A Geometric Approach to Covariance Matrix Estimation and its Applications to Radar Problems

A new class of disturbance covariance matrix estimators for radar signal processing applications is introduced following a geometric paradigm. Each estimator is associated with a given unitary invariant norm and performs the sample covariance matrix projection into a specific set of structured covariance matrices. Regardless of the considered norm, an efficient solution technique to handle the resulting constrained optimization problem is developed. Specifically, it is shown that the new family of distribution-free estimators shares a shrinkagetype form; besides, the eigenvalues estimate just requires the solution of a one-dimensional convex problem whose objective function depends on the considered unitary norm. For the two most common norm instances, i.e., Frobenius and spectral, very efficient algorithms are developed to solve the aforementioned one-dimensional optimization leading to almost closed form covariance estimates. At the analysis stage, the performance of the new estimators is assessed in terms of achievable Signal to Interference plus Noise Ratio (SINR) both for a spatial and a Doppler processing assuming different data statistical characterizations. The results show that interesting SINR improvements with respect to some counterparts available in the open literature can be achieved especially in training starved regimes.Comment: submitted for journal publicatio

Adaptive detection in elliptically distributed noise and under-sampled scenario

The problem of adaptive detection of a signal of interest embedded in elliptically distributed noise with unknown scatter matrix $\R$ is addressed, in the specific case where the number of training samples $T$ is less than the dimension $M$ of the observations. In this under-sampled scenario, whenever $\R$ is treated as an arbitrary positive definite Hermitian matrix, one cannot resort directly to the generalized likelihood ratio test (GLRT) since the maximum likelihood estimate (MLE) of $\R$ is not well-defined, the likelihood function being unbounded. Indeed, inference of $\R$ can only be made in the subspace spanned by the observations. In this letter, we present a modification of the GLRT which takes into account the specific features of under-sampled scenarios. We come up with a test statistic that, surprisingly enough, coincides with a subspace detector of Scharf and Friedlander: the detector proceeds in the subspace orthogonal to the training samples and then compares the energy along the signal of interest to the total energy. Moreover, this detector does not depend on the density generator of the noise elliptical distribution. Numerical simulations illustrate the performance of the test and compare it with schemes based on regularized estimates of $\R$

Knowledge-aided STAP in heterogeneous clutter using a hierarchical bayesian algorithm

This paper addresses the problem of estimating the covariance matrix of a primary vector from heterogeneous samples and some prior knowledge, under the framework of knowledge-aided space-time adaptive processing (KA-STAP). More precisely, a Gaussian scenario is considered where the covariance matrix of the secondary data may differ from the one of interest. Additionally, some knowledge on the primary data is supposed to be available and summarized into a prior matrix. Two KA-estimation schemes are presented in a Bayesian framework whereby the minimum mean square error (MMSE) estimates are derived. The first scheme is an extension of a previous work and takes into account the non-homogeneity via an original relation. {In search of simplicity and to reduce the computational load, a second estimation scheme, less complex, is proposed and omits the fact that the environment may be heterogeneous.} Along the estimation process, not only the covariance matrix is estimated but also some parameters representing the degree of \emph{a priori} and/or the degree of heterogeneity. Performance of the two approaches are then compared using STAP synthetic data. STAP filter shapes are analyzed and also compared with a colored loading technique

Parametric Estimation Techniques for Space-Time Adaptive Processing with Applications for Airborne Bistatic Radar Systems

