Abstract

An approximate bit-error rate (BER) expression of 8-star quadrature amplitude modulation (QAM) in the presence of a phase estimation error is derived. The accuracy of the approximate BER is verified via numerical integration of the conditional BER and the Monte-Carlo (MC) simulations. This approximation allows quick estimation of the BER performance and prediction of laser linewidth tolerance, and also facilitates optimization of the ring ratio.

© 2012 OSA

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References

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  1. K. P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).
  2. K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).
  3. P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993).
    [CrossRef]
  4. Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun.142(5), 292–296 (1995).
    [CrossRef]
  5. M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun.43(2), 1192–1201 (1995).
    [CrossRef]
  6. G. Goldfarb and G. Li, “BER estimation of QPSK homodyne detection with carrier phase estimation using digital signal processing,” Opt. Express14(18), 8043–8053 (2006).
    [CrossRef] [PubMed]
  7. C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express18(12), 12088–12103 (2010).
    [CrossRef] [PubMed]
  8. Y. Li, S. Xu, and H. Yang, “Design of signal constellations in the presence of phase noise,” in Proc. VTC’08-Fall, (Sept. 21–24, 2008, Calgary, Canada), pp. 1–5.
  9. L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun.4(5), 2600–2609 (2005).
    [CrossRef]
  10. 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]
  11. W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall, 1973).
  12. A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding (McGraw-Hill, 1979).

2010

2006

2005

L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun.4(5), 2600–2609 (2005).
[CrossRef]

1995

Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun.142(5), 292–296 (1995).
[CrossRef]

M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun.43(2), 1192–1201 (1995).
[CrossRef]

1993

P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993).
[CrossRef]

1984

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).

Chen, J.

Cramer, R. J. M.

M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun.43(2), 1192–1201 (1995).
[CrossRef]

Dong, X.

L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun.4(5), 2600–2609 (2005).
[CrossRef]

Fitz, M. P.

M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun.43(2), 1192–1201 (1995).
[CrossRef]

Goldfarb, G.

Henmi, N.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).

Kam, P. Y.

C. Yu, S. Zhang, P. Y. Kam, and J. Chen, “Bit-error rate performance of coherent optical M-ary PSK/QAM using decision-aided maximum likelihood phase estimation,” Opt. Express18(12), 12088–12103 (2010).
[CrossRef] [PubMed]

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]

Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun.142(5), 292–296 (1995).
[CrossRef]

P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993).
[CrossRef]

Kikuchi, K.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).

Li, G.

Nagamatsu, M.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).

Okoshi, T.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).

Some, Y. K.

Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun.142(5), 292–296 (1995).
[CrossRef]

P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993).
[CrossRef]

Teo, S. K.

P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993).
[CrossRef]

Tjhung, T. T.

P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993).
[CrossRef]

Xiao, L.

L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun.4(5), 2600–2609 (2005).
[CrossRef]

Yu, C.

Zhang, S.

IEE Proc. Commun.

Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun.142(5), 292–296 (1995).
[CrossRef]

IEEE Trans. Commun.

M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun.43(2), 1192–1201 (1995).
[CrossRef]

P. Y. Kam, S. K. Teo, Y. K. Some, and T. T. Tjhung, “Approximate results for the bit error probability of binary PSK with noisy phase reference,” IEEE Trans. Commun.41(7), 1020–1022 (1993).
[CrossRef]

IEEE Trans. Wireless Commun.

L. Xiao and X. Dong, “The exact transition probability and bit error probability of two-dimensional signaling,” IEEE Trans. Wireless Commun.4(5), 2600–2609 (2005).
[CrossRef]

J. Lightwave Technol.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit error rate in coherent optical communications due to spectral spread of the transmitter and the local oscillator,” J. Lightwave Technol.LT-2, 103–112 (1984).

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]

Opt. Express

Other

W. C. Lindsey and M. K. Simon, Telecommunication Systems Engineering (Prentice-Hall, 1973).

A. J. Viterbi and J. K. Omura, Principles of Digital Communication and Coding (McGraw-Hill, 1979).

K. P. Ho, Phase-Modulated Optical Communication Systems (Springer, 2005).

Y. Li, S. Xu, and H. Yang, “Design of signal constellations in the presence of phase noise,” in Proc. VTC’08-Fall, (Sept. 21–24, 2008, Calgary, Canada), pp. 1–5.

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Figures (7)

Fig. 1
Fig. 1

Constellation map, decision boundaries and Gray code mapping of 8-star QAM with a phase error Δθ.

Fig. 2
Fig. 2

The BERs of 8-star QAM from analysis and MC simulations with different phase error variance.

Fig. 3
Fig. 3

Comparison between numerical BER, MC simulations, and approximate BER under different laser linewidth with optimum memory length.

Fig. 4
Fig. 4

The SNR/bit penalty as a function of linewidth per laser symbol duration product ( Δν T s ) when BER = 10−4 from numerical integration and approximation.

Fig. 5
Fig. 5

Optimal ring ratio as a function of SNR/symbol phase error variance product.

Fig. 6
Fig. 6

BER as a function of ring ratio β from numerical integration given γ b =15dB and σ Δθ 2 =1× 10 3 ra d 2 .

