Abstract

In a nonlinear optical fiber communication (OFC) system with signal power much stronger than noise power, the noise field in the fiber can be described by linearized noise equation (LNE). In this case, the noise impact on the system performance can be evaluated by moment-generating function (MGF) method. Many published MGF calculations were based on the LNE using continuous wave (CW) approximation, where the modulated signal needs to be artificially simplified as an unmodulated signal. Results thus obtained should be treated carefully. Reliable results can be obtained by replacing the CW-based LNE with the accurate LNE proposed by Holzlöhner et al in Ref. [1]. In this work we show that, for the case of linearized noise amplified by EDFAs, its MGF can be obtained by calculating the noise propagator directly from the accurate LNE. Our results agree well with the experimental data of multi-span DPSK systems.

© 2014 Optical Society of America

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References

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  1. R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002).
    [CrossRef]
  2. K. Kikuchi, “Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers,” IEEE Photon. Technol. Lett. 5, 1079–1081 (1993).
  3. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
    [CrossRef]
  4. R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997).
    [CrossRef]
  5. P. Serena, A. Orlandini, A. Bononi, “Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. 24, 2026–2037 (2006).
    [CrossRef]
  6. A. Demir, “Nonlinear phase noise in optical-fiber-communication systems,” J. Lightwave Technol. 25, 2002–2032 (2007).
    [CrossRef]
  7. L. D. Coelho, L. Molle, D. Gross, N. Hanik, R. Freund, C. Caspar, E.-D. Schmidt, B. Spinnler, “Modeling nonlinear phase noise in differentially phase-modulated optical communication systems,” Opt. Express 17, 3226–3241 (2009).
    [CrossRef] [PubMed]
  8. M. Secondini, E. Forestieri, C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems,” J. Lightwave Technol. 27, 3358–3369 (2009).
    [CrossRef]
  9. R. Holzlöhner, “A covariance matrix method for the computation of bit errors in optical transmission systems,” Ph.D. thesis, University of Maryland Baltimore County. Baltimore, Maryland, USA (2003).
  10. R. Holzlöhner, C. R. Menyuk, W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. 14, 1079–1081 (2002).
    [CrossRef]
  11. R. Holzlöhner, C. R. Menyuk, W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. 15, 688–690 (2003).
    [CrossRef]
  12. R. Holzlöhner, C. R. Menyuk, “Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems,” Opt. Lett. 28, 1894–1896 (2003).
    [CrossRef] [PubMed]
  13. A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. 201, 148–171 (2004).
    [CrossRef]
  14. J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007).
    [CrossRef]
  15. Z. Zhang, L. Chen, X. Bao, “A fourth-order Runge-Kutta in the interaction picture method for numerically solving the coupled nonlinear Schrödinger equation,” Opt. Express 18, 8261–8276 (2010).
    [CrossRef] [PubMed]
  16. E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering,” J. Lightwave Technol. 18, 1493–1503 (2000).
    [CrossRef]
  17. Z. Zhang, L. Chen, X. Bao, “Accurate BER evaluation for lumped DPSK and OOK systems with PMD and PDL,” Opt. Express 15, 9418–9433 (2007).
    [CrossRef] [PubMed]
  18. D. Gariepy, G. He, “Measuring OSNR in WDM systemsEffects of resolution bandwidth and optical rejection ratio,” White paper, EXFO Inc. (2009).
  19. P. Serena, A. Bononi, J. C. Antona, S. Bigo, “Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction,” J. Lightwave Technol. 23, 2352–2363 (2006).
    [CrossRef]
  20. A. Bononi, P. Serena, A. Orlandini, “A unified design framework for single-channel dispersion-managed terrestrial systems,” J. Lightwave Technol. 26, 3617–3631 (2008).
    [CrossRef]
  21. E. Ip, J. M. Kahn, “Power spectra of return-to-zero optical signals,” J. Lightwave Technol. 24, 1610–1618 (2006).
    [CrossRef]
  22. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).
  23. E. Forestieri, G. Prati, “Exact analytical evaluation of second-order PMD impact on the outage probability for a compensated system,” J. Lightwave Technol. 22, 988–996 (2004).
    [CrossRef]
  24. Z. Zhang, L. Chen, X. Bao, “The noise propagator in an optical system using EDFAs and its effect on system performance: accurate evaluation based on linear perturbation,” arXiv:physics.optics/1207.3362v1.

