Abstract

New semi-analytic formulas of the power spectral density (PSD) of single- and cross-channel nonlinear interference (NLI) in coherent optical links for which the Gaussian Noise (GN) model applies are presented. From the PSD, a new bound on cross-channel NLI power is obtained. The new formulas are useful to both quickly compute single- and cross-channel NLI power, and to test the accuracy of numerical routines that directly solve the double frequency integral in the GN reference formula.

© 2013 Optical Society of America

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References

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  1. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30(10), 1524–1539 (2012).
    [Crossref]
  2. P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Light-wave Technol. 30(24), 3857–3879 (2012).
    [Crossref]
  3. S. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photonics Technol. Lett. 25(10), 961–964 (2013).
    [Crossref]
  4. V. Curri, A. Carena, P. Poggiolini, G. Bosco, and F. Forghieri, “Extension and validation of the GN model for non-linear interference to uncompensated links using Raman amplification,” Opt. Express 21(3), 3308–3317 (2013).
    [Crossref] [PubMed]
  5. P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
    [Crossref]
  6. A. Bononi and P. Serena, “An alternative derivation of Johannisson’s regular perturbation model,” arXiv:1207.4729v1 [physics.optics] (2012).
  7. P. Serena and A. Bononi, “On the accuracy of the Gaussian nonlinear model for dispersion-unmanaged coherent links,” in Proc. ECOC’13(2013), paper Th1D3.
  8. R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
    [Crossref] [PubMed]
  9. P. Serena and A. Bononi, “An alternative approach to the Gaussian noise model and its system implications,” J. Lightwave Technol. 31(22), 3489–3499 (2013).
    [Crossref]
  10. M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008).
    [Crossref] [PubMed]
  11. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012).
    [Crossref] [PubMed]
  12. X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express 18(18), 19039–19054 (2010).
    [Crossref] [PubMed]
  13. A. Bononi and O. Beucher, “Semi-analytic formulas of single-channel and cross-channel nonlinear interference in highly-dispersed WDM coherent optical links with rectangular signal spectra,” arXiv:1309.0244v1 [physics.optics] (2013).
  14. A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prenctice-Hall, 1989).
  15. “Optilux Toolbox,” http://www.optilux.sourceforge.net .
  16. A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC’11(2011), paper OWO7.
  17. E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011).
    [Crossref] [PubMed]
  18. A. Bononi, N. Rossi, and P. Serena, “On the nonlinear threshold versus distance in long-haul highly-dispersive coherent systems,” Opt. Express 20, B204–B216 (2012).
    [Crossref] [PubMed]
  19. A. Mecozzi and R. J. Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightwave Technol. 30(12), 2011–2024 (2012),
    [Crossref]
  20. O. Rival and K. Mheidly, “Accumulation rate of inter and intra-channel nonlinear distortions in uncompensated 100G PDM-QPSK systems,” in Proc. OFC’12 (2012), paper JW2A.52.
  21. X. Wei, “Power-weighted dispersion distribution function for characterizing nonlinear properties of long-haul optical transmission links,” Opt. Lett. 31(17) 2544–2546 (2006).
    [Crossref] [PubMed]
  22. B. Châtelain, C. Laperle, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, and D. V. Plant, “A family of Nyquist pulses for coherent optical communications,” Opt. Express 20, 8397–8416 (2012).
    [Crossref] [PubMed]
  23. J. G. Proakis and M. Salehi, Communication Systems Engineering, 2 (Prentice-Hall, 2002).
  24. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).
    [Crossref]

2013 (5)

2012 (6)

2011 (2)

E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011).
[Crossref] [PubMed]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).
[Crossref]

2010 (1)

2008 (1)

2006 (1)

Beucher, O.

A. Bononi and O. Beucher, “Semi-analytic formulas of single-channel and cross-channel nonlinear interference in highly-dispersed WDM coherent optical links with rectangular signal spectra,” arXiv:1309.0244v1 [physics.optics] (2013).

