Abstract

A microwave photonic system which can simultaneously realize the functions of rapidly tunable Doppler frequency shift (DFS) and high fidelity storage of broadband RF signals is proposed and verified. Single-sideband carrier-suppression modulation combined with dual-AOM frequency shifting ensures large-range and fast-responding DFS. And time-gated semiconductor amplifier (SOA) based fiber delay loop can realize high-fidelity RF pulse storage with high extinction ratio switching and amplification characteristics of time-gated SOA. A spurious rejection ratio greater than 40 dB, tuning range of DFS greater than ± 3 MHz, response speed of DFS less than 30 ns, and high fidelity storage of 4 GHz-12 GHz RF signals with greater than 381 circulations (corresponding 80 us delay time) are realized by the proposed structure. The maximum signal-to-noise ratio (SNR) is 13.6 dB within 381 circulations. Based on the experimental data, the simulation results show that the delay time also could be extended to 10 times more.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. D. C. Schleher, Electronic Warfare in the Information Age (Artech House, Inc., 1999).
  2. M. Soumekh, “SAR-ECCM using phase-perturbed LFM chirp signals and DRFM repeat jammer penalization,” IEEE International Radar Conference191–205 (2006).
  3. P. Gonzalez-Blanco, E. D. Diego, E. Millan, B. Errasti, and I. Montiel, “Stepped-Frequency Waveform radar demonstrator and its jamming,” International Waveform Diversity & Design Conference (2009).
  4. Q. Yang, Y. Zhang, and X. Gu, “Wide-band chaotic noise signal for velocity estimation and imaging of high-speed moving targets,” Prog. Electromagn. Res. 63, 1–15 (2015).
    [Crossref]
  5. S. D. Berger, “Digital radio frequency memory linear range gate stealer spectrum,” IEEE Trans. Aerosp. Electron. Syst. 39(2), 725–735 (2003).
    [Crossref]
  6. M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
    [Crossref]
  7. A. Partizian, “Airborne Pulse-Doppler Radar,” IRE Trans. Mil. Electron. MIL-5(2), 116–126 (1961).
    [Crossref]
  8. J. Baldwinson and I. Antipov, “Prediction of electronic attack effectiveness against Maritime Patrol Radars,” International Conference on Radar (2008).
  9. M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
    [Crossref]
  10. B. Manz, “DRFM grow to meet new threats,” J. Electron. Defense 33(8), 43 (2010).
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    [Crossref]
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    [Crossref]
  13. Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).
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    [Crossref]
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    [Crossref]
  17. D. Liu, S. Sun, X. Yin, B. Sun, J. Sun, Y. Liu, W. Li, N. Zhu, and M. Li, “Large-capacity and low-loss integrated optical buffer,” Opt. Express 27(8), 11585–11593 (2019).
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  20. Agilent Technologies Application Note 150–2, “Spectrum and signal analysis...pulsed RF,” (2012).
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    [Crossref]

2019 (1)

2018 (1)

2015 (2)

2014 (1)

2013 (1)

E. Chan and R. Minasian, “Single sideband suppressed carrier modulator based frequency shifting recirculating delay line microwave photonic filter,” Opt. Laser Technol. 45, 160–167 (2013).
[Crossref]

2010 (1)

B. Manz, “DRFM grow to meet new threats,” J. Electron. Defense 33(8), 43 (2010).

2009 (1)

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

2007 (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

2006 (2)

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

2003 (1)

S. D. Berger, “Digital radio frequency memory linear range gate stealer spectrum,” IEEE Trans. Aerosp. Electron. Syst. 39(2), 725–735 (2003).
[Crossref]

1961 (1)

A. Partizian, “Airborne Pulse-Doppler Radar,” IRE Trans. Mil. Electron. MIL-5(2), 116–126 (1961).
[Crossref]

Antipov, I.

J. Baldwinson and I. Antipov, “Prediction of electronic attack effectiveness against Maritime Patrol Radars,” International Conference on Radar (2008).

