Abstract

Analyzing the experimental data of the velocity distribution in a fluid flow using Doppler Optical Coherence Tomography (OCT), we compared the Wigner distribution method to the short-time Fourier transform method, the Hilbert-based phase-resolved method and the autocorrelation method. We conclude that the pseudo Wigner-distribution signal processing method is overall more precise than other often-used methods in Doppler OCT for the analysis of cross-sectional velocity distributions.

© 2007 Optical Society of America

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  1. D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
    [Crossref] [PubMed]
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    [Crossref]
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  4. J. A. Izatt, M. D. Kulkami, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomograghy,” Opt. Lett. 22, 1439–1441 (1997).
    [Crossref]
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  17. T. C. M. Claasen and W. F. G. Mecklenbräuker “The Wigner distribution - a tool for time-frequency signal analysis” Part II: Discrete time signals. Philips J. Res. 35, 276–300 (1980).
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    [Crossref]
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    [Crossref]
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2007 (2)

2005 (2)

2004 (2)

L. Wang, W. Xu, M. Bachman, G. P. Li, and Z. P. Chen, “Phase-resolved optical Doppler tomography for imaging flow dynamics in microfluidic channels,” Appl. Phys. Lett. 85, 1855–1857 (2004).
[Crossref]

Wu L., “Simultaneous measurement of flow velocity and Doppler angle by the use of Doppler optical coherence tomography,” Opt. Lasers Eng. 42, 303–313 (2004).
[Crossref]

2003 (1)

2002 (3)

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Daqing Piao, Linda L. Otis, Niloy K. Dutta, and Quing Zhu, “Quantitative assessment of flow velocity-estimation algorithms for optical Doppler tomography imaging”, App. Opt. 41, 6118–6127 (2002).
[Crossref]

Andrew M. Rollins, Siavash Yazdanfar, Jennifer K. Barton, and Joseph A. Izatt, “Real-time in vivo color Doppler optical coherence tomography”, J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

2001 (1)

S. Yazdanfar, A. M. Rollins, and J. A. Izatt “Ultrahigh velocity resolution imaging of the microcirculation in vivo using color Doppler optical coherence tomography”. Proc. SPIE,  4251, 156–164 (2001).
[Crossref]

2000 (1)

1999 (3)

W. Drexler, U. Morgner, F. X. Kartner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fujimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24, 1221–1223 (1999).
[Crossref]

Andrew M. Rollins, Siavash Yazdanfar, Rujchai Ung-arunyawee, and Joseph A. Izatt “Real time color Doppler optical coherence tomography using an autocorrelation technique”, Proc. SPIE. 3598, 168–176 (1999).
[Crossref]

Z. P. Chen, Y. H. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. 5, 1134–1142 (1999).
[Crossref]

1997 (2)

1996 (1)

T. P. Zielinski, “Wigner transform instantaneous phase estimator”, Eusipco-96, Trieste, PDE.10 (1996)

1991 (2)

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

N. Bergmann “New formulation of discrete Wigner-Ville distribution”, Electron. Lett. 27, 111–112 (1991).
[Crossref]

1988 (1)

T. P. Zielinski, “On a software implementation of the Wigner-Ville transform”, Comp. Phys. Comm. 50, 269–272 (1988).
[Crossref]

1980 (1)

T. C. M. Claasen and W. F. G. Mecklenbräuker “The Wigner distribution - a tool for time-frequency signal analysis” Part II: Discrete time signals. Philips J. Res. 35, 276–300 (1980).

Auger, F.

F. Auger, P. Flandrin, P. Gonçalvès, and O. Lemoine, “Time-Frequency Toolbox tutorial”, CNRS (France), Rice U. (U.S.A.), http://tftb.nongnu.org/ (2005) and http://gdr-isis.org/tftb/tutorial/tutorial.html.

Bachman, M.

L. Wang, W. Xu, M. Bachman, G. P. Li, and Z. P. Chen, “Phase-resolved optical Doppler tomography for imaging flow dynamics in microfluidic channels,” Appl. Phys. Lett. 85, 1855–1857 (2004).
[Crossref]

Barton, J. K.

Barton, Jennifer K.

Andrew M. Rollins, Siavash Yazdanfar, Jennifer K. Barton, and Joseph A. Izatt, “Real-time in vivo color Doppler optical coherence tomography”, J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

Bergmann, N.

