Abstract

We derive an analytical formula to estimate the variance of nonlinear phase noise caused by the interaction of amplified spontaneous emission (ASE) noise with fiber nonlinearity such as self-phase modulation (SPM), cross-phase modulation (XPM), and four-wave mixing (FWM) in coherent orthogonal frequency division multiplexing (OFDM) systems. The analytical results agree very well with numerical simulations, enabling the study of the nonlinear penalties in long-haul coherent OFDM systems without extensive numerical simulation. Our results show that the nonlinear phase noise induced by FWM is significantly larger than that induced by SPM and XPM, which is in contrast to traditional WDM systems where ASE-FWM interaction is negligible in quasi-linear systems. We also found that fiber chromatic dispersion can reduce the nonlinear phase noise. The variance of the total phase noise increases linearly with the bit rate, and does not depend significantly on the number of subcarriers for systems with moderate fiber chromatic dispersion.

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  1. W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
    [CrossRef]
  2. A. Lowery, L. Du, and J. Armstrong, “Performance of optical OFDM in ultra-long-haul WDM lightwave systems,” J. Lightwave Technol. 25(1), 131–138 (2007).
    [CrossRef]
  3. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
    [CrossRef]
  4. A. Sano, E. Yamada, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, and Y. Takatori, “No-guard-interval coherent optical ODFM for 100-Gb/s long-haul WDM transmission,” J. Lightwave Technol. 27(16), 3705–3713 (2009).
    [CrossRef]
  5. S. Jasen, I. Morita, T. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral efficiency over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009).
    [CrossRef]
  6. Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009).
    [CrossRef]
  7. A. J. Lowery, S. Wang, and M. Premaratne, “Calculation of power limit due to fiber nonlinearity in optical OFDM systems,” Opt. Express 15(20), 13282–13287 (2007).
    [CrossRef] [PubMed]
  8. 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]
  9. A. J. Lowery, “Fiber nonlinearity pre- and post-compensation for long-haul optical links using OFDM,” Opt. Express 15(20), 12965–12970 (2007).
    [CrossRef] [PubMed]
  10. L. B. Du and A. J. Lowery, “Improved nonlinearity precompensation for long-haul high-data-rate transmission using coherent optical OFDM,” Opt. Express 16(24), 19920–19925 (2008).
    [CrossRef] [PubMed]
  11. X. Liu and F. Buchali, “Intra-symbol frequency-domain averaging based channel estimation for coherent optical OFDM,” Opt. Express 16(26), 21944–21957 (2008).
    [CrossRef] [PubMed]
  12. X. Liu, F. Buchali, and R. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009).
    [CrossRef]
  13. W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
    [CrossRef] [PubMed]
  14. X. Li, X. Chen, G. Goldfarb, E. Mateo, I. Kim, F. Yaman, and G. Li, “Electronic post-compensation of WDM transmission impairments using coherent detection and digital signal processing,” Opt. Express 16(2), 880–888 (2008).
    [CrossRef] [PubMed]
  15. E. Ip and J. Kahn, “Compensation of dispersion and nonlinear effects using digital back-propagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
    [CrossRef]
  16. E. Yamazaki, H. Masuda, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, M. Matsui, and Y. Takatori, “Multi-staged nonlinear compensation in coherent receiver for 16,340-km transmission of 111-Gb/s no-guard-interval co-OFDM,” ECOC2009, paper 9.4.6.
  17. J. Gordon and L. Mollenauer, “Phase noise in photonic communications systems using linear amplifiers,” Opt. Lett. 15(23), 1351–1353 (1990).
    [CrossRef] [PubMed]
  18. P. Winzer and R. Essiambre, “Advanced modulation formats for high capacity optical transport networks,” J. Lightwave Technol. 24(12), 4711–4728 (2006).
    [CrossRef]
  19. A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
    [CrossRef]
  20. K.-P. Ho, “Probability density of nonlinear phase noise,” J. Opt. Soc. Am. B 20(9), 1875–1879 (2003).
    [CrossRef]
  21. K.-P. Ho, “Asymptotic probability density of nonlinear phase noise,” Opt. Lett. 28(15), 1350–1352 (2003).
    [CrossRef] [PubMed]
  22. A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004).
    [CrossRef] [PubMed]
  23. A. G. Green, P. P. Mitra, and L. G. Wegener, “Effect of chromatic dispersion on nonlinear phase noise,” Opt. Lett. 28(24), 2455–2457 (2003).
    [CrossRef] [PubMed]
  24. S. Kumar, “Effect of dispersion on nonlinear phase noise in optical transmission systems,” Opt. Lett. 30(24), 3278–3280 (2005).
    [CrossRef]
  25. C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27(21), 1887–1889 (2002).
    [CrossRef]
  26. C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8(3), 616–625 (2002).
    [CrossRef]
  27. M. Hanna, D. Boivin, P. Lacourt, and J. Goedgebuer, “Calculation of optical phase jitter in dispersion-managed systems by the use of the moment method,” J. Opt. Soc. Am. B 21(1), 24–28 (2004).
    [CrossRef]
  28. K.-P. Ho and H.-C. Wang, “Effect of dispersion on nonlinear phase noise,” Opt. Lett. 31(14), 2109–2111 (2006).
    [CrossRef] [PubMed]
  29. K.-P. Ho, “Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise,” IEEE J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004).
    [CrossRef]
  30. X. Zhu, S. Kumar, and X. Li, “Analysis and comparison of impairments in differential phase-shift keying and on-off keying transmission systems based on the error probability,” Appl. Opt. 45(26), 6812–6822 (2006).
    [CrossRef] [PubMed]
  31. A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007).
    [CrossRef]
  32. S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
    [CrossRef]
  33. G. P. Agrawal, Nonlinear Fiber optics, New York: Academic, 3rd Edition, 2001.
  34. K. Inoue, “Phase-mismatching characteristic of four-wave mixing in fiber lines with multistage optical amplifiers,” Opt. Lett. 17(11), 801–803 (1992).
    [CrossRef] [PubMed]
  35. J. G. Proakis, Digital Communications, New York: McGraw-Hill, 4th Edition, 2000, pp. 269–274.

