Abstract

In this paper, from an information theory point of view, we investigate the performance of a coherent free-space optical (FSO) communication system with multiple receive apertures over atmospheric turbulence channels. Our study builds on a recently introduced statistical model that characterizes the combined effects of turbulence-induced wavefront distortion and amplitude fluctuation in coherent receivers with phase compensation. We investigate the link reliability as quantified by “diversity gain” and the relationship between the link reliability and the spectral efficiency as quantified by “diversity–multiplexing trade-off (DMT).” Our results provide insight into the performance mechanisms of coherent FSO systems and demonstrate significant performance gains that can be obtained through the deployment of multiple receive apertures and phase compensation techniques.

© 2010 Optical Society of America

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  30. H. F. Inman, E. L. Bradley, “The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities,” Commun. Stat: Theory Meth., vol. 18, no. 10, pp. 3851–3874, 1989.
    [CrossRef]
  31. W. Y. Shin, S. Y. Chung, Y. H. Lee, “Diversity-multiplexing tradeoff and outage performance for Rician MIMO channels,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1186–1196, Mar. 2008.
    [CrossRef]

2009 (3)

2008 (5)

A. Belmonte, J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express, vol. 16, no. 18, pp. 14151–14162, Sept. 2008.
[CrossRef] [PubMed]

W. O. Popoola, Z. Ghassemlooy, J. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, May 2008.
[CrossRef]

M. L. B. Riediger, R. Schober, L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec. 2008.
[CrossRef]

W. Y. Shin, S. Y. Chung, Y. H. Lee, “Diversity-multiplexing tradeoff and outage performance for Rician MIMO channels,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1186–1196, Mar. 2008.
[CrossRef]

2007 (1)

S. M. Navidpour, M. Uysal, M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 2813–2819, Aug. 2007.
[CrossRef]

2006 (3)

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett., vol. 18, no. 14, pp. 1491–1493, July 2006.
[CrossRef]

V. W. S. Chan, “Free-space optical communication,” J. Lightwave Technol., vol. 24, no. 12, pp. 4750–4762, Dec. 2006.
[CrossRef]

R. Narasimhan, “Finite-SNR diversity multiplexing tradeoff for correlated Rayleigh and Ricean MIMO channels,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3965–3979, Sept. 2006.
[CrossRef]

2005 (1)

S. G. Wilson, M. Brandt-Pearce, Q. Cao, J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1402–1412, Aug. 2005.
[CrossRef]

2004 (2)

E. Lee, V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

D. Kedar, S. Arnon, “Urban optical wireless communication networks: the main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no. 5, pp. S2–S7, May 2004.
[CrossRef]

2003 (2)

S. M. Haas, J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Commun., vol. 21, no. 8, pp. 1346–1357, Oct. 2003.
[CrossRef]

L. Zheng, D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.
[CrossRef]

2002 (2)

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[CrossRef]

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 3, pp. 1–11, Mar. 2002.

2000 (1)

1998 (1)

E. Biglieri, J. Proakis, S. Shamai, “Fading channels: information theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
[CrossRef]

1989 (1)

H. F. Inman, E. L. Bradley, “The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities,” Commun. Stat: Theory Meth., vol. 18, no. 10, pp. 3851–3874, 1989.
[CrossRef]

1976 (1)

Allen, J.

W. O. Popoola, Z. Ghassemlooy, J. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Alouini, M.-S.

M. K. Simon, M.-S. Alouini, Digital Communication Over Fading Channels. Wiley, 2000.
[CrossRef]

Arnon, S.

D. Kedar, S. Arnon, “Urban optical wireless communication networks: the main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no. 5, pp. S2–S7, May 2004.
[CrossRef]

Belmonte, A.

Biglieri, E.

E. Biglieri, J. Proakis, S. Shamai, “Fading channels: information theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
[CrossRef]

Bowman, F.

F. Bowman, Introduction to Bessel Functions. Dover, New York, 1958.

Bradley, E. L.

H. F. Inman, E. L. Bradley, “The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities,” Commun. Stat: Theory Meth., vol. 18, no. 10, pp. 3851–3874, 1989.
[CrossRef]

Brandt-Pearce, M.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1402–1412, Aug. 2005.
[CrossRef]

Cagigal, M. P.

Canales, V. F.

Cao, Q.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1402–1412, Aug. 2005.
[CrossRef]

Chan, V.

E. Lee, V. Chan, “Diversity coherent and incoherent receivers for free-space optical communication in the presence and absence of interference,” J. Opt. Commun. Netw., vol. 1, no. 5, pp. 463–483, Oct. 2009.
[CrossRef]

E. Lee, V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

E. Lee, V. Chan, “Diversity coherent receivers for optical communication over the clear turbulent atmosphere,” in IEEE ICC, Glasgow, 2007, pp. 2485–2492.

