Detection of signals in chaos

In this paper, we present a new method for the detection of signals in "noise", which is based on the premise that the "noise" is chaotic with at least one positive Lyapunov exponent. The method is naturally rooted in nonlinear dynamical systems and relies on neural networks for its implementation. We first present an introductory review of chaos. The subject matter selected for this part of the paper is written with emphasis on experimental studies of chaos using a time series. Specifically, we discuss the issues involved in the reconstruction of chaotic dynamics, attractor dimensions, and Lyapunov exponents. We describe procedures for the estimation of the correlation dimension and the largest Lyapunov exponent. The need for an adequate data length is stressed. In the second part of the paper we apply the chaos-based method to a difficult task: the radar detection of a small target in sea clutter. >

[1]  Fraser,et al.  Independent coordinates for strange attractors from mutual information. , 1986, Physical review. A, General physics.

[2]  Ronnie Mainieri,et al.  On the equality of Hausdorff and box counting dimensions. , 1993, Chaos.

[3]  L. Chua,et al.  Chaos: A tutorial for engineers , 1987, Proceedings of the IEEE.

[4]  G. Benettin,et al.  Lyapunov Characteristic Exponents for smooth dynamical systems and for hamiltonian systems; a method for computing all of them. Part 1: Theory , 1980 .

[5]  Kunihiko Kaneko,et al.  Spatiotemporal chaos and noise , 1989 .

[6]  P. Grassberger,et al.  On noise reduction methods for chaotic data. , 1993, Chaos.

[7]  A. Wolf,et al.  Determining Lyapunov exponents from a time series , 1985 .

[8]  D. Ruelle,et al.  Ergodic theory of chaos and strange attractors , 1985 .

[9]  Broggi,et al.  Dimension increase in filtered chaotic signals. , 1988, Physical review letters.

[10]  Cris Koutsougeras,et al.  Complex domain backpropagation , 1992 .

[11]  Simon Haykin,et al.  Chaotic detection of small target in sea clutter , 1993, 1993 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[12]  Thomas L. Clarke,et al.  Generalization of neural networks to the complex plane , 1990, 1990 IJCNN International Joint Conference on Neural Networks.

[13]  C. Myers,et al.  Signal separation for nonlinear dynamical systems , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[14]  E. Jakeman.  On the statistics of K-distributed noise , 1980 .

[15]  Ruedi Stoop,et al.  Calculation of Lyapunov exponents avoiding spurious elements , 1991 .

[16]  Martin Casdagli,et al.  Nonlinear prediction of chaotic time series , 1989 .

[17]  Leon Glass,et al.  Coarse-grained embeddings of time series: random walks, Gaussian random processes, and deterministic chaos , 1993 .

[18]  Henry D. I. Abarbanel,et al.  Prediction and system identification in chaotic nonlinear systems: Time series with broadband spectra , 1989 .

[19]  S. Haykin,et al.  Chaotic signal processing: new research directions and novel applications , 1992, [1992] IEEE Sixth SP Workshop on Statistical Signal and Array Processing.

[20]  A. Lapedes,et al.  Nonlinear modeling and prediction by successive approximation using radial basis functions , 1994 .

[21]  A. Provenzale,et al.  Finite correlation dimension for stochastic systems with power-law spectra , 1989 .

[22]  Michael I. Jordan,et al.  A Competitive Modular Connectionist Architecture , 1990, NIPS.

[23]  Biman Das,et al.  Calculating the dimension of attractors from small data sets , 1986 .

[24]  C. Baker.  K-distributed coherent sea clutter , 1991 .

[25]  S. Haykin,et al.  Is there a radar clutter attractor , 1990 .

[26]  E. Lorenz.  Deterministic nonperiodic flow , 1963 .

[27]  C. Baker,et al.  Maritime surveillance radar Part 1 : Radar scattering from the ocean surface , 1990 .

[28]  R. Robinson,et al.  GENERIC PROPERTIES OF CONSERVATIVE SYSTEMS. , 1970 .

[29]  E. Jakeman,et al.  A model for non-Rayleigh sea echo , 1976 .

[30]  Kurt Hornik,et al.  Multilayer feedforward networks are universal approximators , 1989, Neural Networks.

[31]  Farmer,et al.  Predicting chaotic time series. , 1987, Physical review letters.

[32]  Hediger,et al.  Fractal dimension and local intrinsic dimension. , 1989, Physical review. A, General physics.

[33]  A. M. Fraser.  Modeling nonlinear time series , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[34]  Simon Haykin,et al.  A coherent dual-polarized radar for studying the ocean environment , 1991, IEEE Trans. Geosci. Remote. Sens..

[35]  C. Baker,et al.  Maritime surveillance radar. II. Detection performance prediction in sea clutter , 1990 .

[36]  Gregory W. Wornell,et al.  Signal processing in the context of chaotic signals , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[37]  D. Ruelle,et al.  Fundamental limitations for estimating dimensions and Lyapunov exponents in dynamical systems , 1992 .

[38]  Floris Takens,et al.  On the numerical determination of the dimension of an attractor , 1985 .

[39]  J.J. Sidorowich,et al.  Modeling of chaotic time series for prediction, interpolation, and smoothing , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[40]  A. Farina,et al.  A review of CFAR detection techniques in radar systems , 1987 .

[41]  J. Yorke,et al.  Dimension of chaotic attractors , 1982 .

[42]  F. Hausdorff.  Dimension und äußeres Maß , 1918 .

[43]  Patrick Flandrin,et al.  An investigation of chaos-oriented dimensionality algorithms applied to AR(1) processes , 1992, [Proceedings] ICASSP-92: 1992 IEEE International Conference on Acoustics, Speech, and Signal Processing.

[44]  S. Haykin,et al.  Chaotic characterization of sea clutter , 1994 .

[45]  S. E. Newhouse.  Entropy and smooth dynamics , 1983 .

[46]  Jon Wright,et al.  Recognition and classification of nonlinear chaotic signals. , 1993, Chaos.

[47]  D. T. Kaplan,et al.  Direct test for determinism in a time series. , 1992, Physical review letters.

[48]  P. F. Meier,et al.  Evaluation of Lyapunov exponents and scaling functions from time series , 1988 .

[49]  F. Girosi,et al.  Networks for approximation and learning , 1990, Proc. IEEE.

[50]  P. Grassberger,et al.  Measuring the Strangeness of Strange Attractors , 1983 .

[51]  F. Takens,et al.  On the nature of turbulence , 1971 .

[52]  S M Pincus,et al.  Approximate entropy as a measure of system complexity. , 1991, Proceedings of the National Academy of Sciences of the United States of America.

[53]  F. Takens.  Detecting strange attractors in turbulence , 1981 .

[54]  James P. Crutchfield,et al.  Geometry from a Time Series , 1980 .

[55]  P. Grassberger.  Finite sample corrections to entropy and dimension estimates , 1988 .