Abstract

In this paper, a multipoint-to-point system consisting K users and a central node over wireless optical communication (WOC) channel is analyzed. The scenario focused on is that there is simultaneous communication from a number of users to the central node. As a powerful solution, we utilize non-orthogonal multiple access (NOMA) technique in the system. Although the superiority of NOMA in radio frequency (RF) system has been greatly considered, the NOMA in WOC still needs further research due to the special features of WOC, especially the non-broadcast nature of optical beam and the vulnerable turbulence channel. With the special features of WOC in mind, system is evaluated in terms of outage probability, bit error rate and ergodic sum rate. In addition, we theoretically prove that NOMA outperforms orthogonal multiple access (OMA), and the performance gain increases with the increase of turbulence strength. Hence, NOMA is more suitable for WOC, especially in strong turbulence channel. Moreover, we also analyze the user pairing scheme. Monte Carlo simulations have been done, which match quite well with the theoretical analysis.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
    [Crossref]
  2. H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Comm. Surv. and Tutor. 19(1), 57–96 (2017).
    [Crossref]
  3. M. A. Khalighi and M. Uysal, “Survey on free space optical communication: a communication theory perspective,” IEEE Comm. Surv. and Tutor. 16(4), 2231–2258 (2014).
    [Crossref]
  4. B. Born, I. R. Hristovski, S. Geoffroy-Gagnon, and J. F. Holzman, “All-optical retro-modulation for free-space optical communication,” Opt. Express 26(4), 5031–5042 (2018).
    [Crossref] [PubMed]
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    [Crossref] [PubMed]
  6. R. Li, S. Ding, and A. Dang, “Suboptimal maximum likelihood detection of on-off keying for a wireless optical communication system,” J. Opt. Soc. Am. A 34(5), 798–803 (2017).
    [Crossref] [PubMed]
  7. A. Dang, “A closed-form solution of the bit-error rate for optical wireless communication systems over atmospheric turbulence channels,” Opt. Express 19(4), 3494–3502 (2011).
    [Crossref] [PubMed]
  8. A. Jurado-Navas, T. R. Raddo, J. M. Garrido-Balsells, B.-H. V. Borges, J. J. V. Olmos, and I. T. Monroy, “Hybrid optical CDMA-FSO communications network under spatially correlated gamma-gamma scintillation,” Opt. Express 24(15), 16799–16814 (2016).
    [Crossref] [PubMed]
  9. R. Li and A. Dang, “A novel coherent OCDMA scheme over atmospheric turbulence channels,” IEEE Photonics Technol. Lett. 29(5), 427–430 (2017).
    [Crossref]
  10. J. Abouei and K. N. Plataniotis, “Multiuser diversity scheduling in free-space optical communications,” J. Lightwave Technol. 30(9), 1351–1358 (2012).
    [Crossref]
  11. J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
    [Crossref]
  12. Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, “Non-orthogonal multiple access (NOMA) for cellular future radio access,” in 2013 IEEE 77th Vehicular Technology Conference (VTC Spring), 2013), 1–5.
    [Crossref]
  13. R. Li and A. Dang, “Performance analysis of non-orthogonal multiple access in free space optical communication system,” Online version available on arXiv. [Online]. Available: https://arxiv.org/pdf/1707.06571.pdf (2017).
  14. M. Najafi, V. Jamali, P. D. Diamantoulakis, G. K. Karagiannidis, and R. Schober, “Non-orthogonal multiple access for FSO backhauling,” in 2018 IEEE Wireless Communications and Networking Conference (WCNC), 2018), 1–6.
  15. N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink nonorthogonal multiple access in 5G systems,” IEEE Commun. Lett. 20(3), 458–461 (2016).
    [Crossref]
  16. A. Chaaban, O. M. S. Al-Ebraheemy, T. Y. Al-Naffouri, and M. S. Alouini, “Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel,” IEEE Trans. Wirel. Commun. 16(5), 3328–3340 (2017).
    [Crossref]
  17. I. I. Kim, B. McArthur, and E. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Optical Wireless Communications Iii, E.J. Korevaar, ed. (Spie-Int Soc Optical Engineering, Bellingham, 2001), pp. 26–37.
  18. M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
    [Crossref]
  19. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2014).
  20. H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (Wiley, USA, 2003).

