Abstract

An algorithm based on independent component analysis for blind polarization demultiplexing in a coherent transmission system is presented. A comparison with the constant modulus algorithm in terms of the convergence properties is performed, and it is found that the suggested algorithm has a significantly faster convergence rate and does not have any singularity problems. We also demonstrate that the algorithm convergence is strongly dependent on the choice of starting condition and show how this can be exploited to increase the convergence rate.

© 2011 OSA

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References

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  1. M. Tseytlin, O. Ritterbush, and A. Salamon, "Digital, endless polarization control for polarization multiplexed fiber-optic communications," Optical Fiber Communication Conf. (OFC), 2003, MF83.
  2. R. Noé, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing," IEEE Photon. Technol. Lett. 17, (4), 887‒889 (2005).
    [Crossref]
  3. Y. Han and G. Li, "Coherent optical communication using polarization multiple-input–multiple-output," Opt. Express 13, (19), 7527‒7534 (2005).
    [Crossref]
  4. D. N. Godard, "Self-recovering equalization and carrier tracking in two-dimensional data communication systems," IEEE Trans. Commun. 28, (11), 1867‒1875 (1980).
    [Crossref]
  5. K. Kikuchi, "Polarization-demultiplexing algorithm in the digital coherent receiver," IEEE/LEOS Summer Topical Meetings (LEOSST), 2008, MC2.2.
  6. L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.
  7. J.-F. Cardoso and B. H. Laheld, "Equivariant adaptive source separation," IEEE Trans. Signal Process. 44, (12), 3017‒3030 (1996).
    [Crossref]
  8. J.-F. Cardoso, "Blind signal separation: statistical principles," Proc. IEEE 86, (10), 2009‒2025 (1998).
    [Crossref]
  9. H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.
  10. T. Öktem, A. T. Erdogan, and A. Demir, "Adaptive receiver structures for fiber communication systems employing polarization-division multiplexing," J. Lightwave Technol. 27, (23), 5394‒5404 (2009).
    [Crossref]
  11. T. Öktem, A. T. Erdogan, and A. Demir, "Adaptive receiver structures for fiber communication systems employing polarization division multiplexing: high symbol rate case," J. Lightwave Technol. 28, (10), 1536‒1546 (2010).
    [Crossref]
  12. X. Xie, F. Yaman, X. Zhou, and G. Li, "Polarization demultiplexing by independent component analysis," IEEE Photon. Technol. Lett. 22, (11), 805‒807 (2010).
    [Crossref]
  13. X. Zhou and J. Yu, "200-Gb/s PDM-16QAM generation using a new synthesizing method," European Conf. Optical Communication (ECOC), 2009, 10.3.5.
  14. I. Fatadin, D. Ives, and S. J. Savory, "Blind equalization and carrier phase recovery in a 16-QAM optical coherent system," J. Lightwave Technol. 27, (15), 3042‒3049 (2009).
    [Crossref]
  15. P. J. Winzer, A. H. Gnauck, C. R. Doerr, M. Magarini, and L. L. Buhl, "Spectrally efficient long-haul optical networking using 112-Gb/s polarization-multiplexed 16-QAM," J. Lightwave Technol. 28, (4), 547‒556 (2010).
    [Crossref]
  16. P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, 2010.
  17. A. Hjørungnes and D. Gesbert, "Complex-valued matrix differentiation: techniques and key results," IEEE Trans. Signal Process. 55, (6), 2740‒2746 (2007).
    [Crossref]
  18. S. Amari, "Natural gradient works efficiently in learning," Neural Comput. 10, (2), 251‒276 (1998).
    [Crossref]
  19. P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.
  20. I. Roudas, A. Vgenis, C. S. Petrou, D. Toumpakaris, J. Hurley, M. Sauer, J. Downie, Y. Mauro, and S. Raghavan, "Optimal polarization demultiplexing for coherent optical communications systems," J. Lightwave Technol. 28, (7), 1121‒1134 (2010).
    [Crossref]
  21. S. J. Savory, "Digital coherent optical receivers: Algorithms and subsystems," IEEE J. Sel. Top. Quantum Electron. 16, (5), 1164‒1179 (2010).
    [Crossref]
  22. A. Hjørungnes, D. Gesbert, and D. P. Palomar, "Unified theory of complex-valued matrix differentiation," IEEE Int. Conf. Acoustics, Speech and Signal Processing, Vol. 3, 2007.

