| Chaos resides in various physical systems,such as biological systems,electrical systems and mechanical systems.Detecting chaos from the observation data is a key procedure to recognize,analyze and predict these systems,and it’s also an important task in many areas of science and engineering.Usually,the observation data is contaminated by noise.When the noise is strong,the underlying system dynamics could be concealed,which would not only make it very difficult to calculate the invariant chaotic parameters such as the Lyapunov exponents and the fractal dimension,but also affect the chaos behavior detection.The inherent features of chaos,such as aperiodic property and wide band spectrum,present a big challenge for the conventional signal reconstruction methods.Therefore,it is crucial to research on effective signal reconstruction methods for contaminated chaotic signals.Aiming at the limitations of existing chaotic signal reconstruction methods,this paper presents new solutions to reconstruct noise contaminated chaotic signals by exploring the inherent features of chaos.The specific research contents include the following three parts:(1)Unlike the fractal self-similarity nature in the phase space,the time domain self-similarity of chaotic attractors has not yet been recognized and studied.Based on the fractal characteristics of chaos,we observed and analyzed the time domain self-similarity in the waveform of continuous time chaotic systems.Further more,we mathematically proved that the fractal self-similarity in the phase space would lead to time-domain self-similarity.By exploiting this self-similarity feature,a novel noise reduction algorithm based on the collaborative filtering is proposed.In the proposed algorithm,the 1-D denoising is transformed into a 2-D joint filtering by grouping the similar fragments within the observation data.Subsequently,the noises are attenuated by thresholding in the 2-D transforming domain.Finally,the noise-free signals are estimated with an inverse transformation.Since the fragments within a group are closely correlated due to their mutual similarity,the 2-D transforming of the group should be much sparser than the 1-D transforming of the original signal,achieving a much better noise attenuation capability.However,this algorithm relies on self-similarity of the signal,which makes its application limited.(2)When applying conventional SSA denoising method to chaotic signals,it is difficult to identify the number of the singular values corresponding to the signal components due to the noise-like nature of chaos.To address this issue,a novel parameter identification method based on phase space reconstruction is proposed.In this method,the number of the singular values is estimated by comparing the statistical difference between the chaotic signals and the noise in the reconstructed phase space.Accordingly,an adaptive noise reduction algorithm is designed.Compared with the conventional denoising methods,the proposed algorithm shows better adaptivity and noise reduction performance.However,this algorithm is time consuming when processing long data series,since its’computational complexity is O(N~3).(3)Blind source separation is a signal reconstruction procedure which commonly emerges in wireless sensor networks and multiuser communications.The inherent features of non-periodic and wide band spectrum make it quite a challenge to separate mixed chaotic signals blindly.To address this issue,a new blind source separation method is proposed.This method is structured in the phase space of the demixed signals,which is reconstructed from the observations by delay-embedding method.An objective function is designed in the reconstructed phase space so that the blind source separation problem is transformed into an unconstrained optimization problem and be solved by an artificial bee colony optimizer.Different from conventional independent component analyse approaches which concern mainly the statistical features,the proposed blind source separation method utilizes the dynamics reside in the observed mixtures by phase space reconstruction.Therefore,better performance can be achieved when it is used to deal with chaotic signals.Additionally,parameterized representation of orthogonal matrices is adopted to reduce the dimension of the optimization procedure so that the algorithm can converge quickly. |