2010

2009

2008

2007

2006

2004

E. Forestieri, G. Prati, “Exact analytical evaluation of second-order PMD impact on the outage probability for a compensated system,” J. Lightwave Technol. 22, 988–996 (2004).
[CrossRef]

A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. 201, 148–171 (2004).
[CrossRef]

2003

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. 15, 688–690 (2003).
[CrossRef]

R. Holzlöhner, C. R. Menyuk, “Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems,” Opt. Lett. 28, 1894–1896 (2003).
[CrossRef] [PubMed]

2002

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. 14, 1079–1081 (2002).
[CrossRef]

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002).
[CrossRef]

2000

1997

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
[CrossRef]

R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997).
[CrossRef]

1993

K. Kikuchi, “Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers,” IEEE Photon. Technol. Lett. 5, 1079–1081 (1993).

Antona, J. C.

Bao, X.

Benedetto, S.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
[CrossRef]

Bigo, S.

Bononi, A.

Carena, A.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
[CrossRef]

Caspar, C.

Chen, L.

Coelho, L. D.

Curri, V.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
[CrossRef]

Demir, A.

A. Demir, “Nonlinear phase noise in optical-fiber-communication systems,” J. Lightwave Technol. 25, 2002–2032 (2007).
[CrossRef]

A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. 201, 148–171 (2004).
[CrossRef]

Forestieri, E.

Freund, R.

Gariepy, D.

D. Gariepy, G. He, “Measuring OSNR in WDM systemsEffects of resolution bandwidth and optical rejection ratio,” White paper, EXFO Inc. (2009).

Gaudino, R.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
[CrossRef]

Grigoryan, V. S.

Gross, D.

Hanik, N.

He, G.

D. Gariepy, G. He, “Measuring OSNR in WDM systemsEffects of resolution bandwidth and optical rejection ratio,” White paper, EXFO Inc. (2009).

Holzlöhner, R.

R. Holzlöhner, C. R. Menyuk, “Use of multicanonical Monte Carlo simulations to obtain accurate bit error rates in optical communications systems,” Opt. Lett. 28, 1894–1896 (2003).
[CrossRef] [PubMed]

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. 15, 688–690 (2003).
[CrossRef]

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. 14, 1079–1081 (2002).
[CrossRef]

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002).
[CrossRef]

R. Holzlöhner, “A covariance matrix method for the computation of bit errors in optical transmission systems,” Ph.D. thesis, University of Maryland Baltimore County. Baltimore, Maryland, USA (2003).

Hui, R.

R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997).
[CrossRef]

Hult, J.

Ip, E.

Kahn, J. M.

Kath, W. L.

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. 15, 688–690 (2003).
[CrossRef]

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. 14, 1079–1081 (2002).
[CrossRef]

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002).
[CrossRef]

Kikuchi, K.

K. Kikuchi, “Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers,” IEEE Photon. Technol. Lett. 5, 1079–1081 (1993).

Menyuk, C. R.

Molle, L.

O’Sullivan, M.

R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997).
[CrossRef]

Orlandini, A.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Poggiolini, P.

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
[CrossRef]

Prati, G.

Robinson, A.

R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997).
[CrossRef]

Schmidt, E.-D.

Secondini, M.

Serena, P.

Spinnler, B.

Taylor, M.

R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997).
[CrossRef]

Zhang, Z.

IEEE Photon. Technol. Lett.

