Abstract

During its propagation along a fiber, any noise is amplified by the modulated signal through the fiber’s nonlinearity-induced parametric gain. This nonlinear amplification of noise has been previously studied with the assumption that the signal can be approximated as continuous-wave. We present a novel method to analyze the parametric-gain-induced, nonlinear amplification of noise by an arbitrarily modulated signal based on perturbation theory. Because of the nonstationary nature of the output noise after its interaction with the modulated signal, the detailed correlation function is best computed in the frequency domain by assuming a given input bit sequence. The results are validated by split-step Fourier simulation and applied to obtain the probability distribution function of the detector statistics.

© 2004 Optical Society of America

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References

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  1. I. T. Monroy and G. Einarsson, “Bit error evaluation of optically preamplified direct detection receivers with Fabry–Perot optical filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
    [Crossref]
  2. R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
    [Crossref]
  3. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and 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. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
    [Crossref]
  5. G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).
  6. S. Wen, “Bi-end dispersion compensation for ultralong optical communication system,” J. Lightwave Technol. 17, 729–798 (1999).
    [Crossref]
  7. E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, New York, 1998).
  8. K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2232–2241 (1997).
    [Crossref]
  9. B. Xu and M. Brandt-Pearce, “Analysis on noise amplification by a CW pump signal due to fiber nonlinearity,” IEEE Photon. Technol. Lett., to be published.
  10. B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method and applications to fiber-optic communications,” in 2001 35th Asilomar Conference on Signals, Systems, and Computers (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 23–27.
  11. B. Xu, “Study of fiber nonlinear effects on fiber optic communication systems,” Ph.D. dissertation (University of Virginia, Charlottesville, Va., 2003).
  12. G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
    [Crossref]
  13. A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
    [Crossref]

2001 (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

2000 (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

1999 (1)

1998 (1)

1997 (4)

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2232–2241 (1997).
[Crossref]

I. T. Monroy and G. Einarsson, “Bit error evaluation of optically preamplified direct detection receivers with Fabry–Perot optical filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
[Crossref]

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

Benedetto, S.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[Crossref]

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

Bosco, G.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

Brandt-Pearce, M.

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2232–2241 (1997).
[Crossref]

Carena, A.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[Crossref]

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

Chowdhury, D.

R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
[Crossref]

Curri, V.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[Crossref]

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

Einarsson, G.

I. T. Monroy and G. Einarsson, “Bit error evaluation of optically preamplified direct detection receivers with Fabry–Perot optical filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

Gaudino, R.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[Crossref]

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

Hui, R.

R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
[Crossref]

Monroy, I. T.

I. T. Monroy and G. Einarsson, “Bit error evaluation of optically preamplified direct detection receivers with Fabry–Perot optical filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

Newhouse, M.

R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
[Crossref]

O’Sullivan, M.

R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
[Crossref]

Peddanarappagari, K. V.

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2232–2241 (1997).
[Crossref]

Poettcker, M.

R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
[Crossref]

Poggiolini, P.

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[Crossref]

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

Wen, S.

IEEE Photon. Technol. Lett. (3)

R. Hui, D. Chowdhury, M. Newhouse, M. O’Sullivan, and M. Poettcker, “Nonlinear amplification of noise in fibers with dispersion and its impact in optically amplified systems,” IEEE Photon. Technol. Lett. 9, 392–394 (1997).
[Crossref]

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

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “Parametric gain in multiwavelength systems: a new approach to noise enhancement analysis,” IEEE Photon. Technol. Lett. 12, 152–154 (2000).
[Crossref]

IEEE Trans. Commun. (1)

G. Bosco, A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “A novel analytical approach to the evaluation of the impact of fiber parametric gain on the bit error rate,” IEEE Trans. Commun. 49, 2154–2163 (2001).
[Crossref]

J. Lightwave Technol. (4)

K. V. Peddanarappagari and M. Brandt-Pearce, “Volterra series transfer function of single-mode fibers,” J. Lightwave Technol. 15, 2232–2241 (1997).
[Crossref]

