Abstract

We proposed a novel adaptive carrier phase estimator based on the phase information of the received signal only. Through eliminating the perturbation due to the amplitude of the AWGN, the proposed method outperforms the conventional adaptive filter in terms of both the carrier phase estimation and the filter gain tracking. Additionally, a dynamic tracking of both the laser linewidth and SNR is derived based on the proposed carrier phase estimator which requires no prior knowledge of the channel parameters. Numerical simulations and experiments are conducted to verify its feasibility in real applications.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
OSA Recommended Articles
Joint carrier phase and frequency-offset estimation with parallel implementation for dual-polarization coherent receiver

Jianing Lu, Xiang Li, Songnian Fu, Ming Luo, Meng Xiang, Huibin Zhou, Ming Tang, and Deming Liu
Opt. Express 25(5) 5217-5231 (2017)

Decision-aided maximum likelihood detection in coherent optical phase-shift-keying system

S. Zhang, P. Y. Kam, J. Chen, and C. Yu
Opt. Express 17(2) 703-715 (2009)

References

  • View by:
  • |
  • |
  • |

  1. K.-P. Ho, Phase-Modulated Optical Communication (SpringerUS, 2005), 1st ed.
  2. R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technology Letters 17, 887–889 (2005).
    [Crossref]
  3. A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Transactions on Information Theory 29, 543–551 (1983).
    [Crossref]
  4. D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology 24, 12–21 (2006).
    [Crossref]
  5. P. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Transactions on Communications 34, 522–527 (1986).
    [Crossref]
  6. P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of mpsk on the gaussian channel with unknown carrier phase characteristics,” IEEE Transactions on Communications 46, 1275–1279 (1998).
    [Crossref]
  7. S. Zhang, P. Y. Kam, C. Yu, and J. Chen, “Decision-aided carrier phase estimation for coherent optical communications,” Journal of Lightwave Technology 28, 1597–1607 (2010).
    [Crossref]
  8. C. C. Chan, Optical performance monitoring: advanced techniques for next-generation photonic networks (Academic Press, 2010).
  9. D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Transactions on Communications 48, 1681–1691 (2000).
    [Crossref]
  10. Y. Yu, B. Dong, and C. Yu, “Optical signal-to-noise ratio monitoring using a Sagnac interferometer based on fiber birefringence,” IEEE Photonics Technology Letters 27, 1899–1902 (2015).
    [Crossref]
  11. D. Derickson, “Fiber optic test and measurement,” in “Fiber optic test and measurement/edited by Dennis Derickson. Upper Saddle River, NJ: Prentice HallUSA , c1998.”, (1998).
  12. T. Sutili, R. C. Figueiredo, and E. Conforti, “Laser linewidth and phase noise evaluation using heterodyne offline signal processing,” Journal of Lightwave Technology 34, 4933–4940 (2016).
    [Crossref]
  13. H. Fu and P. Y. Kam, “Phase-based, time-domain estimation of the frequency and phase of a single sinusoid in awgn-the role and applications of the additive observation phase noise model,” IEEE Transactions on Information Theory 59, 3175–3188 (2013).
    [Crossref]
  14. E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” Journal of Lightwave Technology 25, 2675–2692 (2007).
    [Crossref]
  15. Q. Wang and P. Y. Kam, “Optimum detection of two-dimensional carrier modulations with linear phase noise using received amplitude and phase information and performance analysis,” Journal of Lightwave Technology 34, 2439–2451 (2016).
    [Crossref]
  16. R. Maher and B. Thomsen, “Dynamic linewidth measurement technique using digital intradyne coherent receivers,” Opt. Express 19, B313–B322 (2011).
    [Crossref]
  17. Z. Zan and A. J. Lowery, “Experimental demonstration of a flexible and stable semiconductor laser linewidth emulator,” Opt. Express 18, 13880–13885 (2010).
    [Crossref] [PubMed]

2016 (2)

T. Sutili, R. C. Figueiredo, and E. Conforti, “Laser linewidth and phase noise evaluation using heterodyne offline signal processing,” Journal of Lightwave Technology 34, 4933–4940 (2016).
[Crossref]

