Knowledge-Aided Covariance Matrix Estimation via Kronecker Product Expansions for Airborne STAP
This letter proposes a new approach for knowledge-aided estimation of structured clutter covariance matrices (CCMs) in airborne radar systems with limited training data. First, we model the CCM in space–time adaptive processing (STAP) as a sum of low-rank Kronecker products. We then apply a permutation operation to convert the Kronecker factors into linear structures and propose a novel CCM estimation method under the maximum-likelihood framework. Employing a proximal gradient algorithm, the proposed method simultaneously exploits the knowledge about the clutter and the Kronecker structure of the CCM. We finally evaluate the performance of the proposed method using real data from airborne STAP.
CFAR Detection in Clutter With a Kronecker Covariance Structure
In this paper, an approach for the design and analysis of coherent constant false alarm rate (CFAR) detectors in clutter and interference with a Kronecker covariance structure is described. In a two-dimensional example considered, the interference-plus-noise matrix <inline-formula><tex-math notation="LaTeX">${\mathbf X} \in {\mathbb{C}}^{N \times L}$</tex-math></inline-formula> is modeled by a doubly correlated, zero-mean multivariate complex Gaussian distribution described by two covariance matrices <inline-formula><tex-math notation="LaTeX">${\mathbf C}$</tex-math> </inline-formula> and <inline-formula><tex-math notation="LaTeX">${\mathbf R}$</tex-math></inline-formula> that are unknown to the receiver. The concatenated columns of <inline-formula><tex-math notation="LaTeX">${\mathbf X}$ </tex-math></inline-formula> has a structured covariance matrix <inline-formula><tex-math notation="LaTeX">$\mathbf \Sigma$</tex-math></inline-formula> given by <inline-formula><tex-math notation="LaTeX">${\mathbf \Sigma } = {\mathbf R}^{*} \otimes {\mathbf C}$</tex-math></inline-formula>. In the approach described, an estimate of <inline-formula> <tex-math notation="LaTeX">${\mathbf R}$</tex-math></inline-formula> is used to “prewhiten” and match filter all the rows of both the training data matrices and the test data matrix. The processing enables one to reduce the detection problem to a one-dimensional case that can be handled by any one of the several adaptive detection algorithms. The proposed algorithm for the doubly correlated clutter is analyzed to show that the detection performance is determined by two statistically independent signal-to-interference-plus-noise loss factors both of which have complex beta distributions. Sample results show that the proposed approach requires training samples that is a multiple of <inline-formula><tex-math notation="LaTeX">$N+L$</tex-math></inline-formula>, while an adaptive detection algorithm that do not explicitly use the Kronecker constraint on the covariance structure requires training samples that is a multiple of <inline-formula><tex-math notation="LaTeX">$N \times L$</tex-math></inline-formula> for comparable detection performance.
Normed units in abelian group rings
We compute explicitly the isomorphic structure of the normalized unit group of an abelian group ring under some minimal natural restrictions on the group basis and the coefficient ring. This enlarges affirmations due to Chatzidakis-Pappas (J. London Math. Soc., 1991) and Mollov (Publ. Math. Debrecen, 1971).