2009 (6)

2008 (6)

2007 (4)

2006 (4)

2005 (1)

2004 (3)

2003 (3)

2002 (2)

C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27(21), 1887–1889 (2002).
[CrossRef]

C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8(3), 616–625 (2002).
[CrossRef]

1994 (1)

A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
[CrossRef]

1992 (1)

1990 (1)

Armstrong, J.

Athaudage, C.

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
[CrossRef]

Bao, H.

Boivin, D.

Buchali, F.

Chen, X.

Cho, P.

Demir, A.

Du, L.

Du, L. B.

Essiambre, R.

Goedgebuer, J.

Goldfarb, G.

Gordon, J.

Green, A. G.

Hanna, M.

Ho, K.-P.

Inoue, K.

Ip, E.

Ishihara, K.

Jasen, S.

Kahn, J.

Karagodsky, V.

Khurgin, J.

Kim, I.

Kobayashi, T.

Kudo, R.

Kumar, S.

Lacourt, P.

Lakoba, T. I.

Li, G.

Li, X.

Liu, X.

Lowery, A.

Lowery, A. J.

Ma, Y.

Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009).
[CrossRef]

Masuda, H.

Mateo, E.

McKinstrie, C.

C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8(3), 616–625 (2002).
[CrossRef]

McKinstrie, C. J.

Mecozzi, A.

A. Mecozzi, “Probability density functions of the nonlinear phase noise,” Opt. Lett. 29(7), 673–675 (2004).
[CrossRef] [PubMed]

A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
[CrossRef]

Meiman, Y.

Mitra, P. P.

Miyamoto, Y.

Mollenauer, L.

Morita, I.

Nazarathy, M.

Noe, R.

Premaratne, M.

Sano, A.

Schenk, T.

Shieh, W.

Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009).
[CrossRef]

W. Shieh, H. Bao, and Y. Tang, “Coherent optical OFDM: theory and design,” Opt. Express 16(2), 841–859 (2008).
[CrossRef] [PubMed]

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
[CrossRef]

Shpantzer, I.

Takatori, Y.

Tanaka, H.

Tang, Y.

Tkach, R.

Wang, H.-C.

Wang, S.

Wegener, L. G.

Weidenfeld, R.

Winzer, P.

Xie, C.

