Abstract

The average bit error rate (ABER) performance of an orbital angular momentum (OAM) multiplexing-based free-space optical (FSO) system with multiple-input multiple-output (MIMO) architecture has been investigated over atmospheric turbulence considering channel estimation and space-time coding. The impact of different types of space-time coding, modulation orders, turbulence strengths, receive antenna numbers on the transmission performance of this OAM-FSO system is also taken into account. On the basis of the proposed system model, the analytical expressions of the received signals carried by the k-th OAM mode of the n-th receive antenna for the vertical bell labs layered space-time (V-Blast) and space-time block codes (STBC) are derived, respectively. With the help of channel estimator carrying out with least square (LS) algorithm, the zero-forcing criterion with ordered successive interference cancellation criterion (ZF-OSIC) equalizer of V-Blast scheme and Alamouti decoder of STBC scheme are adopted to mitigate the performance degradation induced by the atmospheric turbulence. The results show that the ABERs obtained by channel estimation have excellent agreement with those of turbulence phase screen simulations. The ABERs of this OAM multiplexing-based MIMO system deteriorate with the increase of turbulence strengths. And both V-Blast and STBC schemes can significantly improve the system performance by mitigating the distortions of atmospheric turbulence as well as additive white Gaussian noise (AWGN). In addition, the ABER performances of both space-time coding schemes can be further enhanced by increasing the number of receive antennas for the diversity gain and STBC outperforms V-Blast in this system for data recovery. This work is beneficial to the OAM FSO system design.

© 2017 Optical Society of America

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References

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2017 (6)

Z. Qu and I. B. Djordjevic, “High-speed free-space optical continuous-variable quantum key distribution enabled by three-dimensional multiplexing,” Opt. Express 25(7), 7919–7928 (2017).
[Crossref] [PubMed]

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 25(7), 7919–7928 (2017).

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
[Crossref] [PubMed]

M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Spatial division multiplexing for high capacity optical interconnects in modular data centers,” Opt. Commun. Netw. 9(2), A143–A153 (2017).
[Crossref]

L. Zhu, C. Yang, D. Xie, and J. Wang, “Demonstration of km-scale orbital angular momentum multiplexing transmission using 4-level pulse-amplitude modulation signals,” Opt. Lett. 42(4), 763–766 (2017).
[Crossref] [PubMed]

G. Zhu, Y. Chen, Y. Liu, Y. Zhang, and S. Yu, “Characterizing a 14 × 14 OAM mode transfer matrix of a ring-core fiber based on quadrature phase-shift interference,” Opt. Lett. 42(7), 1257–1260 (2017).
[Crossref] [PubMed]

2016 (9)

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Z. Qu and I. B. Djordjevic, “500 Gb/s free-space optical transmission over strong atmospheric turbulence channels,” Opt. Lett. 41(14), 3285–3288 (2016).
[Crossref] [PubMed]

C. Chen, H. Yang, S. Tong, and Y. Lou, “Changes in orbital-angular-momentum modes of a propagated vortex Gaussian beam through weak-to-strong atmospheric turbulence,” Opt. Express 24(7), 6959–6975 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

H. H. Lu, C. Y. Lin, T. C. Lu, C. A. Chu, H. H. Lin, B. R. Chen, C. J. Wu, and W. S. Tsai, “150 m/280 Gbps WDM/SDM FSO link based on OEO-based BLS and afocal telescopes,” Opt. Lett. 41(12), 2835–2838 (2016).
[Crossref] [PubMed]

M. Cheng, L. Guo, J. Li, Q. Huang, Q. Cheng, and D. Zhang, “Propagation of an optical vortex carried by a partially coherent Laguerre-Gaussian beam in turbulent ocean,” Appl. Opt. 55(17), 4642–4648 (2016).
[Crossref] [PubMed]

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

2015 (2)

2014 (3)

2012 (1)

2008 (1)