K. Kikuchi, “Enhancement of optical-amplifier noise by nonlinear refractive index and group-velocity dispersion of optical fibers,” IEEE Photon. Technol. Lett. 5, 1079–1081 (1993).

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, S. Benedetto, “New analytical results on fiber parametric gain and its effects on ASE noise,” IEEE Photon. Technol. Lett. 9, 535–537 (1997).
[CrossRef]

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “Efficient and accurate computation of eye diagrams and bit error rates in a single-channel CRZ system,” IEEE Photon. Technol. Lett. 14, 1079–1081 (2002).
[CrossRef]

R. Holzlöhner, C. R. Menyuk, W. L. Kath, “A covariance matrix method to compute bit error rates in a highly nonlinear dispersion-managed soliton system,” IEEE Photon. Technol. Lett. 15, 688–690 (2003).
[CrossRef]

J. Comput. Phys.

A. Demir, “Non-Monte Carlo formulations and computational techniques for the stochastic nonlinear Schrodinger equation,” J. Comput. Phys. 201, 148–171 (2004).
[CrossRef]

J. Lightwave Technol.

J. Hult, “A fourth-order Runge-Kutta in the interaction picture method for simulating supercontinuum generation in optical fibers,” J. Lightwave Technol. 25, 3770–3775 (2007).
[CrossRef]

E. Forestieri, “Evaluating the error probability in lightwave systems with chromatic dispersion, arbitrary pulse shape and pre- and postdetection filtering,” J. Lightwave Technol. 18, 1493–1503 (2000).
[CrossRef]

P. Serena, A. Bononi, J. C. Antona, S. Bigo, “Parametric gain in the strongly nonlinear regime and its impact on 10-Gb/s NRZ systems with forward-error correction,” J. Lightwave Technol. 23, 2352–2363 (2006).
[CrossRef]

A. Bononi, P. Serena, A. Orlandini, “A unified design framework for single-channel dispersion-managed terrestrial systems,” J. Lightwave Technol. 26, 3617–3631 (2008).
[CrossRef]

E. Ip, J. M. Kahn, “Power spectra of return-to-zero optical signals,” J. Lightwave Technol. 24, 1610–1618 (2006).
[CrossRef]

M. Secondini, E. Forestieri, C. R. Menyuk, “A combined regular-logarithmic perturbation method for signal-noise interaction in amplified optical systems,” J. Lightwave Technol. 27, 3358–3369 (2009).
[CrossRef]

R. Hui, M. O’Sullivan, A. Robinson, M. Taylor, “Modulation instability and its impact in multispan optical amplified IMDD systems: theory and experiments,” J. Lightwave Technol. 15, 1071–1082 (1997).
[CrossRef]

P. Serena, A. Orlandini, A. Bononi, “Parametric-gain approach to the analysis of single-channel DPSK/DQPSK systems with nonlinear phase noise,” J. Lightwave Technol. 24, 2026–2037 (2006).
[CrossRef]

A. Demir, “Nonlinear phase noise in optical-fiber-communication systems,” J. Lightwave Technol. 25, 2002–2032 (2007).
[CrossRef]

E. Forestieri, G. Prati, “Exact analytical evaluation of second-order PMD impact on the outage probability for a compensated system,” J. Lightwave Technol. 22, 988–996 (2004).
[CrossRef]

R. Holzlöhner, V. S. Grigoryan, C. R. Menyuk, W. L. Kath, “Accurate calculation of eye diagrams and bit error rates in optical transmission systems using linearization,” J. Lightwave Technol. 20, 389–400 (2002).
[CrossRef]

Opt. Express

Opt. Lett.

Other

R. Holzlöhner, “A covariance matrix method for the computation of bit errors in optical transmission systems,” Ph.D. thesis, University of Maryland Baltimore County. Baltimore, Maryland, USA (2003).

D. Gariepy, G. He, “Measuring OSNR in WDM systemsEffects of resolution bandwidth and optical rejection ratio,” White paper, EXFO Inc. (2009).

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, 1991).

