Maximum Likelihood Estimation of Compound-Gaussian Clutter and Target Parameters

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

I. Introduction

When a radar system illuminates a large area of the sea, the probability density function (pdf) of the amplitude of the returned signal is well approximated by the Rayleigh distribution [1], i.e., the echo can be modeled as a complex-Gaussian process. That distribution is a good approximation. This can be proved theoretically by the central limit theorem, since the returned signal can be viewed as the sum of the reflection from a large number of randomly phased independent scatterers. However, in high-resolution and low-grazing-angle radar, the real clutter data show significant deviations from the complex Gaussian model, see [2], because only a small sea surface area is illuminated by the narrow radar beam. The behavior of the small patch is nonstationary [1] and the number of scatterers is random, see [3]. Due to the different waveform characteristics and generation mechanism, the sea surface wave, i.e., the roughness of the sea surface, is often modeled in two scales [4], [5]. To take into account different scales of roughness, a two-scale sea surface scattering model was developed, see [6]–[8]. In this two-scale model—a compound-Gaussian model—the fast-changing component, which accounts for local scattering, is referred to as speckle . It is assumed to be a stationary complex Gaussian process with zero mean. The slow-changing component, texture is used to describe the variation of the local power due to the tilting of the illuminated area, and it is modeled as a nonnegative real random process. The complex clutter can be written as the product of these two components $${\mmb e}(t)=\sqrt{u(t)}{\mmb\chi}(t).\eqno{\hbox{(1)}}$$The compound-Gaussian model is a model widely used to characterize the heavy-tailed clutter distributions in radar, especially sea clutter, see [2], [6], [9], and Section II. It belongs to the class of the spherically invariant random process (SIRP), see [10] and [17]. Note that the compound-Gaussian distribution is also often used to model speech waveforms and various radio propagation channel disturbance, see [10] and the references therein.