Abstract

Blind source separation of two electromagnetic fields is investigated. The difficulty of this task lies in the fact that only the power, which is the square of the sum of the electromagnetic fields, can be directly measured; the cross term of the electromagnetic fields is inevitable, and a strong correlation occurs in blind deconvolution. However, the relative phase is physically different from the field intensities, and, hence, extracting the phase during separation seems inconceivable. Our results demonstrate that the intensities and the relative phase of two electromagnetic waves can be determined with eigenvalue problem formalism even when the mixing processes are completely unknown.

© 2002 Optical Society of America

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References

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  1. J. Maddox, Nature 369, 517 (1994).
    [CrossRef]
  2. F. Cristopher and C. Morrisean, Proc. Xeme Coll. CRETSI, 1017 (1985).
  3. S. Williamson, M. Hoke, G. Stroink, and M. Kotani, eds., Advances in Biomagnetism (Plenum, New York, 1990).
  4. H. W. Strube, Signal Process. 3, 355 (1981).
    [CrossRef]
  5. C. Jutten and J. Herault, Signal Process. 24, 1 (1991).
    [CrossRef]
  6. J. J. Hopfield, Proc. Natl. Acad. Sci. USA 88, 6462 (1991).
    [CrossRef]
  7. A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing (Wiley, New York, 1995).
  8. A. J. Bell and T. J. Sejnowski, Neural Comput. 7, 1129 (1995).
    [CrossRef] [PubMed]
  9. P. Comon, Signal Process. 36, 287 (1994).
    [CrossRef]
  10. D. Yellin and E. Weinstein, IEEE Trans. Signal Process. 42, 2158 (1994).
    [CrossRef]
  11. S. Haykin, ed., Blind Deconvolution (Prentice-Hall, Englewood Cliffs, N.J., 1994).
  12. L. Molgedey and H. G. Schuster, Phys. Rev. Lett. 72, 3634 (1994).
    [CrossRef] [PubMed]
  13. J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
    [CrossRef]
  14. S. C. Yang, CDMA RF System Engineering (Artech House, Norwood, Mass., 1998); more-advanced information on the development of code-division multiple access can be found at http://www.3gpp.org .
  15. F. C. Moon, Chaotic and Fractal Dynamics (Wiley, New York, 1992).
    [CrossRef]
  16. V. Doya, O. Legrand, and F. Mortessagne, Opt. Lett. 26, 872 (2001).
    [CrossRef]
  17. K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Kluwer, Dordrecht, The Netherlands, 1999).
  18. Available, e.g., from New Focus, http://www.NewFocus.com .
  19. B. L. Shoop, Photonic Analog-to-Digital Conversion (Springer-Verlag, Berlin, 2001).
    [CrossRef]
  20. F. Ehlers and H. G. Schuster, IEEE Trans. Signal Process. 45, 2608 (1997).
    [CrossRef]

2001 (1)

1998 (1)

J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
[CrossRef]

1997 (1)

F. Ehlers and H. G. Schuster, IEEE Trans. Signal Process. 45, 2608 (1997).
[CrossRef]

1995 (1)

A. J. Bell and T. J. Sejnowski, Neural Comput. 7, 1129 (1995).
[CrossRef] [PubMed]

1994 (4)

P. Comon, Signal Process. 36, 287 (1994).
[CrossRef]

D. Yellin and E. Weinstein, IEEE Trans. Signal Process. 42, 2158 (1994).
[CrossRef]

L. Molgedey and H. G. Schuster, Phys. Rev. Lett. 72, 3634 (1994).
[CrossRef] [PubMed]

J. Maddox, Nature 369, 517 (1994).
[CrossRef]

1991 (2)

C. Jutten and J. Herault, Signal Process. 24, 1 (1991).
[CrossRef]

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 88, 6462 (1991).
[CrossRef]

1985 (1)

F. Cristopher and C. Morrisean, Proc. Xeme Coll. CRETSI, 1017 (1985).

1981 (1)

H. W. Strube, Signal Process. 3, 355 (1981).
[CrossRef]

Bell, A. J.

A. J. Bell and T. J. Sejnowski, Neural Comput. 7, 1129 (1995).
[CrossRef] [PubMed]

Chern, J.-L.

J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
[CrossRef]

Cichocki, A.

A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing (Wiley, New York, 1995).

Comon, P.

P. Comon, Signal Process. 36, 287 (1994).
[CrossRef]

Cristopher, F.

F. Cristopher and C. Morrisean, Proc. Xeme Coll. CRETSI, 1017 (1985).

Doya, V.

Ehlers, F.

F. Ehlers and H. G. Schuster, IEEE Trans. Signal Process. 45, 2608 (1997).
[CrossRef]

Herault, J.

C. Jutten and J. Herault, Signal Process. 24, 1 (1991).
[CrossRef]

Ho, M.-C.

J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
[CrossRef]

Hopfield, J. J.

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 88, 6462 (1991).
[CrossRef]

Hsu, R.-R.

J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
[CrossRef]

Jutten, C.

C. Jutten and J. Herault, Signal Process. 24, 1 (1991).
[CrossRef]

Ko, J.-Y.

J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
[CrossRef]

Legrand, O.

Maddox, J.

J. Maddox, Nature 369, 517 (1994).
[CrossRef]

Molgedey, L.

L. Molgedey and H. G. Schuster, Phys. Rev. Lett. 72, 3634 (1994).
[CrossRef] [PubMed]

Moon, F. C.

