Abstract

We present a novel statistical moments-based method for optical signal-to-noise ratio (OSNR) monitoring in polarization-multiplexed (pol-mux) coherent optical systems. This technique only requires the knowledge of the envelope of the equalized signal before phase correction, which can be achieved by using any two arbitrary statistical moments, and it is suitable for both constant and non-constant modulus modulation formats. The proposed estimation method is experimentally demonstrated for 10-Gbaud pol-mux coherent systems using QPSK and 16-QAM. Additionally, numerical simulations are carried out to demonstrate 20-Gbaud systems using 16-QAM and 64-QAM. The results show that the OSNR can be estimated accurately over a wide range of values for QPSK, 16-QAM or 64-QAM systems up to 1920-km long and with up to 50-ps all-order polarization mode dispersion. By setting a proper reference value for calibration, the proposed algorithm also shows good tolerance when the received signal is not well compensated.

© 2012 OSA

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References

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  1. J. Yu and X. Zhou, “Ultra-high-capacity DWDM transmission system for 100G and beyond,” IEEE Commun. Mag.48(3), S56–S64 (2010).
    [CrossRef]
  2. S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express16(2), 804–817 (2008).
    [CrossRef] [PubMed]
  3. K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
    [CrossRef]
  4. D. C. Kilper, R. Bach, D. J. Blumenthal, D. Einstein, T. Landolsi, L. Ostar, M. Preiss, and A. E. Willner, “Optical performance monitoring,” J. Lightwave Technol.22(1), 294–304 (2004).
    [CrossRef]
  5. W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
    [CrossRef]
  6. Y. C. Chung, “Polarization-nulling method for monitoring OSNR in WDM networks,” in Optical Fiber Communication Conference (OFC), paper OThP3 (2006).
  7. T. S. R. Shen, A. P. T. Lau, and G. N. Liu, “OSNR monitoring for higher order modulation formats using asynchronous amplitude histogram,” IEEE Photon. Technol. Lett.22, 1632–1634 (2010).
  8. S. D. Dods, T. B. Anderson, K. Clarke, M. Bakaul, and A. Kowalczyk, “Asynchronous sampling for optical performance monitoring,” in Optical Fiber Communication Conference (OFC), paper OMM5 (2007).
  9. J. A. Jargon, X. Wu, H. Y. Choi, Y. C. Chung, and A. E. Willner, “Optical performance monitoring of QPSK data channels by use of neural networks trained with parameters derived from asynchronous constellation diagrams,” Opt. Express18(5), 4931–4938 (2010).
    [CrossRef] [PubMed]
  10. F. N. Khan, A. P. T. Lau, C. Lu, and P. K. A. Wai, “Joint OSNR and chromatic dispersion monitoring using empirical moments of asynchronously sampled signal amplitudes,” Opto-Electronics and Communications Conference, Paper 6B4_5 (2011).
  11. M. S. Faruk and K. Kikuchi, “Monitoring of optical signal-to-noise ratio using statistical moments of adaptive-equalizer output in coherent optical receivers,” Opto-Electronics and Communications Conference, 233–234 (2011).
  12. D. J. Ives, B. C. Thomasen, R. Maher, and S. Savory, “Estimating OSNR of equalised QPSK signals,” in European Conference and Exhibition on Optical Communication (ECOC), OThH5 (2011).
  13. W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express15(2), 350–356 (2007).
    [CrossRef] [PubMed]
  14. F. N. Hauske, J. Geyer, M. Kuschnerov, K. Piyawanno, B. Lankl, T. Duthel, C. R. S. Fludger, D. van den Borne, E. D. Schmidt, and B. Spinnler, “Optical performance monitoring from FIR filter coefficients in coherent receivers,” in Proc. Fiber Commun. Conf., paper OthW2 (2008).
  15. M. P. Dlubek, A. J. Phillips, and E. C. Larkins, “Optical signal quality metric based on statistical moments and Laguerre expansion,” Opt. Quantum Electron.40(8), 561–575 (2008).
    [CrossRef]
  16. B. Porat and B. Friedlander, “Blind equalization of digital communication channels using high-order moments,” IEEE Trans. Signal Process.39(2), 522–526 (1991).
    [CrossRef]
  17. B. Spinnler, “Equalizer design and complexity for digital coherent receivers,” IEEE Sel. Top. Quantum Electron.16(5), 1180–1192 (2010).
    [CrossRef]
  18. M. S. Faruk and K. Kikuchi, “Adaptive frequency-domain equalization in digital coherent optical receivers,” Opt. Express19(13), 12789–12798 (2011).
    [CrossRef] [PubMed]
  19. J. J. K. O. Ruanaidh and W. J. Fitzgerald, Numerical Bayesian Methods applied to Signal Processing (Springer-Verlag, 1995).
  20. G. L. Stuber, Principles of Mobile Communication (Norwell, MA: Kluwer, 1996).
  21. G. Ping and C. Tepedelenlioglu, “SNR estimation for nonconstant modulus constellations,” IEEE Trans. Signal Process.53(3), 865–870 (2005).
    [CrossRef]
  22. A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
    [CrossRef]
  23. R. Kudo, T. Kobayashi, K. Ishihara, Y. Takatori, A. Sano, and Y. Miyamoto, “Coherent optical single carrier transmission using overlap frequency domain equalization for long-haul optical systems,” J. Lightwave Technol.27(16), 3721–3728 (2009).
    [CrossRef]
  24. S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Commun.16, 1451–1458 (1998).