2018 (1)

2017 (5)

J. M. Garrido-Balsells, F. Javier Lopez-Martinez, M. Castillo-Vázquez, A. Jurado-Navas, and A. Puerta-Notario, “Performance analysis of FSO communications under LOS blockage,” Opt. Express 25(21), 25278–25294 (2017).
[Crossref] [PubMed]

R. Li, S. Ding, and A. Dang, “Suboptimal maximum likelihood detection of on-off keying for a wireless optical communication system,” J. Opt. Soc. Am. A 34(5), 798–803 (2017).
[Crossref] [PubMed]

H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Comm. Surv. and Tutor. 19(1), 57–96 (2017).
[Crossref]

R. Li and A. Dang, “A novel coherent OCDMA scheme over atmospheric turbulence channels,” IEEE Photonics Technol. Lett. 29(5), 427–430 (2017).
[Crossref]

A. Chaaban, O. M. S. Al-Ebraheemy, T. Y. Al-Naffouri, and M. S. Alouini, “Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel,” IEEE Trans. Wirel. Commun. 16(5), 3328–3340 (2017).
[Crossref]

2016 (2)

2014 (2)

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: a communication theory perspective,” IEEE Comm. Surv. and Tutor. 16(4), 2231–2258 (2014).
[Crossref]

2012 (1)

2011 (2)

J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
[Crossref]

A. Dang, “A closed-form solution of the bit-error rate for optical wireless communication systems over atmospheric turbulence channels,” Opt. Express 19(4), 3494–3502 (2011).
[Crossref] [PubMed]

2001 (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Abouei, J.

Al-Ebraheemy, O. M. S.

A. Chaaban, O. M. S. Al-Ebraheemy, T. Y. Al-Naffouri, and M. S. Alouini, “Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel,” IEEE Trans. Wirel. Commun. 16(5), 3328–3340 (2017).
[Crossref]

Al-Habash, M. A.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Al-Naffouri, T. Y.

A. Chaaban, O. M. S. Al-Ebraheemy, T. Y. Al-Naffouri, and M. S. Alouini, “Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel,” IEEE Trans. Wirel. Commun. 16(5), 3328–3340 (2017).
[Crossref]

Alouini, M. S.

A. Chaaban, O. M. S. Al-Ebraheemy, T. Y. Al-Naffouri, and M. S. Alouini, “Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel,” IEEE Trans. Wirel. Commun. 16(5), 3328–3340 (2017).
[Crossref]

Andrews, J. G.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

Andrews, L. C.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Borges, B.-H. V.

Born, B.

Buzzi, S.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

Castillo-Vázquez, M.

Chaaban, A.

A. Chaaban, O. M. S. Al-Ebraheemy, T. Y. Al-Naffouri, and M. S. Alouini, “Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel,” IEEE Trans. Wirel. Commun. 16(5), 3328–3340 (2017).
[Crossref]

Choi, W.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

Dang, A.

Ding, S.

Fujikawa, C.

J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
[Crossref]

Garrido-Balsells, J. M.

Geoffroy-Gagnon, S.

Hanly, S. V.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

Holzman, J. F.

Hristovski, I. R.

Javier Lopez-Martinez, F.

Jurado-Navas, A.

Kaddoum, G.

H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Comm. Surv. and Tutor. 19(1), 57–96 (2017).
[Crossref]

Kang, G.

N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink nonorthogonal multiple access in 5G systems,” IEEE Commun. Lett. 20(3), 458–461 (2016).
[Crossref]

Kaushal, H.

H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Comm. Surv. and Tutor. 19(1), 57–96 (2017).
[Crossref]

Khalighi, M. A.

