0.1 that is beyond the decoding capacity of the QC-LDPC. In DFT-S-OFDM, the errors are evenly distributed over subcarriers due to the flat SNR profile. However, even for the ROP of −6.6 dBm, the BERs of all subcarriers tend to exceed the decoding threshold of the SD-FEC. It implies that OFDM can be better than DFT-S-OFDM at the SD-FEC decoding threshold if the performance diversity of subcarriers can be exploited properly, although it is generally recognized that DFT-S-OFDM outperforms OFDM in a low pre-FEC BER region, i.e. at the HD-FEC threshold.
Fig. 3. (a) Experimental SNR and (b) pre-FEC BER versus the index of subcarriers for a 16QAM-OFDM signal. The experimental setup is the same as that in Section 3.1. The data rate and the transmission distance are 120 Gb/s and 2 km, respectively.
In this paper, we will exploit the characteristics of the SNR/BER diversity over subcarriers and propose a simple yet effective subcarrier mapping strategy for the FEC-encoded symbols. In each QC-LDPC block, the systematic symbols are more important and should be better protected because they carry the information. In the proposed design, the systematic and parity-check symbols of each block are mapped to subcarriers separately, as shown in Fig. 2(d). Specifically, the systematic symbols are assigned to high-reliability subcarriers at low frequencies while the parity-check symbols are allocated to low-reliability subcarriers at high frequencies. In contrast to Fig. 2(b) and 2(c), the errors in the systematic symbols are less than those in the parity-check symbols before decoding. Even when the errors are not fully corrected after the pre-set number of iterations, the errors are still mainly from the parity-check symbols and do not influence the overall performance significantly.
The coding processing of the proposed design is as follows. The subcarriers are divided into two parts, (N-K) high-reliability subcarriers at low frequencies and K low-reliability subcarriers at high frequencies. The systematic symbols and parity-check symbols in each QC-LDPC block are assigned to these two parts, respectively. As shown later, when (N-K)/N is equal to the code rate R, i.e. the systematic symbols and parity-check symbols are mapped to these two parts exactly, the performance is optimal. For example, if the code rate is 5/6 and the total number of modulated subcarriers N is 240, the optimal design is to assign the systematic symbols to the 200 low-frequency subcarriers and the parity-check symbols to the 40 high-frequency subcarriers. On the other hand, when (N-K) and K are 220 and 20, respectively, the low-reliability subcarriers are filled with parity-check symbols while the systematic symbols and the remaining parity-check symbols are loaded to the 220 high-reliability subcarriers. When (N-K) and K are 180 and 60, respectively, the high-reliability subcarriers include only the systematic symbols while the low-reliability subcarriers include all parity-check symbols and the remaining systematic symbols. The performance of these two designs is better than the conventional mapping but is not as good as the case where (N-K)/N is equal to the code rate.
Figure 2(d) assumes that the low-reliability subcarriers are at high frequencies, which is true for short-reach systems. In this case, the proposed strategy can also be implemented using the method in Fig. 2(e) without separating the systematic and parity-check symbols. However, in longer-distance applications where spectral nulls occur in the signal spectrum, the parity-check symbols should be mapped to the subcarriers around the spectral nulls rather than those at high frequencies. In this case, the implementation of Fig. 2(e) is no longer valid. The proposed strategy can also be combined with interleaving, where the systematic and parity-check symbols should be interleaved separately, and the mapping is still similar to Fig. 2(d) or Fig. 2(e) except that the high- and low-reliability subcarriers are filled with interleaved systematic and parity-check symbols, respectively. In fact, the proposed mapping can be viewed as conventional mapping II when the FEC block length is equal to N, or the interleaving of multiple N-length FEC blocks with the mapping method of Fig. 2(e). Obviously, different from the conventional interleaving, the information of OFDM subcarriers should be considered here. The FEC should have a block length equal to N or be interleaved with the total length equal to an integer multiple of N, which limits the design flexibility. Finally, it is noted that the proposed design needs to know the positions of better/poorer subcarriers to enable the mapping, but it does not require precise channel/SNR estimation that is complicated and time-consuming. This is in contrast to the adaptively-loaded OFDM or OFDM with pre-equalization where the CSI should be obtained at the receiver and fed back to the transmitter.
