Abstract

We experimentally demonstrate blind source separation of chaos generated in Nd:YVO4 microchip solid-state lasers by using independent component analysis. Two chaotic source signals are linearly mixed with randomly selected mixing ratio and independent component analysis is applied for the mixed signals to extract the source signals. We investigate blind source separation of many chaotic laser signals and succeed 100- signal separation of chaotic temporal waveforms. Longer temporal waveforms are required with increase of the number of mixed signals.

© 2008 Optical Society of America

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  1. I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, "High-dimensional chaotic dynamics of an external cavity semiconductor laser," Phys. Rev. Lett. 73,2188-2191 (1994).
    [CrossRef] [PubMed]
  2. J.-Y. Ko, K. Otsuka, and T. Kubota, "Quantum-noise-induced order in lasers placed in chaotic oscillation by frequency-shifted feedback," Phys. Rev. Lett. 86,4025-4028 (2001).
    [CrossRef] [PubMed]
  3. M. P. Kennedy, R. Rovatti, and G. Setti, "Chaotic Electronics in Telecommunications," CRC Press, Boca Raton, 2000.
  4. K. Umeno and A. Yamaguchi, "Construction of optimal chaotic spreading sequence using Lebesgue spectrum filter," IEICE Trans. FundamentalsE 85-A,849-852 (2002).
  5. K. Umeno, "Independent component analysis of mixed chaotic signals for communications systems," Nonlinear Phenom. Complex Syst. 10,170-175 (2007).
  6. Y. Liu and P. Davis, "Dual synchronization of chaos," Phys. Rev. E 61,R2176-R2179 (2000).
    [CrossRef]
  7. E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, "Dual and dual-cross synchronizations in chaotic systems," Opt. Commun. 216,179-183 (2003).
    [CrossRef]
  8. A. Uchida, M. Kawano, and S. Yoshimori, "Dual synchronization of chaos in Colpitts electronic oscillators and its applications for communications," Phys. Rev. E 68, 056207-1—056207-11 (2003).
    [CrossRef]
  9. P. Arena, A. Buscarino, L. Fortuna, and M. Frasca, "Separation and synchronization of piecewise linear chaotic systems," Phys. Rev. E 74, 026212-1—026212-11 (2006).
    [CrossRef]
  10. S. Sano, A. Uchida, S. Yoshimori, and R. Roy, "Dual synchronization of chaos in Mackey-Glass electronic circuits with time-delayed feedback," Phys. Rev. E 75, 016207-1—016207-6 (2007).
    [CrossRef]
  11. A. Uchida, S. Kinugawa, T. Matsuura, and S. Yoshimori, "Dual synchronization of chaos in microchip lasers," Opt. Lett. 28,19-21 (2003).
    [CrossRef] [PubMed]
  12. A. Uchida, S. Kinugawa, T. Matsuura, and S. Yoshimori, "Dual synchronization of chaos in one-way coupled microchip lasers," Phys. Rev. E 67, 026220-1—026220-8 (2003).
    [CrossRef]
  13. L. Molgedey and H. G. Schuster, "Separation of a mixture of independent signals using time delayed correlations," Phys. Rev. Lett. 72,3634-3637 (1994).
    [CrossRef] [PubMed]
  14. A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (John Wiley and Sons, New York, 2001).
    [CrossRef]
  15. A. Hyvärinen and E. Oja, "A fast fixed-point algorithm for independent component analysis," Neural Comput. 9,1483-1492 (1997).
    [CrossRef]

2007 (1)

K. Umeno, "Independent component analysis of mixed chaotic signals for communications systems," Nonlinear Phenom. Complex Syst. 10,170-175 (2007).

2003 (2)

E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, "Dual and dual-cross synchronizations in chaotic systems," Opt. Commun. 216,179-183 (2003).
[CrossRef]

A. Uchida, S. Kinugawa, T. Matsuura, and S. Yoshimori, "Dual synchronization of chaos in microchip lasers," Opt. Lett. 28,19-21 (2003).
[CrossRef] [PubMed]

2002 (1)

K. Umeno and A. Yamaguchi, "Construction of optimal chaotic spreading sequence using Lebesgue spectrum filter," IEICE Trans. FundamentalsE 85-A,849-852 (2002).

