Abstract

In this article we have enhanced the security of an orthogonal frequency division multiplexed passive optical network (OFDM-PON) based on four dimensional (4D) encryption, including constellation, subcarrier, symbol and time, which is proposed for the first time in this paper. 4D-hyperchaotic mapping is used to generate four masking factors to achieve ultra-high security encryption in four different dimensions. During the encryption, dimension coordination optimization is adopted, which effectively reduces the time cost of the system and improves the encryption efficiency by 3 times. At the same time, probabilistic shaping (PS) technology is used to further optimize the system that has effectively improved the bit error performance by about 1 dB. The proposed encryption technique for OFDM-PON has been demonstrated successfully with the help of experiments. The generated OFDM signal is modulated by the quadrature amplitude modulation (QAM) technique, which transmitted 16 Gb/s data rate across a 25 km fiber span of standard single-mode fiber. The values of bit error rate (BER) and peak-to-average-power ratio (PAPR) are analyzed during the experiments, and the obtained results show that the proposed security-enhanced OFDM-PON has high sensitivity and security and can be well compatible with PS and OFDM technologies. The proposed scheme has very reliable security performance and also has excellent benefit improvement, which is very promising in the future PS-OFDM-PON.

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

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    [Crossref]

2020 (2)

T. Wu, C. Zhang, H. Huan, Z. Zhang, H. Wei, H. Wen, and K. Qiu, “Security Improvement for OFDM-PON via DNA Extension Code and Chaotic Systems,” IEEE Access 8, 75119–75126 (2020).
[Crossref]

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

2019 (4)

2018 (4)

K. Zhang, G. Gao, J. Zhang, and A. Fei, “Physical Layer Security Based on Chaotic Spatial Symbol Transforming in Fiber-Optic Systems,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

C. Zhang, W. Zhang, C. Chen, X. He, and K. Qiu, “Physical-enhanced secure strategy for OFDMA-PON using chaos and deoxyribonucleic acid encoding,” J. Lightwave Technol. 36(9), 1706–1712 (2018).
[Crossref]

T. Wu, C. Zhang, W. Zhang, C. Chen, H. Hou, and K. Qiu, “Security enhancement for OFDM-PON using Brownian motion and chaos in cell,” Opt. Express 26(18), 22857–22865 (2018).
[Crossref]

A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic constellation mapping for physical-layer data encryption in OFDM-PON,” IEEE Photonics Technol. Lett. 30(4), 339–342 (2018).
[Crossref]

2017 (4)

J. Zhong, X. Yang, and W. Hu, “Performance-improved secure OFDM transmission using chaotic active constellation extension,” IEEE Photonics Technol. Lett. 29(12), 991–994 (2017).
[Crossref]

B. Liu, X. Li, Y. Zhang, X. Xin, and J. Yu, “Probabilistic shaping for ROF system with heterodyne coherent detection,” APL Photonics 2(5), 056104 (2017).
[Crossref]

W. Zhang, C. Zhang, C. Chen, and K. Qiu, “Experimental demonstration of security-enhanced OFDMA-PON using chaotic constellation transformation and pilot-aided secure key agreement,” J. Lightwave Technol. 35(9), 1524–1530 (2017).
[Crossref]

N. Li, H. Susanto, B. Cemlyn, I. D. Henning, and M. J. Adams, “Secure communication systems based on chaos in optically pumped spin-VCSELs,” Opt. Lett. 42(17), 3494–3497 (2017).
[Crossref]

2016 (1)

2015 (2)

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Y. Liu, C. Yang, and H. Li, “Cost-effective and spectrum-efficient coherent TDM-OFDM-PON aided by blind ICI suppression,” IEEE Photonics Technol. Lett. 27(8), 887–890 (2015).
[Crossref]

2014 (1)

B. Liu, L. Zhang, X. Xin, and Y. Wang, “Physical layer security in ofdm-pon based on dimension-transformed chaotic permutation,” IEEE Photonics Technol. Lett. 26(2), 127–130 (2014).
[Crossref]

2013 (1)

2012 (1)

2011 (1)

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical layer security in fiber-optic networks,” IEEE Trans.Inform.Forensic Secur. 6(3), 725–736 (2011).
[Crossref]

2010 (4)

2004 (1)

D. Raphaeli and A. Gurevitz, “Constellation Shaping for Pragmatic Turbo-Coded Modulation With High Spectral Efficiency,” IEEE Trans. Commun. Technol. 52(3), 341–345 (2004).
[Crossref]

Adams, M. J.

Bocherer, G.

Cemlyn, B.

Chang, J.

Chen, C.

Chen, J.

Chen, L.

Chen, M.

Chen, Q.

Chi, S.

Cho, J.

