Abstract

This paper investigates the asymptotic BER performance of coherent optical wireless communication systems in Gamma-Gamma turbulence when applying the V-BLAST MIMO scheme. A new method is proposed to quantify the performance of the system and mathematical solutions for asymptotic BER performance are derived. Counterintuitive results are shown since the diversity gain of the V-BLAST MIMO system is equal to the number of the receivers. As a consequence, it is shown that when applying the V-BLAST MIMO scheme, the symbol rate per transmission can be equal to the number of transmitters with some cost to diversity gain. This means that we can simultaneously exploit the spatial multiplexing and diversity properties of the MIMO system to achieve a higher data rate than existing schemes in a channel that displays severe turbulence and moderate attenuation.

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

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References

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  • Publication
  1. M. A. Khalighi and M. Uysal, “Survey on free space optical communication: A communication theory perspective,” IEEE Commun. Surv. Tutorials 16, 2231–2258 (2014).
    [Crossref]
  2. V. W. S. Chan, “Free-space optical communications,” J. Light. Technol. 24, 4750–4762 (2006).
    [Crossref]
  3. A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications: A key to gigabit wireless,” J. Light. Technol. 34, 2158–2169 (2016).
  4. E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Transactions on Commun. 57, 3415–3424 (2009).
    [Crossref]
  5. M. R. Bhatnagar and Z. Ghassemlooy, “Performance analysis of gamma-gamma fading FSO MIMO links with pointing errors,” Proc. IEEE 92, 198–218 (2004).
  6. Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
    [Crossref]
  7. J. Y. Wang, J. Dai, R. Guan, L. Jia, Y. Wang, and M. Chen, “Channel capacity and receiver deployment optimization for multi-input multi-output visible light communications,” Opt. Express 24, 13060 (2016).
    [Crossref] [PubMed]
  8. M. Niu, “Coherent optical wireless communications over atmospheric turbulence channels,” Ph. D. thesis, University of British Columbia (2012).
  9. M. Niu, J. Cheng, and J. F. Holzman, “Alamouti-type STBC for atmospheric optical communication using coherent detection,” IEEE Photonics J. 6, 7900217 (2014).
    [Crossref]
  10. Y. Zhang, P. Wang, L. Guo, W. Wang, and H. Tian, “Performance analysis of an OAM multiplexing-based MIMO FSO system over atmospheric turbulence using space-time coding with channel estimation,” Opt. Express 25, 19995–20011 (2017).
    [Crossref] [PubMed]
  11. J. Zhou, J. Wu, and Q. Hu, “Capacity analysis for free space coherent optical MIMO transmission systems: with and without adaptive optics,” Opt. Express 26, 23008–23018 (2018).
    [Crossref] [PubMed]
  12. L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Transactions on Inf. Theory 49, 1073–1096 (2003).
    [Crossref]
  13. S. J. Grant and J. K. Cavers, “Performance enhancement through joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 46, 1038–1049 (1998).
    [Crossref]
  14. S. J. Grant and J. K. Cavers, “Further analytical results on the joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 48, 1788–1792 (2000).
    [Crossref]
  15. M. Seimetz and C.-M. Weinert, “Options, feasibility, and availability of 2 × 4 90° hybrids for coherent optical systems,” J. lightwave technology 24, 1317–1322 (2006).
    [Crossref]
  16. F. Derr, “Comparison of electrical and optical BPSK and QPSK systems,” J. Opt. Commun. 10, 127–131 (1989).
    [Crossref]
  17. R. Barrios and F. Dios, “Exponentiated weibull distribution family under aperture averaging for gaussian beam waves,” Opt. express 20, 13055–13064 (2012).
    [Crossref] [PubMed]
  18. L. C. Andrews and R. L. Phillips, Laser beam propagation through random media(SPIE, 2005).
    [Crossref]
  19. A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv preprint arXiv:1102.1915 (2011).
  20. M. Naboulsi, H. Sizun, and F. Fornel, “Propagation of optical and infrared waves in the atmosphere,” Proc. union radio scientifique internationale (2005).
  21. A. Touati, A. Abdaoui, F. Touati, M. Uysal, and A. Bouallegue, “On the effects of combined atmospheric fading and misalignment on the hybrid FSO/RF transmission,” J. Opt. Commun. Netw. 8, 715725 (2016).
    [Crossref]
  22. J. G. Proakis and M. Salehi, Digital communications(McGraw-Hill, 2008).
  23. M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20, 6515–6520 (2012).
    [Crossref] [PubMed]
  24. A. Jeffrey and D. Zwillinger, Table of integrals, series, and products (Elsevier, 2007).

