基于Fisher信息量的弱信号处理增益问题研究
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摘要
随机共振是研究一些非线性系统中噪声的积极建设性作用的一类物理现象。本文在深入研究随机共振和循环平稳理论的基础上,在弱周期信号条件下,利用信噪比和费舍尔信息量进一步研究随机共振现象,并且找到了二者之间的关联。费舍尔信息量能够描述几个重要非线性处理过程中的性能；一个局部最优处理器能够获得最大输出输入信噪比,最大输出输入信噪比增益是由标准噪声分布的费舍尔信息量给定的,并且最大信噪比增益是静态非线性元素组成的阵列的信噪比增益的上限。在本论文中,又进一步的对静态和动态非线性系统的随机共振现象进行了对比。论文的主要研究成果如下：
1.最初费舍尔信息量是作为参数估计的性能指标。我们将它扩展并且表明费舍尔信息量能够描述几个重要非线性处理过程中的性能。对于加性白噪声中的弱信号,费舍尔信息量能决定如下四个方面：(i)周期信号的最大输出信噪比；(ii)信号检测的最优渐近性能；(iii)信号传输的最优互相关系数；(iv)无偏估计值的最小均方差。通过费舍尔信息量不等式,这个统一的结论用于建立通过噪声改善随机共振是否可行的条件。
2.通过噪声概率密度和噪声强度能精确地决定一个局部最优处理器,并且局部最优处理器的输出输入信噪比增益是由标准噪声分布的费舍尔信息量给定的。基于这个关联,我们发现对于局部最优处理器,能够获得比一任意大的信噪比增益。对于随机共振,考虑向已知信号中加入额外噪声时,我们证明了通过费舍尔信息量不等式,和新噪声完全匹配的更新的局部最优处理器,不能改进输出信噪比以超过无额外噪声时所对应的初始值。这个结果印证了一个以前只对高斯噪声存在的定理。此外,在参数不可调处理器的情况下,比如由噪声概率密度描述的局部最优处理器的结构不能完全适应噪声强度时,表明了可以恢复随机共振的一般条件,通过添加额外声提高输出信噪比的可能性来证明。
3.研究了为传输在加性白噪声中的弱周期信号,由任意的静态非线性元素组成的非耦合并联阵列的输出输入信噪比增益。在小信号的限制条件下,推导出信噪比增益的一个渐近表达式。并且证明了对任意给定的非线性系统和噪声环境,信噪比增益是关于阵列大小的单调递增的函数。由局部最优非线性系统所对应的信噪比增益,是静态非线性元素组成的阵列的信噪比增益的上限。在局部最优非线性系统中,随机共振不能发生,也就是说,在阵列中加入内部噪声不能改善信噪比增益。然而,在一个由次优但易实现的阈值非线性系统组成的阵列中,我们证明了随机共振发生的可行性,也证明了对于各种内部噪声分布,信噪比增益大于一的可能性。
4.利用输出信噪比作为测量方法,比较了静态和动态非线性系统的随机共振现象。对于给定的含噪弱周期信号,通过调谐内部噪声强度,静态和动态非线性并联阵列都能提高输出信噪比。静态非线性系统容易实现,而动态非线性系统有较多参数需要调整,存在不能利用内部噪声的有利作用的风险。并且外部噪声是非高斯类型时,可以观察到动态非线性系统是优于静态非线性系统,可以获得一个更好的输出信噪比,证明了加入额外白噪声以提高输出信噪比的可能性。
Stochastic resonance (SR) is a nonlinear phenomenon where the transmission of a coherent signal by certain nonlinear systems can be improved by the addition of noise. On the basis of stochastic resonance and cyclostationary theory, in the small-signal limit we find the relationship between signal-to-noise (SNR) and fisher information and extend the stochatic resonance theoretical framework. We show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing a weak periodic signal in additive white noise, a locally optimal processor (LOP) achieves the maximal output SNR. It is shown that the output-input SNR gain of a LOP is given by the Fisher information of a standardized noise distribution. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. We compare the SR effects in an array of static and dynamical nonlinearities via the measure of the output SNR. The main results of the thesis are summarized as follows:
1. The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (ⅰ) the maximum output signal-to-noise ratio for a periodic signal;(ⅱ) the optimum asymptotic efficacy for signal detection;(ⅲ) the best cross-correlation coefficient for signal transmission; and (ⅳ) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.
2. For processing a weak periodic signal in additive white noise, a LOP achieves the maximal output SNR. In general, such a LOP is precisely determined by the noise probability density and also by the noise level. It is shown that the output-input SNR gain of a LOP is given by the Fisher information of a standardized noise distribution. Based on this connection, we find that an arbitrarily large SNR gain, for a LOP, can be achieved ranging from the minimal value of unity upwards. For stochastic resonance, when considering adding extra noise to the original signal, we here demonstrate via the appropriate Fisher information inequality that the updated LOP fully matched to the new noise, is unable to improve the output SNR above its original value with no extra noise. This result generalizes a proof that existed previously only for Gaussian noise. Furthermore, in the situation of non-adjustable processors, for instance when the structure of the LOP as prescribed by the noise probability density is not fully adaptable to the noise level, we show general conditions where stochastic resonance can be recovered, manifested by the possibility of adding extra noise to enhance the output SNR.
3. We study the output-input SNR gain of an uncoupled parallel array of static, yet arbitrary, nonlinear elements for transmitting a weak periodic signal in additive white noise. In the small-signal limit, an explicit expression for the SNR gain is derived. It serves to prove that the SNR gain is always a monotonically increasing function of the array size for any given nonlinearity and noisy environment. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. With locally optimal nonlinearity, it is demonstrated that stochastic resonance cannot occur, i.e. adding internal noise into the array never improves the SNR gain. However, in an array of suboptimal but easily implemented threshold nonlinearities, we show the feasibility of situations where stochastic resonance occurs, and also the possibility of the SNR gain exceeding unity for a wide range of input noise distributions.
4. We compare the SR effects in an array of static and dynamical nonlinearities via the measure of the output SNR. For a given noisy periodic signal, parallel arrays of both static and dynamical nonlinearities can enhance the output SNR by optimally tuning the internal noise level. The static nonlinearity is easily implementable, while the dynamical nonlinearity has more parameters to be tuned, at the risk of not exploiting the beneficial role of internal noise terms. More interestingly, as the input signal buried in external Laplacian noise, it is observed that the dynamical nonlinearity is superior to the static nonlinearity in obtaining a better output SNR.
1.最初费舍尔信息量是作为参数估计的性能指标。我们将它扩展并且表明费舍尔信息量能够描述几个重要非线性处理过程中的性能。对于加性白噪声中的弱信号,费舍尔信息量能决定如下四个方面：(i)周期信号的最大输出信噪比；(ii)信号检测的最优渐近性能；(iii)信号传输的最优互相关系数；(iv)无偏估计值的最小均方差。通过费舍尔信息量不等式,这个统一的结论用于建立通过噪声改善随机共振是否可行的条件。
2.通过噪声概率密度和噪声强度能精确地决定一个局部最优处理器,并且局部最优处理器的输出输入信噪比增益是由标准噪声分布的费舍尔信息量给定的。基于这个关联,我们发现对于局部最优处理器,能够获得比一任意大的信噪比增益。对于随机共振,考虑向已知信号中加入额外噪声时,我们证明了通过费舍尔信息量不等式,和新噪声完全匹配的更新的局部最优处理器,不能改进输出信噪比以超过无额外噪声时所对应的初始值。这个结果印证了一个以前只对高斯噪声存在的定理。此外,在参数不可调处理器的情况下,比如由噪声概率密度描述的局部最优处理器的结构不能完全适应噪声强度时,表明了可以恢复随机共振的一般条件,通过添加额外声提高输出信噪比的可能性来证明。
3.研究了为传输在加性白噪声中的弱周期信号,由任意的静态非线性元素组成的非耦合并联阵列的输出输入信噪比增益。在小信号的限制条件下,推导出信噪比增益的一个渐近表达式。并且证明了对任意给定的非线性系统和噪声环境,信噪比增益是关于阵列大小的单调递增的函数。由局部最优非线性系统所对应的信噪比增益,是静态非线性元素组成的阵列的信噪比增益的上限。在局部最优非线性系统中,随机共振不能发生,也就是说,在阵列中加入内部噪声不能改善信噪比增益。然而,在一个由次优但易实现的阈值非线性系统组成的阵列中,我们证明了随机共振发生的可行性,也证明了对于各种内部噪声分布,信噪比增益大于一的可能性。
4.利用输出信噪比作为测量方法,比较了静态和动态非线性系统的随机共振现象。对于给定的含噪弱周期信号,通过调谐内部噪声强度,静态和动态非线性并联阵列都能提高输出信噪比。静态非线性系统容易实现,而动态非线性系统有较多参数需要调整,存在不能利用内部噪声的有利作用的风险。并且外部噪声是非高斯类型时,可以观察到动态非线性系统是优于静态非线性系统,可以获得一个更好的输出信噪比,证明了加入额外白噪声以提高输出信噪比的可能性。
Stochastic resonance (SR) is a nonlinear phenomenon where the transmission of a coherent signal by certain nonlinear systems can be improved by the addition of noise. On the basis of stochastic resonance and cyclostationary theory, in the small-signal limit we find the relationship between signal-to-noise (SNR) and fisher information and extend the stochatic resonance theoretical framework. We show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing a weak periodic signal in additive white noise, a locally optimal processor (LOP) achieves the maximal output SNR. It is shown that the output-input SNR gain of a LOP is given by the Fisher information of a standardized noise distribution. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. We compare the SR effects in an array of static and dynamical nonlinearities via the measure of the output SNR. The main results of the thesis are summarized as follows:
1. The origins of Fisher information are in its use as a performance measure for parametric estimation. We augment this and show that the Fisher information can characterize the performance in several other significant signal processing operations. For processing of a weak signal in additive white noise, we demonstrate that the Fisher information determines (ⅰ) the maximum output signal-to-noise ratio for a periodic signal;(ⅱ) the optimum asymptotic efficacy for signal detection;(ⅲ) the best cross-correlation coefficient for signal transmission; and (ⅳ) the minimum mean square error of an unbiased estimator. This unifying picture, via inequalities on the Fisher information, is used to establish conditions where improvement by noise through stochastic resonance is feasible or not.
