Abstract

Exact error rate performances are studied for coherent free-space optical communication systems under strong turbulence with diversity reception. Equal gain and selection diversity are considered as practical schemes to mitigate turbulence. The exact bit-error rate for binary phase-shift keying and outage probability are developed for equal gain diversity. Analytical expressions are obtained for the bit-error rate of differential phase-shift keying and asynchronous frequency-shift keying, as well as for outage probability using selection diversity. Furthermore, we provide the closed-form expressions of diversity order and coding gain with both diversity receptions. The analytical results are verified by computer simulations and are suitable for rapid error rates calculation.

© 2010 Optical Society of America

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References

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  1. . M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. 143,369–372 (1996).
    [CrossRef]
  2. . T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. 8,951–957 (2009).
    [CrossRef]
  3. . E. Lee and V. Chan, “Diversity coherent receivers for optical communication over the clear turbulent atmosphere,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 2007).
  4. . K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54,604–607 (2006).
    [CrossRef]
  5. . T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. 44,373–375 (2008).
    [CrossRef]
  6. . H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27,4440–4445 (2009).
    [CrossRef]
  7. . A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16,14151–14162 (2008).
    [CrossRef] [PubMed]
  8. . A. Belmonte and J. M. Kahn, “Capacity of coherent free-space optical links using diversity combining techniques,” Opt. Express 17,12601–12611 (2009).
    [CrossRef] [PubMed]
  9. . M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, “Performance analysis of coherent free-space optical communication systems with K-distributed turbulence,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 2009).
  10. . R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71,1440–1445 (1981).
    [CrossRef]
  11. . M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008).
    [CrossRef]
  12. . G. P. Agrawal, Fiber-Optical Communication Systems, (New York: Wiley, 2002).
    [CrossRef]
  13. . L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999).
    [CrossRef]
  14. . E. Jakeman and R. J. A. Tough, “Generalized K distribution: a statistical model for weak scattering,” J. Opt. Soc. Am. A 4,1764–1772 (1987).
    [CrossRef]
  15. . G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. 55,1453–1458 (2007).
    [CrossRef]
  16. . A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. 47,1335–1344 (1999).
    [CrossRef]
  17. . I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (San Diego: Academic Press, 2000).
  18. . Q. T. Zhang, “Outage probability in cellular mobile ratio due to nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol. 45,364–372 (1996).
    [CrossRef]
  19. . Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51,1389–1398 (2003).
    [CrossRef]

2009 (3)

2008 (3)

. A. Belmonte and J. M. Kahn, “Performance of synchronous optical receivers using atmospheric compensation techniques,” Opt. Express 16,14151–14162 (2008).
[CrossRef] [PubMed]

. T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. 44,373–375 (2008).
[CrossRef]

. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008).
[CrossRef]

2007 (1)

. G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. 55,1453–1458 (2007).
[CrossRef]

2006 (1)

. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54,604–607 (2006).
[CrossRef]

2003 (1)

. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51,1389–1398 (2003).
[CrossRef]

1999 (2)

. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. 47,1335–1344 (1999).
[CrossRef]

. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999).
[CrossRef]

1996 (2)

. Q. T. Zhang, “Outage probability in cellular mobile ratio due to nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol. 45,364–372 (1996).
[CrossRef]

. M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. 143,369–372 (1996).
[CrossRef]

1987 (1)

1981 (1)

Al-Habash, M. A.

. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999).
[CrossRef]

Andrews, L. C.

. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999).
[CrossRef]

. R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71,1440–1445 (1981).
[CrossRef]

Annamalai, A.

. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. 47,1335–1344 (1999).
[CrossRef]

Belmonte, A.

Bhargava, V. K.

. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. 47,1335–1344 (1999).
[CrossRef]

Edwards, D. J.

. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008).
[CrossRef]

Giannakis, G. B.

. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51,1389–1398 (2003).
[CrossRef]

Hopen, C. Y.

. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999).
[CrossRef]

Ibrahim, A. M.

. M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. 143,369–372 (1996).
[CrossRef]

Ibrahim, M. M.

. M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. 143,369–372 (1996).
[CrossRef]

Jafar, M.

. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008).
[CrossRef]

Jakeman, E.

Kahn, J. M.

Karagiannidis, G. K.

. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. 8,951–957 (2009).
[CrossRef]

. H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27,4440–4445 (2009).
[CrossRef]

. G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. 55,1453–1458 (2007).
[CrossRef]

Kiasaleh, K.

. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54,604–607 (2006).
[CrossRef]

Mathiopoulos, P. T.

. G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. 55,1453–1458 (2007).
[CrossRef]

O’Brien, D. C.

. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008).
[CrossRef]

Phillips, R. L.

. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999).
[CrossRef]

. R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71,1440–1445 (1981).
[CrossRef]

Sagias, N. C.

. G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. 55,1453–1458 (2007).
[CrossRef]

Sandalidis, H. G.

. H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27,4440–4445 (2009).
[CrossRef]

. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. 8,951–957 (2009).
[CrossRef]

Stevens, C. J.

. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008).
[CrossRef]

Tellambura, C.

. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. 47,1335–1344 (1999).
[CrossRef]

Tough, R. J. A.

Tsiftsis, T. A.

. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. 8,951–957 (2009).
[CrossRef]

. H. G. Sandalidis, T. A. Tsiftsis, and G. K. Karagiannidis, “Optical wireless communications with heterodyne detection over turbulence channels with pointing errors,” J. Lightwave Technol. 27,4440–4445 (2009).
[CrossRef]

. T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. 44,373–375 (2008).
[CrossRef]

Uysal, M.

. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. 8,951–957 (2009).
[CrossRef]

Wang, Z.

. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51,1389–1398 (2003).
[CrossRef]

Zhang, Q. T.

. Q. T. Zhang, “Outage probability in cellular mobile ratio due to nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol. 45,364–372 (1996).
[CrossRef]

Electron. Lett. (1)

. T. A. Tsiftsis, “Performance of heterodyne wireless optical communication systems over Gamma-Gamma atmospheric turbulence channels,” Electron. Lett. 44,373–375 (2008).
[CrossRef]

IEE Proc. Commun. (1)

. M. M. Ibrahim and A. M. Ibrahim, “Performance analysis of optical receivers with space diversity reception,” IEE Proc. Commun. 143,369–372 (1996).
[CrossRef]

IEEE Trans. Commun. (4)

. K. Kiasaleh, “Performance of coherent DPSK free-space optical communication systems in K-distributed turbulence,” IEEE Trans. Commun. 54,604–607 (2006).
[CrossRef]

. G. K. Karagiannidis, N. C. Sagias, and P. T. Mathiopoulos, “N*Nakagami: A Novel Stochastic Model for Cascaded Fading Channels,” IEEE Trans. Commun. 55,1453–1458 (2007).
[CrossRef]

. A. Annamalai, C. Tellambura, and V. K. Bhargava, “Exact evaluation of maximal-ratio and equal-gain diversity receivers for M-ary QAM on Nakagami fading channels,” IEEE Trans. Commun. 47,1335–1344 (1999).
[CrossRef]

. Z. Wang and G. B. Giannakis, “A simple and general parameterization quantifying performance in fading channels,” IEEE Trans. Commun. 51,1389–1398 (2003).
[CrossRef]

IEEE Trans. Veh. Technol. (1)

. Q. T. Zhang, “Outage probability in cellular mobile ratio due to nakagami signal and interferers with arbitrary parameters,” IEEE Trans. Veh. Technol. 45,364–372 (1996).
[CrossRef]

IEEE Trans. Wirel. Commun. (1)

. T. A. Tsiftsis, H. G. Sandalidis, G. K. Karagiannidis, and M. Uysal, “Optical wireless links with spatial diversity over strong atmospheric turbulence channels,” IEEE Trans. Wirel. Commun. 8,951–957 (2009).
[CrossRef]

IET Commun. (1)

. M. Jafar, D. C. O’Brien, C. J. Stevens, and D. J. Edwards, “Evaluation of coverage area for a wide line-of-sight indoor optical free-space communication system empolying coherent detection,” IET Commun. 2,18–26 (2008).
[CrossRef]

J. Lightwave Technol. (1)

J. Opt. Soc. Am. (2)

. R. L. Phillips and L. C. Andrews, “Measured statistics for laser light scattering in atmospheric turbulence,” J. Opt. Soc. Am. 71,1440–1445 (1981).
[CrossRef]

. L. C. Andrews, R. L. Phillips, C. Y. Hopen, and M. A. Al-Habash, “Theory of optical scintillation,” J. Opt. Soc. Am. 16,1417–1429 (1999).
[CrossRef]

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

Opt. Express (2)

Other (4)

. I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and Products, (San Diego: Academic Press, 2000).

