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Golay sequences coded coherent optical OFDM for long-haul transmission
Optics Communications 399 (2017) 52–55
Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Golay sequences coded coherent optical OFDM for long-haul transmission
MARK
⁎
Cui Qin , Xiangrong Ma, Tao Hua, Jing Zhao, Huilong Yu, Jian Zhang Nanjing Institute of Technology, Nanjing, Jiangsu, 211167, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Orthogonal frequency division multiplexing (OFDM) Golay sequences Reed-Muller codes
We propose to use binary Golay sequences in coherent optical orthogonal frequency division multiplexing (COOFDM) to improve the long-haul transmission performance. The Golay sequences are generated by binary Reed-Muller codes, which have low peak-to-average power ratio and certain error correction capability. A lowcomplexity decoding algorithm for the Golay sequences is then proposed to recover the signal. Under same spectral efficiency, the QPSK modulated OFDM with binary Golay sequences coding with and without discrete Fourier transform (DFT) spreading (DFTS-QPSK-GOFDM and QPSK-GOFDM) are compared with the normal BPSK modulated OFDM with and without DFT spreading (DFTS-BPSK-OFDM and BPSK-OFDM) after longhaul transmission. At a 7% forward error correction code threshold (Q2 factor of 8.5 dB), it is shown that DFTSQPSK-GOFDM outperforms DFTS-BPSK-OFDM by extending the transmission distance by 29% and 18%, in non-dispersion managed and dispersion managed links, respectively.
1. Introduction Coherent optical orthogonal frequency-division multiplexing(COOFDM)has shown its capability in high-speed long-haul optical fiber transmission systems [1]. However, the nonlinear noise in CO-OFDM due to relatively high peak-to-average power ratio (PAPR) severely limits the transmission distance [1]. Several methods have been proposed to reduce the PAPR and hence fiber nonlinear effects over long-haul transmission [2–4]. However, these methods are inefficient when the transmission distance is more than 8000-km. Recently, nonlinear noise cancellation or squeezing (NLNC or NLNS)methods have been proposed in both OFDM [5] and singlecarrier [6,7] long-haul transmission systems. Due to the loss of 50% spectral efficiency, the QPSK format is selected in [5–7] to compare with the BPSK format in the demonstration. Through both simulation and experiment, it is shown that the Q2factor of QPSK signal with nonlinear noise cancellation (or squeezing) is ~1.4–2 dB higher than BPSK signal after ~8000-km fiber transmission. However, both NLNC and NLNS require pre-equalization for chromatic dispersion (CD) to achieve a symmetric dispersion map, which reduces the flexibility of optical networks and in turn increases the computational complexity. It has been proved in wireless OFDM that the PAPR of any binary Golay sequences is at most 3 dB [8]. However, the performances of Golay sequences coded OFDM have not been evaluated through
⁎
Corresponding author. E-mail address: [email protected] (C. Qin).
http://dx.doi.org/10.1016/j.optcom.2017.04.030 Received 7 March 2017; Received in revised form 9 April 2017; Accepted 13 April 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
nonlinear channels in both wireless and optical transmission systems. The main problem of Golay sequences is their low code rate at large code length and high modulation order. In this paper, for the first time, we propose to apply binary Golay sequences in CO-OFDM system with QPSK modulation to maintain good spectral efficiency and improve the long-haul transmission performance. We first construct QPSK codes based on binary Golay sequences. The Golay sequences are generated through the Reed-Muller codes, which have low PAPR and certain error correction capability [8]. In our scheme, the length of Golay sequences is 16 and the code rate is 0.5. Therefore, we cascade several Golay sequences in one OFDM symbol to achieve a larger subcarrier number. A low-complexity decoding algorithm for the QPSK codes is then proposed. With the same spectral efficiency, the QPSK modulated OFDM with Golay sequences coding with and without discrete Fourier transform (DFT) spreading (DFTS-QPSK-GOFDM and QPSK-GOFDM) are chosen to compare with the normal BPSK modulated OFDM with and without DFT spreading (DFTS-BPSK-OFDM and BPSK-OFDM) in a wavelength division multiplexing (WDM) polarization division multiplexing (PDM) long-haul transmission system. Through simulation, it is shown that DFTS-QPSK-GOFD Maintains the best performance after both non-dispersion managed (NDM) and dispersion managed (DM) transmission links. The performance of DFTS-QPSK-GOFDM surpasses DFTS-BPSK-OFDM by 29% and 18% in maximum transmission distances for NDM and DM links at 7% forward error correction(FEC) code threshold, respectively.
Optics Communications 399 (2017) 52–55
C. Qin et al.
sequence Sk has the minimum Euler distance with the real part of the received signal R. 6) Repeat the steps (1) to (5) until both the real and imaginary parts of the received signal in all sub-blocks have been decoded.
