Abstract

By using a 200GHz AWG channelized ASE source in connection with a saturable semiconductor optical amplifier (SOA) based noise blocker as the injecting source at the remote node in front of the local optical network units (ONUs), we demonstrate the spectrum-sliced ASE transmitter with greatly suppressed intensity noise performance in WDM-PON network. Such channelized SOA filtering technique effectively reduces the relative intensity noise of the ASE source by at least 4.5 dB. The low-noise WRC-FPLD transmitter improves its extinction-ratio (ER) from 8.9 to 9.6 dB and signal-to-noise ratio (SNR) from 5.9 to 6.3 dB. In comparison with broad-band ASE injection-locked WRC-FPLD transmitter at same power, there is an improvement on receiving power penalty (ΔPReceiver) by 2 dB at BER 10−9 in back-to-back case, and the receiving power of BER 10−9 can achieve −24 dBm even after 25km fiber transmission. With additional AWG filtering, the intraband crosstalk effect between the upstream transmitted data and the reflected ASE signal is significantly reduced by 6.3dB. The compromised effects of ER and SNR on BER performance are also elucidated via the modified SNR model for the WRC-FPLD under ASE injection induced gain-saturation condition. The ΔPReceiver/ΔSNR of 8.89 at same ER condition is more pronounced than the ΔPReceiver/ΔER of 3.17 obtained under same SNR condition, indicating that the SNR plays a more important role than the ER on enhancing the BER performance.

©2010 Optical Society of America

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References

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  1. H. D. Kim, S. Kang, and C. Lee, “A low-cost WDM source with an ASE injected Fabry–Pérot semiconductor laser,” IEEE Photon. Technol. Lett. 12(8), 1067–1069 (2000).
    [Crossref]
  2. S. L. Woodward, “P. P. lannone, K. C. Reichmann, and N. J. Frigo, “A spectrally sliced PON employing Fabry–Pérot lasers,” J. Lightwave Technol. 10, 1337–1339 (1998).
  3. K.-Y. Park and C.-H. Lee, “Intensity Noise in a Wavelength-Locked Fabry–Perot Laser Diode to a Spectrum Sliced ASE,” IEEE J. Quantum Electron. 44(3), 209–215 (2008).
    [Crossref]
  4. A. McCoy, P. Horak, B. C. Thomsen, M. Isben, and D. J. Richardson, “Noise suppression of incoherent light using a gain-saturated SOA: Implications for spectrum-sliced WDM systems,” J. Lightwave Technol. 23(8), 2399–2409 (2005).
    [Crossref]
  5. S. Kim, J. Han, J. Lee, and C. S Park, “Intensity noise suppression in spectrum-sliced incoherent light communication systems using a gain-saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 11(8), 1042–1044 (1999).
    [Crossref]
  6. M. Zhao, G. Morthier, and R. Baets, “Analysis and optimization of intensity noise reduction in spectrum-sliced WDM systems using a saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 14(3), 390–392 (2002).
    [Crossref]
  7. X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
    [Crossref]
  8. D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SOA-based noise reduction,” IEEE Photon. Technol. Lett. 16(2), 680–682 (2004).
    [Crossref]
  9. G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
    [Crossref]
  10. G. P. Agrawal, Fiber-Optic Communication Systems, Third Ed., (Willy Inter-Science, 2002) Chaps. 4–6.
  11. Y.-C. Chang, Y.-H. Lin, J. H. Chen, and G.-R. Lin, “All-optical NRZ-to-PRZ format transformer with an injection-locked Fabry-Perot laser diode at unlasing condition,” Opt. Express 12(19), 4449–4456 (2004).
    [Crossref] [PubMed]
  12. L. Li, “Static and dynamic properties of injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 30(8), 1701–1708 (1994).
    [Crossref]
  13. K. Petermann, Laser Diode Modulation and Noise, (Publishers Dordrecht, The Nitherlands: Kluwer Academic, 1998), Chap. 2.
  14. S. Gee, F. Quinlan, S. Ozharar, and P. Delfyett, “Two-mode beat phase noise of actively modelocked lasers,” Opt. Express 13(11), 3977–3982 (2005).
    [Crossref] [PubMed]
  15. J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for mutichannel WDM application,” IEEE Photon. Technol. Lett. 5(12), 1458–1461 (1993).
    [Crossref]