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: a communication theory perspective,” IEEE Comm. Surv. and Tutor. 16(4), 2231–2258 (2014).
[Crossref]

Kodate, K.

J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
[Crossref]

Li, R.

R. Li, S. Ding, and A. Dang, “Suboptimal maximum likelihood detection of on-off keying for a wireless optical communication system,” J. Opt. Soc. Am. A 34(5), 798–803 (2017).
[Crossref] [PubMed]

R. Li and A. Dang, “A novel coherent OCDMA scheme over atmospheric turbulence channels,” IEEE Photonics Technol. Lett. 29(5), 427–430 (2017).
[Crossref]

Liu, J.

J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
[Crossref]

Liu, Y.

N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink nonorthogonal multiple access in 5G systems,” IEEE Commun. Lett. 20(3), 458–461 (2016).
[Crossref]

Lozano, A.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

Monroy, I. T.

Olmos, J. J. V.

Phillips, R. L.

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Plataniotis, K. N.

Puerta-Notario, A.

Raddo, T. R.

Sando, J.

J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
[Crossref]

Shimamoto, S.

J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
[Crossref]

Soong, A. C. K.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

Uysal, M.

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: a communication theory perspective,” IEEE Comm. Surv. and Tutor. 16(4), 2231–2258 (2014).
[Crossref]

Wang, J.

N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink nonorthogonal multiple access in 5G systems,” IEEE Commun. Lett. 20(3), 458–461 (2016).
[Crossref]

Zhang, J. C.

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

Zhang, N.

N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink nonorthogonal multiple access in 5G systems,” IEEE Commun. Lett. 20(3), 458–461 (2016).
[Crossref]

IEEE Comm. Surv. and Tutor. (2)

H. Kaushal and G. Kaddoum, “Optical communication in space: challenges and mitigation techniques,” IEEE Comm. Surv. and Tutor. 19(1), 57–96 (2017).
[Crossref]

M. A. Khalighi and M. Uysal, “Survey on free space optical communication: a communication theory perspective,” IEEE Comm. Surv. and Tutor. 16(4), 2231–2258 (2014).
[Crossref]

IEEE Commun. Lett. (1)

N. Zhang, J. Wang, G. Kang, and Y. Liu, “Uplink nonorthogonal multiple access in 5G systems,” IEEE Commun. Lett. 20(3), 458–461 (2016).
[Crossref]

IEEE J. Sel. Areas Comm. (1)

J. G. Andrews, S. Buzzi, W. Choi, S. V. Hanly, A. Lozano, A. C. K. Soong, and J. C. Zhang, “What will 5G be?” IEEE J. Sel. Areas Comm. 32(6), 1065–1082 (2014).
[Crossref]

IEEE Photonics Technol. Lett. (1)

R. Li and A. Dang, “A novel coherent OCDMA scheme over atmospheric turbulence channels,” IEEE Photonics Technol. Lett. 29(5), 427–430 (2017).
[Crossref]

IEEE Trans. Consum. Electron. (1)

J. Liu, J. Sando, S. Shimamoto, C. Fujikawa, and K. Kodate, “Experiment on space and time division multiple access scheme over free space optical communication,” IEEE Trans. Consum. Electron. 57(4), 1571–1578 (2011).
[Crossref]

IEEE Trans. Wirel. Commun. (1)

A. Chaaban, O. M. S. Al-Ebraheemy, T. Y. Al-Naffouri, and M. S. Alouini, “Capacity Bounds for the Gaussian IM-DD Optical Multiple-Access Channel,” IEEE Trans. Wirel. Commun. 16(5), 3328–3340 (2017).
[Crossref]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

M. A. Al-Habash, L. C. Andrews, and R. L. Phillips, “Mathematical model for the irradiance probability density function of a laser beam propagating through turbulent media,” Opt. Eng. 40(8), 1554–1562 (2001).
[Crossref]

Opt. Express (4)

Other (6)

I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, 2014).

