Abstract

A wideband microwave phase noise measurement system is proposed based on quadrature phase demodulation of the mixing components of a signal under test (SUT) and its delayed replica. The time delay is introduced by a low-loss optical fiber, which can be sufficiently large to achieve a high phase noise measurement sensitivity, and the quadrature phase demodulation is achieved by photonic-assisted in-phase and quadrate (I/Q) mixing together with digital signal processing. Thanks to the optoelectronic hybrid quadrature phase demodulation, the use of feedback loops, which are usually required in conventional photonic-delay-line-based phase noise measurement systems, is avoided, and the measurable frequency range is expanded. An experiment is implemented. Accurate phase noise measurement of SUTs in a frequency range of 5-35 GHz is demonstrated. With a 2-km single-mode fiber serving as the photonic delay line, the phase noise floor is as low as -131 dBc/Hz at the offset frequency of 10 kHz. The proposed scheme can be applied for evaluating the performance of microwave systems using low-phase-noise and wideband tunable microwave sources.

© 2017 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
High sensitivity microwave phase noise analyzer based on a phase locked optoelectronic oscillator

Huanfa Peng, Yongchi Xu, Rui Guo, Huayang Du, Jingbiao Chen, and Zhangyuan Chen
Opt. Express 27(13) 18910-18927 (2019)

Phase noise measurement of wideband microwave sources based on a microwave photonic frequency down-converter

Dengjian Zhu, Fangzheng Zhang, Pei Zhou, and Shilong Pan
Opt. Lett. 40(7) 1326-1329 (2015)

References

  • View by:
  • |
  • |
  • |

  1. D. B. Leeson, “Oscillator phase noise: a 50-year review,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63(8), 1208–1225 (2016).
    [Crossref] [PubMed]
  2. A. K. Poddar, U. L. Rohde, and A. M. Apte, “How low can they go?: oscillator phase noise model, theoretical, experimental validation, and phase noise measurements,” IEEE Microw. Mag. 14(6), 50–72 (2013).
    [Crossref]
  3. U. L. Rohde, A. K. Poddar, and A. M. Apte, “Getting its measure: oscillator phase noise measurement techniques and limitations,” IEEE Microw. Mag. 14(6), 73–86 (2013).
    [Crossref]
  4. E. Rubiola, E. Salik, S. Huang, N. Yu, and L. Maleki, “Photonic-delay technique for phase-noise measurement of microwave oscillators,” J. Opt. Soc. Am. B 22(5), 987–997 (2005).
    [Crossref]
  5. B. Onillon, S. Constant, and O. Liopis, “Optical links for ultra-low phase noise microwave oscillators measurement,” in IEEE International Frequency Control Symposium and Exposition, 545–550, (2005).
    [Crossref]
  6. P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
    [Crossref]
  7. S. Pan and J. Yao, “Photonics-based broadband microwave measurement,” J. Lightwave Technol. 35(16), 3498–3513 (2017).
    [Crossref]
  8. D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
    [Crossref]
  9. W. Wang, J. Liu, H. Mei, W. Sun, and N. Zhu, “Photonic-assisted wideband phase noise analyzer based on optoelectronic hybrid units,” J. Lightwave Technol. 34(14), 3425–3431 (2016).
    [Crossref]
  10. D. Zhu, F. Zhang, P. Zhou, and S. Pan, “Phase noise measurement of wideband microwave sources based on a microwave photonic frequency down-converter,” Opt. Lett. 40(7), 1326–1329 (2015).
    [Crossref] [PubMed]
  11. F. Zhang, D. Zhu, and S. Pan, “Photonic-assisted wideband phase noise measurement of microwave signal sources,” Electron. Lett. 51(16), 1272–1274 (2015).
    [Crossref]
  12. S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
    [Crossref]
  13. H. Gheidi and A. Banai, “Phase-noise measurement of microwave oscillators using phase-shifterless delay line discriminator,” IEEE Trans. Microw. Theory Tech. 58(2), 468–477 (2010).
    [Crossref]
  14. K. Volyanskiy, J. Cussey, H. Tavernier, P. Salzenstein, G. Sauvage, L. Larger, and E. Rubiola, “Applications of the optical fiber to the generation and measurement of low-phase-noise microwave signals,” J. Opt. Soc. Am. B 25(12), 2140–2150 (2008).
    [Crossref]
  15. E8257D PSG Microwave Analog Signal Generator Data Sheet, Keysight Technologies Co. USA, 17–20 (2016).

