Abstract

The bit-error rate (BER) expressions of 16- phase-shift keying (PSK) and 16- quadrature amplitude modulation (QAM) are analytically obtained in the presence of a phase error. By averaging over the statistics of the phase error, the performance penalty can be analytically examined as a function of the phase error variance. The phase error variances leading to a 1-dB signal-to-noise ratio per bit penalty at BER=10−4 have been found to be 8.7×10−2 rad2, 1.2×10−2 rad2, 2.4×10−3 rad2, 6.0×10−4 rad2 and 2.3×10−3 rad2 for binary, quadrature, 8-, and 16-PSK and 16QAM, respectively. With the knowledge of the allowable phase error variance, the corresponding laser linewidth tolerance can be predicted. We extend the phase error variance analysis of decision-aided maximum likelihood carrier phase estimation in M-ary PSK to 16QAM, and successfully predict the laser linewidth tolerance in different modulation formats, which agrees well with the Monte Carlo simulations. Finally, approximate BER expressions for different modulation formats are introduced to allow a quick estimation of the BER performance as a function of the phase error variance. Further, the BER approximations give a lower bound on the laser linewidth requirements in M-ary PSK and 16QAM. It is shown that as far as laser linewidth tolerance is concerned, 16QAM outperforms 16PSK which has the same spectral efficiency (SE), and has nearly the same performance as 8PSK which has lower SE. Thus, 16-QAM is a promising modulation format for high SE coherent optical communications.

© 2010 Optical Society of America

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2010 (1)

N. Xie, T. Zhang, and E. F. Haratsch, “Improving burst error tolerance of LDPC-centric coding systems in read channel, ” IEEE Trans. Magnetics 46,933–941 (2010).
[CrossRef]

2009 (6)

H. Fu and P. Y. Kam, “A simple bit error probability analysis for square QAM in Rayleigh fading with channel estimation,” IEEE Trans. Commun. 57,2200–2206 (2009).
[CrossRef]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photon. Technol. Lett. 21,1075–1077 (2009).
[CrossRef]

S. Zhang, C. Yu, P. Y. Kam, and J. Chen, “Parallel implementation of decision-aided maximum likelihood phase estimation in coherent M-ary phase-shifted keying systems,” IEEE Photon. Technol. Lett. 21,1471–1473 (2009).
[CrossRef]

Y. Mori, C. Zhang, K. Igarashi, K. Katoh, and K. Kikuchi, “Unrepeated 200-km transmission of 40-Gbit/s 16-QAM signals using digital coherent receiver,” Opt. Express 17,1435–1441 (2009), http://www. opticsinfobase.org/abstract.cfm?uri=oe-17-3-1435.
[CrossRef] [PubMed]

X. Zhou, J. Yu, D. Qian, T. Wang, G. Zhang, and P. D. Magill, “ High-spectral-efficiency 114-Gb/s transmission using PolMux-RZ-8PSK modulation format and single-ended digital coherent detection technique,” J. Lightwave Technol. 27,146–152 (2009).
[CrossRef]

T. Pfau, S. Hoffmann, and R. Noé, “Hardware-efficient coherent digital receiver concept with feedforward carrier recovery for M-QAM constellations,” J. Lightwave Technol. 27,989–999 (2009).
[CrossRef]

2007 (2)

2006 (2)

2005 (2)

2004 (1)

C. Xu, X. Liu, and X. Wei, “Differential phase-shift keying for high spectral efficiency optical transmissions,” IEEE J. Sel. Top. Quantum Electron. 10,281–293 (2004).
[CrossRef]

2003 (1)

J. Lassing, E. Str¨om, E. Agrell, and T. Ottosson, “Computation of the exact bit-error rate of coherent M-ary PSK with gray code bit mapping,” IEEE Trans. Commun. 51,1758–1760 (2003).
[CrossRef]

1996 (1)