J. Lin, “Least-Squares Channel Estimation for Mobile OFDM Communication on Time-Varying Frequency-Selective Fading Channels,” IEEE Trans. Vehicular Technol. 57(6), 3538–3550 (2008).
[Crossref]

2007 (1)

S. J. Lee, “Effect of Least Square channel estimation errors on achievable rate in MIMO fading channels,” IEEE Commun. Lett. 11(11), 862–864 (2007).
[Crossref]

2005 (1)

M. K. Simon and V. A. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wirel. Commun. 4(1), 35–39 (2005).
[Crossref]

1999 (2)

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel. Areas Comm. 17(3), 451–460 (1999).
[Crossref]

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory 45(5), 1456–1467 (1999).
[Crossref]

Ahmed, N.

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Aksenov, V. P.

Ashrafi, N.

Ashrafi, S.

Bao, C.

Bock, R.

Boyd, R. W.

Caire, G.

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Calderbank, A. R.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel. Areas Comm. 17(3), 451–460 (1999).
[Crossref]

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory 45(5), 1456–1467 (1999).
[Crossref]

Cao, Y.

Chen, B. R.

Chen, C.

Chen, H.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Chen, J.

M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Spatial division multiplexing for high capacity optical interconnects in modular data centers,” Opt. Commun. Netw. 9(2), A143–A153 (2017).
[Crossref]

Chen, Y.

Cheng, M.

Cheng, Q.

Cheng, W.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Chi, H.

Chu, C. A.

Ding, J.

Djordjevic, I. B.

Dolinar, S. J.

Fiorani, M.

M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Spatial division multiplexing for high capacity optical interconnects in modular data centers,” Opt. Commun. Netw. 9(2), A143–A153 (2017).
[Crossref]

Fu, S.

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

Gao, C.

S. Fu and C. Gao, “Influences of atmospheric turbulence effects on the orbital angular momentum spectra of vortex beams,” Photonics Res. 4(5), B1–B4 (2016).
[Crossref]

Garrido-Balsells, J. M.

Gong, L.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Gong, L. Y.

Guo, L.

Guo, Z.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 25(7), 7919–7928 (2017).

Haroer, W. W.

J. D. Strasburg and W. W. Haroer, “Impact of atmospheric turbulence on beam propagation,” in Proceedings of Laser Systems Technology (SPIE, 2004), pp. 5413–5424.

Huang, H.

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Huang, P.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 25(7), 7919–7928 (2017).

Huang, Q.

Hui, X.

Jafarkhani, H.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel. Areas Comm. 17(3), 451–460 (1999).
[Crossref]

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory 45(5), 1456–1467 (1999).
[Crossref]

Jin, X.

Jurado-Navas, A.

Kai, C.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 25(7), 7919–7928 (2017).

Kolosov, V. V.

Lavery, M. J.

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Lavery, M. P. J.

Lavery, P. J.

Leach, J.

Lee, S. J.

S. J. Lee, “Effect of Least Square channel estimation errors on achievable rate in MIMO fading channels,” IEEE Commun. Lett. 11(11), 862–864 (2007).
[Crossref]

Li, J.

Li, L.

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Li, S.

S. Li and J. Wang, “Adaptive free-space optical communications through turbulence using self-healing Bessel beams,” Sci. Rep. 7, 43233 (2017).
[Crossref] [PubMed]

Liao, P.

Lin, C. Y.

Lin, H. H.

Lin, J.

J. Lin, “Least-Squares Channel Estimation for Mobile OFDM Communication on Time-Varying Frequency-Selective Fading Channels,” IEEE Trans. Vehicular Technol. 57(6), 3538–3550 (2008).
[Crossref]

Liu, Y.

Lou, Y.

Lu, H. H.

Lu, T. C.

Lu, X.

Molisch, A. F.

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Monroy, I. T.

Mukherjee, B.

M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Spatial division multiplexing for high capacity optical interconnects in modular data centers,” Opt. Commun. Netw. 9(2), A143–A153 (2017).
[Crossref]

Neifeld, M. A.

