## Reconstructing signals via stochastic resonance generated by photorefractive two-wave mixing bistability |

Optics Express, Vol. 22, Issue 4, pp. 4214-4223 (2014)

http://dx.doi.org/10.1364/OE.22.004214

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### Abstract

Stochastic resonance is theoretically investigated in an optical bistable system, which consists of a unidirectional ring cavity and a photorefractive two-wave mixer. It is found that the output properties of stochastic resonance are mainly determined by the applied noise, the crystal length and the applied electric field. The influences of these parameters on the stochastic resonance are also numerically analyzed via cross-correlation, which offers general guidelines for the optimization of recovering noise-hidden signals. A cross-correlation gain of 4 is obtained by optimizing these parameters. This provides a general method for reconstructing signals in nonlinear communications systems.

© 2014 Optical Society of America

## 1. Introduction

1. H. Chen, P. K. Varshney, S. M. Kay, and J. H. Michels, “Theory of the stochastic resonance effect in signal detection: Part I—fixed detectors,” IEEE Trans. Signal Processing **55**(7), 3172–3184 (2007). [CrossRef]

2. R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus **34**(1), 10–16 (1982). [CrossRef]

3. C. Nicolis, “Stochastic aspects of climatic transitions—response to a periodic forcing,” Tellus **34**(1), 1–9 (1982). [CrossRef]

4. S. Fauve and F. Heslot, “Stochastic resonance in a bistable system,” Phys. Lett. A **97**(1–2), 5–7 (1983). [CrossRef]

5. S. M. Bezrukov and I. Vodyanoy, “Stochastic resonance in non-dynamical systems without response thresholds,” Nature **385**(6614), 319–321 (1997). [CrossRef] [PubMed]

6. J. K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, “Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance,” Nature **365**(6444), 337–340 (1993). [CrossRef] [PubMed]

7. A. R. Bulsara, T. C. Elston, C. R. Doering, S. B. Lowen, and K. Lindenberg, “Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics **53**(4), 3958–3969 (1996). [CrossRef] [PubMed]

8. R. L. Badzey and P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” Nature **437**(7061), 995–998 (2005). [CrossRef] [PubMed]

9. H. B. Chan and C. Stambaugh, “Fluctuation-enhanced frequency mixing in a nonlinear micromechanical oscillator,” Phys. Rev. B **73**(22), 224301 (2006). [CrossRef]

11. F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, “Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering,” J. Opt. Soc. Am. B **15**(11), 2674–2680 (1998). [CrossRef]

12. B. M. Jost and B. E. A. Saleh, “Signal-to-noise ratio improvement by stochastic resonance in a unidirectional photorefractive ring resonator,” Opt. Lett. **21**(4), 287–289 (1996). [CrossRef] [PubMed]

13. S. Weiss and B. Fischer, “Photorefractive saturable absorptive and dispersive optical bistability,” Opt. Commun. **70**(6), 515–521 (1989). [CrossRef]

12. B. M. Jost and B. E. A. Saleh, “Signal-to-noise ratio improvement by stochastic resonance in a unidirectional photorefractive ring resonator,” Opt. Lett. **21**(4), 287–289 (1996). [CrossRef] [PubMed]

14. R. Daisy and B. Fischer, “Optical bistability in a nonlinear - linear interface with two-wave mixing,” Opt. Lett. **17**(12), 847–849 (1992). [CrossRef] [PubMed]

12. B. M. Jost and B. E. A. Saleh, “Signal-to-noise ratio improvement by stochastic resonance in a unidirectional photorefractive ring resonator,” Opt. Lett. **21**(4), 287–289 (1996). [CrossRef] [PubMed]

13. S. Weiss and B. Fischer, “Photorefractive saturable absorptive and dispersive optical bistability,” Opt. Commun. **70**(6), 515–521 (1989). [CrossRef]

## 2. Theory model of stochastic resonance in optical bistability

15. R. Bartussek, P. Hänggi, and P. Jung, “Stochastic resonance in optical bistable systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. **49**(5), 3930–3939 (1994). [CrossRef] [PubMed]