H. A. David and H. N. Nagaraja, Order Statistics, 3rd ed. (Wiley, USA, 2003).

I. I. Kim, B. McArthur, and E. Korevaar, “Comparison of laser beam propagation at 785 nm and 1550 nm in fog and haze for optical wireless communications,” in Optical Wireless Communications Iii, E.J. Korevaar, ed. (Spie-Int Soc Optical Engineering, Bellingham, 2001), pp. 26–37.

Y. Saito, Y. Kishiyama, A. Benjebbour, T. Nakamura, A. Li, and K. Higuchi, “Non-orthogonal multiple access (NOMA) for cellular future radio access,” in 2013 IEEE 77th Vehicular Technology Conference (VTC Spring), 2013), 1–5.
[Crossref]

R. Li and A. Dang, “Performance analysis of non-orthogonal multiple access in free space optical communication system,” Online version available on arXiv. [Online]. Available: https://arxiv.org/pdf/1707.06571.pdf (2017).

M. Najafi, V. Jamali, P. D. Diamantoulakis, G. K. Karagiannidis, and R. Schober, “Non-orthogonal multiple access for FSO backhauling,” in 2018 IEEE Wireless Communications and Networking Conference (WCNC), 2018), 1–6.

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Figures (6)

Fig. 1
Fig. 1 Diagram of the NOMA in WOC system. (a) System model of the proposed system. (b) A scenario of the proposed system model.
Fig. 2
Fig. 2 Outage probability of NOMA for different ζ. (a) σ R 2 = 0.1. (b) σ R 2 = 1.
Fig. 3
Fig. 3 Outage probability of NOMA for different targeted data rate. (a) σ R 2 =0.1. (b) σ R 2 =1.
Fig. 4
Fig. 4 BER performance of the proposed NOMA system σ R 2 =0.1. (a) BER of NOMA versus ρ. (b) BER of NOMA versus ζ.
Fig. 5
Fig. 5 Ergodic sum rate performance of NOMA. (a) Ergodic sum rate results of NOMA as the function of ρ. (b) Difference value of ergodic sum rate.
Fig. 6
Fig. 6 Performance gain of different user pairing cases.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