H. Ghafouri-Shiraz, Y. H. Heng, and T. Aruga, “Effect of phase noise on the performance of 10-Gbits/s coherent optical synchronous receivers,” Microwave Opt. Technol. Lett. 11,14–17 (1996).
[CrossRef]

1995 (2)

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

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

1993 (2)

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. 41,1020–1022 (1993).
[CrossRef]

S. Norimatsu and K. Iwashita, “Damping factor influence on linewidth requirements for optical PSK coherent detection systems,” J. Lightwave Technol. 11,1226–1233 (1993).
[CrossRef]

1986 (2)

L. G. Kazovsky, “Performance analysis and laser linewidth requirements for optical PSK heterodyne communications systems,” J. Lightwave Technol. LT-4, 415–425 (1986).
[CrossRef]

P. Y. Kam, “Maximum-likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. COM-34,522-527 (1986).

1984 (1)

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit-error rate in coherent optical communications due to spectrum spread of the transmitter and the local oscillator,” J. Lightwave Technol. LT-2,1024–1033 (1984).
[CrossRef]

1983 (1)

A. J. Viterbi and A. N. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory IT-29,543–551 (1983).
[CrossRef]

Agrell, E.

J. Lassing, E. Str¨om, E. Agrell, and T. Ottosson, “Computation of the exact bit-error rate of coherent M-ary PSK with gray code bit mapping,” IEEE Trans. Commun. 51,1758–1760 (2003).
[CrossRef]

Aruga, T.

H. Ghafouri-Shiraz, Y. H. Heng, and T. Aruga, “Effect of phase noise on the performance of 10-Gbits/s coherent optical synchronous receivers,” Microwave Opt. Technol. Lett. 11,14–17 (1996).
[CrossRef]

Chen, J.

S. Zhang, C. Yu, P. Y. Kam, and J. Chen, “Parallel implementation of decision-aided maximum likelihood phase estimation in coherent M-ary phase-shifted keying systems,” IEEE Photon. Technol. Lett. 21,1471–1473 (2009).
[CrossRef]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photon. Technol. Lett. 21,1075–1077 (2009).
[CrossRef]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” to appear in J. Lightwave Technol.

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. 43,1192–1201 (1995).
[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. 43,1192–1201 (1995).
[CrossRef]

Fu, H.

H. Fu and P. Y. Kam, “A simple bit error probability analysis for square QAM in Rayleigh fading with channel estimation,” IEEE Trans. Commun. 57,2200–2206 (2009).
[CrossRef]

Ghafouri-Shiraz, H.

H. Ghafouri-Shiraz, Y. H. Heng, and T. Aruga, “Effect of phase noise on the performance of 10-Gbits/s coherent optical synchronous receivers,” Microwave Opt. Technol. Lett. 11,14–17 (1996).
[CrossRef]

Gnauck, A. H.

Haratsch, E. F.

N. Xie, T. Zhang, and E. F. Haratsch, “Improving burst error tolerance of LDPC-centric coding systems in read channel, ” IEEE Trans. Magnetics 46,933–941 (2010).
[CrossRef]

Heng, Y. H.

H. Ghafouri-Shiraz, Y. H. Heng, and T. Aruga, “Effect of phase noise on the performance of 10-Gbits/s coherent optical synchronous receivers,” Microwave Opt. Technol. Lett. 11,14–17 (1996).
[CrossRef]

Henmi, N.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit-error rate in coherent optical communications due to spectrum spread of the transmitter and the local oscillator,” J. Lightwave Technol. LT-2,1024–1033 (1984).
[CrossRef]

Hoffmann, S.

Igarashi, K.

Ip, E.

Iwashita, K.

S. Norimatsu and K. Iwashita, “Damping factor influence on linewidth requirements for optical PSK coherent detection systems,” J. Lightwave Technol. 11,1226–1233 (1993).
[CrossRef]

Kahn, J. M.

Kalogerakis, G.