2. For processing a weak periodic signal in additive white noise, a LOP achieves the maximal output SNR. In general, such a LOP is precisely determined by the noise probability density and also by the noise level. It is shown that the output-input SNR gain of a LOP is given by the Fisher information of a standardized noise distribution. Based on this connection, we find that an arbitrarily large SNR gain, for a LOP, can be achieved ranging from the minimal value of unity upwards. For stochastic resonance, when considering adding extra noise to the original signal, we here demonstrate via the appropriate Fisher information inequality that the updated LOP fully matched to the new noise, is unable to improve the output SNR above its original value with no extra noise. This result generalizes a proof that existed previously only for Gaussian noise. Furthermore, in the situation of non-adjustable processors, for instance when the structure of the LOP as prescribed by the noise probability density is not fully adaptable to the noise level, we show general conditions where stochastic resonance can be recovered, manifested by the possibility of adding extra noise to enhance the output SNR.
3. We study the output-input SNR gain of an uncoupled parallel array of static, yet arbitrary, nonlinear elements for transmitting a weak periodic signal in additive white noise. In the small-signal limit, an explicit expression for the SNR gain is derived. It serves to prove that the SNR gain is always a monotonically increasing function of the array size for any given nonlinearity and noisy environment. It also determines the SNR gain maximized by the locally optimal nonlinearity as the upper bound of the SNR gain achieved by an array of static nonlinear elements. With locally optimal nonlinearity, it is demonstrated that stochastic resonance cannot occur, i.e. adding internal noise into the array never improves the SNR gain. However, in an array of suboptimal but easily implemented threshold nonlinearities, we show the feasibility of situations where stochastic resonance occurs, and also the possibility of the SNR gain exceeding unity for a wide range of input noise distributions.
4. We compare the SR effects in an array of static and dynamical nonlinearities via the measure of the output SNR. For a given noisy periodic signal, parallel arrays of both static and dynamical nonlinearities can enhance the output SNR by optimally tuning the internal noise level. The static nonlinearity is easily implementable, while the dynamical nonlinearity has more parameters to be tuned, at the risk of not exploiting the beneficial role of internal noise terms. More interestingly, as the input signal buried in external Laplacian noise, it is observed that the dynamical nonlinearity is superior to the static nonlinearity in obtaining a better output SNR.
引文
1. Benzi R, Sutera A, Vulpiani A. The mechanism of stochastic resonance. Journal of Physics A: Mathematical and General,1981,14:453-457.
2. Jung P, Hanggi P. Ampliation of small signals via stochastic resonance. Physical Review A, 1991,44:8032-8042.
3. Godivier X, Chapeau-Blondeau F. Noise-assisted signal transmission in a nonlinear electronic comparator: Experiment and theory. Signal Processing, 1996,56:293-303.
4. Chapeau-Blondeau F, Godivier X. Theory of stochastic resonance in signal transimission by static nonlinear systems. Physical Review E, 1997,55:1478-1495.
5. Collins J J, Chow C C, Imhoff T T. Stochastic resonance in neuron models. Physical Review E,1995,52:3321-3324.
6. Lindner B, Garcia J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Physics Reports, 2004,392:321-424.
7. Gammaitoni L, Hanggi P, Jung P, Marchesoni F. Stochastic resonance. Reviews of Modern Physics,1998,70:233-287.
8. Dykman M I, McClintock P V E. What can stochastic resonance do? Nature, 1998,391:344.
9. Fauve S, Heslot F. Stochastic resonance in a bistable system. Physics Letters A, 1983,97:5-7.
10. McNamara B, Wiensenfeld K. Theory of stochastic resonance. Physical Review A, 1989,39: 4854-4869.
11. Loerincz K, Gingl Z, Kiss L B. A stochastic resonator is able to greatly improve signal-to-noise ratio. Physics Letters A, 1996,224:63-67.
12. Rousseau D, Rojas-Varela J, Chapeau-BlondeauF. Stochastic resonance for nonlinear sensors with saturation.Physical Review E, 2003,67(no.021102).
13. Comte J C, Morfu S. Stochastic resonance: another way to retrieve subthreshold digital data. Physics Letters A, 2003,309:39-43.
14. Greenwood P E, Muller U U, Ward L M. Soft threshold stochastic resonance. Physical Review E, 2004,70(no.051110).
15. Li J, Xu B. Parameter-induced stochastic resonance with a periodic signal. Chinese Physics, 2006,15:2867-2871.
16. Ray R, Sengupta S. Stochastic resonance in underdamped, bistable systems. Physics Letters A, 2006,353:364-371.
17. Gandhimathi V M, Rajasekar S, Kurths J. Vibrational and stochastic resonances in two coupled overdampedanharmonic oscillators. Physics Letters A, 2006,360:279-286.
18. Perc M, Gosaka M, Kralj S. Stochastic resonance in soft matter systems: combined effects of static and dynamic disorder. Soft Matter, 2008,4:1861-1870.
19. Ward L M, MacLean S E, Kirschner A. Stochastic resonance modulates neural synchronization within and between cortical sources. PLoS ONE, 2010,5(no.e14371).
20. Gosaka M, Perc M, Kralj S. The impact of static disorder on vibrational resonance in a ferro-electric liquid crystal. Molecular Crystals & Liquid Crystals, 2012,553:13-20.
21. Hasegawa Y, Arita M. Escape process and stochastic resonance under noise intensity fluctua-tion. Physics Letters A, 2012,375:3450-3458.
22. Storni R, Ando H, Aihara K, Muralic K, Sinha S. Manipulating potential wells in Logical Stochastic Resonance to obtain XOR logic. Physics Letters A, 2012,376:930-937.
23. McDonnell M D, Abbott D. What is stochastic resonance? definitions, misconceptions, de-bates, and its relevance to biology. PLoS Computational Biology, 2009,5(no.1000348).
24. Wiesenfeld K, Moss F. Stochastic resonance and the benefits of noise:from ice ages to crayfish and SQUIDs. Nature, 1995,373:33-36.
25. Hanggi P. Stochastic resonance in biology how noise can enhance detection of weak signals and help improve biological information processing. ChemPhysChem, 2002,3:285-290.
26. Bezrukov S M, Vodyanoy I. Noise-induced enhancement of signal transduction across voltage-dependent ion channels. Nature, 1995,378:362-364.
27. Moss F, Pei X. Neurons in parallel. Nature, 1995,376:211-212.
28. Russell D F, Wilkens L A, Moss F. Noise-enhanced tactile sensation. Nature, 1999,401:291-294.
29. Collins J J, Priplata A A, Gravelle D D, Niemi J, Harry J, Lipsitz L A. Noise-enhanced human sensorimotor function. IEEE Engineering in Medicine and Biology, 2003,22:76-83.
30. Priplata A A, Patritti B L, Niemi J B, Hughes R. Noise-enhanced balance control in patients with diabetes and patients with stroke. Annals of Neurology, 2006,59:4-12.
31. Kitajo K, Yamanaka K, Ward L M. Stochastic resonance in attention control. Europhysics Letters, 2006,76:1029.
32. Hohmann W. Stochastic resonance in chemistry.3. The mininal-bromate reaction. The Journal of Physical Chemistry. 1996,100:5388-5392.
33. Leonard D S, Reichl L. Stochastic resonance in a chemical reaction. Physical Review E, 1994, 49:1734.
34. Bezrukov S M, Vodyanoy I. Stochastic resonance in thermally activated reactions: Application to biological ion channels. Chaos, 1998,8:557-566.
35. Gluckman B J, Netoff T L, NeelE J, Ditto W L, SpanoM L, SchiffS J. Stochastic resonancein a neuronal network from mammalian brain. Phys. Rev. Lett, 1996,77:4098-4101.
36. Levin J E, Miller J P. Broadband neural encoding in the cricket cercal sensory system enhanced by stochastic resonance. Nature, 1996,380:165-168.
37. Gianni. Modeling eletromagneticfields detectability in a HH-like neuronal system:stochastic resonance and window behavior. Biological Cybernetics, 2006,94:118-127.
38. Bartussek R. Stochastic resonance in optical bistable systems. Physical Review E, 1994,49: 3930.
39. Dykman M. Stochastic resonance in an all-optical passive bistable system.Sov. Phys. JETP Lett., 1991,53(4):194-196.
40. Xu B. A New Approach to Degraded Image Processing Based on Two-Dimensional Parameter-Induced Stochastic Resoance. Computer Modeling in Engineering & Sciences, 2010,57:159-174.
41. Zhang Y. The application of bistable stochastic resonance in image processing via tuning sys-tem parameters. 2008 International Conference on Information and Automation, 2008,489-492.
42. Kay S. Can detectabilty be improved by adding noise? IEEE Signal Processing Letters, 2000, 7:8-10.
43. Chen H, Varshney P K, Michels J H, Kay S M. Theory of the stochastic resonance effect in signal detection:Part I-Fixed detectors. IEEE Transactions on Signal Processing, 2007,55: 3172-3184.
44. Zozor S, Amblard P O. Stochastic resonance in locally optimal detectors.IEEE Transactions on Signal Processing, 2003,51:3177-3183.
45. Chapeau-Blondeau F, Blanchard S, Rousseau D. Fisher information and noise-aided power estimation from one-bit quantizers. Digital Signal Processing, 2008,18:434-443.
46. McDonnel M D, Stocks N G, Pearce C E M, Abbott D. Stochastic resonance: from suprathresh-old stochastic resonance to stochastic signal quantization. Cambridge University Press, Cam-bridge, 2008.
47. Rousseau D, Chapeau-Blondeau F. Noise-improved Bayesian estimation with arrays of one-bit quantizers. IEEE Transactions on Instrumentation and Measurement,2007,56:2658-2662.
