Abstract

Multi-input multi-output (MIMO) technique is attractive for visible light communication (VLC), which exploits the high signal-to-noise ratio (SNR) of a single channel to overcome the capacity limitation due to the small modulation bandwidth of the light emitting diode. This paper establishes a MIMO VLC system under the non-negativity, peak power and dimmable average power constraints. Assume that perfect channel state information at the transmitter is known, the MIMO channel is changed to parallel, non-interfering sub-channels by using the singular value decomposition (SVD). Based on the SVD, the lower bound on the channel capacity for MIMO VLC is derived by employing entropy power inequality and variational method. Moreover, by maximizing the derived lower bound on the capacity under the given constraints, the receiver deployment optimization problem is formulated. The problem is solved by employing the principle of particle swarm optimization. Numerical results verify the derived capacity bound and the proposed deployment optimization scheme.

© 2016 Optical Society of America

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References

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  1. A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
    [Crossref]
  2. L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
    [Crossref]
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    [Crossref]
  4. K.-I. Ahn and J. K. Kwon, “Capacity analysis of M-PAM inverse source coding in visible light communication,” J. Lightwave Technol. 30(10), 1399–1404 (2012).
    [Crossref]
  5. J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, and J.-Y. Wang, “Tight bound on channel capacity for dimmable visible light communications,” J. Lightwave Technol. 31(23), 3771–3779 (2013).
    [Crossref]
  6. J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).
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  8. D. Ding, X. Ke, and L. Xu, “An optimal lights layout scheme for visible-light communication system,” in 8th Int. Conf. Electron. Measurement and Instruments, Xi’an, China, 189–194 (2007).
  9. Z. Wang, C. Yu, W.-D. Zhong, J. Chen, and W. Chen, “Performance of a novel LED lamp arrangement to reduce SNR fluctuation for multi-user visible light communication systems,” Opt. Express 20(4), 4564–4573 (2012).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  13. E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
    [Crossref]
  14. S. Verdu and D. Guo, “A simple proof of the entropy-power inequality,” IEEE Trans. Inf. Theory 52(5), 2165–2166 (2006).
    [Crossref]
  15. M. reznikov, Variational Method, (Lambert Academic Publishing, 2012).
  16. J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in IEEE Int. Conf. Neural Netw. (1995), pp. 1942–1948.
  17. A. E. Forooshani, A. A. Lot-Neyestanak, and D. G. Michelson, “Optimization of antenna placement in distributed MIMO systems for underground mines,” IEEE Trans. Wirel. Commun. 13(9), 4685–4692 (2014).
    [Crossref]

2014 (2)

J.-Y. Wang, J.-B. Wang, X. Song, M. Chen, and J. Zhang, “Network planning for distributed antenna-basedhigh-speed railway mobile communications,” Trans. Emerging Telecommun. Technol. 25(7), 702–722 (2014).

A. E. Forooshani, A. A. Lot-Neyestanak, and D. G. Michelson, “Optimization of antenna placement in distributed MIMO systems for underground mines,” IEEE Trans. Wirel. Commun. 13(9), 4685–4692 (2014).
[Crossref]

2013 (2)

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, and J.-Y. Wang, “Tight bound on channel capacity for dimmable visible light communications,” J. Lightwave Technol. 31(23), 3771–3779 (2013).
[Crossref]

2012 (4)

2009 (1)

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

2006 (1)

S. Verdu and D. Guo, “A simple proof of the entropy-power inequality,” IEEE Trans. Inf. Theory 52(5), 2165–2166 (2006).
[Crossref]

2004 (1)

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consumer Electron.,  50(1), 100–107 (2004).
[Crossref]

Ahn, K.-I.

Biglieri, E.

E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
[Crossref]

Calderbank, R.

E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
[Crossref]

Chen, J.

Chen, M

J.-Y. Wang, J.-B. Wang, M Chen, and J. Wang, “Capacity bounds for dimmable visible light communications using PIN photodiodes with input-dependent Gaussian noise,” in IEEE Global Commun. Conf. (2014), pp. 2107–2112.

Chen, M.

J.-Y. Wang, J.-B. Wang, X. Song, M. Chen, and J. Zhang, “Network planning for distributed antenna-basedhigh-speed railway mobile communications,” Trans. Emerging Telecommun. Technol. 25(7), 702–722 (2014).