D. Gutierrez, J. Cho, and L. G. Kazovsky, “TDM-PON security issues:Upstream encryption is needed,” Proc. OFC/NFOEC, pp.1–3, paper JWA83 (2007).

Cvijetic, N.

Deng, Y.

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical layer security in fiber-optic networks,” IEEE Trans.Inform.Forensic Secur. 6(3), 725–736 (2011).
[Crossref]

Fei, A.

K. Zhang, G. Gao, J. Zhang, and A. Fei, “Physical Layer Security Based on Chaotic Spatial Symbol Transforming in Fiber-Optic Systems,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

Fok, M. P.

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical layer security in fiber-optic networks,” IEEE Trans.Inform.Forensic Secur. 6(3), 725–736 (2011).
[Crossref]

Gao, G.

K. Zhang, G. Gao, J. Zhang, and A. Fei, “Physical Layer Security Based on Chaotic Spatial Symbol Transforming in Fiber-Optic Systems,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

Gurevitz, A.

D. Raphaeli and A. Gurevitz, “Constellation Shaping for Pragmatic Turbo-Coded Modulation With High Spectral Efficiency,” IEEE Trans. Commun. Technol. 52(3), 341–345 (2004).
[Crossref]

Gutierrez, D.

D. Gutierrez, J. Cho, and L. G. Kazovsky, “TDM-PON security issues:Upstream encryption is needed,” Proc. OFC/NFOEC, pp.1–3, paper JWA83 (2007).

Hajomer, A. A. E.

A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic constellation mapping for physical-layer data encryption in OFDM-PON,” IEEE Photonics Technol. Lett. 30(4), 339–342 (2018).
[Crossref]

He, X.

Henning, I. D.

Hou, H.

Hu, W.

A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic constellation mapping for physical-layer data encryption in OFDM-PON,” IEEE Photonics Technol. Lett. 30(4), 339–342 (2018).
[Crossref]

J. Zhong, X. Yang, and W. Hu, “Performance-improved secure OFDM transmission using chaotic active constellation extension,” IEEE Photonics Technol. Lett. 29(12), 991–994 (2017).
[Crossref]

Huan, H.

T. Wu, C. Zhang, H. Huan, Z. Zhang, H. Wei, H. Wen, and K. Qiu, “Security Improvement for OFDM-PON via DNA Extension Code and Chaotic Systems,” IEEE Access 8, 75119–75126 (2020).
[Crossref]

Huang, Z. R.

Jiang, N.

Jiang, W.-J.

Kazovsky, L. G.

D. Gutierrez, J. Cho, and L. G. Kazovsky, “TDM-PON security issues:Upstream encryption is needed,” Proc. OFC/NFOEC, pp.1–3, paper JWA83 (2007).

Li, F.

Li, H.

Y. Liu, C. Yang, and H. Li, “Cost-effective and spectrum-efficient coherent TDM-OFDM-PON aided by blind ICI suppression,” IEEE Photonics Technol. Lett. 27(8), 887–890 (2015).
[Crossref]

Li, N.

Li, N. Q.

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Li, X.

B. Liu, X. Li, Y. Zhang, X. Xin, and J. Yu, “Probabilistic shaping for ROF system with heterodyne coherent detection,” APL Photonics 2(5), 056104 (2017).
[Crossref]

Lin, C.-T.

Liu, B

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Liu, B.

Liu, Y.

Y. Liu, C. Yang, and H. Li, “Cost-effective and spectrum-efficient coherent TDM-OFDM-PON aided by blind ICI suppression,” IEEE Photonics Technol. Lett. 27(8), 887–890 (2015).
[Crossref]

Luo, B.

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Mao, Y

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Mu, P. H.

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Pan, W.

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Prucnal, P. R.

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical layer security in fiber-optic networks,” IEEE Trans.Inform.Forensic Secur. 6(3), 725–736 (2011).
[Crossref]

G. Z. Wang, J. Chang, and P. R. Prucnal, “Theoretical analysis and experimental investigation on the security performance of incoherent optical CDMA code,” J. Lightwave Technol. 28(12), 1761–1769 (2010).
[Crossref]

Qiu, K.

Raphaeli, D.

D. Raphaeli and A. Gurevitz, “Constellation Shaping for Pragmatic Turbo-Coded Modulation With High Spectral Efficiency,” IEEE Trans. Commun. Technol. 52(3), 341–345 (2004).
[Crossref]

Ren, J

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Schulte, P.

Shih, P.-T.

Steiner, F.

Sultan, A.

A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic constellation mapping for physical-layer data encryption in OFDM-PON,” IEEE Photonics Technol. Lett. 30(4), 339–342 (2018).
[Crossref]

Susanto, H.

Tang, J.

Wang, G. Z.