Bononi, A.

P. Serena and A. Bononi, “An alternative approach to the Gaussian noise model and its system implications,” J. Lightwave Technol. 31(22), 3489–3499 (2013).
[Crossref]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012).
[Crossref] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “On the nonlinear threshold versus distance in long-haul highly-dispersive coherent systems,” Opt. Express 20, B204–B216 (2012).
[Crossref] [PubMed]

E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011).
[Crossref] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC’11(2011), paper OWO7.

A. Bononi and O. Beucher, “Semi-analytic formulas of single-channel and cross-channel nonlinear interference in highly-dispersed WDM coherent optical links with rectangular signal spectra,” arXiv:1309.0244v1 [physics.optics] (2013).

A. Bononi and P. Serena, “An alternative derivation of Johannisson’s regular perturbation model,” arXiv:1207.4729v1 [physics.optics] (2012).

P. Serena and A. Bononi, “On the accuracy of the Gaussian nonlinear model for dispersion-unmanaged coherent links,” in Proc. ECOC’13(2013), paper Th1D3.

Borowiec, A.

Bosco, G.

Carena, A.

Chagnon, M.

Châtelain, B.

Chen, X.

Cho, P.

Curri, V.

Dar, R.

Essiambre, R. J.

Feder, M.

Forghieri, F.

Gagnon, F.

Grellier, E.

Johannisson, P.

P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
[Crossref]

Karagodsky, V.

Karlsson, M.

P. Johannisson and M. Karlsson, “Perturbation analysis of nonlinear propagation in a strongly dispersive optical communication system,” J. Lightwave Technol. 31(8), 1273–1282 (2013).
[Crossref]

Khurgin, J.

Laperle, C.

Mecozzi, A.

Meiman, Y.

Mheidly, K.

O. Rival and K. Mheidly, “Accumulation rate of inter and intra-channel nonlinear distortions in uncompensated 100G PDM-QPSK systems,” in Proc. OFC’12 (2012), paper JW2A.52.

Nazarathy, M.

Noe, R.

Oppenheim, A. V.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prenctice-Hall, 1989).

Plant, D. V.

Poggiolini, P.

V. Curri, A. Carena, P. Poggiolini, G. Bosco, and F. Forghieri, “Extension and validation of the GN model for non-linear interference to uncompensated links using Raman amplification,” Opt. Express 21(3), 3308–3317 (2013).
[Crossref] [PubMed]

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Light-wave Technol. 30(24), 3857–3879 (2012).
[Crossref]

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of non-linear propagation effects in uncompensated optical coherent transmission links,” J. Lightwave Technol. 30(10), 1524–1539 (2012).
[Crossref]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).
[Crossref]

Proakis, J. G.

J. G. Proakis and M. Salehi, Communication Systems Engineering, 2 (Prentice-Hall, 2002).

Rival, O.

O. Rival and K. Mheidly, “Accumulation rate of inter and intra-channel nonlinear distortions in uncompensated 100G PDM-QPSK systems,” in Proc. OFC’12 (2012), paper JW2A.52.

Roberts, K.

Rossi, N.

Salehi, M.

J. G. Proakis and M. Salehi, Communication Systems Engineering, 2 (Prentice-Hall, 2002).

Savory, S.

S. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photonics Technol. Lett. 25(10), 961–964 (2013).
[Crossref]

Schafer, R. W.

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prenctice-Hall, 1989).

Serena, P.

P. Serena and A. Bononi, “An alternative approach to the Gaussian noise model and its system implications,” J. Lightwave Technol. 31(22), 3489–3499 (2013).
[Crossref]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012).
[Crossref] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “On the nonlinear threshold versus distance in long-haul highly-dispersive coherent systems,” Opt. Express 20, B204–B216 (2012).
[Crossref] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC’11(2011), paper OWO7.

