Abstract

We propose a coded-modulation scheme suitable for beyond 100Gbs transmission and 100Gbs Ethernet. It is based on polarization multiplexed coded-orthogonal frequency division multiplexing (OFDM). By using 32-QAM-based polarization multiplexed coded-OFDM, we are able to achieve the aggregate rate of 100Gbs, while the OFDM signal bandwidth is only 10GHz, resulting in a spectral efficiency of 10bitssHz. The spectral efficiency of the proposed scheme is twice higher than that of the polarization diversity OFDM scheme. We show that the proposed multicarrier scheme is insensitive to polarization mode dispersion (PMD), while the PMD represents a major source of performance degradation in single carrier systems, in addition to fiber nonlinearities. We also describe how to determine the symbols' log-likelihood ratios in the presence of laser phase noise.

© 2009 Optical Society of America

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References

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  1. I. B. Djordjevic, M. Cvijetic, L. Xu, T. Wang, “Using LDPC-coded modulation and coherent detection for ultra high-speed optical transmission,” J. Lightwave Technol., vol. 25, pp. 3619–3625, 2007.
    [Crossref]
  2. L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
    [Crossref]
  3. I. B. Djordjevic, “Suppression of intrachannel nonlinearities in high-speed WDM systems,” in Advanced Technologies for High-Speed Optical Communications, L. Xu, ed., p. 247–277, Research Signpost, 2007.
  4. N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.
  5. W. Shieh, W. Chen, R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” IEE Electron. Lett., vol. 42, pp. 996–997, 2006.
    [Crossref]
  6. W. Shieh, X. Yi, Y. Ma, Y. Tang, “Theoretical and experimental study on PMD-supported transmission using polarization diversity in coherent optical OFDM systems,” Opt. Express, vol. 15, pp. 9936–9947, 2007.
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]
  9. P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.
  10. I. B. Djordjevic, L. Xu, T. Wang, “Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes,” Opt. Express, vol. 16, pp. 10269–10278, 2008.
    [Crossref] [PubMed]
  11. H. Sun, K.-T. Wu, K. Roberts, “Real-time measurements of a 40 Gb∕s coherent system,” Opt. Express, vol. 16, pp. 873–879, 2008.
    [Crossref] [PubMed]
  12. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express, vol. 16, pp. 804–817, 2008.
    [Crossref] [PubMed]
  13. D. Penninckx, V. Morenás, “Jones matrix of polarization mode dispersion,” Opt. Lett., vol. 24, pp. 875–877, 1999.
    [Crossref]
  14. I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, paper JWA53.
  15. E. Biglieri, R. Calderbank, A. Constantinides, A. Goldsmith, A. Paulraj, H. V. Poor, MIMO Wireless Communications, Cambridge Univ. Press, 2007.
    [Crossref]
  16. J. G. Proakis, Digital Communications, Boston: McGraw-Hill, 2001.
  17. M. Cvijetic, Coherent and Nonlinear Lightwave Communications, Boston: Artech House, 1996.
  18. X. Yi, W. Shieh, Y. Ma, “Phase noise effects on high spectral efficiency coherent optical OFDM transmission,” J. Lightwave Technol., vol. 26, pp. 1309–1316, 2008.
    [Crossref]

2008 (6)

2007 (3)

2006 (1)

W. Shieh, W. Chen, R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” IEE Electron. Lett., vol. 42, pp. 996–997, 2006.
[Crossref]

1999 (1)

Alic, N.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

Batshon, H. G.

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

Benlachtar, Y.

P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.

Biglieri, E.

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

Bosco, G.

P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.

Calderbank, R.

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

Chen, W.

W. Shieh, W. Chen, R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” IEE Electron. Lett., vol. 42, pp. 996–997, 2006.
[Crossref]

Constantinides, A.

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

Cvijetic, M.

I. B. Djordjevic, M. Cvijetic, L. Xu, T. Wang, “Using LDPC-coded modulation and coherent detection for ultra high-speed optical transmission,” J. Lightwave Technol., vol. 25, pp. 3619–3625, 2007.
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

M. Cvijetic, Coherent and Nonlinear Lightwave Communications, Boston: Artech House, 1996.

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, paper JWA53.

Djordjevic, I. B.

I. B. Djordjevic, L. Xu, T. Wang, “Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes,” Opt. Express, vol. 16, pp. 10269–10278, 2008.
[Crossref] [PubMed]

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[Crossref]

I. B. Djordjevic, M. Cvijetic, L. Xu, T. Wang, “Using LDPC-coded modulation and coherent detection for ultra high-speed optical transmission,” J. Lightwave Technol., vol. 25, pp. 3619–3625, 2007.
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

I. B. Djordjevic, “Suppression of intrachannel nonlinearities in high-speed WDM systems,” in Advanced Technologies for High-Speed Optical Communications, L. Xu, ed., p. 247–277, Research Signpost, 2007.

