This paper considers the problem of optimum prediction of noisy chaotic time series using a basis function neural network, in particular, the radial basis function (RBF) network. In the noiseless environment, predicting a chaotic time series is equivalent to approximating a nonlinear function. The optimal generalization is achieved when the number of hidden units of a RBF predictor approaches infinity. When noise exists, it is shown that an optimal RBF predictor should use a finite number of hidden units. To determine the structure of an optimal RBF predictor, we propose a new technique called the cross-validated subspace method to estimate the optimum number of hidden units. While the subspace technique is used to identify a suitable number of hidden units by detecting the dimension of the subspace spanned by the signal eigenvectors, the cross validation method is applied to prevent the problem of overfitting. The effectiveness of this new method is evaluated using simulated noisy chaotic time series as well as real-life oceanic radar signals. Results show that the proposed method can find the correct number of hidden units of an RBF network for an optimal prediction.

Prediction of noisy chaotic time series using an optimal radial basis function neural network

Detection of small objects in a radar or satellite image is an important problem with many applications. Due to a recent discovery that sea clutter, the electromagnetic wave backscatter from a sea surface, is chaotic rather than purely random, computational intelligence techniques such as neural networks have been applied to reconstruct the chaotic dynamic of sea clutter. The reconstructed sea clutter dynamical system which usually takes the form of a nonlinear predictor does not only provide a model of the sea scattering phenomenon, but it can also be used to detect the existence of small targets such as fishing boats and small fragments of icebergs by observing abrupt changes in the prediction error. We applied a genetic algorithm (GA) to obtain an optimal reconstruction of sea clutter dynamic based on a radial basis function (RBF) neural network. This GA-RBF uses a hybrid approach that employes a GA to search for the optimum values of the following RBF parameters: centers, variance, and number of hidden nodes, and uses the least square method to determine the weights. It is shown here that if the functional form of an unknown nonlinear dynamical system can be represented exactly using an RBF net (i.e., no approximation error), this GA-RBF approach can reconstruct the exact dynamic from its time series measurements. In addition to the improved accuracy in modeling sea clutter dynamic, the GA-RBF is also shown to enhance the detectability of small objects embedded in the sea. Using real-life radar data that are collected in the east coast of Canada by two different radar systems: a ground-based radar and a satellite equipped with synthetic aperture radar (SAR), we show that the GA-RBF network is a reliable detector for small surface targets in various sea conditions and is practical for real-life search and rescue, navigation, and surveillance applications.

Detection of small objects in clutter using a GA-RBF neural network

In recent years an abundance of technical papers and books have been written on several topics of radar signal processing. The detection of radar targets against ground and sea clutter is a problem of great interest in the radar community. The fusion of signals from different radar to improve detection performance is another relevant topic of research and application. These are two of several topics that have been extensively described in the technical literature. Needless to say that the consultation of technical references plays a key role in the daily work of researchers and engineers involved in the radar field. We collected almost 700 references in a single document to facilitate our work and the work of other colleagues of the radar community. The collection of references is by no means exhaustive; the period of screening mainly covers the last two decades.

Selected list of references on radar signal processing

A nonlinear operator is presented that is able to effectively attenuate the noise that corrupts an image while introducing small distortions on the image details. It is described by a rational function, i.e., by the ratio of two polynomials in the input variables. Notwithstanding its simplicity, this operator proves to be more powerful than conventional methods for many noise distributions.

The rational filter for image smoothing

The Statistical Signal and Array Processing Technical Committee (SSAP-TC) deals with signals that are random and processes an array of signals simultaneously. The field of SSAP represents both solid theory and practical applications. Starting with research in spectrum estimation and statistical modeling, study in this field is always full of elegant mathematical tools such as statistical analysis and matrix theory. The area of statistical signal processing expands into estimation and detection algorithms, time-frequency domain analysis, system identification, and channel modeling and equalization. The area of array signal processing also extends into multichannel filtering, source localization and separation, and so on. This article represents an endeavor by the members of the SSAT-TC to review all the significant developments in the field of SSAP. To provide readers with pointers for further study of the field, this article includes a very impressive bibliography-close to 500 references are cited. This is just one of the indications that the field of statistical signals has been an extremely active one in the signal processing community. The article also introduces the recent reorganization of three technical committees of the Signal Processing Society.

Highlights of statistical signal and array processing

Proposes a new nonlinear adaptive filter structure based on rational functions. There are several advantages to the use of this filter. First, it is a universal approximator and a good extrapolator. Second, it ran be trained by a linear adaptive algorithm, which makes it suitable for real-time adaptive signal processing. Third, it has a best approximation for a specified function. To demonstrate its utility as a tool for solving adaptive signal processing problems, the authors apply the adaptive rational function filter to the problem of estimation and detection. The estimation problem pertains to the direction of arrival (DOA) estimation problem in array signal processing. For the detection problem, the authors consider the detection of a weak radar target (a small piece of ice) in an ocean environment. >

H. Leung

Papers

Detection and estimation using an adaptive rational function filter