Q. Wang and P. Y. Kam, “Optimum detection of two-dimensional carrier modulations with linear phase noise using received amplitude and phase information and performance analysis,” Journal of Lightwave Technology 34, 2439–2451 (2016).
[Crossref]

2015 (1)

Y. Yu, B. Dong, and C. Yu, “Optical signal-to-noise ratio monitoring using a Sagnac interferometer based on fiber birefringence,” IEEE Photonics Technology Letters 27, 1899–1902 (2015).
[Crossref]

2013 (1)

H. Fu and P. Y. Kam, “Phase-based, time-domain estimation of the frequency and phase of a single sinusoid in awgn-the role and applications of the additive observation phase noise model,” IEEE Transactions on Information Theory 59, 3175–3188 (2013).
[Crossref]

2011 (1)

2010 (2)

Z. Zan and A. J. Lowery, “Experimental demonstration of a flexible and stable semiconductor laser linewidth emulator,” Opt. Express 18, 13880–13885 (2010).
[Crossref] [PubMed]

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

2007 (1)

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” Journal of Lightwave Technology 25, 2675–2692 (2007).
[Crossref]

2006 (1)

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology 24, 12–21 (2006).
[Crossref]

2005 (1)

R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technology Letters 17, 887–889 (2005).
[Crossref]

2000 (1)

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Transactions on Communications 48, 1681–1691 (2000).
[Crossref]

1998 (1)

P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of mpsk on the gaussian channel with unknown carrier phase characteristics,” IEEE Transactions on Communications 46, 1275–1279 (1998).
[Crossref]

1986 (1)

P. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Transactions on Communications 34, 522–527 (1986).
[Crossref]

1983 (1)

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Transactions on Information Theory 29, 543–551 (1983).
[Crossref]

Beaulieu, N. C.

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Transactions on Communications 48, 1681–1691 (2000).
[Crossref]

Chan, C. C.

C. C. Chan, Optical performance monitoring: advanced techniques for next-generation photonic networks (Academic Press, 2010).

Chen, J.

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

Chua, K. H.

P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of mpsk on the gaussian channel with unknown carrier phase characteristics,” IEEE Transactions on Communications 46, 1275–1279 (1998).
[Crossref]

Conforti, E.

T. Sutili, R. C. Figueiredo, and E. Conforti, “Laser linewidth and phase noise evaluation using heterodyne offline signal processing,” Journal of Lightwave Technology 34, 4933–4940 (2016).
[Crossref]

Derickson, D.

D. Derickson, “Fiber optic test and measurement,” in “Fiber optic test and measurement/edited by Dennis Derickson. Upper Saddle River, NJ: Prentice HallUSA , c1998.”, (1998).

Dong, B.

Y. Yu, B. Dong, and C. Yu, “Optical signal-to-noise ratio monitoring using a Sagnac interferometer based on fiber birefringence,” IEEE Photonics Technology Letters 27, 1899–1902 (2015).
[Crossref]

Figueiredo, R. C.

T. Sutili, R. C. Figueiredo, and E. Conforti, “Laser linewidth and phase noise evaluation using heterodyne offline signal processing,” Journal of Lightwave Technology 34, 4933–4940 (2016).
[Crossref]

Fu, H.

H. Fu and P. Y. Kam, “Phase-based, time-domain estimation of the frequency and phase of a single sinusoid in awgn-the role and applications of the additive observation phase noise model,” IEEE Transactions on Information Theory 59, 3175–3188 (2013).
[Crossref]

Ho, K.-P.

K.-P. Ho, Phase-Modulated Optical Communication (SpringerUS, 2005), 1st ed.

Ip, E.

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” Journal of Lightwave Technology 25, 2675–2692 (2007).
[Crossref]

Kahn, J. M.

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” Journal of Lightwave Technology 25, 2675–2692 (2007).
[Crossref]

Kam, P.

P. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Transactions on Communications 34, 522–527 (1986).
[Crossref]

Kam, P. Y.

