Symbolic dynamics for processing chaotic signals. I. Noise reduction of chaotic sequences

Chaotic signals attracted the attention among researchers because of their rich dynamics and their random-like behavior. What has been missing so far is an appropriate characterization of chaotic systems from a signal-processing point of view. This paper demonstrates that the framework of symbolic dynamics gives the possibility to partition the infinite number of finite-length trajectories of the piecewise-linear chaotic system into a countable number of trajectory-sets with common statistical properties. It turns out that this partitioning allows to derive noise-reduction schemes directly from the maximum-likelihood criteria. For the two proposed noise-reduction methods, the upper performance limits are given in an analytical form and the results are verified by applying the schemes to different types of one dimensional piecewise-linear Markov maps.

[1]  Cheng-Fu Chen,et al.  On the absolute stability of nonlinear control systems , 1970 .

[2]  U. Schönwandt,et al.  Approximations to nonlinear observers , 1973 .

[3]  Hao Bai-lin.  Elementary Symbolic Dynamics , 1988 .

[4]  J. Suykens,et al.  Master-slave synchronization using dynamic output feedback , 1997 .

[5]  Schreiber,et al.  Noise reduction in chaotic time-series data: A survey of common methods. , 1993, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[6]  M. Gotz,et al.  Statistical analysis of chaotic communication schemes , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[7]  P. Walters.  Introduction to Ergodic Theory , 1977 .

[8]  Carroll,et al.  Synchronization in chaotic systems. , 1990, Physical review letters.

[9]  Tzyh Jong Tarn,et al.  Exponential Observers for Nonlinear Dynamic Systems , 1975, Inf. Control..

[10]  J. Suykens,et al.  Nonlinear H/sub /spl infin// synchronization of Lur'e systems: dynamic output feedback case , 1997 .

[11]  A. N. Kolmogorov,et al.  Interpolation and extrapolation of stationary random sequences. , 1962 .

[12]  Douglas Lind,et al.  An Introduction to Symbolic Dynamics and Coding , 1995 .

[13]  Michael Peter Kennedy,et al.  Synchronization of chaotic signals , 1993 .

[14]  F. Thau.  Observing the state of non-linear dynamic systems† , 1973 .

[15]  Kennedy,et al.  Predictive Poincaré control: A control theory for chaotic systems. , 1995, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[16]  John G. Proakis,et al.  Digital Communications , 1983 .

[17]  T. Schimming,et al.  Symbolic dynamics for processing chaotic signal. II. Communication and coding , 2001 .

[18]  J. Schweizer.  The performance of chaos shift keying: synchronization versus symbolic backtracking , 1998, ISCAS '98. Proceedings of the 1998 IEEE International Symposium on Circuits and Systems (Cat. No.98CH36187).

[19]  H. Dedieu,et al.  Noise reduction in chaotic time series - an overview , 1998 .

[20]  Guanrong Chen.  Controlling Chaos and Bifurcations in Engineering Systems , 1999 .

[21]  UngerboeckG..  Trellis-coded modulation with redundant signal sets Part II , 1987 .

[22]  R. E. Kalman,et al.  A New Approach to Linear Filtering and Prediction Problems , 2002 .

[23]  G. Ungerboeck,et al.  Trellis-coded modulation with redundant signal sets Part I: Introduction , 1987, IEEE Communications Magazine.

[24]  J. D. Farmer,et al.  Optimal shadowing and noise reduction , 1991 .

[25]  Chungyong Lee,et al.  Generalized iterative methods for enhancing contaminated chaotic signals , 1997 .

[26]  Joerg Schweizer.  Application of chaos to communications , 1999 .

[27]  Grégoire Nicolis,et al.  Probabilistic approach to homoclinic chaos , 1994 .