A. Bononi and P. Serena, “An alternative derivation of Johannisson’s regular perturbation model,” arXiv:1207.4729v1 [physics.optics] (2012).

P. Serena and A. Bononi, “On the accuracy of the Gaussian nonlinear model for dispersion-unmanaged coherent links,” in Proc. ECOC’13(2013), paper Th1D3.

Shieh, W.

Shpantzer, I.

Shtaif, M.

Vacondio, F.

Wei, X.

Weidenfeld, R.

Xu, X.

IEEE Photonics Technol. Lett. (2)

S. Savory, “Approximations for the nonlinear self-channel interference of channels with rectangular spectra,” IEEE Photonics Technol. Lett. 25(10), 961–964 (2013).
[Crossref]

P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-linear propagation in uncompensated optical transmission links,” IEEE Photonics Technol. Lett. 23, 742–744 (2011).
[Crossref]

J. Light-wave Technol. (1)

P. Poggiolini, “The GN model of non-linear propagation in uncompensated coherent optical systems,” J. Light-wave Technol. 30(24), 3857–3879 (2012).
[Crossref]

J. Lightwave Technol. (4)

Opt. Express (8)

R. Dar, M. Feder, A. Mecozzi, and M. Shtaif, “Properties of nonlinear noise in long, dispersion-uncompensated fiber links,” Opt. Express 21(22), 25685–25699 (2013).
[Crossref] [PubMed]

B. Châtelain, C. Laperle, K. Roberts, M. Chagnon, X. Xu, A. Borowiec, F. Gagnon, and D. V. Plant, “A family of Nyquist pulses for coherent optical communications,” Opt. Express 20, 8397–8416 (2012).
[Crossref] [PubMed]

M. Nazarathy, J. Khurgin, R. Weidenfeld, Y. Meiman, P. Cho, R. Noe, I. Shpantzer, and V. Karagodsky, “Phased-array cancellation of nonlinear FWM in coherent OFDM dispersive multi-span links,” Opt. Express 16(20), 15777–15810 (2008).
[Crossref] [PubMed]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling nonlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express 20(7), 7777–7791 (2012).
[Crossref] [PubMed]

X. Chen and W. Shieh, “Closed-form expressions for nonlinear transmission performance of densely spaced coherent optical OFDM systems,” Opt. Express 18(18), 19039–19054 (2010).
[Crossref] [PubMed]

E. Grellier and A. Bononi, “Quality parameter for coherent transmissions with Gaussian-distributed nonlinear noise,” Opt. Express 19(13), 12781–12788 (2011).
[Crossref] [PubMed]

A. Bononi, N. Rossi, and P. Serena, “On the nonlinear threshold versus distance in long-haul highly-dispersive coherent systems,” Opt. Express 20, B204–B216 (2012).
[Crossref] [PubMed]

V. Curri, A. Carena, P. Poggiolini, G. Bosco, and F. Forghieri, “Extension and validation of the GN model for non-linear interference to uncompensated links using Raman amplification,” Opt. Express 21(3), 3308–3317 (2013).
[Crossref] [PubMed]

Opt. Lett. (1)

Other (8)

O. Rival and K. Mheidly, “Accumulation rate of inter and intra-channel nonlinear distortions in uncompensated 100G PDM-QPSK systems,” in Proc. OFC’12 (2012), paper JW2A.52.

J. G. Proakis and M. Salehi, Communication Systems Engineering, 2 (Prentice-Hall, 2002).

A. Bononi and P. Serena, “An alternative derivation of Johannisson’s regular perturbation model,” arXiv:1207.4729v1 [physics.optics] (2012).

P. Serena and A. Bononi, “On the accuracy of the Gaussian nonlinear model for dispersion-unmanaged coherent links,” in Proc. ECOC’13(2013), paper Th1D3.

A. Bononi and O. Beucher, “Semi-analytic formulas of single-channel and cross-channel nonlinear interference in highly-dispersed WDM coherent optical links with rectangular signal spectra,” arXiv:1309.0244v1 [physics.optics] (2013).