Baldwinson, J.

J. Baldwinson and I. Antipov, “Prediction of electronic attack effectiveness against Maritime Patrol Radars,” International Conference on Radar (2008).

Berger, S. D.

S. D. Berger, “Digital radio frequency memory linear range gate stealer spectrum,” IEEE Trans. Aerosp. Electron. Syst. 39(2), 725–735 (2003).
[Crossref]

Bernasconi, P.

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

Buhl, L.

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

Cai, H.

J. Wang, P. Hou, H. Cai, J. Sun, S. Wang, L. Wang, and F. Yang, “Continuous angle steering of an optically- controlled phased array antenna based on differential true time delay constituted by micro-optical components,” Opt. Express 23(7), 9432–9439 (2015).
[Crossref]

Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).

Cao, Y.

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

Capmany, J.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Chan, E.

E. Chan and R. Minasian, “Single sideband suppressed carrier modulator based frequency shifting recirculating delay line microwave photonic filter,” Opt. Laser Technol. 45, 160–167 (2013).
[Crossref]

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

Chan, E. H.

Chen, K.

Diego, E. D.

P. Gonzalez-Blanco, E. D. Diego, E. Millan, B. Errasti, and I. Montiel, “Stepped-Frequency Waveform radar demonstrator and its jamming,” International Waveform Diversity & Design Conference (2009).

Ding, Z.

Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).

Errasti, B.

P. Gonzalez-Blanco, E. D. Diego, E. Millan, B. Errasti, and I. Montiel, “Stepped-Frequency Waveform radar demonstrator and its jamming,” International Waveform Diversity & Design Conference (2009).

Farina, A.

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

Feng, X.

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

Feng, Z.

Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).

Gini, F.

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

Gonzalez-Blanco, P.

P. Gonzalez-Blanco, E. D. Diego, E. Millan, B. Errasti, and I. Montiel, “Stepped-Frequency Waveform radar demonstrator and its jamming,” International Waveform Diversity & Design Conference (2009).

Greco, M.

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

Gripp, J.

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

Gu, X.

Q. Yang, Y. Zhang, and X. Gu, “Wide-band chaotic noise signal for velocity estimation and imaging of high-speed moving targets,” Prog. Electromagn. Res. 63, 1–15 (2015).
[Crossref]

Guan, B.

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

Hou, P.

Hu, Z.

Kong, Y.

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

LeGrange, J. D.

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

Li, M.

Li, W.

Liu, D.

Liu, Y.

Lu, B.

Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).

Manz, B.

B. Manz, “DRFM grow to meet new threats,” J. Electron. Defense 33(8), 43 (2010).

Millan, E.

P. Gonzalez-Blanco, E. D. Diego, E. Millan, B. Errasti, and I. Montiel, “Stepped-Frequency Waveform radar demonstrator and its jamming,” International Waveform Diversity & Design Conference (2009).

Minasian, R.

E. Chan and R. Minasian, “Single sideband suppressed carrier modulator based frequency shifting recirculating delay line microwave photonic filter,” Opt. Laser Technol. 45, 160–167 (2013).
[Crossref]

Minasian, R. A.

Montiel, I.

P. Gonzalez-Blanco, E. D. Diego, E. Millan, B. Errasti, and I. Montiel, “Stepped-Frequency Waveform radar demonstrator and its jamming,” International Waveform Diversity & Design Conference (2009).

Neilson, D. T.

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

Nguyen, L. V.

L. V. Nguyen, “Photonic radio frequency memory-design issues and possible solutions,” DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION SALISBURY (AUSTRALIA) SYSTEM (2003).

Nguyen, T. A.

Novak, D.

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Partizian, A.

A. Partizian, “Airborne Pulse-Doppler Radar,” IRE Trans. Mil. Electron. MIL-5(2), 116–126 (1961).
[Crossref]

Ravenni, V.

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

Schleher, D. C.