2012 (1)

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

2011 (1)

2010 (5)

J. A. Jargon, X. Wu, H. Y. Choi, Y. C. Chung, and A. E. Willner, “Optical performance monitoring of QPSK data channels by use of neural networks trained with parameters derived from asynchronous constellation diagrams,” Opt. Express18(5), 4931–4938 (2010).
[CrossRef] [PubMed]

J. Yu and X. Zhou, “Ultra-high-capacity DWDM transmission system for 100G and beyond,” IEEE Commun. Mag.48(3), S56–S64 (2010).
[CrossRef]

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
[CrossRef]

B. Spinnler, “Equalizer design and complexity for digital coherent receivers,” IEEE Sel. Top. Quantum Electron.16(5), 1180–1192 (2010).
[CrossRef]

T. S. R. Shen, A. P. T. Lau, and G. N. Liu, “OSNR monitoring for higher order modulation formats using asynchronous amplitude histogram,” IEEE Photon. Technol. Lett.22, 1632–1634 (2010).

2009 (1)

2008 (2)

S. J. Savory, “Digital filters for coherent optical receivers,” Opt. Express16(2), 804–817 (2008).
[CrossRef] [PubMed]

M. P. Dlubek, A. J. Phillips, and E. C. Larkins, “Optical signal quality metric based on statistical moments and Laguerre expansion,” Opt. Quantum Electron.40(8), 561–575 (2008).
[CrossRef]

2007 (1)

2005 (2)

G. Ping and C. Tepedelenlioglu, “SNR estimation for nonconstant modulus constellations,” IEEE Trans. Signal Process.53(3), 865–870 (2005).
[CrossRef]

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
[CrossRef]

2004 (1)

1998 (1)

S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Commun.16, 1451–1458 (1998).

1991 (1)

B. Porat and B. Friedlander, “Blind equalization of digital communication channels using high-order moments,” IEEE Trans. Signal Process.39(2), 522–526 (1991).
[CrossRef]

Alamouti, S.

S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Commun.16, 1451–1458 (1998).

Anderson, T.

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

Bach, R.

Beckett, D.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
[CrossRef]

Berthold, J.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
[CrossRef]

Blumenthal, D. J.

Boertjes, D.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
[CrossRef]

Chen, S.

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

Chen, W.

