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The inherent features, such as non-periodic, wide band spectrum, and extreflely sensitive to initial values etc. make it quite a challenge to blindly separate the mixed chaotic signals. A new blind source separation method based on the artificial bee colony algorithm is proposed in this paper. This method can recover chaotic sources from noisy observations on their linear mixtures without any prior information about the source equations. The proposed method is structured in the phase space of the demixed signals, which is reconstructed from the observations by using delay-embedding method. An objective function in the reconstructed phase space is designed so that the blind source separation problem is transformed into an optimization problem. The optimal demixing matrix is obtained by maximize the objective function with an artificial bee colony optimizer and the chaotic sources are then recovered by multiplying the observed mixtures and the optimal demixing matrix. Before the optimization procedure is made, a pre-whitening should be employed. Additionally, the parameterized repreflentation of orthogonal matrices through principal rotation is adopted to reduce the dimension of the optimization procedure so that the proposed blind source separation algorithm can converge quickly. Different from the traditional independent component analysis approaches which concern mainly the statistical features, the proposed blind source separation method utilizes the dynamics in the observed mixtures by means of phase space reconstruction. Therefore, better performance can be achieved when it is used to deal with chaotic signals. In computer simulation, two cases are taken into consideration: namely, the mixture is noiseless or not contaminated by noise. The correlation coefficient criterion and the performance index criterion are adopted to evaluate the separation performances. Simulation result shows that in most cases the proposed approach converges within a few tens of iterations and the chaotic sources can be accurately recovered. The impact of noise level and signal length on the separation performance is investigated in detail. The overall performance of the proposed approach is much better than the traditional independent component analysis approaches. Moreover, the capability of separating the mixed chaotic and Gaussian signals reflealed in the simulation indicates that the proposed approach has the potential to be applied in a wider range of applications.
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Keywords:
- Artificial bee colony algorithm /
- blind source separation /
- chaotic signal /
- phase space reconstruction
[1] Feng J C, Tse C K 2008 Reconstruction of Chaotic Signals with Applications to Chaos-based Communications (Beijing: Tsinghua University Press) pp3-20
[2] Andreflev Y V, Dmitriev A S, Efremova E V, Anagnostopoulos A N 2003 IEEE Trans. Circ. Syst. I 50 613
[3] Hu Z H, Feng J C 2010 Journal of Southwest University (Natural Science) 32 146 (in Chinese) [胡志辉, 冯久超 2010 西南大学学报 (自然科学版) 32 146]
[4] Huang J W, Feng J C, L S X 2014 Acta Phys. Sin. 63 050502 (in Chinese) [黄锦旺, 冯久超, 吕善翔 2014 63 050502]
[5] Wang S Y, Feng J C 2012 Acta Phys. Sin. 61 170508 (in Chinese) [王世元, 冯久超 2012 61 170508]
[6] Chen H B, Feng J C, Fang Y 2008 Chin. Phys. Lett. 25 405
[7] Kuraya M, Uchida A, Yoshimori S, Umeno K 2008 Optics Express 16 725