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, paper JWA53.

Fainman, Y.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

Goldsmith, A.

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

Jiang, R.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

Killey, R. I.

P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.

Kueppers, F.

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

Ma, Y.

Marki, C.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

Minkov, L. L.

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

Morenás, V.

Papen, G. C.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

Paulraj, A.

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

Penninckx, D.

Poggiolini, P.

P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.

Poor, H. V.

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

Prat, J.

P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.

Proakis, J. G.

J. G. Proakis, Digital Communications, Boston: McGraw-Hill, 2001.

Radic, S.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

Roberts, K.

Saperstein, R. E.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

Savory, S.

P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.

Savory, S. J.

Shieh, W.

Sun, H.

Tang, Y.

Tucker, R. S.

W. Shieh, W. Chen, R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” IEE Electron. Lett., vol. 42, pp. 996–997, 2006.
[Crossref]

Wang, T.

I. B. Djordjevic, L. Xu, T. Wang, “Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes,” Opt. Express, vol. 16, pp. 10269–10278, 2008.
[Crossref] [PubMed]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

I. B. Djordjevic, M. Cvijetic, L. Xu, T. Wang, “Using LDPC-coded modulation and coherent detection for ultra high-speed optical transmission,” J. Lightwave Technol., vol. 25, pp. 3619–3625, 2007.
[Crossref]

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, paper JWA53.

Wu, K.-T.

Xu, L.

I. B. Djordjevic, L. Xu, T. Wang, “Simultaneous chromatic dispersion and PMD compensation by using coded-OFDM and girth-10 LDPC codes,” Opt. Express, vol. 16, pp. 10269–10278, 2008.
[Crossref] [PubMed]

I. B. Djordjevic, M. Cvijetic, L. Xu, T. Wang, “Using LDPC-coded modulation and coherent detection for ultra high-speed optical transmission,” J. Lightwave Technol., vol. 25, pp. 3619–3625, 2007.
[Crossref]

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, paper JWA53.

Yang, Q.

Yi, X.

IEE Electron. Lett. (1)

W. Shieh, W. Chen, R. S. Tucker, “Polarization mode dispersion mitigation in coherent optical orthogonal frequency division multiplexed systems,” IEE Electron. Lett., vol. 42, pp. 996–997, 2006.
[Crossref]

IEEE J. Sel. Areas Commun. (1)

I. B. Djordjevic, L. L. Minkov, H. G. Batshon, “Mitigation of linear and nonlinear impairments in high-speed optical networks by using LDPC-coded turbo equalization,” IEEE J. Sel. Areas Commun., vol. 26, no. 6, pp. 73–83, Aug. 2008.
[Crossref]

IEEE Photon. Technol. Lett. (1)

L. L. Minkov, I. B. Djordjevic, H. G. Batshon, L. Xu, T. Wang, M. Cvijetic, F. Kueppers, “Demonstration of PMD compensation by LDPC-coded turbo equalization and channel capacity loss characterization due to PMD and quantization,” IEEE Photon. Technol. Lett., vol. 19, pp. 1852–1854, 2007.
[Crossref]

J. Lightwave Technol. (2)

J. Opt. Netw. (1)

Opt. Express (4)

Opt. Lett. (1)

Other (7)

I. B. Djordjevic, L. Xu, T. Wang, M. Cvijetic, “Large girth low-density parity-check codes for long-haul high-speed optical communications,” in Proc. OFC/NFOEC 2008, paper JWA53.

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

J. G. Proakis, Digital Communications, Boston: McGraw-Hill, 2001.

M. Cvijetic, Coherent and Nonlinear Lightwave Communications, Boston: Artech House, 1996.

I. B. Djordjevic, “Suppression of intrachannel nonlinearities in high-speed WDM systems,” in Advanced Technologies for High-Speed Optical Communications, L. Xu, ed., p. 247–277, Research Signpost, 2007.

N. Alic, G. C. Papen, R. E. Saperstein, R. Jiang, C. Marki, Y. Fainman, S. Radic, “Experimental demonstration of 10 Gb∕s NRZ extended dispersion-limited reach over 600 km-SMF link without optical dispersion compensation,” in Proc. OFC/NFOEC 2006, Mar. 2006, paper OWB7.

