Abstract

This paper proposes a probabilistic shaping orthogonal frequency division multiplexing passive optical network (PS-OFDM-PON) based on chaotic constant composition distribution matching (CCDM). With the implementation of a four-dimensional hyperchaotic Lv system, probabilistic shaping and chaotic encryption are realized with low complexity on the process of signal modulation, so as to enhance the system performance in the presence of bit error rate (BER) and security. An 8.9 Gb/s encrypted PS-16 quadrature amplitude modulation (QAM)-OFDM signal transmission over a 25 km standard single mode fiber (SSMF) is experimentally demonstrated. And experimental results indicate that compared with conventional uniform 16QAM-OFDM, the encrypted PS-16QAM-OFDM can obtain a 1.2 dB gain in receiver sensitivity at a BER of 10−3 under the same bit rate. Moreover, the key space of the proposed scheme is 1.98 × 1073, which is a large enough number to effectively guard against any malicious attacks from illegal optical network units (ONUs). The combined superiority of BER and security performance enables a promising prospect for the proposed PS chaotic encryption scheme in a future low-cost optical access network.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Seen as a promising and futureproof candidate, a passive optical network (PON) is capable of dealing with the rapidly increasing demand for bandwidth and capacity in an optical access network. Whereas due to the characteristics of high spectral efficiency, robust tolerance to chromatic dispersion, universal flexibility for modulated constellation and low cost, orthogonal frequency division multiplexing (OFDM) PON stands out to attract more and more momentum [14]. And the two key indicators for system performance, namely effectiveness and reliability, are given full play in OFDM-PON. Meanwhile, the broadcasting mechanism in PON, where downstream signal is broadcasted to all optical network units (ONUs) without discretion, makes it a must to take the security of OFDM-PON into serious consideration [57].

In order to enhance the effectiveness and reliability of the OFDM-PON, as well as improve the system capacity and signal transmission quality, probabilistic shaping (PS) captures researchers’ attention as it can accomplish Maxwell-Boltzmann distribution for quadrature amplitude modulation (QAM) signals to obtain flexible spectral efficiency. And combined with forward error correction (FEC) technique, PS is able to extend the system capacity to nearly reach Shannon limit [810]. On the other hand, as far as constellation figure of merit (CFM) is concerned, PS can effectively reduce the average power of the signal constellation while maintaining the positions of constellation points unchanged (i.e., constant minimum Euclidean distance), thus obtaining CFM gain and greatly improving the bit error rate (BER) performance of signal transmission [1113]. To implement PS, distribution matching (DM) plays an important role, of which constant composition distribution matching (CCDM) is proposed earlier to be proven rather mature and practical [14], given that numerous investigations have been done both theoretically and experimentally [1517]. However, two major challenges still loom large for implementing CCDM. Firstly, high-precision arithmetical ability is required in CCDM module, resulting in high computing complexity. This huge cost of computational complexity may be acceptable in long-haul coherent optical communication system, but for the short-reach optical access network prioritized in low-cost, it seems not to be an ideal solution. Although the computational complexity can be brought down by reducing block length (BL) n in the CCDM implementation, yet when n is small enough, the caused rate loss increases notably, as the relation of $rate\textrm{ }loss = H(A) - {k / n}$ in finite-length DM is sustained, where$H(A)$ denotes the entropy of the shaped amplitudes A, ${k / n}$ is the ratio of DM input bits to the output length in symbols. This above-mentioned constraint is the second challenge faced by CCDM [18,19]. Therefore, how to implement fast DM operation with low rate loss in low-cost OFDM-PON is considered to be a challenging barrier.

When it comes to the security of OFDM-PON, efficient and convenient digital signal processing (DSP)-based physical layer encryption can fend off data being maliciously accessed in essence. Whereas due to the ergodicity, pseudo-randomness and sensitivity to initial values and control parameters, chaotic encryption has been applied in DSP-based physical layer encryption [20]. And multi-dimensional chaotic encryptions in electrical domain, including chaotic encoding, constellation masking, time-frequency confusion, subcarrier scrambling have been reported, which greatly improve the security performance of OFDM-PON [2125]. However, as far as we are concerned, in most of the researches regarding OFDM-PON, either only effectiveness and reliability are taken into account by employing advanced coded modulations and high-performance DSP algorithms to improve system capacity and signal transmission quality, or the system security is solely obtained by applying chaotic encryption to protect data from access by illegals. There is a vacancy for coordinated and unified orchestration between the system effectiveness, reliability and security, which calls for a better architect aimed at better BER and higher security performance, thus accommodating OFDM-PON that is extremely sensitive to computing complexing, cost, and security.

In this paper, we propose and demonstrate a chaotic CCDM-based PS-OFDM-PON to be applied in optical access network. The adopted 4D hyperchaotic Lv system can generate 4 chaotic sequences at the same time, and the chaotic scrambled encryption for the amplitude mapping is implemented in the processing of QAM mapping, as well as the dynamic chaotic configuration for signs. In doing so, not only can the optimization of probability distribution of the constellation points be realized to improve BER performance, but also the system security be enhanced via the data encryption in physical layer. Through the simultaneous scrambling of multi-dimensional information by the 4D hyperchaotic system, the key space of the proposed scheme is greatly expanded. Moreover, since short BL is used in distribution matching process, the system complexity can be extremely low to facilitate the excellent adaption to OFDM-PON application scenarios. An 8.9 Gb/s encrypted PS-16QAM-OFDM signal transmission over 25 km standard single mode fiber (SSMF) is experimentally demonstrated to verify the feasibility of the proposed scheme.

