Abstract

Polarization dependent loss (PDL) causes imbalanced optical signal to noise ratio (OSNR) of the two polarizations, thus remains one of the major bottlenecks for next-generation polarization-division-multiplexed (PDM) coherent optical transmission systems. In this paper, we investigate Pairwise Coding for adaptive PDL mitigation in PDM coherent optical systems. By pre-coding across two polarizations, the PDL-induced performance degradation can be largely mitigated without any coding overhead. We present details of the coding and de-coding design, and also derive the analytical symbol/bit error rate of the Polarization Pairwise Coding scheme, which can be used to predict the performance gain as well as for optimal rotation angle calculation. Simulation results verify that Pairwise Coding achieves substantial system performance gains over a wide range of PDL values. Compared with other digital coding techniques, Polarization Pairwise Coding shows improved performance than Walsh-Hadamard transform since it maximizes the coordinate diversity; and also Pairwise Coding is computationally much simpler to decode compared with the Golden and Silver Codes, therefore is practical for current 100-Gb/s and future 400-Gb/s and 1-Tb/s digital coherent transceivers.

© 2015 Optical Society of America

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References

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  1. P. J. Winzer, “High-spectral-efficiency optical modulation formats,” J. Lightwave Technol. 30(24), 3824–3835 (2012).
    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  11. W.-R. Peng, T. Tsuritani, and I. Morita, “Modified Walsh-Hadamard transform for PDL mitigation,” in 39th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2013), paper P.3.5.
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    [Crossref]
  16. Y. Hong, A. J. Lowery, and E. Viterbo, “Sensitivity improvement and carrier power reduction in direct-detection optical OFDM systems by subcarrier pairing,” Opt. Express 20(2), 1635–1648 (2012).
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    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
  23. C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
    [Crossref]
  24. S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010).
    [Crossref]
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    [Crossref]
  26. B. Huang, J. Zhang, J. Yu, Z. Dong, X. Li, H. Ou, N. Chi, and W. Liu, “Robust 9-QAM digital recovery for spectrum shaped coherent QPSK signal,” Opt. Express 21(6), 7216–7221 (2013).
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    [Crossref]
  29. O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inf. Theory 48(2), 384–395 (2002).
    [Crossref]

2014 (2)

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

2013 (3)

2012 (5)

2011 (3)

2010 (3)

2009 (1)

2008 (1)

2007 (1)

N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, “Performance of BICM-ID with signal space diversity,” IEEE Trans. Wirel. Commun. 6(5), 1732–1742 (2007).
[Crossref]

2005 (1)

J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2x2 full-rate space-time code with nonvanishing determinants,” IEEE Trans. Inf. Theory 51(4), 1432–1436 (2005).
[Crossref]

2002 (1)

O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inf. Theory 48(2), 384–395 (2002).
[Crossref]

1998 (1)

J. Boutros and E. Viterbo, “Signal space diversity: a power- and bandwidth-efficient diversity technique for the Rayleigh fading channel,” IEEE Trans. Inf. Theory 44(4), 1453–1467 (1998).
[Crossref]

1995 (1)

E. Lichtman, “Limitations imposed by polarization-dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13(5), 906–913 (1995).
[Crossref]

Anderson, T.

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

C. Zhu, A. V. Tran, S. Chen, L. B. Du, C. C. Do, T. Anderson, A. J. Lowery, and E. Skafidas, “Statistical moments-based OSNR monitoring for coherent optical systems,” Opt. Express 20(16), 17711–17721 (2012).
[Crossref] [PubMed]

Andrusier, A.

Antonelli, C.

Awwad, E.

Belfiore, J.-C.

J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2x2 full-rate space-time code with nonvanishing determinants,” IEEE Trans. Inf. Theory 51(4), 1432–1436 (2005).
[Crossref]

Birk, M.

Boutros, J.

J. Boutros and E. Viterbo, “Signal space diversity: a power- and bandwidth-efficient diversity technique for the Rayleigh fading channel,” IEEE Trans. Inf. Theory 44(4), 1453–1467 (1998).
[Crossref]

Chen, J.

Chen, S.

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

C. Zhu, A. V. Tran, S. Chen, L. B. Du, C. C. Do, T. Anderson, A. J. Lowery, and E. Skafidas, “Statistical moments-based OSNR monitoring for coherent optical systems,” Opt. Express 20(16), 17711–17721 (2012).
[Crossref] [PubMed]

Chi, N.

Chockalingam, A.

