Toeplitz Inverse Eigenvalue Problem (ToIEP) and Random Matrix Theory (RMT) Support for the Toeplitz Covariance Matrix Estimation

"Toeplitzification" or "redundancy (spatial) averaging", the well-known routine for deriving the Toeplitz covariance matrix estimate from the standard sample covariance matrix, recently regained new attention due to the important Random Matrix Theory (RMT) findings. The asymptotic consistency in the spectral norm was proven for the Kolmogorov's asymptotics when the matrix dimension N and independent identically distributed (i.i.d.) sample volume T both tended to infinity (N->inf, T->inf, T/N->c > 0). These novel RMT results encouraged us to reassess the well-known drawback of the redundancy averaging methodology, which is the generation of the negative minimal eigenvalues for covariance matrices with big eigenvalues spread, typical for most covariance matrices of interest. We demonstrate that for this type of Toeplitz covariance matrices, convergence in the spectral norm does not prevent the generation of negative eigenvalues, even for the sample volume T that significantly exceeds the covariance matrix dimension (T >> N). We demonstrate that the ad-hoc attempts to remove the negative eigenvalues by the proper diagonal loading result in solutions with the very low likelihood. We demonstrate that attempts to exploit Newton's type iterative algorithms, designed to produce a Hermitian Toeplitz matrix with the given eigenvalues lead to the very poor likelihood of the very slowly converging solution to the desired eigenvalues. Finally, we demonstrate that the proposed algorithm for restoration of a positive definite (p.d.) Hermitian Toeplitz matrix with the specified Maximum Entropy spectrum, allows for the transformation of the (unstructured) Hermitian maximum likelihood (ML) sample matrix estimate in a p.d. Toeplitz matrix with sufficiently high likelihood

Airborne Radar Interference Suppression Using Adaptive Three-Dimensional Techniques

This research advances adaptive interference suppression techniques for airborne radar, addressing the problem of target detection within severe interference environments characterized by high ground clutter levels, levels, noise jammer infiltration, and strong discrete interferers. Two-dimensional (2D) Space-Time Adaptive Processing (STAP) concepts are extended into three-dimensions (3D) by casting each major 2D STAP research area into a 3D framework. The work first develops an appropriate 3D data model with provisions for range ambiguous clutter returns. Adaptive 3D development begins with two factored approaches, 3D Factored Time-Space (3D-FTS) and Elevation-Joint Domain Localized (Elev-JDL). The 3D adaptive development continues with optimal techniques, i.e., joint domain methods. First, the 3D matched Filter (3D-MF) is derived followed by a 3D Adaptive Matched Filter (3D-AMF) discussion focusing on well-established practical limitations consistent with the 2D case. Finally, a 3D-JDL method is introduced. Proposed 3D Hybrid methods extend current state-of-the-art 2D hybrid methods. The initial 3D hybrid, a functional extension of the 2D technique, exhibits distinct performance advantages in heterogeneous clutter. The final 3D hybrid method is virtually impervious to discrete interference

Estimation et détection en milieu non-homogène, application au traitement spatio-temporel adaptatif

Pour un radar aéroporté, la détection de cibles nécessite, de par la nature du fouillis de sol, la mise en place d'un filtre spatio-temporel adaptatif (STAP). Les détecteurs basés sur l'hypothèse d'un milieu homogène sont souvent mis à mal dans un environnement réel, où les caractéristiques du fouillis peuvent varier significativement en distance et en angle. Diverses stratégies existent pour contrer les effets délétères de l'hétérogénéité. La thèse propose d'approfondir deux de ces stratégies. Plus précisément, un nouveau modèle d'environnement est présenté dans un contexte Bayésien : il intègre à la fois une relation originale d'hétérogénéité et de la connaissance a priori. De nouveaux estimateurs de la matrice de covariance du bruit ainsi que de nouveaux détecteurs sont calculés à partir de ce modèle. Ils sont étudiés de manière théorique et par simulations numériques. Les résultats obtenus montrent que le modèle proposé permet d'intégrer de manière intelligente l'information a priori dans le processus de détection. ABSTRACT : Space-time adaptive processing is required in future airborne radar systems to improve the detection of targets embedded in clutter. Performance of detectors based on the assumption of a homogeneous environment can be severely degraded in practical applications. Indeed real world clutter can vary significantly in both angle and range. So far, different strategies have been proposed to overcome the deleterious effect of heterogeneity. This dissertation proposes to study two of these strategies. More precisely a new data model is introduced in a Bayesian framework ; it allows to incorporate both an original relation of heterogeneity and a priori knowledge. New estimation and detection schemes are derived according to the model ; their performances are also studied theoretically and through numerical simulations. Results show that the proposed model and algorithms allow to incorporate in an appropriate way a priori information in the detection schem