2. Principle

Figure 1 depicts the principle of the proposed chaotic CCDM-based QAM mapping scheme, where every 12 bits are grouped to be mapped into the output of 4 complex PS-16QAM symbols. And such a mapping brings the entropy of 3 bits/symbol. Firstly, the input data of original 12 bits are evenly divided into two parts for respective generation of bipolar amplitudes in I and Q parts. Specifically speaking, for each part, out of the input 6 bits, 2 bits are used for chaotic CCDM to obtain 4 unipolar amplitudes, while the other 4 bits undertake an exclusive OR (XOR) operation with chaotic sequence to generate 4 signs. The multiplexing of signs and unipolar amplitudes can give birth to the bipolar amplitudes in each part, which are then converted to complex PS-16QAM symbols after QAM mapping. As shown in Fig. 1, 4D hyperchaotic Lv system is employed to generate 4 chaotic sequences, which can be expressed as follows [26]:

$$\left\{ \begin{array}{l} \dot{x} = a({y - x} )+ u\\ \dot{y} ={-} xz + cy\\ \dot{z} = xy - bz\\ \dot{u} = xz + du \end{array} \right.\textrm{.}$$
where a, b, and c are three constants of Lv system, and d is a control parameter. Their values are set as 36, 3, 20 and 1.2 in this paper, respectively. And the initial value of $({{x_0},{y_0},{z_0},{u_0}} )$ is set as (12,12,33,124). Among the 4 chaotic sequences simultaneously generated by the hyperchaotic system, x and z are used for chaotic CCDM to generate chaotic unipolar amplitudes, while y and u are used for XOR operation to implement dynamic chaotic configuration for the signs. More complex dynamical behaviors endowed by the hyperchaotic system, as compared with the normal chaotic one, will effectively enhance the security performance of the modulation scheme.

 figure: Fig. 1.

Fig. 1. Schematic of the proposed chaotic CCDM-based QAM mapping.

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There are two possible unipolar amplitudes of {1,3} available for PS-16QAM. In order to form a PS constellation in a way that the inner points are given a higher transmitting chances while lowering those in the outer space, the desirable probability distribution can be set as P(1) = 3/4, P(3) = 1/4. And since short BL can bring down system complexity, we set the value as 4 in this paper. The output unipolar amplitudes by CCDM are composed of three ‘1’ and a single ‘3’, resulting in four possible outputs as denoted by A={1,1,1,3}, B={1,1,3,1}, C={1,3,1,1}, D={3,1,1,1}. Similarly, four possible combinations of 00, 01, 10, 11 can be obtained by the 2 input bits. Therefore, as shown in Table 1, a total of $A_4^4 = 24$ can be observed for the mapping rules between input bits and output unipolar amplitudes. In this paper, two chaotic sequences, x and z, are adopted to perform a dynamic choosing among these mapping rules, so as to implement chaotic CCDM to effectively encrypt data for transmission. The detailed procedures for generating scrambling sequences are illustrated as follows:

$$\left\{ \begin{array}{l} k = floor(\bmod (x \cdot {10^{13}},24)) + 1\\ m = floor(\bmod (z \cdot {10^{13}},24)) + 1 \end{array} \right.\textrm{.}$$
Specifically speaking, a series of operations including modulo, rounding, multiplication and addition is performed, and first 13 digits following the decimal point in x and z is used to generate the scrambling sequences of k and m for chaotic CCDM in I and Q parts, respectively. And both of them are integers in the range of [1,24]. Supposing bit stream of ‘00011011’ is input into CCDM, there will be 24 possible output unipolar amplitudes available. Figure 2 displays the 24 possible outputs in a vivid graphic approach, where blue box denotes amplitude of ‘1’ and yellow one ‘3’. Each possible output is represented by a column in this checkerboard-like diagram. As can be seen, the disorderly distribution of these boxes with different colors indicates that the unipolar amplitudes output by the chaotic CCDM takes on a similar disorderly pattern, thus provisioning guarantee for the security of PS-OFDM-PON.

 figure: Fig. 2.

Fig. 2. Possible amplitude outputs when inputting bit stream ‘00011011’.

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Tables Icon

Table 1. Chaotic CCDM mapping rules.

As for the signs of I and Q parts, two other chaotic sequences, y and u, are employed to generate corresponding scrambling sequences according to the parity of the 13th decimal place, as shown in follows:

$$\left\{ \begin{array}{l} l = floor(\bmod (y \cdot {10^{13}},2))\\ n = floor(\bmod (u \cdot {10^{13}},2)) \end{array} \right.\textrm{.}$$
The generated scrambling sequences, l and n, perform XOR operation with the original bits to obtain the new bit sequences, respectively, which are then mapped to the signs in a way that ‘0’ corresponds to negativity while ‘1’ positivity. Here $sign\_I$ and $sign\_Q$ denote the generated sign sequences of I and Q parts, respectively, which can be expressed as follows:
$$\left\{ \begin{array}{l} sign\_I = xor(l,original\textrm{ }bit) \times 2 - 1\\ sign\_Q = xor(n,original\textrm{ }bit) \times 2 - 1 \end{array} \right.\textrm{.}$$
After the joint implementation of chaotic CCDM-based encrypted amplitude mapping and XOR operation-based encrypted signs generating, bipolar amplitudes in both I and Q parts can be obtained for the following QAM mapping to form the encrypted PS-16QAM constellation. Figure 3 depicts the probability distribution and constellation diagram of the encrypted PS-16QAM. It can be observed that inner points are of higher transmitting chances with 9/64 for each of the innermost 4 constellation points, while the probability distribution sees a descending trend outwards with 3/64 for each of the middle 8 constellation points and 1/64 for the outermost 4 ones. This introduction of PS technique lays a solid foundation for the improvement of the system BER performance.

 figure: Fig. 3.

Fig. 3. (a) Probability distribution and (b) constellation diagram of the encrypted PS-16QAM.