A. Carena, V. Curri, R. Gaudino, P. Poggiolini, and S. Benedetto, “On the joint effects of fiber parametric gain and birefringence and their influence on ASE noise,” J. Lightwave Technol. 16, 1149–1157 (1998).
[Crossref]

S. Wen, “Bi-end dispersion compensation for ultralong optical communication system,” J. Lightwave Technol. 17, 729–798 (1999).
[Crossref]

I. T. Monroy and G. Einarsson, “Bit error evaluation of optically preamplified direct detection receivers with Fabry–Perot optical filters,” J. Lightwave Technol. 15, 1546–1553 (1997).
[Crossref]

Other (5)

E. Iannone, F. Matera, A. Mecozzi, and M. Settembre, Nonlinear Optical Communication Networks (Wiley, New York, 1998).

B. Xu and M. Brandt-Pearce, “Analysis on noise amplification by a CW pump signal due to fiber nonlinearity,” IEEE Photon. Technol. Lett., to be published.

B. Xu and M. Brandt-Pearce, “Modified Volterra series transfer function method and applications to fiber-optic communications,” in 2001 35th Asilomar Conference on Signals, Systems, and Computers (Institute of Electrical and Electronics Engineers, New York, 2002), pp. 23–27.

B. Xu, “Study of fiber nonlinear effects on fiber optic communication systems,” Ph.D. dissertation (University of Virginia, Charlottesville, Va., 2003).

G. P. Agrawal, Nonlinear Fiber Optics (Academic, New York, 1995).

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Figures (8)

Fig. 1
Fig. 1

Simplified system model for multispan system with a single ASE noise source. DC is the dispersion compensator.

Fig. 2
Fig. 2

Simplified system model for multispan system with multiple ASE noise sources.

Fig. 3
Fig. 3

Complex noise variance SNN(ω, ω) at each frequency.

Fig. 4
Fig. 4

Noise variance at each frequency for real and imaginary parts: (a) normalized SUU(ω, ω), (b) normalized SVV(ω, ω).

Fig. 5
Fig. 5

Noise correlation between ω1=0 and different ω2: (a) normalized SUU(ω1=0, ω2), (b) normalized SVV(ω1=0, ω2), (c) normalized SUV(ω1=0, ω2).

Fig. 6
Fig. 6

Eigenvalues of the approximate correlation matrix for two spans.

Fig. 7
Fig. 7

Eigenvalues of the approximate correlation matrix for four spans.

Fig. 8
Fig. 8

Comparison between PDFs based on MC simulation and Gaussian approximation.

Tables (3)

Tables Icon

Table 1 Second-Order Nonlinear Terms of the Noise Frequency Correlation Functions from the Second-Order Noise Nonlinear Terms N˜H(ω) to N˜N(ω)

Tables Icon

Table 2 Second-Order Nonlinear Terms of the Noise Frequency Correlation Functions from the Beating of the First-Order Noise Nonlinear Terms N˜X(ω) and N˜F(ω)

Tables Icon

Table 3 Number of Trials Associated with Different Estimation Confidence

Equations (88)