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, and J.-Y. Wang, “Tight bound on channel capacity for dimmable visible light communications,” J. Lightwave Technol. 31(23), 3771–3779 (2013).
[Crossref]

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).

Chen, W.

Choi, S.-I.

Constantinides, A.

E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
[Crossref]

Ding, D.

D. Ding, X. Ke, and L. Xu, “An optimal lights layout scheme for visible-light communication system,” in 8th Int. Conf. Electron. Measurement and Instruments, Xi’an, China, 189–194 (2007).

Eberhart, R. C.

J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in IEEE Int. Conf. Neural Netw. (1995), pp. 1942–1948.

Faulkner, G.

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Forooshani, A. E.

A. E. Forooshani, A. A. Lot-Neyestanak, and D. G. Michelson, “Optimization of antenna placement in distributed MIMO systems for underground mines,” IEEE Trans. Wirel. Commun. 13(9), 4685–4692 (2014).
[Crossref]

Goldsmith, A.

E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
[Crossref]

Guo, D.

S. Verdu and D. Guo, “A simple proof of the entropy-power inequality,” IEEE Trans. Inf. Theory 52(5), 2165–2166 (2006).
[Crossref]

Gyongyosi, L.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Haas, H.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Hanzo, L.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Hu, Q.-S.

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, and J.-Y. Wang, “Tight bound on channel capacity for dimmable visible light communications,” J. Lightwave Technol. 31(23), 3771–3779 (2013).
[Crossref]

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).

Huang, Y.-H.

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).

Imre, S.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Jovicic, A.

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

Jung, D.

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Ke, X.

D. Ding, X. Ke, and L. Xu, “An optimal lights layout scheme for visible-light communication system,” in 8th Int. Conf. Electron. Measurement and Instruments, Xi’an, China, 189–194 (2007).

Kennedy, J.

J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in IEEE Int. Conf. Neural Netw. (1995), pp. 1942–1948.

Komine, T.

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consumer Electron.,  50(1), 100–107 (2004).
[Crossref]

Kwon, J. K.

Lee, K.

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Li, J.

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

Lot-Neyestanak, A. A.

A. E. Forooshani, A. A. Lot-Neyestanak, and D. G. Michelson, “Optimization of antenna placement in distributed MIMO systems for underground mines,” IEEE Trans. Wirel. Commun. 13(9), 4685–4692 (2014).
[Crossref]

Michelson, D. G.

A. E. Forooshani, A. A. Lot-Neyestanak, and D. G. Michelson, “Optimization of antenna placement in distributed MIMO systems for underground mines,” IEEE Trans. Wirel. Commun. 13(9), 4685–4692 (2014).
[Crossref]

Minh, H.

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Nakagawa, M.

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consumer Electron.,  50(1), 100–107 (2004).
[Crossref]

O’Brien, D.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Oh, Y.

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Paulraj, A.

E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
[Crossref]

Poor, H. V.

E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
[Crossref]

reznikov, M.

M. reznikov, Variational Method, (Lambert Academic Publishing, 2012).

Richardson, T.

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

Rupp, M.

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Song, X.

J.-Y. Wang, J.-B. Wang, X. Song, M. Chen, and J. Zhang, “Network planning for distributed antenna-basedhigh-speed railway mobile communications,” Trans. Emerging Telecommun. Technol. 25(7), 702–722 (2014).

Verdu, S.

S. Verdu and D. Guo, “A simple proof of the entropy-power inequality,” IEEE Trans. Inf. Theory 52(5), 2165–2166 (2006).
[Crossref]

Wang, J.

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, and J.-Y. Wang, “Tight bound on channel capacity for dimmable visible light communications,” J. Lightwave Technol. 31(23), 3771–3779 (2013).
[Crossref]

J.-Y. Wang, J.-B. Wang, M Chen, and J. Wang, “Capacity bounds for dimmable visible light communications using PIN photodiodes with input-dependent Gaussian noise,” in IEEE Global Commun. Conf. (2014), pp. 2107–2112.

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).

Wang, J.-B.

J.-Y. Wang, J.-B. Wang, X. Song, M. Chen, and J. Zhang, “Network planning for distributed antenna-basedhigh-speed railway mobile communications,” Trans. Emerging Telecommun. Technol. 25(7), 702–722 (2014).