S. K. Mohammed, E. Viterbo, Y. Hong, and A. Chockalingam, “MIMO precoding with X- and Y-codes,” IEEE Trans. Inf. Theory 57(6), 3542–3566 (2011).
[Crossref]

Do, C. C.

Do Cuong, C.

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

Dong, Z.

Du, L. B.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

C. Zhu, A. V. Tran, S. Chen, L. B. Du, C. C. Do, T. Anderson, A. J. Lowery, and E. Skafidas, “Statistical moments-based OSNR monitoring for coherent optical systems,” Opt. Express 20(16), 17711–17721 (2012).
[Crossref] [PubMed]

Ellis, A. D.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

Feder, M.

Hong, Y.

Hottinen, A.

O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inf. Theory 48(2), 384–395 (2002).
[Crossref]

Huang, B.

Jaouen, Y.

S. Mumtaz, G. Othman, and Y. Jaouen, “Space-time codes for optical fiber communication with polarization multiplexing,” in Proc. IEEE ICC, Cape Town, South Africa, May 2010, pp. 1–5.
[Crossref]

Jaouën, Y.

Kam, P. Y.

Kuschnerov, M.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

Le-Ngoc, T.

N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, “Performance of BICM-ID with signal space diversity,” IEEE Trans. Wirel. Commun. 6(5), 1732–1742 (2007).
[Crossref]

Li, C.

M. Zamani, C. Li, and Z. Zhang, “Polarization-time code and 4 × 4 equalizer-decoder for coherent optical transmission,” IEEE Photonics Technol. Lett. 24(20), 1815–1818 (2012).
[Crossref]

Li, X.

Lichtman, E.

E. Lichtman, “Limitations imposed by polarization-dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13(5), 906–913 (1995).
[Crossref]

Liu, W.

Lowery, A. J.

Magill, P.

Mecozzi, A.

Meron, E.

Mohammed, S. K.

S. K. Mohammed, E. Viterbo, Y. Hong, and A. Chockalingam, “MIMO precoding with X- and Y-codes,” IEEE Trans. Inf. Theory 57(6), 3542–3566 (2011).
[Crossref]

Mumtaz, S.

S. Mumtaz, G. Othman, and Y. Jaouen, “Space-time codes for optical fiber communication with polarization multiplexing,” in Proc. IEEE ICC, Cape Town, South Africa, May 2010, pp. 1–5.
[Crossref]

Napoli, A.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

Nelson, L. E.

Nguyen, H. H.

N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, “Performance of BICM-ID with signal space diversity,” IEEE Trans. Wirel. Commun. 6(5), 1732–1742 (2007).
[Crossref]

Othman, G.

S. Mumtaz, G. Othman, and Y. Jaouen, “Space-time codes for optical fiber communication with polarization multiplexing,” in Proc. IEEE ICC, Cape Town, South Africa, May 2010, pp. 1–5.
[Crossref]

Othman, G. R.

Ou, H.

Poggiolini, P.

Rafique, D.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

Rapp, L.

Rekaya, G.

J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2x2 full-rate space-time code with nonvanishing determinants,” IEEE Trans. Inf. Theory 51(4), 1432–1436 (2005).
[Crossref]

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]

Schex, A.

Shtaif, M.

Skafidas, E.

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

C. Zhu, A. V. Tran, S. Chen, L. B. Du, C. C. Do, T. Anderson, A. J. Lowery, and E. Skafidas, “Statistical moments-based OSNR monitoring for coherent optical systems,” Opt. Express 20(16), 17711–17721 (2012).
[Crossref] [PubMed]

Spinnler, B.

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

Tirkkonen, O.

O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inf. Theory 48(2), 384–395 (2002).
[Crossref]

Tran, A. V.

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

C. Zhu, A. V. Tran, S. Chen, L. B. Du, C. C. Do, T. Anderson, A. J. Lowery, and E. Skafidas, “Statistical moments-based OSNR monitoring for coherent optical systems,” Opt. Express 20(16), 17711–17721 (2012).
[Crossref] [PubMed]

Tran, N. H.

N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, “Performance of BICM-ID with signal space diversity,” IEEE Trans. Wirel. Commun. 6(5), 1732–1742 (2007).
[Crossref]

Viterbo, E.

