Abstract

In this paper, we investigate the performance of intensity modulation with optical carrier reduced by biasing a dual-electrode Mach-Zehnder modulator (DEMZM) below its quadrature point, in the presence of controlled modulation chirp. The effects of the modulation chirp and of the bias point of a DEMZM on the received signal are analytically derived for small-signal operation. The interaction of these two effects is assessed in terms of optical signal to noise ratio (OSNR) required for a BER = 10−9, through a comparison between double sideband (DSB) modulation and double sideband - reduced carrier (DSB-RC) modulation, by numerical simulation. We found that the power of the optical carrier has impact on the optimum value of the α chirp parameter and that a positive value of the α chirp parameter can be the optimum value to drive a DEMZM, depending on the central frequency of the orthogonal frequency division multiplexing (OFDM) signal and on the DEMZM bias point.

© 2013 Optical Society of America

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References

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  1. M. Popov, “The convergence of wired and wireless services delivery in access and home networks” Proc. Optical Fiber Communication conference (2010), paper OWQ6.
    [Crossref]
  2. X. Mazda and F. Mazda, Focal Illustrated Dictionary of Telecommunications (Focal Press, 2013).
  3. R. Hui, Z. Benyuan, H. Renxing, C. T. Allen, K. Demarest, and D. Richards, “Subcarrier multiplexing for high-speed optical transmission,” J. Lightwave Technol. 20(3), 417–427 (2002).
    [Crossref]
  4. M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
    [Crossref]
  5. J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
    [Crossref]
  6. P. Almeida and H. Silva, “Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(3), 283–286 (2013).
    [Crossref]
  7. J. L. Wei, E. Hugues-Salas, R. P. Giddings, X. Q. Jin, X. Zheng, S. Mansoor, and J. M. Tang, “Wavelength reused bidirectional transmission of adaptively modulated optical OFDM signals in WDM-PONs incorporating SOA and RSOA intensity modulators,” Opt. Express 18(10), 9791–9808 (2010).
    [Crossref] [PubMed]
  8. 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]
  9. H. Silva, R. Fyath, and J. O'Reilly, “Sensitivity degradation with laser wavelength chirp for direct-detection optical receivers,” IEE Proc. Optoelectron.– Pt. J,136(4), 209–218, (1989).
  10. C. Sánchez, B. Ortega, J. L. Wei, J. Tang, and J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
    [Crossref] [PubMed]
  11. P. Almeida and H. Silva, “Expressions of the chirp parameter components for intensity modulation with a dual-electrode Mach-Zehnder modulator, ” Proc. International Conference on Transparent Optical Networks (2012).
    [Crossref]
  12. S. Walklin and J. Conradi, “Effect of Mach-Zehnder modulator DC extinction ratio on residual chirp-induced dispersion in 10-Gb/s binary and AM-PSK duobinary lightwave systems,” IEEE Photon. Technol. Lett. 9(10), 1400–1402 (1997).
    [Crossref]
  13. P. Almeida and H. Silva, “Corrections to ”Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(11), 1087–1087 (2013).
    [Crossref]
  14. G. P. Agrawal, Fiber-Optic Communication Systems, (NJ, 2002).
  15. L. V. T. Nguyen and D. B. Hunter, “A photonic technique for microwave frequency measurement,” IEEE Photon. Technol. Lett. 18(10), 1188–1190 (2006).
    [Crossref]
  16. High rate ultra wideband PHY and MAC standard (2007). European Computer Manufacturers Association International Std. ECMA-368.
  17. A. B. Carlson, P. B. Crilly, and J. C. Rutledge, Communications systems- An introduction to signals and noise in electrical communication, (Mc Graw-Hill, 2002).
  18. A. Ali, J. Leibrich, and W. Rosenkranz, “Spectral efficiency and receiver sensitivity in direct detection optical-OFDM,”Proc. Conference on Optical Fiber Communication conference (2009), paper OMT7.
    [Crossref]
  19. W. Peng, K. Feng, A. E. Willner, and S. Chi, “Estimation of the bit error rate for direct-detected OFDM signals with optically preamplified receivers,” IEEE/OSA J. Lightwave Technol. 27(10), 1340–1346 (2009).
    [Crossref]
  20. T. Alves and A. Cartaxo, “Analysis of methods of performance evaluation of direct-detection OFDM communication systems,” Fiber Integr. Opt. 29(3), 170–186 (2010).
    [Crossref]
  21. T. Dennis and P. A. Williams, “Chirp characterization of external modulators with finite extinction ratio using linear optical sampling,” IEEE Photon. Technol. Lett. 22(9), 646–648 (2010).
    [Crossref]
  22. 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]
  23. M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” Proc. Optical Fiber Communication Conference,(2012), paper OW3B.2.
    [Crossref]