17. M. Misono, T. Kohmoto, Y. Fukuda, and M. Kunitomo, “Stochastic resonance in an optical bistable system driven by colored noise,” Opt. Commun. **152**(4), 255–258 (1998). [CrossRef]

_{0.6}Ba

_{0.4}Nb

_{2}O

_{6}(SBN:60) [18]. To describe the process of nonlinear intensity exchange between the pump and the signal light, we consider near-degenerate two-wave mixing in a unidirectional photorefractive ring cavity. The two waves interact in the photorefractive crystal via the photoinduced reversible refractive-index grating. The coupled wave equations for this case can be expressed as [19]where

*α*is the absorption coefficient,

*A*(

_{j}*j*= 1,2) are the complex amplitudes of the pump and signal,

*I*=

_{j}**|**

*A*

_{j}**|**

^{2}, and

*I*

_{0}(z) =

*I*

_{1}(z) +

*I*

_{2}(z) is the total light intensity at

*z*, respectively. The two beams symmetrically incident upon the photorefractive crystal with the incident angles of

*θ*

_{1}

*= θ*

_{2}

*= θ*. The complex coupling coefficient

*γ*is given by [20

20. S. K. Kwong, M. C. Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. **22**(8), 1508–1523 (1986). [CrossRef]

*r*is the relevant electrooptic coefficient,

_{eff}*n*

_{0}is the ordinary refractive index of the crystal,

*τ*

_{0}is the characteristic time, Ω =

*ω*

_{2}-

*ω*

_{1},

*E*

_{0}is the externally applied electric field parallel to the grating wave vector in the crystal,

*E*,

_{μ}*E*, and

_{D}*E*are internal electric fields characteristic of drift, diffusion, and maximum space charge, respectively. With

_{q}*A*=

_{j}*I*

_{j}^{1/2}exp(

*iψ*),

_{j}*ψ*, Γ = 2Re(

_{j}= k_{j}z + φ_{j}*γ*) and Γ′ = Im(

*γ*), the Eq. (1) can be rewritten asThe solutions of Eq. (3) are

*m*is the input intensity ratio

*m*=

*I*

_{2}(0)/

*I*

_{1}(0).

*T*and

*R*are the transmissivity and reflectivity of the beam splitter (

*T*= 1-

*R*), respectively. The cavity detuning

*δ*

_{0}is defined by

*δ*

_{0}= (

*ω*

_{2}-

*ω*)

_{c}**/**(

*c*

**/**

*L*),

*L*is the effective cavity length,

*ω*=

_{c}*q*(2

*πc*

**/**

*L*) is the cavity mode nearest to the frequency of the incident field

*ω*

_{2}and

*q*is an integer.

*φ*

_{2}

*= φ*

_{2}(

*l*)-

*φ*

_{2}(0) is the additional phase due to the nonlinear process and can be obtained from Eq. (4). For convenience, the normalized input intensity and output intensity are simplified as

*Y = I*

_{in}**/**[

*TI*

_{1}(0)] and

*X = I*

_{out}**/**[

*TI*

_{1}(0)], respectively. According to Eq. (7), the steady-state equation of optical bistability in the ring cavity can be rewritten asThe ratio

*m*(

*X*) is acquired from the second equation of Eq. (4)where

*a*and

*b*are defined as

*a*= exp(

*αl*) and

*b*= exp(Γ

*l*), respectively.