2008 (1)

K.-Y. Park and C.-H. Lee, “Intensity Noise in a Wavelength-Locked Fabry–Perot Laser Diode to a Spectrum Sliced ASE,” IEEE J. Quantum Electron. 44(3), 209–215 (2008).
[Crossref]

2006 (1)

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

2005 (2)

2004 (2)

D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SOA-based noise reduction,” IEEE Photon. Technol. Lett. 16(2), 680–682 (2004).
[Crossref]

Y.-C. Chang, Y.-H. Lin, J. H. Chen, and G.-R. Lin, “All-optical NRZ-to-PRZ format transformer with an injection-locked Fabry-Perot laser diode at unlasing condition,” Opt. Express 12(19), 4449–4456 (2004).
[Crossref] [PubMed]

2002 (1)

M. Zhao, G. Morthier, and R. Baets, “Analysis and optimization of intensity noise reduction in spectrum-sliced WDM systems using a saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 14(3), 390–392 (2002).
[Crossref]

2000 (1)

H. D. Kim, S. Kang, and C. Lee, “A low-cost WDM source with an ASE injected Fabry–Pérot semiconductor laser,” IEEE Photon. Technol. Lett. 12(8), 1067–1069 (2000).
[Crossref]

1999 (1)

S. Kim, J. Han, J. Lee, and C. S Park, “Intensity noise suppression in spectrum-sliced incoherent light communication systems using a gain-saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 11(8), 1042–1044 (1999).
[Crossref]

1998 (1)

S. L. Woodward, “P. P. lannone, K. C. Reichmann, and N. J. Frigo, “A spectrally sliced PON employing Fabry–Pérot lasers,” J. Lightwave Technol. 10, 1337–1339 (1998).

1994 (1)

L. Li, “Static and dynamic properties of injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 30(8), 1701–1708 (1994).
[Crossref]

1993 (1)

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for mutichannel WDM application,” IEEE Photon. Technol. Lett. 5(12), 1458–1461 (1993).
[Crossref]

1989 (1)

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Agrawal, G. P.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Baets, R.

M. Zhao, G. Morthier, and R. Baets, “Analysis and optimization of intensity noise reduction in spectrum-sliced WDM systems using a saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 14(3), 390–392 (2002).
[Crossref]

Chang, Y.-C.

Chen, J. H.

Cheng, X.

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

Chung, Y. C.

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for mutichannel WDM application,” IEEE Photon. Technol. Lett. 5(12), 1458–1461 (1993).
[Crossref]

Delfyett, P.

DiGiovanni, D. J.

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for mutichannel WDM application,” IEEE Photon. Technol. Lett. 5(12), 1458–1461 (1993).
[Crossref]

Dong, Y.

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

Gee, S.

Han, J.

S. Kim, J. Han, J. Lee, and C. S Park, “Intensity noise suppression in spectrum-sliced incoherent light communication systems using a gain-saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 11(8), 1042–1044 (1999).
[Crossref]

Horak, P.

Ibsen, M.

D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SOA-based noise reduction,” IEEE Photon. Technol. Lett. 16(2), 680–682 (2004).
[Crossref]

Isben, M.

Kang, S.

H. D. Kim, S. Kang, and C. Lee, “A low-cost WDM source with an ASE injected Fabry–Pérot semiconductor laser,” IEEE Photon. Technol. Lett. 12(8), 1067–1069 (2000).
[Crossref]

Kim, H. D.