2013 (3)

P. Almeida and H. Silva, “Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(3), 283–286 (2013).
[Crossref]

P. Almeida and H. Silva, “Corrections to ”Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(11), 1087–1087 (2013).
[Crossref]

C. Sánchez, B. Ortega, J. L. Wei, J. Tang, and J. Capmany, “Analytical formulation of directly modulated OOFDM signals transmitted over an IM/DD dispersive link,” Opt. Express 21(6), 7651–7666 (2013).
[Crossref] [PubMed]

2010 (4)

2009 (2)

J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
[Crossref]

W. Peng, K. Feng, A. E. Willner, and S. Chi, “Estimation of the bit error rate for direct-detected OFDM signals with optically preamplified receivers,” IEEE/OSA J. Lightwave Technol. 27(10), 1340–1346 (2009).
[Crossref]

2006 (1)

L. V. T. Nguyen and D. B. Hunter, “A photonic technique for microwave frequency measurement,” IEEE Photon. Technol. Lett. 18(10), 1188–1190 (2006).
[Crossref]

2005 (1)

M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
[Crossref]

2002 (1)

1997 (1)

S. Walklin and J. Conradi, “Effect of Mach-Zehnder modulator DC extinction ratio on residual chirp-induced dispersion in 10-Gb/s binary and AM-PSK duobinary lightwave systems,” IEEE Photon. Technol. Lett. 9(10), 1400–1402 (1997).
[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]

Ali, A.

J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
[Crossref]

Allen, C. T.

Almeida, P.

P. Almeida and H. Silva, “Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(3), 283–286 (2013).
[Crossref]

P. Almeida and H. Silva, “Corrections to ”Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(11), 1087–1087 (2013).
[Crossref]

P. Almeida and H. Silva, “Expressions of the chirp parameter components for intensity modulation with a dual-electrode Mach-Zehnder modulator, ” Proc. International Conference on Transparent Optical Networks (2012).
[Crossref]

Alves, T.

T. Alves and A. Cartaxo, “Analysis of methods of performance evaluation of direct-detection OFDM communication systems,” Fiber Integr. Opt. 29(3), 170–186 (2010).
[Crossref]

Attygalle, M.

M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
[Crossref]

Benyuan, Z.

Capmany, J.

Cartaxo, A.

T. Alves and A. Cartaxo, “Analysis of methods of performance evaluation of direct-detection OFDM communication systems,” Fiber Integr. Opt. 29(3), 170–186 (2010).
[Crossref]

Chen, H. Y.

Chen, J.

Chi, S.

W. Peng, K. Feng, A. E. Willner, and S. Chi, “Estimation of the bit error rate for direct-detected OFDM signals with optically preamplified receivers,” IEEE/OSA J. Lightwave Technol. 27(10), 1340–1346 (2009).
[Crossref]

Conradi, J.

S. Walklin and J. Conradi, “Effect of Mach-Zehnder modulator DC extinction ratio on residual chirp-induced dispersion in 10-Gb/s binary and AM-PSK duobinary lightwave systems,” IEEE Photon. Technol. Lett. 9(10), 1400–1402 (1997).
[Crossref]

Demarest, K.

Dennis, T.

