Abstract

A cross-referenced and deadband-free method with photonic harmonic down-conversion is proposed for microwave frequency measurement based on cascaded-four-wave-mixing (CFWM) in semiconductor optical amplifiers. The proposed method enables ultra-wide and accurate frequency measurement with low-frequency spectrum detection, and at the same time achieves deadband-free and multi-tone frequency measurement by cross-referenced frequency discrimination. For a proof of concept, microwave signal measurement is experimentally demonstrated up to the 40 GHz frequency range with an 0.2-MHz measurement error. The frequency measurement features ease of configuration by only changing the low-frequency electrical local oscillators of the CFWM-based photonic harmonic down-converter.

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

Full Article  |  PDF Article
OSA Recommended Articles
Broadband high-resolution microwave frequency measurement based on low-speed photonic analog-to-digital converters

Yangxue Ma, Dong Liang, Di Peng, Zhiyao Zhang, Yali Zhang, Shangjian Zhang, and Yong Liu
Opt. Express 25(3) 2355-2368 (2017)

Ultra-broadband microwave frequency down-conversion based on optical frequency comb

Xiao Fang, Ming Bai, Xiuzhu Ye, Jungang Miao, and Zheng Zheng
Opt. Express 23(13) 17111-17119 (2015)

Optical frequency comb based multi-band microwave frequency conversion for satellite applications

Xinwu Yang, Kun Xu, Jie Yin, Yitang Dai, Feifei Yin, Jianqiang Li, Hua Lu, Tao Liu, and Yuefeng Ji
Opt. Express 22(1) 869-877 (2014)

References

  • View by:
  • |
  • |
  • |

  1. J. Capmany and D. Novak, “Microwave photonics combines two worlds,” Nat. Photonics 1(6), 319–330 (2007).
    [Crossref]
  2. S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
    [Crossref]
  3. J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
    [Crossref]
  4. S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. 35(16), 3498–3513 (2017).
    [Crossref]
  5. X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
    [Crossref]
  6. X. H. Zou, H. Chi, and J. P. Yao, “Microwave frequency measurement based on optical power monitoring using a complementary optical filter pair,” IEEE Trans. Microw. Theory Tech. 57(2), 505–511 (2009).
    [Crossref]
  7. J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
    [Crossref]
  8. S. Fu, M. Tang, and P. Shum, “Instantaneous microwave frequency measurement using optical carrier suppression based DC power monitoring,” Opt. Express 19(24), 24712–24717 (2011).
    [Crossref] [PubMed]
  9. S. Pan, J. Fu, and J. Yao, “Photonic approach to the simultaneous measurement of the frequency, amplitude, pulse width, and time of arrival of a microwave signal,” Opt. Lett. 37(1), 7–9 (2012).
    [Crossref] [PubMed]
  10. M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
    [Crossref] [PubMed]
  11. X. H. Zou, S. L. Pan, and J. P. Yao, “Instantaneous microwave frequency measurement with improved measurement range and resolution based on simultaneous phase modulation and intensity modulation,” J. Lightwave Technol. 27(23), 5314–5320 (2009).
    [Crossref]
  12. S. L. Pan and J. P. Yao, “Instantaneous microwave frequency measurement using a photonic microwave filter pair,” IEEE Photonics Technol. Lett. 22(19), 1437–1439 (2010).
    [Crossref]
  13. D. Marpaung, “On-chip photonic-assisted instantaneous microwave frequency measurement system,” IEEE Photonics Technol. Lett. 25(9), 837–840 (2013).
    [Crossref]
  14. H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
    [Crossref]
  15. Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
    [Crossref]
  16. H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Amplitude independent RF instantaneous frequency measurement system using photonic Hilbert transform,” Opt. Express 16(18), 13707–13712 (2008).
    [Crossref] [PubMed]
  17. H. Emami, N. Sarkhosh, and M. Ashourian, “Reduced cost amplitude independent photonic RF frequency measurement system,” IEEE Microw. Wirel. Compon. Lett. 23(11), 617–619 (2013).
    [Crossref]
  18. M. Pagani, B. Morrison, Y. Zhang, A. Casas-Bedoya, T. Aalto, M. Harjanne, M. Kapulainen, B. J. Eggleton, and D. Marpaung, “Low-error and broadband microwave frequency measurement in a silicon chip,” Optica 2(8), 751–756 (2015).
    [Crossref]
  19. H. Emami and M. Ashourian, “Improved dynamic range microwave photonic instantaneous frequency measurement based on four-wave mixing,” IEEE Trans. Microw. Theory Tech. 62(10), 2462–2470 (2014).
    [Crossref]
  20. B. W. Zhang, X. C. Wang, and S. L. Pan, “Photonics-based instantaneous multi-parameter measurement of a linear frequency modulation microwave signal,” J. Lightwave Technol. 36(13), 2589–2596 (2018).
    [Crossref]
  21. T. Yasui, K. Hayashi, R. Ichikawa, H. Cahyadi, Y. D. Hsieh, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, and K. Minoshima, “Real-time absolute frequency measurement of continuous-wave terahertz radiation based on dual terahertz combs of photocarriers with different frequency spacings,” Opt. Express 23(9), 11367–11377 (2015).
    [Crossref] [PubMed]
  22. G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
    [Crossref] [PubMed]
  23. Y. Ma, D. Liang, D. Peng, Z. Zhang, Y. Zhang, S. Zhang, and Y. Liu, “Broadband high-resolution microwave frequency measurement based on low-speed photonic analog-to-digital converters,” Opt. Express 25(3), 2355–2368 (2017).
    [Crossref] [PubMed]
  24. X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
    [Crossref]
  25. S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).
  26. J. X. Ma, J. J. Yu, C. X. Yu, Z. S. Jia, X. Z. Sang, Z. Zhou, T. Wang, and G. K. Chang, “Wavelength conversion based on four-wave mixing in high-nonlinear dispersion shifted fiber using a cross-pump configuration,” J. Lightwave Technol. 24(7), 2851–2858 (2006).
    [Crossref]
  27. Z. Tong, A. O. J. Wiberg, E. Myslivets, B. P. P. Kuo, N. Alic, and S. Radic, “Spectral linewidth preservation in parametric frequency combs seeded by dual pumps,” Opt. Express 20(16), 17610–17619 (2012).
    [Crossref] [PubMed]
  28. X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
    [Crossref]