2. Golay sequences coded OFDM 2.1. Construction of higher modulation order OFDM codes It has been shown that the Golay sequences with length 2m can be obtained from the summation of second-order cosets and first-order Reed-Muller codes [8]. Therefore, the binary Golay sequences of length 2m is given by m −1
G(1, 2, …, 2m) =
We carried out simulation investigation using Optisystem 13.0 [9]. The simulation setup consists of four WDM-PDM channels with 20GHz channel spacing, and the original data at two polarizations are sampled at 32-GSa/s and mapped onto 128 subcarriers with QPSK or BPSK modulation in each channel. The DFT size is 256, resulting in a filling ratio of 2. The subband number is 2 for DFTS scheme as this achieves almost optimal performance [2]. We choose 4 samples as the cyclic prefix to avoid inter-symbol interference. The laser linewidth is set to be 100-kHz. The fiber link consists of several spans of DM or NDM sections. A DM section consists of 70-km standard single mode fiber (SSMF) and 10-km dispersion compensation fiber (DCF),while a NDM section consists of 80-km SSMF. The CD, polarization mode dispersion (PMD), attenuation coefficient, and nonlinearity coefficient for the SSMF and DCF are: DSSMF=16 ps/nm/km, DDCF=−112 ps/nm/ km, αSSMF=0.2 dB/km, αDCF=0.6 dB/km, γSSMF=1.3 w−1 km−1, γDCF=5.4 w−1 km−1, DPMD=0.5 ps km−1/2. An Erbium doped fiber amplifier is used to fully compensate the fiber loss with a noise figure of 6.0 dB. The CD in NDM transmission link is first compensated digitally at the receiver side before OFDM demodulation. We use 10 training symbols for every 200 data symbols for channel estimation and equalization. 4 pilot subcarriers are used in each OFDM symbols for common phase compensation. Therefore, the raw bitrate for the WDMPDM channel is 32GSa/s×128/256×1b/Sa×4 × 2 = 128 Gb/s. The total number of transmission bits in the simulation is 10 6 . The Q 2 factor is derived directly from bit error rate (BER) based on the formula Q2 [dB]=20 × log10 ( 2 erfcinv(2 × BER)) [7].
m
∑ xπ (k) xπ (k +1) + ∑ ck xk k =1
3. Simulation setup
k =0
(1)
where π is a permutation of the elements[1, 2, …,m]. The Boolean function xi can be specified by a true table [8]. The modulated signals in the OFDM subcarriers are then given by
S(1, 2, …, 2m) = exp[ jπG (1, 2, …, 2m)]
(2)
where only BPSK modulation is considered in (2). It is because the generated Golay sequences in (2) are binary sequences. The higher order modulation can also be achieved based on the forms of (1) and (2), which is also shown in [8]. In (1), the first part in the right hand-side is the second-order cosets. For a certain m, there are m!/2 cosets generated in (1). Generally, w information bits are used to choose the cosets representative, where 2w is the largest number no greater than m!/2 . The second part in right hand-side of (1) is the (m+1) bits ck (k is from 0 to m) mapping to first-order Reed-Muller codes with length 2m . Therefore, we encode [w + (m + 1)] information bits into Golay sequences of length 2m . It is easy to see that the spectral efficiency of Golay sequences is very low, especially when m is very large. In order to solve this problem, we choose m = 4 , which means 8 (w = 3) information bits are encoded into Golay sequences of length 16, corresponding to spectral efficiency of 0.5. We use S1 and S2 to represent two BPSK sequences in the form of (2). Then the generated QPSK sequences SQ will have the same spectral efficiency as normal BPSK sequences (spectral efficiency of 1), which can be expressed as
SQ =
2 2 S1 + j S2 2 2
4. Simulation results and discussions We first investigate and compare the complementary cumulative distribution function (CCDF) of (DFTS-)QPSK-GOFDM, (DFTS-) BPSK-OFDM, and normal QPSK-OFDM. As shown in Fig. 1, the PAPRs of QPSK-GOFDM and DFTS-BPSK-OFDM are similar. The PAPR of QPSK-GOFDM is lower than that of QPSK-OFDM, which proves low PAPR characteristic of Golay sequences. The DFTS-QPSKGOFDM has the lowest PAPR value due to extra DFT spreading at the transmitter side [2–4]. Fig. 2 depicts the BER versus optical signal-to-noise ratio (OSNR)in the back-to-back case for the four schemes in single channel at bit rate
(3)
In (3), the length of QPSK sequences is 16, which is too small to be an OFDM symbol. In order to increase the length of QPSK sequences, we define SQ in (3) as a sub-block and one OFDM symbol consists of several sub-blocks. 2.2. Decoding algorithm for the proposed QPSK sequences The decoding algorithm for binary Reed-Muller codes has been reported in [8]. The algorithm can be modified to decode the binary Golay sequences in (1), where the second-order cosets are considered. In this paper, only 8 cosets are chosen in the generation of Golay sequences. Therefore, we can subtract each possible cosets from the received codeword to obtain 8 Reed-Muller codes for decoding. The best decoding result determines the coset representative. Considering that the real and imaginary parts of the QPSK sequences are two independent binary Golay sequences, the decoding algorithm for the proposed QPSK sequences in one OFDM symbol is summarized below. 1) Input the real part of the received signal R as a binary sequence r of length 16 in one sub-block after decision. 2) Subtract each possible cosets representative to obtain 8 binary sequences rk (k = 1,2, …, 8). 3) Decode the 8 sequences to get the corresponding decoded sequences sk based on the Reed-Muller decoding algorithm [8]. 4) Modulate each decoded sequence sk based on (1) and (2) as Sk . 5) The best decoded result sˆ is chosen as the corresponding modulated
Fig. 1. CCDF for (DFTS-) QPSK-GOFDM, (DFTS-) BPSK-OFDM and QPSK-OFDM, respectively.