D. C. Schleher, Electronic Warfare in the Information Age (Artech House, Inc., 1999).

Simsarian, J. E.

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

Song, Q.

Soumekh, M.

M. Soumekh, “SAR-ECCM using phase-perturbed LFM chirp signals and DRFM repeat jammer penalization,” IEEE International Radar Conference191–205 (2006).

Sun, B.

Sun, J.

Sun, S.

Wang, J.

Wang, L.

Wang, S.

Wang, X.

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

Yang, F.

J. Wang, P. Hou, H. Cai, J. Sun, S. Wang, L. Wang, and F. Yang, “Continuous angle steering of an optically- controlled phased array antenna based on differential true time delay constituted by micro-optical components,” Opt. Express 23(7), 9432–9439 (2015).
[Crossref]

Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).

Yang, Q.

Q. Yang, Y. Zhang, and X. Gu, “Wide-band chaotic noise signal for velocity estimation and imaging of high-speed moving targets,” Prog. Electromagn. Res. 63, 1–15 (2015).
[Crossref]

Yin, X.

Zhang, Y.

Q. Yang, Y. Zhang, and X. Gu, “Wide-band chaotic noise signal for velocity estimation and imaging of high-speed moving targets,” Prog. Electromagn. Res. 63, 1–15 (2015).
[Crossref]

Zhao, J.

Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).

Zheng, R.

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

Zhu, N.

Appl. Opt. (1)

IEE Proc., Radar Sonar Navig. (2)

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

M. Greco, F. Gini, A. Farina, and V. Ravenni, “Effect of phase and range gate pull-off delay quantisation on jammer signal,” IEE Proc., Radar Sonar Navig. 153(5), 454–459 (2006).
[Crossref]

IEEE Photonics Technol. Lett. (1)

J. D. LeGrange, J. E. Simsarian, P. Bernasconi, D. T. Neilson, L. Buhl, and J. Gripp, “Demonstration of an integrated buffer for an all-optical packet router,” IEEE Photonics Technol. Lett. 21(12), 781–783 (2009).
[Crossref]

IEEE Trans. Aerosp. Electron. Syst. (1)

S. D. Berger, “Digital radio frequency memory linear range gate stealer spectrum,” IEEE Trans. Aerosp. Electron. Syst. 39(2), 725–735 (2003).
[Crossref]

IRE Trans. Mil. Electron. (1)

A. Partizian, “Airborne Pulse-Doppler Radar,” IRE Trans. Mil. Electron. MIL-5(2), 116–126 (1961).
[Crossref]

J. Electron. Defense (1)

B. Manz, “DRFM grow to meet new threats,” J. Electron. Defense 33(8), 43 (2010).

J. Lightwave Technol. (1)

Nat. Photonics (1)

J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
[Crossref]

Opt. Express (2)

Opt. Laser Technol. (1)

E. Chan and R. Minasian, “Single sideband suppressed carrier modulator based frequency shifting recirculating delay line microwave photonic filter,” Opt. Laser Technol. 45, 160–167 (2013).
[Crossref]

Prog. Electromagn. Res. (1)

Q. Yang, Y. Zhang, and X. Gu, “Wide-band chaotic noise signal for velocity estimation and imaging of high-speed moving targets,” Prog. Electromagn. Res. 63, 1–15 (2015).
[Crossref]

Other (8)

D. C. Schleher, Electronic Warfare in the Information Age (Artech House, Inc., 1999).

M. Soumekh, “SAR-ECCM using phase-perturbed LFM chirp signals and DRFM repeat jammer penalization,” IEEE International Radar Conference191–205 (2006).

P. Gonzalez-Blanco, E. D. Diego, E. Millan, B. Errasti, and I. Montiel, “Stepped-Frequency Waveform radar demonstrator and its jamming,” International Waveform Diversity & Design Conference (2009).