y = k = 1 K g k ξ P k x k + n ,
P = P a i m / L ,
P k = P a i m L k 10 ( k 1 ) ς 10 = a k P a i m ,
R k = { [ 1 2 log ( 1 + ρ ( μ k g k a k ) 2 ρ i = k + 1 K ( μ i g i a i ) 2 + A ) ε ϕ ] + R ˜ k , k < K [ 1 2 log ( 1 + ρ ( μ K g K a K ) 2 A ) ε ϕ ] + R ˜ K , k = K
f I ( I ) = 2 ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) I ( α + β ) / 2 1 K α β ( 2 α β I ) , I > 0 ,
α = [ exp ( 0.49 σ R 2 ( 1 + 1.11 σ R 12 / 5 ) 7 / 6 ) 1 ] 1 ,
β = [ exp ( 0.51 σ R 2 ( 1 + 0.69 σ R 12 / 5 ) 5 / 6 ) 1 ] 1 .
f h k ( h ) = ( α β ) ( α + β ) / 2 Γ ( α ) Γ ( β ) ( h ) ( α + β ) / 2 2 K α β ( 2 α β h ) .
F h k ( y ) = ( α β ) α + β 2 y α + β 4 4 π Γ ( α ) Γ ( β ) G 1 , 5 4 , 1 ( 1 16 ( α β ) 2 y | 1 ( α + β ) / 4 α β 4 , α β + 2 4 , β α 4 , 2 α + β 4 , α + β 4 ) ,
f h k ( h ) = K ! ( k 1 ) ! ( K k ) ! { F h k ( h ) } K k { 1 F h k ( h ) } k 1 f h k ( h ) .
f h 1 , , h k ( h 1 , , h k ) = k ! i = 1 k f h i ( h i ) , h 1 h k .
P k o u t = 1 P ( E 1 c E k c ) = 1 i = 1 k P ( E i c ) ,
P ( E k c ) = P { [ 1 2 log ( 1 + ( μ k g k P k ) 2 i = k + 1 K ( μ i g i P i ) 2 + A σ 2 ) ε ϕ ] + R ˜ k } = ( a ) 1 P { h k A ϕ k + ϕ k ρ i = k + 1 K L i 2 μ i 2 h i a i 2 ρ μ k 2 a k 2 L k 2 } = 1 y k + 1 υ f h k , , h K ( y k , , y K ) d y k d y K ,
P ( E K c ) = 1 0 ψ K K [ 1 F h k ( y ) ] K 1 f h k ( y ) d y = [ 1 F h k ( ψ K ) ] K ,
P e ( C k | h ) = e k 1 = 0 , 1 , 1 e 1 = 0 , 1 , 1 P e ( C k | e k 1 , , e 1 , h ) × P e ( e k 1 | e k 2 , , e 1 , h ) P e ( e 2 | e 1 , h ) P e ( e 1 | h ) ,
P e ( e k | e k 1 , , e 1 , h ) = { 1 2 P e ( D k | e k 1 , , e 1 , h ) , e k = 1 , 1 P e ( C k | e k 1 , , e 1 , h ) , e k = 0 , 1 2 P e ( E k | e k 1 , , e 1 , h ) , e k = 1 ,
P e ( C k | e k 1 , , e 1 , h ) = 1 2 [ P e ( D k | e k 1 , , e 1 , h ) + P e ( E k | e k 1 , , e 1 , h ) ] ,
P e ( D k | e k 1 , , e 1 , h ) = 1 2 K k + 1 x k e r f c ( γ k 1 ) ,
P e ( E k | e k 1 , , e 1 , h ) = 1 2 K k + 1 x k e r f c ( γ k 2 ) ,
γ k 1 = ( a k g k P a i m + F k τ k ) 2 σ , γ k 2 = ( τ k F k ) 2 σ ,
R s u m N O M A = E { k = 1 K R k } = ( b ) E { 1 2 log ( 1 + ρ k = 1 K ( μ k g k a k ) 2 A ) K ε ϕ } ( c ) 1 2 E { log ( ρ k = 1 K ( μ k g k a k ) 2 A ) } = 1 2 E { log ( k = 1 K ( μ k g k a k ) 2 ) } + 1 2 log ρ 1 2 log A ,
R k O M A = 0.5 [ 1 2 log ( 1 + 2 ρ ( μ k g k a k ) 2 A ) ε ϕ ] + ,
lim ρ E { R i + 1 + R j R i + 1 O M A R j O M A } < lim ρ E { R i + R j R i O M A R j O M A } .
lim ρ E { R i + R j R i O M A R j O M A } < lim ρ E { R i + R j + 1 R i O M A R j + 1 O M A } .
y k = { a k g k P a i m + F k + n , x k = 1 , F k + n , x k = 0 ,