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Fig. 3 for NDM and DM links, respectively. In the case of NDM link, as shown in Fig. 4(a), DFTS-QPSK-GOFDM attains a transmission distance of ~10800 km, which is ~29% and ~42% improvement, compared with DFTS-BPSK-OFDM and BPSK-OFDM, respectively. In the case of DM link as shown in Fig. 4(b), as expected, the maximum transmission distance is reduced to ~4200 km and the improvement of DFTS-QPSK-GOFDM is 18% and 26% compared to DFTS-BPSKOFDM and BPSK-OFDM, respectively. It is noted that QPSKGOFDM also increases the transmission performance by ~19% and ~12%, compared to DFTS-BPSK-OFDM in NDM and DM links, respectively. The performance improvement of (DFTS-) QPSKGOFDM mainly arises from two aspects: (1) OSNR improvement due to certain error correction capability of Reed-Muller codes; (2) low PAPR property of Golay sequences. We also investigate the Golay sequence coded OFDM with 16-QAM format, which is constructed based on two QPSK Golay sequences [10]. The sub-block length of the QPSK Golay sequences is 8 in order to achieve the spectral efficiency of 2. From Fig. 5, although the performance of DFTS-16QAM-GOFDM is better than DFTS-16QAMOFDM with the same symbol rate in NDM transmission link, it is worse than DFTS-QPSK-OFDM due to the much lower receiver sensitivity of 16-QAM as compared to QPSK [5–7]. This indicates the limitation of the Golay sequences coding approach in high-spectral-efficiency systems, which is similar to the NLNC or NLNS methods. Regarding the additional DSP load needed to implement the Golay sequences coding in optical OFDM, the complexity of Golay sequences encoding and decoding is small. The main reason is as follows: (1) all the operations can be processed in a logic way using majority logic circuits [8], where no multiplication operations are needed; (2) the length of Golay sequences is 16 and only binary Golay sequences are considered, which further simplifies the design of logic circuits; (3) the complexity of Golay sequences encoding/decoding method scales only mildly with the number of subcarriers, which is much simpler than digital back propagation [11] or Volterra series transfer function [12] based nonlinear compensation methods, whose complexity is usually over one order of magnitude higher than the number of subcarriers.
Fig. 2. BER performances versus OSNR in back-to-back for (DFTS-) QPSK-GOFDM and (DFTS-) BPSK-OFDM, respectively.
of 32-Gb/s. As can be seen, OFDM system with and without DFTS have similar performance. However, the (DFTS-) QPSK-GOFD Maintains better OSNR performance than (DFTS-) BPSK-OFDM. It is noted that the noise in long-haul transmission includes both amplified spontaneous emission noise and nonlinear noise. From Fig. 1, it is expected that the power of generated nonlinear noise is similar among the three schemes of QPSK-GOFDM, BPSK-OFDM and DFTS-BPSK-OFDM due to their similar PAPR performance at CCDF of 10−1. Therefore, QPSKGOFDM is expected to have better performance than the other two schemes after long-haul transmission due to its better OSNR performance. In addition, DFTS-QPSK-OFDM should have the best long-haul transmission performance considering its lowest PAPR and best OSNR performance at the back-to-back case. Fig. 3(a) and (b) show the Q2 factor of the central channel versus the signal launch power after 8800-km NDM (Fig. 3(a)) and 3600-km DM (Fig. 3(b)) transmission links, respectively. As shown in Fig. 3, DFTS-QPSK-GOFDM achieves the best long-haul transmission performance. At the optimal launch power (2 dBm and −1 dBm for NDM and DM links, respectively), the Q2 factor of DFTS-QPSK-GOFDM is ~1.7 dB higher than that of DFTS-BPSK-OFDM for both NDM and DM links. From Fig. 3, we can also see that the long-haul transmission performance of QPSK-GOFDM is better than that of DFTS-BPSKOFDM and BPSK-OFDM, which agrees well with our analysis above. We next evaluate the maximum fiber transmission distance with 7% FEC limit atQ2 factor of 8.5 dB [2]. Fig. 4 shows the Q2 factor versus the transmission distance at the optimal launch power obtained from
5. Conclusion We have implemented Golay sequences in CO-OFDM system to improve the long-haul transmission performance. The Golay sequences are generated based on binary Reed-Muller codes with length of 16, code rate of 0.5 and certain error correction capability. Alarge OFDM symbol size is achieved by cascading several Golay sequences block in one OFDM symbol. A decoding algorithm is proposed to recover the signal with low-complexity. Under the same spectral efficiency, the
Fig. 3. Q2 factor of the center channel inside the WDM channel versus launch power for (DFTS-) QPSK-GOFDM and (DFTS-) BPSK-OFDM under (a) 8800 km NDM and (b) 3600 km DM transmission links.