H. D. Kim, S. Kang, and C. Lee, “A low-cost WDM source with an ASE injected Fabry–Pérot semiconductor laser,” IEEE Photon. Technol. Lett. 12(8), 1067–1069 (2000).
[Crossref]

Kim, S.

S. Kim, J. Han, J. Lee, and C. S Park, “Intensity noise suppression in spectrum-sliced incoherent light communication systems using a gain-saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 11(8), 1042–1044 (1999).
[Crossref]

Lee, C.

H. D. Kim, S. Kang, and C. Lee, “A low-cost WDM source with an ASE injected Fabry–Pérot semiconductor laser,” IEEE Photon. Technol. Lett. 12(8), 1067–1069 (2000).
[Crossref]

Lee, C.-H.

K.-Y. Park and C.-H. Lee, “Intensity Noise in a Wavelength-Locked Fabry–Perot Laser Diode to a Spectrum Sliced ASE,” IEEE J. Quantum Electron. 44(3), 209–215 (2008).
[Crossref]

Lee, J.

S. Kim, J. Han, J. Lee, and C. S Park, “Intensity noise suppression in spectrum-sliced incoherent light communication systems using a gain-saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 11(8), 1042–1044 (1999).
[Crossref]

Lee, J. S.

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for mutichannel WDM application,” IEEE Photon. Technol. Lett. 5(12), 1458–1461 (1993).
[Crossref]

Li, L.

L. Li, “Static and dynamic properties of injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 30(8), 1701–1708 (1994).
[Crossref]

Lin, G.-R.

Lin, Y.-H.

Lu, C.

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

McCoy, A.

McCoy, D.

D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SOA-based noise reduction,” IEEE Photon. Technol. Lett. 16(2), 680–682 (2004).
[Crossref]

Morthier, G.

M. Zhao, G. Morthier, and R. Baets, “Analysis and optimization of intensity noise reduction in spectrum-sliced WDM systems using a saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 14(3), 390–392 (2002).
[Crossref]

Olsson, N. A.

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

Ozharar, S.

Park, C. S

S. Kim, J. Han, J. Lee, and C. S Park, “Intensity noise suppression in spectrum-sliced incoherent light communication systems using a gain-saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 11(8), 1042–1044 (1999).
[Crossref]

Park, K.-Y.

K.-Y. Park and C.-H. Lee, “Intensity Noise in a Wavelength-Locked Fabry–Perot Laser Diode to a Spectrum Sliced ASE,” IEEE J. Quantum Electron. 44(3), 209–215 (2008).
[Crossref]

Quinlan, F.

Richardson, D. J.

A. McCoy, P. Horak, B. C. Thomsen, M. Isben, and D. J. Richardson, “Noise suppression of incoherent light using a gain-saturated SOA: Implications for spectrum-sliced WDM systems,” J. Lightwave Technol. 23(8), 2399–2409 (2005).
[Crossref]

D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SOA-based noise reduction,” IEEE Photon. Technol. Lett. 16(2), 680–682 (2004).
[Crossref]

Shao, X.

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

Thomsen, B. C.

A. McCoy, P. Horak, B. C. Thomsen, M. Isben, and D. J. Richardson, “Noise suppression of incoherent light using a gain-saturated SOA: Implications for spectrum-sliced WDM systems,” J. Lightwave Technol. 23(8), 2399–2409 (2005).
[Crossref]

D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SOA-based noise reduction,” IEEE Photon. Technol. Lett. 16(2), 680–682 (2004).
[Crossref]

Wang, Y.

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

Wen, Y. J.

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

Woodward, S. L.

S. L. Woodward, “P. P. lannone, K. C. Reichmann, and N. J. Frigo, “A spectrally sliced PON employing Fabry–Pérot lasers,” J. Lightwave Technol. 10, 1337–1339 (1998).

Xu, Z.

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

Zhao, M.