Wang, Y.

B. Liu, L. Zhang, X. Xin, and Y. Wang, “Physical layer security in ofdm-pon based on dimension-transformed chaotic permutation,” IEEE Photonics Technol. Lett. 26(2), 127–130 (2014).
[Crossref]

L. Zhang, B. Liu, X. Xin, Q. Zhang, J. Yu, and Y. Wang, “Theory and Performance Analyses in Secure CO-OFDM Transmission System Based on Two-Dimensional Permutation,” J. Lightwave Technol. 31(1), 74–80 (2013).
[Crossref]

Wang, Z.

M. P. Fok, Z. Wang, Y. Deng, and P. R. Prucnal, “Optical layer security in fiber-optic networks,” IEEE Trans.Inform.Forensic Secur. 6(3), 725–736 (2011).
[Crossref]

Wei, H.

T. Wu, C. Zhang, H. Huan, Z. Zhang, H. Wei, H. Wen, and K. Qiu, “Security Improvement for OFDM-PON via DNA Extension Code and Chaotic Systems,” IEEE Access 8, 75119–75126 (2020).
[Crossref]

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

Wen, H.

T. Wu, C. Zhang, H. Huan, Z. Zhang, H. Wei, H. Wen, and K. Qiu, “Security Improvement for OFDM-PON via DNA Extension Code and Chaotic Systems,” IEEE Access 8, 75119–75126 (2020).
[Crossref]

Wu, T.

Wu, X

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Wu, X.

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Xiao, X.

Xin, X.

Xu, J.

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Xu, M. F.

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Xu, X

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Xue, C.

Yan, L. S.

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Yang, C.

Y. Liu, C. Yang, and H. Li, “Cost-effective and spectrum-efficient coherent TDM-OFDM-PON aided by blind ICI suppression,” IEEE Photonics Technol. Lett. 27(8), 887–890 (2015).
[Crossref]

Yang, X.

A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic constellation mapping for physical-layer data encryption in OFDM-PON,” IEEE Photonics Technol. Lett. 30(4), 339–342 (2018).
[Crossref]

J. Zhong, X. Yang, and W. Hu, “Performance-improved secure OFDM transmission using chaotic active constellation extension,” IEEE Photonics Technol. Lett. 29(12), 991–994 (2017).
[Crossref]

Yu, C.

Yu, J.

Zhang,

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Zhang, C.

Zhang, J.

K. Zhang, G. Gao, J. Zhang, and A. Fei, “Physical Layer Security Based on Chaotic Spatial Symbol Transforming in Fiber-Optic Systems,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

Zhang, K.

K. Zhang, G. Gao, J. Zhang, and A. Fei, “Physical Layer Security Based on Chaotic Spatial Symbol Transforming in Fiber-Optic Systems,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

Zhang, L.

Zhang, Q.

Zhang, W.

Zhang, Y.

B. Liu, X. Li, Y. Zhang, X. Xin, and J. Yu, “Probabilistic shaping for ROF system with heterodyne coherent detection,” APL Photonics 2(5), 056104 (2017).
[Crossref]

Zhang, Z.

T. Wu, C. Zhang, H. Huan, Z. Zhang, H. Wei, H. Wen, and K. Qiu, “Security Improvement for OFDM-PON via DNA Extension Code and Chaotic Systems,” IEEE Access 8, 75119–75126 (2020).
[Crossref]

Zhao, A.

Zhao, J

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

Zhao, M.

L. Zhang and M. Zhao, “Secrecy enhancement for media-based modulation via probabilistic optimization,” IEEE Commun. Lett. 23(7), 1149–1152 (2019).
[Crossref]

Zhong, J.

J. Zhong, X. Yang, and W. Hu, “Performance-improved secure OFDM transmission using chaotic active constellation extension,” IEEE Photonics Technol. Lett. 29(12), 991–994 (2017).
[Crossref]

Zou, X. H.

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

Acta Phys. Sin. (1)

P. H. Mu, W. Pan, N. Q. Li, L. S. Yan, B. Luo, X. H. Zou, and M. F. Xu, “Performance of chaos synchronization and security in dual-chaotic optical multiplexing system,” Acta Phys. Sin. 64(12), 124206 (2015).
[Crossref]

APL Photonics (1)

B. Liu, X. Li, Y. Zhang, X. Xin, and J. Yu, “Probabilistic shaping for ROF system with heterodyne coherent detection,” APL Photonics 2(5), 056104 (2017).
[Crossref]

IEEE Access (1)

T. Wu, C. Zhang, H. Huan, Z. Zhang, H. Wei, H. Wen, and K. Qiu, “Security Improvement for OFDM-PON via DNA Extension Code and Chaotic Systems,” IEEE Access 8, 75119–75126 (2020).
[Crossref]