2018 (4)

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

B. W. Zhang, X. C. Wang, and S. L. Pan, “Photonics-based instantaneous multi-parameter measurement of a linear frequency modulation microwave signal,” J. Lightwave Technol. 36(13), 2589–2596 (2018).
[Crossref]

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

2017 (3)

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Y. Ma, D. Liang, D. Peng, Z. Zhang, Y. Zhang, S. Zhang, and Y. Liu, “Broadband high-resolution microwave frequency measurement based on low-speed photonic analog-to-digital converters,” Opt. Express 25(3), 2355–2368 (2017).
[Crossref] [PubMed]

S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. 35(16), 3498–3513 (2017).
[Crossref]

2016 (3)

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
[Crossref] [PubMed]

2015 (2)

2014 (2)

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

H. Emami and M. Ashourian, “Improved dynamic range microwave photonic instantaneous frequency measurement based on four-wave mixing,” IEEE Trans. Microw. Theory Tech. 62(10), 2462–2470 (2014).
[Crossref]

2013 (2)

D. Marpaung, “On-chip photonic-assisted instantaneous microwave frequency measurement system,” IEEE Photonics Technol. Lett. 25(9), 837–840 (2013).
[Crossref]

H. Emami, N. Sarkhosh, and M. Ashourian, “Reduced cost amplitude independent photonic RF frequency measurement system,” IEEE Microw. Wirel. Compon. Lett. 23(11), 617–619 (2013).
[Crossref]

2012 (2)

2011 (1)

2010 (2)

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

S. L. Pan and J. P. Yao, “Instantaneous microwave frequency measurement using a photonic microwave filter pair,” IEEE Photonics Technol. Lett. 22(19), 1437–1439 (2010).
[Crossref]

2009 (3)

2008 (1)

2007 (1)

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

2006 (2)

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

J. X. Ma, J. J. Yu, C. X. Yu, Z. S. Jia, X. Z. Sang, Z. Zhou, T. Wang, and G. K. Chang, “Wavelength conversion based on four-wave mixing in high-nonlinear dispersion shifted fiber using a cross-pump configuration,” J. Lightwave Technol. 24(7), 2851–2858 (2006).
[Crossref]

Aalto, T.

Alic, N.

Ashourian, M.

H. Emami and M. Ashourian, “Improved dynamic range microwave photonic instantaneous frequency measurement based on four-wave mixing,” IEEE Trans. Microw. Theory Tech. 62(10), 2462–2470 (2014).
[Crossref]

H. Emami, N. Sarkhosh, and M. Ashourian, “Reduced cost amplitude independent photonic RF frequency measurement system,” IEEE Microw. Wirel. Compon. Lett. 23(11), 617–619 (2013).
[Crossref]

Azaña, J.