15. R. Bartussek, P. Hänggi, and P. Jung, “Stochastic resonance in optical bistable systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. **49**(5), 3930–3939 (1994). [CrossRef] [PubMed]

1. H. Chen, P. K. Varshney, S. M. Kay, and J. H. Michels, “Theory of the stochastic resonance effect in signal detection: Part I—fixed detectors,” IEEE Trans. Signal Processing **55**(7), 3172–3184 (2007). [CrossRef]

15. R. Bartussek, P. Hänggi, and P. Jung, “Stochastic resonance in optical bistable systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. **49**(5), 3930–3939 (1994). [CrossRef] [PubMed]

17. M. Misono, T. Kohmoto, Y. Fukuda, and M. Kunitomo, “Stochastic resonance in an optical bistable system driven by colored noise,” Opt. Commun. **152**(4), 255–258 (1998). [CrossRef]

21. R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A **18**(3), 1129–1144 (1978). [CrossRef]

*X*and input intensity

*Y*is given by where the potential function

*V*(

*X*) readsHaving chosen a value for the input pulse sequence intensity

*Y*(

*t*) =

*Y*

_{0}(

*t*) within the region of bistability, we then modulate the value with the additive noise. Both the input and output signals are normalized by the pump intensity, so

*D*is considered to be a normalized intensity. The normalized input intensity

*Y*in Eq. (11) should be substituted bywhere

*D*denotes the noise intensity,

*N*(

*t*) is the additive white Gaussian noise and the noise correlation

*X*can be described by the nonlinear Langevin equation.

## 3. Results and discussion

### 3.1. Properties of the optical bistability

*Y*

_{0}(

*t*) is less than 1. The transmissivity of the beam splitter is

*T =*0.5.

*M*

_{1}and

*M*

_{2}are total reflection mirrors as presented in Fig. 1. The electric field was applied antiparallel to the optic axis. The material parameters for the calculation of the complex coupling coefficient

*γ*are according to reference [18]. Moreover, the ring cavity without an injected signal can oscillate over a large range of cavity detuning [13

13. S. Weiss and B. Fischer, “Photorefractive saturable absorptive and dispersive optical bistability,” Opt. Commun. **70**(6), 515–521 (1989). [CrossRef]

22. P. Yeh, “Theory of unidirectional photorefractive ring oscillators,” J. Opt. Soc. Am. B **2**(12), 1924–1928 (1985). [CrossRef]

*δ*

_{0}= 0.01) and the boundary conditions are not satisfied to allow the oscillation to occur. To determine the optimum operating parameters for the optical bistability in the unidirectional ring cavity, the output property of the optical bistability with different interaction lengths and electric field intensities are numerically simulated and analyzed as illustrated in Fig. 3. Figure 3(a) shows the relationship between

*X*and

*Y*for various interaction lengths with

*E*

_{0}= 0 kV

**/**cm. It is found that there is only one stationary

*X*for a fixed

*Y*when the interaction length is 1 mm. The bistability appears with the increase of the interaction length, and three stationary

*X*appear in the bistable region, which are the stable lower branch, the stable upper branch, and the unstable intermediate branch. As seen from Fig. 3(b), the bistability region moves toward to the larger

*Y*direction when the electric field is applied. The bistability region is enlarged by tuning the interaction length and applied electric field. Starting with a weak signal beam, the photorefractive grating transfers a moderate percentage of the pump energy to the signal beam, and the output signal intensity increases slowly with the increase of the input signal intensity at the beginning. As the signal intensity increases further, the photorefractive grating reaches its saturation, the output signal jump to the upper branch. At present, if the signal intensity begins to reduce, the balance in the cavity would not be broken immediately. However, the output signal will jump to the low branch again when the incident light is faint.

*V*(

*X*), we get two minima of

*V*(

*X*) from Eq. (13) when the system exhibits optical bistability as illustrated in Fig. 4. The dynamic stochastic resonance system has two stable stationary states corresponding to the two minima of the potential, which generally is a double-well potential.

### 3.2. Numerical analysis of the stochastic resonance

*y*

_{0}= 0.05 is located in the bistable region illustrated by the green curve in Fig. 3(b),

*t*/1.777 and

_{g}= t_{p}*t*is the pulse width, respectively. The observation time is about 20 s, and the sampling period is 0.001 s in the calculations.