This thesis considers parametric scenario based methods for Space-Time Adaptive Processing (STAP) in airborne bistatic radar systems. STAP is a multidimensional filtering technique used to mitigate the influence of interference and noise in a target detector. To be able to perform the mitigation, an accurate estimate is required of the associated space-time covariance matrix to the interference and noise distribution. In an airborne bistatic radar system geometry-induced effects due to the bistatic configuration introduces variations in the angle-Doppler domain over the range dimension. As a consequence of this, clutter observations of such systems may not follow the same distribution over the range dimension. This phenomena may affect the estimator of the space-time covariance matrix.\ua0In this thesis, we study a parametric scenario based approach to alleviate the geometry-induced effects. Thus, the considered framework is based on so called radar scenarios. A radar scenario is a description of the current state of the bistatic configuration, and is thus dependent on a few parameters connected to the two radar platforms which comprise the configuration. The scenario description can via a parametric model be used to represent the geometry-induced effects present in the system. In the first topic of this thesis, an investigation is conducted of the effects on scenario parameter residuals on the performance of a detector. Moreover, two methods are presented which estimate unknown scenario parameters from secondary observations. In the first estimation method, a maximum likelihood estimate is calculated for the scenario parameters using the most recent set of secondary data. In the second estimation method, a density is formed by combination of the likelihood associated with the most recent set of radar observations with a prior density obtained by propagation of previously considered scenario parameter estimates through a dynamical model of the scenario platforms motion over time. From the formed density a maximum a posteriori estimate of the scenario parameters can be derived. Thus, in the second estimation method, the radar scenario is tracked over time. Consequently, in the first topic of the thesis, the sensitivity between scenario parameters and detector performance is evaluated in various aspects, and two methods are investigated to estimate unknown scenario parameters from different radar scenarios.\ua0In the second part of the thesis, the scenario description is used to estimate a space-time covariance matrix and to derive a generalized likelihood ratio test for the airborne bistatic radar configuration. Consequently, for the covariance matrix estimate, the scenario description is used to derive a transformation matrix framework which aims to limit the non-stationary behavior of the secondary data observed by a bistatic radar system. Using the scenario based transformation framework, a set of non-stationary secondary data can be transformed to become more stationarily distributed after the transformation. A transformed set of secondary data can then be used in a conventional estimator to estimate the space-time covariance matrix. Furthermore, as the scenario description provides a representation of the geometry-induced effects in a bistatic configuration, the scenario description can be used to incorporate these effects into the design of a detector. Thus, a generalized likelihood ratio test is derived for an airborne bistatic radar configuration. Moreover, the presented detector is adaptive towards the strength of both the clutter interference and the thermal noise

HIGH PERFORMANCE, LOW COST SUBSPACE DECOMPOSITION AND POLYNOMIAL ROOTING FOR REAL TIME DIRECTION OF ARRIVAL ESTIMATION: ANALYSIS AND IMPLEMENTATION

This thesis develops high performance real-time signal processing modules for direction of arrival (DOA) estimation for localization systems. It proposes highly parallel algorithms for performing subspace decomposition and polynomial rooting, which are otherwise traditionally implemented using sequential algorithms. The proposed algorithms address the emerging need for real-time localization for a wide range of applications. As the antenna array size increases, the complexity of signal processing algorithms increases, making it increasingly difficult to satisfy the real-time constraints. This thesis addresses real-time implementation by proposing parallel algorithms, that maintain considerable improvement over traditional algorithms, especially for systems with larger number of antenna array elements. Singular value decomposition (SVD) and polynomial rooting are two computationally complex steps and act as the bottleneck to achieving real-time performance. The proposed algorithms are suitable for implementation on field programmable gated arrays (FPGAs), single instruction multiple data (SIMD) hardware or application specific integrated chips (ASICs), which offer large number of processing elements that can be exploited for parallel processing. The designs proposed in this thesis are modular, easily expandable and easy to implement. Firstly, this thesis proposes a fast converging SVD algorithm. The proposed method reduces the number of iterations it takes to converge to correct singular values, thus achieving closer to real-time performance. A general algorithm and a modular system design are provided making it easy for designers to replicate and extend the design to larger matrix sizes. Moreover, the method is highly parallel, which can be exploited in various hardware platforms mentioned earlier. A fixed point implementation of proposed SVD algorithm is presented. The FPGA design is pipelined to the maximum extent to increase the maximum achievable frequency of operation. The system was developed with the objective of achieving high throughput. Various modern cores available in FPGAs were used to maximize the performance and details of these modules are presented in detail. Finally, a parallel polynomial rooting technique based on Newton’s method applicable exclusively to root-MUSIC polynomials is proposed. Unique characteristics of root-MUSIC polynomial’s complex dynamics were exploited to derive this polynomial rooting method. The technique exhibits parallelism and converges to the desired root within fixed number of iterations, making this suitable for polynomial rooting of large degree polynomials. We believe this is the first time that complex dynamics of root-MUSIC polynomial were analyzed to propose an algorithm. In all, the thesis addresses two major bottlenecks in a direction of arrival estimation system, by providing simple, high throughput, parallel algorithms