Y. Hong, A. J. Lowery, and E. Viterbo, “Sensitivity improvement and carrier power reduction in direct-detection optical OFDM systems by subcarrier pairing,” Opt. Express 20(2), 1635–1648 (2012).
[Crossref] [PubMed]

S. K. Mohammed, E. Viterbo, Y. Hong, and A. Chockalingam, “MIMO precoding with X- and Y-codes,” IEEE Trans. Inf. Theory 57(6), 3542–3566 (2011).
[Crossref]

J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2x2 full-rate space-time code with nonvanishing determinants,” IEEE Trans. Inf. Theory 51(4), 1432–1436 (2005).
[Crossref]

J. Boutros and E. Viterbo, “Signal space diversity: a power- and bandwidth-efficient diversity technique for the Rayleigh fading channel,” IEEE Trans. Inf. Theory 44(4), 1453–1467 (1998).
[Crossref]

Winzer, P. J.

Yu, C.

Yu, J.

Zamani, M.

M. Zamani, C. Li, and Z. Zhang, “Polarization-time code and 4 × 4 equalizer-decoder for coherent optical transmission,” IEEE Photonics Technol. Lett. 24(20), 1815–1818 (2012).
[Crossref]

Zhang, J.

Zhang, S.

Zhang, Z.

M. Zamani, C. Li, and Z. Zhang, “Polarization-time code and 4 × 4 equalizer-decoder for coherent optical transmission,” IEEE Photonics Technol. Lett. 24(20), 1815–1818 (2012).
[Crossref]

Zhu, C.

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

C. Zhu, A. V. Tran, S. Chen, L. B. Du, C. C. Do, T. Anderson, A. J. Lowery, and E. Skafidas, “Statistical moments-based OSNR monitoring for coherent optical systems,” Opt. Express 20(16), 17711–17721 (2012).
[Crossref] [PubMed]

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 Photonics Technol. Lett. (1)

M. Zamani, C. Li, and Z. Zhang, “Polarization-time code and 4 × 4 equalizer-decoder for coherent optical transmission,” IEEE Photonics Technol. Lett. 24(20), 1815–1818 (2012).
[Crossref]

IEEE Signal Process. Mag. (1)

L. B. Du, D. Rafique, A. Napoli, B. Spinnler, A. D. Ellis, M. Kuschnerov, and A. J. Lowery, “Digital fiber nonlinearity compensation: towards 1Tb/s transport,” IEEE Signal Process. Mag. 31(2), 46–56 (2014).
[Crossref]

IEEE Trans. Inf. Theory (4)

J. Boutros and E. Viterbo, “Signal space diversity: a power- and bandwidth-efficient diversity technique for the Rayleigh fading channel,” IEEE Trans. Inf. Theory 44(4), 1453–1467 (1998).
[Crossref]

S. K. Mohammed, E. Viterbo, Y. Hong, and A. Chockalingam, “MIMO precoding with X- and Y-codes,” IEEE Trans. Inf. Theory 57(6), 3542–3566 (2011).
[Crossref]

J.-C. Belfiore, G. Rekaya, and E. Viterbo, “The golden code: a 2x2 full-rate space-time code with nonvanishing determinants,” IEEE Trans. Inf. Theory 51(4), 1432–1436 (2005).
[Crossref]

O. Tirkkonen and A. Hottinen, “Square-matrix embeddable space-time block codes for complex signal constellations,” IEEE Trans. Inf. Theory 48(2), 384–395 (2002).
[Crossref]

IEEE Trans. Wirel. Commun. (1)

N. H. Tran, H. H. Nguyen, and T. Le-Ngoc, “Performance of BICM-ID with signal space diversity,” IEEE Trans. Wirel. Commun. 6(5), 1732–1742 (2007).
[Crossref]

J. Lightwave Technol. (5)

C. Zhu, A. V. Tran, C. Do Cuong, S. Chen, T. Anderson, and E. Skafidas, “Digital signal processing for training-aided coherent optical angle-carrier frequency-domain equalization systems,” J. Lightwave Technol. 32(24), 4712–4722 (2014).
[Crossref]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” J. Lightwave Technol. 28(11), 1597–1607 (2010).
[Crossref]

P. J. Winzer, “High-spectral-efficiency optical modulation formats,” J. Lightwave Technol. 30(24), 3824–3835 (2012).
[Crossref]

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Lightwave Technol. 30(24), 3857–3879 (2012).
[Crossref]

E. Lichtman, “Limitations imposed by polarization-dependent gain and loss on all-optical ultralong communication systems,” J. Lightwave Technol. 13(5), 906–913 (1995).
[Crossref]

Opt. Express (7)

Opt. Lett. (3)

Other (6)

W.-R. Peng, T. Tsuritani, and I. Morita, “Modified Walsh-Hadamard transform for PDL mitigation,” in 39th European Conference and Exposition on Optical Communications, OSA Technical Digest (CD) (Optical Society of America, 2013), paper P.3.5.