_{p}*Y*(

*t*) and output signals

*X*(

*t*). The normalized cross-correlation coefficients are defined as [23

23. J. Ma, D. Zeng, and H. Chen, “Spatial-temporal cross-correlation analysis: a new measure and a case study in infectious disease informatics,” in IEEE International Conference on Intelligence and Security Informatics (2006). [CrossRef]

*C*

_{Y}_{0}

*is close to 0 when the output signal is severely distorted, while it is close to 1 under the condition that the output signal*

_{X}*X*are very similar with the pure input signal

*Y*

_{0}.

*C*

_{Y}_{0}

*has the same variation trend as*

_{Y}*C*

_{Y}_{0}

_{X}.*D*are numerically calculated and analyzed. The normalized intensity of the input signal is set at

*Y*

_{0}= 0.05, and the noise intensity

*D*is gradually increased from zero in our simulations. Figure 6(a) clearly shows that

*C*

_{Y}_{0}

*and*

_{Y}*C*

_{Y}_{0}

*decrease as the noise intensity increases, and*

_{X}*C*

_{Y}_{0}

*decays much faster than*

_{Y}*C*

_{Y}_{0}

*when*

_{X}*D*is small. As seen from Fig. 6(b), the cross-correlation gain increases to a maximum value of 4 firstly, and then begins to drop dramatically. The reason is that the stochastic resonance system becomes increasingly dominated by the noise, which leads to the distortion of extracted signals. Too little noise disables the reversal of the potential well, but too much noise leads to excessive distortion. Therefore, there exists an optimal noise level for acquiring the maximum cross-correlation gain that can be realized by adjusting the input noise intensity. This is the signature of the stochastic resonance effect, which is caused by the interaction between the random noise and weak signal in the potential well.

*γ*at different applied electric fields

*E*

_{0}are acquired as shown in Fig. 8 (a). Figure 8(b) presents the curve of

*-Γ′*

*/**Γ*versus

*E*

_{0}calculated from Eq. (4)(d). The dependence of stochastic resonance on the parameter of

*E*

_{0}can be clearly seen from Figs. 7(b) and 8(b). It is found that the maximum of the cross-correlation gain and the value of

*-Γ′*

*/**Γ*can be achieved at nearly the same

*E*

_{0}. As a result, the stochastic resonance effect performed by tuning the parameter of

*E*

_{0}is mainly depending on

*-Γ′*

*/**Γ*.

## 4. Conclusion

## Acknowledgments

## References and links

1. | H. Chen, P. K. Varshney, S. M. Kay, and J. H. Michels, “Theory of the stochastic resonance effect in signal detection: Part I—fixed detectors,” IEEE Trans. Signal Processing |

2. | R. Benzi, G. Parisi, A. Sutera, and A. Vulpiani, “Stochastic resonance in climatic change,” Tellus |

3. | C. Nicolis, “Stochastic aspects of climatic transitions—response to a periodic forcing,” Tellus |

4. | S. Fauve and F. Heslot, “Stochastic resonance in a bistable system,” Phys. Lett. A |

5. | S. M. Bezrukov and I. Vodyanoy, “Stochastic resonance in non-dynamical systems without response thresholds,” Nature |

6. | J. K. Douglass, L. Wilkens, E. Pantazelou, and F. Moss, “Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance,” Nature |

7. | A. R. Bulsara, T. C. Elston, C. R. Doering, S. B. Lowen, and K. Lindenberg, “Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics |

8. | R. L. Badzey and P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” Nature |

9. | H. B. Chan and C. Stambaugh, “Fluctuation-enhanced frequency mixing in a nonlinear micromechanical oscillator,” Phys. Rev. B |

10. | D. V. Dylov and J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nature |

11. | F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, and F. Chapeau-Blondeau, “Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering,” J. Opt. Soc. Am. B |

12. | B. M. Jost and B. E. A. Saleh, “Signal-to-noise ratio improvement by stochastic resonance in a unidirectional photorefractive ring resonator,” Opt. Lett. |

13. | S. Weiss and B. Fischer, “Photorefractive saturable absorptive and dispersive optical bistability,” Opt. Commun. |