A. V. Oppenheim and R. W. Schafer, Discrete-Time Signal Processing (Prenctice-Hall, 1989).

“Optilux Toolbox,” http://www.optilux.sourceforge.net .

A. Bononi, N. Rossi, and P. Serena, “Transmission limitations due to fiber nonlinearity,” in Proc. OFC’11(2011), paper OWO7.

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Figures (9)

Fig. 1
Fig. 1 Domains over which integrand in Eq. (1) is non-zero when input PSD is a gate over f ∈ [−δ, δ]. Integration over domains I through IV yields the four lines in Eq. (4).
Fig. 2
Fig. 2 NLI normalized spectra vs. frequency (normalized to bandwidth 2δ = 28GHz) for channels with rectangular input PSD at three dispersion values D (ps/nm/km) for 20×100km DU link. Kernel (3) had α = 0.2dB/km, ζ = 1, γ = 1.27(W km)−1. Theory: cyan, yellow, dark green curves. SSF simulations: blue, red, green curves. (a)GNLI(f)/P3 for single channel, theory Eqs. (5) and (6); (b)GXCI(f)/P3 on central channel for M =15 channels at Δ = 50GHz, theory Eqs.(13) and (14).
Fig. 3
Fig. 3 Over-estimation �� in an M-channel Nyquist-WDM N×100km DU system with each channel having rectangular spectrum of bandwidth 2δ. (a) versus fiber dispersion D at 2δ = 28GHz; (b) versus 2δ, at D = 17ps/nm/km. Blue lines: value at D = 0, Eq. (11).
Fig. 4
Fig. 4 Integration domain in Eq. (1) given at f = 0 by intersection of support of G(f + f2) (horizontal stripes), G(f + f1) (vertical) and G(f + f1 + f2) (tilted). Red island = SCI. Yellow islands = XCI. Blue islands = MCI. Red curves: contours of |�� (f1f2)|2.
Fig. 5
Fig. 5 (a) The four integration domains in the (f1, f2)-plane whose integration yields the four v-integral terms in Eq. (15) in the order they appear (see also [13, Fig. 10]). Dashed curves are hyperbolas. (b) Domain �� delimited by hyperbola v = f 1 f 2 = δ Δ m passing at point A, approximating squared kernel integral over domain ��1, and also ��3.
Fig. 6
Fig. 6 NLI coefficient [dB(mW−2)] versus channel bandwidth 2δR [GHz] for SMF DU link with N = 20 spans, span length zA = 100km, M = 81 channels, at a bandwidth efficiency η = 2 δ Δ = 1 (a); and η = 0.56 (b). Crosses: [2, eq. (13), (22)].
Fig. 7
Fig. 7 NLI coefficient [dB(mW−2)] vs. channel spacing Δ [GHz], at fixed channel bandwidth 2δR, for 120×50km SMF DU link, and M = [1, 11, 81] channels. Crosses: [2, eq. (13),(22)].
Fig. 8
Fig. 8 (a) Input WDM spectra at increasing per-channel bandwidth 2δ. (b) NLI coefficient vs. channel bandwidth 2δ, at fixed symbol-rate R = 5Gbaud, for SMF 20x100km DU link, M = 81 channels, Δ = 50GHz spacing. Blue circle at Nyquist-WDM case 2δ = 50GHz gives the analytical WDM value.
Fig. 9
Fig. 9 (a),(c): NLI coefficient versus spans N (log scale) at fixed bandwidth/symbol-rate 2δ = R = 28Gbaud, for SMF DU link with span length zA = 50km, and M = 19 channels. (b),(d): local slope [dB/dB] of log-log plots in (a),(c) versus N. Dash-dotted: average slope. Black dashed: ε-fit Eq. (21). Crosses: [2, eq. (13), (22)].