2010 (5)

2009 (2)

2007 (1)

A. Hjørungnes and D. Gesbert, "Complex-valued matrix differentiation: techniques and key results," IEEE Trans. Signal Process. 55, (6), 2740‒2746 (2007).
[Crossref]

2005 (2)

R. Noé, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing," IEEE Photon. Technol. Lett. 17, (4), 887‒889 (2005).
[Crossref]

Y. Han and G. Li, "Coherent optical communication using polarization multiple-input–multiple-output," Opt. Express 13, (19), 7527‒7534 (2005).
[Crossref]

1998 (2)

J.-F. Cardoso, "Blind signal separation: statistical principles," Proc. IEEE 86, (10), 2009‒2025 (1998).
[Crossref]

S. Amari, "Natural gradient works efficiently in learning," Neural Comput. 10, (2), 251‒276 (1998).
[Crossref]

1996 (1)

J.-F. Cardoso and B. H. Laheld, "Equivariant adaptive source separation," IEEE Trans. Signal Process. 44, (12), 3017‒3030 (1996).
[Crossref]

1980 (1)

D. N. Godard, "Self-recovering equalization and carrier tracking in two-dimensional data communication systems," IEEE Trans. Commun. 28, (11), 1867‒1875 (1980).
[Crossref]

Agrell, E.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

Amari, S.

S. Amari, "Natural gradient works efficiently in learning," Neural Comput. 10, (2), 251‒276 (1998).
[Crossref]

Andrekson, P. A.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

Buhl, L. L.

Cardoso, J.-F.

J.-F. Cardoso, "Blind signal separation: statistical principles," Proc. IEEE 86, (10), 2009‒2025 (1998).
[Crossref]

J.-F. Cardoso and B. H. Laheld, "Equivariant adaptive source separation," IEEE Trans. Signal Process. 44, (12), 3017‒3030 (1996).
[Crossref]

Demir, A.

Doerr, C. R.

Downie, J.

Erdogan, A. T.

Fatadin, I.

Gesbert, D.

A. Hjørungnes and D. Gesbert, "Complex-valued matrix differentiation: techniques and key results," IEEE Trans. Signal Process. 55, (6), 2740‒2746 (2007).
[Crossref]

A. Hjørungnes, D. Gesbert, and D. P. Palomar, "Unified theory of complex-valued matrix differentiation," IEEE Int. Conf. Acoustics, Speech and Signal Processing, Vol. 3, 2007.

Gnauck, A. H.

Godard, D. N.

D. N. Godard, "Self-recovering equalization and carrier tracking in two-dimensional data communication systems," IEEE Trans. Commun. 28, (11), 1867‒1875 (1980).
[Crossref]

Han, Y.

Hjørungnes, A.

A. Hjørungnes and D. Gesbert, "Complex-valued matrix differentiation: techniques and key results," IEEE Trans. Signal Process. 55, (6), 2740‒2746 (2007).
[Crossref]

A. Hjørungnes, D. Gesbert, and D. P. Palomar, "Unified theory of complex-valued matrix differentiation," IEEE Int. Conf. Acoustics, Speech and Signal Processing, Vol. 3, 2007.

Hoshida, T.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.

Hurley, J.

Ives, D.

Johannisson, P.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

Karlsson, M.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

Kikuchi, K.

K. Kikuchi, "Polarization-demultiplexing algorithm in the digital coherent receiver," IEEE/LEOS Summer Topical Meetings (LEOSST), 2008, MC2.2.

Laheld, B. H.

J.-F. Cardoso and B. H. Laheld, "Equivariant adaptive source separation," IEEE Trans. Signal Process. 44, (12), 3017‒3030 (1996).
[Crossref]

Li, G.