Kam, P. Y.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photon. Technol. Lett. 21,1075–1077 (2009).
[CrossRef]

S. Zhang, C. Yu, P. Y. Kam, and J. Chen, “Parallel implementation of decision-aided maximum likelihood phase estimation in coherent M-ary phase-shifted keying systems,” IEEE Photon. Technol. Lett. 21,1471–1473 (2009).
[CrossRef]

H. Fu and P. Y. Kam, “A simple bit error probability analysis for square QAM in Rayleigh fading with channel estimation,” IEEE Trans. Commun. 57,2200–2206 (2009).
[CrossRef]

Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun. 142,292–296 (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. 41,1020–1022 (1993).
[CrossRef]

P. Y. Kam, “Maximum-likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. COM-34,522-527 (1986).

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” to appear in J. Lightwave Technol.

Katoh, K.

Kazovsky, L. G.

L. G. Kazovsky, G. Kalogerakis, and W.-T. Shaw, “Homodyne phase-shift-keying systems: Past challenges and future opportunities,” J. Lightwave Technol. 24,4876–4884 (2006).
[CrossRef]

L. G. Kazovsky, “Performance analysis and laser linewidth requirements for optical PSK heterodyne communications systems,” J. Lightwave Technol. LT-4, 415–425 (1986).
[CrossRef]

Kikuchi, K.

Lassing, J.

J. Lassing, E. Str¨om, E. Agrell, and T. Ottosson, “Computation of the exact bit-error rate of coherent M-ary PSK with gray code bit mapping,” IEEE Trans. Commun. 51,1758–1760 (2003).
[CrossRef]

Liu, X.

C. Xu, X. Liu, and X. Wei, “Differential phase-shift keying for high spectral efficiency optical transmissions,” IEEE J. Sel. Top. Quantum Electron. 10,281–293 (2004).
[CrossRef]

Ly-Gagnon, D.-S.

Magill, P. D.

Mori, Y.

Nagamatsu, M.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit-error rate in coherent optical communications due to spectrum spread of the transmitter and the local oscillator,” J. Lightwave Technol. LT-2,1024–1033 (1984).
[CrossRef]

Noé, R.

Noelle, M.

Norimatsu, S.

S. Norimatsu and K. Iwashita, “Damping factor influence on linewidth requirements for optical PSK coherent detection systems,” J. Lightwave Technol. 11,1226–1233 (1993).
[CrossRef]

Okoshi, T.

K. Kikuchi, T. Okoshi, M. Nagamatsu, and N. Henmi, “Degradation of bit-error rate in coherent optical communications due to spectrum spread of the transmitter and the local oscillator,” J. Lightwave Technol. LT-2,1024–1033 (1984).
[CrossRef]

Ottosson, T.

J. Lassing, E. Str¨om, E. Agrell, and T. Ottosson, “Computation of the exact bit-error rate of coherent M-ary PSK with gray code bit mapping,” IEEE Trans. Commun. 51,1758–1760 (2003).
[CrossRef]

Patzak, E.

Pfau, T.

Qian, D.

Seimetz, M.

Shaw, W.-T.

Some, Y. K.

Y. K. Some and P. Y. Kam, “Bit-error probability of QPSK with noisy phase reference,” IEE Proc. Commun. 142,292–296 (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. 41,1020–1022 (1993).
[CrossRef]

Str¨om, E.

J. Lassing, E. Str¨om, E. Agrell, and T. Ottosson, “Computation of the exact bit-error rate of coherent M-ary PSK with gray code bit mapping,” IEEE Trans. Commun. 51,1758–1760 (2003).
[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. 41,1020–1022 (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. 41,1020–1022 (1993).
[CrossRef]

Tsukamoto, S.

Viterbi, A. J.

A. J. Viterbi and A. N. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory IT-29,543–551 (1983).
[CrossRef]

Viterbi, A. N.

A. J. Viterbi and A. N. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory IT-29,543–551 (1983).
[CrossRef]

Wang, T.

Wei, X.