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

Az=-α2A+j2β2α2Aαt2-jγ|A|2A,
SXY(ω1, ω2)=E{X(ω1)Y*(ω2)},
ΔANL(t, z)=-jγ|A(t, z)|2A(t, z)Δz.
As(t, z)=As,0(t, z)+As,1(t, z)+As,2(t, z)+ ,
N(t, z)=N0(t, z)+N1(t, z)+N2(t, z)+.
As,0(ω, z)=exp(-αz/2)exp(-jβ2ω2z/2)×exp(jDpreω2/2)As(ω, 0).
ΔAs,1(t, z)+ΔN1(t, z)=-jΔz[|As,0(t, z)|2As,0(t, z)+2|As,0(t, z)|2N0(t, z)+As,02(t, z)N0*(t, z)+H.O.T.].
ΔN˜1(ω, z)=expα2Lexpj2Dpostω2exp-α2(L-z)×exp-j2β2ω2(L-z)ΔN1(ω, z).
ΔN˜1(ω, z)=exp(αz/2)exp(-jβ2ω2(LD-z)/2)×ΔN1(ω, z)
N˜X(ω)=0Lexpα2zexp-j2β2ω2(LD-z)×(-jγ)2F{As,0(t, z)As,0*(t, z)N0(t, z)}dz
=0Lexpα2zexp-j2β2ω2(LD-z)×-jγ4π22As,0(ω, z)As,0*(ω, z)×N0(ω-ω1+ω2, z)dω1dω2dz.
N˜F(ω)=0Lexpα2zexp-j2β2ω2(LD-z)×(-jγ)F{As,0(t, z)N0*(t, z)As,0(t, z)}dz.
ΔN2(t, z)=-jγΔz[As,02(t, z)N1*(t, z)+2As,0(t, z)As,1*(t, z)N0(t, z)+2As,0(t, z)N0*(t, z)As,1(t, z)+2As,0(t, z)As,0*(t, z)N1(t, z)+2As,1(t, z)As,0*(t, z)N0(t, z)+H.O.T.].
ΔNH(t, z)=(-jγ)As,0(t, z)NX*(t, z)As,0(t, z)Δz,
ΔNI(t, z)=(-jγ)As,0(t, z)NF*(t, z)As,0(t, z)Δz,
ΔNJ(t, z)=(-jγ)2As,0(t, z)As,0*(t, z)NX(t, z)Δz,
ΔNK(t, z)=(-jγ)2As,0(t, z)As,0*(t, z)NF(t, z)Δz,
ΔNL(t, z)=(-jγ)2N0(t, z)As,0*(t, z)As,1(t, z)Δz,
ΔNM(t, z)=(-jγ)2As,0(t, z)N0*(t, z)As,1(t, z)Δz,
ΔNN(t, z)=(-jγ)2As,0(t, z)As,1*(t, z)N0(t, z)Δz.
N˜2(ω)=N˜H(ω)+N˜I(ω)+N˜J(ω)+N˜K(ω)+N˜L(ω)+N˜M(ω)+N˜N(ω)
N˜(ω)N˜0(ω)+N˜1(ω)+N˜2(ω),
E{U˜(ω1)U˜(ω2)}E{[U˜0(ω1)+U˜1(ω1)+U˜2(ω1)]×[U˜0(ω2)+U˜1(ω2)+U˜1(ω2)]}=E{U˜0(ω1)U˜0(ω2)}
+E{U˜0(ω1)U˜1(ω2)}+E{U˜1(ω1)U˜0(ω2)}
+E{U˜0(ω1)U˜2(ω2)}+E{U˜1(ω1)U˜1(ω2)}+E{U˜2(ω1)U˜0(ω2)}+H.O.T.
SUU,0(ω1, ω2)=E{U˜0(ω1)U˜0(ω2)}=E{V˜0(ω1)V˜0(ω2)}=SVV,0(ω1, ω2)=σω22δ(ω1-ω2),
SUV,0(ω1, ω2)=E{U˜0(ω1)V˜0(ω2)}=0
SUU,1(ω1, ω2)=E{U˜0(ω1)U˜1(ω2)}+E{U˜1(ω1)U˜0(ω2)}.