Olmos, J. J.

Padgett, M.

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Pogutsa, E. C.

Qu, Z.

Ren, Y.

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Shapiro, J. H.

Shen, F.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 25(7), 7919–7928 (2017).

Simon, M. K.

M. K. Simon and V. A. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wirel. Commun. 4(1), 35–39 (2005).
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Strasburg, J. D.

J. D. Strasburg and W. W. Haroer, “Impact of atmospheric turbulence on beam propagation,” in Proceedings of Laser Systems Technology (SPIE, 2004), pp. 5413–5424.

Tarokh, V.

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block codes from orthogonal designs,” IEEE Trans. Inf. Theory 45(5), 1456–1467 (1999).
[Crossref]

V. Tarokh, H. Jafarkhani, and A. R. Calderbank, “Space-time block coding for wireless communications: performance results,” IEEE J. Sel. Areas Comm. 17(3), 451–460 (1999).
[Crossref]

Tatarczak, A.

Tong, S.

Tornatore, M.

M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Spatial division multiplexing for high capacity optical interconnects in modular data centers,” Opt. Commun. Netw. 9(2), A143–A153 (2017).
[Crossref]

Tsai, W. S.

Tur, M.

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Vilnrotter, V. A.

M. K. Simon and V. A. Vilnrotter, “Alamouti-type space-time coding for free-space optical communication with direct detection,” IEEE Trans. Wirel. Commun. 4(1), 35–39 (2005).
[Crossref]

Wang, J.

Wang, L.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Wang, Z.

Willner, A.

Willner, A. E.

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Willner, A. J.

Wosinska, L.

M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Spatial division multiplexing for high capacity optical interconnects in modular data centers,” Opt. Commun. Netw. 9(2), A143–A153 (2017).
[Crossref]

Wu, C. J.

Xie, D.

Xie, G.

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Yan, Y.

Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
[Crossref] [PubMed]

L. Li, G. Xie, Y. Ren, N. Ahmed, H. Huang, Z. Zhao, P. Liao, M. P. J. Lavery, Y. Yan, C. Bao, Z. Wang, A. J. Willner, N. Ashrafi, S. Ashrafi, M. Tur, and A. E. Willner, “Orbital-angular-momentum-multiplexed free-space optical communication link using transmitter lenses,” Appl. Opt. 55(8), 2098–2103 (2016).
[Crossref] [PubMed]

Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
[Crossref] [PubMed]

H. Huang, Y. Cao, G. Xie, Y. Ren, Y. Yan, C. Bao, N. Ahmed, M. A. Neifeld, S. J. Dolinar, and A. E. Willner, “Crosstalk mitigation in a free-space orbital angular momentum multiplexed communication link using 4×4 MIMO equalization,” Opt. Lett. 39(15), 4360–4363 (2014).
[Crossref] [PubMed]

Y. Ren, G. Xie, H. Huang, N. Ahmed, Y. Yan, L. Li, C. Bao, P. J. Lavery, M. Tur, M. A. Neifeld, R. W. Boyd, J. H. Shapiro, and A. E. Willner, “Adaptive-optics-based simultaneous pre- and post-turbulence compensation of multiple orbital-angular-momentum beams in a bidirectional free-space optical link,” Optica 1(6), 376–382 (2014).
[Crossref]

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Yang, C.

Yang, H.

Yu, S.

Zhang, C.

Y. Ren, L. Li, G. Xie, Y. Yan, Y. Cao, H. Huang, N. Ahmed, M. J. Lavery, Z. Zhao, C. Zhang, M. Tur, M. Padgett, G. Caire, A. F. Molisch, and A. E. Willner, “Experimental demonstration of 16Gbit/s millimeter-wave communications using MIMO processing of 2 OAM modes on each of two transmitter/receiver antenna apertures,” inProceedings of Global Communications Conference (IEEE, 2014), pp. 3821–3826.

Zhang, D.