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Fig. 4. Q2 factor of the center channel inside the WDM channel versus transmission distance for (DFTS-) QPSK-GOFDM and (DFTS-) BPSK-OFDM under (a) NDM and (b) DM transmission links.
Natural Science Foundation of China (Grant Nos. 61501222) and the Scientific Research Foundation of Nanjing Institute of Technology (Grant No. YKJ201320 and CKJB201504). References [1] W. Shieh, I.B. Djordjevic, OFDM for Optical Communications, Academic, New York, 2010. [2] X. Li, W. Zhong, A. Alphones, C. Yu, Fiber nonlinearity tolerance of APSK modulated DFT-S OFDM systems, IEEE Photon. Technol. Lett. 25 (23) (. 2013) 2304–2307. [3] H. Wang, Y. Li, X. Yi, De Kong, J. Wu, J. Lin, APSK modulated CO-OFDM system with increased tolerance towards fiber nonlinearity, IEEE Photon. Technol. Lett. 24 (13) (. 2012) 1085–1087. [4] M. Sung, S. Kang, J. Shim, J. Lee, J. Jeong, DFT-precoded coherent optical OFDM with hermitian symmetry for fiber nonlinearity mitigation, J. Lightwave Technol. 30 (17) (.2012) 2757–2763. [5] W. Peng, T. Tsuritana, I. Morita, Digital nonlinear noise cancellation approach for long-haul optical transmission systems, in: Proceedings of ECOC 2013, London, British, Mo.3.D.2, Sep, 2013. [6] X. Liu, A.R. Chraplyvy, P.J. Winzer, R.W. Tkach, S. Chandrasekhar, Phaseconjugated twin waves for communication beyond the Kerr nonlinearity limit, Nat. Photon. 7 (2013) 560–568. [7] X. Liu, S. Chandrasekhar, P.J. Winzer, R.W. Tkach, A.R. Chraplyvy, Fibernonlinearity-tolerant superchannel transmission via nonlinear noise squeezing and generalized phase-conjugated twin waves, J. Lightwave Technol. 32 (4) (.2014) 766–775. [8] J.A. Davis, J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, ”IEEE Trans. Inform. Theory 45 (. 1999) 2397–2417. [9] OptiSystem Version 13.0 Available: 〈http://www.optiwave.com/〉. [10] B. Tarokh, H.R. Sadjadpour, Construction of OFDM M-QAM sequences with low peak-to-average power ratio, IEEE Trans. Commun. 51 (1) (. 2003) 25–28. [11] E. Ip, J.M. Kahn, Fiber communications: time-reversed twin, Nat. Photon. 7 (2013) 507–508. [12] G. Shulkind, M. Nazarathy, Estimating the volterra series transfer function over coherent optical OFDM for efficient monitoring of the fiber channel nonlinearity, Opt. Express 20 (27) (. 2012) 29035–29062.
Fig. 5. Q2 factor of the center channel inside the WDM channel versus transmission distance for DFTS-QPSK, DFTS-16QAM-GOFDM, DFTS-16QAM after NDM transmission link.
