Abstract

A novel scheme for the generation and stabilization of the millimeter-wave (mmW) signal employing a frequency-demultiplication optoelectronic oscillator (FD-OEO) has been theoretically analyzed and experimentally demonstrated. The FD-OEO can keep sustaining without optical first-order sidebands, which would help to simplify the photonic-assisted frequency multiplication process and provide a wide frequency compensation range for the mmW system simultaneously. The stability of the generated 40-GHz mmW signal reaches 1.38 × 10−12 at the average time of 100s. Besides, the measured single-sideband phase noise of the generated mmW signal exhibits as low as −103 dBc/Hz at 10-kHz offset frequency, maintaining a spurious level of −97 dBc.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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2015 (1)

A. Bluestone, D. T. Spencer, S. Srinivasan, D. Guerra, J. E. Bowers, and L. Theogarajan, “An ultra-low phase-noise 20-GHz pll utilizing an optoelectronic voltage-controlled oscillator,” IEEE Transactions on Microw. Theory Tech. 63, 1046–1052 (2015).
[Crossref]

2014 (2)

Y. Zhang, D. Hou, and J. Zhao, “Long-term frequency stabilization of an optoelectronic oscillator using phase-locked loop,” J. Light. Technol. 32, 2408–2414 (2014).
[Crossref]

S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,” Proc. IEEE 102, 366–385 (2014).
[Crossref]

2013 (2)

T. F. Tseng, “High-resolution 3-dimensional radar imaging based on a few-cycle w-band photonic millimeter-wave pulse generator,” Opt. Express 21, 14109–14119 (2013).
[Crossref] [PubMed]

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

2011 (1)

Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband systems,” Commun. Mag. IEEE 49, 101–107 (2011).
[Crossref]

2010 (1)

L. Xia, S. Aditya, P. Shum, and J. Zhou, “Microwave photonic frequency doubling in an rof link DWDM system with four wave mixing,” Microw. Opt. Technol. Lett. 52, 1428–1431 (2010).
[Crossref]

2009 (5)

2008 (2)

H. Chi and J. Yao, “Frequency quadrupling and upconversion in a radio over fiber link,” J. Light. Technol. 26, 2706–2711 (2008).
[Crossref]

H. Ou, B. Chen, K. Zhu, and S. He, “Microwave-photonic frequency doubling utilising phase modulator and fibre bragg grating,” Electron. Lett. 44, 131–132 (2008).
[Crossref]

2007 (2)

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Light. Technol. 24, 4628–4641 (2007).
[Crossref]

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

2006 (1)

A. Wiberg, P. Perez-Millan, M. V. Andres, and P. O. Hedekvist, “Microwave-photonic frequency multiplication utilizing optical four-wave mixing and fiber bragg gratings,” J. Light. Technol. 24, 329–334 (2006).
[Crossref]

2005 (1)

P. D. Grant, R. Dudek, L. Wolfson, M. Buchanan, and H. C. Liu, “Ultra-high frequency monolithically integrated quantum well infrared photodetector up to 75 GHz,” Electron. Lett. 41, 214–215 (2005).
[Crossref]

2003 (1)

X. J. Meng and J. Menders, “Optical generation of microwave signals using ssb-based frequency-doubling scheme,” Electron. Lett. 39, 103–105 (2003).
[Crossref]

2002 (1)

D. H. Chang, H. R. Fetterman, H. Erlig, and H. Zhang, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photonics Technol. Lett. 14, 191–193 (2002).
[Crossref]

2000 (1)

H. Ito, T. Furuta, S. Kodama, and T. Ishibashi, “Inp/ingaas uni-travelling-carrier photodiode with 310 GHz bandwidth,” Electron. Lett. 36, 1809–1810 (2000).
[Crossref]

1996 (2)

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt.Soc. Am. B 13, 1725–1735 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

1994 (1)

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[Crossref]

Aditya, S.

L. Xia, S. Aditya, P. Shum, and J. Zhou, “Microwave photonic frequency doubling in an rof link DWDM system with four wave mixing,” Microw. Opt. Technol. Lett. 52, 1428–1431 (2010).
[Crossref]

Andres, M. V.

A. Wiberg, P. Perez-Millan, M. V. Andres, and P. O. Hedekvist, “Microwave-photonic frequency multiplication utilizing optical four-wave mixing and fiber bragg gratings,” J. Light. Technol. 24, 329–334 (2006).
[Crossref]

Azar, Y.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Bluestone, A.

