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本文建立了分数阶可停振动系统, 其可停振动状态的改变对周期策动力敏感, 对零均值随机微小扰动不敏感, 这事实上为周期未知微弱信号检测提供了一种新的高效检测方法和判别标准. 与现有的利用混沌系统的大尺度周期状态变化检测周期未知弱信号的方法 需逐一尝试设置不同频率内置信号以便期望与待检周期信号发生共振不同, 利用分数阶可停振动系统的可停振动状态变化检测周期未知微弱信号的方法, 除了同样具有因为状态变化对周期信号的敏感性而能够实现极低检测门限的特点外, 还具有混沌系统信号检测所不具有的优点: 1)无需预先估计待检信号的周期; 2)无需计算系统状态的临界阈值; 3)可停振动状态可由本文设计的指数波动函数可靠地进行判断; 4)通过系统微分阶数的变化, 将检测系统层次化, 从而可得到比整数阶检测系统更低的检测门限, 特别是在色噪声环境下, 通过选取合适的微分阶数, 基于分数阶可停振动系统的微弱周期信号检测法能够大幅度的降低检测门限, 在本文的仿真试验中, 检测门限可达-182 dB.In this paper, a new detecting method for weak periodic signals with unknown periods and unknown forms, the so-called fractional stopping oscillation method, is presented. This new detecting method, which is based on the research of some dissipative system of single degree of freedom, is sensitive to periodic signal—even with unknown period and unknown form—and insensitive to noise. Compared with the known chaotic detections in which a built-in signal must be pre-set with the same frequency and the same form as the detected periodic signal, the fractional stopping oscillation method can not only be used even at lower SNR than chaotic detection, but also has some other notable advantages as follows: (1) it need not get the period and the form of detected signal before hand or pre-estimate them; (2) it need not pre-calculate the chaotic threshold value; (3) the existence of periodic signal in system input can be reliably and quantitatively judged by volatility index function, designed in this paper, for stopping oscillation method; (4) a more sensitive detection method can be achieved by the fractionalization of the detection system, especially, the detection threshold can reach -182 dB when the background noise is colored Gaussian noise.
[1] Proakis J G 2003 Digital Communications (4th Edition) (Beijing: Electronic Industry Press) p169
[2] Li Y, Yang B J 2004 Introduction of Detection by Chaotic Oscillator (Beijing: Electronic Industry Press) p55 (in Chinese) [李月, 杨宝俊 2004 混沌振子检测引论 (北京: 电子工业出版社) 第55页]
[3] Wang G Y 2001 IEEE Transaction on Industrial Electronics 46 440
[4] Wang Y S, Jiang W Z, Zhao J J, Fan H D 2008 Acta Phys. Sin. 57 2053 (in Chinese) [王永生, 姜文志, 赵建, 范洪达 2008 57 2053]
[5] Wang J X, Hou C L 2010 International Conference on e-Education, e-Business, e-Management, e-Learning Sanya, Jan, 2010 p387
[6] Zhang Z F, Ding T R, Huang W Z, Dong Z X 1997 Qualitative Theory of Differential Equation (2nd Edition) (Beijing: Science Press) p450 (in Chinese) [张芷芬, 丁同仁, 黄文灶, 董镇喜 1997 微分方程的定性理论(第2版) (北京: 科学出版社) 第450页]
[7] Zhao P D, Zhang X D 2008 Acta Phys. Sin. 58 2791 (in Chinese) [赵品栋, 张晓丹 2008 58 2791 ]
[8] Wang M J, Wang X Y 2010 Acta Phys. Sin. 59 1583 (in Chinese) [王明军, 王兴元 2010 59 1583]
[9] Petráš I 2011 Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing: Higher Education Press) p55
[10] Zhou S, Fu J L, Liu Y S 2010 Chin. Phys. B 19 120301
[11] Zhang Y 2012 Chin. Phys. B 21 084502
[12] Wei H Y, Xia T C 2012 Chin. Phys. B 21 100505
[13] Zhang S H, Chen B Y, Fu J L 2012 Chin. Phys. B 21 100202