C. J. McKinstrie, C. Xie, and T. I. Lakoba, “Efficient modeling of phase jitter in dispersion-managed soliton systems,” Opt. Lett. 27(21), 1887–1889 (2002).
[CrossRef]

C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8(3), 616–625 (2002).
[CrossRef]

Yamada, E.

Yaman, F.

Yamazaki, E.

Yang, Y.

Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009).
[CrossRef]

Yoshida, E.

Zhu, X.

Appl. Opt. (1)

Electron. Lett. (1)

W. Shieh and C. Athaudage, “Coherent optical orthogonal frequency division multiplexing,” Electron. Lett. 42(10), 587–588 (2006).
[CrossRef]

IEEE J. Sel. Top. Quantum Electron. (2)

C. McKinstrie and C. Xie, “Phase jitter in single-channel soliton systems with constant dispersion,” IEEE J. Sel. Top. Quantum Electron. 8(3), 616–625 (2002).
[CrossRef]

K.-P. Ho, “Error probability of DPSK signals with cross-phase modulation induced nonlinear phase noise,” IEEE J. Sel. Top. Quantum Electron. 10(2), 421–427 (2004).
[CrossRef]

IEEE Photon. Technol. Lett. (1)

Y. Yang, Y. Ma, and W. Shieh, “Performance impact of inline chromatic dispersion compensation for 107-Gb/s coherent optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1042–1044 (2009).
[CrossRef]

J. Lightwave Technol. (10)

A. Lowery, L. Du, and J. Armstrong, “Performance of optical OFDM in ultra-long-haul WDM lightwave systems,” J. Lightwave Technol. 25(1), 131–138 (2007).
[CrossRef]

J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
[CrossRef]

A. Sano, E. Yamada, H. Masuda, E. Yamazaki, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, and Y. Takatori, “No-guard-interval coherent optical ODFM for 100-Gb/s long-haul WDM transmission,” J. Lightwave Technol. 27(16), 3705–3713 (2009).
[CrossRef]

S. Jasen, I. Morita, T. Schenk, and H. Tanaka, “121.9-Gb/s PDM-OFDM transmission with 2-b/s/Hz spectral efficiency over 1000 km of SSMF,” J. Lightwave Technol. 27(3), 177–188 (2009).
[CrossRef]

P. Winzer and R. Essiambre, “Advanced modulation formats for high capacity optical transport networks,” J. Lightwave Technol. 24(12), 4711–4728 (2006).
[CrossRef]

A. Mecozzi, “Limits to the long haul coherent transmission set by the Kerr nonlinearity and noise of in-line amplifiers,” J. Lightwave Technol. 12(11), 1993–2000 (1994).
[CrossRef]

X. Liu, F. Buchali, and R. Tkach, “Improving the nonlinear tolerance of polarization-division-multiplexed CO-OFDM in long-haul fiber transmission,” J. Lightwave Technol. 27(16), 3632–3640 (2009).
[CrossRef]

E. Ip and J. Kahn, “Compensation of dispersion and nonlinear effects using digital back-propagation,” J. Lightwave Technol. 26(20), 3416–3425 (2008).
[CrossRef]

A. Demir, “Nonlinear phase noise in optical fiber communication systems,” J. Lightwave Technol. 25(8), 2002–2032 (2007).
[CrossRef]

S. Kumar, “Analysis of nonlinear phase noise in coherent fiber-optic systems based on phase shift keying,” J. Lightwave Technol. 27(21), 4722–4733 (2009).
[CrossRef]

J. Opt. Soc. Am. B (2)

Opt. Express (7)

Opt. Lett. (8)

Other (3)

J. G. Proakis, Digital Communications, New York: McGraw-Hill, 4th Edition, 2000, pp. 269–274.

G. P. Agrawal, Nonlinear Fiber optics, New York: Academic, 3rd Edition, 2001.

E. Yamazaki, H. Masuda, A. Sano, T. Yoshimatsu, T. Kobayashi, E. Yoshida, Y. Miyamoto, R. Kudo, K. Ishihara, M. Matsui, and Y. Takatori, “Multi-staged nonlinear compensation in coherent receiver for 16,340-km transmission of 111-Gb/s no-guard-interval co-OFDM,” ECOC2009, paper 9.4.6.