A. Bluestone, D. T. Spencer, S. Srinivasan, D. Guerra, J. E. Bowers, and L. Theogarajan, “An ultra-low phase-noise 20-GHz pll utilizing an optoelectronic voltage-controlled oscillator,” IEEE Transactions on Microw. Theory Tech. 63, 1046–1052 (2015).
[Crossref]

Blume, B. T.

F. Downs, J. L. Woodputnam, B. T. Blume, and R. M. Smith, “Passive millimeter-wave imaging device for naval special warfare,” in Aerospace/Defense Sensing and Controls, (1998), p. 3378.

Bolton, D. R.

D. R. Bolton, D. A. Robertson, and G. M. Smith, “Phase noise of sources for multiplication to mm-wave frequencies,” in Joint International Conference on Infrared and Millimeter Waves and International Conference on Terahertz Electronics, (2005), pp. 74–75 vol. 1.

Bowers, J. E.

A. Bluestone, D. T. Spencer, S. Srinivasan, D. Guerra, J. E. Bowers, and L. Theogarajan, “An ultra-low phase-noise 20-GHz pll utilizing an optoelectronic voltage-controlled oscillator,” IEEE Transactions on Microw. Theory Tech. 63, 1046–1052 (2015).
[Crossref]

Buchanan, M.

P. D. Grant, R. Dudek, L. Wolfson, M. Buchanan, and H. C. Liu, “Ultra-high frequency monolithically integrated quantum well infrared photodetector up to 75 GHz,” Electron. Lett. 41, 214–215 (2005).
[Crossref]

Capmany, J.

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

Chang, D. H.

D. H. Chang, H. R. Fetterman, H. Erlig, and H. Zhang, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photonics Technol. Lett. 14, 191–193 (2002).
[Crossref]

Chen, B.

H. Ou, B. Chen, K. Zhu, and S. He, “Microwave-photonic frequency doubling utilising phase modulator and fibre bragg grating,” Electron. Lett. 44, 131–132 (2008).
[Crossref]

Chen, J.

Chen, J. J.

Chen, Y. H.

Chi, H.

H. Chi and J. Yao, “Frequency quadrupling and upconversion in a radio over fiber link,” J. Light. Technol. 26, 2706–2711 (2008).
[Crossref]

Chi, S.

Donoho, D. L.

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[Crossref]

Downs, F.

F. Downs, J. L. Woodputnam, B. T. Blume, and R. M. Smith, “Passive millimeter-wave imaging device for naval special warfare,” in Aerospace/Defense Sensing and Controls, (1998), p. 3378.

Dudek, R.

P. D. Grant, R. Dudek, L. Wolfson, M. Buchanan, and H. C. Liu, “Ultra-high frequency monolithically integrated quantum well infrared photodetector up to 75 GHz,” Electron. Lett. 41, 214–215 (2005).
[Crossref]

Erkip, E.

S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,” Proc. IEEE 102, 366–385 (2014).
[Crossref]

Erlig, H.

D. H. Chang, H. R. Fetterman, H. Erlig, and H. Zhang, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photonics Technol. Lett. 14, 191–193 (2002).
[Crossref]

Fetterman, H. R.

D. H. Chang, H. R. Fetterman, H. Erlig, and H. Zhang, “39-GHz optoelectronic oscillator using broad-band polymer electrooptic modulator,” IEEE Photonics Technol. Lett. 14, 191–193 (2002).
[Crossref]

Figueiredo, M. A. T.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Transactions on Signal Process. 57, 2479–2493 (2009).
[Crossref]

Furuta, T.

H. Ito, T. Furuta, S. Kodama, and T. Ishibashi, “Inp/ingaas uni-travelling-carrier photodiode with 310 GHz bandwidth,” Electron. Lett. 36, 1809–1810 (2000).
[Crossref]

Grant, P. D.

P. D. Grant, R. Dudek, L. Wolfson, M. Buchanan, and H. C. Liu, “Ultra-high frequency monolithically integrated quantum well infrared photodetector up to 75 GHz,” Electron. Lett. 41, 214–215 (2005).
[Crossref]

Guerra, D.

A. Bluestone, D. T. Spencer, S. Srinivasan, D. Guerra, J. E. Bowers, and L. Theogarajan, “An ultra-low phase-noise 20-GHz pll utilizing an optoelectronic voltage-controlled oscillator,” IEEE Transactions on Microw. Theory Tech. 63, 1046–1052 (2015).
[Crossref]

Gutierrez, F.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

He, S.