M. Zhao, G. Morthier, and R. Baets, “Analysis and optimization of intensity noise reduction in spectrum-sliced WDM systems using a saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 14(3), 390–392 (2002).
[Crossref]

IEEE J. Quantum Electron. (3)

K.-Y. Park and C.-H. Lee, “Intensity Noise in a Wavelength-Locked Fabry–Perot Laser Diode to a Spectrum Sliced ASE,” IEEE J. Quantum Electron. 44(3), 209–215 (2008).
[Crossref]

G. P. Agrawal and N. A. Olsson, “Self-phase modulation and spectral broadening of optical pulses in semiconductor laser amplifiers,” IEEE J. Quantum Electron. 25(11), 2297–2306 (1989).
[Crossref]

L. Li, “Static and dynamic properties of injection-locked semiconductor lasers,” IEEE J. Quantum Electron. 30(8), 1701–1708 (1994).
[Crossref]

IEEE Photon. Technol. Lett. (6)

J. S. Lee, Y. C. Chung, and D. J. DiGiovanni, “Spectrum-sliced fiber amplifier light source for mutichannel WDM application,” IEEE Photon. Technol. Lett. 5(12), 1458–1461 (1993).
[Crossref]

H. D. Kim, S. Kang, and C. Lee, “A low-cost WDM source with an ASE injected Fabry–Pérot semiconductor laser,” IEEE Photon. Technol. Lett. 12(8), 1067–1069 (2000).
[Crossref]

S. Kim, J. Han, J. Lee, and C. S Park, “Intensity noise suppression in spectrum-sliced incoherent light communication systems using a gain-saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 11(8), 1042–1044 (1999).
[Crossref]

M. Zhao, G. Morthier, and R. Baets, “Analysis and optimization of intensity noise reduction in spectrum-sliced WDM systems using a saturated semiconductor optical amplifier,” IEEE Photon. Technol. Lett. 14(3), 390–392 (2002).
[Crossref]

X. Cheng, Y. J. Wen, Y. Dong, Z. Xu, X. Shao, Y. Wang, and C. Lu, “Optimization of Spectrum-Sliced ASE Source for Injection-Locking a Fabry–Pérot Laser Diode,” IEEE Photon. Technol. Lett. 18(18), 1961–1963 (2006).
[Crossref]

D. McCoy, B. C. Thomsen, M. Ibsen, and D. J. Richardson, “Filtering effects in a spectrum-sliced WDM system using SOA-based noise reduction,” IEEE Photon. Technol. Lett. 16(2), 680–682 (2004).
[Crossref]

J. Lightwave Technol. (2)

S. L. Woodward, “P. P. lannone, K. C. Reichmann, and N. J. Frigo, “A spectrally sliced PON employing Fabry–Pérot lasers,” J. Lightwave Technol. 10, 1337–1339 (1998).

A. McCoy, P. Horak, B. C. Thomsen, M. Isben, and D. J. Richardson, “Noise suppression of incoherent light using a gain-saturated SOA: Implications for spectrum-sliced WDM systems,” J. Lightwave Technol. 23(8), 2399–2409 (2005).
[Crossref]

Opt. Express (2)

Other (2)

K. Petermann, Laser Diode Modulation and Noise, (Publishers Dordrecht, The Nitherlands: Kluwer Academic, 1998), Chap. 2.

G. P. Agrawal, Fiber-Optic Communication Systems, Third Ed., (Willy Inter-Science, 2002) Chaps. 4–6.

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Figures (12)

Fig. 1
Fig. 1 (a). The configuration a conventional ASE injecting transmitter wavelength independent operation WDM-PON. (b)Configuration of the DWDM-PON with WRC-FPLD injection-locked by the source of ASE through SOA at the end of remote node.

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Fig. 2
Fig. 2 Power-current curves of WRC-FPLD operated without and with injection power of −12 and −3 dBm.

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Fig. 3
Fig. 3 SNR of AWG-sliced ASE without (pink-dotted) and with SOA based filter at different biased currents.

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Fig. 4
Fig. 4 Measured the RIN of the WRC-FPLD injection locked by different light sources.

