Abstract

A photonics-based multiple-input-multiple-output (MIMO) radar is proposed and demonstrated based on wavelength-division-multiplexed broadband microwave photonic signal generation and processing. The proposed radar has a large operation bandwidth, which helps to achieve an ultra-high range resolution. Compared with a monostatic radar, improved radar performance and extended radar applications originated from the MIMO architecture can be achieved. In addition, low-speed electronics with real-time signal processing capability is feasible. A photonics-based 2 × 2 MIMO radar is established with a 4-GHz bandwidth in each transmitter and a sampling rate of 100 MSa/s in the receiver. Performance of the photonics-based multi-channel signal generation and processing is evaluated, and an experiment for direction of arrival (DOA) estimation and target positioning is demonstrated, through which the feasibility of the proposed radar system can be verified.

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

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References

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  5. W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
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  18. J. Li and P. Stoica, “MIMO radar with collocated antennas,” IEEE Signal Process. Mag. 24(5), 106–114 (2007).
    [Crossref]
  19. Q. Guo, F. Zhang, P. Zhou, and S. Pan, “Dual-band LFM signal generation by frequency quadrupling and polarization multiplexing,” IEEE Photonics Technol. Lett. 29(16), 1320–1323 (2017).
    [Crossref]
  20. H. Messer, G. Signal, and L. Bialy, “On the achievable DF accuracy of two kinds of active interferometers,” IEEE Trans. Aerosp. Electron. Syst. 32(3), 1158–1164 (1996).
    [Crossref]
  21. S. Rao, “MIMO Radar,” Texas Instruments Application ReportSWRA554, 1–12 (2017).
  22. K. Xu, X. Sun, J. Yin, H. Huang, J. Wu, X. Hong, and J. Lin, “Enabling ROF technologies and integration architectures for in-building optical–wireless access networks,” IEEE Photonics J. 2(2), 102–112 (2010).
    [Crossref]
  23. D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
    [Crossref]

2018 (2)

2017 (6)

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), 16274–16281 (2017).
[Crossref] [PubMed]

F. Zhang, Q. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Zhu, and S. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 112801 (2017).
[Crossref]

O. Aldayel, V. Monga, and M. Rangaswamy, “Tractable transmit MIMO beampattern design under a constant modulus constraint,” IEEE Trans. Signal Process. 65(10), 2588–2599 (2017).
[Crossref]

F. Zhang, Q. Guo, and S. Pan, “Photonics-based real-time ultra-high-range-resolution radar with broadband signal generation and processing,” Sci. Rep. 7(1), 13848 (2017).
[Crossref] [PubMed]

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] [PubMed]

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

2016 (2)

H. Yang and J. Chun, “An improved algebraic solution for moving target localization in noncoherent MIMO radar systems,” IEEE Trans. Signal Process. 64(1), 258–270 (2016).
[Crossref]

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

2015 (2)

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (2015).
[Crossref]

T. Yao, D. Zhu, D. Ben, and S. Pan, “Distributed MIMO chaotic radar based on wavelength-division multiplexing technology,” Opt. Lett. 40(8), 1631–1634 (2015).
[Crossref] [PubMed]

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] [PubMed]

C. Ma, T. S. Yeo, Y. Zhao, and J. Feng, “MIMO radar 3D imaging based on combined amplitude and total variation cost function with sequential order one negative exponential form,” IEEE Trans. Image Process. 23(5), 2168–2183 (2014).
[Crossref] [PubMed]

2013 (1)

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

2012 (1)

D. Grodensky, D. Kravitz, and A. Zadok, “Ultra-wideband microwave-photonic noise radar based on optical waveform generation,” IEEE Photonics Technol. Lett. 24(10), 839–841 (2012).

2010 (1)

K. Xu, X. Sun, J. Yin, H. Huang, J. Wu, X. Hong, and J. Lin, “Enabling ROF technologies and integration architectures for in-building optical–wireless access networks,” IEEE Photonics J. 2(2), 102–112 (2010).
[Crossref]

2007 (1)

J. Li and P. Stoica, “MIMO radar with collocated antennas,” IEEE Signal Process. Mag. 24(5), 106–114 (2007).
[Crossref]

2006 (1)

I. Bekkerman and J. Tabrikian, “Target detection and localization using MIMO radars and sonars,” IEEE Trans. Signal Process. 54(10), 3873–3883 (2006).
[Crossref]

1996 (1)

H. Messer, G. Signal, and L. Bialy, “On the achievable DF accuracy of two kinds of active interferometers,” IEEE Trans. Aerosp. Electron. Syst. 32(3), 1158–1164 (1996).
[Crossref]

Aldayel, O.

O. Aldayel, V. Monga, and M. Rangaswamy, “Tractable transmit MIMO beampattern design under a constant modulus constraint,” IEEE Trans. Signal Process. 65(10), 2588–2599 (2017).
[Crossref]

Bekkerman, I.