H. Ou, B. Chen, K. Zhu, and S. He, “Microwave-photonic frequency doubling utilising phase modulator and fibre bragg grating,” Electron. Lett. 44, 131–132 (2008).
[Crossref]

Hedekvist, P. O.

A. Wiberg, P. Perez-Millan, M. V. Andres, and P. O. Hedekvist, “Microwave-photonic frequency multiplication utilizing optical four-wave mixing and fiber bragg gratings,” J. Light. Technol. 24, 329–334 (2006).
[Crossref]

Hou, D.

Y. Zhang, D. Hou, and J. Zhao, “Long-term frequency stabilization of an optoelectronic oscillator using phase-locked loop,” J. Light. Technol. 32, 2408–2414 (2014).
[Crossref]

Huang, H. S.

Ishibashi, T.

H. Ito, T. Furuta, S. Kodama, and T. Ishibashi, “Inp/ingaas uni-travelling-carrier photodiode with 310 GHz bandwidth,” Electron. Lett. 36, 1809–1810 (2000).
[Crossref]

Ito, H.

H. Ito, T. Furuta, S. Kodama, and T. Ishibashi, “Inp/ingaas uni-travelling-carrier photodiode with 310 GHz bandwidth,” Electron. Lett. 36, 1809–1810 (2000).
[Crossref]

Jiang, W. J.

Johnstone, I. M.

D. L. Donoho and I. M. Johnstone, “Ideal spatial adaptation by wavelet shrinkage,” Biometrika 81, 425–455 (1994).
[Crossref]

Khan, F.

Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband systems,” Commun. Mag. IEEE 49, 101–107 (2011).
[Crossref]

Kodama, S.

H. Ito, T. Furuta, S. Kodama, and T. Ishibashi, “Inp/ingaas uni-travelling-carrier photodiode with 310 GHz bandwidth,” Electron. Lett. 36, 1809–1810 (2000).
[Crossref]

Lin, C. T.

Liu, H. C.

P. D. Grant, R. Dudek, L. Wolfson, M. Buchanan, and H. C. Liu, “Ultra-high frequency monolithically integrated quantum well infrared photodetector up to 75 GHz,” Electron. Lett. 41, 214–215 (2005).
[Crossref]

Maleki, L.

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt.Soc. Am. B 13, 1725–1735 (1996).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic oscillator for photonic systems,” IEEE J. Quantum Electron. 32, 1141–1149 (1996).
[Crossref]

Mayzus, R.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Menders, J.

X. J. Meng and J. Menders, “Optical generation of microwave signals using ssb-based frequency-doubling scheme,” Electron. Lett. 39, 103–105 (2003).
[Crossref]

Meng, X. J.

X. J. Meng and J. Menders, “Optical generation of microwave signals using ssb-based frequency-doubling scheme,” Electron. Lett. 39, 103–105 (2003).
[Crossref]

Novak, D.

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

Nowak, R. D.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Transactions on Signal Process. 57, 2479–2493 (2009).
[Crossref]

Ou, H.

H. Ou, B. Chen, K. Zhu, and S. He, “Microwave-photonic frequency doubling utilising phase modulator and fibre bragg grating,” Electron. Lett. 44, 131–132 (2008).
[Crossref]

Peng, P. C.

Perez-Millan, P.

A. Wiberg, P. Perez-Millan, M. V. Andres, and P. O. Hedekvist, “Microwave-photonic frequency multiplication utilizing optical four-wave mixing and fiber bragg gratings,” J. Light. Technol. 24, 329–334 (2006).
[Crossref]

Pi, Z.

Z. Pi and F. Khan, “An introduction to millimeter-wave mobile broadband systems,” Commun. Mag. IEEE 49, 101–107 (2011).
[Crossref]

Rangan, S.

S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,” Proc. IEEE 102, 366–385 (2014).
[Crossref]

Rappaport, T. S.

S. Rangan, T. S. Rappaport, and E. Erkip, “Millimeter-wave cellular wireless networks: Potentials and challenges,” Proc. IEEE 102, 366–385 (2014).
[Crossref]

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Robertson, D. A.

D. R. Bolton, D. A. Robertson, and G. M. Smith, “Phase noise of sources for multiplication to mm-wave frequencies,” in Joint International Conference on Infrared and Millimeter Waves and International Conference on Terahertz Electronics, (2005), pp. 74–75 vol. 1.