W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express15(2), 350–356 (2007).
[CrossRef] [PubMed]

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
[CrossRef]

Choi, H. Y.

Chung, Y. C.

Dlubek, M. P.

M. P. Dlubek, A. J. Phillips, and E. C. Larkins, “Optical signal quality metric based on statistical moments and Laguerre expansion,” Opt. Quantum Electron.40(8), 561–575 (2008).
[CrossRef]

Do, C. C.

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

Einstein, D.

Evans, J.

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
[CrossRef]

Faruk, M. S.

Friedlander, B.

B. Porat and B. Friedlander, “Blind equalization of digital communication channels using high-order moments,” IEEE Trans. Signal Process.39(2), 522–526 (1991).
[CrossRef]

Hewitt, D.

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

Ishihara, K.

Jargon, J. A.

Kikuchi, K.

Kilper, D. C.

Kobayashi, T.

Kudo, R.

Landolsi, T.

Laperle, C.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
[CrossRef]

Larkins, E. C.

M. P. Dlubek, A. J. Phillips, and E. C. Larkins, “Optical signal quality metric based on statistical moments and Laguerre expansion,” Opt. Quantum Electron.40(8), 561–575 (2008).
[CrossRef]

Lau, A. P. T.

T. S. R. Shen, A. P. T. Lau, and G. N. Liu, “OSNR monitoring for higher order modulation formats using asynchronous amplitude histogram,” IEEE Photon. Technol. Lett.22, 1632–1634 (2010).

Liu, G. N.

T. S. R. Shen, A. P. T. Lau, and G. N. Liu, “OSNR monitoring for higher order modulation formats using asynchronous amplitude histogram,” IEEE Photon. Technol. Lett.22, 1632–1634 (2010).

Miyamoto, Y.

Ostar, L.

Pendock, G.

Phillips, A. J.

M. P. Dlubek, A. J. Phillips, and E. C. Larkins, “Optical signal quality metric based on statistical moments and Laguerre expansion,” Opt. Quantum Electron.40(8), 561–575 (2008).
[CrossRef]

Ping, G.

G. Ping and C. Tepedelenlioglu, “SNR estimation for nonconstant modulus constellations,” IEEE Trans. Signal Process.53(3), 865–870 (2005).
[CrossRef]

Porat, B.

B. Porat and B. Friedlander, “Blind equalization of digital communication channels using high-order moments,” IEEE Trans. Signal Process.39(2), 522–526 (1991).
[CrossRef]

Preiss, M.

Roberts, K.

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
[CrossRef]

Sano, A.

Savory, S. J.

Shen, T. S. R.

T. S. R. Shen, A. P. T. Lau, and G. N. Liu, “OSNR monitoring for higher order modulation formats using asynchronous amplitude histogram,” IEEE Photon. Technol. Lett.22, 1632–1634 (2010).

Shieh, W.

W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express15(2), 350–356 (2007).
[CrossRef] [PubMed]

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
[CrossRef]

Skafidas, E.

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

Spinnler, B.

B. Spinnler, “Equalizer design and complexity for digital coherent receivers,” IEEE Sel. Top. Quantum Electron.16(5), 1180–1192 (2010).
[CrossRef]

Takatori, Y.

Tepedelenlioglu, C.

G. Ping and C. Tepedelenlioglu, “SNR estimation for nonconstant modulus constellations,” IEEE Trans. Signal Process.53(3), 865–870 (2005).
[CrossRef]

Tran, A. V.

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

Tucker, R. S.

W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express15(2), 350–356 (2007).
[CrossRef] [PubMed]

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
[CrossRef]

Willner, A. E.

Wu, X.

Yi, X.

W. Shieh, R. S. Tucker, W. Chen, X. Yi, and G. Pendock, “Optical performance monitoring in coherent optical OFDM systems,” Opt. Express15(2), 350–356 (2007).
[CrossRef] [PubMed]

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
[CrossRef]

Yu, J.