48. Chen H, Varshney P K. Theory of the Stochastic Resonance Effect in Signal Detection:Part Ⅱ-Variable Detectors. IEEE Transactions on Signal Processing,2008,56:5031-5041.
49. Chen H, Varshney P K, Michels J H. Noise enhanced parameter estimation. IEEE Transactions on Signal Processing,2008,56:5074-5081.
50. Patel A, Kosko B. Optimal mean-square noise benefits in quantizer-array linear estimation. IEEE Signal Processing Letters,2010,17:1005-1009.
51. Longtin A, Bulsara A, Moss F. Time-interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons. Physical Review Letters,1991,67, 656-659.
52. Morse R P, Evans E F. Enhancement of vowel coding for cochlear implants by addition of noise. Nat. Med,2:928-932.
53. Morse R P, Roper P. Enhanced coding in a cochlear-implant model using additive noise: Ape-riodstochastic resource with tuning. Phys. Rew. E,2000,61(5):5683-5692.
54. Mori T, Kai S. Noise-induced entrainment and stochastic resonance in human brain waves. Physical Review Letters,2002,88(no.218101).
55. Stacey W C, Durand D M. Synaptic noise improves detection of subthreshold signals in hip-pocampal CA1 neurons. Journal of Neurophysiology,2001,86:1104-1112.
56. Chapeau-Blondeau F, Duan F, Abbott D. Synaptic signal transduction aided by noise in a dy-namical saturating model. Physical Review E,2010,81(no.021124).
57. Bulsara A R, Schmera G. Stochastic resonance in globally coupled nonlinear oscillator. Phys-ical Review E,1993,47:3734-3737.
58. Bulsara A R, Zador A. Threshold detection of wideband signals:A noise-induced maximum in the mutual information. Physical Review E,1996,54:2185-2188.
59. McNamara B, Wiensenfeld K, Roy R. Observation of Stochastic Resonance in a Ring Laser. Physical Review Letters,1988,60(25):2626-2629.
60. Hanggi P, Inchiosa M E, Fogliatti D, Bulsara A R. Nonlinear stochastic resonance:The saga of anomalous output-input gain. Physical Review E,2000,62:6155-6163.
61. Rousseau D, Chapeau-Blondeau F. Suprathreshold stochastic resonance and signal-to-noise ratio imiprovement in arrays of comparators. Physics Letters A,2004,321:280-290.
62. Duan F, Chapeau-Blondeau F, Abbott D. Noise-enhanced SNR gain in parallel array of bistable oscillators. Electronics Letters,2006,42:1008-1009.
63. Chapeau-BlondeauF, Rousseau D. Nonlinear devices acting as SNR amplifiers for a harmonic signal in noise.Circuits, Systems and Signal Processing,2006,25:431-446.
64. Zhou T, Moss F. Analog Simulations of Stochastic Resonance. Physical Review A,1990,41: 4255-4264.
65. Stocks N G. Suprathreshold stochastic resonance in multilevel threshold systems. Physical Review Letters,2000,84:2310-2313.
66. Chapeau-Blondeau F, Rousseau D. Noise-enhanced performance for an optimal Bayesian esti-mator.IEEE Transactions on Signal Processing,2004,52:1327-1334.
67. Jung P. Stochastic resonance and optimal design of threshold detectors. Physics Letter A,1995, 207:93-104.
68. Kosko B, Mitaim S. Robust stochastic resonance: Signal detection and adaptation in impulsive noise. Physical Review E,2001,64(no.051110).
69. Zozor S, Amblard P O. On the use of stochastic resonance in sine detection. Signal Processing, 2002,82:353-367.
70. Chapeau-Blondeau F, Rousseau D. Raising the noise to improve performance in optimal pro-cessing. Journal of Statistical Mechanics: Theory and Experiment,2009, 1(no.P01003).
71. Kondo T, Munakata T. Stochastic resonance and self-tuning:A simple threshold system. Phys-ical Review E,2009,79(no.061121).
72. Patel A, Kosko B. Optimal noise benefits in Neyman-Pearson and inequality-constrained sta-tistical signal detection. IEEE Transactions on Signal Processing,2009,57:1655-1669.
73. Chen H, Varshney P K, Kay S, Michels J H. Noise enhanced nonparametric detection. IEEE Transactions on Information Theory,2009,55:499-506.
74. Chapeau-Blondeau F. Stochastic resonance for an optimal detector with phase noise. Signal Processing,2003,83:665-670.
75. Rousseau D, Chapeau-BlondeauF. Stochastic resonance and improvement by noise in optimal detection strategies. Digital Signal Processing,2005,15:19-32.
76. Guerriero M, Marano S, Matta V, Willett P. Stochastic resonance in sequential detectors. IEEE Transactions on Signal Processing,2009,57:2-15.
77. Patel A, Kosko B. Noise benefits in quantizer-array correlation detection and watermark de-coding. IEEE Transactions on Signal Processing,2011,59:488-505.
78. Bayram S, Gezici S, Poor H V. Noise enhanced hypothesis-testing in the restricted Bayesian framework. IEEE Transactions on Signal Processing,2010,58:3972-3989
79. Hari V N, Anand G V, Premkumar A B, Madhukumar A S. Design and performance analysis of a signal detector based on suprathreshold stochastic resonance. Signal Processing, 2012,92: 1745-1757.
80. Saha A A, Guha V G. Detectors based on stochastic resonance. Part 2:Convergence analysis and perturbative corrections. Signal Processing, 2007,87:134-147.
81. Saberali S M, Amindavar H, Ritcey J A. Blind detection in symmetric non-Gaussian noise with unknown PDF using maximum entropy method with moment generating function constraints. Signal Processing, 2010,90:891-899.
82. Chen H, Varshney P K, Michels J H. Improving sequential detection performance via stochastic resonance. IEEE Signal Processing Letters, 2008,15:685-688.
83. Kay S. Noise enhanced detection as a special case of randomization. IEEE Signal Processing Letters,2008,15:709-712.
84. Hu G, Gong D, Wan X. Stochastic resonance in a nonlinear system driven by an apeoridic force.Phys. Rev. A, 1992,46(no.6)PP,3250-3254.
85. Duan F, Xu B. Parameter-induced stochastic resonance and baseband binary signals transmis-sion over anAWGN channel. Int. J. Bifurcation & Chaos, 2003,13(2).
86. Collins J J. Aperiodic stochastic resonance in excitable sysytems.Physical Review E,1995, 52(P.R3321).
87. Collins J J, Imhoff T T, Grigg P. Noise-enhanced tactile sensation. Nature, 1996,383:770.
88. Amblard P O, Zozor S. Cyclostationarity and stochastic resonance in threshold devices. Phys-ical Review E,1999,59:5009-5020.
89. Wannamaker R A, Lipshitz S R, Vanderkoov J. Stochastic resooance as dithering. Phys. Rev. E,2000,61(10):233-236.
90. Fletcher J E, Havlin S, Weiss G H. J. First passage time problems in time-dependent fields. Journal of Statistical Physics, 1988,51(1-2),215-232.
91. Masoliver J, Robinson A. Coherent stanhasticresonance. Physics Review E,1995,51 (5),4021-4026.
92. Zhou C, Kurths J, Hu B. Array-enhanced coherence resonance: nontrivial effects of hetero-geneity and spatial independence of noise? Phys. Rev. Lett., 2001,87(9),098101.
93. Dykman M I. Comment on "Stochastic resonance in bistable systems". Physical Review Let-ters, 1990,65:2606.
94. HanggiP. Bistability driven by colored noise:Theory and experiment. Physical Review A, 1985,32:695-698.
95. Jung P, Hanggi P. Stochastic nonlinear dynamicas modulated by external periodic forces. Eu-rophysics Letters, 1989,8:505.
96. Hu G. Periodically forced Fokker-Plank equation and stochastic resonance. Physical Review A,1990,42:2030-2041.
97. Anishchenko V, Safonova M. Stochastic resonance in chua's circuit. INT. J. BIFURCATION CHAOS,1992,46:3250-3254.
98. Xu B. Stochastic resonance with tuning system parameters: the application of bistable systems in signal processing. Chaos, Solitons & Fractals, 2002,13:633-644.
99. Mitaim S, Kosiko B. Adaptive stochastic resonance. Proceedings of the IEEE, 1998,86:2152-2183.
100. Lindner J F, Meadows B K, Ditto W L, Inchiosa M E, Bulsara A R. Array enhanced stochastic resonance and spatiotemporal synchronization. Physical Review Letters, 1995,75:3-6.
101. Collins J J, Chow C C, Imhoff T T. Stochastic resonance without tuning. Nature, 1995,376: 236-238.
102. Chapeau-Blondeau F, Rousseau D. Enhancement by noise in parallel arrays of sensors with power-law characteristics.Physical Review E, 2004,70(no.060101(R)).
103. Wang Q, Perc M, Duan Z, Chen G. Delay-induced multiple stochastic resonances on scalefree neuronal networks. Chaos, 2009,19(no.023112).
104. Perc M. Stochastic resonance on excitable small-world networks via a pacemaker. Physical Review E, 2007,76(no.066203).
105. Perc M. Stochastic resonance on weakly paced scale-free networks. Physical Review E, 2008, 78(no.036105).
106. Wang Q, Perc M,Duan Z,Chen G. Delay-induced multiple stochastic resonances on scale-free neuronal networks.Chaos, 2009,19(no.023112).
107. Gan C,Perc M,Wang Q. Delay-aided stochastic multiresonances on scale-free FitzHugh-Nagumo neuronal networks.Chinese Physical B, 2010,19(no.040508).
108. Hutt M T, Jain M K, Hilgetag C C, Lesne A. Stochastic resonance in discrete excitable dynam-ics on graphs.Chaos, Solitons & Fractals, 2012,45:611-618.
109. Gosak M, Perc M, Kralj S. Stochastic resonance in a locally excited system of bistable oscilla-tors. The European Physical Journal B, 2011,80:519-528.
110. Teramae J, Tsubo Y, Fukai T. Optimal spike-based communication in excitable networks with strong-sparse and weak-dense links. Scientific Reports, 2012,2(no.00485).
