Abstract

A photonics-assisted multi-band radar transmitter operating in a wide frequency range has been proposed and experimentally demonstrated. The multi-band radar transmitter incorporates a tunable optoelectronic oscillator (OEO), a low-frequency RF source and a microwave photonic frequency-converting link. In the frequency-converting link, a single tone with ultra-low phase noise and a low-frequency narrow-band RF signal that are generated respectively by the OEO and the RF source, are mixed, frequency converted and bandwidth multiplied to generate multi-band transmission signals. The central frequency, bandwidth and modulation format of transmission signals are reconfigurable. A multi-band radar transmitter with an instantaneous bandwidth of 1.6 GHz is developed. The frequency range of the multi-band radar transmitter covers six bands (from S to Ka), and a moving target detection experiment verifies that the proposed system has potential in multifunctional radar applications.

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

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References

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    [Crossref]
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2019 (1)

2018 (2)

2017 (4)

2016 (1)

2015 (1)

F. Scotti, D. Onori, and F. Laghezza, “Fully coherent S- and X-band photonics-aided radar system demonstration,” IEEE Microw. Wireless Compon. Lett. 25(11), 757–759 (2015).
[Crossref]

2014 (2)

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

X. Liu, W. Pan, X. Hua, L. Yan, B. Luo, and B. Lu, “Investigation on tunable modulation index in the polarization-modulator-based optoelectronic oscillator,” IEEE J. Quantum Electron. 50(2), 68–73 (2014).
[Crossref]

2012 (2)

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Techn. 60(6), 1735–1742 (2012).
[Crossref]

W. Li and J. Yao, “Optically tunable frequency-multiplying optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(10), 812–814 (2012).
[Crossref]

2007 (2)

M. Vespe, C. J. Baker, and H. D. Griffiths, “Automatic target recognition using multi-diversity radar,” IET Radar Sonar Navig. 1(6), 470–478 (2007).
[Crossref]

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

2000 (1)

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36(1), 79–84 (2000).
[Crossref]

1996 (1)

baili, G.

Baker, C. J.

M. Vespe, C. J. Baker, and H. D. Griffiths, “Automatic target recognition using multi-diversity radar,” IET Radar Sonar Navig. 1(6), 470–478 (2007).
[Crossref]

Berger, P.

Berizzi, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Bogoni, A.

P. Ghelfi, D. Onori, F. Laghezza, and A. Bogoni, “Photonics for radars operating on multiple coherent bands,” J. Lightwave Technol. 34(2), 500–507 (2016).
[Crossref]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

S. Melo, A. Bogoni, S. Pinna, F. Laghezza, and F. Scotti, “Dual-use system combining simultaneous active radar & communication, based on a single photonics-assisted transceiver,” in Proceedings of International Radar Symposium (IEEE, 2016), pp. 1–4.

Cao, J.

Capmany, J.

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

Capria, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Chen, J.

J. Chen, W. Zou, and W. Kan, “Reconfigurable microwave photonics radars,” in Proceedings of International Topical Meeting Microwave Photonics (IEEE, 2016), pp. 59–62.

Cong, W.

Crozatier, V.

Cui, Z.

X. Wei, Y. Zheng, Z. Cui, and Q. Wang, “Multi-band SAR images fusion using the EM algorithm in Contourlet domain,” in Proceedings of International Conference on Fuzzy Systems & Knowledge Discovery (IEEE, 2007) pp. 502–506.

Ding, M.

Dong, J.

Dorp, P. V.

P. V. Dorp, R. Ebeling, and A. G. Huizing, “High-resolution radar imaging using coherent multiband processing techniques,” in Proceedings of Radar Conference (IEEE, 2010) pp. 981–986.

Du, P.

Ebeling, R.

P. V. Dorp, R. Ebeling, and A. G. Huizing, “High-resolution radar imaging using coherent multiband processing techniques,” in Proceedings of Radar Conference (IEEE, 2010) pp. 981–986.

Eliyahu, D.

D. Eliyahu, D. Seidel, and L. Maleki, “Phase noise of a high performance OEO and an ultra low noise floor cross-correlation microwave photonic homodyne system,” in Proceedings of International Frequency Control Symposium (IEEE, 2008), pp. 811–814.

Gao, B.

Ghelfi, P.

P. Ghelfi, D. Onori, F. Laghezza, and A. Bogoni, “Photonics for radars operating on multiple coherent bands,” J. Lightwave Technol. 34(2), 500–507 (2016).
[Crossref]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Griffiths, H. D.

M. Vespe, C. J. Baker, and H. D. Griffiths, “Automatic target recognition using multi-diversity radar,” IET Radar Sonar Navig. 1(6), 470–478 (2007).
[Crossref]

Guo, P.

Guo, Q.