M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
[Crossref] [PubMed]

Bai, M.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Bui, L. A.

Burla, M.

M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
[Crossref] [PubMed]

Cahyadi, H.

Capmany, J.

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

Casas-Bedoya, A.

Chang, G. K.

Chazelas, J.

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

Chen, M.

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

Chen, W.

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

Chi, H.

X. H. Zou, H. Chi, and J. P. Yao, “Microwave frequency measurement based on optical power monitoring using a complementary optical filter pair,” IEEE Trans. Microw. Theory Tech. 57(2), 505–511 (2009).
[Crossref]

Chrostowski, L.

M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
[Crossref] [PubMed]

Dai, J.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Dolfi, D.

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

Eggleton, B. J.

Emami, H.

H. Emami and M. Ashourian, “Improved dynamic range microwave photonic instantaneous frequency measurement based on four-wave mixing,” IEEE Trans. Microw. Theory Tech. 62(10), 2462–2470 (2014).
[Crossref]

H. Emami, N. Sarkhosh, and M. Ashourian, “Reduced cost amplitude independent photonic RF frequency measurement system,” IEEE Microw. Wirel. Compon. Lett. 23(11), 617–619 (2013).
[Crossref]

H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Amplitude independent RF instantaneous frequency measurement system using photonic Hilbert transform,” Opt. Express 16(18), 13707–13712 (2008).
[Crossref] [PubMed]

Formont, S.

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

Fu, J.

Fu, S.

Gao, Y. S.

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

Harjanne, M.

Hayashi, K.

Hong, X. B.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Hsieh, Y. D.

Hu, G.

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Hu, G. Q.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

Huignard, J. P.

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

Ichikawa, R.

Inaba, H.

Iwata, T.

Jia, Z. S.

Kapulainen, M.

Kuo, B. P. P.

Li, C.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Li, J. J.

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

Li, M.

M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
[Crossref] [PubMed]

Li, R. X.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

Li, T.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

Li, W.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Liang, D.

Lin, J. T.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Liu, S.

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

Liu, Y.

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

Y. Ma, D. Liang, D. Peng, Z. Zhang, Y. Zhang, S. Zhang, and Y. Liu, “Broadband high-resolution microwave frequency measurement based on low-speed photonic analog-to-digital converters,” Opt. Express 25(3), 2355–2368 (2017).
[Crossref] [PubMed]

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

Lu, B.

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

Lu, R. G.

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

Lv, Q.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Ma, J. X.

Ma, Y.

Marpaung, D.

Minamikawa, T.

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Minoshima, K.

Mitchell, A.

Mizuguchi, T.

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Mizuno, T.

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Mizutani, Y.

Monsterleet, A.

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

Morrison, B.

Myslivets, E.

Niu, J.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Novak, D.

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

Pagani, M.

Pan, S.

Pan, S. L.

Pan, W.

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

Peng, D.

Peng, Z. X.

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

Radic, S.

Sang, X. Z.

Sarkhosh, N.

H. Emami, N. Sarkhosh, and M. Ashourian, “Reduced cost amplitude independent photonic RF frequency measurement system,” IEEE Microw. Wirel. Compon. Lett. 23(11), 617–619 (2013).
[Crossref]

H. Emami, N. Sarkhosh, L. A. Bui, and A. Mitchell, “Amplitude independent RF instantaneous frequency measurement system using photonic Hilbert transform,” Opt. Express 16(18), 13707–13712 (2008).
[Crossref] [PubMed]

Shum, P.

Stöhr, A.

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

Sun, X. Q.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Tang, M.

Tonda-Goldstein, S.

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

Tong, Z.

Tu, Z. Y.

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

Wang, H.

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

Wang, T.

Wang, X.

M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
[Crossref] [PubMed]

Wang, X. C.

Wen, A. J.

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

Wiberg, A. O. J.

Wu, J.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Xu, K.

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

Yamamoto, H.

Yan, L. S.

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

Yang, Y.

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Yao, J.

Yao, J. P.

