Covariance Structure Maximum-Likelihood Estimates in Compound Gaussian Noise: Existence and Algorithm Analysis

Abstract

Recently, a new adaptive scheme [1], [2] has been introduced for covariance structure matrix estimation in the context of adaptive radar detection under non Gaussian noise. This latter has been modelled by compound-Gaussian noise, which is the product c of the square root of a positive unknown variable τ (deterministic or random) and an independent Gaussian vector x, c = √ τ x. Because of the implicit algebraic structure of the equation to solve, we called the corresponding solution, the Fixed Point (FP) estimate. When τ is assumed deterministic and unknown, the FP is the exact Maximum Likelihood (ML) estimate of the noise covariance structure, while when τ is a positive random variable, the FP is an Approximate Maximum Likelihood (AML). This estimate has been already used for its excellent statistical properties without proofs of its existence and uniqueness. The major contribution of this paper is to fill these gaps. Our derivation is based on some Likelihood functions general properties like homogeneity and can be easily adapted to other recursive contexts. Moreover, the corresponding iterative algorithm used for the FP estimate practical determination is also analyzed and we show the convergence of this recursive scheme, ensured whatever the initialization.

Detection of Range Migrating Targets in Compound-Gaussian Clutter

“…In order to find α, we need to maximize the logarithm of (27), which can be done by taking the derivative and setting it to zero. Finally, the amplitude estimationα is written by the transcendental equation in the form (22).…”

“…Consequently both tests (17), (27) reduce to their narrow-band counterparts and amplitude estimation can be obtained directly:…”

“…Note that GLRT in case of texture considered as an unknown parameter (27) reduces to the same test. Under H 0 variable γ follows Beta distribution with parameters γ 0 ∼ β (1, M − 1), for detection γ has to be compared with [32].…”

“…In order to check the ability of the algorithms to keep the designed P F A , detection is applied to a target free scene. Therefore, we initialize both algorithms withα = a H M −1 y/a H M −1 a and apply Alternate Maximization (AM) (27) and Maximum Likelihood (ML) (17) algorithms with a priori known PDF of clutter texture.…”

“…[11], [13], [18], [32], [37]. The modern trend is to model high resolution radar clutter as a compound-Gaussian process (which belongs to the class of spherically invariant random vectors (SIRV) [11], [27], [28]) i.e. a Gaussian process with power varying from one range cell to another, but sharing the same correlation structure in slowtime [11], [36].…”