C. Xie, “Polarization-dependent loss induced penalties in PDM-QPSK coherent optical communication systems,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, OSA Technical Digest Series (CD) (Optical Society of America, 2010), paper OWE6.
[Crossref]

S. Mumtaz, G. Othman, and Y. Jaouen, “Space-time codes for optical fiber communication with polarization multiplexing,” in Proc. IEEE ICC, Cape Town, South Africa, May 2010, pp. 1–5.
[Crossref]

P. M. Krummrich, E. Schmidt, W. Weiershausen, and A. Mattheus, Field trial results on statistics of fast polarization changes in long haul WDM transmission systems,” in Optical Fiber Communication Conference and Exposition and The National Fiber Optic Engineers Conference, Technical Digest (CD) (Optical Society of America, 2005), paper OThT6.

C. Zhu, B. Song, L. Zhuang, B. Corcoran, and A. Lowery, “Pairwise coding to mitigate polarization dependent loss,” in Optical Fiber Communication Conference, OSA Technical Digest (online) (Optical Society of America, 2015), paper W4K.4.
[Crossref]

S. Mumtaz, G. Rekaya-Ben Othman, Y. Jaouen, J. Li, S. Koenig, R. Schmogrow, and J. Leuthold, “Alamouti Code against PDL in Polarization Multiplexed Systems,” in Advanced Photonics, OSA Technical Digest (CD) (Optical Society of America, 2011), paper SPTuA2.

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Figures (9)

Fig. 1
Fig. 1

Conceptual diagram for pairwise pre-coding.

Fig. 2
Fig. 2

Conceptual diagram of pairwise decoding process.

Fig. 3
Fig. 3

Example for maximum likelihood detection, with QPSK and θ = 45°.

Fig. 4
Fig. 4

Optimal rotation angle based on different approaches: (a) 12-dB SNR with different delta SNR values; (b) 9-dB ΔSNR for different SNR values.

Fig. 5
Fig. 5

(a) and (b): Simulated and theoretical BER/SER performance with different rotation angles at 3-dB (left) and 9-dB ΔSNR (right).

Fig. 6
Fig. 6

Simulation results for single-channel OSNR sweep (signal bandwidth 12.5 GHz) with various ΔSNR between two polarizations and different optimal rotation angle search schemes: (a)-(d) BER results for 0/3/6/9 dB SNR difference; (e), (f): SER results for 6/9 dB SNR difference.

Fig. 7
Fig. 7

(a): Simulated performance with a single 4-dB PDL element, sweeping the angle between signals’ SOP and PDL lossy axis from 0 to 360°; (b), (c): unpaired and paired signal performance with 1000 simulation runs of four cascaded PDL elements, each with 1-dB PDL and the signals’ SOPs randomly rotated between the PDL elements.

Fig. 8
Fig. 8

Transmitted constellations with optimal rotation angles at: (a) 3-dB and (b) 6-dB and (c) 9-dB ΔSNR.

Fig. 9
Fig. 9

(a) Performance comparison between 45° and optimal angle; recovered signal constellation 12 dB SNR and 10 dB ΔSNR when using: (b) θ = 45° and (c) optimal angle for Pairwise Coding, and (d) WHT coding scheme.

Equations (19)

