Abstract

A first order perturbation theory is used to develop analytical expressions for the power spectral density (PSD) of the nonlinear distortions due to intra-channel four-wave mixing (IFWM). For non-Gaussian pulses, the PSD can not be calculated analytically. However, using the stationary phase approximations, we found that convolutions become simple multiplications and a simple analytical expression for the PSD of the nonlinear distortion is found. The PSD of the nonlinear distortion is combined with the amplified spontaneous emission (ASE) PSD to obtain the total variance and bit error ratio (BER). The analytically estimated BER is found to be in good agreement with numerical simulations.

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  1. I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
    [CrossRef]
  2. R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett.35, 1576–1578 (1999).
    [CrossRef]
  3. P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett.24, 1454–1456 (1999).
    [CrossRef]
  4. A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett.12, 292–294 (2000).
  5. S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett.13, 800–802 (2001).
    [CrossRef]
  6. 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. Express16, 15777–15810 (2008).
    [CrossRef] [PubMed]
  7. D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol.27, 2916–2923 (2009).
    [CrossRef]
  8. A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling monlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express20, 7777–7791 (2012).
    [CrossRef] [PubMed]
  9. A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol.30, 1524–1539 (2012).
    [CrossRef]
  10. P. Poggiolini, A. Carena, V. Curri, G. Bosco, and F. Forghieri, “Analytical modeling of non-Linear propagation in uncompensated optical transmission links,” IEEE Photon. Technol. Lett.23, 742–744 (2011).
    [CrossRef]
  11. A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol.30, 2011–2024 (2012).
    [CrossRef]
  12. S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett.37, 2931–2933 (2012).
    [CrossRef] [PubMed]
  13. S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol.23, 2073–2080 (2005).
    [CrossRef]
  14. J. D. Jackson, Classical Electrodynamics (Wiley, 1998).
  15. U. Madhow, “Chapter 3: Demodulation,” in Fundamentals of Digital Communication (Cambridge University Press, 2008).
    [CrossRef]
  16. A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.
  17. M. Nazarathy, “Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series,” IEEE Trans. Commun.54, 295–301 (2006).
    [CrossRef]

2012 (4)

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

A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol.30, 2011–2024 (2012).
[CrossRef]

A. Bononi, P. Serena, N. Rossi, E. Grellier, and F. Vacondio, “Modeling monlinearity in coherent transmissions with dominant intrachannel-four-wave-mixing,” Opt. Express20, 7777–7791 (2012).
[CrossRef] [PubMed]

S. Turitsyn, M. Sorokina, and S. Derevyanko, “Dispersion-dominated nonlinear fiber-optic channel,” Opt. Lett.37, 2931–2933 (2012).
[CrossRef] [PubMed]

2011 (1)

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

2009 (1)

D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol.27, 2916–2923 (2009).
[CrossRef]

2008 (1)

2006 (1)

M. Nazarathy, “Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series,” IEEE Trans. Commun.54, 295–301 (2006).
[CrossRef]

2005 (1)

S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol.23, 2073–2080 (2005).
[CrossRef]

2001 (1)

S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett.13, 800–802 (2001).
[CrossRef]

2000 (1)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett.12, 292–294 (2000).

1999 (2)

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett.35, 1576–1578 (1999).
[CrossRef]

P. V. Mamyshev and N. A. Mamysheva, “Pulse-overlapped dispersion managed data transmission and intrachan-nel four-wave mixing,” Opt. Lett.24, 1454–1456 (1999).
[CrossRef]

1998 (1)

I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
[CrossRef]

Bononi, A.

Bosco, G.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol.30, 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 Photon. Technol. Lett.23, 742–744 (2011).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.

Carena, A.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol.30, 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 Photon. Technol. Lett.23, 742–744 (2011).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.

Cho, P.

Clausen, C. B.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett.12, 292–294 (2000).

Curri, V.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol.30, 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 Photon. Technol. Lett.23, 742–744 (2011).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.

Derevyanko, S.

Essiambre, R. J.

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett.35, 1576–1578 (1999).
[CrossRef]

Essiambre, Rene

A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol.30, 2011–2024 (2012).
[CrossRef]

Forghieri, F.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol.30, 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 Photon. Technol. Lett.23, 742–744 (2011).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.