E{U˜0(ω1)U˜1(ω2)}=(1/4)E{[N˜0(ω1)+N˜0*(ω1)][N˜1(ω2)+N˜1*(ω2)]}.
E{N˜0(ω1)N˜1(ω2)}=E{[N˜0(ω1)[N˜X(ω2)+N˜F(ω2)]]}=E{N˜0(ω1)N˜F(ω2)},
E{N˜0(ω1)N˜1(ω2)}=0Lexp-j2β2ω22(LD-z)×-jγ4π2σω2As,0(v, z)×As,0(ω1+ω2-v, z)dvdz,
T1(ω1, ω2).
E{N˜0(ω1)N˜1*(ω2)}=0Lexp-j2β2(ω12-ω22)(LD-z)jγ4π2σω22As,0(v,z)×As,0*(v+ω1-ω2, z)dvdz,
T2(ω1, ω2),
E{N˜0*(ω1)N˜1(ω2)}=T2*(ω1, ω2),
E{N˜0*(ω1)N˜1*(ω2)}=T1*(ω1, ω2).
SUU,1(ω1, ω2)=(1/2){T1(ω1, ω2)+T1*(ω1, ω2)}
=12γ2πσω2I0Lexp-j2β2(ω12+ω22)(LD-z)×C(ω1+ω2, z)dz.
C(ω, z)=12πAs,0(v, z)As,0(w-v, z)dv,
SVV,1(ω, ω2)=-(1/2){T1(ω1, ω2)+T1*(ω1, ω2)}.
SUV,1(ω1, ω2)=(-j/2){T1(ω1, ω2)-T1*(ω1, ωv2)}.
SNN,1(ω, ω)=SUU,1(ω, ω)+SVV,1(ω, ω)=0.
SUU,2(ω1, ω2)=E{U˜0(ω1)U˜2(ω2)}+E{U˜1(ω1)U˜1(ω2)}+E{U˜2(ω1)U˜0(ω2)},
E{U˜0(ω1)U˜2(ω2)}=(1/4)E{[N˜0(ω1)+N˜0*(ω1)]×[N˜2(ω2)+N˜2*(ω2)]}.
E{N˜0(ω1)N˜H(ω2)}
=2γ2σω20LyLexp-j2β2ω12(LD-y)×exp-j2β2ω22(LD-z)×14π2C(ω2+v, z)B(ω1-v, y)×expj2β2v2(z-y)dvdzdy,
TH(ω1, ω2),
E{N˜X(ω1)N˜F(ω2)}=TK(ω2, ω1)-TH(ω1, ω2),
E{N˜X(ω1)N˜X*(ω2)}=-TJ(ω1, ω2)-TJ*(ω2, ω1),
E{N˜F(ω1)N˜F*(ω2)}=TI(ω1, ω2)+TI*(ω2, ω1).
SUU,2(ω1, ω2)=R{TI(ω1, ω2)+TI(ω2, ω1)}+R{TK(ω1, ω2)+TK(ω2, ω1)}+(1/2)R{TM(ω1, ω2)+TM(ω2, ω1)},
SVV,2(ω1, ω2)=R{TI(ω1, ω2)+TI(ω2, ω1)}-R{TK(ω1, ω2)+TK(ω2, ω1)}-(1/2)R{TM(ω1, ω2)+TM(ω2, ω1)},
SUV,2(ω1, ω2)=-I{TI(ω1, ω2)-TI(ω2, ω1)}+I{TK(ω1, ω2)+TK(ω2, ω1)}+(1/2)I{TM(ω1, ω2)+TM(ω2, ω1)}.
SUU,2(ω, ω)+SVV,2(ω, ω)=2[TI(ω, ω)+TI*(ω, ω)]0.
As,0(t, z+mL)=As,0(t, z),
N0(t, z+mL)=N0(t, z),
N˜X,m(ω)=mN˜X(ω),N˜F,m(ω)=mN˜F(ω),
SUU,1,m(ω1, ω2)=(1/2)m{T1(ω1, ω2)+T1*(ω1, ω2)},
SVV,1,m(ω1, ω2)=-(1/2)m{T1(ω1, ω2)+T1*(ω1, ω2)},
SUV,1,m(ω1, ω2)=(-j/2)m{T1(ω1, ω2)-T1*(ω1, ω2)}.
N˜H,m(ω)=0mL expα2z˜exp-j2β2ω2(LD-z˜)×-jγ4π2As,0(v3, z)×As,0(ω-v3+v4, z)×0zexpj2β2v42(z˜-y˜)×exp-α2(z˜-y˜)jγ4π2×2As,0(v1, y)As,0(v2, y)×N0*(v4-v1+v2, y)d4vdydz,