14. | R. Daisy and B. Fischer, “Optical bistability in a nonlinear - linear interface with two-wave mixing,” Opt. Lett. |

15. | R. Bartussek, P. Hänggi, and P. Jung, “Stochastic resonance in optical bistable systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. |

16. | L. Zhang, L. Cao, and D. J. Wu, “Effect of correlated noises in an optical bistable system,” Phys. Rev. A |

17. | M. Misono, T. Kohmoto, Y. Fukuda, and M. Kunitomo, “Stochastic resonance in an optical bistable system driven by colored noise,” Opt. Commun. |

18. | J. E. Ford, J. Ma, Y. Fainman, and S. H. Lee, “Multiplex holography in strontium barium niobate with applied field,” J. Opt. Soc. Am. B |

19. | M. Cronin-Golomb, B. Fischer, J. O. White, and A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. |

20. | S. K. Kwong, M. C. Golomb, and A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. |

21. | R. Bonifacio and L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A |

22. | P. Yeh, “Theory of unidirectional photorefractive ring oscillators,” J. Opt. Soc. Am. B |

23. | J. Ma, D. Zeng, and H. Chen, “Spatial-temporal cross-correlation analysis: a new measure and a case study in infectious disease informatics,” in IEEE International Conference on Intelligence and Security Informatics (2006). [CrossRef] |

24. | V. S. Anishchenko, M. A. Safonova, and L. O. Chua, “Stochastic resonance in the nonautonomous Chua's circuit,” J. Circuits Syst. Comput. |

25. | B. Xu, Z. P. Jiang, X. Wu, and D. W. Repperger, “Theoretical analysis of image processing using parameter-tuning stochastic resonance technique,” in |

**OCIS Codes**

(000.5490) General : Probability theory, stochastic processes, and statistics

(190.0190) Nonlinear optics : Nonlinear optics

(190.1450) Nonlinear optics : Bistability

(190.7070) Nonlinear optics : Two-wave mixing

**ToC Category:**

Nonlinear Optics

**History**

Original Manuscript: October 29, 2013

Revised Manuscript: December 26, 2013

Manuscript Accepted: December 31, 2013

Published: February 18, 2014

**Citation**

Guangzhan Cao, Hongjun Liu, Xuefeng Li, Nan Huang, and Qibing Sun, "Reconstructing signals via stochastic resonance generated by photorefractive two-wave mixing bistability," Opt. Express **22**, 4214-4223 (2014)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-22-4-4214