111. Gluckman B J, Netoff T L, Neel E J, Ditto W L, Spano M L, Schiff S J. Stochastic resonancein a neuronal network from mammalian brain. Phys. Rev. Lett. 1996,77(19):4098-4101.
112. Guderiart A, Dechert G, Zeyer K P, Schneider F W. Stochastic resonance in chemistry. Phys. Chem.,1996, 100:4437-4441.
113. 王利亚,蔡文生,印春生,潘忠孝.一种有效提取信号的新方法.高等学校化学学报,2000,21(1)：53-55.
114.李华峰,徐博侯.随机共振系统输出的一种新的反演方法.力学学报,2003,35(2)：194-198.
115.李蓉,温孝东,胡岗.用同频噪声驱动的双稳态系统分离弱信号.数据采集与处理,2001,16(2)：32-35.
116. 祝恒江,吴锡田.随机共振研究进展.大学物理,1997,16(7)：28-31.
117.王利亚,沈阳,潘忠孝.随机共振应用初步研究.计算机与应用化学,2000,17(1)：79-80.
118.傅祖芸.信息论基础[M].第一版.北京：电子工业出版社,1989.
119.王崇彦,路锦辉.信息论基础[M].第一版.北京：兵器工业出版社,1992.
120. Shannon C E. Newsletter ACM SIGMOBILE Mobile Computing and Communications Re-view. ACM New York,2001.
121.傅祖芸,赵建中.信息论与编码[M].第一版.北京：电子工业出版社,2006.
122. Cover T M, Thomas J A. Elements of Information Theory, Wiley: New York, 1991.
123.赵廷玉,张文字,余飞鸿.应用费希尔信息量评价函数的波前编码系统设计.光学学报,2007,27(6)：1096-1010.
124.黄继伟,李云飞.删失数据下的Fisher信息量.电子科技大学学报,2006,35(3)：423-425.
125. Kay S. Fundamentals of Statistical Signal Processing:Detection Theory. Englewood Cliffs, NJ:Prentice-Hall,1998.
126. Capon J. On the asymptotic efficiency of locally optimum detectors. IRE Transactions on In-formation Theory, 1961, IT-7:67-71.
127. Poor H V, Thomas J B. Locally optimum detection of discrete-time stochastic signals in non-Gaussian noise. Journal of the Acoustical Society of America, 1978,63:75-80.
128. Lu N H, Eisenstein B. Detection of weak signals in non-Gaussian noise. IEEE Transactions on Information Theory, 1981,IT-27:755-771.
129. Kassam S A. Signal Detection in Non-Gaussian Noise. Springer-Verlag:New York, 1988.
130. Song I, Kassam S A. Locally optimum detection of signals in a generalized observation model: The random signal case. IEEE Transactions on Information Theory, 1990,36:516-530.
131. Duan F, Chapeau-Blondeau F, Abbott D. Fisher-information condition for enhanced signal detection via stochastic resonance. Physical Review E, 2011,84(no.051107).
132. Duan F, Chapeau-Blondeau F, Abbott D. Fisher information as a metric of locally optimum processing and stochastic resonance.PLoS ONE, 2012,7(no.e34282).
133. Bennett W R. Statistics of regenerative digital transmission. Bell System Tech.J.,1958, 37: 1501-1542.
134. Gudzenko L I. On periodically nonstationary processes. Radiotekhnika i Electronika, 1959,4: 1062-1064.
135. Gudzenko LI.On periodic nonstationary processes. Radio Eng. Electron, Phys., 1959,4:220-224.
136. Dragan Y P. Periodically correlated random processes, and transformers with periodically vary-ing parameters. Otbor i Peredavcalnformacii,1969, 22:27-33.
137. Dragan Y P. The spectral properties of periodically correlated stochastic processes.Otbor i PeredavcaInformacii,1971,30:16-24.
138. Dragan Y P. Biperiodically correlated random processes. Otbor i Peredavcalnformacii, 1972, 34:12-14.
139. Dragan Y P, Ja R. Properties of counts of periodically correlated random processes. Otbor i Perdavcalnformacii, 1972,33:9-12.
140. Dragan Y P. The representation of a periodically correlated random process by stationary com-ponents. Otbor i Peredavcalnformacii, 1975,45:7-20.
141. Dragan Y P. Harmonizability and spectral disttibution of random processes with finite mean power. Dokl. Akad. Nauk. (Ukrain) SSR Ser. A, 1978,8:679-684.
142. Dragan Y P., Yavorsky I.N. Rythmics of sea waves and underwater scoustic signals. Nauko-vaDumka, Kiev, Ukraine, 1982 (in Russian).
143. Gladyshev E G. Periodically correlated random sequences. Soviet Math.Dokl, 1961,2:385-388.
144. Gladyshev E G. Periodically correlated random sequences. Dokl. Akad. Nauk SSSR, 1961,137: 1026-1029 (in Russian).
145. Gladyshev E G. Periodically and semi-periodically correlated random processes with continu-ous time. Teor.Verojatnost.iPrimenen, 1963,8:184-189.
146. Gladyshev E G. Periodically and almost periodically correlated with continuous time parame-ter. Theory of Probability and Its Applications,1963,8:173-177.
147. Hurd H L, Bloomfield P, Lund R B. Periodic correlation in meteorological time series. Fifth International Meeting on Statistical Meteorology,1992,1-6.
148. Nedoma J. Uber die ergodizitat and r-ergozitatstationarerwahrscheinlichkeitsmasse. Zeitschrift fur Wahrscheinlichkeitstheorie,1963,2:90-97.
149. Franks L E. Signal Theory.Prentice-Hall, Englewood Cliffs, NJ,1969.
150. Gardner W A, Franks L. Characterization of cyclostationary random signal processes. IEEE Trans. Information Theory,1975,21:4-14.
151. Boyles R, Gardner W A. Cycloergodic properties of discrete-parameter nonstationary stochas-tic processes. IEEE Trans. Information Theory,1983,29:105-114.
152. Wang J D, Chen T W, Huang B. On spectral theory of cyclostationary signals in multirate systems. IEEE Transactions on Signal Processing,2005,53(7):2421-2431.
153. Gardner W A. The Spectral Correlation Theory of Cyclostationary Time-Series. Signal Pro-cessing,1986,11(1):13-36.
154. Gardner W A. Spectral Correlation of Modulated Signals: Part I-Analog Modulation.IEEE Transactions on Communications,1987,35(6):584-594.
155. Gardner W A, Brown W, Chen Chih-Kang. Spectral Correlation of Modulated Signals: Part Ⅱ-Digital Modulation. IEEE Transactions on Communications,1987,35(6):595-601.
156. Gardner W A. Measurement of spectral correlation. Acoustics, Speech, and Signal Processing, 1986,34(5):1111-1123.
157. Gardner W A. Signal Interee:ptio: A Unifying Theoretical Framework for Feature Detec-tion. IEEE Trans on communications,1988,36(8):897-906.
158. Gardner W A, Spooner C M. Cyclic Spectral Analysis for Signal Detection andModulation Recognition. Proc. IEEE Conf Military Communications,1988:419-423.
159. Gardner W A, Archer T L. Exploitation of cyclostationarity for identifying the Volterrakernels ofnonlinear systems. IEEE Transactions on Information Theory,1993,39:535-542.
160. Ciblat P, Loubaton P, Serpedin E. Asymptotic analysis ofblind cyclic correlation-basedsymbol-rate estimators. IEEE Trans on information theory,2002,48(7):1922-1934.
161. Ferreol A, Chevalier P, Albera L. Second-order blind separation of first and second-ordercyclostationary sources-application to AM and dcterministic sources. IEEE Trans. Signal Processing,2004,52(4):845-861.
162. Gardner W A. Exploitation of spectral redundancy incyclostationary signals. IEEE.Signal Pro-cessing Magazine,1991,8(2):14-36.
163. 张贤达,保铮.非平稳信号分析与处理.北京：国防工业出版社,2001.
164. 黄知涛,周一宇,姜文利.循环平稳信号处理及应用.北京：科学出版社,2006.
165. Dandawate A V, Giannakis G B. Modeling (almost) periodic moving average processes using cyclic statistics. IEEE Trans. Signal Processing,1996,44:673-684.
166. Giannakis G B. Cyclostationary signal analysis. The Digital Signal Processing Handbook (Eds.: MadisettiV. K. Wiliams D. B.), CRC Press and IEEE Press, Boca Raton, FL and New York,1998 (Chapter 17).
167. Izzo L, Napolitano A. The higher order theory of generalized almost-cyclostationary time se-ries. IEEE Transactions on Signal Processing,1998,46(11):2975-2989.
168. Giannakis G B, Dandawate A V. Polyspectral analysis of (almost) cyclostationary signals: LPTV system identification and related applications.1991 Conference Record of the Twenty-Fifth Asilomar Conference on Signals, Systems andComputers,1991,1:377-382.
169. Kacimov A R, Obnosov Y V, Yakimov N D. Groundwater flow in a medium with a parquet-type conductivity distribution. Journal of Hydrology,1999,226:242-249.
170. Finelli C J, Jenkins J M. A cyclostationary least mean squares algorithm for discrimination of ventticular tachycardia from sinus rhythm. International Conference of the IEEE Engineering in Medicine and Biology Society, Baltimore, MD,1991,13:740-741.
171. Pagano M, Paizen E. An approach to modeling seasonally stationary time-series. Journal of Econometrics,1979,9:137-153.
172. Sherman P J. Random processes from rotating machinery, in: A.G.Miamiee (Ed.). Proceedings of Workshop on Nonstationary Stochastic Processes and their Applications, Word Scientific, New Jersey,1991,211-218.
173. Campbell J, Gibbs A, Smith B. The cyclostationary nature of crosstalk interference from digital signals in multipair cable-Part I: fundamentals. IEEE Trans. Commun.,1983,31:629-637.
174. Tong L, Guanghan X, Kailath T. Blind identification and equalization based on second-order statistics: a time domain approach. IEEE Trans. Information Theory,1994,40:340-349.