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, and J.-Y. Wang, “Tight bound on channel capacity for dimmable visible light communications,” J. Lightwave Technol. 31(23), 3771–3779 (2013).
[Crossref]

J.-Y. Wang, J.-B. Wang, M Chen, and J. Wang, “Capacity bounds for dimmable visible light communications using PIN photodiodes with input-dependent Gaussian noise,” in IEEE Global Commun. Conf. (2014), pp. 2107–2112.

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).

Wang, J.-Y.

J.-Y. Wang, J.-B. Wang, X. Song, M. Chen, and J. Zhang, “Network planning for distributed antenna-basedhigh-speed railway mobile communications,” Trans. Emerging Telecommun. Technol. 25(7), 702–722 (2014).

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, and J.-Y. Wang, “Tight bound on channel capacity for dimmable visible light communications,” J. Lightwave Technol. 31(23), 3771–3779 (2013).
[Crossref]

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).

J.-Y. Wang, J.-B. Wang, M Chen, and J. Wang, “Capacity bounds for dimmable visible light communications using PIN photodiodes with input-dependent Gaussian noise,” in IEEE Global Commun. Conf. (2014), pp. 2107–2112.

Wang, Z.

Won, E. T.

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Xu, L.

D. Ding, X. Ke, and L. Xu, “An optimal lights layout scheme for visible-light communication system,” in 8th Int. Conf. Electron. Measurement and Instruments, Xi’an, China, 189–194 (2007).

Yu, C.

Zeng, L.

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

Zhang, J.

J.-Y. Wang, J.-B. Wang, X. Song, M. Chen, and J. Zhang, “Network planning for distributed antenna-basedhigh-speed railway mobile communications,” Trans. Emerging Telecommun. Technol. 25(7), 702–722 (2014).

Zhong, W.-D.

IEEE Commun. Mag. (1)

A. Jovicic, J. Li, and T. Richardson, “Visible light communication: opportunities, challenges and the path to market,” IEEE Commun. Mag. 51(12), 26–32 (2013).
[Crossref]

IEEE J. Sel. Areas Commun. (1)

L. Zeng, D. O’Brien, H. Minh, G. Faulkner, K. Lee, D. Jung, Y. Oh, and E. T. Won, “High data rate multiple input multiple output (MIMO) optical wireless communications using white LED lighting,” IEEE J. Sel. Areas Commun. 27(9), 1654–1662 (2009).
[Crossref]

IEEE Trans. Consumer Electron (1)

T. Komine and M. Nakagawa, “Fundamental analysis for visible-light communication system using LED lights,” IEEE Trans. Consumer Electron.,  50(1), 100–107 (2004).
[Crossref]

IEEE Trans. Inf. Theory (1)

S. Verdu and D. Guo, “A simple proof of the entropy-power inequality,” IEEE Trans. Inf. Theory 52(5), 2165–2166 (2006).
[Crossref]

IEEE Trans. Wirel. Commun. (1)

A. E. Forooshani, A. A. Lot-Neyestanak, and D. G. Michelson, “Optimization of antenna placement in distributed MIMO systems for underground mines,” IEEE Trans. Wirel. Commun. 13(9), 4685–4692 (2014).
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Soc. Korea (1)

Opt. Express (1)

Proc. IEEE (1)

L. Hanzo, H. Haas, S. Imre, D. O’Brien, M. Rupp, and L. Gyongyosi, “Wireless myths, realities, and futures: from 3G/4G to optical and quantum wireless,” Proc. IEEE 100, 1853–1888 (2012).
[Crossref]

Trans. Emerging Telecommun. Technol. (1)

J.-Y. Wang, J.-B. Wang, X. Song, M. Chen, and J. Zhang, “Network planning for distributed antenna-basedhigh-speed railway mobile communications,” Trans. Emerging Telecommun. Technol. 25(7), 702–722 (2014).

Other (6)

J.-B. Wang, Q.-S. Hu, J. Wang, M. Chen, Y.-H. Huang, and J.-Y. Wang, “Capacity analysis for dimmable visible light communications,” in IEEE Int. Conf. Commun. (2014).