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### References

- H. Chen, P. K. Varshney, S. M. Kay, J. H. Michels, “Theory of the stochastic resonance effect in signal detection: Part I—fixed detectors,” IEEE Trans. Signal Processing 55(7), 3172–3184 (2007). [CrossRef]
- R. Benzi, G. Parisi, A. Sutera, A. Vulpiani, “Stochastic resonance in climatic change,” Tellus 34(1), 10–16 (1982). [CrossRef]
- C. Nicolis, “Stochastic aspects of climatic transitions—response to a periodic forcing,” Tellus 34(1), 1–9 (1982). [CrossRef]
- S. Fauve, F. Heslot, “Stochastic resonance in a bistable system,” Phys. Lett. A 97(1–2), 5–7 (1983). [CrossRef]
- S. M. Bezrukov, I. Vodyanoy, “Stochastic resonance in non-dynamical systems without response thresholds,” Nature 385(6614), 319–321 (1997). [CrossRef] [PubMed]
- J. K. Douglass, L. Wilkens, E. Pantazelou, F. Moss, “Noise enhancement of information transfer in crayfish mechanoreceptors by stochastic resonance,” Nature 365(6444), 337–340 (1993). [CrossRef] [PubMed]
- A. R. Bulsara, T. C. Elston, C. R. Doering, S. B. Lowen, K. Lindenberg, “Cooperative behavior in periodically driven noisy integrate-fire models of neuronal dynamics,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Topics 53(4), 3958–3969 (1996). [CrossRef] [PubMed]
- R. L. Badzey, P. Mohanty, “Coherent signal amplification in bistable nanomechanical oscillators by stochastic resonance,” Nature 437(7061), 995–998 (2005). [CrossRef] [PubMed]
- H. B. Chan, C. Stambaugh, “Fluctuation-enhanced frequency mixing in a nonlinear micromechanical oscillator,” Phys. Rev. B 73(22), 224301 (2006). [CrossRef]
- D. V. Dylov, J. W. Fleischer, “Nonlinear self-filtering of noisy images via dynamical stochastic resonance,” Nature 4, 323–328 (1993).
- F. Vaudelle, J. Gazengel, G. Rivoire, X. Godivier, F. Chapeau-Blondeau, “Stochastic resonance and noise-enhanced transmission of spatial signals in optics: the case of scattering,” J. Opt. Soc. Am. B 15(11), 2674–2680 (1998). [CrossRef]
- B. M. Jost, B. E. A. Saleh, “Signal-to-noise ratio improvement by stochastic resonance in a unidirectional photorefractive ring resonator,” Opt. Lett. 21(4), 287–289 (1996). [CrossRef] [PubMed]
- S. Weiss, B. Fischer, “Photorefractive saturable absorptive and dispersive optical bistability,” Opt. Commun. 70(6), 515–521 (1989). [CrossRef]
- R. Daisy, B. Fischer, “Optical bistability in a nonlinear - linear interface with two-wave mixing,” Opt. Lett. 17(12), 847–849 (1992). [CrossRef] [PubMed]
- R. Bartussek, P. Hänggi, P. Jung, “Stochastic resonance in optical bistable systems,” Phys. Rev. E Stat. Phys. Plasmas Fluids Relat. Interdiscip. Top. 49(5), 3930–3939 (1994). [CrossRef] [PubMed]
- L. Zhang, L. Cao, D. J. Wu, “Effect of correlated noises in an optical bistable system,” Phys. Rev. A 77(1), 015801 (2008). [CrossRef]
- M. Misono, T. Kohmoto, Y. Fukuda, M. Kunitomo, “Stochastic resonance in an optical bistable system driven by colored noise,” Opt. Commun. 152(4), 255–258 (1998). [CrossRef]
- J. E. Ford, J. Ma, Y. Fainman, S. H. Lee, “Multiplex holography in strontium barium niobate with applied field,” J. Opt. Soc. Am. B 9(7), 1183–1192 (1992).
- M. Cronin-Golomb, B. Fischer, J. O. White, A. Yariv, “Theory and applications of four-wave mixing in photorefractive media,” IEEE J. Quantum Electron. 20, 12–30 (1984).
- S. K. Kwong, M. C. Golomb, A. Yariv, “Oscillation with photorefractive gain,” IEEE J. Quantum Electron. 22(8), 1508–1523 (1986). [CrossRef]
- R. Bonifacio, L. A. Lugiato, “Optical bistability and cooperative effects in resonance fluorescence,” Phys. Rev. A 18(3), 1129–1144 (1978). [CrossRef]
- P. Yeh, “Theory of unidirectional photorefractive ring oscillators,” J. Opt. Soc. Am. B 2(12), 1924–1928 (1985). [CrossRef]
- J. Ma, D. Zeng, H. Chen, “Spatial-temporal cross-correlation analysis: a new measure and a case study in infectious disease informatics,” in IEEE International Conference on Intelligence and Security Informatics (2006). [CrossRef]
- V. S. Anishchenko, M. A. Safonova, L. O. Chua, “Stochastic resonance in the nonautonomous Chua's circuit,” J. Circuits Syst. Comput. 3(2), 553–578 (1993). [CrossRef]
- B. Xu, Z. P. Jiang, X. Wu, and D. W. Repperger, “Theoretical analysis of image processing using parameter-tuning stochastic resonance technique,” in IEEE American Control Conference ACC'07 (2007), pp. 1747–1752. [CrossRef]

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