Chan, V. W. S.

Chung, S. Y.

W. Y. Shin, S. Y. Chung, Y. H. Lee, “Diversity-multiplexing tradeoff and outage performance for Rician MIMO channels,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1186–1196, Mar. 2008.
[CrossRef]

Cowley, W.

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, May 2008.
[CrossRef]

Djordjevic, I. B.

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett., vol. 18, no. 14, pp. 1491–1493, July 2006.
[CrossRef]

Gagliardi, R. M.

R. M. Gagliardi, S. Karp, Optical Communications. Wiley, New York, 1995.

Gao, S.

W. O. Popoola, Z. Ghassemlooy, J. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Ghassemlooy, Z.

W. O. Popoola, Z. Ghassemlooy, J. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Goldsmith, A.

A. Goldsmith, Wireless Communications. Cambridge U. Press, New York, 2005.
[CrossRef]

Haas, S. M.

S. M. Haas, J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Commun., vol. 21, no. 8, pp. 1346–1357, Oct. 2003.
[CrossRef]

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 3, pp. 1–11, Mar. 2002.

Holland, I.

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, May 2008.
[CrossRef]

Inman, H. F.

H. F. Inman, E. L. Bradley, “The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities,” Commun. Stat: Theory Meth., vol. 18, no. 10, pp. 3851–3874, 1989.
[CrossRef]

Kahn, J. M.

Karagiannidis, G. K.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[CrossRef]

Karp, S.

R. M. Gagliardi, S. Karp, Optical Communications. Wiley, New York, 1995.

Kavehrad, M.

S. M. Navidpour, M. Uysal, M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 2813–2819, Aug. 2007.
[CrossRef]

Kedar, D.

D. Kedar, S. Arnon, “Urban optical wireless communication networks: the main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no. 5, pp. S2–S7, May 2004.
[CrossRef]

Lampe, L.

M. L. B. Riediger, R. Schober, L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec. 2008.
[CrossRef]

Lee, E.

E. Lee, V. Chan, “Diversity coherent and incoherent receivers for free-space optical communication in the presence and absence of interference,” J. Opt. Commun. Netw., vol. 1, no. 5, pp. 463–483, Oct. 2009.
[CrossRef]

E. Lee, V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

E. Lee, V. Chan, “Diversity coherent receivers for optical communication over the clear turbulent atmosphere,” in IEEE ICC, Glasgow, 2007, pp. 2485–2492.

Lee, Y. H.

W. Y. Shin, S. Y. Chung, Y. H. Lee, “Diversity-multiplexing tradeoff and outage performance for Rician MIMO channels,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1186–1196, Mar. 2008.
[CrossRef]

Leitgeb, E.

W. O. Popoola, Z. Ghassemlooy, J. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Letzepis, N.

N. Letzepis, I. Holland, W. Cowley, “The Gaussian free space optical MIMO channel with Q-ary pulse position modulation,” IEEE Trans. Wireless Commun., vol. 7, no. 5, pp. 1744–1753, May 2008.
[CrossRef]

Leveque, J. H.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1402–1412, Aug. 2005.
[CrossRef]

Narasimhan, R.

R. Narasimhan, “Finite-SNR diversity multiplexing tradeoff for correlated Rayleigh and Ricean MIMO channels,” IEEE Trans. Inf. Theory, vol. 52, no. 9, pp. 3965–3979, Sept. 2006.
[CrossRef]

Navidpour, S. M.

S. M. Navidpour, M. Uysal, M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 2813–2819, Aug. 2007.
[CrossRef]

Neifeld, M. A.

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett., vol. 18, no. 14, pp. 1491–1493, July 2006.
[CrossRef]

Noll, R. J.

Osche, G. R.

G. R. Osche, Optical Detection Theory for Laser Applications. Wiley, New York, 2002.

Popoola, W. O.

W. O. Popoola, Z. Ghassemlooy, J. Allen, E. Leitgeb, S. Gao, “Free-space optical communication employing subcarrier modulation and spatial diversity in atmospheric turbulence channel,” IET Optoelectron., vol. 2, no. 1, pp. 16–23, Feb. 2008.
[CrossRef]

Proakis, J.

E. Biglieri, J. Proakis, S. Shamai, “Fading channels: information theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
[CrossRef]

Proakis, J. G.

J. G. Proakis, Digital Communication, 4th ed.McGraw Hill, 2001.

Riediger, M. L. B.

M. L. B. Riediger, R. Schober, L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec. 2008.
[CrossRef]

Sandalidis, H. G.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[CrossRef]

Schober, R.