Z. Ding, J. Zhao, F. Yang, Z. Feng, B. Lu, and H. Cai, “Wide Bandwidth Reconfigurable and Frequency Tunable Microwave Photonic Filter Based on Tunable Ultra-sharp Roll-off Optical Filter,” 2018 Asia Communications and Photonics Conference (ACP)1–3 (2018).

L. V. Nguyen, “Photonic radio frequency memory-design issues and possible solutions,” DEFENCE SCIENCE AND TECHNOLOGY ORGANISATION SALISBURY (AUSTRALIA) SYSTEM (2003).

R. Zheng, Y. Kong, E. Chan, Y. Cao, X. Wang, X. Feng, and B. Guan, “Photonics based Microwave Frequency Shifter for Doppler Shift Compensation in High-Speed Railways,” Conference on Lasers and Electro-Optics/Pacific Rim F2C. 4 (2018).

Agilent Technologies Application Note 150–2, “Spectrum and signal analysis...pulsed RF,” (2012).

J. Baldwinson and I. Antipov, “Prediction of electronic attack effectiveness against Maritime Patrol Radars,” International Conference on Radar (2008).

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Figures (10)

Fig. 1.
Fig. 1. Experimental diagram of photonic RF pulse DFS and storage system. EDFA: Erbium-doped fiber amplifier; DPMZM: dual parallel Mach-Zender modulator; AOM: acousto-optic frequency shifter; SMF: single model fiber; SOA: semiconductor amplifier; PC: polarization controller; PD: photodetector; OSA: optical spectrum analyzer; ESA: electronic spectrum analyzer; OSC: oscilloscope.

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Fig. 2.
Fig. 2. Spectrum before entering the fiber loop and corresponding output RF signal spectrum (inset).

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Fig. 3.
Fig. 3. (a) Relationship between output optical power of AOM2 and its drive frequency and (b) the spectrum response of frequency shift when the input RF signal was 4 GHz.

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Fig. 4.
Fig. 4. (a) AOM2 frequency shift response speed and drive source low level and (b) its impulse response waveform

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Fig. 5.
Fig. 5. Stored 4 GHz pulse signals and their waveforms after (a) 0, (b) 191, and (c) 381 circulations.

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Fig. 6.
Fig. 6. Time domain display of ESA measuring the output pulse signal from the time-gated SOA based fiber optic loop

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Fig. 7.
Fig. 7. Pulse spectrum of (a) RF pulse signal and (b) accumulated noise under different circulations

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Fig. 8.
Fig. 8. Storage performance of pulse signals at different input RF frequencies.

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Fig. 9.
Fig. 9. Spectrum of output pulse signal with (a) 200 ns pulse width and (b) 2 us pulse width at different Doppler frequency shifts after 40 us delay time.

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Fig. 10.
Fig. 10. The relationship between SNR degradation and the number of cycles for different losses, gains and lengths of fiber in the loop. The parameters settings are as follows: γ=0.5, P0=13 mW, η=0.65, ψ=0.65 A/W, nsp=8, γ*(1—γ1) = 0.601 when the length of fiber is 42 m and γ*(1—γ1) = 0.603 when it is 420 m.

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Tables (2)

Tables Icon

Table 1. Relationship between number of circulations and SNR degradation

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Table 2. Relationship between SNR degradation and number of circulations at different RF frequencies

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Equations (4)

Equations on this page are rendered with MathJax. Learn more.

ω R F − o u t p u t = ω R F + ω A O M 1 − ω A O M 2
P n , R F e l e c = 1 2 ( ( 1 − γ ) 2 γ n − 1 ( 1 − γ 1 ) n G n P 0 η ψ ) 2 R
P n , s − s p e l e c = ( 1 − γ ) 2 γ n − 1 ( 1 − γ 1 ) n G n P 0 ⋅ 2 n n s p ( G − 1 ) h v B d ψ 2 R
S N R n = P n , R F e l e c P n , s − s p e l e c = ( 1 − γ ) 2 γ n − 1 ( 1 − γ 1 ) n G n P 0 ψ 2 4 n n s p ( G − 1 ) h v B d