Q. Wang and P. Y. Kam, “Optimum detection of two-dimensional carrier modulations with linear phase noise using received amplitude and phase information and performance analysis,” Journal of Lightwave Technology 34, 2439–2451 (2016).
[Crossref]

H. Fu and P. Y. Kam, “Phase-based, time-domain estimation of the frequency and phase of a single sinusoid in awgn-the role and applications of the additive observation phase noise model,” IEEE Transactions on Information Theory 59, 3175–3188 (2013).
[Crossref]

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

P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of mpsk on the gaussian channel with unknown carrier phase characteristics,” IEEE Transactions on Communications 46, 1275–1279 (1998).
[Crossref]

Katoh, K.

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology 24, 12–21 (2006).
[Crossref]

Kikuchi, K.

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology 24, 12–21 (2006).
[Crossref]

Lowery, A. J.

Ly-Gagnon, D. S.

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology 24, 12–21 (2006).
[Crossref]

Maher, R.

Noe, R.

R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technology Letters 17, 887–889 (2005).
[Crossref]

Pauluzzi, D. R.

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Transactions on Communications 48, 1681–1691 (2000).
[Crossref]

Sutili, T.

T. Sutili, R. C. Figueiredo, and E. Conforti, “Laser linewidth and phase noise evaluation using heterodyne offline signal processing,” Journal of Lightwave Technology 34, 4933–4940 (2016).
[Crossref]

Thomsen, B.

Tsukamoto, S.

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology 24, 12–21 (2006).
[Crossref]

Viterbi, A. J.

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Transactions on Information Theory 29, 543–551 (1983).
[Crossref]

Viterbi, A. M.

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Transactions on Information Theory 29, 543–551 (1983).
[Crossref]

Wang, Q.

Q. Wang and P. Y. Kam, “Optimum detection of two-dimensional carrier modulations with linear phase noise using received amplitude and phase information and performance analysis,” Journal of Lightwave Technology 34, 2439–2451 (2016).
[Crossref]

Yu, C.

Y. Yu, B. Dong, and C. Yu, “Optical signal-to-noise ratio monitoring using a Sagnac interferometer based on fiber birefringence,” IEEE Photonics Technology Letters 27, 1899–1902 (2015).
[Crossref]

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

Yu, X.

P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of mpsk on the gaussian channel with unknown carrier phase characteristics,” IEEE Transactions on Communications 46, 1275–1279 (1998).
[Crossref]

Yu, Y.

Y. Yu, B. Dong, and C. Yu, “Optical signal-to-noise ratio monitoring using a Sagnac interferometer based on fiber birefringence,” IEEE Photonics Technology Letters 27, 1899–1902 (2015).
[Crossref]

Zan, Z.

Zhang, S.

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

IEEE Photonics Technology Letters (2)

R. Noe, “PLL-free synchronous QPSK polarization multiplex/diversity receiver concept with digital I&Q baseband processing,” IEEE Photonics Technology Letters 17, 887–889 (2005).
[Crossref]

Y. Yu, B. Dong, and C. Yu, “Optical signal-to-noise ratio monitoring using a Sagnac interferometer based on fiber birefringence,” IEEE Photonics Technology Letters 27, 1899–1902 (2015).
[Crossref]

IEEE Transactions on Communications (3)

D. R. Pauluzzi and N. C. Beaulieu, “A comparison of snr estimation techniques for the awgn channel,” IEEE Transactions on Communications 48, 1681–1691 (2000).
[Crossref]

P. Kam, “Maximum likelihood carrier phase recovery for linear suppressed-carrier digital data modulations,” IEEE Transactions on Communications 34, 522–527 (1986).
[Crossref]

P. Y. Kam, K. H. Chua, and X. Yu, “Adaptive symbol-by-symbol reception of mpsk on the gaussian channel with unknown carrier phase characteristics,” IEEE Transactions on Communications 46, 1275–1279 (1998).
[Crossref]

IEEE Transactions on Information Theory (2)

A. J. Viterbi and A. M. Viterbi, “Nonlinear estimation of PSK-modulated carrier phase with application to burst digital transmission,” IEEE Transactions on Information Theory 29, 543–551 (1983).
[Crossref]

H. Fu and P. Y. Kam, “Phase-based, time-domain estimation of the frequency and phase of a single sinusoid in awgn-the role and applications of the additive observation phase noise model,” IEEE Transactions on Information Theory 59, 3175–3188 (2013).
[Crossref]

Journal of Lightwave Technology (5)

