Abstract

This work theoretically studies the transmission performance of a DML-based OFDM system by small-signal approximation, and the model considers both the transient and adiabatic chirps. The dispersion-induced distortion is modeled as subcarrier-to-subcarrier intermixing interference (SSII), and the theoretical SSII agrees with the distortion obtained from large-signal simulation statistically and deterministically. The analysis shows that the presence of the adiabatic chirp will ease power fading or even provide gain, but will increase the SSII to deteriorate OFDM signals after dispersive transmission. Furthermore, this work also proposes a novel iterative equalization to eliminate the SSII. From the simulation, the distortion could be effectively mitigated by the proposed equalization such that the maximum transmission distance of the DML-based OFDM signal is significantly improved. For instance, the transmission distance of a 30-Gbps DML-based OFDM signal can be extended from 10 km to more than 100 km. Besides, since the dispersion-induced distortion could be effectively mitigated by the equalization, negative power penalties are observed at some distances due to chirp-induced power gain.

©2012 Optical Society of America

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References

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  1. J. Armstrong, “OFDM for optical communications,” J. Lightwave Technol. 27(3), 189–204 (2009).
    [Crossref]
  2. M. C. Yuang, P.-L. Tien, D.-Z. Hsu, S.-Y. Chen, C.-C. Wei, J.-L. Shih, and J. Chen, “A high-performance OFDMA PON system architecture and medium access control,” J. Lightwave Technol. 30(11), 1685–1693 (2012).
    [Crossref]
  3. D. F. Hewitt, “Orthogonal frequency division multiplexing using baseband optical single sideband for simpler adaptive dispersion compensation,” in Optical Fiber Communication Conference (2007), Paper OME7.
  4. W.-R. Peng, B. Zhang, K.-M. Feng, X. Wu, A. E. Willner, and S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J. Lightwave Technol. 27(24), 5723–5735 (2009).
    [Crossref]
  5. D.-Z. Hsu, C.-C. Wei, H.-Y. Chen, J. Chen, M. C. Yuang, S.-H. Lin, and W.-Y. Li, “21 Gb/s after 100 km OFDM long-reach PON transmission using a cost-effective electro-absorption modulator,” Opt. Express 18(26), 27758–27763 (2010).
    [Crossref] [PubMed]
  6. A. Gharba, P. Chanclou, M. Ouzzif, J. L. Masson, L. A. Neto, R. Xia, N. Genay, B. Charbonnier, M. Hélard, E. Grard, and V. Rodrigues, “Optical transmission performance for DML considering laser chirp and fiber dispersion using AMOOFDM,” in 2010 International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (2010), pp. 1022–1026.
  7. D.-Z. Hsu, C.-C. Wei, H.-Y. Chen, W.-Y. Li, and J. Chen, “Cost-effective 33-Gbps intensity modulation direct detection multi-band OFDM LR-PON system employing a 10-GHz-based transceiver,” Opt. Express 19(18), 17546–17556 (2011).
    [Crossref] [PubMed]
  8. C.-C. Wei, “Small-signal analysis of OOFDM signal transmission with directly modulated laser and direct detection,” Opt. Lett. 36(2), 151–153 (2011).
    [Crossref] [PubMed]
  9. F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
    [Crossref]
  10. J. Binder and U. Kohn, “10 Gbit/s-dispersion optimized transmission at 1.55 μm wavelength on standard single mode fiber,” IEEE Photon. Technol. Lett. 6(4), 558–560 (1994).
    [Crossref]
  11. B. Wedding, “Analysis of fibre transfer function and determination of receiver frequency response for dispersion supported transmission,” Electron. Lett. 30(1), 58–59 (1994).
    [Crossref]
  12. I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
    [Crossref]
  13. L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).
  14. L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996).
    [Crossref]
  15. K. Sato, S. Kuwahara, and Y. Miyamoto, “Chirp characteristics of 40-Gb/s directly modulated distributed- feedback laser diodes,” J. Lightwave Technol. 23(11), 3790–3797 (2005).
    [Crossref]
  16. T. L. Koch and R. A. Linke, “Effect of nonlinear gain reduction on semiconductor laser wavelength chirping,” Appl. Phys. Lett. 48(10), 613–615 (1986).
    [Crossref]
  17. J. A. P. Morgado and A. V. T. Cartzxo, “Improved model to discriminate adiabatic and transient chirps in directly modulated semiconductor lasers,” J. Mod. Opt. 56(21), 2309–2317 (2009).
    [Crossref]
  18. U. Gliese, S. Nørskov, and T. N. Nielsen, “Chromatic dispersion in fiber-optic microwave and millimieter-wave links,” IEEE Trans. Microw. Theory Tech. 44(10), 1716–1724 (1996).
    [Crossref]
  19. J. M. Tang and K. A. Shore, “30-Gb/s signal transmission over 40-km directly modulated DFB-laser-based single-mode-fiber links without optical amplification and dispersion compensation,” J. Lightwave Technol. 24(6), 2318–2327 (2006).
    [Crossref]
  20. K. Yuksel, V. Moeyaert, M. Wuilpart, and P. Mégret, “Optical layer monitoring in passive optical networks (PONs): a review,” in International Conference on Transparent Optical Networks (2008), Paper Tu.B1.1.
  21. E. O. Brigham, Fast Fourier Transform and Its Applications, 1st ed. (New York: Wiley, 1997).
  22. G. P. Agrawal, Fibre-Optic Communication Systems, 2nd ed. (Prentice Hall, 1988).
  23. W.-R. Peng, “Analysis of laser phase noise effect in direct-detection optical OFDM transmission,” J. Lightwave Technol. 28(17), 2526–2536 (2010).
    [Crossref]
  24. L. Hanzo, W. Webb, and T. Keller, Single- and Multi-Carrier Quadrature Amplitude Modulation, 2nd ed. (Wiley, 2000).