2017 (1)

2016 (2)

2015 (2)

F. Zhang, D. Zhu, and S. Pan, “Photonic-assisted wideband phase noise measurement of microwave signal sources,” Electron. Lett. 51(16), 1272–1274 (2015).
[Crossref]

D. Zhu, F. Zhang, P. Zhou, and S. Pan, “Phase noise measurement of wideband microwave sources based on a microwave photonic frequency down-converter,” Opt. Lett. 40(7), 1326–1329 (2015).
[Crossref] [PubMed]

2014 (1)

D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
[Crossref]

2013 (2)

A. K. Poddar, U. L. Rohde, and A. M. Apte, “How low can they go?: oscillator phase noise model, theoretical, experimental validation, and phase noise measurements,” IEEE Microw. Mag. 14(6), 50–72 (2013).
[Crossref]

U. L. Rohde, A. K. Poddar, and A. M. Apte, “Getting its measure: oscillator phase noise measurement techniques and limitations,” IEEE Microw. Mag. 14(6), 73–86 (2013).
[Crossref]

2012 (1)

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

2010 (1)

H. Gheidi and A. Banai, “Phase-noise measurement of microwave oscillators using phase-shifterless delay line discriminator,” IEEE Trans. Microw. Theory Tech. 58(2), 468–477 (2010).
[Crossref]

2008 (2)

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

K. Volyanskiy, J. Cussey, H. Tavernier, P. Salzenstein, G. Sauvage, L. Larger, and E. Rubiola, “Applications of the optical fiber to the generation and measurement of low-phase-noise microwave signals,” J. Opt. Soc. Am. B 25(12), 2140–2150 (2008).
[Crossref]

2005 (1)

Affes, S.

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

Apte, A. M.

A. K. Poddar, U. L. Rohde, and A. M. Apte, “How low can they go?: oscillator phase noise model, theoretical, experimental validation, and phase noise measurements,” IEEE Microw. Mag. 14(6), 50–72 (2013).
[Crossref]

U. L. Rohde, A. K. Poddar, and A. M. Apte, “Getting its measure: oscillator phase noise measurement techniques and limitations,” IEEE Microw. Mag. 14(6), 73–86 (2013).
[Crossref]

Banai, A.

H. Gheidi and A. Banai, “Phase-noise measurement of microwave oscillators using phase-shifterless delay line discriminator,” IEEE Trans. Microw. Theory Tech. 58(2), 468–477 (2010).
[Crossref]

Bosisio, R. G.

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

Boukari, B.

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

Chembo, Y.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

Cholley, N.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

Constant, S.

B. Onillon, S. Constant, and O. Liopis, “Optical links for ultra-low phase noise microwave oscillators measurement,” in IEEE International Frequency Control Symposium and Exposition, 545–550, (2005).
[Crossref]

Cussey, J.

Galliou, S.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

Gheidi, H.

H. Gheidi and A. Banai, “Phase-noise measurement of microwave oscillators using phase-shifterless delay line discriminator,” IEEE Trans. Microw. Theory Tech. 58(2), 468–477 (2010).
[Crossref]

Hmima, A.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

Huang, S.

Larger, L.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

K. Volyanskiy, J. Cussey, H. Tavernier, P. Salzenstein, G. Sauvage, L. Larger, and E. Rubiola, “Applications of the optical fiber to the generation and measurement of low-phase-noise microwave signals,” J. Opt. Soc. Am. B 25(12), 2140–2150 (2008).
[Crossref]

Leeson, D. B.

D. B. Leeson, “Oscillator phase noise: a 50-year review,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63(8), 1208–1225 (2016).
[Crossref] [PubMed]

Liopis, O.

B. Onillon, S. Constant, and O. Liopis, “Optical links for ultra-low phase noise microwave oscillators measurement,” in IEEE International Frequency Control Symposium and Exposition, 545–550, (2005).
[Crossref]

Liu, J.

Maleki, L.

Mei, H.

Moldovan, E.

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

Onillon, B.

B. Onillon, S. Constant, and O. Liopis, “Optical links for ultra-low phase noise microwave oscillators measurement,” in IEEE International Frequency Control Symposium and Exposition, 545–550, (2005).
[Crossref]

Pan, S.

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

D. Zhu, F. Zhang, P. Zhou, and S. Pan, “Phase noise measurement of wideband microwave sources based on a microwave photonic frequency down-converter,” Opt. Lett. 40(7), 1326–1329 (2015).
[Crossref] [PubMed]

F. Zhang, D. Zhu, and S. Pan, “Photonic-assisted wideband phase noise measurement of microwave signal sources,” Electron. Lett. 51(16), 1272–1274 (2015).
[Crossref]

D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
[Crossref]

Pavlyuchenko, E.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

Poddar, A. K.

U. L. Rohde, A. K. Poddar, and A. M. Apte, “Getting its measure: oscillator phase noise measurement techniques and limitations,” IEEE Microw. Mag. 14(6), 73–86 (2013).
[Crossref]

A. K. Poddar, U. L. Rohde, and A. M. Apte, “How low can they go?: oscillator phase noise model, theoretical, experimental validation, and phase noise measurements,” IEEE Microw. Mag. 14(6), 50–72 (2013).
[Crossref]

Rohde, U. L.

