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混沌信号的压缩感知去噪

李广明 吕善翔

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混沌信号的压缩感知去噪

李广明, 吕善翔

Chaotic signal denoising in a compressed sensing perspective

Li Guang-Ming, Lü Shan-Xiang
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  • 对非线性时间序列进行噪声抑制是从中提取有效信息的前提. 混沌信号的去噪算法不仅要使滤波后的信号具有较高的信噪比, 也要具有较好的不确定性. 从压缩感知的角度出发,提出了一种新的噪声抑制方法. 该方法包括估计噪声方差, 以及依据动态的稀疏度将观测值往确定的过完备字典上投影. 仿真实验表明, 该方法比常用的小波阈值法和局部曲线拟合法具有更高的输出信噪比, 而原始信号的混沌特性也能得到较大程度的恢复.
    Nonlinear time series denoising is the premise for extracting useful information from an observable, for the applications in analyzing natural chaotic signals or achieving chaotic signal synchronizations. A good chaotic signal denoising algorithm processes not only a high signal-to-noise ratio (SNR), but also a good unpredictability of a signal. Starting from the compressed sensing perspective, in this work we provide a novel filtering algorithm for chaotic flows. The first step is to estimate the strength of the noise variance, which is not explicitly provided by any blind algorithm. Then the second step is to construct a deterministic projection matrix, whose columns are polynomials of different orders, which are sampled from the Maclaurin series. Since the noise variance is provided from the first step, then a sparsity level with regard to this signal can be fully constructed, and this sparsity value in conjunction with the orthogonal matching pursuit algorithm is used to recover the original signal. Our method can be regarded as an extension to the local curve fitting algorithm, where the extension lies in allowing the algorithm to choose a wider range of polynomial orders, not just those of low orders. In the analysis of our algorithm, the correlation coefficient of the proposed projection matrix is given, and the reason for shrinking the sparsity when the noise variance increases is also presented, which emphasizes that there is a larger probability of error column selection with larger noise variance. In the simulation, we compare the denoising performance of our algorithm with those of the wavelet shrinking algorithm and the local curve fitting algorithm. In terms of SNR improvement for the Lorenz signal, the proposed algorithm outperforms the local curve fitting method in an input SNR range from 0 dB to 20 dB. And this superiority also exists if the input SNR is larger than 9 dB when compared with the wavelet methods. A similar performance also exists concerning the Rössler chaotic system. The last simulation shows that the chaotic properties of the originals are largely recovered by using our algorithm, where the quantity for "chaotic degree" is described by using the proliferation exponent.
    • 基金项目: 国家自然科学基金(批准号: 61170216, 61372082)资助的课题.
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 61170216, 61372082).
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    [2]

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    [3]

    Lü S X, Feng J C 2013 Acta Phys. Sin. 62 230503 (in Chinese) [吕善翔, 冯久超 2013 物理学报 62 230503]

    [4]

    Feng J C 2005 Chin. Phys. Lett. 22 1851

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    Feng J C, Tse C K 2001 Phys. Rev. E 63 026202

    [6]

    Constantine W L B, Reinhall P G 2001 Int. J. Bifurcat. Chaos 11 483

    [7]

    Han M, Liu Y H, Xi J H, Guo W 2007 IEEE Signal Process. Lett. 14 62

    [8]

    Gao J B, Sultan H, Hu J, Tung W W 2010 IEEE Signal Process. Lett. 17 237

    [9]

    Tung W W, Gao J B, Hu J, Yang L 2011 Phys. Rev. E 83 046210

    [10]

    Wang W B, Zhang X D, Wang X L 2013 Acta Phys. Sin. 62 050201 (in Chinese) [王文波, 张晓东, 汪祥莉 2013 物理学报 62 050201]

    [11]

    Candes E, Romberg J, Tao T 2006 Commun. Pure Appl. Math. 59 1207

    [12]

    Lustig M, Donoho D, Pauly J 2007 Magn. Reson. Med. 58 1182

    [13]

    Lustig M, Donoho D, Santos J, Pauly J 2008 IEEE Signal Process. Mag. 25 72

    [14]