simulation results have shown that (DFTS-) QPSK-GOFDM has better long-haul transmission performance than(DFTS-) BPSK-OFDM. It is also revealed that, DFTS-QPSK-GOFDM can increase the maximum transmission reach by ~29% and ~18% compared with DFTS-BPSKOFDM in NDM and DM transmission links, respectively. Considering the benefits of (DFTS-) QPSK-GOFDM, we believe (DFTS-) QPSKGOFDM is very promising for the implementation of future long haul/ under sea optical network over 10,000-km fiber transmission. Acknowledgement This work is supported by the Project supported by the National
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Contents lists available at ScienceDirect
Optics Communications journal homepage: www.elsevier.com/locate/optcom
Golay sequences coded coherent optical OFDM for long-haul transmission
MARK
⁎
Cui Qin , Xiangrong Ma, Tao Hua, Jing Zhao, Huilong Yu, Jian Zhang Nanjing Institute of Technology, Nanjing, Jiangsu, 211167, China
A R T I C L E I N F O
A BS T RAC T
Keywords: Orthogonal frequency division multiplexing (OFDM) Golay sequences Reed-Muller codes
We propose to use binary Golay sequences in coherent optical orthogonal frequency division multiplexing (COOFDM) to improve the long-haul transmission performance. The Golay sequences are generated by binary Reed-Muller codes, which have low peak-to-average power ratio and certain error correction capability. A lowcomplexity decoding algorithm for the Golay sequences is then proposed to recover the signal. Under same spectral efficiency, the QPSK modulated OFDM with binary Golay sequences coding with and without discrete Fourier transform (DFT) spreading (DFTS-QPSK-GOFDM and QPSK-GOFDM) are compared with the normal BPSK modulated OFDM with and without DFT spreading (DFTS-BPSK-OFDM and BPSK-OFDM) after longhaul transmission. At a 7% forward error correction code threshold (Q2 factor of 8.5 dB), it is shown that DFTSQPSK-GOFDM outperforms DFTS-BPSK-OFDM by extending the transmission distance by 29% and 18%, in non-dispersion managed and dispersion managed links, respectively.
1. Introduction Coherent optical orthogonal frequency-division multiplexing(COOFDM)has shown its capability in high-speed long-haul optical fiber transmission systems [1]. However, the nonlinear noise in CO-OFDM due to relatively high peak-to-average power ratio (PAPR) severely limits the transmission distance [1]. Several methods have been proposed to reduce the PAPR and hence fiber nonlinear effects over long-haul transmission [2–4]. However, these methods are inefficient when the transmission distance is more than 8000-km. Recently, nonlinear noise cancellation or squeezing (NLNC or NLNS)methods have been proposed in both OFDM [5] and singlecarrier [6,7] long-haul transmission systems. Due to the loss of 50% spectral efficiency, the QPSK format is selected in [5–7] to compare with the BPSK format in the demonstration. Through both simulation and experiment, it is shown that the Q2factor of QPSK signal with nonlinear noise cancellation (or squeezing) is ~1.4–2 dB higher than BPSK signal after ~8000-km fiber transmission. However, both NLNC and NLNS require pre-equalization for chromatic dispersion (CD) to achieve a symmetric dispersion map, which reduces the flexibility of optical networks and in turn increases the computational complexity. It has been proved in wireless OFDM that the PAPR of any binary Golay sequences is at most 3 dB [8]. However, the performances of Golay sequences coded OFDM have not been evaluated through
⁎
Corresponding author. E-mail address: [email protected] (C. Qin).
http://dx.doi.org/10.1016/j.optcom.2017.04.030 Received 7 March 2017; Received in revised form 9 April 2017; Accepted 13 April 2017 0030-4018/ © 2017 Elsevier B.V. All rights reserved.
nonlinear channels in both wireless and optical transmission systems. The main problem of Golay sequences is their low code rate at large code length and high modulation order. In this paper, for the first time, we propose to apply binary Golay sequences in CO-OFDM system with QPSK modulation to maintain good spectral efficiency and improve the long-haul transmission performance. We first construct QPSK codes based on binary Golay sequences. The Golay sequences are generated through the Reed-Muller codes, which have low PAPR and certain error correction capability [8]. In our scheme, the length of Golay sequences is 16 and the code rate is 0.5. Therefore, we cascade several Golay sequences in one OFDM symbol to achieve a larger subcarrier number. A low-complexity decoding algorithm for the QPSK codes is then proposed. With the same spectral efficiency, the QPSK modulated OFDM with Golay sequences coding with and without discrete Fourier transform (DFT) spreading (DFTS-QPSK-GOFDM and QPSK-GOFDM) are chosen to compare with the normal BPSK modulated OFDM with and without DFT spreading (DFTS-BPSK-OFDM and BPSK-OFDM) in a wavelength division multiplexing (WDM) polarization division multiplexing (PDM) long-haul transmission system. Through simulation, it is shown that DFTS-QPSK-GOFD Maintains the best performance after both non-dispersion managed (NDM) and dispersion managed (DM) transmission links. The performance of DFTS-QPSK-GOFDM surpasses DFTS-BPSK-OFDM by 29% and 18% in maximum transmission distances for NDM and DM links at 7% forward error correction(FEC) code threshold, respectively.
Optics Communications 399 (2017) 52–55
C. Qin et al.
sequence Sk has the minimum Euler distance with the real part of the received signal R. 6) Repeat the steps (1) to (5) until both the real and imaginary parts of the received signal in all sub-blocks have been decoded.