. G. P. Agrawal, Fiber-Optical Communication Systems, (New York: Wiley, 2002).
[CrossRef]

. E. Lee and V. Chan, “Diversity coherent receivers for optical communication over the clear turbulent atmosphere,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 2007).

. M. Niu, J. Cheng, J. F. Holzman, and L. McPhail, “Performance analysis of coherent free-space optical communication systems with K-distributed turbulence,” in Proceedings of IEEE International Conference on Communications (Institute of Electrical and Electronics Engineers, New York, 2009).

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Figures (6)

Fig. 1.
Fig. 1.

The exact BERs and asymptotic BERs of BPSK for coherent EGC diversity operating on L-branch K-distributed turbulence channel with α = 1.8.

Fig. 2.
Fig. 2.

Exact BERs of BPSK for coherent MRC and EGC diversity operating on L-branch K-distributed turbulence channel with α = 1.8.

Fig. 3.
Fig. 3.

Outage probabilities of coherent MRC, EGC and SD operating on L-branch K-distributed turbulence channel with α = 1.8.

Fig. 4.
Fig. 4.

The exact BERs and asymptotic BERs of asynchronous DPSK for SD operating on L-branch K-distributed turbulence channel with α = 1.8.

Fig. 5.
Fig. 5.

The exact BERs and asymptotic BERs of asynchronous FSK for SD operating on L-branch K-distributed turbulence channel with α = 1.8.

Fig. 6.
Fig. 6.

The impact of scintillation index σ2si on BPSK BERs for MRC and EGC diversity operating with three diversity branches on K-distributed turbulence channels.

Equations (39)