Equations (28)

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I ( f ) | 𝒦 ( f 1 f 2 ) | 2 G ( f + f 1 ) G ( f + f 2 ) G ( f + f 1 + f 2 ) d f 1 d f 2
𝒦 ( v ) 0 L γ ( s ) 𝒢 ( s ) e j ( 2 π ) 2 C ( s ) v d s
𝒦 ( v ) = 𝒦 ( 0 ) χ ( v ) η 1 ( v )
χ ( v ) = 1 N k = 1 N e j ( k 1 ) β 2 z A ζ ( 2 π ) 2 v , and η 1 ( v ) = 1 e α z A ( 1 j β 2 α ( 2 π ) 2 v ) ( 1 e α z A ) ( 1 j β 2 α ( 2 π ) 2 v ) .
I ( f ) = 0 | 𝒦 ( v ) | 2 [ 0 1 u G ( f + u ) G ( f + v u ) G ( f + u + v u ) d u + 0 1 u G ( f u ) G ( f + v u ) G ( f u + v u ) d u + 0 1 u G ( f u ) G ( f v u ) G ( f u v u ) d u + 0 1 u G ( f + u ) G ( f v u ) G ( f + u v u ) d u ] d v
( f ) = 0 ( δ f 2 ) 2 | 𝒦 ( v ) | 2 ln ( δ f 2 + ( δ f 2 ) 2 v δ f 2 ( δ f 2 ) 2 v ) d v + 2 0 δ 2 f 2 | 𝒦 ( v ) | 2 ln ( δ 2 f 2 v ) d v + 0 ( δ + f 2 ) 2 | 𝒦 ( v ) | 2 ln ( δ + f 2 + ( δ + f 2 ) 2 v δ + f 2 ( δ + f 2 ) 2 v ) d v
( f ) = ( | f | δ ) 2 2 δ ( | f | δ ) | 𝒦 ( v ) | 2 ln ( v ( | f | δ ) 2 ) d v + 2 δ ( | f | δ ) ( δ + | f | 2 ) 2 | 𝒦 ( v ) | 2 ln ( δ + | f | 2 + ( δ + | f | 2 ) 2 v δ + | f | 2 ( δ + | f | 2 ) 2 v ) d v
( 0 ) = 2 0 ( δ 2 ) 2 | 𝒦 ( v ) | 2 ln ( δ 2 + ( δ 2 ) 2 v δ 2 ( δ 2 ) 2 v ) d v + 2 0 δ 2 | 𝒦 ( v ) | 2 ln ( δ 2 v ) d v .
( 0 ) 4 0 δ 2 | 𝒦 ( v ) | 2 ln ( δ 2 v ) d v
( f ) = { 3 δ 2 f 2 if | f | δ ( 3 δ | f | ) 2 / 2 if δ | f | < 3 δ
P N L I = 16 27 ( P 2 δ ) 3 δ δ ( f ) d f 16 27 P 3 ¯ ( 2 δ ) 2 .
𝒪 = 1 / ( 1 1 9 M 2 )
I m ( f ) | 𝒦 ( f 1 f 2 ) | 2 G 0 ( f + f 1 ) G m ( f + f 2 ) G m ( f + f 1 + f 2 ) d f 1 d f 2
m ( f ) = 0 η ε m | 𝒦 ( v ) | 2 ln ( v ε m η m + 2 ( η m + 2 ) 2 v ) d v + η ε m η m Δ | 𝒦 ( v ) | 2 ln ( η η m + 2 ( η m + 2 ) 2 v ) d v + 0 η m Δ | 𝒦 ( v ) | 2 ln ( η m 2 + ( η m 2 ) 2 + v v ε m + ) d v + η m Δ η ε m + | 𝒦 ( v ) | 2 ln ( η ε m + v ) d v + 0 ε m Δ | 𝒦 ( v ) | 2 ln ( ε m 2 + ( ε m 2 ) 2 + v v η m + ) d v + ε m Δ ε η m + | 𝒦 ( v ) | 2 ln ( ε η m + v ) d v + 0 ε η m | 𝒦 ( v ) | 2 ln ( v η m ε m + 2 ( ε m + 2 ) 2 v ) d v + ε η m ε m Δ | 𝒦 ( v ) | 2 ln ( ε ε m + 2 ( ε m + 2 ) 2 v ) d v