Zhang, X.

Zhang, Y.

Zhao, S.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Zhao, S. M.

Zhao, Z.

Zheng, B.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
[Crossref]

Zheng, B. Y.

Zheng, S.

Zhou, H.

C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 25(7), 7919–7928 (2017).

Zhu, G.

Zhu, J.

Zhu, L.

Zou, L.

S. Zhao, L. Wang, L. Zou, L. Gong, W. Cheng, B. Zheng, and H. Chen, “Both channel coding and wavefront correction on the turbulence mitigation of optical communications using orbital angular momentum multiplexing,” Opt. Commun. 376, 92–98 (2016).
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C. Kai, P. Huang, F. Shen, H. Zhou, and Z. Guo, “Orbital angular momentum shift keying based optical communication system,” IEEE Photonics J. 25(7), 7919–7928 (2017).

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M. Fiorani, M. Tornatore, J. Chen, L. Wosinska, and B. Mukherjee, “Spatial division multiplexing for high capacity optical interconnects in modular data centers,” Opt. Commun. Netw. 9(2), A143–A153 (2017).
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Y. Ren, Z. Wang, G. Xie, L. Li, A. J. Willner, Y. Cao, Z. Zhao, Y. Yan, N. Ahmed, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, and A. E. Willner, “Atmospheric turbulence mitigation in an OAM-based MIMO free-space optical link using spatial diversity combined with MIMO equalization,” Opt. Lett. 41(11), 2406–2409 (2016).
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Y. Ren, Z. Wang, P. Liao, L. Li, G. Xie, H. Huang, Z. Zhao, Y. Yan, N. Ahmed, A. Willner, M. P. J. Lavery, N. Ashrafi, S. Ashrafi, R. Bock, M. Tur, I. B. Djordjevic, M. A. Neifeld, and A. E. Willner, “Experimental characterization of a 400 Gbit/s orbital angular momentum multiplexed free-space optical link over 120 m,” Opt. Lett. 41(3), 622–625 (2016).
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[Crossref]

J. D. Strasburg and W. W. Haroer, “Impact of atmospheric turbulence on beam propagation,” in Proceedings of Laser Systems Technology (SPIE, 2004), pp. 5413–5424.

J. D. Schmidt, Numerical Simulation of Optical Wave Propagation (2010).

M. Jankiraman, Space-Time Codes and MIMO Systems (Artech House, 2004).

N. Petrellis, “STBC-OFDM communication systems with sub-sampling support,” in Proceedings of IEEE 5th International Conference on Modern Circuits and Systems Technologies (MOCAST 2016) (IEEE 2016), pp. 1–4.
[Crossref]

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Figures (6)

Fig. 1
Fig. 1 The proposed OAM multiplexing-based MIMO FSO system using space-time coding combined with channel estimation
Fig. 2
Fig. 2 Simulation results of two OAM multiplexing propagation under moderate atmospheric turbulence (a) the intensity and phase distributions of these two OAM beams; (b) the intensity and phase distributions of these two OAM beams superposition; (c) turbulence phase screens; (d) the intensity and phase distributions of multiplexed OAM beams impaired by atmospheric turbulence; (e) the intensity and phase distributions of each OAM beam after demultiplexing.
Fig. 3
Fig. 3 ABER performance comparison of signal propagation through atmospheric turbulence between turbulence phase screen simulations and channel estimation.
Fig. 4
Fig. 4 (a) ABER comparison between V-Blast scheme and STBC scheme with different atmospheric turbulence. The constellations of (b) QPSK signals propagation under moderate turbulence condition, (c) recovered signal with V-Blast scheme, and (d) recovered signal with STBC scheme.
Fig. 5
Fig. 5 ABER performance of OAM multiplexing-based 2 × 2 MIMO system with different space-time coding schemes in (a) weak, (b) moderate, and (c) strong turbulence conditions using BPSK, QPSK, 8PSK modulations.
Fig. 6
Fig. 6 ABER performance comparison between different space-time coding schemes over atmospheric turbulence with different receive antennas. The space-time coding scheme of (a) is V-Blast. The space-time coding scheme of (b) is STBC.

