GLRT-based threshold detection-estimation performance improvement and application to uniform circular antenna arrays

©2006 IEEE. Personal use of this material is permitted. However, permission to reprint/republish this material for advertising or promotional purposes or for creating new collective works for resale or redistribution to servers or lists, or to reuse any copyrighted component of this work in other works must be obtained from the IEEE."This material is presented to ensure timely dissemination of scholarly and technical work. Copyright and all rights therein are retained by authors or by other copyright holders. All persons copying this information are expected to adhere to the terms and constraints invoked by each author's copyright. In most cases, these works may not be reposted without the explicit permission of the copyright holder."The problem of estimating the number of independent Gaussian sources and their parameters impinging upon an antenna array is addressed for scenarios that are problematic for standard techniques, namely, under "threshold conditions" (where subspace techniques such as MUSIC experience an abrupt and dramatic performance breakdown). We propose an antenna geometry-invariant method that adopts the generalized-likelihood-ratio test (GLRT) methodology, supported by a maximum-likelihood-ratio lower-bound analysis that allows erroneous solutions ("outliers") to be found and rectified. Detection-estimation performance in both uniform circular and linear antenna arrays is shown to be significantly improved compared with conventional techniques but limited by the performance-breakdown phenomenon that is intrinsic to all such maximum-likelihood (ML) techniques.Yuri I. Abramovich, Nicholas K. Spencer, and Alexei Y. Gorokho

Optimal sensor arrangements in Angle of Arrival (AoA) and range based localization with linear sensor arrays

This paper investigates the linear separation requirements for Angle-of-Arrival (AoA) and range sensors, in order to achieve the optimal performance in estimating the position of a target from multiple and typically noisy sensor measurements. We analyse the sensor-target geometry in terms of the Cramer–Rao inequality and the corresponding Fisher information matrix, in order to characterize localization performance with respect to the linear spatial distribution of sensors. Here in this paper, we consider both fixed and adjustable linear sensor arrays

Order estimation and discrimination between stationary and time-varying (TVAR) autoregressive models

Copyright © 2007 IEEEFor a set of T independent observations of the same N-variate correlated Gaussian process, we derive a method of estimating the order of an autoregressive (AR) model of this process, regardless of its stationary or time-varying nature. We also derive a test to discriminate between stationary AR models of order m,AR(m), and time-varying autoregressive models of order m,TVAR(m). We demonstrate that within this technique the number T of independent identically distributed data samples required for order estimation and discrimination just exceeds the maximum possible order mmax, which in many cases is significantly fewer than the dimension of the problem NYuri I. Abramovich, Nicholas K. Spencer, and Michael D. E. Turle

Nested Arrays: A Novel Approach to Array Processing With Enhanced Degrees of Freedom

A new array geometry, which is capable of significantly increasing the degrees of freedom of linear arrays, is proposed. This structure is obtained by systematically nesting two or more uniform linear arrays and can provide O(N^2) degrees of freedom using only physical sensors when the second-order statistics of the received data is used. The concept of nesting is shown to be easily extensible to multiple stages and the structure of the optimally nested array is found analytically. It is possible to provide closed form expressions for the sensor locations and the exact degrees of freedom obtainable from the proposed array as a function of the total number of sensors. This cannot be done for existing classes of arrays like minimum redundancy arrays which have been used earlier for detecting more sources than the number of physical sensors. In minimum-input–minimum-output (MIMO) radar, the degrees of freedom are increased by constructing a longer virtual array through active sensing. The method proposed here, however, does not require active sensing and is capable of providing increased degrees of freedom in a completely passive setting. To utilize the degrees of freedom of the nested co-array, a novel spatial smoothing based approach to DOA estimation is also proposed, which does not require the inherent assumptions of the traditional techniques based on fourth-order cumulants or quasi stationary signals. As another potential application of the nested array, a new approach to beamforming based on a nonlinear preprocessing is also introduced, which can effectively utilize the degrees of freedom offered by the nested arrays. The usefulness of all the proposed methods is verified through extensive computer simulations

Estimation of the Number of Sources in Unbalanced Arrays via Information Theoretic Criteria

Estimating the number of sources impinging on an array of sensors is a well known and well investigated problem. A common approach for solving this problem is to use an information theoretic criterion, such as Minimum Description Length (MDL) or the Akaike Information Criterion (AIC). The MDL estimator is known to be a consistent estimator, robust against deviations from the Gaussian assumption, and non-robust against deviations from the point source and/or temporally or spatially white additive noise assumptions. Over the years several alternative estimation algorithms have been proposed and tested. Usually, these algorithms are shown, using computer simulations, to have improved performance over the MDL estimator, and to be robust against deviations from the assumed spatial model. Nevertheless, these robust algorithms have high computational complexity, requiring several multi-dimensional searches. In this paper, motivated by real life problems, a systematic approach toward the problem of robust estimation of the number of sources using information theoretic criteria is taken. An MDL type estimator that is robust against deviation from assumption of equal noise level across the array is studied. The consistency of this estimator, even when deviations from the equal noise level assumption occur, is proven. A novel low-complexity implementation method avoiding the need for multi-dimensional searches is presented as well, making this estimator a favorable choice for practical applications.Comment: To appear in the IEEE Transactions on Signal Processin