Grellier, E.

Jackson, J. D.

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

Karagodsky, V.

Kawanishi, S.

I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
[CrossRef]

Khurgin, J.

Kumar, S.

D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol.27, 2916–2923 (2009).
[CrossRef]

S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol.23, 2073–2080 (2005).
[CrossRef]

S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett.13, 800–802 (2001).
[CrossRef]

Madhow, U.

U. Madhow, “Chapter 3: Demodulation,” in Fundamentals of Digital Communication (Cambridge University Press, 2008).
[CrossRef]

Mamyshev, P. V.

Mamysheva, N. A.

Mecozzi, A.

A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol.30, 2011–2024 (2012).
[CrossRef]

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett.12, 292–294 (2000).

Meiman, Y.

Mikkelsen, B.

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett.35, 1576–1578 (1999).
[CrossRef]

Mori, K.

I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
[CrossRef]

Nazarathy, M.

Noe, R.

Poggiolini, P.

A. Carena, V. Curri, G. Bosco, P. Poggiolini, and F. Forghieri, “Modeling of the impact of nonlinear propagation effects in uncompensated optical coherent transmission links,” J. Lightw. Technol.30, 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 Photon. Technol. Lett.23, 742–744 (2011).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.

Raybon, G.

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett.35, 1576–1578 (1999).
[CrossRef]

Rossi, N.

Serena, P.

Shake, I.

I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
[CrossRef]

Shpantzer, I.

Shtaif, M.

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett.12, 292–294 (2000).

Sorokina, M.

Taiba, M. T.

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.

Takara, H.

I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
[CrossRef]

Turitsyn, S.

Vacondio, F.

Weidenfeld, R.

Yamabayashi, Y.

I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
[CrossRef]

Yang, D.

D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol.27, 2916–2923 (2009).
[CrossRef]

S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol.23, 2073–2080 (2005).
[CrossRef]

Electron. Lett. (2)

I. Shake, H. Takara, K. Mori, S. Kawanishi, and Y. Yamabayashi, “Influence of inter-bit four-wave mixing in optical TDM transmission,” Electron. Lett.34, 1600–1601 (1998).
[CrossRef]

R. J. Essiambre, B. Mikkelsen, and G. Raybon, “Intra-channel crossphase modulation and four-wave mixing in high-speed TDM systems,” Electron. Lett.35, 1576–1578 (1999).
[CrossRef]

IEEE Photon. Technol. Lett. (3)

A. Mecozzi, C. B. Clausen, and M. Shtaif, “Analysis of intrachannel nonlinear effects in highly dispersed optical pulse transmission,” IEEE Photon. Technol. Lett.12, 292–294 (2000).

S. Kumar, “Intrachannel four-wave mixing in dispersion managed RZ systems,” IEEE Photon. Technol. Lett.13, 800–802 (2001).
[CrossRef]

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

IEEE Trans. Commun. (1)

M. Nazarathy, “Accurate evaluation of bit-error rates of optical communication systems using the Gram-Charlier series,” IEEE Trans. Commun.54, 295–301 (2006).
[CrossRef]

J. Lightw. Technol. (4)

D. Yang and S. Kumar, “Intra-channel four-wave mixing impairments in dispersion-managed coherent fiber-optic systems based on binary phase-shift keying,” J. Lightw. Technol.27, 2916–2923 (2009).
[CrossRef]

A. Mecozzi and Rene Essiambre, “Nonlinear Shannon limit in pseudolinear coherent systems,” J. Lightw. Technol.30, 2011–2024 (2012).
[CrossRef]

S. Kumar and D. Yang, “Second-order theory for self-phase modulation and cross-phase modulation in optical fibers,” J. Lightw. Technol.23, 2073–2080 (2005).
[CrossRef]

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

Opt. Express (2)

Opt. Lett. (2)

Other (3)

J. D. Jackson, Classical Electrodynamics (Wiley, 1998).

U. Madhow, “Chapter 3: Demodulation,” in Fundamentals of Digital Communication (Cambridge University Press, 2008).
[CrossRef]

A. Carena, G. Bosco, V. Curri, P. Poggiolini, M. T. Taiba, and F. Forghieri, “Statistical characterization of PM-QPSK signals after propagation in uncompensated fiber links,” in Proceedings ECOC, (2010), pp. 1–3.