M. L. B. Riediger, R. Schober, L. Lampe, “Multiple-symbol detection for photon-counting MIMO free-space optical communications,” IEEE Trans. Wireless Commun., vol. 7, no. 12, pp. 5369–5379, Dec. 2008.
[CrossRef]

Shamai, S.

E. Biglieri, J. Proakis, S. Shamai, “Fading channels: information theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
[CrossRef]

Shapiro, J. H.

S. M. Haas, J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Commun., vol. 21, no. 8, pp. 1346–1357, Oct. 2003.
[CrossRef]

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 3, pp. 1–11, Mar. 2002.

Shin, W. Y.

W. Y. Shin, S. Y. Chung, Y. H. Lee, “Diversity-multiplexing tradeoff and outage performance for Rician MIMO channels,” IEEE Trans. Inf. Theory, vol. 54, no. 3, pp. 1186–1196, Mar. 2008.
[CrossRef]

Simon, M. K.

M. K. Simon, M.-S. Alouini, Digital Communication Over Fading Channels. Wiley, 2000.
[CrossRef]

Tarokh, V.

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 3, pp. 1–11, Mar. 2002.

Tse, D. N. C.

L. Zheng, D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.
[CrossRef]

Tsiftsis, T. A.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[CrossRef]

Tyson, R. K.

R. K. Tyson, Principles of Adaptive Optics. Academic, San Diego, 1991.

Uysal, M.

T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wireless Commun., vol. 8, no. 2, pp. 951–957, Feb. 2009.
[CrossRef]

S. M. Navidpour, M. Uysal, M. Kavehrad, “BER performance of free-space optical transmission with spatial diversity,” IEEE Trans. Wireless Commun., vol. 6, no. 8, pp. 2813–2819, Aug. 2007.
[CrossRef]

Vasic, B.

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett., vol. 18, no. 14, pp. 1491–1493, July 2006.
[CrossRef]

Wilson, S. G.

S. G. Wilson, M. Brandt-Pearce, Q. Cao, J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1402–1412, Aug. 2005.
[CrossRef]

Zheng, L.

L. Zheng, D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.
[CrossRef]

Zhu, X.

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[CrossRef]

Commun. Stat: Theory Meth. (1)

H. F. Inman, E. L. Bradley, “The overlapping coefficient as a measure of agreement between probability distributions and point estimation of the overlap of two normal densities,” Commun. Stat: Theory Meth., vol. 18, no. 10, pp. 3851–3874, 1989.
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

S. M. Haas, J. H. Shapiro, V. Tarokh, “Space-time codes for wireless optical communications,” EURASIP J. Appl. Signal Process., vol. 3, pp. 1–11, Mar. 2002.

IEEE Commun. Mag. (1)

D. Kedar, S. Arnon, “Urban optical wireless communication networks: the main challenges and possible solutions,” IEEE Commun. Mag., vol. 42, no. 5, pp. S2–S7, May 2004.
[CrossRef]

IEEE J. Sel. Areas Commun. (2)

S. M. Haas, J. H. Shapiro, “Capacity of wireless optical communications,” IEEE J. Sel. Areas Commun., vol. 21, no. 8, pp. 1346–1357, Oct. 2003.
[CrossRef]

E. Lee, V. Chan, “Part 1: Optical communication over the clear turbulent atmospheric channel using diversity,” IEEE J. Sel. Areas Commun., vol. 22, no. 9, pp. 1896–1906, Nov. 2004.
[CrossRef]

IEEE Photon. Technol. Lett. (1)

I. B. Djordjevic, B. Vasic, M. A. Neifeld, “Multilevel coding in free-space optical MIMO transmission with q-ary PPM over the atmospheric turbulence channel,” IEEE Photon. Technol. Lett., vol. 18, no. 14, pp. 1491–1493, July 2006.
[CrossRef]

IEEE Trans. Commun. (2)

X. Zhu, J. M. Kahn, “Free-space optical communication through atmospheric turbulence channels,” IEEE Trans. Commun., vol. 50, no. 8, pp. 1293–1300, Aug. 2002.
[CrossRef]

S. G. Wilson, M. Brandt-Pearce, Q. Cao, J. H. Leveque, “Free-space optical MIMO transmission with Q-ary PPM,” IEEE Trans. Commun., vol. 53, no. 8, pp. 1402–1412, Aug. 2005.
[CrossRef]

IEEE Trans. Inf. Theory (4)

E. Biglieri, J. Proakis, S. Shamai, “Fading channels: information theoretic and communications aspects,” IEEE Trans. Inf. Theory, vol. 44, no. 6, pp. 2619–2692, Oct. 1998.
[CrossRef]

L. Zheng, D. N. C. Tse, “Diversity and multiplexing: a fundamental tradeoff in multiple antenna channels,” IEEE Trans. Inf. Theory, vol. 49, no. 5, pp. 1073–1096, May 2003.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Block diagram of a coherent FSO system with multiple heterodyne receivers.