    Candes E, Wakin M 2008 IEEE Signal Process. Mag. 25 21

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    Candes E 2008 C. R. Math. 346 589

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    Chen S, Donoho D, Saunders M 1998 SIAM J. Sci. Comput. 20 33

    [17]

    Donoho D, Huo X 2001 IEEE Trans. Inf. Theory 47 2845

    [18]

    Figueiredo M, Nowak R, Wright S 2007 IEEE J. Sel. Topics Signal Process. 1 586

    [19]

    Cai T T, Wang L 2011 IEEE Trans. Inf. Theory 57 4680

    [20]

    Donoho D, Drori I, Starck J L 2012 IEEE Trans. Inf. Theory 58 1094

    [21]

    Needell D, Vershynin R 2009 Found. Comput. Math. 9 317

    [22]

    Lü S X, Wang Z S, Hu Z H, Feng J C 2014 Chin. Phys. B 23 010506

    [23]

    Holger K, Thomas S 2004 Nonlinear Time Series Ana- lysis (Cambridge: Cambridge University Press) pp65-74

  • [1]

    Wang S Y, Feng J C 2012 Acta Phys. Sin. 61 170508 (in Chinese) [王世元, 冯久超 2012 物理学报 61 170508]

    [2]

    Feng J C 2012 Chaotic Signals and Information Processing (Beijing: Tsinghua Univ. Press) pp32-35 (in Chinese) [冯久超 2012 混沌信号与信息处理 (清华大学出版社)第32–35页]

    [3]

    Lü S X, Feng J C 2013 Acta Phys. Sin. 62 230503 (in Chinese) [吕善翔, 冯久超 2013 物理学报 62 230503]

    [4]

    Feng J C 2005 Chin. Phys. Lett. 22 1851

    [5]

    Feng J C, Tse C K 2001 Phys. Rev. E 63 026202

    [6]

    Constantine W L B, Reinhall P G 2001 Int. J. Bifurcat. Chaos 11 483

    [7]

    Han M, Liu Y H, Xi J H, Guo W 2007 IEEE Signal Process. Lett. 14 62

    [8]

    Gao J B, Sultan H, Hu J, Tung W W 2010 IEEE Signal Process. Lett. 17 237

    [9]

    Tung W W, Gao J B, Hu J, Yang L 2011 Phys. Rev. E 83 046210

    [10]

    Wang W B, Zhang X D, Wang X L 2013 Acta Phys. Sin. 62 050201 (in Chinese) [王文波, 张晓东, 汪祥莉 2013 物理学报 62 050201]

    [11]

    Candes E, Romberg J, Tao T 2006 Commun. Pure Appl. Math. 59 1207

    [12]

    Lustig M, Donoho D, Pauly J 2007 Magn. Reson. Med. 58 1182

    [13]

    Lustig M, Donoho D, Santos J, Pauly J 2008 IEEE Signal Process. Mag. 25 72

    [14]

    Candes E, Wakin M 2008 IEEE Signal Process. Mag. 25 21

    [15]

    Candes E 2008 C. R. Math. 346 589

    [16]

    Chen S, Donoho D, Saunders M 1998 SIAM J. Sci. Comput. 20 33

    [17]

    Donoho D, Huo X 2001 IEEE Trans. Inf. Theory 47 2845

    [18]

    Figueiredo M, Nowak R, Wright S 2007 IEEE J. Sel. Topics Signal Process. 1 586

    [19]

    Cai T T, Wang L 2011 IEEE Trans. Inf. Theory 57 4680

    [20]

    Donoho D, Drori I, Starck J L 2012 IEEE Trans. Inf. Theory 58 1094

    [21]

    Needell D, Vershynin R 2009 Found. Comput. Math. 9 317

    [22]

    Lü S X, Wang Z S, Hu Z H, Feng J C 2014 Chin. Phys. B 23 010506

    [23]

    Holger K, Thomas S 2004 Nonlinear Time Series Ana- lysis (Cambridge: Cambridge University Press) pp65-74

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出版历程
  • 收稿日期:  2015-03-25
  • 修回日期:  2015-05-23
  • 刊出日期:  2015-08-05

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