J. Yu and X. Zhou, “Ultra-high-capacity DWDM transmission system for 100G and beyond,” IEEE Commun. Mag.48(3), S56–S64 (2010).
[CrossRef]

Zhou, X.

J. Yu and X. Zhou, “Ultra-high-capacity DWDM transmission system for 100G and beyond,” IEEE Commun. Mag.48(3), S56–S64 (2010).
[CrossRef]

Zhu, C.

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

IEEE Commun. Mag. (2)

J. Yu and X. Zhou, “Ultra-high-capacity DWDM transmission system for 100G and beyond,” IEEE Commun. Mag.48(3), S56–S64 (2010).
[CrossRef]

K. Roberts, D. Beckett, D. Boertjes, J. Berthold, and C. Laperle, “100G and beyond with digital coherent signal processing,” IEEE Commun. Mag.48(7), 62–69 (2010).
[CrossRef]

IEEE J. Commun. (1)

S. Alamouti, “A simple transmit diversity technique for wireless communications,” IEEE J. Commun.16, 1451–1458 (1998).

IEEE Photon. Technol. Lett. (3)

W. Chen, R. S. Tucker, X. Yi, W. Shieh, and J. Evans, “Optical signal-to-noise ratio monitoring using uncorrelated beat noise,” IEEE Photon. Technol. Lett.17(11), 2484–2486 (2005).
[CrossRef]

T. S. R. Shen, A. P. T. Lau, and G. N. Liu, “OSNR monitoring for higher order modulation formats using asynchronous amplitude histogram,” IEEE Photon. Technol. Lett.22, 1632–1634 (2010).

A. V. Tran, C. Zhu, C. C. Do, S. Chen, T. Anderson, D. Hewitt, and E. Skafidas, “8×40-Gb/s optical coherent pol-mux single carrier system with frequency domain equalization and training sequences,” IEEE Photon. Technol. Lett.24(11), 885–887 (2012).
[CrossRef]

IEEE Sel. Top. Quantum Electron. (1)

B. Spinnler, “Equalizer design and complexity for digital coherent receivers,” IEEE Sel. Top. Quantum Electron.16(5), 1180–1192 (2010).
[CrossRef]

IEEE Trans. Signal Process. (2)

B. Porat and B. Friedlander, “Blind equalization of digital communication channels using high-order moments,” IEEE Trans. Signal Process.39(2), 522–526 (1991).
[CrossRef]

G. Ping and C. Tepedelenlioglu, “SNR estimation for nonconstant modulus constellations,” IEEE Trans. Signal Process.53(3), 865–870 (2005).
[CrossRef]

J. Lightwave Technol. (2)

Opt. Express (4)

Opt. Quantum Electron. (1)

M. P. Dlubek, A. J. Phillips, and E. C. Larkins, “Optical signal quality metric based on statistical moments and Laguerre expansion,” Opt. Quantum Electron.40(8), 561–575 (2008).
[CrossRef]

Other (8)

J. J. K. O. Ruanaidh and W. J. Fitzgerald, Numerical Bayesian Methods applied to Signal Processing (Springer-Verlag, 1995).

G. L. Stuber, Principles of Mobile Communication (Norwell, MA: Kluwer, 1996).

S. D. Dods, T. B. Anderson, K. Clarke, M. Bakaul, and A. Kowalczyk, “Asynchronous sampling for optical performance monitoring,” in Optical Fiber Communication Conference (OFC), paper OMM5 (2007).

F. N. Khan, A. P. T. Lau, C. Lu, and P. K. A. Wai, “Joint OSNR and chromatic dispersion monitoring using empirical moments of asynchronously sampled signal amplitudes,” Opto-Electronics and Communications Conference, Paper 6B4_5 (2011).

M. S. Faruk and K. Kikuchi, “Monitoring of optical signal-to-noise ratio using statistical moments of adaptive-equalizer output in coherent optical receivers,” Opto-Electronics and Communications Conference, 233–234 (2011).