Sensor Array Processing with Manifold Uncertainty

<p>The spatial spectrum, also known as a field directionality map, is a description of the spatial distribution of energy in a wavefield. By sampling the wavefield at discrete locations in space, an estimate of the spatial spectrum can be derived using basic wave propagation models. The observable data space corresponding to physically realizable source locations for a given array configuration is referred to as the array manifold. In this thesis, array manifold ambiguities for linear arrays of omni-directional sensors in non-dispersive fields are considered. </p><p>First, the problem of underwater a hydrophone array towed behind a maneuvering platform is considered. The array consists of many hydrophones mounted to a flexible cable that is pulled behind a ship. The towed cable will bend or distort as the ship performs maneuvers. The motion of the cable through the turn can be used to resolve ambiguities that are inherent to nominally linear arrays. The first significant contribution is a method to estimate the spatial spectrum using a time-varying array shape in a dynamic field and broadband temporal data. Knowledge of the temporal spectral shape is shown to enhance detection performance. The field is approximated as a sum of uncorrelated planewaves located at uniform locations in angle, forming a gridded map on which a maximum likelihood estimate for broadband source power is derived. Uniform linear arrays also suffer from spatial aliasing when the inter-element spacing exceeds a half-wavelength. Broadband temporal knowledge is shown to significantly reduce aliasing and thus, in simulation, enhance target detection in interference dominated environments. </p><p>As an extension, the problem of towed array shape estimation is considered when the number and location of sources are unknown. A maximum likelihood estimate of the array shape using the field directionality map is derived. An acoustic-based array shape estimate that exploits the full 360$^\circ$ field via field directionality mapping is the second significant contribution. Towed hydrophone arrays have heading sensors in order to estimate array shape, but these sensors can malfunction during sharp turns. An array shape model is described that allows the heading sensor data to be statistically fused with heading sensor. The third significant contribution is method to exploit dynamical motion models for sharp turns for a robust array shape estimate that combines acoustic and heading data. The proposed array shape model works well for both acoustic and heading data and is valid for arbitrary continuous array shapes.</p><p>Finally, the problem of array manifold ambiguities for static under-sampled linear arrays is considered. Under-sampled arrays are non-uniformly sampled with average spacing greater than a half-wavelength. While spatial aliasing only occurs in uniformly sampled arrays with spacing greater than a half-wavelength, under-sampled arrays have increased spatial resolution at the cost of high sidelobes compared to half-wavelength sampled arrays with the same number of sensors. Additionally, non-uniformly sampled arrays suffer from rank deficient array manifolds that cause traditional subspace based techniques to fail. A class of fully agumentable arrays, minimally redundant linear arrays, is considered where the received data statistics of a uniformly spaced array of the same length can be reconstructed in wide sense stationary fields at the cost of increased variance. The forth significant contribution is a reduced rank processing method for fully augmentable arrays to reduce the variance from augmentation with limited snapshots. Array gain for reduced rank adaptive processing with diagonal loading for snapshot deficient scenarios is analytically derived using asymptotic results from random matrix theory for a set ratio of sensors to snapshots. Additionally, the problem of near-field sources is considered and a method to reduce the variance from augmentation is proposed. In simulation, these methods result in significant average and median array gains with limited snapshots.</p>Dissertatio

Structured Sub-Nyquist Sampling with Applications in Compressive Toeplitz Covariance Estimation, Super-Resolution and Phase Retrieval

Sub-Nyquist sampling has received a huge amount of interest in the past decade. In classical compressed sensing theory, if the measurement procedure satisfies a particular condition known as Restricted Isometry Property (RIP), we can achieve stable recovery of signals of low-dimensional intrinsic structures with an order-wise optimal sample size. Such low-dimensional structures include sparse and low rank for both vector and matrix cases. The main drawback of conventional compressed sensing theory is that random measurements are required to ensure the RIP property. However, in many applications such as imaging and array signal processing, applying independent random measurements may not be practical as the systems are deterministic. Moreover, random measurements based compressed sensing always exploits convex programs for signal recovery even in the noiseless case, and solving those programs is computationally intensive if the ambient dimension is large, especially in the matrix case. The main contribution of this dissertation is that we propose a deterministic sub-Nyquist sampling framework for compressing the structured signal and come up with computationally efficient algorithms. Besides widely studied sparse and low-rank structures, we particularly focus on the cases that the signals of interest are stationary or the measurements are of Fourier type. The key difference between our work from classical compressed sensing theory is that we explicitly exploit the second-order statistics of the signals, and study the equivalent quadratic measurement model in the correlation domain. The essential observation made in this dissertation is that a difference/sum coarray structure will arise from the quadratic model if the measurements are of Fourier type. With these observations, we are able to achieve a better compression rate for covariance estimation, identify more sources in array signal processing or recover the signals of larger sparsity. In this dissertation, we will first study the problem of Toeplitz covariance estimation. In particular, we will show how to achieve an order-wise optimal compression rate using the idea of sparse arrays in both general and low-rank cases. Then, an analysis framework of super-resolution with positivity constraint is established. We will present fundamental robustness guarantees, efficient algorithms and applications in practices. Next, we will study the problem of phase-retrieval for which we successfully apply the sparse array ideas by fully exploiting the quadratic measurement model. We achieve near-optimal sample complexity for both sparse and general cases with practical Fourier measurements and provide efficient and deterministic recovery algorithms. In the end, we will further elaborate on the essential role of non-negative constraint in underdetermined inverse problems. In particular, we will analyze the nonlinear co-array interpolation problem and develop a universal upper bound of the interpolation error. Bilinear problem with non-negative constraint will be considered next and the exact characterization of the ambiguous solutions will be established for the first time in literature. At last, we will show how to apply the nested array idea to solve real problems such as Kriging. Using spatial correlation information, we are able to have a stable estimate of the field of interest with fewer sensors than classic methodologies. Extensive numerical experiments are implemented to demonstrate our theoretical claims