Abstract

A spectral liquid signature analyzer system including a miniature fiber-optic spectrometer is described in this paper. Through the combination of the spectral method and liquid signature analysis technology, the analytical function of the liquid signature analyzer will be improved obviously. With this spectral liquid signature analyzer system, the physical, chemical, and mechanical characteristics of the tested sample can be obtained at the same time. In detail, that is the light intensity signal, the equivalent drop volume signal, and the spectroscopic data from the miniature spectrometer. And after merging the information data, the three-dimensional fingerprint that is unique to the liquid sample is attainable. This three-dimensional fingerprint includes the full information we gain from the experiment, and it provides visualized differences in various samples. The application of spectral analysis makes it possible for further research into the chemical compositions of liquid samples and supplies discrimination evidence to different liquids. Therefore, the liquid signature analyzer can be used in more fields related to liquid.

© 2011 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. Z. Qiu, G. Zhang, C. Wang, and X. Li, “A study on the technology of drop analysis,” Trans. Tianjin Univ. 34, 587–590 (2001).
  2. Z. Qiu, “A study on the technology of drop analysis,” doctoral dissertation (Tianjin University of China, 2000).
  3. C. H. Wang, A. T. Augousti, J. Mason, and N. D. McMillan, “The capacitive drop tensiometer—a novel multianalysing technique for measuring the properties of liquids,” Meas. Sci. Technol. 10, 19–24 (1999).
    [CrossRef]
  4. N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
    [CrossRef]
  5. N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
    [CrossRef]
  6. N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
    [CrossRef]
  7. Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
    [CrossRef]
  8. Z. Qiu, H. Chen, and G. Zhang, “Photoelectric multiple-wavelength liquid drop sensor,” Instrum. Sci. Technol. 31, 305–315 (2003).
    [CrossRef]
  9. M. Wu, Instrument Analysis (Academic, 2003).
  10. J. Shi, M. Hui, D. Wang, and W. Mao, “Micromation and application of spectrometer,” Opt. Tech. 29, 13–20 (2003).

2004 (1)

Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
[CrossRef]

2003 (2)

Z. Qiu, H. Chen, and G. Zhang, “Photoelectric multiple-wavelength liquid drop sensor,” Instrum. Sci. Technol. 31, 305–315 (2003).
[CrossRef]

J. Shi, M. Hui, D. Wang, and W. Mao, “Micromation and application of spectrometer,” Opt. Tech. 29, 13–20 (2003).

2001 (1)

Z. Qiu, G. Zhang, C. Wang, and X. Li, “A study on the technology of drop analysis,” Trans. Tianjin Univ. 34, 587–590 (2001).

1999 (1)

C. H. Wang, A. T. Augousti, J. Mason, and N. D. McMillan, “The capacitive drop tensiometer—a novel multianalysing technique for measuring the properties of liquids,” Meas. Sci. Technol. 10, 19–24 (1999).
[CrossRef]

1994 (2)

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

1992 (1)

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Augousti, A. T.

C. H. Wang, A. T. Augousti, J. Mason, and N. D. McMillan, “The capacitive drop tensiometer—a novel multianalysing technique for measuring the properties of liquids,” Meas. Sci. Technol. 10, 19–24 (1999).
[CrossRef]

Breen, L.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

Chen, H.

Z. Qiu, H. Chen, and G. Zhang, “Photoelectric multiple-wavelength liquid drop sensor,” Instrum. Sci. Technol. 31, 305–315 (2003).
[CrossRef]

Dalton, M. J.

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Daly, D.

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Feeney, F. E.

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

Fingleton, M.

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Finlayson, O.

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Fortune, F.

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Hammil, C.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

Hui, M.

J. Shi, M. Hui, D. Wang, and W. Mao, “Micromation and application of spectrometer,” Opt. Tech. 29, 13–20 (2003).

Kelly, M. P.

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

Kinsella, S. M.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

Li, X.

Z. Qiu, G. Zhang, C. Wang, and X. Li, “A study on the technology of drop analysis,” Trans. Tianjin Univ. 34, 587–590 (2001).

Mao, W.

J. Shi, M. Hui, D. Wang, and W. Mao, “Micromation and application of spectrometer,” Opt. Tech. 29, 13–20 (2003).

Mason, J.

C. H. Wang, A. T. Augousti, J. Mason, and N. D. McMillan, “The capacitive drop tensiometer—a novel multianalysing technique for measuring the properties of liquids,” Meas. Sci. Technol. 10, 19–24 (1999).
[CrossRef]

McMillan, D. D. G.

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

McMillan, D. G. E.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

McMillan, N. D.

C. H. Wang, A. T. Augousti, J. Mason, and N. D. McMillan, “The capacitive drop tensiometer—a novel multianalysing technique for measuring the properties of liquids,” Meas. Sci. Technol. 10, 19–24 (1999).
[CrossRef]

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

O’Dea, J. P.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

O’Mongain, E.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

Orr, D.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

Power, M. J.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

Power, M. J. P.

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

Qiu, Z.

Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
[CrossRef]

Z. Qiu, H. Chen, and G. Zhang, “Photoelectric multiple-wavelength liquid drop sensor,” Instrum. Sci. Technol. 31, 305–315 (2003).
[CrossRef]

Z. Qiu, G. Zhang, C. Wang, and X. Li, “A study on the technology of drop analysis,” Trans. Tianjin Univ. 34, 587–590 (2001).

Z. Qiu, “A study on the technology of drop analysis,” doctoral dissertation (Tianjin University of China, 2000).

Shi, J.