I. Bekkerman and J. Tabrikian, “Target detection and localization using MIMO radars and sonars,” IEEE Trans. Signal Process. 54(10), 3873–3883 (2006).
[Crossref]

Ben, D.

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] [PubMed]

Bialy, L.

H. Messer, G. Signal, and L. Bialy, “On the achievable DF accuracy of two kinds of active interferometers,” IEEE Trans. Aerosp. Electron. Syst. 32(3), 1158–1164 (1996).
[Crossref]

Bogoni, A.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

Capmany, J.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[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] [PubMed]

Chen, J.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Cheng, R.

Chun, J.

H. Yang and J. Chun, “An improved algebraic solution for moving target localization in noncoherent MIMO radar systems,” IEEE Trans. Signal Process. 64(1), 258–270 (2016).
[Crossref]

Cui, Y.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Ding, M.

Dong, Y.

Feng, J.

C. Ma, T. S. Yeo, Y. Zhao, and J. Feng, “MIMO radar 3D imaging based on combined amplitude and total variation cost function with sequential order one negative exponential form,” IEEE Trans. Image Process. 23(5), 2168–2183 (2014).
[Crossref] [PubMed]

Gao, B.

Ghelfi, P.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

Grodensky, D.

D. Grodensky, D. Kravitz, and A. Zadok, “Ultra-wideband microwave-photonic noise radar based on optical waveform generation,” IEEE Photonics Technol. Lett. 24(10), 839–841 (2012).

Guo, P.

Guo, Q.

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), 16274–16281 (2017).
[Crossref] [PubMed]

F. Zhang, Q. Guo, and S. Pan, “Photonics-based real-time ultra-high-range-resolution radar with broadband signal generation and processing,” Sci. Rep. 7(1), 13848 (2017).
[Crossref] [PubMed]

F. Zhang, Q. Guo, Y. Zhang, Y. Yao, P. Zhou, D. Zhu, and S. Pan, “Photonics-based real-time and high-resolution ISAR imaging of non-cooperative target,” Chin. Opt. Lett. 15(11), 112801 (2017).
[Crossref]

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

Han, D.

Heideman, R.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Hong, X.

K. Xu, X. Sun, J. Yin, H. Huang, J. Wu, X. Hong, and J. Lin, “Enabling ROF technologies and integration architectures for in-building optical–wireless access networks,” IEEE Photonics J. 2(2), 102–112 (2010).
[Crossref]

Huang, H.

K. Xu, X. Sun, J. Yin, H. Huang, J. Wu, X. Hong, and J. Lin, “Enabling ROF technologies and integration architectures for in-building optical–wireless access networks,” IEEE Photonics J. 2(2), 102–112 (2010).
[Crossref]

Kravitz, D.

D. Grodensky, D. Kravitz, and A. Zadok, “Ultra-wideband microwave-photonic noise radar based on optical waveform generation,” IEEE Photonics Technol. Lett. 24(10), 839–841 (2012).

Laghezza, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

Lazzeri, E.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

Leinse, A.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Li, J.

J. Li and P. Stoica, “MIMO radar with collocated antennas,” IEEE Signal Process. Mag. 24(5), 106–114 (2007).
[Crossref]

Li, R.

Li, S.

Li, W.

Li, Y.

Liang, X.

Lin, J.

K. Xu, X. Sun, J. Yin, H. Huang, J. Wu, X. Hong, and J. Lin, “Enabling ROF technologies and integration architectures for in-building optical–wireless access networks,” IEEE Photonics J. 2(2), 102–112 (2010).
[Crossref]

Liu, Z.

Long, X.

W. Zou, H. Zhang, X. Long, S. Zhang, Y. Cui, and J. Chen, “All-optical central-frequency-programmable and bandwidth-tailorable radar,” Sci. Rep. 6(1), 19786 (2016).
[Crossref] [PubMed]

Luan, Y.

Ma, C.

C. Ma, T. S. Yeo, Y. Zhao, and J. Feng, “MIMO radar 3D imaging based on combined amplitude and total variation cost function with sequential order one negative exponential form,” IEEE Trans. Image Process. 23(5), 2168–2183 (2014).
[Crossref] [PubMed]

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] [PubMed]

Marpaung, D.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Messer, H.

H. Messer, G. Signal, and L. Bialy, “On the achievable DF accuracy of two kinds of active interferometers,” IEEE Trans. Aerosp. Electron. Syst. 32(3), 1158–1164 (1996).
[Crossref]

Monga, V.

O. Aldayel, V. Monga, and M. Rangaswamy, “Tractable transmit MIMO beampattern design under a constant modulus constraint,” IEEE Trans. Signal Process. 65(10), 2588–2599 (2017).
[Crossref]

Onori, D.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

Pan, S.