2012 (1)

2011 (2)

2010 (2)

2009 (4)

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

W.-R. Peng, B. Zhang, K.-M. Feng, X. Wu, A. E. Willner, and S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J. Lightwave Technol. 27(24), 5723–5735 (2009).
[Crossref]

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

J. A. P. Morgado and A. V. T. Cartzxo, “Improved model to discriminate adiabatic and transient chirps in directly modulated semiconductor lasers,” J. Mod. Opt. 56(21), 2309–2317 (2009).
[Crossref]

2006 (1)

2005 (1)

1996 (2)

U. Gliese, S. Nørskov, and T. N. Nielsen, “Chromatic dispersion in fiber-optic microwave and millimieter-wave links,” IEEE Trans. Microw. Theory Tech. 44(10), 1716–1724 (1996).
[Crossref]

L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996).
[Crossref]

1994 (2)

J. Binder and U. Kohn, “10 Gbit/s-dispersion optimized transmission at 1.55 μm wavelength on standard single mode fiber,” IEEE Photon. Technol. Lett. 6(4), 558–560 (1994).
[Crossref]

B. Wedding, “Analysis of fibre transfer function and determination of receiver frequency response for dispersion supported transmission,” Electron. Lett. 30(1), 58–59 (1994).
[Crossref]

1993 (1)

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[Crossref]

1986 (1)

T. L. Koch and R. A. Linke, “Effect of nonlinear gain reduction on semiconductor laser wavelength chirping,” Appl. Phys. Lett. 48(10), 613–615 (1986).
[Crossref]

Armstrong, J.

Binder, J.

J. Binder and U. Kohn, “10 Gbit/s-dispersion optimized transmission at 1.55 μm wavelength on standard single mode fiber,” IEEE Photon. Technol. Lett. 6(4), 558–560 (1994).
[Crossref]

Birbas, A. N.

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

Bjerkan, L.

L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996).
[Crossref]

Cartzxo, A. V. T.

J. A. P. Morgado and A. V. T. Cartzxo, “Improved model to discriminate adiabatic and transient chirps in directly modulated semiconductor lasers,” J. Mod. Opt. 56(21), 2309–2317 (2009).
[Crossref]

Chen, H.-Y.

Chen, J.

Chen, S.-Y.

Chi, S.

Devaux, F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[Crossref]

Feng, K.-M.

Gliese, U.

U. Gliese, S. Nørskov, and T. N. Nielsen, “Chromatic dispersion in fiber-optic microwave and millimieter-wave links,” IEEE Trans. Microw. Theory Tech. 44(10), 1716–1724 (1996).
[Crossref]

Hafskjær, L.

L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996).
[Crossref]

Hsu, D.-Z.

Kerdiles, J. F.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[Crossref]

Klonidis, D.

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

Koch, T. L.

T. L. Koch and R. A. Linke, “Effect of nonlinear gain reduction on semiconductor laser wavelength chirping,” Appl. Phys. Lett. 48(10), 613–615 (1986).
[Crossref]

Kohn, U.