N. Bergmann “New formulation of discrete Wigner-Ville distribution”, Electron. Lett. 27, 111–112 (1991).
[Crossref]

Boppart, S.

Boppart, S. A.

Bouma, B. E.

Bouma, Brett E.

Cense, Barry

Chang, W.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Chen, Teresa C.

Chen, Y.

Y. Chen, P. Willett, and Q. Zhu, “Frequency tracking in optical Doppler tomography using an adaptive notch filter”, J. Biomed. Opt. 12, 014018-1 to 014018-9 (2007).
[Crossref]

Chen, Z.

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Chen, Z. P.

L. Wang, W. Xu, M. Bachman, G. P. Li, and Z. P. Chen, “Phase-resolved optical Doppler tomography for imaging flow dynamics in microfluidic channels,” Appl. Phys. Lett. 85, 1855–1857 (2004).
[Crossref]

Z. P. Chen, Y. H. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. 5, 1134–1142 (1999).
[Crossref]

Z. P. Chen, T. E. Milner, D. Dave, and J. S. Nelson, “Optical Doppler tomographic imaging of fluid flow velocity in highly scattering media,” Opt. Lett. 22, 64–66 (1997).
[Crossref] [PubMed]

Chen, Zhongping

Claasen, T. C. M.

T. C. M. Claasen and W. F. G. Mecklenbräuker “The Wigner distribution - a tool for time-frequency signal analysis” Part II: Discrete time signals. Philips J. Res. 35, 276–300 (1980).

Cobbold, R. S. C.

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Cohen, Leon

Leon Cohen, Time-frequency analysis, Prentice Hall, Englewood Cliffs, N.J. (1995).

Dave, D.

de Boer, J. F.

de Boer, Johannes F.

Drexler, W.

Dutta, Niloy K.

Daqing Piao, Linda L. Otis, Niloy K. Dutta, and Quing Zhu, “Quantitative assessment of flow velocity-estimation algorithms for optical Doppler tomography imaging”, App. Opt. 41, 6118–6127 (2002).
[Crossref]

Flandrin, P.

F. Auger, P. Flandrin, P. Gonçalvès, and O. Lemoine, “Time-Frequency Toolbox tutorial”, CNRS (France), Rice U. (U.S.A.), http://tftb.nongnu.org/ (2005) and http://gdr-isis.org/tftb/tutorial/tutorial.html.

Flotte, T.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Frostig, R. D.

Z. P. Chen, Y. H. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. 5, 1134–1142 (1999).
[Crossref]

Fujimoto, J. G.

W. Drexler, U. Morgner, F. X. Kartner, C. Pitris, S. A. Boppart, X. D. Li, E. P. Ippen, and J. G. Fujimoto, “In vivo ultrahigh-resolution optical coherence tomography,” Opt. Lett. 24, 1221–1223 (1999).
[Crossref]

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Gonçalvès, P.

F. Auger, P. Flandrin, P. Gonçalvès, and O. Lemoine, “Time-Frequency Toolbox tutorial”, CNRS (France), Rice U. (U.S.A.), http://tftb.nongnu.org/ (2005) and http://gdr-isis.org/tftb/tutorial/tutorial.html.

Gordon, M. L

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Gregory, K.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Hee, M. R.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Huang, D.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Ippen, E. P.

Izatt, J. A.

S. Yazdanfar, A. M. Rollins, and J. A. Izatt “Ultrahigh velocity resolution imaging of the microcirculation in vivo using color Doppler optical coherence tomography”. Proc. SPIE,  4251, 156–164 (2001).
[Crossref]

J. A. Izatt, M. D. Kulkami, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomograghy,” Opt. Lett. 22, 1439–1441 (1997).
[Crossref]

Izatt, Joseph A.

Andrew M. Rollins, Siavash Yazdanfar, Jennifer K. Barton, and Joseph A. Izatt, “Real-time in vivo color Doppler optical coherence tomography”, J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

Andrew M. Rollins, Siavash Yazdanfar, Rujchai Ung-arunyawee, and Joseph A. Izatt “Real time color Doppler optical coherence tomography using an autocorrelation technique”, Proc. SPIE. 3598, 168–176 (1999).
[Crossref]

Kamalabadi, F.

Kartner, F. X.

Kolios, M. C.

Kulkami, M. D.

L., Wu

Wu L., “Simultaneous measurement of flow velocity and Doppler angle by the use of Doppler optical coherence tomography,” Opt. Lasers Eng. 42, 303–313 (2004).
[Crossref]

Lemoine, O.