Samimi, M.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Schulz, J. K.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Seeds, A. J.

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Light. Technol. 24, 4628–4641 (2007).
[Crossref]

Shih, P. T.

Shum, P.

L. Xia, S. Aditya, P. Shum, and J. Zhou, “Microwave photonic frequency doubling in an rof link DWDM system with four wave mixing,” Microw. Opt. Technol. Lett. 52, 1428–1431 (2010).
[Crossref]

Smith, G. M.

D. R. Bolton, D. A. Robertson, and G. M. Smith, “Phase noise of sources for multiplication to mm-wave frequencies,” in Joint International Conference on Infrared and Millimeter Waves and International Conference on Terahertz Electronics, (2005), pp. 74–75 vol. 1.

Smith, R. M.

F. Downs, J. L. Woodputnam, B. T. Blume, and R. M. Smith, “Passive millimeter-wave imaging device for naval special warfare,” in Aerospace/Defense Sensing and Controls, (1998), p. 3378.

Spencer, D. T.

A. Bluestone, D. T. Spencer, S. Srinivasan, D. Guerra, J. E. Bowers, and L. Theogarajan, “An ultra-low phase-noise 20-GHz pll utilizing an optoelectronic voltage-controlled oscillator,” IEEE Transactions on Microw. Theory Tech. 63, 1046–1052 (2015).
[Crossref]

Srinivasan, S.

A. Bluestone, D. T. Spencer, S. Srinivasan, D. Guerra, J. E. Bowers, and L. Theogarajan, “An ultra-low phase-noise 20-GHz pll utilizing an optoelectronic voltage-controlled oscillator,” IEEE Transactions on Microw. Theory Tech. 63, 1046–1052 (2015).
[Crossref]

Sun, S.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Theogarajan, L.

A. Bluestone, D. T. Spencer, S. Srinivasan, D. Guerra, J. E. Bowers, and L. Theogarajan, “An ultra-low phase-noise 20-GHz pll utilizing an optoelectronic voltage-controlled oscillator,” IEEE Transactions on Microw. Theory Tech. 63, 1046–1052 (2015).
[Crossref]

Tseng, T. F.

Wang, K.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Wen, H.

Wiberg, A.

A. Wiberg, P. Perez-Millan, M. V. Andres, and P. O. Hedekvist, “Microwave-photonic frequency multiplication utilizing optical four-wave mixing and fiber bragg gratings,” J. Light. Technol. 24, 329–334 (2006).
[Crossref]

Williams, K. J.

A. J. Seeds and K. J. Williams, “Microwave photonics,” J. Light. Technol. 24, 4628–4641 (2007).
[Crossref]

Wolfson, L.

P. D. Grant, R. Dudek, L. Wolfson, M. Buchanan, and H. C. Liu, “Ultra-high frequency monolithically integrated quantum well infrared photodetector up to 75 GHz,” Electron. Lett. 41, 214–215 (2005).
[Crossref]

Wong, E. Z.

Wong, G. N.

T. S. Rappaport, S. Sun, R. Mayzus, H. Zhao, Y. Azar, K. Wang, G. N. Wong, J. K. Schulz, M. Samimi, and F. Gutierrez, “Millimeter wave mobile communications for 5G cellular: It will work!” IEEE Access 1, 335–349 (2013).
[Crossref]

Woodputnam, J. L.

F. Downs, J. L. Woodputnam, B. T. Blume, and R. M. Smith, “Passive millimeter-wave imaging device for naval special warfare,” in Aerospace/Defense Sensing and Controls, (1998), p. 3378.

Wright, S. J.

S. J. Wright, R. D. Nowak, and M. A. T. Figueiredo, “Sparse reconstruction by separable approximation,” IEEE Transactions on Signal Process. 57, 2479–2493 (2009).
[Crossref]

Xia, L.

