Heavy-tailed Noise Research Articles

Multiuser Detection in Heavy Tailed Noise

Zoubir, A. M., and Brcich, R. F., Multiuser Detection in Heavy Tailed Noise, Digital Signal Processing 12 (2002) 262–273 We consider the problem of multiuser detection in impulsive noise channels. Multiuser detection methods have been shown to effectively combat multiple access interference in Gaussian noise, but are highly vulnerable to impulsive noise common in urban and indoor areas. Many multiuser detectors proposed for impulsive noise are based on a specific parametric noise model. While such a detector will perform well near the chosen model, performance is uncertain under larger deviations from the model. Robust detectors seek to minimize this loss, though they still rely on a static, albeit broader, model. We propose a nonparametric detector which makes minimal a priori assumptions on the noise model, requiring only a symmetric density. The detector is based on a nonparametric estimate of the noise density, obtained from the observations without the need for training data. Simulations show the nonparametric detector offers improved performance over existing methods when the noise is highly impulsive.

Image denoising: a nonlinear robust statistical approach

Nonlinear filtering techniques based on the theory of robust estimation are introduced. Some deterministic and asymptotic properties are derived. The proposed denoising methods are optimal over the Huber /spl epsi/-contaminated normal neighborhood and are highly resistant to outliers. Experimental results showing a much improved performance of the proposed filters in the presence of Gaussian and heavy-tailed noise are analyzed and illustrated.

The Kalman–Lévy filter

The Kalman filter combines forecasts and new observations to obtain an estimation which is optimal in the sense of a minimum average quadratic error. The Kalman filter has two main restrictions: (i) the dynamical system is assumed linear and (ii) forecasting errors and observational noises are projected onto Gaussian distributions. Here, we offer an important generalization to the case where errors and noises have heavy tail distributions such as power laws and Lévy laws. The main tool needed to solve this “Kalman–Lévy” filter is the “tail-covariance” matrix which generalizes the covariance matrix in the case where it is mathematically ill-defined (i.e. for power law tail exponents μ≤2). We present the general solution and discuss its properties on pedagogical examples. The standard Kalman–Gaussian filter is recovered for the case μ=2. The optimal Kalman–Lévy filter is found to deviate substantially from the standard Kalman–Gaussian filter as μ deviates from 2. As μ decreases, the filter favors more strongly the better one of the forecast and the observation, based on the tail-covariance matrix because a small exponent μ implies large errors with significant probabilities. In terms of implementation, the price-to-pay associated with the presence of heavy tail noise distributions is that the standard linear formalism valid for the Gaussian case is transformed into a nonlinear matrix equation for the Kalman–Lévy filter. Direct numerical experiments in the univariate case confirms our theoretical predictions.

High-resolution autofocus techniques for SAR imaging based on fractional lower-order statistics

The authors address the autofocusing problem in synthetic aperture radar imagery by introducing techniques that achieve robust image formation in the presence of severe heavy-tailed clutter and noise. Current state-of-the-art methods are extended, which are based on second-order moment theory, by employing fractional lower-order statistics (FLOS) of the phase-history data. The introduced FLOS-based methods perform a nonlinear transformation of the radar measurements, can mitigate the effects of impulsive additive clutter, and include the conventional algorithms as special cases. The benefits of the proposed approach are quantified by means of simulations, and the new FLOS-based methods are compared to current state-of-the-art processing with real synthetic aperture radar imagery data.

Subspace-based direction-of-arrival estimation using nonparametric statistics

The problem of subspace estimation using multivariate nonparametric statistics is addressed. We introduce new high-resolution direction-of-arrival (DOA) estimation methods that have almost optimal performance in nominal conditions and are robust in the face of heavy-tailed noise. The extensions of the techniques for the case of coherent sources are considered as well. The proposed techniques are based on spatial sign and rank concepts. We show that spatial sign and rank covariance matrices can be used to obtain convergent estimates of the signal and noise subspaces. In the proofs, the noise is assumed to be spherically symmetric. Moreover, we illustrate how the number of signals may be determined using the proposed covariance matrix estimates and a robust estimator of variance. The performance of the algorithms is studied using simulations in a variety of noise conditions including noise that is not spherically symmetric. The results show that the algorithms perform near optimally in the case of Gaussian noise and highly reliably if the noise is non-Gaussian.

