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

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References

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  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)

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]

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)

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]

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]

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]

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]

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).

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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)

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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 ) .

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