Equations (26)

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X V B l a s t - O A M = [ x 11 l 1 x 1 N l 1 x M 1 l 1 x M N l 1 x 11 l K x 1 N l K x M 1 l K x M N l K ] K M × N ,
H V B l a s t O A M = [ h 11 l 1 h N M l 1 h N 1 l 1 h N M l 1 h 11 l K h 1 M l K h N 1 l K h N M l K ] K N × M .
[ y 1 l 1 y N l 1 y 1 l K y N l K ] = [ h 11 l 1 h N M l 1 h N 1 l 1 h N M l 1 h 11 l K h 1 M l K h N 1 l K h N M l K ] [ x 1 l 1 x M l 1 x 1 l K x M l K ] + [ w 1 w N w 1 w N ] ,
Y V B l a s t O A M = H V B l a s t O A M X V B l a s t O A M + W ,
X S T B C O A M = [ x 1 l 1 ( x 2 l 1 ) * x 2 l 1 ( x 1 l 1 ) * x 1 l K ( x 2 l K ) * x 2 l K ( x 1 l K ) * ] .
H S T B C O A M = [ h 1 l 1 h 2 l 1 h 1 l K h 2 l K ] ,
y 1 l 1 = h 1 l 1 x 1 l 1 + h 1 l 1 x 2 l 1 + w 1 l 1 y 2 l 1 = h 2 l 1 ( x 2 l 1 ) * + h 2 l 1 ( x 1 l 1 ) * + w 2 l 1 y 1 l K = h 1 l K x 1 l K + h 1 l K x 2 l K + w 1 l K y 2 l K = h 2 l K ( x 2 l K ) * + h 2 l K ( x 1 l K ) * + w 2 l K ,
[ y 1 l 1 y 2 l 1 y 1 l K y 2 l K ] = [ h 1 l 1 h 2 l 1 ( h 2 l 1 ) * ( h 1 l 1 ) * h 1 l K h 2 l K ( h 2 l K ) * ( h 1 l K ) * ] [ x 1 l 1 x 2 l 1 x 1 l K x 2 l K ] + [ w 1 l 1 w 2 l 1 w 1 l K w 2 l K ] .
H ¯ ¯ S T B C O A M = [ h 1 l 1 h 2 l 1 ( h 2 l 1 ) * ( h 1 l 1 ) * h 1 l K h 2 l K ( h 2 l K ) * ( h 1 l K ) * ] .
[ s 1 l 1 s 2 l 1 s 1 l K s 2 l K ] = [ h 1 l 1 h 2 l 1 ( h 2 l 1 ) * ( h 1 l 1 ) * h 1 l K h 2 l K ( h 2 l K ) * ( h 1 l K ) * ] [ y 1 l 1 y 2 l 1 y 1 l K y 2 l K ] .
[ s 1 l 1 s 2 l 1 s 1 l K s 2 l K ] = [ ρ 1 l 1 ε l 1 ( ε l 1 ) * ρ 2 l 1 ρ 1 l K ε l K ( ε l K ) * ρ 2 l K ] [ x 1 l 1 x 2 l 1 x 1 l K x 2 l K ] + [ h 1 l 1 h 2 l 1 ( h 2 l 1 ) * ( h 1 l 1 ) * h 1 l K h 2 l K ( h 2 l K ) * ( h 1 l K ) * ] [ w 1 l 1 w 2 l 1 w 1 l K w 2 l K ] ,
ρ 1 l 1 = | h 1 l 1 | 2 + | h 2 l 1 | 2 , ρ 2 l 1 = | h 2 l 1 | 2 + | h 1 l 1 | 2 , ε l 1 = h 1 l 1 ( h 1 l 1 ) * h 2 l 1 ( h 2 l 1 ) * ρ 1 l K = | h 1 l K | 2 + | h 2 l K | 2 , ρ 2 l K = | h 2 l K | 2 + | h 1 l K | 2 , ε l K = h 1 l K ( h 1 l K ) * h 2 l K ( h 2 l K ) * .