175. Tong L, Xu G, HassibiB, et al. Blind channel identification based on second-order statistics: a frequency-domain approach. IEEE Trans. Information Theory,1995,41:329-334.
176. Tugnait J K. On blind identifiability of multipath channels using fractional sampling and second-order cyclostationary statistics. IEEE Trans. Information Theory,1995,41:308-311.
177. Ding Z, Li Y. On channel identification based on second-order cyclic spectra. IEEE Trans. Signal Processing, 1994,42:1260-1264.
178. Bezrukov S M, Vodyanoy I. Stochastic resonance in non-dynamical systems without response thresholds. Nature, 1997,385:319-321.
179. Zaikin A A, Kurths J, Schimansky-Geier L. Doubly stochastic resonance. Physical Review E, 2000,85:227-231.
180. Zeng L, Li J. Residual aperiodic stochastic resonance in Levy noise. Fluctuation and Noise Letters,2011,10:223-229.
181. Hanggi P, Jung P, Zerbe C, Moss F. Can colored noise improve stochastic resonance? Journal of Statistical Physics, 1993,70:25-47.
182. DeWeese M, Bialek W. Information flow in sensory neurons.NuovoCimento, 1995,17:733-741.
183. Brown L D. A proof of the central limit theorem motivated by the Cramer-Rao inequality. Statistics and Probability: Essays in Honor of C. R. Rao, eds. by G. Kallianpur, P. R. Krishnaiah and J. K. Ghosh, North-Holland Publishing Company, 1982,141-148.
184. Luchinsky D G, Mannella R, McClintock P V E, Stocks N G. Stochastic resonance in electrical circuits-I:coventional stochastic Resonance. IEEE Transactions on Circuits and Systems-II: Analog and Digtial Signal Processing. 1999,46:1205-1214.
185. Zozor S, Amblar P O. Stochastic resonance in discrete time nonlinear AR(1) models. IEEE Transactions on Signal Processing, 1999,47:108-122.
186. Greenwood P E, Ward L M, Wefelmeyer W. Statistical analysis of stochastic resonance in a simple setting. Physical Review E, 1999,60:4687-4695.
187. Casado-Pascual J, Gomez-Ordonez J, Morillo M. Two-state theory of nonlinear stochastic res-onance. Physical Review Letters, 2003,91(no.210601).
188. Stam A J. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control,1959,2:101-112.
189. Madiman M, Barron A. Generalized entropy power inequalities and monotonicity properties of information. IEEE Transactions on Information Theory, 2007,53:2317-2329.
190. Johnson O, Barron A. Fisher information inequalities and the central limit theorem. Probability Theory and Related Fields, 2004,129:391-409.
191. Stocks N G. Information transmission in parallel threshold arrays: Suprathreshold stochastic resonance. Physical Review E,2001,63:0411141-9.
192. Rousseau D, Chapeau-Blondeau F. Stochastic resonance and improvement by noise in optimal detection strategies. Digital Signal Processing, 2005,15:19-32.
193. Inchiosa M E, Bulsara A R. Signal detection statistics of stochastic resonators. Physical Review E,1996,53:R2021-R2024.
194. Duan F, Chapeau-Blondeau F, Abbott D. Stochastic resonance in a parallel array of nonlinear dynamical elements. Physics Letters A, 2008,372:2159-2166.
195. Gailey P C, Neiman A, Collins J J, Moss F. Stochastic resonance in ensembles of nondynamical elements:the role of internal noise. Physical Review E, 1997, 79:4701-4704.
196. Newman M E J, Barabasi A L, Watts D J. The Structure and Dynamics of Networks. Princeton University Press, Princeton,2006.
197. Chen L, Wang R, Zhou T, Aihara K. Noise-induced cooperative behavior in a multicell system. Oxford Univ Press, Oxford, 2005.
198. Albert R, Barabasi A L. Statistical mechanics of complex networks. Reviews of modern physics, 2002,74:47.
199. Williams R J, Berlow E L, Dunne J A, Barabasi A L, Martinez N D. Two degrees of separation in complex food webs. Proceedings of the National Academy of Sciences, 2002,99:12913-12916.
200. Newman M E J. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, 2001,98:404-409.
201. Barabasi A L, Oltvai Z N. Network biology: understanding the cell's functional organization. Nature Reviews Genetics,2004,5:101-113.
202. Sporns O, Chialvo D R, Kaiser M, Hilgetag C C. Organization, development and function of complex brain networks. TRENDS in Cognitive Sciences, 2004,8:418-425.
203. Zhou C, Zemanova L, Zamora G, Hilgetag C C, Kurths J. Hierarchical organization unveiled by functional connectivity in complex brain networks. Physical review letters, 2006,97:238103.
204. Benzi R. Stochastic resonance in complex systems. Journal of Statistical Mechanics:Theory and Experiment, 2009,1 (no.P01052).
205. Chapeau-Blondeau F, Rousseau D. Noise-aided SNR amplification by parallel arrays of sensors with saturation. Physics Letters A, 2006,351:231-237.
206. Ichiki A, Tadokoro Y. Signal-to-noise ratio improvement by stochastic resonance in moments in non-dynamical systems with multiple states. Physics Letters A, 2013,377:185-188.
207. Ma Y, Duan F, Chapeau-Blondeau F, Abbott D. Weak-periodic stochastic resonance in a paral-lel array of static nonlinearities. PLoS ONE, 2013,8(no.e58507).
208. Xu B. Comparison of aperiodic stochastic resonance in a bistable system realized by adding noise and by tuning system parameters. Physical Review E, 2004,69(no.061110).
209. Apostolico F, Gammaitoni L, Marcheson F, Santucci S. Resonant trapping:a failure mecha-nism in switch transitions. Physics Review E, 1997,55:36-39.
210. Bulsara A R, In V, Kho A, Longhini P, Palacios A, Rappel W-J, AcebronJ, Baglio S, Ando B. Emergent oscillations in unidirectionally coupled overdamped bistable systems. Physical Review E, 2004,70(no.036103).
211. Bulsara A R, Lindner J F, In V, Kho A, Baglio S, Sacco V, Ando B, LonghiniP, Palacios A, Rappel W-J. Coupling-induced cooperative behaviour in dynamic ferromagnetic cores in the presence of a noise floor. Physics Letters A, 2005,353:4-10.
2. Jung P, Hanggi P. Ampliation of small signals via stochastic resonance. Physical Review A, 1991,44:8032-8042.
3. Godivier X, Chapeau-Blondeau F. Noise-assisted signal transmission in a nonlinear electronic comparator: Experiment and theory. Signal Processing, 1996,56:293-303.
4. Chapeau-Blondeau F, Godivier X. Theory of stochastic resonance in signal transimission by static nonlinear systems. Physical Review E, 1997,55:1478-1495.
5. Collins J J, Chow C C, Imhoff T T. Stochastic resonance in neuron models. Physical Review E,1995,52:3321-3324.
6. Lindner B, Garcia J, Neiman A, Schimansky-Geier L. Effects of noise in excitable systems. Physics Reports, 2004,392:321-424.
7. Gammaitoni L, Hanggi P, Jung P, Marchesoni F. Stochastic resonance. Reviews of Modern Physics,1998,70:233-287.
8. Dykman M I, McClintock P V E. What can stochastic resonance do? Nature, 1998,391:344.
9. Fauve S, Heslot F. Stochastic resonance in a bistable system. Physics Letters A, 1983,97:5-7.
10. McNamara B, Wiensenfeld K. Theory of stochastic resonance. Physical Review A, 1989,39: 4854-4869.
11. Loerincz K, Gingl Z, Kiss L B. A stochastic resonator is able to greatly improve signal-to-noise ratio. Physics Letters A, 1996,224:63-67.
12. Rousseau D, Rojas-Varela J, Chapeau-BlondeauF. Stochastic resonance for nonlinear sensors with saturation.Physical Review E, 2003,67(no.021102).
13. Comte J C, Morfu S. Stochastic resonance: another way to retrieve subthreshold digital data. Physics Letters A, 2003,309:39-43.
14. Greenwood P E, Muller U U, Ward L M. Soft threshold stochastic resonance. Physical Review E, 2004,70(no.051110).
15. Li J, Xu B. Parameter-induced stochastic resonance with a periodic signal. Chinese Physics, 2006,15:2867-2871.
16. Ray R, Sengupta S. Stochastic resonance in underdamped, bistable systems. Physics Letters A, 2006,353:364-371.
17. Gandhimathi V M, Rajasekar S, Kurths J. Vibrational and stochastic resonances in two coupled overdampedanharmonic oscillators. Physics Letters A, 2006,360:279-286.
18. Perc M, Gosaka M, Kralj S. Stochastic resonance in soft matter systems: combined effects of static and dynamic disorder. Soft Matter, 2008,4:1861-1870.
19. Ward L M, MacLean S E, Kirschner A. Stochastic resonance modulates neural synchronization within and between cortical sources. PLoS ONE, 2010,5(no.e14371).
20. Gosaka M, Perc M, Kralj S. The impact of static disorder on vibrational resonance in a ferro-electric liquid crystal. Molecular Crystals & Liquid Crystals, 2012,553:13-20.
21. Hasegawa Y, Arita M. Escape process and stochastic resonance under noise intensity fluctua-tion. Physics Letters A, 2012,375:3450-3458.
22. Storni R, Ando H, Aihara K, Muralic K, Sinha S. Manipulating potential wells in Logical Stochastic Resonance to obtain XOR logic. Physics Letters A, 2012,376:930-937.
23. McDonnell M D, Abbott D. What is stochastic resonance? definitions, misconceptions, de-bates, and its relevance to biology. PLoS Computational Biology, 2009,5(no.1000348).
24. Wiesenfeld K, Moss F. Stochastic resonance and the benefits of noise:from ice ages to crayfish and SQUIDs. Nature, 1995,373:33-36.
25. Hanggi P. Stochastic resonance in biology how noise can enhance detection of weak signals and help improve biological information processing. ChemPhysChem, 2002,3:285-290.