Peng, S.

Pinna, S.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

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] [PubMed]

Qin, J.

Rangaswamy, M.

O. Aldayel, V. Monga, and M. Rangaswamy, “Tractable transmit MIMO beampattern design under a constant modulus constraint,” IEEE Trans. Signal Process. 65(10), 2588–2599 (2017).
[Crossref]

Rao, S.

S. Rao, “MIMO Radar,” Texas Instruments Application ReportSWRA554, 1–12 (2017).

Roeloffzen, C.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[Crossref]

Sales, S.

D. Marpaung, C. Roeloffzen, R. Heideman, A. Leinse, S. Sales, and J. Capmany, “Integrated microwave photonics,” Laser Photonics Rev. 7(4), 506–538 (2013).
[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] [PubMed]

Scotti, F.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

Serafino, G.

P. Ghelfi, F. Laghezza, F. Scotti, G. Serafino, S. Pinna, D. Onori, E. Lazzeri, and A. Bogoni, “Photonics in radar systems,” IEEE Microw. Mag. 16(8), 74–83 (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] [PubMed]

Signal, G.

H. Messer, G. Signal, and L. Bialy, “On the achievable DF accuracy of two kinds of active interferometers,” IEEE Trans. Aerosp. Electron. Syst. 32(3), 1158–1164 (1996).
[Crossref]

Stoica, P.

J. Li and P. Stoica, “MIMO radar with collocated antennas,” IEEE Signal Process. Mag. 24(5), 106–114 (2007).
[Crossref]

Sun, J.

Sun, X.

K. Xu, X. Sun, J. Yin, H. Huang, J. Wu, X. Hong, and J. Lin, “Enabling ROF technologies and integration architectures for in-building optical–wireless access networks,” IEEE Photonics J. 2(2), 102–112 (2010).
[Crossref]

Tabrikian, J.

I. Bekkerman and J. Tabrikian, “Target detection and localization using MIMO radars and sonars,” IEEE Trans. Signal Process. 54(10), 3873–3883 (2006).
[Crossref]

Tian, Y.

Tong, Y.

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Figures (10)
Fig. 1
Fig. 1 Schematic diagram of the proposed photonics-based M × N MIMO radar. LD: laser diode; OC: optical coupler; DPMZM: dual-parallel Mach-Zehnder modulator; WDM: wavelength division multiplexer; EDFA: erbium-doped fiber amplifier; PD: photodetector; EA: electrical amplifier; PA: power amplifier; LNA: low noise amplifier; MZM: Mach-Zehnder modulator; LPF: electrical low-pass filter; ADC: analog-to-digital converter.

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Fig. 2
Fig. 2 Illustration of the optical spectrum in the mth output channel of the WDM in a receiver.

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Fig. 3
Fig. 3 Experimental setup of the photonics-based 2 × 2 MIMO radar.

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Fig. 4
Fig. 4 Spectra of the frequency quadrupling modulated optical signal after the WDM (measured at point a in Fig. 3).

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Fig. 5
Fig. 5 (a) Waveform and (b) spectrum of the generated LFM signal in 17.5-21.5 GHz (RBW = 50 kHz, measured at point b in Fig. 3); (c) waveform and (d) spectrum of the generated LFM signal in 22-26 GHz (RBW = 50 kHz, measured at point c in Fig. 3).

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Fig. 6
Fig. 6 Spectra of the de-multiplexed two optical signals after the WDM in the receiver: (a) the 1550.92 nm channel and (b) the 1552.52 nm channel (measured at point d and point e in Fig. 3, respectively).

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Fig. 7
Fig. 7 (a) Waveform and (b) power spectrum of the de-chirped signal S11, (c) waveform and (d) power spectrum of the de-chirped signal S21.

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Fig. 8
Fig. 8 (a) Picture of the antennas and targets in the experiment, (b) power spectrum of the de-chirped signal S11.

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Fig. 9
Fig. 9 Illustration of the antennas and target in the Cartesian coordinate when performing the DOA estimation and positioning.

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Fig. 10
Fig. 10 Calculated power spectra of (a) signal S11, (b) signal S12, (c) signal S21, and (d) signal S22.

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Equations (6)

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

f IF_ m ( t ) = u m + k t ( 0 ≤ t ≤ T )
f LFM_ m ( t ) = 4 u m + 4 k t ( 0 ≤ t ≤ T )
4 ( u m + 1 − u m ) > > Δ f
f LFM _ m ( t ) − 4 ( u m + 1 − u m ) > > Δ f
| T 1 T 2 | sin ( α ) = | T 1 T | − | T 2 T |
{ | T 1 T | + | T R 2 | = y cos ( α ) + y cos ( β ) x y = tan ( α )

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