T. Dennis and P. A. Williams, “Chirp characterization of external modulators with finite extinction ratio using linear optical sampling,” IEEE Photon. Technol. Lett. 22(9), 646–648 (2010).
[Crossref]

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]

Edvell, G.

M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
[Crossref]

Feng, K.

W. Peng, K. Feng, A. E. Willner, and S. Chi, “Estimation of the bit error rate for direct-detected OFDM signals with optically preamplified receivers,” IEEE/OSA J. Lightwave Technol. 27(10), 1340–1346 (2009).
[Crossref]

Giddings, R. P.

Hsu, D. Z.

Hugues-Salas, E.

Hui, R.

Hunter, D. B.

L. V. T. Nguyen and D. B. Hunter, “A photonic technique for microwave frequency measurement,” IEEE Photon. Technol. Lett. 18(10), 1188–1190 (2006).
[Crossref]

Jin, X. Q.

Kammeyer, K. D.

J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
[Crossref]

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]

Leibrich, J.

J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
[Crossref]

Li, W. Y.

Lim, C.

M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
[Crossref]

Lin, S. H.

Mansoor, S.

Nguyen, L. V. T.

L. V. T. Nguyen and D. B. Hunter, “A photonic technique for microwave frequency measurement,” IEEE Photon. Technol. Lett. 18(10), 1188–1190 (2006).
[Crossref]

Nirmalathas, A.

M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
[Crossref]

Ortega, B.

Paul, H.

J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
[Crossref]

Pendock, G. J.

M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
[Crossref]

Peng, W.

W. Peng, K. Feng, A. E. Willner, and S. Chi, “Estimation of the bit error rate for direct-detected OFDM signals with optically preamplified receivers,” IEEE/OSA J. Lightwave Technol. 27(10), 1340–1346 (2009).
[Crossref]

Renxing, H.

Richards, D.

Rosenkranz, W.

J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
[Crossref]

Sánchez, C.

Silva, H.

P. Almeida and H. Silva, “Corrections to ”Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(11), 1087–1087 (2013).
[Crossref]

P. Almeida and H. Silva, “Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(3), 283–286 (2013).
[Crossref]

P. Almeida and H. Silva, “Expressions of the chirp parameter components for intensity modulation with a dual-electrode Mach-Zehnder modulator, ” Proc. International Conference on Transparent Optical Networks (2012).
[Crossref]

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.

Tang, J. M.

Walklin, S.

S. Walklin and J. Conradi, “Effect of Mach-Zehnder modulator DC extinction ratio on residual chirp-induced dispersion in 10-Gb/s binary and AM-PSK duobinary lightwave systems,” IEEE Photon. Technol. Lett. 9(10), 1400–1402 (1997).
[Crossref]

Wei, C. C.

Wei, J. L.

Williams, P. A.

T. Dennis and P. A. Williams, “Chirp characterization of external modulators with finite extinction ratio using linear optical sampling,” IEEE Photon. Technol. Lett. 22(9), 646–648 (2010).
[Crossref]

Willner, A. E.

W. Peng, K. Feng, A. E. Willner, and S. Chi, “Estimation of the bit error rate for direct-detected OFDM signals with optically preamplified receivers,” IEEE/OSA J. Lightwave Technol. 27(10), 1340–1346 (2009).
[Crossref]

Yuang, M. C.

Zheng, X.

Fiber Integr. Opt. (1)

T. Alves and A. Cartaxo, “Analysis of methods of performance evaluation of direct-detection OFDM communication systems,” Fiber Integr. Opt. 29(3), 170–186 (2010).
[Crossref]

IEEE Photon. Technol. Lett. (7)

T. Dennis and P. A. Williams, “Chirp characterization of external modulators with finite extinction ratio using linear optical sampling,” IEEE Photon. Technol. Lett. 22(9), 646–648 (2010).
[Crossref]

M. Attygalle, C. Lim, G. J. Pendock, A. Nirmalathas, and G. Edvell, “Transmission improvement in fiber wireless links using fiber Bragg gratings,” IEEE Photon. Technol. Lett. 17(1), 190–192 (2005).
[Crossref]