2. Golay sequences coded OFDM 2.1. Construction of higher modulation order OFDM codes It has been shown that the Golay sequences with length 2m can be obtained from the summation of second-order cosets and first-order Reed-Muller codes [8]. Therefore, the binary Golay sequences of length 2m is given by m −1
G(1, 2, …, 2m) =
We carried out simulation investigation using Optisystem 13.0 [9]. The simulation setup consists of four WDM-PDM channels with 20GHz channel spacing, and the original data at two polarizations are sampled at 32-GSa/s and mapped onto 128 subcarriers with QPSK or BPSK modulation in each channel. The DFT size is 256, resulting in a filling ratio of 2. The subband number is 2 for DFTS scheme as this achieves almost optimal performance [2]. We choose 4 samples as the cyclic prefix to avoid inter-symbol interference. The laser linewidth is set to be 100-kHz. The fiber link consists of several spans of DM or NDM sections. A DM section consists of 70-km standard single mode fiber (SSMF) and 10-km dispersion compensation fiber (DCF),while a NDM section consists of 80-km SSMF. The CD, polarization mode dispersion (PMD), attenuation coefficient, and nonlinearity coefficient for the SSMF and DCF are: DSSMF=16 ps/nm/km, DDCF=−112 ps/nm/ km, αSSMF=0.2 dB/km, αDCF=0.6 dB/km, γSSMF=1.3 w−1 km−1, γDCF=5.4 w−1 km−1, DPMD=0.5 ps km−1/2. An Erbium doped fiber amplifier is used to fully compensate the fiber loss with a noise figure of 6.0 dB. The CD in NDM transmission link is first compensated digitally at the receiver side before OFDM demodulation. We use 10 training symbols for every 200 data symbols for channel estimation and equalization. 4 pilot subcarriers are used in each OFDM symbols for common phase compensation. Therefore, the raw bitrate for the WDMPDM channel is 32GSa/s×128/256×1b/Sa×4 × 2 = 128 Gb/s. The total number of transmission bits in the simulation is 10 6 . The Q 2 factor is derived directly from bit error rate (BER) based on the formula Q2 [dB]=20 × log10 ( 2 erfcinv(2 × BER)) [7].
m
∑ xπ (k) xπ (k +1) + ∑ ck xk k =1
3. Simulation setup
k =0
(1)
where π is a permutation of the elements[1, 2, …,m]. The Boolean function xi can be specified by a true table [8]. The modulated signals in the OFDM subcarriers are then given by
S(1, 2, …, 2m) = exp[ jπG (1, 2, …, 2m)]
(2)
where only BPSK modulation is considered in (2). It is because the generated Golay sequences in (2) are binary sequences. The higher order modulation can also be achieved based on the forms of (1) and (2), which is also shown in [8]. In (1), the first part in the right hand-side is the second-order cosets. For a certain m, there are m!/2 cosets generated in (1). Generally, w information bits are used to choose the cosets representative, where 2w is the largest number no greater than m!/2 . The second part in right hand-side of (1) is the (m+1) bits ck (k is from 0 to m) mapping to first-order Reed-Muller codes with length 2m . Therefore, we encode [w + (m + 1)] information bits into Golay sequences of length 2m . It is easy to see that the spectral efficiency of Golay sequences is very low, especially when m is very large. In order to solve this problem, we choose m = 4 , which means 8 (w = 3) information bits are encoded into Golay sequences of length 16, corresponding to spectral efficiency of 0.5. We use S1 and S2 to represent two BPSK sequences in the form of (2). Then the generated QPSK sequences SQ will have the same spectral efficiency as normal BPSK sequences (spectral efficiency of 1), which can be expressed as
SQ =
2 2 S1 + j S2 2 2
4. Simulation results and discussions We first investigate and compare the complementary cumulative distribution function (CCDF) of (DFTS-)QPSK-GOFDM, (DFTS-) BPSK-OFDM, and normal QPSK-OFDM. As shown in Fig. 1, the PAPRs of QPSK-GOFDM and DFTS-BPSK-OFDM are similar. The PAPR of QPSK-GOFDM is lower than that of QPSK-OFDM, which proves low PAPR characteristic of Golay sequences. The DFTS-QPSKGOFDM has the lowest PAPR value due to extra DFT spreading at the transmitter side [2–4]. Fig. 2 depicts the BER versus optical signal-to-noise ratio (OSNR)in the back-to-back case for the four schemes in single channel at bit rate
(3)
In (3), the length of QPSK sequences is 16, which is too small to be an OFDM symbol. In order to increase the length of QPSK sequences, we define SQ in (3) as a sub-block and one OFDM symbol consists of several sub-blocks. 2.2. Decoding algorithm for the proposed QPSK sequences The decoding algorithm for binary Reed-Muller codes has been reported in [8]. The algorithm can be modified to decode the binary Golay sequences in (1), where the second-order cosets are considered. In this paper, only 8 cosets are chosen in the generation of Golay sequences. Therefore, we can subtract each possible cosets from the received codeword to obtain 8 Reed-Muller codes for decoding. The best decoding result determines the coset representative. Considering that the real and imaginary parts of the QPSK sequences are two independent binary Golay sequences, the decoding algorithm for the proposed QPSK sequences in one OFDM symbol is summarized below. 1) Input the real part of the received signal R as a binary sequence r of length 16 in one sub-block after decision. 2) Subtract each possible cosets representative to obtain 8 binary sequences rk (k = 1,2, …, 8). 3) Decode the 8 sequences to get the corresponding decoded sequences sk based on the Reed-Muller decoding algorithm [8]. 4) Modulate each decoded sequence sk based on (1) and (2) as Sk . 5) The best decoded result sˆ is chosen as the corresponding modulated
Fig. 1. CCDF for (DFTS-) QPSK-GOFDM, (DFTS-) BPSK-OFDM and QPSK-OFDM, respectively.