Q. Guo, P. Zhou, and S. Pan, “Dual-band linear frequency modulation signal generation by optical frequency quadrupling and polarization multiplexing,” IEEE Photonics Technol. Lett. 29(16), 1320–1323 (2017).
[Crossref]

F. Zhang, Q. Guo, Z. Wang, P. Zhou, G. Zhang, J. Sun, and S. Pan, “Photonics-based broadband radar for high resolution and real-time inverse synthetic aperture imaging,” Opt. Express 25(14), 16274 (2017).
[Crossref]

Hua, X.

X. Liu, W. Pan, X. Hua, L. Yan, B. Luo, and B. Lu, “Investigation on tunable modulation index in the polarization-modulator-based optoelectronic oscillator,” IEEE J. Quantum Electron. 50(2), 68–73 (2014).
[Crossref]

Huizing, A. G.

P. V. Dorp, R. Ebeling, and A. G. Huizing, “High-resolution radar imaging using coherent multiband processing techniques,” in Proceedings of Radar Conference (IEEE, 2010) pp. 981–986.

Ji, Y.

X. S. Yao, L. Maleki, Y. Ji, G. Lutes, and M. Tu, “Dual-loop optoelectronic oscillator,” in Proceedings of International Frequency Control Symposium (IEEE, 1998), pp. 545–549.

Jiang, W.

Kan, W.

J. Chen, W. Zou, and W. Kan, “Reconfigurable microwave photonics radars,” in Proceedings of International Topical Meeting Microwave Photonics (IEEE, 2016), pp. 59–62.

Laghezza, F.

P. Ghelfi, D. Onori, F. Laghezza, and A. Bogoni, “Photonics for radars operating on multiple coherent bands,” J. Lightwave Technol. 34(2), 500–507 (2016).
[Crossref]

F. Scotti, D. Onori, and F. Laghezza, “Fully coherent S- and X-band photonics-aided radar system demonstration,” IEEE Microw. Wireless Compon. Lett. 25(11), 757–759 (2015).
[Crossref]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

S. Melo, A. Bogoni, S. Pinna, F. Laghezza, and F. Scotti, “Dual-use system combining simultaneous active radar & communication, based on a single photonics-assisted transceiver,” in Proceedings of International Radar Symposium (IEEE, 2016), pp. 1–4.

Lan, J.

Lazzeri, E.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Lelièvre, O.

Li, R.

Li, S.

Li, W.

J. Cao, R. Li, J. Yang, Z. Mo, J. Dong, X. Zhang, W. Jiang, and W. Li, “Photonic deramp receiver for dual-band LFM-CW radar,” J. Lightwave Technol. 37(10), 2403–2408 (2019).
[Crossref]

R. Li, W. Li, M. Ding, Z. Wen, Y. Li, L. Zhou, S. Yu, T. Xing, B. Gao, Y. Luan, Y. Zhu, P. Guo, Y. Tian, and X. liang, “Demonstration of a microwave photonic synthetic aperture radar based on photonic-assisted signal generation and stretch processing,” Opt. Express 25(13), 14334–14340 (2017).
[Crossref]

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Techn. 60(6), 1735–1742 (2012).
[Crossref]

W. Li and J. Yao, “Optically tunable frequency-multiplying optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(10), 812–814 (2012).
[Crossref]

Li, Y.

liang, X.

Liu, X.

X. Liu, W. Pan, X. Hua, L. Yan, B. Luo, and B. Lu, “Investigation on tunable modulation index in the polarization-modulator-based optoelectronic oscillator,” IEEE J. Quantum Electron. 50(2), 68–73 (2014).
[Crossref]

Lu, B.

X. Liu, W. Pan, X. Hua, L. Yan, B. Luo, and B. Lu, “Investigation on tunable modulation index in the polarization-modulator-based optoelectronic oscillator,” IEEE J. Quantum Electron. 50(2), 68–73 (2014).
[Crossref]

Luan, Y.

Luo, B.

X. Liu, W. Pan, X. Hua, L. Yan, B. Luo, and B. Lu, “Investigation on tunable modulation index in the polarization-modulator-based optoelectronic oscillator,” IEEE J. Quantum Electron. 50(2), 68–73 (2014).
[Crossref]

Luo, X.

Lutes, G.

X. S. Yao, L. Maleki, Y. Ji, G. Lutes, and M. Tu, “Dual-loop optoelectronic oscillator,” in Proceedings of International Frequency Control Symposium (IEEE, 1998), pp. 545–549.

Malacarne, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Maleki, L.