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Figures (10)

Fig. 1
Fig. 1

Structure of coherent OFDM transmission systems

Fig. 2
Fig. 2

OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is 8, with one subcarrier carrying data.

Fig. 3
Fig. 3

Variance of the total phase noise as a function of propagation distance for SPM effect only. Total number of subcarrier is 8 with only one subcarrier carrying data.

Fig. 4
Fig. 4

OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is 64, with 8 subcarriers carrying data.

Fig. 5
Fig. 5

Variance of the total phase noise as a function of propagation distance considering the ASE interaction with SPM, XPM and FWM effects. Total number of subcarriers is 64 with 8 subcarriers carrying data.

Fig. 6
Fig. 6

OFDM signal spectrum before entering into fiber spans. Total number of subcarriers is 128 with 2-folder oversampling (64 subcarriers carrying data).

Fig. 8
Fig. 8

Variance of the total phase noise as a function of bit rate in Gb/s. The total number of subcarriers is 128 with two-fold oversampling, total channel power is −3 dBm, and transmission distance is 1000 km.

Fig. 7
Fig. 7

Variance of the total phase noise as a function of channel power. The total number of subcarriers is 128 with two-fold oversampling, bit rate is 10 Gb/s, and transmission distance is 1000 km.

Fig. 9
Fig. 9

Variance of the total phase noise as a function of number of subcarriers, obtained analytically. Two-folder oversampling is used in the simulation. Bit rate is 10 Gb/s, total channel power is −3 dBm, and transmission distance is 1000 km.

Fig. 10
Fig. 10

Variance of the nonlinear phase noise due to separate effects of SPM, XPM, and FWM, as a function of propagation distance, obtained analytically. Total number of subcarriers is 128 with 2-folder oversampling. Bit rate is 10 Gb/s with −3 dBm launch power.

Equations (36)