P. Poggiolini, G. Bosco, S. Savory, Y. Benlachtar, R. I. Killey, J. Prat, “1,040 km uncompensated IMDD transmission over G.652 fiber at 10 Gbit∕s using a reduced-state SQRT-metric MLSE receiver,” in Proc. ECOC 2006, Sept. 2006, Cannes, France, Post-Deadline Paper Th4.4.6.

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Figures (5)

Fig. 1
Fig. 1

Magnitude response of h x x and h x y Jones matrix coefficients against the normalized frequency for (a) θ = π 2 and ε = 0 , and (b) θ = π 3 and ε = 0 .

Fig. 2
Fig. 2

The architecture of the polarization-multiplexed LDPC-coded OFDM scheme: (a) transmitter architecture, (b) OFDM transmitter architecture (x- or y-polarization), (c) receiver architecture, and (d) balanced coherent detector configuration. DFB, distributed feedback laser; PBS(C), polarization beam splitter (combiner); MZM, dual-drive Mach–Zehnder modulator; APP, a posteriory probability; LLRs, log-likelihood ratios.

Fig. 3
Fig. 3

Equivalent OFDM channel model. ϕ C D ( k ) denotes the phase distortion of the k th subcarrier due to chromatic dispersion.

Fig. 4
Fig. 4

The constellation diagrams for M = 32 QAM (the aggregate data rate is 100 Gb s ) after 1200 ps of DGD observing the worst-case scenario ( θ = π 2 , ε = 0 ) (a) before PMD compensation and (b) after PMD compensation.

Fig. 5
Fig. 5

BER performance of the proposed polarization multiplexed coded-OFDM scheme, for DGD of 1200 ps and the worst-case scenario ( θ = π 2 , ε = 0 ). R D denotes the aggregate data rate.

Equations (15)

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

H ( ω ) = [ h x x ( ω ) h x y ( ω ) h y x ( ω ) h y y ( ω ) ] = R 1 P ( ω ) R ,
P ( ω ) = [ e j ω τ 2 0 0 e j ω τ 2 ] ,
R = [ cos ( θ 2 ) e j ε 2 sin ( θ 2 ) e j ε 2 sin ( θ 2 ) e j ε 2 cos ( θ 2 ) e j ε 2 ] ,
r i , k = H ( k ) s i , k e j ϕ P N + n i , k ,
r x , i , k = h x x ( k ) s x , i , k + h x y ( k ) s y , i , k + n x , i , k ,
r y , i , k = h y x ( k ) s x , i , k + h y y ( k ) s y , i , k + n y , i , k ,
s x , i , k = h x x * ( k ) h x x ( k ) 2 [ r x , i , k h x y ( k ) h y y * ( k ) h y y ( k ) 2 r y , i , k ] 1 h x x * ( k ) h x y ( k ) h x x ( k ) 2 h y x ( k ) h y y * ( k ) h y y ( k ) 2 ,
s y , i , k = h y y * ( k ) h y y ( k ) 2 r y , i , k h y x ( k ) h y y * ( k ) h y y ( k ) 2 s x , i , k ,
s i , k = ( h x x * ( k ) r x , i , k + h x y * ( k ) r y , i , k ) ( h x x ( k ) 2 + h x y ( k ) 2 ) .
s x , i , k = h x x * ( k ) e j ϕ P N h x x ( k ) 2 [ r x , i , k h x y ( k ) h y y * ( k ) h y y ( k ) 2 r y , i , k ] 1 h x x * ( k ) h x y ( k ) h x x ( k ) 2 h y x ( k ) h y y * ( k ) h y y ( k ) 2 ,
s y , i , k = h y y * ( k ) e j ϕ P N h y y ( k ) 2 r y , i , k h y x ( k ) h y y * ( k ) h y y ( k ) 2 s x , i , k .
λ x ( y ) ( s ϕ P N ) = ( R [ s x ( ( y ) , i , k ) ( ϕ P N ) ] R [ QAM ( map ( s ) ) ] ) 2 2 σ 2 ( I [ s x ( ( y ) , i , k ) ( ϕ P N ) ] I [ QAM ( map ( s ) ) ] ) 2 2 σ 2 ;
s = 0 , 1 , , 2 n b 1
λ x ( y ) ( s ) = log e λ x ( y ) ( s ϕ P N ) 1 σ P N 2 π e ϕ P N 2 2 σ P N 2 d ϕ P N .
L ( b j , x ( y ) ) = log Σ s : b j = 0 exp { λ x ( y ) ( s ) } Σ s : b j = 1 exp { λ x ( y ) ( s ) } .