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Meanwhile, the 4D hyperchaotic Lv system adopted in this paper can generate four chaotic sequences of $\{{x,y,z,u} \}$ simultaneously. When the initial value of $({{x_0},{y_0},{z_0},{u_0}} )$ is set as (12,12,33,124), the phase diagrams of the chaotic system in various phase planes are illustrated in Fig. 4, where complex dynamics of bifurcation and chaos with high security performance are exhibited. Figure 5 gives a clear picture of the deviation of the generated four chaotic scrambling sequences as the consequence of the tiny change of the ${10^{ - 11}}$ of initial values, so as to demonstrate the initial values sensitivity of the chaotic system. The deviation values of k, l, m and n at the initial values of $({{x_0} + {{10}^{ - 11}},{y_0},{z_0},{u_0}} )$, $({{x_0},{y_0} + {{10}^{ - 11}},{z_0},{u_0}} )$, $({{x_0},{y_0},{z_0} + {{10}^{ - 11}},{u_0}} )$, $({{x_0},{y_0},{z_0},{u_0} + {{10}^{ - 11}}} )$ are shown in Figs. 5(a)–5(d) respectively. It can be seen that for k and m, $\Delta k$ and $\Delta m$ randomly fall into the range of [-24,24] with little chance of no deviation, while due to the values of either ‘0’ or ‘1’ for l and n, $\Delta l$ and $\Delta n$ are randomly distributed in {-1,0,1}. These observations clearly display the excellent initial values sensitivity of the proposed hyperchaotic scheme. It is worth mentioning that, compared with the conventional CCDM-based probabilistic shaping scheme, the generation of chaotic sequences requires the 4th Runge-kutta method to solve the partial differential equations in Eq. (1). A series of multiplication and addition operations will inevitably increase the computational complexity of the system to a certain extent. That said, the introduction of chaotic system has brought a fundamental guarantee for the physical layer security of the OFDM-PON system. It is worthwhile to increase the security performance at the cost of a certain computational complexity.

 figure: Fig. 4.

Fig. 4. Phase diagram of 4D hyperchaotic Lv system.

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 figure: Fig. 5.

Fig. 5. Initial values sensitivity of 4D hyperchaotic Lv system.

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3. Experiment and results

To verify the BER and security performance of the proposed chaotic CCDM-based encrypted PS-OFDM-PON, an experiment is carried out as shown in Fig. 6. The secure sharing of keys between ONUs and optical line terminal (OLT) is implemented as follows: when an ONU is authorized by the OLT, it randomly generates a key named KEY1 and sends to OLT. After receiving KEY1, the OLT would send the control information and another key named KEY2 to the ONU, which is encrypted with KEY1. Then the ONU will use KEY2 as the key to encrypt and transmit data information. In the OLT, digitally-encrypted PS-16QAM-OFDM signal is generated via offline DSP in the electrical domain. And a total of 512 subcarriers are used to carry the encrypted PS-16QAM data, while other 512 subcarriers are adopted to carry the corresponding complex conjugates to fulfil Hermitian symmetry, so as to ensure the output of the real values of the signal, which are then used for intensity modulation and direct detection. The size of IFFT is 2048. The 1/8 of the OFDM symbol length is added as cyclic prefix (CP) to avoid inter-symbol interference (ISI). And the generated digitally-encrypted PS-16QAM-OFDM signal undergoes digital-to-analog converting procedure by an arbitrary waveform generator (AWG, TekAWG70002A). Then after the amplification of an electrical amplifier (EA), electrically-encrypted PS-16QAM-OFDM signal is injected into Mach-Zehnder modulator (MZM) for intensity modulation to generate modulated optical signal for 25 km SSMF transmission. In the experiment, a continuous wave (CW) laser with wavelength of 1550 nm and optical power of 10 dBm is functioned as the light source. In the ONU, a variable optical attenuator (VOA) is placed to adjust the received optical power for further optical signal detection by photodiode (PD) to obtain the electrical PS-16QAM-OFDM signal. And following the analog-to-digital converting functioned by a mixed signal oscilloscope (MSO, TekMSO73304DX), the offline DSP reversed to the OLT performs signal demodulation and decryption to recover the original information.

 figure: Fig. 6.

Fig. 6. Experimental setup (AWG: arbitrary waveform generator; EA: electrical amplifier; MZM: Mach-Zehnder modulator; PS: power splitter; VOA: variable optical attenuator; PD: photodiode; MSO: mixed signal oscilloscope).

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Figure 7 gives a comparison of BER performance between the proposed encrypted PS-16QAM-OFDM and several other 16QAM-OFDM schemes. As far as OFDM signal is concerned, the total bit rate can be tantamount to the expression of (subcarrier number × entropy × AWG sampling rate/IFFT size/(1+CP)). As these four modulation schemes are of different entropies of 3, 3, 3.6226, and 4 bits/symbol, respectively. It is necessary to adjust their AWG sampling rates to ensure the same bit rate of 8.9 Gb/s for fair comparison, which are set as 13.33, 13.33, 11.04, and 10 GSa/s, respectively. As can be seen, the BER curves of the proposed encrypted PS-16QAM-OFDM and the unencrypted PS-16QAM-OFDM with short BL basically overlap, indicating little impact of the encrypted scheme on BER performance. However, if long BL is applied, the proposed chaotic PS encryption scheme witnesses certain rate loss, resulting in the 0.1 dB deficiency in receiver sensitivity at a BER of ${10^{ - 3}}$. That said, the huge advantage of significantly reducing complexity can be obtained when short BL is employed. In addition, a 1.2 dB gain in receiver sensitivity compared with the conventional uniform 16QAM-OFDM shows that PS can enable better BER performance under the same bit rate even with a certain rate loss. In a nutshell, the experimental results in Fig. 7 can show that the proposed encrypted PS-16QAM-OFDM scheme employing short BL can not only satisfy cost-and-complexity-sensitive optical access network, but also excel in BER performance. The received constellations of the encrypted PS-16QAM-OFDM and uniform 16QAM-OFDM at the received optical power of -13 dBm are also embedded in Fig. 7, of which the former one is more clearly-viewed and displays an optimized probability distribution pattern where inner points are given higher transmitting chances.

 figure: Fig. 7.