Z. Zhang, L. Chen, X. Bao, “The noise propagator in an optical system using EDFAs and its effect on system performance: accurate evaluation based on linear perturbation,” arXiv:physics.optics/1207.3362v1.

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Figures (4)

Fig. 1
Fig. 1

Low-pass equivalent optical model. The receiver consists of optical and electrical filters and DPSK balance detection. In this work, calculations, except those in Sec. 5.1, are based on the assumptions N0k = N0 and Gk = G [k = 1, 2,...(K + 1)]. Also, the fiber in each span is assumed to have same length (L km) and same loss (α dB/km). Nin is the ASE noise added at the transmitter. Changing Nin will change the OSNR at the receiver. In a typical balanced DPSK receiver, the delay in one branch of the interferometer is Tb = 1/Rb. Here, Rb = 20 Gb/s.

Fig. 2
Fig. 2

BER versus received OSNR for a 20Gb/s 20-span RZ-DPSK system with Φ̄NL = 0.2π. Solid: obtained using CMM of Ref. [1]. Dashed (dotted): improved CW approach of Ref. [5] with CW power being peak power (average power), respectively. Dash-dotted: RK4IP approach. All curves, except the dash-dotted, are obtained from Fig. 8 of Ref. [5].

Fig. 3
Fig. 3

BER versus received OSNR for the multi-span RZ-50% DPSK systems with Φ̄NL = 0.9. Each span consists of a SMF fiber (42 km long) and a DCF file (7 km long). Other fiber parameters were detailed in Table 1 of Ref. [7]. According to Fig. 7(b) in Ref. [7], where the experimental curves for 5-span, 10-span, and 25-span systems were almost the same, here we replot these three curves using a thick solid curve.

Fig. 4
Fig. 4

BER vs RX-OSNR for the 5-span RZ-50% DPSK system with Φ̄NL = 0.9. Other parameters are same as those used in Fig. 3. Solid: experimental results. Dotted (Dash-dotted): numerical calculation using RK4IP with (without) phase shift Δ.

Equations (43)