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36(1), 79–84 (2000).
[Crossref]

X. S. Yao and L. Maleki, “Optoelectronic microwave oscillator,” J. Opt. Soc. Am. B 13(8), 1725–1735 (1996).
[Crossref]

D. Eliyahu, D. Seidel, and L. Maleki, “Phase noise of a high performance OEO and an ultra low noise floor cross-correlation microwave photonic homodyne system,” in Proceedings of International Frequency Control Symposium (IEEE, 2008), pp. 811–814.

X. S. Yao, L. Maleki, Y. Ji, G. Lutes, and M. Tu, “Dual-loop optoelectronic oscillator,” in Proceedings of International Frequency Control Symposium (IEEE, 1998), pp. 545–549.

Melo, S.

S. Melo, A. Bogoni, S. Pinna, F. Laghezza, and F. Scotti, “Dual-use system combining simultaneous active radar & communication, based on a single photonics-assisted transceiver,” in Proceedings of International Radar Symposium (IEEE, 2016), pp. 1–4.

Mo, Z.

Novak, D.

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

Onori, D.

P. Ghelfi, D. Onori, F. Laghezza, and A. Bogoni, “Photonics for radars operating on multiple coherent bands,” J. Lightwave Technol. 34(2), 500–507 (2016).
[Crossref]

F. Scotti, D. Onori, and F. Laghezza, “Fully coherent S- and X-band photonics-aided radar system demonstration,” IEEE Microw. Wireless Compon. Lett. 25(11), 757–759 (2015).
[Crossref]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Pan, S.

Q. Guo, P. Zhou, and S. Pan, “Dual-band linear frequency modulation signal generation by optical frequency quadrupling and polarization multiplexing,” IEEE Photonics Technol. Lett. 29(16), 1320–1323 (2017).
[Crossref]

F. Zhang, Q. Guo, Z. Wang, P. Zhou, G. Zhang, J. Sun, and S. Pan, “Photonics-based broadband radar for high resolution and real-time inverse synthetic aperture imaging,” Opt. Express 25(14), 16274 (2017).
[Crossref]

Pan, W.

X. Liu, W. Pan, X. Hua, L. Yan, B. Luo, and B. Lu, “Investigation on tunable modulation index in the polarization-modulator-based optoelectronic oscillator,” IEEE J. Quantum Electron. 50(2), 68–73 (2014).
[Crossref]

Peng, S.

Pillet, G.

Pinna, S.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

S. Melo, A. Bogoni, S. Pinna, F. Laghezza, and F. Scotti, “Dual-use system combining simultaneous active radar & communication, based on a single photonics-assisted transceiver,” in Proceedings of International Radar Symposium (IEEE, 2016), pp. 1–4.

Porzi, C.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Scaffardi, M.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Scotti, F.

F. Scotti, D. Onori, and F. Laghezza, “Fully coherent S- and X-band photonics-aided radar system demonstration,” IEEE Microw. Wireless Compon. Lett. 25(11), 757–759 (2015).
[Crossref]

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

S. Melo, A. Bogoni, S. Pinna, F. Laghezza, and F. Scotti, “Dual-use system combining simultaneous active radar & communication, based on a single photonics-assisted transceiver,” in Proceedings of International Radar Symposium (IEEE, 2016), pp. 1–4.

Seidel, D.

D. Eliyahu, D. Seidel, and L. Maleki, “Phase noise of a high performance OEO and an ultra low noise floor cross-correlation microwave photonic homodyne system,” in Proceedings of International Frequency Control Symposium (IEEE, 2008), pp. 811–814.

Serafino, G.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Sun, J.

Tian, Y.

Tu, M.

X. S. Yao, L. Maleki, Y. Ji, G. Lutes, and M. Tu, “Dual-loop optoelectronic oscillator,” in Proceedings of International Frequency Control Symposium (IEEE, 1998), pp. 545–549.

Vercesi, V.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, A. Capria, S. Pinna, D. Onori, C. Porzi, M. Scaffardi, A. Malacarne, V. Vercesi, E. Lazzeri, F. Berizzi, and A. Bogoni, “A fully photonics-based coherent radar system,” Nature 507(7492), 341–345 (2014).
[Crossref]

Vespe, M.

M. Vespe, C. J. Baker, and H. D. Griffiths, “Automatic target recognition using multi-diversity radar,” IET Radar Sonar Navig. 1(6), 470–478 (2007).
[Crossref]

Wang, A.

Wang, Q.

X. Wei, Y. Zheng, Z. Cui, and Q. Wang, “Multi-band SAR images fusion using the EM algorithm in Contourlet domain,” in Proceedings of International Conference on Fuzzy Systems & Knowledge Discovery (IEEE, 2007) pp. 502–506.

Wang, Y.

Wang, Z.

Wei, X.

X. Wei, Y. Zheng, Z. Cui, and Q. Wang, “Multi-band SAR images fusion using the EM algorithm in Contourlet domain,” in Proceedings of International Conference on Fuzzy Systems & Knowledge Discovery (IEEE, 2007) pp. 502–506.