3. Experimental setup and results
3.1 Experimental setup
We conducted experiments to verify the advantage of the proposed design. Figure 4 shows the experimental setup. At the transmitter, a bit sequence was sent to the QC-LDPC encoder to insert the parity-check bits. The code rate was 5/6 unless otherwise stated, and the size of the corresponding check matrix was 1400×8400. The encoded bits were mapped to 16QAM symbols using gray mapping. The symbol sequence was divided into training and payload symbols, which were used to estimate the channel response and calculate the BER, respectively. The generated symbols were mapped to OFDM subcarriers either in conventional ways or by using the proposed design, as shown in Fig. 2. Hermitian extension and an M-point inverse fast Fourier transform (IFFT) were applied to generate a real time-domain signal. The signal was added with a CP to combat ISI/ICI and a start-of-frame symbol was used to facilitate synchronization. The peak-to-average power ratio (PAPR) was optimized as 9 dB. After the parallel-series (P/S) conversion, the time-domain signal was downloaded to a 64-GS/s arbitrary waveform generator (AWG) with a 3-dB bandwidth of ∼23 GHz. The electronic signal was amplified by an electric amplifier (EA) with a 40-GHz bandwidth and a 16-dB gain and modulated a CW light using a Mach-Zehnder modulator (MZM). The DC bias of the MZM and the peak-to-peak voltage of the signal were optimized. The power into the fiber was ∼4 dBm. After transmitting a 2-km or 5-km single mode fiber (SMF), the optical signal was detected by a 40-GHz photodiode (PD). The received power was controlled by a variable optical attenuator (VOA). Finally, the received electric signal was amplified by a 40-GHz EA with a 30-dB gain and sampled by an 80-GS/s oscilloscope. The vertical scale of the scope was optimized so that the signal was sampled with the full quantization resolution. The receiver DSP included synchronization, removing CP, FFT, equalization and demapping. The bits were re-ordered after the demapping according to the design at the transmitter before sent to the QC-LDPC decoder. The decoding of the QC-LDPC was based on the sum-product algorithm. The number of systematic bits in the BER calculation was larger than 1 million for all code rates. The size of FFT, M, was 512. The DC subcarrier was set as zero. The number of modulated subcarriers was 240 and 200 for 2-km and 5-km transmissions, respectively. Therefore, the corresponding gross bit rates were 120 Gb/s and 100 Gb/s, respectively.
For comparison, we also implemented DFT-S-OFDM which utilized an additional pair of DFT/IDFT for spreading/de-spreading and OFDM with pre-equalization. The bias of the MZM and the peak-to-peak voltage of the transmitted signal were also optimized. Other parameters were the same as those in OFDM. In OFDM with pre-equalization, the SNR was measured at the receiver and the signal at the transmitter was equalized accordingly to enable a flat response.
3.2 Experimental results
We firstly investigate the proposed design in a 120-Gb/s system over 2 km. Figure 5(a) shows the BER versus the ROP for the OFDM system without the FEC decoding and with decoding using different mapping strategies. The BERs of DFT-S-OFDM without and with FEC decoding are also given for comparison. Note that because DFT-S-OFDM has a flat SNR profile as shown in Fig. 3(a), different mapping strategies result in the same performance and so only that using the conventional mapping II is given for simplicity. The maximal number of decoding iterations is 8. The degree distribution of the QC-LDPC is optimized under 8 iterations by using the method in Section 2.1. The figure shows that the performance of DFT-S-OFDM before decoding is poorer than that of OFDM. This is because the investigated ROP region is chosen such that the pre-FEC BER is 10−2∼10−1, which is around the SD-FEC decoding threshold. DFT-S-OFDM has a poorer performance in this region although it exhibits better performance than OFDM in a lower BER region [6]. Consequently, its performance after decoding is also not good. On the other hand, in OFDM, the performance of the conventional method I is similar as that in the uncoded case. In this method, each QC-LDPC block is mapped to the same subcarrier and the decoding performance of a QC-LDPC block strongly depends on the reliability of the subcarrier this block is mapped to. In the bandwidth-limited system, the errors before decoding are mainly from high-frequency subcarriers. The performance of these subcarriers can be very poor with a BER of 0.1∼0.3 as shown in Fig. 3(b), which is beyond the decoding capacity of the QC-LDPC. Consequently, the errors for the QC-LDPC blocks mapped to these subcarriers cannot be corrected and the overall performance cannot be improved after decoding. In the conventional method II, the symbols in each QC-LDPC block are spread over all subcarriers. For each QC-LDPC block, although the symbols allocated to high-frequency subcarriers may have a lot of errors, those allocated to low-frequency subcarriers have good performance and can be used to rectify the errors in high-frequency subcarriers. Consequently, the decoding performance is much better than that of the conventional mapping I. However, this method does not differentiate the systematic symbols and the parity-check symbols. In the proposed design, the systematic symbols of each block are mapped to low-frequency subcarrier while the parity-check symbols are mapped to high-frequency subcarriers. Even when the errors are not fully corrected after decoding, the remaining errors do not influence the performance significantly, thus resulting in further performance improvement compared to the conventional mapping II.