A. K. Poddar, U. L. Rohde, and A. M. Apte, “How low can they go?: oscillator phase noise model, theoretical, experimental validation, and phase noise measurements,” IEEE Microw. Mag. 14(6), 50–72 (2013).
[Crossref]

U. L. Rohde, A. K. Poddar, and A. M. Apte, “Getting its measure: oscillator phase noise measurement techniques and limitations,” IEEE Microw. Mag. 14(6), 73–86 (2013).
[Crossref]

Rubiola, E.

Salik, E.

Salzenstein, P.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

K. Volyanskiy, J. Cussey, H. Tavernier, P. Salzenstein, G. Sauvage, L. Larger, and E. Rubiola, “Applications of the optical fiber to the generation and measurement of low-phase-noise microwave signals,” J. Opt. Soc. Am. B 25(12), 2140–2150 (2008).
[Crossref]

Sauvage, G.

Sun, W.

Tatu, S. O.

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

Tavernier, H.

Volyanskiy, K.

Wang, W.

Wu, K.

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

Yao, J.

Yu, N.

Zarubin, M.

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

Zhang, F.

F. Zhang, D. Zhu, and S. Pan, “Photonic-assisted wideband phase noise measurement of microwave signal sources,” Electron. Lett. 51(16), 1272–1274 (2015).
[Crossref]

D. Zhu, F. Zhang, P. Zhou, and S. Pan, “Phase noise measurement of wideband microwave sources based on a microwave photonic frequency down-converter,” Opt. Lett. 40(7), 1326–1329 (2015).
[Crossref] [PubMed]

D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
[Crossref]

Zhou, P.

D. Zhu, F. Zhang, P. Zhou, and S. Pan, “Phase noise measurement of wideband microwave sources based on a microwave photonic frequency down-converter,” Opt. Lett. 40(7), 1326–1329 (2015).
[Crossref] [PubMed]

D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
[Crossref]

Zhu, D.

F. Zhang, D. Zhu, and S. Pan, “Photonic-assisted wideband phase noise measurement of microwave signal sources,” Electron. Lett. 51(16), 1272–1274 (2015).
[Crossref]

D. Zhu, F. Zhang, P. Zhou, and S. Pan, “Phase noise measurement of wideband microwave sources based on a microwave photonic frequency down-converter,” Opt. Lett. 40(7), 1326–1329 (2015).
[Crossref] [PubMed]

D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
[Crossref]

D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
[Crossref]

Zhu, N.

Electron. Lett. (1)

F. Zhang, D. Zhu, and S. Pan, “Photonic-assisted wideband phase noise measurement of microwave signal sources,” Electron. Lett. 51(16), 1272–1274 (2015).
[Crossref]

IEEE Microw. Mag. (2)

A. K. Poddar, U. L. Rohde, and A. M. Apte, “How low can they go?: oscillator phase noise model, theoretical, experimental validation, and phase noise measurements,” IEEE Microw. Mag. 14(6), 50–72 (2013).
[Crossref]

U. L. Rohde, A. K. Poddar, and A. M. Apte, “Getting its measure: oscillator phase noise measurement techniques and limitations,” IEEE Microw. Mag. 14(6), 73–86 (2013).
[Crossref]

IEEE Photonics Technol. Lett. (1)

D. Zhu, F. Zhang, P. Zhou, D. Zhu, and S. Pan, “Wideband phase noise measurement using a multifunctional microwave photonic processor,” IEEE Photonics Technol. Lett. 26(24), 2434–2437 (2014).
[Crossref]

IEEE Trans. Microw. Theory Tech. (2)

S. O. Tatu, E. Moldovan, S. Affes, B. Boukari, K. Wu, and R. G. Bosisio, “Six-port interferometric technique for accurate W-band phase-noise measurements,” IEEE Trans. Microw. Theory Tech. 56(6), 1372–1379 (2008).
[Crossref]

H. Gheidi and A. Banai, “Phase-noise measurement of microwave oscillators using phase-shifterless delay line discriminator,” IEEE Trans. Microw. Theory Tech. 58(2), 468–477 (2010).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

D. B. Leeson, “Oscillator phase noise: a 50-year review,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 63(8), 1208–1225 (2016).
[Crossref] [PubMed]

J. Lightwave Technol. (2)

J. Opt. Soc. Am. B (2)

Opt. Lett. (1)

Phys. Scr. T (1)

P. Salzenstein, E. Pavlyuchenko, A. Hmima, N. Cholley, M. Zarubin, S. Galliou, Y. Chembo, and L. Larger, “Estimation of the uncertainty for a phase noise optoelectronic metrology system,” Phys. Scr. T 149, 014025 (2012).
[Crossref]

Other (2)

B. Onillon, S. Constant, and O. Liopis, “Optical links for ultra-low phase noise microwave oscillators measurement,” in IEEE International Frequency Control Symposium and Exposition, 545–550, (2005).
[Crossref]

E8257D PSG Microwave Analog Signal Generator Data Sheet, Keysight Technologies Co. USA, 17–20 (2016).