2001 (1)

J.-Y. Ko, K. Otsuka, and T. Kubota, "Quantum-noise-induced order in lasers placed in chaotic oscillation by frequency-shifted feedback," Phys. Rev. Lett. 86,4025-4028 (2001).
[CrossRef] [PubMed]

2000 (1)

Y. Liu and P. Davis, "Dual synchronization of chaos," Phys. Rev. E 61,R2176-R2179 (2000).
[CrossRef]

1997 (1)

A. Hyvärinen and E. Oja, "A fast fixed-point algorithm for independent component analysis," Neural Comput. 9,1483-1492 (1997).
[CrossRef]

1994 (2)

L. Molgedey and H. G. Schuster, "Separation of a mixture of independent signals using time delayed correlations," Phys. Rev. Lett. 72,3634-3637 (1994).
[CrossRef] [PubMed]

I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, "High-dimensional chaotic dynamics of an external cavity semiconductor laser," Phys. Rev. Lett. 73,2188-2191 (1994).
[CrossRef] [PubMed]

Davis, P.

Y. Liu and P. Davis, "Dual synchronization of chaos," Phys. Rev. E 61,R2176-R2179 (2000).
[CrossRef]

Elsäßer, W.

I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, "High-dimensional chaotic dynamics of an external cavity semiconductor laser," Phys. Rev. Lett. 73,2188-2191 (1994).
[CrossRef] [PubMed]

Fischer, I.

I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, "High-dimensional chaotic dynamics of an external cavity semiconductor laser," Phys. Rev. Lett. 73,2188-2191 (1994).
[CrossRef] [PubMed]

Göbel, E.

I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, "High-dimensional chaotic dynamics of an external cavity semiconductor laser," Phys. Rev. Lett. 73,2188-2191 (1994).
[CrossRef] [PubMed]

Hess, O.

I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, "High-dimensional chaotic dynamics of an external cavity semiconductor laser," Phys. Rev. Lett. 73,2188-2191 (1994).
[CrossRef] [PubMed]

Hyvärinen, A.

A. Hyvärinen and E. Oja, "A fast fixed-point algorithm for independent component analysis," Neural Comput. 9,1483-1492 (1997).
[CrossRef]

Kinugawa, S.

Ko, J.-Y.

J.-Y. Ko, K. Otsuka, and T. Kubota, "Quantum-noise-induced order in lasers placed in chaotic oscillation by frequency-shifted feedback," Phys. Rev. Lett. 86,4025-4028 (2001).
[CrossRef] [PubMed]

Kubota, T.

J.-Y. Ko, K. Otsuka, and T. Kubota, "Quantum-noise-induced order in lasers placed in chaotic oscillation by frequency-shifted feedback," Phys. Rev. Lett. 86,4025-4028 (2001).
[CrossRef] [PubMed]

Liu, Y.

Y. Liu and P. Davis, "Dual synchronization of chaos," Phys. Rev. E 61,R2176-R2179 (2000).
[CrossRef]

Matsuura, T.

Molgedey, L.

L. Molgedey and H. G. Schuster, "Separation of a mixture of independent signals using time delayed correlations," Phys. Rev. Lett. 72,3634-3637 (1994).
[CrossRef] [PubMed]

Oja, E.

A. Hyvärinen and E. Oja, "A fast fixed-point algorithm for independent component analysis," Neural Comput. 9,1483-1492 (1997).
[CrossRef]

Otsuka, K.

J.-Y. Ko, K. Otsuka, and T. Kubota, "Quantum-noise-induced order in lasers placed in chaotic oscillation by frequency-shifted feedback," Phys. Rev. Lett. 86,4025-4028 (2001).
[CrossRef] [PubMed]

Schuster, H. G.

L. Molgedey and H. G. Schuster, "Separation of a mixture of independent signals using time delayed correlations," Phys. Rev. Lett. 72,3634-3637 (1994).
[CrossRef] [PubMed]

Shahverdiev, E. M.

E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, "Dual and dual-cross synchronizations in chaotic systems," Opt. Commun. 216,179-183 (2003).
[CrossRef]

Shore, K. A.

E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, "Dual and dual-cross synchronizations in chaotic systems," Opt. Commun. 216,179-183 (2003).
[CrossRef]

Sivaprakasam, S.

E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, "Dual and dual-cross synchronizations in chaotic systems," Opt. Commun. 216,179-183 (2003).
[CrossRef]

Uchida, A.

Umeno, K.

K. Umeno, "Independent component analysis of mixed chaotic signals for communications systems," Nonlinear Phenom. Complex Syst. 10,170-175 (2007).

K. Umeno and A. Yamaguchi, "Construction of optimal chaotic spreading sequence using Lebesgue spectrum filter," IEICE Trans. FundamentalsE 85-A,849-852 (2002).

Yamaguchi, A.

K. Umeno and A. Yamaguchi, "Construction of optimal chaotic spreading sequence using Lebesgue spectrum filter," IEICE Trans. FundamentalsE 85-A,849-852 (2002).

Yoshimori, S.

E (1)

K. Umeno and A. Yamaguchi, "Construction of optimal chaotic spreading sequence using Lebesgue spectrum filter," IEICE Trans. FundamentalsE 85-A,849-852 (2002).