Fig. 7
Fig. 7

SNR penalty due to ring ratio shift

Equations (24)

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P( e B )= π π P( e B |Δθ)p(Δθ)dΔθ ,
R in 2 + R out 2 2 = α 2 × γ s + β 2 × α 2 × γ s 2 = γ s .
α 2 ( β 2 +1) 2 =1, β>1 and 0<α<1.
P( e B |Δθ)=4× 1 8 P( e B |Δθ,000)+4× 1 8 P( e B |Δθ,100)= 1 2 P( e B |Δθ,000)+ 1 2 P( e B |Δθ,100),
P( e B |Δθ,000)= 1 3 [P( A 1 |Δθ,000)+2P( A 2 |Δθ,000)+P( A 3 |Δθ,000)+P( A 4 |Δθ,000) +2P( A 5 |Δθ,000)+3P( A 6 |Δθ,000)+2P( A 7 |Δθ,000)],
P( e B |Δθ,000)= 1 3 [ P( U 1 |Δθ,000)+P( U 2 |Δθ,000)+P( A 4 + A 5 + A 6 + A 7 |Δθ,000) ].
P( U 1 |Δθ,000)=P[ n>α γ s sin( π 4 Δθ ) ]= 1 2 erfc[ α γ s sin( π 4 Δθ ) ],
P( U 2 |Δθ,000)=P[ n>α γ s cos( π 4 Δθ ) ]= 1 2 erfc[ α γ s cos( π 4 Δθ ) ],
P( A 4 + A 5 + A 6 + A 7 |Δθ,000)=1P( A 1 + A 2 + A 3 + A 0 |Δθ,000) =1P[ γ s α 2 ( 2cosΔθβ1 )< n 1 < γ s α 2 ( 2cosΔθ+β+1 ), γ s α 2 (2sinΔθ+β+1)< n 2 < γ s α 2 (2sinΔθ+β+1) ] =1 1 2 erfc[ γ s α 2 ( 2cosΔθβ1 ) ][ 1 1 2 erfc[ γ s α 2 ( 2cosΔθ+β+1 ) ] ] 1 2 erfc[ γ s α 2 (2sinΔθ+β+1) ][ 1 1 2 erfc[ γ s α 2 (2sinΔθ+β+1) ] ].
P( e B |Δθ,100)= 1 3 [ P( U 3 |Δθ,100)+P( U 4 |Δθ,100)+P( A 1 + A 2 + A 3 + A 0 |Δθ,100) ],
P( U 3 |Δθ,100)= 1 2 erfc[ αβ γ s sin( π 4 Δθ ) ],
P( U 4 |Δθ,100)= 1 2 erfc[ αβ γ s cos( π 4 Δθ ) ],
P( A 1 + A 2 + A 3 + A 0 |Δθ,100)= 1 2 erfc[ γ s α 2 (2βcosΔθβ1) ][ 1 1 2 erfc[ γ s α 2 (2βcosΔθ+β+1) ] ] 1 2 erfc[ γ s α 2 (2βsinΔθ+β+1) ][ 1 1 2 erfc[ γ s α 2 (2βsinΔθ+β+1) ] ].
σ Δθ 2 = 2 L 2 +3L+1 6L σ p 2 + 1 2L σ n' 2 ,
P( e B |Δθ) 1 12 { erfc[ α γ s sin( π 4 Δθ ) ]+erfc[ α γ s cos( π 4 Δθ ) ] +erfc[ γ s α 2 ( 2cosΔθ+β+1 ) ] } + 1 12 { erfc[ αβ γ s sin( π 4 Δθ ) ]+erfc[ αβ γ s cos( π 4 Δθ ) ] +erfc[ γ s α 2 (2βcosΔθβ1) ] }.
P ( e B ) 1st = π π erfc[ α γ s sin( π 4 Δθ ) ] 1 2π σ Δθ 2 exp( Δ θ 2 2 σ Δθ 2 ) dΔθ.
P ( e B ) 1st 1 2 π 2 σ Δθ 2 exp[ 1 2 α 2 γ s + 1 2 α 4 ( γ s ) 2 σ Δθ 2 ] π π exp[ 1 2 σ Δθ 2 ( Δθ α 2 γ s σ Δθ 2 ) 2 ] dΔθ = 1 π exp[ 1 2 α 2 γ s + 1 2 α 4 ( γ s ) 2 σ Δθ 2 ],
P( e B ) 1 12 1 π { exp[ 1 2 α 2 γ s ( α 2 γ s σ Δθ 2 1 ) ]+exp[ 1 2 α 2 β 2 γ s ( α 2 β 2 γ s σ Δθ 2 1 ) ] + 1 2 { [ 1+ α 2 ( β1 ) σ Δθ 2 γ s ] 1 2 + [ 1+ α 2 β( β1 ) σ Δθ 2 γ s ] 1 2 }exp[ γ s α 2 ( β1 ) 2 4 ] }.
1 12 1 π { exp[ 1 2 α 2 γ s ( α 2 γ s σ Δθ 2 1 ) ]+exp[ γ s α 2 ( β1 ) 2 4 ] }.
2 2 γ s σ Δθ 2 α 3 α 2 2 α+1=0.
A= 2 48 ( γ s σ Δθ 2 ) 2 + 2 864 ( γ s σ Δθ 2 ) 3 2 8 γ s σ Δθ 2 ,
B= A 2 [ 1 6 γ s σ Δθ 2 + 1 72 ( γ s σ Δθ 2 ) 2 ] 3 ,
α opt = 1 6 2 γ s σ Δθ 2 + 1 3 i 2 A+ B 3 + 1+ 3 i 2 A B 3 ,
β opt = 2 α opt 2 1 .

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