C. Xu, X. Liu, and X. Wei, “Differential phase-shift keying for high spectral efficiency optical transmissions,” IEEE J. Sel. Top. Quantum Electron. 10,281–293 (2004).
[CrossRef]

Winzer, P. J.

Xie, N.

N. Xie, T. Zhang, and E. F. Haratsch, “Improving burst error tolerance of LDPC-centric coding systems in read channel, ” IEEE Trans. Magnetics 46,933–941 (2010).
[CrossRef]

Xu, C.

C. Xu, X. Liu, and X. Wei, “Differential phase-shift keying for high spectral efficiency optical transmissions,” IEEE J. Sel. Top. Quantum Electron. 10,281–293 (2004).
[CrossRef]

Yu, C.

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photon. Technol. Lett. 21,1075–1077 (2009).
[CrossRef]

S. Zhang, C. Yu, P. Y. Kam, and J. Chen, “Parallel implementation of decision-aided maximum likelihood phase estimation in coherent M-ary phase-shifted keying systems,” IEEE Photon. Technol. Lett. 21,1471–1473 (2009).
[CrossRef]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” to appear in J. Lightwave Technol.

Yu, J.

Zhang, C.

Zhang, G.

Zhang, S.

S. Zhang, C. Yu, P. Y. Kam, and J. Chen, “Parallel implementation of decision-aided maximum likelihood phase estimation in coherent M-ary phase-shifted keying systems,” IEEE Photon. Technol. Lett. 21,1471–1473 (2009).
[CrossRef]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photon. Technol. Lett. 21,1075–1077 (2009).
[CrossRef]

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” to appear in J. Lightwave Technol.

Zhang, T.

N. Xie, T. Zhang, and E. F. Haratsch, “Improving burst error tolerance of LDPC-centric coding systems in read channel, ” IEEE Trans. Magnetics 46,933–941 (2010).
[CrossRef]

Zhou, X.

IEE Proc. Commun. (1)

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

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

C. Xu, X. Liu, and X. Wei, “Differential phase-shift keying for high spectral efficiency optical transmissions,” IEEE J. Sel. Top. Quantum Electron. 10,281–293 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (2)

S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Laser linewidth tolerance of decision-aided maximum likelihood phase estimation in coherent optical M-ary PSK and QAM systems,” IEEE Photon. Technol. Lett. 21,1075–1077 (2009).
[CrossRef]

S. Zhang, C. Yu, P. Y. Kam, and J. Chen, “Parallel implementation of decision-aided maximum likelihood phase estimation in coherent M-ary phase-shifted keying systems,” IEEE Photon. Technol. Lett. 21,1471–1473 (2009).
[CrossRef]

IEEE Trans. Commun. (5)

P. Y. Kam, “Maximum-likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Trans. Commun. COM-34,522-527 (1986).

J. Lassing, E. Str¨om, E. Agrell, and T. Ottosson, “Computation of the exact bit-error rate of coherent M-ary PSK with gray code bit mapping,” IEEE Trans. Commun. 51,1758–1760 (2003).
[CrossRef]

H. Fu and P. Y. Kam, “A simple bit error probability analysis for square QAM in Rayleigh fading with channel estimation,” IEEE Trans. Commun. 57,2200–2206 (2009).
[CrossRef]

M. P. Fitz and R. J. M. Cramer, “A performance analysis of a digital PLL-based MPSK demodulator,” IEEE Trans. Commun. 43,1192–1201 (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. 41,1020–1022 (1993).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. J. Viterbi and A. N. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Trans. Inf. Theory IT-29,543–551 (1983).
[CrossRef]

IEEE Trans. Magnetics (1)

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Opt. Express (1)

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Figures (7)

Fig. 1.
Fig. 1.

Constellation map and bit mapping of 16PSK. Note that r = E s ( k ) exp [ j ( ϕ s ( k ) + θ ( k ) ) ] .

Fig. 2.
Fig. 2.

Constellation map and bit mapping of 16QAM.