Fig. 7. BER curves of the encrypted PS-16QAM-OFDM, PS-16QAM-OFDM with short/long BL, and uniform 16QAM-OFDM under the same bit rate after 25 km transmission.

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In addition, the security performance of the proposed encrypted PS-16QAM-OFDM is appraised and vindicated. The BER curves of two legal ONUs and one illegal ONU after 25 km transmission are displayed in Fig. 8. And the legal ONUs are equipped with the correct key, including initial values and control parameters of the chaotic system. As the received optical power tops -13.6 dBm, the BER of these two legal ONUs both descends to ${10^{ - 3}}$. However, as for the illegal ONU with incorrect key, even with the received optical power setting large enough, the BER still maintained so high at about 0.5 that makes the cracking on the transmitted signal impossible. The received constellations of the legal ONU and illegal one are also illustrates as insets in Fig. 8 when the received optical power is -13 dBm. It is worth mentioning that even though these constellations seem identical, original information still can’t be deciphered in illegal ONU.

 figure: Fig. 8.

Fig. 8. BER curves of legal ONUs and illegal ONU after 25 km transmission.

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We attempt to transmit a cute panda-featured image employing the proposed chaotic PS encryption scheme, and the received images and their histograms for legal ONU and illegal one are shown in Fig. 9. It can be seen that the received image of legal ONU is very clear with uneven histogram indicating correct information of gray-scale value. While in illegal ONU we can’t see anything but total fuzz, leaving no information of gray-scale value with even histogram.

 figure: Fig. 9.

Fig. 9. (a) Image and (b) histogram received by the legal ONU; (c) Image and (d) histogram received by the illegal ONU.

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Finally, we further evaluate the anti-attack capability of the proposed chaotic PS encryption scheme. Figure 10 depicts the BER curves of an illegal ONU under the circumstances that any three initial values out of $\{{x,y,z,u} \}$ are already breached. It can be seen that BER fluctuates at around 0.5, meaning that even though the received constellation is clearly viewed, original information can’t be deciphered and the received panda image is blurry. In the 4D hyberchaotic Lv system, the ranges of initial values of x, y, z, and u are (-25,25), (-40,40), (0,60) and (-200,300), respectively. And the system can only sustain a state of hyperchaos when the control parameter d is in the range of (-0.35,1.3). If only taking d and four initial values into consideration, the key space of the proposed scheme can be conservatively calculated as (50×${10^{13}}$)×(80×${10^{13}}$)×(60×${10^{13}}$)×(500×${10^{13}}$)×(1.65×${10^{13}}$) = 1.98×${10^{\textrm{73}}}$, which is large enough to effectively prevent any violent attacks from illegal ONUs.

 figure: Fig. 10.

Fig. 10. BER curves of an illegal ONU with any three initial values known.

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4. Conclusion

Aimed at realizing double enhancement of both BER and security performance in OFDM-PON, this paper proposes a chaotic CCDM-based PS-OFDM-PON scheme to perform chaotic data encryption and probabilistic constellation shaping via four chaotic sequences produced by 4D hyperchaotic Lv system. An 8.9 Gb/s encrypted PS-16QAM-OFDM signal transmission over 25 km SSMF is experimentally demonstrated. With regards to the BER performance, the proposed encrypted PS-16QAM-OFDM outperforms the uniform 16QAM-OFDM counterpart by 1.2 dB receiver sensitivity gain at a BER of ${10^{ - 3}}$ under the same bit rate. When it comes to the security enhancement, the proposed scheme possesses a key space of $1.98 \times {10^{73}}$, which is so enormous that even with enough received optical power the illegal ONU still can’t decipher the original information due to the worsening BER of about 0.5. Experimental results indicate that the proposed chaotic PS encryption scheme can effectively bring about the double improvement in both BER and security performance in OFDM-PON, proving to be an efficient solution for future low-cost optical access network.

Funding

National Key Research and Development Program of China (2018YFB1800901); National Natural Science Foundation of China (61675004, 61705107, 61720106015, 61727817, 61775098, 61822507, 61835005, 61875248, 61935005, 61935011, 61975084); BUPT Excellent Ph.D. Students Foundation (CX2020301); Postgraduate Research & Practice Innovation Program of Jiangsu Province (KYCX20_0963); Open Fund of IPOC (BUPT); Jiangsu talent of innovation and entrepreneurship; Jiangsu team of innovation and entrepreneurship.

Disclosures

The authors declare no conflicts of interest.