J. Binder and U. Kohn, “10 Gbit/s-dispersion optimized transmission at 1.55 μm wavelength on standard single mode fiber,” IEEE Photon. Technol. Lett. 6(4), 558–560 (1994).
[Crossref]

Kuwahara, S.

Li, W.-Y.

Lin, S.-H.

Linke, R. A.

T. L. Koch and R. A. Linke, “Effect of nonlinear gain reduction on semiconductor laser wavelength chirping,” Appl. Phys. Lett. 48(10), 613–615 (1986).
[Crossref]

Miyamoto, Y.

Morgado, J. A. P.

J. A. P. Morgado and A. V. T. Cartzxo, “Improved model to discriminate adiabatic and transient chirps in directly modulated semiconductor lasers,” J. Mod. Opt. 56(21), 2309–2317 (2009).
[Crossref]

Myhre, D.

L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996).
[Crossref]

Nielsen, T. N.

U. Gliese, S. Nørskov, and T. N. Nielsen, “Chromatic dispersion in fiber-optic microwave and millimieter-wave links,” IEEE Trans. Microw. Theory Tech. 44(10), 1716–1724 (1996).
[Crossref]

Nørskov, S.

U. Gliese, S. Nørskov, and T. N. Nielsen, “Chromatic dispersion in fiber-optic microwave and millimieter-wave links,” IEEE Trans. Microw. Theory Tech. 44(10), 1716–1724 (1996).
[Crossref]

Papagiannakis, I.

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

Peng, W.-R.

Rosenkranz, W.

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

Røyset, A.

L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996).
[Crossref]

Sato, K.

Shih, J.-L.

Shore, K. A.

Sorel, Y.

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[Crossref]

Tang, J. M.

Tien, P.-L.

Tomkos, I.

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

Wedding, B.

B. Wedding, “Analysis of fibre transfer function and determination of receiver frequency response for dispersion supported transmission,” Electron. Lett. 30(1), 58–59 (1994).
[Crossref]

Wei, C.-C.

Willner, A. E.

Wu, X.

Xia, C.

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

Yuang, M. C.

Zhang, B.

Appl. Phys. Lett. (1)

T. L. Koch and R. A. Linke, “Effect of nonlinear gain reduction on semiconductor laser wavelength chirping,” Appl. Phys. Lett. 48(10), 613–615 (1986).
[Crossref]

Electron. Lett. (1)

B. Wedding, “Analysis of fibre transfer function and determination of receiver frequency response for dispersion supported transmission,” Electron. Lett. 30(1), 58–59 (1994).
[Crossref]

IEEE Photon. Technol. Lett. (1)

J. Binder and U. Kohn, “10 Gbit/s-dispersion optimized transmission at 1.55 μm wavelength on standard single mode fiber,” IEEE Photon. Technol. Lett. 6(4), 558–560 (1994).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

U. Gliese, S. Nørskov, and T. N. Nielsen, “Chromatic dispersion in fiber-optic microwave and millimieter-wave links,” IEEE Trans. Microw. Theory Tech. 44(10), 1716–1724 (1996).
[Crossref]

IET Optoelectron. (1)

I. Papagiannakis, C. Xia, D. Klonidis, W. Rosenkranz, A. N. Birbas, and I. Tomkos, “Electronic distortion equalisation by using decision-feedback/feed-forward equaliser for transient and adiabatic chirped directly modulated lasers at 2.5 and 10 Gb/s,” IET Optoelectron. 3(1), 18–29 (2009).
[Crossref]

J. Lightwave Technol. (8)

F. Devaux, Y. Sorel, and J. F. Kerdiles, “Simple measurement of fiber dispersion and of chirp parameter of intensity modulated light emitter,” J. Lightwave Technol. 11(12), 1937–1940 (1993).
[Crossref]

L. Bjerkan, A. Røyset, L. Hafskjær, and D. Myhre, “Measurement of laser parameters for simulation of high-speed fiberoptic systems,” J. Lightwave Technol. 14(5), 839–850 (1996).
[Crossref]

K. Sato, S. Kuwahara, and Y. Miyamoto, “Chirp characteristics of 40-Gb/s directly modulated distributed- feedback laser diodes,” J. Lightwave Technol. 23(11), 3790–3797 (2005).
[Crossref]