[8] Gong Y R, He D, He C 2012 Acta Phys. Sin. 61 120502 (in Chinese) [宫蕴瑞, 何迪, 何晨 2012 61 120502]
[9] Comon P, Jutten C 2010 Handbook of Blind Source Separation (Waltham: Academic Press) pp6-15
[10] Olsson R K, Hansen L K 2006 Journal of Machine Learning Research 7 2585
[11] Galka A, Wong K, Ozaki T 2010 Modeling Phase Transitions in the Brain (Berlin:Springer) pp27-52
[12] Galka A, Wong K, Stephani U, Ozaki T. Siniatchkin M 2013 International Journal of Bifurcation and Chaos 23 1350165
[13] Wang B Y, Zheng W X 2006 IEEE Trans. Circ. Syst. II 53 143
[14] Karaboga D, Basturk B 2007 Journal of Global Optimization 39 459
[15] Karaboga D, Basturk B 2008 Applied Soft Computing 8 687
[16] Takens F 1981 Lecture Notes in Mathematics (Berlin:Springer) pp366-381
[17] L S X, Wang Z S, Hu Z H, Feng J C 2014 Chin. Phys. B 23 010506
[18] Mavaddaty S, Ebrahimzadeh A 2012 20th Iranian Conference on Electrical Engineering Tehran, Iran May 15-17, 2012 p1172
[19] Ebrahimzadeh A, Mavaddaty S 2014 Swarm and Evolutionary Computation 14 15
[20] Schaub H, Tsiotras P, Junkins J L 1995 International Journal of Engineering Science 33 2277
[21] Yang H H, Amari S I 1997 Neural Computation 9 1457
[22] Cardoso J F 1999 Neural Computation 11 157
[23] Hyvärinen A 1999 IEEE Trans. Neural Networks 10 626
[24] Yang H H, Amari S 1997 Neural Computation 9 1457
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[1] Feng J C, Tse C K 2008 Reconstruction of Chaotic Signals with Applications to Chaos-based Communications (Beijing: Tsinghua University Press) pp3-20
[2] Andreflev Y V, Dmitriev A S, Efremova E V, Anagnostopoulos A N 2003 IEEE Trans. Circ. Syst. I 50 613
[3] Hu Z H, Feng J C 2010 Journal of Southwest University (Natural Science) 32 146 (in Chinese) [胡志辉, 冯久超 2010 西南大学学报 (自然科学版) 32 146]
[4] Huang J W, Feng J C, L S X 2014 Acta Phys. Sin. 63 050502 (in Chinese) [黄锦旺, 冯久超, 吕善翔 2014 63 050502]
[5] Wang S Y, Feng J C 2012 Acta Phys. Sin. 61 170508 (in Chinese) [王世元, 冯久超 2012 61 170508]
[6] Chen H B, Feng J C, Fang Y 2008 Chin. Phys. Lett. 25 405
[7] Kuraya M, Uchida A, Yoshimori S, Umeno K 2008 Optics Express 16 725
[8] Gong Y R, He D, He C 2012 Acta Phys. Sin. 61 120502 (in Chinese) [宫蕴瑞, 何迪, 何晨 2012 61 120502]
[9] Comon P, Jutten C 2010 Handbook of Blind Source Separation (Waltham: Academic Press) pp6-15
[10] Olsson R K, Hansen L K 2006 Journal of Machine Learning Research 7 2585
[11] Galka A, Wong K, Ozaki T 2010 Modeling Phase Transitions in the Brain (Berlin:Springer) pp27-52
[12] Galka A, Wong K, Stephani U, Ozaki T. Siniatchkin M 2013 International Journal of Bifurcation and Chaos 23 1350165
[13] Wang B Y, Zheng W X 2006 IEEE Trans. Circ. Syst. II 53 143
[14] Karaboga D, Basturk B 2007 Journal of Global Optimization 39 459
[15] Karaboga D, Basturk B 2008 Applied Soft Computing 8 687
[16] Takens F 1981 Lecture Notes in Mathematics (Berlin:Springer) pp366-381
[17] L S X, Wang Z S, Hu Z H, Feng J C 2014 Chin. Phys. B 23 010506
[18] Mavaddaty S, Ebrahimzadeh A 2012 20th Iranian Conference on Electrical Engineering Tehran, Iran May 15-17, 2012 p1172
[19] Ebrahimzadeh A, Mavaddaty S 2014 Swarm and Evolutionary Computation 14 15
[20] Schaub H, Tsiotras P, Junkins J L 1995 International Journal of Engineering Science 33 2277
[21] Yang H H, Amari S I 1997 Neural Computation 9 1457
[22] Cardoso J F 1999 Neural Computation 11 157
[23] Hyvärinen A 1999 IEEE Trans. Neural Networks 10 626
[24] Yang H H, Amari S 1997 Neural Computation 9 1457
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