Sample correlation behavior for the heavy tailed general bilinear process

In this paper, we consider the class of general bilinear models given by , where is an i.i.d. sequence of heavy tailed noise variables, and where . By means of a point process analysis, we show that the sample correlation function converges in distribution to a nondegenerate random variable. Thus standard model selection and fitting tools when applied to nonlinear heavy tailed models will be misleading. Also, consistency of the Hill estimator as an estimate of the tail index for this class of bilinear models is proved

Minimax description length for signal denoising and optimized representation

Approaches to wavelet-based denoising (or signal enhancement) have generally relied on the assumption of normally distributed perturbations. To relax this assumption, which is often violated in practice, we derive a robust wavelet thresholding technique based on the minimax description length (MMDL) principle. We first determine the least favorable distribution in the /spl epsiv/-contaminated normal family as the member that maximizes the entropy. We show that this distribution, and the best estimate based upon it, namely the maximum-likelihood estimate, together constitute a saddle point. The MMDL approach results in a thresholding scheme that is resistant to heavy tailed noise. We further extend this framework and propose a novel approach to selecting an adapted or best basis (BB) that results in optimal signal reconstruction. Finally, we address the practical case where the underlying signal is known to be bounded, and derive a two-sided thresholding technique that is resistant to outliers and has bounded error.

Alpha-stable modeling of noise and robust time-delay estimation in the presence of impulsive noise

A new representation of audio noise signals is proposed, based on symmetric /spl alpha/-stable (S/spl alpha/S) distributions in order to better model the outliers that exist in real signals. This representation addresses a shortcoming of the Gaussian model, namely, the fact that it is not well suited for describing signals with impulsive behavior. The /spl alpha/-stable and Gaussian methods are used to model measured noise signals. It is demonstrated that the /spl alpha/-stable distribution, which has heavier tails than the Gaussian distribution, gives a much better approximation to real-world audio signals. The significance of these results is shown by considering the time delay estimation (TDE) problem for source localization in teleimmersion applications. In order to achieve robust sound source localization, a novel time delay estimation approach is proposed. It is based on fractional lower order statistics (FLOS), which mitigate the effects of heavy-tailed noise. An improvement in TDE performance is demonstrated using FLOS that is up to a factor of four better than what can be achieved with second-order statistics.

Data-adaptive algorithms for signal detection in sub-Gaussian impulsive interference

We address the problem of coherent detection of a signal embedded in heavy-tailed noise modeled as a sub-Gaussian, alpha-stable process. We assume that the signal is a complex-valued vector of length L, known only within a multiplicative constant, while the dependence structure of the noise, i.e. the underlying matrix of the sub-Gaussian process, is not known. We implement a generalized likelihood ratio detector that employs robust estimates of the unknown noise underlying matrix and the unknown signal strength. The performance of the proposed adaptive detector is compared with that of an adaptive matched filter that uses Gaussian estimates of the noise-underlying matrix and the signal strength and is found to be clearly superior. The proposed new algorithms are theoretically analyzed and illustrated in a Monte-Carlo simulation.

Limit theory for bilinear processes with heavy-tailed noise

We consider a simple stationary bilinear model $X_t = cX_{t-1} Z_{t-1} + Z_t, t = 0, \pm 1, \pm 2, \dots,$ generated by heavy-tailed noise variables ${Z_t}$. A complete analysis of weak limit behavior is given by means of a point process analysis. A striking feature of this analysis is that the sample correlation converges in distribution to a nondegenerate limit. A warning is sounded about trying to detect nonlinearities in heavy-tailed models by means of the sample correlation function.

Blind equalization of constant modulus signals using an adaptive observer approach

The paper addresses the problem of blind equalization of constant modulus signals which are degraded by frequency selective multipath propagation and additive white noise. An adaptive observer is used to update the weights of an FIR equalizer in order to restore the signal's constant modulus property. The observer gain is selected using fake algebraic Riccati methods in order to guarantee local stability. The performance of this method is compared to the constant modulus algorithm for simulated FM-FDM signals and exhibits significantly better convergence properties, particularly for heavy-tailed noise.

Asymptotically robust detection and estimation for very heavy-tailed noise

The asymptotically robust detection of signals and the analogous problem of estimation of location are considered for noise distributions with heavier than exponential tails. The solution to both problems is identical and is obtained when using efficacy or asymptotic variance as the performance measure. The density of the noise is assumed to belong to a E-contamination class where the contaminant is either (1) from the class of unimodal densities, (2) from the class of densities limited in support to an interval about the mode of the nominal density, or (3) a mixture of a member of (1) with a member of (2). The robust detector or estimator is similar to a censored version of the nonlinearity used for maximum-likelihood estimation or locally optimal detection in the presence of noise with the nominal distribution. >