Neural Comput. (1)

A. Hyvärinen and E. Oja, "A fast fixed-point algorithm for independent component analysis," Neural Comput. 9,1483-1492 (1997).
[CrossRef]

Nonlinear Phenom. Complex Syst. (1)

K. Umeno, "Independent component analysis of mixed chaotic signals for communications systems," Nonlinear Phenom. Complex Syst. 10,170-175 (2007).

Opt. Commun. (1)

E. M. Shahverdiev, S. Sivaprakasam, and K. A. Shore, "Dual and dual-cross synchronizations in chaotic systems," Opt. Commun. 216,179-183 (2003).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. E (1)

Y. Liu and P. Davis, "Dual synchronization of chaos," Phys. Rev. E 61,R2176-R2179 (2000).
[CrossRef]

Phys. Rev. Lett. (3)

I. Fischer, O. Hess, W. Elsäßer, and E. Göbel, "High-dimensional chaotic dynamics of an external cavity semiconductor laser," Phys. Rev. Lett. 73,2188-2191 (1994).
[CrossRef] [PubMed]

J.-Y. Ko, K. Otsuka, and T. Kubota, "Quantum-noise-induced order in lasers placed in chaotic oscillation by frequency-shifted feedback," Phys. Rev. Lett. 86,4025-4028 (2001).
[CrossRef] [PubMed]

L. Molgedey and H. G. Schuster, "Separation of a mixture of independent signals using time delayed correlations," Phys. Rev. Lett. 72,3634-3637 (1994).
[CrossRef] [PubMed]

Other (6)

A. Hyvärinen, J. Karhunen, and E. Oja, Independent Component Analysis (John Wiley and Sons, New York, 2001).
[CrossRef]

M. P. Kennedy, R. Rovatti, and G. Setti, "Chaotic Electronics in Telecommunications," CRC Press, Boca Raton, 2000.

A. Uchida, M. Kawano, and S. Yoshimori, "Dual synchronization of chaos in Colpitts electronic oscillators and its applications for communications," Phys. Rev. E 68, 056207-1—056207-11 (2003).
[CrossRef]

P. Arena, A. Buscarino, L. Fortuna, and M. Frasca, "Separation and synchronization of piecewise linear chaotic systems," Phys. Rev. E 74, 026212-1—026212-11 (2006).
[CrossRef]

S. Sano, A. Uchida, S. Yoshimori, and R. Roy, "Dual synchronization of chaos in Mackey-Glass electronic circuits with time-delayed feedback," Phys. Rev. E 75, 016207-1—016207-6 (2007).
[CrossRef]

A. Uchida, S. Kinugawa, T. Matsuura, and S. Yoshimori, "Dual synchronization of chaos in one-way coupled microchip lasers," Phys. Rev. E 67, 026220-1—026220-8 (2003).
[CrossRef]

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Figures (5)

Fig. 1.
Fig. 1.

Experimental setup of Nd:YVO4 microchip solid-state lasers with external modulation for blind source separation by ICA. BS, beam splitter; GP, glass plate; L, lens; M, mirror; and PD, photodiode.

Fig. 2.
Fig. 2.

Experimental results of blind source separation of chaotic laser signals by using ICA. (a) Mixed signals 1 and 2, (b) separated signal 1 and source signal 1, and (c) separated signal 2 and source signal 2.

Fig. 3.
Fig. 3.

Correlation plots between one of the separated signals and one of the source signals. (a) Source signal 1 and separated signal 1, (b) source signal 1 and separated signal 2, (c) source signal 2 and separated signal 1, and (d) source signal 2 and separated signal 2.

Fig. 4.
Fig. 4.

(a) The length of temporal waveform T L and (b) the number of data point N d (=T L f s) required for successful signal separation as a function of the number of mixed signals N m at different data-sampling frequencies f s for the microchip lasers. The curves correspond to the condition at which the average value of cross correlations between the source and successfully separated signals is 0.95. The fundamental frequency of chaotic laser signals is 3.25 MHz. Solid black curve; f s=100 MHz, dashed blue curve; f s=50 MHz, dotted red curve; f s=25 MHz, and dotted-dashed green curve; f s=12.5 MHz.

Fig. 5.
Fig. 5.

Probability density function (PDF) of the source, two-mixed, and 50-mixed signals obtained from the microchip lasers. Solid black curve; source signal, dashed blue curve; twomixed signal, dotted red curve; 50-mixed signal, and dotted-dashed green curve; Gaussian distribution for reference. The kurtosis is 7.858 for the source signal, 4.647 for the two mixed signal, and 0.0841 for 50 mixed signals.

Equations (1)

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C = ( I 1 I ¯ 1 ) ( I 2 I ¯ 2 ) σ 1 σ 2

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