J. M. Tang and K. A. Shore, “30-Gb/s signal transmission over 40-km directly modulated DFB-laser-based single-mode-fiber links without optical amplification and dispersion compensation,” J. Lightwave Technol. 24(6), 2318–2327 (2006).
[Crossref]

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

W.-R. Peng, B. Zhang, K.-M. Feng, X. Wu, A. E. Willner, and S. Chi, “Spectrally efficient direct-detected OFDM transmission incorporating a tunable frequency gap and an iterative detection techniques,” J. Lightwave Technol. 27(24), 5723–5735 (2009).
[Crossref]

W.-R. Peng, “Analysis of laser phase noise effect in direct-detection optical OFDM transmission,” J. Lightwave Technol. 28(17), 2526–2536 (2010).
[Crossref]

M. C. Yuang, P.-L. Tien, D.-Z. Hsu, S.-Y. Chen, C.-C. Wei, J.-L. Shih, and J. Chen, “A high-performance OFDMA PON system architecture and medium access control,” J. Lightwave Technol. 30(11), 1685–1693 (2012).
[Crossref]

J. Mod. Opt. (1)

J. A. P. Morgado and A. V. T. Cartzxo, “Improved model to discriminate adiabatic and transient chirps in directly modulated semiconductor lasers,” J. Mod. Opt. 56(21), 2309–2317 (2009).
[Crossref]

Opt. Express (2)

Opt. Lett. (1)

Other (7)

K. Yuksel, V. Moeyaert, M. Wuilpart, and P. Mégret, “Optical layer monitoring in passive optical networks (PONs): a review,” in International Conference on Transparent Optical Networks (2008), Paper Tu.B1.1.

E. O. Brigham, Fast Fourier Transform and Its Applications, 1st ed. (New York: Wiley, 1997).

G. P. Agrawal, Fibre-Optic Communication Systems, 2nd ed. (Prentice Hall, 1988).

A. Gharba, P. Chanclou, M. Ouzzif, J. L. Masson, L. A. Neto, R. Xia, N. Genay, B. Charbonnier, M. Hélard, E. Grard, and V. Rodrigues, “Optical transmission performance for DML considering laser chirp and fiber dispersion using AMOOFDM,” in 2010 International Congress on Ultra Modern Telecommunications and Control Systems and Workshops (2010), pp. 1022–1026.

L. Hanzo, W. Webb, and T. Keller, Single- and Multi-Carrier Quadrature Amplitude Modulation, 2nd ed. (Wiley, 2000).

D. F. Hewitt, “Orthogonal frequency division multiplexing using baseband optical single sideband for simpler adaptive dispersion compensation,” in Optical Fiber Communication Conference (2007), Paper OME7.

L. A. Coldren and S. W. Corzine, Diode Lasers and Photonic Integrated Circuits (Wiley, 1995).

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Figures (9)

Fig. 1
Fig. 1 Relative power of subcarriers with the dispersion of (a) 320 ps/nm and (b) 1600 ps/nm.

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Fig. 2
Fig. 2 Relative power of the SSII with the dispersion of (a) 320 ps/nm and (b) 1600 ps/nm.

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Fig. 3
Fig. 3 Theoretical SIR with the dispersion of (a) 320 ps/nm and (b) 1600 ps/nm.

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Fig. 4
Fig. 4 (a) The relative subcarrier powers with 320- and 1600-ps/nm dispersion, (b) the SIRs with 320-ps/nm dispersion, and (c) the SIRs with 1600-ps/nm dispersion of the SSM and the LSS; (d) the correlation coefficients between the SSII of the SSM and the distortion of the LSS.

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Fig. 5
Fig. 5 The concept of the proposed iterative equalization

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Fig. 6
Fig. 6 The SINRs with and without the iterative equalization after (a) 20-km SSMF and (b) 100-km SSMF. (0-dBm received power, 30-mA IB and 30-mA Ipp).

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Fig. 7
Fig. 7 The BERs with and without the iterative equalization after (a) 20-km SSMF and (b) 100-km SSMF. (30-mA IB and 30-mA Ipp)

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Fig. 8
Fig. 8 The sensitivities as functions of the transmission distance with the drive currents of (a) IB = 15 mA, Ipp = 12.3 mA, (b) IB = 20 mA, Ipp = 18.1 mA, (c) IB = 30 mA, Ipp = 30 mA, (d) IB = 40 mA, Ipp = 42 mA, (e) IB = 30 mA, Ipp = 20 mA, and (f) IB = 30 mA, Ipp = 45 mA.