F k = P a i m i = 1 k 1 g i a i e i + P a i m i = k + 1 K g i a i x i ,
τ k = a k g k P a i m + P a i m i = k + 1 K a i g i 2 .
P e ( D k | e k 1 , , e 1 , x k , h ) = - τ k 1 2 π σ 2 exp [ ( x ( a k g k P a i m + F k ) ) 2 2 σ 2 ] d x = 1 2 e r f c ( γ k 1 ) .
P e ( E k | e k 1 , , e 1 , x k , h ) = τ k + 1 2 π σ 2 exp [ ( x F k ) 2 2 σ 2 ] d x = 1 2 e r f c ( γ k 2 ) .
R s u m O M A = E { k = 1 K η k [ 1 2 log ( 1 + ρ ( μ k g k a k ) 2 η k A ) ε ϕ ] + } ( d ) 1 2 k = 1 K η k E [ log ( ρ ( μ k g k a k ) 2 η k A ) ] = 1 2 log ρ + 1 2 k = 1 K η k E [ log ( ( μ k g k a k ) 2 ) ] 1 2 k = 1 K η k log ( η k A ) ,
R s u m O M A σ R 2 = 1 2 σ R 2 { k = 1 K η k E [ log ( ( μ k g k a k ) 2 ) ] } = 1 2 g k = 1 K η k log ( ( μ k g k a k ) 2 ) σ R 2 f ( g , σ R 2 ) d g 1 2 g log ( k = 1 K η k ( μ k g k a k ) 2 ) σ R 2 f ( g , σ R 2 ) d g 1 2 g log ( k = 1 K ( μ k g k a k ) 2 ) σ R 2 f ( g , σ R 2 ) d g ,
R s u m N O M A σ R 2 = 1 2 σ R 2 E { log ( k = 1 K ( μ k g k a k ) 2 ) } = 1 2 σ R 2 g log ( k = 1 K ( μ k g k a k ) 2 ) f ( g , σ R 2 ) d g = 1 2 g log ( k = 1 K ( μ k g k a k ) 2 ) σ R 2 f ( g , σ R 2 ) d g .
R j R j O M A 1 2 log ( 1 + ρ ( μ j g j a j ) 2 A ) 1 4 log ( 1 + 2 ρ ( μ j g j a j ) 2 A ) = 1 4 log ( A 2 + 2 A ρ ( μ j g j a j ) 2 + [ ρ ( μ j g j a j ) 2 ] 2 A ( A + 2 ρ ( μ j g j a j ) 2 ) ) = 1 4 log ( 1 + [ ρ ( μ j g j a j ) 2 ] 2 A ( A + 2 ρ ( μ j g j a j ) 2 ) ) > 0 ,
1 2 log ( 1 + ρ ( μ i g i a i ) 2 ρ ( μ j g j a j ) 2 + A ) ε ϕ > 0.5 { 1 2 log ( 1 + ρ ( μ i g i a i ) 2 0.5 × A ) ε ϕ } .
lim ρ E { R i + 1 R i I 1 ( R i + 1 O M A R i O M A ) I 2 } < 0.
I 1 = 1 2 log ( 1 + ρ ( μ i + 1 g i + 1 a i + 1 ) 2 ρ ( μ j g j a j ) 2 + A ) ε ϕ 1 2 log ( 1 + ρ ( μ i g i a i ) 2 ρ ( μ j g j a j ) 2 + A ) + ε ϕ = 1 2 log ( ρ ( μ j g j a j ) 2 + A + ρ ( μ i + 1 g i + 1 a i + 1 ) 2 ρ ( μ j g j a j ) 2 + A + ρ ( μ i g i a i ) 2 ) .
I 2 = 0.5 [ 1 2 log ( 1 + 2 ρ ( μ i + 1 g i + 1 a i + 1 ) 2 A ) ε ϕ ] 0.5 [ 1 2 log ( 1 + 2 ρ ( μ i g i a i ) 2 A ) ε ϕ ] = 1 4 [ log ( A + 2 ρ ( μ i + 1 g i + 1 a i + 1 ) 2 A + 2 ρ ( μ i g i a i ) 2 ) ] .
lim ρ E { R i + 1 R i R i + 1 O M A + R i O M A } = 1 2 E { log ( ( μ j g j a j ) 2 + ( μ i + 1 g i + 1 a i + 1 ) 2 ( μ j g j a j ) 2 + ( μ i g i a i ) 2 μ i g i a i μ i + 1 g i + 1 a i + 1 ) I 3 } .
lim ρ E { R i R i + R j + 1 R j R j + 1 O M A + R j O M A } > 0.
lim ρ E { R i R i + R j + 1 R j R j + 1 O M A + R j O M A } = 1 2 E { log ( ( μ j + 1 g j + 1 a j + 1 ) 2 + ( μ i g i a i ) 2 μ j + 1 g j + 1 a j + 1 μ j g j a j ( μ j g j a j ) 2 + ( μ i g i a i ) 2 ) I 4 } > 0.

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