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 (5)

Fig. 1
Fig. 1 Schematic diagram of the proposed phase noise measurement system. SUT: signal under test; LD: laser diode; PM: phase modulator; SMF: single mode fiber; DP-MZM: dual-polarization Mach-Zehnder modulator; PR: polarization rotator; PBC: polarization beam combiner; OBPF: optical band-pass filter; PC: polarization controller; PBS: polarization beam splitter; PD: photodetector; ADC: analog-to-digital converter.
Fig. 2
Fig. 2 The measured optical spectrum after the DP-MZM (blue solid curve) and the optical spectra of the selected −1st-order sidebands in the two polarizations (black dash-dotted curve and red dashed curve) when the frequency of the SUT is 10 GHz.
Fig. 3
Fig. 3 Phase noise of a 10-GHz clock signal measured by the proposed system (red dashed curve) and by a commercial signal analyzer R&S FSV40 (black solid curve).
Fig. 4
Fig. 4 (a) Phase noise floor of the established system at 10 GHz frequency, and (b) the phase noise of a 10-GHz signal from a microwave signal generator according, measured by the proposed system and by the signal analyzer R&S FSV40 (the blue marker are the values provided by the datasheet).
Fig. 5
Fig. 5 Phase noises at offset frequency of 10 kHz of a wideband signal source (Keysight E8257D-567) measured by the proposed system (red-circle marker) and provided by the datasheet (black-square marker).

Equations (14)

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

v(t)=[ V 0 +ε( t ) ]cos[ ω s t+φ( t ) ].
E 1 ( t )= E 0 cos{ ω c t+βcos[ ω s t+φ( t ) ] }.
E 2 ( t )= E 0 cos{ ω c (tτ)+βcos[ ω s (tτ)+φ( tτ ) ] }.
v 3 (t)= 1 2 [ V 0 +ε( t ) ]cos[ ω s t+φ( t ) ]. v 4 (t)= 1 2 [ V 0 +ε( t ) ]sin[ ω s t+φ( t ) ].
E 3 ( t )= x ^ E X1 ( t )+ y ^ E Y1 ( t ) = x ^ { E 2 ( t )cos[ αcos( ω s t+φ( t ) )+ π 4 ] } + y ^ { E 2 ( t )cos[ αsin( ω s t+φ( t ) )+ π 4 ] }.
E 4 ( t )= x ^ E X2 ( t )+ y ^ E Y2 ( t ) = x ^ 2 2 E 0 { J 0 ( β ) J 1 ( α )cos[ ( ω c ω s )( tτ ) ω s τφ( t ) ] + J 0 ( α ) J 1 ( β )sin[ ( ω c ω s )( tτ )φ( tτ ) ] } + y ^ 2 2 E 0 { J 0 ( β ) J 1 ( α )sin[ ( ω c ω s )( tτ ) ω s τφ( t ) ] + J 0 ( α ) J 1 ( β )sin[ ( ω c ω s )( tτ )φ( tτ ) ] }.
v 5 ( t )=R Z L | E X2 | 2 = V DC +Q( t ). v 6 ( t )=R Z L | E Y2 | 2 = V DC +I( t ).
V DC = E 0 2 R Z L 4 [ J 0 2 ( β ) J 1 2 ( α )+ J 0 2 ( α ) J 1 2 ( β ) ]. Q( t )= E 0 2 R Z L J 0 ( β ) J 1 ( α ) J 0 ( α ) J 1 ( β ) 2 sin[ ω s τ+φ( t )φ( tτ ) ]. I( t )= E 0 2 R Z L J 0 ( β ) J 1 ( α ) J 0 ( α ) J 1 ( β ) 2 cos[ ω s τ+φ( t )φ( tτ ) ].
v 7 ( t )= V DC +Q(t). v 8 ( t )= V DC Q(t).
V DC = v 7 ( t )+ v 8 ( t ) 2 .
Q( t )= v 5 ( t ) V DC . I( t )= v 6 ( t ) V DC .
θ( t )= ω s τ+φ( t )φ( tτ )= tan 1 [ Q( t ) I( t ) ].
[ φ( t )φ( tτ ) ] PSD = S θ ( f m )= | + θ( t ) e j2π f m t dt | 2 ,for f m >0.
L( f m )= S θ ( f m ) 2 | 1 e j2π f m τ | 2 = S θ ( f m ) 8 sin 2 ( π f m τ ) .

Metrics