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

d a ˜ I d z = N ^ I a ˜ I
a ˜ n + 1 = e L ^ h / 2 [ a ˜ n I + h k 1 6 + h k 2 3 + h k 3 3 + h k 4 6 ] a ˜ n I = e L ^ h / 2 a ˜ n k 1 = N ^ n I a ˜ n I = e L ^ h / 2 N ^ ( z n ) a ˜ n k ^ 1 a ˜ n k 2 = N ^ n + 1 / 2 I [ a ˜ n I + h k 1 2 ] = N ^ ( z n + h / 2 ) [ e L ^ h / 2 + h k ^ 1 2 ] a ˜ n k ^ 2 a ˜ n k 3 = N ^ n + 1 / 2 I [ a ˜ n I + h k 2 2 ] = N ^ ( z n + h / 2 ) [ e L ^ h / 2 + h k ^ 2 2 ] a ˜ n k ^ 3 a ˜ n k 4 = N ^ n + 1 I [ a ˜ n I + h k 3 ] = e L ^ h / 2 N ^ ( z n + 1 ) e L ^ h / 2 [ e L ^ h / 2 + h k ^ 3 ] a ˜ n k ^ 4 a ˜ n
a ˜ n + 1 = ( e L ^ h / 2 [ e L ^ h / 2 + h k ^ 1 6 + h k ^ 2 3 + h k ^ 3 3 ] + h 6 N ^ ( z n + 1 ) e L ^ h / 2 ( e L ^ h / 2 + h k ^ 3 ) ) a ˜ n ,
H ( z n + 1 , z n ) = e L ^ h / 2 [ e L ^ h / 2 + 1 3 ( h 2 k ^ 1 + 2 h 2 k ^ 2 + h k ^ 3 ) ] + h 6 N ^ ( z n + 1 ) e L ^ h / 2 [ e L ^ h / 2 + h k ^ 3 ]
p n ( L , 0 ) = H ( L , L h L ) H ( h 1 , 0 ) .
PG 1 = p n ( L , 0 ) σ 2 I p n T ( L , 0 ) = σ 2 p n ( L , 0 ) p n T ( L , 0 ) ,
σ 2 = N 0 / ( 2 T 0 ) [ N 0 = n s p ( G 1 ) h ¯ ω ]
P G ^ = ( G σ in 2 + σ 2 ) P n ( K , 0 ) P n T ( K , 0 ) + σ 2 k = 1 K P n ( K , k ) P n T ( K , k ) , P n ( K , k ) = p n ( K L , ( K 1 ) L ) p n ( ( k + 1 ) L , k L ) , ( k = 0 , 1 , , K 1 ) P n ( K , K ) = I ,
P G ^ = σ 2 P n , eq P n , eq T ,
E [ exp ( s ( λ ˜ c 2 + 2 c b ˜ ) ) ] = d c 2 π σ 2 exp ( c 2 2 σ 2 ) exp [ s ( λ ˜ c 2 + 2 c b ˜ ) ] = exp [ 2 σ 2 s 2 b ˜ 2 1 2 σ 2 s λ ˜ ] 1 2 σ 2 s λ ˜ ,
Ψ t s ( s ) = E [ exp [ s I ( t s ) ] ] = exp [ s I ss ( t s ) ] i = 1 4 M n + 2 exp [ 2 σ 2 s 2 b ˜ i 2 ( t s ) 1 s β i ] ( 1 s β i ) ξ , ( β i = 2 σ 2 λ ˜ i )
OSNR 0.1 nm = P ¯ s P ASE ( B r ) .
OSNR L , 0.1 nm = P ¯ P ASE ( B m ) × B m B r ( P ASE ( B m ) = [ GN in + N 0 ( K + 1 ) ] B m ) ,
OSNR N L , 0.1 nm = P ¯ s P ASE ( B m ) × B m B r ( P ASE ( B m ) = T r ( O m T P G ^ O m ) σ 2 = T r ( P G ^ O m O m T ) σ 2 ) .
P ASE ( B m , ± Δ λ ) = T r [ P G ^ ( O m ( Δ λ ) O m T ( Δ λ ) + O m ( + Δ λ ) O m T ( + Δ λ ) ) ] σ 2 2 ,
D l l s s = e j 2 π l N + e j 2 π l N 2 , D m m n n = e j 2 π m T b T 0 + e j 2 π m T b T 0 2 , D m l n s = e j 2 π m T b T 0 j Δ + e j 2 π l N + j Δ 2 ,
Δ = ϕ 0 < δ ϕ > ,
u z = j β ω ω 2 2 u t 2 + β ω ω ω 6 3 u t 3 j γ | u | 2 u α 2 u ,
v z = j β ω ω 2 2 v t 2 + β ω ω ω 6 3 v t 3 j e α z γ | v | 2 v .
v z = j β ω ω 2 2 v t 2 + β ω ω ω 6 3 v t 3 j e α z γ | v | 2 v j w ( z , t ) ,