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Figures (6)

Fig. 1
Fig. 1

Classification of intrachannel nonlinear impairments (N = 6).

Fig. 2
Fig. 2

Analytical and numerical variances vs. peak power for (a) 5-spans, and (b) 20-spans system (β2 = −21 ps2/km).

Fig. 3
Fig. 3

Analytical and numerical variances vs. dispersion parameter for (a) 5-spans, and (b) 20-spans system (Ppeak = 0 dBm).

Fig. 4
Fig. 4

Validation of the stationary phase approximation. Analytical and numerical variances vs. launch peak power for (a) 5-spans, and (b) 20-spans system. Raised-cosine pulses are used. β2 = −21 ps2/km.

Fig. 5
Fig. 5

Validation of the stationary phase approximation. Analytical and numerical variances vs. dispersion parameter for (a) 5-spans, and (b) 20-spans system. Raised-cosine pulses are used. P = 0 dBm.

Fig. 6
Fig. 6

Analytical and numerical BER vs. average launch power. Gaussian pulses are used. Number of spans=20 and β2 = −9.1 ps2/km.

Tables (1)

Tables Icon

Table 1 Computational cost of nonlinear impairments per frequency

Equations (106)

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u ( t , 0 ) = P n = N / 2 N / 2 a n p ( t n T s , 0 ) ,
a n = x n + i y n 2 ,
i u z β 2 2 2 u t 2 + γ a 2 ( z ) | u | 2 u = 0 ,
u = u 0 + γ u 1 ( t , z ) + .
i u 0 z β 2 2 2 u 0 t 2 = 0 .
i u 1 z β 2 2 2 u 1 t 2 = a 2 ( z ) | u 0 | 2 u 0 .
i d u ˜ 1 d z β 2 2 ( 2 π f ) 2 u ˜ 1 = a 2 ( z ) b ˜ ( f , z ) ,
b ˜ ( f , z ) = [ | u 0 | 2 u 0 ] ,
u ˜ 1 ( f , z ) = [ u 1 ( t , z ) ] ,
u ˜ 1 ( f , L tot ) = i 0 L tot a 2 ( z ) b ˜ ( f , z ) exp [ i β 2 ( 2 π f ) 2 z / 2 ] d z ,
u 0 ( t , z ) = P n = N / 2 N / 2 a n p ( t n T s , z ) ,
p ( t , z ) = 1 [ p ˜ ( f , z ) ] ,
p ˜ ( f , z ) = p ˜ ( f , 0 ) exp [ i β 2 ( 2 π f ) 2 z / 2 ] .
b ˜ ( f , z ) = [ u 0 u 0 u 0 * ] = u ˜ 0 ( f , z ) * u ˜ 0 ( f , z ) * u ˜ 0 * ( f , z ) ,
b ˜ ( f , z ) = P 3 / 2 l = N / 2 N / 2 m = N / 2 N / 2 n = N / 2 N / 2 a l a m a n * { [ p ˜ ( f , z ) exp ( i 2 π f l T s ) ] * [ p ˜ ( f , z ) exp ( i 2 π f m T s ) ] * [ p ˜ * ( f , z ) exp ( i 2 π f n T s ) ] } = P 3 / 2 l m n a l a m a n * X ˜ l , m , n ( f , z ) ,