J. Shi, M. Hui, D. Wang, and W. Mao, “Micromation and application of spectrometer,” Opt. Tech. 29, 13–20 (2003).

Shi, Q.

Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
[CrossRef]

Song, Q.

Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
[CrossRef]

Thompson, K. W.

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

Townsend, D.

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Walsh, J.

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

Wang, C.

Z. Qiu, G. Zhang, C. Wang, and X. Li, “A study on the technology of drop analysis,” Trans. Tianjin Univ. 34, 587–590 (2001).

Wang, C. H.

C. H. Wang, A. T. Augousti, J. Mason, and N. D. McMillan, “The capacitive drop tensiometer—a novel multianalysing technique for measuring the properties of liquids,” Meas. Sci. Technol. 10, 19–24 (1999).
[CrossRef]

Wang, D.

J. Shi, M. Hui, D. Wang, and W. Mao, “Micromation and application of spectrometer,” Opt. Tech. 29, 13–20 (2003).

Wu, M.

M. Wu, Instrument Analysis (Academic, 2003).

Zhang, A.

Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
[CrossRef]

Zhang, G.

Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
[CrossRef]

Z. Qiu, H. Chen, and G. Zhang, “Photoelectric multiple-wavelength liquid drop sensor,” Instrum. Sci. Technol. 31, 305–315 (2003).
[CrossRef]

Z. Qiu, G. Zhang, C. Wang, and X. Li, “A study on the technology of drop analysis,” Trans. Tianjin Univ. 34, 587–590 (2001).

Instrum. Sci. Technol. (2)

N. D. McMillan, F. E. Feeney, M. J. P. Power, S. M. Kinsella, M. P. Kelly, C. Hammil, K. W. Thompson, and J. P. O’Dea, “A qualitative study of fibre drop traces on mechanically excited damped vibrations of drops of pure liquids using FFT analysis,” Instrum. Sci. Technol. 22, 375–395 (1994).
[CrossRef]

Q. Song, G. Zhang, Z. Qiu, A. Zhang, and Q. Shi, “Liquid signature analyzer based on fiber-capacitive drop analysis,” Instrum. Sci. Technol. 32, 43–59 (2004).
[CrossRef]

Meas. Sci. Technol. (2)

C. H. Wang, A. T. Augousti, J. Mason, and N. D. McMillan, “The capacitive drop tensiometer—a novel multianalysing technique for measuring the properties of liquids,” Meas. Sci. Technol. 10, 19–24 (1999).
[CrossRef]

N. D. McMillan, O. Finlayson, F. Fortune, M. Fingleton, D. Daly, D. Townsend, D. D. G. McMillan, and M. J. Dalton, “The fiber drop analyser: a new multianalyser analytical instrument with applications in sugar processing and for the analysis of pure liquids,” Meas. Sci. Technol. 3, 746–764(1992).
[CrossRef]

Opt. Eng. (1)

N. D. McMillan, E. O’Mongain, J. Walsh, L. Breen, D. G. E. McMillan, M. J. Power, J. P. O’Dea, S. M. Kinsella, M. P. Kelly, C. Hammil, and D. Orr, “Preliminary investigation into the analytical potential of a multiwavelength fiber drop analyzer with special reference to applications in medial diagnostics,” Opt. Eng. 33, 3871–3890 (1994).
[CrossRef]

Opt. Tech. (1)

J. Shi, M. Hui, D. Wang, and W. Mao, “Micromation and application of spectrometer,” Opt. Tech. 29, 13–20 (2003).

Trans. Tianjin Univ. (1)

Z. Qiu, G. Zhang, C. Wang, and X. Li, “A study on the technology of drop analysis,” Trans. Tianjin Univ. 34, 587–590 (2001).

Other (3)

Z. Qiu, “A study on the technology of drop analysis,” doctoral dissertation (Tianjin University of China, 2000).

Z. Qiu, H. Chen, and G. Zhang, “Photoelectric multiple-wavelength liquid drop sensor,” Instrum. Sci. Technol. 31, 305–315 (2003).
[CrossRef]

M. Wu, Instrument Analysis (Academic, 2003).

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 (9)

Fig. 1
Fig. 1

Schematic diagram of spectral liquid signature analyzer. I/O, input/output; analog/digital, A/D; F/V, fiber/voltage.

Fig. 2
Fig. 2

Liquid drop sensor.

Fig. 3
Fig. 3

Volume-based liquid drop trace of water.

Fig. 4
Fig. 4

Spectrogram of water.

Fig. 5
Fig. 5

Three-dimensional liquid fingerprint of water.

Fig. 6
Fig. 6

Three-dimensional liquid fingerprint of Kuer drink.

Fig. 7
Fig. 7

Three-dimensional liquid fingerprint of Jinjiu wine.

Fig. 8
Fig. 8

(a) Spectrogram of ethanol; (b) spectrogram of 2-propanol.

Fig. 9
Fig. 9

(a) Volume-based liquid drop trace of ethanol; (b) volume-based liquid drop trace of 2-propanol.

Equations (5)

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

W = ( w 1 w 2 w n ) T ,
A = ( a 1 a 2 a n ) T .
F = ( f 1 f 2 f m ) T ,
C = ( c 1 c 2 c m ) T ,
Z = F A T = ( f 1 a 1 f 1 a 2 f 1 a n f 2 a 1 f 2 a 2 f 2 a n f m a 1 f m a 2 f m a n ) .

Metrics