D. J. Ives, B. C. Thomasen, R. Maher, and S. Savory, “Estimating OSNR of equalised QPSK signals,” in European Conference and Exhibition on Optical Communication (ECOC), OThH5 (2011).

F. N. Hauske, J. Geyer, M. Kuschnerov, K. Piyawanno, B. Lankl, T. Duthel, C. R. S. Fludger, D. van den Borne, E. D. Schmidt, and B. Spinnler, “Optical performance monitoring from FIR filter coefficients in coherent receivers,” in Proc. Fiber Commun. Conf., paper OthW2 (2008).

Y. C. Chung, “Polarization-nulling method for monitoring OSNR in WDM networks,” in Optical Fiber Communication Conference (OFC), paper OThP3 (2006).

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Figures (11)

Fig. 1
Fig. 1

Constellation diagram of (a) QPSK, (b) 16-QAM and (c) 64-QAM.

Fig. 2
Fig. 2

Experimental setup, LPF: low-pass filter, AMP: amplifier, PBS: polarization beam splitter.

Fig. 3
Fig. 3

DSP procedures at the (a) transmitter and (b) receiver.

Fig. 4
Fig. 4

Constellation diagrams of equalized QPSK signals after 400-km with different OSNR: (a) 20.4 dB (b) 14.1dB and (c) 10.5 dB.

Fig. 5
Fig. 5

Constellation diagrams of equalized 16-QAM signals after 400-km with different OSNR: (a) 21.8 dB (b) 17.3 dB and (c) 13 dB.

Fig. 6
Fig. 6

Estimation error versus true OSNR for (a) QPSK and (b) 16-QAM.

Fig. 7
Fig. 7

Estimation performance for 16-QAM transmission with: (a) different mean DGD values with 1120-km transmission, (b) different transmission distances with 30-ps mean DGD.

Fig. 8
Fig. 8

Estimation performance for 16-QAM with different length of equalizer.

Fig. 9
Fig. 9

Estimation performance with different launched powers for 16-QAM transmission.

Fig. 10
Fig. 10

(a) Fully equalized X-polarization constellation diagrams for 64-QAM signal after 1120-km transmission and 30-ps DGD; (b) Estimation performance.

Fig. 11
Fig. 11

Estimation performance using training-aided and blind equalization techniques.

Tables (2)

Tables Icon

Table 1 BER for Different Equalizer Length for 16-QAM System

Tables Icon

Table 2 BER for Different Equalizer Length for 64-QAM System

Equations (11)

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

r k = S s n +N
η=SNR= n=1 m p n S Z n 2 2 σ 2 = S 2 σ 2 .
P PSK ( r k | Z n ,θ,σ)= 1 2π σ 2 exp( | r k Z n e jθ | 2 2 σ 2 )
P PSK (| r k |)= | r k | σ 2 exp(ηZ | r k | 2 2 σ 2 ) I 0 ( | r k | σ 2ηZ )
ρ i,PSK (σ,η)=E[| r k | i ]= (2 σ 2 ) i/2 Γ(i/2+1)exp(η Z 2 ) 1 F 1 (i/2+1;1;η Z 2 )
ρ i (σ,η)= n=1 m p n (2 σ 2 ) i/2 Γ(i/2+1)exp(η Z n 2 ) 1 F 1 (i/2+1;1;η Z n 2 ) .
ξ f,g (η)= ρ f μ/f ρ g μ/g
η f,g = ξ f,g 1 ( ρ f μ/f ρ g μ/g )
ρ 2 =(2 σ 2 )(η+1) and ρ 4 = (2 σ 2 ) 2 (α η 2 +4η+2)
η 2,4 = ρ 4 2 ρ 2 2 (2α)(2 ρ 2 4 ρ 2 2 ρ 4 ) α ρ 2 2 ρ 4 .
OSN R dB =10 log 10 ( B ref R s )10 log 10 ( 1 η 1 SN R RF )

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