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j u ( t , z ) z β 2 ( z ) 2 2 u ( t , z ) t 2 + γ exp [ w ( z ) ] | u ( t , z ) | 2 u ( t , z ) = 0 ,
u ( t , z ) = l = N / 2 N / 2 1 u l ( t , z ) exp ( j ω l t ) ,
j ( u l z β 2 ω l u l t ) β 2 2 ω l 2 u l t 2 + β 2 2 ω l 2 u l = γ exp [ w ( z ) ] ( | u l | 2 + 2 m l | u m | 2 ) u l .
u l ( t , z ) = u l ( t , 0 ) exp [ j β 2 2 ω l 2 z + j γ L e f f ( z ) ( | u l | 2 + 2 m l | u m | 2 ) ] ,
L e f f ( z ) = 1 exp ( α z ) α .
n ( t ) = l = N / 2 N / 2 1 n l exp ( j ω l t ) .
u + ( t , κ L s ) = l = N / 2 N / 2 1 ( u l ( t , κ L s ) + n l ) exp ( j ω l t ) .
u l + ( t , κ L s ) = u l ( t , κ L s ) + n l = [ u l ( t , 0 ) + n l ' ] exp [ j β 2 ω l 2 κ L s / 2 + j γ L e f f ( κ L s ) ( | u l | 2 + 2 m l | u m | 2 ) ] .
n l ' n k ' * = n l n k * = ρ A S E T b l o c k δ l , k , n l ' n k ' = 0 ,
δ l , k = { 1 if l = k 0 otherwise . .
u l + ( t , M L s ) = [ u l + n l ' ] exp { j Φ D + j γ ( M κ ) L e f f ( L s ) ( u l n l ' * + u l * n l ' ) } ,
Φ D = β 2 ω l 2 M L s / 2 + γ M L e f f ( L s ) ( | u l | 2 + 2 m l | u m | 2 ) .
δ Φ S P M + X P M , κ = γ ( M κ ) L e f f ( L s ) ( u l n l ' * + u l * n l ' ) .
δ Φ S P M + X P M , κ 2 = 2 γ 2 ( M κ ) 2 L e f f 2 ( L s ) ( | u l | 2 + 2 m l | u m | 2 ) ρ A S E T b l o c k .
δ Φ S P M + X P M 2 = 1 3 M ( M 1 ) ( 2 M 1 ) γ 2 L e f f 2 ( L s ) ( 2 N e 1 ) P s c ρ A S E T b l o c k ,
u l z j β 2 2 ω l 2 u l = j γ exp [ w ( z ) ] p + q r = l p l , q r u p u q u r * exp { j ( ω p 2 + ω q 2 ω r 2 ) β 2 2 z } .
u l ( M L s ) = u ' l , z 0 + j p + q r = l z 0 M L s γ e α z ' u p , z 0 u p , z 0 u r , z 0 * exp [ j Δ β p , q , r , l ( z ' ) ] d z ' = u ' l , z 0 + j p + q r = l u p , z 0 u q , z 0 u r , z 0 * Y p , q , r , l ( z 0 , M L s ) ,
u ' l , z 0 = u l , z 0 exp ( j β 2 2 ω l 2 z 0 ) ,
Δ β p , q , r , l ( z ) = ( ω p 2 + ω q 2 ω r 2 ω l 2 ) β 2 z 2 .
Y p , q , r , l ( z 0 , M L s ) = z 0 M L s γ e α z ' exp [ j Δ β p , q , r , l ( z ' ) ] d z ' .
u l + ( M L s , κ ) = u l + ( κ L s ) exp ( j β 2 2 ω l 2 κ L s ) + j p + q r = l u p , κ + u q , κ + u r , κ + * Y p , q , r , l ( κ L s , M L s ) = ( u l , κ + n l ) exp ( j β 2 2 ω l 2 κ L s ) + j p + q r = l ( u p , κ + n p ) ( u q , κ + n q ) ( u r , κ * + n r * ) Y p , q , r , l ( κ L s , M L s ) .
u l + ( M L s , κ ) ( u l , κ + n l ) exp ( j β 2 2 ω l 2 κ L s ) + j p + q r = l ( u p , κ u q , κ u r , κ * + n p u q , κ u r , κ * + n q u p , κ u r , κ * + n r * u p , κ u q , κ ) Y p , q , r , l ( κ L s , M L s ) .
u l + ( M L s , κ ) = u l , κ exp ( j β 2 2 ω l 2 κ L s ) + u F W M , κ + δ u l ( M L s , κ ) ,
u F W M , κ = j p + q r = l u p , κ u q , κ u r , κ * Y p , q , r , l ( κ L s , M L s ) .
δ u l ( M L s , κ ) = n l exp ( j β 2 2 ω l 2 κ L s ) + j q = 1 N ( n q A q + n q * B q ) ,
A q = 2 p = N / 2 N / 2 1 u p + l q , κ u p * Y q , p + l q , p , l ( κ L s , M L s ) , p q , l p + l q
B q = p = N / 2 N / 2 1 u q + l p , κ u p Y q + l p , p , q , l ( κ L s , M L s ) , p q , l q + l p
| δ u l | 2 = | n l | 2 + q = 1 N | n q | 2 ( | A q | 2 + | B q | 2 ) ,
δ u l 2 = j | n l | 2 2 B l exp ( j β 2 2 ω l 2 κ L s ) q = N / 2 N / 2 1 | n q | 2 2 A q B q .
δ Φ l , κ Im ( δ u l ) | u l | = δ u l δ u l * 2 j | u l | ,
δ Φ l , κ 2 ( δ u l δ u l * ) 2 4 | u l | 2 = 2 | δ u l | 2 ( δ u l 2 + δ u l * 2 ) 4 | u l | 2 .
δ Φ l , κ 2 = ρ A S E 2 P s c T b l o c k + ρ A S E 4 P s c T b l o c k { 2 q = 1 N | A q + B q | 2 + 4 Im ( B l exp ( j β 2 2 ω l 2 κ L s ) ) } .
δ Φ l i n e a r , l 2 = κ = 1 M δ Φ l i n e a r , l , κ 2 = M ρ A S E 2 P s c T b l o c k .
δ Φ F W M , l 2 = κ = 1 M δ Φ F W M , l , κ 2 = ρ A S E 4 P s c T b l o c k { 2 κ = 1 M q = 1 N | A q + B q | 2 + κ = 1 M 4 Im ( B l exp ( j β 2 2 ω l 2 κ L s ) ) } .
δ Φ l 2 = δ Φ l i n e a r , l 2 + δ Φ S P M + X P M 2 + δ Φ F W M , l 2 ,
B E R = 1 2 e r f c ( 1 2 σ 2 ) ,

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