Fig. 3.
Fig. 3.

The BERs of 16QAM and 16PSK from analysis and MC simulations with different phase error variance σ 2 Δθ .

Fig. 4.
Fig. 4.

The SNR penalty (γb ) as a function of phase error variances (σ 2 Δθ ) in different M-ary PSK and 16QAM @BER=10−4.

Fig. 5.
Fig. 5.

The phase error variance (σ 2 Δθ ) of DA ML from analysis Eq. (23) and MC simulations in 16QAM @40Gb/s. The phase error variance from Eq. (24) for M-ary PSK is also plotted for comparison.

Fig. 6.
Fig. 6.

A BER performance comparison of DA ML PE in M-ary PSK and 16QAM through numerical integration, MC simulations and BER approximation with optimum memory length L. σ 2 p = 1×10−4 rad2 (blue) and 5×10−5 rad2 (red).

Fig. 7.
Fig. 7.

The SNR per bit penalty versus the ratio of linewidth per laser to the symbol rate (ΔνTs ) at different BER levels using one-symbol-lag and parallel DA ML (p=10) in 16QAM format.

Tables (3)

Tables Icon

Table 1. The Phase Error Tolerance of M-ary PSK and 16QAM with Coherent Detection

Tables Icon

Table 2. The ratio of linewidth per laser to the symbol rate (ΔνTs ) leading to a 1-dB γb penalty at BER=10−4 for DA ML (Analysis and MC Simulations)

Tables Icon

Table 3. The comparison between the ratio of linewidth per laser to the symbol rate(ΔνTs ) leading to a 1-dB γb penalty at BER=10−3 using DA ML (one-symbol-lag and parallelism) and phase-searching PE in [30]

Equations (64)