2018 (1)

2017 (1)

2016 (4)

A. Touati, A. Abdaoui, F. Touati, M. Uysal, and A. Bouallegue, “On the effects of combined atmospheric fading and misalignment on the hybrid FSO/RF transmission,” J. Opt. Commun. Netw. 8, 715725 (2016).
[Crossref]

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

J. Y. Wang, J. Dai, R. Guan, L. Jia, Y. Wang, and M. Chen, “Channel capacity and receiver deployment optimization for multi-input multi-output visible light communications,” Opt. Express 24, 13060 (2016).
[Crossref] [PubMed]

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications: A key to gigabit wireless,” J. Light. Technol. 34, 2158–2169 (2016).

2014 (2)

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

M. Niu, J. Cheng, and J. F. Holzman, “Alamouti-type STBC for atmospheric optical communication using coherent detection,” IEEE Photonics J. 6, 7900217 (2014).
[Crossref]

2012 (2)

2009 (1)

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Transactions on Commun. 57, 3415–3424 (2009).
[Crossref]

2006 (2)

V. W. S. Chan, “Free-space optical communications,” J. Light. Technol. 24, 4750–4762 (2006).
[Crossref]

M. Seimetz and C.-M. Weinert, “Options, feasibility, and availability of 2 × 4 90° hybrids for coherent optical systems,” J. lightwave technology 24, 1317–1322 (2006).
[Crossref]

2004 (1)

M. R. Bhatnagar and Z. Ghassemlooy, “Performance analysis of gamma-gamma fading FSO MIMO links with pointing errors,” Proc. IEEE 92, 198–218 (2004).

2003 (1)

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Transactions on Inf. Theory 49, 1073–1096 (2003).
[Crossref]

2000 (1)

S. J. Grant and J. K. Cavers, “Further analytical results on the joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 48, 1788–1792 (2000).
[Crossref]

1998 (1)

S. J. Grant and J. K. Cavers, “Performance enhancement through joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 46, 1038–1049 (1998).
[Crossref]

1989 (1)

F. Derr, “Comparison of electrical and optical BPSK and QPSK systems,” J. Opt. Commun. 10, 127–131 (1989).
[Crossref]

Abdaoui, A.

Andrews, L. C.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media(SPIE, 2005).
[Crossref]

Barrios, R.

Bayaki, E.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Transactions on Commun. 57, 3415–3424 (2009).
[Crossref]

Bhatnagar, M. R.

M. R. Bhatnagar and Z. Ghassemlooy, “Performance analysis of gamma-gamma fading FSO MIMO links with pointing errors,” Proc. IEEE 92, 198–218 (2004).

Bolcskei, H.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications: A key to gigabit wireless,” J. Light. Technol. 34, 2158–2169 (2016).

Bouallegue, A.

Cavers, J. K.

S. J. Grant and J. K. Cavers, “Further analytical results on the joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 48, 1788–1792 (2000).
[Crossref]

S. J. Grant and J. K. Cavers, “Performance enhancement through joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 46, 1038–1049 (1998).
[Crossref]

Chan, V. W. S.

V. W. S. Chan, “Free-space optical communications,” J. Light. Technol. 24, 4750–4762 (2006).
[Crossref]

Chen, M.

Cheng, J.

M. Niu, J. Cheng, and J. F. Holzman, “Alamouti-type STBC for atmospheric optical communication using coherent detection,” IEEE Photonics J. 6, 7900217 (2014).
[Crossref]

M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20, 6515–6520 (2012).
[Crossref] [PubMed]

Dai, J.

Derr, F.