X ˜ l , m , n ( f , z ) = [ p ˜ ( f , z ) exp ( i 2 π f l T s ) ] * [ p ˜ ( f , z ) exp ( i 2 π f m T s ) ] * [ p ˜ * ( f , z ) exp ( i 2 π f n T s ) ] .
u ˜ 1 ( f , L tot ) = i P 3 / 2 l m n a l a m a n * Y ˜ l , m , n ( f ) ,
Y ˜ l , m , n ( f ) = 0 L tot a 2 ( z ) exp ( i β 2 ( 2 π f ) 2 z / 2 ) X ˜ l , m , n ( f , z ) d z .
ρ N L ( f ) = lim N 1 ( N + 1 ) T s E { | δ u ˜ N L ( f ) | 2 } ,
ρ N L ( f ) = lim N γ 2 P 3 ( N + 1 ) T s l m n l n m E { a l a l * a m a m * a n * a n } Y ˜ l , m , n ( f ) Y ˜ l , m , n * ( f ) .
E { a l a l * } = K 1 δ l l ,
E { a l a l } = 0 ,
K 1 = E { | a l | 2 } = 1 , E { a l a l * a m a m * a n * a n } = [ δ l l δ m m δ n n + δ l m δ l m δ n n ] .
ρ N D I F W M ( f ) = lim N 2 γ 2 P 3 ( N + 1 ) T s l m n | Y ˜ l , m , n ( f ) | 2 .
ρ N D I F W M ( f ) = lim N 2 γ 2 P 3 ( N + 1 ) T s { l m n l m , l + m n = N / 2 | Y ˜ l , m , n ( f ) | 2 + l m n l m , l + m n = N / 2 + 1 | Y ˜ l , m , n ( f ) | 2 + + l m n l m , l + m n = N / 2 | Y ˜ l , m , n ( f ) | 2 } .
ρ N D I F W M ( f ) = 2 γ 2 P 3 T s l m n l m , l + m n = 0 | Y ˜ l , m , n ( f ) | 2 = 2 γ 2 P 3 T s l m l m Z ˜ l , m ( f ) ,
Z ˜ l , m ( f ) = | Y ˜ l , m , l + m ( f ) | 2 .
E { a l 2 ( a l * ) 2 a n * a n } = K 1 K 2 { δ l l δ n n } ,
K 2 = E { | a l | 4 } = 1 .
ρ D I F W M = lim N γ 2 P 3 ( N + 1 ) T s l n | Y ˜ l , l , n | 2 .
ρ D I F W M = γ 2 P 3 T s l Z ˜ l , l ( f ) .
E { a l a l * a m a m * a n * a n } = E { a l 2 a l * . a m * a n * a n } = 0 ,
E { a l 2 a l * } = { 0 , if l l E { | a l | 2 a l } = 0 , if l = l
ρ N L ( f ) = ρ N D I F W M ( f ) + ρ D I F W M ( f ) ,
X ˜ l , m , l + m = D exp ( A f 2 + B f + C ) ,
A = ( ξ 2 + δ 2 ) 3 ξ + i δ ,
B = i 4 π f T s ( l + m ) ξ 3 ξ + i δ ,
C = 2 π 2 T s 2 [ ( l 2 + m 2 ) ξ l m ( ξ + i δ ) ] ( 3 ξ + i δ ) ( ξ i δ ) ,
D = k 3 π ( ξ i δ ) ( 3 ξ + i δ ) ,
δ = 2 π 2 β 2 z , k = 2 π T 0 , ξ = 2 π 2 T 0 2 .
I = G ( x ) e i y ( x ) d x ,
y ( x ) = y ( x 0 ) + 1 2 y ( x 0 ) ( x x 0 ) 2 .
I G ( x 0 ) e i y ( x 0 ) e i y ( x 0 ) ( x x 0 ) 2 / 2 d x ,
G ( x 0 ) e i y ( x 0 ) 2 π i y ( x 0 )
X ˜ l , m , l + m ( f , z ) = π | δ | p ˜ ( f π l T s δ , 0 ) p ˜ ( f π m T s δ , 0 ) p ˜ ( f + π ( l + m ) T s δ , 0 ) exp [ i ( δ f 2 + 2 π 2 T s 2 l m δ ) ] .