26. Bezrukov S M, Vodyanoy I. Noise-induced enhancement of signal transduction across voltage-dependent ion channels. Nature, 1995,378:362-364.
27. Moss F, Pei X. Neurons in parallel. Nature, 1995,376:211-212.
28. Russell D F, Wilkens L A, Moss F. Noise-enhanced tactile sensation. Nature, 1999,401:291-294.
29. Collins J J, Priplata A A, Gravelle D D, Niemi J, Harry J, Lipsitz L A. Noise-enhanced human sensorimotor function. IEEE Engineering in Medicine and Biology, 2003,22:76-83.
30. Priplata A A, Patritti B L, Niemi J B, Hughes R. Noise-enhanced balance control in patients with diabetes and patients with stroke. Annals of Neurology, 2006,59:4-12.
31. Kitajo K, Yamanaka K, Ward L M. Stochastic resonance in attention control. Europhysics Letters, 2006,76:1029.
32. Hohmann W. Stochastic resonance in chemistry.3. The mininal-bromate reaction. The Journal of Physical Chemistry. 1996,100:5388-5392.
33. Leonard D S, Reichl L. Stochastic resonance in a chemical reaction. Physical Review E, 1994, 49:1734.
34. Bezrukov S M, Vodyanoy I. Stochastic resonance in thermally activated reactions: Application to biological ion channels. Chaos, 1998,8:557-566.
35. Gluckman B J, Netoff T L, NeelE J, Ditto W L, SpanoM L, SchiffS J. Stochastic resonancein a neuronal network from mammalian brain. Phys. Rev. Lett, 1996,77:4098-4101.
36. Levin J E, Miller J P. Broadband neural encoding in the cricket cercal sensory system enhanced by stochastic resonance. Nature, 1996,380:165-168.
37. Gianni. Modeling eletromagneticfields detectability in a HH-like neuronal system:stochastic resonance and window behavior. Biological Cybernetics, 2006,94:118-127.
38. Bartussek R. Stochastic resonance in optical bistable systems. Physical Review E, 1994,49: 3930.
39. Dykman M. Stochastic resonance in an all-optical passive bistable system.Sov. Phys. JETP Lett., 1991,53(4):194-196.
40. Xu B. A New Approach to Degraded Image Processing Based on Two-Dimensional Parameter-Induced Stochastic Resoance. Computer Modeling in Engineering & Sciences, 2010,57:159-174.
41. Zhang Y. The application of bistable stochastic resonance in image processing via tuning sys-tem parameters. 2008 International Conference on Information and Automation, 2008,489-492.
42. Kay S. Can detectabilty be improved by adding noise? IEEE Signal Processing Letters, 2000, 7:8-10.
43. Chen H, Varshney P K, Michels J H, Kay S M. Theory of the stochastic resonance effect in signal detection:Part I-Fixed detectors. IEEE Transactions on Signal Processing, 2007,55: 3172-3184.
44. Zozor S, Amblard P O. Stochastic resonance in locally optimal detectors.IEEE Transactions on Signal Processing, 2003,51:3177-3183.
45. Chapeau-Blondeau F, Blanchard S, Rousseau D. Fisher information and noise-aided power estimation from one-bit quantizers. Digital Signal Processing, 2008,18:434-443.
46. McDonnel M D, Stocks N G, Pearce C E M, Abbott D. Stochastic resonance: from suprathresh-old stochastic resonance to stochastic signal quantization. Cambridge University Press, Cam-bridge, 2008.
47. Rousseau D, Chapeau-Blondeau F. Noise-improved Bayesian estimation with arrays of one-bit quantizers. IEEE Transactions on Instrumentation and Measurement,2007,56:2658-2662.
48. Chen H, Varshney P K. Theory of the Stochastic Resonance Effect in Signal Detection:Part Ⅱ-Variable Detectors. IEEE Transactions on Signal Processing,2008,56:5031-5041.
49. Chen H, Varshney P K, Michels J H. Noise enhanced parameter estimation. IEEE Transactions on Signal Processing,2008,56:5074-5081.
50. Patel A, Kosko B. Optimal mean-square noise benefits in quantizer-array linear estimation. IEEE Signal Processing Letters,2010,17:1005-1009.
51. Longtin A, Bulsara A, Moss F. Time-interval sequences in bistable systems and the noise-induced transmission of information by sensory neurons. Physical Review Letters,1991,67, 656-659.
52. Morse R P, Evans E F. Enhancement of vowel coding for cochlear implants by addition of noise. Nat. Med,2:928-932.
53. Morse R P, Roper P. Enhanced coding in a cochlear-implant model using additive noise: Ape-riodstochastic resource with tuning. Phys. Rew. E,2000,61(5):5683-5692.
54. Mori T, Kai S. Noise-induced entrainment and stochastic resonance in human brain waves. Physical Review Letters,2002,88(no.218101).
55. Stacey W C, Durand D M. Synaptic noise improves detection of subthreshold signals in hip-pocampal CA1 neurons. Journal of Neurophysiology,2001,86:1104-1112.
56. Chapeau-Blondeau F, Duan F, Abbott D. Synaptic signal transduction aided by noise in a dy-namical saturating model. Physical Review E,2010,81(no.021124).
57. Bulsara A R, Schmera G. Stochastic resonance in globally coupled nonlinear oscillator. Phys-ical Review E,1993,47:3734-3737.
58. Bulsara A R, Zador A. Threshold detection of wideband signals:A noise-induced maximum in the mutual information. Physical Review E,1996,54:2185-2188.
59. McNamara B, Wiensenfeld K, Roy R. Observation of Stochastic Resonance in a Ring Laser. Physical Review Letters,1988,60(25):2626-2629.
60. Hanggi P, Inchiosa M E, Fogliatti D, Bulsara A R. Nonlinear stochastic resonance:The saga of anomalous output-input gain. Physical Review E,2000,62:6155-6163.
61. Rousseau D, Chapeau-Blondeau F. Suprathreshold stochastic resonance and signal-to-noise ratio imiprovement in arrays of comparators. Physics Letters A,2004,321:280-290.
62. Duan F, Chapeau-Blondeau F, Abbott D. Noise-enhanced SNR gain in parallel array of bistable oscillators. Electronics Letters,2006,42:1008-1009.
63. Chapeau-BlondeauF, Rousseau D. Nonlinear devices acting as SNR amplifiers for a harmonic signal in noise.Circuits, Systems and Signal Processing,2006,25:431-446.
64. Zhou T, Moss F. Analog Simulations of Stochastic Resonance. Physical Review A,1990,41: 4255-4264.
65. Stocks N G. Suprathreshold stochastic resonance in multilevel threshold systems. Physical Review Letters,2000,84:2310-2313.
66. Chapeau-Blondeau F, Rousseau D. Noise-enhanced performance for an optimal Bayesian esti-mator.IEEE Transactions on Signal Processing,2004,52:1327-1334.
67. Jung P. Stochastic resonance and optimal design of threshold detectors. Physics Letter A,1995, 207:93-104.
68. Kosko B, Mitaim S. Robust stochastic resonance: Signal detection and adaptation in impulsive noise. Physical Review E,2001,64(no.051110).
69. Zozor S, Amblard P O. On the use of stochastic resonance in sine detection. Signal Processing, 2002,82:353-367.
70. Chapeau-Blondeau F, Rousseau D. Raising the noise to improve performance in optimal pro-cessing. Journal of Statistical Mechanics: Theory and Experiment,2009, 1(no.P01003).
71. Kondo T, Munakata T. Stochastic resonance and self-tuning:A simple threshold system. Phys-ical Review E,2009,79(no.061121).
72. Patel A, Kosko B. Optimal noise benefits in Neyman-Pearson and inequality-constrained sta-tistical signal detection. IEEE Transactions on Signal Processing,2009,57:1655-1669.
73. Chen H, Varshney P K, Kay S, Michels J H. Noise enhanced nonparametric detection. IEEE Transactions on Information Theory,2009,55:499-506.
74. Chapeau-Blondeau F. Stochastic resonance for an optimal detector with phase noise. Signal Processing,2003,83:665-670.
75. Rousseau D, Chapeau-BlondeauF. Stochastic resonance and improvement by noise in optimal detection strategies. Digital Signal Processing,2005,15:19-32.
76. Guerriero M, Marano S, Matta V, Willett P. Stochastic resonance in sequential detectors. IEEE Transactions on Signal Processing,2009,57:2-15.
77. Patel A, Kosko B. Noise benefits in quantizer-array correlation detection and watermark de-coding. IEEE Transactions on Signal Processing,2011,59:488-505.
78. Bayram S, Gezici S, Poor H V. Noise enhanced hypothesis-testing in the restricted Bayesian framework. IEEE Transactions on Signal Processing,2010,58:3972-3989
79. Hari V N, Anand G V, Premkumar A B, Madhukumar A S. Design and performance analysis of a signal detector based on suprathreshold stochastic resonance. Signal Processing, 2012,92: 1745-1757.
80. Saha A A, Guha V G. Detectors based on stochastic resonance. Part 2:Convergence analysis and perturbative corrections. Signal Processing, 2007,87:134-147.
81. Saberali S M, Amindavar H, Ritcey J A. Blind detection in symmetric non-Gaussian noise with unknown PDF using maximum entropy method with moment generating function constraints. Signal Processing, 2010,90:891-899.
82. Chen H, Varshney P K, Michels J H. Improving sequential detection performance via stochastic resonance. IEEE Signal Processing Letters, 2008,15:685-688.
83. Kay S. Noise enhanced detection as a special case of randomization. IEEE Signal Processing Letters,2008,15:709-712.
84. Hu G, Gong D, Wan X. Stochastic resonance in a nonlinear system driven by an apeoridic force.Phys. Rev. A, 1992,46(no.6)PP,3250-3254.
85. Duan F, Xu B. Parameter-induced stochastic resonance and baseband binary signals transmis-sion over anAWGN channel. Int. J. Bifurcation & Chaos, 2003,13(2).