E. Ip and J. M. Kahn, “Feedforward carrier recovery for coherent optical communications,” Journal of Lightwave Technology 25, 2675–2692 (2007).
[Crossref]

Q. Wang and P. Y. Kam, “Optimum detection of two-dimensional carrier modulations with linear phase noise using received amplitude and phase information and performance analysis,” Journal of Lightwave Technology 34, 2439–2451 (2016).
[Crossref]

T. Sutili, R. C. Figueiredo, and E. Conforti, “Laser linewidth and phase noise evaluation using heterodyne offline signal processing,” Journal of Lightwave Technology 34, 4933–4940 (2016).
[Crossref]

D. S. Ly-Gagnon, S. Tsukamoto, K. Katoh, and K. Kikuchi, “Coherent detection of optical quadrature phase-shift keying signals with carrier phase estimation,” Journal of Lightwave Technology 24, 12–21 (2006).
[Crossref]

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

Opt. Express (2)

Other (3)

D. Derickson, “Fiber optic test and measurement,” in “Fiber optic test and measurement/edited by Dennis Derickson. Upper Saddle River, NJ: Prentice HallUSA , c1998.”, (1998).

C. C. Chan, Optical performance monitoring: advanced techniques for next-generation photonic networks (Academic Press, 2010).

K.-P. Ho, Phase-Modulated Optical Communication (SpringerUS, 2005), 1st ed.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

Fig. 1
Fig. 1 Inverse MSE of the phase estimation versus the symbol SNR γs with symbol rate R = 50 G Symbols/second.
Fig. 2
Fig. 2 Inverse NMSE of measured α versus (a) combined laser linewidth Δν and, (b) symbol SNR γs.
Fig. 3
Fig. 3 Performance comparison of different (a) laser linewidth estimation methods, and (b) symbol SNR estimation techniques.
Fig. 4
Fig. 4 Performance investigation of (a) decision and, (b) filter gain based laser linewidth estimators.
Fig. 5
Fig. 5 Performance investigation of (a) filter gain and, (b) decision based SNR estimators.
Fig. 6
Fig. 6 Experimental setup of the 25 G Symbols/second 16QAM coherent optical transmission system.
Fig. 7
Fig. 7 Experimental evaluation of the performance of the decision based (a) laser linewidth and, (b) SNR estimation.

Equations (43)