Fig. 2
Fig. 2

Simulated and Rician pdfs for different numbers of receive apertures.

Fig. 3
Fig. 3

Outage probability for different numbers of receive apertures.

Fig. 4
Fig. 4

Finite-SNR DMT for various values of SNR ( M = 2 ) .

Fig. 5
Fig. 5

Diversity gain (at a fixed transmission rate) for various numbers of receive apertures.

Fig. 6
Fig. 6

Effect of the modal compensation on the finite-SNR diversity gain.

Equations (29)

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E r k ( t , r ) = u s ( t ) e j ( 2 π f c t + θ s ( t ) ) e [ χ k ( r ) + j φ k ( r ) ] ,
σ φ 2 = Δ J ( D r 0 ) 2 ,
y k ( t ) = x k ( t ) + n k ( t ) ,
P n T = B s e 2 η π 2 h f c D k 2 A L 2 .
x T ( t ) = e η π 2 h f c D k 2 A L u s ( t ) Re { e j ( 2 π f IF t + θ s ( t ) θ L ) k = 1 M α k λ k * } ,
γ = P x T P n T = η I s B s h f c ( π 4 D 2 M ) | k = 1 M α k λ k * | 2 ,
λ k , opt = α k i = 1 M | α i | 2 .
γ = ρ M k = 1 M a k 2 ,
G = { 1.09 ( ρ 0 D k ) 2 Γ [ 1.2 , 1.08 ( D k ρ 0 ) 5 3 ] } 1 ,
f a k ( a k ) = 2 a k ( 1 + K ) a 2 ¯ e K ( 1 + K ) a k 2 a 2 ¯ I 0 ( 2 a k K ( 1 + K ) a 2 ¯ ) ,
K = [ a 2 ¯ α ¯ r 4 + 2 α ¯ r 2 ( σ i 2 σ r 2 ) ( σ i 2 σ r 2 ) 2 1 ] 1 .
d = lim ρ log P out ( R , ρ ) log ρ ,
r = lim ρ R ( ρ ) log ρ .
d f ( R , ρ ) = log P out ( R , ρ ) log ρ .
r f = R ( ρ ) log ( 1 + ρ ) .
d f ( r f , ρ ) = ρ P out ( r f , ρ ) P out ( r f , ρ ) ρ .
P out ( R ) = Pr { I ( γ ) < R } ,
P out ( R ) = Pr { γ < γ R } ,
P out ( R ) = Pr { y < M γ R ρ } ,
f Y ( y ) = ( 1 + K ) a 2 ¯ ( ( 1 + K ) y M K a 2 ¯ ) ( M 1 ) 2 × e K M ( 1 + K ) y a 2 ¯ I M 1 ( 2 K ( 1 + K ) M y a 2 ¯ ) ,
P out ( R , ρ ) = 0 M γ R ρ f y ( y ) d y = 1 Q M ( 2 K M , 2 ( 1 + K ) M γ R ρ a 2 ¯ ) ,
γ R = 2 R 1 = ( 1 + ρ ) r f 1 .
d f ( r f , ρ ) = ρ 1 Q M ( ζ , β ) Q M ( ζ , β ) β β ρ ,
β = 2 ( 1 + K ) M [ ( 1 + ρ ) r f 1 ] ρ a 2 ¯ .
d f ( r f , ρ ) = β M + 1 e ( β 2 + ζ 2 ) 2 I M 1 ( ζ β ) 2 ζ M 1 [ 1 Q M ( ζ , β ) ] ( 1 r f ρ ( 1 + ρ ) r f 1 ( 1 + ρ ) r f 1 ) .
d f ( r f , ρ ) = k = 0 1 k ! ( M + k 1 ) ! ( ζ β 2 ) 2 k j = 0 ( β 2 2 ) j [ k = 0 1 k ! ( j + k + M ) ! ( ζ β 2 ) 2 k ] × ( 1 r f ρ ( 1 + ρ ) r f 1 ( 1 + ρ ) r f 1 ) .
lim ρ d ( r f , ρ ) = M ( 1 r ) .
d f ( R , ρ ) = β M + 1 e ( β 2 + ζ 2 ) 2 I M 1 ( ζ β ) 2 ζ M 1 [ 1 Q M ( ζ , β ) ] .
lim ρ d f ( R , ρ ) = M ,