S. L. Pan and J. P. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. 35(16), 3498–3513 (2017).
[Crossref]

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

S. L. Pan and J. P. Yao, “Instantaneous microwave frequency measurement using a photonic microwave filter pair,” IEEE Photonics Technol. Lett. 22(19), 1437–1439 (2010).
[Crossref]

X. H. Zou, S. L. Pan, and J. P. Yao, “Instantaneous microwave frequency measurement with improved measurement range and resolution based on simultaneous phase modulation and intensity modulation,” J. Lightwave Technol. 27(23), 5314–5320 (2009).
[Crossref]

X. H. Zou, H. Chi, and J. P. Yao, “Microwave frequency measurement based on optical power monitoring using a complementary optical filter pair,” IEEE Trans. Microw. Theory Tech. 57(2), 505–511 (2009).
[Crossref]

J. P. Yao, “Microwave photonics,” J. Lightwave Technol. 27(3), 314–335 (2009).
[Crossref]

Yasui, T.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

T. Yasui, K. Hayashi, R. Ichikawa, H. Cahyadi, Y. D. Hsieh, Y. Mizutani, H. Yamamoto, T. Iwata, H. Inaba, and K. Minoshima, “Real-time absolute frequency measurement of continuous-wave terahertz radiation based on dual terahertz combs of photocarriers with different frequency spacings,” Opt. Express 23(9), 11367–11377 (2015).
[Crossref] [PubMed]

Yu, C. X.

Yu, J. J.

Zhang, B. W.

Zhang, S.

Zhang, S. J.

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

Zhang, Y.

Zhang, Y. L.

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

Zhang, Z.

Zhang, Z. Y.

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

Zhao, X.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Zheng, Z.

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Zhou, Z.

Zou, X. H.

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

X. H. Zou, H. Chi, and J. P. Yao, “Microwave frequency measurement based on optical power monitoring using a complementary optical filter pair,” IEEE Trans. Microw. Theory Tech. 57(2), 505–511 (2009).
[Crossref]

X. H. Zou, S. L. Pan, and J. P. Yao, “Instantaneous microwave frequency measurement with improved measurement range and resolution based on simultaneous phase modulation and intensity modulation,” J. Lightwave Technol. 27(23), 5314–5320 (2009).
[Crossref]

IEEE J. Sel. Top. Quant. (1)

X. Zhao, C. Li, T. Li, G. Q. Hu, R. X. Li, M. Bai, T. Yasui, and Z. Zheng, “Dead-band-free, high-resolution microwave frequency measurement using a free-running triple-comb fiber laser,” IEEE J. Sel. Top. Quant. 24(3), 1101008 (2018).
[Crossref]

IEEE Microw. Wirel. Compon. Lett. (1)

H. Emami, N. Sarkhosh, and M. Ashourian, “Reduced cost amplitude independent photonic RF frequency measurement system,” IEEE Microw. Wirel. Compon. Lett. 23(11), 617–619 (2013).
[Crossref]

IEEE Photonics J. (2)

S. J. Zhang, H. Wang, X. H. Zou, Y. L. Zhang, R. G. Lu, and Y. Liu, “Calibration-free electrical spectrum analysis for microwave characterization of optical phase modulators using frequency-shifted heterodyning,” IEEE Photonics J. 6(4), 5501008 (2014).

X. H. Zou, S. J. Zhang, H. Wang, Z. Y. Zhang, J. J. Li, Y. L. Zhang, S. Liu, and Y. Liu, “Microwave photonic harmonic down-conversion based on cascaded four-wave mixing in a semiconductor optical amplifier,” IEEE Photonics J. 10(1), 5500308 (2018).
[Crossref]

IEEE Photonics Technol. Lett. (5)

S. L. Pan and J. P. Yao, “Instantaneous microwave frequency measurement using a photonic microwave filter pair,” IEEE Photonics Technol. Lett. 22(19), 1437–1439 (2010).
[Crossref]

D. Marpaung, “On-chip photonic-assisted instantaneous microwave frequency measurement system,” IEEE Photonics Technol. Lett. 25(9), 837–840 (2013).
[Crossref]

H. Wang, S. J. Zhang, X. H. Zou, Z. Y. Zhang, Y. L. Zhang, and Y. Liu, “Photonic microwave frequency measurement based on frequency-configurable pilot tones,” IEEE Photonics Technol. Lett. 30(4), 363–366 (2018).
[Crossref]

Z. Y. Tu, A. J. Wen, Y. S. Gao, W. Chen, Z. X. Peng, and M. Chen, “A photonic technique for instantaneous microwave frequency measurement utilizing a phase modulator,” IEEE Photonics Technol. Lett. 28(24), 2795–2798 (2016).
[Crossref]

J. Dai, K. Xu, X. Q. Sun, J. Niu, Q. Lv, J. Wu, X. B. Hong, W. Li, and J. T. Lin, “A simple photonic-assisted microwave frequency measurement system based on MZI with tunable measurement range and high resolution,” IEEE Photonics Technol. Lett. 22(15), 1162–1164 (2010).
[Crossref]