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Figures (8)

Fig. 1
Fig. 1 Schematic configuration of the proposed mmW generation and stabilization. LD: laser diode; MZM: Mach-Zehnder modulator; DSF: dispersion shifted fiber; TOF: tunable optical filter; EDFA: Erbium-doped fiber amplifier; OSA: optical spectrum analyzer; PD: photodetector; EA: electrical amplifier; BPF: band-pass filter; EFD: electronic frequency divider; VPS: voltage-controlled phase shifter; PID: proportion integration differentiation regulator; EPD: electrical phase detector; FC: frequency counter; ESA: electrical spectrum analyzer.
Fig. 2
Fig. 2 Open loop model for the setup.
Fig. 3
Fig. 3 Experimental measurements of the EFD and the OEO loop. (a) The output of the EFD with one input signal; (b) The loop gain of the OEO with one input signal; (c) The output of the EFD with an extra injected-signal; (d) The loop gain of the OEO with an extra injected-signal.
Fig. 4
Fig. 4 The specific frequency conversion of the proposed scheme.
Fig. 5
Fig. 5 Optical and electrical spectrum. (a) Dash curve, the optical spectra without the TOF; line curve, the optical spectra after filtering; (b),(c) and (d) The electrical spectrum for 10 GHz, 20 GHz and 40 GHz, respectively.
Fig. 6
Fig. 6 Stability test of the generated signals. (a) The ADEV of the generated 10 GHz and 40 GHz with or without PLL; (b) Frequency drift of the PLL-locked and free-running mmW signals.
Fig. 7
Fig. 7 (a) Spectra for free-running mmW signal; (b) Spectra for PLL-locked mmW signal.
Fig. 8
Fig. 8 Comparison of phase noise for the system.

Equations (13)

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P ( t , ω ) = ( γ P 0 / 2 ) { 1 + η cos π [ V in ( t , ω ) / V π ] }
V 2 ( t , ω ) = V ph η cos π [ | V in | / V π cos ( ω t ) ]
V 2 ( t , ω ) = V ph η cos π [ | V in | / V π cos ( ω t ) ] = 2 V ph η n = 1 ( 1 ) n J 2 n ( π | V in | / V π ) cos ( 2 n ω t )
F H ( V ( t , ω ) , V T ) = { V H ( t , ω / 2 ) , | V ( t , ω ) | > V T 0 , | V ( t , ω ) | < V T
V 3 = G 2 F H ( V 2 ( t , ω ) , V T ) = G 2 V H ( t , ω / 2 ) { | V 2 ( t , ω | > V T }
G s = d V 3 d V in | | V in | = 0 = G 2 d F H d V 2 d V 2 d V in | | V in | = 0 = 0
V input = | V E | / V π cos ( ω t + Δ ω t ) + | V in | / V π cos ( ω t ) = [ | V E | / V π cos ( Δ ω t ) + | V in | / V π ] cos ( ω t ) | V E | / V π sin ( ω t ) sin ( Δ ω t )
V 2 ( t , ω ) = V ph η cos [ α cos ( ω t ) β sin ( ω t ) ] = V ph η cos [ α cos ( ω t ) ] cos [ β sin ( ω t ) ] + V ph η sin [ α cos ( ω t ) ] sin [ β sin ( ω t ) ]
V 2 ( t , ω ) = V ph η [ J 0 ( α ) + 2 n = 1 ( 1 ) n J 2 n ( α ) cos ( 2 n ω t ) ] [ J 0 ( β ) + 2 n = 1 J 2 n ( β ) cos ( 2 n ω t ) ] + V ph η { 2 k = 1 ( 1 ) k J 2 k 1 ( α ) cos [ ( 2 k 1 ) ω t ] } { 2 k = 1 J 2 k 1 ( β ) sin [ ( 2 k 1 ) ω t ] } 2 V ph η [ ( 1 α 2 4 ) β 2 8 ( 1 β 2 4 ) α 2 8 ] cos ( 2 ω t ) + V ph η α β 2 sin ( 2 ω t )
V 2 ( t , ω ) = 2 V ph η ( π | V E | / V π sin ( Δ ω t ) ) 2 ( π | V E | / V π cos ( Δ ω t ) + π | V in | / V π ) 2 8 cos ( 2 ω t ) + V ph η ( π | V E | / V π cos ( Δ ω t ) + π | V in | / V π ) π | V E | / V π sin ( Δ ω t ) 2 sin ( 2 ω t )
V 2 ( t , ω ) = V ph η π 2 | V in V E | 2 V π 2 cos ( 2 ω + Δ ω ) t
F H ( V in ( t , 2 ω + Δ ω ) , V T ) = { G H | V in | cos ( ω t ) , | V E | > V T 0 , | V E | < V T
G s = d V 3 d V in | | V in | = 0 = G 2 d F H d V in | | V in | = 0 = G 2 G H d V 2 d V in | | V in | = 0 = G 2 G H V ph η π 2 | V E | 2 V π 2

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