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r ( k ) = E s ( k ) exp [ j ( ϕ s ( k ) + θ ( k ) ) ] + n ( k ) ,
P b ( e ) = π π P b ( e Δ θ ) p ( Δ θ ) d Δ θ ,
P b ( e Δ θ ) = 1 2 erfc ( γ b cos Δ θ )
P b ( e Δ θ ) = 1 4 [ erfc ( γ s sin ( π 4 Δ θ ) ) + erfc ( γ s sin ( π 4 + Δ θ ) ) ]
P b ( e Δ θ ) + P b ( e Δ θ ) = 1 3 erfc [ γ s sin ( π 8 Δ θ ) ] + 1 3 erfc [ γ s sin ( π 8 + Δ θ ) ]
+ 1 6 erfc [ γ s cos ( π 8 Δ θ ) ] · erfc [ γ s sin ( Δ θ π 8 ) ]
+ 1 6 erfc [ γ s cos ( π 8 + Δ θ ) ] · erfc [ γ s sin ( Δ θ + π 8 ) ]
P b ( e Δ θ ) = 1 4 [ P ( A 1 Δ θ ) + 2 P ( A 3 Δ θ ) + P ( A 2 Δ θ ) + 2 P ( A 6 Δ θ ) + 3 P ( A 7 Δ θ )
+ 2 P ( A 5 Δ θ ) + P ( A 4 Δ θ ) + 2 P ( A 12 Δ θ ) + 3 P ( A 13 Δ θ ) + 4 P ( A 15 Δ θ )
+ 3 P ( A 14 Δ θ ) + 2 P ( A 10 Δ θ ) + 3 P ( A 11 Δ θ ) + 2 P ( A 9 Δ θ ) + P ( A 8 Δ θ ) ] ,
P b ( e Δ θ ) = 1 4 [ P ( U 1 Δ θ ) + P ( U 2 Δ θ ) + P ( A 3 Δ θ ) + P ( A 6 Δ θ ) + 2 P ( A 7 Δ θ ) + P ( A 5 Δ θ )
+ 2 P ( A 13 Δ θ ) + 3 P ( A 15 Δ θ ) + 2 P ( A 14 Δ θ ) + P ( A 10 Δ θ ) + 2 P ( A 11 Δ θ )
+ P ( A 9 Δ θ ) ] .
P b ( e Δ θ ) + P b ( e Δ θ ) = 1 2 [ P ( U 1 Δ θ ) + P ( U 2 Δ θ ) + P ( U 3 Δ θ ) + P ( U 4 Δ θ )
+ P ( A 3 + A 2 Δ θ ) + P ( A 7 + A 5 Δ θ ) + P ( A 15 + A 14 Δ θ )
+ P ( A 11 + A 9 Δ θ ) ] ,
P ( U 1 Δ θ ) = 1 2 erfc [ γ s sin ( π 16 Δ θ ) ] ,
P ( U 2 Δ θ ) = 1 2 erfc [ γ s sin ( π 16 + Δ θ ) ] ,
P ( U 3 Δ θ ) = 1 4 erfc [ γ s sin ( π 16 + Δ θ ) ] · erfc [ γ s cos ( π 16 + Δ θ ) ] ,
P ( U 4 Δ θ ) = 1 4 erfc [ γ s sin ( π 16 Δ θ ) ] · erfc [ γ s cos ( π 16 Δ θ ) ] .
P ( A 3 + A 2 Δ θ ) = P ( x 0 , y 0 y y 0 + x )
= 0 + 1 π N 0 exp [ ( x x 0 ) 2 N 0 ] d x · y 0 y 0 + x 1 π N 0 exp [ y 2 N 0 ] d y
= 1 2 0 + [ erfc ( y 0 N 0 ) erfc ( y 0 + x N 0 ) ] f ( x x 0 ) d x ,
P ( A 9 + A 11 Δ θ ) = 1 2 0 + [ erfc ( y 1 N 0 ) erfc ( y 1 + x N 0 ) ] · f ( x x 1 ) d x ,
P ( A 14 + A 15 Δ θ ) = 1 2 0 + [ erfc ( y 2 N 0 ) erfc ( y 2 + x N 0 ) ] · f ( x + x 2 ) d x ,
P ( A 5 + A 7 Δ θ ) = 1 2 0 + [ erfc ( y 3 N 0 ) erfc ( y 3 + x N 0 ) ] · f ( x + x 3 ) d x ,
P b ( e Δ θ ) = 1 2 [ P MSB ( e Δ θ ) + P LSB ( e Δ θ ) ] ,
P MSB ( e S 0 , Δ θ ) = 1 2 erfc ( 2 d · cos ( π 4 + Δ θ ) N 0 ) ,
P MSB ( e S 1 , Δ θ ) = 1 2 erfc ( 10 d · cos ( arctan ( 3 ) + Δ θ ) N 0 ) ,
P MSB ( e S 2 , Δ θ ) = 1 2 erfc ( 18 d · cos ( π 4 + Δ θ ) N 0 ) ,