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Fig. 3 for NDM and DM links, respectively. In the case of NDM link, as shown in Fig. 4(a), DFTS-QPSK-GOFDM attains a transmission distance of ~10800 km, which is ~29% and ~42% improvement, compared with DFTS-BPSK-OFDM and BPSK-OFDM, respectively. In the case of DM link as shown in Fig. 4(b), as expected, the maximum transmission distance is reduced to ~4200 km and the improvement of DFTS-QPSK-GOFDM is 18% and 26% compared to DFTS-BPSKOFDM and BPSK-OFDM, respectively. It is noted that QPSKGOFDM also increases the transmission performance by ~19% and ~12%, compared to DFTS-BPSK-OFDM in NDM and DM links, respectively. The performance improvement of (DFTS-) QPSKGOFDM mainly arises from two aspects: (1) OSNR improvement due to certain error correction capability of Reed-Muller codes; (2) low PAPR property of Golay sequences. We also investigate the Golay sequence coded OFDM with 16-QAM format, which is constructed based on two QPSK Golay sequences [10]. The sub-block length of the QPSK Golay sequences is 8 in order to achieve the spectral efficiency of 2. From Fig. 5, although the performance of DFTS-16QAM-GOFDM is better than DFTS-16QAMOFDM with the same symbol rate in NDM transmission link, it is worse than DFTS-QPSK-OFDM due to the much lower receiver sensitivity of 16-QAM as compared to QPSK [5–7]. This indicates the limitation of the Golay sequences coding approach in high-spectral-efficiency systems, which is similar to the NLNC or NLNS methods. Regarding the additional DSP load needed to implement the Golay sequences coding in optical OFDM, the complexity of Golay sequences encoding and decoding is small. The main reason is as follows: (1) all the operations can be processed in a logic way using majority logic circuits [8], where no multiplication operations are needed; (2) the length of Golay sequences is 16 and only binary Golay sequences are considered, which further simplifies the design of logic circuits; (3) the complexity of Golay sequences encoding/decoding method scales only mildly with the number of subcarriers, which is much simpler than digital back propagation [11] or Volterra series transfer function [12] based nonlinear compensation methods, whose complexity is usually over one order of magnitude higher than the number of subcarriers.
Fig. 2. BER performances versus OSNR in back-to-back for (DFTS-) QPSK-GOFDM and (DFTS-) BPSK-OFDM, respectively.
of 32-Gb/s. As can be seen, OFDM system with and without DFTS have similar performance. However, the (DFTS-) QPSK-GOFD Maintains better OSNR performance than (DFTS-) BPSK-OFDM. It is noted that the noise in long-haul transmission includes both amplified spontaneous emission noise and nonlinear noise. From Fig. 1, it is expected that the power of generated nonlinear noise is similar among the three schemes of QPSK-GOFDM, BPSK-OFDM and DFTS-BPSK-OFDM due to their similar PAPR performance at CCDF of 10−1. Therefore, QPSKGOFDM is expected to have better performance than the other two schemes after long-haul transmission due to its better OSNR performance. In addition, DFTS-QPSK-OFDM should have the best long-haul transmission performance considering its lowest PAPR and best OSNR performance at the back-to-back case. Fig. 3(a) and (b) show the Q2 factor of the central channel versus the signal launch power after 8800-km NDM (Fig. 3(a)) and 3600-km DM (Fig. 3(b)) transmission links, respectively. As shown in Fig. 3, DFTS-QPSK-GOFDM achieves the best long-haul transmission performance. At the optimal launch power (2 dBm and −1 dBm for NDM and DM links, respectively), the Q2 factor of DFTS-QPSK-GOFDM is ~1.7 dB higher than that of DFTS-BPSK-OFDM for both NDM and DM links. From Fig. 3, we can also see that the long-haul transmission performance of QPSK-GOFDM is better than that of DFTS-BPSKOFDM and BPSK-OFDM, which agrees well with our analysis above. We next evaluate the maximum fiber transmission distance with 7% FEC limit atQ2 factor of 8.5 dB [2]. Fig. 4 shows the Q2 factor versus the transmission distance at the optimal launch power obtained from
5. Conclusion We have implemented Golay sequences in CO-OFDM system to improve the long-haul transmission performance. The Golay sequences are generated based on binary Reed-Muller codes with length of 16, code rate of 0.5 and certain error correction capability. Alarge OFDM symbol size is achieved by cascading several Golay sequences block in one OFDM symbol. A decoding algorithm is proposed to recover the signal with low-complexity. Under the same spectral efficiency, the
Fig. 3. Q2 factor of the center channel inside the WDM channel versus launch power for (DFTS-) QPSK-GOFDM and (DFTS-) BPSK-OFDM under (a) 8800 km NDM and (b) 3600 km DM transmission links.