N˜H,m(ω)=m2+m20L0zΔNH(ω, y, z)dydz+m2-m20LzLΔNH(ω, y, z)dydz,
E{N˜0(ω1)N˜H,m(ω2)}=m2+m2TH(ω1, ω2)+m2-m2TH(ω1, ω2),
TH(ω1, ω2)=-TK(ω2, ω1).
TI(ω1, ω2)=TI *(ω2, ω1),
TJ(ω1, ω2)=TJ*(ω2, ω1),
TK(ω1, ω2)=TH(ω2, ω1),
TL(ω1, ω2)=-TN*(ω2, ω1).
SUU,2,m(ω1, ω2)=142m2T˜I(ω1, ω2)-(m2-m)T˜H(ω1, ω2)+(m2-m)T˜K(ω1, ω2)+m2+m2T˜M(ω1, ω2)+m2-m2T˜M(ω1, ω2),
SVV,2,m(ω1, ω2)=142m2T˜I(ω1, ω2)+(m2-m)T˜H(ω1, ω2)-(m2-m)T˜K(ω1, ω2)-m2+m2T˜M(ω1, ω2)-m2-m2T˜M(ω1, ω2),
SUV,2,m(ω1, ω2)=j42m2T˜˜I(ω1, ω2)+(m2-m)T˜˜H(ω1, ω2)-(m2-m)T˜˜K(ω1, ω2)-m2+m2T˜˜M(ω1, ω2)-m2-m2T˜˜M(ω1, ω2),
T˜i(ω1, ω2)=Ti(ω1, ω2)+Ti*(ω1, ω2)+Ti(ω2, ω1)+Ti*(ω2, ω1),i=I, H, K, M, M,
T˜˜I(ω1, ω2)=TI(ω1, ω2)=TI*(ω1, ω2)-TI(ω2, ω1)+TI*(ω2, ω1),
T˜˜i(ω1, ω2)=Ti(ω1, ω2)-Ti*(ω1, ω2)+Ti(ω2, ω1)-Ti*(ω2, ω1),i=H, K, M, M.
SUU,2,m(ω, ω)+SVV,2,m(ω, ω)=m2T˜I(ω, ω).
SSXY,i,M(ω1, ω2)=k=1MSXY,i,M-k(ω1, ω2),i=0, 1, 2,
N˜H,m(ω)=0mLG(z, mL)exp-j2D(z, mL)ω2×-jγ4π2As,0(v3, z)×As,0(ω-v3+v4, z)×0zexpj2D(y, z)v42×G(y, z)jγ4π22As,0(v1, y)As,0(v2, y)×N0*(v4-v1+v2, y)d4vdydz,
T1(ω1, ω2)=0Lexp-j2β2ω22(LD-z)×-jγ2πσω2C(ω1+ω2, z)dz,
FF-1exp-j2β2v2(z-y)C(v+ω1, y)×F-1{C*(v+ω2, z)}.
TI(ω1, ω2)=γ2σω20L expj2β2ω22(LD-z)×14π2C*(v+ω2, z)exp-j2β2v2z×0zexp-j2β2ω12(LD-y)×expj2β2v2yC(v+ω1, y)dydvdz.
0zexp-j2β2ω12(LD-y)×expj2β2v2yC(v+ω1, y)dy
P(|(1/M)Σyi2-σ2|βσ2)>α
SXY(ω1, ω2)=1/MXi(ω1)Yi*(ω2)-1/MXi(ω1)1/MYi*(ω2),
SUU(ω1, ω2)=SUU(ω1, ω2)-(σω2/2)δ(ω1-ω2),
SVV(ω1, ω2)=SVV(ω1, ω2)-(σω2/2)δ(ω1-ω2),
A(t, mL)=P0s(t)×exp-jγmP0|s(t)|2α[1-exp(-αL)].
Am(t)=A0(t)1-jγmP0αs(t) 2-12γmP0αs(t)22+j6γmP0αs(t)23+,
fY(y|b0=0or1)=12N-1b˜0{0,1}N-1N [m(b0, b˜0),σ2(b0, b˜0)]

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