IEEE Commun. Lett. (1)

L. Zhang and M. Zhao, “Secrecy enhancement for media-based modulation via probabilistic optimization,” IEEE Commun. Lett. 23(7), 1149–1152 (2019).
[Crossref]

IEEE Photonics J. (1)

K. Zhang, G. Gao, J. Zhang, and A. Fei, “Physical Layer Security Based on Chaotic Spatial Symbol Transforming in Fiber-Optic Systems,” IEEE Photonics J. 10(3), 1–10 (2018).
[Crossref]

IEEE Photonics Technol. Lett. (5)

B. Liu, L. Zhang, X. Xin, and Y. Wang, “Physical layer security in ofdm-pon based on dimension-transformed chaotic permutation,” IEEE Photonics Technol. Lett. 26(2), 127–130 (2014).
[Crossref]

Y. Liu, C. Yang, and H. Li, “Cost-effective and spectrum-efficient coherent TDM-OFDM-PON aided by blind ICI suppression,” IEEE Photonics Technol. Lett. 27(8), 887–890 (2015).
[Crossref]

J Zhao, B Liu, Y Mao, J Ren, X Xu, X Wu, X. Wu, J. Xu, and Zhang, “High-Security Physical Layer in CAP-PON System Based on Floating Probability Disturbance,” IEEE Photonics Technol. Lett. 32(7), 367–370 (2020).
[Crossref]

A. Sultan, X. Yang, A. A. E. Hajomer, and W. Hu, “Chaotic constellation mapping for physical-layer data encryption in OFDM-PON,” IEEE Photonics Technol. Lett. 30(4), 339–342 (2018).
[Crossref]

J. Zhong, X. Yang, and W. Hu, “Performance-improved secure OFDM transmission using chaotic active constellation extension,” IEEE Photonics Technol. Lett. 29(12), 991–994 (2017).
[Crossref]

IEEE Trans. Commun. Technol. (1)

D. Raphaeli and A. Gurevitz, “Constellation Shaping for Pragmatic Turbo-Coded Modulation With High Spectral Efficiency,” IEEE Trans. Commun. Technol. 52(3), 341–345 (2004).
[Crossref]

IEEE Trans.Inform.Forensic Secur. (1)

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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).
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Figures (10)

Fig. 1.
Fig. 1. The proposed model based on 4D-hyperchaotic permutation.
Fig. 2.
Fig. 2. Block diagram of permutation/de-permutation.
Fig. 3.
Fig. 3. Constellation diagram (a) before masking, (b) after masking based on constellation rotation, (c) the original constellation received, (d) with correct demasking.
Fig. 4.
Fig. 4. Schematic diagram of unit division scrambling.
Fig. 5.
Fig. 5. Experimental setup (AWG: arbitrary waveform generator; MZM: Mach-Zehnder modulator; EDFA: Erbium doped fiber amplifier; VOA: variable optical attenuator; PD: photodiode; MSO: mixed signal oscilloscope).
Fig. 6.
Fig. 6. (a) Encryption time under different schemes (total time for 100 runs); (b) BER of illegal ONU (average after 100 runs).
Fig. 7.
Fig. 7. Phase diagram of 4D-hyperchaotic map.
Fig. 8.
Fig. 8. BER curves of various ONUs with a tiny change in initial value.
Fig. 9.
Fig. 9. BER curves of normal and illegal ONUs.
Fig. 10.
Fig. 10. Comparison of the CCDFs for the OFDM signal.

Equations (7)

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P X ( x i ) = e μ | x i | 2 / x i X e μ | x i | 2 ,
{ x ^ = a ( y x )  +  w y ^ = d x + c y x z z ^ = x y b z w ^ = y z + r w ,
X k = log ( k + 1 ) log N x k θ , Q l , k = Q l , k ( cos X k + j sin X k ) ,
s t = k = 1 M Q l , k × exp ( j 2 π f k ( t 1 ) T M ) , t = 1 , 2 , , T ,
m = k ( mod ( M , k ) = 0 ) , k = 1 , 2 , , M , l = w ( mod ( L , w ) = 0 ) , w = 1 , 2 , , L ,
Y m = M a t { mod ( y m , 1 ) [ mod ( y m , 1 ) ] } , m = 1 , 2 , , N c a r , Z l = M a t { mod ( z l , 1 ) [ mod ( z l , 1 ) ) ] } , l = 1 , 2 , , N s y m , W t = M a t { mod ( w t , 1 ) [ mod ( w t , 1 ) ) ] } , t = 1 , 2 , , T ,
s t = { k = 1 M Q l , k ( cos X k + j sin X k ) × Z l exp ( j 2 π f k ( t 1 ) T M ) × Y m } × W t ,

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