86. Collins J J. Aperiodic stochastic resonance in excitable sysytems.Physical Review E,1995, 52(P.R3321).
87. Collins J J, Imhoff T T, Grigg P. Noise-enhanced tactile sensation. Nature, 1996,383:770.
88. Amblard P O, Zozor S. Cyclostationarity and stochastic resonance in threshold devices. Phys-ical Review E,1999,59:5009-5020.
89. Wannamaker R A, Lipshitz S R, Vanderkoov J. Stochastic resooance as dithering. Phys. Rev. E,2000,61(10):233-236.
90. Fletcher J E, Havlin S, Weiss G H. J. First passage time problems in time-dependent fields. Journal of Statistical Physics, 1988,51(1-2),215-232.
91. Masoliver J, Robinson A. Coherent stanhasticresonance. Physics Review E,1995,51 (5),4021-4026.
92. Zhou C, Kurths J, Hu B. Array-enhanced coherence resonance: nontrivial effects of hetero-geneity and spatial independence of noise? Phys. Rev. Lett., 2001,87(9),098101.
93. Dykman M I. Comment on "Stochastic resonance in bistable systems". Physical Review Let-ters, 1990,65:2606.
94. HanggiP. Bistability driven by colored noise:Theory and experiment. Physical Review A, 1985,32:695-698.
95. Jung P, Hanggi P. Stochastic nonlinear dynamicas modulated by external periodic forces. Eu-rophysics Letters, 1989,8:505.
96. Hu G. Periodically forced Fokker-Plank equation and stochastic resonance. Physical Review A,1990,42:2030-2041.
97. Anishchenko V, Safonova M. Stochastic resonance in chua's circuit. INT. J. BIFURCATION CHAOS,1992,46:3250-3254.
98. Xu B. Stochastic resonance with tuning system parameters: the application of bistable systems in signal processing. Chaos, Solitons & Fractals, 2002,13:633-644.
99. Mitaim S, Kosiko B. Adaptive stochastic resonance. Proceedings of the IEEE, 1998,86:2152-2183.
100. Lindner J F, Meadows B K, Ditto W L, Inchiosa M E, Bulsara A R. Array enhanced stochastic resonance and spatiotemporal synchronization. Physical Review Letters, 1995,75:3-6.
101. Collins J J, Chow C C, Imhoff T T. Stochastic resonance without tuning. Nature, 1995,376: 236-238.
102. Chapeau-Blondeau F, Rousseau D. Enhancement by noise in parallel arrays of sensors with power-law characteristics.Physical Review E, 2004,70(no.060101(R)).
103. Wang Q, Perc M, Duan Z, Chen G. Delay-induced multiple stochastic resonances on scalefree neuronal networks. Chaos, 2009,19(no.023112).
104. Perc M. Stochastic resonance on excitable small-world networks via a pacemaker. Physical Review E, 2007,76(no.066203).
105. Perc M. Stochastic resonance on weakly paced scale-free networks. Physical Review E, 2008, 78(no.036105).
106. Wang Q, Perc M,Duan Z,Chen G. Delay-induced multiple stochastic resonances on scale-free neuronal networks.Chaos, 2009,19(no.023112).
107. Gan C,Perc M,Wang Q. Delay-aided stochastic multiresonances on scale-free FitzHugh-Nagumo neuronal networks.Chinese Physical B, 2010,19(no.040508).
108. Hutt M T, Jain M K, Hilgetag C C, Lesne A. Stochastic resonance in discrete excitable dynam-ics on graphs.Chaos, Solitons & Fractals, 2012,45:611-618.
109. Gosak M, Perc M, Kralj S. Stochastic resonance in a locally excited system of bistable oscilla-tors. The European Physical Journal B, 2011,80:519-528.
110. Teramae J, Tsubo Y, Fukai T. Optimal spike-based communication in excitable networks with strong-sparse and weak-dense links. Scientific Reports, 2012,2(no.00485).
111. Gluckman B J, Netoff T L, Neel E J, Ditto W L, Spano M L, Schiff S J. Stochastic resonancein a neuronal network from mammalian brain. Phys. Rev. Lett. 1996,77(19):4098-4101.
112. Guderiart A, Dechert G, Zeyer K P, Schneider F W. Stochastic resonance in chemistry. Phys. Chem.,1996, 100:4437-4441.
113. 王利亚,蔡文生,印春生,潘忠孝.一种有效提取信号的新方法.高等学校化学学报,2000,21(1)：53-55.
114.李华峰,徐博侯.随机共振系统输出的一种新的反演方法.力学学报,2003,35(2)：194-198.
115.李蓉,温孝东,胡岗.用同频噪声驱动的双稳态系统分离弱信号.数据采集与处理,2001,16(2)：32-35.
116. 祝恒江,吴锡田.随机共振研究进展.大学物理,1997,16(7)：28-31.
117.王利亚,沈阳,潘忠孝.随机共振应用初步研究.计算机与应用化学,2000,17(1)：79-80.
118.傅祖芸.信息论基础[M].第一版.北京：电子工业出版社,1989.
119.王崇彦,路锦辉.信息论基础[M].第一版.北京：兵器工业出版社,1992.
120. Shannon C E. Newsletter ACM SIGMOBILE Mobile Computing and Communications Re-view. ACM New York,2001.
121.傅祖芸,赵建中.信息论与编码[M].第一版.北京：电子工业出版社,2006.
122. Cover T M, Thomas J A. Elements of Information Theory, Wiley: New York, 1991.
123.赵廷玉,张文字,余飞鸿.应用费希尔信息量评价函数的波前编码系统设计.光学学报,2007,27(6)：1096-1010.
124.黄继伟,李云飞.删失数据下的Fisher信息量.电子科技大学学报,2006,35(3)：423-425.
125. Kay S. Fundamentals of Statistical Signal Processing:Detection Theory. Englewood Cliffs, NJ:Prentice-Hall,1998.
126. Capon J. On the asymptotic efficiency of locally optimum detectors. IRE Transactions on In-formation Theory, 1961, IT-7:67-71.
127. Poor H V, Thomas J B. Locally optimum detection of discrete-time stochastic signals in non-Gaussian noise. Journal of the Acoustical Society of America, 1978,63:75-80.
128. Lu N H, Eisenstein B. Detection of weak signals in non-Gaussian noise. IEEE Transactions on Information Theory, 1981,IT-27:755-771.
129. Kassam S A. Signal Detection in Non-Gaussian Noise. Springer-Verlag:New York, 1988.
130. Song I, Kassam S A. Locally optimum detection of signals in a generalized observation model: The random signal case. IEEE Transactions on Information Theory, 1990,36:516-530.
131. Duan F, Chapeau-Blondeau F, Abbott D. Fisher-information condition for enhanced signal detection via stochastic resonance. Physical Review E, 2011,84(no.051107).
132. Duan F, Chapeau-Blondeau F, Abbott D. Fisher information as a metric of locally optimum processing and stochastic resonance.PLoS ONE, 2012,7(no.e34282).
133. Bennett W R. Statistics of regenerative digital transmission. Bell System Tech.J.,1958, 37: 1501-1542.
134. Gudzenko L I. On periodically nonstationary processes. Radiotekhnika i Electronika, 1959,4: 1062-1064.
135. Gudzenko LI.On periodic nonstationary processes. Radio Eng. Electron, Phys., 1959,4:220-224.
136. Dragan Y P. Periodically correlated random processes, and transformers with periodically vary-ing parameters. Otbor i Peredavcalnformacii,1969, 22:27-33.
137. Dragan Y P. The spectral properties of periodically correlated stochastic processes.Otbor i PeredavcaInformacii,1971,30:16-24.
138. Dragan Y P. Biperiodically correlated random processes. Otbor i Peredavcalnformacii, 1972, 34:12-14.
139. Dragan Y P, Ja R. Properties of counts of periodically correlated random processes. Otbor i Perdavcalnformacii, 1972,33:9-12.
140. Dragan Y P. The representation of a periodically correlated random process by stationary com-ponents. Otbor i Peredavcalnformacii, 1975,45:7-20.
141. Dragan Y P. Harmonizability and spectral disttibution of random processes with finite mean power. Dokl. Akad. Nauk. (Ukrain) SSR Ser. A, 1978,8:679-684.
142. Dragan Y P., Yavorsky I.N. Rythmics of sea waves and underwater scoustic signals. Nauko-vaDumka, Kiev, Ukraine, 1982 (in Russian).
143. Gladyshev E G. Periodically correlated random sequences. Soviet Math.Dokl, 1961,2:385-388.
144. Gladyshev E G. Periodically correlated random sequences. Dokl. Akad. Nauk SSSR, 1961,137: 1026-1029 (in Russian).
145. Gladyshev E G. Periodically and semi-periodically correlated random processes with continu-ous time. Teor.Verojatnost.iPrimenen, 1963,8:184-189.
146. Gladyshev E G. Periodically and almost periodically correlated with continuous time parame-ter. Theory of Probability and Its Applications,1963,8:173-177.
147. Hurd H L, Bloomfield P, Lund R B. Periodic correlation in meteorological time series. Fifth International Meeting on Statistical Meteorology,1992,1-6.
148. Nedoma J. Uber die ergodizitat and r-ergozitatstationarerwahrscheinlichkeitsmasse. Zeitschrift fur Wahrscheinlichkeitstheorie,1963,2:90-97.
149. Franks L E. Signal Theory.Prentice-Hall, Englewood Cliffs, NJ,1969.
150. Gardner W A, Franks L. Characterization of cyclostationary random signal processes. IEEE Trans. Information Theory,1975,21:4-14.
151. Boyles R, Gardner W A. Cycloergodic properties of discrete-parameter nonstationary stochas-tic processes. IEEE Trans. Information Theory,1983,29:105-114.
152. Wang J D, Chen T W, Huang B. On spectral theory of cyclostationary signals in multirate systems. IEEE Transactions on Signal Processing,2005,53(7):2421-2431.
153. Gardner W A. The Spectral Correlation Theory of Cyclostationary Time-Series. Signal Pro-cessing,1986,11(1):13-36.
154. Gardner W A. Spectral Correlation of Modulated Signals: Part I-Analog Modulation.IEEE Transactions on Communications,1987,35(6):584-594.
