The problem of identifying digital transmitted symbols over nonlinear communication channels is addressed. The equalization scenario is considered from the decision point of view, and constitutes a joint identification and estimation situation due to incomplete knowledge of the system model. A new class of multilinearization algorithms for nonlinear systems is derived according to partitioning theory concepts. The procedure targets on adaptively selecting the best reference points for linearization from an ensemble of generated trajectories that span the whole state space of the desired signal. In the various simulations examined, the partitioned-based equalizer is found superior to the classical extended Kalman filter.

Deconvolution is one of the most important aspects of seismic signal processing. The objective of the deconvolution procedure is to remove the obscuring effect of the wavelet's replica making up the seismic trace and therefore obtain an estimate of the reflection coefficient sequence. This paper introduces a new deconvolution algorithm. Optimal distributed estimators and smoothers are utilized in the proposed solution. The new distributed methodology, perfectly suitable for a multisensor environment, such as the seismic signal processing, is compared to the centralized approach, with respect to computational complexity and architectural efficiency. It is shown that the distributed approach greatly outperforms the currently used centralized methodology offering flexibility in the design of the data fusion network.

The problem of estimating the return power in a LIDAR system in the presence of multiplicative noise (speckle) is addressed. A significant class of the partitioning approach is applied and comparisons are made with the extended Kalman filter (EKF) in the case where model parameter uncertainty exists. Through extensive simulations, the partitioned filter is shown to be significantly superior to the EKF algorithm.

The problem of state estimation for partially unknown, time-varying, linear systems with non-Gaussian initial conditions is addressed. It is shown that the optimal estimator for this problem is an adaptive Lainiotis (1989) filter with nonlinear Lainiotis filters for non-Gaussian initial conditions as elemental filters. Closed-form solutions for several explicit cases of the initial state PDFs are given. Simulation experiments show the superiority of the proposed algorithm over an adaptive Lainiotis filter with Kalman filters as elemental filters.>

In a previous paper [1], Bayes-optimal recursive supervised learning structure and parameter adaptive pattern recognition systems were derived for continuous "lumped" Gaussian processes. In this paper, the discrete data case is considered, and the discrete data version of the partition theorem is derived. Several examples are also presented of the application of the adaptive detectors, and computational results are given indicating their learning capacity and convergence rate.

In problems of pattern recognition and signal detection, one of the most important tasks is that of finding practical ways of preprocessing the data to eliminate complexity, redundancy, and irrelevancy. In this paper it is assumed that a vector wavefonn is received during an interval[t_o, t_f]. The waveform is considered to be a sample of a nonstationary vector random process containing a signal process and a noise process consisting of both white and colored noise. The optimum set of weighting functions is found for integrating the received waveform to extract those features that best reveal the presence of the signal. The solution is also shown to be the optimum one for estimating signal strength. A practical scheme for obtaining the optimum weighting functions is derived using state variables, and worked examples are presented.

Bayes optimal sequential structure and parameter-adaptive pattern-recognition systems for continuous data are derived. Both off-line (or prior to actual operation) and on-line (while in operation) supervised learning is considered. The concept of structure adaptation is introduced and both structure as well as parameter-adaptive optimal pattern-recognition systems are obtained. Specifically, for the class of supervised-learning pattern-recognition problems with Gaussian process models and linear dynamics, the adaptive pattern-recognition systems are shown to be decomposable ("partition theorem") into a linear nonadaptive part consisting of recursive matched Kalman filters, a nonlinear part--a set of probability computers--that incorporates the adaptive nature of the system, and finally a part of the correlator-estimator (Kailath) form. Extensions of the above results to theM-ary hypotheses case whereM \geq 2are given.

The problem considered is that of finding the best linear transformation to reduce a random-data vectorzto a vector of smaller dimension. It is assumed that the original data are Gaussian under either of two hypotheses, and that one wishes to use the transformed data to distinguish the hypotheses. The Bhattacharya distance is used to measure the information carried by the transformed data. A compromise solution is obtained for the case in which the data have both different means and different covariances under the alternative hypotheses.

A general and fundamental relationship between estimation, detection/pattern recognition, and the Bhattacharyya coefficient is established. Specifically, it is shown that a functional of the likelihood ratio necessary for Bayes optimal detection/pattern recognition is the Bayes-optimal mean-square estimate of the indicator variable\beta, where\beta = 1, 0, for hypothesesH_{1}, H_{0}, respectively. The Bhattacharyya coefficient is then shown to be the expected value of the estimation error standard deviation.

A class of upper bounds on the probability of error for the general multihypotheses pattern recognition problem is obtained. In particular, an upper bound in the class is shown to be a linear functional of the pairwise Bhattacharya coefficients. Evaluation of the bounds requires knowledge of a priori probabilities and of the hypothesis-conditional probability density functions. A further bound is obtained that is independent of a priori probabilities. For the case of unknown a priori probabilities and conditional probability densities, an estimate of the latter upper bound is derived using a sequence of classified samples and Kernel functions to estimate the unknown densities.

A Bayesian decision theory approach is applied to the solution of the problem of unsupervised parametric pattern recognition. The parametric model for this investigation includes the cases where both constant and time-varying unknown parameters are present, and, most significantly, the unknown hypotheses do not constitute a statistically independent sequence. They are restricted only to be from a source with finite-order Markov dependence. The resulting optimal learning system is found and shown to grow initially in size and memory until theNth observation (whereNis the highest Markov order), and subsequently to remain of fixed size and memory. It can, therefore, operate indefinitely and continue to improve its ability to recognize patterns utilizing only a fixed size memory. In summary, the main contributions of this paper are the following: \begin{enumerate} \item the extension of previous investigations of the unsupervised parametric pattern recognition problem to include cases where both constant and time-varying unknown parameter vectors are simultaneously present; \item that the a priori probabilities of the hypotheses, the time-varying parameters, and their transition laws may, if constant, be expressed as functions of the constant unknown parameter and, thus, also be learned; and \item the removal of the assumption of statistical independence between hypotheses for the sequence of observations. \end{enumerate}

Recursive relations are given for updating the conditional densityp(\theta_{k} | X_{k-1}, \cdots X_{1})(also forp(\theta_{k} | X_{k}, \cdots, X_{1})), where\theta_{k}is a parameter of the density ofX_{k}. The observationsX_{1}, X_{2}, \cdotsare assumed to be conditionally independent (i.e., for known parameters), and the sequence of time-varying parameters\theta_{1}, \theta_{2}, \cdotsconstitutes a Markov-M sequence. The result requires the storage of an intermediate function of(\theta_{k-1}, \cdots , \theta_{k-M}).

In this paper, coincidence detection procedures with invariant or distribution-free false alarm rates are proposed and investigated. The only information concerning the channel statistics required by these coincidence procedures is the median of the noise under no-signal conditions. The coincidence procedures are subsequently modified so that the detectors constitute learning systems with respect to time-varying and/or unknown medians. It is shown that their false alarm rates remain distribution free for wide classes of detection problems. The distribution-free coincidence detectors are then applied to various detection problems of practical importance, and their performance evaluated and compared to the performance of comparable likelihood detectors. It is shown that the distribution-free detectors are reasonably efficient, though suboptimal, for channels with Gaussian statistics, and highly efficient for channels with a combination of Gaussian and impulse noise.