Abstract

We evaluate the Brillouin frequency shift (BFS) determination error when utilizing the Brillouin phase spectrum (BPS) instead of the Brillouin gain spectrum (BGS) in BOTDA systems. Systems based on the BPS perform the determination of the BFS through a linear fit around the zero de-phase frequency region. An analytical expression of the error obtained in the BFS determination as a function of the different experimental parameters is provided and experimentally validated. The experimental results show a good agreement with the theoretical predictions as a function of the number of sampling points, signal-to-noise ratio (SNR) and Brillouin spectral linewidth. For an equal SNR and linewidth, the phase response only provides a better BFS estimation than the gain response when the fit is performed over a restricted frequency range around the center of the spectral profile. This may reduce the measurement time of specific BOTDA systems requiring a narrow frequency scanning. When the frequency scan covers most of the Brillouin spectral profile, gain and phase responses give very similar estimations of the BFS and the BPS offers no crucial benefit.

© 2016 Optical Society of America

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References

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  1. T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7(8), 1170–1176 (1989).
    [Crossref]
  2. G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007), Ch. 9.
  3. M. Nikles, L. Thévenaz, and P. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
    [Crossref]
  4. M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
    [Crossref] [PubMed]
  5. J. Urricelqui, A. Zornoza, M. Sagues, and A. Loayssa, “Dynamic BOTDA measurements based on Brillouin phase-shift and RF demodulation,” Opt. Express 20(24), 26942–26949 (2012).
    [Crossref] [PubMed]
  6. M. Dossou, D. Bacquet, and P. Szriftgiser, “Vector Brillouin optical time-domain analyzer for high-order acoustic modes,” Opt. Lett. 35(22), 3850–3852 (2010).
    [Crossref] [PubMed]
  7. X. Lu, M. A. Soto, M. Gonzalez-Herraez, and L. Thévenaz, “Brillouin distributed fibre sensing using phase modulated probe,” Proc. SPIE 8794, 87943P (2013).
    [Crossref]
  8. X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
    [Crossref]
  9. A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Exploiting nonreciprocity in BOTDA systems,” Opt. Lett. 40(10), 2193–2196 (2015).
    [Crossref] [PubMed]
  10. A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).
  11. A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
    [Crossref]
  12. X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
    [Crossref]
  13. E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol. 7(1), 171–174 (1989).
    [Crossref]
  14. A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in 12th International Conference on Optical Fiber Sensors, OSA Technical Digest Series (Optical Society of America, 1997), paper OWD3.
    [Crossref]
  15. P. H. Richter, “Estimating errors in least-squares fitting,” Telecommun. Data Acquisition Prog. Rep. 42(122), 107–137 (1995).
  16. L. Thévenaz, S. F. Mafang, and J. Lin, “Effect of pulse depletion in a Brillouin optical time-domain analysis system,” Opt. Express 21(12), 14017–14035 (2013).
    [Crossref] [PubMed]

2016 (1)

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
[Crossref]

2015 (3)

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Exploiting nonreciprocity in BOTDA systems,” Opt. Lett. 40(10), 2193–2196 (2015).
[Crossref] [PubMed]

2014 (1)

X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
[Crossref]

2013 (3)

2012 (1)

2010 (1)

1997 (1)

M. Nikles, L. Thévenaz, and P. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

1995 (1)

P. H. Richter, “Estimating errors in least-squares fitting,” Telecommun. Data Acquisition Prog. Rep. 42(122), 107–137 (1995).

1989 (2)

T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7(8), 1170–1176 (1989).
[Crossref]

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol. 7(1), 171–174 (1989).
[Crossref]

Andres, M. V.

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

Angulo-Vinuesa, X.

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
[Crossref]

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Exploiting nonreciprocity in BOTDA systems,” Opt. Lett. 40(10), 2193–2196 (2015).
[Crossref] [PubMed]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).

Bacquet, D.

Chen, M.

X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
[Crossref]

Chen, W.

X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
[Crossref]

Cruz, J. L.

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

Dominguez-Lopez, A.

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
[Crossref]

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Exploiting nonreciprocity in BOTDA systems,” Opt. Lett. 40(10), 2193–2196 (2015).
[Crossref] [PubMed]

Dominguez-López, A.

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).

Dossou, M.

Friesem, A. A.

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol. 7(1), 171–174 (1989).
[Crossref]

Gonzalez-Herraez, M.

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Exploiting nonreciprocity in BOTDA systems,” Opt. Lett. 40(10), 2193–2196 (2015).
[Crossref] [PubMed]

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).