References

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References

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  1. M. Chen, X. Xiao, Z. R. Huang, J. Yu, F. Li, Q. Chen, and L. Chen, “Experimental demonstration of an IFFT/FFT size efficient DFT-spread OFDM for short reach optical transmission systems,” J. Lightwave Technol. 34(9), 2100–2105 (2016).
    [Crossref]
  2. S.-Y. Jung, C.-H. Kim, S.-M. Jung, and S.-K. Han, “Optical pulse division multiplexing-based OBI reduction for single wavelength uplink multiple access in IM/DD OFDMA-PON,” Opt. Express 24(25), 29198–29208 (2016).
    [Crossref]
  3. N. Cvijetic, “OFDM for next-generation optical access networks,” J. Lightwave Technol. 30(4), 384–398 (2012).
    [Crossref]
  4. N. Cvijetic, M. Cvijetic, M.-F. Huang, E. Ip, Y.-K. Huang, and T. Wang, “Terabit optical access networks based on WDM-OFDMA-PON,” J. Lightwave Technol. 30(4), 493–503 (2012).
    [Crossref]
  5. T. Wu, C. Zhang, H. Wei, and K. Qiu, “PAPR and security in OFDM-PON via optimum block dividing with dynamic key and 2D-LASM,” Opt. Express 27(20), 27946–27961 (2019).
    [Crossref]
  6. M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
    [Crossref]
  7. L. Zhang, X. Xin, B. Liu, and Y. Wang, “Secure OFDM-PON based on chaos scrambling,” IEEE Photonics Technol. Lett. 23(14), 998–1000 (2011).
    [Crossref]
  8. G. Böcherer, P. Schulte, and F. Steiner, “Probabilistic shaping and forward error correction for fiber-optic communication systems,” J. Lightwave Technol. 37(2), 230–244 (2019).
    [Crossref]
  9. J.-X. Cai, H. G. Batshon, M. V. Mazurczyk, O. V. Sinkin, D. Wang, M. Paskov, W. W. Patterson, C. R. Davidson, P. C. Corbett, G. M. Wolter, T. E. Hammon, M. A. Bolshtyansky, D. G. Foursa, and A. N. Pilipetskii, “70.46 Tb/s over 7,600 km and 71.65 Tb/s over 6,970 km transmission in C + L band using coded modulation with hybrid constellation shaping and nonlinearity compensation,” J. Lightwave Technol. 36(1), 114–121 (2018).
    [Crossref]
  10. G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
    [Crossref]
  11. G. D. Forney and L.-F. Wei, “Multidimensional constellations—Part I: Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun. 7(6), 877–892 (1989).
    [Crossref]
  12. J. Ren, B. Liu, X. Xu, L. Zhang, Y. Mao, X. Wu, Y. Zhang, L. Jiang, and X. Xin, “A probabilistically shaped star-CAP-16/32 modulation based on constellation design with honeycomb-like decision regions,” Opt. Express 27(3), 2732–2746 (2019).
    [Crossref]
  13. J. Ren, B. Liu, X. Wu, L. Zhang, Y. Mao, X. Xu, Y. Zhang, L. Jiang, J. Zhang, and X. Xin, “Three-dimensional probabilistically shaped CAP modulation based on constellation design using regular tetrahedron cells,” J. Lightwave Technol. 38(7), 1728–1734 (2020).
    [Crossref]
  14. P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
    [Crossref]
  15. J. Ma, J. He, M. Chen, K. Wu, and J. He, “Performance enhancement of probabilistically shaped OFDM enabled by precoding technique in an IM-DD System,” J. Lightwave Technol. 37(24), 6063–6071 (2019).
    [Crossref]
  16. C. Xie, Z. Chen, S. Fu, W. Liu, Z. He, L. Deng, M. Tang, and D. Liu, “Achievable information rate enhancement of visible light communication using probabilistically shaped OFDM modulation,” Opt. Express 26(1), 367–375 (2018).
    [Crossref]
  17. J. Shi, J. Zhang, X. Li, N. Chi, Y. Zhang, Q. Zhang, and J. Yu., “Improved performance of high-order QAM OFDM based on probabilistically shaping in the Datacom,” in Opt. Fiber Commun. Conf. (OFC)2018, paper W4G.6.
  18. J. Cho and P. J. Winzer, “Multi-rate prefix-free code distribution matching,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.7.
  19. T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Partition-based probabilistic shaping for fiber-optic communication systems,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.3.
  20. J. Ren, B. Liu, X. Wu, X. Xu, Y. Mao, Y. Wu, X. Song, L. Jiang, J. Zhang, Y. Zhang, and X. Xin, “Security-enhanced 3D-CAP-PON based on two-stage spherical constellation masking,” IEEE Access 8, 111966–111973 (2020).
    [Crossref]
  21. L. Zhang, B. Liu, X. Xin, and Y. Wang, “Joint robustness security in optical OFDM access system with Turbo-coded subcarrier rotation,” Opt. Express 23(1), 13–18 (2015).
    [Crossref]
  22. B. Liu, L. Zhang, X. Xin, and J. Yu, “Constellation-masked secure communication technique for OFDM-PON,” Opt. Express 20(22), 25161–25168 (2012).
    [Crossref]
  23. C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
    [Crossref]
  24. C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
    [Crossref]
  25. J. Zhao, B. Liu, Y. Mao, R. Ullah, J. Ren, S. Chen, L. Jiang, S. Han, J. Zhang, and J. Shen, “High security OFDM-PON with a physical layer encryption based on 4D-hyperchaos and dimension coordination optimization,” Opt. Express 28(14), 21236–21246 (2020).
    [Crossref]
  26. A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Phys. A 364, 103–110 (2006).
    [Crossref]

2020 (3)

2019 (4)

2018 (3)

2017 (2)

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

2016 (3)

2015 (2)

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

L. Zhang, B. Liu, X. Xin, and Y. Wang, “Joint robustness security in optical OFDM access system with Turbo-coded subcarrier rotation,” Opt. Express 23(1), 13–18 (2015).
[Crossref]

2012 (3)

2011 (1)

L. Zhang, X. Xin, B. Liu, and Y. Wang, “Secure OFDM-PON based on chaos scrambling,” IEEE Photonics Technol. Lett. 23(14), 998–1000 (2011).
[Crossref]

2006 (1)

A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Phys. A 364, 103–110 (2006).
[Crossref]

1989 (1)

G. D. Forney and L.-F. Wei, “Multidimensional constellations—Part I: Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun. 7(6), 877–892 (1989).
[Crossref]

Batshon, H. G.