Fig. 5. (a) BER versus ROP for DFT-S-OFDM and OFDM using different mapping methods. (b) BER versus ROP for OFDM and DFT-S-OFDM using interleaved QC-LDPC (Inter) or pre-equalization (Pre) + QC-LDPC. In (a) and (b), the QC-LDPC code rate is 5/6 and the length of CP is 12. The number of decoding iterations is 8 and the data rate is 120 Gb/s.
In order to further verify the benefit of the proposed design, Fig. 5(b) shows the BER performance of OFDM and DFT-S-OFDM using interleaved FEC or pre-equalization, denoted as “inter” or “pre” in the figure, respectively. In the interleaved scenario, 6 QC-LDPS blocks are combined for joint encoding/decoding. Comparison of Fig. 5(a) and Fig. 5(b) shows that interleaving has little influence on the performance of both DFT-S-OFDM and OFDM. This is because interleaving only tries to evenly spread the errors over multiple FEC blocks but does not consider the performance diversity of subcarriers. Taking the conventional mapping II in Fig. 2(b) as an example, the systematic symbols and the parity-check symbols of each block traverse all subcarriers so that the pre-FEC errors have been evenly distributed over not only the systematic and parity-check symbols within a block but also different blocks. Interleaving in Fig. 2(c) does not change the error distributions and so cannot improve the performance. On the other hand, pre-equalization results in similar performance regardless of the mapping strategy due to a flat channel response and its performance is also comparable to that of conventional mapping II without pre-equalization. However, the BER is still higher than that using the proposed mapping, which further exploits the subcarrier diversity and can protect the systematic symbols better by allocating less errors to these symbols.
Figure 6(a) shows the performance of OFDM using the conventional mapping II and the proposed mapping design when the variable degree distribution of the QC-LDPC is optimized under an infinite number of iterations or 8 iterations. The code rate is 5/6 and the number of the decoding iterations is 8. The check degree distribution is fixed for fair comparison. It is seen that the proposed mapping is better than the conventional mapping II regardless of the degree distribution of the QC-LDPC. In both mapping methods, the QC-LDPC based on the 8-iteration degree distribution exhibits a penalty in the high post-FEC BER region but is advantageous for low BERs compared to that using the infinite-iteration degree distribution [11]. In the following results, unless otherwise stated, we adopt the degree distribution optimized under n iterations to reduce the latency, where n is the number of iterations in the QC-LDPC decoding.
Fig. 6. (a) BER versus ROP for OFDM using the conventional mapping II and the proposed mapping when the degree distribution of the QC-LDPC is optimized under an infinite number of iterations or 8 iterations. The number of decoding iterations is 8. (b) BER versus the number of iterations for the systematic and parity-check bits under the conventional mapping II and the proposed mapping when the ROP is −7.2 dBm. In (a) and (b), the QC-LDPC code rate is 5/6 and the length of CP is 12. The data rate is 120 Gb/s.