J.-Y. Wang, J.-B. Wang, M Chen, and J. Wang, “Capacity bounds for dimmable visible light communications using PIN photodiodes with input-dependent Gaussian noise,” in IEEE Global Commun. Conf. (2014), pp. 2107–2112.

D. Ding, X. Ke, and L. Xu, “An optimal lights layout scheme for visible-light communication system,” in 8th Int. Conf. Electron. Measurement and Instruments, Xi’an, China, 189–194 (2007).

M. reznikov, Variational Method, (Lambert Academic Publishing, 2012).

J. Kennedy and R. C. Eberhart, “Particle swarm optimization,” in IEEE Int. Conf. Neural Netw. (1995), pp. 1942–1948.

E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, and H. V. Poor, MIMO Wireless Communications(Cambridge University Press, 2007).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 An indoor MIMO VLC system
Fig. 2
Fig. 2 The distance and angle relationship between the j-th LED and the i-th PD
Fig. 3
Fig. 3 MIMO VLC channel decomposition
Fig. 4
Fig. 4 Lower capacity bound versus peak power of the LED A for different numbers of the receivers when A = 2P and ξ = 0.3
Fig. 5
Fig. 5 Lower capacity bound versus dimming target ξ for different APPRs when A = 60 dB and NR = 3
Fig. 6
Fig. 6 Lower capacity bound versus the iteration number
Fig. 7
Fig. 7 The distribution of the receivers when NR=1, 2, 3 and 4
Fig. 8
Fig. 8 The distribution of the receivers when NR=15, 20, 15 and 30

Tables (3)

Tables Icon

Algorithm 1. PSO-based iterative searching algorithm

Tables Icon

Table 1 Main simulation parameters

Tables Icon

Table 2 Simulation parameters for the PSO algorithm

Equations (54)