F. Derr, “Comparison of electrical and optical BPSK and QPSK systems,” J. Opt. Commun. 10, 127–131 (1989).
[Crossref]

Dios, F.

Fornel, F.

M. Naboulsi, H. Sizun, and F. Fornel, “Propagation of optical and infrared waves in the atmosphere,” Proc. union radio scientifique internationale (2005).

Garrido-Balsells, J. M.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv preprint arXiv:1102.1915 (2011).

Ghassemlooy, Z.

M. R. Bhatnagar and Z. Ghassemlooy, “Performance analysis of gamma-gamma fading FSO MIMO links with pointing errors,” Proc. IEEE 92, 198–218 (2004).

Gore, D. A.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications: A key to gigabit wireless,” J. Light. Technol. 34, 2158–2169 (2016).

Grant, S. J.

S. J. Grant and J. K. Cavers, “Further analytical results on the joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 48, 1788–1792 (2000).
[Crossref]

S. J. Grant and J. K. Cavers, “Performance enhancement through joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 46, 1038–1049 (1998).
[Crossref]

Guan, R.

Guo, L.

Holzman, J. F.

M. Niu, J. Cheng, and J. F. Holzman, “Alamouti-type STBC for atmospheric optical communication using coherent detection,” IEEE Photonics J. 6, 7900217 (2014).
[Crossref]

M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20, 6515–6520 (2012).
[Crossref] [PubMed]

Hu, Q.

Jeffrey, A.

A. Jeffrey and D. Zwillinger, Table of integrals, series, and products (Elsevier, 2007).

Jia, L.

Jurado-Navas, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv preprint arXiv:1102.1915 (2011).

Khalighi, M. A.

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

Li, X.-S.

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

Mallik, R. K.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Transactions on Commun. 57, 3415–3424 (2009).
[Crossref]

Nabar, R. U.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications: A key to gigabit wireless,” J. Light. Technol. 34, 2158–2169 (2016).

Naboulsi, M.

M. Naboulsi, H. Sizun, and F. Fornel, “Propagation of optical and infrared waves in the atmosphere,” Proc. union radio scientifique internationale (2005).

Niu, M.

M. Niu, J. Cheng, and J. F. Holzman, “Alamouti-type STBC for atmospheric optical communication using coherent detection,” IEEE Photonics J. 6, 7900217 (2014).
[Crossref]

M. Niu, X. Song, J. Cheng, and J. F. Holzman, “Performance analysis of coherent wireless optical communications with atmospheric turbulence,” Opt. Express 20, 6515–6520 (2012).
[Crossref] [PubMed]

M. Niu, “Coherent optical wireless communications over atmospheric turbulence channels,” Ph. D. thesis, University of British Columbia (2012).

Paris, J. F.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv preprint arXiv:1102.1915 (2011).

Paulraj, A. J.

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications: A key to gigabit wireless,” J. Light. Technol. 34, 2158–2169 (2016).

Phillips, R. L.

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media(SPIE, 2005).
[Crossref]

Proakis, J. G.

J. G. Proakis and M. Salehi, Digital communications(McGraw-Hill, 2008).

Puerta-Notario, A.

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv preprint arXiv:1102.1915 (2011).

Salehi, M.

J. G. Proakis and M. Salehi, Digital communications(McGraw-Hill, 2008).

Schober, R.

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Transactions on Commun. 57, 3415–3424 (2009).
[Crossref]

Seimetz, M.

M. Seimetz and C.-M. Weinert, “Options, feasibility, and availability of 2 × 4 90° hybrids for coherent optical systems,” J. lightwave technology 24, 1317–1322 (2006).
[Crossref]

Sizun, H.

M. Naboulsi, H. Sizun, and F. Fornel, “Propagation of optical and infrared waves in the atmosphere,” Proc. union radio scientifique internationale (2005).

Song, X.

Tian, H.

Touati, A.

Touati, F.

Tse, D. N. C.

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Transactions on Inf. Theory 49, 1073–1096 (2003).
[Crossref]

Uysal, M.