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T X n =( X θ,n )+( Y θ,n )= a n cosθ b n sinθ+( c n cosθ d n sinθ )j T Y n =( X θ,n )+( Y θ,n )= a n sinθ+ b n cosθ+( c n sinθ+ d n cosθ )j
[ ( T X n ) ( T Y n ) ( T X n ) ( T Y n ) ]= [ cosθ sinθ sinθ cosθ ] [ a n b n c n d n ].
[ RX( f ) RY( f ) ]= H linear ( f ) H PDL ( f )[ TX( f ) TY( f ) ]+[ NX( f ) NY( f ) ]
H PDL = i=1 N R α,i ( 1 γ i 0 0 1+ γ i ) R β,i
[ RX( f ) RY( f ) ]= H linear ' ( f ) [ 1η 0 0 1+η ][ TX( f ) TY( f ) ]+[ NX( f ) NY( f ) ]
[ E X n E Y n ]=[ T X n T Y n ]+[ N X n ' / 1η N Y n ' / 1+η ]
E X θ,n = 1η ( a n cosθ b n sinθ )+( N X n ' )+( 1+η ( a n sinθ+ b n cosθ )+( N Y n ' ) )j E Y θ,n = 1η ( c n cosθ d n sinθ )+( N X n ' )+( 1+η ( c n sinθ+ d n cosθ )+( N Y n ' ) )j.
D X n =arg min C k { | E X θ,n ζ k | 2 },D Y n =arg min C k { | E Y θ,n ζ k | 2 }, ζ k =( C k e jθ ) 1η +j( C k e jθ ) 1+η
SE R pairwise,QPSK k=1 4 0.25×Prob( D X n C k | X n = C k ) =0.5×Prob( D X n C 1 | X n = C 1 )+0.5×Prob( D X n C 2 | X n = C 2 ) =0.5×( Prob( D 1 > D 2 | X n = C 1 )+Prob( D 1 > D 3 | X n = C 1 )+Prob( D 1 > D 4 | X n = C 1 ) ) +0.5×( Prob( D 2 > D 1 | X n = C 2 )+Prob( D 2 > D 3 | X n = C 2 )+Prob( D 2 > D 4 | X n = C 2 ) ).
D 1 = | ( N X n ' )+( N Y n ' )j | 2 = ( N X n ' ) 2 + ( N Y n ' ) 2 D 2 =| 2 1η sinθ+( N X n ' )+( 2 1+η cosθ+( N Y n ' ) )j | 2 =( 4( 1η ) sin 2 θ4 1η sinθ( N X n ' )+ ( N X n ' ) 2 ) +( 4( 1+η ) cos 2 θ+4 1+η cosθ( N Y n ' )+ ( N Y n ' ) 2 ) Prob( D 1 > D 2 | X n = C 1 ) =Prob( 4 1+η cosθ( N Y n ' )4 1η sinθ( N X n ' )<4( 1η ) sin 2 θ4( 1+η ) cos 2 θ ) =0.5( 1+erf( 1+η( sin 2 θ cos 2 θ ) 2+2η( cos 2 θ sin 2 θ ) σ ) ).
SE R pairwise,QPSK =0.5( 1+erf( 1+η( sin 2 θ cos 2 θ ) 2+2η( cos 2 θ sin 2 θ ) σ ) )+0.5( 1+erf( 1+η( cos 2 θ sin 2 θ ) 2+2η( sin 2 θ cos 2 θ ) σ ) ) +0.25( 1+erf( 1+2ηsinθcosθ σ ) )+0.25( 1+erf( 12ηsinθcosθ σ ) )
SE R pairwise,QPSK, 45 o =( 1+erf( 1 2 σ ) )+0.25( 1+erf( 1+η σ ) )+0.25( 1+erf( 1η σ ) ).
SE R QPSK =0.5( 1erf( 1η 2 σ 2 ) )+0.5( 1erf( 1+η 2 σ 2 ) ).
BE R pairwise,QPSK =0.25( 1+erf( 1+η( sin 2 θ cos 2 θ ) 2+2η( cos 2 θ sin 2 θ ) σ ) )+0.25( 1+erf( 1+η( cos 2 θ sin 2 θ ) 2+2η( sin 2 θ cos 2 θ ) σ ) ) +0.25( 1+erf( 1+2ηsinθcosθ σ ) )+0.25( 1+erf( 12ηsinθcosθ σ ) ).
θ opt ={ π/4ΔSNR3 tan 1 [ ( ΔSNR1 ) ( ΔSNR1 ) 2 ΔSNR ]ΔSNR>3
[ T X n T Y n ]= 1 2 [ 1 1 1 1 ][ X n Y n ].
[ T X n1 T Y n1 T X n T Y n ]= [ X n1 Y n1 conj( Y n1 ) conj( X n1 ) ].
[ T X n1 T Y n1 T X n T Y n ]= 1 5 [ ( 1+jjα )( X n1 +α X n ) ( j1+ α ¯ )( Y n1 + α ¯ Y n ) ( 1+jjα )( Y n1 +α Y n ) ( 1+jj α ¯ )( X n1 + α ¯ X n ) ]
[ T X n1 T Y n1 T X n T Y n ]= [ X n1 + X n ¯ Y n1 Y n ¯ conj( Y n1 )conj( Y n ¯ ) conj( X n1 )conj( X n ¯ ) ] [ X n ¯ Y n ¯ ]= 1 7 [ 1+j 1+2j 1+2j 1j ][ X n Y n ].

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