P MSB ( e S 3 , Δ θ ) = 1 2 erfc ( 10 d · cos ( arctan ( 1 3 ) + Δ θ ) N 0 ) .
P LSB = 0 ( e S 0 , Δ θ ) = 1 2 erfc ( 2 d + 2 d · cos ( π 4 + Δ θ ) N 0 )
+ 1 2 erfc ( 2 d 2 d · cos ( π 4 + Δ θ ) N 0 ) ,
P LSB = 0 ( e S 1 , Δ θ ) = 1 2 erfc ( 2 d + 10 d · cos ( arctan ( 3 ) + Δ θ ) N 0 )
+ 1 2 erfc ( 2 d 10 d · cos ( arctan ( 3 ) + Δ θ ) N 0 ) ,
P LSB = 1 ( e S 3 , Δ θ ) = 1 2 erfc ( 18 d · cos ( π 4 + Δ θ ) 2 d N 0 )
1 2 erfc ( 18 d · cos ( π 4 + Δ θ ) + 2 d N 0 ) ,
P LSB = 1 ( e S 2 , Δ θ ) = 1 2 erfc ( 10 d · cos ( arctan ( 1 3 ) + Δ θ ) 2 d N 0 )
1 2 erfc ( 10 d · cos ( arctan ( 1 3 ) + Δ θ ) + 2 d N 0 ) .
P b ( e Δ θ ) = 1 16 [ erfc ( 2 d · cos ( π 4 + Δ θ ) N 0 ) + erfc ( 10 d · cos ( arctan ( 3 ) + Δ θ ) N 0 )
+ erfc ( 18 d · cos ( π 4 + Δ θ ) N 0 ) + erfc ( 10 d · cos ( arctan ( 1 3 ) + Δ θ ) N 0 )
+ erfc ( 2 d + 2 d · cos ( π 4 + Δ θ ) N 0 ) + erfc ( 2 d 2 d · cos ( π 4 + Δ θ ) N 0 )
+ erfc ( 2 d + 10 d · cos ( arctan ( 3 ) + Δ θ ) N 0 )
+ erfc ( 2 d 10 d · cos ( arctan ( 3 ) + Δ θ ) N 0 )
+ erfc ( 18 d · cos ( π 4 + Δ θ ) 2 d N 0 ) erfc ( 18 d · cos ( π 4 + Δ θ ) + 2 d N 0 )
+ erfc ( 10 d · cos ( arctan ( 1 3 ) + Δ θ ) 2 d N 0 )
erfc ( 10 d · cos ( arctan ( 1 3 ) + Δ θ ) + 2 d N 0 ) ] .
γ b = 1 log 2 M [ erfc 1 ( log 2 M · BER sin ( π M ) ] 2
γ b = 1 log 2 M · 2 ( M 1 ) 3 [ erfc 1 ( log 2 M · ( 1 1 BER 1 1 M ) ] 2
θ ( k ) = Σ m = k v ( m ) ,
V ( k ) = U ( k ) 1 Σ l = k L k 1 r ( l ) m ̂ * ( l ) ,
θ ̂ ( k ) arg V ( k ) = θ ( k ) + ψ ( k ) + Im [ z ( k ) ] ,
ψ ( k ) Σ l = k L k 1 E s ( l ) [ θ ( l ) θ ( k ) ] Σ l = k L k 1 E s ( l )
z ( k ) Σ l = k L k 1 E s ( l ) n ( l ) Σ l = k L k 1 E s ( l ) ,
σ Δ θ 2 2 L 2 + 3 L + 3.88 6 ( L + 0.32 ) σ p 2 + 1 2 ( L + 0.32 ) σ n 2 .
σ Δ θ 2 2 L 2 + 3 L + 1 6 L σ p 2 + 1 2 L σ n 2 .
σ Δ θ 2 = ( 6 q 1 ) p 2 + 1 6 p σ p 2 + 1 2 p σ n 2 .
P b ( e ) = α π π π erfc [ c 1 + c 2 cos ( ϕ + Δ θ ) ] · exp [ α ( Δ θ ) 2 ] d ( Δ θ ) ,
erfc ( x ) 1 π exp ( x 2 ) ,
P b ( e ) α π 2 exp [ c 1 2 1 2 c 2 2 ( 1 + cos ( 2 ϕ ) ) 2 c 1 c 2 cos ϕ + A 1 2 4 A 0 ]
· π π exp [ A 0 ( Δ θ A 1 2 A 0 ) 2 ] d ( Δ θ ) ,
P b ( e ) α π A 0 exp [ c 1 2 1 2 c 2 2 ( 1 + cos 2 ϕ ) 2 c 1 c 2 cos ϕ + A 1 2 4 A 0 ] ,
P ( A 3 + A 2 + A 7 + A 5 Δ θ ) P ( A 3 + A 2 + A 7 + A 6 Δ θ )
P ( A 9 + A 11 + A 14 + A 15 Δ θ ) P ( A 9 + A 11 + A 14 + A 10 Δ θ ) ,

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