p ˜ ( f ) = { 1 , | f | B s / 2 0 , otherwise
X ˜ l , m , l + m ( f , z ) = π | δ | p ˜ l , m ( f ) exp [ i ( δ f 2 + 2 π 2 T s 2 l m δ ) ] ,
p ˜ l , m ( f ) = { 1 , lt f rt 0 , otherwise
lt = max ( l , m , l + m ) π T s δ B s 2 ,
rt = min ( l , m , l + m ) π T s δ + B s 2 .
X l , m , l + m ( f ) = X m , l , l + m ( f ) .
X l , m , l + m ( f ) = X m , l , ( l + m ) ( f ) .
σ N L 2 = + ρ N L ( f ) H rec ( f ) d f ,
σ N L 2 = σ N D I F W M 2 + σ D I F W M 2 ,
σ r 2 = + ρ r ( f ) H rec ( f ) d f ,
ρ r ( f ) = ρ r ( f ) , r = N D I F W M , D I F W M .
σ r 2 = + ρ r ( f ) [ H rec ( f ) + H rec ( f ) ] d f .
p ˜ ( f ) = { 1 , | f | 1 a 2 T s 1 2 [ 1 sin ( π T s a ( | f | 1 2 T s ) ) ] , 1 a 2 T s < | f | 1 + a 2 T s 0 , | f | > 1 + a 2 T s
ρ tot ( f ) = ρ N L ( f ) + ρ A S E ( f ) ,
ρ A S E = j = 1 N A ( e α L a , j 1 ) h f ¯ n s p ,
P e = 2 Q ( 2 S N R ) Q 2 ( 2 S N R ) ,
S N R = P a v σ tot 2 ,
P a v = P 1.88 ,
σ tot 2 = ρ tot ( f ) H rec ( f ) d f .
Q ( lin ) = 2 erfc 1 ( 2 P e ) ,
Q ( d B Q 20 ) = 20 log 10 Q ( lin ) .
p ( t , 0 ) = exp ( t 2 2 T 0 2 ) ,
p ˜ ( f , 0 ) = k exp ( ξ f 2 ) ,
k = 2 π T 0 , ξ = 2 π 2 T 0 2 .
X ˜ l , m , l + m ( f , z ) = p ˜ 1 ( f 1 ) exp ( i 2 π f 1 l T s ) p ˜ 2 ( f 2 f 1 ) exp [ i 2 π ( f 2 f 1 ) m T s ] p ˜ 3 ( f f 2 ) exp [ i 2 π ( f f 2 ) n T s ] d f 1 d f 2 ,
n = l + m ,
p ˜ 1 ( f ) = p ˜ ( f , z ) = p ˜ ( f , 0 ) exp [ i ( 2 π f ) 2 β 2 z / 2 ] ,
p ˜ 2 ( f ) = p ˜ 1 ( f ) ,
p ˜ 3 ( f ) = p ˜ 1 * ( f ) = k exp [ ξ f 2 i ( 2 π f 2 ) β 2 z / 2 ] .
X l , m , n ( f , z ) = M ( f 2 ) p ˜ 3 ( f f 2 ) exp [ i 2 π ( f f 2 ) n T s ] d f 2 ,
M ( f 2 ) = p ˜ ( f 1 , 0 ) exp [ i 2 π f 1 l T s + i δ f 1 2 ] p ˜ ( f 2 f 1 , 0 ) exp [ i 2 π ( f 2 f 1 ) m T s + i δ ( f 2 f 1 ) 2 ] d f 1 ,
M ( f 2 ) = k 2 exp [ ξ f 1 2 + i δ f 1 2 ξ ( f 2 f 1 ) 2 + i 2 π f 1 l T s + i 2 π ( f 2 f 1 ) m T S + i δ ( f 2 f 1 ) 2 ] d f 1 , = k 2 exp [ ξ f 2 2 + i δ f 2 2 + i 2 π f 2 m T s ] exp { 2 [ ξ i δ ] f 1 2 + i 2 π f 1 ( l m ) T s + 2 ξ f 1 f 2 i 2 δ f 1 f 2 } d f 1 .
M ( f 2 ) = k 2 exp [ ( ξ i δ ) f 2 2 + i 2 π f 2 m T s ] I ( f 2 ) ,