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Fig. 9
Fig. 9 The sensitivity penalty as a function of the estimation errors of the chirp parameters with the fix dispersion errors of (a) –20%, (b) –10%, (c) 0, (d) + 10% and (e) + 20%.

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Tables (1)

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Table 1 Laser parameters

View Table

Equations (23)

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dN dt = I qV − v g gS−( AN+B N 2 +C N 3 )
dS dt =Γ v g gS+Γ β sp B N 2 − S τ p
dφ dt = αΓ v g σ 2 ( N− N th )
g= σ( N− N tr ) 1+εS
Δν= 1 2π ⋅ dφ dt ≅ α 4π ( 1 S dS dt + ε τ p S )= α 4π ( 1 P dP dt +κP )
P= P B + ∑ m=1 M ℜ{ p m e jmωt } = P B ⋅[ 1+ ∑ m=−M M x m e jmωt ]
φ≅ α 2 [ ln( P B ( 1+X ) )+κ P B ∫ ( 1+X ) dt ]= α 2 ln( 1+X )+ Ω 2ω X ˜ + α 2 ln P B + Ω 2 t
E ′ ≅1+ ( 1−jα 2 X−j δ 2 X ˜ ) ︸ S 1 + ( − 1+ α 2 8 X 2 − δ 2 8 X ˜ 2 −j δ( 1−jα ) 4 X X ˜ ) ︸ S 2
s m = 1 2 ( 1+ α 2   e −j θ α − δ m ) x m e j m 2 θ D
| E ′ | 2 ≅1+( S 1 + S 1 ∗ )+   S 1 × S 1 ∗ ︸ D R + ( S 2 + S 2 ∗ ) ︸ D T
w m 〈| x m | 2 〉 =( 1+ α 2 ) cos 2 ( m 2 θ D − θ α )+ δ 2 m 2 sin 2 ( m 2 θ D )
d z,m>0 = 1 2 ξ z ( m 2 ,m ) x m/2 2 + ∑ k=⌊ m 2 +1 ⌋ M ξ z ( k,m ) x k x k−m ∗
ξ R ( k,m )=[ 1+ α 2 2 + δ 2 +jmαδ 2k( k−m ) ]cos( ( 2k−m )m θ D )−j ( 2k−m )δ 2k( k−m ) sin( ( 2k−m )m θ D )
ξ T ( k,m )=−[ 1+ α 2 2 + δ 2 +jmαδ 2k( k−m ) ]cos( m 2 θ D )+j mδ 2k( k−m ) sin( m 2 θ D )
d m>0 = 1 2 [ ξ R ( m 2 ,m )+ ξ T ( m 2 ,m ) ] x m/2 2 + ∑ k=⌊ m 2 +1 ⌋ M [ ξ R ( k,m )+ ξ T ( k,m ) ] x k x m−k
μ m>0 ≅ ∑ k=⌊ m 2 +1 ⌋ M [ ξ R ( k,m )+ ξ T ( k,m ) ] 2 w 2
y m>0 = ∑ k=m−M M u k v m−k = u m/2 v m/2 + ∑ k=⌊ m 2 +1 ⌋ M u k v m−k + u m−k v k
d R,m>0 = s m/2 s −m /2 ∗ + ∑ k=⌊ m 2 +1 ⌋ M s k s k−m ∗ + s m−k s −k ∗
x ¯ ¯ m>0 = x m/2 2 + ∑ k=⌊ m 2 +1 ⌋ M 2 x k x m−k
x ˜ ˜ m>0 =− 4 m 2 x m/2 2 − ∑ k=⌊ m 2 +1 ⌋ M 2 k(m−k) x k x m−k
x ˜ ¯ m>0 =−j 2 m x m/2 2 −j ∑ k=⌊ m 2 +1 ⌋ M m k(m−k) x k x m−k
s ^ m>0 =[ − 1+ α 2 8 x ¯ ¯ m − δ 2 8 x ˜ ˜ m −j δ( 1−jα ) 4 x ˜ ¯ m ] e j m 2 θ D
s ^ m<0 =[ − 1+ α 2 8 x ¯ ¯ −m ∗ − δ 2 8 x ˜ ˜ −m ∗ −j δ( 1−jα ) 4 x ˜ ¯ −m ∗ ] e j m 2 θ D

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