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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.
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Keywords:
- chaotic signal /
- denoising /
- compressed sensing /
- sparsity
[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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[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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