F. C. Moon, Chaotic and Fractal Dynamics (Wiley, New York, 1992).
[CrossRef]

Morrisean, C.

F. Cristopher and C. Morrisean, Proc. Xeme Coll. CRETSI, 1017 (1985).

Mortessagne, F.

Otsuka, K.

K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Kluwer, Dordrecht, The Netherlands, 1999).

Schuster, H. G.

F. Ehlers and H. G. Schuster, IEEE Trans. Signal Process. 45, 2608 (1997).
[CrossRef]

L. Molgedey and H. G. Schuster, Phys. Rev. Lett. 72, 3634 (1994).
[CrossRef] [PubMed]

Sejnowski, T. J.

A. J. Bell and T. J. Sejnowski, Neural Comput. 7, 1129 (1995).
[CrossRef] [PubMed]

Shoop, B. L.

B. L. Shoop, Photonic Analog-to-Digital Conversion (Springer-Verlag, Berlin, 2001).
[CrossRef]

Strube, H. W.

H. W. Strube, Signal Process. 3, 355 (1981).
[CrossRef]

Unbehauen, R.

A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing (Wiley, New York, 1995).

Wang, C.-S.

J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
[CrossRef]

Weinstein, E.

D. Yellin and E. Weinstein, IEEE Trans. Signal Process. 42, 2158 (1994).
[CrossRef]

Yang, S. C.

S. C. Yang, CDMA RF System Engineering (Artech House, Norwood, Mass., 1998); more-advanced information on the development of code-division multiple access can be found at http://www.3gpp.org .

Yellin, D.

D. Yellin and E. Weinstein, IEEE Trans. Signal Process. 42, 2158 (1994).
[CrossRef]

IEEE Trans. Signal Process. (2)

D. Yellin and E. Weinstein, IEEE Trans. Signal Process. 42, 2158 (1994).
[CrossRef]

F. Ehlers and H. G. Schuster, IEEE Trans. Signal Process. 45, 2608 (1997).
[CrossRef]

Nature (1)

J. Maddox, Nature 369, 517 (1994).
[CrossRef]

Neural Comput. (1)

A. J. Bell and T. J. Sejnowski, Neural Comput. 7, 1129 (1995).
[CrossRef] [PubMed]

Opt. Lett. (1)

Phys. Rev. E (1)

J.-Y. Ko, M.-C. Ho, J.-L. Chern, R.-R. Hsu, and C.-S. Wang, Phys. Rev. E 58, 4872 (1998).
[CrossRef]

Phys. Rev. Lett. (1)

L. Molgedey and H. G. Schuster, Phys. Rev. Lett. 72, 3634 (1994).
[CrossRef] [PubMed]

Proc. Natl. Acad. Sci. USA (1)

J. J. Hopfield, Proc. Natl. Acad. Sci. USA 88, 6462 (1991).
[CrossRef]

Proc. Xeme Coll. CRETSI (1)

F. Cristopher and C. Morrisean, Proc. Xeme Coll. CRETSI, 1017 (1985).

Signal Process. (3)

H. W. Strube, Signal Process. 3, 355 (1981).
[CrossRef]

C. Jutten and J. Herault, Signal Process. 24, 1 (1991).
[CrossRef]

P. Comon, Signal Process. 36, 287 (1994).
[CrossRef]

Other (8)

A. Cichocki and R. Unbehauen, Neural Networks for Optimization and Signal Processing (Wiley, New York, 1995).

S. Williamson, M. Hoke, G. Stroink, and M. Kotani, eds., Advances in Biomagnetism (Plenum, New York, 1990).

S. C. Yang, CDMA RF System Engineering (Artech House, Norwood, Mass., 1998); more-advanced information on the development of code-division multiple access can be found at http://www.3gpp.org .

F. C. Moon, Chaotic and Fractal Dynamics (Wiley, New York, 1992).
[CrossRef]

K. Otsuka, Nonlinear Dynamics in Optical Complex Systems (Kluwer, Dordrecht, The Netherlands, 1999).

Available, e.g., from New Focus, http://www.NewFocus.com .

B. L. Shoop, Photonic Analog-to-Digital Conversion (Springer-Verlag, Berlin, 2001).
[CrossRef]

S. Haykin, ed., Blind Deconvolution (Prentice-Hall, Englewood Cliffs, N.J., 1994).

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Figures (2)

Fig. 1
Fig. 1

Phase retrieval for two periodic functions with quasi-periodic relative phase. (a), (b), (c) Differences between the separated and the original signals, in terms of E1, E2, and ψ, respectively. (d), (e), (f) Detected power, Pi (where i=1, i=2, and i=3, respectively). (g) Average mismatch versus the variation in time delay.

Fig. 2
Fig. 2

Phase retrieval for complex waveforms of the Lorenz model. (a)–(c) Original waveforms: (a) E1, (b) E2, (c) relative phase. (d), (e), (f) Detected power Pi (where i=1, i=2, and i=3, respectively). (g), (h), (i) Corresponding differences between the original and the separated signals.

Equations (4)

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

P1P2P3=C112C122C11C12C212C222C21C22C312C322C31C32E12E222E1E2cos ϕ.
P1-P1P2-P2P3-P3=C112C122C11C12C212C222C21C22C312C322C31C32×E12-E12E22-E222E1E2cos ϕ,
i=13PiCi12x1min+Ci22x2min20.
C112C122C11C122=1=C212C222C21C222=C312C322C31C322.

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