In order to observe the evolution of the decoding performance, Fig. 6(b) shows the BER performance versus the number of iterations for the systematic and parity-check bits using the conventional mapping II and the proposed mapping. The ROP is −7.2 dBm. It is seen that the BER decreases as the number of iterations increases for all cases. In the proposed mapping, the performance of the systematic bits is always better than that of the parity-check bits regardless of the number of iterations. Note that only the systematic bits are used in the ultimate performance evaluation. It is also confirmed that the advantage of the proposed mapping over the conventional mapping II maintains for a larger number of decoding iterations.
Figure 7(a) shows the performance of OFDM using the proposed design with different K at the code rate of 5/6, where K is the number of low-reliability subcarriers allocated to the parity-check symbols. For instance, when K is 20, the 221st to 240th subcarriers are used to transmit the parity-check symbols, while the systematic symbols and remaining parity-check symbols are loaded to the 1st to 220th subcarriers. The conventional mapping II in the figure can be viewed as a special case of the proposed design with K=0. It is clearly seen that the performance improves as K increases from 20 to 40 but degrades as K further increases. Because the number of modulated subcarriers is 240 and the code rate is 5/6, K=40 corresponds to the case where the systematic and parity-check symbols are exactly loaded to the (N-K) high-reliability subcarriers and K low-reliability subcarriers, respectively. On the other hand, there is a penalty when K is too large or too small where some systematic symbols are mapped to high-frequency subcarriers or (N-K) high-reliability subcarriers consist of more high-frequency subcarriers. Additional result shows that this conclusion is also valid at other code rates. Figure 7(b) depicts the BER versus the length of CP. It is shown that the performance advantage of the proposed design maintains as the length of CP varies. The BER improvement is one order of magnitude compared to conventional mapping II. Unless otherwise stated, K will be design as floor(N(1-R)), where floor is the maximal integer closest to the input argument and R is the code rate.
Fig. 7. (a) BER versus ROP for OFDM using the proposed design with different number of K, where (N-K) low-frequency subcarriers and K high-frequency subcarriers are allocated to the systematic and parity-check symbols, respectively. (b) BER versus the length of CP. In (a) and (b), the QC-LDPC code rate is 5/6 and the number of decoding iterations is 8. The data rate is 120 Gb/s. In (a), the length of CP is 12. In (b), the ROP is −6.8 dBm.
Figure 8 shows (a) BER and (b) frame error rate (FER) versus ROP for OFDM at different code rates, where the FER is defined as the ratio of the number of error-free blocks after decoding to the total block number. The size of the check matrices at the code rate of 3/4, 4/5 and 5/6 is 1000×4000, 1200×6000, and 1400×8400, respectively. The number of low-reliability subcarriers K are 60, 48, 40, respectively. It is seen that the proposed method exhibits better BER and FER performance regardless of the code rates. In Fig. 8(a), by using the proposed design, the required ROP to achieve a BER of 10−4 is reduced by around 0.3 dBm, 0.4 dBm and 0.4 dBm at the code rate of 3/4, 4/5 and 5/6, respectively.
Fig. 8. (a) BER (b) FER versus ROP for OFDM at different code rates. The length of CP is 12 and the number of decoding iterations is 8.
Fig. 9. (a) BER versus ROP for OFDM when the degree distribution of the QC-LDPC is optimized under an infinite number of iterations or 4 iterations. (b) BER versus ROP at different code rates. In (a) and (b), the maximal number of decoding iterations is 4 and the length of CP is 12. The data rate is 120 Gb/s. In (a), the code rate is 5/6.
In the above results, the maximal number of decoding iterations is 8. Figure 9(a) shows the performance of OFDM using the conventional method II and the proposed mapping when the maximal number of iterations is 4. Accordingly, the degree distribution of the QC-LDPC is optimized under either an infinite number of iterations or 4 iterations. Note that the optimal degree distribution at a fixed code rate may change when the maximal number of decoding iterations changes. It is seen that similar to Fig. 6(a), the proposed mapping is better than the conventional mapping II regardless of the degree distribution. The QC-LDPC based on the 4-iteration degree distribution is superior to that using the infinite-iteration degree distribution for BERs less than 10−4. Figure 9(b) depicts the BER versus the ROP at different code rates. It is confirmed that the proposed design exhibits better performance than the conventional mapping II at all code rates. The required ROP to achieve a BER of 10−4 is reduced by 0.4 dBm, 0.3 dBm and 0.4 dBm at the code rate of 3/4, 4/5 and 5/6, respectively. It is also shown that the required ROP to achieve the same BER is higher than that in Fig. 8(a), because a smaller number of iterations results in more remaining errors and thus degraded performance.