A. Touati, A. Abdaoui, F. Touati, M. Uysal, and A. Bouallegue, “On the effects of combined atmospheric fading and misalignment on the hybrid FSO/RF transmission,” J. Opt. Commun. Netw. 8, 715725 (2016).
[Crossref]

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

Wang, J. Y.

Wang, J.-L.

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

Wang, P.

Wang, W.

Wang, Y.

Weinert, C.-M.

M. Seimetz and C.-M. Weinert, “Options, feasibility, and availability of 2 × 4 90° hybrids for coherent optical systems,” J. lightwave technology 24, 1317–1322 (2006).
[Crossref]

Wu, J.

Yu, H.-Y.

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

Zhang, J.-K.

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

Zhang, Y.

Zhang, Y.-Y.

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

Zheng, L.

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Transactions on Inf. Theory 49, 1073–1096 (2003).
[Crossref]

Zhou, J.

Zhu, Y.-J.

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

Zwillinger, D.

A. Jeffrey and D. Zwillinger, Table of integrals, series, and products (Elsevier, 2007).

IEEE Commun. Lett. (1)

Y.-Y. Zhang, H.-Y. Yu, J.-K. Zhang, Y.-J. Zhu, J.-L. Wang, and X.-S. Li, “On the optimality of spatial repetition coding for MIMO optical wireless communications,” IEEE Commun. Lett. 20, 846–849 (2016).
[Crossref]

IEEE Commun. Surv. Tutorials (1)

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

IEEE Photonics J. (1)

M. Niu, J. Cheng, and J. F. Holzman, “Alamouti-type STBC for atmospheric optical communication using coherent detection,” IEEE Photonics J. 6, 7900217 (2014).
[Crossref]

IEEE Transactions on Commun. (3)

E. Bayaki, R. Schober, and R. K. Mallik, “Performance analysis of MIMO free-space optical systems in gamma-gamma fading,” IEEE Transactions on Commun. 57, 3415–3424 (2009).
[Crossref]

S. J. Grant and J. K. Cavers, “Performance enhancement through joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 46, 1038–1049 (1998).
[Crossref]

S. J. Grant and J. K. Cavers, “Further analytical results on the joint detection of cochannel signals using diversity arrays,” IEEE Transactions on Commun. 48, 1788–1792 (2000).
[Crossref]

IEEE Transactions on Inf. Theory (1)

L. Zheng and D. N. C. Tse, “Diversity and multiplexing: A fundamental tradeoff in multiple-antenna channels,” IEEE Transactions on Inf. Theory 49, 1073–1096 (2003).
[Crossref]

J. Light. Technol. (2)

V. W. S. Chan, “Free-space optical communications,” J. Light. Technol. 24, 4750–4762 (2006).
[Crossref]

A. J. Paulraj, D. A. Gore, R. U. Nabar, and H. Bolcskei, “An overview of MIMO communications: A key to gigabit wireless,” J. Light. Technol. 34, 2158–2169 (2016).

J. lightwave technology (1)

M. Seimetz and C.-M. Weinert, “Options, feasibility, and availability of 2 × 4 90° hybrids for coherent optical systems,” J. lightwave technology 24, 1317–1322 (2006).
[Crossref]

J. Opt. Commun. (1)

F. Derr, “Comparison of electrical and optical BPSK and QPSK systems,” J. Opt. Commun. 10, 127–131 (1989).
[Crossref]

J. Opt. Commun. Netw. (1)

Opt. Express (4)

Proc. IEEE (1)

M. R. Bhatnagar and Z. Ghassemlooy, “Performance analysis of gamma-gamma fading FSO MIMO links with pointing errors,” Proc. IEEE 92, 198–218 (2004).

Other (6)

M. Niu, “Coherent optical wireless communications over atmospheric turbulence channels,” Ph. D. thesis, University of British Columbia (2012).

L. C. Andrews and R. L. Phillips, Laser beam propagation through random media(SPIE, 2005).
[Crossref]

A. Jurado-Navas, J. M. Garrido-Balsells, J. F. Paris, and A. Puerta-Notario, “A unifying statistical model for atmospheric optical scintillation,” arXiv preprint arXiv:1102.1915 (2011).