I ( f 2 ) = exp { 2 ( ξ i δ ) f 1 2 + 2 f 1 [ i π ( l m ) T s + ( ξ i δ ) f 2 ] } d f 1 , = exp { [ ( ξ i δ ) f 2 + i π ( l m ) T s ] 2 2 ( ξ i δ ) } π 2 ( ξ i δ ) .
M ( f 2 ) = J exp ( β f 2 2 + i μ f 2 ) ,
J = k 2 π 2 ( ξ i δ ) exp ( π 2 ( l m ) 2 T s 2 2 ( ξ i δ ) ) ,
β = ξ i δ 2 ,
μ = π T s ( l + m ) .
p ˜ 3 ( f f 2 ) = k exp [ ( ξ + i δ ) ( f f 2 ) 2 ] ,
X l , m , l + m = J exp [ ( ξ + i δ ) f 2 + i 2 π f n T s ] exp [ ( ξ i δ ) f 2 2 / 2 ( ξ + i δ ) f 2 2 + ( ξ + i δ ) 2 f f 2 ] d f 2 , = D exp ( A f 2 + B f + C ) ,
A = ( ξ 2 + δ 2 ) f 2 3 ξ + i δ ,
B = i 4 π f T s ( l + m ) ξ 3 ξ + i δ ,
C = 2 π 2 T s 2 [ ( l 2 + m 2 ) ξ l m ( ξ + i δ ) ] ( 3 ξ + i δ ) ( ξ i δ ) ,
D = k 3 π ( ξ i δ ) ( 3 ξ + i δ ) .
M ( f 2 ) = p ˜ ( f 1 , 0 ) p ˜ ( f 2 f 1 , 0 ) exp [ i 2 π f 1 l T s + i δ f 1 2 + i δ ( f 2 f 1 ) 2 + i 2 π ( f 2 f 1 ) m T s ] d f 1 = exp ( i 2 π f 2 m T s + i δ f 2 2 ) I ( f 2 ) ,
I ( f 2 ) = p ˜ ( f 1 , 0 ) p ˜ ( f 2 f 1 , 0 ) exp [ i 2 δ f 1 2 + i 2 π f 1 ( l m ) T s 2 i δ f 1 f 2 ] d f 1 .
θ ( f 1 ) = 2 δ f 1 2 2 δ f 1 f 2 + f 1 2 π ( l m ) T s ,
I ( f 2 ) = p ˜ ( f 1 , 0 ) p ˜ ( f 2 f 1 , 0 ) exp ( i θ ( f 1 ) ) d f 1 .
d θ d f 1 = 0 ,
f 1 , opt = π ( m l ) T s 2 δ + f 2 2 .
θ ( f 1 , opt ) = δ 2 [ f 2 + π ( m l ) T s δ ] 2 .
I ( f 2 ) p ˜ ( f 1 , opt , 0 ) p ˜ ( f 2 f 1 , opt , 0 ) exp [ i θ ( f 1 , opt ) ] l 1 ,
l 1 = 2 π | θ ( f 1 , opt ) | exp [ sgn ( θ ( f 1 , opt ) ) i π / 4 ] , = π 2 | δ | exp ( i sgn ( δ ) π / 4 ) ,
I ( f 2 ) = l 1 p ˜ ( π ( m l ) T s + f 2 δ 2 δ , 0 ) p ˜ ( δ f 2 π ( m l ) T s 2 δ , 0 ) exp { i δ 2 [ f 2 + π ( m l ) T s δ ] } .
M ( f 2 ) = s 1 ( f 2 ) exp [ i δ f 2 2 2 i π 2 ( m l ) 2 T s 2 2 δ + i π f 2 ( m + l ) T s ] ,
s 1 ( f 2 ) = p ˜ ( π ( m l ) T s + f 2 δ 2 δ , 0 ) p ˜ ( δ f 2 π ( m l ) T s 2 δ , 0 ) .
X ˜ l , m , l + m ( f ) = exp [ i π 2 ( m l ) 2 T s 2 2 δ i δ f 2 ] s 2 ( f 2 ) exp ( i θ 2 ( f 2 ) ) d f 2 ,
s 2 ( f 2 ) = s 1 ( f 2 ) p ˜ ( f f 2 , 0 ) ,
θ 2 ( f 2 ) = δ f 2 2 2 + 2 δ f f 2 π T s f 2 ( l + m ) .
f 2 , opt = 2 f π T s ( l + m ) δ .
X ˜ l , m , l + m ( f ) = π | δ | p ˜ ( f π l T s δ , 0 ) p ˜ ( f π m T s δ , 0 ) p ˜ ( f + π ( l + m ) T s δ , 0 ) exp [ i ( δ f 2 + 2 π 2 T s 2 l m δ ) ] .

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