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

r ( k ) = m ( k ) e j θ ( k ) + n ( k ) .
A ( i ) = ρ i E s ,
θ ( k ) = m = k ν ( m ) ,
σ p 2 = 2 π Δ ν T .
V ( k + 1 ) = α V ( k ) + ( 1 α ) r ( k ) / m ^ ( k ) ,
R ( k ) = E [ l = 1 k | r ( l ) V ( l ) m ^ ( l ) | 2 | { r ( l ) } l = 1 k ] ,
α ( k ) = A ( k ) B ( k ) ,
A ( k ) = A ( k 1 ) + l = 1 k | m ^ ( l ) | 2 [ | g ( l 1 ) | 2 [ V ( l 1 ) g * ( l 1 ) g ( l ) [ V * ( l 1 ) g * ( l 1 ) ] ] ] B ( k ) = B ( k 1 ) + l = 1 k | m ^ ( l ) | 2 | V ( l 1 ) g ( l 1 ) | 2 g ( k ) = r ( k ) m ^ ( k ) .
σ e 2 = σ p 2 1 α 2 + η 2 γ s 1 α 1 + α .
α o = 1 + γ s σ p 2 η γ s σ p 2 η 2 η γ s σ p 2 + 1 .
V ( k + 1 ) = α ( k ) V ( k ) + [ 1 α ( k ) ] [ e j θ ( k ) + n ( k ) m ( k ) ] .
V E ( k + 1 ) = α E V E ( k ) + ( 1 α E ) r ( k ) m ( k ) .
α E ( k ) = A E ( k ) B E ( k ) A E ( k ) = A E ( k 1 ) + l = 1 k | m ( l ) | 2 [ | g E ( l 1 ) | 2 [ V E ( l 1 ) g E * ( l 1 ) g E ( l ) [ V E * ( l 1 ) g E * ( l 1 ) ] ] ] B E ( k ) = B E ( k 1 ) + l = 1 k | m ( l ) | 2 | V E ( l 1 ) g E ( l 1 ) | 2 g E ( k ) = r ( k ) m ( k ) .
V E ( k + 1 ) = α E V E ( k ) + ( 1 α E ) e j [ θ ( k ) + ( k ) ] ,
σ 2 = η / 2 γ s .
e j ( θ ( k + 1 ) θ e ( k + 1 ) ) = α E e j ( θ ( k ) θ e ( k ) ) + ( 1 α E ) e j [ θ ( k ) + ( k ) ] .
θ e ( k + 1 ) = ν ( k + 1 ) + α E θ e ( k ) ( 1 α E ) ( k ) .
E [ θ e ( k + 1 ) ] = E [ ν ( k + 1 ) ] + α E E [ θ e ( k ) ] ( 1 α E ) E [ ( k ) ] = α E E [ θ e ( k ) ] .
E [ θ e ( 0 ) ] = E [ θ ( 0 ) ] E [ θ ( 0 ) ^ ] = E [ ν ( 0 ) ] E [ [ V ( 0 ) ] ] .
E [ θ e ( 0 ) ] = E [ ν ( 0 ) ] E [ [ 1 ] ] = 0 .
var [ θ e ( k + 1 ) ] = α E 2 var [ θ e ( k ) ] + σ p 2 + ( 1 α E ) 2 σ 2 .
σ e 2 = σ p 2 1 α E 2 + η 2 γ s 1 α E 1 + α E .
σ p 2 = α o 2 2 α o + 1 2 α o η γ s .
γ s = α o 2 2 α o + 1 2 α o η σ p 2 .
D ( k ) = r ( k ) e j [ ϕ ( k ) + θ ^ ( k ) ] .
D Re ( k ) = [ D ( k ) ] = [ ρ ( k ) E s e j θ e ( k ) + n ( k ) e j ( θ ^ ( k ) + ϕ ( k ) ) ] ,
D Re ( k ) ρ ( k ) E s + [ n ( k ) ] ,
E [ D Re ( k ) ] = ρ ¯ E s + E [ [ n ( k ) ] ] = ρ ¯ E s .
E s = [ E [ D Re ( k ) ] / ρ ¯ ] 2 .
N 0 = 2 × var [ D Re ( k ) ρ ( k ) E s ] .
γ s = [ E [ D Re ( k ) ] / ρ ¯ ] 2 2 × var [ D Re ( k ) ρ ( k ) E [ D Re ( k ) ] / ρ ¯ ]
D Im ( k ) = [ D ( k ) ] = [ ρ ( k ) E s e j θ e ( k ) + n ( k ) ] ,
D Im ( k ) = E s ρ ( k ) θ e ( k ) + [ n ( k ) ] .
E [ D Im ( k ) ] = E s E [ ρ ( k ) θ e ( k ) ] + E [ [ n ( k ) ] ] = E s E [ ρ ( k ) ] E [ θ e ( k ) ] + E [ [ n ( k ) ] ] = 0 .
σ I 2 = var [ D Im ( k ) ] = E [ D Im ( k ) 2 ] .
D Im ( k ) 2 = E s ρ ( k ) 2 θ e ( k ) 2 + [ n ( k ) ] 2 + 2 E s ρ ( k ) θ e ( k ) [ n ( k ) ] .
σ I 2 = E s E [ ρ ( k ) 2 ] E [ θ e ( k ) 2 ] + E [ [ n ( k ) ] 2 ] + 2 E s E [ ρ ( k ) ] E [ θ e ( k ) ] E [ [ n ( k ) ] ] .
E [ θ e ( k ) 2 ] = σ e 2 , E [ [ n ( k ) ] 2 ] = var [ [ n ( k ) ] ] = N 0 2 .
σ I 2 = σ e 2 E s + N 0 2 .
σ p 2 = ( 1 α E 2 ) [ σ I 2 E s η 2 γ s 1 α E 1 + α E 1 2 γ s ] .
Δ ν = σ p 2 / 2 π T .
MSE = 1 N i = 1 N ( θ ( i ) θ ^ ( i ) ) 2
NMSE = 1 N i = 1 N ( α ^ i α o α o ) 2 ,

Metrics