X. Xie, F. Yaman, X. Zhou, and G. Li, "Polarization demultiplexing by independent component analysis," IEEE Photon. Technol. Lett. 22, (11), 805‒807 (2010).
[Crossref]

Y. Han and G. Li, "Coherent optical communication using polarization multiple-input–multiple-output," Opt. Express 13, (19), 7527‒7534 (2005).
[Crossref]

Liu, L.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.

Magarini, M.

Mauro, Y.

Noé, R.

R. Noé, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing," IEEE Photon. Technol. Lett. 17, (4), 887‒889 (2005).
[Crossref]

Oda, S.

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.

Öktem, T.

Palomar, D. P.

A. Hjørungnes, D. Gesbert, and D. P. Palomar, "Unified theory of complex-valued matrix differentiation," IEEE Int. Conf. Acoustics, Speech and Signal Processing, Vol. 3, 2007.

Petrou, C. S.

Raghavan, S.

Rasmussen, J. C.

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.

Ritterbush, O.

M. Tseytlin, O. Ritterbush, and A. Salamon, "Digital, endless polarization control for polarization multiplexed fiber-optic communications," Optical Fiber Communication Conf. (OFC), 2003, MF83.

Roudas, I.

Salamon, A.

M. Tseytlin, O. Ritterbush, and A. Salamon, "Digital, endless polarization control for polarization multiplexed fiber-optic communications," Optical Fiber Communication Conf. (OFC), 2003, MF83.

Sauer, M.

Savory, S. J.

S. J. Savory, "Digital coherent optical receivers: Algorithms and subsystems," IEEE J. Sel. Top. Quantum Electron. 16, (5), 1164‒1179 (2010).
[Crossref]

I. Fatadin, D. Ives, and S. J. Savory, "Blind equalization and carrier phase recovery in a 16-QAM optical coherent system," J. Lightwave Technol. 27, (15), 3042‒3049 (2009).
[Crossref]

Scharf, L. L.

P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, 2010.

Schreier, P. J.

P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, 2010.

Sjödin, M.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

Tan, A. S.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

Tao, Z.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.

Toumpakaris, D.

Tseytlin, M.

M. Tseytlin, O. Ritterbush, and A. Salamon, "Digital, endless polarization control for polarization multiplexed fiber-optic communications," Optical Fiber Communication Conf. (OFC), 2003, MF83.

Vgenis, A.

Winzer, P. J.

Wymeersch, H.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

Xie, X.

X. Xie, F. Yaman, X. Zhou, and G. Li, "Polarization demultiplexing by independent component analysis," IEEE Photon. Technol. Lett. 22, (11), 805‒807 (2010).
[Crossref]

Yaman, F.

X. Xie, F. Yaman, X. Zhou, and G. Li, "Polarization demultiplexing by independent component analysis," IEEE Photon. Technol. Lett. 22, (11), 805‒807 (2010).
[Crossref]

Yan, W.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.

Yu, J.

X. Zhou and J. Yu, "200-Gb/s PDM-16QAM generation using a new synthesizing method," European Conf. Optical Communication (ECOC), 2009, 10.3.5.

Zhang, H.

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.

Zhou, X.

X. Xie, F. Yaman, X. Zhou, and G. Li, "Polarization demultiplexing by independent component analysis," IEEE Photon. Technol. Lett. 22, (11), 805‒807 (2010).
[Crossref]

X. Zhou and J. Yu, "200-Gb/s PDM-16QAM generation using a new synthesizing method," European Conf. Optical Communication (ECOC), 2009, 10.3.5.

IEEE J. Sel. Top. Quantum Electron. (1)

S. J. Savory, "Digital coherent optical receivers: Algorithms and subsystems," IEEE J. Sel. Top. Quantum Electron. 16, (5), 1164‒1179 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (2)

R. Noé, "PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing," IEEE Photon. Technol. Lett. 17, (4), 887‒889 (2005).
[Crossref]

X. Xie, F. Yaman, X. Zhou, and G. Li, "Polarization demultiplexing by independent component analysis," IEEE Photon. Technol. Lett. 22, (11), 805‒807 (2010).
[Crossref]

IEEE Trans. Commun. (1)

