Complex elliptically symmetric (CES) distributions have been widely used in various engineering applications for which non-Gaussian models are needed. In this overview, circular CES distributions are surveyed, some new results are derived and their applications e.g., in radar and array signal processing are discussed and illustrated with theoretical examples, simulations and analysis of real radar data. The maximum likelihood (ML) estimator of the scatter matrix parameter is derived and general conditions for its existence and uniqueness, and for convergence of the iterative fixed point algorithm are established. Specific ML-estimators for several CES distributions that are widely used in the signal processing literature are discussed in depth, including the complex t -distribution, K-distribution, the generalized Gaussian distribution and the closely related angular central Gaussian distribution. A generalization of ML-estimators, the M-estimators of the scatter matrix, are also discussed and asymptotic analysis is provided. Applications of CES distributions and the adaptive signal processors based on ML- and M-estimators of the scatter matrix are illustrated in radar detection problems and in array signal processing applications for Direction-of-Arrival (DOA) estimation and beamforming. Furthermore, experimental validation of the usefulness of CES distributions for modelling real radar data is given.

Complex Elliptically Symmetric Distributions: Survey, New Results and Applications

This introduction to the expectation–maximization (EM) algorithm provides an intuitive and mathematically rigorous understanding of EM. Two of the most popular applications of EM are described in detail: estimating Gaussian mixture models (GMMs), and estimating hidden Markov models (HMMs). EM solutions are also derived for learning an optimal mixture of fixed models, for estimating the parameters of a compound Dirichlet distribution, and for dis-entangling superimposed signals. Practical issues that arise in the use of EM are discussed, as well as variants of the algorithm that help deal with these challenges.

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Theory and Use of the Em Algorithm

We address high dimensional covariance estimation for elliptical distributed samples, which are also known as spherically invariant random vectors (SIRV) or compound-Gaussian processes Specifically we consider shrinkage methods that are suitable for high dimensional problems with a small number of samples (large p small n) We start from a classical robust covariance estimator [Tyler (1987)], which is distribution-free within the family of elliptical distribution but inapplicable when n <; p Using a shrinkage coefficient, we regularize Tyler's fixed-point iterations We prove that, for all n and p , the proposed fixed-point iterations converge to a unique limit regardless of the initial condition Next, we propose a simple, closed-form and data dependent choice for the shrinkage coefficient, which is based on a minimum mean squared error framework Simulations demonstrate that the proposed method achieves low estimation error and is robust to heavy-tailed samples Finally, as a real-world application we demonstrate the performance of the proposed technique in the context of activity/intrusion detection using a wireless sensor network

Robust Shrinkage Estimation of High-Dimensional Covariance Matrices

This paper deals with the problem of detecting a radar target signal against correlated non-Gaussian clutter, which is modeled by the compound-Gaussian distribution. We prove that if the texture of compound-Gaussian clutter is modeled by an inverse-gamma distribution, the optimum detector is the optimum Gaussian matched filter detector compared to a data-dependent threshold that varies linearly with a quadratic statistic of the data. We call this optimum detector a linear-threshold detector (LTD). Then, we show that the compound-Gaussian model presented here varies parametrically from the Gaussian clutter model to a clutter model whose tails are evidently heavier than any K-distribution model. Moreover, we show that the generalized likelihood ratio test (GLRT), which is a popular suboptimum detector because of its constant false-alarm rate (CFAR) property, is an optimum detector for our clutter model in the limit as the tails get extremely heavy. The GLRT-LTD is tested against simulated high-resolution sea clutter data to investigate the dependence of its performance on the various clutter parameters.

Coherent Radar Target Detection in Heavy-Tailed Compound-Gaussian Clutter

The inverse gamma distributed texture is important for modeling compound-Gaussian clutter (e.g. for sea reflections), due to the simplicity of estimating its parameters. We develop maximum-likelihood (ML) and method of fractional moments (MoFM) estimates to find the parameters of this distribution. We compute the Cramer-Rao bounds (CRBs) on the estimate variances and present numerical examples. We also show examples demonstrating the applicability of our methods to real lake-clutter data. Our results illustrate that, as expected, the ML estimates are asymptotically efficient, and also that the real lake-clutter data can be very well modeled by the inverse gamma distributed texture compound-Gaussian model.

Maximum likelihood estimation for compound-gaussian clutter with inverse gamma texture

Compound-Gaussian models are used in radar signal processing to describe heavy-tailed clutter distributions. The important problems in compound-Gaussian clutter modeling are choosing the texture distribution, and estimating its parameters. Many texture distributions have been studied, and their parameters are typically estimated using statistically suboptimal approaches. We develop maximum likelihood (ML) methods for jointly estimating the target and clutter parameters in compound-Gaussian clutter using radar array measurements. In particular, we estimate i) the complex target amplitudes, ii) a spatial and temporal covariance matrix of the speckle component, and iii) texture distribution parameters. Parameter-expanded expectation-maximization (PX-EM) algorithms are developed to compute the ML estimates of the unknown parameters. We also derived the Cramer-Rao bounds (CRBs) and related bounds for these parameters. We first derive general CRB expressions under an arbitrary texture model then simplify them for specific texture distributions. We consider the widely used gamma texture model, and propose an inverse-gamma texture model, leading to a complex multivariate t clutter distribution and closed-form expressions of the CRB. We study the performance of the proposed methods via numerical simulations

https://lib.dr.iastate.edu/cgi/viewcontent.cgi?article=1127&context=ece_pubs

Maximum Likelihood Estimation of Compound-Gaussian Clutter and Target Parameters

Adaptive polarimetry design for a target in compound-Gaussian clutter

We develop optimal adaptive design of radar waveform polarizations for a target in compound-Gaussian clutter. We present the maximum likelihood (ML) estimates of the target's scattering matrix and clutter parameters using a parameter-expanded expectation-maximization (PX-EM) algorithm. We compute the Cramer-Rao bound (CRB) on the target's scattering matrix and use it as the optimization cost function. We illustrate the performance of our algorithm and compare it with traditional ones through numerical examples.

https://www.ese.wustl.edu/~nehorai/DARPA/publications/WDD06-9sp_neho.pdf

We compute Crame/spl acute/r-Rao bounds (CRBs) for the target and compound-Gaussian clutter parameters using radar array measurements. In particular, we compute CRBs for (i) complex target amplitudes, (ii) the spatial covariance matrix of the speckle component, (iii) texture distribution parameters. We first derive general CRB expressions under an arbitrary texture model and simplify them for gamma and inverse gamma texture distributions. We use the generalized Gauss-Laguerre quadrature to compute the CRBs for gamma texture, whereas the CRBs for inverse-gamma texture do not require numerical integration. We use numerical simulations to validate our results.

Jian Wang

Papers

Maximum likelihood estimation for compound-gaussian clutter with inverse gamma texture

Maximum Likelihood Estimation of Compound-Gaussian Clutter and Target Parameters

Adaptive polarimetry design for a target in compound-Gaussian clutter

Adaptive polarimetry design for a target in compound-Gaussian clutter

Crame/spl acute/r-Rao bounds for compound-Gaussian clutter and target parameters