Wen, Z.

Wo, J.

Wu, D.

Xiao, X.

Xing, T.

Xue, X.

Yan, L.

X. Liu, W. Pan, X. Hua, L. Yan, B. Luo, and B. Lu, “Investigation on tunable modulation index in the polarization-modulator-based optoelectronic oscillator,” IEEE J. Quantum Electron. 50(2), 68–73 (2014).
[Crossref]

Yang, J.

Yao, J.

W. Li and J. Yao, “Optically tunable frequency-multiplying optoelectronic oscillator,” IEEE Photonics Technol. Lett. 24(10), 812–814 (2012).
[Crossref]

W. Li and J. Yao, “A wideband frequency tunable optoelectronic oscillator incorporating a tunable microwave photonic filter based on phase-modulation to intensity-modulation conversion using a phase-shifted fiber Bragg grating,” IEEE Trans. Microw. Theory Techn. 60(6), 1735–1742 (2012).
[Crossref]

J. Yao, “Microwave photonics,” in Proceedings of International Workshop on Electromagnetics; Applications and Student Innovation (iWEM) (IEEE2012), pp.314–335.

Yao, X. S.

X. S. Yao and L. Maleki, “Multi-loop optoelectronic oscillator,” IEEE J. Quantum Electron. 36(1), 79–84 (2000).
[Crossref]

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Figures (10)

Fig. 1.
Fig. 1. Schematic of the microwave photonic multi-band radar transmitter; CW: continuous-wave light source; PolM: polarization modulator; OC: optical coupler; PC: polarization controller; MZM: Mach-Zehnder modulator; PM: phase modulator; TBPF: optical tunable bandpass filter; LNA: low noise amplifier; PA: power amplifier; PS: phase shifter; ATT: attenuator; EDFA: erbium-doped fiber amplifier; Div: power divider; PD: photodetector; RF SG: RF signal generator.
Fig. 2.
Fig. 2. Schematic of the microwave photonic frequency-converting link; A, B (red): Optical spectra at different locations in the microwave photonic link; C, D (black): Electrical spectra at different locations in the microwave photonic link.
Fig. 3.
Fig. 3. (a) Optical spectrum of light wave after being CS-DSB modulated by the oscillation signal; (b) optical spectrum of the light wave at output of the MZM.
Fig. 4.
Fig. 4. (a) Measured oscillation frequency from the tunable OEO; (b) phase noise comparison among an RF source (in blue curve), a triple-loop OEO (in red curve), and two simulation results with the actual (in green curve) and ideal (in purple bold curve) fiber length when generating signals with frequency of 13 GHz.
Fig. 5.
Fig. 5. Spectra of the generated signals in six frequency bands.
Fig. 6.
Fig. 6. (a) Waveform of the generated X-band signal in the time domain; (b) the calculated instantaneous frequency-time diagram of the X-band signal; (c) a vertical-section of the instantaneous frequency-time diagram of the X-band signal; (d) waveform of the generated Ku-band signal in the time domain; (e) the calculated instantaneous frequency-time diagram of the Ku-band signal; (f) a vertical-section of the instantaneous frequency-time diagram of the Ku-band signal.
Fig. 7.
Fig. 7. (a) Result of the X-band signal after digital autocorrelation; (b) Result of the Ku-band signal after digital autocorrelation; the insets show the peak of the compressed transmission signals.
Fig. 8.
Fig. 8. Photograph of the experimental setup.
Fig. 9.
Fig. 9. Schematic diagram of the experimental test scenario and photograph of the target.
Fig. 10.
Fig. 10. (a) Measured TCR’s horizontal range variation with respect to time; inset shows the de-chirped result of the echo at a certain moment); (b) measured TCR’s horizontal velocity variation with respect to time (yellow curve); a simulation result of the TCR’s horizontal velocity variation with respect to time (white curve).

Equations (5)

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E P A ( t ) E 1 exp ( j ω c t + j π 2 ) × cos ( ω e t )
E MZM ( t ) E 2 exp ( j ω c t ) × cos ( ω e t ) × cos ( ω 0 t + π k t 2 ) ,   T / 2 < t < T / 2
V PD ( t ) cos [ 2 ω e t 2( ω 0 t + π k t 2 ) ] + 2 cos [ 2 ( ω 0 t + π k t 2 ) ] + cos [ 2 ω e t + 2( ω 0 t + π k t 2 ) ] + 2 cos ( 2 ω e t )
ω e ω 0 + π k T / 2 < ω 0 π k T / 2 ω 0 + π k T / 2 < ω e ω e < ω e + ω 0 π k T / 2
ω 0 + π k T / 2 < ω e < 2 ω 0 π k T ω 0 > π k T / 2

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