Bi, M.

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Böcherer, G.

G. Böcherer, P. Schulte, and F. Steiner, “Probabilistic shaping and forward error correction for fiber-optic communication systems,” J. Lightwave Technol. 37(2), 230–244 (2019).
[Crossref]

P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
[Crossref]

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

Bolshtyansky, M. A.

Cai, J.-X.

Chen, A.

A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Phys. A 364, 103–110 (2006).
[Crossref]

Chen, C.

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Chen, L.

Chen, M.

Chen, Q.

Chen, S.

Chen, Z.

Chi, N.

J. Shi, J. Zhang, X. Li, N. Chi, Y. Zhang, Q. Zhang, and J. Yu., “Improved performance of high-order QAM OFDM based on probabilistically shaping in the Datacom,” in Opt. Fiber Commun. Conf. (OFC)2018, paper W4G.6.

Cho, J.

J. Cho and P. J. Winzer, “Multi-rate prefix-free code distribution matching,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.7.

Corbett, P. C.

Cvijetic, M.

Cvijetic, N.

Davidson, C. R.

Deng, L.

Fehenberger, T.

T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Partition-based probabilistic shaping for fiber-optic communication systems,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.3.

Forney, G. D.

G. D. Forney and L.-F. Wei, “Multidimensional constellations—Part I: Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun. 7(6), 877–892 (1989).
[Crossref]

Foursa, D. G.

Fu, S.

Fu, X.

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Hammon, T. E.

Han, S.

Han, S.-K.

He, J.

He, X.

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

He, Z.

Hu, W.

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Huang, M.-F.

Huang, Y.-K.

Huang, Z. R.

Ip, E.

Jiang, L.

Jung, S.-M.

Jung, S.-Y.

Kim, C.-H.

Koike-Akino, T.

T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Partition-based probabilistic shaping for fiber-optic communication systems,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.3.

Kojima, K.

T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Partition-based probabilistic shaping for fiber-optic communication systems,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.3.

Li, F.

Li, X.

J. Shi, J. Zhang, X. Li, N. Chi, Y. Zhang, Q. Zhang, and J. Yu., “Improved performance of high-order QAM OFDM based on probabilistically shaping in the Datacom,” in Opt. Fiber Commun. Conf. (OFC)2018, paper W4G.6.

Liu, B.

J. Ren, B. Liu, X. Wu, X. Xu, Y. Mao, Y. Wu, X. Song, L. Jiang, J. Zhang, Y. Zhang, and X. Xin, “Security-enhanced 3D-CAP-PON based on two-stage spherical constellation masking,” IEEE Access 8, 111966–111973 (2020).
[Crossref]

J. Zhao, B. Liu, Y. Mao, R. Ullah, J. Ren, S. Chen, L. Jiang, S. Han, J. Zhang, and J. Shen, “High security OFDM-PON with a physical layer encryption based on 4D-hyperchaos and dimension coordination optimization,” Opt. Express 28(14), 21236–21246 (2020).
[Crossref]

J. Ren, B. Liu, X. Wu, L. Zhang, Y. Mao, X. Xu, Y. Zhang, L. Jiang, J. Zhang, and X. Xin, “Three-dimensional probabilistically shaped CAP modulation based on constellation design using regular tetrahedron cells,” J. Lightwave Technol. 38(7), 1728–1734 (2020).
[Crossref]

J. Ren, B. Liu, X. Xu, L. Zhang, Y. Mao, X. Wu, Y. Zhang, L. Jiang, and X. Xin, “A probabilistically shaped star-CAP-16/32 modulation based on constellation design with honeycomb-like decision regions,” Opt. Express 27(3), 2732–2746 (2019).
[Crossref]

L. Zhang, B. Liu, X. Xin, and Y. Wang, “Joint robustness security in optical OFDM access system with Turbo-coded subcarrier rotation,” Opt. Express 23(1), 13–18 (2015).
[Crossref]

B. Liu, L. Zhang, X. Xin, and J. Yu, “Constellation-masked secure communication technique for OFDM-PON,” Opt. Express 20(22), 25161–25168 (2012).
[Crossref]

L. Zhang, X. Xin, B. Liu, and Y. Wang, “Secure OFDM-PON based on chaos scrambling,” IEEE Photonics Technol. Lett. 23(14), 998–1000 (2011).
[Crossref]

Liu, D.

Liu, W.

Lu, J.

A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Phys. A 364, 103–110 (2006).
[Crossref]

Lü, J.

A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Phys. A 364, 103–110 (2006).
[Crossref]

Ma, J.

Mao, Y.

Mazurczyk, M. V.

Millar, D. S.

T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Partition-based probabilistic shaping for fiber-optic communication systems,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.3.

Parsons, K.

T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Partition-based probabilistic shaping for fiber-optic communication systems,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.3.

Paskov, M.

Patterson, W. W.

Pilipetskii, A. N.

Qiu, K.

T. Wu, C. Zhang, H. Wei, and K. Qiu, “PAPR and security in OFDM-PON via optimum block dividing with dynamic key and 2D-LASM,” Opt. Express 27(20), 27946–27961 (2019).
[Crossref]

C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
[Crossref]

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Ren, J.

Schulte, P.

G. Böcherer, P. Schulte, and F. Steiner, “Probabilistic shaping and forward error correction for fiber-optic communication systems,” J. Lightwave Technol. 37(2), 230–244 (2019).
[Crossref]

P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
[Crossref]

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

Shen, J.

Shi, J.

J. Shi, J. Zhang, X. Li, N. Chi, Y. Zhang, Q. Zhang, and J. Yu., “Improved performance of high-order QAM OFDM based on probabilistically shaping in the Datacom,” in Opt. Fiber Commun. Conf. (OFC)2018, paper W4G.6.

