Chair: Thomas F. Quatieri, MIT Lincoln Laboratory (USA)
A. S. Asdi, University of Minnesota (USA)
A. H. Tewfik, University of Minnesota (USA)
We present a novel nonlinear filtering approach for detecting weak signals in heavy noise from short data records. Such detection problems arise in many applications including communications, radar, sonar, medical imaging, seismology, industrial measurements, etc. The performance of a matched filter detector of a weak signal in heavy noise is directly proportional to the observation time. We discuss an alternative detection approach that relies on a nonlinear filtering of the input signal using a bistable system. We show that by adaptively selecting the parameters of the system, it is possible to increase the ratio of the square of the amplitude of a sinusoid to that of the noise intensity around the frequency of the sinusoid (stochastic resonance). The sinusoid can then be reliably detected at the output of the nonlinear system using a suitable matched filter even when the data record is short.
Andrew C. Singer, Massachusetts Institute of Technology (USA)
Solitons and the nonlinear evolution equations that support them arise in the description of a wide range of nonlinear physical phenomena including shallow water waves, piezo-electrics, and optical transmission in nonlinear fibers. Although such systems are nonlinear, they are exactly solvable and possess a class of remarkably robust solutions, known as solitons, which satisfy a nonlinear form of superposition. By exploiting the properties of solitons, such nonlinear systems may be attractive for a variety of signal processing problems including multiple access communications, private or low power transmission, and multi- resolution transmission. We outline a number of modulation techniques using solitons, and explore some of the properties of such systems in the presence of additive channel corruption.
Ted W. Frison, Randle Inc.
Henry D.I. Abarbanel, University of California (USA)
Lev S. Tsimring, University of California (USA)
Methods for analyzing chaotic time series are extended to two-dimensional images. The motivation for this work is to develop new tools for understanding ocean surface dynamics and filters for ocean surface features. This paper discusses the computation of the average mutual information in a scene. Average mutual information provides a crucial parameter needed to reconstruct an attractor for the image and conduct further processing using chaotic methods. It also provides insights into energy transport and loss rate of information. The spectral content of the image can be determined along with the decorrelation rate and some sense of direction of movement. The technique is demonstrated on an ocean surface optical image.
Henry Leung, Defence Research Establishment Ottawa
Xinping Huang, Applied Silicon Inc. Canada (CANADA)
The problem of sinusoidal frequency estimation in chaotic noise is considered in this paper. Since the chaotic noise is inherently deterministic, a new complexity measure called the phase space volume ( PSV ) is introduced. The PSV quantifies the complexity of a signal by measuring its volume in a reconstructed phase space. To estimate the sinusoidal frequencies, an autoregressive ( AR ) model is applied to the received signal and the coefficients are estimated by minimizing the PSV of the prediction error. It is shown here that the frequencies can indeed be obtained by this MPSV-AR spectral estimator. To illustrate the efficiency of this new technique, simulated chaotic noise and real-life radar clutter ( radar backscatters ) are used as background noise for sinusoidal frequency estimation. Basically, we assume that chaos is a good model of background noise and apply the MPSV-AR technique to estimate the frequencies. The usefulness of this approach is evaluated using real-life measurement noise ( radar clutter ). In both simulated and real noise environments, we observe that the MPSV-AR spectral estimator provides an efficient frequency estimates in terms of both the mean squares errors and frequency resolution.
Chungyong Lee, Georgia Institute of Technology (USA)
Douglas B. Williams, Georgia Institute of Technology (USA)
An iterative method for reducing noise in contaminated chaotic signals is proposed. This method estimates the deviation of the observed signal from the nearest noise-free signal satisfying the system dynamics in order to get a noise-reduced (or enhanced) signal. To calculate the deviation we minimize a cost function composed of two parts: one containing information that represents how close the enhanced signal is to the observed signal and another including constraints that fit the dynamics of the system. This method has a simple structure and is flexible in the choice of the parts of the cost function. The proposed method is compared with Farmer's method which is known to have good performance in mild signal-to-noise ratios but has a more complex structure.
Steven H. Isabelle, Gregory W. Wornell Massachusetts Institute of Technology (USA)
Signals arising out of nonlinear dynamical systems are compelling models for a wide range of phenomena. We develop several properties of signals obtained from Markov maps, an important family of such systems, and present analytical techniques for computing their statistics. Among other results, we demonstrate that all Markov maps produce signals with rational spectra, and can therefore be viewed as chaotic ARMA processes. Finally, we demonstrate how Markov maps can approximate to arbitrary precision any of a broad class of chaotic maps and their statistics.
F.M. Coetzee, Carnegie Mellon University (USA)
V.L. Stonick, Carnegie Mellon University (USA)
Homotopy methods have achieved significant success in solving systems of nonlinear equations for which the number of solutions are known and the homotopy paths are bounded. We present a two-stage homotopy process which does not require a- priori knowledge of the number of solutions to a system of nonlinear equations. This approach makes use of compact manifolds to find solutions sequentially along disconnected homotopy paths. The procedure is tested on two standard optimization and neural network benchmark problems.
Sumit A. Talwalkar, University of Rhode Island (USA)
Steven M. Kay, University of Rhode Island (USA)
We show how to realize one-dimensional chaotic signals with the given first order autoregressive (AR(1)) autocorrelation function (ACF) of the form $r[k] = r[0] a^k$, where $0~<~a~<~1$. We consider the class of piece-wise linear and ``piece-wise onto'' maps defined from the unit interval $[0, 1]$ onto itself. We prove that these maps are chaotic using their topological conjugacy with known chaotic maps. The autocovariance function of sequences generated by these maps can be calculated analytically as an ensemble average. The expression for the autocovariance function leads to a way of designing chaotic signals with the given ACF. ACF estimates (temporal averages) of typical signals are close to the theoretical values (ensemble averages).
M. Vinson, Shippensburg University (USA)
L. Khadra, Jordan University of Science & Technology (JORDAN)
T. Maayah, Jordan University of Science & Technology (JORDAN)
H. Dickhaus, University of Heidelberg (GERMANY)
An investigation of chaotic behavior in the heart rate variability (HRV) signals of a normal person and a heart transplant recipient is presented. A statistical approach of chaos identification in time series is described and applied to the HRV signals. The method compares the short-term predictability for a given time series to an ensemble of random data which has the same Fourierspectrum as the original time series. The forecasting error is computed as a statistic for performing statistical hypothesis testing. The results suggest that the HRV signal of the transplant recipient recorded three months after the transplanting shows the same dynamical behavior as that of the HRV signal for a normal person.
Mototsugu Abe, The University of Tokyo (JAPAN)
Shigeru Ando, The University of Tokyo (JAPAN)
In this paper, we propose a method for decomposing instantaneous changes of sounds into three energy components, i.e., loudness, pitch, and timbre. These operators are derived from an eigenstructure analysis of the time-frequency gradient space (a 3-D space spanned by a modulus and partial derivatives of a wavelet transform). By several experiments, we found that they have superior resolution and sensitivity for segmenting speech into phonemes, characterizing dynamical nature of musical sounds, and so on.