M. Naboulsi, H. Sizun, and F. Fornel, “Propagation of optical and infrared waves in the atmosphere,” Proc. union radio scientifique internationale (2005).

A. Jeffrey and D. Zwillinger, Table of integrals, series, and products (Elsevier, 2007).

J. G. Proakis and M. Salehi, Digital communications(McGraw-Hill, 2008).

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Figures (7)

Fig. 1
Fig. 1 V-BLAST MIMO coherent OWC system.

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Fig. 2
Fig. 2 Effective integral area of the MISO system in the complex plane.

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Fig. 3
Fig. 3 BER of coherent V-BLAST MIMO systems: M = 2, 3, N = 1, 2, 3, α = 4, β = 2.

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Fig. 4
Fig. 4 BER of coherent V-BLAST MIMO systems: M = 2, N = 2.

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Fig. 5
Fig. 5 BER comparison of different coding schemes: M = 2, N = 2, α = 4, β = 2.

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Fig. 6
Fig. 6 Comparison of different coding schemes: M = 2, N = 2, α = 4, β = 2.

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Fig. 7
Fig. 7 Modified integral area of the MISO system in the complex plane.

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Equations (41)

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

e n m ( t ) = P n m ⋅ A s , m exp ( j ( ω t + ϕ n m + ϕ s , m ) ) ,
e L O ( t ) = P L O exp ( j ( ω L O t + ϕ L O ) ) ,
{ i 1 ( t ) = 1 4 ⋅ R o e | e L O ( t ) + ∑ m = 1 M e n m ( t ) | 2 i 2 ( t ) = 1 4 ⋅ R o e | j e L O ( t ) + ∑ m = 1 M e n m ( t ) | 2 i 3 ( t ) = 1 4 ⋅ R o e | − e L O ( t ) + ∑ m = 1 M e n m ( t ) | 2 i 4 ( t ) = 1 4 ⋅ R o e | − j e L O ( t ) + ∑ m = 1 M e n m ( t ) | 2 ,
i I ( t ) = i 1 ( t ) − i 3 ( t ) .
i I ( t ) = ∑ m = 1 M η n m ⋅ A s , m cos ( ϕ n m − ϕ L O + ϕ s , m ) ,
i Q ( t ) = ∑ m = 1 M η n m ⋅ A s , m sin ( ϕ n m − ϕ L O + ϕ s , m ) .
y n ( t ) = i I ( t ) + i Q ( t ) + n n ( t ) = ∑ m = 1 M η n m ⋅ A s , m exp ( j ( Δ ϕ n m + ϕ s , m ) ) + n n ( t ) ,
H = [ h 11 h 12 ⋯ h 1 M h 21 h 22 ⋯ h 2 M ⋮ ⋮ ⋱ ⋮ h N 1 h N 2 … h N M ] ,
y = Hs + n ,
S N R = 2 T b R o e I ¯ s h v ,
s ^ = arg min s | | y − Hs | | 2 ,
f I ( I ) = 2 Γ ( α ) Γ ( β ) ( α β ) α + β 2 I α + β 2 − 1 K α − β ( 2 α β I ) ,
P ( s → s ′ ) = P { | | y − Hs | | > | | y − H s ′ | | } .
P ( s → s ′ ) = P { | | n | | > | | H ( s − s ′ ) + n | | } ,
P ( s → s ′ ) = P { n > 1 2 | | H ( s − s ′ ) | | } = P { n > 1 2 | | H Δ s | | } ,