Sinkin, O. V.

Song, X.

J. Ren, B. Liu, X. Wu, X. Xu, Y. Mao, Y. Wu, X. Song, L. Jiang, J. Zhang, Y. Zhang, and X. Xin, “Security-enhanced 3D-CAP-PON based on two-stage spherical constellation masking,” IEEE Access 8, 111966–111973 (2020).
[Crossref]

Steiner, F.

G. Böcherer, P. Schulte, and F. Steiner, “Probabilistic shaping and forward error correction for fiber-optic communication systems,” J. Lightwave Technol. 37(2), 230–244 (2019).
[Crossref]

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

Tang, M.

Ullah, R.

Wang, D.

Wang, T.

Wang, Y.

L. Zhang, B. Liu, X. Xin, and Y. Wang, “Joint robustness security in optical OFDM access system with Turbo-coded subcarrier rotation,” Opt. Express 23(1), 13–18 (2015).
[Crossref]

L. Zhang, X. Xin, B. Liu, and Y. Wang, “Secure OFDM-PON based on chaos scrambling,” IEEE Photonics Technol. Lett. 23(14), 998–1000 (2011).
[Crossref]

Wei, H.

Wei, L.-F.

G. D. Forney and L.-F. Wei, “Multidimensional constellations—Part I: Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun. 7(6), 877–892 (1989).
[Crossref]

Wen, G.

C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
[Crossref]

Winzer, P. J.

J. Cho and P. J. Winzer, “Multi-rate prefix-free code distribution matching,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.7.

Wolter, G. M.

Wu, K.

Wu, T.

T. Wu, C. Zhang, H. Wei, and K. Qiu, “PAPR and security in OFDM-PON via optimum block dividing with dynamic key and 2D-LASM,” Opt. Express 27(20), 27946–27961 (2019).
[Crossref]

C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
[Crossref]

Wu, X.

Wu, Y.

J. Ren, B. Liu, X. Wu, X. Xu, Y. Mao, Y. Wu, X. Song, L. Jiang, J. Zhang, Y. Zhang, and X. Xin, “Security-enhanced 3D-CAP-PON based on two-stage spherical constellation masking,” IEEE Access 8, 111966–111973 (2020).
[Crossref]

Xiao, S.

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Xiao, X.

Xie, C.

Xin, X.

Xu, X.

Yan, Y.

C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
[Crossref]

Yang, G.

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Yang, X.

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

Yu, J.

Yu, S.

A. Chen, J. Lu, J. Lü, and S. Yu, “Generating hyperchaotic Lü attractor via state feedback control,” Phys. A 364, 103–110 (2006).
[Crossref]

Zhang, C.

T. Wu, C. Zhang, H. Wei, and K. Qiu, “PAPR and security in OFDM-PON via optimum block dividing with dynamic key and 2D-LASM,” Opt. Express 27(20), 27946–27961 (2019).
[Crossref]

C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
[Crossref]

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Zhang, H.

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Zhang, J.

J. Zhao, B. Liu, Y. Mao, R. Ullah, J. Ren, S. Chen, L. Jiang, S. Han, J. Zhang, and J. Shen, “High security OFDM-PON with a physical layer encryption based on 4D-hyperchaos and dimension coordination optimization,” Opt. Express 28(14), 21236–21246 (2020).
[Crossref]

J. Ren, B. Liu, X. Wu, X. Xu, Y. Mao, Y. Wu, X. Song, L. Jiang, J. Zhang, Y. Zhang, and X. Xin, “Security-enhanced 3D-CAP-PON based on two-stage spherical constellation masking,” IEEE Access 8, 111966–111973 (2020).
[Crossref]

J. Ren, B. Liu, X. Wu, L. Zhang, Y. Mao, X. Xu, Y. Zhang, L. Jiang, J. Zhang, and X. Xin, “Three-dimensional probabilistically shaped CAP modulation based on constellation design using regular tetrahedron cells,” J. Lightwave Technol. 38(7), 1728–1734 (2020).
[Crossref]

J. Shi, J. Zhang, X. Li, N. Chi, Y. Zhang, Q. Zhang, and J. Yu., “Improved performance of high-order QAM OFDM based on probabilistically shaping in the Datacom,” in Opt. Fiber Commun. Conf. (OFC)2018, paper W4G.6.

Zhang, L.

Zhang, Q.

J. Shi, J. Zhang, X. Li, N. Chi, Y. Zhang, Q. Zhang, and J. Yu., “Improved performance of high-order QAM OFDM based on probabilistically shaping in the Datacom,” in Opt. Fiber Commun. Conf. (OFC)2018, paper W4G.6.

Zhang, W.

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

Zhang, X.

C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
[Crossref]

Zhang, Y.

J. Ren, B. Liu, X. Wu, X. Xu, Y. Mao, Y. Wu, X. Song, L. Jiang, J. Zhang, Y. Zhang, and X. Xin, “Security-enhanced 3D-CAP-PON based on two-stage spherical constellation masking,” IEEE Access 8, 111966–111973 (2020).
[Crossref]

J. Ren, B. Liu, X. Wu, L. Zhang, Y. Mao, X. Xu, Y. Zhang, L. Jiang, J. Zhang, and X. Xin, “Three-dimensional probabilistically shaped CAP modulation based on constellation design using regular tetrahedron cells,” J. Lightwave Technol. 38(7), 1728–1734 (2020).
[Crossref]

J. Ren, B. Liu, X. Xu, L. Zhang, Y. Mao, X. Wu, Y. Zhang, L. Jiang, and X. Xin, “A probabilistically shaped star-CAP-16/32 modulation based on constellation design with honeycomb-like decision regions,” Opt. Express 27(3), 2732–2746 (2019).
[Crossref]

J. Shi, J. Zhang, X. Li, N. Chi, Y. Zhang, Q. Zhang, and J. Yu., “Improved performance of high-order QAM OFDM based on probabilistically shaping in the Datacom,” in Opt. Fiber Commun. Conf. (OFC)2018, paper W4G.6.