J. Leibrich, A. Ali, H. Paul, W. Rosenkranz, and K. D. Kammeyer, “Impact of modulator bias on the OSNR requirement of direct-detection optical OFDM,” IEEE Photon. Technol. Lett. 21(15), 1033–1035 (2009).
[Crossref]

P. Almeida and H. Silva, “Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(3), 283–286 (2013).
[Crossref]

L. V. T. Nguyen and D. B. Hunter, “A photonic technique for microwave frequency measurement,” IEEE Photon. Technol. Lett. 18(10), 1188–1190 (2006).
[Crossref]

S. Walklin and J. Conradi, “Effect of Mach-Zehnder modulator DC extinction ratio on residual chirp-induced dispersion in 10-Gb/s binary and AM-PSK duobinary lightwave systems,” IEEE Photon. Technol. Lett. 9(10), 1400–1402 (1997).
[Crossref]

P. Almeida and H. Silva, “Corrections to ”Impact of the modulation chirp of a DEMZM on the transmission of signals based on OFDM,” IEEE Photon. Technol. Lett. 25(11), 1087–1087 (2013).
[Crossref]

IEEE/OSA J. Lightwave Technol. (1)

W. Peng, K. Feng, A. E. Willner, and S. Chi, “Estimation of the bit error rate for direct-detected OFDM signals with optically preamplified receivers,” IEEE/OSA J. Lightwave Technol. 27(10), 1340–1346 (2009).
[Crossref]

J. Lightwave Technol. (2)

R. Hui, Z. Benyuan, H. Renxing, C. T. Allen, K. Demarest, and D. Richards, “Subcarrier multiplexing for high-speed optical transmission,” J. Lightwave Technol. 20(3), 417–427 (2002).
[Crossref]

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]

Opt. Express (3)

Other (9)

P. Almeida and H. Silva, “Expressions of the chirp parameter components for intensity modulation with a dual-electrode Mach-Zehnder modulator, ” Proc. International Conference on Transparent Optical Networks (2012).
[Crossref]

M. Popov, “The convergence of wired and wireless services delivery in access and home networks” Proc. Optical Fiber Communication conference (2010), paper OWQ6.
[Crossref]

X. Mazda and F. Mazda, Focal Illustrated Dictionary of Telecommunications (Focal Press, 2013).

H. Silva, R. Fyath, and J. O'Reilly, “Sensitivity degradation with laser wavelength chirp for direct-detection optical receivers,” IEE Proc. Optoelectron.– Pt. J,136(4), 209–218, (1989).

G. P. Agrawal, Fiber-Optic Communication Systems, (NJ, 2002).

High rate ultra wideband PHY and MAC standard (2007). European Computer Manufacturers Association International Std. ECMA-368.

A. B. Carlson, P. B. Crilly, and J. C. Rutledge, Communications systems- An introduction to signals and noise in electrical communication, (Mc Graw-Hill, 2002).

A. Ali, J. Leibrich, and W. Rosenkranz, “Spectral efficiency and receiver sensitivity in direct detection optical-OFDM,”Proc. Conference on Optical Fiber Communication conference (2009), paper OMT7.
[Crossref]

M. Morant, T. Alves, A. Cartaxo, and R. Llorente, “Transmission impairment compensation using broadband channel sounding in multi-format OFDM-based long-reach PONs,” Proc. Optical Fiber Communication Conference,(2012), paper OW3B.2.
[Crossref]

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Figures (8)

Fig. 1
Fig. 1 Normalized RF powers as a function of fiber length, for an OFDM signal centered at 3.96 GHz.

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Fig. 2
Fig. 2 Relationship between the optical powers and the RF power as functions of the Vbias voltage.

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Fig. 3
Fig. 3 Simulation transmission scheme of an OHS over a hybrid long-reach FTTH network with controlled modulation chirp.