F. Auger, P. Flandrin, P. Gonçalvès, and O. Lemoine, “Time-Frequency Toolbox tutorial”, CNRS (France), Rice U. (U.S.A.), http://tftb.nongnu.org/ (2005) and http://gdr-isis.org/tftb/tutorial/tutorial.html.

Li, G. P.

L. Wang, W. Xu, M. Bachman, G. P. Li, and Z. P. Chen, “Phase-resolved optical Doppler tomography for imaging flow dynamics in microfluidic channels,” Appl. Phys. Lett. 85, 1855–1857 (2004).
[Crossref]

Li, X. D.

Lin, C. P.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Mecklenbräuker, W. F. G.

T. C. M. Claasen and W. F. G. Mecklenbräuker “The Wigner distribution - a tool for time-frequency signal analysis” Part II: Discrete time signals. Philips J. Res. 35, 276–300 (1980).

Milner, T. E.

Mok, A.

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Morgner, U.

Morofke, D.

Nassif, Nader

Nelson, J. S.

Otis, Linda L.

Daqing Piao, Linda L. Otis, Niloy K. Dutta, and Quing Zhu, “Quantitative assessment of flow velocity-estimation algorithms for optical Doppler tomography imaging”, App. Opt. 41, 6118–6127 (2002).
[Crossref]

Park, B. Hyle

Piao, Daqing

Daqing Piao, Linda L. Otis, Niloy K. Dutta, and Quing Zhu, “Quantitative assessment of flow velocity-estimation algorithms for optical Doppler tomography imaging”, App. Opt. 41, 6118–6127 (2002).
[Crossref]

Pierce, Mark C.

Pitris, C.

Prakash, N.

Z. P. Chen, Y. H. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. 5, 1134–1142 (1999).
[Crossref]

Puliafito, C. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Rollins, A. M.

S. Yazdanfar, A. M. Rollins, and J. A. Izatt “Ultrahigh velocity resolution imaging of the microcirculation in vivo using color Doppler optical coherence tomography”. Proc. SPIE,  4251, 156–164 (2001).
[Crossref]

Rollins, Andrew M.

Andrew M. Rollins, Siavash Yazdanfar, Jennifer K. Barton, and Joseph A. Izatt, “Real-time in vivo color Doppler optical coherence tomography”, J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

Andrew M. Rollins, Siavash Yazdanfar, Rujchai Ung-arunyawee, and Joseph A. Izatt “Real time color Doppler optical coherence tomography using an autocorrelation technique”, Proc. SPIE. 3598, 168–176 (1999).
[Crossref]

Saxer, C.

Schuman, J. S.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Srinivas, S. M.

Z. P. Chen, Y. H. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. 5, 1134–1142 (1999).
[Crossref]

Stinson, W. G.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Swanson, E. A.

D. Huang, E. A. Swanson, C. P. Lin, J. S. Schuman, W. G. Stinson, W. Chang, M. R. Hee, T. Flotte, K. Gregory, C. A. Puliafito, and J. G. Fujimoto, “Optical Coherence Tomography,” Science 254, 1178–1181 (1991).
[Crossref] [PubMed]

Tearney, G. J.

Tearney, Guillermo J.

Ung-arunyawee, Rujchai

Andrew M. Rollins, Siavash Yazdanfar, Rujchai Ung-arunyawee, and Joseph A. Izatt “Real time color Doppler optical coherence tomography using an autocorrelation technique”, Proc. SPIE. 3598, 168–176 (1999).
[Crossref]

Vakoc, B. J.

Vitkin, I. A.

Vitkin, I. Alex.

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Wang, L.

L. Wang, W. Xu, M. Bachman, G. P. Li, and Z. P. Chen, “Phase-resolved optical Doppler tomography for imaging flow dynamics in microfluidic channels,” Appl. Phys. Lett. 85, 1855–1857 (2004).
[Crossref]

Welch, A. J.

White, Brian R.

Willett, P.

Y. Chen, P. Willett, and Q. Zhu, “Frequency tracking in optical Doppler tomography using an adaptive notch filter”, J. Biomed. Opt. 12, 014018-1 to 014018-9 (2007).
[Crossref]

Wilson, Brian C.

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Xiang, Shaohua

Xu, C.

Xu, W.