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

Y = r GX + Z
G = [ h 1 , 1 h 1 , 2 h 1 , N T h 2 , 1 h 2 , 2 h 2 , N T h N R , 1 h N R , 2 h N R , N T ]
Y = GX + Z
h i , j = { ( m + 1 ) A r 2 π D i , j 2 T s g cos m ( φ i , j ) cos ( ψ i , j ) , if 0 ψ i , j Ψ c , i 0 , if ψ i , j Ψ c , i
cos ( φ i , j ) = cos ( ψ i , j ) = c c R ( x j T x i R ) 2 + ( y j T y i R ) 2 + ( c c R ) 2
h i , j = ( m + 1 ) A r T s g ( c c R ) m + 1 2 π [ ( x j T x i R ) 2 + ( x j T x j R ) 2 + ( c c R ) 2 ] m + 3 2
0 N T × 1 X A I N T × 1
E ( X ) = ξ P I N T × 1
G = UD V H
Y ˜ = U H Y = D X ˜ + Z ˜
X ^ = D V H X = ( λ 1 j = 1 N T v j , 1 X j , , λ Γ j = 1 N T v j , Γ X j , 0 , , 0 ) T
Y ˜ = X ^ + Z ˜
Y ˜ n = X ^ n + Z ˜ n
X ^ n = λ n j = 1 N T v j , n X j , n = 1 , 2 , , Γ
A min { 0 , v j , n } v j , n X j A max { 0 , v j , n }
A ^ n X ^ n B ^ n , n = 1 , 2 , , Γ
E ( X ^ n ) = ξ P λ n j = 1 N T v j , n
I ( X ^ n ; Y ˜ n ) = H ( Y ˜ n ) H ( Y ˜ n | X ^ n ) = H ( X ^ n + Z ˜ n ) H ( Z ˜ n )
e 2 H ( X ^ n + Z ˜ n ) e 2 H ( X ^ n ) + e 2 H ( Z ˜ n )
I ( X ^ n ; Y ˜ n ) 1 2 ln ( e 2 H ( X ^ n ) + e 2 H ( Z ˜ n ) ) H ( Z ˜ n ) = 1 2 ln ( 1 + exp { 2 A ^ n B ^ n f X ^ n ( x ) ln [ f X ^ n ( x ) ] d x } 2 π e σ 2 )
J [ f X ^ n ( x ) ] = A ^ n B ^ n f X ^ n ( x ) ln [ f X ^ n ( x ) ] d x
s . t . min f X ^ n ( x ) J [ f X ^ n ( x ) ] A n B ^ n f X ^ n ( x ) d x = 1 A n B ^ n x f X ^ n ( x ) d x = ξ P λ n j = 1 N T v j , n
f X ^ n ( x ) = 1 B ^ n A ^ n , A ^ n x B ^ n
f X ^ n ( x ) = c 0 e c 0 B ^ n e c 0 A ^ n e c 0 x , A ^ n x B ^ n
α n = B ^ n e c 0 B ^ n A ^ n e c 0 A ^ n ( e c 0 B ^ n e c 0 A ^ n ) ( A ^ n + B ^ n ) 1 c 0 ( A ^ n + B ^ n )
C n 1 2 ln [ 1 + ( λ n A j = 1 N T | v j , n | ) 2 2 π e σ 2 ]
C n 1 2 ln { 1 + [ exp ( c 0 ξ P λ n j = 1 N T v j , n ) c 0 2 π e σ 2 × ( exp ( c 0 λ n A j = 1 N T max { 0 , v j , n } ) exp ( c 0 λ n A j = 1 N T min { 0 , v j , n } ) ) ] 2 }
C MIMO = n = 1 Γ C n
C MIMO C Low = 1 2 n = 1 Γ ln [ 1 + ( λ n A j = 1 N T | v j , n | 2 π e σ 2 ) 2 ]
C MIMO C Low = 1 2 n = 1 Γ ln { 1 + [ 1 + exp ( c 0 P λ n j = 1 N T v j , n ) c 0 2 π e σ 2 × ( exp ( c 0 λ n A j = 1 N T max { 0 , v j , n } ) exp ( c 0 λ n A j = 1 N T min { 0 , v j , n } ) ) ] 2 }
s . t . max W C Low 0 x i R a , i = 1 , 2 , , N R 0 y i R b , i = 1 , 2 , , N R
V k ( t + 1 ) = ω t V k ( t ) + c 1 φ [ Q j L W k ( t ) ] + c 2 ψ [ Q * W k ( t ) ]
W k ( t + 1 ) = W k ( t ) + V k ( t + 1 )
ω t = ω max ω max ω min T max × t
f k fit ( t ) = C Low ( W k ( t ) )
f ˜ X ^ n ( x ) = f X ^ n ( x ) + ε η ( x )
A ^ n B ^ n η ( x ) d x = 0 A ^ n B ^ n x η ( x ) d x = 0
ρ ( ε ) = J [ f X ^ n ( x ) + ε η ( x ) ]
d ρ ( ε ) d ε | ε = 0 = A ^ n B ^ n η ( x ) { ln [ f X ^ n ( x ) ] + 1 } d x = 0
ln [ f X ^ n ( x ) ] + 1 = c 0
ln [ f X ^ n ( x ) ] + 1 = c 0 x
f X ^ n ( x ) = e c 0 1 , A ^ n x B ^ n
f X ^ n ( x ) = e c 0 x 1 , A ^ n x B ^ n
e c 0 1 = 1 B ^ n A ^ n
ξ P = A 2
e 1 = c 0 e c 0 B ^ n e c 0 A ^ n
B ^ n e c 0 B ^ n A ^ n e c 0 A ^ n e c 0 B ^ n e c 0 A ^ n 1 c 0 = ξ P λ n j = 1 N T v j , n
A ^ n + B ^ n = λ n A j = 1 N T v j , n
H ( X ^ n ) = ln ( B ^ n A ^ n )
C n 1 2 ln ( 1 + ( B ^ n A ^ n ) 2 2 π e σ 2 )
B ^ n A ^ n = λ n A j = 1 N T | v j , n |
H ( X ^ n ) = ln ( e c 0 B ^ n e c 0 A ^ n c 0 ) c 0 ( B ^ n e c 0 B ^ n A ^ n e c 0 A ^ n e c 0 B ^ n e c 0 A ^ n 1 c 0 )
H ( X ^ n ) = ln [ exp ( c 0 ξ P λ n j = 1 N T v j , n ) ( e c 0 B ^ n e c 0 A ^ n ) c 0 ]
H ( X ^ n ) = ln [ ( exp ( c 0 ξ P λ n j = 1 N T v j , n ) × ( exp ( c 0 λ n A j = 1 N T max { 0 , v j , n } ) ( c 0 λ n A j = 1 N T max { 0 , v j , n } ) ) c 0

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