155. Gardner W A, Brown W, Chen Chih-Kang. Spectral Correlation of Modulated Signals: Part Ⅱ-Digital Modulation. IEEE Transactions on Communications,1987,35(6):595-601.
156. Gardner W A. Measurement of spectral correlation. Acoustics, Speech, and Signal Processing, 1986,34(5):1111-1123.
157. Gardner W A. Signal Interee:ptio: A Unifying Theoretical Framework for Feature Detec-tion. IEEE Trans on communications,1988,36(8):897-906.
158. Gardner W A, Spooner C M. Cyclic Spectral Analysis for Signal Detection andModulation Recognition. Proc. IEEE Conf Military Communications,1988:419-423.
159. Gardner W A, Archer T L. Exploitation of cyclostationarity for identifying the Volterrakernels ofnonlinear systems. IEEE Transactions on Information Theory,1993,39:535-542.
160. Ciblat P, Loubaton P, Serpedin E. Asymptotic analysis ofblind cyclic correlation-basedsymbol-rate estimators. IEEE Trans on information theory,2002,48(7):1922-1934.
161. Ferreol A, Chevalier P, Albera L. Second-order blind separation of first and second-ordercyclostationary sources-application to AM and dcterministic sources. IEEE Trans. Signal Processing,2004,52(4):845-861.
162. Gardner W A. Exploitation of spectral redundancy incyclostationary signals. IEEE.Signal Pro-cessing Magazine,1991,8(2):14-36.
163. 张贤达,保铮.非平稳信号分析与处理.北京：国防工业出版社,2001.
164. 黄知涛,周一宇,姜文利.循环平稳信号处理及应用.北京：科学出版社,2006.
165. Dandawate A V, Giannakis G B. Modeling (almost) periodic moving average processes using cyclic statistics. IEEE Trans. Signal Processing,1996,44:673-684.
166. Giannakis G B. Cyclostationary signal analysis. The Digital Signal Processing Handbook (Eds.: MadisettiV. K. Wiliams D. B.), CRC Press and IEEE Press, Boca Raton, FL and New York,1998 (Chapter 17).
167. Izzo L, Napolitano A. The higher order theory of generalized almost-cyclostationary time se-ries. IEEE Transactions on Signal Processing,1998,46(11):2975-2989.
168. Giannakis G B, Dandawate A V. Polyspectral analysis of (almost) cyclostationary signals: LPTV system identification and related applications.1991 Conference Record of the Twenty-Fifth Asilomar Conference on Signals, Systems andComputers,1991,1:377-382.
169. Kacimov A R, Obnosov Y V, Yakimov N D. Groundwater flow in a medium with a parquet-type conductivity distribution. Journal of Hydrology,1999,226:242-249.
170. Finelli C J, Jenkins J M. A cyclostationary least mean squares algorithm for discrimination of ventticular tachycardia from sinus rhythm. International Conference of the IEEE Engineering in Medicine and Biology Society, Baltimore, MD,1991,13:740-741.
171. Pagano M, Paizen E. An approach to modeling seasonally stationary time-series. Journal of Econometrics,1979,9:137-153.
172. Sherman P J. Random processes from rotating machinery, in: A.G.Miamiee (Ed.). Proceedings of Workshop on Nonstationary Stochastic Processes and their Applications, Word Scientific, New Jersey,1991,211-218.
173. Campbell J, Gibbs A, Smith B. The cyclostationary nature of crosstalk interference from digital signals in multipair cable-Part I: fundamentals. IEEE Trans. Commun.,1983,31:629-637.
174. Tong L, Guanghan X, Kailath T. Blind identification and equalization based on second-order statistics: a time domain approach. IEEE Trans. Information Theory,1994,40:340-349.
175. Tong L, Xu G, HassibiB, et al. Blind channel identification based on second-order statistics: a frequency-domain approach. IEEE Trans. Information Theory,1995,41:329-334.
176. Tugnait J K. On blind identifiability of multipath channels using fractional sampling and second-order cyclostationary statistics. IEEE Trans. Information Theory,1995,41:308-311.
177. Ding Z, Li Y. On channel identification based on second-order cyclic spectra. IEEE Trans. Signal Processing, 1994,42:1260-1264.
178. Bezrukov S M, Vodyanoy I. Stochastic resonance in non-dynamical systems without response thresholds. Nature, 1997,385:319-321.
179. Zaikin A A, Kurths J, Schimansky-Geier L. Doubly stochastic resonance. Physical Review E, 2000,85:227-231.
180. Zeng L, Li J. Residual aperiodic stochastic resonance in Levy noise. Fluctuation and Noise Letters,2011,10:223-229.
181. Hanggi P, Jung P, Zerbe C, Moss F. Can colored noise improve stochastic resonance? Journal of Statistical Physics, 1993,70:25-47.
182. DeWeese M, Bialek W. Information flow in sensory neurons.NuovoCimento, 1995,17:733-741.
183. Brown L D. A proof of the central limit theorem motivated by the Cramer-Rao inequality. Statistics and Probability: Essays in Honor of C. R. Rao, eds. by G. Kallianpur, P. R. Krishnaiah and J. K. Ghosh, North-Holland Publishing Company, 1982,141-148.
184. Luchinsky D G, Mannella R, McClintock P V E, Stocks N G. Stochastic resonance in electrical circuits-I:coventional stochastic Resonance. IEEE Transactions on Circuits and Systems-II: Analog and Digtial Signal Processing. 1999,46:1205-1214.
185. Zozor S, Amblar P O. Stochastic resonance in discrete time nonlinear AR(1) models. IEEE Transactions on Signal Processing, 1999,47:108-122.
186. Greenwood P E, Ward L M, Wefelmeyer W. Statistical analysis of stochastic resonance in a simple setting. Physical Review E, 1999,60:4687-4695.
187. Casado-Pascual J, Gomez-Ordonez J, Morillo M. Two-state theory of nonlinear stochastic res-onance. Physical Review Letters, 2003,91(no.210601).
188. Stam A J. Some inequalities satisfied by the quantities of information of Fisher and Shannon. Information and Control,1959,2:101-112.
189. Madiman M, Barron A. Generalized entropy power inequalities and monotonicity properties of information. IEEE Transactions on Information Theory, 2007,53:2317-2329.
190. Johnson O, Barron A. Fisher information inequalities and the central limit theorem. Probability Theory and Related Fields, 2004,129:391-409.
191. Stocks N G. Information transmission in parallel threshold arrays: Suprathreshold stochastic resonance. Physical Review E,2001,63:0411141-9.
192. Rousseau D, Chapeau-Blondeau F. Stochastic resonance and improvement by noise in optimal detection strategies. Digital Signal Processing, 2005,15:19-32.
193. Inchiosa M E, Bulsara A R. Signal detection statistics of stochastic resonators. Physical Review E,1996,53:R2021-R2024.
194. Duan F, Chapeau-Blondeau F, Abbott D. Stochastic resonance in a parallel array of nonlinear dynamical elements. Physics Letters A, 2008,372:2159-2166.
195. Gailey P C, Neiman A, Collins J J, Moss F. Stochastic resonance in ensembles of nondynamical elements:the role of internal noise. Physical Review E, 1997, 79:4701-4704.
196. Newman M E J, Barabasi A L, Watts D J. The Structure and Dynamics of Networks. Princeton University Press, Princeton,2006.
197. Chen L, Wang R, Zhou T, Aihara K. Noise-induced cooperative behavior in a multicell system. Oxford Univ Press, Oxford, 2005.
198. Albert R, Barabasi A L. Statistical mechanics of complex networks. Reviews of modern physics, 2002,74:47.
199. Williams R J, Berlow E L, Dunne J A, Barabasi A L, Martinez N D. Two degrees of separation in complex food webs. Proceedings of the National Academy of Sciences, 2002,99:12913-12916.
200. Newman M E J. The structure of scientific collaboration networks. Proceedings of the National Academy of Sciences, 2001,98:404-409.
201. Barabasi A L, Oltvai Z N. Network biology: understanding the cell's functional organization. Nature Reviews Genetics,2004,5:101-113.
202. Sporns O, Chialvo D R, Kaiser M, Hilgetag C C. Organization, development and function of complex brain networks. TRENDS in Cognitive Sciences, 2004,8:418-425.
203. Zhou C, Zemanova L, Zamora G, Hilgetag C C, Kurths J. Hierarchical organization unveiled by functional connectivity in complex brain networks. Physical review letters, 2006,97:238103.
204. Benzi R. Stochastic resonance in complex systems. Journal of Statistical Mechanics:Theory and Experiment, 2009,1 (no.P01052).
205. Chapeau-Blondeau F, Rousseau D. Noise-aided SNR amplification by parallel arrays of sensors with saturation. Physics Letters A, 2006,351:231-237.
206. Ichiki A, Tadokoro Y. Signal-to-noise ratio improvement by stochastic resonance in moments in non-dynamical systems with multiple states. Physics Letters A, 2013,377:185-188.
207. Ma Y, Duan F, Chapeau-Blondeau F, Abbott D. Weak-periodic stochastic resonance in a paral-lel array of static nonlinearities. PLoS ONE, 2013,8(no.e58507).
208. Xu B. Comparison of aperiodic stochastic resonance in a bistable system realized by adding noise and by tuning system parameters. Physical Review E, 2004,69(no.061110).
209. Apostolico F, Gammaitoni L, Marcheson F, Santucci S. Resonant trapping:a failure mecha-nism in switch transitions. Physics Review E, 1997,55:36-39.
210. Bulsara A R, In V, Kho A, Longhini P, Palacios A, Rappel W-J, AcebronJ, Baglio S, Ando B. Emergent oscillations in unidirectionally coupled overdamped bistable systems. Physical Review E, 2004,70(no.036103).
211. Bulsara A R, Lindner J F, In V, Kho A, Baglio S, Sacco V, Ando B, LonghiniP, Palacios A, Rappel W-J. Coupling-induced cooperative behaviour in dynamic ferromagnetic cores in the presence of a noise floor. Physics Letters A, 2005,353:4-10.