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Fig. 4
Fig. 4 Required OSNR for a BER = 10−9 as a function of the modulation index for different bias voltages, for: (a) OFDM-GbE signal; (b) OFDM-UWB #1 signal; (c) OFDM-UWB #2 signal; (d) OFDM-UWB #3 signal.

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Fig. 5
Fig. 5 Required OSNR for a BER = 10−9 as a function of the modulation index for two different bias point of DEMZM: −0.5Vπ and −0.8Vπ.

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Fig. 6
Fig. 6 Absolute error between the desired α chirp parameter at the output of the DEMZM and the value obtained for different bias points of the DEMZM: −0.5Vπ and −0.8Vπ.

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Fig. 7
Fig. 7 Required OSNR for a BER = 10−9 for different distances from the CO to the ONU, as a function of the α chirp parameter, for the following signals: (a) GbE signal, (b) UWB#1 signal, (c) UWB#2 signal, (d) UWB#3 signal, modulated in DSB-RC (Vbias = −0.8Vπ) and DSB (Vbias = −0.5Vπ).

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Fig. 8
Fig. 8 Frequency response magnitude of a dispersive optical link.

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Tables (1)

Tables Icon

Table 1 Main Parameters Used to Generate the Baseband OFDM Signals

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Equations (16)

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x( t )= x i ( t )cos( ω rf t )− x q ( t )sin( ω rf t )
E MZM ( t )= 2P e j ω c t 2 { e j π V π ( V 1 x( t )− V bias1 ) +γ e j π V π ( V 2 x( t )+ V bias2 ) },
V 1 =2 γcos( π V bias / V π )+ γ 2 −αγsin( π V bias / V π ) 2γcos( π V bias / V π )+1+ γ 2 m V π
V 2 =−2 γcos( π V bias / V π )+1+αγsin( π V bias / V π ) 2γcos( π V bias / V π )+1+ γ 2 m V π
E MZM ( t )≅ 2P 2 { E − e j( ω c − ω rf )t + E 0 e j ω c t + E + e j( ω c + ω rf )t },
E − =−[ Γ 1 Λ 1 * +γ Γ 2 Λ 2 * ) ] E 0 =[ Γ 1 A 1 +γ Γ 2 A 2 ) ] E + =[ Γ 1 Λ 1 +γ Γ 2 Λ 2 ) ]
β( ω )≈ β 0 + β 1 ( ω− ω c )+ β 2 /2 ( ω− ω c ) 2 ,
β( ω ± c ω rf )= β ± = β 0 ± β 1 ω rf + β 2 /2 ω rf 2 .
E( t,L )≅ 2P 2 { E − e −j β − L e j( ω c − ω rf )t + E 0 e −j β 0 L e j ω c t + E + e −jβ+L e j( ω c + ω rf )t }.
i( t )=ℜ{ E( t,L )E ( t,L ) * }= i DC + i s ( t )
i s ( t )≈2ℜPπγm F bias 1+ α 2 cos( πD λ c 2 f rf 2 L c +arctan( α ) )[ x i ( t )+j x q ( t ) ]cos( ω rf ( t−τ ) )
P rf =〈 i s ( t ) 2 〉 R L ≈2 R L ( ℜPπγm ) 2 F bias 2 ( σ xi 2 + σ xq 2 )( 1+ α 2 )cos ( πD λ c 2 f rf 2 L c +arctan( α ) ) 2
P OC = ( 2P 2 ) 2 | E 0 | 2 =4P( 1+cos( π V π ( V bias1 + V bias2 ) ) ),
P OSB = ( 2P 2 ) 2 〈 ( | E − | 2 + | E + | 2 ) 〉=2 ( 2P 2 ) 2 〈 | E + | 2 〉 = P 2 ( mπ ) 2 ( σ xi 2 + σ xq 2 )( 1−cos( π V π ( V bias1 + V bias2 ) ) ).
P rf = R L ℜ 2 P OC P OSB .
I sn ( f )=2 e −2 α f 1+ α 2 | cos[ πD λ c 2 f 2 L c +arctan( α ) ] |,

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