U l ( r , θ , z ) = 2 π | l | ! 1 ω ( z ) [ 2 r ω ( z ) ] | l | exp [ r 2 ω 2 ( z ) ] exp [ i k r 2 z 2 ( z 2 + z R 2 ) ] × exp [ i ( | l | + 1 ) tan 1 z z R ] exp ( i l θ ) ,
Φ n ( k ) = 0.033 C n 2 ( k λ 2 + 1 / L 0 2 ) 11 6 exp ( k λ 2 / k l 2 ) ,
Φ φ ( k ) = 2 π k 2 Δ z Φ n ( k ) .
φ ( x , y ) = FFT [ C 2 π N Δ x Φ φ ( k ) ] ,
U 1 ( x , y ) = F F T 1 [ FFT [ U l ( x , y ) ] U p r o p ( k x , k y ) ] .
U 1 + ( x , y ) = U 1 ( x , y ) e x p [ i φ ( x , y ) ] ,
U l ' ( r , θ , z ) = U l ( r , θ , z ) exp ( ψ ( r ) ) ,
e x 2 = ( x U l ' ( r , θ , z ) h x μ x ) H ( x U l ' ( r , θ , z ) h x μ x ) ,
h x L S = ( μ x H μ x ) 1 ( μ x H x U l ' ( r , θ , z ) ) .
U n V B l a s t R x ( r , θ , t ) = m = 1 M U m R x ( r , θ , t ) = m = 1 M k m = 1 K x k m ( t ) U k m R x ( r , θ , z ) = m = 1 M k m = 1 K x k ( t ) A k m R x ( r , z ) e x p ( i l k m θ ' ) = m = 1 M k m = 1 K x k ( t ) η k m A k m ( r , z ) e x p ( i l k m θ ) ,
η k m = k = 1 K A k m R x ( r , z ) A k m ( r , z ) exp ( i l k m ( θ ' θ ) = k = 1 K A k m R x ( r , z ) A k m ( r , z ) exp ( i l k m Δ θ ) .
y k m V B l a s t n ( t ) = U n R x ( r , θ , t ) U k m * ( r , q , t ) r d r d θ + W k m ( t ) = U k m * ( r , θ , t ) m = 1 M k m = 1 K x k m ( t ) η k m A k m ( r , z ) e x p ( i l k m θ ) r d r d θ + W k m ( t ) = ( A k m ( r , z ) ) 2 x k m ( t ) η k m k m + A k m ( r , z ) A k ( r , z ) m = 1 M k = 1 , k k m K η k x m ( t ) + W k m ( t ) = x k m ( t ) ζ k m + i n t e r f e r e n c e k m + W k m ( t ) ,
U n S T B C R x ( r , θ , t ) = k = 1 K x k ( t ) U k R x ( r , θ , z ) = k = 1 K x k ( t ) A k R x ( r , z ) e x p ( i l k θ ' ) = k = 1 K x k ( t ) η k A k ( r , z ) e x p ( i l k θ ) ,
y k S T B C k n ( t ) = n = 1 2 U n R x ( r , θ , t ) U k n * ( r , θ , t ) r d r d θ + W k n ( t ) = m = 1 2 U k n * ( r , θ , t ) k = 1 K x k ( t ) η k A k ( r , z ) e x p ( i l k θ ) r d r d θ + W k n ( t ) = n = 1 2 ( ( A k n ( r , z ) ) 2 x k n ( t ) η k n k n + A k n ( r , z ) A k ( r , z ) k = 1 , k k n K η k k n x k ( t ) ) + W k n ( t ) = n = 1 2 ( S k n ( t ) ζ k n + interference k n ) + W k n ( t ) .

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