r ( z , z , t , t ) = E { w ( z , t ) w * ( z , t ) } = δ ( t t ) δ ( z z ) k = 1 K + 1 N 0 k δ ( z ( k 1 ) L ) .
N 0 k = N 0 = n s p ( G 1 ) h ¯ ω ( k = 1 , 2 , , K , ( K + 1 ) )
v 0 z = j β ω ω 2 2 v 0 t 2 + β ω ω ω 6 3 v 0 t 3 j e α z γ | v 0 | 2 v 0
δ v z = j β ω ω 2 2 δ v t 2 + β ω ω ω 6 3 δ v t 3 j 2 e α z γ | v 0 | 2 δ v j e α z γ v 0 2 δ v * j w ( z , t ) .
ν l = ν ( ω l ) = e α z | v 0 | 2 e j ω l t d t , μ l = μ ( ω l ) = e α z v 0 2 e j ω l t d t ,
d a l d z = j β ω ω 2 ω l 2 a l j β ω ω ω 6 ω l 3 a l j 2 γ ( z ) [ M ν ] l m a m j γ ( z ) [ M μ ] l m a m * j W l ,
d a d z = L ¯ a + ν a + μ a * j W
L ¯ = j L C D , ν = 2 j γ ( z ) ( M ν R + j M ν I ) , μ = j γ ( z ) ( M μ R + j M μ I )
d a ˜ d z = ( L ^ + ν ^ + μ ^ ) a ˜ + W ˜
L ^ = ( 0 L C D L C D 0 ) , ν ^ = ( ν A A ν S S ν S S ν A A ) , μ ^ = ( μ R μ I μ I μ R ) ,
E { W ˜ l ( z ) W ˜ l * ( z ) } = δ z , z δ l , l k = 0 K N 0 2 T 0 δ z , k L ( l = 1 , , 4 M n + 2 ) ,
d a ˜ d z = ( L ^ + N ^ ) a ˜ ; ( N ^ = ν ^ + μ ^ )
E { W ˜ ( z f ) W ˜ * ( z f ) } = N 0 2 T 0 I = σ 2 I ,
y s s ( t s ) = [ s o ( t s + T b ) | R s s | s o ( t s ) + c . c . ] / 2 = s o ( t s ) | R s s D | s o ( t s ) y n n ( t s ) = [ n o ( t s + T b ) | R n n | n o ( t s ) + c . c . ] / 2 = N o | R n n D | N o = Z | Λ | Z y n s ( t s ) = [ n o ( t s + T b ) | R n s | s o ( t s ) + n o ( t s ) | R n s | s o ( t s + T b ) + c . c . ] / 2 = [ N in | O n n R n s D | s o ( t s ) + c . c . ] = [ Z | b D ( t s ) + c . c . ]
D l l s s = e j 2 π l N + e j 2 π l N 2 , D m m n n = e j 2 π m T b T 0 + e j 2 π m T b T 0 2 , D m l n s = e j 2 π m T b T 0 + e j 2 π l N 2 .
L s = η N T b B o , M n = η B o T 0 , T 0 = μ ( 1 B o + 1 B r ) .
| s ˜ o = [ Re { | s o } Im { | s o } ] , | a ˜ 0 = [ Re { | a 0 } Im { | a 0 } ] , | Z ˜ = [ Re { | Z } Im { | Z } ] ,
y n n ( t s ) = a ˜ 0 | U ˜ Λ ˜ U ˜ T | a ˜ 0 = Z ˜ | Λ ˜ | Z ˜ ( Λ ˜ = U ˜ T P n , eq T O ˜ n m T P n , eq U ˜ = diag { λ ˜ 1 , , λ ˜ 4 M n + 2 } ) y n s ( t s ) = Z ˜ | U ˜ T P ˜ n , eq T O ˜ n m T R ˜ n s D O ˜ s s | s ˜ o ( t s ) Z ˜ | B ˜ | s ˜ o ( t s ) Z ˜ | b ˜ ( t s ) ( B ˜ = U ˜ T P n , eq T O ˜ n n T R ˜ n s D O ˜ s s ) ,
ϕ N L = ( P ¯ + δ P ¯ ) γ K L .
< δ ϕ 2 > = < ϕ N L 2 Φ ¯ N L 2 > 2 P ¯ < δ P ¯ > ( γ K L ) 2 .
< δ ϕ > 2 P ¯ γ K L / P ¯ / < δ P ¯ > = 2 Φ ¯ N L / OSNR ,
< δ ϕ 1 / OSNR π / 2 arctan ( OSNR ) ,
< δ ϕ 2 > = 2 [ G N in B in + N 0 B l k K 3 ( 1 + 1 K ) ( 1 + 1 2 K ) ] Φ ¯ N L 2 / P ¯ ,

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