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Fig. 5
Fig. 5 WRC-FPLD up-stream BER under the injection of AWG-sliced ASE with SOA filter at different biases.

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Fig. 6
Fig. 6 BER and corresponding eye-diagrams of the SOA filtered ASE injection-locked WRC-FPLD at different biases of (a) 28 mA, (b) 32 mA, (c) 36 mA, (d) 42 mA.

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Fig. 7
Fig. 7 SNR and ER of the ASE injection-locked WRC-FPLD at different currents.

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Fig. 8
Fig. 8 The calculated Q factor of WRC-FPLD at different currents

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Fig. 9
Fig. 9 SNR and ER of WRC-FPLD injection-locked by ASE with changing power levels in different systems.

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Fig. 10
Fig. 10 BER of filtered ASE with SOA (new system), filtered ASE (new system without SOA) and broadband ASE injection-locked WRC-FPLD based WDM-PON (inset: back-to-back eye diagram of (a) old system, (b) new system without SOA, (c) new system with SOA).

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Fig. 11
Fig. 11 BER analysis of AWG-sliced and SOA-bleached ASE injection-locked WRC-FPLD transmitter with changing SNR (left) and changing ER (right).

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Fig. 12
Fig. 12 BER of the WRC-FPLD transmitted data in the DWDM-PON systems with changing ASE-Demux-Mux linewidths of (a) 1.0-1.0-1.0 nm, (b) 0.5-1.0-1.0 nm, and (c) 0.5-0.5-0.5 nm.

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Equations (8)

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S N R ≅ [ 1 2 π ∫ − ∞ + ∞ R I N ( ω ) d ω ] − 1 2                         ( 1 )
η ∝ Γ g ( N 0 ) Γ g ' P 0 ( z ) h ν A ( 1 τ c + Γ g ' P 0 ( z ) h ν A ) ( 1 τ c + Γ g ' P 0 ( z ) h ν A ) 2 = Γ g ( N 0 ) Γ g ' P 0 ( z ) h ν A ( 1 τ c + Γ g ' P 0 ( z ) h ν A )       ( 2 )
SNR o u t =  < I > 2 σ 2  =  ( R G P i n j ) 2 σ 2 ≈ G P i n j 4 S s p Δ f ,                  (3)
G = G 0 exp [ ( P o u t / P i n j ) − 1 ( P o u t / P i n j ) P o u t P s a t ]                         ( 4 )
G = G 0 − Δ G = 1 τ p − R s p h ν P m τ p − 4 β c 2 k c 2 ( 1 + β c 2 ) P i n j P m                           ( 5 )
P o u t = η d h ν q ( I − I t h , i n j ) = η d h ν q ( I − q η i τ c N t h , i n j ) = η d h ν q ( I − q η i τ c [ G ( N i n j , P i n j ) g' + N t r ] )                         ( 6 ) = η d h ν q { I − q η i τ c ( 1 g ' τ p + N t r ) + q η i τ c g ' ( R s p h ν P m τ p + 4 β c 2 k c 2 ( 1 + β c 2 ) P i n j P m ) }
SNR o u t ≈ P i n j 4 S s p Δ f [ 1 τ p − R s p h ν P m τ p − 4 β c 2 k c 2 ( 1 + β c 2 ) P i n j P m ]                   ( 7 )
SNR o u t ≈ 1 4 S s p Δ f η d h ν q { I − q η i τ c ( 1 g ' τ p + N t r ) + q η i τ c g ' ( R s p h ν P m τ p + 4 β c 2 k c 2 ( 1 + β c 2 ) P i n j P m )    }                exp { 1 P s a t [ η d h ν q { I − q η i τ c ( 1 g ' τ p + N t r ) + q η i τ c g ' ( R s p h ν P m τ p + 4 β c 2 k c 2 ( 1 + β c 2 ) P i n j P m )    } − P i n j ] }                 ( 8 )

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