X. Lu, M. A. Soto, M. Gonzalez-Herraez, and L. Thévenaz, “Brillouin distributed fibre sensing using phase modulated probe,” Proc. SPIE 8794, 87943P (2013).
[Crossref]

Horiguchi, T.

T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7(8), 1170–1176 (1989).
[Crossref]

Lichtman, E.

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol. 7(1), 171–174 (1989).
[Crossref]

Lin, J.

Loayssa, A.

Lopez-Gil, A.

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
[Crossref]

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Exploiting nonreciprocity in BOTDA systems,” Opt. Lett. 40(10), 2193–2196 (2015).
[Crossref] [PubMed]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).

Lu, X.

X. Lu, M. A. Soto, M. Gonzalez-Herraez, and L. Thévenaz, “Brillouin distributed fibre sensing using phase modulated probe,” Proc. SPIE 8794, 87943P (2013).
[Crossref]

Mafang, S. F.

Martin-Lopez, S.

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Exploiting nonreciprocity in BOTDA systems,” Opt. Lett. 40(10), 2193–2196 (2015).
[Crossref] [PubMed]

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).

Meng, Z.

X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
[Crossref]

Nikles, M.

M. Nikles, L. Thévenaz, and P. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Richter, P. H.

P. H. Richter, “Estimating errors in least-squares fitting,” Telecommun. Data Acquisition Prog. Rep. 42(122), 107–137 (1995).

Robert, P.

M. Nikles, L. Thévenaz, and P. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Sagues, M.

Soto, M. A.

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

X. Lu, M. A. Soto, M. Gonzalez-Herraez, and L. Thévenaz, “Brillouin distributed fibre sensing using phase modulated probe,” Proc. SPIE 8794, 87943P (2013).
[Crossref]

Sun, Q.

X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
[Crossref]

Szriftgiser, P.

Tateda, M.

T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7(8), 1170–1176 (1989).
[Crossref]

Thévenaz, L.

L. Thévenaz, S. F. Mafang, and J. Lin, “Effect of pulse depletion in a Brillouin optical time-domain analysis system,” Opt. Express 21(12), 14017–14035 (2013).
[Crossref] [PubMed]

X. Lu, M. A. Soto, M. Gonzalez-Herraez, and L. Thévenaz, “Brillouin distributed fibre sensing using phase modulated probe,” Proc. SPIE 8794, 87943P (2013).
[Crossref]

M. A. Soto and L. Thévenaz, “Modeling and evaluating the performance of Brillouin distributed optical fiber sensors,” Opt. Express 21(25), 31347–31366 (2013).
[Crossref] [PubMed]

M. Nikles, L. Thévenaz, and P. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

Tu, X.

X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
[Crossref]

Urricelqui, J.

Waarts, R. G.

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol. 7(1), 171–174 (1989).
[Crossref]

Zornoza, A.

IEEE Photonics J. (1)

X. Tu, Q. Sun, W. Chen, M. Chen, and Z. Meng, “Vector brillouin optical time-domain analysis with heterodyne detection and IQ demodulation algorithm,” IEEE Photonics J. 6(2), 6800908 (2014).
[Crossref]

IEEE Photonics Technol. Lett. (1)

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-Lopez, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple baseband method for the distributed analysis of Brillouin phase-shift spectra,” IEEE Photonics Technol. Lett. 28(13), 1379–1382 (2016).
[Crossref]

J. Lightwave Technol. (3)

T. Horiguchi and M. Tateda, “BOTDA-nondestructive measurement of single-mode optical fiber attenuation characteristics using Brillouin interaction: theory,” J. Lightwave Technol. 7(8), 1170–1176 (1989).
[Crossref]

M. Nikles, L. Thévenaz, and P. Robert, “Brillouin gain spectrum characterization in single-mode optical fibers,” J. Lightwave Technol. 15(10), 1842–1851 (1997).
[Crossref]

E. Lichtman, R. G. Waarts, and A. A. Friesem, “Stimulated Brillouin scattering excited by a modulated pump wave in single-mode fibers,” J. Lightwave Technol. 7(1), 171–174 (1989).
[Crossref]

Opt. Express (3)

Opt. Lett. (2)

Proc. SPIE (3)

X. Lu, M. A. Soto, M. Gonzalez-Herraez, and L. Thévenaz, “Brillouin distributed fibre sensing using phase modulated probe,” Proc. SPIE 8794, 87943P (2013).
[Crossref]

X. Angulo-Vinuesa, A. Lopez-Gil, A. Dominguez-Lopez, J. L. Cruz, M. V. Andres, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simultaneous gain and phase profile determination on an interferometric BOTDA,” Proc. SPIE 9634, 963419 (2015).
[Crossref]

A. Lopez-Gil, X. Angulo-Vinuesa, A. Dominguez-López, S. Martin-Lopez, and M. Gonzalez-Herraez, “Simple BOTDA temperature sensor based on distributed Brillouin phase-shift measurements within a Sagnac interferometer,” Proc. SPIE 9634, 96346L (2015).