D. N. Godard, "Self-recovering equalization and carrier tracking in two-dimensional data communication systems," IEEE Trans. Commun. 28, (11), 1867‒1875 (1980).
[Crossref]

IEEE Trans. Signal Process. (2)

J.-F. Cardoso and B. H. Laheld, "Equivariant adaptive source separation," IEEE Trans. Signal Process. 44, (12), 3017‒3030 (1996).
[Crossref]

A. Hjørungnes and D. Gesbert, "Complex-valued matrix differentiation: techniques and key results," IEEE Trans. Signal Process. 55, (6), 2740‒2746 (2007).
[Crossref]

J. Lightwave Technol. (5)

Neural Comput. (1)

S. Amari, "Natural gradient works efficiently in learning," Neural Comput. 10, (2), 251‒276 (1998).
[Crossref]

Opt. Express (1)

Proc. IEEE (1)

J.-F. Cardoso, "Blind signal separation: statistical principles," Proc. IEEE 86, (10), 2009‒2025 (1998).
[Crossref]

Other (8)

H. Zhang, Z. Tao, L. Liu, S. Oda, T. Hoshida, and J. C. Rasmussen, "Polarization demultiplexing based on independent component analysis in optical coherent receivers," European Conf. Optical Communication (ECOC), 2008, Mo.3.D.5.

K. Kikuchi, "Polarization-demultiplexing algorithm in the digital coherent receiver," IEEE/LEOS Summer Topical Meetings (LEOSST), 2008, MC2.2.

L. Liu, Z. Tao, W. Yan, S. Oda, T. Hoshida, and J. C. Rasmussen, "Initial tap setup of constant modulus algorithm for polarization de-multiplexing in optical coherent receivers," Optical Fiber Communication Conf. (OFC), 2009, OMT2.

P. Johannisson, H. Wymeersch, M. Sjödin, A. S. Tan, E. Agrell, P. A. Andrekson, and M. Karlsson, "Convergence comparison of CMA and ICA for blind polarization demultiplexing of QPSK and 16-QAM signals," European Conf. Optical Communication (ECOC), 2010, Th.9.A.3.

M. Tseytlin, O. Ritterbush, and A. Salamon, "Digital, endless polarization control for polarization multiplexed fiber-optic communications," Optical Fiber Communication Conf. (OFC), 2003, MF83.

P. J. Schreier and L. L. Scharf, Statistical Signal Processing of Complex-Valued Data, Cambridge University Press, 2010.

X. Zhou and J. Yu, "200-Gb/s PDM-16QAM generation using a new synthesizing method," European Conf. Optical Communication (ECOC), 2009, 10.3.5.

A. Hjørungnes, D. Gesbert, and D. P. Palomar, "Unified theory of complex-valued matrix differentiation," IEEE Int. Conf. Acoustics, Speech and Signal Processing, Vol. 3, 2007.

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Figures (6)

Fig. 1
Fig. 1

System model showing the addition of noise, the phase drift, and the polarization change before the signal is sampled as x k .

Fig. 2
Fig. 2

(Color online) Contour plots of the objective functions Λ and J CMA . Top row: QPSK. Bottom row: 16-QAM. Left column: ICA. Right column: CMA. The solution is indicated by the dot.

Fig. 3
Fig. 3

(Color online) Probability for being above 1 dB penalty for the CMA and the ICA algorithm on 200 symbols of QPSK data. The thick lines are symbol-by-symbol updates and the thin lines use the cumulative approach. The step size, μ, is indicated next to each curve.

Fig. 4
Fig. 4

(Color online) Probability for being above 1 dB penalty for the CMA and the ICA algorithm on 2000 symbols of 16-QAM data. The thick lines are symbol-by-symbol updates and the thin lines use the cumulative approach. The step size, μ, is indicated next to each curve.

Fig. 5
Fig. 5

(Color online) Probability for being above 1 dB penalty for the CMA and the ICA algorithm on 200 symbols of QPSK data. The thick lines show three parallel runs and the thin lines show the combined result. The step size, μ, is indicated next to each curve.