IEEE Trans. Microw. Theory Tech. (3)

X. H. Zou, H. Chi, and J. P. Yao, “Microwave frequency measurement based on optical power monitoring using a complementary optical filter pair,” IEEE Trans. Microw. Theory Tech. 57(2), 505–511 (2009).
[Crossref]

S. Tonda-Goldstein, D. Dolfi, A. Monsterleet, S. Formont, J. Chazelas, and J. P. Huignard, “Optical signal processing in radar systems,” IEEE Trans. Microw. Theory Tech. 54(2), 847–853 (2006).
[Crossref]

H. Emami and M. Ashourian, “Improved dynamic range microwave photonic instantaneous frequency measurement based on four-wave mixing,” IEEE Trans. Microw. Theory Tech. 62(10), 2462–2470 (2014).
[Crossref]

J. Lightwave Technol. (5)

Laser Photonics Rev. (1)

X. H. Zou, B. Lu, W. Pan, L. S. Yan, A. Stöhr, and J. P. Yao, “Photonics for microwave measurements,” Laser Photonics Rev. 10(5), 711–734 (2016).
[Crossref]

Nat. Commun. (1)

M. Burla, X. Wang, M. Li, L. Chrostowski, and J. Azaña, “Wideband dynamic microwave frequency identification system using a low-power ultracompact silicon photonic chip,” Nat. Commun. 7(1), 13004 (2016).
[Crossref] [PubMed]

Nat. Photonics (1)

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

Opt. Express (5)

Opt. Lett. (1)

Optica (1)

Sci. Rep. (1)

G. Hu, T. Mizuguchi, X. Zhao, T. Minamikawa, T. Mizuno, Y. Yang, C. Li, M. Bai, Z. Zheng, and T. Yasui, “Measurement of absolute frequency of continuous-wave terahertz radiation in real time using a free-running, dual-wavelength mode-locked, erbium-doped fibre laser,” Sci. Rep. 7(1), 42082 (2017).
[Crossref] [PubMed]

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1 Schematic diagram of the proposed frequency measurement method and relative positions of fs, nfr1, nfr2, (n + 1)fr1 and (n + 1)fr2. LD, laser diode; OS, optical splitter; OC, optical coupler; IM, intensity modulator; SOA: semiconductor optical amplifier; OHI, optical harmonic intensifier; BPD, balanced photodetector; ADC, analog to digital converter; DSP, digital signal processing.
Fig. 2
Fig. 2 Flow chart of microwave frequency discrimination algorithm.
Fig. 3
Fig. 3 Measured optical spectra of intensified optical harmonic signals in the cases of different LO frequencies.
Fig. 4
Fig. 4 Spectra of harmonic down-conversion tones for a two-tone microwave signal under test.
Fig. 5
Fig. 5 Measured fundamental down-converted tone and third-order intermodulation distortions (IMD3) as a function of input RF power.
Fig. 6
Fig. 6 Measured RF frequency and error versus the input RF frequency.

Tables (2)

Tables Icon

Table 1 Frequency Measurement Results of Two-Tone Microwave Signal

Tables Icon

Table 2 Performance Comparison between Electrical Spectrum Analysis and Our Work

Equations (9)

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

E 1 ( t )= e j2π f 0 t q= N 1 + N 1 B q e j2πq f r1 t+j ξ q
E 1 ( n f r1 ;t )= B n e j2π( f 0 +n f r1 )t+j ξ n + p= + χ( Δ f pq )[ B p e j2π( f 0 +q f r1 )t+j ξ p ] [ B q e j2π( f 0 +q f r1 )t+j ξ q ] [ B p e j2π( f 0 +q f r1 )t+j ξ p ] =[ B n e j ξ n + p= + χ( Δ f pq ) B p 2 B q e j( 2 ξ p ξ q ) ] e j2π( f 0 +n f r1 )t
E C1 ( t )= n= N 2 + N 2 E 1 ( n f r1 ;t ) = n= N 2 + N 2 F n e j2π( f 0 +n f r1 )t+j ϕ n
E C2 ( t )= n= N 2 + N 2 F n e j2π( f 0 +n f r2 )t+j ϕ n
{ f a = f s n f r1 f b =( n+1 ) f r1 f s f a =n f r2 f s f b = f s ( n1 ) f r2
n=( f a + f a )/Δf
{ f a = f s n f r1 f b =( n+1 ) f r1 f s f a = f s n f r2 f b =( n+1 ) f r2 f s
n=( f a f a )/Δf=( f b f b )/Δf1
f s = f a +n f r1

Metrics