m ( f ) = η ( ε m η ) η η m + | 𝒦 ( v ) | 2 ln ( ε m 2 ( ε m 2 ) 2 + v η ) d v + η η m + 2 δ η m + | 𝒦 ( v ) | 2 ln ( ε m 2 + ( ε m 2 ) 2 + v v η m + ) d v + η η m η ( ε m + + η ) | 𝒦 ( v ) | 2 ln ( v η η m ) d v + η ( ε m + + η ) 2 δ η m | 𝒦 ( v ) | 2 ln ( v η m ε m + 2 ( ε m + 2 ) 2 v ) d v
m ( 0 ) = 2 { 0 δ Δ m | 𝒦 ( v ) | 2 ln ( v Δ m Δ m + 2 ( Δ m + 2 ) 2 v ) d v + δ Δ m δ m Δ | 𝒦 ( v ) | 2 ln ( δ Δ m + 2 ( Δ m + 2 ) 2 v ) d v + 0 δ m Δ | 𝒦 ( v ) | 2 ln ( Δ m 2 + ( Δ m 2 ) 2 + v v Δ m + ) d v + δ m Δ δ Δ m + | 𝒦 ( v ) | 2 ln ( δ Δ m + v ) d v } .
m ( 0 ) 4 𝒟 | 𝒦 ( f 1 f 2 ) | 2 d f 1 d f 2 = 4 v = 0 δ Δ m u = v / Δ m + v / Δ m | 𝒦 ( v ) | 2 1 u d u d v = 4 ln ( Δ m + Δ m ) 0 δ Δ m | 𝒦 ( v ) | 2 d v
m ( 0 ) 4 ln ( 1 + η / 2 m 1 η / 2 m ) 0 | 𝒦 ( v ) | 2 d v
a S C I = 16 27 R ( 2 δ ) 3 ( 0 )
a X C I = 16 27 R ( 2 δ ) 3 2 m = 1 N c m ( 0 )
a X C I U B = 16 27 R δ 3 [ Γ ( N c + 1 + η 2 ) Γ ( 1 η 2 ) Γ ( N c + 1 η 2 ) Γ ( 1 + η 2 ) ] 0 | 𝒦 ( v ) | 2 d v
ε = 1 ln ( N ) + 1 2 ln ( 4 μ 5 ( α z A ) 2 | β 2 | α R 2 )
S N R = 2 ( N 0 2 + G N L I ( 0 ) ) P R = P ( N 0 2 + G N L I ( 0 ) ) R
I 0 | 𝒦 ( v ) | 2 d v = 1 2 0 | 𝒦 ( w 2 π ) | 2 d w 2 π = 𝒦 ( 0 ) 2 2 J 2 ( c ) d c 2 π .
J 2 ( c ) d c 2 π = 1 N 2 k = 1 N J 1 2 ( c ( k 1 ) β 2 z A ) d c 2 π = 1 N J 1 2 ( c ) d c 2 π .
I = N 𝒦 1 ( 0 ) 2 2 J 1 2 ( c ) d c 2 π N 0 | 𝒦 1 ( v ) | 2 d v .
J 1 ( c ) = { e α c β 2 ( 1 e α z A ) α | β 2 | for { 0 < c < z A β 2 if β 2 < 0 z A β 2 < c < 0 if β 2 > 0 0 else
I = N ( γ 2 ) 2 ( 1 e α z A ) 2 2 [ α 4 π | β 2 | 1 e 2 α z A ( 1 e α z A ) 2 ] = N γ 2 8 π 1 e 2 α z A α | β 2 | .

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