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Fig. 4. Q2 factor of the center channel inside the WDM channel versus transmission distance for (DFTS-) QPSK-GOFDM and (DFTS-) BPSK-OFDM under (a) NDM and (b) DM transmission links.
Natural Science Foundation of China (Grant Nos. 61501222) and the Scientific Research Foundation of Nanjing Institute of Technology (Grant No. YKJ201320 and CKJB201504). References [1] W. Shieh, I.B. Djordjevic, OFDM for Optical Communications, Academic, New York, 2010. [2] X. Li, W. Zhong, A. Alphones, C. Yu, Fiber nonlinearity tolerance of APSK modulated DFT-S OFDM systems, IEEE Photon. Technol. Lett. 25 (23) (. 2013) 2304–2307. [3] H. Wang, Y. Li, X. Yi, De Kong, J. Wu, J. Lin, APSK modulated CO-OFDM system with increased tolerance towards fiber nonlinearity, IEEE Photon. Technol. Lett. 24 (13) (. 2012) 1085–1087. [4] M. Sung, S. Kang, J. Shim, J. Lee, J. Jeong, DFT-precoded coherent optical OFDM with hermitian symmetry for fiber nonlinearity mitigation, J. Lightwave Technol. 30 (17) (.2012) 2757–2763. [5] W. Peng, T. Tsuritana, I. Morita, Digital nonlinear noise cancellation approach for long-haul optical transmission systems, in: Proceedings of ECOC 2013, London, British, Mo.3.D.2, Sep, 2013. [6] X. Liu, A.R. Chraplyvy, P.J. Winzer, R.W. Tkach, S. Chandrasekhar, Phaseconjugated twin waves for communication beyond the Kerr nonlinearity limit, Nat. Photon. 7 (2013) 560–568. [7] X. Liu, S. Chandrasekhar, P.J. Winzer, R.W. Tkach, A.R. Chraplyvy, Fibernonlinearity-tolerant superchannel transmission via nonlinear noise squeezing and generalized phase-conjugated twin waves, J. Lightwave Technol. 32 (4) (.2014) 766–775. [8] J.A. Davis, J. Jedwab, Peak-to-mean power control in OFDM, Golay complementary sequences, and Reed-Muller codes, ”IEEE Trans. Inform. Theory 45 (. 1999) 2397–2417. [9] OptiSystem Version 13.0 Available: 〈http://www.optiwave.com/〉. [10] B. Tarokh, H.R. Sadjadpour, Construction of OFDM M-QAM sequences with low peak-to-average power ratio, IEEE Trans. Commun. 51 (1) (. 2003) 25–28. [11] E. Ip, J.M. Kahn, Fiber communications: time-reversed twin, Nat. Photon. 7 (2013) 507–508. [12] G. Shulkind, M. Nazarathy, Estimating the volterra series transfer function over coherent optical OFDM for efficient monitoring of the fiber channel nonlinearity, Opt. Express 20 (27) (. 2012) 29035–29062.
Fig. 5. Q2 factor of the center channel inside the WDM channel versus transmission distance for DFTS-QPSK, DFTS-16QAM-GOFDM, DFTS-16QAM after NDM transmission link.
simulation results have shown that (DFTS-) QPSK-GOFDM has better long-haul transmission performance than(DFTS-) BPSK-OFDM. It is also revealed that, DFTS-QPSK-GOFDM can increase the maximum transmission reach by ~29% and ~18% compared with DFTS-BPSKOFDM in NDM and DM transmission links, respectively. Considering the benefits of (DFTS-) QPSK-GOFDM, we believe (DFTS-) QPSKGOFDM is very promising for the implementation of future long haul/ under sea optical network over 10,000-km fiber transmission. Acknowledgement This work is supported by the Project supported by the National
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