[14] Chen X R 2000 Probability and Statistics (Beijing: Science Press) p141 (in Chinese) [陈希孺 2000 概率论与数理统计(北京:科学出版社) 第141页]
[15] Zhu W Q 2003 Nonlinear Stochastic Dynamical Systems and Control p122 (in Chinese) [朱卫秋 2003 非线性随机动力系统与控制(北京: 科学出版社) 第122页]
[16] Arnold V I 1961 Sov. Math. Dokl. 2 247
[17] Tavazoei M S, Haeri M 2007 Phys. Lett. A 367 102
[18] Tavazoei M S, Haeri M 2008 Nonlinear Analysis 69 1299
[19] Tavazoei M S, Haeri M 2010 Automatic 46 94
[20] Wang Z H, Hu H Y 2010 Science China: Physics, Mechanics {& Astronomy} 53 345
[21] Sabattier J, Moze M, Farges C 2010 Comput. Math. Appl. 59 1594
[22] Tavazoei M S, Haeri M 2008 Physica D 237 2628
[23] Tavazoei M S, Haeri M 2009 Math. Comput. Simul. 79 1566
[24] Podlubny I 1999 Fractional Differential Equations (San Diego USA: Acadamic Press) p78
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[1] Proakis J G 2003 Digital Communications (4th Edition) (Beijing: Electronic Industry Press) p169
[2] Li Y, Yang B J 2004 Introduction of Detection by Chaotic Oscillator (Beijing: Electronic Industry Press) p55 (in Chinese) [李月, 杨宝俊 2004 混沌振子检测引论 (北京: 电子工业出版社) 第55页]
[3] Wang G Y 2001 IEEE Transaction on Industrial Electronics 46 440
[4] Wang Y S, Jiang W Z, Zhao J J, Fan H D 2008 Acta Phys. Sin. 57 2053 (in Chinese) [王永生, 姜文志, 赵建, 范洪达 2008 57 2053]
[5] Wang J X, Hou C L 2010 International Conference on e-Education, e-Business, e-Management, e-Learning Sanya, Jan, 2010 p387
[6] Zhang Z F, Ding T R, Huang W Z, Dong Z X 1997 Qualitative Theory of Differential Equation (2nd Edition) (Beijing: Science Press) p450 (in Chinese) [张芷芬, 丁同仁, 黄文灶, 董镇喜 1997 微分方程的定性理论(第2版) (北京: 科学出版社) 第450页]
[7] Zhao P D, Zhang X D 2008 Acta Phys. Sin. 58 2791 (in Chinese) [赵品栋, 张晓丹 2008 58 2791 ]
[8] Wang M J, Wang X Y 2010 Acta Phys. Sin. 59 1583 (in Chinese) [王明军, 王兴元 2010 59 1583]
[9] Petráš I 2011 Fractional-Order Nonlinear Systems: Modeling, Analysis and Simulation (Beijing: Higher Education Press) p55
[10] Zhou S, Fu J L, Liu Y S 2010 Chin. Phys. B 19 120301
[11] Zhang Y 2012 Chin. Phys. B 21 084502
[12] Wei H Y, Xia T C 2012 Chin. Phys. B 21 100505
[13] Zhang S H, Chen B Y, Fu J L 2012 Chin. Phys. B 21 100202
[14] Chen X R 2000 Probability and Statistics (Beijing: Science Press) p141 (in Chinese) [陈希孺 2000 概率论与数理统计(北京:科学出版社) 第141页]
[15] Zhu W Q 2003 Nonlinear Stochastic Dynamical Systems and Control p122 (in Chinese) [朱卫秋 2003 非线性随机动力系统与控制(北京: 科学出版社) 第122页]
[16] Arnold V I 1961 Sov. Math. Dokl. 2 247
[17] Tavazoei M S, Haeri M 2007 Phys. Lett. A 367 102
[18] Tavazoei M S, Haeri M 2008 Nonlinear Analysis 69 1299
[19] Tavazoei M S, Haeri M 2010 Automatic 46 94
[20] Wang Z H, Hu H Y 2010 Science China: Physics, Mechanics {& Astronomy} 53 345
[21] Sabattier J, Moze M, Farges C 2010 Comput. Math. Appl. 59 1594
[22] Tavazoei M S, Haeri M 2008 Physica D 237 2628
[23] Tavazoei M S, Haeri M 2009 Math. Comput. Simul. 79 1566
[24] Podlubny I 1999 Fractional Differential Equations (San Diego USA: Acadamic Press) p78
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