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i l ( t ) = i dc , l + i ac , l ( t ) + n l ( t )
γ EGC = R ( Σ l = 1 L P s , l ) 2 Lq Δ f = 2 RP LO ( Σ l = 1 L I N , l ) 2 L 2 E b N 0
γ EGC = RA ( Σ l = 1 L I s , l ) 2 Lq Δ f = g 2 ( Σ l = 1 L I s , l ) 2
γ SD = max { γ 1 , γ 2 , , γ L } = R q Δ f max { P s , l ; l = 1 , , L } .
f ( I s ) = 2 Γ ( α ) η α + 1 α α + 1 2 I s α 1 2 K α 1 ( 2 η α I s ) , I s > 0
σ si 2 E [ I s 2 ] ( E [ I s ] ) 2 1
α = 2 σ si 2 1 .
σ R 2 = 1.23 C n 2 k 7 6 d 11 6
f ( z ) = 4 Γ ( α ) η α + 1 α α + 1 2 z α K α 1 ( 2 α η z ) , z > 0
f ( x ) = 2 η 2 xe x 2 η 2 , x > 0
f ( y ) = 2 α α y 2 α 1 e α y 2 Γ ( α ) , y > 0 .
ϕ z ( ω ) = { ϕ z ( ω ) } + j { ϕ z ( ω ) }
{ ϕ z ( ω ) } = 2 F 1 ( α , 1 , 1 2 ; ω 2 η 2 4 α )
{ ϕ z ( ω ) } = ω Γ ( α + 1 2 ) Γ ( 3 2 ) Γ ( α ) ( η 2 α ) 1 2 2 F 1 ( α + 1 2 , 3 2 , 3 2 ; ω 2 η 2 4 α ) .
P e = 0 P e ( z s ) f ( z s ) d z s
P e ( z s ) = 1 2 erfc ( γ EGC 2 ) = 1 2 erfc ( g z s 2 )
f ( z s ) = 1 2 π ϕ z s ( ω ) e j ω z s d ω = 1 2 π l = 1 L ϕ z l ( ω ) e j ω z s d ω
P e = 1 2 π ϕ z s * ( ω ) 0 P e ( z s ) e j ω z s d z s d ω
𝘍 ( ω ) = 0 erfc ( z s ) e j ω z s d z s = 1 π 1 F 1 ( 1 , 3 2 , ω 2 4 ) + j ω ( 1 e ω 2 4 ) ,
P e = 1 2 π l = 1 L ϕ z l * ( g ω 2 ) 𝘍 ( ω ) 2 d ω .
P e = 1 2 π π 2 π 2 sec 2 θ l = 1 L ϕ z l * ( g tan θ 2 ) 𝘍 ( tan θ ) 2 d θ .
F z s ( z o ) = 1 2 1 π 0 { ϕ z s ( ω ) exp [ j ω z o ] } ω d ω .
ϕ z l ( ω ) = 2 { ϕ z l ( ω ) } + 2 { ϕ z l ( ω ) } exp [ j tan 1 ( { ϕ z l ( ω ) } { ϕ z l ( ω ) } ) ]
{ ϕ z s ( ω ) exp [ j ω z o ] } = l = 1 L 2 { ϕ z l ( ω ) } + 2 { ϕ z l ( ω ) } sin [ Σ l = 1 L tan 1 ( { ϕ z l ( ω ) } { ϕ z l ( ω ) } ) ω z o ] .
P o , EGC ( γ * ) = 1 2 2 π 0 π 2 l = 1 L 2 { ϕ z l ( tan θ ) } + 2 { ϕ z l ( tan θ ) } sin 2 θ csc [ Σ l = 1 L tan 1 ( { ϕ z l ( tan θ ) } { ϕ z l ( tan θ ) } ) tan θ ω γ * g ] d θ
F I s , l ( I ) = 1 2 ( α l I ) α 2 Γ ( α l ) η l α l K α l ( 2 η l α l I ) , I > 0 .
P o , SD ( γ * ) = l = 1 L F I s , l ( γ * g 2 ) .
f I s , SD ( I s ) = L [ F I s , l ( I s ) ] L 1 f ( I s ) .
P e ( I s ) = 1 2 exp [ β γ ¯ I s ]
P e = 1 2 0 exp [ β γ ¯ I s ] f I s , SD ( I s ) dI s = 1 2 0 π 2 sec 2 θ exp [ β γ ¯ tan θ ] f I s , SD ( tan θ ) d θ .
G c , SD = [ 2 L 1 Γ ( Σ l = 1 L G dl ) Σ l = 1 L ( G dl ) l = 1 L G cl G dl Γ ( G dl ) G dl ] 1 Σ l = 1 L G dl .
i ac , l ( t ) = c l [ cos ϕ cos ( ω IF t + ϕ s , l ) sin ϕ sin ( ω IF t + ϕ s , l ) ] .
y c , l ( t ) = 2 { c l [ cos ϕ cos ( ω IF t + ϕ s , l ) sin ϕ sin ( ω IF t + ϕ s , l ) ] } cos ( ω IF t )
y s , l ( t ) = 2 { c l [ cos ϕ cos ( ω IF t + ϕ s , l ) sin ϕ sin ( ω IF t + ϕ s , l ) ] } sin ( ω IF t ) .
y ˜ c , l ( t ) = c l 2 [ cos ϕ cos ϕ s , l sin ϕ sin ϕ s , l ]
y ˜ s , l ( t ) = c l 2 [ cos ϕ cos ϕ s , l + sin ϕ sin ϕ s , l ] .
y ˜ l ( t ) = y ˜ c , l ( t ) + j y ˜ s , l ( t ) = c l 2 [ cos ϕ + j sin ϕ ] e j ϕ s , l .
y ˜ l ( t ) e j ϕ s , l = 2 R P LO P s , l e j ϕ .
γ EGC = ( Σ l = 1 L 2 R P LO P s , l ) 2 Σ l = 1 L σ n , l 2 = ( Σ l = 1 L 2 R P LO P s , l ) 2 L ( 2 q R P LO Δ f ) = R ( Σ l = 1 L P s , l ) 2 Lq Δ f

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