Fig. 10. (a) BER versus ROP for OFDM using the conventional mapping II and the proposed mapping when the degree distribution of the QC-LDPC is optimized under an infinite number of iterations and 8 iterations. (b) BER versus ROP for OFDM using the proposed design with different number of K. In (a) and (b), the code rate is 5/6 and the length of CP is 12. The number of decoding iterations is 8. The data rate and the fiber length are 100 Gb/s and 5 km, respectively.
Next, we investigate the proposed mapping design in a 100-Gb/s OFDM system over 5-km single mode fiber. Figure 10(a) shows the performance of OFDM using the conventional mapping II and the proposed design when the degree distribution of the QC-LDPC is optimized under an infinite number of iterations or 8 iterations. The code rate is 5/6 and the number of the decoding iterations is 8. Similar to the results in Fig. 6(a), the proposed mapping is superior to the conventional mapping II regardless of the degree distribution. In both mapping strategies, the QC-LDPC based on the 8-iteration degree distribution is better than that using the infinite-iteration degree distribution for BERs below 10−4. Figure 10(b) shows the performance of OFDM using the proposed design with different number of K at the code rate of 5/6. It is seen that the conclusion is the same as that in Fig. 7(a). The optimal performance is achieved for K = floor(200/6) = 33, in which the systematic and parity check symbols are exactly loaded to the (N-K) high-reliability subcarriers and K low-reliability subcarriers, respectively.
Figure 11 shows the BER versus ROP at different code rates when the number of decoding iterations is (a) 8 and (b) 4. The degree distribution is also optimized under 8 and 4 iterations in (a) and (b). The number of low-reliability subcarriers K are 50, 40, and 33 at the code rate of 3/4, 4/5, and 5/6, respectively. It is seen that proposed method exhibits better BER performance than the conventional mapping regardless of the code rate and the number of iterations. At a fixed mapping strategy and a code rate, the system under 4 iterations requires a higher ROP to achieve the same BER. By using the proposed design, the required ROP to achieve a BER of 10−4 is reduced by 0.1∼0.4 dBm and 0.3∼0.8 dBm in Figs. 11(a) and 11(b), respectively.
Fig. 11. BER versus ROP for OFDM at different code rates when the number of decoding iterations is (a) 8 and (b) 4. The data rate and the fiber length are 100 Gb/s and 5 km, respectively.
4. Summary
We have proposed a simple yet effective subcarrier mapping strategy for FEC-encoded symbols in bandwidth-limited IM/DD OFDM systems. The design exploits the performance diversity of subcarriers and allocates systematic and parity-check symbols according to the reliability of subcarriers. It is found that the performance of the proposed mapping is optimal when the systematic and parity-check symbols are exactly loaded to low-frequency and high-frequency subcarriers, respectively. We investigate the performance of the proposed design in QC-LDPC encoded OFDM systems with different numbers of decoding iterations. Experimental results of a 120-Gb/s OFDM system over 2 km and a 100-Gb/s system over 5 km show that the OFDM system using the proposed mapping is superior to both OFDM and DFT-S-OFDM using conventional mappings or pre-equalization, regardless of the ROP, the code rate, the length of the CP, the transmission distance, the number of decoding iterations, and the degree distribution of the QC-LDPC optimized under either an infinite or a limited number of iterations. The investigation is based on QC-LDPC and OFDM, but the proposed mapping strategy can also be migrated to other SD-FEC and multicarrier systems with subcarrier diversity.
Funding
National Natural Science Foundation of China (61971199); Natural Science Foundation of Guangdong Province (2021A1515012309); Science and Technology Planning Project of Guangdong Province (2019A050503003).
Disclosures
The author declares 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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