Zhao, J.

Zhou, X.

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

IEEE Access (1)

J. Ren, B. Liu, X. Wu, X. Xu, Y. Mao, Y. Wu, X. Song, L. Jiang, J. Zhang, Y. Zhang, and X. Xin, “Security-enhanced 3D-CAP-PON based on two-stage spherical constellation masking,” IEEE Access 8, 111966–111973 (2020).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

G. D. Forney and L.-F. Wei, “Multidimensional constellations—Part I: Introduction, figures of merit, and generalized cross constellations,” IEEE J. Sel. Areas Commun. 7(6), 877–892 (1989).
[Crossref]

IEEE Photonics J. (3)

M. Bi, X. Fu, X. Zhou, L. Zhang, G. Yang, X. Yang, S. Xiao, and W. Hu, “A key space enhanced chaotic encryption scheme for physical layer security in OFDM-PON,” IEEE Photonics J. 9(1), 1–10 (2017).
[Crossref]

C. Zhang, Y. Yan, T. Wu, X. Zhang, G. Wen, and K. Qiu, “Phase masking and time-frequency chaotic encryption for DFMA-PON,” IEEE Photonics J. 10(4), 1–9 (2018).
[Crossref]

C. Zhang, W. Zhang, X. He, C. Chen, H. Zhang, and K. Qiu, “Physically secured optical OFDM-PON by employing chaotic pseudorandom RF subcarriers,” IEEE Photonics J. 9(5), 1–8 (2017).
[Crossref]

IEEE Photonics Technol. Lett. (1)

L. Zhang, X. Xin, B. Liu, and Y. Wang, “Secure OFDM-PON based on chaos scrambling,” IEEE Photonics Technol. Lett. 23(14), 998–1000 (2011).
[Crossref]

IEEE Trans. Commun. (1)

G. Böcherer, F. Steiner, and P. Schulte, “Bandwidth efficient and rate-matched low-density parity-check coded modulation,” IEEE Trans. Commun. 63(12), 4651–4665 (2015).
[Crossref]

IEEE Trans. Inf. Theory (1)

P. Schulte and G. Böcherer, “Constant composition distribution matching,” IEEE Trans. Inf. Theory 62(1), 430–434 (2016).
[Crossref]

J. Lightwave Technol. (7)

J. Ma, J. He, M. Chen, K. Wu, and J. He, “Performance enhancement of probabilistically shaped OFDM enabled by precoding technique in an IM-DD System,” J. Lightwave Technol. 37(24), 6063–6071 (2019).
[Crossref]

J. Ren, B. Liu, X. Wu, L. Zhang, Y. Mao, X. Xu, Y. Zhang, L. Jiang, J. Zhang, and X. Xin, “Three-dimensional probabilistically shaped CAP modulation based on constellation design using regular tetrahedron cells,” J. Lightwave Technol. 38(7), 1728–1734 (2020).
[Crossref]

G. Böcherer, P. Schulte, and F. Steiner, “Probabilistic shaping and forward error correction for fiber-optic communication systems,” J. Lightwave Technol. 37(2), 230–244 (2019).
[Crossref]

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Opt. Express (7)

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T. Fehenberger, D. S. Millar, T. Koike-Akino, K. Kojima, and K. Parsons, “Partition-based probabilistic shaping for fiber-optic communication systems,” in Opt. Fiber Commun. Conf. (OFC)2019, paper M4B.3.

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Figures (10)

Fig. 1.
Fig. 1. Schematic of the proposed chaotic CCDM-based QAM mapping.
Fig. 2.
Fig. 2. Possible amplitude outputs when inputting bit stream ‘00011011’.
Fig. 3.
Fig. 3. (a) Probability distribution and (b) constellation diagram of the encrypted PS-16QAM.
Fig. 4.
Fig. 4. Phase diagram of 4D hyperchaotic Lv system.
Fig. 5.
Fig. 5. Initial values sensitivity of 4D hyperchaotic Lv system.
Fig. 6.
Fig. 6. Experimental setup (AWG: arbitrary waveform generator; EA: electrical amplifier; MZM: Mach-Zehnder modulator; PS: power splitter; VOA: variable optical attenuator; PD: photodiode; MSO: mixed signal oscilloscope).
Fig. 7.
Fig. 7. BER curves of the encrypted PS-16QAM-OFDM, PS-16QAM-OFDM with short/long BL, and uniform 16QAM-OFDM under the same bit rate after 25 km transmission.
Fig. 8.
Fig. 8. BER curves of legal ONUs and illegal ONU after 25 km transmission.
Fig. 9.
Fig. 9. (a) Image and (b) histogram received by the legal ONU; (c) Image and (d) histogram received by the illegal ONU.
Fig. 10.
Fig. 10. BER curves of an illegal ONU with any three initial values known.

Tables (1)

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Table 1. Chaotic CCDM mapping rules.

Equations (4)

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{ x ˙ = a ( y x ) + u y ˙ = x z + c y z ˙ = x y b z u ˙ = x z + d u .
{ k = f l o o r ( mod ( x 10 13 , 24 ) ) + 1 m = f l o o r ( mod ( z 10 13 , 24 ) ) + 1 .
{ l = f l o o r ( mod ( y 10 13 , 2 ) ) n = f l o o r ( mod ( u 10 13 , 2 ) ) .
{ s i g n _ I = x o r ( l , o r i g i n a l   b i t ) × 2 1 s i g n _ Q = x o r ( n , o r i g i n a l   b i t ) × 2 1 .

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