Telecommun. Data Acquisition Prog. Rep. (1)

P. H. Richter, “Estimating errors in least-squares fitting,” Telecommun. Data Acquisition Prog. Rep. 42(122), 107–137 (1995).

Other (2)

A. Fellay, L. Thévenaz, M. Facchini, M. Niklès, and P. Robert, “Distributed sensing using stimulated Brillouin scattering: towards ultimate resolution,” in 12th International Conference on Optical Fiber Sensors, OSA Technical Digest Series (Optical Society of America, 1997), paper OWD3.
[Crossref]

G. P. Agrawal, Nonlinear Fiber Optics (Academic, 2007), Ch. 9.

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Figures (8)

Fig. 1
Fig. 1

Schematic representation of the BGS/BLS (gain - loss) and BPS (phase) including their dependence with increased strain Ɛ or temperature T.

Fig. 2
Fig. 2

Graphical representation of experimental gain and phase profiles obtained simultaneously through the SI-BOTDA presented in [12]. In the picture, the position of the BFS is represented with a red dot (at the maximum of the BGS and the zero-cross for the BPS measurement) as well as the most convenient mathematical fits to determine it; quadratic and linear for the gain and phase curves respectively. The relevant experimental parameters to determine the BFS determination error are also shown: frequency sampling step δ, gain FWHM ΔνB and normalized noise σ.

Fig. 3
Fig. 3

Temporal normalized gain trace obtained with a standard BOTDA sensor. Noisy and ideal gain (b) and phase (c) profiles. In all cases, it has been defined the signal and the noise to calculate the SNR.

Fig. 4
Fig. 4

Linear and quadratic BFS determination error vs. the number of fitting points. The analysis has been performed for different frequency sampling steps (δ = 0.25, 0.5, 1, 1.5, 2, and 3 MHz), with experimental results retrieved with 20 ns pulses and a 13.4 SNR equivalent to 300 average number. The results show theoretical simulations as well as experimental results. In all the cases, the highest N corresponds to the use of the full ΔνB frequency range around the BFS position.

Fig. 5
Fig. 5

Error comparison as a function of δ for 20 ns pulses (ΔνB = 56.7 MHz) and N = 18. As observed, the linear error has no basic dependence upon the sampling step, as predicted by Eq. (8).

Fig. 6
Fig. 6

Linear and quadratic BFS determination error vs ΔνB. The analysis has been performed for δ = 1 MHz, N = 35, and a SNR of 10.6. It is compatible with the quadratic tendency in the quadratic error while the linear fit trend follows a straight line.

Fig. 7
Fig. 7

Linear and quadratic BFS determination error vs. SNR. The analysis has been performed with 20 ns pulses, a δ = 1 MHz and N = 44.

Fig. 8
Fig. 8

Logarithmic representation of the ratio among quadratic and linear errors as a function of ξ when δ = 1 MHz and ΔνB = 56.7 MHz. As can be seen, for ξ values greater than 3 /2 , the quadratic fit suits better. This condition is generally fulfilled for standard frequency sweeps, when the tested number of frequencies is around two times the ΔνB of the system.

Equations (14)

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

y( x )=ax+b
y( ν B )=a ν B +b=0         ν B = b a
σ ν 2 = | ν B a | 2 σ a 2 + | ν B b | 2 σ b 2 +2 ν B a ν B b co v a,b
σ a 2 = 12 σ 2 N( N 2 1 ) δ 2
σ b 2 = σ 2 N
σ ν 2 = σ 2 a 2 N [ 1+ 12 b 2 a 2 N 2 δ 2 ]
σ ν 2 = σ 2 a 2 N
σ νlinear ( z )=σ( z )Δ ν B 1 N = 1 SNR( z ) Δ ν B 1 N
σ νparabolic ( z )=σ( z ) 3δΔ ν B 8 2 ( 1η ) 3/2
η=1 N 2 δ 2 2Δ ν B 2    ξ= Nδ Δ ν B   η=1 ξ 2 2
σ νparabolic ( z )=σ( z ) Δ ν B 2 2Nδ 3 N =σ( z )Δ ν B 1 2ξ 3 N
σ νparabolic ( z ) σ νlinear ( z ) = 3 2ξ
σ νlinear (z)=σ(z)Δ ν B 1 ξ δ Δ ν B
σ νparabolic (z)=σ(z)Δ ν B 3 4 ξ 3 δ Δ ν B

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