The subject here is generalized (i.e., non-Gaussian) noise models, and specifically their first-order probability density functions (PDFs). Attention is focused primarily on the author's canonical statistical-physical Class A and Class B models. In particular, Class A noise describes the type of electromagnetic interference (EMI) often encountered in telecommunication applications, where this ambient noise is largely due to other, "intelligent" telecommunication operations. On the other hand, ambient Class B noise usually represents man-made or natural "nonintelligent"-i.e., nonmessage-bearing noise-and is highly impulsive. Class A noise is not an /spl alpha/-stable process, nor is it reducible to such, except in the limiting Gaussian cases of high-density noise (by the central limit theorem). Class B noise is also asymptotically normal (before model approximation). Under rather broad conditions, principally governed by the source propagation and distribution scenarios, the PDF of Class B noise alone (no Gaussian component) can usually be approximated by (1) a symmetric Gaussian /spl alpha/-stable (S/spl alpha/S) model in the case of narrowband reception, or when the PDF /spl omega//sub 1/(/spl alpha/) of the amplitude is symmetric; and (2) a nonsymmetric /spl alpha/-stable (NS/spl alpha/S) model (no Gaussian component) can be constructed in broadband regimes. New results here include: (i) counting functional methods for constructing the general qth-order characteristic functions (CFs) of Class A and Class B noise, from which (all) moments and (in principle), the PDFs follow; (ii) the first-order CFs, PDFs, and cumulative probabilities (APDs) of nonsymmetric broadband Class B noise, extended to include additive Gauss noise (AGN); (iii) proof of the existence of all moments in the basic Class A and Class B models; (iv) the key physical role of AGN and the fact that AGN removes /spl alpha/-stability; (v) the explicit roles of the propagation and distribution scenarios; and (vi) extension to noise fields. Although telecommunication applications are emphasized, Class A and Class B noise models apply selectively, but equally well, to other physical regimes, e.g., underwater acoustics and EM (radar, optics, etc.). Supportive empirical data are included.

Non-Gaussian noise models in signal processing for telecommunications: new methods an results for class A and class B noise models

The successful optimisation of the detection performance of maritime surveillance radars requires a detailed knowledge and understanding of both forward and backscattering from the ocean surface. Such an understanding enables the development of suitable signal processing techniques. In part I the authors discuss radar scattering from an ocean environment. It is shown, through the analysis of full-scale measurements, how the amplitude and correlation properties of high-resolution radar backscatter (sea clutter) can be accurately represented by the compound K-distribution model which has the unique characteristic of providing realistic performance predictions for a wide range of signal processing techniques. In addition to modelling noncoherent clutter, it is shown how the spectral and polarisation characteristics of coherently detected sea clutter relate to those observed in the noncoherent case. Results are also presented of forward-scattering and multipath propagation. This includes a consideration of the importance of the spatial and temporal coherence of the forward-scattered wavefront.

Maritime surveillance radar Part 1 : Radar scattering from the ocean surface

This paper addresses non-Gaussian statistical modeling of interference as a superposition of a large number of small effects from terminals/scatterers distributed in the plane/volume according to a Poisson point process. This problem is relevant to multiple access communication systems without power control and radar. Assuming that the signal strength is attenuated over distance r as 1/r/m, we show that the interference/clutter could be modeled as a spherically symmetric /spl alpha/-stable noise. A novel approach to stable noise modeling is introduced based on the LePage series representation. This establishes grounds to investigate practical constraints in the system model adopted, such as the finite number of interferers and nonhomogeneous Poisson fields of interferers. In addition, the formulas derived allow us to predict noise statistics in environments with lognormal shadowing and Rayleigh fading. The results obtained are useful for the prediction of noise statistics in a wide range of environments with deterministic and stochastic power propagation laws. Computer simulations are provided to demonstrate the efficiency of the /spl alpha/-stable noise model in multiuser communication systems. The analysis presented will be important in the performance evaluation of complex communication systems and in the design of efficient interference suppression techniques.

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Analytic alpha-stable noise modeling in a Poisson field of interferers or scatterers

Interest in describing and modeling envelope distributions of sea-floor backscatter has increased recently, particularly with regard to high-resolution active sonar systems. Sea-floor scattering that results in heavy-tailed-matched-filter-envelope probability distribution functions (i.e., non-Rayleigh distributions exemplified by the K, Weibull, Rayleigh mixture, or log-normal distributions) is often the limiting factor in the performance of these types of sonar systems and in this context is referred to as reverberation or acoustic clutter analogous to radar clutter. Modeling of reverberation has traditionally entailed fitting various candidate distributions to time samples of the envelope of the scattered sonar (or radar) returns. This type of descriptive analysis and the asymptotic (infinite number of scatterers) analysis defining the K-distribution yield little insight into the environmental mechanisms responsible for heavy-tailed distributions (e.g., distributions and, clustering of discrete scatterers, patchiness in geo-acoustic properties, scattering strength of scatterers, etc.) and do not allow evaluation of the effect of changing sonar system parameters such as bandwidth and beamwidth. In contrast, we derive the envelope distribution for the scattered returns starting from simple physical descriptions of the environment with a finite number of scatterers. It is shown that plausible descriptions of the environment can lead to K-distributed reverberation. This result explains, at least partially, the success of the K-distribution in the modeling of radar clutter and sonar reverberation at a variety of frequencies and scales. The finite-number-of-scatterers model is then used to predict how the shape parameter of the K-distribution will change as the beamwidth of a towed-array receiver is varied. Analysis of reverberation data from a low-frequency (450-700 Hz) active sonar system illustrates that, within our ability to estimate it, the shape parameter of the K-distribution is proportional to the beamwidth of the towed-array receiver, a result important for sonar simulation and performance prediction models. These results should prove useful in developing methods for modeling, predicting and mitigating reverberation on high-resolution sonar systems.

Novel physical interpretations of K-distributed reverberation

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

This paper addresses problems associated with the development of widely applicable non-Gaussian noise models, particularly with reference to the statistical properties of scattered waves. A combination of phenomenological arguments and exact solutions of specific scattering problems are used to elucidate the significance of one model—K-distributed noise—which has several attractive features and has already found many applications. A full statistical-mechanical formulation is developed for non-Gaussian compound Markov processes, with this model as a special case. The implications for numerical simulation of correlated non-Gaussian noise are explored and comparisons made with experimental data. A brief review of current applications of the K-distribution model is given.

R.J.A. Tough

Papers

Non-Gaussian models for the statistics of scattered waves