L. Wang, W. Xu, M. Bachman, G. P. Li, and Z. P. Chen, “Phase-resolved optical Doppler tomography for imaging flow dynamics in microfluidic channels,” Appl. Phys. Lett. 85, 1855–1857 (2004).
[Crossref]

Yang, V. X. D.

Yang, Victor X. D.

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Yazdanfar, S.

S. Yazdanfar, A. M. Rollins, and J. A. Izatt “Ultrahigh velocity resolution imaging of the microcirculation in vivo using color Doppler optical coherence tomography”. Proc. SPIE,  4251, 156–164 (2001).
[Crossref]

J. A. Izatt, M. D. Kulkami, S. Yazdanfar, J. K. Barton, and A. J. Welch, “In vivo bidirectional color Doppler flow imaging of picoliter blood volumes using optical coherence tomograghy,” Opt. Lett. 22, 1439–1441 (1997).
[Crossref]

Yazdanfar, Siavash

Andrew M. Rollins, Siavash Yazdanfar, Jennifer K. Barton, and Joseph A. Izatt, “Real-time in vivo color Doppler optical coherence tomography”, J. Biomed. Opt. 7, 123–129 (2002).
[Crossref] [PubMed]

Andrew M. Rollins, Siavash Yazdanfar, Rujchai Ung-arunyawee, and Joseph A. Izatt “Real time color Doppler optical coherence tomography using an autocorrelation technique”, Proc. SPIE. 3598, 168–176 (1999).
[Crossref]

Yun, S. H.

Zhao, Y.

Victor X. D. Yang, M. L Gordon, A. Mok, Y. Zhao, Z. Chen, R. S. C. Cobbold, Brian C. Wilson, and I. Alex. Vitkin “Improved phase-resolved optical Doppler tomography using the Kasai velocity estimator and histogram segmentation”, Opt. Comm. 208, 209–214 (2002).
[Crossref]

Zhao, Y. H.

Z. P. Chen, Y. H. Zhao, S. M. Srinivas, J. S. Nelson, N. Prakash, and R. D. Frostig, “Optical Doppler Tomography,” IEEE J. Sel. Top. Quantum Electron. 5, 1134–1142 (1999).
[Crossref]

Zhao, Yonghua

Zhu, Q.

Y. Chen, P. Willett, and Q. Zhu, “Frequency tracking in optical Doppler tomography using an adaptive notch filter”, J. Biomed. Opt. 12, 014018-1 to 014018-9 (2007).
[Crossref]

Zhu, Quing

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Figures (8)

Fig. 1.
Fig. 1.

Experimental setup for the Doppler frequency-shift measurements. FC: fiber coupler, PC: polarization controller, MO: microscope objective

Fig. 2.
Fig. 2.

OCT image of the tube profile. The superimposed contour map indicates the frequency calculated using experimental values and the pseudo Wigner method. The white line indicates the A-scan that is studied afterwards in this work.

Fig. 3.
Fig. 3.

Small segment of the detected signal at a 100 kHz sampling rate. The boxed region represents the STFT window size.

Fig. 4.
Fig. 4.

STFT time-frequency distribution of the sample signal.

Fig. 5.
Fig. 5.

Time-frequency distribution of the sample signal obtained with the Wigner distribution.

Fig. 6.
Fig. 6.

Comparison of the frequency spectra calculated with two time-frequency methods.

Fig. 7
Fig. 7

Spectra calculated at each sampling point of the signal using the Wigner distribution.

Fig. 8
Fig. 8

Computed flow profiles using a quadratic fit.

Tables (1)

Tables Icon

Table 1. Main parameters of the plots in Fig. 8

Equations (9)

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Δ f D = 1 2 π ( k s k i ) × v .
v ODT = λ 0 Δ f D 2 n ¯ cos ( θ ) .
STFT ( f , τ ) = y ( t ) w ( t + τ ) e j 2 πft dt .
f c = m f m STFT ( f m , τ i ) m STFT ( f m , τ i ) .
W ( t , ω ) = 1 2 π y * ( t 1 2 τ ) y ( t + 1 2 τ ) e j ωτ .
W ( t , ω ) dt = Y ( ω ) 2 .
W ( t , ω ) = y ( t ) 2 .
PW ( t , ω ) = 1 2 π h ( τ ) y * ( t 1 2 τ ) y ( t + 1 2 τ ) e jωτ .
V ( r ) = V c [ 1 ( r d 2 ) 2 ] .

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