P ( s → s ′ ) = P { n > 1 2 | ∑ 1 ≤ j ≤ k h 1 m j Δ s m j | } ,
F 1 ( r ) = C r r 2 ,
f 1 ( r ) = d F 1 ( r ) d r = 2 C r r .
C r = ∫ I 4 | Δ s m k | 2 f I ( 4 | X | 2 | Δ s m k | 2 ) f I ( I ) d I ,
C r = 1 π ∫ 0 + ∞ f I 2 ( I 11 ) d I 11 .
C r = Γ ( α + β − 1 ) Γ ( 2 β − 1 ) Γ ( 2 α − 1 ) Γ 2 ( α ) Γ 2 ( β ) Γ ( α + β − 1 / 2 ) ⋅ 2 3 − 2 α − 2 β π ⋅ α β .
F N ( r ) = P { 1 2 | | H Δ s | | < r } .
F N ( r ) = C r N N ! r 2 N ,
f N ( r ) = d F N ( r ) d r = 2 C r N ( N − 1 ) ! r 2 N − 1 .
F N ( r ) = ∫ 0 r P { | 1 2 h N Δ s | 2 < r 2 − r 0 2 } f N − 1 ( r 0 ) d r 0 = ∫ 0 r C r ( r 2 − r 0 2 ) ⋅ 2 C r N − 1 ( N − 2 ) ! r 2 N − 3 d r 0 = C r N N ! r 2 N ,
lim S N R → ∞ P ( s → s ′ ) = ∫ 0 r x Q ( t 2 ⋅ S N R ) ⋅ f N ( t ) d t .
lim S N R → ∞ ∫ r x + ∞ Q ( t 2 ⋅ S N R ) ⋅ f N ( t ) d t = 0 .
lim S N R → ∞ P ( s → s ′ ) = ∫ 0 + ∞ Q ( t 2 ⋅ S N R ) ⋅ f N ( t ) d t = ∫ 0 + ∞ 1 2 erfc ( x 2 ) ⋅ x 2 N − 1 ⋅ 2 C r N ( N − 1 ) ! ⋅ 1 ( 2 ⋅ S N R ) N d x = C r N 2 π ⋅ N ! ⋅ Γ ( N + 1 2 ) ⋅ 1 ( S N R ) N .
1 q N max s ≠ s ′ { σ s s ′ P ( s → s ′ ) } ≤ P e ≤ 1 q N ∑ s ≠ s ′ σ s s ′ P ( s → s ′ ) ,
{ lim S N R → ∞ log P e log S N R lim S N R → ∞ log ( 1 q N max s ≠ s ′ { σ s s ′ P ( s → s ′ ) } ) log S N R = − N lim S N R → ∞ log P e log S N R lim S N R → ∞ log ( 1 q N ∑ s ≠ s ′ σ s s ′ P ( s → s ′ ) ) log S N R = − N
d * = − lim S N R → ∞ log P e log S N R = N ,
lim S N R → ∞ P e = C x ⋅ S N R − N .
P ( 11 → 00 ) ≤ P e ≤ P ( 11 → 00 ) + 1 2 P ( 11 → 10 ) + 1 2 P ( 11 → 01 ) .
lim S N R → ∞ P e = C r N 2 π ⋅ N ! ⋅ Γ ( N + 1 2 ) ⋅ 1 ( S N R ) N ,
P r e c t ≜ ∫ I P { | arg ( h 1 m k Δ s m k ) | < θ 0 | I } P { | 1 2 | | h 1 m k Δ s m k | − | X | | < r | I } f I ( I ) d I ,
P { | arg ( h 1 m k Δ s m k ) | < θ 0 | I } = 2 r 2 π | X | ,
P { | 1 2 | h 1 m k Δ s m k | − | X | | < | I } = ∫ 2 ( | X | − r ) / | Δ s m k | 2 ( | X | + r ) / | Δ s m k | f h ( h 1 m k ) d h 1 m k = ∫ 2 ( | X | − r ) / | Δ s m k | 2 ( | X | + r ) / | Δ s m k | f I ( I ) d I = 16 | X | r | Δ s m k | 2 f I ( 4 | X | 2 | Δ s m k | 2 ) ,
P r e c t = ∫ I 2 r 2 π | X | ⋅ 16 | X | r | Δ s m k | 2 f I ( 4 | X | 2 | Δ s m k | 2 ) f I ( I ) d I = ∫ I 16 r 2 π | Δ s m k | 2 f I ( 4 | X | 2 | Δ s m k | 2 ) f I ( I ) d I .
F 1 ( r ) = P c i r c l e = π 4 P r e c t = C r r 2 ,
C r = ∫ I 4 | Δ s m k | 2 f I ( 4 | X | 2 | Δ s m k | 2 ) f I ( I ) d I .
f 1 ( r ) = d F 1 ( r ) d r = 2 C r r .

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