Fig. 6
Fig. 6

(Color online) Probability for being above 1 dB penalty for the CMA and the ICA algorithm on 2000 symbols of 16-QAM data. The thick lines show three parallel runs and the thin lines show the combined result. The step size, μ, is indicated next to each curve.

Equations (38)

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

J CMA = E ( | y k ( X ) | 2 ρ 2 ) 2 + ( | y k ( Y ) | 2 ρ 2 ) 2 ,
B k + 1 = B k μ ψ ( y k ) x k H ,
p X ( x k | A k ) = | det A k 1 | 2 p S ( A k 1 x k ) .
p X ( x k | B k ) = | det B k | 2 p S ( B k x k ) .
Λ ( B k ) = log p X ( x k | B k ) = log | det B k | 2 + log p S ( y k ) .
B k + 1 = B k + μ G ( B k ) , G ( B ) = Λ B ,
B k + 1 = B k + μ G ( B k ) B k H B k .
Λ B = [ I + f ( y ) y H ] B H ,
f ( y ) = 1 p S ( y ) p S ( y ) s
G ̃ ( B ) = [ I y y H + f ( y ) y H y f ( y ) H ] B H .
p S ( s ) p S ( s ( X ) ) p S ( s ( Y ) ) .
s = ( c l + n ) e i ϕ ,
p S | C ( s | c l ) = 1 2 π σ 2 exp | c l | 2 + | s | 2 2 σ 2 I 0 | c l s | σ 2 ,
p S ( s ) = 1 M l = 1 M p S | C ( s | c l ) .
f ( s ) = 1 2 σ 2 l = 1 M p S | C ( s | c l ) I 1 ( | c l s | / σ 2 ) I 0 ( | c l s | / σ 2 ) | c l | e i s s l = 1 M p S | C ( s | c l ) .
f M PSK ( s ) 1 2 σ 2 ( D | c 1 | e i s s ) ,
D = I 1 ( | c 1 | 2 / σ 2 ) I 0 ( | c 1 | 2 / σ 2 ) .
A = u v v u e i ϕ common
B ( θ , ϕ ) = cos θ sin θ e i ϕ sin θ e i ϕ cos θ .
V = e i η ( X ) 0 0 e i η ( Y ) ,
y ( X ) y ( Y ) = C 11 C 12 C 21 C 22 a ( X ) + n ( X ) a ( Y ) + n ( Y ) e i ϕ .
SNR ( X ) = E | C 11 a ( X ) | 2 E | C 11 n ( X ) + C 12 a ( Y ) + C 12 n ( Y ) | 2 ,
SNR ( X ) = | C 11 | 2 E s | C 12 | 2 E s + 2 σ 2 ( | C 11 | 2 + | C 12 | 2 ) ,
SNR k pen = SNR nom min ( SNR k ( X ) , SNR k ( Y ) ) ,
log | det B | 2 = log det ( B B ) ,
B log | det B | 2 = B H .
H ( Z , Z ) = G ( F ( Z , Z ) , F ( Z , Z ) ) ,
D Z H = ( D F G ) ( D Z F ) + ( D F G ) ( D Z F ) .
p S ( B x ) B = p S ( B x ) s ( B x ) B = 0 + p S ( B x ) s ( B x ) B = p S ( B x ) s x H = p S ( y ) s y H B H ,
Λ B = I + 1 p S ( y ) p S ( y ) s y H B H .
S = ( a + X ) e i Φ ,
p X ( x ) = 1 2 π σ 2 exp | x | 2 2 σ 2 ,
X = S e i Φ a ,
p S | Φ , A ( s | ϕ , a ) = 1 2 π σ 2 exp | s e i ϕ a | 2 2 σ 2 .
p S | A ( s | a ) = 0 2 π p S | Φ , A ( s | ϕ , a ) p Φ ( ϕ ) d ϕ = 1 4 π 2 σ 2 exp | a | 2 + | s | 2 2 σ 2 f a s σ 2 ,
f ( ξ ) 0 2 π exp [ Re ( ξ e i ϕ ) ] d ϕ .
f ( ξ ) = 0 2 π exp | ξ | Re e i ( ξ ϕ ) d ϕ ,
f ( ξ ) = 2 π I 0 | ξ | ,